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---
author:
- 'Quentin De Mourgues\'
title: |
\
\
of the KZB Classification Theorem
---
Rauzy-type dynamics are group actions on a collection of combinatorial objects. The first and best known example (the Rauzy dynamics) concerns an action on permutations, associated to interval exchange transformations (IET) for the Poincaré map on compact orientable translation surfaces. The equivalence classes on the objects induced by the group action have been classified by Kontsevich and Zorich, and by Boissy through methods involving both combinatorics algebraic geometry, topology and dynamical systems. Our precedent paper [@DS17] as well as the one of Fickenscher [@Fic16] proposed an ad hoc combinatorial proof of this classification.
However, unlike those two previous combinatorial proofs, we develop in this paper a general method, called the labelling method, which allows one to classify Rauzy-type dynamics in a much more systematic way. We apply the method to the Rauzy dynamics and obtain a third combinatorial proof of the classification. The method is versatile and will be used to classify three other Rauzy-type dynamics in follow-up articles.
Another feature of this paper is to introduce an algorithmic method to work with the sign invariant of the Rauzy dynamics. With this method we can prove most of the identities appearing in the literature so far ([@KZ03],[@Del13] [@Boi13] [@DS17]...) in an automatic way.
The sign invariant {#sec.signinv}
==================
Arf functions for permutations {#ssec.arf_inv}
------------------------------
For $\s$ a permutation in $\kS_n$, let () = \# { 1i<j n | (i)<(j) } i.e. $\chi(\s)$ is the number of pairs of non-crossing edges in the diagram representation of $\s$.
Let $E=E(\s)$ be the subset of $n$ edges in $\cK_{n,n}$ described by $\s$. For any $I \subseteq E$ of cardinality $k$, the permutation $\s|_{I} \in \kS_k$ is defined in the obvious way, as the one associated to the subgraph of $\cK_{n,n}$ with edge-set $I$, with singletons dropped out, and the inherited total ordering of the two vertex-sets.
Define the two functions $$\begin{aligned}
A(\s)
&:=
\sum_{I \subseteq E(\s)} (-1)^{\chi(\s|_I)}
\ef;
&
\Abar(\s)
&:=
\sum_{I \subseteq E(\s)} (-1)^{|I|+\chi(\s|_I)}
\ef.\end{aligned}$$ When $\s$ is understood, we will just write $\chi_I$ for $\chi(\s|_I)$. The quantity $A$ is accessory in the forthcoming analysis, while the crucial fact for our purpose is that the quantity $\Abar$ is invariant in the $\perms_n$ dynamics.
In the following section, we define a technique to demonstrate identities of the arf invariant involving differents configurations.
Automatic proofs of Arf identitites {#ssec.arfcalcseasy}
-----------------------------------
We will *not* try to evaluate Arf functions of large configurations starting from scratch. We will rather compare the Arf functions of two (or more) configurations, which differ by a finite number of edges, and establish linear relations among their Arf functions. The method we develop here, gives an algorithm to find and check Arf identities.
In order to have the appropriate terminology for expressing this strategy, let us define the following: Given a permutation $\s$ define $\s_{k,\ell}$ to be a permutation with $k$ marks on its bottom line and $\ell$ marks on its top line. The marks are all at distinct positions and do not touch the corners of the permutation. These marks break the bottom (respectively top) line into $k+1$ open interval $P_{-,1},\ldots,P_{-,k+1}$ (respectively $\ell+1$ open interval $P_{+,1},\ldots,P_{+,\ell+1}$).
For example if $k=1,\ell=3$: $$\put(50,-30){$P_{-,1}$}\put(100,-30){$P_{-,2}$}\put(26,25){$P_{+,1}$}\put(60,25){$P_{+,2}$}\put(90,25){$P_{+,3}$}\put(125,25){$P_{+,4}$}
\s_{k,\ell}=\raisebox{-20pt}{\includegraphics[scale=2.5]{P05_Arf/figure/fig_arf_ex1.pdf}}$$
Let $\s_{k,\ell,E'}$ be the permutation obtained by adding a set of edges $E'$ on the marks of permutation $\s_{k,\ell}$ with the following convention: an edge $e\in E'$ is a pair $(i.x,j.y)$. The edge connects the $i$th bottom mark and the $j$th top mark, and it is ordered as the $x$th edge within the bottom mark and the $y$th edge within the top mark. Note that if $i=0$ of $i=k+1$ (likewise of $j$) this implies that the edge is connected to a corner of the permutation.
For example if $k=1,\ell=3$ and $E'=\{(0.1,2.2),(1.1,3.1),(1.2,1.1),(1.3,2.1),(2,1.2)\}$: $$\put(60,-30){$P_{-,1}$}\put(110,-30){$P_{-,2}$}\put(36,25){$P_{+,1}$}\put(70,25){$P_{+,2}$}\put(105,25){$P_{+,3}$}\put(135,25){$P_{+,4}$}
\s_{k,\ell,E'}=\raisebox{-20pt}{\includegraphics[scale=2.5]{P05_Arf/figure/fig_arf_ex2.pdf}}$$ We will define an algorithm that allows one to check if, for all $\s$, we have $\sum^n_{i=1} K_i\Abar(\s_{k,\ell,E^i})=0$ or $\sum^n_{i=1} K_iA(\s_{k,\ell,E^i})=0$ for some $k,\ell,(E^i)_i,(K_i)_i,n.$
\[def.637647\] Let $\s_{k,\ell,E'}$, $P_{-,1},\ldots,P_{-,k+1}$ and $P_{+,1},\ldots,P_{+,\ell+1}$ be as defined above. Then define the $m \times (k+1)(\ell+1)$ matrix valued in $\gf_2$ Q\_[e,ij]{} := {
[ll]{} 1 &\
0 &
. For $v \in \gf_2^{(k+1)(\ell+1)}$, let $|v|$ be the number of entries equal to $1$. Similarly, identify $v$ with the corresponding subset of $[(k+1)(\ell+1)]$. Given such a construction, introduce the following functions on $(\gf_2)^{(k+1)(\ell+1)}$ $$\begin{aligned}
A_{k,\ell,E'}(v)
&:=
\sum_{u \in (\gf_2)^{E'}}
(-1)^{\chi_u + (u,Qv)}
\ef;
&
\Abar_{k,\ell,E'}(v)
&:=
\sum_{u \in (\gf_2)^{E'}}
(-1)^{|u|+\chi_u + (u,Qv)}
\ef.\end{aligned}$$
The construction is illustrated in Figure \[fig.arf\_ex\_def\].
![\[fig.arf\_ex\_def\]The permutation $\s_{1,4,\{(0.1,0.1),(0.2,3.1),(0.3,1.1),(1.1,4.1),(1.2,2.1)\}}$. We cannot show the full matrix $Q$ for such a big example, but we can give one row, for the edge which has the label $e$ in the drawing. The row $Q_e$ reads $(Q_e)_{11, 12, \ldots, 15, 21, \ldots, 25}=(1,1,0,0,0,\,0,0,1,1,1)$.](FigFol/Figure4_fig_arf_ex_def.pdf)
Let us comment on the reasons for introducing such a definition. The quantities $A_{k,\ell,E'}(v)$ (respectively $\Abar_{k,\ell,E'}(v)$) do not depend on $\s$ and allows to sum together many contributions to the function $A(\s_{k,\ell,E'})$. Our goal is to have $E'$ of fixed size, while $E$ (the edge set of $\s$) is arbitrary and of unbounded size, so that the verification of our properties, as it is confined to the matrix $Q$, involves a finite data structure. Thus the algorithm will be exponential in $|E'|$ which will not be a problem for small sizes.
Indeed, let us split in the natural way the sum over subsets $I$ of $E\bigcup E'$ the edge set of $\s_{k,\ell,E'}$ namely $$\sum_{I_0 \subseteq E\bigcup E'} f(I_0)
=
\sum_{I \subseteq E}
\sum_{I' \subseteq E'}
f(I \cup I')$$ For $I$ and $J$ two disjoint sets of edges, call $\chi_{I,J}$ the number of pairs $(i,j) \in I \times J$ which do not cross. Then clearly $$\chi_{I \cup J} = \chi_{I} + \chi_{J} + \chi_{I,J}$$ Now let $u(I') \in \{0,1\}^{E'}$ be the vector with entries $u_e=1$ if $e \in I'$ and $0$ otherwise. Let $m(I)=\{m_{ij}(I)\}$ be the $(k+1) \times (\ell+1)$ matrix describing the number of edges connecting the intervals $P_{-,i}$ to $P_{+,j}$ in $\s$, and let $v(I)=\{v_{ij}(I)\}$, $v_{ij} \in \{0,1\}$ be the parities of the $m_{ij}$’s. Call $I_{ij}$ the restriction of $I$ to edges connecting $P_{-,i}$ and $P_{+,j}$. Clearly, $$\chi_{I',I}=\sum_{ij} \chi_{I',I_{ij}}
= \sum_{e,ij} u_e Q_{e,ij} m_{ij}=(u(I'),Qm(I)),$$ which has the same parity as the analogous expression with $v$’s instead of $m$’s. I.e. we have $$(-1)^{\chi_{I',I}}=(-1)^{(u(I'),Qv(I))}.$$ Now, while the $m$’s are in $\bN$, the vector $v$ is in a linear space of finite cardinality, which is crucial for allowing a finite analysis of our expressions.
As a consequence, $$\begin{aligned}
\label{eq.arf_explain_def}
A(\s_{k,\ell,E^i})
&=
\sum_{I \subseteq E}(-1)^{\chi_I}A_{k,\ell,E'}(v(I))
\ef;
\\\label{eq.arf_explain_def_1}
\Abar(\s_{k,\ell,E^i})
&=\sum_{I \subseteq E}(-1)^{|I|+\chi_I}\Abar_{k,\ell,E'}(v(I))
\ef.\end{aligned}$$
Thus we have the following proposition:
\[prop.ArfProp\]\[prop.Arf0\] Let $k,\ell \in \N$ and let $(E^i)_{1\leq i\leq n}$ be a family of edge set. Then the two following statements are equivalent:
1. For all $v \in GF_2^{(k+1)(\ell+1)},$ we have $\sum_{i=1}^n K_i A_{k,\ell,E^i}(v)=0$.
2. for all $\s,$ we have $\sum_{i=1}^n K_i A(\s_{k,\ell,E^i})=0.$
The same statement holds for $\Abar$.
Statement 1 implies 2 due to equation (\[eq.arf\_explain\_def\]).\
Let us show the converse: If $\s$ has no edge then $\sum_{i=1}^n K_i A(\s_{k,\ell,E^i})=0$ is equivalent to $\sum_{i=1}^n K_i A_{k,\ell,E^i}(v)=0$ with $v$ being the zero vector of $(GF_2)^{(k+1)(\ell+1)}$.
Then we choose the family of permutations $(\s^{a,b})_{a,b}$ with exactly one edge connecting $P_{-,a}$ to $P_{+,b}$. Then if we define $v_{a,b}$ to be the vector $(GF_2)^{(k+1)(\ell+1)}$ with exactly one 1 at position $ab$ we have
&\_[i=1]{}\^n K\_i A(\^[a,b]{}\_[k,,E\^i]{})=0\
& \_[i=1]{}\^n K\_i ( A\_[k,,E\^i]{}(0) + A\_[k,,E\^i]{}(v\_[a,b]{})) =0\
& \_[i=1]{}\^n K\_i A\_[k,,E\^i]{}(v\_[a,b]{}) =0\
in the last line we have used that $\sum_{i=1}^n K_i A_{k,\ell,E^i}(0)=0$.
Thus inductively we show that $\sum_{i=1}^n K_i A_{k,\ell,E^i}(v)=0$ for any $v\in (GF_2)^{(k+1)(\ell+1)}$ with at most $p\leq n$ ones.
This theorem is very important since it reduces the problem of calculating Arf identities for permutations of any size to a check of an Arf identity for $2^{(k+1)(\ell+1)}$ values. Thus in exponential time in $k\ell$ we can calculate $A_{k,\ell,E'}(v)$ for every $v \in GF_2^{(k+1)(\ell+1)}$ and decide if a given Arf identity is correct.
We can even do better:
Let $k,\ell \in \N$ and let $(E^i)_{1\leq i\leq n}$ be a family of edge set. We can decide (in exponential time in $k\ell$) if there exists $x_1,\ldots,x_n$ such that $\sum_{i=1}^n x_i A(\s_{k,\ell,E'})=0$.
For every $v$ we have an equation $\sum_{i=1}^n x_i A_{k,\ell,E^i}(v) =0$ with the $n$ unknown variables. So there are $2^{(k+1)(\ell+1)}$ equations. We can find the subspace of solution in time exponential in $k\ell$.
The previous proposition can be used when we suspect a relation between a few configurations without knowing the coefficients. The algorithm demands little more than the previous one for the verification so it remains usable for small $|E'|$.
Finally we can actually enumerate all the possible Arf identities:
There is an algorithm that enumerate all the arf identities with at most $n$ terms and on an edge set $E'$ of size at most $h$.
This algorithm is really not praticable. However it can be used in the following case: we have two terms and we want to find an arf identity relating them to one another but the previous algorithm failed (i.e there are no identity containing only those two terms). Then we use this algorithm to find a third term (or a fourth etc...) for which an identity exists.
We can even propose a generalisation of this framework: let us choose two permutations $\pi_{-}$ and $\pi_{+}$ of size $k+1$ and $\ell+1$ respectively then $\s_{k,\ell,E',\pi_{-},\pi_{+}}$ is obtained from $\s_{k,\ell,E'}$ by permuting the $P_{-,i}$ with $\pi_{-}$ and $P_{+,i}$ with $\pi_{+}$.
For example if $k=1,\ell=3$ and $E'=\{(0.1,2.2),(1.1,3.1),(1.2,1.1),(1.3,2.1),(2,1.2)\}$: $$\put(60,-30){$P_{-,1}$}\put(110,-30){$P_{-,2}$}\put(36,25){$P_{+,1}$}\put(70,25){$P_{+,2}$}\put(105,25){$P_{+,3}$}\put(135,25){$P_{+,4}$}
\s_{k,\ell,E'}=\raisebox{-20pt}{\includegraphics[scale=2.5]{P05_Arf/figure/fig_arf_ex2.pdf}}
\put(130,-30){$P_{-,\pi_{-}1}$}\put(180,-30){$P_{-,\pi_{-}2}$}\put(100,25){$P_{+,\pi_{+}1}$}\put(140,25){$P_{+,\pi_{+}2}$}\put(175,25){$P_{+,\pi_{+}3}$}\put(210,25){$P_{+,\pi_{+}4}$}
\qquad \qquad\s_{k,\ell,E',\pi_{-},\pi_{+}}=\raisebox{-20pt}{\includegraphics[scale=2.5]{P05_Arf/figure/fig_arf_ex2.pdf}}.$$ For an example with $\pi_{-}=id_2$ and $\pi_{+}=(2,1)$ (the reversing permutation $\omega$ at size 2) we have: $$\put(55,-30){$P_{-,1}$}\put(95,-30){$P_{-,2}$}\put(55,25){$P_{+,1}$}\put(85,25){$P_{+,2}$}
\s_{1,1,E'}=\raisebox{-20pt}{\includegraphics[scale=2.5]{P05_Arf/figure/fig_arfproof_s11-eps-converted-to.pdf}}
\put(125,-30){$P_{-,1}$}\put(165,-30){$P_{-,2}$}\put(120,50){$P_{+,2}$}\put(160,50){$P_{+,1}$}
\qquad \qquad\s_{1,1,E',\pi_{-},\pi_{+}}=\raisebox{-20pt}{\includegraphics[scale=2.5]{FigFol/fig_arfproof_general_model_ex.pdf}}$$ It is easily checked that the previous theorems continue to hold for this generalisation once we introduce for $v \in \gf_2^{(k+1)(\ell+1)}$ the following function on $(\gf_2)^{(k+1)(\ell+1)}$ (similar definition for $\Abar_{k,\ell,E',\pi_{-},\pi_{+}}$) $$\begin{aligned}
A_{k,\ell,E',\pi_{-},\pi_{+}}(v)
&:=
\sum_{u \in (\gf_2)^{E'}}
(-1)^{\chi_u + (u,Q(P^{-1}_{\pi_{-}}vP_{\pi_{+}}))}
\ef;\end{aligned}$$ Where in the expression $(P^{-1}_{\pi_{-}}vP_{\pi_{+}})$, $v$ is identified to the matrix of size $(k+1)\times (\ell+1)$ and $P_{\pi_{-}}$ and $P_{\pi_{+}}$ are the permutation matrices associated to $\pi_{-}$ and $\pi_{+}$.
The framework of automatic proofs of Arf identities we have developped is rather general. Most of the identities found in the litterature (see [@KZ03], [@Boi13], [@DS17], [@Del13], [@Gut17]) can be obtained in this setting.
Let us now apply the algorithm to find Arf identities.
It is convenient to introduce the notation $\vec{A}(\s)=\begin{pmatrixsm} \Abar(\s) \\ A(\s) \end{pmatrixsm}$. We have
\[prop.fingerred\] $$\begin{aligned}
\label{prop.signinvdyn}
\Abar\bigg(\,\tau=\raisebox{-12pt}{\includegraphics[scale=1.8]{FigFol/Figure4_fig_arfproof_t1.pdf}}\,\bigg)
&=
\Abar\bigg(\,\s=\raisebox{-12pt}{\includegraphics[scale=1.8]{FigFol/Figure4_fig_arfproof_s1.pdf}}\,\bigg)
\\
\label{eq.546455a}
\vec{A}\bigg(\,
%\tau=
\raisebox{-12pt}{\includegraphics[scale=1.8]{FigFol/Figure4_fig_arfproof_t3.pdf}}\,\bigg)
&=
\begin{pmatrix}
1 & -1 \\ 1 & 1
\end{pmatrix}
\vec{A}\bigg(\,
%\s=
\raisebox{-12pt}{\includegraphics[scale=1.8]{FigFol/Figure4_fig_arfproof_s3.pdf}}\,\bigg)
\ef;
\\
\label{eq.546455b}
\vec{A}\bigg(\,
%\tau=
\raisebox{-12pt}{\includegraphics[scale=1.8]{FigFol/Figure4_fig_arfproof_t4.pdf}}\,\bigg)
&=
\begin{pmatrix}
0 & 0 \\ 0 & 2
\end{pmatrix}
\vec{A}\bigg(\,
%\s=
\raisebox{-12pt}{\includegraphics[scale=1.8]{FigFol/Figure4_fig_arfproof_s3.pdf}}\,\bigg)
\ef;\end{aligned}$$
Clearly equation (\[prop.signinvdyn\]) prove the invariance of the arf invariant for the dynamics since $\s=L(\tau)$ and the case $R$ is deduced by symmetry.
Exceptional classes
===================
In this appendix, when using a matrix representation of configurations, it is useful to adopt the following notation: The symbol $\epsilon$ denotes the $0 \times 0$ empty matrix. The symbol denotes a square block in a matrix (of any size $\geq 0$), filled with an identity matrix. A diagram, containing these special symbols and the ordinary bullets used through the rest of the appendix, describes the set of all configurations that could be obtained by specifying the sizes of the identity blocks. In such a syntax, we can write equations of the like $$\begin{aligned}
\id
&:=
\raisebox{-1mm}{\includegraphics[width=4.8mm]{FigFol/FigureA2_fig_matr_id.pdf}}
=
\epsilon \cup
\raisebox{-3mm}{\includegraphics[width=8.8mm]{FigFol/FigureA2_fig_matr_Bid.pdf}}
=
\epsilon \cup
\raisebox{5.8mm}{\includegraphics[width=8.8mm,
angle=180]{FigFol/FigureA2_fig_matr_Bid.pdf}}
\ef;
% \\
% \intertext{and}
&
\id'
&:=
\raisebox{-7mm}{\includegraphics[width=16.8mm]{FigFol/fig_matr_idp.pdf}}
\ef.\end{aligned}$$ The sets $\id$ and $\id'$ contain one element per size, $\id_n$ and $\id'_n$, for $n \geq 0$ and $n \geq 3$ respectively.
The two exceptional classes $\Id_n$ and $\tree_n$ contain the configurations $\id_n$ and $\id'_n$, respectively.
We have the following proposition:
\[pro\_excep\_std\] The permutation $\s=\id_n$ (respectively $\s=\id'_n)$ is the only permutation of $\Id_n$ (respectively $\Id'_n$) with $\s(1)=1$ and $\s(2)=2$.
The structure of the classes $\Id_n$ is summarised by the following relation: := \_n \_n = ( \_[k 1]{} (X\_[RL]{}\^[k]{} X\_[LR]{}\^[k]{} X\_[LL]{}\^[k]{} X\_[RR]{}\^[k]{}) ) where the configurations $X_{\cdot \cdot}^{k}$ are defined as in figure \[fig.struct\_id\] (discard colours for the moment).\
$$\begin{aligned}
X_{LL}^{(k)}
&=
\quad
\makebox[0pt][l]{\raisebox{-16mm}{\includegraphics[width=33mm]{FigFol/FigureA2_fig_matr_CidGen1.pdf}}}
\hspace{-16pt}
\raisebox{-30pt}{$\rotatebox{45}{$k \left\{\rule{0pt}{50pt}\right. $}$}
&
X_{RR}^{(k)}
&=
\makebox[0pt][l]{\raisebox{-16mm}{\includegraphics[width=33mm]{FigFol/FigureA2_fig_matr_CidGen2.pdf}}}
\hspace{28pt}\raisebox{14pt}{$\rotatebox{45}{$\left. \rule{0pt}{50pt}\right\} k $}$}
\\
X_{RL}^{(k)}
&=
\makebox[0pt][l]{\raisebox{-16mm}{\includegraphics[width=38mm]{FigFol/FigureA2_fig_matr_CidGen3.pdf}}}
\hspace{-2pt}
\raisebox{-30pt}{$\rotatebox{45}{$k \left\{\rule{0pt}{50pt}\right. $}$}
&
X_{LR}^{(k)}
&=
\makebox[0pt][l]{\raisebox{-16mm}{\includegraphics[width=38mm]{FigFol/FigureA2_fig_matr_CidGen4.pdf}}}
\hspace{42pt}\raisebox{14pt}{$\rotatebox{45}{$\left. \rule{0pt}{50pt}\right\} k $}$}\end{aligned}$$
We can now prove the lemma \[lem.StFamNotmanyId\_tobeproveninAppC\] that we introduced in section 10.4 !!!!!!!!!.\
This is equivalent to say that there are no pairs of permutations $\s_1, \s_2 \in \Id_n$ which allow for a block decomposition $$\begin{aligned}
\s_1 &= \begin{array}{|c|}
\hline
\rule{5pt}{0pt}
A
\rule{5pt}{0pt}
\\
\hline
B \\
\hline
\end{array}
&
\s_2 &= \begin{array}{|c|}
\hline
\rule{5pt}{0pt}
B
\rule{5pt}{0pt}
\\
\hline
A \\
\hline
\end{array}\end{aligned}$$ If the block $A$ has $\ell$ rows, we say that $\s_2$ is the result of shifting $\s_1$ by $\ell$.
Clearly, at the light of the structure of configurations that we have presented (refer in particular to Figure \[fig.struct\_id\]), this pattern is incompatible with $\s_1$ or $\s_2$ being $\id_n$ (as a non-trivial shift produces a configuration which is not even irreducible), so we have excluded the cases in which, still with reference to the figure, we have only one violet block, and the number of black points is at least 3, for $X^{(k)}_{LL}$ and $X^{(k)}_{RR}$, and at least 4, for $X^{(k)}_{LR}$ and $X^{(k)}_{RL}$. Note that the black points are the positions in the grid which are south-west or north-east extremal (i.e., positions $(i,j) \in \s$ such that there is no $(i',j') \in \s$ with $i'<i$ and $j'<j$, or the analogous statement with $i'>i$ and $j'>j$). Let us call *number of records*, $\rho(\s)$, this parameter. Thus we have that configurations in $X^{(k)}_{LL}$ and $X^{(k)}_{RR}$ have $\rho=2k+1$, and configurations in $X^{(k)}_{LR}$ and $X^{(k)}_{RL}$ have $\rho=2k+2$.
Now, if we perform a shift within one block of consecutive ascents, it is easily seen, by investigation of the sub-configuration at the right of the entry of the new configuration in the bottom-most row, or the one at the left of the entry of the new configuration in the top-most row, that the resulting structure is incompatible with the structure of $\Id$. The same reasoning apply if we perform the shift at the beginning/end of a non-empty diagonal block, which is not the one at the bottom-right/top-left. On the other side, if we perform a shift in any other configuration, we have a new configuration in which $\rho$ has strictly decreased. As $\s_2$ is a non-trivial shift of $\s_1$, and $\s_1$ is a non-trivial shift of $\s_2$, we can thus conclude.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We describe a cooperad structure on the simplicial bar construction on a reduced operad of based spaces or spectra and, dually, an operad structure on the cobar construction on a cooperad. We also show that if the homology of the original operad (respectively, cooperad) is Koszul, then the homology of the bar (respectively, cobar) construction is the Koszul dual. We use our results to construct an operad structure on the partition poset models for the Goodwillie derivatives of the identity functor on based spaces and show that this induces the ‘Lie’ operad structure on the homology groups of these derivatives. We also extend the bar construction to modules over operads (and, dually, to comodules over cooperads) and show that a based space naturally gives rise to a left module over the operad formed by the derivatives of the identity.'
address: |
Department of Mathematics, Room 2-089\
Massachusetts Institute of Technology\
Cambridge, MA 02139, USA
author:
- Michael Ching
title: |
Bar constructions for topological operads and\
the Goodwillie derivatives of the identity
---
Introduction {#introduction .unnumbered}
============
The motivation for this paper was an effort to construct an operad structure on the derivatives (in the sense of Tom Goodwillie’s homotopy calculus [@goodwillie:1990; @goodwillie:1991; @goodwillie:2003]) of the identity functor $I$ on the category of based spaces. Such an operad structure has been ‘known’ intuitively by experts for some time but, as far as the author knows, no explicit construction has previously been given. One piece of evidence for such a structure is the calculation, due to various people, of the homology of these derivatives. This homology is the suspension of the standard Lie operad and so is itself an operad. It is reasonable to ask, therefore, if there is an operad structure on the derivatives themselves[^1] that induces this structure on the homology.
Our construction is based on the partition poset model for the derivatives $\partial_*I$ described by Arone and Mahowald in [@arone/mahowald:1999]. They show that the derivatives are the dual spectra associated to certain finite complexes known as the partition poset complexes. In the present work we notice that these complexes are precisely the simplicial bar construction[^2] on the operad $P$ in based spaces with $P(n) = S^0$ for all $n$. Most of the paper is concerned with showing that such a bar construction has a natural cooperad structure.[^3] We do this by reinterpreting the bar construction in terms of spaces of trees. The cooperad structure then comes from a natural way to break trees apart. Taking duals, we get the required operad structure on the derivatives of the identity. In fact, we can view the derivatives of the identity as a cobar construction on the cooperad $Q$ in spectra with $Q(n) = S$, the sphere spectrum, for all $n$.
In the final part of the paper (Section \[sec:alg\]) we show that by taking homology we do indeed recover the ‘Lie’ operad structure on $H_*(\partial_*I)$. We do this by introducing spectral sequences for calculating the homology of the topological bar and cobar constructions. The $E^1$ terms of these spectral sequences can be identified with algebraic versions of the bar and cobar constructions, which in turn are related to the theory of Koszul duality for operads introduced by Ginzburg and Kapranov in [@ginzburg/kapranov:1994]. Our main result on this connection is that if the homology of a topological operad $P$ is Koszul, then the homology of the bar construction $B(P)$ is its Koszul dual cooperad. In our case of interest, we deduce that the induced operad structure on the homology of the derivatives of the identity is that of the Koszul dual of the cocommutative cooperad. This is precisely the ‘Lie’ operad structure referred to above.
Outline of the paper {#outline-of-the-paper .unnumbered}
--------------------
We now give a more detailed description of the paper. The first two sections are concerned with preliminaries. In Section \[sec:monoidal\] we recall the notions of symmetric monoidal and enriched categories and specify the categories we will be working with in this paper. These are symmetric monoidal categories that are enriched, tensored and cotensored over the category ${\mathcal{T}_{}}$ of based compactly-generated spaces (where ${\mathcal{T}_{}}$ is a symmetric monoidal category with respect to the smash product). It is to operads in these categories that we refer in the title when we say ‘topological operads’. We also require an extra condition that relates the symmetric monoidal structure to the tensoring over ${\mathcal{T}_{}}$. This condition (see Definition \[def:axiom\]) is crucial to our later constructions. The two main examples of categories satisfying our requirements are: based spaces themselves, and a suitable symmetric monoidal category of spectra, such as that of EKMM [@elmendorf/kriz/mandell/may:1997].
In Section \[sec:operads\] we recall the definitions of operads and cooperads. We should stress that the constructions of this paper apply only to what we call *reduced* operads and cooperads. These are $P$ with $P(0) = {\ast}$ and $P(1) = S$ the unit of the symmetric monoidal structure. The bar construction can still be defined for more general operads, but the cooperad structure described here does not seem to extend to such cases. In this section we also define modules and comodules over operads and cooperads respectively.
The real substance of the paper starts in Section \[sec:trees\]. Here we define the trees that will form the combinatorial heart of our description of the bar and cobar constructions. It is not a coincidence that these trees are the same species used by, for example, Getzler and Jones in their work [@getzler/jones:1994] on the bar constructions for algebraic operads and Koszul duality. We also describe what we call a *weighting* on a tree (Definition \[def:weighting\]), that is, a suitable assignment of lengths to the edges of the tree. The spaces $w(T)$ of weightings are at the heart of everything we do in this paper.
In Section \[sec:bardef\] we give our description of the bar construction on an operad in terms of such trees. If $P$ is an operad of based spaces, we can think of a point in the bar construction $B(P)$ as a weighted tree (that is, a tree with lengths assigned to the edges) with vertices labelled by points coming from the spaces $P(n)$. See Definition \[def:bar(operad)\] for a precise statement and Definition \[def:formal\_bar\] for a more formal approach. In Section \[sec:simpbar\] we show that what we have defined is isomorphic to the standard simplicial bar construction on an operad.
In Section \[sec:cooperad\] we concern ourselves with the cooperad structure on $B(P)$. This is given by the process of ‘ungrafting’ trees (see Definition \[def:grafting\] and beyond). This involves taking a weighted, labelled tree and breaking it up into smaller trees. Finding the right way to weight and label these smaller trees gives us the required cooperad structure maps.
One of the advantages of the way we have set up the theory is that the cobar construction on a cooperad is strictly dual to the bar construction on an operad. In Section \[sec:cobar\] we go through the definitions and results dual to those of Section \[sec:bar\].
The short section Section \[sec:dual\] is devoted to a simple but key result (Proposition \[prop:duality\]) that relates the bar and cobar constructions via a duality functor that reduces to Spanier–Whitehead duality in the case of spectra. This result says that, under the right circumstances, the dual of the bar construction on an operad $P$ is isomorphic to the cobar construction on the dual of $P$. This allows us, later on, to identify the derivatives of the identity as the cobar construction on a cooperad of spectra.
Before turning to our main example and application, we deal in Section \[sec:bar(modules)\] with the two-sided bar and cobar constructions. These include the bar construction for a module over an operad and, dually, the cobar construction for a comodule over a cooperad. To describe these requires a fairly simple generalization of much of the work we did in Sections \[sec:trees\]–\[sec:bar\], in particular, a more general notion of tree (see Definition \[def:gen\_trees\]).
Finally, in Section \[sec:application\] we are able to complete the main aim of this paper. We identify the partition poset complexes with a bar construction and deduce the existence of an operad structure on the derivatives of the identity functor (Corollary \[cor:operad\]). We also give examples of modules over the resulting operad, including, in particular, a module $M_X$ naturally associated to a based space $X$.
The last section of the paper Section \[sec:alg\] is concerned with the relationship of our work to the algebraic bar construction and Koszul operads. As promised, we construct a spectral sequence (Proposition \[prop:specseq\]) relating the two and deduce the result on Koszul duality (Proposition \[prop:koszul\]).
Future Work {#future-work .unnumbered}
-----------
The work of this paper raises various questions that seem to the author to warrant further attention:
- What is the homotopy theory of the topological bar and cobar constructions? In particular, how do they relate to known model structures on the categories of operads and cooperads (see, for example, Berger–Moerdijk [@berger/moerdijk:2003])?
- Is there a deeper relationship between Goodwillie’s homotopy calculus and the theory of operads? The present paper does not do any calculus, the only connection being via the partition poset complexes. One might ask, for example, if the derivatives of other functors can be described and/or treated using these ideas.
- What object is described by an algebra or module over the operad formed by the derivatives of the identity? In Remark \[rem:modules\] we show that a based space $X$ gives rise to such a module. How much of (the homotopy theory of) the space $X$ is retained by this module?
Acknowledgements {#acknowledgements .unnumbered}
----------------
The work of this paper forms the author’s PhD thesis written at the Massachusetts Institute of Technology under the supervision of Haynes Miller, to whom the greatest thanks are due for his constant support, encouragement and advice. The idea that the derivatives of the identity might be related to a cobar construction was suggested by work of Kristine Bauer, Brenda Johnson and Jack Morava. The observation that the partition poset complexes (and hence the derivatives of the identity) can be described in terms of spaces of trees was mentioned to the author by Tom Goodwillie, who heard it from Greg Arone. The work of Benoit Fresse [@fresse:2004] on the algebraic side of the theory was invaluable to the present paper. The author has also benefited greatly from conversations with Mark Behrens and Andrew Mauer-Oats while writing this paper, and finally would like to thank the referee for some helpful comments and suggestions.
Symmetric monoidal and enriched categories {#sec:monoidal}
==========================================
On the one hand, the bar and cobar constructions are most easily defined (and understood) in the category of based spaces. On the other hand, our main application is in a category of spectra. We will develop the theory in a general setting that encompasses both cases. This approach will also allow us to appreciate more readily the duality between the bar and cobar constructions.
In this section we recall the basic theory of symmetric monoidal and enriched categories (see [@borceux:1994(2) Section 6] for a detailed account). We state precisely (Definition \[def:axiom\]) the structure we will require of a category to make the bar and cobar constructions in it. The only material in this chapter that is not standard is the definition of enriched symmetric monoidal categories or ‘symmetric monoidal ${\mathcal{V}}$–categories’ as we have called them (Definition \[def:axiom\]). The ‘distributivity’ morphism described there is a key component of the constructions made later in the paper and so we draw the reader’s attention to it now.
\[def:sym mon\] A *monoidal category* consists of
- a (locally small) category ${\mathcal{V}}$,
- a functor $- {\wedge}- \co {\mathcal{V}} \times {\mathcal{V}} \to {\mathcal{V}}$,
- a *unit* object $I$ in ${\mathcal{V}}$ together with natural isomorphisms $X {\wedge}I {\cong}X {\cong}I {\wedge}X$,
- a natural *associativity* isomorphism $X {\wedge}(Y {\wedge}Z)
{\cong}(X {\wedge}Y) {\wedge}Z$,
such that the appropriate three coherence diagrams commute [@maclane:1971 Section VII]. A *symmetric monoidal category* is a monoidal category together with
- a natural *symmetry* isomorphism $X {\wedge}Y {\cong}Y {\wedge}X$,
such that four additional coherence diagrams also commute. We will denote such a symmetric monoidal category by $({\mathcal{V}},{\wedge},I)$, or just ${\mathcal{V}}$ with the rest of the structure understood.
\[rem:symmetry\] We will not give names to the associativity and symmetry isomorphisms in a symmetric monoidal category. When we write unbracketed expressions such as $$X {\wedge}Y {\wedge}Z$$ or unordered expressions such as $$\bigwedge_{a \in A} X_a$$ we mean any one particular choice of ordering and bracketing. Different choices are related by the appropriate associativity and commutativity isomorphisms between them. A map to or from a particular choice determines a map to or from any other choice by composing with the relevant isomorphism.
\[def:closed sym mon\] A *closed symmetric monoidal category* is a symmetric monoidal category $({\mathcal{V}},{\wedge},I)$ together with a functor $${\mathcal{V}}^\text{op} \times {\mathcal{V}} \to {\mathcal{V}}; \: (X,Y) \mapsto \operatorname{Map}(X,Y)$$ and a natural isomorphism of sets $$\operatorname{Hom}_{{\mathcal{V}}}(X {\wedge}Y,Z) {\cong}\operatorname{Hom}_{{\mathcal{V}}}(X,\operatorname{Map}(Y,Z)),$$ where $\operatorname{Hom}_{{\mathcal{V}}}(X,Y)$ is the set of morphisms from $X$ to $Y$ in the category ${\mathcal{V}}$.
The natural isomorphism of sets in Definition \[def:closed sym mon\] can be made into an isomorphism within ${\mathcal{V}}$. That is, in any closed symmetric monoidal category there is a natural isomorphism $$\operatorname{Map}(X {\wedge}Y, Z) {\cong}\operatorname{Map}(X, \operatorname{Map}(Y,Z)).$$ See [@borceux:1994(2) Section 6.5.3] for details.
Let $({\mathcal{V}},{\wedge},I)$ be a given closed symmetric monoidal category. A *${\mathcal{V}}$–category* or *category enriched over ${\mathcal{V}}$* consists of
- a class ${\mathcal{C}}$,
- for each pair of elements $C,D \in {\mathcal{C}}$, an object $\operatorname{Map}_{{\mathcal{V}}}(C,D)$ of ${\mathcal{V}}$,
- composition morphisms $$\operatorname{Map}_{{\mathcal{V}}}(C,D) {\wedge}\operatorname{Map}_{{\mathcal{V}}}(D,E) \to \operatorname{Map}_{{\mathcal{V}}}(C,E)$$ for each $C,D,E \in {\mathcal{C}}$,
- identity morphisms $$I \to \operatorname{Map}_{{\mathcal{V}}}(C,C)$$ for each $C \in {\mathcal{C}}$,
that satisfy the appropriate conditions [@borceux:1994(2) Section 6.2.1]. We will denote such a ${\mathcal{V}}$–category by ${\mathcal{C}}$ with the rest of the structure understood.
\[rem:enriched\] We include some basic observations about enriched categories from [@borceux:1994(2) Section 6.2].
1. Let $(\mathsf{Set},\times,{\ast})$ be the symmetric monoidal category of sets under cartesian product. A $\mathsf{Set}$–category is then the same thing as a (locally small) category.
2. A ${\mathcal{V}}$–category ${\mathcal{C}}$ has an underlying category whose objects are the elements of ${\mathcal{C}}$ and whose morphisms $C \to D$ are the elements of the set $\operatorname{Hom}_{{\mathcal{V}}}(I,\operatorname{Map}_{{\mathcal{V}}}(C,D))$, where $I$ is the unit object of ${\mathcal{V}}$. We often therefore think of a ${\mathcal{V}}$–category ${\mathcal{C}}$ as a normal category with extra structure given by the objects $\operatorname{Map}_{{\mathcal{V}}}(C,D)$.
3. A closed symmetric monoidal category ${\mathcal{V}}$ is enriched over itself with $$\operatorname{Map}_{{\mathcal{V}}}(X,Y) := \operatorname{Map}(X,Y).$$
Let ${\mathcal{C}}$ be a ${\mathcal{V}}$–category. A *tensoring* of ${\mathcal{C}}$ over ${\mathcal{V}}$ is a functor $${\mathcal{V}} \times {\mathcal{C}} \to {\mathcal{C}}; \; (X,C) \mapsto X \otimes C$$ together with a natural isomorphism $$\operatorname{Map}_{{\mathcal{V}}}(X \otimes C, D) {\cong}\operatorname{Map}(X,\operatorname{Map}_{{\mathcal{V}}}(C,D)).$$ A category ${\mathcal{C}}$ *tensored* over ${\mathcal{V}}$ is a ${\mathcal{V}}$–category together with a chosen tensoring.
A *cotensoring* of ${\mathcal{C}}$ over ${\mathcal{V}}$ is a functor $${\mathcal{V}}^\text{op} \times {\mathcal{C}} \to {\mathcal{C}}; \; (X,D) \mapsto
\operatorname{Map}_{{\mathcal{C}}}(X,D)$$ together with a natural isomorphism $$\operatorname{Map}_{{\mathcal{V}}}(C, \operatorname{Map}_{{\mathcal{C}}}(X,D)) {\cong}\operatorname{Map}(X,\operatorname{Map}_{{\mathcal{V}}}(C,D)).$$ A category ${\mathcal{C}}$ *cotensored* over ${\mathcal{V}}$ is a ${\mathcal{V}}$–category together with a chosen cotensoring.
\[rem:tensor\] Here are some basic observations about tensorings and cotensorings.
1. A closed symmetric monoidal category $({\mathcal{V}},{\wedge},I)$ is tensored and cotensored over itself with $X \otimes Y := X {\wedge}Y$ and $\operatorname{Map}_{{\mathcal{V}}}(X,Y) := \operatorname{Map}(X,Y)$.
2. If ${\mathcal{C}}$ is tensored over ${\mathcal{V}}$, we have natural isomorphisms $$(X {\wedge}Y) \otimes C {\cong}X \otimes (Y \otimes C)$$ for $X,Y \in {\mathcal{V}}$ and $C \in {\mathcal{C}}$. If ${\mathcal{C}}$ is cotensored over ${\mathcal{V}}$, we have natural isomorphisms $$\operatorname{Map}_{{\mathcal{C}}}(X {\wedge}Y, C) {\cong}\operatorname{Map}_{{\mathcal{C}}}(X,
\operatorname{Map}_{{\mathcal{C}}}(Y,C))$$ for $X,Y \in {\mathcal{V}}$ and $C \in {\mathcal{C}}$.
\[prop:dual1\] Let ${\mathcal{C}}$ be a ${\mathcal{V}}$–category. Then ${\mathcal{C}}^\text{op}$ has a natural enrichment over ${\mathcal{V}}$.[^4] If ${\mathcal{C}}$ is tensored, then ${\mathcal{C}}^\text{op}$ is naturally cotensored and vice versa.
We define an enrichment on ${\mathcal{C}}^\text{op}$ by $$\operatorname{Map}_{{\mathcal{V}}}(C^\text{op},D^\text{op}) := \operatorname{Map}_{{\mathcal{V}}}(D,C)$$ where $C^\text{op}$ is the object in ${\mathcal{C}}^\text{op}$ corresponding to $C \in
{\mathcal{C}}$. If $- \otimes -$ is a tensoring for ${\mathcal{C}}$ then we get a cotensoring for ${\mathcal{C}}^\text{op}$ by setting $$\operatorname{Map}_{{\mathcal{C}}^\text{op}}(X,D^\text{op}) := (X \otimes D)^\text{op}.$$ The required natural isomorphism comes from $$\begin{split}
\operatorname{Map}_{{\mathcal{V}}}(C^\text{op}, \operatorname{Map}_{{\mathcal{C}}^\text{op}}(X,D^\text{op})) &=
\operatorname{Map}_{{\mathcal{V}}}(X \otimes D, C) \\
&{\cong}\operatorname{Map}(X,\operatorname{Map}_{{\mathcal{V}}}(D,C)) \\
&= \operatorname{Map}(X,\operatorname{Map}_{{\mathcal{V}}}(C^\text{op},D^\text{op})). \\
\end{split}$$ The vice versa part is similar.
We are interested in categories that both are themselves symmetric monoidal categories and are enriched over another symmetric monoidal category. The following definition contains the properties of these that we require in this paper.
\[def:axiom\] Let $({\mathcal{V}},{\wedge},I)$ be a closed symmetric monoidal category. A *symmetric monoidal ${\mathcal{V}}$–category* consists of
- a symmetric monoidal category $({\mathcal{C}},\barwedge,S)$ with ${\mathcal{C}}$ enriched, tensored and cotensored over ${\mathcal{V}}$,
- a natural transformation $$d\co (X {\wedge}Y) \otimes (C \barwedge D) \to (X \otimes C) \barwedge
(Y \otimes D)$$
satisfying the following axioms:
- (Associativity) The diagram $$\begin{diagram} \dgARROWLENGTH=2.4em
\node{(X {\wedge}Y {\wedge}Z) \otimes (C \barwedge D \barwedge E)}
\arrow{e,t}{d} \arrow{s,l}{d}
\node{((X {\wedge}Y) \otimes (C \barwedge D)) \barwedge (Z \otimes
E)} \arrow{s,r}{id \, \barwedge \, d} \\
\node{(X \otimes C) \barwedge ((Y {\wedge}Z) \otimes (D \barwedge E))}
\arrow{e,t}{id \, \barwedge \, d}
\node{(X \otimes C) \barwedge (Y \otimes D) \barwedge (Z
\otimes E)}
\end{diagram}$$ commutes for all $X,Y,Z \in {\mathcal{V}}$ and $C,D,E \in {\mathcal{C}}$.
- (Unit) The composite $$X \otimes C {\cong}(X {\wedge}I) \otimes (C \barwedge S) \arrow{e,t}{d}
(X \otimes C) \barwedge (I \otimes S) {\cong}X \otimes C$$ is the identity, for any $X \in {\mathcal{V}}$ and $C \in {\mathcal{C}}$. Recall that $I,S$ are the units of the symmetric monoidal structures on ${\mathcal{V}},
{\mathcal{C}}$ respectively.
The transformation $d$ (for ‘distribute’) is our way of relating the symmetric monoidal structures in the two categories. It will be essential in constructing the cooperad structure on the bar construction of an operad (see Definition \[def:formal\_cooperad\_maps\]).
A closed symmetric monoidal category ${\mathcal{V}}$ is itself a symmetric monoidal ${\mathcal{V}}$–category with the transformation $d$ given by the symmetry and associativity isomorphism: $$(X {\wedge}Y) {\wedge}(C {\wedge}D) {\cong}(X {\wedge}C) {\wedge}(Y {\wedge}D)$$
\[prop:dual\] Let ${\mathcal{C}}$ be a symmetric monoidal ${\mathcal{V}}$–category. Then ${\mathcal{C}}^\text{op}$ is naturally also a symmetric monoidal ${\mathcal{V}}$–category.
We already know from Proposition \[prop:dual1\] that ${\mathcal{C}}^\text{op}$ is enriched, tensored and cotensored over ${\mathcal{V}}$ and there is a canonical symmetric monoidal structure on ${\mathcal{C}}^\text{op}$ given by that on ${\mathcal{C}}$. It therefore only remains to construct the map $d$. The tensoring in ${\mathcal{C}}^\text{op}$ is given by the cotensoring in ${\mathcal{C}}$. Therefore $d$ for ${\mathcal{C}}^\text{op}$ corresponds to the following map in ${\mathcal{C}}$: $$\operatorname{Map}_{{\mathcal{C}}}(X,C) \barwedge \operatorname{Map}_{{\mathcal{C}}}(Y,D) \to \operatorname{Map}_{{\mathcal{C}}}(X
{\wedge}Y, C \barwedge D)$$ This is adjoint to a map $$(X {\wedge}Y) \otimes (\operatorname{Map}_{{\mathcal{C}}}(X,C) \barwedge \operatorname{Map}_{{\mathcal{C}}}(Y,D))
\to C \barwedge D$$ constructed by first using $d$ for ${\mathcal{C}}$ to get to $$(X \otimes \operatorname{Map}_{{\mathcal{C}}}(X,C)) \barwedge (Y \otimes
\operatorname{Map}_{{\mathcal{C}}}(Y,D))$$ and then using the evaluation maps $$X \otimes \operatorname{Map}_{{\mathcal{C}}}(X,C) \to C \text{ and } Y \otimes
\operatorname{Map}_{{\mathcal{C}}}(Y,D) \to D. \eqno{\Box}$$ An important property of the categories that we work with in this paper is that they are *pointed*, that is, they have a *null* object ${\ast}$ that is both initial and terminal. The following proposition describes how null objects interact with symmetric monoidal structures and enrichments.
\[prop:null\] Let $({\mathcal{V}},{\wedge},I)$ be a closed symmetric monoidal category that is pointed with null object ${\ast}$. Then $${\ast}{\wedge}X {\cong}{\ast}{\cong}\operatorname{Map}({\ast},X) {\cong}\operatorname{Map}(X,{\ast})$$ for all $X \in {\mathcal{V}}$.
Moreover, let ${\mathcal{C}}$ be a category enriched over ${\mathcal{V}}$. If ${\mathcal{C}}$ is tensored then ${\ast}\otimes C$ is an initial object in ${\mathcal{C}}$ for all $C \in {\mathcal{C}}$. If ${\mathcal{C}}$ is cotensored then $\operatorname{Map}_{{\mathcal{C}}}({\ast},D)$ is a terminal object in ${\mathcal{C}}$ for all $D \in {\mathcal{C}}$.
Finally, if ${\mathcal{C}}$ is both tensored and cotensored over ${\mathcal{V}}$, then the initial and terminal objects are isomorphic and so ${\mathcal{C}}$ is itself pointed.
We observe that $$\operatorname{Hom}_{{\mathcal{V}}}({\ast}{\wedge}X, Y) {\cong}\operatorname{Hom}_{{\mathcal{V}}}({\ast},\operatorname{Map}(X,Y))$$ which has one element for any $X,Y$. This tells us that ${\ast}{\wedge}X$ is initial and hence isomorphic to ${\ast}$. The other isomorphisms in the first part of the proposition are similar.
Next, the tensoring functor $- \otimes C \co {\mathcal{V}} \to {\mathcal{C}}$ is a left adjoint so preserves an initial object. Dually, the cotensoring functor $\operatorname{Map}_{{\mathcal{C}}}(-,D) \co {\mathcal{V}}^\text{op} \to {\mathcal{C}}$ is a right adjoint so preserves the terminal object. This gives us the second part.
Finally, if ${\mathcal{C}}$ is both tensored and cotensored, we get a map from the terminal object to the initial object by $$\operatorname{Map}_{{\mathcal{C}}}({\ast},D) \to I \otimes \operatorname{Map}_{{\mathcal{C}}}({\ast},D)
\to {\ast}\otimes \operatorname{Map}_{{\mathcal{C}}}({\ast},D).$$ The first map here is an example of a general isomorphism $C \to I \otimes
C$ where $I$ is the unit object of ${\mathcal{V}}$. The second map comes from $I \to {\ast}$. A map from a terminal object to an initial object must be an isomorphism. Therefore ${\mathcal{C}}$ is pointed.
\[ex:categories\] The categories with which we will mainly be concerned in this paper are the following.
1. Let ${\mathcal{T}_{}}$ be the category of compactly generated based spaces and basepoint-preserving continuous maps of [@lewis/may/steinberger:1986]. Then ${\mathcal{T}_{}}$ is a pointed closed symmetric monoidal category under the usual smash product ${\wedge}$, with unit the $0$–sphere $S^0$ and $\operatorname{Map}(X,Y)$ equal to the space of basepoint-preserving maps $X \to Y$.
2. Let ${\mathcal{S}p}$ be the category of $S$–modules of EKMM [@elmendorf/kriz/mandell/may:1997]. Then $({\mathcal{S}p},{\wedge}_S,S)$ is a symmetric monoidal ${\mathcal{T}_{}}$–category, where $S$ is the sphere spectrum and ${\wedge}_S$ is the smash product of $S$–modules [@elmendorf/kriz/mandell/may:1997 Section II.1.1]. The enrichment, tensoring and cotensoring are described in [@elmendorf/kriz/mandell/may:1997 Section VII.2.8]. For the distributivity map $d$ we have a natural isomorphism $$d\co (X {\wedge}Y) {\wedge}(E {\wedge}_S F) \arrow{e,t}{{\cong}} (X {\wedge}E)
{\wedge}_S (Y {\wedge}F)$$ given by the fact that $X {\wedge}E {\cong}(X {\wedge}S) {\wedge}_S E$ (see [@elmendorf/kriz/mandell/may:1997 Section II.1.4]).
We will usually work with a general symmetric monoidal ${\mathcal{T}_{}}$–category denoted $({\mathcal{C}},\barwedge,S)$, but these examples will be foremost in our minds.
Operads and cooperads {#sec:operads}
=====================
In this section $({\mathcal{C}},\barwedge,S)$ denotes a pointed symmetric monoidal category with null object ${\ast}$. We will assume that ${\mathcal{C}}$ has all necessary limits and colimits and write the coproduct in ${\mathcal{C}}$ as a wedge product using $\vee$.
\[def:symseq\] A *symmetric sequence* in ${\mathcal{C}}$ is a functor $F$ from the category of nonempty finite sets and bijections to ${\mathcal{C}}$. For each nonempty finite set $A$, the symmetric group $\Sigma_A$ acts on $F(A)$. We will write $F(n)$ for $F(\{1,\dots,n\})$. Note that our symmetric sequences (and hence our operads) do not have an $F(0)$ term because our indexing sets are nonempty. We will often write ‘finite set’ when we mean ‘nonempty finite set’ and these will usually be labelled $A,B,\dots$. We write ${\mathcal{C}}^{\Sigma}$ for the category of symmetric sequences in ${\mathcal{C}}$ (whose morphisms are the natural transformations).
There are several different but equivalent ways to define operads (see Markl–Shnider–Stasheff [@markl/shnider/stasheff:2002] for a comprehensive guide). We will use the following definition.
\[def:operad\] An *operad* in the symmetric monoidal category $({\mathcal{C}},\barwedge,S)$ is a symmetric sequence $P$ together with *partial composition maps* $$- \circ_a - \co P(A) \barwedge P(B) \to P(A \cup_a B)$$ for each pair of finite sets $A,B$, and each $a \in A$ (where $A \cup_a B$ denotes $(A \setminus \{a\}) \amalg B$), and a *unit map* $$\eta \co S \to P(1).$$ The composition maps must be natural in $A$ and $B$ and must satisfy the following four axioms:
1. The diagram $$\begin{diagram}
\node{P(A) \barwedge P(B) \barwedge P(C)} \arrow{e,t}{id \, \barwedge
\, \circ_b} \arrow{s,l}{\circ_a \, \barwedge \, id} \node{P(A)
\barwedge P(B \cup_b C)} \arrow{s,r}{\circ_a} \\
\node{P(A \cup_a B) \barwedge P(C)} \arrow{e,t}{\circ_b} \node{P(A
\cup_a B \cup_b C)}
\end{diagram}$$ commutes for all $a \in A$ and $b \in B$. (Notice that $(A \cup_a B)
\cup_b C = A \cup_a (B \cup_b C)$.)
2. The diagram $$\begin{diagram}
\node{P(A) \barwedge P(B) \barwedge P(C)} \arrow{s,l}{{\cong}}
\arrow{e,t}{\circ_a \, \barwedge \, id} \node{P(A \cup_{a} B)
\barwedge P(C)} \arrow[2]{s,r}{\circ_{a'}} \\
\node{P(A) \barwedge P(C) \barwedge P(B)} \arrow{s,l}{\circ_{a'} \,
\barwedge \, id} \\
\node{P(A \cup_{a'} C) \barwedge P(B)} \arrow{e,t}{\circ_a} \node{P(A
\cup_a B \cup_{a'} C)}
\end{diagram}$$ commutes for all $a \neq a' \in A$. (Notice that $(A \cup_a B) \cup_{a'}
C = (A \cup_{a'} C) \cup_a B$.)
3. The diagram $$\begin{diagram}
\node{P(A)} \arrow{e,t}{\eta \, \barwedge \, id} \arrow{se,b}{id}
\node{P(1) \barwedge P(A)} \arrow{s,r}{\circ_1} \\
\node[2]{P(\{1\} \cup_{1} A)}
\end{diagram}$$ commutes for all $A$.
4. The diagram $$\begin{diagram}
\node{P(A)} \arrow{e,t}{id \, \barwedge \, \eta} \arrow{se,b}{{\cong}}
\node{P(A) \barwedge P(1)} \arrow{s,r}{\circ_a} \\
\node[2]{P(A \cup_a \{1\})}
\end{diagram}$$ commutes for all $a \in A$. (The diagonal map here is induced by the obvious bijection $A \to A \cup_a \{1\}$.)
A *morphism* of operads $P \to P'$ is a morphism of symmetric sequences that commutes with the composition and unit maps.
\[def:reduced\] An *augmentation* of an operad $P$ is a map $\varepsilon\co P(1)
\to S$ such that the composite $$\begin{diagram} \node{S} \arrow{e,t}{\eta} \node{P(1)}
\arrow{e,t}{\varepsilon} \node{S} \end{diagram}$$ is the identity on $S$. An *augmented operad* is an operad together with an augmentation. An operad $P$ is *reduced* if the unit map $\eta\co S \to P(1)$ is an isomorphism. A reduced operad has a unique augmentation given by the inverse of the unit map. A *morphism* of augmented operads is a morphism of operads that commutes with the augmentation.
\[rem:operad\] Operads are a generalization of monoids for the symmetric monoidal category $({\mathcal{C}},\barwedge,S)$. A monoid $X$ in ${\mathcal{C}}$ gives rise to an operad $P_X$ with $P_X(1) = X$ and $P_X(n) = {\ast}$ for $n >
1$. Conversely, given an operad $P$ in the symmetric monoidal category ${\mathcal{C}}$, $P(1)$ is a monoid in ${\mathcal{C}}$.
An alternative definition of an operad is based on a monoidal structure on the category of symmetric sequences. We define this monoidal structure now.
\[def:compprod\] Let the *composition product* of the two symmetric sequences $M,N$ be the symmetric sequence $M \circ N$ with $$(M \circ N)(A) := \bigvee_{A = \coprod_{j \in J} A_j} M(J) \barwedge
\operatorname*{\overline{\bigwedge}}_{j \in J} N(A_j).$$ The coproduct here is taken over all unordered partitions of $A$ into a collection of nonempty subsets $\{A_j\}_{j \in J}$. The particular choice of indexing set is not important in the sense that we do not sum over different $J$ that index the same partition. A bijection $A \to A'$ determines a bijection between partitions of $A$ and partitions of $A'$ in an obvious way. Thus we match up the terms in the coproducts that define $(M \circ N)(A)$ and $(M \circ N)(A')$. If $J$ and $J'$ index two corresponding partitions of $A$ and $A'$ respectively, then we get a natural choice of bijection $J \to J'$. Moreover, if $j \in J$ and $j'
\in J'$ correspond under this bijection then we get a bijection $A_j
\to A'_{j'}$ by restricting the bijection $A \to A'$. The actions of $M$ and $N$ on these bijections together give us an isomorphism $$(M \circ N)(A) \to (M \circ N)(A').$$ Thus $M \circ N$ becomes a symmetric sequence in ${\mathcal{C}}$.
The *unit symmetric sequence* in the pointed symmetric monoidal category $({\mathcal{C}},\barwedge,S)$ is the symmetric sequence $I$ given by $$I(A) := \begin{cases} S & \text{if $|A| = 1$}; \\ {\ast}&
\text{otherwise}; \end{cases}$$ where ${\ast}$ is the null object of ${\mathcal{C}}$.
\[lem:circ\_unit\] Let $({\mathcal{C}},\barwedge,S)$ be a pointed symmetric monoidal category. Then for any symmetric sequence $M$ there are natural isomorphisms $$M \circ I {\cong}M {\cong}I \circ M.$$
For the finite set $A$, the only term that contributes to $(M \circ I)(A)$ comes from the partition of $A$ into singleton subsets. This makes it clear that $M \circ I {\cong}M$. The only term that contributes to $(I
\circ M)(A)$ comes from the trivial partition of $A$ into one subset, that is $A$ itself. From this we see that $I \circ M {\cong}M$.
To get a monoidal structure on the category of symmetric sequences, we also need an associativity isomorphism. This does not exist in general, although it does in the case of the following lemma.
\[lem:circ\_assoc\] Let $({\mathcal{C}},\barwedge,S)$ be a pointed symmetric monoidal category in which $\barwedge$ commutes with finite coproducts. Then there are natural isomorphisms $$L \circ (M \circ N) {\cong}(L \circ M) \circ N$$ for symmetric sequences $L,M,N$ in ${\mathcal{C}}$.
Using the hypothesis that $\barwedge$ commutes with finite coproducts, it is not hard to see that each side is naturally isomorphic to the symmetric sequence $(L \circ M \circ N)$ given by $$(L \circ M \circ N)(A) := \bigvee_{A = \coprod_{b \in B} A_b, \; B =
\coprod_{c \in C} B_c} L(C) \barwedge \operatorname*{\overline{\bigwedge}}_{c \in C} M(B_c)
\barwedge \operatorname*{\overline{\bigwedge}}_{b \in B} N(A_b).$$ The coproduct here is over all partitions of $A$ into nonempty subsets indexed by some set $B$, together with a partition of $B$ into subsets indexed by some $C$. Equivalently, the coproduct is indexed of pairs of partitions of $A$, one (indexed by $B$) a refinement of the other (indexed by $C$).
The following description of operads is due to Smirnov. See [@markl/shnider/stasheff:2002 Theorem 1.68] for further details.
\[prop:comp\_monoidal\] Let $({\mathcal{C}},\barwedge,S)$ be a pointed symmetric monoidal category in which $\barwedge$ commutes with finite coproducts. Then the composition product $\circ$ is a monoidal product on the category of symmetric sequences in ${\mathcal{C}}$ with unit object $I$ and unit and associativity isomorphisms given by Lemmas \[lem:circ\_unit\] and \[lem:circ\_assoc\] respectively. In this case, an operad in ${\mathcal{C}}$ is precisely a monoid for this monoidal product.
One can easily check that the axioms for a monoidal structure are satisfied. If $P$ is an operad in ${\mathcal{C}}$, the operad compositions make up a map $$P \circ P \to P$$ and the unit map $\eta$ gives a map of symmetric sequences $$I \to P.$$ The operad axioms then translate into associativity and unit axioms that give $P$ the structure of a monoid under $\circ$.
\[rem:triple\_product\] If ${\mathcal{C}}$ is a *closed* symmetric monoidal category then $\barwedge$ has a right adjoint and so preserves all colimits. In particular, the hypothesis of Lemma \[lem:circ\_assoc\] holds and so we get a true monoidal structure on the symmetric sequences in ${\mathcal{C}}$.
Unfortunately, even when ${\mathcal{C}}$ is closed symmetric monoidal, its opposite category ${\mathcal{C}}^\text{op}$ (with the standard symmetric monoidal structure) is unlikely to be closed. Since we will want to dualize most of the results of this paper to be able to deal with cooperads as well as operads, we need to get round this hypothesis. For this, we notice that in general there are natural maps of symmetric sequences $$(L \circ M \circ N) \to L \circ (M \circ N)$$ and $$(L \circ M \circ N) \to (L \circ M) \circ N$$ where $(L \circ M \circ N)$ is defined as in the proof of Lemma \[lem:circ\_assoc\]. In general these are not isomorphisms so we do not get a monoidal structure on the category of symmetric sequences. However, it is possible to define monoids in this more general case (see [@ching:2005c] for more details), and we get the following alternative characterization of an operad.
\[prop:comp\_general\] Let $({\mathcal{C}},\barwedge,S)$ be a pointed symmetric monoidal category. An operad in ${\mathcal{C}}$ is equivalent to a symmetric sequence $P$ together with maps $$m \co P \circ P \to P ; \; \eta\co I \to P$$ of symmetric sequences such that the following diagrams commute:
1. Associativity: $$\begin{diagram} \node[2]{(P \circ P) \circ P} \arrow{e,t}{m \, \circ \,
id} \node{P \circ P} \arrow{se,t}{m} \\
\node{(P \circ P \circ P)} \arrow{ne} \arrow{se} \node[3]{P}
\\
\node[2]{P \circ (P \circ P)} \arrow{e,t}{id \, \circ \, m}
\node{P \circ P} \arrow{ne,t}{m}
\end{diagram}$$ where the two initial arrows are the maps mentioned in Remark \[rem:triple\_product\].
2. Left unit: $$\begin{diagram} \node{P} \arrow{se,b}{id} \arrow{e,t}{id \, \circ \,
\eta} \node{P \circ P} \arrow{s,r}{m} \\ \node[2]{P} \end{diagram}$$
3. Right unit: $$\begin{diagram} \node{P} \arrow{se,b}{id} \arrow{e,t}{\eta \, \circ \,
id} \node{P \circ P} \arrow{s,r}{m} \\ \node[2]{P} \end{diagram}$$
We will refer to an operad $P$ as a *monoid* with respect to the composition product, even when we do not in fact have a monoidal structure. There are similarly defined notions of an object with a right or left action of a monoid in this generalized setting. These give us right and left modules over our operads which we now define.
\[def:module\] A *left module* over the operad $P$ is a symmetric sequence $M$ together with a left action of the monoid $P$, that is, a map $$l\co P \circ M \to M$$ such that the diagrams $$\begin{diagram} \node[2]{(P \circ P) \circ M} \arrow{e,t}{m \, \circ \,
id} \node{P \circ M} \arrow{se,t}{l} \\
\node{(P \circ P \circ M)} \arrow{se} \arrow{ne} \node[3]{M}
\\
\node[2]{P \circ (P \circ M)} \arrow{e,t}{id \, \circ \, l}
\node{P \circ M} \arrow{ne,t}{l}
\end{diagram}$$ and $$\begin{diagram} \node{M} \arrow{e,t}{\eta \, \circ \, id}
\arrow{se,b}{id} \node{P \circ M} \arrow{s,r}{l} \\ \node[2]{M}
\end{diagram}$$ commute.
A *right module* over $P$ is a symmetric sequence $M$ with a right action of $P$, that is a map $$M \circ P \to M$$ satisfying corresponding axioms. A *$(P,P)$–bimodule* is a symmetric sequence $M$ that is both a right and a left module over $P$ such that $$\begin{diagram} \node[2]{(P \circ M) \circ P} \arrow{e} \node{M \circ
P} \arrow{se} \\
\node{P \circ M \circ P} \arrow{ne} \arrow{se} \node[3]{M} \\
\node[2]{P \circ (M \circ P)} \arrow{e} \node{P \circ M}
\arrow{ne}
\end{diagram}$$ commutes. Clearly, $P$ itself is a $(P,P)$–bimodule.
It’s useful to have a slightly more explicit description of a module over an operad. The action map for a left $P$–module $M$ consists of maps $$P(r) \barwedge M(A_1) \barwedge \dots \barwedge M(A_r) \to M(A)$$ for every partition $A = \coprod_{i = 1}^{r} A_i$ of a finite set $A$ into nonempty subsets. Conversely, giving maps of this form that satisfy appropriate conditions uniquely determines a left $P$–module. Similarly, a right module structure consists of maps of the form $$M(r) \barwedge P(A_1) \barwedge \dots \barwedge P(A_r) \to M(A).$$
In the same way that operads are a generalization of monoids in ${\mathcal{C}}$, modules over those operads are generalization of modules over the monoids. A module $M$ over the monoid $X$ gives rise to a module $P_M$ over the operad $P_X$ described in Remark \[rem:operad\], with $P_M(n)
= {\ast}$ if $n > 1$ and $P_M(1) = M$.
An augmentation for the operad $P$ is equivalent to either a left or right module structure on the unit symmetric sequence $I$.
The standard notion of an algebra over an operad is closely related to that of a module. We briefly describe how this works.
An *algebra* over the operad $P$ is an object $C \in {\mathcal{C}}$ together with maps $$P(A) \barwedge \operatorname*{\overline{\bigwedge}}_{a \in A} C \to C$$ that satisfy appropriate naturality, associativity and unit axioms.
The following result allows us to construct a left $P$–module from a $P$–algebra.[^5]
\[lem:algebra\] Let $C$ be an algebra over the operad $P$. Then there is a natural left $P$–module structure on the constant symmetric sequence $\underline{C}$ with $\underline{C}(A) = C$ for all finite sets $A$.[^6]
The components of the module structure map $P \circ \underline{C} \to
\underline{C}$ are given by the algebra structure maps as follows: $$P(r) \barwedge \underline{C}(A_1) \barwedge \dots \barwedge
\underline{C}(A_r) = P(r) \barwedge C^{\barwedge r} \to C =
\underline{C}(A) \eqno{\Box}$$
\[def:cooperad\] The notion of a cooperad is dual to that of an operad. That is, a *cooperad* in ${\mathcal{C}}$ is an operad in the opposite category ${\mathcal{C}}^\text{op}$ with the canonical symmetric monoidal structure determined by that in ${\mathcal{C}}$. More explicitly, a cooperad consists of a symmetric sequence $Q$ in ${\mathcal{C}}$ together with *cocomposition maps* $$Q(A \cup_a B) \to Q(A) \barwedge Q(B)$$ and a *counit map* $$Q(1) \to S$$ satisfying axioms dual to (1)–(4) of Definition \[def:operad\]. A *morphism of cooperads* is a morphism of symmetric sequences that commutes with the cocomposition and counit maps. A *coaugmentation* for a cooperad is a map $S \to Q(1)$ left inverse to the counit map. A cooperad $Q$ is *reduced* if the counit map is an isomorphism.
\[rem:cooperads\] The description of an operad as a monoid for the composition product of symmetric sequences naturally dualizes to cooperads. We define the *dual composition product* ${\mathbin{\widehat{\circ}}}$ of two symmetric sequences by replacing the coproduct in Definition \[def:compprod\] with a product. That is: $$M {\mathbin{\widehat{\circ}}}N (A) := \prod_{A = \coprod_{j \in J} A_j} M(J) \barwedge
\operatorname*{\overline{\bigwedge}}_{j \in J} N(A_j).$$ If $\barwedge$ commutes with finite products (which is in general not likely) this is a monoidal product of symmetric sequences (the result dual to Proposition \[prop:comp\_monoidal\]) and a cooperad is precisely a comonoid for this product. In general we can define the triple product $(L {\mathbin{\widehat{\circ}}}M {\mathbin{\widehat{\circ}}}N)$ by replacing coproduct with product in the definition given in the proof of Lemma \[lem:circ\_assoc\]. We then have natural maps $$(L {\mathbin{\widehat{\circ}}}M) {\mathbin{\widehat{\circ}}}N \to (L {\mathbin{\widehat{\circ}}}M {\mathbin{\widehat{\circ}}}N)
\quad\mbox{and}\quad
L {\mathbin{\widehat{\circ}}}(M {\mathbin{\widehat{\circ}}}N) \to (L {\mathbin{\widehat{\circ}}}M {\mathbin{\widehat{\circ}}}N)$$ which allow us to say what we mean by a comonoid in general. Thus we get the result dual to Proposition \[prop:comp\_general\], that a cooperad in ${\mathcal{C}}$ is a symmetric sequence $Q$ together with maps $$Q \to Q {\mathbin{\widehat{\circ}}}Q \quad\mbox{and}\quad Q \to I$$ such that the corresponding diagrams commute. In particular we have a coassociativity diagram: $$\begin{diagram} \node[2]{Q {\mathbin{\widehat{\circ}}}Q} \arrow{e} \node{(Q {\mathbin{\widehat{\circ}}}Q)
{\mathbin{\widehat{\circ}}}Q} \arrow{se} \\
\node{Q} \arrow{ne} \arrow{se} \node[3]{(Q {\mathbin{\widehat{\circ}}}Q {\mathbin{\widehat{\circ}}}Q)} \\
\node[2]{Q {\mathbin{\widehat{\circ}}}Q} \arrow{e} \node{Q {\mathbin{\widehat{\circ}}}(Q {\mathbin{\widehat{\circ}}}Q)} \arrow{ne}
\end{diagram}$$
In [@getzler/jones:1994] Getzler and Jones define a cooperad to be a comonoid for the composition product $\circ$. In their case, $\circ$ and ${\mathbin{\widehat{\circ}}}$ are equal because finite products are isomorphic to finite coproducts in the category of chain complexes.
\[def:comodule\] A *left comodule* $C$ over the cooperad $Q$ is a left module over $Q$ considered as an operad in ${\mathcal{C}}^\text{op}$. More explicitly, $C$ is a symmetric sequence together with a left coaction of the comonoid $Q$, that is, a map $C \to Q {\mathbin{\widehat{\circ}}}C$. Equivalently, we have a suitable collection of *cocomposition maps* $$C(A) \to Q(r) \barwedge C(A_1) \barwedge \dots \barwedge C(A_r)$$ for partitions $A = \coprod_{i=1}^{r} A_i$. Similarly a *right comodule* is a symmetric sequence $C$ with a right coaction $C \to C
{\mathbin{\widehat{\circ}}}Q$, or equivalently, *cocomposition maps* $$C(A) \to C(r) \barwedge Q(A_1) \barwedge \dots \barwedge Q(A_r).$$ A *bicomodule* is a symmetric sequence with compatible left and right comodule structures. The cooperad $Q$ is itself a $(Q,Q)$–bicomodule.
A *coalgebra* over a cooperad is the dual concept of an algebra over an operad and the constant symmetric sequence with value equal to a $Q$–coalgebra is a left $Q$–comodule.
Spaces of trees {#sec:trees}
===============
As mentioned in the introduction to the paper, the key to finding a cooperad structure on the bar construction on an operad is its reinterpretation in terms of trees. These are the same sorts of trees used in many other places to work with operads. See Getzler–Jones [@getzler/jones:1994], Ginzburg–Kapranov [@ginzburg/kapranov:1994] and Markl–Shnider–Stasheff [@markl/shnider/stasheff:2002] for many examples.
\[def:tree\] A typical tree of the sort we want is shown in Figure \[fig:trees\]. It has a root element at the base, a single edge attached to the root, and no other vertices with only one incoming edge. We encode these geometric requirements in the following combinatorial definition. A *tree* $T$ is a finite poset satisfying the following conditions:
1. $T$ has at least two elements: an initial (or minimal) element $r$, the *root*, and another element $b$ such that $b \leq t$ for all $t \in T$, $t \neq r$.
2. For any elements $t,u,v \in T$, if $u \leq t$ and $v \leq t$, then either $u \leq v$ or $v \leq u$.
3. For any $t < u$ in $T$ with $t \neq r$, there is some $v \in T$ such that $t < v$ but $u \nleq v$.
We picture a tree by its *graph*, whose vertices are the elements of $T$ with an edge between $t$ and $u$ if $t < u$ and there is no $v$ with $t < v < u$. An *incoming* edge to a vertex $t$ is an edge corresponding to some relation $t < u$. Condition (1) above ensures that the tree has a root $r$ with exactly one incoming edge (that connects it to $b$). The second condition ensures that this graph is indeed a tree in the usual sense. The third condition ensures that no vertices except the root have exactly one incoming edge.
More terminology: the maximal elements of the tree $T$ will be called *leaves*. From now on, by a *vertex*, we mean an element other than the root or a leaf (see Figure \[fig:trees\]). A tree is *binary* if each vertex has precisely two incoming edges. The *root edge* is the edge connected to the root element. The *leaf edges* are the edges connected to the leaves. The other edges in the tree are *internal edges*. Given a vertex $v$ of a tree, we write $i(v)$ for the set of incoming edges of the vertex $v$. We generally denote trees with the letters $T,U,\dots$.
\[rem:reduced\_trees\] We stress that our trees are *not* allowed to have vertices with only one incoming edge, as guaranteed by condition (3) of the definition. This reflects the fact that we will deal only with *reduced* operads in this paper.
\[def:labelling\] A *labelling* of the tree $T$ by a finite set $A$ is a bijection between $A$ and the set of leaves of $T$. An *isomorphism* of $A$–labelled trees is an isomorphism of the underlying trees that preserves the labelling. We denote the set of isomorphism classes of $A$–labelled trees by $\mathsf{T}(A)$. For a finite set $A$, $\mathsf{T}(A)$ is also finite. For a positive integer $n$, we write $\mathsf{T}(n)$ for the set $\mathsf{T}(\{1,\dots,n\})$.
There is up to isomorphism only one tree with one leaf. It has a single edge whose endpoints are the root and the leaf. Thus $\mathsf{T}(1)$ has one element. It is easy to see that $\mathsf{T}(2)$ also only has one element: the tree with one vertex that has two input edges. Figure \[fig:tree\_examples\] shows $\mathsf{T}(1),\mathsf{T}(2),\mathsf{T}(3)$.
Given a tree $T$ and an internal edge $e$, denote by $T/e$ the tree obtained by collapsing the edge $e$, identifying its endpoints. (In poset terms, this is equivalent to removing from the poset the element corresponding to the upper endpoint of the edge.) If $u$ and $v$ are those endpoints, write $u \circ v$ for the resulting vertex of $T/e$. Note that $T/e$ has the same leaves as $T$ so retains any labelling. See Figure \[fig:edge\_collapse\] for an example.
The process of collapsing edges gives us a partial order on the set $\mathsf{T}(A)$ of isomorphism classes of $A$–labelled trees. We say that $T \leq T'$ if $T$ can be obtained from $T'$ be collapsing a sequence of edges. We think of the resulting poset as a category.
We now give our trees topological significance by introducing ‘weightings’ on them.
\[def:weighting\] A *weighting* on a tree $T$ is an assignment of nonnegative ‘lengths’ to the edges of $T$ in such a way that the ‘distance’ from the root to each leaf is exactly $1$. The set of weightings on a tree $T$ is a subset of the space of functions from the set of edges of $T$ to the unit interval $[0,1]$ and we give it the subspace topology. We denote the resulting space by $w(T)$. A tree together with a weighting is a *weighted tree*.
There is only one way to weight the unique tree $T \in \mathsf{T}(1)$ (the single edge must have length $1$), so $w(T) = {\ast}$. For any $n$, $\mathsf{T}(n)$ contains a tree $T_n$ with a single vertex that has $n$ incoming edges. For this tree we have $w(T_n) = \Delta^1$ the topological $1$–simplex or unit interval. Figure \[fig:tree\_examples\] displays another shape of tree with three leaves, one that has two vertices. For such a tree $U$, we have $w(U) = \Delta^2$, the topological $2$–simplex. Not all spaces of weightings are simplices, but we do have the following result.
\[lem:W\] Let $T$ be a tree with $n$ (internal) vertices. Then $w(T)$ is homeomorphic to the $n$–dimensional disc $D^n$. If $n \geq 1$, the boundary $\partial w(T)$ is the subspace of weightings for which at least one edge has length zero.
Suppose $T$ has $l$ leaves. Then it has $n+l$ total edges and using the lengths of the edges as coordinates we can think of $w(T)$ as a subset of $\mathbb{R}^{n+l}$. For each leaf $l_i$ of $T$ there is a condition on the lengths of the edges in a weighting that translates into an affine hyperplane $H_i$ in $\mathbb{R}^{n+l}$. Then $w(T)$ is the intersection of all these hyperplanes with $[0,1]^{n+l}$.
Now these hyperplanes all pass through the point that corresponds to the root edge having length $1$ and all other edges length zero. Therefore their intersection is another affine subspace of $\mathbb{R}^{n+l}$. To see that they intersect transversely, we check that each $H_i$ does not contain the intersection of the $H_j$ for $j \neq i$. Consider the point $p_i$ in $\mathbb{R}^{n+l}$ that assigns length $1$ to each leaf edge except that corresponding to leaf $l_i$, and length $0$ to all other edges (including the leaf edge for $l_i$). Since the equation for the hyperplane $H_j$ contains the length of exactly one leaf edge, this point $p_i$ is in $$\bigcap_{j \neq i} H_j$$ but not in $H_i$. This shows that the $H_i$ do indeed intersect transversely. Therefore their intersection is an $n$–dimensional affine subspace $V$ of $\mathbb{R}^{n+l}$.
Finally, notice that, as long as $n > 0$, $V$ passes through an interior point of $[0,1]^{n+l}$, for example, the point where all edges except the leaf edges have length $\varepsilon$ for some small $\varepsilon >
0$ and the leaf edges then have whatever lengths they must have to obtain a weighting. It then follows that $w(T) = V \cap [0,1]^{n+l}$ is homeomorphic to $D^n$. If $n = 0$, there is only one tree and its space of weightings is a single point, that is, $D^0$.
For the second statement, notice that the boundary of $w(T)$ is the intersection of $V$ with the boundary of the cube $[0,1]^{n+l}$. If a weighting includes an edge of length zero, it lies in this boundary. Conversely, a weighting in this boundary must have some edge with length either $0$ or $1$. If the root edge has length $1$, all other edges must have length $0$. If some other edge has length $1$, the root edge must have length $0$. In any case, some edge has length $0$.
\[def:w\_functor\] For each finite set $A$, the assignment $T \mapsto w(T)$ determines a functor $$w(-)\co \mathsf{T}(A) \to {\mathcal{U}_{}}$$ where ${\mathcal{U}_{}}$ is the category of unbased spaces. To see this we must define maps $$w(T/e) \to w(T)$$ whenever $e$ is an internal edge in the $A$–labelled tree $T$. Given a weighting on $T/e$ we define a weighting on $T$ by giving edges in $T$ their lengths in $T/e$ with the edge $e$ having length zero. This is an embedding of $w(T/e)$ as a ‘face’ of the ‘simplex’ $w(T)$. It’s easy to check that this defines a functor as claimed.
Let $w_0(T)$ be the subspace of $w(T)$ containing weightings for which either the root edge or some leaf edge has length zero. We set $$\overline{w}(T) := w(T)/w_0(T).$$ This is a based space with basepoint given by the point to which $w_0(T)$ has been identified. If $T$ is the tree with only one edge then $w_0(T)$ is empty. We use the convention that taking the quotient by the empty set is equivalent to adjoining a disjoint basepoint. So in this case, $\overline{w}(T) = S^0$.
The maps $w(T/e) \to w(T)$ clearly map $w_0(T/e)$ to $w_0(T)$ and so give us maps $$\overline{w}(T/e) \to \overline{w}(T).$$ For each finite set $A$, these form a functor $$\overline{w}(-) \co \mathsf{T}(A) \to {\mathcal{T}_{}}$$ where ${\mathcal{T}_{}}$ is the category of based spaces.
\[ex:weightings\] Figure \[fig:w\_functor\] displays the spaces $w(T)$ for $T \in
\mathsf{T}(3)$ and how the functor $w(-)$ fits them together. Recall that the poset $\mathsf{T}(3)$ has four objects: one minimal object (the tree with one vertex and three incoming edges) and three maximal objects (three binary trees with two vertices). As the picture shows, the functor $w(-)$ embeds a 1–simplex for the minimal object as one of the 1–dimensional faces of a 2–simplex for each of the maximal objects. The subspaces $w_0(T)$ are outlined in bold. Collapsing these we get the functor $\overline{w}(-)$ which embeds $S^1$ (for the minimal object) as the boundary of $D^2$ (for each maximal object).
Bar constructions for reduced operads {#sec:bar}
=====================================
This section forms the heart of the paper. We show that by giving an explicit description of the simplicial bar construction in terms of trees, we can construct a cooperad structure on it. In Section \[sec:bardef\] we give our definition of the bar construction $B(P)$ for an operad $P$ in ${\mathcal{C}}$. In Section \[sec:simpbar\] we show that this is isomorphic to the standard simplicial reduced bar construction on $P$. Then in Section \[sec:cooperad\] we prove the main result of this paper: that $B(P)$ admits a natural cooperad structure.
We will work in a fixed symmetric monoidal ${\mathcal{T}_{}}$–category $({\mathcal{C}},\barwedge,S)$ where ${\mathcal{T}_{}}$ is the category of based compactly-generated spaces and basepoint preserving maps. Since ${\mathcal{T}_{}}$ is pointed, Proposition \[prop:null\] implies that ${\mathcal{C}}$ too is pointed. We denote the null object in ${\mathcal{C}}$ also by ${\ast}$. We assume that ${\mathcal{C}}$ has all limits and colimits. The examples to bear in mind are ${\mathcal{C}} = {\mathcal{T}_{}}$ itself and ${\mathcal{C}} = {\mathcal{S}p}$, which we take to be the category of $S$–modules of EKMM [@elmendorf/kriz/mandell/may:1997], although other categories of spectra could be used. We will use the notation developed in Section 1 for the enrichment, tensoring and cotensoring of ${\mathcal{C}}$ over ${\mathcal{T}_{}}$.
Before we start we should stress that the constructions in this paper only apply to *reduced* operads and cooperads. That is, those for which the unit (or counit) map is an isomorphism. This is reflected in several places, most notably in the fact that our trees are not allowed to have vertices with only one incoming edge (see Remark \[rem:reduced\_trees\]). It is a necessary condition for our construction of the cooperad structure on $B(P)$.
Definition of the bar construction {#sec:bardef}
----------------------------------
We give two definitions of the bar construction for an operad. The first is somewhat informal and relies on ${\mathcal{C}}$ being the category of based spaces, but captures how we really think about these objects. The second is a precise formal definition as a coend in the category ${\mathcal{C}}$.
\[def:bar(operad)\] Let $P$ be a reduced operad in ${\mathcal{T}_{}}$. The *bar construction* on $P$ is the symmetric sequence $B(P)$ defined as follows. A general point $p$ in $B(P)(A)$ consists of
- an isomorphism class of $A$–labelled trees: $T \in \mathsf{T}(A)$,
- a weighting on $T$ and,
- for each (internal) vertex $v$ of $T$, a point $p_v$ in the based space $P(i(v))$ (recall that $i(v)$ is the set of incoming edges of the vertex $v$),
subject to the following identifications:
- If $p_v$ is the basepoint in $P(i(v))$ for any $v$ then $p$ is identified with the basepoint ${\ast}\in B(P)(A)$.
- If the internal edge $e$ has length zero, we identify $p$ with the point $q$ given by
- the tree $T/e$,
- the weighting on $T/e$ in which an edge has the same length as the corresponding edge of $T$ in the weighting that makes up $p$,[^7]
- $q_{u \circ v}$ given by the image under the composition map $$P(i(u)) {\wedge}P(i(v)) \to P(i(u \circ v))$$ of $(p_u,p_v)$ (notice that $i(u \circ v) = i(u) \circ_v i(v)$),
- $q_t = p_t$ for the other vertices $t$ of $T/e$.
- If a root or leaf edge has length zero, $p$ is identified with ${\ast}\in B(P)(A)$.
A bijection $\sigma\co A \to A'$ gives us an isomorphism $\sigma_*\co B(P)(A)
\to B(P)(A')$ by relabelling the leaves of the underlying trees. In this way, $B(P)$ becomes a symmetric sequence in ${\mathcal{T}_{}}$.
Consider $B(P)(1)$. There is only one tree with a single leaf and only one weighting on it. It has no vertices so $B(P)(1)$ does not depend at all on $P$. With the basepoint (which is disjoint in this case because nothing is identified to it) we get $B(P)(1) = S^0$.
Next consider $B(P)(2)$. Again there is only one tree, but this time it has a vertex (with two incoming edges) and the space of ways to weight the tree is the $1$–simplex $\Delta^1$. Making all the identifications we see that $$B(P)(2) = \Sigma P(2),$$ the reduced suspension of $P(2)$.
\[def:P\] A key ingredient of the general definition of the bar construction is that an operad $P$ in ${\mathcal{C}}$ determines a functor $$P_A(-)\co \mathsf{T}(A)^\text{op} \to {\mathcal{C}}.$$ where $\mathsf{T}(A)$, as always, is the poset of isomorphism classes of $A$–labelled trees ordered by edge collapse. For a tree $T$ we define $$P_A(T) := \operatorname*{\overline{\bigwedge}}_{\text{vertices $v$ in $T$}} P(i(v))$$ where we recall that $i(v)$ is the set of incoming edges to the vertex $v$. If $e$ is an internal edge in $T$ with endpoints $u$ and $v$ then there is a partial composition map $$P(i(u)) \barwedge P(i(v)) \to P(i(u \circ v)).$$ Using this we get a map $$P_A(T) \to P_A(T/e).$$ The associativity axioms for the operad $P$ ensure that these maps make $P_A(-)$ into a functor as claimed.
Recall from Definition \[def:w\_functor\] that we have a functor $$\overline{w}(-) \co \mathsf{T}(A) \to {\mathcal{T}_{}}$$ given by taking the space of weightings on a tree, modulo those for which a root or leaf edge has length zero.
\[def:formal\_bar\] Let the *bar construction* of the reduced operad $P$ be the symmetric sequence $B(P)$ defined by $$B(P)(A) := \overline{w}(-) \otimes_{\mathsf{T}(A)} P_A(-) = \int^{T \in
\mathsf{T}(A)} \overline{w}(T) \otimes P_A(T).$$ This is the coend in ${\mathcal{C}}$ of the bifunctor $$\overline{w}(-) \otimes P_A(-) \co \mathsf{T}(A) \times
\mathsf{T}(A)^\text{op}
\to {\mathcal{C}}.$$ (See [@maclane:1971] for the theory of coends.) The definition of the coend is a colimit over a category whose objects are morphisms in $\mathsf{T}(A)$ and we will write the coend above as $$\operatorname*{colim}_{T \leq T' \in \mathsf{T}(A)} \overline{w}(T) \otimes P_A(T')$$ when we need to manipulate it as such.
A bijection $A \to A'$ induces an isomorphism of categories $\mathsf{T}(A)
\to \mathsf{T}(A')$ by the relabelling of trees. If $T \mapsto T'$ under this isomorphism then $P_A(T) = P_A(T')$ and $\overline{w}(T) =
\overline{w}(T')$. Therefore we get an induced isomorphism $B(P)(A) \to
B(P)(A')$. This makes $B(P)$ into a symmetric sequence in ${\mathcal{C}}$.
To see that our two definitions of the bar construction are equivalent when ${\mathcal{C}} = {\mathcal{T}_{}}$, recall that the coend is a quotient of the coproduct $${\bigvee}_{T \in \mathsf{T}(A)} \overline{w}(T) \otimes P_A(T).$$ That is, a point consists of a weighted tree together with elements of the $P(i(v))$ for vertices $v$ subject to some identifications. The maps $P_A(T) \to P_A(T/e)$ and $\overline{w}(T/e) \to \overline{w}(T)$ encode the identifications made in Definition \[def:bar(operad)\].
Our definition of the bar construction is rather reminiscent of the geometric realization of simplicial sets or spaces. This line of thought leads to the definition of an *arboreal object* in ${\mathcal{C}}$ as a functor $$\mathsf{T}(A)^\text{op} \to {\mathcal{C}}$$ in which $\mathsf{T}(A)$ plays the role of the simplicial indexing category $\Delta$ for simplicial sets. With the spaces of weightings $\overline{w}(T)$ playing the role of the topological simplices, the bar construction $B(P)$ can be thought of as the geometric realization of the arboreal object $P_A(-)$. We will formalize and extend these ideas in future work [@ching:2005b].
\[rem:BV\] The $W$–construction of Boardman and Vogt (also sometimes called the bar construction) is defined in a very similar manner to $B(P)$. It uses slightly different spaces of trees and produces an operad instead of a cooperad. See [@vogt:2003] for details. Benoit Fresse has noticed a relationship between $W(P)$ and $B(P)$, namely that $$B(P) = \Sigma\operatorname{Indec}(W(P))$$ where $\Sigma$ is a single suspension (that is, tensoring with $S^1$) and $\operatorname{Indec}$ denotes the ‘operadic indecomposables functor’. It is the cooperad structure on $\Sigma\operatorname{Indec}(W(P))$, corresponding to that on $B(P)$, that was described by Salvatore in [@salvatore:1998].
\[ex:bars\] Let $\mathcal{A}ss$ be the operad for associative monoids in unbased spaces. This is given by $$\mathcal{A}ss(n) := \Sigma_n$$ (with the discrete topology and regular $\Sigma_n$–action). The composition maps are the inclusions given by identifying $$\Sigma_r \times \Sigma_{n_1} \times \dots \times \Sigma_{n_r}$$ with a subgroup of $\Sigma_{n_1 + \dots + n_r}$. We obtain an operad $\mathcal{A}ss_+$ in ${\mathcal{T}_{}}$ by adding a disjoint basepoint to each of the terms of $\mathcal{A}ss$. Let us calculate $B(\mathcal{A}ss_+)$.
The points $p_v \in \mathcal{A}ss_+(i(v))$ required by Definition \[def:bar(operad)\] can be thought of as determining an order on the incoming edges to vertices of a tree. This allows us to identify a point in $B(\mathcal{A}ss_+)(n)$ with a *planar* weighted tree with leaves labelled $1,\dots,n$. This breaks $B(\mathcal{A}ss_+)(n)$ up into a wedge of $n!$ terms, each corresponding to an ordering of the leaves of the trees involved.
As we now show, each of these terms is an $(n-1)$–sphere. Think of constructing a planar weighted tree with leaves labelled in a fixed order (say, $1,\dots,n$) by the following method. Connect the first leaf to the root with an edge of length $1$. Then attach the second leaf at some point along the edge already drawn. Attach the third leaf at some point along the path from the second leaf to the root, and so on. The space of choices made in doing all this is $[0,1]^{n-1}$ and we obtain precisely the planar weighted trees we want in this manner (see Figure \[fig:planar\]). The root edge or a leaf edge will have length zero if and only if at least one of our choices was either $0$ or $1$. Hence the space we want is obtained by identifying the boundary of $[0,1]^{n-1}$ to a basepoint. This gives $S^{n-1}$.
Therefore we have $$B(\mathcal{A}ss_+)(n) {\cong}S^{n-1} {\wedge}(\Sigma_n)_+$$ where $\Sigma_n$ acts trivially on the $S^{n-1}$ term and by translation on the non-basepoints of $(\Sigma_n)_+$.
We can also picture what happens for $n = 3$ in terms of sticking together the spaces $\overline{w}(T) {\wedge}\mathcal{A}ss_3(T)_+$ for $T \in
\mathsf{T}(3)$. The $\overline{w}(T)$ are the quotients of the spaces pictured in Figure \[fig:w\_functor\] by the subspaces outlined in bold. To make up $B(\mathcal{A}ss_+)(3)$ we need six copies of the 1–simplex (corresponding to the points in $\mathcal{A}ss(3)$) and twelve copies of the 2–simplex. (There are four points in $\mathcal{A}ss(2) \times
\mathcal{A}ss(2)$ and three trees of this type.) These fit together to form six disjoint copies of the space of Figure \[fig:ass\], one for each permutation of $1,2,3$. The type of tree used to form each part is shown.
When we collapse the bold subspaces to the basepoint we get a wedge of six copies of $S^2$ as expected.
Relation to the simplicial bar construction {#sec:simpbar}
-------------------------------------------
In this section we show that $B(P)$ is isomorphic to the geometric realization of the standard simplicial bar construction on the reduced operad $P$. This simplicial bar construction can be defined for any augmented monoid in a monoidal category.[^8] We have seen (Proposition \[prop:comp\_monoidal\]) that under the right conditions an operad is just a monoid for the monoidal product on the category of symmetric sequences given by the composition product $\circ$. To define the simplicial bar construction in general (that is, without the assumption that $\barwedge$ commutes with finite coproducts) we must say what we mean by higher iterates of $\circ$. For this we use the following natural extension of the three-way product introduced in the proof of Lemma \[lem:circ\_assoc\].
\[def:iterated\] The *composition product* of the symmetric sequences $M_1,\dots,M_r$ is the symmetric sequence given by $$(M_1 \circ \dots \circ M_r)(A) :=\!\! \bigvee_{A_i = \coprod_{a \in
A_{i-1}} A_{i,a}}\!\! M_1(A_1) \barwedge \operatorname*{\overline{\bigwedge}}_{a \in A_1}
M_2(A_{2,a}) \barwedge \dots \barwedge \operatorname*{\overline{\bigwedge}}_{a \in A_{r-1}}
M_r(A_{r,a})$$ for each finite set $A = A_r$. Here we are taking the coproduct over partitions of $A$ into subsets $A_{r,a}$ indexed over $a \in A_{r-1}$, partitions of $A_{r-1}$ indexed over $A_{r-2}$, and so on. Equivalently we can view this coproduct as indexed over sequences of $r-1$ partitions of $A$, each a refinement of the next.
There is a natural map from $(M_1 \circ \dots \circ M_r)$ to any of the symmetric sequences obtained by choosing ways to bracket this expression. All the ‘obvious’ diagrams relating these maps commute. If $\barwedge$ commutes with finite coproducts in ${\mathcal{C}}$ then all these maps are isomorphisms and reflect the associativity isomorphisms of the monoidal product $\circ$.
\[def:simplicial\_bar\] Let $P$ be a reduced operad in ${\mathcal{C}}$. The *simplicial bar construction* ${\mathcal{B}}_{\bullet}(P)$ is the simplicial object in the category of symmetric sequences on ${\mathcal{C}}$ with $${\mathcal{B}}_k(P) = \underset{k}{\underbrace{P \circ \dots \circ P}}.$$ For $i = 1,\dots,k-1,$ face maps $$d_i\co \underset{k}{\underbrace{P \circ \dots \circ P}} \to
\underset{k-1}{\underbrace{P \circ \dots \circ P}}$$ are given by $$\dots \circ P \circ P \circ \dots \to \dots \circ (P \circ P) \circ
\dots \to \dots \circ P \circ \dots$$ where we are using the operad composition $P \circ P \to P$ to compose the [$i$^th^]{} and [$i{+}1$^th^]{} factors. The maps $d_0$ and $d_k$ are given by applying the augmentation map $P \to I$ to the first and last copies of $P$ respectively. Degeneracy maps $$s_j\co \underset{k}{\underbrace{P \circ \dots \circ P}} \to
\underset{k+1}{\underbrace{P \circ \dots \circ P}}$$ are given for $j = 0,\dots, k$ by using the unit map $I \to P$ to insert a copy of $P$ between the [$j$^th^]{} and [$(j+1)$^th^]{} factors: $$\cdots \circ P \circ P \circ \cdots {\cong}\cdots \circ P \circ I
\circ P \circ \cdots \to \cdots \circ P \circ P \circ P \circ \cdots.$$
It is sufficient for this definition that $P$ be augmented. However, we need $P$ to be reduced to make the following identification of the simplicial bar construction with $B(P)$ as defined previously.
\[prop:simp\] Let $P$ be a reduced operad in ${\mathcal{C}}$. Then the geometric realization[^9] of ${\mathcal{B}}_{\bullet}(P)$ is isomorphic to the bar construction $B(P)$.
We give the proof for ${\mathcal{C}} = {\mathcal{T}_{}}$ (which is the only case we require in this paper) based on the informal description of $B(P)$ in Definition \[def:bar(operad)\]. The same idea could be used to write a proof that works for any ${\mathcal{C}}$ using the formal definition of $B(P)$ as a coend.
The idea is that the iterated composition products that make up the simplicial bar construction can be thought of in terms of sequences of partitions which in turn are related to trees of the type we are using to define $B(P)$.
We first give an explicit description of the $n$–simplices in ${\mathcal{B}}_{\bullet}(P)(A)$. These are given by the object $$\underset{n}{\underbrace{P \circ \dots \circ P}}(A).$$ Enlarging on the last sentence of Definition \[def:iterated\], we can write this as a coproduct over all sequences of partitions $$\widehat{0} = \lambda_0 \leq \lambda_1 \leq \dots \leq \lambda_{n-1}
\leq \lambda_n = \widehat{1}$$ of the set $A$, where $\lambda \leq \mu$ if $\lambda$ is *finer* than $\mu$ (if two elements of $A$ are in the same block in $\lambda$, they are also in the same block in $\mu$) and $\widehat{0},\widehat{1}$ are the minimal and maximal partitions with respect to this order. The terms in the coproduct are appropriate smash products of the $P(r)$. We get a factor of $P(r)$ every time one of the blocks of one of the partitions breaks up into $r$ blocks in the next partition along.
A point in the geometric realization $|{\mathcal{B}}_{\bullet}(P)|$ can be represented by a point in the topological $n$–simplex $\Delta^n$ together with a choice of sequence of partitions as described above and a point in the appropriate smash product of the spaces $P(r)$.
A sequence of partitions determines an $A$–labelled tree $T$ as follows. (See Figure \[fig:simplicial\] for an example when $n = 3$.) Take a vertex for each block of each $\lambda_i$ for $i =
1,\dots,n$. Add a root, and a leaf for each element of $A$. Two vertices are joined by an edge if they come from consecutive partitions of the sequence and the block for one is contained in the block for the other. Finally we add a root edge from the $\lambda_n$ vertex to the root and a leaf edge from each leaf to the corresponding $\lambda_1$ vertex. (Notice that vertices in this tree might have only one input edge – let’s allow this for the moment.)
A point in $\Delta^n$ determines a weighting on the tree we have just constructed. Thinking of $\Delta^n$ as the subspace of $\mathbb{R}^{n+1}$ with $x_0 + \dots + x_n = 1$ and $x_i \geq 0$, we get a weighting by giving the root edge length $x_0$, the edges connecting the vertices for $\lambda_{i-1}$ to the vertices for $\lambda_i$ length $x_i$ and the leaf edges length $x_n$. We can now remove the vertices with only one input edge, connecting their input and output edges. This gives us a point in $w(T)$ for some tree $T$ in the sense of Definition \[def:tree\].
Finally notice that because $P(1) = S^0$ (as $P$ is reduced), the smash product of spaces $P(r)$ determined by the sequence of partitions is precisely $P_A(T)$. Therefore we actually obtain a point in $B(P)(A)$.
It remains to show that this process sets up a homeomorphism between the geometric realization $|{\mathcal{B}}_{\bullet}(P)(A)|$ and $B(P)(A)$. There are a couple of key steps. Firstly the degeneracy maps in the simplicial bar construction are isomorphisms on terms in the coproduct. These correspond to inserting lots of vertices with one input edge in our trees, which are then removed by our construction. So we only have to worry about the identifications made by the face maps. The face maps are given by removing partitions from the sequences, which corresponds to edge collapse. Hence the identifications made in defining $B(P)$ are the same as those in defining the realization of ${\mathcal{B}}_{\bullet}(P)$. This completes the proof.
Cooperad structure on the bar construction {#sec:cooperad}
------------------------------------------
Up to this point, all we have done is identify the simplicial bar construction on a reduced operad in terms of trees. The main point of this paper is that this identification allows us to see that there is a cooperad structure on the bar construction. In this section we describe that structure. The key to getting the cooperad cocomposition maps is the process of grafting (or rather *un*grafting) trees.
\[def:grafting\] Let $T$ be an $A$–labelled tree, let $U$ be a $B$–labelled tree and let $a$ be an element of $A$. We define the *grafting of $U$ onto $T$ at $a$* to be the tree $T \cup_a U$ obtained by identifying the root edge of $U$ to the leaf edge of $T$ corresponding to $a$. Figure \[fig:grafting\] below illustrates this process.
We denote the newly identified edge by $e_a$. Every other edge of $T
\cup_a U$ comes either from $T$ or from $U$. The vertices of $T \cup_a U$ are the vertices of $T$ together with the vertices of $U$ (and they have the same number of incoming edges). Finally there is a natural $A \cup_a
B$–labelling of $T \cup_a U$, given by combining the labellings of $T$ and of $U$.
We say that an $A \cup_a B$–labelled tree is *of type $(A,B)$* if it is of the form $T \cup_a U$ for an $A$–labelled tree $T$ and a $B$–labelled tree $U$. The next lemma says that an $A \cup_a B$–labelled tree is a grafting in at most one way. This is trivial but crucial to the construction of the cooperad structure maps below.
\[lem:ungrafting\] For any $A \cup_a B$–labelled tree $V$ there is at most one pair $(T,U)$ such that $V = T \cup_a U$.
In the grafted tree $T \cup_a U$ the ‘upper’ endpoint of the edge $e_a$ is a vertex whose ‘parent leaves’ are labelled precisely by the elements of $B$. There can be at most one such vertex $v$ in $V$ and cutting along the edge immediately below $v$ produces the trees $T,U$ that make up $V$.
\[def:cooperad\_maps\] To give $B(P)$ a cooperad structure we have to define maps $$B(P)(A \cup_a B) \to B(P)(A) \barwedge B(P)(B)
\label{eq:cooperad_maps}$$ for finite sets $A,B$ and $a \in A$. A point $p$ in $B(P)(A \cup_a B)$ consists of a weighted tree $V$ labelled by $A \cup_a B$ together with elements of $p_v \in P(i(v))$ for vertices $v$ of $V$. We treat two cases:
1. If $V$ is not of the form $T \cup_a U$ for an $A$–labelled tree $T$ and a $B$–labelled tree $U$, then we will map $p$ to the basepoint on the right-hand side of (\[eq:cooperad\_maps\]).
2. If $V$ is of this form (that is, it is of type $(A,B)$) then things are more interesting. Below we describe how the map (\[eq:cooperad\_maps\]) is defined in this case.
Since $V$ is of type $(A,B)$, Lemma \[lem:ungrafting\] tells us that there is a unique $A$–labelled tree $T$ and a unique $B$–labelled tree $U$ such that $V = T \cup_a U$. We use these trees as the basis for elements $q \in B(P)(A)$ and $r \in B(P)(B)$ respectively. What remains to be seen is how the weighting and vertex labels of $V$ determine weightings and vertex labels for $T$ and $U$.
The vertex labels are easy because the vertices of $T \cup_a U$ consist of the vertices of each of $T$ and $U$ with the same numbers of input edges. Therefore we take $$q_v := p_v \in P(i(v))$$ for vertices $v$ of $T$ and $$r_u := p_u \in P(i(v))$$ for vertices $u$ of $U$.
The way in which a weighting on $T \cup_a U$ determines weightings on $T$ and $U$ is the key part of our construction. This comes about via a map $$\overline{w}(T \cup_a U) \to \overline{w}(T) {\wedge}\overline{w}(U)
\label{eq:key}$$ (recall that $\overline{w}(-)$ is the space of weightings on a tree with those that have zero length root or leaf edges identified to a basepoint).
So take a weighting of $T \cup_a U$. Define a weighting on $T$ by giving the edges the same lengths they had in $T \cup_a U$ and giving the leaf edge for $a$ the necessary length to make the root-leaf distances equal to $1$.[^10] Next define a weighting on $U$ by taking the lengths from $T \cup_a U$ and scaling up by a constant factor to make the root-leaf distances equal to $1$ (the length of the root edge of $U$ comes from the length of the edge $e_a$ in $T \cup_a U$). The scaling factor is the inverse of the total length of the $U$ part of $T \cup_a U$. The only time this doesn’t work is if all the $U$–edges in $T \cup_a U$ (including $e_a$) are of length zero. However in that case the weighting we just defined on $T$ has a leaf edge of length zero and so is the basepoint in $\overline{w}(T)$. This is almost enough to define a map of the form (\[eq:key\]). The only thing left to check is that if a leaf or root edge of $T \cup_a U$ is of length zero then the same is true of either of the chosen weightings on $T$ and $U$. This is clear. Figure \[fig:key\] illustrates a particular case of the map (\[eq:key\]).
This completes the definition of the cooperad structure maps (\[eq:cooperad\_maps\]): $$B(P)(A \cup_a B) \to B(P)(A) \barwedge B(P)(B)$$ given, in summary, by: $$p = (V,\{p_v\}) \mapsto \begin{cases} q = (T,\{p_v\}_{v \in T}), \;
r = (U,\{p_v\}_{v \in U}) & \text{if $V = T \cup_a U$}; \\ {\ast}&
\text{otherwise}. \end{cases}$$ with the weightings on $T,U$ given by the map (\[eq:key\]) just constructed.
We still have to check that these maps are well-defined. To see this we have to look at the identifications made in the definition of $B(P)(A
\cup_a B)$:
- If $p_v$ equals the basepoint in $P(i(v))$ for any vertex $v \in
V$ then the same will be true of the corresponding vertex in either $T$ or $U$. Hence such a $p$ maps to the basepoint.
- If an interior edge $e$ of the tree $V$ underlying the point $p$ is of length zero, $p$ is identified with another point $p'$ as described in Definition \[def:bar(operad)\]. We have various possibilities:
1. $V$ is not of the form $T \cup_a U$ in which case neither is $V/e$ and both $p$ and $p'$ map to the basepoint.
2. $V = T \cup_a U$ and $e$ corresponds to an internal edge of $T$. In this case, the points $q$ and $q'$ will be identified via the collapse of that edge, and the points $r$ and $r'$ will be equal. So $p$ and $p'$ map to the same element of $B(P)(A) \barwedge B(P)(B)$.
3. $V = T \cup_a U$ and $e$ corresponds to an internal edge of $U$. This is similar to case (2).
4. $V = T \cup_a U$ and $e$ is the edge $e_a$ obtained from identifying the root edge of $U$ with the $a$–leaf edge of $T$. In this case $V/e$ is no longer of the form $T \cup_a U$ and so $p'$ maps to the basepoint. But in the weighting on $U$ determined by that on $T \cup_a
U$ the root edge has length scaled up from the length of $e_a$ which is therefore zero. So the point $r$ is the basepoint in $B(P)(B)$ and so $p$ also maps to the basepoint.
- We have already checked in the definition of the map (\[eq:key\]) that if a root or leaf edge in $p$ is of length zero, then the same is true of at least one of $q$ and $r$. Therefore such a $p$ maps to the basepoint in $B(P)(A) \barwedge B(P)(B)$.
This completes the check that our maps (\[eq:cooperad\_maps\]) are well-defined. The final piece of the cooperad structure for $B(P)$ is a counit map $B(P)(1) \to S^0$. But we already saw that $B(P)(1) {\cong}S^0$ (in the based space case) so our counit is this isomorphism. Note that this means $B(P)$ turns out to be a reduced cooperad.
The map $$B(P)(\{1,2,3\}) \to B(P)(\{a,3\}) {\wedge}B(P)(\{1,2\})$$ is pictured in Figure \[fig:key\]. The left-hand side (with vertices labelled by elements of $P(2)$) represents a point $p$ of $B(P)(\{1,2,3\})$. The two trees on the right-hand side (with vertices labelled by those same elements in the obvious way) represent the image of $p$ in $B(P)(\{a,3\}) \barwedge B(P)(\{1,2\})$. In this example, all points that are based on trees of shapes other than that shown are mapped to the basepoint.
We will save for later the task of checking that these maps do indeed give us a cooperad structure. First we translate Definition \[def:cooperad\_maps\] into the category-theoretic language needed to define the cocomposition maps for a general ${\mathcal{C}}$. To do this, we notice that the ‘ungrafting’ process more-or-less makes our categories $\mathsf{T}(A)$ into a cooperad of categories. To make this precise, we describe an ‘add a disjoint basepoint’ functor for categories.
Write $\mathsf{Cat}_+$ for the category in which an object is a (small) category $\mathsf{C}_+$ together with an initial object ${\ast}$ such that $\operatorname{Hom}_{\mathsf{C}_+}(X,{\ast})$ is empty for all $X \neq
{\ast}$. The morphisms in $\mathsf{Cat}_+$ are functors that preserve the initial objects.
There is a functor from the category $\mathsf{Cat}$ of all (small) categories to $\mathsf{Cat}_+$ given by adding an initial object with the correct morphisms to a category $\mathsf{C}$ to obtain $\mathsf{C}_+$. Note that every object in $\mathsf{Cat}_+$ can be obtained in this way, but not every morphism in $\mathsf{Cat}_+$ is given by adding an initial object to a morphism in $\mathsf{Cat}$.
Define a symmetric monoidal product ${\wedge}$ on $\mathsf{Cat}_+$ by $$\mathsf{C}_+ {\wedge}\mathsf{D}_+ := \mathsf{C}_+ \times
\mathsf{D}_+/\mathsf{C}_+ {\vee}\mathsf{D}_+,$$ where the wedge product is the disjoint union with the initial objects identified and the quotient identifies this wedge product to the initial object of the smash product. Notice that if $\mathsf{C},\mathsf{D}
\in \mathsf{Cat}$ then $$\mathsf{C}_+ {\wedge}\mathsf{D}_+ = (\mathsf{C} \times \mathsf{D})_+.$$ The unit for this product is the category $1_+$ with two objects and a single morphism between them.
In particular we write $\mathsf{T}(A)_+$ for the category formed by adding an initial object to our poset of $A$–labelled trees $\mathsf{T}(A)$. The reason for making all these new definitions is then the following result.
\[prop:category\_cooperad\] The categories $\mathsf{T}(A)_+$ form a reduced cooperad in $\mathsf{Cat}_+$.
The cocomposition maps have the form $$\mathsf{T}(A \cup_a B)_+ \to \mathsf{T}(A)_+ {\wedge}\mathsf{T}(B)_+ =
(\mathsf{T}(A) \times \mathsf{T}(B))_+$$ and are given by ‘ungrafting’ trees. Take $V \in \mathsf{T}(A \cup_a
B)$. If $V$ is a tree of type $(A,B)$ we map it to the pair $(T,U)$ where $T,U$ are the unique trees that graft together to give $V$ (see Lemma \[lem:ungrafting\]). If $V$ is not of type $(A,B)$ (or is the initial object) we map it to the initial object of the right-hand side.
First we must check that we have indeed given a functor here. Suppose that $V \leq V'$ in $\mathsf{T}(A \cup_A B)$. The only interesting case is when $V$ is of type $(A,B)$, so maps to a pair $(T,U)$ on the right-hand side. We have to show two things: that $V'$ is also of type $(A,B)$ with decomposition $(T',U')$ and then that $T \leq T'$ and $U
\leq U'$. Well, let $e_a$ be the edge in $V$ at which the grafting took place. Since $V$ is obtained from $V'$ by a sequence of edge collapses, $e_a$ must come from an edge $e_{a'}$ in $V'$ that is not collapsed in this sequence. This edge breaks $V'$ into two parts and we can write $V'
= T' \cup_{a'} U'$ for some trees $T',U'$ with some labellings (a priori, not necessarily by $A$ and $B$). But it is now clear that $U'$ must yield $U$ after undergoing some edge collapses. So $U' \in \mathsf{T}(B)$ and $U \leq U'$. Similarly, $T' \in \mathsf{T}(A)$ and $T \leq T'$ (after relabelling $a'$ by $a$).
Notice that $\mathsf{T}(1)_+$ is isomorphic to the unit $1_+$ for the symmetric monoidal structure on $\mathsf{Cat}_+$. We take as unit map the (unique) isomorphism $1_+ \to \mathsf{T}(1)_+$.
It still remains to check that the cooperad axioms do indeed hold for our cocomposition maps. This is simple and we leave it to the reader.
The original categories $\mathsf{T}(A)$ in fact already form an *operad* in $\mathsf{Cat}$ with composition maps given by grafting rather than ungrafting. This operad structure is effectively what is used by Boardman and Vogt to define their $W$–construction.
The next step is to show that the bar construction can be defined as a coend in $\mathsf{T}(A)_+$ instead of $\mathsf{T}(A)$.
Let $P$ be a reduced operad in ${\mathcal{C}}$. The functors $\overline{w}(-)$ and $P_A(-)$ on $\mathsf{T}(A)$ naturally extend to functors $$\overline{w}(-)\co \mathsf{T}(A)_+ \to {\mathcal{T}_{}}$$ and $$P_A(-)\co (\mathsf{T}(A)_+)^\text{op} \to {\mathcal{C}}$$ and we have $$B(P)(A) = \int^{T \in \mathsf{T}(A)_+} \overline{w}(T) \otimes P_A(T).$$
We set $\overline{w}({\ast}) = {\ast}_{{\mathcal{T}_{}}}$ and $P_A({\ast}) =
{\ast}_{{\mathcal{C}}}$ with the necessary definition on morphisms (given by the fact that ${\ast}_{{\mathcal{T}_{}}}$ is an initial object in ${\mathcal{T}_{}}$ and ${\ast}_{{\mathcal{C}}}$ is a terminal object in ${\mathcal{C}}$). It is then clear that ${\ast}\in \mathsf{T}(A)_+$ does not contribute anything to the coend which therefore reduces to the previous definition of $B(P)(A)$.
The maps (\[eq:key\]) of Definition \[def:cooperad\_maps\] are still the key ingredients in constructing the cooperad maps for $B(P)$.
\[lem:trans1\] The maps $$\overline{w}(T \cup_a U) \to \overline{w}(T) {\wedge}\overline{w}(U)$$ previously defined form part of a natural transformation $$\begin{diagram} \node{\mathsf{T}(A \cup_a B)_+} \arrow[2]{s}
\arrow{ese,t}{\overline{w}(-)} \\
\node[2]{\Downarrow} \node{{\mathcal{T}_{}}} \\
\node{\mathsf{T}(A)_+ {\wedge}\mathsf{T}(B)_+} \arrow{ene,b}{\overline{w}(-)
{\wedge}\overline{w}(-)} \\
\end{diagram}.$$
The bottom functor here is defined in the obvious way on $\mathsf{T}(A)
\times \mathsf{T}(B)$ and sends ${\ast}$ to ${\ast}$. For $V \in
\mathsf{T}(A \cup_a B)$ not of type $(A,B)$, the corresponding part of the natural transformation is $$\overline{w}(V) \to {\ast}.$$
The only really interesting naturality square comes from $V \leq V'$ with $V'$ of type $(A,B)$ and $V$ not. The square that must commute in this case is $$\begin{diagram} \node{\overline{w}(V)} \arrow{s} \arrow{e}
\node{\overline{w}(V')} \arrow{s} \\
\node{{\ast}} \arrow{e} \node{\overline{w}(T') {\wedge}\overline{w}(U').}
\end{diagram}$$ This is the content of part (4) of the checking we did towards the end of Definition \[def:cooperad\_maps\]: from any weighting on $V$, the weighting we get on $V'$ will have length zero for the edge connecting the $T'$–part to the $U'$–part. Hence the root edge of the corresponding weighting on $U'$ will have length zero. So we map into the basepoint of $\overline{w}(T') {\wedge}\overline{w}(U')$.
We have a corresponding result for the functors $P_A(-)$ of Definition \[def:formal\_bar\].
\[lem:trans2\] Let $P$ be a reduced operad in ${\mathcal{C}}$. Then there is a natural transformation $$\begin{diagram} \node{\mathsf{T}(A \cup_a B)_+^\text{op}} \arrow[2]{s}
\arrow{ese,t}{P_{A \cup_a B}(-)} \\
\node[2]{\Downarrow} \node{{\mathcal{C}}} \\
\node{(\mathsf{T}(A)_+ {\wedge}\mathsf{T}(B)_+)^\text{op}}
\arrow{ene,b}{P_A(-) \barwedge P_B(-)}
\end{diagram}$$
In other words, given $V \in \mathsf{T}(A \cup_a B)$ we have maps $$P_{A \cup_a B}(V) \to P_A(T) \barwedge P_B(U)$$ when $V = T \cup_a U$. There are obvious isomorphisms that we take for these maps. The naturality squares are easily seen to commute. Again the only one that seems like it might be interesting is for $V \leq V'$ with $V'$ of type $(A,B)$ and $V$ not. But in fact this square just turns out to be $$\begin{diagram} \node{P_{A \cup_a B}(V')} \arrow{s} \arrow{e}
\node{P_{A \cup_a B}(V)} \arrow{s} \\
\node{P_A(T') \barwedge P_B(U')} \arrow{e} \node{{\ast}}
\end{diagram}$$ which is not so interesting after all.
Finally, we can give the formal construction of the cocomposition maps for the cooperad $B(P)$.
\[def:formal\_cooperad\_maps\] Let $P$ be a reduced operad in ${\mathcal{C}}$ and let $B(P)$ be the symmetric sequence of Definition \[def:formal\_bar\]. The cocomposition map $$B(P)(A \cup_a B) \to B(P)(A) \barwedge B(P)(B)$$ is given by the following sequence of maps: $$\begin{split}
B(P)&(A \cup_a B) = \operatorname*{colim}_{V \leq V' \in \mathsf{T}(A \cup_a B)_+}
\overline{w}(V) \otimes P_{A \cup_a B}(V') \\
\longrightarrow & \operatorname*{colim}_{(T,U) \leq (T',U') \in \mathsf{T}(A)_+ {\wedge}\mathsf{T}(B)_+} (\overline{w}(T) {\wedge}\overline{w}(U)) \otimes (P_A(T')
\barwedge P_B(U')) \\
\longrightarrow & \operatorname*{colim}_{(T,U) \leq (T',U') \in \mathsf{T}(A)_+ {\wedge}\mathsf{T}(B)_+} (\overline{w}(T) \otimes P_A(T')) \barwedge
(\overline{w}(U) \otimes P_B(U')) \\
\longrightarrow & \left(\operatorname*{colim}_{T \leq T' \in \mathsf{T}(A)_+} \overline{w}(T)
\otimes P_A(T') \right) \barwedge \left(\operatorname*{colim}_{U \leq U' \in
\mathsf{T}(B)_+} \overline{w}(U) \otimes P_B(U') \right) \\
= & \; B(P)(A) \barwedge B(P)(B). \\
\end{split} \label{eq:cocomp}$$ The first map here comes from combining the natural transformations of Lemmas \[lem:trans1\] and \[lem:trans2\]. The second is given by the transformation $d$ of Definition \[def:axiom\]. It is for precisely this reason that the axiom giving us $d$ is necessary. The third map is given by universal properties of colimits. This completes the construction of the cooperad structure maps for $B(P)$.
The next task is to check that the maps we have described actually do make $B(P)$ into a cooperad. That is, we must check the duals of axioms (1)–(4) from Definition \[def:operad\]. The key step is to see that the maps (\[eq:key\]) satisfy corresponding conditions.
\[lem:keyassoc\] Let $T,U,V$ be $A$–, $B$– and $C$–labelled trees respectively and let $a,a' \in A$ and $b \in B$. Let $I$ denote the unique ${\ast}$–labelled tree. Recall that $\overline{w}(I) = S^0$. Then the following diagrams commute:
1. $ \qquad\qquad\begin{diagram}
\node{\overline{w}(T \cup_a U \cup_b V)} \arrow{s} \arrow{e}
\node{\overline{w}(T \cup_a U) {\wedge}\overline{w}(V)} \arrow{s} \\
\node{\overline{w}(T) {\wedge}\overline{w}(U \cup_b V)} \arrow{e}
\node{\overline{w}(T)
{\wedge}\overline{w}(U) {\wedge}\overline{w}(V)}
\end{diagram} $
2. $ \qquad\qquad\begin{diagram}
\node{\overline{w}(T \cup_a U \cup_{a'} V)} \arrow{s} \arrow{e}
\node{\overline{w}(T \cup_a U) {\wedge}\overline{w}(V)} \arrow{s} \\
\node{\overline{w}(T \cup_{a'} V) {\wedge}\overline{w}(U)} \arrow{e}
\node{\overline{w}(T) {\wedge}\overline{w}(U) {\wedge}\overline{w}(V)}
\end{diagram} $
3. $ \qquad\qquad\qquad\begin{diagram}
\node{\overline{w}(T \cup_a I)} \arrow{e} \arrow{se,b}{{\cong}}
\node{\overline{w}(T) {\wedge}\overline{w}(I)} \arrow{s} \\
\node[2]{\overline{w}(T)}
\end{diagram} $
4. $ \qquad\qquad\qquad\begin{diagram}
\node{\overline{w}(I \cup_{\ast}T)} \arrow{e} \arrow{se,b}{{\cong}}
\node{\overline{w}(I) {\wedge}\overline{w}(T)} \arrow{s} \\
\node[2]{\overline{w}(T)}
\end{diagram} $
The argument for diagram (1) is contained in Figure \[fig:assoc1\]. A point in $\overline{w}(T \cup_a U \cup_b V)$ comes from a weighting of the grafted tree $T \cup_a U \cup_b V$. The top-left corner of Figure \[fig:assoc1\] shows a generic version of such a tree with some lengths labelled:
- $u$ is the length of the root edge.
- $v$ is the distance from the root vertex to the lower vertex of the edge that joins $U$ to $T$ (there may be intermediate vertices along this route, we let $v$ denote the total distance).
- $w$ is the length of the edge that joins $U$ to $T$.
- $x$ is the distance from the upper vertex of that edge to the lower vertex of the edge that joins $V$ to $U$.
- $y$ is the length of the edge that joins $V$ to $U$.
- $z$ is the remaining distance to any of the leaves of $V$.
Figure \[fig:assoc1\] shows that whichever way we map our weighted tree around diagram (1) we get the same result. (Note that if $y+z$ or $w+x+y+z$ equals to zero, then $z = 0$ and we are the basepoint in every corner of diagram (1).) We therefore conclude that diagram (1) commutes.
Diagram (2) is similar to (1) but easier. For diagram (3), notice that the image in $\overline{w}(T)$ of a weighting of $T \cup_a I$ will be effectively the same weighting. The image in $\overline{w}(I) = S^0$ will be the non-basepoint unless the leaf edge for $a$ has length zero. But if this is the case our starting point was the basepoint in $\overline{w}(T
\cup_a I)$. This shows that the diagram commutes.
For diagram (4), the image in $\overline{w}(T)$ of a weighting of $I
\cup_{\ast}T$ will again be the very same weighting (no scaling up is necessary). The image in $\overline{w}(I) = S^0$ will always be the non-basepoint. Therefore this diagram also commutes.
We are now in a position to state the main result of this paper.
\[thm:bar=cooperad\] Let $P$ be a reduced operad in the symmetric monoidal ${\mathcal{T}_{}}$–category ${\mathcal{C}}$. The maps of Definition \[def:cooperad\_maps\] make $B(P)$ into a reduced cooperad in ${\mathcal{C}}$.
We give the formal argument for the maps of Definition \[def:formal\_cooperad\_maps\]. To fit the relevant diagrams onto a page we need some new notation. Let’s write $$\overline{w}(T,U) := \overline{w}(T) {\wedge}\overline{w}(U)$$ and $$P_{A,B}(T',U') := P_A(T') \barwedge P_B(U').$$ Figure \[fig:assoc2\] then shows the diagram that has to commute for the dual of axiom (1) of Definition \[def:operad\] to hold for $B(P)$.
$$\begin{diagram} \dgARROWLENGTH=.7em
\node{\operatorname*{colim}_{S \leq S' \in \mathsf{T}(A \cup_a B \cup_b C)_+}
\overline{w}(S) \otimes P_{A \cup_a B \cup_b C}(S')} \arrow{s} \arrow{e}
\node{\operatorname*{colim}_{\substack{Q \leq Q' \in \mathsf{T}(A \cup_a B)_+ \\ V
\leq V' \in \mathsf{T}(C)_+}} \begin{matrix} \overline{w}(Q,V) \otimes \\
P_{A \cup_a B,C}(Q',V') \end{matrix}} \arrow{s} \arrow{e}
\node{\operatorname*{colim}_{\substack{Q \leq Q' \in \mathsf{T}(A \cup_a B)_+ \\
V \leq V' \in \mathsf{T}(C)_+}} \begin{matrix} (\overline{w}(Q) \otimes
P_{A \cup_a B}(Q')) \barwedge \\ (\overline{w}(V) \otimes P_C(V'))
\end{matrix}} \arrow{s} \\
\node{\operatorname*{colim}_{\substack{T \leq T' \in \mathsf{T}(A)_+ \\ R \leq R'
\in \mathsf{T}(B \cup_b C)_+}} \begin{matrix} \overline{w}(T,R) \otimes \\
P_{A,B \cup_b C}(T',R') \end{matrix}} \arrow{s} \arrow{e}
\node{\operatorname*{colim}_{\substack{T \leq T' \in \mathsf{T}(A)_+ \\ U \leq U'
\in \mathsf{T}(B)_+ \\ V \leq V' \in \mathsf{T}(C)_+}} \begin{matrix}
\overline{w}(T,U,V) \otimes \\ P_{A,B,C}(T',U',V') \end{matrix}} \arrow{e}
\arrow{s}
\node{\operatorname*{colim}_{\substack{T \leq T' \in \mathsf{T}(A)_+ \\ U \leq U'
\in \mathsf{T}(B)_+ \\ V \leq V' \in \mathsf{T}(C)_+}} \begin{matrix}
(\overline{w}(T,U) \otimes P_{A,B}(T',U')) \barwedge \\ (\overline{w}(V)
\otimes P_C(V')) \end{matrix}} \arrow{s} \\
\node{\operatorname*{colim}_{\substack{T \leq T' \in \mathsf{T}(A)_+ \\ R \leq R'
\in \mathsf{T}(B \cup_b C)_+}} \begin{matrix} (\overline{w}(T) \otimes
P_A(T')) \barwedge \\ (\overline{w}(R) \otimes P_{B \cup_b C}(R'))
\end{matrix}} \arrow{e}
\node{\operatorname*{colim}_{\substack{T \leq T' \in \mathsf{T}(A)_+ \\ U \leq U'
\in \mathsf{T}(B)_+ \\ V \leq V' \in \mathsf{T}(C)_+}} \begin{matrix}
(\overline{w}(T) \otimes P_A(T')) \barwedge \\ (\overline{w}(U,V) \otimes
P_{B,C}(U',V')) \end{matrix}} \arrow{e}
\node{\operatorname*{colim}_{\substack{T \leq T' \in \mathsf{T}(A)_+ \\ U \leq U'
\in \mathsf{T}(B)_+ \\ V \leq V' \in \mathsf{T}(C)_+}} \begin{matrix}
(\overline{w}(T) \otimes P_A(T')) \barwedge \\ (\overline{w}(U) \otimes P_B(U'))
\barwedge \\ (\overline{w}(V) \otimes P_C(V')) \end{matrix}}
\end{diagram}$$
The key to showing that this commutes is putting $$\operatorname*{colim}_{\substack{T \leq T' \in \mathsf{T}(A)_+ \\ U \leq U' \in
\mathsf{T}(B)_+ \\ V \leq V' \in \mathsf{T}(C)_+}} \overline{w}(T,U,V)
\otimes P_{A,B,C}(T',U',V')$$ into the center of the square. We’ve connected this to the top and left sides of the square using maps similar to the first map in Definition \[def:formal\_cooperad\_maps\]. We’ve connected it to the right and bottom sides using maps of the form $d$ from Definition \[def:axiom\]. It’s then enough to show that the four smaller squares commute.
The top-left square commutes because of diagram (1) in Lemma \[lem:keyassoc\]. The bottom-left and top-right squares commute because of the naturality of the transformations $d$. The bottom-right square commutes because it is an example of the associativity axiom we required of our $d$ transformations in \[def:axiom\].
This completes the verification of the dual of axiom (1) of Definition \[def:operad\]. For axiom (2) the argument is similar, but using diagram (2) of Lemma \[lem:keyassoc\]. For the duals of axioms (3) and (4) we use the unit axiom for the transformations $d$ together with diagrams (3) and (4) of Lemma \[lem:keyassoc\]. We leave the reader to fill in the details of these proofs.
Cobar constructions for reduced cooperads {#sec:cobar}
=========================================
We now dualize to cooperads. The cobar construction for a cooperad is strictly dual to the bar construction for an operad. More precisely, recall that a cooperad $Q$ in a category ${\mathcal{C}}$ is the same thing as an operad $Q^\text{op}$ in the opposite category ${\mathcal{C}}^\text{op}$. The cobar construction on $Q$ is then defined to be the bar construction on $Q^\text{op}$. This bar construction is a cooperad in ${\mathcal{C}}^\text{op}$ and hence an operad in ${\mathcal{C}}$. In symbols, the *cobar construction on $Q$* is $$\Omega(Q) := B(Q^\text{op})^\text{op}.$$ It can be useful to have a more explicit description of this.
\[def:cobar(cooperad)\] The cobar construction, being dual to the bar construction, is defined as an end rather than a coend. Let $Q$ be a cooperad in ${\mathcal{C}}$. Then for each finite set $A$, $Q$ determines a functor $$Q_A(-) \co \mathsf{T}(A) \to {\mathcal{C}}$$ by $$Q_A(T) = Q(i(v_1)) \barwedge \dots \barwedge Q(i(v_n))$$ where $v_1,\dots,v_n$ are the vertices of $T$. This is a functor because the cocomposition maps for $Q$ give us maps $$Q_A(T/e) \to Q_A(T).$$ (Recall that the corresponding functor for an operad was defined on $\mathsf{T}(A)^\text{op}$.) The *cobar construction* $\Omega(Q)$ is then the symmetric sequence with $$\Omega(Q)(A) := \operatorname{Map}_{\mathsf{T}(A)}(\overline{w}(-),Q_A(-)) = \int_{T
\in \mathsf{T}(A)} \operatorname{Map}_{{\mathcal{C}}}(\overline{w}(T),Q_A(T)).$$ This is the end of the bifunctor $$\mathsf{T}(A)^\text{op} \times \mathsf{T}(A) \to {\mathcal{C}}$$ given by $$(T,U) \mapsto \operatorname{Map}_{{\mathcal{C}}}(\overline{w}(T),Q_A(U))$$ where $\operatorname{Map}_{{\mathcal{C}}}$ denotes the cotensoring structure for ${\mathcal{C}}$ over ${\mathcal{T}_{}}$ (and hence the tensoring structure for ${\mathcal{C}}^\text{op}$).
The cobar construction $\Omega(Q)$ on a reduced cooperad $Q$ in based spaces is isomorphic to the totalization of a cosimplicial cobar construction that is dual to the simplicial bar construction. The terms in this cosimplicial construction are iterated versions of the dual composition product of Remark \[rem:cooperads\]. The fact that $\Omega(Q)$ is the totalization of this is dual to the result that $B(P)$ is the realization of the simplicial bar construction.
The operad structure maps for $\Omega(Q)$ are dual to the cooperad maps for $B(P)$. The following result is the dual of Proposition \[thm:bar=cooperad\].
\[cor:cobar=operad\] Let $Q$ be a reduced cooperad in a symmetric monoidal ${\mathcal{T}_{}}$–category ${\mathcal{C}}$. Then the cobar construction $\Omega(Q)$ is a reduced operad in ${\mathcal{C}}$.
Duality for operads and cooperads {#sec:dual}
=================================
In this section we examine how the bar and cobar constructions relate to the ‘duality’ functor $${\mathbb{D}}\co {\mathcal{T}_{}}^\text{op} \to {\mathcal{C}}; \; X \mapsto \operatorname{Map}_{{\mathcal{C}}}(X,S)$$ where $S$ is the unit of the symmetric monoidal structure on ${\mathcal{C}}$. The case to keep in mind is ${\mathcal{C}} = {\mathcal{S}p}$ in which case $S$ is the sphere spectrum and this duality functor is Spanier–Whitehead duality.
\[lem:dual\] Let $Q$ be a cooperad of based spaces. Then ${\mathbb{D}}Q$ is an operad in the category ${\mathcal{C}}$.
The composition maps for ${\mathbb{D}}Q$ are given by $$\begin{split} \operatorname{Map}_{{\mathcal{C}}}(Q(A),S) \barwedge \operatorname{Map}_{{\mathcal{C}}}(Q(B),S)
&\to \operatorname{Map}_{{\mathcal{C}}}(Q(A) {\wedge}Q(B),S) \\ &\to \operatorname{Map}_{{\mathcal{C}}}(Q(A \cup_a
B),S). \\ \end{split}$$ The first map is the natural transformation constructed in Proposition \[prop:dual\] (it’s the distributive map $d$ for ${\mathcal{C}}^\text{op}$). The second comes from the corresponding cocomposition map for $Q$.
The dual of an operad need not in general be a cooperad because the map $d$ need not in general have an inverse. However when it does we have a nice duality result connecting the bar and cobar constructions. For this to work we need to put the following condition on the spaces that make up our operad.
Two based spaces $X,Y$ are *compatibly dualizable in ${\mathcal{C}}$* if the map $$d\co \operatorname{Map}_{{\mathcal{C}}}(X,S) \barwedge \operatorname{Map}_{{\mathcal{C}}}(Y,S) \to
\operatorname{Map}_{{\mathcal{C}}}(X {\wedge}Y,S)$$ is an isomorphism.
\[prop:duality\] Let $P$ be an operad in based spaces whose terms (that is, the $P(A)$ for finite sets $A$) are pairwise compatibly dualizable. Then ${\mathbb{D}}P$ has a natural cooperad structure. Moreover, we have an isomorphism $${\mathbb{D}}B(P) {\cong}\Omega({\mathbb{D}}P)$$ of operads in ${\mathcal{C}}$.
The cooperad structure maps for ${\mathbb{D}}P$ are constructed in the same way as the operad structure maps for ${\mathbb{D}}Q$ in \[lem:dual\] but using the inverse of the relevant map $d$ provided by the ‘compatibly dualizable’ hypothesis.
The second part relies on the descriptions of the bar and cobar constructions as coends and ends respectively. The coend $B(P)$ is a colimit: $$B(P)(A) = \operatorname*{colim}_{T \leq T'} \overline{w}(T) {\wedge}P_A(T')$$ where the colimit is taken over all inequalities of trees in $\mathsf{T}(A)$. Therefore $$\begin{split}
{\mathbb{D}}B(P)(A) & = \operatorname{Map}_{{\mathcal{C}}}( \operatorname*{colim}\overline{w}(T) {\wedge}P_A(T'),
S) \\
& {\cong}\lim \operatorname{Map}_{{\mathcal{C}}}(\overline{w}(T) {\wedge}P_A(T'), S) \\
& {\cong}\lim \operatorname{Map}_{{\mathcal{C}}}(\overline{w}(T), \operatorname{Map}_{{\mathcal{C}}}(P_A(T'),
S)) \\
& {\cong}\lim \operatorname{Map}_{{\mathcal{C}}}(\overline{w}(T), ({\mathbb{D}}P)_A(T')). \\
\end{split}$$ The last identity again uses the ‘compatibly dualizable’ hypothesis in the form $$\operatorname{Map}_{{\mathcal{C}}}(P(i(v_1)) {\wedge}\ldots {\wedge}P(i(v_n)), S)
{\cong}\operatorname{Map}_{{\mathcal{C}}}(P(i(v_1)), S) \barwedge \ldots \barwedge
\operatorname{Map}_{{\mathcal{C}}}(P(i(v_n)),S).$$ The final line of this calculation is precisely the limit that defines $\Omega({\mathbb{D}}P)$. We leave the reader to check that this is an isomorphism of operads.
The only case of this result we will use in this paper is when all the terms of the operad $P$ are $S^0$. These are pairwise compatibly dualizable in any ${\mathcal{C}}$ because $$\operatorname{Map}_{{\mathcal{C}}}(S^0,C) {\cong}C$$ for any $C \in {\mathcal{C}}$.
Replacing ${\mathcal{C}}$ with ${\mathcal{C}}^\text{op}$ we obtain dual results. These concern the functor $\mathbb{S}\co X \mapsto X \otimes S$, the ‘suspension spectrum’ functor. We find that if $Q$ is a cooperad in based spaces then $\mathbb{S}Q$ is a cooperad in ${\mathcal{C}}$. If $P$ is an operad whose terms are pairwise compatibly dualizable then $\mathbb{S}P$ is an operad in ${\mathcal{C}}$ and $\mathbb{S}B(P) {\cong}B(\mathbb{S}P)$.
We have now reached the stage where we can apply our constructions to Goodwillie’s calculus of functors (see Section \[sec:application\]). Before doing so, we extend our bar and cobar constructions to modules and comodules. This will then allow us to construct modules over the derivatives of the identity.
Bar constructions for modules and comodules {#sec:bar(modules)}
===========================================
In this section we extend the bar and cobar constructions to modules and comodules. We show that there is a bar construction on left (respectively right) modules over a reduced operad $P$ that yields left (respectively right) comodules over the cooperad $B(P)$. Dually, there is a cobar construction on left (respectively right) comodules over a reduced cooperad $Q$ that yields left (respectively right) modules over the operad $\Omega(Q)$. These are special cases of two-sided bar and cobar constructions. Given a reduced operad $P$ with right module $R$ and left module $L$, we will define a two-sided bar construction $B(R,P,L)$. Taking either $R$ or $L$ to be the unit symmetric sequence $I$ will yield the promised one-sided constructions for individual modules. The two-sided construction is isomorphic to the standard simplicial two-sided bar construction (see Definition \[def:two-sided\_simp\]) but, in order to get the comodule structure, we have reinterpreted this in terms of trees.
Most of the material in this section is a straightforward generalization of that of Sections \[sec:trees\]–\[sec:cobar\]. First, in Section \[sec:gen\_trees\] we describe the more general species of tree necessary for the definitions of the two-sided constructions. In Section \[sec:two-sided\] we give these definitions and show that the bar construction of Section \[sec:bardef\] is a special case. In Section \[sec:bar(modules)\_maps\] we construct the maps that make the bar construction on a module into a comodule, and dually, the cobar construction on a comodule into a module.
As previously, ${\mathcal{C}}$ denotes a symmetric monoidal ${\mathcal{T}_{}}$–category with null object ${\ast}$ and which has all necessary limits and colimits.
Generalized trees {#sec:gen_trees}
-----------------
To accommodate the presence of the $P$–modules $R$ and $L$ in the two-sided bar construction, we need to make two changes to our notion of tree, one at the root level and one at the leaf level:
1. We allow the root element of a tree to have more than one incoming edge.
2. We allow the leaves of a tree to have repeated labels, that is, an $A$–labelling is a surjection from $A$ to the set of leaves, rather than a bijection.
We will refer to this notion as a ‘generalized tree’, or sometimes just a ‘tree’ if the context makes it clear that we mean the generalized version. The following definition makes things precise.
\[def:gen\_trees\] Let $A$ be a finite set. A *generalized $A$–labelled tree* consists of
- a poset $T$ with a unique minimal element $r$ (the *root*) satisfying conditions (2) and (3) of Definition \[def:tree\], and
- a surjection $\iota$ from the finite set $A$ to the set of maximal elements (the *leaves*) of $T$.
We use letters $T,U,\dots$ to denote generalized trees, usually taking the labelling map $\iota$ for granted. We write $\mathsf{Tree}(A)$ for the set of isomorphism classes of generalized $A$–labelled trees. All the terminology of Definition \[def:tree\] applies equally well to generalized trees.
Edge collapse for generalized trees is defined in exactly the same way as for the trees of Section \[sec:trees\] except that now we allow ourselves to collapse root edges as well as internal edges. To get the right category structure on $\mathsf{Tree}(A)$ we need a way to collapse leaf edges as well. The following definition provides this.
\[def:bud\_collapse\] A *bud* in a generalized tree $T$ is a vertex all of whose incoming edges are leaf edges. Equivalently, a bud is a maximal vertex. If $b$ is a bud in $T$, a *$b$–leaf* is a leaf of $T$ that is attached to $b$.
Given a generalized $A$–labelled tree $T$ and a bud $b \in T$, we define a generalized $A$–labelled tree $T_b$ which is obtained from $T$ by *bud collapse*. The underlying poset of $T_b$ is obtained from $T$ by removing the $b$–leaves. This makes $b$ into a leaf in $T_b$. The $A$–labelling on $T_b$ is that of $T$ for the leaves that still remain, with $b$ inheriting the labels of its old leaves. Formally, we are composing the $A$–labelling on $T$ with the surjection from the leaves of $T$ to the leaves of $T_b$ that sends the $b$–leaves in $T$ to $b$. Visually, we can think of this process as collapsing all the leaf edges attached to $b$ (see Figure \[fig:bud\_collapse\]).
\[def:Tree(A)\] If $T$ and $T'$ are generalized $A$–labelled trees, we say that $T \leq
T'$ if $T$ can be obtained from $T'$ by a sequence of edge collapses (of either internal or root edges) or bud collapses. This makes the set $\mathsf{Tree}(A)$ of isomorphism classes of generalized $A$–labelled trees into a poset and hence a category. Standard $A$–labelled trees (as defined in Section \[sec:trees\]) are also generalized $A$–labelled trees and $\mathsf{T}(A)$ is a full subcategory of $\mathsf{Tree}(A)$. See Figure \[fig:gen\_trees\] for pictures of $\mathsf{Tree}(1)$ and $\mathsf{Tree}(2)$.
\[def:gen\_weighting\] We don’t need to change the definition of a *weighting* for generalized trees: it is an assignment of lengths to the edges of a tree such that the root-leaf distances all equal $1$. As before, we write $w(T)$ for the space of weightings on the generalized tree $T$. The following result generalizes Lemma \[lem:W\].
\[lem:gen\_W\] Let $T$ be a generalized $A$–labelled tree with $n$ (internal) vertices. Then $w(T)$ is homeomorphic to $D^n$ and the boundary $\partial
w(T) {\cong}S^{n-1}$ consists of those points in which some edge of $T$ has length zero.
The labelling plays no role in the space of weightings so we can ignore it. Picture $T$ as a collection of (non-generalized) trees $T_1,\dots,T_k$ attached at their roots. Suppose $T_j$ has $n_j$ vertices so that $n =
\sum n_j$. Then we have $$w(T) {\cong}w(T_1) \times \dots \times w(T_k) {\cong}D^{n_1} \times
\dots \times D^{n_k} {\cong}D^n.$$ Under this decomposition, a point is in the boundary of $w(T)$ if and only if any of it is in the boundary of any of the $w(T_j)$. That is, if and only if any of the edges of $T$ has length zero.
\[def:gen\_w\_functor\] The ‘space of weightings’ functor $w(-)\co \mathsf{T}(A) \to
{\mathcal{U}_{}}$ of Definition \[def:w\_functor\] can be extended to all of $\mathsf{Tree}(A)$. To do this, we have to say what happens when we apply $w(-)$ to a morphism $T_b \to T$ coming from a bud collapse (for $b$ a bud in a tree $T$). Given a weighting of $T_b$ we get a weighting of $T$ by giving length zero to all the leaf edges attached to $b$. This defines a map $$w(T_b) \to w(T)$$ and it is not hard to see that this does indeed give us a functor $$w(-)\co \mathsf{Tree}(A) \to {\mathcal{U}_{}}$$ as claimed. Adding a disjoint basepoint we get a functor $$w(-)_+\co \mathsf{Tree}(A) \to {\mathcal{T}_{}}.$$
The two-sided bar construction {#sec:two-sided}
------------------------------
We now update Definition \[def:formal\_bar\] to the two-sided case. Along with the spaces of weightings the key parts of this definition were functors $$P_A(-) \co \mathsf{T}(A)^\text{op} \to {\mathcal{C}}.$$ The appropriate generalizations of these to functors on $\mathsf{Tree}(A)^\text{op}$ are as follows.
\[def:(R,P,L)\] Let $P$ be a reduced operad in ${\mathcal{C}}$ with right module $R$ and left module $L$. We define functors $(R,P,L)_A \co \mathsf{Tree}(A)^\text{op}
\to {\mathcal{C}}$ by [^11] $$(R,P,L)_A(T) := R(i(r)) \barwedge \operatorname*{\overline{\bigwedge}}_{\text{vertices }
v \in T} P(i(v)) \barwedge \operatorname*{\overline{\bigwedge}}_{\text{leaves } l \in T}
L(\iota^{-1}l).$$ Recall that $i(v)$ denotes the set of incoming edges to a vertex $v \in
T$. Here $\iota$ denotes the labelling surjection from $A$ to the set of leaves of $T$, so that $\iota^{-1}l$ is the set of labels attached to the leaf $l$.
To complete the definition, we have to give the effect of $(R,P,L)_A(-)$ on morphisms in $\mathsf{Tree}(A)$. Notice that $\mathsf{Tree}(A)$ is generated by the morphisms corresponding to
1. collapse of root edges,
2. collapse of internal edges, and
3. bud collapse.
We will describe the effect of $(R,P,L)_A(-)$ on each of these types of generating morphism and then check that they are compatible.
(1)Suppose first that $e$ is a root edge of the generalized $A$–labelled tree $T$. Then we have a morphism $T/e \to T$ that collapses $e$. The corresponding morphism $$(R,P,L)_A(T) \to (R,P,L)_A(T/e)$$ is given by the map $$R(i(r)) \barwedge P(i(v)) \to R(i(r \circ v))$$ that comes from the right $P$–module structure on $R$. Here $v$ is the upper endpoint of the edge $e$ in $T$. Notice that $r \circ v$ is the root element in $T/e$.
(2)Now suppose that $e$ is an internal edge of $T$. The morphism $$(R,P,L)_A(T) \to (R,P,L)_A(T/e)$$ is then given (as in Definition \[def:formal\_bar\]) by the partial composition map $$P(i(u)) \barwedge P(i(v)) \to P(i(u \circ v))$$ for the operad $P$ where $u,v$ are the endpoints of $e$.
(3)Finally, suppose that $b$ is a bud in the generalized $A$–labelled tree $T$. The required map $$(R,P,L)_A(T) \to (R,P,L)_A(T_b)$$ comes from the map $$P(i(b)) \barwedge L(\iota^{-1}l_1) \barwedge \dots \barwedge
L(\iota^{-1}l_r) \to L(\iota_1^{-1}b)$$ that is part of the left $P$–module structure on $L$. Here $l_1,\dots,l_r$ are the $b$–leaves in $T$ and we have $$\iota^{-1}b = \coprod_{i=1}^{r} \iota^{-1}l_i$$ from the definition of bud collapse, where $\iota_1$ is the $A$–labelling of $T_b$. The associativity conditions for $P$ to be an operad and for $R$ and $L$ to be $P$–modules ensure that these choices indeed determine a functor $\mathsf{Tree}(A)^\text{op} \to {\mathcal{C}}$.
\[def:two-sided\_bar\] Let $P$ be a reduced operad in ${\mathcal{C}}$ with right module $R$ and left module $L$ as above. The *bar construction on $P$ with coefficients in $R$ and $L$* is the symmetric sequence $B(R,P,L)$ defined by the coends $$B(R,P,L)(A) := \int^{T \in \mathsf{Tree}(A)} w(T)_+ \otimes
(R,P,L)_A(T)$$ for finite sets $A$. A bijection $A \to A'$ determines an isomorphism of categories $\mathsf{Tree}(A) \to \mathsf{Tree}(A')$ under which the pairs of functors $w_A(-)$, $w_{A'}(-)$ and $(R,P,L)_A$, $(R,P,L)_{A'}$ correspond. It therefore induces an isomorphism $$B(R,P,L)(A) \to B(R,P,L)(A').$$ So we obtain a symmetric sequence $B(R,P,L)$.
There is a more informal description of this bar construction that generalizes that of $B(P)$ from Definition \[def:bar(operad)\]. For a finite set $A$, a point in $B(R,P,L)(A)$ consists of
- a weighted generalized $A$–labelled tree $T$,
- a point in $R(i(r))$ where $r$ is the root of $T$,
- a point in $P(i(v))$ for each vertex $v \in T$, and
- a point in $L(\iota^{-1}l)$ for each leaf $l \in T$.
These are subject to identifications that tell us what happens when the lengths of some of the edges tend to zero. When a root edge tends to zero we use the right $P$–module structure map for $R$. When an internal edge tends to zero we use the composition map for $P$. When a collection of leaf edges attached to a bud tend to zero (note that the leaf edges attached to a particular bud must all have the same length in a weighting) we use the left $P$–module structure for $L$. Finally, of course, we identify to the basepoint in $B(R,P,L)(A)$ if any of the chosen points in $R(i(r)),P(i(v)),L(\iota^{-1}l)$ are the basepoint there.
We now recall the simplicial version of the two-sided bar construction for an operads and modules over them.
\[def:two-sided\_simp\] Let $P$ be an operad in ${\mathcal{C}}$ with right module $R$ and left module $L$. The *simplicial bar construction on $P$ with coefficients in $L$ and $R$* is the simplicial object $\mathcal{B}_{\bullet}(R,P,L)$ in the category of symmetric sequences in ${\mathcal{C}}$ with $$\mathcal{B}_n(R,P,L) := R \circ \underset{n}{\underbrace{P \circ
\dots \circ P}} \circ L.$$ The face maps $$d_i \co \mathcal{B}_n(R,P,L) \to \mathcal{B}_{n-1}(R,P,L)$$ for $i = 1,\dots,n-1$ are given by the operad composition map $P \circ
P \to P$ applied to the [$i$^th^]{} and [$i+1$^th^]{} factors. The face map $d_0$ is given by the right module structure $R \circ P \to R$ and $d_n$ is given by the left module structure $P \circ L \to L$. The degeneracy map $$s_j \co \mathcal{B}_n(R,P,L) \to \mathcal{B}_{n+1}(R,P,L)$$ is given by using the unit map $I \to P$ to insert an extra copy of $P$ between the [$j$^th^]{} and [$j+1$^th^]{} factors.
\[prop:two-sided\_simp\] Let $P$ be a reduced operad in ${\mathcal{C}}$ with right module $R$ and left module $L$. The bar construction of Definition \[def:two-sided\_bar\] is isomorphic to the geometric realization of the simplicial bar construction: $$B(R,P,L) {\cong}|\mathcal{B}_{\bullet}(R,P,L)|.$$
This is a straightforward extension of the argument used to prove Proposition \[prop:simp\].
Our first example of the two-sided bar construction is that the reduced bar construction a lone operad is a special case.
\[ex:old\_bar=new\] Let $P$ be a reduced operad in ${\mathcal{C}}$ and take $R = L = I$ the unit symmetric sequence. Recall that $I$ is a left and right module over any augmented operad. It is easy to see from the definitions that for the simplicial bar constructions we have $$\mathcal{B}_{\bullet}(I,P,I) {\cong}\mathcal{B}_{\bullet}(P).$$ This tells us that $$B(I,P,I) {\cong}B(P),$$ but we can see this directly as well. First notice that $$(I,P,I)_A(T) {\cong}\begin{cases} P_A(T) & \text{if $T \in
\mathsf{T}(A)$}; \\ {\ast}& \text{otherwise}. \end{cases}$$ The reason for this is as follows. Because $I(n) = {\ast}$ for $n > 1$, we have $(I,P,I)_A(T) = {\ast}$ whenever $T$ has more than one root edge, or when any leaf has more than one label. These are precisely the generalized $A$–labelled trees not in $\mathsf{T}(A)$. For $T \in
\mathsf{T}(A)$ we have $$(I,P,I)_A(T) = I(1) \barwedge P_A(T) \barwedge I(1) \barwedge \dots
\barwedge I(1) {\cong}P_A(T).$$
This calculation means that only the objects $T \in \mathsf{T}(A)$ contribute to the calculation of the coend in Definition \[def:two-sided\_bar\]. However, we still have to take into account morphisms $U \to T$ with $U \notin \mathsf{T}(A)$. This amounts to collapsing to the basepoint those weighted trees in which either the root edge or a leaf edge has length zero (since these are the images of the maps $w(U) \to w(T)$). All together this tells us that $B(I,P,I)(A)$ is equal to the coend $$\int^{T \in \mathsf{T}(A)} \overline{w}(T) \otimes P_A(T)$$ where $\overline{w}(T)$ is the quotient of $w(T)$ by the weightings which have either root or leaf edge of length zero. This is precisely $B(P)(A)$. Therefore we have $B(I,P,I) {\cong}B(P)$ as claimed.
\[ex:two-sided\_bar\] It is easy to see that $B(R,P,L)(1) = R(1) \barwedge L(1)$. We have already seen (Figure \[fig:gen\_trees\]) that there are three objects in $\mathsf{Tree}(2)$. From this we see that $B(R,P,L)(2)$ is the homotopy pushout of the following diagram $$\begin{diagram} \node{R(1) \barwedge P(2) \barwedge L(1) \barwedge
L(1)} \arrow{s} \arrow{e} \node{R(1) \barwedge L(2)} \\
\node{R(2) \barwedge L(1) \barwedge L(1)}
\end{diagram}$$ If $R = L = I$, the bottom-left and top-right objects are ${\ast}$ and the top-left object is $P(2)$. So we recover $$B(P)(2) = B(I,P,I)(2) = \Sigma P(2).$$
Let $P$ be a reduced operad in ${\mathcal{C}}$ and let $R$ be a right $P$–module. We define the *bar construction on $R$* by $$B(R) := B(R,P,I)$$ where $I$, as previously, is the unit for the composition product of symmetric sequences. If $L$ is a left $P$–module, its *bar construction* is $$B(L) := B(I,P,L).$$ We trust that it will not be confusing to use the same notation for the bar construction of right and left modules.
\[ex:two-sided\_modules\] Applying Example \[ex:two-sided\_bar\] to the one-sided case we see that $$B(R)(1) {\cong}R(1); \;\; B(R)(2) {\cong}\operatorname*{hocofib}(R(1) \barwedge P(2)
\to R(2))$$ and $$B(L)(1) {\cong}L(1); \;\; B(L)(2) {\cong}\operatorname*{hocofib}(P(2) \barwedge L(1)
\barwedge L(1) \to L(2)).$$
All the constructions of this section can be applied to operads and modules in ${\mathcal{C}}^\text{op}$, that is, to cooperads and comodules in ${\mathcal{C}}$. We summarize the results.
If $Q$ is a reduced cooperad in ${\mathcal{C}}$ with left comodule $L$ and right comodule $R$, the formula of Definition \[def:(R,P,L)\] defines functors $$(R,Q,L)_A(-)\co \mathsf{Tree}(A) \to {\mathcal{C}}$$ for each finite set $A$ and we define the *cobar construction* on $Q$ with coefficients in $R$ and $L$ to be the symmetric sequence $\Omega(R,Q,L)$ with $$\Omega(R,Q,L)(A) := \int_{T \in \mathsf{Tree}(A)}
\operatorname{Map}_{{\mathcal{C}}}(w(T)_+,(R,Q,L)_A(T)).$$ This is isomorphic to the totalization of the two-sided cosimplicial cobar construction on $Q$ with coefficients in $R$ and $L$. The *cobar construction on $R$* is $$\Omega(R) := \Omega(R,Q,I)$$ and the *cobar construction on $L$* is $$\Omega(L) := \Omega(I,Q,L).$$
Taking $R = L = I$ we recover the cobar construction of Section \[sec:cobar\]: $$\Omega(I,Q,I) {\cong}\Omega(Q).$$
\[ex:two-sided\_cobar\] Taking the duals of the results of Example \[ex:two-sided\_bar\] we see that $$\Omega(R,Q,L)(1) {\cong}R(1) \barwedge L(1)$$ and that $\Omega(R,Q,L)(2)$ is the homotopy pullback of $$\begin{diagram}
\node[2]{R(1) \barwedge L(2)} \arrow{s} \\
\node{R(2) \barwedge L(1) \barwedge L(1)} \arrow{e} \node{R(1)
\barwedge Q(2) \barwedge L(1) \barwedge L(1).}
\end{diagram}$$ In particular, $$\Omega(R)(1) {\cong}R(1); \;\; \Omega(R)(2) {\cong}\operatorname*{hofib}(R(2) \to R(1)
\barwedge Q(2))$$ and $$\Omega(L)(1) {\cong}L(1); \;\; \Omega(L)(2) {\cong}\operatorname*{hofib}(L(2) \to Q(2)
\barwedge L(1) \barwedge L(1)).$$
Structure maps for bar constructions on modules {#sec:bar(modules)_maps}
-----------------------------------------------
In this section we use similar methods to Section \[sec:cooperad\] to show that the bar construction on a $P$–module (that is, a single left or right module) is a comodule over the cooperad $B(P)$. In fact, we will construct maps of the form $$B(R,P,L) \to B(R,P,I) {\mathbin{\widehat{\circ}}}B(I,P,L)
\label{eq:general}$$ where ${\mathbin{\widehat{\circ}}}$ is the composition of symmetric sequences defined using the product in ${\mathcal{C}}$ rather than the coproduct (see Remark \[rem:cooperads\]). Taking $R = I$ and recalling that $B(I,P,I) = B(P)$ we obtain a left $B(P)$–comodule structure on $B(L) = B(I,P,L)$. Similarly, taking $L = I$ we get a right $B(P)$–comodule structure on $B(R)$. Notice that taking $R = L = I$ we recover the cooperad structure on $B(P)$.
The definition of the map (\[eq:general\]) is a relatively straightforward generalization of the cooperad structure on $B(P)$. We start by describing the grafting and ungrafting processes for generalized trees.
Let $T$ be a generalized $A$–labelled tree and $U$ a generalized $B$–labelled tree and let $a$ be an element of $A$. We will define the *grafting of $U$ onto $T$* only if $T$ and $U$ satisfy the following conditions:
- The root of $U$ has only one incoming edge.
- The leaf of $T$ labelled by $a$ is labelled only by $a$ and no other elements of $A$.
In this case, the grafted tree $T \cup_a U$ is defined exactly as in Definition \[def:grafting\] by identifying the root edge of $U$ to the $a$–leaf edge of $T$. Figure \[fig:ungraft\] gives an example.
To define the maps (\[eq:general\]) we will need to graft trees onto all of the leaf edges of the base tree $T$. To do this, we must assume that all the leaves of $T$ only have one label, so that $T$ satisfies the stronger condition for a labelling we required in Definition \[def:labelling\]. Notice also that the trees $U$ we are to graft onto $T$ satisfy the stronger root condition of Definition \[def:tree\]. The following definitions will help us talk about trees of these types.
\[def:more\_trees\] For a finite set $A$, we define the following full subcategories of $\mathsf{Tree}(A)$: $$\begin{gathered}
\mathsf{T}_\text{root}(A) := \{ T \in \mathsf{Tree}(A) |
\text{ the root of $T$ has only one incoming edge}\} \\
\mathsf{T}_\text{leaf}(A) := \{ T \in \mathsf{Tree}(A) | \text{ the
leaves of $T$ are labelled bijectively by $A$}\}. \end{gathered}$$ Notice that $\mathsf{T}(A) = \mathsf{T}_\text{root}(A) \cap
\mathsf{T}_\text{leaf}(A)$.
Let $A = \coprod_{j \in J} A_j$ be a partition of $A$ into nonempty subsets. Given trees $U_j \in \mathsf{T}_\text{root}(A_j)$ and $T \in
\mathsf{T}_\text{leaf}(J)$, we denote the tree obtained by grafting all the $U_j$ onto $T$ at the appropriate places by $$T \cup_J U_j.$$ We say that a generalized $A$–labelled tree is *of type* $\{A_j\}$ if it is of the form $T \cup_J U_j$ for some such $T$ and $U_j$. The correct generalization of the functor of Proposition \[prop:category\_cooperad\] is then a functor $$\mathsf{Tree}(A)_+ \to \mathsf{T}_\text{leaf}(J)_+
{\wedge}\mathsf{T}_\text{root}(A_{j_1})_+ {\wedge}\dots {\wedge}\mathsf{T}_\text{leaf}(A_{j_r})_+$$ that breaks the tree $(T \cup_J U_j)$ into its components $T$ and the $U_j$ and sends a tree not of type $\{A_j\}$ to the initial object on the right-hand side. This ‘ungrafting’ functor is the basis of the map (\[eq:general\]).
Our new categories of trees can be used as the base categories for defining the one-sided bar constructions. For this we need the appropriate spaces of weightings.
For each finite set $A$ we define a functor $$w_\text{leaf}(-)\co \mathsf{T}_\text{leaf}(A) \to {\mathcal{T}_{}}$$ where $w_\text{leaf}(T)$ is the quotient of $w(T)$ by the space of weightings in which some leaf edge has length zero, and a functor $$w_\text{root}(-)\co \mathsf{T}_\text{root}(A) \to {\mathcal{T}_{}}$$ where $w_\text{root}(T)$ is the quotient of $w(T)$ by the space of weightings in which the root edge has length zero.
Let $P$ be a reduced operad in ${\mathcal{C}}$ with right module $R$ and left module $L$. Then the one-sided bar constructions are given by $$B(R)(A) = B(R,P,I)(A) {\cong}\int^{T \in \mathsf{T}_\text{leaf}(A)}
w_\text{leaf}(T) \otimes (R,P,I)_A(T)$$ and $$B(L)(A) = B(I,P,L)(A) {\cong}\int^{T \in \mathsf{T}_\text{root}(A)}
w_\text{root}(T) \otimes (I,P,L)_A(T).$$
These calculations are similar to that in Example \[ex:old\_bar=new\] where we showed that $B(P) = B(I,P,I)$. They use the facts that $$(R,P,I)_A(T) = {\ast}\text{ for $T \notin \mathsf{T}_\text{leaf}(A)$}$$ and $$(I,P,L)_A(T) = {\ast}\text{ for $T \notin \mathsf{T}_\text{root}(A)$}.
\eqno{\Box}$$
The final piece of the puzzle is the construction of a map analogous to (\[eq:key\]) that tells us how to weight the trees obtained from ungrafting.
\[def:gen\_key\] Let $A = \coprod_{j \in J} A_j$ be a partition of the finite set $A$ into nonempty subsets. Given trees $T \in \mathsf{T}_\text{leaf}(J)$ and $U_j
\in \mathsf{T}_\text{root}(A_j)$ we define a map $$w(T \cup_J U_j)_+ \to w_\text{leaf}(T) {\wedge}w_\text{root}(U_{j_1}) {\wedge}\dots
{\wedge}w_\text{root}(U_{j_r})$$ by the obvious generalization of the construction of the maps $\overline{w}(T \cup_a U) \to \overline{w}(T) {\wedge}\overline{w}(U)$ in Definition \[def:cooperad\_maps\].
\[def:general\] Putting together all these ingredients we construct maps $$B(R,P,L)(A) \to B(R,P,I)(J) \barwedge B(I,P,L)(A_{j_1}) \barwedge
\dots \barwedge B(I,P,L)(A_{j_r}).$$ In an analogous way to Definition \[def:formal\_cooperad\_maps\], these come from the maps of Definition \[def:gen\_key\] together with the isomorphisms $$(R,P,L)_A(T \cup_J U_j) \to (R,P,I)_J(T) \barwedge
(I,P,L)_{A_{j_1}}(U_{j_1}) \barwedge \dots \barwedge
(I,P,L)_{A_{j_r}}(U_{j_r}).$$ Together these maps make up the map of symmetric sequences $$B(R,P,L) \to B(R,P,I) {\mathbin{\widehat{\circ}}}B(I,P,L)$$ as promised.
\[prop:general\] Let $P$ be a reduced operad in ${\mathcal{C}}$ with right module $R$ and left module $L$. The maps of Definition \[def:general\] determine a right $B(P)$–comodule structure on $B(R)$ and a left $B(P)$–comodule structure on $B(L)$.
Taking $L = I$ in \[def:general\] we get the right comodule structure on $B(R)$. Taking $R = I$ we get the left comodule structure on $B(L)$. We have to check the appropriate associativity and unit axioms. This is a generalization of the work of Section \[sec:cooperad\]. We leave the reader to write out all the details, including the diagrams corresponding to Figure \[fig:assoc2\].
Dually, suppose that $Q$ is a reduced cooperad in ${\mathcal{C}}$ with right comodule $R$ and left comodule $L$. Then there is a map $$\Omega(R,Q,I) \circ \Omega(I,Q,L) \to \Omega(R,Q,L)$$ that makes $\Omega(R)$ into a right $\Omega(Q)$–module (by taking $L =
I$) and $\Omega(L)$ into a left $\Omega(Q)$–module (by taking $R = I$).
Apply Proposition \[prop:general\] to $Q$ considered as an operad in ${\mathcal{C}}^\text{op}$.
This completes our descriptions of the bar and cobar constructions for operads, cooperads, modules and comodules. We turn now to our main application of this theory – the Goodwillie derivatives of the identity functor.
Application to the calculus of functors {#sec:application}
=======================================
In this section we describe our application of bar and cobar constructions to Goodwillie’s calculus of homotopy functors. The main result is that the derivatives of the identity form an operad in spectra. We now assume that ${\mathcal{C}}$ is a suitable category ${\mathcal{S}p}$ of spectra, for example, the $S$–modules of EKMM [@elmendorf/kriz/mandell/may:1997] (see Example \[ex:categories\](2)).
Let $I\co {\mathcal{T}_{}}\to {\mathcal{T}_{}}$ be the identity functor on based spaces. The Goodwillie derivatives of $I$ can be described in terms of the partition poset complexes [@arone/mahowald:1999]. We recall one of the ways to define these.
A *partition* of a finite set $A$ is an equivalence relation on $A$. Let $K(A)$ be the poset formed by the partitions of $A$ with $\lambda \leq \mu$ if $\lambda$ is *finer* than $\mu$, that is, if the set of relations for $\lambda$ is contained in the set of relations for $\mu$. The category $K(A)$ has an initial object $\widehat{0}$ and a terminal object $\widehat{1}$. Let $K_0(A) = K(A) - \widehat{0}$, the category of *proper* partitions, and $K_1(A) = K(A) - \widehat{1}$, the category of *non-trivial* partitions. Note that the group $\Sigma_A$ of permutations of $A$ acts on all of these categories in an obvious way.
For a finite set $A$, the *partition poset complex* $\Delta(A)$ is the geometric realization of the following simplicial set $T(A)_{\bullet}$ formed from the nerves of these categories of partitions: $$T(A)_{\bullet} = \frac{N_{\bullet}K(A)}{N_{\bullet}K_0(A) {\cup}N_{\bullet}K_1(A)}$$ So the $n$–simplices in $T(A)_{\bullet}$ are sequences of $n+1$ partitions $$\lambda_0 \leq \lambda_1 \leq \dots \leq \lambda_n$$ with a sequence identified to the basepoint if it does not have both $\lambda_0 = \widehat{0}$ and $\lambda_n = \widehat{1}$. The face and degeneracy maps are given by respectively removing partitions from the sequence and repeating terms in the usual way for the nerve of a category. The simplicial set $T(A)_{\bullet}$ is pointed and so its geometric realization $\Delta(A)$ is a based space. A bijection $A \to
A'$ induces an isomorphism $\Delta(A) \to \Delta(A')$ that makes $\Delta$ into a symmetric sequence in ${\mathcal{T}_{}}$.
What we are calling the partition poset complex is the suspension of the complex $K_n$ of [@arone/mahowald:1999]. The simplicial set $T(n)_{\bullet}$ is isomorphic to that called $T_n$ in Definition 1.1 of [@arone/mahowald:1999].
\[prop:AM\] The derivatives of the identity are modelled by the dual spectra of the finite complexes $\Delta(n) = \Delta({\{1,\dots,n\}})$: $$\partial_nI \simeq \operatorname{Map}_{{\mathcal{S}p}}(\Delta(n),S)$$ The action of the symmetric group $\Sigma_n$ on $\Delta(n)$ induces an action on the dual spectrum and this agrees with the action that comes with the spectrum $\partial_nI$.
The key observation (apparently due to Greg Arone) is that the partition poset complexes can be described as spaces of trees. We can interpret these as the spaces of a bar construction.
Let $\underline{S^0}$ be the operad in based spaces with $$\underline{S^0}(A) := S^0$$ for all finite sets $A$ and with all composition maps equal to the identity on $S^0$. This is the operad for commutative monoids of based spaces.
\[lem:ppc=bar\] The partition poset complex $\Delta(A)$ is homeomorphic to the bar construction $B(\underline{S^0})(A)$.
We have already seen that $B(\underline{S^0})$ is homeomorphic to the realization of the simplicial bar construction on $\underline{S^0}$. It is therefore enough to show that the simplicial set $T(A)_{\bullet}$ used to define $\Delta(A)$ is also given by this simplicial bar construction.
A non-basepoint $n$–simplex in $T(A)$ is an increasing sequence of partitions of $A$ of length $n-1$. On the other hand the based set of $n$–simplices in the simplicial bar construction is $$\underset{n}{\underbrace{\underline{S^0} \circ \dots \circ
\underline{S^0}}}(A).$$ But this is equal to the wedge over increasing sequences of partitions of length $n-1$ of $S^0$. Hence we see that the two sets of $n$–simplices are the same. The face and degeneracy maps in each case correspond to removing a partition and repeating a partition respectively. We therefore have isomorphic simplicial sets.
\[rem:vallette\] In [@vallette:2004], Bruno Vallette describes the notion of a *$P$–partition* for an operad $P$ in $\mathsf{Set}$. The $P$–partitions form a poset whose nerve (or *order complex* in [@vallette:2004]) is isomorphic to the bar construction $B(P_+)$ (where we are considering $P$ as a discrete operad in unbased spaces and adding a disjoint basepoint). Lemma \[lem:ppc=bar\] is the special case of this fact when $P$ is the ‘commutative operad’ in $\mathsf{Set}$, that is, with $P(n) = {\ast}$ for all $n$.
\[cor:operad\] Let $\partial_nI$ denote the model of the [$n$^th^]{} derivative of the identity given by $$\partial_nI = \operatorname{Map}_{{\mathcal{S}p}}(\Delta(n),S).$$ Then we have $$\partial_nI = \Omega({\mathbb{D}}\underline{S^0})(n).$$ In particular, the derivatives of the identity form an operad in spectra. We denote this operad by $\partial_*I$.
We have $$\partial_nI = \operatorname{Map}(\Delta(n),S) = {\mathbb{D}}B(\underline{S^0})(n) =
\Omega({\mathbb{D}}\underline{S^0})(n)$$ by Lemma \[lem:ppc=bar\] and Proposition \[prop:duality\] (which applies since all the spaces in $\underline{S^0}$ are $S^0$).
The derivatives of the identity are the cobar construction on the cooperad $\underline{S}$ in spectra with $$\underline{S}(A) = {\mathbb{D}}\underline{S^0}(A) = S$$ where $S$ is the sphere spectrum, for all finite sets $A$ and with all cocomposition maps the canonical isomorphisms. This is the analogue for spectra of the cooperad for cocommutative coalgebras.
\[rem:modules\] We can use the constructions of Section \[sec:bar(modules)\] to get modules over the operad $\partial_*I$. If $C$ is a comodule over $\underline{S}$ then its cobar construction $\Omega(C)$ is a $\partial_*I$–module. We give two examples: (1)Let $X$ be a based space. Then the suspension spectrum $\Sigma^{\infty}X$ is a $\underline{S}$–coalgebra (that is, just a commutative coalgebra) with comultiplication given by the (reduced) diagonal map on $X$: $$\Sigma^{\infty}X \to \Sigma^{\infty}(X {\wedge}X) {\cong}\Sigma^{\infty}X
{\wedge}\Sigma^{\infty}X.$$ As remarked in Definition \[def:comodule\], a coalgebra over a cooperad $Q$ determines a left $Q$–comodule. Thus we obtain a left $\underline{S}$–comodule $\underline{\Sigma^{\infty}X}$. We now take the cobar construction to get a left $\partial_*I$–module $$M_X := \Omega(\underline{\Sigma^{\infty}X}) =
\Omega(I,\underline{S},\underline{\Sigma^{\infty}X})$$ (where $I$ in this formula denotes the unit symmetric sequence of Definition \[def:compprod\]). From the calculations of \[ex:two-sided\_cobar\] we find that $$M_X(1) = \Sigma^{\infty}X$$ and $$\begin{split} M_X(2) &{\cong}\operatorname*{hofib}(\Sigma^{\infty}X \to
\Sigma^{\infty}X {\wedge}\Sigma^{\infty}X) \\
&\simeq \Sigma^{-1} \operatorname*{hocofib}(\Sigma^{\infty}X \to \Sigma^{\infty}X
{\wedge}\Sigma^{\infty}X) \\
&\simeq \Sigma^{-1} \Sigma^{\infty} \operatorname*{hocofib}(X \to X {\wedge}X) \\
\end{split}$$ So $M_X(2)$ is (up to homotopy and a desuspension) the mapping cone of the reduced diagonal on $X$. Further work is needed to analyze the spectra $M_X(n)$ for larger $n$. In Section \[sec:modules\_specseq\] we will look at ways to calculate the homology of these spectra. (2)A moment’s thought will reveal that a right $\underline{S}$–comodule is precisely the same thing as a functor $$(\mathsf{FinSets},\twoheadrightarrow) \longrightarrow {\mathcal{S}p}$$ where the left-hand side is the category of finite sets with morphisms given by the surjections. Work in progress by Greg Arone has demonstrated a relationship between such functors and the Goodwillie calculus of homotopy functors $F$ from based spaces to spectra.
The derivatives of any homotopy functor $F$ form a symmetric sequence in spectra and it is natural to ask how these symmetric sequences might be related for different functors. We conjecture that there is in general a map of symmetric sequences $$\partial_*F \circ \partial_*G \to \partial_*(FG)$$ for any two homotopy functors $F,G\co {\mathcal{T}_{}}\to {\mathcal{T}_{}}$ such that $F({\ast}) = {\ast}$, where $FG$ denotes the composite of $F$ and $G$. These maps should have suitable associativity properties that taking $F =
G = I$ would recover an operad structure on $\partial_*I$ equivalent to the one we have constructed in this section. Similarly, taking $F = I$ would yield the structure of a left $\partial_*I$–module on $\partial_*G$ and taking $G = I$ a right $\partial_*I$–module structure on $\partial_*F$. The main obstacle at present for constructing these maps is finding good models for the derivatives of a general functor in a symmetric monoidal category ${\mathcal{S}p}$ of spectra. In the case of the identity functor we were fortunate that such models naturally arose from the partition poset complexes.
Homology of the bar and cobar constructions and Koszul duality {#sec:alg}
==============================================================
In this section we look at spectral sequences for calculating the homology of the bar and cobar constructions on operads and cooperads in based spaces or spectra. It turns out that we can relate the $E^1$–term of these spectral sequences to the algebraic bar and cobar constructions described in, for example, [@getzler/jones:1994] and [@fresse:2004]. This leads to a link with Koszul duality which says, briefly, that if the homology of the reduced operad $P$ is Koszul, then the homology of $B(P)$ is its Koszul dual cooperad, and dually, if the homology of the cooperad $Q$ is Koszul then the homology of $\Omega(Q)$ is its Koszul dual operad. This supports the point-of-view that the bar construction for an operad in based spaces or spectra is the analogue of the Koszul dual for an algebraic operad.
Here is a summary of this section. We start in Section \[sec:homology\] by recalling how the homology (with coefficients in the commutative ring $k$) of an operad in based spaces or spectra has the structure of an operad in graded $k$–modules. Then in Section \[sec:filter\], the main work of the chapter begins and we describe the filtration of the bar construction that gives rise to our spectral sequence and identify the ‘filtration quotients’. This filtration is based on the number of vertices in the trees that underlie the bar construction. We deal immediately with the two-sided construction of Section \[sec:two-sided\], recalling that the construction for a lone operad is a special case of this. As usual, for the cobar construction, we just dualize everything. That is, we get a cofiltration, or tower, whose inverse limit is the cobar construction and we identify the fibres of the stages in this tower. In Section \[sec:cofibrations\] we give conditions under which the inclusion maps of the filtrations are cofibrations, thus ensuring that our ‘filtration quotients’ are actually the homotopy cofibres of filtration. This will allow us later to use our identification of these quotients to calculate the $E^1$ term in the spectral sequence. This $E^1$ term turns out to be given by the algebraic bar construction which we describe in Section \[sec:alg\_bar\]. We give a definition of this that emphasizes its similarity to the topological version and show that this definition is equivalent to that given by Getzler and Jones [@getzler/jones:1994] and Fresse [@fresse:2004]. Then in Section \[sec:specseq\] we finally set up the spectral sequence and identify its $E^1$ term with the algebraic bar construction as claimed. In Section \[sec:koszul\] we look at Koszul operads and prove the result identifying the homology of the bar construction on $P$ with the Koszul dual of the homology of $P$. Finally, in Section \[sec:modules\_specseq\] we use our spectral sequences to investigate the homology of the $\partial_*I$–modules $M_X$ constructed in Remark \[rem:modules\](1).
Homology of topological operads {#sec:homology}
-------------------------------
Throughout the chapter we fix a commutative ring $k$ and consider the categories $\mathsf{Mod}_k$ of graded $k$–modules and $\mathsf{Ch}_k$ of chain complexes over $k$. First we describe the symmetric monoidal structure on these categories.
\[def:ch\_k\] The tensor product determines a symmetric monoidal structure on graded $k$–modules with $$(M \otimes N)_r := \bigoplus_{p + q = r} M_p \otimes N_q$$ where the *graded* symmetry isomorphism $$M \otimes N \to N \otimes M$$ is given by $$m \otimes n \mapsto (-1)^{|m||n|} n \otimes m$$ and the unit object is the graded module $k$ concentrated in degree $0$. If $M$ and $N$ are chain complexes with differentials $d_M$ and $d_N$ respectively, we define a differential on $M \otimes N$ by $$d_{M \otimes N}(m \otimes n) := d_M(m) \otimes n + (-1)^{|m|}m \otimes
d_N(n).$$ This makes $\otimes$ into a symmetric monoidal structure on $\mathsf{Ch}_k$ with the same unit $k$ endowed with the trivial differential.
Throughout this section we will use $H_*(-)$ to denote the homology with coefficients in the commutative ring $k$ of an object in ${\mathcal{C}}$ when ${\mathcal{C}}$ is either ${\mathcal{T}_{}}$ or ${\mathcal{S}p}$. If ${\mathcal{C}}$ is the category ${\mathcal{T}_{}}$ of based spaces, this is the *reduced* homology.[^12] If ${\mathcal{C}}$ is a category ${\mathcal{S}p}$ of spectra, it is the spectrum homology $H_*(E) = \pi_*(Hk {\wedge}E)$. We recall the Künneth maps for these homology theories.
\[prop:kunneth\] Let ${\mathcal{C}} = {\mathcal{T}_{}}$ or ${\mathcal{S}p}$ and take $C,D \in {\mathcal{C}}$. Then there is a natural map $$H_*(C) \otimes H_*(D) \to H_*(C \barwedge D)$$ that is an isomorphism if either $H_*(C)$ or $H_*(D)$ consists of flat $k$–modules. These maps are symmetric monoidal in the sense that they commute with the associativity and commutativity isomorphisms in the categories ${\mathcal{C}}$ and $\mathsf{Mod}_k$.
Let $M$ be a symmetric sequence in ${\mathcal{T}_{}}$ or ${\mathcal{S}p}$. Then we denote by $H_*M$ the symmetric sequence of graded $k$–modules given by $$H_*M(A) := H_*(M(A)).$$
The main result of this section is that the homology of a topological operad or cooperad is, under suitable conditions, an operad or cooperad in $\mathsf{Mod}_k$.
Let $P$ be an operad in ${\mathcal{T}_{}}$ or ${\mathcal{S}p}$. Then $H_*P$ is an operad of graded $k$–modules. If $P$ is reduced then so is $H_*P$. If $M$ is a left (respectively, right) $P$–module, then $H_*M$ is a left (respectively, right) $H_*P$–module.
Let $Q$ be a cooperad in ${\mathcal{T}_{}}$ or ${\mathcal{S}p}$ such that the homology groups $H_*(Q(A))$ are flat $k$–modules. Then $H_*(Q)$ is a cooperad of graded $k$–modules that is reduced if $Q$ is. If $C$ is a right $Q$–comodule then $H_*(C)$ is a right $H_*(Q)$–comodule. If $C$ is a left $Q$–comodule such that the $H_*(C(A))$ are flat $k$–modules then $H_*(C)$ is a left $H_*(Q)$–comodule.
The operad structure maps are given by the composites $$H_*(P(A)) \otimes H_*(P(B)) \to H_*(P(A) {\wedge}P(B)) \to H_*(P(A
\cup_a B))$$ and the unit by the map $$k {\cong}H_*(S) \to H_*(P)(1)$$ where $S$ denotes either $S^0$, the unit of ${\mathcal{T}_{}}$, or the unit of ${\mathcal{S}p}$. To check the operad axioms we use the associativity and commutativity of the Künneth formula as stated in Proposition \[prop:kunneth\]. Clearly, if $P$ is reduced (so that the unit map $S
\to P(1)$ is an isomorphism) then so is $H_*P$. The structure maps for $H_*M$ are defined similarly.
In the cooperad case we need the flatness condition. It allows us to define cocomposition maps by $$H_*(Q(A \cup_a B)) \to H_*(Q(A) {\wedge}Q(B)) {\cong}H_*(Q(A)) \otimes
H_*(Q(B))$$ using the inverse of the Künneth map. The counit map is the composite $$H_*Q(1) \to H_*(S) {\cong}k$$ and again, if $Q$ is reduced, so is $H_*Q$. In the case of a right comodule $C$ we similarly get comodule structure maps $$H_*(C(A \cup_a B)) \to H_*(C(A) {\wedge}Q(B)) {\cong}H_*(C(A)) \otimes
H_*(Q(B))$$ where the Künneth map is an isomorphism without any condition on $H_*(C(A))$ (we are still assuming that the $H_*(Q(B))$ are flat). For a left comodule, we do still need the flatness assumption.
We can consider cohomology instead of homology in which case the Künneth isomorphism also requires a finiteness hypothesis. We get the following results. If $Q$ is a cooperad in based spaces or spectra then $H^*(Q)$ is an operad of graded $k$–modules. If $P$ is an operad with the cohomology groups $H^*(P)$ finitely-generated flat $k$–modules then $H^*(P)$ is a cooperad of graded $k$–modules. Similar results hold for comodules and modules.
Filtering the bar construction {#sec:filter}
------------------------------
The spectral sequence we want to construct comes from a filtration on the bar construction by the number of vertices in the underlying trees. In this section we construct this filtration and calculate the filtration quotients.
Write $\mathsf{Tree}_s(A)$ for the subcategory of $\mathsf{Tree}(A)$ whose objects are the (isomorphism classes of) trees with less than or equal to $s$ (internal) vertices. We then have $$\mathsf{Tree}_0(A) \subset \mathsf{Tree}_1(A) \subset \dots \subset
\mathsf{Tree}_{|A|-1}(A) = \mathsf{Tree}(A).$$ Each $\mathsf{Tree}_s(A)$ is an *initial* subcategory of $\mathsf{Tree}(A)$. That is, if $U \leq T$ and $T \in \mathsf{Tree}_s(A)$ then $U \in \mathsf{Tree}_s(A)$. The filtration ‘quotients’ are the discrete categories $$\mathsf{Q}_s(A) := \mathsf{Tree}_s(A) - \mathsf{Tree}_{s-1}(A)$$ whose objects are the trees with precisely $s$ vertices. For each tree $T \in \mathsf{Tree}(A)$ we write $|T|$ for the number of vertices of $T$.
\[def:filter\] For a reduced operad $P$ in ${\mathcal{C}}$ with right module $R$ and left module $L$, define $$B(R,P,L)_s(A) := \int^{T \in \mathsf{Tree}_s(A)} w(T)_+ \otimes
(R,P,L)_A(T).$$ For varying finite sets $A$ these form a symmetric sequence in ${\mathcal{C}}$. From the inclusion of categories $\mathsf{Tree}_{s-1}(A)
\subset \mathsf{Tree}_s(A)$ we get natural maps $$B(R,P,L)_{s-1}(A) \to B(R,P,L)_s(A).$$ In the case ${\mathcal{C}} = {\mathcal{T}_{}}$, it is easy to see that the resulting sequence of maps is a filtration of $B(R,P,L)(A)$ by subspaces. The subspace $B(R,P,L)_s(A)$ consists of those points in $B(R,P,L)(A)$ that can be represented by trees with less than or equal to $s$ vertices.
The generalized $A$–labelled trees with no vertices (i.e. only a root and some leaves) correspond one-to-one with (unordered) partitions of $A$. We therefore see that $$B(R,P,L)_0 = R \circ L$$ where $\circ$ is the composition product of symmetric sequences.
Take $R = L = I$ so that $B(R,P,L) = B(P)$. We then have $B(P)_0 = I$ by the previous example. If $|A| > 1$ there is precisely one (non-generalized) $A$–labelled tree with only one vertex and we therefore get $$B(P)_1(A) = \begin{cases} S^1 \otimes P(A) & \text{if $|A| > 1$}; \\
B(P)(1) {\cong}S & \text{if $|A| = 1$}; \end{cases}$$ where $S$ is the unit of the symmetric monoidal category ${\mathcal{C}}$.
We can think of the sequence $$B(R,P,L)_0(A) \to B(R,P,L)_1(A) \to \dots \to B(R,P,L)(A)$$ as a kind of ‘cellular’ filtration. That is, we obtain $B(R,P,L)_s(A)$ by attaching ‘cells’ to $B(R,P,L)_{s-1}(A)$, one for each generalized $A$–labelled tree $T$ with exactly $s$ vertices. The following proposition makes this precise.
\[prop:cells\] There is a pushout square in ${\mathcal{C}}$ of the form $$\begin{diagram}
\node{\bigvee_{T \in \mathsf{Q}_s(A)} \partial w(T)_+ \otimes
(R,P,L)_A(T)} \arrow{s} \arrow{e} \node{B(R,P,L)_{s-1}(A)} \arrow{s}
\\
\node{\bigvee_{T \in \mathsf{Q}_s(A)} w(T)_+ \otimes (R,P,L)_A(T)}
\arrow{e} \node{B(R,P,L)_s(A)}
\end{diagram}$$ where $\partial w(T)$ denotes the boundary of the space $w(T)$.
To identify the top horizontal map in this diagram we use the following simple but important lemma.
\[lem:boundary\] Let $T$ be a generalized $A$–labelled tree. Then $$\partial w(T)_+ {\cong}\operatorname*{colim}_{U < T} w(U)_+.$$ The indexing category of the colimit is the full subcategory of $U \in
\mathsf{Tree}(A)$ with $U < T$.
This is a categorical reflection of that fact (Lemma \[lem:gen\_W\]) that the boundary $\partial w(T)$ consists precisely of those weightings of $T$ in which some edge has length zero.
The top horizontal map in the diagram is given by $$\begin{aligned}
\bigvee_{T \in \mathsf{Q}_s(A)} \partial w(T)_+ \otimes
(R,P,L)_A(T) & {\cong}& \bigvee_{T \in \mathsf{Q}_s(A)} \operatorname*{colim}_{U < T}
\left[w(U)_+ \otimes (R,P,L)_A(T)\right] \\
& \longrightarrow & \bigvee_{T \in \mathsf{Q}_s(A)} \operatorname*{colim}_{U <
T} \left[w(U)_+ \otimes (R,P,L)_A(U)\right] \\
& \longrightarrow & B(R,P,L)_{s-1}(A).\end{aligned}$$ Here we’ve used the fact that $- \otimes C$ is a left adjoint so commutes with colimits. If $T \in \mathsf{Q}_s(A)$ and $U < T$ then $U
\in \mathsf{Tree}_{s-1}(A)$ so there are compatible maps from $w(U)_+
\otimes (R,P,L)_A(U)$ to the coend defining $B(R,P,L)_{s-1}(A)$.
With this definition, it is a simple exercise in naturality and colimits to see that the square commutes. To see that it is a pushout, take a commutative diagram $$\begin{diagram}
\node{\bigvee_{T \in \mathsf{Q}_s(A)} \partial w(T)_+ \otimes
(R,P,L)_A(T)} \arrow{s} \arrow{e} \node{B(R,P,L)_{s-1}(A)} \arrow{s}
\\
\node{\bigvee_{T \in \mathsf{Q}_s(A)} w(T)_+ \otimes (R,P,L)_A(T)}
\arrow{e} \node{X}
\end{diagram} \tag{$*$}$$ We have to show that this factors via a unique map $$B(R,P,L)_s(A) \to X.$$ Since $B(R,P,L)_s(A)$ is a coend and hence a colimit, it is enough to get a unique set of compatible maps $$w(U)_+ \otimes (R,P,L)_A(T) \to X$$ for $U \leq T$ in $\mathsf{Tree}_s(A)$. If $T \notin \mathsf{Q}_s(A)$ the required map comes from the right-hand edge of diagram $(*)$. So suppose that $T \in \mathsf{Q}_s(A)$. Then we have $$w(U)_+ \otimes (R,P,L)_A(T) \to w(T)_+ \otimes (R,P,L)_A(T) \to X$$ where the second map comes from the bottom edge of diagram $(*)$. We leave the reader to check that these maps are compatible in the appropriate way and suitably unique. We conclude that $B(R,P,L)_s(A)$ is the claimed pushout.
We use this result to identify the quotients of our filtration of the bar construction.
\[cor:pushout\] Let $P$ be a reduced operad in ${\mathcal{C}}$ with right module $R$ and left module $L$. The following is a pushout square in ${\mathcal{C}}$: $$\begin{diagram}
\node{B(R,P,L)_{s-1}(A)} \arrow{s} \arrow{e} \node{B(R,P,L)_s(A)}
\arrow{s} \\
\node{{\ast}} \arrow{e} \node{\bigvee_{T \in \mathsf{Q}_s(A)}
w(T)_+/\partial w(T)_+ \otimes (R,P,L)_A(T)}
\end{diagram}$$
Since $- \otimes C$ preserves colimits, the following is a pushout square in ${\mathcal{C}}$: $$\begin{diagram}
\node{\partial w(T)_+ \otimes (R,P,L)_A(T)} \arrow{s} \arrow{e}
\node{w(T)_+ \otimes (R,P,L)_A(T)} \arrow{s} \\
\node{{\ast}} \arrow{e} \node{w(T)_+/\partial w(T)_+ \otimes
(R,P,L)_A(T)}
\end{diagram}$$ The corollary now follows from Proposition \[prop:cells\] and the universal properties of colimits.
\[rem:w\_sphere\] Recall from Lemma \[lem:gen\_W\] that for any generalized $A$–labelled tree $T$ with $s$ vertices, $w(T) {\cong}D^s$. Therefore, $w(T)_+/\partial
w(T)_+ {\cong}S^s$. We will be talking a lot about these spaces in the coming sections, so we will give them some more compact notation: $$\underline{w}(T) := w(T)_+/\partial w(T)_+ {\cong}w(T)/\partial w(T)
{\cong}S^s$$
The results for the cobar construction are, as usual, just the duals of those for the bar construction. We summarize these briefly.
\[def:cofilter\] Let $Q$ be a reduced cooperad in a symmetric monoidal ${\mathcal{T}_{}}$–category ${\mathcal{C}}$ with right comodule $R$ and left comodule $L$. Then the two-sided cobar construction $\Omega(R,Q,L)$ has a ‘cofiltration’, that is, there is a sequence $$\Omega(R,Q,L)(A) \to \dots \to \Omega(R,Q,L)^s(A) \to
\Omega(R,Q,L)^{s-1}(A) \to \cdots$$ where $$\Omega(R,Q,L)^s(A) := \int_{T \in \mathsf{Tree}_s(A)}
\operatorname{Map}_{{\mathcal{C}}}(w(T)_+,(R,Q,L)_A(T)),$$ and the ‘projection’ map $$\Omega(R,Q,L)^s(A) \to \Omega(R,Q,L)^{s-1}(A)$$ comes from the inclusion of categories $\mathsf{Tree}_{s-1}(A) \to
\mathsf{Tree}_s(A)$ for $s \geq 1$.
With $Q,R,L$ as in Definition \[def:cofilter\], the following is a pullback square: $$\begin{diagram}
\node{\Omega(R,Q,L)^s(A)} \arrow{s} \arrow{e} \node{\prod_{T \in
\mathsf{Q}_s(A)} \operatorname{Map}_{{\mathcal{C}}}(w(T)_+,(R,Q,L)_A(T))} \arrow{s} \\
\node{\Omega(R,Q,L)^{s-1}(A)} \arrow{e} \node{\prod_{T \in
\mathsf{Q}_s(A)} \operatorname{Map}_{{\mathcal{C}}}(\partial w(T)_+,(R,Q,L)_A(T))}
\end{diagram}$$ We can identify the fibres of the projections by the pullback squares $$\begin{diagram}
\node{\prod_{T \in \mathsf{Q}_s(A)}
\operatorname{Map}_{{\mathcal{C}}}(\underline{w}(T),(R,Q,L)_A(T))} \arrow{e} \arrow{s}
\node{\Omega(R,Q,L)^s(A)} \arrow{s} \\
\node{{\ast}} \arrow{e} \node{\Omega(R,Q,L)^{s-1}(A),}
\end{diagram}$$ where $\underline{w}(T) = w(T)/\partial w(T)$.
Conditions for the inclusion maps of the filtration to be cofibrations {#sec:cofibrations}
----------------------------------------------------------------------
In the case that ${\mathcal{C}}$ is either ${\mathcal{T}_{}}$ or ${\mathcal{S}p}$, the filtration of Section \[sec:filter\] allows us to construct a spectral sequence converging to the homology of $B(R,P,L)$. The $E^1$ term of this spectral sequence is given by the homologies of the homotopy cofibres of the inclusion maps of the filtration. In this section we give conditions under which these inclusions are cofibrations (in the standard model category structures on ${\mathcal{T}_{}}$ and ${\mathcal{S}p}$) and which therefore ensure that the homotopy cofibres are given by the strict cofibres, or filtration quotients, that we have already calculated.
We state the main result of this section (Proposition \[prop:cofibrations\] below) for a general symmetric monoidal ${\mathcal{T}_{}}$–category ${\mathcal{C}}$ with a compatible model structure. Definition \[def:model\] says what we mean by ‘compatible’ here. We use Mark Hovey’s book [@hovey:1999] as our basic reference for model categories.
\[def:model\] A *symmetric monoidal ${\mathcal{T}_{}}$–model category* is a symmetric monoidal ${\mathcal{T}_{}}$–category ${\mathcal{C}}$ (as in Section \[sec:monoidal\]) together with a model structure (in the sense of [@hovey:1999 Definition 1.1.3]) such that the tensoring makes ${\mathcal{C}}$ into a ${\mathcal{T}_{}}$–model category (in the sense of [@hovey:1999 Definition 4.2.18]). That is, if $X \to Y$ is a cofibration in ${\mathcal{T}_{}}$ and $C \to D$ is a cofibration in ${\mathcal{C}}$ then the induced map $$(X \otimes D) \amalg_{X \otimes C} (Y \otimes C) \to Y \otimes D$$ is a cofibration in ${\mathcal{C}}$ that is trivial if either of our original cofibrations is. (The domain of this map is the pushout of $X \otimes D$ and $Y \otimes C$ over $X \otimes C$.)
\[rem:model\] We should say a few words about this definition. Firstly, we are *not* requiring that ${\mathcal{C}}$ be a monoidal model category in its own right (in the sense of [@hovey:1999 Section 4.2.6]). That is, we are not insisting that the symmetric monoidal structure $\barwedge$ on ${\mathcal{C}}$ in any way respect the model structure. Our reason for doing this is to preserve the self-duality of Definition \[def:model\] (see Lemma \[lem:model\_dual\] below). In general, the opposite category of a monoidal model category is not another monoidal model category and we wish to dualize our theory to obtain results on the cobar construction.
On the other hand, the hypotheses we need to prove Proposition \[prop:cofibrations\] are natural consequences of the assumption that ${\mathcal{C}}$ *is* a monoidal model category, *and* the categories we are most interested in, ${\mathcal{T}_{}}$ and ${\mathcal{S}p}$, satisfy this assumption. This suggests that a breaking of the symmetry between bar and cobar is necessary when we come to study the homotopy theory of these constructions. In this paper, we do not pretend to give the beginnings of such a theory and, in particular, we do not claim that Definition \[def:model\] is the philosophically correct way to mix model category theory into this paper. For us, it serves the purposes of allowing us to make calculations with our spectral sequence in cases that are of interest.
\[lem:cofibration\] Let ${\mathcal{C}}$ be a symmetric monoidal ${\mathcal{T}_{}}$–model category. If $C
\in {\mathcal{C}}$ is cofibrant and $X \to Y$ is a cofibration in ${\mathcal{T}_{}}$ then $X \otimes C \to Y \otimes C$ is a cofibration in ${\mathcal{C}}$.
Apply the definition of ${\mathcal{T}_{}}$–model category to the cofibrations $X
\to Y$ and ${\ast}\to C$.
\[prop:cofibrations\] Let ${\mathcal{C}}$ be a symmetric monoidal ${\mathcal{T}_{}}$–model category such that if $C,D$ are cofibrant then $C \barwedge D$ is also cofibrant. Let $P$ be a reduced operad in ${\mathcal{C}}$ with right module $R$ and left module $L$ such that, for all $A$, the objects $P(A),R(A),L(A)$ are cofibrant. Then, for all $s \geq 1$ and all finite sets $A$, the map $$B(R,P,L)_{s-1}(A) \to B(R,P,L)_s(A)$$ of Definition \[def:filter\] is a cofibration in ${\mathcal{C}}$.
The cofibrancy conditions on the $P(A),L(A),R(A)$ together with the extra condition on ${\mathcal{C}}$ ensure that the objects $(R,P,L)_A(T)$ are all cofibrant. For any generalized tree $T$, the map $$\partial w(T)_+ \to w(T)_+$$ is a cofibration in ${\mathcal{T}_{}}$ (it is the inclusion of the boundary of a ball). Therefore, by Lemma \[lem:cofibration\], $$\partial w(T)_+ \otimes (R,P,L)_A(T) \to w(T)_+ \otimes (R,P,L)_A(T)$$ is a cofibration. Proposition \[prop:cells\] tells us that the filtration map $$B(R,P,L)_{s-1}(A) \to B(R,P,L)_s(A)$$ is a pushout of a coproduct of such maps so it too is a cofibration.
As we commented in Remark \[rem:model\] above, if ${\mathcal{C}}$ is a symmetric monoidal model category in its own right, we get for free that $C$ and $D$ cofibrant imply $C \barwedge D$ cofibrant. In particular this is the case for ${\mathcal{T}_{}}$ and ${\mathcal{S}p}$ (that is, the $S$–modules of EKMM [@elmendorf/kriz/mandell/may:1997]).
As promised, our definition of symmetric monoidal ${\mathcal{T}_{}}$–model category is self-dual.
\[lem:model\_dual\] Let ${\mathcal{C}}$ be a symmetric monoidal ${\mathcal{T}_{}}$–model category. Then ${\mathcal{C}}^\text{op}$ is also a symmetric monoidal ${\mathcal{T}_{}}$–model category with the standard dual symmetric monoidal and model structures.
We already know from Proposition \[prop:dual\] that ${\mathcal{C}}^\text{op}$ is a symmetric monoidal ${\mathcal{T}_{}}$–category. Recall that the tensoring for ${\mathcal{C}}^\text{op}$ is given by the cotensoring for ${\mathcal{C}}$, the cofibrations in ${\mathcal{C}}^\text{op}$ are the fibrations in ${\mathcal{C}}$ and a pushout in ${\mathcal{C}}^\text{op}$ is a pullback in ${\mathcal{C}}$. The weak equivalences in ${\mathcal{C}}^\text{op}$ are the same as those in ${\mathcal{C}}$.
To see that ${\mathcal{C}}^\text{op}$ is a ${\mathcal{T}_{}}$–model category we have to show that if $X \to Y$ is a cofibration in ${\mathcal{T}_{}}$ and $D \to C$ a fibration in ${\mathcal{C}}$ then $$\operatorname{Map}_{{\mathcal{C}}}(Y,D) \to \operatorname{Map}_{{\mathcal{C}}}(Y,C)
\times_{\operatorname{Map}_{{\mathcal{C}}}(X,C)} \operatorname{Map}_{{\mathcal{C}}}(X,D)$$ is a fibration in ${\mathcal{C}}^\text{op}$ that is trivial if either of our original maps is a weak equivalence. This result is given by Lemma 4.2.2 of [@hovey:1999].
The result dual to Proposition \[prop:cofibrations\] is then the following.
\[cor:fibrations\] Let ${\mathcal{C}}$ be a symmetric monoidal ${\mathcal{T}_{}}$–model category such that if $C,D$ are fibrant then $C \barwedge D$ is also fibrant. Let $Q$ be a reduced cooperad in ${\mathcal{C}}$ with right comodule $R$ and left comodule $L$ such that all the objects $Q(A),R(A),L(A)$ are fibrant. Then the map $$\Omega(R,Q,L)^s(A) \to \Omega(R,Q,L)^{s-1}(A)$$ of Definition \[def:cofilter\] is a fibration in ${\mathcal{C}}$.
In these circumstances, then, the fibres of the maps in the tower for $\Omega(R,Q,L)$ are also the homotopy fibres and so can be used to calculate the $E^1$ term of the associated spectral sequence.
In our categories of interest, ${\mathcal{T}_{}}$ and ${\mathcal{S}p}$, all objects are fibrant and so the conditions of Corollary \[cor:fibrations\] hold for any cooperad and any comodules over it.
The algebraic bar and cobar constructions {#sec:alg_bar}
-----------------------------------------
So far we have constructed (under suitable conditions) a filtration of the two-sided bar construction $B(R,P,L)$ by a sequence of cofibrations. This filtration yields a homology spectral sequence whose $E^1$ term turns out to be given by an algebraic version of our bar construction. In fact, it was this algebraic version, previously studied by Ginzburg–Kapranov [@ginzburg/kapranov:1994], Getzler–Jones [@getzler/jones:1994] and Fresse [@fresse:2004] among others, that inspired our definition of the bar construction for operads in topological settings. This section is devoted to the description of this algebraic bar construction. As in the topological case, we will only deal with *reduced* operads, that is, those for the unit map $k \to P(1)$ is an isomorphism.
Our definition of the algebraic bar construction emphasizes its similarity to the topological versions of Section \[sec:bar\] and Section \[sec:bar(modules)\] and it will follow the same pattern.
Let $P$ be a reduced operad in the category $\mathsf{Ch}_k$ of chain complexes over the commutative ring $k$ (with the symmetric monoidal structure of Definition \[def:ch\_k\]). Let $R$ be a right $P$–module and $L$ a left $P$–module. More or less repeating Definition \[def:(R,P,L)\], we define a functor $$(R,P,L)_A \co \mathsf{Tree}(A)^\text{op} \to \mathsf{Ch}_k$$ for each nonempty finite set $A$ by the formula $$(R,P,L)_A(T) := R(i(r)) \otimes \bigotimes_{\text{vertices $v \in T$}}
P(i(v)) \otimes \bigotimes_{\text{leaves $l \in T$}} L(\iota^{-1}l).$$ The composition maps for $R$, $P$ and $L$ make $(R,P,L)_A$ into a functor as claimed. In making explicit calculations we have to be careful with the signs involved in the symmetry isomorphism for $\otimes$ but for theoretical purposes we can treat $(R,P,L)_A(T)$ as an unordered tensor product (see Remark \[rem:symmetry\]).
We now wish to define the bar construction $B(R,P,L)$ by the same coend formula as in Definition \[def:two-sided\_bar\]. For this we need chain complex versions of the spaces $w(T)$ of weightings on trees $T \in
\mathsf{Tree}(A)$. As in the topological case, the structures of these ‘spaces’, and how they fit together for different trees, are the key parts of the definition of the bar construction.
Let $T$ be a generalized $A$–labelled tree. The chain complex $C_*w(T)$ representing the space of weightings on $T$ will be the cellular chain complex for a certain cellular decomposition of the space $w(T)$. The cells in this decomposition correspond one-to-one with the trees $U \in
\mathsf{Tree}(A)$ with $U \leq T$. The $r$–skeleton of $w(T)$ is given by $$\operatorname{sk}_r w(T) := \operatorname*{colim}_{U < T : \; U \in
\mathsf{Tree}_r(A)}w(U).$$ The attaching map for the cell corresponding to the tree $U$ with $r+1$ vertices is the map $$S^r {\cong}\partial w(U) {\cong}\operatorname*{colim}_{V < U} w(V) \to \operatorname*{colim}_{V <
T : \; V \in \mathsf{Tree}_r(A)} w(V) = \operatorname{sk}_r w(T).$$ The cellular chain complex for this cell structure then has $$C_r w(T) = \bigoplus_{U \leq T : \; U \in \mathsf{Q}_r(A)}
H_r(w(U),\partial w(U)) {\cong}\bigoplus_{U \leq T : \; U \in
\mathsf{Q}_r(A)} \widetilde{H}_r(\underline{w}(U)).$$ Recall from Remark \[rem:w\_sphere\] that $\underline{w}(U)$ denotes the quotient $w(U)/\partial w(U)$. The differential $$C_r w(T) \to C_{r-1} w(T)$$ is given by summing the maps[^13] $$\widetilde{H}_r(\underline{w}(U)) {\cong}H_r(w(U),\partial w(U)) \to
\widetilde{H}_{r-1}(\partial w(U)_+) \to \widetilde{H}_{r-1}(\underline{w}(V))$$ for pairs $(U,V)$ with $V < U$, $|U| = r$ and $|V| = r-1$. An example of this chain complex for a particular tree is shown in Figure \[fig:chains\].
The inclusion $w(U) \to w(T)$ is cellular and so we have inclusions $$C_*w(U) \to C_*w(T)$$ for $U < T$. These make $C_*w(-)$ into a functor $$C_*w(-) \co \mathsf{Tree}(A) \to \mathsf{Ch}_k.$$ This is the chain complex analogue of the functor $w(-)$ of Definition \[def:gen\_w\_functor\].
\[def:alg\_bar\] With our ‘chain complexes of weighted trees’ $C_*w(T)$, we now define the *two-sided algebraic bar construction* on the reduced operad $P$ with coefficients in $R$ and $L$ to be the symmetric sequence $B(R,P,L)$ with $$B(R,P,L)(A) := \int^{T \in \mathsf{Tree}(A)} C_*w(T) \otimes
(R,P,L)_A(T).$$ This coend is calculated in the category of chain complexes on $k$ and results in a chain complex $B(R,P,L)(A)$. However, it will be useful to consider a bicomplex structure on $B(R,P,L)(A)$ for which this chain complex is the total complex. The bicomplex structure comes about by considering the tensor product of the chain complexes $C_*w(T)$ and $(R,P,L)_A(T)$ as a bicomplex with gradings and differentials coming from these separate terms. We will write $$B(R,P,L)_{*,*}(A)$$ to emphasize this bigrading with the first index denoting the grading that comes from $C_*w(T)$ (we’ll call this the *tree grading*) and the second the grading that comes from $(R,P,L)_A(T)$ (which we’ll call the *internal grading*). We then have two separate differentials on $B(R,P,L)_{*,*}$: $$\partial \co B(R,P,L)_{*,*} \to B(R,P,L)_{*-1,*}$$ coming from the differentials on the chain complexes $C_*w(T)$ which will refer to as the *tree differential* on the bar construction, and $$d\co B(R,P,L)_{*,*} \to B(R,P,L)_{*,*-1}$$ coming from the differentials on the $(R,P,L)_A(T)$ which we will call the *internal differential*.
In later sections, we will be applying the algebraic bar construction to operads of graded $k$–modules, that is, chain complexes with zero differential. In this case, the internal differential of $B(R,P,L)(A)$ will be zero.
We can give a more explicit description of $B(R,P,L)$ as follows.
\[lem:explicit\_bar\] Let $P$ be a reduced operad in $\mathsf{Ch}_k$ with right module $R$ and left module $L$. Then we have[^14] $$B(R,P,L)_{s,*}(A) {\cong}\bigoplus_{T \in \mathsf{Q}_s(A)}
\widetilde{H}_s(\underline{w}(T)) \otimes (R,P,L)_A(T)$$ as chain complexes of $k$–modules with respect to the internal grading and differential.
Under these isomorphisms, the tree differential $$\partial\co B(R,P,L)_{s,*}(A) \to B(R,P,L)_{s-1,*}(A)$$ is given by summing, over all pairs $(T,U)$ with $U < T$, $|T| = s$ and $|U| = s-1$, the maps $$\widetilde{H}_s(\underline{w}(T)) \otimes (R,P,L)_A(T) \to
\widetilde{H}_{s-1}(\underline{w}(U)) \otimes (R,P,L)_A(U)$$ obtained by combining the maps $$(R,P,L)_A(T) \to (R,P,L)_A(U)$$ with the terms $$\widetilde{H}_s(\underline{w}(T)) \to \widetilde{H}_{s-1}(\underline{w}(U))$$ from the top differential of the chain complex $C_*w(T)$.
We consider a filtration of the algebraic bar construction analogous to that of Section \[sec:filter\] for the topological version. Virtually the same analysis applies and we get short exact sequences of chain complexes[^15] $$B(R,P,L)_{s-1}(A) \rightarrow B(R,P,L)_s(A) \rightarrow \bigoplus_{T
\in \mathsf{Q}_s(A)} C_*w/C_*\partial w(T) \otimes (R,P,L)_T(A).$$ where $C_*\partial w(T)$ is the cellular chain complex for the subcomplex $\partial w(T) \subset w(T)$ (that is, everything except the top-dimension cell). Notice that $$C_*w(T)/C_*\partial w(T) {\cong}\widetilde{H}_s(\underline{w}(T))$$ for $T \in \mathsf{Q}_s(A)$. We construct a splitting of the above short exact sequence (with respect to the internal differential but *not* the tree differential) using the obvious splittings (as $k$–modules) of the sequences $$C_*\partial w(T) \to C_*w(T) \to \widetilde{H}_*(\underline{w}(T)).$$ We get by induction on $s$ that $$B(R,P,L)(A) {\cong}\bigoplus_{T \in \mathsf{Tree}(A)}
\widetilde{H}_{|T|}(\underline{w}(T)) \otimes (R,P,L)_A(T)$$ which splits, by tree degree, into the isomorphisms of the lemma. We leave the reader to check that the tree differential has the promised formula.
\[rem:fresse\] Choosing generators of the groups $H_{|T|}(w(T),\partial w(T)) {\cong}k$ determines an isomorphism $$B(R,P,L)(A) {\cong}\bigoplus_{T \in \mathsf{Tree}(A)} (R,P,L)_A(T)$$ which is the definition of the algebraic bar construction given by Fresse in [@fresse:2004 Section 4.4]. Such choices determine choices of the coefficients (in fact, signs) for the maps that make up the differential $\partial$ on $B(R,P,L)(A)$.
Fresse shows that this bar construction is a representative of the derived composition product of $R$ and $L$ as $P$–modules, that is, $$B(R,P,L) \simeq R \circ^{\mathbb{L}}_{P} L$$ and so the homology groups of $B(R,P,L)$, with respect to the tree differential, are $\operatorname{Tor}$ groups of $P$–modules.
The relationship between this algebraic bar construction and the simplicial bar construction was analyzed by Fresse. His proof of the following proposition uses a ‘levelization’ process analogous to that we used in the proof of Proposition \[prop:simp\].
[([@fresse:2004 Theorem 4.1.8])]{}The algebraic two-sided bar construction $B(R,P,L)$ is quasi-isomorphic to the normalized chain complex of the simplicial bar construction on $P$ with coefficients in $R$ and $L$ (the algebraic version of Definition \[def:two-sided\_simp\]).
As usual, we have the dual constructions and results.
Let $Q$ be a reduced cooperad of chain complexes of $k$–modules with right comodule $R$ and left comodule $L$. Then there are functors $(R,Q,L)_A$ from $\mathsf{Tree}(A)$ to $\mathsf{Ch}_k$ and we can define the *algebraic cobar construction on $Q$ with coefficients in $R$ and $L$* by the same formula $$\Omega(R,Q,L)(A) := \int_{T \in \mathsf{Tree}(A)}
\operatorname{Hom}(C_*w(T),(R,Q,L)_A(T))$$ as in Definition \[def:cobar(cooperad)\], where, for chain complexes $M,N$, $\operatorname{Hom}(M,N)$ denotes the chain complex of maps of graded modules $M
\to N$. The cobar construction is a bicomplex with an internal grading and differential coming from the $(R,Q,L)_A(T)$ and a tree grading and differential $\partial^*$ coming from the $C_*w(T)$. We follow the convention that $\operatorname{Hom}(M,N)_{s,t} = \operatorname{Hom}(M_{-s},N_t)$ so that the tree grading on the cobar construction is concentrated in negative degrees.
There is an explicit description of the cobar construction analogous to that of Lemma \[lem:explicit\_bar\] for the bar construction.
\[lem:explicit\_cobar\] With $Q,R,L$ as above: $$\Omega(R,Q,L)_{-s,*}(A) := \bigoplus_{T \in \mathsf{Q}_s(A)}
\operatorname{Hom}(\widetilde{H}_s(\underline{w}(T)),(R,Q,L)_A(T))$$ which again is just isomorphic to $$\bigoplus_{T \in \mathsf{Tree}(A)} (R,Q,L)_A(T)$$ after choosing generators of the groups $\widetilde{H}_s(\underline{w}(T))$. The internal grading and differential correspond in the obvious way under this isomorphism. The explicit form of the tree differential $\partial^*$ is given by the maps $$(R,Q,L)_A(U) \to (R,Q,L)_A(T)$$ with coefficients again given by the components $$\widetilde{H}_s(\underline{w}(T)) \to \widetilde{H}_{s-1}(\underline{w}(U))$$ of the top differential on the chain complex $C_*w(T)$.
When $P$ is a reduced operad in the category of graded $k$–modules, the unit symmetric sequence $I$ defined by $$I(A) := \begin{cases} k & \text{if $|A| = 1$}; \\ 0 &
\text{otherwise}. \end{cases}$$ is both a left and right $P$–module. The *reduced bar construction on $P$* is then given by the two-sided bar construction with coefficients in $I$ on both sides: $$B(P) := B(I,P,I)$$ The definition of the algebraic bar construction reduces in this case to $$B(P)(A) = \int^{T \in \mathsf{T}(A)} C_*\overline{w}(T) \otimes P_A(T).$$ Recall that the space $\overline{w}(T)$ is the quotient of $w(T)$ by the subspace $w_0(T)$ of weightings that give length $0$ to either the root edge or a leaf edge of $T$. This subspace is in fact a subcomplex with respect to our chosen cellular structure on $w(T)$.[^16] Therefore we obtain a cell structure on $\overline{w}(T)$ and in the above formula, $C_*\overline{w}(T)$ denotes the relative cellular chain complex for the pair $(\overline{w}(T),{\ast})$, or equivalently, for the pair $(w(T),w_0(T))$.[^17] It is clear that $C_*\overline{w}(T)$ is a quotient of $C_*w(T)$.
It’s also easy to check that by Lemma \[lem:explicit\_bar\] we have $$B(P)(A) {\cong}\bigoplus_{T \in \mathsf{T}(A)}
\widetilde{H}_{|T|}(\underline{w}(T)) \otimes P_A(T)$$ which (after choosing isomorphisms $\widetilde{H}_{|T|}(\underline{w}(T))
{\cong}k$) is the original definition of the algebraic bar construction given in Getzler–Jones [@getzler/jones:1994 Section 2.1].
Similarly, if $Q$ is a reduced cooperad then $I$ is both a left and right $Q$–comodule and the *reduced cobar construction on $Q$* is $$\Omega(Q) := \Omega(I,Q,I)$$ and is given by a formula analogous to that of Definition \[def:cobar(cooperad)\].
As in the topological case, the reduced algebraic bar construction on a reduced operad $P$ of chain complexes has a cooperad structure. We now describe this.
\[def:alg\_cooperad(bar)\] The required maps $$B(P)(A \cup_a B) \to B(P)(A) \otimes B(P)(B)$$ are defined in exactly the same way as the corresponding maps in the topological case (Definition \[sec:cooperad\]). To do this we must construct the algebraic versions of the key maps (\[eq:key\]): $$C_*\overline{w}(T \cup_a U) \to C_*\overline{w}(T) \otimes C_*\overline{w}(U)$$ for $A$–labelled trees $T$ and $B$–labelled trees $U$. We get this by taking the map of cellular chain complexes induced by the topological map $$\overline{w}(T \cup_a U) \to \overline{w}(T) {\wedge}\overline{w}(U)$$ of Definition \[def:cooperad\_maps\]. For this to work, we need the following lemma.
Let $T$ be an $A$–labelled tree, $U$ a $B$–labelled tree and let $a
\in A$. The map $$\overline{w}(T \cup_a U) \to \overline{w}(T) {\wedge}\overline{w}(U)$$ is cellular, that is, it preserves skeleta.
A point $p$ in $\overline{w}(T \cup_a U)$ is in the $s$–skeleton if and only if it is in the subspace $\overline{w}(V)$ for some tree $V$ with $s$ vertices. If this tree $V$ is not of type $(A,B)$ then $p$ is mapped to the basepoint which is certainly in the $s$–skeleton of the right-hand side. If $V$ is of type $(A,B)$ (that is, obtained by grafting an $A$–labelled tree $T'$ and a $B$–labelled tree $U'$) then the point $p$ maps to a pair consisting of a point in some $\overline{w}(T') \subset \overline{w}(T)$ and a point in some $\overline{w}(U') \subset \overline{w}(U)$. The first point is in the $s'$–skeleton of $\overline{w}(T)$ where $T'$ has $s'$ vertices. The second point is in the $s''$–skeleton of $\overline{w}(U)$ where $U'$ has $s''$ vertices. Therefore the pair is in the $s'+s''$–skeleton of $\overline{w}(T)
{\wedge}\overline{w}(U)$. However, since $V$ only had $s$ vertices, we must have $s'+s'' \leq s$. So the image of $p$ is in the $s$–skeleton of $\overline{w}(T) {\wedge}\overline{w}(U)$ as required.
It is easy to describe explicitly the resulting map of chain complexes $$C_*\overline{w}(T \cup_a U) \to C_*\overline{w}(T) \otimes C_*\overline{w}(U).$$ Recall that the left-hand side is given by the direct sum over $V \in
\mathsf{Tree}(A \cup_a B)$ with $V \leq T \cup_a U$ of the homology groups $\widetilde{H}_*(\underline{w}(V))$. The above map sends the term corresponding to a tree $V$ that is not of type $(A,B)$, to zero. If $V
= T' \cup_a U'$, then $T' \leq T$ and $U' \leq U$ and the corresponding term maps to the right-hand side via the isomorphism $$\widetilde{H}_*(\underline{w}(T' \cup_a U')) \to
\widetilde{H}_*(\underline{w}(T')) \otimes
\widetilde{H}_*(\underline{w}(U')),$$ which is induced by the homeomorphism[^18] $$\underline{w}(T' \cup_a U') \to \underline{w}(T') {\wedge}\underline{w}(U'),$$ which in turn is a quotient of the map $$\overline{w}(T' \cup_a U') \to \overline{w}(T') {\wedge}\overline{w}(U').$$
With this key map in place, the rest of the formal definition of the cooperad structure maps for the topological bar construction (Definition \[def:formal\_cooperad\_maps\]) carries over to the algebraic case.
\[lem:explicit\_cooperad\] Let $P$ be a reduced operad in $\mathsf{Ch}_k$. Under the isomorphism of Lemma \[lem:explicit\_bar\], the cooperad structure on $B(P)$ corresponds to the cooperad structure on the chain complexes $$\bigoplus_{T \in \mathsf{T}(A)} \widetilde{H}_{|T|}(\underline{w}(T))
\otimes P_A(T)$$ whose cocomposition maps are given by summing over the maps obtained by combining the isomorphisms $$H_*(\underline{w}(T \cup_a U)) \to H_*(\underline{w}(T)) \otimes
H_*(\underline{w}(U))$$ with the isomorphisms $$P_{A \cup_a B}(T \cup_a U) \to P_A(T) \otimes P_B(U).$$
This is a simple check using the definition of the isomorphism in Lemma \[lem:explicit\_bar\] by splittings of short exact sequences.
Choosing generators for the groups $\widetilde{H}_*(\underline{w}(T))$, we see that this is equivalent to the cooperad structure defined by Getzler–Jones [@getzler/jones:1994] and by Fresse [@fresse:2004].
Dually, if $Q$ is a reduced cooperad of chain complexes, there is an operad structure on the reduced algebraic cobar construction $\Omega(Q)$. The corresponding operad structure under the isomorphism of Lemma \[lem:explicit\_cobar\] is built from the isomorphisms $$Q_A(T) \otimes Q_B(U) \to Q_{A \cup_a B}(T \cup_a U)$$ and the same maps $$\widetilde{H}_*(\underline{w}(T \cup_a U)) \to
\widetilde{H}_*(\underline{w}(T)) \otimes \widetilde{H}_*(\underline{w}(U)).$$
\[rem:extend\] It does not take much more effort to extend the cooperad and operad structure above to maps $$B(R,P,L) \to B(R,P,I) {\mathbin{\widehat{\circ}}}B(I,P,L)$$ and $$\Omega(R,Q,I) \circ \Omega(I,Q,L) \to \Omega(R,Q,L)$$ following the same sort of generalization that we did in Section \[sec:bar(modules)\_maps\].
A spectral sequence for the homology of the bar construction {#sec:specseq}
------------------------------------------------------------
We now turn our attention directly to the homology spectral sequences born from the filtration of the bar construction and cofiltration of the cobar construction.[^19] The work we have done in the last few sections allows us to identify the $E^1$ terms of these spectral sequences, under suitable conditions, with the algebraic bar and cobar constructions.
A quick word on notation: from now on, the only topological categories ${\mathcal{C}}$ we are interested in are ${\mathcal{T}_{}}$ and ${\mathcal{S}p}$. We will therefore drop the notation $\barwedge$ for the monoidal product and $\otimes$ for the tensoring over ${\mathcal{T}_{}}$, replacing both with the standard notation ${\wedge}$. We will reserve $\otimes$ for the tensor product of graded $k$–modules.
\[prop:specseq\] Let $P$ be a reduced operad in ${\mathcal{T}_{}}$ or ${\mathcal{S}p}$ with right module $R$ and left module $L$ such that all the objects $P(A),R(A),L(A)$ are cofibrant and all homology groups $H_*P,H_*R,H_*L$ flat $k$–modules. Then for each finite set $A$ there is a spectral sequence converging to $H_*B(R,P,L)(A)$ with $E^1$–term and first differential given by the algebraic bar construction: $$(E^1,d^1) {\cong}(B(H_*R,H_*P,H_*L)(A),\partial) \implies
H_*B(R,P,L)(A).$$ Let $Q$ be a reduced cooperad in ${\mathcal{S}p}$ with right comodule $R$ and left comodule $L$ such that all the objects $Q(A),R(A),L(A)$ are fibrant[^20] and all the homology groups $H_*Q,H_*R,H_*L$ are flat $k$–modules. Then for each finite set $A$ there is a spectral sequence converging to $H_*(\Omega(R,Q,L)(A))$ with $E^1$–term and first differential given by the algebraic cobar construction: $$(E^1,d^1) {\cong}(\Omega(H_*R,H_*Q,H_*L)(A),\partial^*) \implies
H_*\Omega(R,Q,L)(A).$$
By the comments of Remark \[rem:fresse\], the work of Fresse allows us to identify the $E^2$ terms of these spectral sequences as suitable $\operatorname{Tor}$ groups. That is, our spectral sequence take the form $$E^2 = \operatorname{Tor}^{H_*P}(H_*R,H_*L) \implies H_*B(R,P,L)$$ and $$E^2 = \operatorname{Tor}^{H_*Q}(H_*R,H_*L) \implies
H_*\Omega(R,Q,L).$$ This suggests that the topological bar and cobar constructions should have an interpretation as topological $\operatorname{Tor}$ objects. We have not yet studied the homotopy theory of these constructions sufficiently to make this precise.
By Proposition \[prop:cofibrations\], we have cofibre sequences $$B(R,P,L)_{s-1}(A) \to B(R,P,L)_s(A) \to \bigvee_{T \in \mathsf{Q}_s(A)}
\underline{w}(T) {\wedge}(R,P,L)_A(T) \tag{$*$}$$ Summing these over $s$ we obtain an exact couple and hence a spectral sequence. The $E^1$ term of this spectral sequence is $$\begin{split}
E^1_{s,t} &:= H_{s+t} (\bigvee_{T \in \mathsf{Q}_s(A)}
\underline{w}(T) {\wedge}(R,P,L)_A(T)) \\
&{\cong}\bigoplus_{T \in \mathsf{Q}_s(A)} H_{s+t} (\underline{w}(T)
{\wedge}(R,P,L)_A(T)) \\
&{\cong}\bigoplus_{T \in \mathsf{Q}_s(A)}
\widetilde{H}_s(\underline{w}(T)) \otimes H_t((R,P,L)_A(T)) \\
&{\cong}\bigoplus_{T \in \mathsf{Q}_s(A)}
\widetilde{H}_s(\underline{w}(T)) \otimes (H_*R,H_*P,H_*L)_A(T)_t \\
&{\cong}B(H_*R,H_*P,H_*L)_{s,t}(A). \\
\end{split}$$ where we have made plentiful use of the Künneth formula. In particular, we need the flatness assumptions to get $$H_*((R,P,L)_A(T)) {\cong}(H_*R,H_*P,H_*L)_A(T).$$ The final isomorphism is that of Lemma \[lem:explicit\_bar\]. Since the filtration of each individual $B(R,P,L)(A)$ is finite, this spectral sequence certainly converges to $H_*B(R,P,L)(A)$. It remains to be shown that $d^1$ is given by the differential $\partial$ of the algebraic bar construction.
The differential $d^1$ is the composite $$\begin{split} H_*\left(\bigvee_{T \in \mathsf{Q}_s(A)} \underline{w}(T)
{\wedge}(R,P,L)_A(T)\right) & \to H_{*-1}B(R,P,L)_{s-1}(A) \\
\to & H_{*-1}\left(\bigvee_{U \in \mathsf{Q}_{s-1}(A)}
\underline{w}(U) {\wedge}(R,P,L)_A(U)\right) \end{split}$$ of the boundary map in the long exact sequence associated to one of the cofibre sequences $(*)$, with the projection map from another one. To analyze this, fix for the moment a generalized $A$–labelled tree $T$ with $s$ vertices and consider the following map of cofibre sequences: $$\begin{diagram} \dgARROWLENGTH=.84em
\node{\partial w(T)_+ {\wedge}(R,P,L)_A(T)} \arrow{s} \arrow{e}
\node{w(T)_+ {\wedge}(R,P,L)_A(T)} \arrow{s} \arrow{e}
\node{\underline{w}(T) {\wedge}(R,P,L)_A(T)} \arrow{s} \\
\node{B(R,P,L)_{s-1}(A)} \arrow{e}
\node{B(R,P,L)_s(A)} \arrow{e}
\node{\bigvee_{T \in \mathsf{Q}_s(A)} \underline{w}(T) {\wedge}(R,P,L)_A(T)}
\end{diagram}$$ This induces a map of long exact sequences in homology, and in particular we have a commutative diagram $$\begin{diagram}
\node{H_*(\underline{w}(T) {\wedge}(R,P,L)_A(T))} \arrow{s} \arrow{e}
\node{H_{*-1}(\partial w(T)_+ {\wedge}(R,P,L)_A(T))} \arrow{s} \\
\node{H_*(\bigvee_{T \in \mathsf{Q}_s(A)} \underline{w}(T) {\wedge}(R,P,L)_A(T))} \arrow{e}
\node{H_{*-1}B(R,P,L)_{s-1}(A).}
\end{diagram}$$ On the other hand, using the identity $$\partial w(T)_+ {\cong}\operatorname*{colim}_{U < T} w(U)_+$$ we also have a commutative diagram $$\begin{diagram}
\node{\partial w(T)_+ {\wedge}(R,P,L)_A(T)} \arrow{s} \arrow{e}
\node{\bigvee_{U \in \mathsf{Q}_{s-1}(A):\; U < T}
\underline{w}(U) {\wedge}(R,P,L)_A(U)} \arrow{s} \\
\node{B(R,P,L)_{s-1}(A)} \arrow{e}
\node{\bigvee_{U \in \mathsf{Q}_{s-1}(A)} \underline{w}(U) {\wedge}(R,P,L)_A(U),}
\end{diagram}$$ where the top horizontal map is constructed from the quotient maps $$\partial w(T)_+ \to \underline{w}(U),$$ for $U \in \mathsf{Tree}_{s-1}(A)$ such that $U < T$, together with the operad composition maps $$(R,P,L)_A(T) \to (R,P,L)_A(U).$$ Taking the homology of this diagram, combining it with our other diagram of homology groups, throwing in the Künneth formula, summing the top lines over all $T \in \mathsf{Q}_s(A)$ and using Lemma \[lem:explicit\_bar\], we get the big commutative diagram of Figure \[fig:big\_diagram\] in which the top row is the differential $\partial$ on the algebraic bar construction $B(H_*R,H_*P,H_*L)(A)$ (under the isomorphism of Lemma \[lem:explicit\_bar\]) and the bottom row is the differential $d^1$ of our spectral sequence. The left and right sides of the diagram are the isomorphisms described at the beginning of this proof that identify $E^1$ with the algebraic bar construction.
=.64em $$\begin{diagram}
\node{B(H_*R,H_*P,H_*L)_{s,*}(A)} \arrow[2]{e,t}{\partial}
\arrow{s,l}{{\cong}}
\node[2]{B(H_*R,H_*P,H_*L)_{s-1,*}(A)} \arrow{s,r}{{\cong}} \\
\node{\bigoplus_{T \in \mathsf{Q}_s(A)} \widetilde{H}_s(\underline{w}(T))
\otimes (H_*...)_A(T)} \arrow{s,l}{{\cong}} \arrow{e}
\node{\bigoplus_{T \in \mathsf{Q}_s(A)} \widetilde{H}_{s-1}(\partial w(T)_+)
\otimes (H_*...)_A(T)} \arrow{s,l}{{\cong}} \arrow{e}
\node{\bigoplus_{U \in \mathsf{Q}_{s-1}(A)}
\widetilde{H}_{s-1}(\underline{w}(U)) \otimes (H_*...)_A(U)}
\arrow{s,r}{{\cong}} \\
\node{\bigoplus_{T \in \mathsf{Q}_s(A)} H_{s+*}(\underline{w}(T) {\wedge}(...)_A(T))} \arrow{s,l}{{\cong}} \arrow{e}
\node{\bigoplus_{T \in \mathsf{Q}_s(A)} H_{s+*-1}(\partial w(T)_+ {\wedge}(...)_A(T))} \arrow{e} \arrow{s}
\node{\bigoplus_{U \in \mathsf{Q}_{s-1}(A)} H_{s+*-1}(\underline{w}(U)
{\wedge}(...)_A(U))} \arrow{s,r}{{\cong}} \\
\node{H_{s+*}(\bigvee_{T \in \mathsf{Q}_s(A)} \underline{w}(T) {\wedge}(R,P,L)_A(T))} \arrow{s,l}{{\cong}} \arrow{e}
\node{H_{s+*-1}B(R,P,L)_{s-1}(A)} \arrow{e}
\node{H_{s+*-1}(\bigvee_{U \in \mathsf{Q}_{s-1}(A)} \underline{w}(U)
{\wedge}(R,P,L)_A(U))} \arrow{s,r}{{\cong}} \\
\node{E^1_{s,*}} \arrow[2]{e,b}{d^1}
\node[2]{E^1_{s-1,*}}
\end{diagram}$$
The argument for the cobar construction is dual but only applies when we are working in a category of spectra. The sequence of isomorphisms that identifies the $E^1$ term then takes the form $$\begin{split}
E^1_{-s,t} &:=
H_{-s+t}(\prod_{T\in\mathsf{Q}_s(A)}\operatorname{Map}_{{\mathcal{S}p}}(\underline{w}(T),(R,Q,L)_A(T)))\\
&{\cong}\bigoplus_{T\in\mathsf{Q}_s(A)}H_{-s+t}\operatorname{Map}_{{\mathcal{S}p}}(\underline{w}(T),(R,Q,L)_A(T)))
\\
&{\cong}\bigoplus_{T\in\mathsf{Q}_s(A)}\operatorname{Hom}(H_s(\underline{w}(T)),H_t(R,Q,L)_A(T))
\\
&{\cong}\bigoplus_{T\in\mathsf{Q}_s(A)}
\operatorname{Hom}(H_s(\underline{w}(T)),(H_*R,H_*Q,H_*L)_A(T)_t) \\
&{\cong}\Omega(H_*R,H_*Q,H_*L)_{s,t}(A). \\
\end{split}$$ In particular we use the fact that we are working with spectra and not based spaces to get the isomorphism $$H_{-s+t}\operatorname{Map}(\underline{w}(T),X) {\cong}H_{-s+t}(\Sigma^{-s}X) {\cong}H_t X {\cong}\operatorname{Hom}(H_s(\underline{w}(T)),H_t X)$$ that replaces an application of the Künneth formula in the bar construction case.
Notice that the spectral sequence for the bar construction lies in the right half-plane (and the first quadrant if the objects $R(A),P(A),L(A)$ only have non-negative homology). That for the cobar construction lies in the left half-plane (and the second quadrant if the objects $R(A),Q(A),L(A)$ only have non-negative homology).
The link to Koszul duality {#sec:koszul}
--------------------------
We now use our spectral sequence to look at the relationship between the bar construction on an operad in based spaces or spectra and Koszul duality. Koszul duality for operads initially appeared in Ginzburg–Kapranov [@ginzburg/kapranov:1994]. Further references include Getzler–Jones [@getzler/jones:1994] and Fresse [@fresse:2004].
The main result of this section is that if $P$ is a reduced operad in based spaces or spectra such that $H_*P$ is a Koszul operad in graded $k$–modules, then the spectral sequence for calculating $H_*B(P)$ collapses at the $E^2$–term and we conclude that $H_*B(P)$ is the Koszul dual cooperad of $H_*P$. This result is a simple consequence of the definitions of a Koszul operad and its Koszul dual cooperad. The dual result holds for cooperads in spectra.
Let $P$ be a reduced operad in the category $\mathsf{Mod}_k$ of graded $k$–modules. We say $P$ is *Koszul* if the homology of the reduced bar construction on $P$ is concentrated in the top tree degree. We explain exactly what we mean by this. The reduced bar construction $B(P)$ is given by $$B(P)_{s,*}(A) {\cong}\bigoplus_{T \in \overline{\mathsf{Q}}_s(A)}
\widetilde{H}_s(\underline{w}(T)) \otimes P_A(T).$$ where $\overline{\mathsf{Q}}_s(A) = \mathsf{Q}_s(A) \cap \mathsf{T}_s(A)$ is the set of $A$–labelled trees (in the sense of Section \[sec:trees\], that is, *not* generalized trees) with exactly $s$ vertices. If $|A|
= 1$, this is concentrated in the column $s = 0$. If $|A| > 1$, it is concentrated in $1 \leq s \leq |A|-1$. We say that $P$ is *Koszul* if, for all $A$, $$H_{s,*}(B(P)(A),\partial) = 0 \text{ for } s \neq |A|-1$$ where $\partial$ denotes the tree differential on $B(P)$.
Let $P$ be a Koszul operad in graded $k$–modules. The *Koszul dual* of $P$ is the symmetric sequence $K(P)$ given by the homology of the reduced bar construction on $P$. We grade $K(P)$ according to the total degree (that is, internal degree plus tree degree) of $B(P)$: $$K(P)_r(A) = H_{|A|-1,r+1-|A|}(B(P)(A),\partial).$$ Notice that $K(P)(A)$ is the kernel of the differential $$B(P)_{|A|-1,*}(A) \to B(P)_{|A|-2,*}(A),$$ so there is a natural inclusion $$K(P) \to B(P).$$
\[prop:koszul\_cooperad\] Let $P$ be a Koszul operad in graded $k$–modules such that each $K(P)(A)$ is a flat $k$–module. Then the Koszul dual $K(P)$ has a natural cooperad structure.
We already know from Definition \[def:alg\_cooperad(bar)\] that the bar construction $B(P)$ has a cooperad structure. We get the structure for $K(P)$ by taking homology. So cocomposition maps for $K(P)$ are given by $$H(B(P)(A \cup_a B)) \to H(B(P)(A) \otimes B(P)(B)) {\cong}H(B(P)(A))
\otimes H(B(P)(B))$$ where we use the flatness assumption to get the isomorphism.
We dually define the Koszul property and Koszul dual for cooperads of graded $k$–modules.
Let $Q$ be a reduced cooperad of graded $k$–modules. Then $Q$ is *Koszul* if the homology of the reduced cobar construction is concentrated in the lowest[^21] tree degree. In this case, the *Koszul dual* of $Q$ is the symmetric sequence $K(Q)$ of graded $k$–modules with $$K(Q)_r(A) := H_{1-|A|,r+|A|-1}(\Omega(Q)(A),\partial^*),$$ where $\partial^*$ is the tree differential on $\Omega(Q)$. Since $K(Q)$ is the bottom homology group of $\Omega(Q)$ there is a natural surjection $$\Omega(Q) \to K(Q).$$
Let $Q$ be a Koszul cooperad of graded $k$–modules. Then the Koszul dual $K(Q)$ has a natural operad structure.
The composition maps for $K(Q)$ are given by $$H(\Omega(Q)(A)) \otimes H(\Omega(Q)(B)) \to H(\Omega(Q)(A) \otimes
\Omega(Q)(B)) \to H(\Omega(Q)(A \cup_a B).$$ Notice that we don’t need a flatness assumption here.
Fresse [@fresse:2004] gives various fundamental results for Koszul duality of operads and cooperads, in particular, the following.
Let $P$ be a Koszul operad of graded $k$–modules such that the $k$–modules $P(A)$ and $K(P)(A)$ are flat. Then $K(P)$ is a Koszul cooperad and $$K(K(P)) {\cong}P$$ as operads. Dually, let $Q$ be a Koszul cooperad of graded $k$–modules such that the modules $Q(A)$ and $K(Q)(A)$ are flat. If $Q$ is Koszul then its Koszul dual operad $K(Q)$ is also Koszul and $$K(K(Q)) {\cong}Q$$ as cooperads.
We now give the main result of this section.
\[prop:koszul\] Let $P$ be a reduced operad in ${\mathcal{T}_{}}$ or ${\mathcal{S}p}$ such that each object $P(A)$ is cofibrant and all homology groups $H_*P(A)$ and $H_*B(P)(A)$ are flat $k$–modules. If $H_*P$ is a Koszul operad then $$H_*B(P) {\cong}K(H_*P)$$ as cooperads.
Dually, let $Q$ be a reduced cooperad in ${\mathcal{S}p}$ such that each object $Q(A)$ is fibrant and the homology groups $H_*Q(A)$ are flat $k$–modules. If $H_*Q$ is a Koszul cooperad then $$H_*\Omega(Q) {\cong}K(H_*Q)$$ as operads.
The cofibrancy and flatness conditions ensure that the spectral sequence of Proposition \[prop:specseq\] exists for each finite set $A$ and that $H_*B(P)$ is a cooperad in $\mathsf{Mod}_k$. We have already seen that the spectral sequence has the form $$(E^1_{*,*},d^1) = (B(H_*P)_{*,*}(A),\partial) \implies H_*B(P).$$ Because $H_*P$ is Koszul, the homology of the bar construction is concentrated in the $s = |A|-1$ column. Therefore, the $E^2$–term is concentrated in this column and so the spectral sequence collapses. We then see that $$H_r B(P)(A) {\cong}E^2_{|A|-1,r-|A|+1} {\cong}H_{|A|-1,r-|A|+1}(B(H_*P)(A),\partial) {\cong}K(H_*P)_r(A)$$ and so $$H_*B(P) {\cong}K(H_*P)$$ as claimed. It follows that the modules $K(H_*P)(A)$ are flat so, by Proposition \[prop:koszul\_cooperad\], $K(H_*P)$ has a cooperad structure. It remains to show that this cooperad structure agrees with that on $H_*B(P)$.
The first thing to notice is that the above identification of $H_*B(P)(A)$ with the submodule $K(H_*P)(A)$ on $B(H_*P)(A)$ is realized by an edge homomorphism of our spectral sequence. This edge homomorphism comes from applying homology to the quotient map $$B(P)(A) \to \bigvee_{T \in \mathsf{Q}_s(A)} \underline{w}(T) {\wedge}P_A(T)$$ where $s = |A|-1$. The key property of these maps is that they fit into commutative diagrams $$\begin{diagram} \dgARROWLENGTH=2.4em
\node{B(P)(A \cup_a B)} \arrow{s} \arrow{e}
\node{\bigvee_{V \in \mathsf{Q}_{s+s'}(A \cup_a B)} \underline{w}(V)
{\wedge}P_{A \cup_a B}(V)} \arrow{s} \\
\node{B(P)(A) {\wedge}B(P)(B)} \arrow{e}
\node{\bigvee_{T \in \mathsf{Q}_s(A)} \bigvee_{U \in
\mathsf{Q}_{s'}(B)} \underline{w}(T) {\wedge}\underline{w}(U) {\wedge}P_A(T) {\wedge}P_B(U)}
\end{diagram}$$ where the map on the right-hand side is built from the familiar maps $$\underline{w}(T \cup_a U) \to \underline{w}(T) {\wedge}\underline{w}(U)$$ and the isomorphisms $$P_{A \cup_a B}(T \cup_a U) \to P_A(T) \otimes P_B(U)$$ with terms for trees $V$ not of type $(A,B)$ mapping to the basepoint.
Taking homology of this diagram, the right-hand side map gives the cooperad structure on $B(H_*P)$ as described in Lemma \[lem:explicit\_cooperad\]. This shows that the edge homomorphisms of the spectral sequence identify the cooperad structure on $H_*B(P)$ with the restriction of that on $B(H_*P)$. Since the cooperad structure on $K(H_*P)$ is also the restriction of that on $B(H_*P)$, it follows that $$H_*B(P) {\cong}K(H_*P)$$ is an isomorphism of cooperads. The dual result is proved similarly.
Proposition \[prop:koszul\] extends a result of Vallette [@vallette:2004] for discrete operads. Recall from Remark \[rem:vallette\] that he constructs the ‘order complex’ for an operad $P$ in $\mathsf{Set}$. His main result then is that $H_*P$ is Koszul if and only if the homology of the order complex is concentrated in top degree. This follows immediately from our spectral sequence argument by identifying the order complex with the bar construction.
\[ex:derivatives\_specseq\] We finally return to the Goodwillie derivatives of the identity functor. Recall that $$\partial_* I {\cong}\Omega(\underline{S})$$ where $\underline{S}$ is the cooperad of spectra with $\underline{S}(A)
= S$ for all $A$. The homology of this cooperad is given by $$H_*(\underline{S})(A) = \begin{cases} k & \text{if $* = 0$}; \\ 0 &
\text{otherwise}; \end{cases}$$ for all finite sets $A$. This is the cooperad of commutative coalgebras in the category of graded $k$–modules. Fresse shows in [@fresse:2004 Section 6] (by updating a result of Ginzburg and Kapranov [@ginzburg/kapranov:1994]) that this cooperad is Koszul (for $k = \mathbb{Q},\mathbb{F}_p,\mathbb{Z}$) with Koszul dual given by a suspension of the Lie operad. Proposition \[prop:koszul\] therefore applies and we recover the homology of the derivatives of the identity: $$H_*(\partial_n I) = \begin{cases} \mathsf{Lie}(n) \otimes sgn_n &
\text{if $* = 1-n$}; \\ 0 & \text{otherwise}. \end{cases}$$ Moreover, we now know that the induced operad structure on this homology of the derivatives is equal to the operad structure on the Koszul dual of the commutative cooperad, that is, the desuspended Lie structure. This completes the main goal set out in the introduction to this paper.
Homology of modules over the derivatives of the identity {#sec:modules_specseq}
--------------------------------------------------------
In this final section, we use our spectral sequence to investigate the homology of the left $\partial_*I$–module $M_X$ associated to a based space $X$ as described in Remark \[rem:modules\](1). Recall that this module is given by a cobar construction: $$M_X := \Omega(I,\underline{S},\underline{\Sigma^{\infty}X}).$$ We can describe explicitly the spectral sequence for calculating $H_*M_X(2)$. The cobar construction is one-sided and we only have to consider trees for which the root has a single incoming edge. There are two $2$–labelled trees of this type with zero and one vertices respectively and a morphism between them. The $E^1$ term in the spectral sequence therefore only has nonzero entries in the columns $s = 0$ and $s = -1$. These entries are respectively $H_*X$ and $H_*(X {\wedge}X) {\cong}H_*X \otimes H_*X$ with the differential given by the reduced diagonal $X
\to X {\wedge}X$. The spectral sequence therefore takes the following form.
This reduces to the long exact sequence of homology determined by the cofibre sequence $$X \to X {\wedge}X \to \operatorname*{hocofib}(X \to X {\wedge}X)$$ which is consistent with the calculation of $M_X(2)$ made in Remark \[rem:modules\].
Things become more interesting (and much more complicated) for $M_X(n)$ when $n > 2$. For $n = 3$ there are eight trees of interest:
and the $E^1$ term of the spectral sequence takes the form
The differential $d^1$ is built from the reduced diagonal (between pairs of terms corresponding to bud collapse) and isomorphisms (between pairs of terms corresponding to collapse of an internal edge).
We will close the paper by looking at $X = S^r$, the $r$–sphere (for $r \geq 2$). In this situation the reduced diagonal is zero on homology and there can be no higher differentials or extensions in the spectral sequence. This allows us to calculate $H_*M_{S^r}$ with $\mathbb{Z}$ coefficients in its entirety.
Let $S^r$ denote the $r$–sphere for $r \geq 2$. Then we have $$H_*(M_{S^r}) {\cong}H_*(\partial_*I) \circ H_*(\underline{S^r})$$ where $H_*(\underline{S^r})$ is the symmetric sequence with $$H_*(\underline{S^r})(n) = \begin{cases} \mathbb{Z} & \text{if $* =
r$}; \\ 0 & \text{otherwise}. \end{cases}$$ The left action of $H_*(\partial_*I)$ on $H_*(M_{S^r})$ is given by the operad structure on $H_*(\partial_*I)$.
The $E^1$ term of the spectral sequence for the homology of $M_X$ is in this case the algebraic cobar construction $$\Omega(I,H_*(\underline{S}),H_*(\underline{S^r})).$$ The coaction of $H_*(\underline{S})$ on $H_*(\underline{S^r})$ is trivial in the sense that the only nonzero cocomposition maps are $$H_*(\underline{S^r})(n) \to H_*(\underline{S})(1) \otimes
H_*(\underline{S^r})(n).$$ This is equivalent to saying that $$H_*(\underline{S^r}) {\cong}I \circ H_*(\underline{S^r})$$ as left $H_*(\underline{S})$–comodules, where the coaction of $H_*(\underline{S})$ on the right-hand side is via the coaugmentation action on $I$. It follows that the $E^1$ term of our spectral sequence can be written $$\Omega(I,H_*(\underline{S}),I \circ H_*(\underline{S^r})) {\cong}\Omega(I,H_*(\underline{S}),I) \circ H_*(\underline{S^r})$$ where the differential on the right-hand side comes solely from the cobar construction and not from $H_*(\underline{S^r})$. This isomorphism can be seen by working through the definition of the algebraic bar construction in this case.
It now follows that the $E^2$ term of our spectral sequence is given by $$H_*(\partial_*I) \circ H_*(\underline{S^r}).$$ In the $E^2$ term for calculating $H_*M_{S^r}(n)$, we only have nonzero entries in bidegrees $(-k,r(k+1))$ for integers $k \geq 0$. Since $r \geq
2$ there can be no further differentials or extensions and so we see that $$H_*(M_{S^r}) {\cong}H_*(\partial_*I) \circ H_*(\underline{S^r}).$$ The proof of Proposition \[prop:koszul\] extends to show that the left action of $H_*(\partial_*I)$ is as claimed.
The functor $P \circ -$ from symmetric sequences to left $P$–modules is left adjoint to the forgetful functor and so can rightfully be called the *free left $P$–module* functor. Hence the homology of $M_{S^r}$ is the free left $P$–module on $H_*(\underline{S^r})$.
Explicitly, there is a generator $x_A$ in $H_r(M_{S^r})(A)$ for each finite set $A$. The entire homology group $H_*(M_{S^r})(A)$ then has a basis given by all possible iterated brackets of the form $$[\dots [[x_{A_1},x_{A_2}],x_{A_3}] \dots,x_{A_k}]$$ where $A_1,\dots,A_k$ is a partition of $A$ into nonempty finite subsets, and $[-,-]$ is a Lie bracket of degree $-1$. This Lie bracket also represents the action of $H_*(\partial_*I)$ on $H_*(M_{S^r})$.
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[^1]: The Goodwillie derivatives of a homotopy functor are a sequence of spectra with actions by the symmetric groups, but are only defined up to homotopy. By an operad structure on these derivatives, we mean choices of models for these spectra in a suitable symmetric monoidal category, such as the category of $S$–modules of EKMM [@elmendorf/kriz/mandell/may:1997], together with an operad structure on those models.
[^2]: See, for example, [@markl/shnider/stasheff:2002 Section II.2.3] for the general form of the two-sided simplicial bar construction.
[^3]: \[foot:new\]After this paper was written, the author learnt that this result had already been proved in unpublished work of Salvatore [@salvatore:1998] using an alternative definition of the bar construction on an operad. See Remark \[rem:BV\].
[^4]: Here ${\mathcal{C}}^\text{op}$ denotes the opposite category of the category underlying ${\mathcal{C}}$ described in Remark \[rem:enriched\](2).
[^5]: There is a more basic way to view algebras over an operad as modules. This requires us to introduce an $M(0)$ term to our modules (that is, our symmetric sequences become functors from the category of all finite sets, not just nonempty finite sets). With a corresponding generalization of the composition product, and hence of the notion of module, a $P$–algebra is equivalent to a left $P$–module concentrated in the $M(0)$ term. The reason we do not allow our modules to have this extra term is that the comodule structure on the bar construction (see Section \[sec:two-sided\]) would not then exist in general.
[^6]: The obvious converse to this Lemma is not true. That is, a constant symmetric sequence together with a left $P$–module structure need not arise from a $P$–algebra. The construction given in the proof of this lemma forces different components of the module structure map to be the same which need not be the same in general.
[^7]: This is the inverse image under the injective map $$w(T/e) \to w(T)$$ of the weighting corresponding to $p$. The condition that $e$ has length zero says precisely that the weighting for $p$ is in the image of this map.
[^8]: See [@markl/shnider/stasheff:2002 Section II.2.3] for a discussion of different forms of the simplicial bar construction.
[^9]: The geometric realization of a simplicial symmetric sequence is defined pointwise: $|X|(A) = |X(A)|$. Note that a simplicial symmetric sequence is the same thing as a symmetric sequence of simplicial objects.
[^10]: Intuitively, we have collapsed the $U$ part of the tree to a single edge with the same overall length.
[^11]: It is a serendipitous fact of our terminology for trees that the **r**ight module $\mathbf{R}$ relates to the **r**oots of our trees and the **l**eft module $\mathbf{L}$ relates to the **l**eaves.
[^12]: We stress that any homology group of a based space in this paper is meant to be the *reduced* homology.
[^13]: The last part of this composite comes from the map $$\partial w(U)_+ {\cong}\operatorname*{colim}_{V < U} w(V)_+ \to w(V)_+/\partial w(V)_+
= \underline{w}(V)$$ given by collapsing to the basepoint everything except the interior of the ‘face’ $w(V)$ of $\partial w(U)$.
[^14]: Here, as elsewhere, the reduced homology of the quotient $\underline{w}(T) = w(T)/\partial w(T)$ can be replaced with the homology of the pair $(w(T),\partial w(T))$. Both of these are isomorphic to the graded module $k$ concentrated in degree $|T|$.
[^15]: The notation here is probably rather confusing. We are using $B(R,P,L)_s(A)$ to denote the part of the filtration of $B(R,P,L)(A)$ obtained via the chain complex version of Definition \[def:filter\]. This is not to be confused with $B(R,P,L)_{s,*}$ which is the graded summand of tree degree $s$. In fact, it’s a consequence of the proof of this lemma that $$B(R,P,L)_s(A) {\cong}\bigoplus_{r \leq s} B(R,P,L)_{r,*}(A).$$
[^16]: It is the union of the cells corresponding to $U < T$ that are not in the original category $\mathsf{T}(A)$, that is, that are generalized trees, but not trees in the sense of Section \[sec:trees\].
[^17]: The tensor product of chain complexes is here playing the role of the smash product of based spaces so we need the reduced version of the cellular chain complex. Strictly speaking, the chain complex $C_*w(T)$ is the relative chain complex of the pair $(w(T)_+,{\ast})$.
[^18]: It is easy to check that this map is a bijection. The spaces involved are all spheres which are compact Hausdorff, so it is a homeomorphism.
[^19]: If ${\mathcal{C}}$ is the category of based spaces, we only get a spectral sequence for the bar construction and not for the cobar construction. This is because a fibre sequence in ${\mathcal{T}_{}}$ does not immediately yield a long exact sequence in homology.
[^20]: This is really automatic since we have chosen ${\mathcal{S}p}$ to be the category of $S$–modules of EKMM [@elmendorf/kriz/mandell/may:1997] in which all objects are fibrant. If we want to work with other categories of spectra, however, we need this condition.
[^21]: Recall that the tree grading for the cobar construction is concentrated in negative degrees. ‘Lowest’ here means ‘most negative’.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this article, we study the axialvector-diquark-axialvector-antidiquark type scalar, axialvector, tensor and vector $ss\bar{s}\bar{s}$ tetraquark states with the QCD sum rules. The predicted mass $m_{X}=2.08\pm0.12\,\rm{GeV}$ for the axialvector tetraquark state is in excellent agreement with the experimental value $(2062.8 \pm 13.1 \pm 4.2) \,\rm{MeV}$ from the BESIII collaboration and supports assigning the new $X$ state to be a $ss\bar{s}\bar{s}$ tetraquark state with $J^{PC}=1^{+-}$. The predicted mass $m_{X}=3.08\pm0.11\,\rm{GeV}$ disfavors assigning the $\phi(2170)$ or $Y(2175)$ to be the vector partner of the new $X$ state. As a byproduct, we obtain the masses of the corresponding $qq\bar{q}\bar{q}$ tetraquark states. The light tetraquark states lie in the region about $2\,\rm{GeV}$ rather than $1\,\rm{GeV}$.'
---
\
Zhi-Gang Wang [^1]\
Department of Physics, North China Electric Power University, Baoding 071003, P. R. China
PACS number: 12.39.Mk, 12.38.Lg
Key words: Tetraquark state, QCD sum rules
Introduction
============
Recently, the BESIII collaboration studied the process $J/\psi \to \phi \eta \eta^\prime$ and observed a structure $X$ in the $\phi\eta^\prime$ mass spectrum [@BES-2000]. The fitted mass and width are $m_X=(2002.1\pm 27.5 \pm 15.0)\,\rm{MeV}$ and $\Gamma_X=(129 \pm 17 \pm 7)\,\rm{MeV}$ respectively with assumption of the spin-parity $J^P=1^-$, the corresponding significance is $5.3\sigma$; while the fitted mass and width are $m_X=((2062.8 \pm 13.1 \pm 4.2)
\,\rm{MeV}$ and $\Gamma_X=(177 \pm 36 \pm 20)\,\rm{MeV}$ respectively with assumption of the spin-parity $J^P=1^+$, the corresponding significance is $4.9\sigma$. The $X$ state was observed in the $\phi\eta^\prime$ decay model rather than in the $\phi\eta$ decay model, they maybe contain a large $ss\bar{s}\bar{s}$ component, in other words, it maybe have a large tetraquark component. In Ref.[@Wang-Luo-Liu], Wang, Luo and Liu assign the $X$ state to be the second radial excitation of the $h_1(1380)$. In Ref.[@Cui-etal], Cui et al assign the $X$ to be the partner of the tetraquark state $Y(2175)$ with the $J^{PC}=1^{+-}$.
We usually assign the lowest scalar nonet mesons $\{f_0(500),a_0(980),\kappa_0(800),f_0(980) \}$ to be tetraquark states, and assign the higher scalar nonet mesons $\{f_0(1370),a_0(1450),K^*_0(1430),f_0(1500) \}$ to be the conventional ${}^3P_0$ quark-antiquark states [@Close2002; @ReviewAmsler2; @Maiani-Scalar]. In Ref.[@WangScalarNonet], we take the nonet scalar mesons below $1\,\rm{ GeV}$ as the two-quark-tetraquark mixed states and study their masses and pole residues with the QCD sum rules in details, and observe that the dominant Fock components of the nonet scalar mesons below $1\,\rm{ GeV}$ are conventional two-quark states. The light tetraquark states maybe lie in the region about $2\,\rm{GeV}$ rather than lie in the region about $1\,\rm{GeV}$.
In this article, we take the axialvector diquark operators as the basic constituents to construct the tetraquark current operators to study the scalar ($S$), axialvector ($A$), tensor ($T$) and vector ($V$) tetraquark states with the QCD sum rules, explore the possible assignments of the new $X$ state. We take the axialvector diquark operators as the basic constituents because the favored configurations from the QCD sum rules are the scalar and axialvector diquark states [@WangLDiquark; @Dosch-Diquark-1989], the current operators or quark structures chosen in the present work differ from that in Ref.[@Cui-etal] completely.
The article is arranged as follows: we derive the QCD sum rules for the masses and pole residues of the $ss\bar{s}\bar{s}$ tetraquark states in section 2; in section 3, we present the numerical results and discussions; section 4 is reserved for our conclusion.
QCD sum rules for the $ss\bar{s}\bar{s}$ tetraquark states
==========================================================
We write down the two-point correlation functions $\Pi_{\mu\nu\alpha\beta}(p)$ and $\Pi(p)$ firstly, $$\begin{aligned}
\Pi_{\mu\nu\alpha\beta}(p)&=&i\int d^4x e^{ip \cdot x} \langle0|T\left\{J_{\mu\nu}(x)J_{\alpha\beta}^{\dagger}(0)\right\}|0\rangle \, , \\
\Pi(p)&=&i\int d^4x e^{ip \cdot x} \langle0|T\left\{J_0(x)J_0^{\dagger}(0)\right\}|0\rangle \, ,\end{aligned}$$ where $J_{\mu\nu}(x)=J_{2,\mu\nu}(x)$, $J_{1,\mu\nu}(x)$, $$\begin{aligned}
J_{2,\mu\nu}(x)&=&\frac{\varepsilon^{ijk}\varepsilon^{imn}}{\sqrt{2}}\Big\{s^{Tj}(x)C\gamma_\mu s^k(x) \bar{s}^{m}(x)\gamma_\nu C \bar{s}^{Tn}(x)+s^{Tj}(x)C\gamma_\nu s^k(x)\bar{s}^m(x)\gamma_\mu C \bar{s}^{Tn}(x) \Big\} \, , \nonumber\\
J_{1,\mu\nu}(x)&=&\frac{\varepsilon^{ijk}\varepsilon^{imn}}{\sqrt{2}}\Big\{s^{Tj}(x)C\gamma_\mu s^k(x) \bar{s}^{m}(x)\gamma_\nu C \bar{s}^{Tn}(x)-s^{Tj}(x)C\gamma_\nu s^k(x)\bar{s}^m(x)\gamma_\mu C \bar{s}^{Tn}(x) \Big\} \, , \nonumber\\
J_0(x)&=&\varepsilon^{ijk}\varepsilon^{imn}s^{Tj}(x)C\gamma_\mu s^k(x) \bar{s}^m(x)\gamma^\mu C \bar{s}^{Tn}(x) \, ,\end{aligned}$$ where the $i$, $j$, $k$, $m$, $n$ are color indexes, the $C$ is the charge conjugation matrix. Under charge conjugation transform $\widehat{C}$, the currents $J_{\mu\nu}(x)$ and $J_0(x)$ have the properties, $$\begin{aligned}
\widehat{C}\,J_{2,\mu\nu}(x)\,\widehat{C}^{-1}&=&+ \,J_{2,\mu\nu}(x)\, , \nonumber \\
\widehat{C}\,J_{1,\mu\nu}(x)\,\widehat{C}^{-1}&=&- \,J_{1,\mu\nu}(x)\, , \nonumber \\
\widehat{C}\,J_0(x)\,\widehat{C}^{-1}&=& +J_0(x) \, .\end{aligned}$$
The doubly-strange diquark operators $$\begin{aligned}
s^{Tj} C\Gamma s^k&=&\frac{1}{2}\Big(s^{Tj} C\Gamma s^k-s^{Tk} C\Gamma s^j \Big)=\frac{1}{2}\varepsilon^{ijk}s^{Tj} C\Gamma s^k\end{aligned}$$ with $\Gamma=\gamma_\mu$, $\sigma_{\mu\nu}$ in color antitriplet $\bar{3}_c$ and $$\begin{aligned}
s^{Tj} C\Gamma s^k&=&\frac{1}{2}\Big(s^{Tj} C\Gamma s^k+s^{Tk} C\Gamma s^j \Big)\end{aligned}$$ with $\Gamma=1$, $\gamma_{5}$, $\gamma_{\mu}\gamma_5$ in color sextet $6_c$ satisfy Fermi-Dirac statistics. On the other hand, the scattering amplitude for one-gluon exchange is proportional to $$\begin{aligned}
\left(\frac{\lambda^a}{2}\right)_{ij}\left(\frac{\lambda^a}{2}\right)_{kl}&=&-\frac{1}{3}\left(\delta_{ij}\delta_{kl}-\delta_{il}\delta_{kj}\right)
+\frac{1}{6}\left(\delta_{ij}\delta_{kl}+\delta_{il}\delta_{kj}\right) \, ,\end{aligned}$$ where $$\begin{aligned}
\varepsilon_{mik}\varepsilon_{mjl} &=&\delta_{ij}\delta_{kl}-\delta_{il}\delta_{kj}\, ,\end{aligned}$$ the $\lambda^a$ is the Gell-Mann matrix. The negative sign in front of the antisymmetric antitriplet $\bar{3}_c$ indicates the interaction is attractive, which favors formation of the diquarks in color antitriplet. The positive sign in front of the symmetric sextet $6_c$ indicates the interaction is repulsive, which disfavors formation of the diquarks in color sextet. The diquark states which couple potentially to the $s^{Tj} C s^k$, $s^{Tj} C\gamma_5 s^k$ and $s^{Tj} C\gamma_\mu\gamma_5 s^k$ operators in color sextet $6_c$ are expected to have larger masses than the diquark states which couple potentially to the $s^{Tj} C\gamma_\mu s^k$ and $s^{Tj} C\sigma_{\mu\nu} s^k$ operators in color antitriplet $\bar{3}_c$. We prefer the diquark operators in color antitriplet $\bar{3}_c$ to the diquark operators in color sextet $6_c$ in constructing the tetraquark current operators. Up to now, the scalar and axialvector diquark states in color antitriplet $\bar{3}_c$ have been studied with the QCD sum rules [@WangLDiquark; @Dosch-Diquark-1989]. In our previous studies, we observed that the pseudoscalar and vector diquark states in color antitriplet $\bar{3}_c$ are not favored configurations, and cannot lead to stable QCD sum rules, which are not included in Ref.[@WangLDiquark]. The tensor diquark states, which have both $J^P=1^+$ and $1^-$ components, have not been studied with the QCD sum rules yet. We can draw the conclusion tentatively that the most favored quark configuration is the axialvector diquark operator $\varepsilon^{ijk}s^{Tj} C\gamma_\mu s^k$. In Ref.[@Cui-etal], Cui et al choose the pseudoscalar diquark operator in color sextet $6_c$ and vector antidiquark operator in color antisextet $\bar{6}_c$, and axialvector diquark operator in color antitriplet $\bar{3}_c$ and tensor antidiquark operator in color triplet $3_c$ to construct the axialvector currents to study the axialvector tetraquark states. In Ref.[@Wang2007NPA], we choose the color octet-octet type vector four-quark current to study the $Y(2175)$, Fierz rearrangement of this current cannot lead to a diquark-antidiquark type tensor component. In the present work, we choose the axialvector diquark (antidiquark) operators in color antitriplet $\bar{3}_c$ (triplet $3_c$) to construct the tensor current, which is expected to couple potentially to the lowest tetraquark states, to study both the axialvector and vector tetraquark states. The quark configuration in the present work differs completely from that in Ref.[@Cui-etal] and Ref.[@Wang2007NPA], it is interesting to study the new quark configuration. Furthermore, the conclusion of the present work differs completely from that of Ref.[@Cui-etal].
At the hadronic side, we can insert a complete set of intermediate hadronic states with the same quantum numbers as the current operators $J_{\mu\nu}(x)$ and $J_0(x)$ into the correlation functions $\Pi_{\mu\nu\alpha\beta}(p)$ and $\Pi(p)$ to obtain the hadronic representation [@SVZ79; @Reinders85]. After isolating the ground state contributions of the scalar, axialvector, vector and tensor tetraquark states, we get the results, $$\begin{aligned}
\Pi_{2,\mu\nu\alpha\beta}(p)&=&\frac{\lambda_{ X_T}^2}{m_{X_T}^2-p^2}\left( \frac{\widetilde{g}_{\mu\alpha}\widetilde{g}_{\nu\beta}+\widetilde{g}_{\mu\beta}\widetilde{g}_{\nu\alpha}}{2}-\frac{\widetilde{g}_{\mu\nu}\widetilde{g}_{\alpha\beta}}{3}\right) +\cdots \nonumber\\
&=&\Pi_{2^+}(p)\left( \frac{\widetilde{g}_{\mu\alpha}\widetilde{g}_{\nu\beta}+\widetilde{g}_{\mu\beta}\widetilde{g}_{\nu\alpha}}{2}-\frac{\widetilde{g}_{\mu\nu}\widetilde{g}_{\alpha\beta}}{3}\right) +\cdots \, ,\end{aligned}$$
$$\begin{aligned}
\Pi_{1,\mu\nu\alpha\beta}(p)&=&\frac{\widetilde{\lambda}_{ X_A}^2}{m_{X_A}^2-p^2}\left(p^2g_{\mu\alpha}g_{\nu\beta} -p^2g_{\mu\beta}g_{\nu\alpha} -g_{\mu\alpha}p_{\nu}p_{\beta}-g_{\nu\beta}p_{\mu}p_{\alpha}+g_{\mu\beta}p_{\nu}p_{\alpha}+g_{\nu\alpha}p_{\mu}p_{\beta}\right) \nonumber\\
&&+\frac{\widetilde{\lambda}_{ X_V}^2}{m_{X_V}^2-p^2}\left( -g_{\mu\alpha}p_{\nu}p_{\beta}-g_{\nu\beta}p_{\mu}p_{\alpha}+g_{\mu\beta}p_{\nu}p_{\alpha}+g_{\nu\alpha}p_{\mu}p_{\beta}\right) +\cdots \nonumber\\
&=&\Pi_{1^+}(p^2)\left(p^2g_{\mu\alpha}g_{\nu\beta} -p^2g_{\mu\beta}g_{\nu\alpha} -g_{\mu\alpha}p_{\nu}p_{\beta}-g_{\nu\beta}p_{\mu}p_{\alpha}+g_{\mu\beta}p_{\nu}p_{\alpha}+g_{\nu\alpha}p_{\mu}p_{\beta}\right) \nonumber\\
&&+\Pi_{1^-}(p^2)\left( -g_{\mu\alpha}p_{\nu}p_{\beta}-g_{\nu\beta}p_{\mu}p_{\alpha}+g_{\mu\beta}p_{\nu}p_{\alpha}+g_{\nu\alpha}p_{\mu}p_{\beta}\right) \, ,\end{aligned}$$
$$\begin{aligned}
\Pi(p)&=&\Pi_{0^+}(p^2)=\frac{\lambda_{ X_S}^2}{m_{X_S}^2-p^2} +\cdots \, \, ,\end{aligned}$$
where $\widetilde{g}_{\mu\nu}=g_{\mu\nu}-\frac{p_\mu p_\nu}{p^2}$, the subscripts $2^+$, $1^+$, $1^-$ and $0^+$ denote the spin-parity $J^P$ of the corresponding tetraquark states. The pole residues $\lambda_{X}$ and $\widetilde{\lambda}_{X}$ are defined by $$\begin{aligned}
\langle 0|J_{2,\mu\nu}(0)|X_{T}(p)\rangle &=& \lambda_{X_T} \, \varepsilon_{\mu\nu} \, , \nonumber\\
\langle 0|J_{1,\mu\nu}(0)|X_{A}(p)\rangle &=& \widetilde{\lambda}_{X_A} \, \varepsilon_{\mu\nu\alpha\beta} \, \varepsilon^{\alpha}p^{\beta}\, , \nonumber\\
\langle 0|J_{1,\mu\nu}(0)|X_{V}(p)\rangle &=& \widetilde{\lambda}_{X_V} \left(\varepsilon_{\mu}p_{\nu}-\varepsilon_{\nu}p_{\mu} \right)\, , \nonumber\\
\langle 0|J_0(0)|X_{S}(p)\rangle &=& \lambda_{X_S} \, ,\end{aligned}$$ where the $\varepsilon_{\mu\nu}$ and $\varepsilon_\mu$ are the polarization vectors of the tetraquark states.
Now we contract the $s$ quarks in the correlation functions with Wick theorem, there are four $s$-quark propagators, if two $s$-quark lines emit a gluon by itself and the other two $s$-quark lines contribute a quark pair by itself, we obtain a operator $GG\bar{s}s\bar{s}s$, which is of order ${\mathcal{O}}(\alpha_s^k)$ with $k=1$ and of dimension $10$. In this article, we take into account the vacuum condensates up to dimension $10$ and $k\leq 1$ in a consistent way. For the technical details, one can consult Refs.[@WangScalarNonet; @WangHuangtao-2014]. Once the analytical expressions of the QCD spectral densities are obtained, we take the quark-hadron duality below the continuum thresholds $s_0$ and perform Borel transform with respect to the variable $P^2=-p^2$ to obtain the QCD sum rules: $$\begin{aligned}
\label{QCDSR}
\lambda^2_{X}\, \exp\left(-\frac{m^2_{X}}{T^2}\right)= \int_{0}^{s_0} ds\, \rho(s) \, \exp\left(-\frac{s}{T^2}\right) \, ,\end{aligned}$$ where $\rho(s)=\rho_S(s)$, $\rho_A(s)$, $\rho_V(s)$ and $\rho_T(s)$, $$\begin{aligned}
\rho_{S}(s)&=& \frac{s^4}{3840\pi^6} -\frac{13s\,m_s\langle\bar{s}g_{s}\sigma Gs\rangle}{384\pi^4}
+\frac{2s\langle\bar{s}s\rangle^2}{3\pi^2} -\frac{17\langle\bar{s}s\rangle \langle\bar{s}g_{s}\sigma Gs\rangle}{48\pi^2}+\frac{s^2}{192\pi^4}\langle\frac{\alpha_{s}GG}{\pi}\rangle \nonumber\\
&& +\frac{19m_s\langle\bar{s}s\rangle}{96\pi^2} \langle\frac{\alpha_{s}GG}{\pi}\rangle-\frac{16m_s\langle\bar{s}s\rangle^3}{3}\delta(s)+\frac{\langle\bar{s}g_{s}\sigma Gs\rangle^2}{192\pi^2}\delta(s) -\frac{\langle\bar{s}s\rangle^2}{24}\langle\frac{\alpha_{s}GG}{\pi}\rangle\delta(s) \, ,\end{aligned}$$
$$\begin{aligned}
\rho_{A}(s)&=& \frac{s^4}{11520\pi^6}-\frac{s^2\,m_s\langle\bar{s}s\rangle}{12\pi^4}
+\frac{s\,m_s\langle\bar{s}g_{s}\sigma Gs\rangle}{9\pi^4} +\frac{4s\,\langle\bar{s}s\rangle^2}{9\pi^2}
-\frac{5\langle\bar{s}s\rangle \langle\bar{s}g_{s}\sigma Gs\rangle}{18\pi^2} \nonumber\\
&&-\frac{s^2}{2304\pi^4}\langle\frac{\alpha_{s}GG}{\pi}\rangle+\frac{3m_s\langle\bar{s}s\rangle}{64\pi^2}\langle\frac{\alpha_{s}GG}{\pi}\rangle-\frac{32m_s\langle\bar{s}s\rangle^3}{9}\delta(s) -\frac{2\langle\bar{s}s\rangle^2}{27}\langle\frac{\alpha_{s}GG}{\pi}\rangle\delta(s) \, ,\end{aligned}$$
$$\begin{aligned}
\rho_{V}(s)&=& \frac{s^4}{11520\pi^6}+\frac{s^2\,m_s\langle\bar{s}s\rangle}{12\pi^4}-\frac{7s\,m_s\langle\bar{s}g_{s}\sigma Gs\rangle}{72\pi^4}
-\frac{2s\langle\bar{s}s\rangle^2}{9\pi^2} +\frac{5\langle\bar{s}s\rangle \langle\bar{s}g_{s}\sigma Gs\rangle}{18\pi^2} \nonumber\\
&&+\frac{s^2}{768\pi^4}\langle\frac{\alpha_{s}GG}{\pi}\rangle-\frac{79m_s\langle\bar{s}s\rangle}{1728\pi^2}\langle\frac{\alpha_{s}GG}{\pi}\rangle +\frac{16m_s\langle\bar{s}s\rangle^3}{9}\delta(s) \nonumber\\
&&-\frac{2\langle\bar{s}s\rangle^2}{81}\langle\frac{\alpha_{s}GG}{\pi}\rangle\delta(s) -\frac{\langle\bar{s}g_{s}\sigma Gs\rangle^2}{18\pi^2}\delta(s) \, ,\end{aligned}$$
$$\begin{aligned}
\rho_T(s)&=&\frac{s^4}{5376\pi^6}-\frac{3s^2\,m_s\langle\bar{s}s\rangle}{20\pi^4} +\frac{29s\,m_s\langle\bar{s}g_{s}\sigma Gs\rangle}{96\pi^4}
+\frac{8s\langle\bar{s}s\rangle^2}{9\pi^2}-\frac{37\langle\bar{s}s\rangle \langle\bar{s}g_{s}\sigma Gs\rangle}{48\pi^2} \nonumber\\
&&-\frac{11s^2}{1920\pi^4}\langle\frac{\alpha_{s}GG}{\pi}\rangle+\frac{43m_s\langle\bar{s}s\rangle}{864\pi^2}\langle\frac{\alpha_{s}GG}{\pi}\rangle
-\frac{64m_s\langle\bar{s}s\rangle^3}{9}\delta(s) -\frac{4\langle\bar{s}s\rangle^2}{27}\langle\frac{\alpha_{s}GG}{\pi}\rangle\delta(s) \, , \nonumber\\\end{aligned}$$
and $\lambda_{X_{A/V}}=m_{X_{A/V}}\widetilde{\lambda}_{X_{A/V}}$.
We derive Eq. with respect to $\tau=\frac{1}{T^2}$, then obtain the QCD sum rules for the masses of the tetraquark states through a fraction, $$\begin{aligned}
\label{mass-QCDSR}
m^2_{X}&=& -\frac{\int_{0}^{s_0} ds\frac{d}{d \tau}\rho(s)\exp\left(-\tau s \right)}{\int_{0}^{s_0} ds \rho(s)\exp\left(-\tau s\right)}\, .\end{aligned}$$
Numerical results and discussions
=================================
We take the standard values of the vacuum condensates $\langle
\bar{q}q \rangle=-(0.24\pm 0.01\, \rm{GeV})^3$, $\langle
\bar{q}g_s\sigma G q \rangle=m_0^2\langle \bar{q}q \rangle$, $m_0^2=(0.8 \pm 0.1)\,\rm{GeV}^2$, $\langle\bar{s}s \rangle=(0.8\pm0.1)\langle\bar{q}q \rangle$, $\langle\bar{s}g_s\sigma G s \rangle=m_0^2\langle \bar{s}s \rangle$, $\langle \frac{\alpha_s
GG}{\pi}\rangle=(0.012\pm0.004)\,\rm{GeV}^4 $ at the energy scale $\mu=1\, \rm{GeV}$ [@SVZ79; @Reinders85; @Colangelo-Review], and choose the $\overline{MS}$ mass $m_s(\mu=2\,\rm{GeV})=0.095\pm 0.005\,\rm{GeV}$ from the Particle Data Group [@PDG], and evolve the $s$-quark mass to the energy scale $\mu=1\,\rm{GeV}$ with the renormalization group equation, furthermore, we neglect the small $u$ and $d$ quark masses.
We choose suitable Borel parameters and continuum threshold parameters to warrant the pole contributions (PC) are larger than $40\%$, i.e. $$\begin{aligned}
\text{PC}&=&\frac{\int_{0}^{s_{0}}ds\,\rho\left(s\right)\exp\left(-\frac{s}{T^{2}}\right)} {\int_{0}^{\infty}ds\,\rho\left(s\right)\exp\left(-\frac{s}{T^{2}}\right)}\geq 40\%\ ,\end{aligned}$$ and convergence of the operator product expansion. The contributions of the vacuum condensates $D(n)$ in the operator product expansion are defined by, $$\begin{aligned}
D(n)&=&\frac{\int_{0}^{s_{0}}ds\,\rho_{n}(s)\exp\left(-\frac{s}{T^{2}}\right)}
{\int_{0}^{s_{0}}ds\,\rho\left(s\right)\exp\left(-\frac{s}{T^{2}}\right)}\ ,\end{aligned}$$ where the subscript $n$ in the QCD spectral density $\rho_{n}(s)$ denotes the dimension of the vacuum condensates. We choose the values $|D(10)|\sim 1\%$ to warrant the convergence of the operator product expansion. In Table \[mass-residue\], we present the ideal Borel parameters, continuum threshold parameters, pole contributions and contributions of the vacuum condensates of dimension $10$. In Fig.\[OPE\], we plot the absolute contributions of the vacuum condensates of dimension $n$ for the central values of the input parameters in the operator product expansion. Although in some cases, the contributions of the perturbative terms $D(0)$ are not the dominant contributions, the contributions of the vacuum condensates of dimensions $6$ and $8$ are very large, the hierarchy $|D(6)|\gg |D(8)|$ warrants the good convergent behavior of the operator product expansion, furthermore, the contributions $D(7)$, $D(9)$ and $D(10)$ are very small. From Table \[mass-residue\] and Fig.\[OPE\], we can see that the pole dominance is well satisfied and the operator product expansion is well convergent, we expect to make reliable predictions.
$T^2 (\rm{GeV}^2)$ $\sqrt{s_0}(\rm{GeV})$ pole $|D(10)|$ $m_{X}(\rm{GeV})$ $\lambda_{X}(10^{-2}\rm{GeV}^5)$
---------------------- -------------------- ------------------------ ------------- ----------- ------------------- ---------------------------------- --
$ss\bar{s}\bar{s}_S$ $1.4-1.8$ $2.65\pm0.10$ $(40-73)\%$ $\ll1\%$ $2.08\pm0.13$ $2.73\pm 0.56$
$ss\bar{s}\bar{s}_A$ $1.5-1.9$ $2.65\pm0.10$ $(41-72)\%$ $<1\%$ $2.08\pm0.12$ $1.87\pm 0.34$
$ss\bar{s}\bar{s}_T$ $1.5-1.9$ $2.75\pm0.10$ $(41-72)\%$ $<1\%$ $2.22\pm0.11$ $3.02\pm 0.53$
$ss\bar{s}\bar{s}_V$ $2.1-2.7$ $3.60\pm0.10$ $(42-73)\%$ $\leq1\%$ $3.08\pm0.11$ $6.47\pm 1.07$
$qq\bar{q}\bar{q}_S$ $1.2-1.6$ $2.40\pm0.10$ $(40-76)\%$ $\ll1\%$ $1.86\pm0.11$ $1.95\pm 0.38$
$qq\bar{q}\bar{q}_A$ $1.3-1.7$ $2.40\pm0.10$ $(40-73)\%$ $\leq1\%$ $1.87\pm0.10$ $1.30\pm 0.22$
$qq\bar{q}\bar{q}_T$ $1.4-1.8$ $2.65\pm0.10$ $(42-74)\%$ $\leq1\%$ $2.13\pm0.10$ $2.58\pm 0.42$
$qq\bar{q}\bar{q}_V$ $1.9-2.5$ $3.40\pm0.10$ $(41-74)\%$ $\leq2\%$ $2.86\pm0.11$ $4.94\pm 0.93$
: The Borel parameters, continuum threshold parameters, pole contributions, contributions of the vacuum condensates of dimension $10$, masses and pole residues of the tetraquark states, where the subscripts $S$, $A$, $T$ and $V$ denote the scalar, axialvector, tensor and vector tetraquark states, respectively. []{data-label="mass-residue"}
We take into account all uncertainties of the input parameters, and obtain the values of the masses and pole residues of the $ss\bar{s}\bar{s}$ tetraquark states, which are shown explicitly in Fig.\[mass-ssss\] and Table \[mass-residue\]. In this article, we have assumed that the energy gaps between the ground state and the first radial state is about $0.6\,\rm{GeV}$ [@Wang-Energy-gap]. In Fig.\[mass-ssss\], we plot the masses of the scalar, axialvector, tensor and vector $ss\bar{s}\bar{s}$ tetraquark states with variations of the Borel parameters at larger regions than the Borel windows shown in Table \[mass-residue\]. From the figure, we can see that there appear platforms in the Borel windows.
From Table \[mass-residue\], we can see that the uncertainties of the masses $\delta M_X$ are small, while the uncertainties of the pole residues $\delta \lambda_X$ are large, for example, $\frac{\delta M_X}{M_X}=6\%$ and $\frac{\delta \lambda_X}{\lambda_X}=21\%$ for the scalar $ss\bar{s}\bar{s}$ tetraquark state. We obtain the tetraquark masses from a fraction, see Eq., the uncertainties originate from the input parameters in the numerator and denominator are almost canceled out with each other, so the net uncertainties of the tetraquark masses are very small. In this article, we have neglected the perturbative $\mathcal{O}(\alpha_s)$ corrections. For the traditional two-quark light mesons, the perturbative $\mathcal{O}(\alpha_s)$ corrections amount to multiplying the perturbative terms with a factor $1+\frac{11}{3}\frac{\alpha_s}{\pi}$ for the $J^{PC}=0^{+-}$, $0^{++}$ mesons, $1+\frac{\alpha_s}{\pi}$ for the $J^{PC}=1^{--}$, $1^{++}$, $1^{+-}$ mesons, and $1-\frac{\alpha_s}{\pi}$ for the $J^{PC}=2^{++}$ mesons [@Reinders85]. Now we estimate the possible uncertainties due to neglecting the perturbative $\mathcal{O}(\alpha_s)$ corrections by multiplying the perturbative terms with a factor $1+(-1\sim 4)\frac{\alpha_s}{\pi}$. The additional uncertainties $\delta M_X$ and $\delta \lambda_X$ are shown in Table \[mass-afs\]. From the Table, we can see again that the uncertainties of the mass $\delta M_X$ are small, while the uncertainties of the pole residues $\delta \lambda_X$ are large, for example, $\frac{\delta M_X}{M_X}={}^{+2\%}_{-1\%}$ and $\frac{\delta \lambda_X}{\lambda_X}={}^{+23\%}_{-7\%}$ for the scalar $ss\bar{s}\bar{s}$ tetraquark state. In the QCD sum rules for the $X$, $Y$, $Z$ states, which are excellent candidates for the compact tetraquark states or loosely bound molecular states, the uncertainties of the masses are less than or about $6\%$ [@Nielsen-Review]. Ref.[@Nielsen-Review] is the most recent review.
![The absolute contributions of the vacuum condensates of dimension $n$ for the central values of the input parameters in the operator product expansion, where the $S$, $A$, $T$ and $V$ denote the scalar, axialvector, tensor and vector tetraquark states, respectively, the (I) and (II) denote the $ss\bar{s}\bar{s}$ and $qq\bar{q}\bar{q}$ quark constituents, respectively.[]{data-label="OPE"}](OPE-ssss.EPS "fig:"){width="7cm"} ![The absolute contributions of the vacuum condensates of dimension $n$ for the central values of the input parameters in the operator product expansion, where the $S$, $A$, $T$ and $V$ denote the scalar, axialvector, tensor and vector tetraquark states, respectively, the (I) and (II) denote the $ss\bar{s}\bar{s}$ and $qq\bar{q}\bar{q}$ quark constituents, respectively.[]{data-label="OPE"}](OPE-uuuu.EPS "fig:"){width="7cm"}
![The masses with variations of the Borel parameters $T^2$, where the $A$, $B$, $C$ and $D$ denote the scalar, axialvector, tensor and vector tetraquark states, respectively.[]{data-label="mass-ssss"}](mass-S.EPS "fig:"){width="7cm"} ![The masses with variations of the Borel parameters $T^2$, where the $A$, $B$, $C$ and $D$ denote the scalar, axialvector, tensor and vector tetraquark states, respectively.[]{data-label="mass-ssss"}](mass-A.EPS "fig:"){width="7cm"} ![The masses with variations of the Borel parameters $T^2$, where the $A$, $B$, $C$ and $D$ denote the scalar, axialvector, tensor and vector tetraquark states, respectively.[]{data-label="mass-ssss"}](mass-T.EPS "fig:"){width="7cm"} ![The masses with variations of the Borel parameters $T^2$, where the $A$, $B$, $C$ and $D$ denote the scalar, axialvector, tensor and vector tetraquark states, respectively.[]{data-label="mass-ssss"}](mass-V.EPS "fig:"){width="7cm"}
The predicted mass $m_{X}=2.08\pm0.12\,\rm{GeV}$ for the axialvector tetraquark state is in excellent agreement with the experimental value $(2062.8 \pm 13.1 \pm 4.2)
\,\rm{MeV}$ from the BESIII collaboration [@BES-2000], which supports assigning the new $X$ state to be an axialvector-diquark-axialvector-antidiquark type $ss\bar{s}\bar{s}$ tetraquark state. The predicted mass $m_{X}=3.08\pm0.11\,\rm{GeV}$ for the vector tetraquark state lies above the experimental value of the mass of the $\phi(2170)$ or $Y(2175)$, $m_{\phi}=2188\pm10\,\rm{MeV}$, from the Particle Data Group, and disfavors assigning the $\phi(2170)$ or $Y(2175)$ to be vector partner of the new $X$ state. If the $\phi(2170)$ have tetraquark component, it maybe have color octet-octet component [@Wang2007NPA]. As a byproduct, we obtain the masses and pole residues of the corresponding $qq\bar{q}\bar{q}$ tetraquark states, which are shown in Table \[mass-residue\]. The present predictions can be confronted to the experimental data in the future.
Now we perform Fierz rearrangement to the currents both in the color and Dirac-spinor spaces, $$\begin{aligned}
J_0 &=& 2 \bar{s}s\,\bar{s}s+2\bar{s}i\gamma_5s\,\bar{s}i\gamma_5s+ \bar{s}\gamma_{\alpha} s\,\bar{s}\gamma^{\alpha}s- \bar{s}\gamma_{\alpha}\gamma_5 s\,\bar{s}\gamma^{\alpha}\gamma_5s \, , \nonumber\\
J_{1,\mu\nu} &=&\sqrt{2}\Big\{\,i\bar{s}s\, \bar{s}\sigma_{\mu\nu}s -\bar{s}\sigma_{\mu\nu}\gamma_5s\,\bar{s}i\gamma_5s +i\varepsilon_{\mu\nu\alpha\beta}\bar{s}\gamma^\alpha\gamma_5s\, \bar{s}\gamma^\beta s\,\Big\} \, , \nonumber\\
J_{2,\mu\nu} &=&\frac{1}{\sqrt{2}}\Big\{\,2 \bar{s}\gamma_\mu\gamma_5s\, \bar{s}\gamma_\nu\gamma_5s -2\bar{s}\gamma_\mu s\, \bar{s}\gamma_\nu s
+2g^{\alpha\beta} \bar{s}\sigma_{\mu\alpha}s\, \bar{s}\sigma_{\nu\beta}s+g_{\mu\nu}\Big( \bar{s}s\,\bar{s}s \nonumber\\
&&+\bar{s}i\gamma_5s\,\bar{s}i\gamma_5s+\bar{s}\gamma_{\alpha} s\,\bar{s}\gamma^{\alpha}s-\bar{s}\gamma_{\alpha}\gamma_5 s\,\bar{s}\gamma^{\alpha}\gamma_5s-\frac{1}{2}\bar{s}\sigma_{\alpha\beta} s\,\bar{s}\sigma^{\alpha\beta}s \Big) \Big\} \, .\end{aligned}$$ The diquark-antidiquark type currents can be re-arranged into currents as special superpositions of color singlet-singlet type currents, which couple potentially to the meson-meson pairs or molecular states, the diquark-antidiquark type tetraquark states can be taken as special superpositions of meson-meson pairs, and embodies the net effects. The decays to their components are Okubo-Zweig-Iizuka supper-allowed, we can search for those tetraquark states in the decays, $$\begin{aligned}
X_{S} &\to& \eta^\prime \eta^\prime\, ,\,\, f_0(980) f_0(980)\, ,\,\,\phi(1020)\phi(1020)\, , \nonumber\\
X_{A/V} &\to& f_0(980) h_1(1380)\, ,\,\, \phi(1020) \eta^\prime\, ,\,\,\phi(1020)\phi(1020)\, , \nonumber\\
X_{T} &\to&\eta^\prime \eta^\prime\, ,\,\, f_0(980) f_0(980)\, ,\,\,\phi(1020)\phi(1020)\, .\end{aligned}$$
$\delta m_{X}(\rm{GeV})$ $\delta \lambda_{X}(10^{-2}\rm{GeV}^5)$
---------------------- -------------------------- ----------------------------------------- -- -- -- -- --
$ss\bar{s}\bar{s}_S$ ${}^{+0.04}_{-0.02}$ ${}^{+0.64}_{-0.18}$
$ss\bar{s}\bar{s}_A$ ${}^{+0.03}_{-0.02}$ ${}^{+0.33}_{-0.09}$
$ss\bar{s}\bar{s}_T$ ${}^{+0.03}_{-0.01}$ ${}^{+0.63}_{-0.18}$
$ss\bar{s}\bar{s}_V$ ${}^{+0.03}_{-0.06}$ ${}^{+1.62}_{-0.45}$
$qq\bar{q}\bar{q}_S$ ${}^{+0.04}_{-0.01}$ ${}^{+0.35}_{-0.10}$
$qq\bar{q}\bar{q}_A$ ${}^{+0.03}_{-0.01}$ ${}^{+0.18}_{-0.05}$
$qq\bar{q}\bar{q}_T$ ${}^{+0.03}_{-0.01}$ ${}^{+0.51}_{-0.14}$
$qq\bar{q}\bar{q}_V$ ${}^{+0.02}_{-0.02}$ ${}^{+1.27}_{-0.37}$
: The possible uncertainties induced by the perturbative $\mathcal{O}(\alpha_s)$ corrections, where the subscripts $S$, $A$, $T$ and $V$ denote the scalar, axialvector, tensor and vector tetraquark states, respectively. []{data-label="mass-afs"}
Conclusion
==========
In this article, we construct the axialvector-diquark-axialvector-antidiquark type currents to interpolate the scalar, axialvector, tensor and vector $ss\bar{s}\bar{s}$ tetraquark states, then calculate the contributions of the vacuum condensates up to dimension-10 in the operator product expansion, and obtain the QCD sum rules for the masses and pole residues of those tetraquark states. The predicted mass $m_{X}=2.08\pm0.12\,\rm{GeV}$ for the axialvector tetraquark state is in excellent agreement with the experimental value, $m_X=(2062.8 \pm 13.1 \pm 4.2)
\,\rm{MeV}$, from the BESIII collaboration and supports assigning the new $X$ state to be an axialvector-diquark-axialvector-antidiquark type $ss\bar{s}\bar{s}$ tetraquark state. The predicted mass $m_{X}=3.08\pm0.11\,\rm{GeV}$ for the vector tetraquark state lies above the experimental value of the mass of the $\phi(2170)$, $m_{\phi}=2188\pm10\,\rm{MeV}$, from the Particle Data Group, and disfavors assigning the $\phi(2170)$ to be the vector partner of the new $X$ state. As a byproduct, we also obtain the masses and pole residues of the corresponding $qq\bar{q}\bar{q}$ tetraquark states. The present predictions can be confronted to the experimental data in the future.
Acknowledgements {#acknowledgements .unnumbered}
================
This work is supported by National Natural Science Foundation, Grant Number 11775079.
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[^1]: E-mail: zgwang@aliyun.com.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
Let $w$ be a word in the free group of rank $n \in \mathbb{N}$ and let $\mathcal{V}(w)$ be the variety of groups defined by the law $w=1$. Define $\mathcal{V}(w^*)$ to be the class of all groups $G$ in which for any infinite subsets $X_1, \dots, X_n$ there exist $x_i \in X_i$, $1\leq i\leq n$, such that $w(x_1, \dots,
x_n)=1$. Clearly, $\mathcal{V}(w) \cup \mathcal{F} \subseteq
\mathcal{V}(w^*)$; $\mathcal{F}$ being the class of finite groups. In this paper, we investigate some words $w$ and some certain classes $\mathcal{P}$ of groups for which the equality $\left(\mathcal{V}(w) \cup \mathcal{F}\right)\cap \mathcal{P}=
\mathcal{P} \cap \mathcal{V}(w^*)$ holds.
address: 'Department of Mathematics,University of Isfahan,Isfahan 81746-73441, Iran.'
author:
- Alireza Abdollahi
title: A combinatorial problem in infinite groups
---
[**Introduction and results**]{}
Let $w$ be a word in the free group of rank $n \in \mathbb{N}$ and let $\mathcal{V}(w)$ be the variety of groups defined by the law $w=w(x_1,\dots,x_n)=1$. P. Longobardi, M. Maj and A. Rhemtulla in [@LMR] defined $\mathcal{V}(w^*)$ to be the class of all groups $G$ in which for any infinite subsets $X_1, \dots, X_n$ there exist $x_i \in X_i$, $1\leq i\leq n$, such that $w(x_1,
\dots, x_n)=1$ and raised the question of whether $\mathcal{V}(w)
\cup \mathcal{F} =\mathcal{V}(w^*)$ is true; $\mathcal{F}$ being the class of finite groups. There is no example, so far, of an infinite group in $\mathcal{V}(w^*)\backslash\mathcal{V}(w)$. In fact the origin of this problem is the following observation:\
Let $G$ be an infinite group such that in every two infinite subsets of $G$ there exist two commuting elements, then $G$ is abelian. This is an immediate consequence of the answer of B. H. Neumman to a question of P. Erdös; B. H. Neumman proved that an infinite group $G$ is centre-by-finite if and only if every infinite subset of $G$ contains two distinct commuting elements [@N]. Since this first paper, problems of a similar nature have been the object of several articles (for example [@A2], [@A3], [@AT1], [@D1], [@D3], [@DRS], [@G], [@LW], [@LM2], [@LMMR], [@RH]).\
As far as we know, the equality $\mathcal{V}(w)\cup \mathcal{F}=\mathcal{V}(w^*)$ is known for the following words: $w=x^m$, $w=[x_1, \dots, x_n]$ [@LMR], $w=[x,y]^2$ [@LM], $w=[x,y,y]$ [@S1], $w=[x,y,y,y]$ [@S2], $w=(xy)^{-3}x^3y^3$ [@A1], $w=x_1^{\alpha_1}\cdots x_m^{\alpha_m}$ where $\alpha_1, \dots, \alpha_m$ are non-zero integers [@AT2], $w=(xy)^2(yx)^{-2}$ or $w=[x^m,y]$ where $m\in\{3,6\}\cup \{2^k
\;|\; k\in\mathbb{N}\}$ [@AT3], $w=[x^n,y][x,y^n]^{-1}$ where $n\in\{\pm 2,3\}$ [@Taeri] and $w=[x^m,y^m]$ or $w=(x_1^mx_2^m\cdots x_n^m)^2$ where $m\in\{2^k \;|\; k\in \mathbb{N}\}$ [@Bouk].\
In [@PS], P. Puglisi and L. S. Spiezia proved that every infinite locally finite group (or locally soluble group) in $\mathcal{V}([x,_ky]^*)$ is a $k$-Engel group; (recall that $[x,_ky]$ is defined inductively by $[x,
_0y]=x$ and $[x,_ky]=[[x,_{k-1}y],y]$ for $k\in\mathbb{N}$). In [@D2], C. Delizia proved the equality $\mathcal{V}(w)\cup
\mathcal{F}=\mathcal{V}(w^*)$ on the classes of hyperabelian, locally soluble and locally finite groups where $w=[x_1, \dots,
x_k, x_1]$ and $k$ is an integer greater than 2. Later G. Endimioni generalized these results by proving that every infinite locally finite or locally soluble group in $\mathcal{V}(w^*)$ belongs to the variety $\mathcal{V}(w)$, where $w$ is a word in a free group such that finitely generated soluble groups in $\mathcal{V}(w)$ are nilpotent (see Theorem 3 of [@E]) (recall that the variety $\mathcal{V}([x_1, \dots,
x_k, x_1])$ ($k>2$) is exactly the variety of nilpotent groups of nilpotency class at most $k$ [@Mac] and every finitely generated soluble Engel group is nilpotent [@Gru].)\
We say that a group $G$ is locally graded if and only if every finitely generated non-trivial subgroup of $G$ has a non-trivial finite quotient. We proved in Theorem 4 of [@A3] that an infinite locally graded group in $\mathcal{V}([x_1,_kx_2]^*)$ is a $k$-Engel group. We generalize this result as Theorem A, below. In order to state our first result we need the following definition. Following [@KR] we say that a group $G$ is restrained if and only if $\left<x\right>^{\left<y\right>}=\left<x^{y^i}
\;|\; i\in \mathbb{Z}\right>$ is finitely generated for all $x,y\in G$. We show by Proposition 1 below, why the following theorem improves the above mentioned results.\
[**Theorem A.**]{} [*Let $w$ be a word in a free group such that every finitely generated residually finite group in $\mathcal{V}(w)$ is polycyclic-by-finite. Then every infinite finitely generated locally graded restrained group in $\mathcal{V}(w^*)$ belongs to the variety $\mathcal{V}(w)$.*]{}\
G. Endimioni proved that every infinite locally nilpotent group in $\mathcal{V}(w^*)$ belongs to the variety $\mathcal{V}(w)$, where $w$ is a word in a free group (see Theorem 1 of [@E]). The following theorem generalizes Theorem 1 of [@E].\
[**Theorem B.**]{} [*Let $w$ be a word in a free group and let $\mathcal{P}$ be a class of groups which satisfies the following conditions:\
(1) the class $\mathcal{P}$ is closed under taking subgroups.\
(2) every $\mathcal{P}$-group is soluble.\
(3) every infinite finitely generated ($\mathcal{P}$-by-finite)-group in $\mathcal{V}(w^*)$ belongs to the variety $\mathcal{V}(w)$.\
Then every infinite residually \[(locally $\mathcal{P}$)-by-finite\] group in $\mathcal{V}(w^*)$ belongs to $\mathcal{V}(w).$\
*]{}
For example, the classes of nilpotent groups, polycyclic groups, abelian-by-nilpotent groups and soluble residually finite groups satisfy the assumptions of Theorem B.\
Here we also obtain some reductions in investigation of the equality $\mathcal{V}(w) \cup \mathcal{F}=\mathcal{V}(w^*)$ on certain classes of groups and certain words $w$. For example\
[**Theorem C.**]{} [*Let $w$ be a non-trivial word in a free group. Then every non-linear simple locally finite group does not belong to the class $\mathcal{V}(w^*)$.*]{}\
In [@E], G. Endimioni proved that if $w$ be a word in a free group such that finitely generated soluble groups in $\mathcal{V}(w)$ are polycyclic, then every finitely generated soluble group in $\mathcal{V}(w^*)$ belongs to the variety $\mathcal{V}(w)$. Before stating our next result, we need a notation (see [@G1]). Let $\alpha$ be a non-zero element of some field of characteristic $p$. Denote the group generated by the matrices $\left\{ \begin{bmatrix} 1&0\\1&1\end{bmatrix},
\begin{bmatrix} \alpha&0\\0&1\end{bmatrix}\right\}$ by $M(\alpha,p)$.\
[**Theorem D.**]{} [*Let $w$ be a word in a free group such that every infinitely presented $M(\alpha, p) \not\in \mathcal{V}(w)$ for all $p\geq 0$ or $C_q\text{wr} C_{\infty}\not\in \mathcal{V}(w)$ for all primes $q$. Then every infinite locally soluble group in $\mathcal{V}(w^*)$ belongs to the variety $\mathcal{V}(w)$.*]{}\
We note that the group $M(\alpha,p)$ is finitely presented if and only if\
(i) $p\not = 0$ and $\alpha$ is algebraic over the prime field, or\
(ii) $p=0$ and at least one of $\alpha$ or $\alpha^{-1}$ is an algebraic integer (see Lemma 11 of [@G1]).\
Theorem D generalizes Theorems 2 and 3 of [@E], since we note that if $\mathcal{V}$ is a variety of groups in which every finitely generated soluble group in $\mathcal{V}$ is polycyclic then $\mathcal{V}$ contains no infinitely presented $M(\alpha, p)$ since $M(\alpha, p)$ is finitely generated metabelian; the subgroup $C_q^{(C_{\infty})}$ of $C_q \text{wr} C_{\infty}$ is not finitely generated and, $C_q \text{wr} C_{\infty}$ is not polycyclic for any prime $q$.\
[**Proofs**]{}
We start the proof of Theorem A.\
[*Proof of Theorem A.*]{} Let $G$ be an infinite finitely generated locally graded restrained group in $\mathcal{V}(w^*)$ and let $R$ be the finite residual of $G$. Then $G/R$ is a finitely generated residually finite group in $\mathcal{V}(w^*)$ and so, by Lemma 1 of [@E], it belongs to $\mathcal{V}(w)$. Thus by hypothesis, $G/R$ is polycyclic-by-finite. Therefore by repeated use of Lemma 3 of [@KR], $R$ is finitely generated. If $R$ is finite then $G$ is residually finite and so by Lemma 1 of [@E], $G$ belongs to the variety $\mathcal{V}(w)$. Now suppose, for a contradiction, that $R$ is infinite. By hypothesis, $R$ has a normal proper subgroup of finite index in $R$, then the finite residual subgroup $T$ of $R$ is proper in $R$. Therefore $R/T$ is a residullay finite group in $\mathcal{V}(w)$ and so $G/T$ is polycylic-by-finite. Thus $G/T$ is residually finite and $R\subseteq T$, a contradiction. This completes the proof. $\Box$\
The following proposition generalizes the result of [@BP].\
[**Proposition 1.**]{} [*Finitely generated residually finite groups in a variety $\mathcal{V}$ in which every finite group is nilpotent, are nilpotent.\
*]{}
[*Proof.*]{} We first prove that there exists a positive integer $k$ depending only on the variety $\mathcal{V}$ such that for all primes $p$, $C_p \text{wr} C_{p^k} \not\in
\mathcal{V}$. By the Lemma of [@E1], there exists an integer $t$ depending only on $\mathcal{V}$ such that every $2$-generated metabelian group in $\mathcal{V}$ is nilpotent of class at most $t$. Now suppose that $C_p \text{wr} C_{p^m} \in \mathcal{V}$ for some prime $p$ and positive integer $m$. Since $C_p \text{wr} C_{p^m}$ is a $2$-generated metabelian group then it is nilpotent of class at most $t$. But the nilpotency class of $C_p \text{wr} C_{p^m}$ is exactly $p^m$, by a result of Liebeck (see [@L] or Theorem 2.5 in page 76 of [@M]) and so $p^m\leq t$. Now the same argument as in Theorem 2 of [@W] completes the proof. $\Box$\
Theorem A improves Theorem 3 of [@E] since by the result of [@BP], in a variety, all finite groups are nilpotent if and only if all finitely generated soluble groups are nilpotent. Therefore by Proposition 1, every variety in which all finitely generated soluble groups are nilpotent is contained in a variety in which all finitely generated residually finite groups are polycyclic-by-finite.\
[**Corollary 2.**]{} [*Let $w$ be a word in a free group such that finitely generated soluble groups in $\mathcal{V}(w)$ are nilpotent. Then every infinite locally graded restrained group in $\mathcal{V}(w^*)$ belongs to the variety $\mathcal{V}(w)$.*]{}\
[*Proof.*]{} As noticed before, every finitely generated residually finite group in $\mathcal{V}(w)$ is polycyclic-by-finite. Let $G$ be an infinite locally graded restrained group in $\mathcal{V}(w^*)$ and assume that $w$ is a word in the free group of rank $n\in\mathbb{N}$. Let $x_1,
\dots, x_n \in G$, we must prove that $w(x_1, \dots, x_n)=1$. Assume that there exists an infinite finitely generated subgroup $H$ of $G$ which contains $x_1, \dots, x_n$. Then by Theorem A, $H\in \mathcal{V}(w)$. Now, we may assume that every finitely generated subgroup of $G$ containing $x_1, \dots, x_n$ is finite. Thus there exists an infinite locally finite subgroup $L$ which contains $x_1, \dots, x_n$ and so by Theorem 3 of [@E], $L$ belongs to the variety $\mathcal{V}(w)$. This completes the proof. $\Box$\
In the following lemmas we use some notion: we say that a word $w\not=1$ in a free group is a semigroup word if $w$ is of the form $uv^{-1}$, where $u$ and $v$ are words in a free semigroup and we say, following [@LMR2], that a group $G$ has no free subsemigroups if and only if for every pair (a,b) of elements of $G$, the subsemigroup generated by $a, b$ has a relation of the form $$(1)\;\;\;\;\;\;\; a^{r_1}b^{s_1}\cdots
a^{r_j}b^{s_j}=b^{m_1}a^{n_1}\cdots b^{m_k}a^{n_k}$$ where $r_i$, $s_i$, $m_i$ $n_i$ are all non-negative and $r_1$ and $m_1$ are positive integers. If $(a,b)$ is a pair of elements in $G$ satsfying a relation of type (1), then we call $j+k$ the width of the relation and the sum $r_1+\cdots +r_j+n_1+\cdots +n_k$ the exponent of $a$ (denoted $\text{exp}(a)$) in the relation.\
We say that a word $w$ in a free group $F$ generated by $x_1, \dots x_n$, is a commutator word whenever $w$ belongs to the derived subgroup of $F$. In the following we study infinite groups in $\mathcal{V}(w^*)$ where $w$ is not a commutator word. We note that if $w$ is not a commutator word then there is a positive integer $e$ depending only on $w$ such that every group in the variety $\mathcal{V}(w)$ is of exponent dividing $e$; for let $G$ be a group in the variety generated by a non-commutator word $w$, since $w$ is not a commuatator word, for some $i$ the sum of the exponents of $x_i$ in $w$ is non-zero: let this sum be $r$ and let $g\in G$. If we replace $x_i$ by $g$ and $x_j$ by $1$ when $j\not =i$, then $w$ assumes the value $g^r$. Thus $g$ has a finite order $r$ and $G$ is of finite exponent.\
[**Lemma 3.**]{} [*Let $w$ be a semigroup word in the free group of rank 2. Then every group in $\mathcal{V}(w^*)$ has no free subsemigroups, and there exist positive integers $M$ and $N$ depending only on $w$ such that for all pairs $(a,b)$ of elements in $G$ there is a relation of the form (1) whose width and $\text{exp}(a)$ is at most $M$ and $N$, respectively.*]{}\
[*Proof.*]{} Let $a, b$ be in $G$. If $b$ is of finite order $m$ then $ab^m=b^ma$, $\text{exp}(a)=2$ and the width is $2$. Now, assume that $b$ is of infinite order and consider the two sets $X=\{a^{b^n} \;|\; n\in \mathbb{N}\}$ and $Y=\{b^m \;|\; m\in \mathbb{N}\}$. If $X$ is finite then the centre of $H=\left<a,b\right>$ is infinite and so by Lemma 3 of [@E], $H$ belongs to the variety $\mathcal{V}(w)$. Therefore $w(a,b)=w(b,a)=1$ and so the pair $(a,b)$ satisfies a relation of the form (1) whose width and $\text{exp}(a)$ is at most $M_1$ and $N_1$, respectively, where $M_1$ and $N_1$ are positive fixed integers depending only on $w$. Now we may assume that $X$ is infinite, then by the property $\mathcal{V}(w^*)$, there exists a relation of the form $$(a^{b^t})^{r_1}b^{s_1}\cdots
(a^{b^t})^{r_j}b^{s_j}=b^{m_1}(a^{b^t})^{n_1}\cdots
b^{m_k}(a^{b^t})^{n_k}$$ where $r_i, s_i, m_i, n_i$ are non-negative integers and $r_1, m_1, t$ are positive integers; also the sum $r_1+\cdots +r_j+n_1+\cdots +n_k$ is the same $N_1$ and $j+k=M_1$. Therefore the pair $(a,b)$ satisfies a relation of the form (1) whose width is at most $M:=\max\{2,M_1\}$ and $\text{exp}(a)$ is at most $N=\max\{2,N_1\}$. $\Box$\
Recall that a group $G$ is right orderable if there exists a total order relation $\geq$ on $G$ such that for all $a, b, g$ in $G$, $a\geq b$ implies $ag\geq bg$, equivalenty, if there exists a subset $P$ in $G$ such that $PP=P$, $P\cup P^{-1}=G$, and $P\cap P^{-1}={1}$.\
[**Proposition 4.**]{} [*Let $w$ be a semigroup word in the free group of rank 2. Then every right orderable group in $\mathcal{V}(w^*)$, belongs to the variety $\mathcal{V}(w)$.*]{}\
[*Proof.*]{} By Theorem 5 of [@LMR2] and Lemma 3, $G$ is locally nilpotent-by-finite. Let $x_1, \dots, x_n\in G$. Since $G$ is right orderable, $G$ is torsion-free. Thus every finitely generated subgroup of $G$ is an infinite finitely generated nilpotent-by-finite group and so residually finite. Therefore by Lemma 1 of [@E], $G$ belongs to the variety $\mathcal{V}(w)$. $\Box$\
[**Lemma 5.**]{} [*Let $w$ be a semigroup word in a free group. Then every group in $\mathcal{V}(w^*)$ is restrained.*]{}\
[*Proof.*]{} Let $G$ be a group in $\mathcal{V}(w^*)$ and let $x,
y$ in $G$. We must prove that $H=\left<x\right>^{\left<y\right>}$ is finitely generated. We may assume that $y$ is of infinite order. Suppose that $w$ is in the free group of rank $n>0$. Consider a partition of the set $X=\{xy^{-1}, xy^{-2},
\dots,\}$ in $n$ infinite subsets $X_1,X_2,\dots, X_n$. Then by the property $\mathcal{V}(w^*)$, there exist negative integers $t_1, \dots, t_n$ such that $$xy^{t_{f(1)}}\cdots xy^{t_{f(m)}}=xy^{t_{g(1)}}\cdots xy^{t_{g(s)}}$$ for some functions $f$ from $\{1,2,\dots, m\}$ to $\{1,2,\dots,n\}$ and $g$ from $\{1,2,\dots,s\}$ to $\{1,2,\dots,n\}$, where $m$ and $s$ depend only on $w$. Now, arguing as in Lemma 1(ii) of [@KR], $H$ is finitely generated. This completes the proof. $\Box$\
[**Lemma 6.**]{} [*Let $w$ be a word in a free group such that $w$ is not a commutator word. Then every group in $\mathcal{V}(w^*)$ is torsion. In particular, $G$ is restrained.*]{}\
[*Proof.*]{} Let $G$ be a group. Suppose, for a contradiction, that $G$ has an element $a$ of infinite order, then, by Lemma 3 of [@E], $\left<a\right>$ belongs to the variety $\mathcal{V}(w)$ and so $a$ is of finite order, a contradiction. $\Box$\
We note that Theorem A can be applied for the following words $w$ in a free group: by Proposition 1 and the result of [@BP], any word $w$ such that every finitely generated soluble group in the variety $\mathcal{V}(w)$ is nilpotent; by Zelmanov’s positive solution to the restricted Burnside problem (see [@Z1] and [@Z2]), any non-commutator word $w$ and by Theorem A of [@KR], every semigroup word $w$.\
By Theorem A and Lemmas 5 and 6 and the above remarks we have\
[**Corollary 7.**]{} [*Let $w$ be a non-commutator word or a semigroup word in a free group. Then every infinite finitely generated locally graded group in $\mathcal{V}(w^*)$ belongs to the variety $\mathcal{V}(w)$.*]{}\
[**Lemma 8.**]{} [*Let $G$ be an infinite group in $\mathcal{V}(w^*)$ and $H$ be a finite subgroup of $G$. If $G$ has an infinite normal locally soluble subgroup, then $H$ belongs to $\mathcal{V}(w)$.*]{}\
[*Proof.*]{} Let $S$ be a normal locally soluble infinite subgroup of $G$. If $S$ is $\Check{\text{C}}$ernikov, then $S$ has an infinite normal characteristic abelian subgroup (see [@R1] vol. I page 68) so $G$ has an infinite normal abelian subgroup whence $G$ belongs to $\mathcal{V}(w)$ by Lemma 3 of [@E].\
Therefore, we may assume that $S$ is not $\Check{\text{C}}$ernikov. By a result of Zaicev (see [@Z]), there is an infinite abelian subgroup $B$ of $S$ such that $H$ normalizes $B$. Hence $B$ is an infinite normal subgroup of the group $BH$ and so again by Lemma 3 of [@E], $H$ belongs to $\mathcal{V}(w)$. $\Box$\
[*Proof of Theorem B.*]{} It suffices to prove that an infinite \[(locally $\mathcal{P}$)-by-finite\] group in $\mathcal{V}(w^*)$ belongs to the variety $\mathcal{V}(w)$. Let $H$ be a normal locally $\mathcal{P}$-subgroup of $G$ of finite index. If $G$ is torsion, then $G$ is locally finite and $H$ is a locally soluble infinite normal subgroup of $G$, so by Lemma 8, $G \in \mathcal{V}(w)$. Therefore we may assume that $G$ has an element $a$ of infinite order. Let $x_1, \dots, x_n$ be arbitrary elements of $G$. Then $K=\left<a, x_1, \dots,
x_n\right>$ is a finitely generated $\mathcal{P}$-by-finite infinite group and so by condition (3), $K \in \mathcal{V}(w)$. $\Box$\
[**Corollary 9.**]{} [*Let $G$ be an infinite locally finite $\mathcal{V}(w^*)$ group. If $G$ satisfies one of the following conditions, then $G$ belongs to the variety $\mathcal{V}(w)$.\
(1) $G$ has an infinite locally soluble normal subgroup.\
(2) $G$ contains an element with finite centralizer.\
(3) $G$ contains an element of prime power order with $\Check{\text{C}}$ernikov centralizer in $G$.\
*]{}
[*Proof.*]{} Let $x_1, \dots, x_n$ be arbitrary elements of $G$, we must prove that $w(x_1, \dots, x_n)=1$. Since $G$ is locally finite, $H=\left<x_1, \dots, x_n\right>$ is finite.\
If $G$ has an infinite locally soluble normal subgroup, then, by Theorem B, $H \in \mathcal{V}(w)$.\
If $G$ satisfies the conditions (2) or (3) then by Hartley’s results of [@H] and [@H2] $G$ is (locally soluble)-by-finite and so by part (1), the proof is complete. $\Box$\
Let $w$ be a word in a free group. Now we state some reductions in investigation of the equality $\mathcal{V}(w) \cup
\mathcal{F} =\mathcal{V}(w^*)$ on the class of locally soluble groups and locally finite groups.\
Let $G$ be an infinite locally soluble group in $\mathcal{V}(w^*)$. If $G$ is torsion then by Corollary 9(1), $G$ belongs to the variety $\mathcal{V}(w)$. Therefore we may assume that $G$ has an element $g$ of infinite order and so in order to prove that $G \in \mathcal{V}(w)$ it suffices to show that for all $x_1, \dots, x_n$, the infinite finitely generated soluble subgroup $\left<x_1, \dots, x_n, g\right>$ belongs to the variety $\mathcal{V}(w)$. Therefore we have\
[**Remark 10.**]{} [*Let $w$ be a word in a free group. Then the following are equivalent:\
(1) any infinite locally soluble group in $\mathcal{V}(w^*)$ belongs to the variety $\mathcal{V}(w)$.\
(2) any infinite finitely generated soluble group in $\mathcal{V}(w^*)$ belongs to $\mathcal{V}(w)$.\
*]{}
We note that, by Lemma 6, every finitely generated soluble group in $\mathcal{V}(w^*)$, where $w$ is not a commutator word, is finite.\
Let $G$ be an infinite locally finite group in $\mathcal{V}(w^*)$. In order to prove that $G \in \mathcal{V}(w)$, we must show that $\left<x_1,
\dots, x_n\right> \in \mathcal{V}(w)$ for all $x_1, \dots, x_n \in
G$, therefore we may assume that $G$ is countable. Fix $x_1,
\dots, x_n \in G$ and let $H=\left<x_1, \dots, x_n\right>$. If $C_G(H)$ is infinite, then there is an infinite abelian subgroup $A$ in $C_G(H)$, as $G$ is locally finite (see Theorem 3.43 of [@R1]). therefore the centre of $K=\left<A, H\right>$ is infinite and so by Lemma 3 of [@E], $K \in \mathcal{V}(w)$. Thus we may assume that $C_G(H)$ is finite. Also, by Lemma 4 of [@E] and Corollary 9 we may assume that $H$ is not supersoluble and the centralizer of any element in $G$ is infinite and the centralizer of every element of prime power order is not $\Check{\text{C}}$ernikov. These conditions on a locally finite group lead us to the following defenitions.\
We say that a group $G$ is an $\mathcal{L}$-group whenever $G$ is an infinite countable locally finite group and there exists a finite subgroup $H$ of $G$ such that\
(1) $H$ is not supersoluble and $C_G(H)$ is finite.\
(2) $C_G(x)$ is infinite for all $x \in G$.\
(3) $C_G(g)$ is not $\Check{\text{C}}$ernikov for all elements $g
\in G$ of prime power order.\
(4) the largest normal locally soluble subgroup of $G$ is finite.\
In this case, We say that $G$ is an $\mathcal{L}$-group with respect to $H$. Also, we say that $G$ is an $\mathcal{L}^*$-group with respect to $H$ whenever every infinite subgroup of $G$ which contains $H$, is an $\mathcal{L}$-group with respect to $H$. By these discussions we have\
[**Remark 11.**]{} [*Let $w$ be a word in a free group. Then the following are equivalent:\
1) an infinite locally finite group in $\mathcal{V}(w^*)$, belongs to the variety $\mathcal{V}(w)$.\
2) an infinite $\mathcal{L}^*$-group in $\mathcal{V}(w^*)$, belongs to the variety $\mathcal{V}(w)$.\
*]{}
We use Remark 11 for the study of an infinite locally finite group $G$ in $\mathcal{V}(w^*)$ where $w$ is not a commutator word in the free group of rank $n>0$, and obtain another condition on such groups $G$. We prove that $G$ is of finite exponent dividing $e$, where $e$ is a positive integer depending only on $w$ such that every group in the variety $\mathcal{V}(w)$ is of exponent dividing $e$. For, let $a$ be an element of $G$, then $C_G(a)$ is infinite and by Theorem 3.43 of [@R1] there exists an infinite abelian subgroup $A$ in $C_G(a)$. By Lemma 3 of [@E], $A\in \mathcal{V}(w)$. Consider infinite subsets $X_1=\dots=X_n=aA$. Therefore, by the property $\mathcal{V}(w^*)$, there exist $a_1, \dots, a_n\in A$ such that $w(aa_1, \dots,
aa_n)=1$. Thus $w(a, \dots, a)w(a_1, \dots, a_n)=1$. But $w(a_1,
\dots, a_n)=1$ and so $w(a, \dots, a)=1$ and $a^e=1$. Therefore we have:\
[**Remark 12.**]{} [*Let $w$ be a non-commutator word in a free group and $e$ be a positive integer depending only on $w$ such that every group in the variety $\mathcal{V}(w)$ is of exponent dividing $e$. Then the following are equivalent:\
1) any infinite locally finite group in $\mathcal{V}(w^*)$ belongs to the variety $\mathcal{V}(w)$.\
2) any infinite $\mathcal{L}^*$-group of exponent dividing $e$ belongs to the variety $\mathcal{V}(w)$.*]{}\
A natural question which arises is the following: Is there an infinite $\mathcal{L}^*$-group of finite exponent?\
We only know that such a group is not simple. For by a result of L. G. Kov$\acute{\text{a}}$cs [@Ko], any infinite, simple, locally finite group $G$ involves infinitely many non-isomorphic non-abelian finite simple groups; hence, if $G$ satisfies non-trivial laws, then according to a result of G. A. Jones (see Theorem of [@J]), the variety generated by infinitely many finite simple groups is the variety of all groups. But the variety generated by $G$ is a proper variety, a contradiction.\
Now we study infinite simple locally finite groups in $\mathcal{V}(w^*)$, where $w$ is a non-trivial word in a free group. As we have seen earlier, there is no infinite simple locally finite group which satisfies a non-trivial identity. Call a simple locally finite group an $S$-group. The $S$-groups fall into two classes with widely different properties—the linear groups and the non-linear groups. Every linear $S$-group is a group of Lie type over an infinite locally finite field (see [@HS]).\
[*Proof of Theorem C.*]{} Suppose, for a contradiction, there exists a non-linear $S$-group $G$ in $\mathcal{V}(w^*)$. By a result of Hartley [@H3], there exists a section $C/D$ of $G$ such that $C/D$ is a direct product of finite alternating groups of unbounded orders. Thus $C/D$ is an infinite residually finite group in $\mathcal{V}(w^*)$ and so $C/D$ belongs to the variety $\mathcal{V}(w)$. Since $C/D$ is a direct product of finite alternating groups of unbounded orders, the variety $\mathcal{V}(w)$ contains infinitely many non-isomorphic finite alternating groups. Therefore, by Theorem of [@J], $\mathcal{V}(w)$ is the variety of all groups and so $w$ is the trivial word, a contradiction. This completes the proof. $\Box$\
P. S. Kim in [@Kim] studied $\mathcal{V}(w_2^*)$ on the class of locally soluble groups, where $w_2=[[x_1,x_2],[x_3,x_4]]$. For this word the variety $\mathcal{V}(w_2)$ is the variety of metabelian groups. It is proved in [@Kim], that every infinite locally soluble group in $\mathcal{V}(w_2^*)$ is metabelian and also it is proved that any infinite group belonging to $\mathcal{V}(w_2^*)$ is metabelian if and only if there is no infinite simple group in $\mathcal{V}(w_2^*)$. We study $\mathcal{V}(w^*)$ on the class of locally finite groups, where $w$ is a soluble word that is $w=w_d$ for some $d\in \mathbb{N}$ where $w_0=x$, $w_i=[w_{i-1},w_{i-1}]$ and $w_{i-1}$ is the word on $2^{i-1}$ distinct letters which has been defined inductively, for all $i\in\mathbb{N}$.\
[**Corollary 13.**]{} [*Let $w$ be a soluble word and let $G$ be an infinite locally finite $\mathcal{V}(w^*)$-group. Then the following are equivalent:\
(1) $G\in\mathcal{V}(w)$.\
(2) $G$ has no infinite linear simple locally finite section.\
*]{}
[*Proof.*]{} Suppose that (1) is true. Then $G$ is soluble and (2) is clear. Now suppose that (2) is true and $w=w_d$ for some positive integer $d$. Suppose, for a contradiction, that $G\not\in\mathcal{V}(w)$. Thus $G$ is not soluble of derived length at most $d$. Suppose, if possible, that $K=G^{(d+1)}$ is finite. Then $H=G^{(d)}$ is an FC-group and so $H$ is soluble by applying suitably Lemma 1 of [@AT2]. Thus $G$ is a torsion soluble group and so by Theorem B, $G$ is soluble of derived length at most $d$, a contradiction. Hence $K$ is infinite and so $G/K$ is a soluble group of derived length $d$. Therefore $G^{(d)}=G^{(d+1)}$ that is $H=H'$, which implies that $H$ is a perfect group. Suppose that $H$ has an infinite proper normal subgroup $N$, then $H/N$ is soluble of derived length at most $d$, this implies $H=H^{(d)}\leq N$ since $H$ is perfect, a contradiction. Let $N$ be a finite normal subgroup of $H$, then $C_H(N)$ has finite index in $H$. Since $H$ has no infinite normal proper subgroups, $C_H(N)=H$. Hence the centre $Z$ of $H$ is the unique maximal normal subgroup of $H$ so that $S=H/Z$ is simple. By Theorem C, $S$ is an infinite linear simple locally finite group, which is a contradiction. $\Box$\
Now, we start proving Theorem D, for this we need the following lemma:\
[**Lemma 14.**]{} [*Every infinite locally soluble group of finite rank in $\mathcal{V}(w^*)$ belongs to the variety $\mathcal{V}(w)$.*]{}\
[*Proof.*]{} Let $G$ be an infinite locally soluble group of finite rank in $\mathcal{V}(w^*)$. By Remark 10, we may assume that $G$ is finitely generated. Therefore $G$ is a minimax group, and so by Theorem 10.33 of [@R1], the finite residual $R$ of $G$ is the direct product of finitely many quasicyclic subgroups of $G$, thus $G$ is residually finite or $G$ has an infinite normal abelian subgroup, then, by Lemma 1 or Lemma 3 of [@E] respectively, the proof is complete.\
[*Proof of Theorem D.*]{} Let $G$ be an infinite locally soluble group in $\mathcal{V}(w^*)$. By Remark 10, we may assume that $G$ is a finitely generated infinite soluble group. Firstly, suppose that $w$ is a word such that $C_p\text{wr} C_{\infty}\not\in \mathcal{V}(w)$ for all primes $p$. We prove that $G$ is a minimax group and so $G$ is of finite rank, then Lemma 14 completes the proof.\
By a deep result of Kropholler (see [@K]), which asserts that every finitely generated soluble group having no sections of type $C_p \text{wr} C_{\infty}$ is minimax, it suffices to show that if $C_p \text{wr} C_{\infty} \in \mathcal{V}(w^*)$ then $C_p
\text{wr} C_{\infty} \in \mathcal{V}(w)$. But $C_p \text{wr}
C_{\infty}$ has an infinite normal abelian subgroup, therefore by Lemma 3 of [@E], $C_p \text{wr} C_{\infty} \in
\mathcal{V}(w)$, which is a contradiction.\
Now, suppose that $w$ is a word such that every infinitely presented $M(\alpha, p) \not\in \mathcal{V}(w)$. If $G$ is not semi-polycyclic group (see [@G1]), then there exists a subgroup $H$ of a quotient group of $G$ which is isomorphic to an infinitely presented $M(\alpha,
p)$. But $M(\alpha, p)$ is an infinite residually finite group in $\mathcal{V}(w^*)$ and so $M(\alpha, p) \in \mathcal{V}(w)$, a contradiction. Therefore $G$ is semi-polycyclic and so is of finite rank (see [@G1]). Thus, by Lemma 14, $G\in\mathcal{V}(w)$. This completes the proof. $\Box$\
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In recent years, a number of experimental studies have been conducted to investigate the mechanical behavior and damage mechanisms of articular cartilage under impact loading. Some experimentally observed results have been explained using a non-linear viscoelastic impact model. At the same time, there is the need of simple mathematical models, which allow comparing experimental results obtained in drop impact testing with impact loads of different weights and incident velocities. The objective of this study was to investigate theoretically whether the main features of articular impact could be qualitatively predicted using a linear viscoelastic theory or the linear biphasic theory. In the present paper, exact analytical solutions are obtained for the main parameters of the Kelvin–Voigt and Maxwell impact models. Perturbation analysis of the impact process according to the standard viscoelastic solid model is performed. Asymptotic solutions are obtained for the drop weight impact test. The dependence of the coefficient of restitution on the impactor parameters has been studied in detail.'
address: 'Institute of Mathematics and Physics, Aberystwyth University, Ceredigion SY23 3BZ, Wales, UK'
author:
- 'I.I. Argatov'
title: ' Mathematical modeling of linear viscoelastic impact: Application to drop impact testing of articular cartilage'
---
Impact contact problem ,blunt indenter ,asymptotic model ,coefficient of restitution
Nomenclature {#nomenclature .unnumbered}
============
--------------- ------------------------------------------------
$b$ damper coefficient
$D$ discriminant of the characteristic equation
$E_{\rm dyn}$ incremental dynamic modulus
$E_{\rm max}$ maximum incremental dynamic modulus
$E_{10}$ modulus at stresses of 10 MPa
$e_*$ coefficient of restitution
$F$ contact force
$F_M$ maximum contact force
$g$ gravitational acceleration
$h$ cartilage layer thickness
$h_0$ drop height of the impactor
$H_A$ aggregate modulus
$k$ stiffness coefficient
$k_1$, $k_2$ spring stiffnesses in the standard solid model
$k_0$ instantaneous stiffness
$k_\infty$ long-term stiffness
$m$ impactor mass
$t$ time variable
$t_c$ impact duration
$t_m$ time to maximum displacement
$t_M$ time to maximum contact force
$v_0$ initial impact velocity
$x$ displacement
$\dot{x}$ velocity
$\ddot{x}$ acceleration
$x_m$ maximum displacement
--------------- ------------------------------------------------
$${}$$
------------------------------ -------------------------------------------------------------------
$\beta$ damping coefficient in the Kelvin–Voigt model
$\beta_1$ real part of complex roots of the characteristic equation
$\Delta m$ percentage increase in mass of cartilage sample
$\epsilon$ strain
$\varepsilon_0$ non-dimensional parameter accounting for the gravitational effect
$\zeta$ loss factor in the Maxwell model
$\zeta_1$ imaginary part of complex roots of the characteristic equation
$\eta$ loss factor in the Kelvin–Voigt model
$\kappa$ cartilage permeability
$\varkappa_1$, $\varkappa_2$ spring stiffnesses in the standard solid model
$\lambda$ Lamé coefficient
$\lambda_1$ root of the characteristic equation
$\Lambda$ non-dimensional parameter in the standard solid model
$\mu$ Lamé coefficient
$\xi$ non-dimensional displacement
$\rho$ ratio of the long-term and instantaneous stiffnesses
$\sigma$ stress
$\tau$ non-dimensional time
$\tau_D$ typical diffusion time
$\tau_R$ relaxation time
$\Psi(\tau)$ dimensionless relaxation function
$\omega$ angular frequency of damped oscillations
$\omega_0$ angular frequency of undamped oscillations
------------------------------ -------------------------------------------------------------------
Introduction {#1dsSectionI}
============
Articular cartilage is a soft hydrated tissue covering the end of each bone at the joints. Cartilage has no known function other than maintaining mechanical competence of joints, allowing bones to move against one another without friction. But there is no need to underline its significance to health of a human body, since almost all the load transmitted by a human joint goes through the articular cartilage, and it prevents biomechanical damage caused by severe loading including impact loading. It is believed that severe articular impact can initiate post-traumatic arthritis [@JeffreyGregoryAspden1995; @QuinnAllen2001]. An impact loading of the joint constitutes the action of extremely high non-physiological loads applied very rapidly (for instance, due to a car accident, sports injury, or a fall from a height).
In recent years, a number of experimental studies have been conducted to investigate the mechanical behavior and damage mechanisms of articular cartilage under impact loading [@AtkinsonHautAltiero1998; @VerteramoSeedhom2007; @BurginAspden2008]. In particular, the experimental data on relative dissipation of the impact energy $\Delta E/E_0$ versus overall impactor energy $E_0$ obtained in [@Varga2007] were fitted with quadratic curves. Here, $E_0=mv_0^2/2$, $\Delta E=m(v_1^2-v_0^2)/2$, $v_0$ and $v_1$ are the initial impact and rebound velocities, respectively, $m$ is the impactor mass. Since, $v_1=-e_* v_0$, where $e_*$ is the coefficient of restitution, we easily get $\Delta E/E_0=1-e_*^2$. Thus, the experimental data and fitting curves for dissipation of the impact energy [@Varga2007] can be recalculated in terms of the coefficient of restitution as presented in Fig. \[Varga2007.pdf\], which shows a non-monotonic dependence of $e_*$ on $v_0$. Some experimentally observed results have been explained using a non-linear viscoelastic impact model [@Edelsten2010]. At the same time, there is the need of a simple mathematical model, which allows comparing experimental results obtained in drop impact testing with impact loads of different weights and incident velocities.
![Coefficient of restitution $e_*$ versus the impact velocity $v_0$ for articular cartilage samples of different thicknesses. Based on the experimental data and fitting curves obtained in [@Varga2007]. []{data-label="Varga2007.pdf"}](Varga2007.pdf)
A variety of mathematical models were suggested to describe the stress-strain response of articular cartilage that represents a multiphasic, structurally complex material possessing viscoelastic properties. It is long known that articular cartilage possesses viscoelastic properties [@HayesMockros1971; @Lau_et_al_2008], though there is no direct correspondence between viscoelastic parameters and parameters of the biphasic/poroelastic models of cartilage. The biphasic theory [@MowKueiLaiArmstrong1980], which models the tissue as a mixture of a solid phase and a fluid phase, has demonstrated very good agreement with experimental results in the creep and stress relaxation tests [@SoltzAteshian2000]. The objective of this study was to investigate theoretically whether the main features of articular impact observed in [@Varga2007; @Edelsten2010] could be qualitatively predicted using a linear viscoelastic theory or the linear biphasic theory.
The rest of the paper is organized as follows. In Sections \[1dsSection1\] and \[1dsSection2\], we consider in detail the viscoelastic Kelvin–Voigt and Maxwell impact models, respectively. Since some elements of the presented solutions are known in the literature, we pay a particular attention to the evaluation of the contact force, $F(t)$, and impactor displacement, $x(t)$, at the time moments $t_M$ and $t_m$, when the force and displacement reach their maxima, $F_M$ and $x_m$, respectively. In Section \[1dsSection3\], we outline a closed form solution of the impact equation in the case of standard solid model. In order to get analytical approximations, we consider the standard solid model as a perturbation of the Kelvin–Voigt (Section \[1dsSection4\]) or the Maxwell model (Section \[1dsSection5\]). In particular, simple analytical approximations are derived for the impact duration, $t_c$, and for the coefficient of restitution, $e_*$. In Sections \[1dsSection05\] and \[1dsSection06\], we consider the influence of the gravity effect on these parameters in the framework of the Kelvin–Voigt and Maxwell models for drop weight impact. In Section \[1dsSection07\], we develop an asymptotic model for the force-displacement relationship in the indentation problem for a thin biphasic layer corresponding to the conditions of the so-called blunt impact, when the specimen thickness is much smaller than the radius of a flat-ended cylindrical impactor. An example of application of the developed linear theory of viscoelastic impact for analyzing experimental data is given in Section \[1dsSection10\]. Finally, in Sections \[1dsSectionD\] and \[1dsSectionC\], we outline a discussion of the results obtained and formulate our conclusions.
Viscoelastic Kelvin–Voigt impact model {#1dsSection1}
======================================
In this section, the deformation of articular cartilage layer is modeled schematically as a parallel combination of linear spring $k$ and dashpot $b$ (Fig. \[Kelvin-Voigt\_Impact.pdf\]). Dynamic balance between the force of cartilage reaction $$F=kx+b\dot{x}
\label{1vI(1.0)}$$ and the force of body inertia $m\ddot{x}$ governs the development of collision. According to Newton’s second law, the differential equation of the impact has the form $$m\ddot{x}+b\dot{x}+kx=0, \quad t\in[0,t_c],
\label{1vI(1.1)}$$ where $t_c$ is the contact duration, that is $t_c$ denotes the instant, when the cartilage reaction force changes its sign, or the indenter acceleration vanishes.
![Impact viscoelastic Kelvin–Voigt model.[]{data-label="Kelvin-Voigt_Impact.pdf"}](Kelvin-Voigt_Impact.pdf)
The initial conditions for Eq. (\[1vI(1.1)\]) are as follows: $$x(0)=0, \quad \dot{x}(0)=v_0.
\label{1vI(1.2)}$$
The impact duration is determined by the condition $$kx+b\dot{x}\bigr\vert_{t=t_c}=0,
\label{1vI(1.3)}$$ or, in view of Eq. (\[1vI(1.1)\]), by the condition $$\ddot{x}\bigr\vert_{t=t_c}=0.
\label{1vI(1.4)}$$
The impact problem (\[1vI(1.1)\]), (\[1vI(1.2)\]) has the following well-known solution [@WinemanRajagopal2000]: $$x(t)=\frac{v_0}{\omega}e^{-\beta t}\sin\omega t, \quad t\in[0,t_c].
\label{1vI(1.5)}$$ Here we used the notation $$\omega_0^2=\frac{k}{m},\quad \omega^2=\omega_0^2-\beta^2, \quad \beta=\frac{b}{2m}.
\label{1vI(1.6)}$$ We assume that $\omega_0>\beta$.
Fig. \[Kelvin-Voigt\_exact.pdf\] shows the behavior of the dimensionless quantities $\omega_0 x/v_0$, $F/(mv_0\omega_0)$, and $\dot{x}/v_0$ with respect to time. Observe that the time moment $t_M$, when the contact force reaches its maximum, approaches the initial moment of impact as the damping ratio $\eta$ increases.
![Viscoelastic Kelvin–Voigt impact model. Behavior of the main impact variables with time for the following values of the damping ratio: $\eta=0{.}1$ (a), $\eta=0{.}3$ (b), $\eta=0{.}5$ (c).[]{data-label="Kelvin-Voigt_exact.pdf"}](Kelvin-Voigt_exact_01.pdf "fig:") ![Viscoelastic Kelvin–Voigt impact model. Behavior of the main impact variables with time for the following values of the damping ratio: $\eta=0{.}1$ (a), $\eta=0{.}3$ (b), $\eta=0{.}5$ (c).[]{data-label="Kelvin-Voigt_exact.pdf"}](Kelvin-Voigt_exact_03.pdf "fig:") ![Viscoelastic Kelvin–Voigt impact model. Behavior of the main impact variables with time for the following values of the damping ratio: $\eta=0{.}1$ (a), $\eta=0{.}3$ (b), $\eta=0{.}5$ (c).[]{data-label="Kelvin-Voigt_exact.pdf"}](Kelvin-Voigt_exact_05.pdf "fig:")
By differentiating the both sides of Eq. (\[1vI(1.5)\]) with respect to $t$, we get $$\dot{x}(t)=\frac{v_0}{\omega}e^{-\beta t}\bigl(
\omega\cos\omega t-\beta\sin\omega t\bigr),
\label{1vI(1.7a)}$$ $$\ddot{x}(t)=-\frac{v_0}{\omega}e^{-\beta t}\bigl[
(\omega^2-\beta^2)\sin\omega t+2\beta\omega\cos\omega t\bigr].
\label{1vI(1.7b)}$$
After solving Eq. (\[1vI(1.4)\]) for $t_c$ in view of (\[1vI(1.7b)\]), the following expression for the impact duration can be obtained [@ButcherSegalman2000]: $$t_c=\frac{1}{\omega}{\rm Atan\,}\frac{-2\beta\omega}{\omega^2-\beta^2},
\label{1vI(1.8)}$$ where the first positive value of the many-valued ${\rm Atan}$ function should be taken.
Using properties of the ${\rm Atan}$ function, we rewrite Eq. (\[1vI(1.8)\]) as follows [@Popov2010]: $$t_c=\frac{1}{\omega}
\left\{
\begin{array}{l}
\displaystyle
\pi-{\,\rm atan\,}\frac{2\beta\omega}{\omega^2-\beta^2},\quad \beta<\omega, \\
\displaystyle
{\,\rm atan\,}\frac{2\beta\omega}{\omega^2-\beta^2}, \quad \omega>\beta.
\end{array}
\right.
\label{1vI(1.9)}$$ Here, ${\,\rm atan\,}(z)$ is the principal branch of the arctangent function ${\,\rm Atan\,}(z)$.
Finally, using properties of the ${\,\rm atan\,}$ function, we can rewrite formula (\[1vI(1.9)\]) in a more simple form as $$t_c=\frac{2}{\omega}{\,\rm atan\,}\frac{\omega}{\beta}.
\label{1vI(1.9a)}$$
Let $\eta$ denote the loss factor, i.e., $$\eta=\frac{\beta}{\omega_0}.
\label{1vI(1.10a)}$$ Then, Eq. (\[1vI(1.9a)\]) can be rewritten as $$t_c=\frac{2}{\omega_0\sqrt{1-\eta^2}}{\,\rm atan\,}\frac{\sqrt{1-\eta^2}}{\eta}.
\label{1vI(1.10)}$$ Recall that we assume that $\eta\in[0,1]$. Also, note that in view of the notation (\[1vI(1.6)\]), we have $$\eta=\frac{b}{2\sqrt{km}}.
\label{1vI(1.10b)}$$
The velocity of the indenter at separation can be obtained by the substitution of (\[1vI(1.10)\]) into (\[1vI(1.7a)\]) in the following form: $$\begin{aligned}
\dot{x}(t_c) & = & -v_0\exp(-\beta t_c)
\label{1vI(1.11)} \\
{} & = & -v_0\exp\biggl\{- \frac{2\eta}{\sqrt{1-\eta^2}}{\,\rm atan\,}\frac{\sqrt{1-\eta^2}}{\eta}\biggr\}.
\label{1vI(1.12)}\end{aligned}$$
From Eq. (\[1vI(1.11)\]), it follows that the coefficient of restitution, $e_*$, which is defined as the ratio of the velocity at separation $\vert \dot{x}(t_c)\vert$ to the velocity of the indenter at incidence $\vert \dot{x}(0)\vert=v_0$, is given by $$\begin{aligned}
e_* & = & \exp\Bigl\{- \frac{2\beta}{\omega}{\,\rm atan\,}\frac{\omega}{\beta}\Bigr\}
\label{1vI(1.13)} \\
{} & = & \exp\biggl\{- \frac{2\eta}{\sqrt{1-\eta^2}}{\,\rm atan\,}\frac{\sqrt{1-\eta^2}}{\eta}\biggr\}.
\label{1vI(1.14)}\end{aligned}$$
The peak value of the indenter penetration occurs at the instant $t=t_m$, when $\dot{x}(t_m)=0$. In view of (\[1vI(1.7a)\]), we have $$\begin{aligned}
t_m & = & \frac{1}{\omega}{\,\rm atan\,}\frac{\omega}{\beta}
\label{1vI(1.15)} \\
{} & = & \frac{1}{\omega_0\sqrt{1-\eta^2}}{\,\rm arcsin\,}\sqrt{1-\eta^2}.
\label{1vI(1.16)}\end{aligned}$$ Substituting the expression (\[1vI(1.15)\]) into Eq. (\[1vI(1.5)\]), we obtain the maximum penetration $x_m=x(t_m)$ in the form $$\begin{aligned}
x_m & = & \frac{v_0}{\omega_0}\exp\Bigl(-\frac{\beta}{\omega}{\,\rm atan\,}\frac{\omega}{\beta}\Bigr)
\label{1vI(1.17)} \\
{} & = & \frac{v_0}{\omega_0}\exp\Bigl(-\frac{\eta}{\sqrt{1-\eta^2}}{\,\rm arcsin\,}\sqrt{1-\eta^2}
\Bigr).
\label{1vI(1.18)}\end{aligned}$$
Note that from (\[1vI(1.10)\]) and (\[1vI(1.16)\]), it is readily seen that $t_m=t_c/2$.
The peak value of the contact force $F$ occurs at the instant $t=t_M$, when $\dot{F}(t_M)=0$. According to Eqs. (\[1vI(1.5)\]), (\[1vI(1.7a)\]), we obtain $$\begin{aligned}
F(t) & = & \frac{mv_0}{\omega}\exp(-\beta t)\bigl[(\omega^2-\beta^2)\sin\omega t+2\beta\omega\cos\omega t\bigr]
\nonumber \\
{} & = & m v_0\omega_0 \exp(-\eta\omega_0 t)\biggl(\frac{(1-2\eta^2)}{\sqrt{1-\eta^2}}
\sin\omega_0\sqrt{1-\eta^2}t
+2\eta\cos\omega_0\sqrt{1-\eta^2}t
\biggr).
\label{1vI(1.19)}\end{aligned}$$
Differentiating the previous expression, we can reduce the equation $\dot{F}(t_M)=0$ to the following one: $$\sqrt{1-\eta^2}(1-4\eta^2)\cos(t_M\omega_0\sqrt{1-\eta^2})+
\eta(4\eta^2-3)\sin(t_M\omega_0\sqrt{1-\eta^2})=0.$$
Thus, for $\eta\in(0,0{.}5)$, we obtain $$\begin{aligned}
t_M & = & \frac{1}{\omega}{\,\rm atan\,}\frac{\omega(\omega^2-3\beta^2)}{\beta
(3\omega^2-\beta^2)}
\nonumber \\
{} & = & \frac{1}{\omega_0\sqrt{1-\eta^2}}{\,\rm atan\,}\frac{\sqrt{1-\eta^2}(1-4\eta^2)}{\eta(3-4\eta^2)}.
\label{1vI(1.20)}\end{aligned}$$ For $\eta\in(0{.}5,1)$, the maximum value of the contact force $F_M=F(t_M)$ takes place at the initial instant $t=0$.
Substituting (\[1vI(1.20)\]) into Eq. (\[1vI(1.19)\]), we get $$\begin{aligned}
F_M & = & m v_0\omega_0 \exp\biggl(-\frac{\eta}{\sqrt{1-\eta^2}}
{\,\rm atan\,}\frac{\sqrt{1-\eta^2}(1-4\eta^2)}{\eta(3-4\eta^2)}\biggr),\quad \eta\in(0,0{.}5), \\
\label{1vI(1.21)}
F_M & = & m v_0\omega_0 2\eta,\quad \eta\in(0{.}5,1).
\label{1vI(1.22)}\end{aligned}$$
Note that the function $F_M(\eta)$ defined by Eqs. (\[1vI(1.21)\]) and (\[1vI(1.22)\]) is continuously differentiable.
![Viscoelastic Kelvin–Voigt impact model. Behavior of the main impact parameters $t_m$, $t_c$, $t_M$ (a) and $x_m$, $F_M$ (b) with the damping ratio.[]{data-label="Kelvin-Voigt_tm+tM.pdf"}](Kelvin-Voigt_tm+tM.pdf "fig:") ![Viscoelastic Kelvin–Voigt impact model. Behavior of the main impact parameters $t_m$, $t_c$, $t_M$ (a) and $x_m$, $F_M$ (b) with the damping ratio.[]{data-label="Kelvin-Voigt_tm+tM.pdf"}](Kelvin-Voigt_xm+FM.pdf "fig:")
Fig. \[Kelvin-Voigt\_tm+tM.pdf\]a shows the monotonic behavior of the dimensionless characteristic time moments $t_m/\omega_0$, $t_c/(2\omega_0)$, $t_M/\omega_0$ with the damping ratio $\eta$. Recall that $t_m=t_c/2$. The variations of the relative maximum contact force $F_M/(mv_0\omega_0)$ and displacement $\omega_0 x_m/v_0$ are presented in Fig. \[Kelvin-Voigt\_tm+tM.pdf\]b. It is interesting to observe the non monotonic behavior of $F_M$ with the minimum at $\eta\approx 0{.}26493$.
Viscoelastic Maxwell impact model {#1dsSection2}
=================================
Assuming that the cartilage layer’s response to impact loading is modeled schematically as a serial combination of linear spring $k$ and dashpot $b$ (Fig. \[Maxwell\_Impact.pdf\]). The force-displacement relation is given by the following differential equation [@WinemanRajagopal2000]: $$\frac{\dot{F}}{k}+\frac{F}{b}=\dot{x}.
\label{1vI(2.1)}$$
![Impact viscoelastic Maxwell model.[]{data-label="Maxwell_Impact.pdf"}](Maxwell_Impact.pdf)
From (\[1vI(2.1)\]), it follows that $$F=k\int\limits_0^t\exp\Bigl\{-\frac{k}{b}(t-\tau)\Bigr\}\frac{dx}{d\tau}(\tau)\,d\tau.
\label{1vI(2.2)}$$
The differential equation of the impact $m\ddot{x}+F=0$ in view of (\[1vI(2.1)\]) results in the third-order equation $$\dddot{x}+\frac{k}{b}\ddot{x}+\frac{k}{m}\dot{x}=0
\label{1vI(2.3)}$$ with the initial conditions $$x(0)=0, \quad \dot{x}(0)=v_0,\quad \ddot{x}(0)=0.
\label{1vI(2.4)}$$
The impact problem (\[1vI(2.3)\]), (\[1vI(2.4)\]) has the following solution [@ButcherSegalman2000; @Stronge2000]: $$x(t)=\frac{v_0}{\omega_0}\exp(-\zeta\omega_0 t)\Bigl\{\frac{\omega_0(1-2\zeta^2)}{\omega}\sin\omega t
-2\zeta\cos\omega t\Bigr\}+\frac{2\zeta v_0}{\omega_0},
\label{1vI(2.5)}$$ $$\dot{x}(t)=v_0\exp(-\zeta\omega_0 t)\Bigl\{
\cos\omega t+\frac{\zeta\omega_0}{\omega}\sin\omega t\Bigr\}.
\label{1vI(2.5a)}$$ Here we used the notation $$\omega_0^2=\frac{k}{m},\quad \omega=\omega_0\sqrt{1-\zeta^2}, \quad \zeta=\frac{k}{2\omega_0 b}.
\label{1vI(2.6)}$$
The variation of the contact force during the impact interaction is $$F=\frac{kv_0}{\omega}\exp(-\zeta\omega_0 t)\sin\omega t.
\label{1vI(2.7)}$$
The impact duration $t_c$ is determined by the condition $F\bigr\vert_{t=t_c}=0$. Thus, according to (\[1vI(2.7)\]), the following relation takes place [@ButcherSegalman2000; @Stronge2000]: $$t_c=\frac{\pi}{\omega}.
\label{1vI(2.8)}$$
Substituting the value (\[1vI(2.8)\]) into Eq. (\[1vI(2.5a)\]), one gets the coefficient of restitution in the form $$e_*=\exp\Bigl(-\frac{\pi\zeta}{\sqrt{1-\zeta^2}}\Bigr).
\label{1vI(2.9)}$$
![Viscoelastic Maxwell impact model. Behavior of the main impact variables with time for the following values of the damping ratio: $\zeta=0{.}1$ (a), $\zeta=0{.}3$ (b), $\zeta=0{.}5$ (c).[]{data-label="Maxwell_exact.pdf"}](Maxwell_exact_01.pdf "fig:") ![Viscoelastic Maxwell impact model. Behavior of the main impact variables with time for the following values of the damping ratio: $\zeta=0{.}1$ (a), $\zeta=0{.}3$ (b), $\zeta=0{.}5$ (c).[]{data-label="Maxwell_exact.pdf"}](Maxwell_exact_03.pdf "fig:") ![Viscoelastic Maxwell impact model. Behavior of the main impact variables with time for the following values of the damping ratio: $\zeta=0{.}1$ (a), $\zeta=0{.}3$ (b), $\zeta=0{.}5$ (c).[]{data-label="Maxwell_exact.pdf"}](Maxwell_exact_05.pdf "fig:")
Fig. \[Maxwell\_exact.pdf\] shows the behavior of the dimensionless quantities $\omega_0 x/v_0$, $F/(mv_0\omega_0)$, and $\dot{x}/v_0$ with respect to time. Observe that the time moment $t_m$, when the indenter displacement reaches its maximum, approaches the final moment of impact as the damping ratio $\zeta$ increases.
According to Eq. (\[1vI(2.5a)\]), the peak value of the indenter penetration occurs at the instant $$t_m=\frac{\pi}{2\omega}\Bigl(1+\frac{2}{\pi}{\,\rm arcsin\,}\zeta\Bigr).
\label{1vI(2.10)}$$ The substitution of the value (\[1vI(2.10)\]) into Eqs. (\[1vI(2.5)\]) and (\[1vI(2.7)\]) gives the maximum penetration $$x_m=\frac{v_0}{\omega_0}\biggl(
2\zeta+\exp\biggl\{-\frac{\pi\zeta}{2\sqrt{1-\zeta^2}}\Bigl(1+\frac{2}{\pi}{\,\rm arcsin\,}\zeta\Bigr)
\biggr\}\biggr)
\label{1vI(2.11)}$$ and the corresponding force $$F_m=\frac{kv_0}{\omega_0}
\exp\biggl\{-\frac{\pi\zeta}{2\sqrt{1-\zeta^2}}\Bigl(1+\frac{2}{\pi}{\,\rm arcsin\,}\zeta\Bigr)
\biggr\}
\label{1vI(2.11a)}$$
From Eq. (\[1vI(2.7)\]), it follows that the peak value $F_M$ of the contact force occurs at the instant $$t_M=\frac{1}{\omega}{\,\rm atan\,}\frac{\sqrt{1-\zeta^2}}{\zeta}.
\label{1vI(2.12)}$$
Substituting (\[1vI(2.12)\]) into Eqs. (\[1vI(2.7)\]) and (\[1vI(2.5)\]), we obtain the maximum contact force $$F_M=\frac{kv_0}{\omega_0}
\exp\biggl\{-\frac{\zeta}{\sqrt{1-\zeta^2}}{\,\rm atan\,}\Bigl(
\frac{\sqrt{1-\zeta^2}}{\zeta}\Bigr)\biggr\}
\label{1vI(2.13)}$$ and the corresponding displacement $$x_M=\frac{v_0}{\omega_0}\biggl(2\zeta+
(1-4\zeta^2)
\exp\biggl\{-\frac{\zeta}{\sqrt{1-\zeta^2}}{\,\rm atan\,}
\frac{\sqrt{1-\zeta^2}}{\zeta}\biggr\}\biggr).
\label{1vI(2.14)}$$
![Viscoelastic Maxwell impact model. Behavior of the main impact parameters $t_m$, $t_c$, $t_M$ (a) and $x_m$, $F_M$ (b) with the damping ratio.[]{data-label="Maxwell_tm+tM.pdf"}](Maxwell_tm+tM.pdf "fig:") ![Viscoelastic Maxwell impact model. Behavior of the main impact parameters $t_m$, $t_c$, $t_M$ (a) and $x_m$, $F_M$ (b) with the damping ratio.[]{data-label="Maxwell_tm+tM.pdf"}](Maxwell_xm+FM.pdf "fig:")
Fig. \[Maxwell\_tm+tM.pdf\] shows the monotonic behavior of the dimensionless characteristic time moments $t_m/\omega_0$, $t_c/(2\omega_0)$, $t_M/\omega_0$ with the damping ratio $\zeta$. The variations of the relative maximum contact force $F_M/(mv_0\omega_0)$ and displacement $\omega_0 x_m/v_0$ are presented in Fig. \[Maxwell\_tm+tM.pdf\]b.
Finally, as it was observed [@ButcherSegalman2000], although certain quantities of the Maxwell impact model are equivalent to the so-called half-period Kelvin–Voigt impact model, the inherent physics of these models are completely different.
Standard solid model {#1dsSection3}
====================
There are two schematic representations of the standard linear solid model (Figs. \[SSM\_model-1.pdf\] and \[SSM\_model-2.pdf\]). The force-displacement relationship is given by the following two equations: $$(k_1+k_2)F+b\dot{F}=k_1 k_2 x+k_1 b\dot{x},
\label{1vI(3.1)}$$ $$\varkappa_1 F+\beta\dot{F}=\varkappa_1 \varkappa_2 x+\beta(\varkappa_1+\varkappa_2)\dot{x}.
\label{1vI(3.2)}$$
![Standard solid model. Configuration based on the Kelvin–Voigt model.[]{data-label="SSM_model-1.pdf"}](SSM_model-1.pdf)
The instantaneous and long-term moduli are $$k_0=k_1=\varkappa_1+\varkappa_2,\quad
k_\infty=\varkappa_1=\frac{k_1 k_2}{k_1+k_2}.
\label{1vI(3.3)}$$
![Standard solid model. Configuration based on the Maxwell model.[]{data-label="SSM_model-2.pdf"}](SSM_model-2.pdf)
The relaxation time is equal to $$\tau_R=\frac{b}{k_1+k_2}=\frac{\beta}{\varkappa_2}.
\label{1vI(3.4)}$$
The differential equations (\[1vI(3.1)\]) and (\[1vI(3.2)\]) are equivalent to the force-displacement relationship $$F=\int\limits_0^t k(t-\tau)\frac{dx}{d\tau}(\tau)\,d\tau
\label{1vI(3.5)}$$ with the relaxation stiffness $$k(t)=k_\infty+(k_0-k_\infty)\exp\Bigl(-\frac{t}{\tau_R}\Bigr).
\label{1vI(3.6)}$$
Let us also introduce the notation $$\rho=\frac{k_\infty}{k_0}.
\label{1vI(3.7)}$$ Note that $\rho\in(0,1)$ in view of (\[1vI(3.3)\]).
The differential equation of impact $$m\ddot{x}+F=0, \quad t\in[0,t_c],
\label{1vI(3.8)}$$ where the contact force $F$ is determined by Eq. (\[1vI(3.1)\]), can be written as $$\dddot{x}+\frac{(k_1+k_2)}{b}\ddot{x}+\frac{k_1}{m}\dot{x}+\frac{k_1 k_2}{mb}x=0.
\label{1vI(3.9)}$$
By introducing the non-dimensional time $$\tau=\frac{t}{\tau_R},
\label{1vI(3.10)}$$ Eq. (\[1vI(3.9)\]) can be reduced to the following equation: $$x^{\prime\prime\prime}+x^{\prime\prime}+\Lambda x^\prime+\Lambda\rho x=0.
\label{1vI(3.11)}$$ Here we introduced the notation $$\Lambda=\frac{k_0}{m}\tau_R^2.
\label{1vI(3.12)}$$
The initial conditions for Eq. (\[1vI(3.9)\]) are as follows: $$x(0)=0, \quad \dot{x}(0)=v_0,\quad \ddot{x}(0)=0.
\label{1vI(3.13)}$$
The solution to the problem (\[1vI(3.9)\]), (\[1vI(3.13)\]) is given by the following formula: $$\begin{aligned}
x(t) & = & \frac{\tau_R v_0}{\zeta_1[(\beta_1-\lambda_1)^2+\zeta_1^2]}
\biggl\{[(1-\beta_1)(\lambda_1-\beta_1)+\zeta_1^2]\sin\frac{\zeta_1 t}{\tau_R}
\nonumber \\
{} & { } & {}-\zeta_1(1-\lambda_1)\cos\frac{\zeta_1 t}{\tau_R}\biggr\}
\exp\Bigl(-\frac{\beta_1 t}{\tau_R}\Bigr)
%\nonumber \\
%{} & { } & {}
+\frac{(1-\lambda_1)\tau_R v_0}{(\beta_1-\lambda_1)^2+\zeta_1^2}
\exp\Bigl(-\frac{\lambda_1 t}{\tau_R}\Bigr).
\label{1vI(3.14)}\end{aligned}$$ Here, $-\lambda_1$ and $-(\beta_1\pm{\rm i}\zeta_1)$ are the roots of the characteristic equation corresponding to Eq. (\[1vI(3.11)\]). In other words, the following factorization takes place: $$z^3+z^2+\Lambda z+\Lambda\rho =(z+\lambda_1)
(z^2+2\beta_1 z +\beta_1^2+\zeta_1^2).
\label{1vI(3.15)}$$ The discriminant of the characteristic equation is $$D=4\Lambda(\Lambda^2+\rho)-\Lambda^2(1+18\rho-27\rho^2).
\label{1vI(3.16)}$$
We underline that formula (\[1vI(3.14)\]) holds true when $D>0$. In this case, we have $$\begin{aligned}
\lambda_1 & = & \frac{1}{3}+\frac{C_1}{3}+\frac{(1-3\Lambda)}{3C_1},
\nonumber \\
\beta_1 & = & \frac{1}{3}-\frac{C_1}{6}-\frac{(1-3\Lambda)}{6C_1},
\nonumber \\
\zeta_1 & = & \frac{\sqrt{3}C_1}{6}-\frac{\sqrt{3}(1-3\Lambda)}{6C_1},
\nonumber\end{aligned}$$ where $$C_1=\sqrt[3]{\frac{1}{2}(Q_1+2-9\Lambda+27\Lambda\rho)},\quad
Q_1=\sqrt{(2-9\Lambda+27\Lambda\rho)^2-4(1-3\Lambda)^3}.$$
Perturbation of the Kelvin–Voigt model {#1dsSection4}
======================================
Taking into account (\[1vI(3.3)\]) and (\[1vI(3.7)\]), we rewrite Eq. (\[1vI(3.1)\]) in the following form: $$k_\infty F+\rho(1-\rho)b\dot{F}=k_\infty^2 x+(1-\rho)k_\infty b\dot{x}.
\label{1vI(4.1)}$$
Now, letting $\rho\to 0$, we arrive at the equation $$F=k_\infty x+b\dot{x},
\label{1vI(4.2)}$$ which coincides with Eq. (\[1vI(1.0)\]). Thus, for small values of $\rho$, the standard solid model (\[1vI(4.1)\]) is a perturbation of the Kelvin–Voigt model (\[1vI(4.2)\]).
Let us introduce the notation $$\omega_0^2=\frac{k_\infty}{m},\quad
\beta=\frac{b}{2m},\quad
\eta=\frac{\beta}{\omega_0}.
\label{1vI(4.3)}$$
Then, the parameters (\[1vI(3.4)\]) and (\[1vI(3.12)\]) can be evaluated as $$\tau_R=2\eta\rho(1-\rho)\frac{1}{\omega_0},\quad
\Lambda=4\eta^2\rho(1-\rho)^2.
\label{1vI(4.4)}$$
In view of (\[1vI(4.4)\]), the discriminant (\[1vI(3.16)\]) and the roots of the characteristic equation (\[1vI(3.15)\]) can be asymptotically evaluated as follows: $$\begin{aligned}
D & = & 16\eta^2(1-\eta^2)\rho^2+O(\rho^3),\quad \rho\to 0,
\nonumber \\
\lambda_1 & = & 1-4\eta^2\rho+O(\rho^2),
\nonumber \\
\beta_1 & = & 2\eta^2\rho+O(\rho^2), \quad
\zeta_1 = 2\eta\sqrt{1-\eta^2}\rho+O(\rho^2).
\nonumber\end{aligned}$$
Consequently, we obtain the following asymptotic formulas for the impact duration, $t_c$, and the coefficient of restitution, $e_*$: $$\omega_0 t_c \simeq \frac{2}{\sqrt{1-\eta^2}}{\,\rm atan\,}\frac{\sqrt{1-\eta^2}}{\eta}+
\rho\biggl\{4\eta-\frac{8\eta^2}{\sqrt{1-\eta^2}}{\,\rm atan\,}\frac{\sqrt{1-\eta^2}}{\eta}
\biggr\},
\label{1vI(4.5)}$$ $$e_* \simeq \exp\biggl(-\frac{2\eta}{\sqrt{1-\eta^2}}{\,\rm atan\,}\frac{\sqrt{1-\eta^2}}{\eta}
\biggr)\biggl\{1+\frac{4\rho\eta}{\sqrt{1-\eta^2}}{\,\rm atan\,}\frac{\sqrt{1-\eta^2}}{\eta}
\biggr\}.
\label{1vI(4.6)}$$
![Perturbation of the Kelvin–Voigt model. Relative errors of the asymptotic formulas (\[1vI(4.5)\]) and (\[1vI(4.6)\]) for the duration of impact (a) and the coefficient of restitution (b).[]{data-label="SSM_Kelvin-Voigt.pdf"}](SSM_Kelvin-Voigt_e.pdf "fig:") ![Perturbation of the Kelvin–Voigt model. Relative errors of the asymptotic formulas (\[1vI(4.5)\]) and (\[1vI(4.6)\]) for the duration of impact (a) and the coefficient of restitution (b).[]{data-label="SSM_Kelvin-Voigt.pdf"}](SSM_Kelvin-Voigt_time.pdf "fig:")
The accuracy of the asymptotic approximations (\[1vI(4.5)\]) and (\[1vI(4.6)\]) is presented in Fig. \[SSM\_Kelvin-Voigt.pdf\]. Note that the asymptotic formulas (\[1vI(4.5)\]) and (\[1vI(4.6)\]) are not uniformly valid as $\eta\to 1$.
Perturbation of the Maxwell model {#1dsSection5}
=================================
Now, taking into account (\[1vI(3.3)\]) and (\[1vI(3.7)\]), we rewrite Eq. (\[1vI(3.1)\]) as follows: $$k_0 F+(1-\rho)b\dot{F}=\rho k_0^2 x+(1-\rho)k_0 b\dot{x}.
\label{1vI(5.1)}$$
Again, by letting $\rho\to 0$, we obtain the limit equation $$\frac{F}{b}+\frac{\dot{F}}{k_0}=\dot{x},
\label{1vI(5.2)}$$ which coincides with Eq. (\[1vI(2.1)\]). Thus, for small values of $\rho$, the standard solid model (\[1vI(5.1)\]) can be regarded as a perturbation of the Maxwell model (\[1vI(5.2)\]).
Let us introduce the notation $$\omega_0^2=\frac{k_0}{m},\quad
\zeta=\frac{k_0}{2\omega_0 b}.
\label{1vI(5.3)}$$
In view of (\[1vI(5.3)\]), the parameters (\[1vI(3.4)\]) and (\[1vI(3.12)\]) can be evaluated as $$\tau_R=\frac{1-\rho}{2\zeta\omega_0},\quad
\Lambda=\frac{(1-\rho)^2}{4\zeta^2}.
\label{1vI(5.4)}$$
Now, taking into account (\[1vI(5.4)\]), we expand the discriminant (\[1vI(3.16)\]) and the roots of the characteristic equation (\[1vI(3.15)\]) as follows: $$\begin{aligned}
D & = & \frac{(1-\zeta^2)}{16\zeta^6}-\frac{\rho(7\zeta^2-8\zeta^4+3)}{8\zeta^6}+O(\rho^2),\quad \rho\to 0,
\nonumber \\
\lambda_1 & = & \rho+O(\rho^2),\quad
\beta_1 = \frac{1}{2}-\frac{\rho}{2}+O(\rho^2),
\nonumber \\
\zeta_1 & = & \frac{\sqrt{1-\zeta^2}}{2\zeta}-
\rho\frac{\sqrt{1-\zeta^2}(2+\zeta^2)}{2\zeta(1-\zeta^2)}+O(\rho^2).
\nonumber\end{aligned}$$
Consequently, we obtain the following asymptotic approximations for the impact duration, $t_c$, and the coefficient of restitution, $e_*$: $$\omega_0 t_c \simeq \frac{\pi}{\sqrt{1-\zeta^2}}+
\frac{2\pi\rho\zeta^2}{(1-\zeta^2)^{3/2}},
\label{1vI(5.5)}$$ $$e_* \simeq \exp\Bigl(-\frac{\pi\zeta}{\sqrt{1-\zeta^2}}\Bigr)
+4\rho\zeta^2
\biggl\{1+\biggl(1-\frac{\pi\zeta}{2(1-\zeta^2)^{3/2}}\biggr)
\exp\Bigl(-\frac{\pi\zeta}{\sqrt{1-\zeta^2}}\Bigr)\biggr\}.
\label{1vI(5.6)}$$
![Perturbation of the Maxwell model. Relative errors of the asymptotic formulas (\[1vI(5.5)\]) and (\[1vI(5.6)\]) for the duration of impact (a) and the coefficient of restitution (b).[]{data-label="SSM_Maxwell.pdf"}](SSM_Maxwell_e.pdf "fig:") ![Perturbation of the Maxwell model. Relative errors of the asymptotic formulas (\[1vI(5.5)\]) and (\[1vI(5.6)\]) for the duration of impact (a) and the coefficient of restitution (b).[]{data-label="SSM_Maxwell.pdf"}](SSM_Maxwell_time.pdf "fig:")
The accuracy of the asymptotic approximations (\[1vI(5.5)\]) and (\[1vI(5.6)\]) is presented in Fig. \[SSM\_Maxwell.pdf\]. Note that the asymptotic formulas (\[1vI(5.5)\]) and (\[1vI(5.6)\]) are not uniformly valid as $\eta\to 1$.
Drop weight impact. Viscoelastic Kelvin–Voigt model {#1dsSection05}
===================================================
Due to Newton’s second law, the differential equation of the drop weight impact has the form $$m\ddot{x}+b\dot{x}+kx=mg, \quad t\in[0,t_c],
\label{1vI(05.1)}$$ where $g$ is the gravitational acceleration.
The initial conditions for Eq. (\[1vI(05.1)\]) are $$x(0)=0, \quad \dot{x}(0)=v_0.
\label{1vI(05.2)}$$
The drop weight impact problem (\[1vI(05.1)\]), (\[1vI(05.2)\]) has the following solution: $$x(t)=\frac{g}{\omega_0^2}\bigl(1-e^{-\beta t}\cos\omega t\bigr)+
\frac{1}{\omega}\Bigl(v_0-\frac{g\beta}{\omega_0^2}\Bigr)e^{-\beta t}\sin\omega t,
\label{1vI(05.3)}$$ $$\dot{x}(t)=v_0 e^{-\beta t}\cos\omega t+
\frac{(g-\beta v_0)}{\omega}e^{-\beta t}\sin\omega t.
\label{1vI(05.4)}$$ Here we used the notation (\[1vI(1.6)\]).
According to Eqs. (\[1vI(05.3)\]), (\[1vI(05.4)\]), the reaction force $F(x,\dot{x})=kx+b\dot{x}$ is given by $$\begin{aligned}
\frac{F}{mv_0\omega_0} & = & \varepsilon_0\biggl\{1+e^{-\beta t}\biggl(
\frac{\eta}{\sqrt{1-\eta^2}}\sin\omega t-\cos\omega t\biggr)\biggr\}
\nonumber \\
{} & {} & {}+e^{-\beta t}\biggl(
\frac{1-2\eta^2}{\sqrt{1-\eta^2}}\sin\omega t+\frac{2\eta}{\sqrt{1-\eta^2}}\cos\omega t\biggr),
\label{1vI(05.5)}\end{aligned}$$ where we introduced the notation $$\varepsilon_0=\frac{g}{\omega_0 v_0}.
\label{1vI(05.5e)}$$
The problem (\[1vI(05.1)\]), (\[1vI(05.2)\]) was studied in [@Ivanov1997], where the existence of the parameter domain of “plastic impact” was established. This means that for any $\eta>0$, there exists a unique value of $\varepsilon_0^*$ such that for all $\varepsilon_0>\varepsilon_0^*$ we have $F(x,\dot{x})>0$ in the time interval $t\in(0,+\infty)$. The critical value $\varepsilon_0^*$ of the parameter $\varepsilon_0$ determines the critical value $v_0^*$ of the initial velocity $v_0$ below which there is no rebound effect.
With the aim of application to the drop weight impact testing, we consider the problem (\[1vI(05.1)\]), (\[1vI(05.2)\]) for small values of the dimensionless parameter $\varepsilon_0$ and construct an asymptotic solution for the coefficient of restitution.
Let $t_c^0$ and $e_*^0$ be the impact duration and the coefficient of restitution for the Kelvin–Voigt impact model, correspondingly. According to Eqs. (\[1vI(1.9a)\]) and (\[1vI(1.14)\]), we have $$t_c^0=\frac{2}{\omega_0\sqrt{1-\eta^2}}{\,\rm atan\,}\frac{\sqrt{1-\eta^2}}{\eta},\quad
e_*^0=\exp\biggl\{- \frac{2\eta}{\sqrt{1-\eta^2}}{\,\rm atan\,}\frac{\sqrt{1-\eta^2}}{\eta}\biggr\}.
\label{1vI(05.6)}$$
Now, solving the transcendental equation $F(x,\dot{x})\bigr\vert_{t=t_c}=0$ by a perturbation method to terms of the first order inclusive, we obtain $$t_c\simeq t_c^0+\varepsilon_0\frac{(1+e_*^0)}{e_*^0\omega_0},
\label{1vI(05.7)}$$ $$e_*\simeq e_*^0(1-2\varepsilon_0\eta),
\label{1vI(05.8)}$$ where $t_c^0$ and $e_*^0$ are given by Eqs. (\[1vI(05.6)\]).
From the asymptotic formulas (\[1vI(05.7)\]) and (\[1vI(05.8)\]), it is clearly seen that the gravitational effect increases the duration of the impact process and decreases the coefficient of restitution. But it is more interesting to observe that the coefficient of restitution $e_*$ increases with velocity $v_0$, since the parameter $\varepsilon_0$ is inversely proportional to $v_0$ . That is why the effect of decrease in the coefficient of restitution in the drop weight impact test experimentally observed in [@Edelsten2010] for the velocity range $v_0\in(0{.}7,1{.}4)$ m/s and extrapolated for the low velocity region by means of the nonlinear Kelvin–Voigt model $F(x,\dot{x})=kx+c\vert x\vert\dot{x}$ with no account for the impactor weight cannot be explained by the linear viscoelastic Kelvin–Voigt model considered in this section.
Drop weight impact. Viscoelastic Maxwell model {#1dsSection06}
==============================================
By applying the approach [@Stronge2000], the differential equation of motion $m\ddot{x}+F=mg$ with the initial conditions $x(0)=0$ and $\dot{x}(0)=v_0$ in view of the constitutive relationship (\[1vI(2.1)\]) can be reduced to the following problem: $$\dddot{x}+\frac{k}{b}\ddot{x}+\frac{k}{m}\dot{x}-\frac{kg}{b}=0,
\label{1vI(06.1)}$$ $$x(0)=0, \quad \dot{x}(0)=v_0,\quad \ddot{x}(0)=g.
\label{1vI(06.2)}$$
The drop weight impact problem (\[1vI(06.1)\]), (\[1vI(06.2)\]) has the following exact solution: $$\begin{aligned}
x(t) & = & \frac{v_0}{\omega_0}e^{-\zeta\omega_0 t}\biggl\{
\frac{\omega_0}{\omega}(1-2\zeta^2-\varepsilon_0\zeta)\sin\omega t
-(2\zeta+\varepsilon_0)\cos\omega t\biggr\}
\nonumber \\
{} & {} & {}+\frac{v_0}{\omega_0}\bigl(2\zeta+\varepsilon_0(1-2\zeta\omega_0 t)\bigr),
\label{1vI(06.3)}\end{aligned}$$ $$\begin{aligned}
\dot{x}(t) & = & v_0 e^{-\zeta\omega_0 t}\biggl\{
\frac{\omega_0}{\omega}(\zeta+\varepsilon_0)\sin\omega t+\cos\omega t\biggr\}
\nonumber \\
{} & {} & {}+2\zeta\varepsilon_0 v_0.
\label{1vI(06.4)}\end{aligned}$$ Here we used the notation (\[1vI(2.6)\]), (\[1vI(05.5e)\]).
According to Eqs. (\[1vI(06.3)\]) and (\[1vI(06.4)\]), the reaction force $F=mg-\ddot{x}$ is given by $$\frac{F}{mv_0\omega_0}=\frac{\omega_0}{\omega} e^{-\zeta\omega_0 t}\biggl\{
(1+\varepsilon_0\zeta)\sin\omega t-\frac{\omega}{\omega_0}\varepsilon_0\cos\omega t\biggr\}
+\varepsilon_0
\label{1vI(06.5)}$$
Now, let $t_c^0$ and $e_*^0$ be the impact duration and the coefficient of restitution for the Maxwell impact model, correspondingly. According to Eqs. (\[1vI(2.8)\]) and (\[1vI(2.9)\]), we have $$t_c^0=\frac{\pi}{\omega_0\sqrt{1-\zeta^2}},\quad
e_*^0=\exp\biggl(- \frac{\pi\zeta}{\sqrt{1-\zeta^2}}\biggr).
\label{1vI(06.6)}$$
Applying a perturbation method, we find $$t_c\simeq t_c^0+\varepsilon_0\frac{(1+e_*^0)}{e_*^0\omega_0},
\label{1vI(06.7)}$$ $$e_*\simeq e_*^0-2\zeta\varepsilon_0,
\label{1vI(06.8)}$$ where $t_c^0$ and $e_*^0$ are given by Eqs. (\[1vI(06.6)\]).
It is interesting to note that the asymptotic formulas (\[1vI(05.7)\]) and (\[1vI(06.7)\]) coincide. Clearly, the same conclusions can be drawn about the influence of the gravitational effect in the framework of the viscoelastic Maxwell drop weight impact model as those that were formulated in Section \[1dsSection05\].
Short-time asymptotic solution of the indentation problem for a thin biphasic layer {#1dsSection07}
===================================================================================
We assume that the deformational behavior of articular cartilage is modeled in the framework of linear biphasic theory [@MowKueiLaiArmstrong1980], which represents the biological tissue as a mixture consisting of a porous solid phase and a fluid phase (mobile interstitial water). The constitution equations for the solid and fluid phase stresses, $\mbox{$\boldsymbol\sigma$}^s$ and $\mbox{$\boldsymbol\sigma$}^f$, are given by $$\mbox{$\boldsymbol\sigma$}^s=-\phi^s p{\bf I}+\lambda^s {\rm tr}(\mbox{$\boldsymbol\varepsilon$}){\bf I}
+2\mu^s \mbox{$\boldsymbol\varepsilon$},\quad
\mbox{$\boldsymbol\sigma$}^f=-\phi^f p{\bf I}.$$ Here, $\phi^f$ is the fluid volume fraction (porosity), $\phi^s=1-\phi^f$ is the solid volume fraction, $p$ is the true pressure of the fluid, $\lambda^s$ and $\mu^s$ are the Lamé constants, which together define the aggregate modulus $H_A=\lambda^s+2\mu^s$, $\mbox{$\boldsymbol\varepsilon$}$ is the strain tensor of the solid phase, and $\bf I$ is the identity tensor. Note that the fluid phase is assumed to be intrinsically incompressible and inviscid.
The continuity equation for the mixture and the momentum equations for each phase are given by $${\rm div}(\phi^s{\bf v}^s+\phi^f{\bf v}^f)=0,$$ $${\rm div}\mbox{$\boldsymbol\sigma$}^s+\frac{(\phi^f)^2}{\kappa}({\bf v}^f-{\bf v}^s)={\bf 0}, \quad
{\rm div}\mbox{$\boldsymbol\sigma$}^f-\frac{(\phi^f)^2}{\kappa}({\bf v}^f-{\bf v}^s)={\bf 0},$$ where ${\bf v}^s$ and ${\bf v}^f$ are the solid and fluid velocities, respectively, and $\kappa$ is the permeability of the solid phase.
Let us consider an axisymmetric contact problem for a thin biphasic layer indented without friction by a rigid impermeable cylindrical indenter. It is assumed that the contact radius $a$ is much larger than the cartilage layer thickness $h$ (i.e., $h/a\ll 1$). In this case, according to [@Ateshian1994], the vertical displacements $w(r,t)$ of the boundary points of the articular cartilage tissue at the contact zone can be approximated by the following asymptotic formula: $$w(r,t) = \frac{h^3}{3\mu_s}\biggl\{
\frac{1}{3r}\frac{\partial}{\partial r}\Bigl(r\frac{\partial P}{\partial r}(r,t)\Bigr)
+\frac{\mu_s \kappa}{h^2}\int\limits_0^t
\frac{1}{r}\frac{\partial}{\partial r}\Bigl(r\frac{\partial P}{\partial r}(r,\tau)\Bigr)d\tau\biggr\}.
\label{1vI(9.1)}$$ Here, $P(r,t)$ is the contact pressure. It is assumed that the cartilage layer is bonded to a rigid impermeable substrate, that is there is no solid displacement at the cartilage-bone interface and no fluid flow through the bone [@Ateshian1994].
In view of (\[1vI(9.1)\]), the contact condition that the boundary points of the cartilage layer acquire a constant vertical displacement $-\delta_0(t)$ (due to the action of the indenter) can be written as $$w(r,t) = -\delta_0(t), \quad r\leq a.
\label{1vI(9.2)}$$
The substitution of (\[1vI(9.1)\]) into Eq. (\[1vI(9.2)\]) results in an integro-differential equation $$\frac{1}{r}\frac{\partial}{\partial r}\Bigl(r\frac{\partial P}{\partial r}(r,t)\Bigr)
+\frac{3\mu_s \kappa}{h^2}\int\limits_0^t
\frac{1}{r}\frac{\partial}{\partial r}\Bigl(r\frac{\partial P}{\partial r}(r,\tau)\Bigr)d\tau
=-\frac{3\mu_s}{h^3}\delta_0(t),
\label{1vI(9.2a)}$$ which requires imposing a suitable boundary condition at the edge of the contact zone, i.e., at $r=a$ .
In order to impose the mentioned boundary condition, we note that at the initial moment of contact $t=0$, formula (\[1vI(9.1)\]) simplifies as follows: $$w(r,0) = \frac{h^3}{3\mu_s}\frac{1}{3r}\frac{\partial}{\partial r}\Bigl(r\frac{\partial P}{\partial r}(r,0)\Bigr).
\label{1vI(9.3)}$$ Comparing formula (\[1vI(9.3)\]) with the known asymptotic solutions for thin elastic layers [@Barber1990; @Chadwick2002; @ArgatovMishuris2011ve], we conclude that the instantaneous deformational response of a thin biphasic layer coincides with the response of a thin bonded incompressible elastic layer. Thus, by this analogy, we will require that $P(r,t)\to 0$ as $r\to a$, that is the contact pressure is assumed to vanish at the edge of the contact area.
As a result of integration of Eq. (\[1vI(9.2a)\]) with respect to the radial coordinate, we arrive at the following integral equation: $$P(r,t)+\frac{3\mu_s \kappa}{h^2}\int\limits_0^t P(r,\tau)\,d\tau
=\frac{3\mu_s}{4h^3}\delta_0(t)(a^2-r^2).
\label{1vI(9.4)}$$
Now, in order to derive the relationship between the indenter displacement and the contact force $$F(t)=2\pi\int\limits_0^a P(r,t)r\,dr,$$ we multiply both sides of Eq. (\[1vI(9.4)\]) by $2\pi r$ and after that we integrate the equation obtained with respect to $r$ from $0$ to $a$. As a results of this operation, we get $$F(t)+\frac{3\mu_s \kappa}{h^2}\int\limits_0^t F(\tau)\,d\tau
=\frac{3\mu_s a^4}{16h^3}\delta_0(t).
\label{1vI(9.5)}$$
Further, by inverting the Volterra integral operator on the right-hand side of Eq. (\[1vI(9.5)\]), we obtain $$F(t)=\frac{3\mu_s a^4}{16h^3}\biggl\{\delta_0(t)-
\chi\int\limits_0^t e^{-\chi(t-\tau)}\delta_0(\tau)\,d\tau\biggr\},
\label{1vI(9.6)}$$ where we introduced the shorthand notation $$\chi=\frac{3\mu_s \kappa}{h^2}.
\label{1vI(9.6b)}$$
Finally, after integrating by parts, Eq. (\[1vI(9.6)\]) yields $$F(t)=\frac{3\mu_s a^4}{16h^3}\biggl\{\delta_0(0)+
\int\limits_0^t e^{-\chi(t-\tau)}\frac{d\delta_0}{d\tau}(\tau)\,d\tau\biggr\}.
\label{1vI(9.7)}$$
In impact problems, under the assumption that $$\delta_0(0)=0,$$ the force-displacement relationship (\[1vI(9.7)\]) takes the form $$F(t)=\frac{3\mu_s a^4}{16h^3}\int\limits_0^t
\exp\Bigl\{-\frac{(t-\tau)}{\tau_R}\Bigr\}\frac{d\delta_0}{d\tau}(\tau)\,d\tau\biggr\}.
\label{1vI(9.8)}$$ Here we introduced the notation $\tau_R=1/\chi$. In view of (\[1vI(9.6b)\]), we have $$\tau_R=\frac{h^2}{3\mu_s \kappa},
\label{1vI(9.9)}$$ while comparing (\[1vI(9.9)\]) with (\[1vI(2.2)\]), we get the stiffness coefficient $$k=\frac{3\mu_s a^4}{16h^3}.
\label{1vI(9.90)}$$
It should be emphasized that Eq. (\[1vI(9.8)\]) represents a short-time asymptotic approximation, which is valid for moments of time $t$ such that $H_A\kappa t/h^2\ll 1$. For typical human cartilage material properties, $H_A=0{.}5$ MPa and $\kappa=2\times 10^{-15}$ ${\rm m}^4/{\rm Ns}$. Thus, assuming a typical cartilage thickness $h=1$ mm, we get $h^2/(H_A\kappa)=10^3$ s; thus, the asymptotic model (\[1vI(9.8)\]) certainly remains valid for up to 100 s, which is well in the range of usual values of impact durations.
Comparing Eq. (\[1vI(9.8)\]) with Eq. (\[1vI(2.2)\]), we see that the short-time deformational response of a thin biphasic layer bonded to a rigid impermeable substrate under the action of a frictionless flat-ended indenter is mathematically equivalent to that of a thin incompressible layer following the Maxwell viscoelastic model. Note that the Maxwell’s model based perturbation model considered in Section \[1dsSection5\] could be useful for modeling the impact response of articular cartilage (or artificial tissues for its replacement) in the whole time range, i.e. in the short-, medium- and long-time range.
We also emphasize that the biphasic model is not equivalent to a viscoelastic model, because the biomechanical response of a poroelastic material such as articular cartilage is crucially dependent on the boundary conditions for the sample. In particular, viscoelastic equivalents of the deformational response of an articular cartilage sample subjected to the same simple loading protocols in confined and unconfined conditions will be essentially different, especially in the short-time range. Thus, in comparing experimental results from different sources, a particular attention should be paid to the fixation conditions for tissue samples.
Key features of non-linear impact {#1dsSection10}
=================================
To illustrate the application of the developed linear theory of viscoelastic impact, let us analyze the experimental data obtained in [@BurginAspden2008] for drop-weight impact testing (with the impactor mass $m=100$ g) of isolated bovine articular cartilage samples of 5 mm diameter (correspondingly, the radius of the samples is $a=2{.}5$ mm). In [@BurginAspden2008], the force data, $F(t)$, were converted to engineering stress, $\sigma(t)$, by dividing them by the original cross-section area of the sample, $\pi a^2$, i.e., $$\sigma(t)=\frac{F(t)}{\pi a^2}.$$ The effective strain, $\epsilon(t)$, was evaluated by dividing the measured impactor displacement, $x(t)$, by the sample thickness, $h$, which is assumed to be $0{.}5\pm 0{.}11$ mm, as follows: $$\epsilon(t)=\frac{x(t)}{h}.$$ (Here, stress and strain are assumed to be positive in compression.) The stress-strain relationship was differentiated to obtain the incremental dynamic modulus $$E_{\rm dyn}=\frac{d\sigma}{d\epsilon}.$$ The maximum incremental dynamic modulus, $E_{\rm max}$, was found, and the modulus $E_{10}$ at stresses of 10 MPa was determined to enable comparison of dynamic moduli at constant value of stress. The initial impact velocity was calculated from the drop height, $h_0$, by the well-known formula $v_0=\sqrt{2gh_0}$.
The incremental dynamic modulus can be evaluated as a function of time in the form $$E_{\rm dyn}(t)=\frac{\dot{\sigma}(t)}{\dot{\epsilon}(t)}=\frac{h}{\pi a^2}\frac{\dot{F}(t)}{\dot{x}(t)}.
\label{1vI(E.1)}$$ In the case of the Maxwell model (see, Section \[1dsSection2\]), we will have $$\frac{\dot{F}(t)}{\dot{x}(t)}=k\frac{\cos\omega t-(\zeta\omega_0/\omega)\sin\omega t
}{\cos\omega t+(\zeta\omega_0/\omega)\sin\omega t},
\label{1vI(E.2)}$$ where $k$ is the stiffness coefficient.
First of all, observe that in view of (\[1vI(E.1)\]) and (\[1vI(E.2)\]), the variation of $E_{\rm dyn}(t)$ does not depend on the impact velocity $v_0$. In other words, the time variation of the incremental dynamic stiffness in the linear viscoelastic impact tests remains the same for different initial impact velocities. We emphasize that this conclusion is valid for a general linear viscoelastic law. Second, from (\[1vI(E.1)\]) and (\[1vI(E.2)\]), it follows that the value of $E_{\rm dyn}(t)$ gradually decreases to zero with increasing contact force $F(t)$ (when $\dot{F}(t)>0$). Thus, we arrive at the formula $$E_{\rm max}=E_{\rm dyn}(0).
\label{1vI(E.0)}$$
Further, in order to evaluate $E_{10}$, we need first solve the equation $$F(t_{10})=\pi a^2\sigma_{10},
\label{1vI(E.3)}$$ where $\sigma_{10}=10$ MPa. In view of (\[1vI(2.7)\]), Eq. (\[1vI(E.3)\]) takes the form $$\exp(-\zeta\omega_0 t_{10})\sin\omega t_{10}=\frac{\pi a^2\omega\sigma_{10}}{kv_0}.
\label{1vI(E.4)}$$ Here, $\zeta$, $\omega_0$, and $\omega$ are independent of $v_0$, and are determined by formulas (\[1vI(2.6)\]).
Now, from (\[1vI(E.4)\]), it is seen that the value of the time moment $t_{10}$ depends on the initial velocity $v_0$. Thus, the Maxwell impact model (and generally speaking, any linear viscoelastic model of impact) predicts that the value of $E_{10}$ increases with increasing impact velocity $v_0$.
$h_0$ (mm) $v_0$ (m/s) $E_{\rm max}$ (MPa) $E_{10}$ (MPa) $\sigma_{\rm max}$ (MPa) $\epsilon_{\max}$ $e_*$ $\Delta m$ (%)
------------ ------------- --------------------- ---------------- -------------------------- -------------------- -------------------- ----------------
25 $0{.}70$ $86\pm 22$ $75\pm 13$ $15{.}6\pm 2{.}9$ $0{.}48\pm 0{.}06$ $0{.}64\pm 0{.}08$ $2{.}2$
50 $0{.}99$ $100\pm 32$ $71\pm 16$ $24{.}5\pm 3{.}5$ $0{.}60\pm 0{.}13$ $0{.}46\pm 0{.}14$ $2{.}5$
80 $1{.}25$ $118\pm 33$ $73\pm 12$ $34{.}2\pm 5{.}0$ $0{.}62\pm 0{.}11$ $0{.}47\pm 0{.}05$ $5{.}7$
100 $1{.}40$ $128\pm 28$ $72\pm 13$ $40{.}5\pm 4{.}6$ $0{.}68\pm 0{.}09$ $0{.}41\pm 0{.}08$ $9{.}9$
: Impact parameters for isolated bovine articular cartilage samples [@BurginAspden2008][]{data-label="1vITable1"}
Table \[1vITable1\] shows that the impact testing [@BurginAspden2008] was performed in the non-linear regime with maximum compressive strains of 50–60%. That is why, the prediction of the linear impact model concerning $E_{\rm max}$ are not fulfilled. Furthermore, the linear theories of impact predict that the maximum contact force $F_M$ (correspondingly, the maximum contact stress $\sigma_{\rm max}=F_M/(\pi a^2)$) and the maximum displacement $x_m$ (correspondingly, the maximum strain $\epsilon_{\rm max}=x_m/h$) are proportional to $v_0$. On the other hand, the data from Table 1 show that the ratio $\sigma_{\rm max}/\epsilon_{\rm max}$ increases with increasing $v_0$. This fact also clearly indicates the non-linear deformational behavior of cartilage at high level of strain. Note here that the ratio $\sigma_{\rm max}/\epsilon_{\rm max}$ is ralted to the so-called pulsatile dynamic modulus (see, in particular, [@Argatov2012sine]).
Concerning the coefficient of restitution $e_*$ note that it is not constant, as it would be if the cartilage deformation were described by the Maxwell model (see formula (\[1vI(2.9)\])).
Finally, the last column of Table \[1vITable1\] gives the values of percentage increase in mass of each sample after 24 h immersed in PBS following impact loading. This is indicative of increasing amounts of damage in the cartilage samples [@BurginAspden2008].
Discussion {#1dsSectionD}
==========
Consider now the general case of linear viscoelastic force-displacement relationship $$F=\int\limits_0^t k(t-s)\frac{dx}{ds}(s)\,ds
\label{1vI(7.1)}$$ with the relaxation stiffness $$k(t)=k_0\Psi\Bigl(\frac{t}{\tau_R}\Bigr).$$ Here, $k_0$ is the initial stiffness, $\tau_R$ is the characteristic relaxation time, $\Psi(\tau)$ is the dimensionless relaxation function with $\tau$ being a dimensionless independent time-like variable.
Making use of the change of variables $$t=\tau_R\tau,\quad x=v_0\tau_R\xi,
\label{1vI(7.2)}$$ we transform the impact equation $m\ddot{x}+F=0$ and the initial conditions $x(0)=0$, $\dot{x}=v_0$ into the following problem: $$\xi^{\prime\prime}+\alpha\int\limits_0^\tau \Psi(\tau-\sigma)\frac{d\xi}{d\sigma}(\sigma)\,d\sigma=0,
\label{1vI(7.3)}$$ $$\xi(0)=0,\quad \xi^\prime(0)=1.
\label{1vI(7.4)}$$ Here prime denotes differentiation with respect to $\tau$, and we introduced the notation $$\alpha=\frac{k_0\tau_R^2}{m}.
\label{1vI(7.5)}$$ Note that for the Maxwell model (see Section \[1dsSection2\], Eq. (\[1vI(2.2)\])) we have $k_0=k$, $\tau_R=b/k$, and $\alpha=1/(4\zeta^2)$.
Furthermore, according to Eqs. (\[1vI(7.2)\])), the variable impact velocity is $$\dot{x}(t)=v_0\xi^\prime(\tau).$$
Let $\tau_c$ be the dimensionless duration of the impact process. Then, the coefficient of restitution can be found as $$e_*=-\xi^\prime(\tau_c).
\label{1vI(7.6)}$$
From Eqs. (\[1vI(7.3)\])) and (\[1vI(7.4)\])), it is evident that $\tau_c$ is a function of $\alpha$ only and does not depend on $v_0$. Thus, in view of (\[1vI(7.6)\])), we conclude that the coefficient of restitution $e_*$ is constant with respect to the initial impact velocity $v_0$.
It can be shown that the same qualitative conclusions are drawn from the linear biphasic model [@MowKueiLaiArmstrong1980] for articular cartilage deformation. In this case, the parameter $\tau_R$, which enters Eqs. (\[1vI(7.2)\])), can be defined as a typical diffusion time $\tau_D=h^2/(\kappa H_A)$, where $h$ is the cartilage layer thickness, $\kappa$ is the cartilage permeability, and $H_A$ is the aggregate modulus.
Observe that the biphasic theory incorporating Lamé parameters assumes that the material of solid phase is linearly elastic in order for these to have unique values. But if the material is viscoelastic these parameters are difficult to define and they become functions of deformation and/or time, if they are meaningful at all. Furthermore, there is an intrinsic circularity problem associated with using the aggregate modulus $H_A$, which is evaluated at equilibrium after the interstitial water is squeezed out, to define the mechanical properties that are then assumed to pertain during the impact deformation. Thus, the fact that the biphasic theory provides a good fit to measured curves in the creep and stress relaxation tests can be basically considered as a consequence of a curve-fitting procedure with a minimum of three free parameters rather than a derivation from first principles. In other words, it remains to be an open question on the efficiency of mixture models for articular cartilage at high strain rates.
In the present study we addressed the question of whether the main features of articular impact observed in [@Varga2007; @Edelsten2010] could be qualitatively predicted using a linear viscoelastic theory or the linear biphasic theory. It is to note that the deformations encountered in impact tests should be small enough for the linear theories to apply. With respect to engineering polymers note that the linear theory of viscoelasticity may hold reasonably well even up to some $5-10\%$ extension, in particular for certain rubbers [@Tschoegl1997].
Conclusions {#1dsSectionC}
===========
The results of this study based on the linear viscoelasticity imply the following properties of the linear impact models:
1\. The coefficient of restitution $e_*$ is a function of the damping ratio $\zeta$ alone. This means that $e_*$ does not depend on the impact velocity $v_0$, but it depends on the impactor mass $m$ and the sample thickness (through the stiffness $k$).
2\. The impact duration $t_c$ is inversely proportional to $\omega_0$, that is $t_c$ is proportional to $\sqrt{m}$, and depends on the damping ratio as well. The impact duration does not depend on the impact velocity $v_0$.
3\. The maximum displacement, $x_m$, and the maximum contact force, $F_M$, are proportional to $\sqrt{m}$ and $v_0$.
4\. The time variation of the incremental dynamic stiffness $dF/dx$ remains the same for different initial impact velocities.
5\. In the drop weight impact test, the gravitational effect increases the impact duration $t_c$ and decreases the coefficient of restitution $e_*$. At that, the coefficient of restitution increases with the impact velocity $v_0$.
Acknowledgment {#acknowledgment .unnumbered}
==============
The financial support from the European Union Seventh Framework Programme under contract number PIIF-GA-2009-253055 is gratefully acknowledged. The author also would like to express his gratitude to the Referees for their helpful comments and discussions.
[99]{}
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We first show that the pions produced at high $p_T$ in heavy-ion collisions over a wide range of high energies exhibit a scaling behavior when the distributions are plotted in terms of a scaling variable. We then use the recombination model to calculate the scaling quark distribution just before hadronization. From the quark distribution it is then possible to calculate the proton distribution at high $p_T$, also in the framework of the recombination model. The resultant $p/\pi$ ratio exceeds one in the intermediate $p_T$ region where data exist, but the scaling result for the proton distribution is not reliable unless $p_T$ is high enough to be insensitive to the scale-breaking mass effects.'
author:
- 'Rudolph C. Hwa$^1$ and C. B. Yang$^{1,2}$'
date: January 2003
title: 'Scaling Behavior at High $p_T$ and the $p/\pi$ Ratio'
---
Introduction
============
There are three separate and independent aspects about the hadrons produced at large transverse momentum $(p_T)$ in heavy-ion collisions at high energies that collectively contribute to a coherent picture to be addressed in this paper. One is the existence of a scaling behavior at large $p_T$ that we have found by presenting the data in terms of a new variable. Another is the issue about the surprisingly large proton-to-pion ratio at moderate $p_T$ ($\sim$ 2 - 3 GeV/c) discovered by PHENIX [@ts] in central $AuAu$ reactions at $\sqrt{s} =$ 130 and 200 GeV. The third issue concerns the hadronization process relevant for the formation of hadrons at large $p_T$ and the applicability of the recombination model [@dh]. It is our goal to show that, in light of the scaling behavior of the $\pi^0$ produced, the recombination mechanism naturally gives rise to a $p/\pi$ ratio that exceeds 1 in the $2 < p_T < 3$ GeV/c range.
Particle production in heavy-ion collisions at very high energies is usually described in terms of hydrodynamical flow [@hyd], jet production at high $p_T$ [@jet], thermal statistical model [@the], or a combination of various hadronization mechanisms [@all]. In none of the conventional approaches does one expect protons to be produced at nearly the same rate as the pions. If all hadrons with $p_T > 2$ GeV/c are regarded as products of jet fragmentation, then the known fragmentation functions of quark or gluon jets would suppress proton relative to pion by the sheer weight of the proton mass. Such a discrepancy from the observed data led some to regard the situation as an anomaly and proposed the gluonic baryon junction as a mechanism to enhance the proton production rate [@vg]. Their predictions remain to be checked by experiments.
The parton fragmentation functions have been used even at low $p_T$ in string models where the production of particles in hadronic collisions is treated as the fragmentation of diquarks, as done in the dual parton model [@dpm]. There has been a long-standing dichotomy on whether particle production in the fragmentation region can better be described by fragmentation [@dpm; @lpy] or recombination [@dh; @hy]. It is possible that the two pictures might be unified in a more comprehensive treatment of hadronization in the future. Here we extend the recombination model to the central region at large $p_T$. It should be recognized that an essential part of the recombination model is the determination of the distribution functions of the quarks and antiquarks that are to recombine. In the case of large-$p_T$ hadrons the underlying physics is undoubtedly hard collisions of partons and the associated radiation of gluons. If the parton distributions can be calculated just before hadronization, then the final step of recombination can readily be completed. If those distributions cannot be determined in pQCD, then the step between the initiating large-$p_T$ parton and the resultant hadrons may efficiently be described by a fragmentation function, determined phenomenologically from experiments. Thus in that sense the two approaches, recombination and fragmentation, are not contradictory, but complementary.
We state from the outset that no attempt will be made here to perform a first-principle calculation of the parton distributions at large $p_T$ before recombination. However, from the observed data on pion production in central $AuAu$ collisions at the Relativistic Heavy-Ion Collider (RHIC), it is possible to work backwards in the recombination model to determine the quark (and antiquark) distribution at large $p_T$. On the basis of the quark distributions inferred, it is then possible to calculate the proton distribution in the recombination model. The basic idea is that if there is a dense system of quarks and antiquarks produced in a heavy-ion collision whatever the dynamics responsible for them may have been (gluons having been converted to $q\bar{q}$ pairs before hadronization), then the formation of pions and protons (and whatever else) is prescribed by the recombination model without any arbitrariness in normalization and momentum dependence.
One limitation of the recombination model as it stands at present is that it is formulated in a frame-independent way in terms of momentum fractions and is therefore inapplicable to a system where the particle momenta are low and the mass effects are large. The physics of recombination is still valid at low momentum, but the details of the wave functions of the constituent quarks become important; they have not been built into the recombination function that takes the simplest form in the infinite momentum frame. Thus our calculation of particles produced at midrapidity is not reliable when $p_T$ is of the order of the masses of the hadrons under consideration. For protons we can trust the results only for $p_T > 3$ GeV/c. For pions the lower limit of validity can be pushed much lower.
Since our approach makes crucial use of the experimental data on the pion spectrum as the input, it is essential to relate the spectra determined at different energies to an invariant distribution so that the scale-invariant recombination model can be applied. To discover the existence of an invariant distribution with no theoretical prejudices is a problem worthy in its own right. Fortunately, that turns out to be possible. The analysis for that part of the study will be presented below first to emphasize its independence from the theoretical modeling of hadronization. It should be mentioned that the scaling of transverse mass spectra has been investigated recently [@sb]. The emphasis there has been on the dependences on the particle species and centrality for $m_T<3.8$ GeV, while our focus is on the dependence on energy ($17<\sqrt s<200
$ GeV) for $p_T<8$ GeV/c. Thus the two studies are complementary to each other.
A Universal Scaling Distribution
================================
The preliminary data of the $p_T$ distributions of $\pi^0$ produced at RHIC at $\sqrt{s} = 130$ and 200 GeV were shown by the PHENIX Collaboration at Quark Matter 2002 [@ddl] for central $AuAu$ collisions together with the WA98 data for $PbPb$ collisions at $\sqrt{s} = 17$ GeV [@rey]. They show that the level of the tail at large $p_T$ rises , as $\sqrt{s}$ is increased. We want to consider the possibility that the three sets of data points can be combined to form a universal curve.
The $\pi^0$ inclusive distributions at midrapidity are integrated over $\eta$ for a range of $\Delta \eta = 1$ so that the data points are given for the following quantity [@ddl]: $$\begin{aligned}
f(p_T, s) = {1 \over 2 \pi p_T}{dN \over dp_T}=
\int_{\Delta\eta} d\eta \left(2 \pi p_TN_{evt} \right)^{-1}
{d^2N_{\pi^0}
\over d p_T d\eta} .
\label{1}\end{aligned}$$ In comparing the PHENIX data with those of WA98 one should recognize that in addition to the difference in the colliding nuclei there is a slight mismatch in centrality (top 10% for PHENIX and top 12.7% for WA98) [@we].
To unify the three data sets it is natural to first consider a momentum fraction variable similar to $x_F$ in longitudinal momentum. However, so much momenta are taken by the other particles outside the $\Delta \eta = 1$ range, it is unwise to also use $\sqrt{s}/2$ as the scale to calculate the transverse momentum fraction. We assume that for every $\sqrt{s}$ there is a relevant scale $K$ to describe the $p_T$ behavior relative to that scale. Let us define $$\begin{aligned}
z = p_T/K,
\label{2}\end{aligned}$$ and transform $f(p_T, s)$ to a new function $\Phi (z,K)$, where $$\begin{aligned}
\Phi (z,K) = K^2 f(p_T, s) = {1 \over 2 \pi z} {dN \over dz} .
\label{3}\end{aligned}$$ We adjust $K$ for each $s$ and check whether all three data sets coalesce into one universal dependence on $z$, which we would simply label as $\Phi(z)$, if it is possible.
In Fig. 1 we show $\Phi(z)$, where the three symbols represent the three data sets for the three energies. Evidently, the universality exists and is striking. While this behavior needs to be confirmed by more data, and the theoretical implication remains to be explored, the existence of this scaling behavior is a significant phenomenological property of the $p_T$ distributions that suggests some underlying simplicity. It is like the KNO scaling of the multiplicity distributions $P(n, s)$ in $pp$ collisions, where for $\sqrt{s} < 200$ GeV they can be expressed by one universal scaling function $\psi (z)$, with $z =
n/\left<n\right>$ [@kno; @lat].
The values of $K$ that are used for the plot in Fig. 1 are in units of GeV: $K = 1 \, (200)$, $0.9 \, (130)$ and $0.717 \, (17)$, the quantities in the parentheses being the values of $\sqrt{s}$. The $\sqrt{s}$ dependence of $K$ forms nearly a straight line, as shown in Fig.2. Since the high and low energy data differ both in colliding nuclei and in centrality, one does not expect strict regularity in how $K$ depends on $\sqrt{s}$. Nevertheless, an approximate linear dependence is a simple behavior expected on dimensional grounds. The straight line in Fig. 2 corresponds to the best fit $$\begin{aligned}
K(s) = 0.69 + 1.55 \times 10^{-3} \sqrt{s} ,
\label{4}\end{aligned}$$ where $\sqrt{s} $ is in units of GeV. It should be recognized that the normalization of $K(s)$ is arbitrary; it is chosen to be $1$ at $\sqrt{s} = 200$ GeV for simplicity. If it is normalized to some other value at that point, the linear behavior in Fig. 2 is unchanged, only the scale of the vertical axis is shifted accordingly. The scaling property in Fig. 1 is also unchanged, the only modifications being the scales of the horizontal and vertical axes. Thus the absolute magnitude of the dimensionless variable $z$ has no significance.
If the $z$ dependence of $\Phi(z)$ in Fig. 1 were strictly linear, so that it is a power-law dependence $$\begin{aligned}
\Phi(z) \propto z^{\alpha} ,
\label{5}\end{aligned}$$ then there would be no relevant scale in the problem. The fact that it is not a straight line implies that there is an intrinsic scale in the $p_T$ problem, which is hardly surprising. What is significant is that while there is no strict scaling in $z$, there is no explicit dependence on $s$. That is, at any energy we have the same universal function $\Phi (z)$, which will be referred to as the scaling behavior in $s$. That function can be parametrized by $$\begin{aligned}
\Phi (z) = 1500 \left(z^2 + 2 \right)^{-4.9} ,
\label{6}\end{aligned}$$ which is represented by the smooth curve in Fig. 1. For large enough $z$ Eq. (\[6\]) does have the form of the power law given in Eq. (\[5\]) with $\alpha = 9.8$. It is a succinct statement of the universal properties at high $p_T$. The departure from Eq. (\[5\]) at small $z$ reflects the physics at low $p_T$. Since there is no data on $\pi^0$ for $p_T<1$ GeV/c, the extrapolation of $\Phi(z)$ to $z<1$ is not reliable. However, there is a more accurate determination of $\Phi(z)$ that includes the low $z$ region when the charge $\pi^+$ data are considered; it is given in [@hy4], and is not needed here.
![Scaled transverse momentum distribution of produced $\pi^0$. Data are from Ref. [@ddl; @rey]. The solid line is a fit of the data by Eq. (\[6\]).](fig1.eps){width="55.00000%"}
Note that there is no fixed scale in $p_T$ that separates the high- and low-$p_T$ physics. Equation (\[6\]) gives a smooth transition from one to the other in the variable $z$, thus implying different ranges of values of the transition $p_T$ at different $s$.
While Eq. (\[6\]) gives a good parametrization of the scaling function $\Phi (z)$ throughout the whole range of $z$, one notices, however, that the WA98 data at 17 GeV shows a slight departure from $\Phi (z)$ at the high $z$ end of that data set. It should be recognized that those data points have $p_T >3$ GeV/c, which represents a huge fraction of the available energy at $\sqrt{s} = 17$ GeV. In fact, one expects the violation of universality to be more severe at higher $z$ at that $\sqrt{s}$, since energy conservation would suppress the inclusive cross section at higher $p_T$. What is amazing is that most of the WA98 data points are well described by $\Phi (z)$, even though the corresponding $p_T$ values take up a much larger fraction of the available energy than the other data points from RHIC. It demonstrates the significance of the variable $z$ in revealing the scaling property.
{width="55.00000%"}
Pion and Quark Distributions in the Recombination Model
=======================================================
Having found a scaling distribution for the produced $\pi^0$ independent of $s$, we now consider the hadronization process in the recombination model in search for an origin of such a scaling behavior. In previous investigations the recombination model has been applied only to the fragmentation region where the longitudinal momenta are large and the transverse momenta are either held fixed at low $p_T$ or integrated over [@dh; @hy; @hy2]. We now consider the creation of pions in the central region of $AA$ collisions and study the $p_T$ dependence. Unlike the former case where the longitudinal momentum fractions of the partons are essentially known (from the structure functions), the $p_T$ distributions of the partons in the latter case are essentially unknown. Indeed, it is the aim of this section to determine the parton $p_T$ distributions from the $\pi^0$ distribution found in the previous section.
Let us start by writing down the basic equation for recombination in the 3-space $$\begin{aligned}
E{d^3N_{\pi}\over d^3p} = \int {d^3p_1 \over E_1}{d^3p_2
\over E_2}\ {\cal F}(\vec{p}_1, \vec{p}_2)\,{\cal
R}_{\pi}(\vec{p}_1,
\vec{p}_2, \vec{p})
\label{7}\end{aligned}$$ where the left-hand side (LHS) is the inclusive distribution of pion with energy-momentum $(E, p)$. ${\cal F}(\vec{p}_1,
\vec{p}_2)$ is the probability of having a quark at $p^{\mu}_1$ and an antiquark at $p^{\mu}_2$ just before hadronization. ${\cal R}_{\pi}(\vec{p}_1,
\vec{p}_2, \vec{p})$ is the invariant distribution, $E{d^3N_{\pi}^{q\bar{q}} /d^3p}$, of producing a pion at $p^{\mu}$ given a $q$ at $p^{\mu}_1$ and a $\bar{q}$ at $p^{\mu}_2$. Note that ${\cal R}_{\pi}$ has the dimension (momentum)$^{-2}$, same as the LHS.
Writing the phase-space density in the form $$\begin{aligned}
{d^3p\over E} = dy \, d\phi \, p_T \, dp_T ,
\label{8}\end{aligned}$$ we define the inclusive distribution in $p_T$, averaged over $y$ and $\phi$, $$\begin{aligned}
{d^3N_{\pi} \over p_T \,dp_T} = {1 \over \Delta y}
\int_{\Delta y } dy \ {1 \over 2 \pi} \int^{2\pi}_0 \, d\phi \
E{d^3N_{\pi}\over d^3p} ,
\label{9}\end{aligned}$$ where $\Delta y$ is limited to one unit of rapidity in the central region. Our focus will be on the $p_T$ distribution at high $p_T$. For the recombination distribution ${\cal
R}_{\pi}(\vec{p}_1, \vec{p}_2, \vec{p})$ we need only consider the partons in the same transverse plane that contains $\vec{p}$, since at high $p_T$ the partons with different $y_i$ are not likely to recombine. Indeed, we assume not only $y_1 = y_2 = y$, but also $\phi _1 = \phi _2 = \phi$ so that the partons and the pion are all colinear, and the kinematics can be reduced to that of a 1-dimensional problem. As in the usual parton model, the parton momentum fractions in the hadron can vary between 0 and 1, but the deviation in the momentum components of the partons transverse to the hadron $\vec{p}$ must be severely limited because of the limited transverse size of the hadron. Thus we write $$\begin{aligned}
{\cal R}_{\pi}(\vec{p}_1, \vec{p}_2,\vec{p}) = {\cal
R}_{\pi} ^0 \ \delta \left(y_1 - y_2\right) \delta \left(\phi _1 -
\phi _2\right)\nonumber\\ \delta \left({y_1 + y_2 \over 2} -y
\right) \delta^2 (\vec{p}_{1_T} + \vec{p}_{2_T} -
\vec{p}_T ) ,
\label{10}\end{aligned}$$ where ${\cal R}_{\pi}^0 $ is dimensionless, since $\delta^2 (\vec{p}_{1_T} + \vec{p}_{2_T} -
\vec{p}_T)$ has the dimension of ${\cal
R}_{\pi} (\vec{p}_1, \vec{p}_2, \vec{p})$. If this $\delta$-function is further written in the colinear form due to the $\delta(\phi _1 - \phi _2)$ in Eq. (\[10\]) $$\begin{aligned}
\delta^2 (\vec{p}_{1_T} + \vec{p}_{2_T} -
\vec{p}_T) &=& \delta \left({\phi _1 +
\phi _2 \over 2} - \phi\right) {1 \over p_T} \,
\nonumber\\ &&\delta\left(p_{1_T} + p_{2_T} -
p_T\right) ,
\label{11}\end{aligned}$$ then Eq. (\[7\]) can be reduced to the 1D form $$\begin{aligned}
{dN_{\pi}\over p_T dp_T} &=& \int
dp_{1_T} dp_{2_T}p_{1_T}p_{2_T}\ {\cal F}(p_{1_T},
p_{2_T})\,{\cal R}_{\pi} ^0 \, p_T^{\ -2}\nonumber\\ &&\delta\left( {
p_{1_T} + p_{2_T} \over p_T} -1 \right),
\label{12}\end{aligned}$$ where ${\cal F}(p_{1_T}, p_{2_T})$ is the $q\bar{q}$ distribution in $p_{i_T}$ averaged over $y$ and $\phi$.
We can reexpress this equation in terms of the scaling variable $z
= p_T/K$, introduced in Eq. (\[2\]), and obtain $$\begin{aligned}
{dN_{\pi}\over zdz} = \int dz_1 dz_2\,z_1\,z_2\ F(z_1, z_2)\ R_{\pi}
(z_1, z_2, z)
\label{13}\end{aligned}$$ where $$\begin{aligned}
F(z_1, z_2) = K^4\ {\cal F}( p_{1_T} , p_{2_T})
\label{14}\end{aligned}$$ $$\begin{aligned}
R_{\pi} (z_1, z_2, z) = {\cal R}_{\pi}^0 \ z^{-2}\ \delta\left( { z_1
+ z_2 \over z}-1
\right) .
\label{15}\end{aligned}$$ Since ${\cal F}( p_{1_T} , p_{2_T} )$ is the parton density in $p_{1_T} dp_{1_T} p_{2_T} dp_{2_T}$, $F(z_1,
z_2)$ is the corresponding dimensionless density in $z_1
dz_1 z_2 dz_2$. Equation (\[13\]) is now our basic formula for recombination in the scaled transverse-momentum variable. The total number of $q$ and $\bar{q}$ is $\int dz_1 dz_2\,z_1z_2
F(z_1, z_2)$, which should be invariant under a change of scale $$\begin{aligned}
z = \lambda x
\label{16}\end{aligned}$$ so that $$\begin{aligned}
x = p_T/K^{\prime}, \qquad \qquad K^{\prime} = \lambda K .
\label{17}\end{aligned}$$ The corresponding change on $F(z_1, z_2)$ is that it becomes $$\begin{aligned}
F^{\prime} (x_1, x_2 ) = \lambda^4 F(z_1, z_2).
\label{18}\end{aligned}$$ Thus the normalization of $F(z_1, z_2)$ is scale dependent, as it should in view of Eq. (\[14\]).
So far the recombination function $R_{\pi}(z_1, z_2, z)$ is not fully specified because ${\cal R}_{\pi}^0$ has not been. In Eq.(\[15\]) the factor $z^{-2}$ is associated with the dimension of the pion density, and the $\delta$-function with momentum conservation. To introduce the pion wave function in terms of the constituent quarks, we rewrite Eq. (\[15\]) as $$\begin{aligned}
R_{\pi}(z_1, z_2, z )=R_{\pi}^0\ z^{-2}\ G_{\pi}(\xi_1,\xi _2),
\label{19}\end{aligned}$$ where $R_{\pi}^0$ is a normalization constant to be determined and $G_{\pi}(\xi _1,\xi _2)$ is the valon distribution of the pion [@dh; @hy]. Since the recombination of a $q$ and $\bar{q}$ into a pion is the time-reversed process of displaying the pion structure, the dependence of $R_{\pi}(z_1, z_2, z)$ on the pion structure is expected. During hadronization the initiating $q$ and $\bar{q}$ dress themselves and become the valons of the produced hadron without significant change in their momenta. The variable $\xi _i$ in Eq. (\[19\]) denotes the momentum fraction of the $i$th valon, i.e., $$\begin{aligned}
\xi _i = z_i/z ,
\label{20}\end{aligned}$$ which is denoted by $y_i$ in the valon model [@dh; @hy], a notation that cannot be repeated here on account of the rapidity variables already used in Eq. (\[10\]). In general, the valon distribution of a hadron $h$ has a part specifying the wave-function squared, $\tilde{G}_h$, and a part specifying momentum conservation $$\begin{aligned}
G_h(\xi _1,\cdots ) = \tilde{G}_h(\xi _1,\cdots)\ \delta\left(\sum_i
\xi _i -1 \right) ,
\label{21}\end{aligned}$$ where the functional form of $\tilde{G}_h$ is determined phenomenologically. Although for proton $\tilde{G}_p$ is found to be highly nontrivial [@hy3], for pion $\tilde{G}_{\pi}$ turns out to be very simple [@hy] $$\begin{aligned}
\tilde{G}_{\pi} (\xi _1, \xi _2) = 1 ,
\label{22}\end{aligned}$$ which is a reflection of the fact that the pion mass is much lower than the constituent quark masses, so tight binding results in large uncertainty in the momentum fractions of the valons. Equation (\[22\]) implies that the valon momenta of the pion is uniformly distributed in the range $0 < \xi _i < 1$.
What remains in Eq. (\[19\]) for us to determine is $R_{\pi}^0$. At this point we need to be more specific about the quark and antiquark that recombine. If the colors of $q$ and $\bar{q}$ are considered, then the probability of forming a color singlet pion is $1/9$ in $3 \times \bar{3}$. Similarly, for three quarks forming a proton the probability is $1/27$ in $3
\times 3
\times 3$. In the parton distributions, $F_{q\bar{q}}$ for pion production involves two color triplets and $F_{qqq}$ for proton production involves three color triplets so the color factors work out just right in that the factors of $9$ for $q\bar{q}$ and $27$ for $qqq$ are cancelled by the corresponding inverse factors in the recombination probabilities. In other words, for the $p/\pi$ ratio to be considered later, we can ignore the factors associated with the color degrees of freedom and proceed with the determination of $F_{q\bar{q}}$ without specifying the quark colors and summing over them.
The situation with flavor is not the same. For a $u \bar{u}$ pair and a $d\bar{d}$ pair, they can form $\pi^0$ and $\eta$ in the flavor octet. The branching ratio of $\eta$ to $3 \pi^0$ is 32.5% and to $\pi ^+ \pi ^- \pi ^0$ is 22.6%. Thus for every $\eta$ produced there is on average $1.2 \pi ^0$. Due to the higher mass of $\eta$ we make the approximation that the rate of indirect production of $\pi ^0$ via $\eta$ is roughly the same as the direct production from $u \bar{u}$ and $d \bar{d}$. If we now use $q\bar{q}$ to denote either $u \bar{u}$ or $d
\bar{d}$, but not both $u \bar{u}$ and $d \bar{d}$, then each pair of $q\bar{q}$ leads to one $\pi ^0$. Since in a heavy-ion collision there are many quarks and antiquarks produced in the central region, it is reasonable to assume that the $q$ distribution is independent of the $\bar{q}$ distribution so that we can write $F_{q\bar{q}}$ in the factorizable form $$\begin{aligned}
F_{q\bar{q}} \left(z _1, z _2\right) = F_q (z _1)\
F_{\bar{q}} ( z _2) ,
\label{23}\end{aligned}$$ where $F_q$ stands for either $u$ or $d$ distributions, and similarly for $F_{\bar{q}} $, but for $\pi ^0$ production $\bar{q}$ should be the antiquark partner of $q$.
The fact that we consider $\eta$ production above, but not the vector meson $\rho$ requires an explanation. We defer that discussion until the next section, after we have presented the formalism for the production of protons.
Returning now to the normalization of $R_{\pi} \left(z _1, z
_2, z\right)$, we note that, using Eqs. (\[19\]), (\[21\]) and (\[22\]), $$\begin{aligned}
\int dzzR_{\pi} (z _1, z_2, z) &=& \int {dz\over z} R_{\pi}^0
\delta \left({z _1 + z_2\over z}-1\right)\nonumber\\&& = R_{\pi}^0
\label{24}\end{aligned}$$ is the probability that a $q$ at $z_1$ and a $\bar{q}$ at $z_2$ recombine to form a pion at any $z$. According to our counting in the second paragraph above, the total probability for $q\bar{q}
\rightarrow \pi ^0$ integrated over all momenta is $$\begin{aligned}
\int^Z_0 {d z _1 \over Z} \int^Z_0 {d z _2 \over Z} \int dz\ z\,
R_{\pi}(z _1, z_2, z) = 1 ,
\label{25}\end{aligned}$$ where $Z$ is the maximum $z_i$, whatever it is. This normalization condition is scale invariant, and we find, using Eq. (\[24\]), that $$\begin{aligned}
R_{\pi}^0 = 1 .
\label{26}\end{aligned}$$
Putting Eqs. (\[19\]) - (\[23\]) and (\[25\]) in (\[15\]) we obtain $$\begin{aligned}
{dN_\pi \over zdz} = \int dz_1 dz_2{z_1z_2\over z}F_q(z_1)\,
F_{\bar{q}}(z_2)\,\delta(z_1 + z_2 - z).
\label{27}\end{aligned}$$ This is obtained from Eq. (\[9\]) where $y$ and $\phi$ are both explicitly averaged over. The LHS is to be identified with $\Phi(z)$. Note that the $1/2\pi$ factors in Eqs. (\[1\]) and (\[3\]), where $\Phi(z)$ is defined, are there to render $f(p_T,
s)$ an average distribution in $\phi$; that is the notation for the experimental distribution, defined in [@ddl]. The distribution defined by us in Eq. (\[9\]) already includes the $1/2\pi$ factor, so our $dN_\pi/z\,dz$ is just the experimental $\Phi(z)$. As we have mentioned earlier, the normalization of $z$ has no significance. By means of a scale change in Eq. (\[16\]) we can move from $z$ to $x$, or vice-versa, without changing the scale invariant form of Eq.(\[27\]). In Eq. (\[6\]) we found $\Phi (z)$ to have the form $$\begin{aligned}
\Phi (z) = A \left(z^2 + c \right)^{-n} .
\label{28}\end{aligned}$$ If we change $z$ to $x$ according to Eq. (\[16\]), then by keeping the total number of pions invariant, i.e., $$\begin{aligned}
\int dz\, z \,\Phi (z, K) = \int dx\, x \,\Phi^{\prime} (x,
K^{\prime})
,
\label{29}\end{aligned}$$ we have $$\begin{aligned}
\Phi^{\prime} (x, \lambda K) = \lambda^2\Phi (\lambda x, K).
\label{30}\end{aligned}$$ It thus follows that $$\begin{aligned}
\Phi^{\prime} (x) = \lambda^{2(1-n)} A \left(x^2 +
c/\lambda^2\right)^{-n}.
\label{31}\end{aligned}$$ Similarly, in the $x$ variable the transformed quark distributions is $$\begin{aligned}
F^{\prime}_q (x_1, K^{\prime}) = \lambda^2 F_q (z_1, K).
\label{32}\end{aligned}$$ Without having to specify the arbitrary scale factor $\lambda$, let us work with the $z$ variable and rewrite Eq. (\[27\]) as $$\begin{aligned}
\Phi(z) = \int^z_0 dz_1\ z_1 \left(1 - { z_1 \over z} \right)\ F_q
(z_1)\ F_{\bar{q}}(z - z_1) .
\label{33}\end{aligned}$$
We must now consider how the $q$ and $\bar{q}$ distributions differ. Unlike the structure functions of the nucleon, where $q$ and $\bar{q}$ have widely different distributions, we are here dealing with the partons at high $p_T$ in heavy-ion collisions just before recombination. The dynamics underlying their $p_T$ dependences is complicated. Many subprocesses are involved, which include hard scattering, gluon radiation, jet quenching, gluon conversion to quark pairs, thermalization, hydrodynamical expansion, to name a few familiar ones. At very large $p_T$ there are far more quark jets than antiquark jets, since the valence quarks have larger longitudinal momentum fractions than the sea quarks. By hard scattering the quarks therefore can acquire larger $p_T$ than the antiquarks. Thus in that way one would expect the $p_T$ distribution of the quarks to be very different from that of the antiquarks. However, that view does not apply to our problem. Those are the $q$ and $\bar{q}$ that initiate jets, along with jets initiated by gluons. The conventional approach is to follow the jet production by jet fragmentation, which can be modified by the dense matter that the initiating partons traverse. As discussed earlier, our approach is not to delve into the dynamical origins of the $q$ and $\bar{q}$ distributions, but to consider the recombination of $q$ and $\bar{q}$ just at the point of hadronization. Such $q$ and $\bar{q}$ are not the partons that initiate jets, but are the parton remnants after the hard partons radiate gluons which subsequently convert to $q\bar{q}$ pairs. Those parton remnants have similar momentum distribution for $q$ and $\bar{q}$, since gluon conversion creates $q$ and $\bar{q}$ on equal basis; those partons are the ones that recombine to form hadrons. They are not to be confused with the jet-initiating hard partons that fragment into hadrons in the fragmentation model. In the recombination picture those hard partons that acquire large $p_T$ immediately after hard scattering are not ready for recombination; they lose momenta and virtuality through gluon radiation until a large body of low-virtuality quarks and antiquarks are assembled for recombination — a view that is complementary to the fragmentation picture. Of course, there are more quarks than antiquarks, since the number of valence quarks of the participating nucleons cannot diminish. For that reason we allow $F_q(z)$ to differ in normalization from $F_{\bar{q}}(z)$. However, as a first approximation we assume that their $z$ dependences are the same.
There is some indirect experimental evidence in support of our assumption. In Ref. [@ts] the $\bar{p}/p$ ratio for central collisions is reported to be essentially constant within errors; more precisely, it ranges between $0.6$ and $0.8$ for $p_T$ in the range $0.5 < p_T < 3.8$ GeV/c. Since $\bar{p}$ is formed by the recombination of three $\bar{q}$, while $p$ is formed from three $q$, a quick estimate of the $\bar{q}/q$ ratio is that it varies between $0.6^{1/3}$ and $0.8^{1/3}$, i.e., from $0.843$ to $0.928$. Such a narrow range of variation is sufficient for us to assume that $F_{\bar{q}}(z)$ has the same $z$ dependence as $F_q(z)$. For their relative normalization we take the mean $\bar{p}/p$ ratio to be $0.7$. Thus we adopt the $\bar{q}/q$ ratio to be $$\begin{aligned}
F_{\bar{q}}(z)/F_q(z) = F^{\prime}_{\bar{q}}(x)/F^{\prime}_q(x)
=0.7^{1/3} .
\label{34}\end{aligned}$$ With this input we are finally ready to infer the quark distribution from the pion distribution.
{width="55.00000%"}
We parameterize $F_q(z)$ by $$\begin{aligned}
F_q(z) = a \left(z^2 + z + z_0 \right)^{-m}
\label{35}\end{aligned}$$ and adjust the three parameters $a$, $z_0$ and $m$ to fit $\Phi (z)$ by using Eq. (\[33\]). We obtain an excellent fit with the values $$\begin{aligned}
a = 90 , \qquad z_0 = 1, \qquad m= 4.65 .
\label{36}\end{aligned}$$ In Fig. 3 we show in solid line the data represented by the formula in Eq. (\[6\]) and in dashed line the result of the theoretical calculation using Eqs. (\[33\])-(\[36\]). They coalesce nearly completely in the interval $1 < z < 8$. The quark distribution $F_q(z)$ is shown in Fig. 4. To appreciate the $p_T$ range corresponding to $z$ in Fig. 4, recall Eq. (\[2\]), $p_T = zK$, and Fig. 2 for $K$. Thus at $\sqrt{s} = 200$ GeV, $p_T$ is $z$ in GeV. Equations (\[35\]) and (\[36\]) represent a main result of this study. What is important is that we have found a scaling quark distribution that is independent of $s$ from SPS to RHIC, and perhaps to LHC. It is a succinct summary of the effects of all the dynamical subprocesses in heavy-ion collisions. The non-trivial $z$ dependence in Eq. (\[35\]) indicates that there are intrinsic scales in the low-$p_T$ problem.
{width="55.00000%"}
The $p/\pi$ Ratio
=================
The quark distribution obtained in the preceding section cannot be checked directly. Since it is the distribution at the end of its evolution, massive dileptons would not be sensitive to it due to their production at the early stages. Proton production provides the most appropriate test, since hadronization occurs near the end. We shall therefore calculate the proton distribution at high $p_T$ and compare with the data on the $p/\pi$ ratio. This is not a completely satisfactory venture, since the proton mass is large, so only at very high $p_T$ can our scale invariant calculation be valid without explicit consideration of the mass effect. Present data on the $p/\pi$ ratio do not extend beyond $p_T \sim 3.8$ GeV/c [@ts]. Nevertheless, our calculation should provide some sense on the magnitude of the rate of proton production at the high $p_T$ end.
The inclusive distributions in the scaled $p_T$ variable can be obtained in the recombination model by generalizing Eq.(\[13\]) to the recombination of three quarks $$\begin{aligned}
{dN_p \over zdz} &=& \int dz_1 dz_2 dz_3\ z_1\,z_2\,z_3\nonumber\\
&& F(z_1, z_2, z_3)\ R_p(z_1, z_2, z_3, z)
\label{37}\end{aligned}$$ where $F(z_1, z_2, z_3)$ is given the factorizable form $$\begin{aligned}
F(z_1, z_2, z_3) = F_u (z_1) F_u (z_2) F_d (z_3).
\label{38}\end{aligned}$$ As in Eq. (\[19\]) we relate the recombination function $R_p$ to the valon distribution, $G_p$, of the proton $$\begin{aligned}
R_p(z_1, z_2, z_3, z) = R_p^0 \ z^{-2}\,G_p(\xi_1,\xi_2, \xi_3) ,
\label{39}\end{aligned}$$ where $G_p$ has the general form given in Eq. (\[21\]), and $R_p^0$ remains to be determined. In Ref. [@hy3] a detailed study of the proton structure functions has been carried out in deriving the valon distribution from the parton distributions that fit the deep inelastic scattering data. It is $$\begin{aligned}
\tilde{G}_p (\xi_1, \xi_2, \xi_3) = g \ (\xi_1\,
\xi_2 ) ^\alpha\ \xi_3^{\beta} ,
\label{40}\end{aligned}$$ where $$\begin{aligned}
\alpha = 1.755, \qquad \beta = 1.05 ,
\label{41}\end{aligned}$$ $$\begin{aligned}
g = \left[B \left(\alpha + 1, \beta +1 \right) B \left(
\alpha + 1, \alpha + \beta +2\right)\right]^{-1} .
\label{42}\end{aligned}$$ Single-valon distributions $G_p(\xi_i)$ can be obtained from the three-valon distribution by integration and are peaked around $\xi = 1/3$, indicating that each of the three valons carries on average roughly $1/3$ the momentum of the proton, their sum being strictly 1. Details of the valon model, described in [@hy3], are not needed for the following. It is only necessary to recognize that the recombination of two $u$ quarks with a $d$ quark to form a proton has a probability proportional to the proton’s valon distribution that accounts for the proton structure. The other point to bear in mind is that the valon distribution in the proton is obtained in the frame where the proton momentum is infinitely large so the finite masses of the proton and valons are unimportant. However, the validity of that result when the proton momentum is only two or three times larger than its mass is questionable. With that caveat we proceed with our scale invariant calculation and see what can emerge.
As discussed in the preceding section, there is no need to consider the color factors for either pion or proton formation since hadrons are color singlets, but the flavor octets for these hadrons do introduce some factors. The $\left.|uud\right>$ state appears in $10 + 8 + 8 ^{\prime}$ of $3 \times 3 \times 3$; among them the first two contain $\Delta ^+$ and $p$. Thus the flavor parts of $|\left<\Delta ^+\left|uud\right>|^2\right.$ and $|\left<p \left| uud\right>|^2\right.$ are $1/3$ for each. Since $\Delta ^+$ decays to $p + \pi^0$ and $n + \pi^+$, $|\left<p\left| \Delta ^+\right>|^2\right.$ gives another factor $1/2$. The spin decomposition of $2 \times 2 \times 2$ is $4
+ 2
+ 2$, among which the $\Delta ^+$ component is $4/8$ and $p$ is $2/8$. Putting the flavor and spin factors together, we have $$\begin{aligned}
&&\left| \left<p \left| uud \right. \right>\right|^2 +
\left| \left<p\, | \Delta ^+\right>\left< \Delta
^+ \left| uud \right. \right>\right|^2 \nonumber\\
&&\, \, \, \, \,= {1 \over 3}\times {1
\over 4} + {1 \over 3}\times {1 \over 2} \times {1 \over 2} =
{1 \over 6} .
\label{43}\end{aligned}$$ We thus normalize $R_p$, as we have done in Eq. (\[25\]), by $$\begin{aligned}
\int^Z_0 \prod^3_{i = 1} {dz_i \over Z} \int dz\, z \,R_p
(z_1, z_2, z_3, z) = {1 \over 6} .
\label{44}\end{aligned}$$ In view of Eqs. (\[21\]) and (\[39\]) we have $$\begin{aligned}
R_p^0\int^Z_0 \prod^3_{i = 1} {dz_i \over Z}\ \tilde{G}_p
\left({z_1 \over z_t}, {z_2\over z_t}, {z_3\over z_t}\right) = {1
\over 6},
\label{45}\end{aligned}$$ where $z_t = \sum_i z_i$. Using Eq. (\[40\]), the above integral can be transformed to $$\begin{aligned}
g \int^1_0 \prod^3_{i = 1} d \zeta _i \left({ \zeta_1 \zeta_2
\over \zeta ^2_t} \right)^{\alpha} \left({ \zeta_3
\over \zeta _t} \right)^{\beta} = 2.924
\label{46}\end{aligned}$$ with $\zeta _i = z_i/Z$ and $\zeta _t = \sum _i \zeta _i$. There is no explicit dependence on $Z$, and Eqs. (\[41\]) and (\[42\]) have been used in getting the numerical value in Eq.(\[46\]). It thus follows that $$\begin{aligned}
R_p^0 = 0.057 .
\label{47}\end{aligned}$$
At this point we should address the question why we consider $\Delta^+$ production above, but not $\rho$ production in the preceding section. For the production of $\pi^0$, if we are to consider the contribution from $\rho^\pm$ (since $\rho^0$ does not decay strongly into $\pi^0$), we would be extending our scope to other flavored states besides $u\bar u$ and $d\bar d$. Then other vector mesons and higher resonances, such as $K^*$, that can decay into $\pi^0$ must also be included. Similarly, for $p$ production the consideration of other states beside $uud$ would involve many resonances that can decay into $p$. The system is not closed without more phenomenological input beside $\pi^0$. Thus for a closed system in which a prediction can be made, we limit ourselves to only the $u\bar u$ and $d\bar d$ in the meson states and $uud$ in the baryon states; hence, only $\pi^0, \eta, p$ and $\Delta^+$ are considered. To include $u\bar d$ and $d\bar u$, we must also include $uuu$ and $udd$, and so on. We surmise that if more resonances are included in both the meson and baryon sectors, the $p/\pi$ ratio to be determined below would change somewhat; however, the result is not likely to differ by a factor greater than 2.
With the recombination function $R_p$ completely determined, and the quark distribution $F_q \left(z_i\right)$ given by Eqs.(\[35\]) and (\[36\]), we can now use Eq. (\[37\]) to calculate the proton distribution in $z$. The result is shown by the solid line in Fig. 5, where only the portion $z > 2$ is exhibited. We have stated at the outset that the scale invariant form of $dN_p/zdz$ cannot be expected to be valid when the mass effect is important. The relevant value of $z$ corresponding to the proton mass (let alone the $\Delta ^+$ mass) is $$\begin{aligned}
z_m = m_p/K ,
\label{48}\end{aligned}$$ which ranges from 1.3 at $\sqrt{s} = 17$ GeV to 0.94 at 200 GeV. As expected, the scaling violating effects are energy dependent. Thus we should not regard the calculated result to be reliable for $z < 3$. At very low $p_T$ the distributions of all hadrons can be given exponential fits in the transverse mass. The STAR data for most central collisions at $\sqrt{s}= 130$ GeV [@pj] give for $\bar{p}$ production for $p_T < 0.6$ GeV $$\begin{aligned}
{1 \over 2 \pi m_T}{d^2N_{\bar{p}} \over dm_T dy} = 4 \exp
\left[-\left(m_T-m_p\right)/T_p\right]
\label{49}\end{aligned}$$ where $m_T = \left(m^2_p + p^2_T\right)^{1/2}$ and $T_p =
565$ MeV. To convert this distribution to that for $p$ we assume that only the normalization at $p_T = 0$ needs to be adjusted. The $\bar{p}/p$ ratio at low $p_T$ is 0.6 [@ts]. Since $m_Tdm_T = p_T dp_T$ and the distribution in $p_T$ changes by a scale factor $K^2$ given in Eq. (\[2\]), where $K = 0.9$ for $\sqrt{s} = 130$ GeV, the factor 4 in Eq. (\[49\]) should therefore be changed to $4 \times 0.81/0.6 = 5.4$. Expressing $m_T$ in terms of $z$ by use of Eq. (\[2\]) with $K = 0.9$, we show the $z$ dependence of the distribution for $p$ in Fig. 5 by the short dashed line. The region $0.5 < z <2$ is left blank because our scaling result cannot be reliably extended into that region. Nevertheless, it is gratifying to observe that the theoretical calculation without any free parameters produces a proton distribution at large $z$ that is reasonable in normalization and shape and can smoothly be connected with the low-$p_T$ distribution by interpolation.
![Proton distribution in $z$. Solid line is the theoretical result; the dashed line is the fit of data at low-$p_T$ [@pj].](fig5.eps){width="55.00000%"}
With the proton distribution now at hand, we can calculate the $p/\pi$ ratio. For the pion distribution we use $\Phi (z)$ given in Eq. (\[6\]). For proton we use the calculated result based on Eq. (\[37\]). Their ratio, defined by $$\begin{aligned}
R_{p/\pi}(z) = {dN_p \over zdz} / \Phi (z)
\label{50}\end{aligned}$$ is shown by the solid line in Fig. 6. The preliminary data on the $p/\pi$ ratio were reported in Ref. [@ts], which we show also in Fig. 6 for both $\sqrt{s} = 130$ and $200$ GeV. Note that because it is a ratio there is no change in the normalizations of $R_{p/\pi}$ for the two energies, but in transforming from $p_T$ to $z$ the factor $K$ in Eq. (\[2\]) must be taken into account. Unlike the pion case the effects of the proton mass are not negligible for $p_T
\stackrel{<}{\sim} 3$ GeV/c, and one sees no scaling in $s$ or $z$ in Fig. 6. Our scale invariant calculation is unreliable for $z < 3$ and shows a result that is obviously too high at $z
\stackrel{<}{\sim} 2$. There seems to be a good chance that the theory and experiment can agree well for $z > 4$. In Fig. 6 we show two curves that can connect our scaling result with the data. The dotted curve is an eyeball fit of the 130 GeV data with a connection at $z= 3.5$, while the dashed curve fits the 200 GeV data with a connection at $z = 4$. In the absence of a theoretical study that takes the mass-dependent effects into account in the intermediate $p_T$ region, the only point we can make here is that it is not hard to produce a $p/\pi$ ratio that exceeds 1 in the scale invariant calculation in the recombination model, but it does so in a region where both theory and experiment need refinement. Judging by what is self-evident in Fig. 6, we see no strong need for any exotic mechanism for proton production (as proposed in [@vg]) beyond the conventional subprocess where three quarks recombine to form a proton.
![Proton-to-pion ratio: solid line is the scaling distribution from calculation; data (preliminary) are from Ref.[@ts]. The dotted and dashed lines are eyeball fits of the data as extrapolations from the scaling result.](fig6.eps){width="55.00000%"}
Conclusion
==========
The discovery of a scale invariant distribution $\Phi (z)$ for pion production at intermediate and high values of $p_T$ in heavy-ion collisions ranging over energies in excess of an order of magnitude of variation is an important phenomenology observation that should be checked experimentally in great detail. Additional energy points should be added not only to strengthen the validity of the scaling behavior, but also to find the onset of scaling violation, if it exists.
The phenomenological properties of hadron production provide useful insights into the hadronization process and into the nature of the quark system just before they turn into hadrons. The usual approach to the study of heavy-ion collisions is from inside out, following the evolution of the dense matter, either in terms of hydrodynamical flow or of hard parton scattering and subsequent hadronization by fragmentation [@gvw]. Our approach pursued here is from outside in, by starting from the observed scaling behavior of the pions produced and deriving the momentum distribution of the quarks that can give rise to such a behavior. That is accomplished by use of the recombination model. There is no direct way to check the validity of the quark distribution thus obtained. However, we have used it to determine the proton distribution at high $p_T$ where the mass effects are unimportant. The data on proton production have not yet reached that regime where the predicted scaling distribution can be checked. In the region where data exist on the $p/\pi$ ratio we find that our calculated result, though not reliable, is in rough agreement with the imprecise data to the extent that the ratio exceeds 1, a feature that is notable.
While the recombination model needs further work to take the proton mass into account at intermediate $p_T$, its formulation in the invariant form has been developed here to treat the very high $p_T$ region. We have made the assumption that the quark and antiquark distributions are the same, apart from normalization, just before recombination. That assumption is supported by the constancy of the $\bar{p}/p$ ratio in the PHENIX data in the central region. That experimental fact can also be used to lend credence to our general approach to hadronization that is treated as a recombination process, for which we have given arguments why the distributions of quarks and antiquarks should be similar before they recombine. In contrast, the fragmentation model would suggest a decreasing function of $\bar{p}/p$ in $p_T$ because of the dominance of quark jets over antiquark jets at large $p_T$ [@jet].
In this paper we have only considered the energy dependence of the $p_T$ spectrum at fixed maximum centrality. It is natural to ask what the dependence is on centrality. We have investigated that problem by making a phenomenological analysis of the data on centrality dependence without using any hadronization model, and found a scaling behavior very similar to what is reported here. The scaling distribution found there [@hy4] includes the very small $p_T$ region in the fit, and is therefore more accurate. But the fits in the intermediate and large $p_T$ region are the same. The implication on the centrality dependence of the $p/\pi$ ratio in the recombination model is still under study.
To have an invariant quark distribution independent of $s$ just before hadronization provides an unexpected picture of the quark system. It suggests that the evolution of the system proceeds toward a universal form whatever the collision energy may be. We expect that universal form to depend on rapidity. The origin of such a scaling distribution in $z$ is not known at this point and can form the focus of a program of future theoretical investigations.
Acknowledgment {#acknowledgment .unnumbered}
==============
We wish to thank D. d’Enterria for a helpful communication. This work was supported, in part, by the U. S. Department of Energy under Grant No. DE-FG03-96ER40972.
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M. Gyulassy, I. Vitev, and X.-N. Wang, Phys.Rev. Lett. [**86**]{}, 2537 (2001).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We study the ground-state spin correlations in the gapless incommensurate regime of a $S=1/2$ $XXZ$ chain and a two-leg antiferromagnetic ladder under a magnetic field, in which the gapless excitations form a Tomonaga-Luttinger (TL) liquid. We calculate numerically the two-spin correlation functions and the local magnetization in the two models using the density-matrix renormalization-group method. By fitting the numerical results for an open $XXZ$ chain of 100 spins to correlation functions of a Gaussian model, we determine the TL-liquid parameter $K$ and the amplitudes of the correlation functions. The value of $K$ estimated from the fits is in excellent agreement with the exact value obtained from the Bethe ansatz. We apply the same method to the open ladder consisting of 200 spins and determine the dependence of $K$ on the magnetization $M$. The $K$-$M$ relation changes drastically depending on the ratio of the coupling constants in the leg and rung directions. We also discuss implications of these results to experiments on the nuclear spin relaxation rate $1/T_1$ and dynamical spin structure factors.'
address:
- |
Department of Earth and Space Science, Graduate School of Science, Osaka University,\
Toyonaka, Osaka 560-0043, Japan
- 'Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan'
author:
- 'T. Hikihara'
- 'A. Furusaki'
date: 'November 20, 2000'
title: 'Spin correlations in the two-leg antiferromagnetic ladder in a magnetic field'
---
[2]{}
INTRODUCTION
============
Spin ladder systems have been studied extensively over the past decade.[@review] There are reasons why the ladders have attracted so much attention. Firstly, they naturally interpolate one- and two-dimensional systems and may provide some hints to better understand the high-temperature superconductivity which occurs in square lattice ${\rm CuO_2}$ planes. Secondly, spin ladders themselves have interesting physics and deserve through investigation in their own right. One of their most surprising properties is that low-energy physics of spin ladders depends drastically on the number of legs. Spin-$1/2$ antiferromagnetic (AF) ladders, for example, have a finite gap in the spin excitation spectrum in the even-leg case, whereas they have no gap in the odd-leg case. The ground state of an even-leg ladder is a spin singlet and its properties can be understood from the short-range resonating-valence-bond picture.[@RVB] This spin-gap behavior has been observed experimentally on $S=1/2$ two-leg ladder compounds,[@exp1; @exp2] such as ${\rm SrCu_2O_3}$ and ${\rm Cu_2(C_5H_{12}N_2)_2Cl_4}$.
A gapless phase can appear in even-leg ladders when an external field $h$ is applied. If the field $h$ is larger than a critical field $h_{c1}$, which is equal to the spin gap, and if it is smaller than the saturation field $h_{c2}$, then the ground state has a nonzero magnetization $M$ and the energy gap between the ground state and the first-excited states vanishes. The gapless mode has been shown to be described as a Tomonaga-Luttinger (TL) liquid both in the strong- and weak-coupling limit,[@Chi-Gia; @Gia-Tsv; @Furu-Zhn] where the coupling in the rung-direction $J_\perp$ is much larger or much smaller than the one in the leg-direction $J_\parallel$, respectively. However, the $M$ dependence of the TL-liquid parameter $K$, which governs the spin correlations in long wave length, has been obtained analytically only in the strong-coupling limit and it remains as a nontrivial problem to determine $K$ for general $J_\perp/J_\parallel$. In the gapless phase the system shows incommensurate spin correlations since the Fermi wavenumber of Jordan-Wigner fermions is shifted from $\pi/2$ in the presence of the magnetic field which acts as a chemical potential for the fermions. The wavenumber $Q$ characterizing the incommensurability of the gapless mode varies continuously as $h$ increases. This gapless incommensurate (IC) phase is in fact in the same universality class as the one-dimensional $S = 1/2$ $XXZ$ model in a magnetic field, as we will see.
In this paper, we study low-energy properties of the $S = 1/2$ two-leg AF ladder in the gapless IC regime for broad range of $J_\perp / J_\parallel$. We show that the system is a TL liquid for arbitrary $J_\perp / J_\parallel$ and determine the $M$ dependence of $K$ numerically. To this end, we compute numerically the ground-state spin-correlation functions and the local magnetization in the open ladders using the density-matrix renormalization-group (DMRG) method [@White1; @White2] and extract the TL-liquid parameter by fitting the data to correlation functions obtained from the Abelian bosonization. This method was applied in our previous work [@CorAm] to the $S=1/2$ $XXZ$ chain at $h = 0$ and proved to be effective in determining both the TL-liquid parameter and amplitudes of correlation functions. In order to demonstrate the validity of the analysis in the gapless IC phase, we first apply it to the $S = 1/2$ $XXZ$ chain for $h > 0$. The model is exactly solvable by the Bethe ansatz and the TL-liquid parameter $K$ can be calculated for arbitrary value of $M$. It thus provides a good test ground to check accuracy of our method. We find that $K$ estimated from the DMRG data is in excellent agreement with the exact calculation. We then apply the same method to the two-leg ladders in a magnetic field. Our numerical data of correlation functions are fitted well for broad range of $J_\perp/J_\parallel$ to the formulas based on the bosonization approach, confirming that the gapless modes are in fact in the universality class of a TL liquid for arbitrary $J_\perp/J_\parallel$. The $M$ dependence of $K$ obtained in the large $J_\perp/J_\parallel$ limit agrees with the analytic result obtained through mapping to the $XXZ$ chain. In this limit $K$ is less than 1 for $0 < M < 1$. As $J_\perp/J_\parallel$ decreases, $K$ increases and become larger than 1 for intermediate values of $M$. Our numerical result indicates that $K$ takes a universal value 1 for any $J_\perp/J_\parallel$ in the limits $M\to0$ and $M\to1$.
The plan of the paper is as follows. We first review the Abelian bosonization approach to the $S=1/2$ $XXZ$ chain under a magnetic field in Sec. II A. The formulas of the spin correlations and the local magnetization in finite open chains are presented. In Sec. II B, we show numerical data for the $XXZ$ chain of $L=100$ sites obtained from the DMRG calculation and fit the data to the functions given in Sec. II A. In Sec. III A, we briefly review some relevant results of the previous analytic studies on the two-leg ladders in the strong- and weak-coupling limits. The DMRG data and the results of fitting on the open ladders with $L = 200$ sites are shown in Sec. III B. The $M$ dependence of $K$ for various values of $J_\perp/J_\parallel$ is obtained. Its implications to NMR and neutron scattering experiments are briefly mentioned. Finally, our results are summarized in Sec. IV.
$XXZ$ CHAIN
===========
Bosonization approach
---------------------
In this section, we consider spin-1/2 $XXZ$ chains with open ends in a magnetic field $h$. The Hamiltonian is $${\cal H}_{\rm ch} = {\cal H}_0 + {\cal H}_h \label{eq:Hchn}$$ with $$\begin{aligned}
{\cal H}_0 &=&
J \sum_{l=1}^{L-1} (\bbox{S}_l, \bbox{S}_{l+1} )_\Delta,
\nonumber \\
{\cal H}_h &=& - h \sum_{l=1}^L S^z_l, \nonumber\end{aligned}$$ where $\bbox{S}_l$ are $S = 1/2$ spin operators and $(\bbox{S}_l, \bbox{S}_{l'} )_\Delta =
S^x_l S^x_{l'} + S^y_l S^y_{l'} + \Delta S^z_l S^z_{l'}$. We assume the system size $L$ to be even throughout this paper and treat only the case where $J > 0$ and $0 \le \Delta \le 1$. We note that the Hamiltonian (\[eq:Hchn\]) for $-1 < \Delta\le 1$ can be solved exactly by Bethe ansatz for arbitrary values of $h$.[@Bethe0; @Bethe1; @Bethe2]
We use the standard Abelian bosonization techniques to analyze spin-spin correlation functions at zero temperature. We basically follow the scheme presented in Ref. and generalize it to the case of open chains in magnetic fields. The bosonization formulas in the absence of magnetic fields are reported in Ref. .
The low-energy dynamics of $XXZ$ chains is described by the Gaussian model,[@Bethe1] $$\widetilde{{\cal H}}_{\rm ch} =
\frac{v}{2} \int_0^{L+1} dx
\left[ \left( \frac{d\phi}{dx}\right)^2
+ \left( \frac{d\tilde{\phi}}{dx}\right)^2 \right] ,
\label{eq:HchnBos}$$ where $v$ is the spin-wave velocity. The continuous variable $x$ is identified with the site index $l$ under the assumption that the lattice spacing equals unity. The bosonic fields $\phi(x)$ and $\tilde{\phi}(x)$ obey the commutation relation $[\phi(x), \tilde{\phi}(y)] = -i \Theta(x-y)$, where $\Theta(x)$ is the step function. The spin operators in the original Hamiltonian (\[eq:Hchn\]) are related to the bosonic fields by the relations, $$\begin{aligned}
S^z_l &=&
\frac{1}{2 \pi R}\frac{d\phi}{dx}
+ a (-1)^l \sin\left(\frac{\phi(l)}{R}\right),
\label{eq:Szchn} \\
S^{-}_l &=&
\exp[- i 2\pi R\tilde{\phi}(l)]
\left[ b \sin\left(\frac{\phi(l)}{R}\right) + c(-1)^l \right],
\label{eq:S-chn}\end{aligned}$$ with $a$, $b$, and $c$ being real constants. The parameter $R$ determines the exponents of correlation functions. We also introduce the TL-liquid parameter $K$ by $K = 1/(4\pi R^2)$; $K=1$ in the $XY$ case ($\Delta=0$), and $K=1/2$ in the Heisenberg case ($\Delta=1$) at $h=0$. From Eq. (\[eq:S-chn\]), $S^x_l$ is written as $$S^x_l = c(-1)^l \cos[2\pi R\tilde{\phi}(l)]
- i b \sin[2\pi R\tilde{\phi}(l)]
\sin\left(\frac{\phi(l)}{R} \right).
\label{eq:Sxchn}$$ The second term with the coefficient $ib$ is Hermitian due to the commutation relation $[\phi(l),\tilde\phi(l)]=-i/2$. The open boundary conditions are translated to the boundary conditions on the bosonic fields at the two phantom sites $l=0$ and $l=L+1$:[@Eggert] $\phi(0)=0$ and $\phi(L+1)=2\pi RLM_{\rm ch}$, where $M_{\rm ch}$ is the magnetization per site, $$M_{\rm ch} = \frac{1}{L} \sum_{l=1}^L S^z_l.
\label{eq:Mch}$$ The total magnetization $LM_{\rm ch}$ is an integer for even $L$. These boundary conditions lead to the mode expansion, $$\begin{aligned}
\phi(x) &=&
\frac{x}{L+1} \phi_0
+ \sum_{n=1}^\infty \frac{\sin(q_n x)}{\sqrt{\pi n}}
\left( a_n + a_n^\dagger \right) , \label{eq:mode1} \\
\tilde{\phi}(x) &=&
\tilde{\phi}_0
+ i \sum_{n=1}^\infty \frac{\cos(q_n x)}{\sqrt{\pi n}}
\left( a_n - a_n^\dagger \right) , \label{eq:mode2}\end{aligned}$$ where $q_n = \pi n /(L+1)$, $[\tilde{\phi}_0, \phi_0] = i$, and $a_m$ are boson annihilation operators obeying $[a_m, a_n^\dagger] = \delta_{m,n}$. Note that the commutation relation between $\phi(x)$ and $\tilde{\phi}(y)$ mentioned above is satisfied. The lowest energy state $|M_{\rm ch}\rangle$ in the subspace in which the magnetization per spin is $M_{\rm ch}$ is a vacuum of $a_n$ $$a_n|M_{\rm ch}\rangle = 0$$ and an eigenstate of $\phi_0$ $$\phi_0 |M_{\rm ch}\rangle = 2\pi R L M_{\rm ch}|M_{\rm ch}\rangle.
\label{eq:phi_0}$$ We may regard $\tilde\phi_0$ as a coordinate variable along a fictitious ring of radius $1/2\pi R$ and take $\phi_0=-id/d\tilde\phi_0$ to be its momentum conjugate. The state $|M_{\rm ch}\rangle$ is then proportional to $\exp(i2\pi RLM_{\rm ch}\tilde\phi_0)$. The bosonization formulas (\[eq:Szchn\]) and (\[eq:S-chn\]) represent only the leading contributions. In the next order $S^-_l$ has a term of the form $(-1)^l\exp[-2\pi iR\tilde\phi(l)]\cos[2\phi(l)/R]$. We will, however, ignore this contribution because it yields only subleading corrections that disappear quickly for large $|l-l'|$.
Using Eqs. (\[eq:Szchn\])–(\[eq:phi\_0\]), one can evaluate the two-spin correlation functions $\langle S^\alpha_l S^\alpha_{l'} \rangle$ $(\alpha = x, z)$ and the local magnetization $\langle S^z_l \rangle$ in open chains, where $\langle \cdots \rangle$ denotes the expectation value in the state $|M_{\rm ch}\rangle$. Brief account of their derivation is given in Appendix. Here we present the final results:
$$\begin{aligned}
\langle S^x_l S^x_{l'} \rangle &\equiv&
X(l,l';q) \nonumber\\
&=&
\frac{f_{\eta/2}(2l) f_{\eta/2}(2l')}{f_\eta(l-l') f_\eta(l+l')}
\left[ \frac{c^2}{2} (-1)^{l-l'}
% \right. \nonumber \\ &&
+ \frac{bc}{2} {\rm sgn}(l-l') \left(
\frac{(-1)^l \cos(q l')}{f_{1/2\eta}(2l')}
- \frac{(-1)^{l'} \cos(q l)}{f_{1/2\eta}(2l)} \right)
\right. \nonumber \\
&& \left.\hspace*{3cm}
- \frac{b^2}{4 f_{1/2\eta}(2l) f_{1/2\eta}(2l')}
\left( \cos\left[ q (l+l')\right]
\frac{f_{1/\eta}(l-l')}{f_{1/\eta}(l+l')}
% \right. \nonumber \\ && \left.
+ \cos\left[ q (l-l')\right]
\frac{f_{1/\eta}(l+l')}{f_{1/\eta}(l-l')} \right)
\right],
\label{eq:Cxechn} \\
\langle S^z_l S^z_{l'} \rangle &\equiv&
Z(l,l';q) \nonumber\\
&=&
\frac{q}{2\pi}
\left(\frac{q}{2\pi}
+ a \frac{(-1)^l \sin(q l)}{f_{1/2\eta}(2l)}
+ a \frac{(-1)^{l'} \sin(q l')}{f_{1/2\eta}(2l')}
\right)
% \nonumber \\ &&
- \frac{1}{4\pi^2 \eta} \left(
\frac{1}{f_2(l-l')} + \frac{1}{f_2(l+l')} \right)
\nonumber \\
&& + \frac{a^2}{2}\frac{(-1)^{l+l'}}{f_{1/2\eta}(2l)f_{1/2\eta}(2l')}
\left( \cos\left[q (l-l')\right]
\frac{f_{1/\eta}(l+l')}{f_{1/\eta}(l-l')}
% \right. \nonumber\\ &&
- \cos\left[q (l+l')\right]
\frac{f_{1/\eta}(l-l')}{f_{1/\eta}(l+l')} \right)
\nonumber \\
&& + \frac{a}{2\pi \eta} \left(
\frac{(-1)^l \cos(q l)}{f_{1/2\eta}(2l)}
[g(l+l')+g(l-l')]
% \right. \nonumber \\ & & \left.~~~~~~~~
+ \frac{(-1)^{l'} \cos(q l')}{f_{1/2\eta}(2l')}
[g(l+l')-g(l-l')] \right) ,
\label{eq:Czechn} \\
\langle S^z_l \rangle &\equiv&
z(l;q)=
\frac{q}{2\pi}
+ a \frac{(-1)^l \sin(q l)}{f_{1/2\eta}(2l)},
\label{eq:Szechn}\end{aligned}$$
[2]{} where $$\begin{aligned}
&&
\eta=2\pi R^2=\frac{1}{2K},\\
&&
f_\alpha(x) =
\left[
\frac{2(L+1)}{\pi}\sin\left(\frac{\pi |x|}{2(L+1)} \right)
\right]^\alpha,
\label{eq:fx} \\
&&
g(x) = \frac{\pi}{2(L+1)} \cot\left( \frac{\pi x}{2(L+1)}\right).
\label{eq:gx}\end{aligned}$$ The wavenumber $q$, characterizing the IC character of the spin correlations in a magnetic field, is related to $M_{\rm ch}$ by $$q = \frac{2\pi M_{\rm ch}L}{L+1}.
\label{eq:Qch}$$ The factor $L/(L+1)$ appears as a result of the open boundary conditions. Under the periodic boundary conditions $q$ should be simply equal to $2\pi M_{\rm ch}$, because the first term in Eq. (\[eq:mode1\]) is $\phi_0 x/L$ in this case. This term must be $\phi_0 x/(L+1)$ in the open-boundary case in order for $\phi(x)$ and $\tilde\phi(x)$ to satisfy the commutation relation in the interval $[0,L+1]$. We emphasize that Eqs. (\[eq:Czechn\]) and (\[eq:Szechn\]) reproduce the exact results for the $XY$ chain when $\eta=1/2$ and $a=-1/\pi$. As is well known, the $XXZ$ spin chain is equivalent to a model of spinless fermions with nearest-neighbor interaction. In this model, $S^z_l$ is none but the fermion density, and the oscillating term in Eq. (\[eq:Szechn\]) corresponds to the Friedel oscillations near the open ends.[@Friedel1; @Friedel2; @Friedel3; @Friedel4]
In the thermodynamic limit ($L \to \infty$) with $|l-L/2| \ll L$ and $|l'-L/2| \ll L$, the spin correlations have the asymptotic forms $$\begin{aligned}
\langle S^x_l S^x_{l'} \rangle &=&
A_x\frac{(-1)^{l-l'}}{|l-l'|^\eta}
- \widehat{A}_x \frac{\cos\left[q (l-l')\right]}
{|l-l'|^{\eta+1/\eta}},
\label{eq:CxLchn} \\
\langle S^z_l S^z_{l'} \rangle &=& {M_{\rm ch}}^2
+ A_z (-1)^{l-l'} \frac{\cos\left[q (l-l')\right]}
{|l-l'|^{1/\eta}} \nonumber \\
&&- \frac{1}{4\pi^2 \eta |l-l'|^2},
\label{eq:CzLchn}\end{aligned}$$ where the correlation amplitudes $A_x$, ${\widehat{A}_x}$, and $A_z$ are related to the numerical constants $a$, $b$, and $c$ by $A_x = c^2/2$, ${\widehat{A}_x} = b^2/4$, and $A_z = a^2/2$. We can therefore estimate the TL-liquid parameter and the correlation amplitudes in the thermodynamic limit by extracting the fitting parameters $R$, $a$, $b$, and $c$ from the numerical data on a finite system with use of Eqs. (\[eq:Cxechn\]), (\[eq:Czechn\]), and (\[eq:Szechn\]). At the same time, the TL-liquid parameter $K$ can be calculated exactly for any $M_{\rm ch}$ by solving an integral equation obtained from the Bethe ansatz.[@TLpr1; @TLpr2] We will compare our estimates of $K$ obtained from the fitting procedure with the Bethe ansatz results in the next subsection.
Numerical results
-----------------
Using the DMRG method,[@White1; @White2] we computed the two-spin correlation functions $\langle S^\alpha_l S^\alpha_{l'} \rangle$ $(\alpha=x,z)$ and the local magnetization $\langle S^z_l \rangle$ in the $L=100$ open chains. The two-point functions were calculated for $l = r_0 - r/2$ and $l' = r_0 + r/2$, where $r_0 = L/2$ for even $r$ and $r_0 = (L+1)/2$ for odd $r$. The calculation was performed for each lowest-energy state of ${\cal H}_0$ in the subspace of various values of $M_{\rm ch}$. We employed the finite system algorithm of improved version.[@White3] The maximum number of kept states $m$ is $100$. We estimate the numerical error due to the truncation of the Hilbert space from the difference between the data with $m = 100$ and those with $m = 70$. The estimated errors for $\langle S^x_l S^x_{l'} \rangle$, $\langle S^z_l S^z_{l'} \rangle$, and $\langle S^z_l \rangle$ are, at largest, of order $10^{-5}$,$10^{-6}$, and $10^{-6}$, respectively.
In Fig. \[fig:chn\], we show the spin correlations $\langle S_l^\alpha S_{l'}^\alpha \rangle$ ($\alpha = x,z$) and the local magnetization $\langle S_l^z \rangle$ at $\Delta = 0.5$ for three different values of $M_{\rm ch}$. The DMRG data are shown by open symbols whose sizes are larger than the truncation error mentioned above. Taking $R$, $a$, $b$, and $c$ as fitting parameters, we fit the numerical data to Eqs. (\[eq:Cxechn\])–(\[eq:Szechn\]). The results of the fitting using the DMRG data of $\langle S_l^\alpha S_{l'}^\alpha \rangle$ for $10 \le r \le 90$ and of $\langle S_l^z \rangle$ for $1 \le l \le 100$ are also plotted in the figure by the small solid symbols. One can see that the fits are in excellent agreement with the DMRG data, proving the validity of the bosonization formulas (\[eq:Cxechn\])–(\[eq:Szechn\]).
For various values of $M_{\rm ch}$ and $\Delta$, we determined the parameters $R$, $a$, $b$, and $c$ by the fitting procedure. In doing so we used numerical data of several ranges, $10 \le r \le 80$, $10 \le r \le 90$, $20 \le r \le 80$, and $20 \le r \le 90$ for the two-spin correlation functions and $1 \le l \le 100$ and $10 \le l \le 90$ for the local magnetization. We take the mean and the variance of the fitting parameters obtained for the different ranges of $r$ and $l$ as the estimated value and the error of the estimates, respectively. The TL-liquid parameter $K \equiv 1/(4\pi R^2)$ estimated from the numerical data of $\langle S^x_l S^x_{l'} \rangle$ is plotted as a function of $M_{\rm ch}$ in Fig. \[fig:Kchn\]. The exact values obtained from the Bethe ansatz method [@TLpr1; @TLpr2] are also shown as dotted lines. The agreement is excellent. We also estimated $K$ from the fitting of $\langle S^z_l S^z_{l'}\rangle$ and $\langle S^z_l\rangle$ and obtained similar results as Fig. \[fig:Kchn\]. We found, however, that the estimates from the last two correlators show some deviations from the Bethe ansatz results when $K$ is small. We do not exactly know why they deviate. One possible reason might be the effect of the leading irrelevant operator neglected in the Gaussian model that becomes marginal at $K=1/2$. In the $XY$ regime of our interest, spins have stronger correlations in the $S^x$ and $S^y$ components than in $S^z$, and thus we may expect that $\langle S^x_l S^x_{l'}\rangle$ should give us most reliable estimates.
For the correlation amplitudes $A_x$ and $A_z$, Lukyanov and Zamolodchikov conjectured the exact formulas which are valid at $h=0$,[@Lu-Za; @Luky]
$$\begin{aligned}
A_x^{\rm LZ} &=&
\frac{1}{8(1-\eta)^2}
\left[\frac{\Gamma(\frac{\eta}{2(1-\eta)})}
{2\sqrt{\pi}\,\Gamma(\frac{1}{2(1-\eta)})}
\right]^\eta
% \nonumber \\ &&\times
\exp\left[
-\int^\infty_0\frac{dt}{t}
\left(\frac{\sinh(\eta t)}{\sinh(t)\cosh[(1-\eta)t]}
-\eta e^{-2t}\right)\right],
\label{eq:LZx} \\
A_z^{\rm L} &=&
\frac{2}{\pi^2}
\left[\frac{\Gamma(\frac{\eta}{2(1-\eta)})}
{2\sqrt{\pi}\,\Gamma(\frac{1}{2(1-\eta)})}
\right]^{1/\eta}
% \nonumber \\ &&\times
\exp\left[
\int^\infty_0\frac{dt}{t}
\left(\frac{\sinh[(2\eta-1) t]}{\sinh(\eta t)\cosh[(1-\eta)t]}
-\frac{2\eta -1}{\eta} e^{-2t}\right)\right],
\label{eq:LZz}\end{aligned}$$
[2]{}where $\Gamma(x)$ is the Gamma function. Equations (\[eq:LZx\]) and (\[eq:LZz\]) have been confirmed numerically.[@CorAm; @Luky] In Table \[tab:Achn\], we give our estimates of the correlation amplitudes $A_x = c^2/2$ and $A_z = a^2/2$ obtained from the fitting of $\langle S^x_l S^x_{l'}\rangle$ and $\langle S^z_l \rangle$ for $0 < M_{\rm ch} < 0.5$, together with the exact values (\[eq:LZx\]) and (\[eq:LZz\]) at $M_{\rm ch} = 0$. [@no; @hat; @Ax] As can be seen in Table \[tab:Achn\] (a), $A_x$ decreases monotonically from the value given by Eq. (\[eq:LZx\]) to zero as $M_{\rm ch}$ increases from 0 to $1/2$. Thus, $A_x$ depends not only on $K$ but also on $M_{\rm ch}$. (See, for example, the data for $\Delta=0$ where $K$ takes a constant value 1 for any $M_{\rm ch}$.) When $M_{\rm ch}$ approaches $1/2$, where $K\to1$, $A_x$ seems to go to zero linearly for any $\Delta$. This can be easily understood once we consider one-magnon contribution to the correlation function. On the other hand, $A_z$ decreases monotonically from the number given by Eq. (\[eq:LZz\]) to the universal value $A_z = 1/(2\pi^2) \simeq 0.05066$ as $M_{\rm ch}$ increases from 0 to $1/2$. [@Ampnote] An exception is the case $\Delta = 0$, where $A_z = 1/(2\pi^2)$ for any $M_{\rm ch}$. The convergence of $A_z$ to $1/(2\pi^2)$ at $M_{\rm ch} \to 1/2$ is consistent with the fact that the correlator $\langle S^z_l S^z_{l'}\rangle$ must take a constant value $1/4$ at $M_{\rm ch}=1/2$. The right-hand side of Eq. (\[eq:CzLchn\]) equals ${M_{\rm ch}}^2$ when $\eta = 1/2$, $q = \pi$, and $A_z = 1/(2\pi^2)$.
TWO-LEG AF LADDER
=================
Encouraged by the success in the last section, we study the two-leg AF ladders in a magnetic field using the same method. We begin with a brief review of the analytic results on the ladder in the strong- and weak-coupling limits.
Review of Analytic Results
--------------------------
The Hamiltonian of the open two-leg ladder studied in this section is given by $$\begin{aligned}
{\cal H} & = &
J_\parallel \sum_{\mu = 1,2} \sum_{l=1}^{L-1}
(\bbox{S}_{\mu,l},\bbox{S}_{\mu,l+1} )_\Delta
+ J_\perp \sum_{l=1}^L (\bbox{S}_{1,l},\bbox{S}_{2,l} )_\Delta
\nonumber \\ &&
- h \sum_{\mu=1,2} \sum_{l=1}^L S_{\mu,l}^z. \label{eq:Hlad}\end{aligned}$$ The anisotropy $\Delta$ is introduced for generality. We assume that the coupling in the leg- and rung-direction, $J_\parallel$ and $J_\perp$, are positive (antiferromagnetic). The spin ladder has an excitation gap in weak magnetic fields $h < h_{c1}$. We concentrate on the ladder in the gapless IC regime, i.e., in the case $h_{c1} \le h \le h_{c2}$. We denote the ratio $J_\perp / J_\parallel$ by $j$ hereafter.
We begin with the strong-coupling limit ($j \gg 1$), for which a simple intuitive picture is available. It is known that the system in this limit can be mapped to an effective $S=1/2$ $XXZ$ chain,[@Furu-Zhn; @Mila; @Totsu] as we explain below. Let us first assume $J_\parallel = 0$. In this case, an eigenstate of ${\cal H}$ is written as a direct product of rung states. At each rung two spins $\bbox{S}_{1,l}$ and $\bbox{S}_{2,l}$ are either in a singlet state $|s_l \rangle = ( |\uparrow \downarrow \rangle
- |\downarrow \uparrow \rangle ) / \sqrt{2}$ or in one of the triplet states, $|t_l^+ \rangle = |\uparrow \uparrow \rangle$, $|t_l^0 \rangle = ( |\uparrow \downarrow \rangle
+ |\downarrow \uparrow \rangle ) / \sqrt{2}$, and $|t_l^- \rangle = |\downarrow \downarrow \rangle$. When $h$ is small, the ground state consists of a product of the singlet rungs. As the field $h$ increases, the energy of the state $|t_l^+ \rangle$ becomes lower, and at $h = J_\perp(1+\Delta)/2$, the state degenerates with $|s_l\rangle$. We can thus analyze the low-energy properties of the system for $h \simeq J_\perp(1+\Delta)/2$ by retaining only the two lowest-energy states $|s_l\rangle$ and $|t_l^+ \rangle$ for each rung. We may regard the two states as \`\`down“ and \`\`up” states of an effective $S = 1/2$ spin, $$\begin{aligned}
\widetilde{S}^x_l&=&-\frac{1}{\sqrt{2}} (S^x_{1,l}-S^x_{2,l}),
\quad
\widetilde{S}^y_l=-\frac{1}{\sqrt{2}} (S^y_{1,l}-S^y_{2,l}),\\
\widetilde{S}^z_l&=&S^z_{1,l}+S^z_{2,l}-\frac{1}{2}.
\label{tilde S^z}\end{aligned}$$ The effective spins $\widetilde{\bbox{S}}_l$ are the only low-energy degrees of freedom, and their dynamics is governed by $$\begin{aligned}
\widetilde{{\cal H}}
&=&
J_\parallel \sum_{l=1}^{L-1}
(\widetilde{\bbox{S}}_l, \widetilde{\bbox{S}}_{l+1} )_{\Delta/2}
- \frac{\Delta}{4} J_\parallel
\left( \widetilde{S}_1^z + \widetilde{S}_L^z \right)
\nonumber\\ &&
- \left( h - \frac{1+\Delta}{2} J_\perp
- \frac{\Delta}{2} J_\parallel
\right) \sum_{l=1}^L \widetilde{S}_l^z
+ {\rm const}.
\label{eq:Hefstr}\end{aligned}$$ This mapping is derived in lowest order in $J_\parallel/J_\perp$ and valid for the entire IC region of $0<M<1$, where $M$ is magnetization per rung $$M=\frac{1}{L}\sum_{l=1}^L\left(S^z_{1,l}+S^z_{2,l}\right).
\label{M}$$ Note that the anisotropy of the effective $S=1/2$ $XXZ$ chain is a half of the anisotropy of the original ladder, $\Delta/2$. Besides the bulk effective magnetic field $h - (1+\Delta) J_\perp /2 - \Delta J_\parallel /2$, there is an additional field $- \Delta J_\parallel/4$ applied only to the boundary spins $\widetilde{S}_1^z$ and $\widetilde{S}_L^z$. It induces oscillating magnetization near the boundaries superposed on the Friedel oscillation which is already present at any $M\ne0$ without the boundary field. The effect of the boundary field may be cancelled by adding an extra term $${\cal H}' =
h' \sum_{\mu = 1,2} \left( S^z_{\mu,1} + S^z_{\mu,L} \right)
\label{eq:H'}$$ to the original ladder Hamiltonian (\[eq:Hlad\]), where $h' = \Delta J_\parallel/4$ for $j \gg 1$. Now we define $$\begin{aligned}
\bbox{S}_{0,l} &=& \bbox{S}_{1,l} + \bbox{S}_{2,l},
\label{eq:S0} \\
\bbox{S}_{\pi,l} &=& \bbox{S}_{1,l} - \bbox{S}_{2,l}.
\label{eq:Spi}\end{aligned}$$ From the mapping explained above, we conclude that the two-spin correlations $\langle S_{\pi,l}^x S_{\pi,l'}^x \rangle$ and $\langle S_{0,l}^z S_{0,l'}^z \rangle$ and the local magnetization $\langle S_{0,l}^z\rangle$ in the open ladder ${\cal H} + {\cal H}'$ in the limit $j\gg1$ are given by the corresponding correlators in the $XXZ$ chain. We thus obtain $$\begin{aligned}
\langle S_{\pi,l}^x S_{\pi,l'}^x \rangle &=&
2 X(l,l';Q),
\label{eq:Cxlad} \\
\langle S_{0,l}^z S_{0,l'}^z \rangle &=&
\left\langle
\left(\frac{1}{2}+\widetilde{\bbox{S}}_l^z\right)
\left(\frac{1}{2}+\widetilde{\bbox{S}}_{l'}^z\right)
\right\rangle
\nonumber\\
&=& \frac{1}{4}+\frac{1}{2}[z(l;Q)+z(l';Q)]+Z(l,l';Q),
\label{eq:Czlad} \\
\langle S_{0,l}^z\rangle &=&
\frac{1}{2}+z(l,Q), \label{eq:Szlad}\end{aligned}$$ where the wavenumber is $$Q = \frac{2\pi L}{L+1}\left(M-\frac{1}{2}\right). \label{eq:Qlad}$$ In the limit $L\to\infty$ the two-spin correlation functions reduce to $$\begin{aligned}
\langle S_{\pi,l}^x S_{\pi,l'}^x\rangle &=&
2A_x\frac{(-1)^{l-l'}}{|l-l'|^{1/2K}} \nonumber\\
&&
-2\widehat{A}_x(-1)^{l-l'}
\frac{\cos[2\pi M(l-l')]}{|l-l'|^{2K+(1/2K)}},
\\
\langle S_{0,l}^z S_{0,l'}^z\rangle &=&
M^2-\frac{1}{4\pi^2\eta|l-l'|^2} \nonumber\\
&&
+A_z\frac{\cos[2\pi M(l-l')]}{|l-l'|^{2K}}.\end{aligned}$$ Note that they can be obtained from Eqs. (\[eq:CxLchn\]) and (\[eq:CzLchn\]) by replacing $q$ and $M_{\rm ch}$ with $2\pi(M-1/2)$ and $M$, respectively. On the other hand, the correlations $\langle S_{0,l}^x S_{0,l'}^x \rangle$ and $\langle S_{\pi,l}^z S_{\pi,l'}^z \rangle$ decay exponentially because $S^x_{0,l}$ and $S^z_{\pi,l}$ always create the high-energy rung states $|t_l^0 \rangle$ and $|t_l^- \rangle$ as a virtual excited state.
Next, we consider the opposite case, the weak-coupling limit ($j\ll1$). The system in this limit has been investigated with the Abelian bosonization method.[@Chi-Gia; @Gia-Tsv; @Furu-Zhn] In these studies, two chains are first bosonized independently, and then the interchain coupling $J_\perp$ is treated perturbatively.[@Shel] Four bosonic fields $\phi_\pm(x)$ and $\tilde{\phi}_\pm(x)$ are introduced, where $\phi_+$ and $\tilde{\phi}_+$ ($\phi_-$ and $\tilde{\phi}_-$) are the symmetric (antisymmetric) combinations of bosonic fields of each chain. All the fields are massive[@Shel] when $h < h_{c1}$. In the IC regime of $h_{c1} \le h \le h_{c2}$, on the other hand, the fields $\phi_+$ and $\tilde{{\phi}}_+$ become massless while the fields $\phi_-$ and $\tilde{{\phi}}_-$ remain massive. The low-energy effective Hamiltonian for the gapless modes has the same form as that of the $S=1/2$ $XXZ$ chain, Eq. (\[eq:HchnBos\]). Furthermore, the spin correlation functions $\langle S_{\pi,l}^x S_{\pi,l'}^x \rangle$ and $\langle S_{0,l}^z S_{0,l'}^z \rangle$ have the same $r$ dependence as in the strong-coupling limit (but with different values of $K$, $a$, $b$, and $c$).[@Furu-Zhn; @note] The correlators $\langle S_{0,l}^x S_{0,l'}^x \rangle$ and $\langle S_{\pi,l}^z S_{\pi,l'}^z \rangle$ decay exponentially,[@Furu-Zhn] because they involve the massive fields. This result also matches the strong-coupling limit. Moreover, the incommensurate wavenumber for the short-ranged correlators is $\tilde q=\pi M$, which is different from the IC wavenumber for the quasi-long-ranged correlators $q=2\pi(M-1/2)$. For example, it was found that[@Furu-Zhn] $$\langle S_{\pi,l}^z S_{\pi,l'}^z\rangle = \widetilde{A}_z
(-1)^{l-l'}e^{-|l-l'|/\xi}\frac{\cos[\pi M(l-l')]}{|l-l'|^{1/2+1/4\eta}},
\label{SpiSpi}$$ where $\xi$ is a correlation length for a massive mode and $\widetilde{A}_z$ is a constant.
We have seen that, both in the strong- and weak-coupling limits, the low-energy physics of the two-leg ladder in the gapless regime is in the same universality class as the $XXZ$ chain in a magnetic field. In particular, the spin correlation functions $\langle S^x_{\pi,l}S^x_{\pi,l'}\rangle$ and $\langle S^z_{0,l}S^z_{0,l'}\rangle$ and the local magnetization $\langle S^z_{0,l}\rangle$ have the same forms as the corresponding functions in the $XXZ$ chain, but with the shifted wavenumber $2\pi(M-1/2)$ and with different values of $K$, $a$, $b$, and $c$.[@nonuniversal] It is then very natural to postulate that the universality is not restricted to the two limits but holds for any $j$. This allows us to use Eqs. (\[eq:Cxlad\])–(\[eq:Szlad\]) for analyzing the correlation functions in the ladder for any $j$ and $M$ ($0<M<1$). We can thus determine the TL-liquid parameter $K$ of the ladder by fitting the numerical data to Eqs. (\[eq:Cxlad\])–(\[eq:Szlad\]) in the same way as we did for the $XXZ$ chain. The result is presented in the next subsection. Finally, we may expect that the $|l-l'|$ dependence of the short-ranged correlators $\langle S_{0,l}^x S_{0,l'}^x\rangle$ and $\langle S_{\pi,l}^z S_{\pi,l'}^z\rangle$ obtained in the weak-coupling analysis, such as Eq. (\[SpiSpi\]), should also be valid for any $j$ and $M$ ($0<M<1$).
Numerical Results
-----------------
Here we present the result of the DMRG calculation of the two-spin correlations $\langle S_{\pi,l}^x S_{\pi,l'}^x \rangle$ and $\langle S_{0,l}^z S_{0,l'}^z \rangle$ and the local magnetization $\langle S_{0,l}^z\rangle$ in the ladder. Taking $R$, $a$, $b$, and $c$ as free parameters, we fit the data to the formulas (\[eq:Cxlad\])–(\[eq:Szlad\]) for several values of $j$ and estimate the $M$ dependence of $K = 1/(4 \pi R^2)$. The numerical calculations were performed for the open ladder ${\cal H} + {\cal H}'$ of $L=100$ rungs ($200$ sites) for $j = 10.0, 2.0, 1.0, 0.5$ and $\Delta = 1.0, 0.5, 0.0$ using the finite-system DMRG method of the improved version. We calculated the two-spin correlations for $l = r_0 - r/2$ and $l' = r_0 + r/2$, where $r_0 = L/2$ for even $r$ and $r_0 = (L+1)/2$ for odd $r$. The maximum value of the kept states $m$ is $160$. From the difference between the data with $m = 160$ and those with $m = 120$, we estimate the numerical error due to the truncation. The estimated errors for $\langle S^x_{\pi,l} S^x_{\pi,l'} \rangle$, $\langle S^z_{0,l} S^z_{0,l'} \rangle$, and $\langle S^z_l \rangle$ are, at largest, of order $10^{-4}$, $10^{-6}$, and $10^{-6}$, which is almost negligible. In the course of the calculation, we optimized the value of the extra boundary field $h'$ to minimize the effect of the boundary field.[@addH] For finite $j$, however, the boundary effect cannot be eliminated completely since it can be represented by the form of ${\cal H}'$ only in the strong-coupling limit $j\gg1$. As a result, the two-spin correlation functions and the local magnetization in the open ladder ${\cal H} + {\cal H}'$ might deviate from the expected form, Eqs. (\[eq:Cxlad\])–(\[eq:Szlad\]), near the boundaries. For this reason we used data of smaller range of $r$ for the fitting than in Sec. II to reduce the unwanted boundary effect. We chose the regions $10 \le r \le 70$, $10 \le r \le 80$, $20 \le r \le 70$, and $20 \le r \le 80$ for the fitting of the two-spin correlators and $10 \le l \le 90$ and $20 \le l \le 80$ for the local magnetization. As in Sec. II, we regard the mean and the variance of the fitting parameters obtained for these different ranges of $r$ and $l$ as the estimated value and the error of the estimates, respectively. Incidentally, we have also checked for $(j,M)=(10.0, 0.5)$ that, without the boundary field $h'$, the local magnetization $\langle S^z_{0,l}\rangle$ has the Friedel oscillations induced by the effective boundary field. We found that the oscillations decay algebraically into the bulk with the exponent $K$, as expected from the bosonization analysis.[@Affleck]
The numerical data of $\langle S_{\pi,l}^x S_{\pi,l'}^x \rangle$, $\langle S_{0,l}^z S_{0,l'}^z \rangle$, and $\langle S_{0,l}^z\rangle$ for $j = 10.0$ and $1.0$ with $\Delta = 1.0$ (Heisenberg case) are shown in Figs. \[fig:lad100\] and \[fig:lad010\] by open symbols, whose sizes are larger than the truncation error mentioned above. The small solid symbols in the figures are the fits to the DMRG data of the two-spin correlations for $10 \le r \le 80$ and to those of the local magnetization for $10 \le l \le 90$, respectively. It is clearly seen that the fitting works extremely well for $j = 10.0$, confirming the validity of the formulas (\[eq:Cxlad\]), (\[eq:Czlad\]), and (\[eq:Szlad\]). Furthermore, the agreement between the numerical data and the fits at $j = 1.0$ is also quite good except some deviations near the boundary, indicating that the formulas are accurate for the intermediate-coupling regime of $j$ as well. We note that the quality of the fitting is also good for other values of $j$ and $\Delta$ that we have examined. We therefore conclude that the gapless mode of the two-leg ladders is a TL liquid for arbitrary $j$, and accordingly, the properties of the strong- and weak-coupling ladders are smoothly connected.
Next we show in Fig. \[fig:Klad\] the $M$ dependence of $K$ estimated from the data of $\langle S_{\pi,l}^x S_{\pi,l'}^x \rangle$ for various $j$ in both the Heisenberg ($\Delta = 1.0$) and $XY$ ($\Delta = 0$) cases. In the earlier study[@Usami] the exponent $\eta$ was obtained for $j=5.0$ in the Heisenberg case only. We note that the estimation from $\langle S_{\pi,l}^x S_{\pi,l'}^x \rangle$ is more reliable than that from $\langle S_{0,l}^z S_{0,l'}^z \rangle$ or $\langle S_{0,l}^z\rangle$ as we have seen in the $XXZ$ chain. Theoretically[@Furu-Zhn] it is expected that $K$ should approach the universal value $K=1$ when $M\to0$ as well as when $M\to1$, since the system is equivalent to the dilute limit of hard-core bosons. Although the data for $M\to0$ have large error bars, we may conclude that our results for the Heisenberg case are consistent with the theoretical prediction. Our results for the $XY$ case show a more subtle feature. At first sight the results for weaker couplings ($j=1.0$ and 0.5) do not seem to approach $K=1$ as $M\to0$. We think, however, that $K$ changes very rapidly at small $M$ to approach $K=1$, in view of the data for $j=10.0$ and $2.0$, which are consistent with the theory. Unfortunately, it is difficult to numerically estimate $K$ for small $M$ with high accuracy to resolve this issue.
In the strong-coupling limit $j\gg1$, the ladder system with anisotropy $\Delta$ is equivalent to the $S=1/2$ $XXZ$ chain with anisotropy $\Delta/2$, as explained in the previous subsection. Figure \[fig:Klad\] clearly shows that the estimated value of $K$ for $j = 10.0$ is consistent with the anticipated behavior shown as the dotted curves in both the Heisenberg and $XY$ cases. As $j$ decreases, $K$ increases monotonically for any $M$ ($0<M<1$). Thus, $K$ is always larger than 1 in the $XY$ ladder because $K\to1$ for any $M$ in the large $j$ limit. In the Heisenberg ladder, on the other hand, $K$ is smaller than 1 in the strong-coupling limit, as expected from the mapping to the $XXZ$ chain. Upon decreasing the interchain coupling $j$, $K$ starts to increase and the $K$-$M$ relation changes from a concave curve to a convex one. We note that the similar behavior is observed also in the ladder with $\Delta = 0.5$: As $j$ decreases, the $K$-$M$ relation changes from a concave curve at $j\gg1$, corresponding to the behavior of the $XXZ$ chain with anisotropy $\Delta/2 = 0.25$, to a convex one. We thus consider that this behavior of the $K$-$M$ curve is an universal feature for $0 < \Delta \le 1$.
The TL-liquid parameter $K$ determines the long-distance behavior of correlation functions in the thermodynamic limit. For example, the leading term of the correlator $\langle S^z_{0,l} S^z_{0,l'}\rangle - M^2$ decays as $\cos[2\pi M(l-l')]/|l-l'|^{2K}$ for $K<1$, while it decays like $|l-l'|^{-2}$ for $K>1$. Hence, in the ladder with $0<\Delta\le1$ the leading term of the correlator changes from $\cos[2\pi M(l-l')]/|l-l'|^{2K}$ to $|l-l'|^{-2}$ at a critical value $j_c(M)$ as $j$ decreases, while in the $XY$ ladder the leading term is always $|l-l'|^{-2}$. On the other hand, the correlator $\langle S^x_{\pi,l} S^x_{\pi,l'}\rangle$ decays as $(-1)^{l-l'}/|l-l'|^{1/2K}$ in the whole range of $K$ covered in Figs. \[fig:Klad\] (a) and \[fig:Klad\] (b). The temperature dependence of the spin-lattice relaxation rate $1/T_1$ in NMR experiments is directly related to the TL-liquid parameter $K$ through the decay exponent of the most slowly decaying correlation.[@Chi-Gia] From the behavior of $K$-$M$ relation obtained above, we find that the correlator $\langle S^x_{\pi,l} S^x_{\pi,l'}\rangle$ decays most slowly for any $j$, $M$, and $0 \le \Delta \le 1$. We therefore conclude that at low temperatures the relaxation rate of the ladder in the gapless regime always shows a power-law divergence $1/T_1\propto T^{-1+(1/2K)}$.
Figure \[fig:SzSz\] shows the numerical result of $\langle S^z_{\pi,l}S^z_{\pi,l'}\rangle$ for the Heisenberg ladder at $j=0.5$. It exhibits exponentially decaying oscillatory behavior. From the period $\lambda$ of oscillations, we obtain the IC wavenumber $\tilde q = 2\pi/\lambda$ as a function of $M$; see the inset figure. The result confirms the theoretical prediction $\tilde q=\pi M$. This IC wavenumber $\tilde q$ tells us that the massive magnon dispersion has a minimum excitation energy at[@Furu-Zhn] $q=\pi-\tilde q=\pi(1-M)$. Accordingly, the dynamical spin structure factor $S^{zz}_\pi(q,\omega)$ should have a power-law divergence along the energy dispersion which is roughly shifted by $\pi M$ from that of the triplet magnon dispersion in the absence of the magnetic field. It would be interesting if this feature is observed by inelastic neutron scattering experiments.
CONCLUSIONS
===========
In this paper we have studied the ground-state spin correlations in the gapless IC regime of the $S=1/2$ $XXZ$ chain and the two-leg AF ladder in a magnetic field. We have used the $S=1/2$ $XXZ$ chain as a first test ground to apply the method we developed in our previous work: We numerically computed the two-spin correlation functions and the local magnetization by the DMRG method and fit the results to functions which are obtained using the bosonization technique. The fitting parameters are the TL-liquid parameter $K$ and the amplitudes of bosonic operators. We found good agreement between $K$ estimated from the fitting and $K$ calculated from the Bethe ansatz. As a byproduct we obtained the amplitudes of the dominant terms in $\langle S^x_l S^x_{l'}\rangle$ and $\langle S^z_l S^z_{l'}\rangle$.
We have applied the same technique to the two-leg AF ladder in the gapless IC regime. It has been known that in both the strong- and weak-coupling limits the low-energy excitations in the ladder are regarded as a TL liquid like the $XXZ$ chain in a field. We fit our DMRG data of the two-spin correlation functions and the local magnetization of the ladder to the same bosonization formulas we used in the analysis of the $XXZ$ chain. The fitting worked very well not only in the strong- and weak-coupling limits but for broad range of the interchain coupling strength $j$. We thereby confirmed that the low-energy gapless excitations are indeed described as the TL liquid for any $j$ and the properties of the strong- and weak-coupling ladders are smoothly connected. For several values of $j$, we have determined $K$, which shows nontrivial $j$ and $M$ dependences (Fig. \[fig:Klad\]). It turned out that, for any $M$ ($0<M<1$), $K$ is a monotonically decreasing function of $j$. In the ladder with anisotropy $0 < \Delta \le 1$ the $K$-$M$ relation changes from a concave curve at $j\gg1$ to a convex one as $j$ decreases, while in the $XY$ ladder ($\Delta=0$) it changes from a line $K=1$ at $j\gg1$ to a convex curve. We also found that the spin-lattice relaxation rate in NMR measurement shows a power-law divergence $1/T_1 \propto T^{-1+(1/2K)}$ at low temperature for any $j$.
Numerical computations were performed at the Yukawa Institute Computing Facility. The work of AF was in part supported by Grant-in-Aid for Scientific Research on Priority Areas (A) from the Ministry of Education, Science, Sports and Culture (No. 12046238) and by Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of Science (No. 11740199).
Derivation of Correlators {#derivation-of-correlators .unnumbered}
=========================
We briefly explain the derivation of Eqs. (\[eq:Cxechn\]) and (\[eq:Czechn\]). As is always the case with the bosonization, we need to introduce a short-distance cutoff to obtain finite results. The lattice spacing in the original Hamiltonian serves as the natural cutoff scale.
When we use Eqs. (\[eq:Szchn\]) and (\[eq:S-chn\]) with the mode expansions (\[eq:mode1\]) and (\[eq:mode2\]), we encounter the summation $$\sum^\infty_{n=1}\frac{1}{n}
\left[1-\cos\left(\frac{\pi nl}{L+1}\right)\right],$$ which is formally divergent. We regularize it by inserting an exponential factor $e^{-\pi n/(L+1)}$: $$\sum^\infty_{n=1}\frac{1}{n}e^{-\pi n/(L+1)}
\left[1-\cos\left(\frac{\pi nl}{L+1}\right)\right]
=\ln[f(l)],
\label{ln(f)}$$ where $f(l)\equiv f_1(l)$ defined in Eq. (\[eq:fx\]). We note that Eq. (\[ln(f)\]) is a very good approximation except near the points where $f_1(l)$ is divergent. Taking derivatives with respect to $l$, we obtain $$\sum^\infty_{n=1}e^{-\pi n/(L+1)}\sin\left(\frac{\pi nl}{L+1}\right)
=\frac{1}{2}\cot\left(\frac{\pi l}{2(L+1)}\right)$$ and $$\sum^\infty_{n=1}n e^{-\pi n/(L+1)}\cos\left(\frac{\pi nl}{L+1}\right)
=-\frac{1}{4\sin^2\left(\frac{\pi l}{2(L+1)}\right)}.$$ Another point to note is that for $\varepsilon_i=\pm1$ $$\begin{aligned}
\langle e^{i2\pi R\epsilon_1\tilde\phi(l)}
e^{i2\pi R\epsilon_2\tilde\phi(l')}\rangle
&\propto&
\int^{1/R}_0 R e^{i2\pi R(\epsilon_1+\epsilon_2)\tilde\phi_0}
d\tilde\phi_0 \nonumber\\
&\propto&
\delta_{\epsilon_1+\epsilon_2,0}.\end{aligned}$$
With the above-mentioned formulas, it is straightforward to obtain $$\begin{aligned}
&&
\langle\cos[2\pi R\tilde\phi(l)]\cos[2\pi R\tilde\phi(l')]\rangle
=\frac{[f(2l)f(2l')]^{\eta/2}}{2[f(l-l')f(l+l')]^\eta},
\\
&&
\langle e^{i2\pi R\epsilon_1\tilde\phi(l)}
e^{-i2\pi R\epsilon_1\tilde\phi(l')}e^{i\epsilon_2\phi(l')/R}\rangle
\\
&&
=i\epsilon_1\epsilon_2 e^{iq\epsilon_2 l'}
\frac{{\rm sgn}(l-l')[f(2l)f(2l')]^{\eta/2}}
{[f(l-l')f(l+l')]^\eta [f(2l')]^{1/2\eta}},
\\
&&
\langle e^{i2\pi R\epsilon_0\tilde\phi(l)}e^{i\epsilon_1\phi(l)/R}
e^{-i2\pi R\epsilon_0\tilde\phi(l')}e^{i\epsilon_2\phi(l')/R}\rangle
\\
&&
=-\epsilon_1\epsilon_2 e^{iq(\epsilon_1l+\epsilon_2l')}
\frac{[f(2l)f(2l')]^{\eta/2-1/2\eta}}
{[f(l-l')f(l+l')]^\eta}
\left(\frac{f(l-l')}{f(l+l')}\right)^{\epsilon_1\epsilon_2/\eta},
\\
&&
\left\langle\frac{d\phi}{dl}\frac{d\phi}{dl'}\right\rangle
=-\frac{1}{2\pi}\left(\frac{1}{f_2(l-l')}+\frac{1}{f_2(l+l')}\right)
+\left(\frac{q}{2\pi}\right)^2,
\\
&&
\left\langle\left(\frac{d\phi}{dl}-\frac{q}{2\pi}\right)
\sin\frac{\phi(l')}{R}\right\rangle
=\frac{\cos(ql')}{2\pi R}\frac{g(l+l')-g(l-l')}{[f(2l')]^{1/2\eta}},
\\
&&
\langle e^{i\epsilon_1\phi(l)/R} e^{i\epsilon_2\phi(l')/R}\rangle
=\frac{e^{iq(\epsilon_1l+\epsilon_2l')}}{[f(2l)f(2l')]^{1/2\eta}}
\left(\frac{f(l-l')}{f(l+l')}\right)^{\epsilon_1\epsilon_2/\eta}.
\\\end{aligned}$$
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We are not sure about the accuracy of our estimates for the amplitude $\widehat{A}_x$ of the subleading term, and thereby we do not show them here.
The estimates of $A_z$ for small $M>0$ shown in Table \[tab:Achn\] (b) do not seem to approach smoothly the exact value at $M_{\rm ch}=0$ as $M_{\rm ch} \to 0$. We do not know exactly the reason why this happens. This might be due to the same reason as the one for the deviation of $K$ estimated from $\langle S^z_l \rangle$ mentioned in the text.
F. Mila, Eur. Phys. J. B [**6**]{}, 201 (1998).
K. Totsuka, Phys. Rev. B [**57**]{}, 3454 (1998).
D.G. Shelton, A.A. Nersesyan, and A.M. Tsvelik, Phys. Rev. B [**53**]{}, 8521 (1996).
We note that the notation has changed from Ref. . The parameter $\eta$ in the present paper is equal to the inverse of the $\eta$ used in Ref. .
In the strong-coupling limit, the parameters $K$, $a$, $b$, and $c$ in the ladder with the anisotropy $\Delta$ are related to those in the $XXZ$ chain with the anisotropy $\Delta/2$; see Eqs. (\[eq:Cxlad\])-(\[eq:Qlad\]).
To be more concrete, we optimized the value of $h'$ to minimize the deviation of $\langle S^z_{0,l} \rangle$ for $M = 1/2$ from the constant value, $1/2$; see Eq. (\[eq:Szlad\]).
I. Affleck, J. Phys. A: Math. Gen. [**31**]{}, 2761 (1998).
M. Usami and S. Suga, Phys. Rev. B [**58**]{}, 14401 (1998).
(a)$A_x$
$M_{\rm ch}$ 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45
--------------- --------- ------------ ------------ ----------- ------------ ------------ ------------ ----------- ---------- -----------
$\Delta= 0.0$ 0.14709 0.14626(1) 0.14364(6) 0.1390(1) 0.13262(1) 0.12410(6) 0.1132(3) 0.0993(7) 0.081(2) 0.0594(7)
$\Delta= 0.1$ 0.14451 0.14369(7) 0.1413(1) 0.1371(1) 0.13101(7) 0.12293(2) 0.1125(3) 0.0991(7) 0.081(2) 0.0597(7)
$\Delta= 0.2$ 0.14187 0.1408(4) 0.1390(2) 0.1351(2) 0.1294(1) 0.12174(5) 0.1111(7) 0.0988(6) 0.081(2) 0.0600(7)
$\Delta= 0.3$ 0.13921 0.1384(3) 0.1366(3) 0.1330(3) 0.1278(2) 0.12053(9) 0.1111(2) 0.0985(6) 0.081(2) 0.0601(7)
$\Delta= 0.4$ 0.13656 0.1358(3) 0.1342(4) 0.1310(3) 0.1261(3) 0.1193(1) 0.1104(1) 0.0982(6) 0.081(2) 0.0603(7)
$\Delta= 0.5$ 0.13400 0.1332(4) 0.1318(5) 0.1289(4) 0.1245(3) 0.1182(2) 0.10973(9) 0.0979(5) 0.081(2) 0.0605(7)
$\Delta= 0.6$ 0.13164 0.1310(5) 0.1294(6) 0.1268(5) 0.1229(4) 0.1170(2) 0.10905(7) 0.0976(5) 0.081(2) 0.0606(7)
$\Delta= 0.7$ 0.12973 0.1281(6) 0.1270(7) 0.1248(5) 0.1213(4) 0.1159(3) 0.10839(6) 0.0973(5) 0.081(2) 0.0607(7)
$\Delta= 0.8$ 0.12896 0.1257(8) 0.1247(8) 0.1227(6) 0.1197(5) 0.1148(3) 0.10775(7) 0.0970(5) 0.081(1) 0.0609(7)
$\Delta= 0.9$ 0.13214 0.1233(9) 0.1223(9) 0.1207(7) 0.1182(6) 0.1137(3) 0.10714(8) 0.0967(4) 0.081(1) 0.0610(8)
$\Delta= 1.0$ 0.121(1) 0.120(1) 0.1188(8) 0.1177(9) 0.1127(4) 0.1065(1) 0.0958(6) 0.081(1) 0.0610(8)
: The correlation amplitudes; (a) $A_x = c^2/2$ estimated from the data of $\langle S^x_l S^x_{l'} \rangle$; (b) $A_z = a^2/2$ estimated from the data of $\langle S^z_l \rangle$. The figures in parentheses indicate the error bar on the last quoted digits. The error bars of $A_z$ for $\Delta=0$ and $0.05 \le M \le 0.45$ are smaller than $10^{-5}$. The exact values for $M_{\rm ch} = 0$ given by Eqs. (\[eq:LZx\]) and (\[eq:LZz\]) are also listed. []{data-label="tab:Achn"}
(b)$A_z$
$M_{\rm ch}$ 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45
--------------- --------- ----------- ------------ ------------ ----------- ----------- ----------- ----------- ---------- -----------
$\Delta= 0.0$ 0.05066 0.05066 0.05066 0.05066 0.05066 0.05066 0.05066 0.05066 0.05066 0.05066
$\Delta= 0.1$ 0.05929 0.0599(7) 0.0581(3) 0.0567(6) 0.0544(1) 0.0537(7) 0.0513(6) 0.0516(6) 0.049(1) 0.0510(8)
$\Delta= 0.2$ 0.06891 0.071(1) 0.0662(6) 0.063(1) 0.0580(2) 0.056(1) 0.052(1) 0.052(1) 0.048(2) 0.051(1)
$\Delta= 0.3$ 0.07978 0.083(2) 0.0748(7) 0.069(1) 0.0614(4) 0.059(1) 0.052(1) 0.053(1) 0.048(2) 0.052(2)
$\Delta= 0.4$ 0.09231 0.097(3) 0.0838(6) 0.075(1) 0.0645(6) 0.060(1) 0.053(2) 0.053(2) 0.047(3) 0.052(2)
$\Delta= 0.5$ 0.10713 0.113(5) 0.093(4) 0.080(1) 0.0674(8) 0.062(2) 0.053(2) 0.053(2) 0.046(3) 0.052(3)
$\Delta= 0.6$ 0.12539 0.132(6) 0.10263(5) 0.0854(9) 0.070(1) 0.063(2) 0.054(2) 0.053(2) 0.046(3) 0.052(3)
$\Delta= 0.7$ 0.14930 0.153(8) 0.1121(5) 0.0903(4) 0.072(1) 0.065(2) 0.054(2) 0.053(2) 0.045(4) 0.052(3)
$\Delta= 0.8$ 0.18414 0.176(10) 0.121(1) 0.09486(6) 0.074(2) 0.066(2) 0.054(2) 0.054(2) 0.045(4) 0.052(4)
$\Delta= 0.9$ 0.24844 0.20(1) 0.131(2) 0.0990(7) 0.076(2) 0.067(1) 0.054(2) 0.054(2) 0.047(4) 0.052(4)
$\Delta= 1.0$ 0.23(1) 0.139(3) 0.103(1) 0.078(2) 0.067(1) 0.054(3) 0.054(2) 0.044(4) 0.052(4)
: The correlation amplitudes; (a) $A_x = c^2/2$ estimated from the data of $\langle S^x_l S^x_{l'} \rangle$; (b) $A_z = a^2/2$ estimated from the data of $\langle S^z_l \rangle$. The figures in parentheses indicate the error bar on the last quoted digits. The error bars of $A_z$ for $\Delta=0$ and $0.05 \le M \le 0.45$ are smaller than $10^{-5}$. The exact values for $M_{\rm ch} = 0$ given by Eqs. (\[eq:LZx\]) and (\[eq:LZz\]) are also listed. []{data-label="tab:Achn"}
[2]{}
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We provide a quantitative version of the isoperimetric inequality for the fundamental tone of a biharmonic Neumann problem. Such an inequality has been recently established by Chasman adapting Weinberger’s argument for the corresponding second order problem. Following a scheme introduced by Brasco and Pratelli for the second order case, we prove that a similar quantitative inequality holds also for the biharmonic operator. We also prove the sharpness of both such an inequality and the corresponding one for the biharmonic Steklov problem.'
author:
- 'D. Buoso'
- 'L.M. Chasman'
- 'L. Provenzano'
bibliography:
- 'bibliography.bib'
title: On the stability of some isoperimetric inequalities for the fundamental tones of free plates
---
Introduction
============
The stability of isoperimetric inequalities is an important question that has gained significant interest in recent decades. For example, the celebrated Faber-Krahn inequality for the smallest eigenvalue of the Dirichlet Laplacian, $$\lambda_1(\Omega)\ge\lambda_1(\Omega^*),$$ can be improved in the following quantitative form: $$\lambda_1(\Omega)\ge\lambda_1(\Omega^*)(1+C\mathcal{A}(\Omega)^2),
\label{quantfk}$$ for some constant $C>0$. Here $\Omega\subset\mathbb{R}^N$ is a bounded open set, $N\geq2$, $\Omega^*$ is a ball such that $|\Omega|=|\Omega^*|$, and $\mathcal A(\Omega)$ is the so-called Fraenkel asymmetry of the domain $\Omega$ (see for the definition). Quantitative versions of the type have also been established for other isoperimetric inequalities involving eigenvalues of the Laplace operator, see, e.g., [@brascosteklov; @brasco2015; @brascopratelli].
Fewer isoperimetric inequalities have been established for eigenvalues of the biharmonic operator, namely for the first nontrivial eigenvalue of the Dirichlet (“clamped plate”) problem [@ashbaugh; @nadirashvili], of the Neumann (“free plate”) problem [@chasmanpreprint; @chasman], and of the Steklov problem introduced in [@buosoprovenzano] (see also [@buosoprovenzano0]). An isoperimetric inequality is still missing for another Steklov problem introduced in [@kuttler68], the conjectured optimizer being the regular pentagon (see, e.g., [@antunesgazzola; @bucurgazzola11] and the references therein).
Among these inequalities, the first one that has been given in quantitative form is the inequality for Steklov problem in [@buosoprovenzano], namely $$\lambda_2(\Omega)\le\lambda_2(\Omega^*)(1-C\mathcal{A}(\Omega)^2),
\label{quantitative_bp}$$ where $\lambda_2(\Omega)$ is the first nontrivial eigenvalue of the biharmonic Steklov problem $$\label{SteklovPDE}
\begin{cases}\Delta^2u-\tau\Delta u=0 &\text{in $\Omega$,}\\
\frac{\partial^2 u}{\partial \nu^2}= 0 &\text{on $\partial\Omega$,}\\
\tau\frac{\partial u}{\partial \nu}
-{\rm div}_{\partial\Omega}\Big(D^2u\cdot \nu\Big)-\frac{\partial\Delta u}{\partial \nu} = \lambda u &\text{on $\partial\Omega$,}
\end{cases}$$ where $\tau$ is a strictly positive constant.
In this paper we provide a quantitative form for the isoperimetric inequality for the first non-trivial eigenvalue of the following biharmonic Neumann problem: $$\label{NeumannPDE}
\begin{cases}\Delta^2u-\tau\Delta u=\lambda u &\text{in $\Omega$,}\\
\frac{\partial^2 u}{\partial \nu^2}= 0 &\text{on $\partial\Omega$,}\\
\tau\frac{\partial u}{\partial \nu}
-{\rm div}_{\partial\Omega}\Big(D^2u\cdot \nu\Big)-\frac{\partial\Delta u}{\partial \nu} = 0 &\text{on $\partial\Omega$.}
\end{cases}$$ We recall that for $N=2$, problem describes the transverse vibrations of an unconstrained thin elastic plate with shape $\Omega\subset \mathbb{R}^2$ when at rest. The constant $\tau$ represents the ratio of lateral tension to lateral rigidity and is taken to be non-negative.
When $\tau>0$ and $\Omega\subset\mathbb{R}^N$ is a smooth connected bounded open set, it is known that the spectrum of the Neumann biharmonic operator $\Delta^2-\tau\Delta$ consists entirely of non-negative eigenvalues of finite multiplicity, repeated according to their multiplicity: $$0=\lambda_1(\Omega)<\lambda_2(\Omega)\leq\cdots\leq\lambda_j(\Omega)\leq\cdots.$$ Note that since constant functions satisfy problem with eigenvalue $\lambda=0$, the first positive eigenvalue is $\lambda_2$, which is usually called the “fundamental tone” of the plate. In [@chasman], the author proved that $$\label{iso_neumann}
\lambda_2(\Omega)\leq \lambda_2(\Omega^*)$$ with equality if and only if $\Omega=\Omega^*$. The proof of inequality is based on Weinberger’s argument for the Neumann Laplacian, taking suitable extensions of the eigenfunctions of the ball as trial functions (see [@weinberger]). In [@brascopratelli], the authors carry out a more careful analysis of such an argument, improving Weinberger’s inequality to a quantitative form. In a similar way, we start from the proof of and improve the result to the quantitative inequality by means of this finer analysis.
The question of sharpness is another important issue that has to be addressed when dealing with quantitative isoperimetric inequalities. More precisely, given an inequality of the form $$%\label{quantitative_general}
\lambda_2(\Omega)\leq\lambda_2(\Omega^*)\left(1-\Phi({\rm dist}(\Omega,\mathcal B))\right),$$ where $\Phi$ is some modulus of continuity, ${\rm dist}(\cdot,\cdot)$ is a suitable distance between open sets and $\mathcal B$ is the family of all balls in $\mathbb R^N$, we say that it is sharp if there exists a family $\lbrace\Omega_{\varepsilon}\rbrace_{\varepsilon\in(0,\varepsilon_0)}$ such that ${\rm dist}(\Omega_{\varepsilon},\mathcal B)\rightarrow 0$, $\lambda_2(\Omega_{\varepsilon})\rightarrow\lambda_2(\Omega^*)$ as $\varepsilon\rightarrow 0$, and there exists contants $c_1,c_2>0$ which do not depend on $\varepsilon>0$ and $\Omega^*$ such that [$$c_1\Phi({\rm dist}(\Omega_{\varepsilon},\mathcal B))\leq 1-\frac{\lambda_2(\Omega_{\varepsilon})}{\lambda_2(\Omega^*)}\leq c_2\Phi({\rm dist}(\Omega_{\varepsilon},\mathcal B)),$$]{} as $\varepsilon\rightarrow 0$. Note that, in our case, the distance function is given by the Fraenkel asymmetry ${\rm dist}(\Omega,\mathcal B)=\mathcal A(\Omega)$ while the modulus of continuity is $\Phi(t)=Kt^2$, for some $K>0$. By means of the construction introduced in [@brascosteklov; @brascopratelli], we prove in Section \[sharpness\_neumann\] that the quantitative Neumann inequality is sharp.
It is worth noting that in the Neumann Laplacian case in [@brascopratelli], the authors try, as a first guess, to consider ellipsoids as the family $\lbrace\Omega_{\varepsilon}\rbrace_{\varepsilon\in(0,\varepsilon_0)}$, with the ball $\Omega_0$ being the maximizer. Unfortunately, this is not a good family to prove sharpness; this can be explained observing that different directions of perturbation behave in a different way with respect to the fundamental tone. In particular, some directions are not “good enough” to see the sharpness (cf. [@brascopratelli Remark 5.2]). This phenomenon can be observed in our case as well: therefore we need to restrict our analysis by excluding some directions. See (\[perturbation\]) and Remark \[directions\].
The Steklov problem is of particular interest despite its recent introduction, since in [@buosoprovenzano] the authors show that it has a very strict relationship with the Neumann problem . Using a mass perturbation argument, they prove that the Steklov problem can in fact be viewed as a limiting Neumann problem where the mass is distributed only on the boundary. Note that this construction was already performed in [@lambertiprozisaac] for the Laplace operator, obtaining similar results (see also [@dallarivaproz; @lambertiproz] for the computation of the topological derivative). Moreover, this justifies the fact of thinking of Steklov problems in terms of vibrating objects (plates or membranes) where the mass lies only on the boundary (see [@steklov]). The authors also prove the quantitative inequality by adapting an argument due to Brock (see [@brock]) for the Steklov Laplacian to the biharmonic case in the refined version of [@brascosteklov]. However, they do not discuss its sharpness. The similarity of the variational characterization of Neumann and Steklov eigenvalues allows us to prove that inequality is sharp by an easy adaptation of the arguments used in the Neumann case.
The paper is organized as follows. In Section \[preliminaries\], we give some preliminary results and introduce the notation. Section \[proof\_neumann\_quantitative\] is devoted to the Neumann quantitative isoperimetric inequality , the sharpness of which we prove in Section \[sharpness\_neumann\]. Finally, in Section \[sharpness\_steklov\] we prove that the Steklov inequality is sharp.
Preliminaries and notation {#preliminaries}
==========================
We introduce here the notation used throughout the paper and recall some fundamental results proved in [@chasman].
Let $B$ be the unit ball in $\mathbb{R}^N$ centered at the origin and $\omega_N$ be the Lebesgue measure $|B|$ of $B$.
We denote by $j_1$ and $i_1$ the ultraspherical and modified ultraspherical Bessel functions of the first kind and order $1$ respectively. They can be expressed in terms of standard Bessel and modified Bessel functions of the first kind $J_{\nu}, I_{\nu}$ as follows: $$j_1(z)=z^{1-N/2}J_{N/2}(z),\qquad i_1(z)=z^{1-N/2}I_{N/2}(z).$$ For more information on Bessel and modified Bessel functions, see, e.g., [@abram §9].
We will define trial functions in terms of the eigenfunctions corresponding to $\lambda_2(B)$ of the Neumann problem. For a fixed $\tau>0$, we take positive constants $a,b$ satisfying $a^2b^2=\lambda_2(B)$ and $b^2-a^2=\tau$. We set $$R(r)=j_1(ar)+\gamma i_1(br),\qquad\text{where}\qquad \gamma=-\frac{a^2 j_1''(a)}{b^2 i_1''(b)}.$$ We then define the function $\rho:[0,+\infty)\to[0,+\infty)$ as $$%\label{rho}
\rho(r)=\begin{cases}
R(r),&r\in[0,1)\\
R(1)+(r-1)R'(1),&r\in[1,+\infty).
\end{cases}$$
Let $u_k:\mathbb{R}^N\to\mathbb{R}$ be defined by $$\label{uk}
u_k(x):=\rho(|x|)\frac{x_k}{|x|},$$ for $k=1,\dots,N$. The functions ${u_k}_{|_{B}}$ are in fact the eigenfunctions associated with the eigenvalue $\lambda_2(B)$ of the Neumann problem on the unit ball $B$. Recall that $\lambda_2(B)$ has multiplicity $N$ (see [@chasman11 Theorem 3]). Moreover, we have (see [@chasman p. 437]) $$\begin{aligned}
%\label{relation_1}
\sum_{k=1}^N |u_k|^2&=\rho(|x|)^2,\\
%\label{relation_2}
\sum_{k=1}^N|D u_k|^2&=\frac{N-1}{|x|^2}\rho(|x|)^2+(\rho'(|x|))^2,\\
%\label{relation_3}
\sum_{k=1}^N|D^2 u_k|^2&=(\rho''(|x|))^2+\frac{3(N-1)}{|x|^4}(\rho(|x|)-|x|\rho'(|x|))^2.\end{aligned}$$ We denote by $N[\rho]$ the quantity $$N[\rho]:=\sum_{k=1}^N |D^2u_k|^2+\tau|D u_k|^2.$$
We recall some properties enjoyed by the functions $\rho$ and $N[\rho]$ which were proved in [@chasman].
\[pro\] The function $\rho$ satisfies the following properties.
i) $\rho''(r)\leq 0$ for all $r\geq 0$, therefore $\rho'$ is non-increasing.
ii) $\rho(r)-r\rho'(r)\geq 0$, equality holding only for $r=0$.
iii) The function $\rho(r)^2$ is strictly increasing.
iv) The function ${\rho(r)^2}/{r^2}$ is decreasing.
v) The function ${3(\rho(r)-r\rho'(r))^2}/{r^4}+\tau{\rho^2(r)}/{r^2}$ is decreasing.
vi) $N[\rho(r_1)]>N[\rho(r_2)]$ for any $r_1\in [0,1)$, $r_2\in [1,+\infty)$.
vii) For all $r\geq 0$ we have $$N[\rho(r)]=(\rho''(r))^2+\frac{3(N-1)(\rho(r)-r\rho'(r))^2}{r^4}+\tau(N-1)\frac{\rho^2(r)}{r^2}+\tau(\rho'(r))^2.$$
viii) For all $r\geq 1$, $N[\rho(r)]$ is decreasing.
To conclude this section, let us recall the definition of the Fraenkel asymmetry $\mathcal A(\Omega)$ of a set $\Omega\subset\mathbb{R}^N$: $$\label{fra}
\mathcal A(\Omega):=\inf\left\{\frac{|\Omega\triangle {B}|}{|\Omega|}: {B}\text{\ is a ball such that}\ |{B}|=|\Omega|\right\}.$$
Quantitative isoperimetric inequality for the Neumann problem {#proof_neumann_quantitative}
=============================================================
In this section we state and prove the [quantitative isoperimetric inequality for the fundamental tone of the Neumann problem ]{}.
\[NeumannQI\] For every bounded domain $\Omega$ in $\mathbb{R}^N$ of class $C^1$ the following estimate holds $$\label{quantitative_neumann}
\lambda_2(\Omega)\leq\lambda_2(\Omega^*)\left(1-\eta_{N,\tau,|\Omega|}\mathcal A(\Omega)^2\right),$$ where $\eta_{N,\tau,|\Omega|}>0$, $\Omega^*$ is a ball such that $|\Omega^*|=|\Omega|$, and $\lambda_2(\Omega)$, $\lambda_2(\Omega^*)$ are the first positive eigenvalues of problem on $\Omega$, $\Omega^*$ respectively.
Let $\Omega$ be a bounded domain in $\mathbb{R}^N$ of class $C^1$ with the same measure as the unit ball $B$. We recall the variational characterization of the second eigenvalue $\lambda_2(\Omega)$ of on $\Omega$: $$\label{ray}
\lambda_2(\Omega)=\inf_{\substack{0\ne u\in H^2(\Omega)\\ \int_{\Omega}u dx=0}}\frac{\int_{\Omega}|D^2u|^2+\tau|Du|^2dx}{\int_{\Omega}u^2dx}.$$ Let $u_k(x)$, for $k=1,\dots,N$, be the eigenfunctions corresponding to $\lambda_2(B)$ defined in . Clearly ${u_k}_{|_{\Omega}}\in H^2(\Omega)$ by construction. It is possible to choose the origin of the coordinate axes in $\mathbb{R}^N$ in such a way that $\int_{\Omega}u_k dx=0$ for all $k=1,\dots,N$. With this choice, the functions $u_k$ are suitable trial functions for the Rayleigh quotient . Once we have fixed the origin, let $$\alpha:=\frac{|\Omega\triangle B|}{|\Omega|}.$$ By definition of Fraenkel asymmetry, we have $$\label{A}
\mathcal A(\Omega)\leq\alpha\leq 2.$$ From the variational characterization , it follows that for each $k=1,\dots,N$, $$\lambda_2(\Omega)\leq\frac{\int_{\Omega}|D^2u_k|^2+\tau|D u_k|^2dx}{\int_{\Omega}u_k^2 dx}.$$ We multiply both sides by $\int_{\Omega}u_k^2 dx$ and sum over $k=1,\dots,N$, obtaining $$\label{n1}
\lambda_2(\Omega)\leq\frac{\int_{\Omega}N[\rho]dx}{\int_{\Omega}\rho^2 dx}.$$ The same procedure for $\lambda_2(B)$ clearly yields $$\label{b1}
\lambda_2(B)=\frac{\int_{B}N[\rho]dx}{\int_{B}\rho^2 dx}.$$ From and , it follows that $$\label{ineq1}
\lambda_2(B)\int_B \rho^2 dx-\lambda_2(\Omega)\int_{\Omega}\rho^2 dx\geq\int_B N[\rho]dx-\int_{\Omega}N[\rho]dx\geq 0,$$ where the last inequality follows from Lemma \[pro\], [*iv)*]{} and [@chasman Lemma 14].
Now we consider the two balls $B_1$ and $B_2$ centered at the origin with radii $r_1,r_2$ taken such that $|\Omega\cap B|=|B_1|=\omega_N r_1^N$ and $|\Omega\setminus B|=|B_2\setminus B|=\omega_N(r_2^N-1)$. Then $|B_2|=\omega_N r_2^N$, and by construction $$\label{aster}
1-r_1^N=\frac{\alpha}{2}=r_2^N-1.$$ This is due to the fact that $|\Omega|+|B|=|\Omega\triangle B|+2 |\Omega\cap B|$, and then $1-r_1^N=\alpha/2$. Similarly, $|\Omega\setminus B|+|\Omega\cap B|=|\Omega|$, hence $r_1^N=2-r_2^N$, and then $r_2^N-1=\alpha/2$.
Now we observe, again by Lemma \[pro\], [*vi)*]{} and [*viii)*]{}, that $$\int_{\Omega}N[\rho]dx\leq\int_{B_1}N[\rho]dx+\int_{B_2\setminus B}N[\rho]dx.$$ From this and , we obtain $$\begin{aligned}
\label{ineq2}
\lambda_2(B)\int_B\rho^2 dx-\lambda_2(\Omega)\int_{\Omega}\rho^2 dx&\geq\int_B N[\rho]dx-\int_{\Omega}N[\rho]dx\\
&\geq\int_{B\setminus B_1}N[\rho]dx-\int_{B_2\setminus B}N[\rho]dx.\nonumber\end{aligned}$$ Since the function $\rho(r)^2$ is strictly increasing by Lemma \[pro\], [*iii)*]{}, we have $$%\label{den}
\int_{\Omega}\rho^2 dx\geq\int_B\rho^2 dx=N\omega_N\int_0^1\rho^2(r)r^{N-1}dr=:C^{(1)}_{N,\tau},$$ hence, $$\begin{aligned}
\label{passoA}
\lambda_2(B)&\int_B\rho^2 dx-\lambda_2(\Omega)\int_{\Omega}\rho^2 dx\\
&\leq\left(\lambda_2(B)-\lambda_2(\Omega)\right)\int_B\rho^2 dx +\lambda_2(\Omega)\left(\int_B\rho^2 dx-\int_{\Omega}\rho^2 dx\right)\nonumber\\
&\leq C^{(1)}_{N,\tau}\left(\lambda_2(B)-\lambda_2(\Omega)\right).\nonumber\end{aligned}$$
Now we consider the right-hand side of . We write $N[\rho]$ more explicitly in terms of $\rho$, obtaining: $$\begin{aligned}
\label{passoB1}
\int_{B\setminus B_1}&N[\rho]dx=N\omega_N\int_{r_1}^1\Big((\rho''(r))^2+\frac{3(N-1)(\rho(r)-r\rho'(r))^2}{r^4}\\
&\qquad\qquad\qquad\qquad+\tau(\rho'(r))^2+\frac{\tau (N-1)}{r^2}\rho(r)^2\Big)r^{N-1}dr\nonumber\\
&\geq N\omega_N\int_{r_1}^1\left(\frac{3(N-1)(\rho(r)-r\rho'(r))^2}{r^4}+\tau(\rho'(r))^2+\frac{\tau (N-1)}{r^2}\rho(r)^2\right)r^{N-1}dr\nonumber\\
&\geq \omega_N\left(3(N-1)(R(1)-R'(1))^2+\tau R'(1)^2+\tau (N-1)R(1)^2\right)(1-r_1^N),\nonumber\end{aligned}$$ where in the last inequality, we used the fact that $N[\rho]-(\rho'')^2$ is non-increasing in $r$ (see Lemma \[pro\], [*i)*]{} and [*v)*]{}). Moreover, $$\begin{aligned}
\label{passoB2}
\int_{B_2\setminus B}&N[\rho]dx\\
&=N\omega_N\int_1^{r_2}\left(\frac{3(N-1)}{r^4}(R(1)-R'(1))^2+\tau R'(1)^2\right.\nonumber\\
&\qquad\qquad+\frac{\tau(N-1)}{r^2}\Big((R(1)-R'(1))^2+2rR'(1)(R(1)-R'(1))\Big)\nonumber\\
&\qquad\qquad\left.+\frac{\tau(N-1)}{r^2}\Big(r^2R'(1)^2\Big)\right)r^{N-1}dr\nonumber\\
&\leq N\omega_N\int_1^{r_2}\left(N\tau R'(1)^2+\frac{N-1}{r}\left((3+\tau)(R(1)-R'(1))^2\right.\right.\nonumber\\
&\qquad\qquad\left.+2\tau R'(1)(R(1)-R'(1))\right)\Big)r^{N-1}dr\nonumber\\
&=N\omega_N\tau R'(1)^2(r_2^N-1)+N\omega_N\left((3+\tau)(R(1)-R'(1))^2\right.\nonumber\\
&\qquad\left.+2\tau R'(1)(R(1)-R'(1))\right)(r_2^{N-1}-1),\nonumber\end{aligned}$$ where we have estimated the quantities ${1}/{r^2}$ and ${1}/{r^4}$ by ${1}/{r}$. We note that $r_2=\left(1+{\alpha}/{2}\right)^{{1}/{N}}$ and $0\leq\alpha\leq 2$. Using the Taylor expansion up to order $1$ and remainder in Lagrange form, we obtain $$\begin{aligned}
\label{1n}
r_2^{N-1}&=1+\frac{N-1}{N}\frac{\alpha}{2}-\frac{(N-1)\left(1+\frac{\xi}{2}\right)^{\frac{N-1}{N}-2}}{8N^2}\alpha^2\\
&\leq 1+\frac{N-1}{N}\frac{\alpha}{2}-\frac{(N-1)2^{\frac{N-1}{N}-2}}{8N^2}\alpha^2=1+\frac{N-1}{N}\frac{\alpha}{2}-c_{N}\alpha^2,\nonumber\end{aligned}$$ for some $\xi\in(0,\alpha)$, where $c_{N}$ is a positive constant which depends only on $N$. Using , , , and , in the right-hand side of , we obtain: $$\begin{aligned}
\label{tofinal1}
\int_{B\setminus B_1}&N[\rho]dx-\int_{B_2\setminus B}N[\rho]dx \\
&\ge-N\omega_N\Big((3+\tau)(R(1)-R'(1))^2+2\tau R'(1)(R(1)-R'(1))\Big)\left(\frac{N-1}{N}\frac{\alpha}{2}- c_{N}\alpha^2\right)\nonumber\\
&\qquad+\omega_N\left(3(N-1)(R(1)-R'(1))^2+\tau R'(1)^2+\tau (N-1) R(1)^2\right)\frac{\alpha}{2}\nonumber\\
&\qquad -N\omega_N\tau R'(1)^2\frac{\alpha}{2}\nonumber\\
&=:C^{(2)}_{N,\tau}\alpha^2,\nonumber\end{aligned}$$ where the constant $C^{(2)}_{N,\tau}>0$ is given by $$C^{(2)}_{N,\tau}=N\omega_N\left((3+\tau)(R(1)-R'(1))^2+2\tau R'(1)(R(1)-R'(1))\right)c_{N}.$$ From , , , and , it follows that $$\lambda_2(B)-\lambda_2(\Omega)\geq\frac{C^{(2)}_{N,\tau}}{C^{(1)}_{N,\tau}}\mathcal A(\Omega)^2,$$ and therefore, $$\label{quantN-1}
\lambda_2(\Omega)\leq\lambda_2(B)\left(1-\frac{C_{N,\tau}^{(2)}}{\lambda_2(B)C_{N,\tau}^{(1)}}\mathcal A(\Omega)^2\right).$$ The isoperimetric inequality is thus proved in the case of $\Omega$ with the same measure as the unit ball. The inequality for a generic domain $\Omega$ follows from scaling properties of the eigenvalues of problem . Writing our eigenvalues as $\lambda_2(\tau,\Omega)$ to make explicit the dependence on the parameter $\tau$, we have $$\label{scaling}
\lambda_2(\tau,\Omega)=s^4\lambda_2(s^{-2}\tau,s\Omega),$$ for all $s>0$. From and taking $s=(\omega_N/|\Omega|)^{1/N}$ in , it follows that for every $\Omega$ in $\mathbb R^N$ of class $C^1$ we have $$\begin{aligned}
\lambda_2(\tau,\Omega)&=s^4\lambda_2(s^{-2}\tau,s\Omega)\\
&\leq s^4\lambda_2(s^{-2}\tau,B)\left(1-\frac{C^{(2)}_{N,s^{-2}\tau}}{\lambda_2(s^{-2}\tau,B)C^{(1)}_{N,s^{-2}\tau}}\mathcal A(s\Omega)\right)\\
&=\lambda_2(\tau,\Omega^*)\left(1-\frac{C^{(2)}_{N,s^{-2}\tau}}{\lambda_2(s^{-2}\tau,B)C^{(1)}_{N,s^{-2}\tau}}\mathcal A(\Omega)\right).\end{aligned}$$ We set $$\eta_{N,\tau,|\Omega|}:=\frac{C^{(2)}_{N,s^{-2}\tau}}{\lambda_2(s^{-2}\tau,B)C^{(1)}_{N,s^{-2}\tau}}.$$ This concludes the proof of the theorem.
One generalization of the [biharmonic Neumann problem ]{} is to consider the case where the plate is made of a material with a nonzero Poisson’s ratio $\sigma$, which replaces the term $|D^2u|^2$ in the Rayleigh quotient by $(1-\sigma)|D^2u|^2+\sigma(\Delta u)^2$. A partial result towards the non-quantitative form of the isopermetric inequality has been obtained for certain values of $\tau>0$ and $\sigma\in(-1/(N-1),1)$, proved by the second author in [@chasmanpreprint] (see also [@buoso15; @prozkalamata]). In this case, the proof of Theorem \[NeumannQI\] can be easily adapted, yielding $$\lambda_2(B)-\lambda_2(\Omega)\geq \frac{C^{(3)}_{N,\tau}}{C^{(1)}_{N,\tau}}\mathcal{A}(\Omega)+\frac{C^{(2)}_{N,\tau}}{C^{(1)}_{N,\tau}}\mathcal{A}(\Omega)^2,$$ where $C^{(1)}_{N,\tau}$, $C^{(2)}_{N,\tau}$ are as in the proof of Theorem \[NeumannQI\], and $$C^{(3)}_{N,\tau}=\frac{1}{2}(R(1)-R'(1))^2(N-1)\sigma\Big(\sigma(N-1)(\sigma-2)+N-2\Big).$$ This result is not particularly satisfying, since it carries all of the same limitations of the non-quantitative result (only being valid for certain $\tau$ and $\sigma$), and in some cases it is strictly worse, since $C^{(3)}_{N,\tau}$ is non-negative only when $0\leq \sigma\leq 1-1/\sqrt{N-1}$.
Even though we are able to give a quantitative isoperimetric inequality for the fundamental tone of problem , very little is known in this regard for higher eigenvalues. To the best of our knowledge, only criticality results are available in the literature, where the ball is shown to be a critical domain under volume constraint (see, e.g., [@buoso15; @buosolamberti15; @buosoprovenzano]). However, as in the second-order case, the ball is not expected to be an optimizer for higher eigenvalues.
Sharpness of the Neumann inequality {#sharpness_neumann}
===================================
In this section, we prove the sharpness of inequality .
\[theorem2\] Let $B$ be the unit ball in $\mathbb{R}^N$ centered at zero. There exist a family $\left\{\Omega_{\epsilon}\right\}_{\epsilon>0}$ of smooth domains and positive constants $c_1,c_2,c_3,c_4$ and $r_1, r_2,r_3,r_4$ independent of $\epsilon>0$ such that $$\label{sharp1}
r_1\epsilon^2\le\Big||\Omega_{\epsilon}|-|B|\Big|\leq r_2\epsilon^2,$$ $$\label{sharp2}
c_1\epsilon\leq c_2\mathcal A(\Omega_{\epsilon})\leq\frac{|\Omega_{\epsilon}\triangle B|}{|\Omega_{\epsilon}|}\leq c_3\mathcal A(\Omega_{\epsilon})\leq c_4\epsilon,$$ and $$\label{sharp3}
r_3\epsilon^2\le\left|\lambda_2(\Omega_{\epsilon})-\lambda_2(B)\right|\leq r_4\epsilon^2,$$ for all $\epsilon\in(0,\epsilon_0)$, where $\epsilon_0>0$ is sufficiently small, [and $\lambda_2(\Omega_\epsilon)$, $\lambda_2(B)$ are the first positive eigenvalues of problem on $\Omega_\epsilon$, $B$ respectively]{}.
In order to prove Theorem \[theorem2\], we start by defining the family of domains $\left\{\Omega_{\epsilon}\right\}_{\epsilon>0}$ as follows (see Figure \[figura\]): $$\label{family}
\Omega_{\epsilon}=\left\{x\in\mathbb R^N: x=0 {\text\ or\ }|x|<1+\epsilon\psi\left(\frac{x}{|x|}\right)\right\},$$ where $\psi$ is a function belonging to the following class: $$\label{perturbation}
\mathcal{P}=\left\{\psi\in C^{\infty}(\partial B):\int_{\partial B}\psi d\sigma=\int_{\partial B}(a\cdot x)\psi d\sigma=\int_{\partial B}(a\cdot x)^2\psi d\sigma=0,\ \forall a\in\mathbb{R}^N\right\}.$$
![Domains $\Omega_{\varepsilon}$ defined by with a given $\psi\in\mathcal P$.[]{data-label="figura"}](good_domains.pdf){width="\textwidth"}
Under our choice of $\Omega_\epsilon$, the existence of constants $r_1,r_2,c_1,\dots,c_4$ satisfying inequalities and follow immediately from [@brascosteklov Lemma 6.2]. Thus, we need only prove .
Let $\lambda_2(\Omega_{\epsilon})$ be the first positive eigenvalue of the Neumann problem on $\Omega_{\epsilon}$, and let $u_{\epsilon}$ be an associated eigenfunction normalized by $\|u_{\epsilon}\|_{L^2(\Omega_{\epsilon})}=1$, so that $$\int_{\Omega_{\epsilon}}|D^2u_{\epsilon}|^2+\tau|Du_{\epsilon}|^2\,dx=\lambda_2(\Omega_{\epsilon}).$$ By standard elliptic regularity (see e.g., [@gazzola §2.4.3]), since $\Omega_{\epsilon}$ is of class $C^{\infty}$ by construction, we may take a sufficiently small $\epsilon_0>0$ so that $u_{\epsilon} \in C^{\infty}(\overline\Omega_{\epsilon})$ for all $\epsilon\in(0,\epsilon_0)$. Moreover, for all $k\in\mathbb N$, the sets $\Omega_{\epsilon}$ are of class $C^k$ uniformly in $\epsilon\in(0,\epsilon_0)$, which means that there exist constants $H_k>0$ independent of $\epsilon$ that satisfy $$\label{regularity}
\|u_{\epsilon} \|_{C^k(\overline{\Omega_{\epsilon}})}\leq H_k.$$ Now let $\tilde{u}_{\epsilon} $ be a $C^4$ extension of $u_{\epsilon}$ to some open neighborhood $A$ of $B\cup\Omega_{\epsilon}$. Then, there exists $K_A>0$ independent of $\epsilon>0$ for which $$\label{regularity_extension}
\|\tilde{u}_{\epsilon} \|_{C^4(\overline A)}\leq K_A\|u_{\epsilon} \|_{C^4(\overline{\Omega_{\epsilon}})}\leq K_A H_4.$$ From the fact that $\int_{\Omega_{\epsilon}}u_{\epsilon} \,dx=0$ and $|B\setminus\Omega_{\epsilon}|,|\Omega_{\epsilon}\setminus B|\in O(\epsilon)$ as $\epsilon\rightarrow 0$, it follows that the quantity $\delta:=\frac{1}{|B|}\int_B\tilde{u}_{\epsilon}\,dx$ satisfies $$\label{delta_bound}
\delta=\frac{1}{|B|}\int_B\tilde{u}_{\epsilon} \,dx=\frac{1}{|B|}\left(\int_{B\setminus \Omega_{\epsilon}}\tilde{u}_{\epsilon}\,dx-\int_{\Omega_{\epsilon}\setminus B}u_{\epsilon} \,dx\right)\leq c\epsilon,$$ where $c>0$ does not depend on $\epsilon\in(0,\epsilon_0)$. Now let us set $$\label{test}
v_{\epsilon} :={\tilde u_{\epsilon|_B}}-\delta.$$ The function $v_{\epsilon}$ is of class $C^4(\overline B)$ with $\int_B v_{\epsilon} \,dx=0$ and $$\label{regularity_v}
\|v_{\epsilon} \|_{C^4(\overline B)}\leq K_1$$ for some constant $K_1>0$ independent of $\epsilon\in(0,\epsilon_0)$. Therefore, $v_{\epsilon}$ is a suitable trial function for the Rayleigh quotient of $\lambda_2(B)$ (see formula ). Thus, $$\label{minmax_1}
\lambda_2(B)\leq\frac{\int_B |D^2 v_{\epsilon}|^2+\tau|Dv_{\epsilon}|^2\,dx}{\int_B {v_{\epsilon} }^2\,dx}.$$ We now consider the quantity $\left|\int_B v_{\epsilon} ^2-\tilde{u}_{\epsilon} ^2\,dx\right|$. We have $$\begin{aligned}
\label{ineq_1}
\left|\int_B v_{\epsilon} ^2-\tilde{u}_{\epsilon} ^2\,dx\right|&=\left|\int_B \delta^2-2\delta\tilde{u}_{\epsilon}\,dx\right|=\left|\int_B\delta(v_{\epsilon}-\tilde{u}_{\epsilon})\,dx\right|\\
%&=\left|\int_B\frac{1}{|B|^2}\left(\int_B \tilde{u}_{\epsilon} dy\right)^2-\frac{2}{|B|}\left(\int_B \tilde{u}_{\epsilon} dy\right)\tilde{u}_{\epsilon}\,dx\right|=\frac{1}{|B|}\left(\int_B \tilde{u}_{\epsilon} \right)^2\leq K_2\epsilon^2,\nonumber
&=\frac{1}{|B|}\left(\int_B \tilde{u}_{\epsilon}\,dx \right)^2\leq K_2\epsilon^2,\nonumber\end{aligned}$$ where $K_2>0$ is a positive constant independent of $\epsilon\in(0,\epsilon_0)$. Moreover, by and , we have that $$\begin{aligned}
\label{ineq_2}
\left|\int_{B\setminus\Omega_{\epsilon}}v_{\epsilon} ^2-\tilde{u}_{\epsilon} ^2dx\right|&\leq \int_{B\setminus\Omega_{\epsilon}}|v_{\epsilon} ^2-\tilde{u}_{\epsilon} ^2|dx\leq K_3\int_{B\setminus\Omega_{\epsilon}}|v_{\epsilon} -\tilde{u}_{\epsilon} |dx\\
&=K_3\frac{|B\setminus\Omega_{\epsilon}|}{|B|}\left|\int_B\tilde{u}_{\epsilon}\,dx\right|\leq K_4\epsilon^2,\nonumber\end{aligned}$$ where $K_3,K_4>0$ are positive constants independent of $\epsilon\in(0,\epsilon_0)$. Therefore, from , , and , it follows that $$\begin{aligned}
\label{estimate_1}
\lambda_2(B)&\leq\frac{\int_{B\cap\Omega_{\epsilon}}|D^2u_{\epsilon}|^2+\tau|Du_{\epsilon}|^2dx+\int_{B\setminus\Omega_{\epsilon}}|D^2 v_{\epsilon}|^2+\tau|D v_{\epsilon}|^2\,dx}{\int_B\tilde{u}_{\epsilon} ^2dx-K_2\epsilon^2}\\
&\le\frac{\lambda_2(\Omega_{\epsilon})+\int_{B\setminus\Omega_{\epsilon}}|D^2v_{\epsilon}|^2+\tau|Dv_{\epsilon}|^2dx-\int_{\Omega_{\epsilon}\setminus B}|D^2 u_{\epsilon}|^2+\tau|Du_{\epsilon}|^2\,dx}{1+\int_{B\setminus\Omega_{\epsilon}}v_{\epsilon} ^2dx-\int_{\Omega_{\epsilon}\setminus B}u_{\epsilon}^2dx-(K_2+K_4)\epsilon^2}.\nonumber\end{aligned}$$ We introduce now the two error terms $R_1(\epsilon)$ and $R_2(\epsilon)$ defined by $$R_1(\epsilon):=\int_{B\setminus\Omega_{\epsilon}}|D^2v_{\epsilon}|^2+\tau|Dv_{\epsilon}|^2dx-\int_{\Omega_{\epsilon}\setminus B}|D^2u_{\epsilon}|^2+\tau|Du_{\epsilon}|^2\,dx$$ and $$R_2(\epsilon):=\int_{B\setminus\Omega_{\epsilon}}v_{\epsilon} ^2dx-\int_{\Omega_{\epsilon}\setminus B}u_{\epsilon}^2dx.$$ Then inequality can be rewritten as $$\label{estimate_2}
\lambda_2(B)\leq\frac{\lambda_2(\Omega_{\epsilon})+R_1(\epsilon)}{1+R_2(\epsilon)-K_5\epsilon^2}.$$ From the uniform estimates and on $u_{\epsilon}$ and $v_{\epsilon}$, it easily follows that $R_1,R_2\in O(\epsilon)$ as $\epsilon\rightarrow 0$, which together with immediately yields $\lambda_2(B)\leq\lambda_2(\Omega_{\epsilon})+C\epsilon$ for some constant $C>0$ independent of $\epsilon\in(0,\epsilon_0)$ (taking $\epsilon_0>0$ smaller if necessary).
We observe that, due to the strict relation of $R_1(\epsilon)$ and $R_2(\epsilon)$ with the difference $\lambda_2(B)-\lambda_2(\Omega_{\epsilon})$, a better estimate for $R_1(\epsilon)$ and $R_2(\epsilon)$ provides a better estimate for $\lambda_2(B)-\lambda_2(\Omega_{\epsilon})$. More precisely, we have the following
\[lemma\_refinement\] Let $\omega:[0,1]\rightarrow[0,+\infty)$ be a continuous function such that $t^2/K\leq\omega(t)\leq K t$, for some $K>0$. If there exists a constant $C>0$ such that $|R_1(\epsilon)|$, $|R_2(\epsilon)|\leq C\omega(\epsilon)$, then there exists a constant $C'>0$ such that $$\lambda_2(B)\leq\lambda_2(\Omega_{\epsilon})+C'\omega(\epsilon)$$ for every sufficient small $\epsilon>0$.
We refer to [@brascopratelli Lemma 6.2] for the proof (see also [@brascosteklov Lemma 6.7]).
We also need the following
\[lemma2\] Let $\omega$ be a function as in Lemma \[lemma\_refinement\], and let $v_{\epsilon}$ be as in . Suppose that there exists $C>0$ such that for all $\epsilon>0$ sufficiently small we have $|R_1(\epsilon)|, |R_2(\epsilon)|\leq C\omega(\epsilon)$. Then there exists an eigenfunction $\xi_{\epsilon}$ associated with $\lambda_2(B)$ such that $$\|v_{\epsilon} -\xi_{\epsilon}\|_{C^3(\overline B)}\leq\tilde C\sqrt{\omega(\epsilon)}$$ for some $\tilde C>0$ independent of $\epsilon>0$.
Take $\{\xi_n\}_{n\geq 1}$ to be an orthonormal basis of $L^2(B)$ consisting of eigenfunctions of problem on the unit ball $B$. Note that from such a normalization, we have $$\int_B|D^2\xi_n|^2+\tau|D\xi_n|^2\,dx=\lambda_n(B)\,\quad\forall n\in\mathbb{N}.$$
We may write $v_{\epsilon} =\sum_{n=1}^{+\infty}a_n(\epsilon)\xi_n$. Note that $a_1(\epsilon)\equiv 0$, since $v_{\epsilon}$ has zero integral mean over $B$ and $\xi_1$ is a constant. We have $$\begin{aligned}
\sum_{n=2}^{+\infty}a_n(\epsilon)^2-1&=\|v_{\epsilon} \|_{L^2(B)}^2-1=\int_Bv_{\epsilon} ^2dx-\int_{\Omega_{\epsilon}}u_{\epsilon}^2dx\\
&=\int_B(v_{\epsilon}^2-\tilde{u}_{\epsilon}^2)dx-\int_{B\setminus\Omega_{\epsilon}}(v_{\epsilon} ^2-\tilde{u}_{\epsilon} ^2)dx+R_2(\epsilon).\end{aligned}$$ Then by using , , we obtain $$\label{asterisco}
\left|\sum_{n=2}^{+\infty}a_n(\epsilon)^2-1\right|\leq K_5\epsilon^2+C\omega(\epsilon)\leq C_1\omega(\epsilon).$$ We may now write $$\begin{aligned}
\lambda_2(\Omega_{\epsilon})&=\int_{\Omega_{\epsilon}}|D^2u_{\epsilon}|^2+\tau|Du_{\epsilon}|^2dx\\
&=\int_B|D^2v_{\epsilon}|^2+\tau|Dv_{\epsilon}|^2\,dx+\int_{\Omega_{\epsilon}\setminus B}|D^2u_{\epsilon}|^2+\tau|Du_{\epsilon}|^2dx\\
&\qquad-\int_{B\setminus\Omega_{\epsilon}}|D^2v_{\epsilon}|^2+\tau|Dv_{\epsilon}|^2\,dx\\
&=\sum_{n=2}^{+\infty}a_n(\epsilon)^2\lambda_n(B)-R_1(\epsilon). \end{aligned}$$ From Lemma \[lemma\_refinement\], it follows that $$|\lambda_2(B)-\lambda_2(\Omega_{\epsilon})|\leq C'\omega(\epsilon),$$ and therefore, $$\label{star1}
\left|\sum_{n=2}^{+\infty}a_n(\epsilon)^2\lambda_n(B)-\lambda_2(B)\right|=|\lambda_2(\Omega_{\epsilon})+R_1(\epsilon)-\lambda_2(B)|\leq C_2\omega(\epsilon).$$
By the symmetry of the ball, the first nonzero eigenvalue $\lambda_2(B)$ has multiplicity $N$, and so $\lambda_2(B)=\lambda_3(B)=\cdots=\lambda_{N+1}(B)<\lambda_{N+2}(B)$. Therefore, $$\begin{aligned}
C_2\omega(\epsilon)
&\geq\left|\sum_{n=2}^{N+1}a_n(\epsilon)^2\lambda_2(B)+\sum_{n=N+2}^{+\infty}a_n(\epsilon)^2\lambda_n(B)-\lambda_2(B)\right|\\
&=\left|\lambda_2(B)\left(\sum_{n=2}^{+\infty}a_n(\epsilon)^2-1\right)+\sum_{n=N+2}^{+\infty}a_n(\epsilon)^2\left(\lambda_n(B)-\lambda_2(B)\right)\right|\\
&\geq\left(\lambda_{N+2}(B)-\lambda_2(B)\right)\sum_{n=N+2}^{+\infty}a_n(\epsilon)^2-\lambda_2(B)C_1\omega(\epsilon),\end{aligned}$$ which yields $$\label{C3}
\sum_{n=N+2}^{+\infty}a_n(\epsilon)^2\leq C_3\omega(\epsilon),$$ and hence by , $$\label{C4}
\left|\sum_{n=2}^{N+1}a_n(\epsilon)^2-1\right|\leq C_4\omega(\epsilon).$$ Revisiting , we see that $$\begin{aligned}
C_2\omega(\varepsilon)&\geq\left|\sum_{n=2}^{N+1}a_n(\epsilon)^2\lambda_2(B)+\sum_{n=N+2}^{+\infty}a_n(\epsilon)^2\lambda_n(B)-\lambda_2(B)\right|\\
&=\left|\lambda_2(B)\left(\sum_{n=2}^{N+1}a_n(\epsilon)^2-1\right)+\sum_{n=N+2}^{+\infty}a_n(\epsilon)^2\lambda_n(B)\right|\\
&\geq\lambda_2(B)\left(\sum_{n=2}^{N+1}a_n(\epsilon)^2-1\right)+\sum_{n=N+2}^{+\infty}a_n(\epsilon)^2\lambda_n(B),\end{aligned}$$ which, together with and , yields $$\label{star2}
\sum_{n=N+2}^{+\infty}a_n(\epsilon)^2\lambda_n(B)\leq C_2\omega(\epsilon)-\lambda_2(B)\left(\sum_{n=2}^{N+1}a_n(\epsilon)^2-1\right)\leq C_5\omega(\epsilon).$$
Now set $\varphi:=\sum_{n=2}^{N+1}a_n(\epsilon)\xi_n$ and define the norm $\|\cdot\|_{H^2_{\tau}(B)}$ by $$\|h\|_{H^2_{\tau}(B)}^2:=\int_B |D^2h|^2+\tau|Dh|^2+h^2\,dx,\qquad \forall h\in H^2(B).$$ This norm is equivalent to the standard $H^2(B)$-norm by coercivity of the bilinear form.
We now estimate the quantity $\|v_{\epsilon} -\varphi\|_{H^{2}_{\tau}(B)}$. We have $$\begin{aligned}
\|v_{\epsilon} -\varphi\|^2_{H^2_{\tau}(B)}&=\int_B|D^2(v_{\epsilon} -\varphi)|^2+\tau|D(v_{\epsilon} -\varphi)|^2+(v_{\epsilon} -\varphi)^2dx\\
&=\int_B\sum_{n=N+2}^{+\infty}a_n(\epsilon)^2(|D^2\xi_n|^2+\tau|D\xi_n|^2+\xi_n^2)dx\\
&=\sum_{n=N+2}^{+\infty}a_n(\epsilon)^2(1+\lambda_n(B))\leq C_6\omega(\epsilon),\end{aligned}$$ where the last inequality follows from and . Thus the function $v_{\epsilon}$ is $\sqrt{\omega(\epsilon)}$-close to $\varphi$ in the $H^2_{\tau}(B)$-norm.
We want to bound the $C^3(\overline B)$-norm with the $H^2_{\tau}(B)$-norm. To do so, we use standard elliptic regularity estimates for the biharmonic operator. We have that, in $B\cap\Omega_{\epsilon}$, $$\Delta^2 v_{\epsilon} -\tau\Delta v_{\epsilon} =\Delta^2u_{\epsilon} -\tau\Delta u_{\epsilon} =\lambda_2(\Omega_{\epsilon})u_{\epsilon} =\lambda_2(\Omega_{\epsilon})(v_{\epsilon} +\delta).$$ Recall that $\delta\in O(\epsilon)$ as $\epsilon\rightarrow 0$ from . We set $$%\label{feps}
f_{\epsilon}:=\Delta^2v_{\epsilon} -\tau\Delta v_{\epsilon} .$$ Note that, in particular, $f_{\epsilon}=\lambda_2(\Omega_{\epsilon})(v_{\epsilon} +\delta)$ on $B\cap\Omega_{\epsilon}$. Then defining the functions $g_{\epsilon}^{(1)}$ and $g_{\epsilon}^{(2)}$ on $\partial B$ by $g_{\epsilon}^{(1)}:=\frac{\partial^2v_{\epsilon} }{\partial\nu^2}$ and $g_{\epsilon}^{(2)}:=\tau\frac{\partial v_{\epsilon} }{\partial\nu}-{\rm div}_{\partial B}(D^2 v_{\epsilon} \cdot\nu)-\frac{\partial\Delta v_{\epsilon} }{\partial\nu}$, we see that the function $v_{\epsilon}$ uniquely solves the problem $$%\label{auxiliarypb}
\begin{cases}
\Delta^2u-\tau\Delta u=f_{\epsilon}, & {\rm in}\ B,\\
\frac{\partial^2u}{\partial\nu^2}=g_{\epsilon}^{(1)}, & {\rm on}\ \partial B,\\
\tau\frac{\partial u}{\partial\nu}-{\rm div}_{\partial B}(D^2u\cdot\nu)-\frac{\partial\Delta u}{\partial\nu}=g_{\epsilon}^{(2)}, & {\rm on}\ \partial B,\\
\int_{B}udx=0.
\end{cases}$$ Now let $f:=\lambda_2(B)\varphi$. Then by definition, the function $\varphi$ is the unique solution of $$\begin{cases}
\Delta^2 u-\tau\Delta u=f, & {\rm in}\ B,\\
\frac{\partial^2u}{\partial\nu^2}=0, & {\rm on}\ \partial B,\\
\tau\frac{\partial u}{\partial\nu}-{\rm div}_{\partial B}(D^2u\cdot\nu)-\frac{\partial\Delta u}{\partial\nu}=0, & {\rm on}\ \partial B,\\
\int_B u\,dx=0.
\end{cases}$$ Finally, define the function $w:=v_{\epsilon} -\varphi$, which is the unique solution of $$\begin{cases}
\Delta^2 w-\tau\Delta w=f_{\epsilon}-f, & {\rm in}\ B,\\
\frac{\partial^2w}{\partial\nu^2}=g_{\epsilon}^{(1)}, & {\rm on}\ \partial B,\\
\tau\frac{\partial w}{\partial\nu}-{\rm div}_{\partial B}(D^2w\cdot\nu)-\frac{\partial\Delta w}{\partial\nu}=g_{\epsilon}^{(2)}, & {\rm on}\ \partial B,\\
\int_B w\,dx=0.
\end{cases}$$ For any $p>N$, we have (see e.g., [@gazzola Theorem 2.20]) $$\label{gazzola_estimate}
\|w\|_{W^{4,p}(B)}\leq C\left(\|f_{\epsilon}-f\|_{L^p(B)}+\|g_{\epsilon}^{(1)}\|_{W^{2-\frac{1}{p},p}(\partial B)}+\|g_{\epsilon}^{(2)}\|_{W^{1-\frac{1}{p},p}(\partial B)}\right).$$ We consider separately the three summands in the right-hand side of . We start from the first summand. Recall that for any $x\in B\cap \Omega_{\epsilon}$, we have (see ) $$f_{\epsilon}(x)=\lambda_2(\Omega_{\epsilon})(v_{\epsilon} (x)+\delta).$$ Since $\delta\in O(\epsilon)$ and $\lambda_2(\Omega_\epsilon)$ is bounded from above and from below, we have that $f_{\epsilon}(x)=\lambda_2(\Omega_{\epsilon})v_{\epsilon}(x)+O(\epsilon)$, and thus, as $\epsilon\rightarrow 0$, for any $p>N$, we have (cf. Lemma \[lemma\_refinement\]) $$\begin{aligned}
\label{gaz1}
\|f_{\epsilon}-f\|_{L^p(B)}&=\|\lambda_2(\Omega_{\epsilon})v_{\epsilon} -\lambda_2(B)\varphi\|_{L^p(B)}+O(\epsilon)\\
&\leq |\lambda_2(\Omega_{\epsilon})-\lambda_2(B)|\|v_{\epsilon} \|_{L^p(B)}+|\lambda_2(B)|\|v_{\epsilon} -\varphi\|_{L^p(B)}+O(\epsilon)\nonumber\\
&\leq C_7\omega(\epsilon)+C_8\sqrt{\omega(\epsilon)}+O(\epsilon)\leq C_9\sqrt{\omega(\epsilon)}.\nonumber\end{aligned}$$
Now we consider the second summand in the right-hand side of . Since $g_{\epsilon}^{(1)}=\frac{\partial^2v_{\epsilon} }{\partial\nu^2}$ and $v_{\epsilon}$ is an extension of $u_{\epsilon}$, by the regularity of both $u_{\epsilon}$ and $v_{\epsilon}$ (see , ) and from the fact that $\frac{\partial^2u_{\epsilon} }{\partial\nu^2}=0$ on $\partial\Omega_{\epsilon}$, we may conclude $$\label{gaz2}
\|g_{\epsilon}^{(1)}\|_{W^{2-\frac{1}{p},p}(\partial B)}\leq C\epsilon.$$ For the same reason, for the third summand in the right-hand side of we have $$\label{gaz3}
\|g_{\epsilon}^{(2)}\|_{W^{1-\frac{1}{p},p}(\partial B)}\leq C\epsilon.$$
From and the bounds , , and , it follows that, for any $p>N$, $$\|v_{\epsilon} -\varphi\|_{W^{4,p}(B)}\leq C_{10}\sqrt{\omega(\epsilon)},$$ and thus, from the Sobolev embedding theorem, $$\|v_{\epsilon} -\varphi\|_{C^{3}(\overline B)}\leq \tilde C\sqrt{\omega(\epsilon)}.$$ The proof is concluded by setting $\xi_{\varepsilon}=\varphi$.
The next lemma gives us refined bounds on $|R_1(\epsilon)|$ and $|R_2(\epsilon)|$.
\[lemma3\] Let $\omega(t), v_{\epsilon}$ be as in Lemma \[lemma\_refinement\]. Suppose that for all $\epsilon>0$ small enough there exists an eigenfunction $\xi_{\epsilon}$ associated with $\lambda_2(B)$ such that $$\label{hypo}
\|v_{\epsilon} -\xi_{\epsilon}\|_{C^3(\overline B)}\leq C \sqrt{\omega(\epsilon)},$$ for some $C>0$ which does not depend on $\epsilon>0$. Then there exists $\tilde C>0$ which does not depend on $\epsilon$ such that $|R_1(\epsilon)|,|R_2(\epsilon)|\leq\tilde C\epsilon\sqrt{\omega(\epsilon)}$.
It is convenient to use spherical coordinates $(r,\theta)\in\mathbb{R}_{+}\times\mathbb{S}^{N-1}$ in $\mathbb R^N$ and the corresponding change of variables $x=\phi(r,\theta)$. We denote by $\mathcal D$ and $\tilde{\mathcal D}$ the sets $\mathcal D:=\partial(\Omega_{\epsilon}\setminus B)\cap\partial B$ and $\tilde{\mathcal D}=\partial(B\setminus \Omega_{\epsilon})\cap\partial B$. Observe that $\psi\geq0$ on $\mathcal D$ and $\psi\le0$ on $\tilde{\mathcal D}$.
Thanks to the regularity of $u_{\epsilon}$ and $\tilde{u}_{\epsilon}$ by , on $\Omega_{\epsilon}\setminus B$ we have $$\begin{aligned}
D^2u_{\epsilon} \circ\phi(1+\epsilon\psi,\theta)&=&D^2u_{\epsilon} \circ\phi(1,\theta)+O(\epsilon),\\
Du_{\epsilon} \circ\phi(1+\epsilon\psi,\theta)&=&Du_{\epsilon} \circ\phi(1,\theta)+O(\epsilon),\end{aligned}$$ as $\epsilon\rightarrow 0$. Therefore, integrating with respect to the radius $r$ and applying the definition of $v_{\epsilon}$ , we see $$\begin{aligned}
\int_{\Omega_{\epsilon}\setminus B}|D^2u_{\epsilon}|^2+\tau|Du_{\epsilon}|^2dx
&=\epsilon\int_{\mathcal D }\psi\left(\left|D^2u_{\epsilon} \right|^2+\tau \left|Du_{\epsilon} \right|^2\right)d\sigma+O(\epsilon^2)\\
&=\epsilon\int_{\mathcal D }\psi\left(\left|D^2v_{\epsilon} \right|^2+\tau \left|Dv_{\epsilon} \right|^2\right)d\sigma+O(\epsilon^2),\end{aligned}$$ as $\epsilon\rightarrow 0$. Similarly, $$\int_{B\setminus\Omega_{\epsilon}}|D^2v_{\epsilon}|^2+\tau|Dv_{\epsilon}|^2dx
=-\epsilon\int_{\tilde{\mathcal D}}\psi\left(\left|D^2v_{\epsilon} \right|^2+\tau \left|Dv_{\epsilon} \right|^2\right)d\sigma+O(\epsilon^2),$$ as $\epsilon\rightarrow 0$. From these and hypothesis , we see $$\begin{aligned}
\label{ineqR1}
|R_1(\epsilon)|&\leq \epsilon\left|\int_{\partial B}\psi\left(\left|D^2v_{\epsilon} \right|^2+\tau \left|Dv_{\epsilon} \right|^2\right)d\sigma\right|+O(\epsilon^2)\\
&\leq \epsilon\left|\int_{\partial B}\psi\left(\left|D^2\xi_{\epsilon}\right|^2+\tau \left|D\xi_{\epsilon}\right|^2\right)d\sigma\right|+C\epsilon\sqrt{\omega(\epsilon)}+O(\epsilon^2)\nonumber\\
&\le \tilde{C}\epsilon\sqrt{\omega(\epsilon)},\nonumber\end{aligned}$$ as $\epsilon\rightarrow 0$. In the last inequality, we have used the following identity for eigenfunctions of $\lambda_2(B)$: $$\label{spherical_harmonic}
\left.\left(\left|D^2\xi_{\epsilon}\right|^2+\tau \left|D\xi_{\epsilon}\right|^2\right)\right|_{\partial B}=(a\cdot x)^2$$ for some $a\in\mathbb R^N$ (cf. ).
By following the same scheme, we can prove the analogue of for $R_2(\epsilon)$. This concludes the proof.
We can now proceed to complete the proof of Theorem \[theorem2\].
Let $\omega_0(\epsilon):=|R_1(\epsilon)|+|R_2(\epsilon)|$. This function is continuous in $\epsilon$ and, moreover, has the property $$\frac{\epsilon^2}{K}\leq\omega_0(\epsilon)\leq K\epsilon.$$ The first inequality follows from Theorem \[NeumannQI\], while the latter follows from the fact that $R_1,R_2\in O(\epsilon)$. By Lemma \[lemma2\], it follows that there exists an eigenfunction $\xi_{\epsilon}$ of the Neumann problem on $B$ associated with eigenvalue $\lambda_2(B)$ such that $$\|v_{\epsilon} -\xi_{\epsilon}\|_{C^3(\overline{B})}\leq C\sqrt{\omega_0(\epsilon)}.$$ Now we apply Lemma \[lemma3\], obtaining $$\omega_0(\epsilon)\leq2\tilde C\epsilon\sqrt{\omega_0(\epsilon)},$$ and therefore $$\sqrt{\omega_0(\epsilon)}=\frac{|R_1(\epsilon)|+|R_2(\epsilon)|}{\sqrt{\omega_0(\epsilon)}}\leq 2\tilde C\epsilon.$$ From this, it follows that $\omega_0(\epsilon)\leq 4\tilde C^2\epsilon^2$, and hence both $|R_1(\epsilon)|,|R_2(\epsilon)|\leq 4\tilde C^2\epsilon^2 $. Finally, we apply Lemma \[lemma\_refinement\] and obtain $$\lambda_2(B)\leq\lambda_2(\Omega_{\epsilon})+\mathcal C\epsilon^2$$ for a constant $\mathcal C>0$ independent of $\epsilon\in(0,\epsilon_0)$. This concludes the proof of Theorem \[theorem2\].
\[directions\] In [@brascopratelli], the authors provided an explicit construction of a family $\{\Omega_{\epsilon}\}_{\epsilon}$ in $\mathbb R^2$ suitable for proving the sharpness of their [quantitative isoperimetric inequality for the fundamental tone of the Neumann Laplacian]{}. On the other hand, in [@brascosteklov], the authors gave only sufficient conditions to generate the family $\{\Omega_{\epsilon}\}_{\epsilon}$, which are exactly those we apply in . We observe that the first two conditions, namely $$\label{ellipses}
\int_{\partial B}\psi d\sigma=\int_{\partial B}(a\cdot x)\psi d\sigma=0,$$ have a purely geometrical meaning, and are used to prove inequalities and (cf. [@brascosteklov Lemma 6.2]). The latter has a stricter relation with the problem, since any function $\xi$ belonging to the eigenspace associated with $\lambda_2(B)$ satisfies equality . This is due to the fact that $\xi$ can be expressed as a radial part times a spherical harmonic polynomial of degree $1$. This also tells us that the correct conditions to impose when considering the Steklov problem are still . In particular, as pointed out in [@brascosteklov Remark 6.9], ellipsoids satisfy conditions , and hence inequalities and hold, but miss the final condition, and therefore are not a suitable family for this problem. Note that for the Dirichlet Laplacian case in [@brasco2015], ellipsoids are a suitable family for proving the sharpness, and therefore conditions are sufficient.
We also observe that in [@brasco2015], the construction is somewhat more general (cf. [@brasco2015 Theorem 3.3, pp. 1788-1789]), while the perturbation used in [@brascopratelli] does not belong to . This means that it is possible to state less-restrictive conditions which would produce families of domains achieving the sharpness.
Sharpness of the Steklov inequality {#sharpness_steklov}
===================================
In this section, we prove the sharpness of inequality . Due to the strong similarities between the Steklov problem and the Neumann problem , we shall maintain the same notation as in the previous section.
Let $B$ be the unit ball in $\mathbb{R}^N$ centered at zero. There exist a family $\left\{\Omega_{\epsilon}\right\}_{\epsilon>0}$ of smooth domains and positive constants $c_1,c_2,c_3,c_4$ and $r_1, r_2,r_3,r_4$ independent of $\epsilon>0$ such that $$%\label{sharp1}
r_1\epsilon^2\le\Big||\Omega_{\epsilon}|-|B|\Big|\leq r_2\epsilon^2,$$ $$%\label{sharp2}
c_1\epsilon\leq c_2\mathcal A(\Omega_{\epsilon})\leq\frac{|\Omega_{\epsilon}\triangle B|}{|\Omega_{\epsilon}|}\leq c_3\mathcal A(\Omega_{\epsilon})\leq c_4\epsilon,$$ and $$\label{sharp4}
r_3\epsilon^2\le\left|\lambda_2(\Omega_{\epsilon})-\lambda_2(B)\right|\leq r_4\epsilon^2,$$ for all $\epsilon\in(0,\epsilon_0)$, where $\epsilon_0>0$ is sufficiently small, [and $\lambda_2(\Omega_\epsilon)$, $\lambda_2(B)$ is the first positive eigenvalue of problem on $\Omega_\epsilon$, $B$ respectively]{}.
To prove this theorem, we begin by defining the family $\left\{\Omega_{\epsilon} \right\}_{\epsilon>0}$ as in . Thus it remains only to prove .
We remind the reader of the variational characterization of the first positive eigenvalue of the Steklov problem on a domain $\Omega$: $$\label{steklov-ray}
\lambda_2(\Omega)=\inf_{\substack{0\ne u\in H^2(\Omega)\\ \int_{\partial\Omega}u \,d\sigma=0}}\frac{\int_{\Omega}|D^2u|^2+\tau|Du|^2\,dx}{\int_{\partial\Omega}u^2\,d\sigma}.$$
We take the first positive eigenvalue $\lambda_2(\Omega_{\epsilon} )$ of the Steklov problem on $\Omega_{\epsilon} $, and let $u_{\epsilon} $ be an associated eigenfunction, normalized by $$%\label{normalization1}
\int_{\partial\Omega_{\epsilon} }u_{\epsilon} ^2dx=1.$$ Then by the variational characterization , $$%\label{normalization2}
\int_{\Omega_{\epsilon} }|D^2u_{\epsilon} |^2+\tau|\nabla u_{\epsilon} |^2 dx=\lambda_2(\Omega_{\epsilon} ).$$ By standard elliptic regularity (see e.g., [@gazzola §2.4.3]), since $\Omega_{\epsilon} $ is of class $C^{\infty}$ by construction, we have that $u_{\epsilon} \in C^{\infty}(\overline{\Omega_{\epsilon}})$ for all $\epsilon\in(0,\epsilon_0)$. Moreover, for all $k\in\mathbb N$, the sets $\Omega_{\epsilon} $ are of class $C^k$ uniformly in $\epsilon\in(0,\epsilon_0)$, which means that there exist constants $H_k>0$ independent of $\epsilon$ such that $$%\label{regularity_s}
\|u_{\epsilon} \|_{C^k(\overline{\Omega_{\epsilon}} )}\leq H_k.$$ Let now $\tilde{u}_{\epsilon} $ be a $C^4$ extension of $u_{\epsilon} $ to an open neighborhood $A$ of $B\cup\Omega_{\epsilon} $. Then, there exists $K_A>0$ independent of $\epsilon>0$ such that $$%\label{regularity_extension_s}
\|\tilde{u}_{\epsilon} \|_{C^4(\overline A)}\leq K_A\|u_{\epsilon} \|_{C^4(\overline{\Omega_{\epsilon}} )}\leq K_A H_4.$$ Analogous to the Neumann case, take $\delta:=\frac{1}{|\partial B|}\int_{\partial B}\tilde{u}_{\epsilon} \,d\sigma$ to be the mean of $\tilde{u}_{\epsilon}$ over $\partial B$. From the fact that $\int_{\partial\Omega_{\epsilon} }u_{\epsilon} dx=0$ and $|B\setminus\Omega_{\epsilon} |,|\Omega_{\epsilon} \setminus B|\in O(\epsilon)$ as $\epsilon\rightarrow 0$, it follows that, as $\epsilon\rightarrow0$ (see also [@brascosteklov formula (6.15)]), $$\delta=\frac{1}{|\partial B|}\int_{\partial B}\tilde{u}_{\epsilon} \,d\sigma\in O(\epsilon).$$ Now let us set $v_{\epsilon}:=\tilde u_{\epsilon|_B}-\delta$. This function is of class $C^4(\overline B)$, satisfies $\int_{\partial B} v_{\epsilon} \,d\sigma=0$, and $$%\label{regularity_v_s}
\|v_{\epsilon} \|_{C^4(\overline B)}\leq K'$$ for a constant $K'>0$ independent of $\epsilon\in(0,\epsilon_0)$. Therefore, $v_{\epsilon} $ is a suitable trial function for the Rayleigh quotient of $\lambda_2(B)$, hence, $$%\label{minmax_1_s}
\lambda_2(B)\leq\frac{\int_B |D^2 v_{\epsilon} |^2+\tau|\nabla v_{\epsilon} |^2 dx}{\int_{\partial B} {v_{\epsilon} }^2 \,d\sigma}.$$ On the other hand, $$%\label{ineq_1}
\left|\int_{\partial B} v_{\epsilon} ^2-\tilde{u}_{\epsilon} ^2 \,d\sigma\right|=\left|\int_{\partial B} \delta^2-2\delta\tilde{u}_{\epsilon} \,d\sigma\right|\leq K''\epsilon^2,$$ where $K''>0$ is a positive constant independent of $\epsilon\in(0,\epsilon_0)$. Therefore, we may write $$%\label{estimate_2}
\lambda_2(B)\leq\frac{\lambda_2(\Omega_{\epsilon} )+R_1(\epsilon)}{1+R_2(\epsilon)-\tilde K\epsilon^2},$$ where we have once again defined the error terms $$%\label{R1}
R_1(\epsilon):=\int_{B\setminus\Omega_{\epsilon} }|D^2v_{\epsilon} |^2+\tau|\nabla v_{\epsilon} |^2dx-\int_{\Omega_{\epsilon} \setminus B}|D^2u_{\epsilon} |^2+\tau|\nabla u_{\epsilon} |^2 dx,$$ and $$R_2(\epsilon):=\int_{\partial B}v_{\epsilon} ^2\,d\sigma-\int_{\partial \Omega_{\epsilon} }u_{\epsilon} ^2\,d\sigma.$$
At this point, we note that the observations made in Section \[sharpness\_neumann\] remain valid here. Therefore, in order to conclude the proof of , we need only a few lemmas.
Let $\omega$ be as in Lemma \[lemma\_refinement\]. If there exists a constant $C>0$ such that $|R_1(\epsilon)|, |R_2(\epsilon)|\leq C\omega(\epsilon)$, then there exists a constant $C'>0$ such that $$\lambda_2(B)\leq\lambda_2(\Omega_{\epsilon} )+C'\omega(\epsilon)$$ for every $\epsilon>0$ sufficiently small.
See [@brascosteklov Lemma 6.7].
Let $\omega$ be as in Lemma \[lemma\_refinement\]. Suppose that there exists $C>0$ such that for all $\epsilon>0$ sufficiently small we have $|R_1(\epsilon)|, |R_2(\epsilon)|\leq C\omega(\epsilon)$. Then there exists an eigenfunction $\xi_{\epsilon}$ associated with $\lambda_2(B)$ such that $$\|v_{\epsilon} -\xi_{\epsilon}\|_{C^3(\overline B)}\leq\tilde C\sqrt{\omega(\epsilon)},$$ for some $\tilde C>0$ independent of $\epsilon>0$.
The proof is essentially identical to that of Lemma \[lemma2\] and hence the details are omitted. Some small changes are necessary since $L^2(\Omega)$-norms have to be replaced by $L^2(\partial\Omega)$-norms, since we are considering the Steklov problem.
Let $\omega$ be as in Lemma \[lemma\_refinement\]. Suppose that for all $\epsilon>0$ sufficiently small there exists an eigenfunction $\xi_{\epsilon}$ associated with $\lambda_2(B)$ such that $$%\label{hypo}
\|v_{\epsilon} -\xi_{\epsilon}\|_{C^3(\overline B)}\leq C \sqrt{\omega(\epsilon)},$$ for some $C>0$ independent of $\epsilon>0$. Then there exists $\tilde C>0$ independent of $\epsilon$ such that $|R_1(\epsilon)|,|R_2(\epsilon)|\leq\tilde C\epsilon\sqrt{\omega(\epsilon)}$.
Regarding the bound on $R_1$, we refer to the proof of Lemma \[lemma3\]. For $R_2$, we refer to [@brascosteklov Lemma 6.8, p. 4701], observing that if $\xi_{\epsilon}$ is an eigenfunction associated with $\lambda_2(B)$, then on $\partial B$, $${\rm div}_{\partial B}(D^2\xi_{\epsilon}\cdot\nu)+\frac{\partial\Delta\xi_{\epsilon}}{\partial\nu}=0,$$ and therefore the second boundary condition in reads as $\partial\xi_{\epsilon}/\partial\nu=\lambda_2(B)\xi_{\epsilon}$.
Acknowledgments {#acknowledgments .unnumbered}
===============
The first and the third author wish to thank Berardo Ruffini for discussions on his paper [@brascosteklov]. The first author has been partially supported by the research project FIR (Futuro in Ricerca) 2013 ‘Geometrical and qualitative aspects of PDE’s’. The third author acknowledges financial support from the research project ‘Singular perturbation problems for differential operators’ Progetto di Ateneo of the University of Padova and from the research project ‘INdAM GNAMPA Project 2015 - Un approccio funzionale analitico per problemi di perturbazione singolare e di omogeneizzazione’. The first and the third author are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Observations of nearby galaxies have firmly established, over a broad range of galactic environments and metallicities, that star formation occurs exclusively in the molecular phase of the interstellar medium (ISM). Theoretical models show that this association results from the correlation between chemical phase, shielding, and temperature. Interstellar gas converts from atomic to molecular only in regions that are well shielded from interstellar ultraviolet (UV) photons, and since UV photons are also the dominant source of interstellar heating, only in these shielded regions does the gas become cold enough to be subject to Jeans instability. However, while the equilibrium temperature and chemical state of interstellar gas are well-correlated, the time scale required to reach chemical equilibrium is much longer than that required to reach thermal equilibrium, and both timescales are metallicity-dependent. Here I show that the difference in time scales implies that, at metallicities below a few percent of the Solar value, well-shielded gas will reach low temperatures and proceed to star formation before the bulk of it is able to convert from atomic to molecular. As a result, at extremely low metallicities, star formation will occur in a cold atomic phase of the ISM rather than a molecular phase. I calculate the observable consequences of this result for star formation in low metallicity galaxies, and I discuss how some current numerical models for H$_2$-regulated star-formation may need to be modified.'
author:
- 'Mark R. Krumholz'
title: Star Formation in Atomic Gas
---
Introduction
============
In present day galaxies, star formation is very well-correlated with the molecular phase of the interstellar medium (ISM) [@wong02a; @kennicutt07a; @leroy08a; @bigiel08a]. In contrast, in the inner parts of disks where there are significant molecular fractions, star formation correlates very poorly or not at all with the atomic ISM. At large galctocentric radii where the ISM becomes atomic-dominated star formation does begin to correlate with H <span style="font-variant:small-caps;">i</span>, but this appears to be only because H$_2$ itself becomes correlated with H <span style="font-variant:small-caps;">i</span>, and the H$_2$ forms stars in the same way regardless of where it is found within a galaxy [@bigiel10a; @schruba11a]. Strong association between star formation and H$_2$ and a lack of association with H <span style="font-variant:small-caps;">i</span> is also found down the lowest metallicity systems that have been measured, at roughly 20% of Solar [@bolatto11a]. In summary, all available observational data indicates that star formation occurs only where the hydrogen in the ISM has converted to H$_2$.
Theoretical models have explained these observations as resulting from a correlation between chemistry and temperature [@schaye04a; @krumholz11b; @glover12a]. Molecular hydrogen is not an important coolant in modern-day galaxies, and while carbon monoxide (which forms only when it is catalyzed by H$_2$ – @van-dishoeck86a) is, the C <span style="font-variant:small-caps;">ii</span> found in H <span style="font-variant:small-caps;">i</span> regions is almost as effective. However, H$_2$ is an excellent proxy for the presence of cold gas because both are sensitive to destruction by UV photons, which photodissociate H$_2$ and increase the temperature through the grain photoelectric effect. As a result, both H$_2$ and low temperature gas are found only in regions of high extinction where the UV photon density is far below its mean value in the ISM, and, conversely, any region that where the photodissociation rate is high enough to convert the bulk of the ISM to H <span style="font-variant:small-caps;">i</span> is also likely to be warm. Since low temperatures that remove thermal pressure support are a prerequisite for collapse into stars, this correlation between temperature and chemical state in turn induces a correlation between star formation and chemical state.
However, the correlation between H$_2$ and star formation must break down at sufficiently low metallicities. Before the first stars formed in the universe, and for a short time thereafter, there were no or very few heavy elements. As a result, forming H$_2$ was extremely difficult due to a lack of dust grain surfaces to catalyze the ${\rm H~\textsc{i}} \rightarrow {\rm H}_2$ reaction. Theoretical models of star formation in such environments indicate that H$_2$ fractions remain extremely small until the density rises so high (${\protect\raisebox{-0.5ex}{$\:\stackrel{\textstyle >}
{\sim}\:$}}10^9$ cm$^{-3}$) that H$_2$ can form via three-body reactions [@palla83a; @lepp84a; @ahn07a; @omukai10a]. The underlying physical basis for this result is a disconnect of timescales: the equilibrium chemical state the gas would reach after a very long time would be H$_2$-dominated, but the cooling and star formation times are short enough that the gas does not reach equilibrium before collapsing into a star.
While this result has been known for zero and extremely low metallicity systems for some time, the relationship between chemical state and star formation in intermediate metallicity regime, for which observations are at least in principle possible in the local universe, has received fairly little attention. @omukai10a consider the chemical evolution of collapsing gas cores with metallicities from 0 to Solar, and investigate under what circumstances they can form H$_2$. However, because their calculation starts with unstable, collapsing cores, it does not address the question of in what phase of the ISM one expects to find such collapsing regions in the first place, which is the central problem for understanding the observed galactic-scale correlation between ISM chemical state and star formation. @glover12b simulate the non-equilibrium chemical and thermal behavior of clouds with metallicities from 1% of Solar to Solar. They find that the bulk of the cloud material converts to H$_2$ before star formation in the high metallicity clouds but not in the lowest metallicity ones, indicating that the star formation - H$_2$ correlation should begin to break down at metallicities observable in nearby galaxies. However, given the computational cost of their simulations, they are able to explore a very limited number of cases, and it is unclear how general their results might be.
The goal of this paper is to go beyond the studies of @omukai10a and @glover12b by deriving general results about the correlation between chemical state and star formation over a wide range of environments and metallicities. I do not perform detailed simulations, such as those of @omukai10a and @glover12b, for every case. Instead, I rely on fairly simple models that can be integrated semi-analytically. The benefit of this approach is that it is the only way to survey a large parameter space, and thereby to answer the central questions with which I am concerned: under what conditions do we expect the correlation between star formation and H$_2$ to break down? When such a breakdown occurs, what is the governing physical mechanism that causes it? What are the resulting observational signatures? What are the implications of this breakdown for the models of star formation commonly adopted in studies of galaxy formation? In the remainder of this paper, I seek to answer these questions.
Model
=====
Consider spatially uniform gas characterized by a mean number density of H nuclei ${\overline{n}_{\rm H}}$, column density of hydrogen nuclei ${N_{\rm H}}$, metallicity $Z'$ relative to Solar, and temperature $T$. A fraction $f_{\rm H_2}$ of the H nuclei are locked in H$_2$ molecules. It is generally more convenient to characterize models by values of the visual extinction $A_V$ instead of ${N_{\rm H}}$. These two are related by $A_V/{N_{\rm H}}\approx 4.0 \times 10^{-22} Z'$ mag cm$^2$, with the normalization chosen as a compromise between the values for Milky Way extinction and the extinction curves of the Large and Small Magellanic Clouds adjusted to Milky Way metallicity.[^1]
Timescale Estimates
-------------------
We are interested in following the behavior of initially warm, atomic gas, and considering whether it will be able to cool to temperatures low enough to allow star formation, and how its chemical state will evolve as it does so. Before computing detailed evolutionary histories, it is helpful first to make a rough estimate of the timescales involved. In interstellar gas that is dense enough to be a candidate for star formation, but that is not yet molecular or forming stars, the dominant cooling process is emission in the \[C <span style="font-variant:small-caps;">ii</span>\] 158 $\mu$m line, which removes energy at a rate $$\Lambda_{\rm CII} \approx k_{\rm CII-H} {\delta_{\rm C}}k_B {T_{\rm CII}}{\mathcal{C}}{\overline{n}_{\rm H}}$$ per H atom, where $k_{\rm CII-H}\approx 8\times 10^{-10} e^{-{T_{\rm CII}}/T}$ cm$^3$ s$^{-1}$ is the rate coefficient for collisional excitation of C <span style="font-variant:small-caps;">ii</span> by H atoms, ${\delta_{\rm C}}\approx 1.1\times 10^{-4} Z'$ is the gas phase carbon abundance, ${T_{\rm CII}}=91$ K is the energy of the excited C <span style="font-variant:small-caps;">ii</span> level over $k_B$, and ${\mathcal{C}}= \langle n_{\rm H}^2 \rangle / {\overline{n}_{\rm H}}^2$ is a clumping factor that accounts for clumping of the medium on size scales below that on which we are computing the average. This expression assumes that C <span style="font-variant:small-caps;">ii</span> collisional excitation is dominated by H rather than by free electrons, that the gas is optically thin, and that the density is far below the critical density for the line; I show below that all these assumptions are valid. The time required for the gas to reach thermal equilibrium is of order $$\begin{aligned}
t_{\rm therm} & \equiv & \frac{k_B T}{\Lambda_{\rm CII}} =
\frac{T}{k_{\rm CII-H} \delta_{\rm C} {T_{\rm CII}}{\mathcal{C}}{\overline{n}_{\rm H}}}
\nonumber \\
& = & 0.036 \left(\frac{T}{{T_{\rm CII}}}\right)e^{{T_{\rm CII}}/T} Z'^{-1} {\mathcal{C}}_{1}^{-1} n_0^{-1}\mbox{ Myr},\end{aligned}$$ where ${\mathcal{C}}_{1} = {\mathcal{C}}/10$ and $n_0 = {\overline{n}_{\rm H}}/1$ cm$^{-3}$. Note that the value of ${\mathcal{C}}$ will depend on the size scale over which the average density is defined; the fiducial value ${\mathcal{C}}= 10$ is intermediate between the values ${\mathcal{C}}\approx 2$ and ${\mathcal{C}}\approx 30$ that numerical experiments indicate are best for $\sim 10$ pc and $\sim 100$ pc scale, respectively [@gnedin09a; @mac-low12a].
Conversion of the gas from atomic to molecular form occurs primarily on the surface of dust grains down to metallicities as low as $\sim 10^{-5}$ of Solar [@omukai10a]. This process occurs at a rate per H atom ${\overline{n}_{\rm H}}{\mathcal{R}}{\mathcal{C}}$, where $\mathcal{R}\approx 3\times 10^{-17} Z'$ cm$^3$ s$^{-1}$ is the rate coefficient for H$_2$ formation on grain surfaces [@wolfire08a]. The associated timescale for conversion of the gas to molecular form is $$t_{\rm chem} \equiv \frac{1}{{\overline{n}_{\rm H}}{\mathcal{R}}{\mathcal{C}}} = 105 Z'^{-1} n_0^{-1} {\mathcal{C}}_{1}^{-1} \mbox{ Myr}.$$ The ratio of the two timescales is $$\frac{t_{\rm chem}}{t_{\rm therm}} = \frac{k_{\rm CII-H} \delta_{\rm C} {T_{\rm CII}}}{{\mathcal{R}}T} = 2900 \left(\frac{{T_{\rm CII}}}{T}\right) e^{-{T_{\rm CII}}/T},$$ indicating that the gas will reach thermal equilibrium vastly before it reaches chemical equilibrium.
This difference in timescale is only important if the cooling of gas toward thermal equilibrium is followed by star formation on a timescale that is too short for the conversion of atomic to molecular gas to keep up. It is therefore helpful to consider a third timescale: the free-fall time $$t_{\rm ff} = \sqrt{\frac{3\pi}{32 G {\overline{n}_{\rm H}}\mu_{\rm H} m_{\rm H}}} = 43 n_0^{-1/2}\mbox{ Myr},$$ where $\mu_{\rm H} \approx 1.4$ is the mean mass per H nucleus in units of the hydrogen mass $m_{\rm H}$. This is the timescale over which star formation should begin once gas is gravitationally unstable. The time for which a cloud survives after the onset of star formation is significantly uncertain, but even the longest modern estimates are $\sim 10 t_{\rm ff}$, while some are as short as $\sim 1 t_{\rm ff}$ [@elmegreen00a; @tan06a; @kawamura09a; @goldbaum11a]. Comparing this timescale to the two previously computed gives $$\begin{aligned}
\frac{t_{\rm therm}}{t_{\rm ff}} & = & 8.3\times 10^{-4} \left(\frac{T}{{T_{\rm CII}}}\right)e^{{T_{\rm CII}}/T} Z'^{-1} {\mathcal{C}}_{1}^{-1} n_0^{-1/2}
\\
\frac{t_{\rm chem}}{t_{\rm ff}} & = & 2.4 Z'^{-1} {\mathcal{C}}_{1}^{-1} n_0^{-1/2}.\end{aligned}$$ Thus we see that the thermal timescale will be smaller than the free-fall timescale down to extremely low metallicities for reasonable ISM densities and temperatures, but the same cannot be said of the chemical timescale. For example, at $n_0 = 100$, $T = 1000$ K, $Z' = 10^{-3}$, and ${\mathcal{C}}_1=1$, the above equation gives $t_{\rm therm}/t_{\rm ff} = 1.0$, while $t_{\rm chem}/t_{\rm ff} = 240$. A cloud with these properties could cool and proceed to star formation on a free-fall timescale without difficulty, but would not build up a substantial amount of H$_2$ until more than 100 free-fall times. If such a cloud were anything like the observed star-forming clouds in the Milky Way, it would likely have been destroyed by stellar feedback well before this point. Note that individual overdense regions within the cloud in the process of collapsing to stars would still convert to H$_2$, since $t_{\rm chem}/t_{\rm ff}$ is a decreasing function of density, dramatically so once three-body reactions begin to occur; however, since the star formation efficiency is low, the non-star-forming bulk of the cloud material would not, and thus the amount of H$_2$ present per unit star formation would be greatly reduced.
Chemical and Thermal Evolution Models
-------------------------------------
The argument above is based on simple timescale estimates. To check whether the result is robust, it is necessary to construct more sophisticated cooling and chemistry models. Below I describe a more detailed model, and how it may be evaluated numerically to follow the thermal and chemical behavior of a cloud.
### Chemical Evolution
The H <span style="font-variant:small-caps;">i</span> to H$_2$ transition is governed by two main processes: formation of H$_2$ on the surfaces of dust grains and destruction of H$_2$ by photodissociation. The former occurs at a rate per H atom $n_{\rm H} {\mathcal{R}}$, where $n_{\rm H}$ is the local (rather than average) number density and ${\mathcal{R}}$ is the metallicity-dependent rate coefficient given in the main text. In a region of mean density ${\overline{n}_{\rm H}}$, the mean rate per H nucleus at which H <span style="font-variant:small-caps;">i</span> converts to H$_2$ is simply the number density-weighted average of the rate given above, which is ${\mathcal{C}}{\overline{n}_{\rm H}}{\mathcal{R}}$. The photodissociation rate per H$_2$ molcule is $$\zeta_{\rm diss} = \zeta_{\rm diss,0} e^{-\sigma_d {N_{\rm H}}} f_{\rm shield}({N_{\rm H_2}}),$$ where $\zeta_{\rm diss,0}\approx 5\times 10^{-11}$ s$^{-1}$ is the dissociation rate for unshielded gas [@draine96a], $\sigma_d\approx 10^{-21} Z'$ cm$^{-2}$ is the dust cross section per H nucleus for Lyman-Werner band photons, ${N_{\rm H}}$ is the column density of H nuclei, ${N_{\rm H_2}}$ is the column density of H$_2$ molecules, and $f_{\rm shield}({N_{\rm H_2}})$ is the shielding function that describes H$_2$ self-shielding. For the latter quantity, I use the approximate form of @draine96a, $$f_{\rm shield} \approx \frac{0.965}{(1+0.1x/b_6)^2} + \frac{0.035}{(1+x)^{0.5}} e^{-8.5\times 10^{-4}(1+x)^{0.5}},$$ where $x={N_{\rm H_2}}/5\times 10^{14}$ cm$^{-2}$ and $b_6$ is the Doppler parameter for the gas in units of $10^6$ cm s$^{-1}$; I use $b_6 = 0.71$, corresponding to a velocity dispersion of a 5 km s$^{-1}$, roughly that observed in molecular clouds in nearby galaxies, but the results are quite insensitive to this choice. The value of $\zeta_{\rm diss,0}$ is that appropriate for the Milky Way’s radiation field, and this value will vary from galaxy to galaxy and within galaxies. However, I show below that the dependence of the results on this choice is also quite weak, because of the exponential dependence of the dissociation rate on column density: a relatively large change in $\zeta_0$ can be compensated for by a far smaller change in ${N_{\rm H}}$ or ${N_{\rm H_2}}$ [also see @krumholz11b].
Given these processes, the rate of change of the H$_2$ fraction is given by $$\label{eq:dfh2dt}
\frac{d}{dt}f_{\rm H_2} = (1-f_{\rm H_2}) {\overline{n}_{\rm H}}{\mathcal{C}}{\mathcal{R}}- f_{\rm H_2} \zeta_{\rm diss}({N_{\rm H}},{N_{\rm H_2}}).$$ Consistent with the simple uniform cloud assumption, I adopt ${N_{\rm H_2}}= f_{\rm H_2} {N_{\rm H}}/2$. Note that there is no factor of 2 or $1/2$ in the second term due to a cancellation: there is one H$_2$ molecule per two H nuclei bound as H$_2$ (multiplying by a factor of $1/2$), but each dissociation generates two free H nuclei (multiplying by a factor of two).
### Thermal Evolution
The thermal evolution depends on heating and cooling processes. For heating, the dominant mechanisms are cosmic ray heating and the grain photoelectric effect. The photoelectric heating rate per H nucleus is $$\Gamma_{\rm PE} = \Gamma_{\rm PE,0} Z' e^{-\sigma_d {N_{\rm H}}}$$ where $\Gamma_{\rm PE,0} \approx 4\times 10^{-26}$ erg s$^{-1}$ is the grain photoelectric heating rate in free space, and the numerical value is for a Milky Way radiation field. Note that, since photoelectric heating is dominated by photons with energies similar to the Lyman-Werner bands, the dust cross section $\sigma_d$ here is the same as that used in the H$_2$ formation calculation. The heating rate per H nucleus from cosmic rays is $$\Gamma_{\rm CR} = \zeta_{\rm CR} q_{\rm CR}$$ where $\zeta_{\rm CR}$ is the primary cosmic ray ionization rate and $q_{\rm CR}$ is the energy added per primary cosmic ray ionization. For atomic gas, $q_{\rm CR} \approx 6.5$ eV [@dalgarno72a]. The observed primary cosmic ray ionization rate in the Milky Way varies sharply between diffuse sightlines and dark clouds; the former show $\zeta_{\rm CR} \approx 3\times 10^{-16}$ s$^{-1}$, with roughly a dex dispersion, while those in dark clouds are an order of magnitude lower, $\zeta_{\rm CR} \approx 2 \times 10^{-17}$ s$^{-1}$ [@wolfire10a; @neufeld10a; @indriolo12a]. Since cosmic ray heating is only significant compared to photoelectric heating in dark clouds, it seems more reasonable to adopt the latter as a fiducial value, although I verify below that this choice does not significantly affect the results. Observations also indicate that cosmic ray ionization rates vary roughly linearly with galactic star formation rates [@abdo10a]. For these reasons I adopt $\zeta_{\rm CR} = 2\times 10^{-17} Z'$ s$^{-1}$ as a fiducial value; the metallicity scaling is a very rough way of accounting for the lower cosmic ray flux in galaxies with lower metallicities, masses, and star formation rates [@krumholz11b].
Cooling for interstellar gas with temperatures $\ll 10^4$ K is dominated by the 158 $\mu$m fine structure line of C <span style="font-variant:small-caps;">ii</span> and the $63$ and $145$ $\mu$m fine structure lines of O <span style="font-variant:small-caps;">i</span>. The critical densities for these transitions are approximately $4\times 10^3$, $6\times 10^4$, and $3\times 10^5$ cm$^{-3}$, respectively assuming the dominant collision partner is H (see below). These critical densities are significantly higher than the densest cases I consider, and so I neglect collisional de-excitation compared to radiative de-excitation. Assuming all cloud atoms are in the ground state, the line-center optical depth of a cloud to photons emitted in one of these lines is $$\tau_0 = \frac{g_u}{g_\ell} \frac{A_{u\ell} \lambda_{u\ell}^3}{8 \pi^{3/2}b} \delta_X {N_{\rm H}}= [0.11, 0.38, 0.51] Z' A_{V,0} b_6^{-1}$$ where $g_u$ and $g_{\ell}$ are the degeneracies of the upper and lower states, $A_{u\ell}$ and $\lambda_{u\ell}$ are the Einstein $A$ and wavelength for the transition, $b$ is the Doppler parameter, $\delta_X$ is the abundance relative to hydrogen for the element in question, and $A_{V,0} = A_V/1$ mag. The three numbers given in square brackets are the numerical results for the lines \[C <span style="font-variant:small-caps;">ii</span>\] 158 $\mu$m, \[O <span style="font-variant:small-caps;">i</span>\] 63 $\mu$m, and \[O <span style="font-variant:small-caps;">i</span>\] 145 $\mu$m, respectively, using abundances $\delta_{\rm C} = 1.1\times 10^{-4} Z'$ and $\delta_{\rm O} = 5.0\times 10^{-4} Z'$. This implies that, for the great majority of the cases I consider, optical depths effects will have at most a marginal effect on the cooling rate and can thus be neglected. For optically thin cooling at densities well below the critical density, the radiative cooling rate per H nucleus is simply $$\Lambda_{\rm line} = \sum_i \frac{g_{u,i}}{g_{\ell,i}} \delta_i E_{u,i} e^{-E_{u,i}/k_B T} {\mathcal{C}}(k_{i-\rm H} {\overline{n}_{\rm H}}+ k_{i-e} \overline{n}_e),$$ where the sum runs over the three upper states for the cooling lines (the $^2P_{3/2}^o$ state of C <span style="font-variant:small-caps;">ii</span> and the $^3P_1$ and $^3P_0$ states of O <span style="font-variant:small-caps;">i</span>), $g_{u,i}$ and $g_{\ell,i}$ are the degeneracies of the upper states and the corresponding ground states, $E_{u,i}$ is the energy of the upper state relative to ground, $\delta_i$ is the abundance of the relevant species, $\overline{n}_e$ is the free electron density, and $k_{i-\rm H}$ and $k_{i-e}$ are the rate coefficients for collisional de-excitation of the level by H and by free electrons, respectively. The clumping factor ${\mathcal{C}}$ appears for the same reason as for H$_2$ formation on dust grains: these are collisional processes whose rates vary as the square of the local volume density. Obviously once H$_2$ becomes dominant over H <span style="font-variant:small-caps;">i</span> one should consider collisions with H$_2$ as well, but since conversion to H$_2$ only happens long after the gas has reached thermal equilibrium, I ignore this complication.
I take the free electron density to be $\overline{n}_e = \delta_{\rm C} {\overline{n}_{\rm H}}$, which assumes that singly ionized carbon is the dominant source of free electrons; with this choice, excitation by H generally dominates. At low column densities, free electrons produced by ionization of hydrogen by soft x-rays outnumber those coming from carbon, but in regions of column density ${N_{\rm H}}{\protect\raisebox{-0.5ex}{$\:\stackrel{\textstyle >}
{\sim}\:$}}10^{20}$ cm$^{-2}$ this source of electrons becomes subdominant [@wolfire03a]. Since most of the cases with which I am concerned are in this regime, I adopt the carbon-dominated limit.
Combining all heating and cooling processes, the temperature evolution obeys $$\begin{aligned}
\lefteqn{\frac{3}{2}k_B \frac{d}{dt} T = \Gamma_{\rm PE,0} Z' e^{-\sigma_d {N_{\rm H}}} + \zeta_{\rm CR} q_{\rm CR}}
\nonumber\\
& & {} - \sum_i \frac{g_{u,i}}{g_{\ell,i}} (k_{i-\rm H} + \delta_{\rm C} k_{i-e}) \delta_i E_{u,i} e^{-E_{u,i}/k_B T} {\mathcal{C}}{\overline{n}_{\rm H}}.\quad
\label{eq:dTdt}\end{aligned}$$
Model Results
=============
Fiducial Parameters
-------------------
In order to survey parameter space, I consider a grid of model clouds of density ${\overline{n}_{\rm H}}= 10^0 - 10^3$ cm$^{-3}$ in steps of 0.1 dex, extinction $A_V = 10^{-2} - 10^1$ mag in steps of 0.1 dex, and metallicity $Z' = 10^{-4} - 10^0$ in steps of 0.05 dex, using the fiducial values for radiation and cosmic rays specified above. All model clouds start with $f_{\rm H_2} = 0$ and $T = 1000$ K.[^2] I integrate each model for 15 Gyr and record the properties at $t=t_{\rm ff}$, $t=10t_{\rm ff}$, and at $t=15$ Gyr, with the latter representing the equilibrium state attained after long times; obviously this is longer than the age of the Universe, but these models are useful to help build intuition for the importance of non-equilibirum effects. From the temperatures produced in the models, I compute the Bonnor-Ebert mass $M_{\rm BE} = 1.18 c_s^3/\sqrt{G^3 \mu_{\rm H} m_{\rm H} n}$, where $c_s =\sqrt{k_B T/\mu m_{\rm H}}$ is the sound speed and $\mu \approx 1.3$ is the mean mass per particle (as opposed to the mean mass per H nucleus $\mu_{\rm H}$). Values of $M_{\rm BE}$ should serve a rough proxies for where star formation can occur, with values much larger than the mass of any star indicating little star formation, and small values indicating star formation.
Figure \[fig:tchemevol\] shows the thermal and chemical evolution of some example clouds drawn from the model grid. The figure is consistent with the qualitative timescale estimates above: gas at an initially high, non-equilibrium temperature will cool to a thermal equilibrium temperature of order 10 K and thus proceed to star formation in less than a free-fall time, even for metallicities as low as $\log Z' \approx -4$. On the other hand, at metallicities below $\log Z' = -1$ the gas will be less than half converted to molecules at one free-fall time, and at metallicities of $\log Z' = -2$ or less the gas will not reach 50% molecular until more than $10t_{\rm ff}$. This result is consistent with numerical experiments in full cosmological simulations which show that equilibrium models of the H$_2$ fraction begin to fail due to non-equilibirum effects at metallicites below $\log Z'\approx -2$ [@krumholz11a].
Figure \[fig:fH2Zgrid\] shows the H$_2$ fraction as a function of metallicity for star-forming, non-star-forming, and intermediate clouds at $t=t_{\rm ff}$, $t=10 t_{\rm ff}$, and $t=15$ Gyr (long enough that nearly all models have reached chemical equilibrium). In clouds that are very old and thus have reached equilibrium, the figure shows that star-forming clouds (those with low $M_{\rm BE}$, indicated in blue) lie almost exclusively at high H$_2$ fractions, and non-star-forming ones (those with high $M_{\rm BE}$, indicated in red) almost exclusively at low H$_2$ fractions, consistent with earlier work indicating a strong correlation between equilibrium gas temperature and chemical state [@krumholz11b; @glover12a]. Out of equilibrium, the figure indicates that the correlation between low M$_{\rm BE}$ and high molecular fraction continues to hold for high metallicities. At lower metallicities, however, all models are displaced to smaller H$_2$ fractions, and at metallicities $\log Z' {\protect\raisebox{-0.5ex}{$\:\stackrel{\textstyle <}
{\sim}\:$}}-2$ star-forming clouds are likely to have H$_2$ fractions well below unity even at $t=10t_{\rm ff}$. The implication is that star-formation will be complete before the gas is significantly converted into H$_2$. The precise transition metallicity below which equilibrium is not achieved will depend on the value of the clumping factor and the timescale for which star-forming clouds typically survive.
Sensitivity to Parameter Choices
--------------------------------
To determine the sensitivity of these results to the choices of radiation field and cosmic ray intensity, I also compute the grid with a radiation field increased by a factor of 30 compared to the Milky Way value ($\Gamma_{\rm PE,0} = 1.2\times 10^{-25}$ erg s$^{-1}$ and $\zeta_{\rm diss,0} = 1.5\times 10^{-10}$ s$^{-1}$) and with a cosmic ray flux increased by a factor of 10 to match the observed diffuse cloud value ($\zeta_{\rm CR} = 2\times 10^{-17}Z'$ s$^{-1}$). The results are shown in Figures \[fig:fH2Zgridhi\] and \[fig:fH2Zgridhicr\]. Comparison with Figure \[fig:fH2Zgrid\] clearly indicates that the qualitative results are not substantially altered.
Finally, note that in the thermal evolution calculation I have neglected heating due to cosmic microwave background (CMB) photons. These will impose a temperature floor $T=2.73 (1+z)$ K, where $z$ is the redshift. A priori one would not expect the CMB to become significant until very high redshifts. The temperature reached by C <span style="font-variant:small-caps;">ii</span> cooling does not fall below $\sim 20$ K over most of the model grid, and the CMB temperature does not exceed this value until $z>6.3$. To confirm this intuition, in Figures \[fig:fH2ZgridCMBz5\] and \[fig:fH2ZgridCMBz10\] I show the results of imposing a minimum temperature $T=2.73 (1+z)$ K on the temperature used to evaluate $M_{\rm BE}$, for $z = 5$ and $z = 10$. The changes in the results from Figure \[fig:fH2Zgrid\] are essentially invisible at $z=5$. At $z=10$, the higher CMB temperature raises the temperature in some models such that there are fewer models with small values of $M_{\rm BE}$, and these cluster at even higher molecular fractions. Qualitatively, however, the results are the same as at lower $z$.
Discussion
==========
Observational Implications and Tests
------------------------------------
The disconnect in timescales between H$_2$ formation and cooling has two major observable consequences, which can be used as a test of the above calculations. The first of these is a drop in H$_2$ fractions below the levels predicted by equilibrium models in star-forming clouds. Figure \[fig:h2ratio\] shows the ratio of the H$_2$ fraction at $t=t_{\rm ff}$ and $t=10t_{\rm ff}$ to the equilibrium H$_2$ fraction for star-forming clouds in the model grid. Clearly we expect equilibrium models to provide good predictions for galaxies down to metallicity $Z' \approx 0.1$ or even somewhat less. This is consistent with observations to date, which show that chemical equilibrium models provide excellent fits to observed H$_2$ to H <span style="font-variant:small-caps;">i</span> ratios in the Milky Way [@krumholz09a; @lee12a] and even the Small Magellanic Cloud (SMC; $Z'\approx 0.2$, @bolatto11a). However, the Figure indicates that at metallicities of $\log Z' = -3$, the H$_2$ fraction in a given star-forming cloud will be at most $\sim 10\%$ of its equilibrium value, and could be less than 1% of that value.
Second, the onset of star formation before the gas has time to fully transform to H$_2$ in low metallicity galaxies should manifest as a reduction in the H$_2$ depletion time $t_{\rm dep-H_2}$, defined as the ratio of the H$_2$ mass to the star formation rate. In Solar metallicity, non-starbursting local galaxies, $t_{\rm dep-H_2} \approx 2$ Gyr [@bigiel08a], although lower values are possible in starbursts. This value should be lower in low metallicity galaxies by a factor of the mean H$_2$ fraction in cold, star-forming clouds, since these clouds will only partially convert to H$_2$ before forming stars and being destroyed by feedback. Figure \[fig:tdep\] illustrates this effect for clouds that live 1 and 10 free-fall times. Note that @glover12b qualitatively suggested the existence of this effect, and Figure \[fig:tdep\] represents a quantitative extension of this prediction.
Observational tests of these predictions are complicated by the fact that H$_2$ is extremely difficult to observe at low metallicities, because CO, the traditional H$_2$ proxy, ceases to track H$_2$ at metallicities below a few tenth of Solar [@krumholz11b; @bolatto11a; @leroy11a; @shetty11b; @narayanan12a; @feldmann12a]. Thus direct observational tests will require the detection of H$_2$ by other means, such as dust or C <span style="font-variant:small-caps;">ii</span> emission that is not associated with observed H <span style="font-variant:small-caps;">i</span>. While observationally challenging, surveys of this sort have already been completed in the closest galaxies like the SMC [@bolatto11a], and with the observational power provided by the Atacama Large Millimeter Telescope (ALMA) should begin to be possible in even lower metallicity nearby galaxies. In particular, 850 $\mu$m observations are an excellent probe of dust and thus all gas including H$_2$, because at 850 $\mu$m dust is generally optically thin, the emission is not very sensitive to dust temperature, and ALMA can achieve both high spatial resolution and excellent sensitivity. Prime targets for such a campaign include IZw18, SBS 0335-052 (both $Z' \approx 0.02$, and probably even lower dust metallicities, @izotov99a [@herrera-camus12a]), and Leo T ($Z'{\protect\raisebox{-0.5ex}{$\:\stackrel{\textstyle <}
{\sim}\:$}}0.01$, @simon07a). The ALMA observations will have to be coupled with high resolution, high sensitivity H <span style="font-variant:small-caps;">i</span> maps to measure the atomic content. Fortunately, sub-kpc resolution H <span style="font-variant:small-caps;">i</span> maps of IZw18 [@van-zee98a] and SBS0335-052 [@ekta09a] are already available in the literature.
Implications for Simulations and Semi-Analytic Models
-----------------------------------------------------
These results have important theoretical implications as well. Many galaxy simulation models allow star formation only in regions where the gas has converted to H$_2$; some of these models include non-equilibrium chemistry for H$_2$ formation and destruction [@pelupessy09a; @gnedin09a; @gnedin10a; @christensen12a], while others assume equilibrium [@fu10a; @lagos11a; @kuhlen12a; @krumholz12d]. The non-equilibrium models on average yield less H$_2$ and thus less star formation at low metallicity, because often gas clouds are not able to build up significant H$_2$ fractions before being destroyed by galactic shear or similar kinematic processes [@krumholz11a]. However, if the relevant timescale is the cooling time and not the H$_2$ formation time, and this effect should be far less significant. As a result, star formation should in fact occur even in gas with low H$_2$ fractions, provided that the [*equilibrium*]{} H$_2$ fraction is high – it is the equilibrium H$_2$ fraction and not the instantaneous one that correlates with gas temperature and thus is a good predictor of where star formation will occur. This suggests that, ironically, models in which the H$_2$ is assumed to be in equilibrium, while they are less accurate in predicting the actual H$_2$ fraction, may in fact be more accurate that the non-equilibrium models in predicting where star formation should occur. More generally, the calculations presented here suggest that star formation thresholds in simulations should be based on the instantaneous density and extinction, which determine the temperature, and not on non-equilibrium chemical abundances.
Summary
=======
I explore under what conditions and for what physical reasons the observed correlation between star formation and molecular gas in the ISM is likely to break down. I show that the breakdown occurs at metallicities below a few percent of Solar, and that the physical mechanism for this breakdown is a disconnect between the thermal and chemical equilibration timescales. Carbon in the ISM is able to cool gas on a timescale shorter by a factor of several thousand than that required for dust grains to convert the H <span style="font-variant:small-caps;">i</span> to H$_2$. As long as both the thermal and chemical equilibration timescales are short compared to cloud free-fall times, which is the case at Solar metallicity, this does not have any practical effect and non-equilibium chemistry is unimportant. However, both the thermal and chemical timescales scale linearly with the metallicity, while the free-fall time time does not. At metallicities below a few percent of Solar, the free-fall time becomes intermediate between the thermal and chemical timescales, and clouds cool and proceed to star formation before molecules form, breaking the H$_2$-star formation connection.
This result has three major implications, two observational and one theoretical. The observational implications are that the equilibrium chemistry models that perform extremely well in the Milky Way and the SMC should begin to overpredict H$_2$ abundances in very low metallicity galaxies, and that star formation should occur in atomic-dominated regions of such galaxies as well, leading to a lower H$_2$ depletion time. These predictions are not trivial to check, given the difficulty of measuring H$_2$ in low metallicity environments, but combining high resolution dust and H <span style="font-variant:small-caps;">i</span> maps to infer the presence of H$_2$ constitutes a viable strategy. The theoretical implication is that galaxy evolution simulations and semi-analytic models that link star formation to the chemical state of the gas, and that treat that chemistry using non-equilibrium models, are likely to underpredict star formation rates in circumstances where the gas should reach thermal but not chemical equilibrium. It is the former that matters for star formation, not the latter.
I thank A. Bolatto, L. Hunt, and A. Leroy for helpful conversations, and C. McKee for helpful comments on the manuscript. Funding for this work as provided by the National Science Foundation through grant CAREER-0955300, by NASA through Astrophysics Theory and Fundamental Physics Grant NNX09AK31G and through a Chandra Space Telescope Grant, and by the Alfred P. Sloan Foundation.
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[^1]: The value of $A_V/{N_{\rm H}}$, and all other parameters in the following discussion, are taken from @krumholz11b unless stated otherwise; arguments for these choices are given in that paper. Atomic data (Einstein coefficients, collision rates) are all taken from @schoier05a, and abundances from @draine11a.
[^2]: Obviously some of these initial conditions (e.g. clouds with ${\overline{n}_{\rm H}}=10^3$ cm$^{-3}$ and $T=1000$ K) are unlikely to be found in real galaxies, but such models constitute a small fraction of the parameter space, and the goal of this study is to perform a broad survey rather than trying to focus on particular assumed sets of “reasonable" initial conditions in galaxies where the actual conditions are poorly determined.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We introduce the notion of Bonnet-Myers and Lichnerowicz sharpness in the Ollivier Ricci curvature sense. Our main result is a classification of all self-centered Bonnet-Myers sharp graphs (hypercubes, cocktail party graphs, even-dimensional demi-cubes, Johnson graphs $J(2n,n)$, the Gosset graph and suitable Cartesian products). We also present a purely combinatorial reformulation of this result. We show that Bonnet-Myers sharpness implies Lichnerowicz sharpness. We also relate Bonnet-Myers sharpness to an upper bound of Bakry-Émery $\infty$-curvature, which motivates a general conjecture about Bakry-Émery $\infty$-curvature.'
author:
- 'D. Cushing'
- 'S. Kamtue'
- 'J. Koolen'
- 'S. Liu'
- 'F. Münch'
- 'N. Peyerimhoff'
title: 'Rigidity of the Bonnet-Myers inequality for graphs with respect to Ollivier Ricci curvature'
---
Introduction and statement of results
=====================================
A fundamental question in geometry is in which way local properties determine the global structure of a space. A famous result of this kind is the Bonnet-Myers Theorem [@My41] for complete $n$-dimensional Riemannian manifolds $M$ with $K = \inf {\rm Ric}_M(v) > 0$ (a condition on the local invariant ${\rm Ric}_M ={\rm Tr}(R_M)$), where the infimum is taken over all unit tangent vectors $v$ of $M$. Under this condition, $M$ is compact and its diameter satisfies $$\label{eq:BM_RG_ineq}
{\rm diam}(M) \le \pi \sqrt{\frac{n-1}{K}}.$$ Moreover, Cheng’s Rigidity Theorem [@Cheng75] states that this diameter estimate is sharp if and only if $M$ is the $n$-dimensional round sphere. Note that inequality can be reformulated as an upper bound on the infimum of the Ricci curvature in terms of the diameter, and this reformulation is the viewpoint we will assume in this paper.
In the discrete setting of graphs there are several analogs of Ricci curvature notions providing Bonnet-Myers type theorems (see, e.g., [@FS; @HLLY; @LMP; @LLY11; @Ol09], ...). In view of Cheng’s rigidity result, it is natural to ask for which graphs the Bonnet-Myers estimates is sharp. We call such graphs *Bonnet-Myers sharp* graphs. For example, in the case of Bakry-Émery $\infty$-curvature, Bonnet-Myers sharp graphs have been fully characterised and are only the hypercubes (see [@LMP2]).
The motivation of this paper is to study Bonnet-Myers sharpness with respect to another curvature notion, namely, *Ollivier Ricci curvature*. (In fact, we will consider a modification of Ollivier’s definition introduced in [@LLY11].) Henceforth, all graphs $G = (V,E)$ with vertex set $V$ and edge set $E$ will be simple (loopless without multiple edges) and edges can be identified with $2$-element subsets of $V$. In this paper, we will only formulate and derive our results for regular graphs, that is, all vertices have the same valency, even though similar questions can be posed for non-regular graphs.
Ollivier Ricci curvature $\kappa(x,y)$ is a notion based on optimal transport and is defined on pairs of different vertices $x,y \in
V$. The precise definition requires a longer introduction and is given in Subsection \[sec:OllivKant\]. Generally, $\kappa(x,y)$ is positive if the average distance between corresponding neighbours of $x$ and $y$ is smaller than $d(x,y)$. For now, we confine ourselves to provide a useful connection of this curvature with a particular combinatorial property, to provide the readers with some understanding of this notion. Note that this Proposition follows directly from Proposition \[prop:curvcalc0\] by choosing $m = \frac{2D}{L} - 2$.
\[prop:curvcalc\] Let $G=(V,E)$ be a $D$-regular graph of diameter $L$ and $e=\{x,y\} \in E$. Assume that $e$ is contained in precisely $\frac{2D}{L}-2$ triangles and there is a perfect matching between the neighbours of $x$ and the neighbours of $y$ which are not involved in these triangles. Then we have $$\kappa(x,y) = \frac{2}{L}.$$
Let us now state the discrete Bonnet-Myers Theorem for Ollivier Ricci curvature and introduce the associated notion of Bonnet-Myers sharpness for this curvature notion:
\[thm:DBM\] Let $G= (V,E)$ be a connected $D$-regular graph and $\inf_{x \sim y}
\kappa(x,y) > 0$. Then $G$ has finite diameter $L = {\rm diam}(G) <
\infty$ and $$\label{eq:BM_est}
\inf_{x \sim y} \kappa(x,y) \le \frac{2}{L}.$$ We say that such a graph $G$ is [**[*$\boldsymbol{(D,L)}$-Bonnet-Myers sharp*]{}**]{} (with respect to Ollivier Ricci curvature) if holds with equality.
Many of our results require the additional condition of self-centeredness. Note that a graph $G=(V,E)$ is called self-centered if, for every vertex $x \in V$, there exists a vertex $\overline{x} \in V$ such that $d(x,\overline{x}) = {\rm diam}(G)$ (see Subsection \[sec:graph\_notation\] for its definition). Let us now state the main results of this paper.
- *Cartesian products:* $G_1 \times G_2 \times \cdots \times G_k$ is Bonnet-Myers sharp if and only if all factors $G_i$ are Bonnet-Myers sharp and satisfy $$\label{eq:cartprod_cond0}
\frac{D_1}{L_1} = \frac{D_2}{L_2} =\cdots = \frac{D_k}{L_k},$$ where $D_i$ and $L_i$ are the vertex degrees and the diameters of the graphs $G_i$, respectively (see Theorem \[thm:cartprod\]).
- Every Bonnet-Myers sharp graph is Lichnerowicz sharp (see Theorem \[thm:lichn\]).
- *Classification of self-centered Bonnet-Myers sharp graphs:* Self-centered Bonnet-Myers sharp graphs are precisely the following ones: Hypercubes, cocktail party graphs, the Johnson graphs $J(2n,n)$, even-dimensional demi-cubes, the Gosset graph and Cartesian products of them satisfying (see Theorem \[thm:main\]).
- Self-centered $(D,L)$-Bonnet-Myers sharp graphs are Bakry-[É]{}mery $\infty$-curvature sharp in all vertices with normalized $\infty$-curvature value $\frac{1}{D} + \frac{1}{L}$ (see Theorem \[thm:aBM-BEcurv\]).
We provide more detailed information about these results in the next subsection. In particular, result (c) above is based on another result which can be reformulated in purely combinatorial terms. This combinatorial reformulation is derived in Subsection \[subsec:comb\_res\].
Our results on Bonnet-Myers sharp graphs
----------------------------------------
It is useful to know that the vertex degree $D$ and the diameter $L$ of a Bonnet-Myers sharp graph cannot be arbitrary:
\[thm:DL\_rel\] Any $(D,L)$-Bonnet-Myers sharp graph satisfies $L \le D$. Moreover $L$ must divide $2D$.
This theorem is proved in Section \[sec:self-centBMsh\].
Another fundamental result between local and global properties of a closed $n$-dimensional Riemannian manifold $M$ is Lichnerowicz’ Theorem [@Li58 p. 135] which states that, under the condition $K = \inf {\rm Ric}_M(v) > 0$, the smallest positive Laplace-Beltrami eigenvalue $\lambda_1(M)$ satisfies $$\frac{n}{n-1}K \le \lambda_1.$$ The associated rigidity result is Obata’s Theorem [@Ob62], which states that this eigenvalue estimate is sharp if and only if $M$ is the $n$-dimensional round sphere. There is a discrete analogue of Lichnerowicz’ Theorem for Ollivier Ricci Curvature and the normalized Laplacian $\Delta_G = D^{-1} A_G - {\rm Id}$ of an arbitrary graph $G=(V,E)$, where $A_G$ denotes the adjacency matrix of $G$ and $D$ is here a diagonal matrix whose entries are the valencies $d_x$ of the vertices $x \in V$. In the case of a $D$-regular graph this matrix is just given by $D \cdot {\rm Id}$. Alternatively, $\Delta_G$ can be written as an operator acting on functions $f$ defined on the vertices $V$ via $$\label{eq:Deltanorm}
\Delta_G f(x) = \frac{1}{d_x} \sum_{y \sim x} (f(y)-f(x)).$$ The discrete Lichnerowicz’ Theorem gives naturally rise to the definition of Lichnerowicz sharpness:
\[thm:Lich\] Let $G=(V,E)$ be a finite connected $D$-regular graph. Then we have for the smallest positive solution $\lambda_1$ of $\Delta_G f + \lambda f = 0$ with $\Delta_G$ given by , $$\label{eq:L_est}
\inf_{x \sim y} \kappa(x,y) \le \lambda_1.$$ We say that the graph $G$ is [**[*Lichnerowicz sharp*]{}**]{} (with respect to Ollivier Ricci curvature) if holds with equality.
We have the following relation between these two sharpness properties, proved in Section \[sec:curvandeig\]:
\[thm:lichn\] Every Bonnet-Myers sharp graph is Lichnerowicz sharp.
Note, however, that the family of all Lichnerowicz sharp graphs is much larger than the family of all Bonnet-Myers sharp graphs (see Remark \[rem:lichcartsharp\]). In Section \[sec:curvandeig\], we classify a special subclass of Lichnerowicz sharp graphs, namely, all distance regular Lichnerowicz sharp graphs with an additional spectral condition.
Our main theorem is the following classification result of self-centered Bonnet-Myers sharp graphs:
\[thm:main\] Self-centered Bonnet-Myers sharp graphs are precisely the following graphs:
1. hypercubes $Q^n$, $n \ge 1$;
2. cocktail party graphs $CP(n)$, $n \ge 3$;
3. the Johnson graphs $J(2n,n)$, $n \ge 3$;
4. even-dimensional demi-cubes $Q^{2n}_{(2)}$, $n \ge 3$;
5. the Gosset graph;
and Cartesian products of 1.-5. satisfying the condition .
Let us explain the proof of Theorem \[thm:main\]: In Section \[sec:examples\] we show that all graphs in the list of Theorem \[thm:main\] are self-centered Bonnet-Myers sharp. In Sections \[sec:transpgeod\] and \[sec:antBMstrspher\] we show that every self-centered Bonnet-Myers sharp graph is strongly spherical (this is the statement of Theorem \[thm\_spherical\]; the notion of strongly spherical graphs is introduced in Definition \[def:strspher\]). Then we use the classification result [@Koo] (see Theorem \[thm:strongspher\_class\] below), which states that strongly spherical graphs are precisely the Cartesian products in the list provided in Theorem \[thm:main\], thus completing the proof.
Finally, we like to present connections between Bonnet-Myers sharpness with respect to Ollivier Ricci curvature and normalized Bakry-[É]{}mery $\infty$-curvature. Generally, relations between different curvature notions of discrete spaces are a very interesting and challenging topic. Normalized Bakry-[Émery]{} $\infty$-curvature is defined on the vertices of a graph $G=(V,E)$ and denoted by ${\mathcal K}^{\rm n}_{G,x}(\infty)$, $x \in V$. For further details and the precise definition, we refer the readers to Subsection \[sec:BE-curvature\]. We have the following results:
\[thm:BMsharp-BEeq\] Every $(D,L)$-Bonnet-Myers sharp graph $G=(V,E)$ satisfies $$\label{eq:BM-BakryEm}
\inf_{x \in V} {\mathcal K}^{\rm n}_{G,x}(\infty) \le
\frac{1}{D} + \frac{1}{L}.$$
This result follows immediately from Theorem \[thm:curv\_relation\] later in the paper.
As a consequence of Theorem \[thm:main\] we also obtain
\[thm:aBM-BEcurv\] Let $G =(V,E)$ be a self-centered $(D,L)$-Bonnet-Myers sharp graph. Then $G$ is Bakry-[É]{}mery $\infty$-curvature sharp at all vertices $x \in V$ and we have $${\mathcal K}^{\rm n}_{G,x}(\infty) = \frac{1}{D} + \frac{1}{L}.$$
The proof of this theorem is given in Subsection \[sec:BEcurvBMsharp\].
These last two results provide a better understanding why Bonnet-Myers sharpness with respect to Bakry-[É]{}mery $\infty$-curvature is much more restrictive: the only graphs with this property are the hypercubes (see [@LMP2]). This result classifies all $D$-regular graphs of diameter $L$ satisfying the condition $$\inf_{x \in V}{\mathcal K}^{\rm n}_{G,x}(\infty) = \frac{2}{L},$$ which is much stronger than the equality condition of , that is $$\label{eq:BM-BakryEmeq}
\inf_{x \in V}{\mathcal K}^{\rm n}_{G,x}(\infty) = \frac{1}{D} + \frac{1}{L},$$ since $L \le D$, by Theorem \[thm:DL\_rel\]. By considering the weaker condition we encounter other graphs like the ones given in the list of Theorem \[thm:main\]. It is an open question whether these graphs and suitable Cartesian products of them are the only self-centered examples of $D$-regular graphs of diameter $L$ satisfying .
Moreover, we are not aware of any $D$-regular graph $G$ of diameter $L$ which violates the condition . This led us to formulate the following conjecture:
\[conj:BM\] Let $G=(V,E)$ be a finite connected $D$-regular graph of diameter $L$. Then we have $$\inf_{x \in V} {\mathcal K}^{\rm n}_{G,x}(\infty) \le
\frac{1}{D} + \frac{1}{L}.$$
Let us compare this conjecture with the combinatorial Bonnet-Myers Theorem for normalized Bakry-[É]{}mery $\infty$-curvature proved in [@LMP], namely, $$\label{eq:BM_BE}
\inf_{x \in V} {\mathcal K}^{\rm n}_{G,x}(\infty) \le \frac{2}{L}.$$ Our conjecture can be viewed as a strengthening of in the case $L \le D$.
A combinatorial result {#subsec:comb_res}
----------------------
For readers interested in combinatorial graph theory, this subsection provides a combinatorial reformulation of Theorem \[thm\_spherical\]. The combinatorial version is given in Theorem \[thm:main\_comb\] below and we think that it is of own interest.
We start with a combinatorial property closely related to Ollivier Ricci curvature (see Proposition \[prop:curvcalc0\] below for the connection).
\[def:Lambda\] Let $G=(V,E)$ be a regular graph. We say $G$ *satisfies $\Lambda(m)$* at an edge $e = \{x,y\} \in E$ if the following holds:
- $e$ is contained in at least $m$ triangles and
- there is a perfect matching between the neighbours of $x$ and the neighbours of $y$ not involved in these triangles.
We say that $G$ satisfies $\Lambda(m)$ if it satisfies $\Lambda(m)$ at each edge.
Next, we introduce the notions of antipodal and strongly spherical graphs, which are based on intervals $[x,y]$ which are the set of all vertices lying on geodesics from $x$ to $y$ (see Subsection \[sec:graph\_notation\] for the precise definition of intervals).
\[def:strspher\] A graph $G = (V,E)$ is called *antipodal* if, for every vertex $x \in V$, there exists another vertex $\overline{x} \in V$ satisfying $[x,\overline{x}]=V$.
A graph $G = (V,E)$ is called *strongly spherical* if $G$ and the induced subgraphs of all its intervals are antipodal. That is, the following two properties are necessary and sufficient conditions for strongly spherical graphs $G = (V,E)$:
- For every $x \in V$ there exists $\overline{x} \in V$ such that $ G = [x,\overline{x}]$.
- For every pair $x,y \in V$, $x \neq y$, and every $z \in [x,y]$ there exists $\overline{z} \in [x,y]$ such that $[x,y] = [z,\overline{z}]$.
To reformulate Theorem \[thm\_spherical\] in purely combinatorial terms we need the following result which allows us to replace the curvature condition by a combinatorial condition.
\[prop:antBMcomb\] Let $G$ be a $D$-regular finite connected graph of diameter $L$. The following are equivalent:
- $G$ is self-centered Bonnet-Myers sharp.
- $G$ is self-centered and satisfies $\Lambda\left(\frac{2D}{L}-2\right)$.
Moreover, if any of these equivalent properties holds, then every edge of $G$ lies in precisely $\frac{2D}{L}-2$ triangles.
This proposition is a consequence of Corollary \[cor:numtriangle\] and Proposition \[prop:curvcalc0\] later in the paper.
Using Proposition \[prop:antBMcomb\], Theorem \[thm\_spherical\] translates then into the following equivalent combinatorial result:
\[thm:main\_comb\] Let $G$ be a $D$-regular finite connected graph of diameter $L$. Assume that $G$ is self-centered and satisfies $\Lambda\left(\frac{2D}{L}-2\right)$. Then $G$ is strongly spherical.
It is an interesting question whether the condition of self-centeredness in Theorem \[thm:main\_comb\] can be removed. If this were possible, we could view this result as another example where *local* properties have a strong *global* implication.
Finally, we would like to present the following classification of strongly spherical graphs.
\[thm:strongspher\_class\] Strongly spherical graphs are precisely the Cartesian products $G_1 \times G_2 \times \cdots \times G_k$, where each factor $G_i$ is either a hypercube, a cocktail party graph, a Johnson graph $J(2n,n)$, an even dimensional demi-cube, or the Gosset graph.
Outline of the paper {#subsec:outline}
--------------------
Here is an overview about the content of the following sections:
- Section \[sec:defn\]: Basic graph theoretical notation and introduction into Ollivier Ricci curvature.
- Section \[sec:cart\]: Cartesian products preserve Bonnet-Myers sharpness under particular conditions.
- Section \[sec:examples\]: Presentation of all known examples of Bonnet-Myers sharp graphs.
- Section \[sec:genfacts\]: Discussion of various consequences of a useful relation between the Laplacian and Ollivier Ricci curvature. For example, all vertices of a Bonnet-Myers sharp graph lie on geodesics between any pair of antipoles.
- Section \[sec:curvandeig\]: Proof that Bonnet-Myers sharp graphs are Lichnerowicz sharp and classification of all distance-regular Lichnerowicz sharp graphs under an additional spectral condition.
- Sections \[sec:transpgeod\] and \[sec:antBMstrspher\]: Proof of our main result: Classification of all self-centered Bonnet-Myers sharp graphs.
- Section \[sec:BMextrdiam\]: Classification of all Bonnet-Myers sharp graphs with extremal diameters $L=2$ and $L=D$.
- Section \[sec:BM-BE-curvature\]: Relations between Bonnet-Myers sharpness and an upper bound on the Bakry-Émery $\infty$-curvature.
Basic definitions, concepts and notation {#sec:defn}
========================================
Throughout this paper, we restrict our graphs $G=(V,E)$ to be undirected, simple, unweighted, finite, and connected.
Graph theoretical notation {#sec:graph_notation}
--------------------------
We write $x \sim y$ if there exists an edge between the vertices $x$ and $y$. The degree of a vertex $x \in V$ is denoted by $d_x$. For a set of vertices $A\subseteq V$, the induced subgraph ${\rm Ind}_G(A)$ is the subgraph of $G$ whose vertex set is $A$ and whose edge set consists of all edges in $G$ that have both endpoints in $A$. For any two vertices $x,y\in V$, the (combinatorial) distance $d(x,y)$ is the length (i.e. the number of edges) in a shortest path from $x$ to $y$. Such paths of minimal length are also called [*geodesics*]{} from $x$ to $y$. An *interval* $[x,y]$ is the set of all vertices lying on geodesics from $x$ to $y$, that is $$[x,y] = \{ z \in V \mid d(x,z) + d(z,y) = d(x,y) \}.$$
The [*diameter*]{} of $G$ is denoted by ${\rm diam}(G)= \max_{x,y\in V} d(x,y)$. A vertex $x\in V$ is called a [*pole*]{} if there exists a vertex $y\in V$ such that $d(x,y)={\rm diam}(G)$, in which case $y$ will be called an [*antipole*]{} of $x$ (with respect to $G$). A graph $G$ is called [*self-centered*]{} if every vertex is a pole.
For $k\in\mathbb{N}$ and $x\in V$ we define the [*$k$-sphere*]{} of $x$ as $S_k(x) = \{z : d(z,x) = k\}$ and the [*$k$-ball*]{} of $x$ as $B_k(x) = \{z: d(z,x) \le k\}$. In particular, $S_1(x)$ is also denoted as $N_x$, the set of all neighbors of $x$. Denote also $N_{xy}=S_1(x)\cap S_1(y)$, the common neighbors of $x$ and $y$. Especially when $d(x,y)=2$, the induced subgraph ${\rm Ind}_G (N_{xy})$ is called $\mu$-graph of $x$ and $y$. In case that $\mu$-graphs are isomorphic to a graph $H$ for all $x,y$ with $d(x,y)=2$, we call the graph $H$ the [*$\mu$-graphs*]{} of $G$.
For a vertex $x\in V$, let $\#_{\Delta}(x)$ denote the number of triangles containing $x$. Similarly, for an edge $e = \{x,y\} \in E$, let $\#_{\Delta}(x,y)$ denote the number of triangles containing $e$. We have the following relation: $$\label{eq:triangles}
\#_{\Delta}(x) = \frac{1}{2} \sum_{y: y \sim x} \#_{\Delta}(x,y).$$
For $x,y\in V$, we also define $$\begin{aligned}
d_{x}^{-}(y) &= |\{z\sim y : d(x,y) = d(x,z) + 1\}|,\\
d_{x}^{0}(y) &= |\{z\sim y : d(x,y) = d(x,z) \}|,\\
d_{x}^{+}(y) &= |\{z\sim y : d(x,y) = d(x,z) - 1\}|,\end{aligned}$$ which we call the [*in degree, spherical degree, out degree*]{} of $y$, respectively. The averages of these degrees are then defined as: $$\begin{aligned}
\label{eq:averdeg}
av_k^-(x) = \frac{1}{|S_k(x)|}\sum_{y \in S_k(x)}d_{x}^{-}(y),\nonumber\\
av_k^0(x) = \frac{1}{|S_k(x)|}\sum_{y \in S_k(x)}d_{x}^{0}(y),\\ av_k^+(x) = \frac{1}{|S_k(x)|}\sum_{y \in S_k(x)}d_{x}^{+}(y). \nonumber\end{aligned}$$
By abuse of notation, we sometimes identify $S_k(x)$ and $B_k(x)$ with the induced subgraphs ${\rm Ind}_G(S_k(x))$ and ${\rm Ind}_G(B_k(x))$, respectively.
Ollivier Ricci curvature and Kantorovich duality {#sec:OllivKant}
------------------------------------------------
A fundamental concept in Optimal Transport Theory is the Wasserstein distance defined on probability measures.
Let $G = (V,E)$ be a graph. Let $\mu_{1},\mu_{2}$ be two probability measures on $V$. The *Wasserstein distance* $W_1(\mu_{1},\mu_{2})$ between $\mu_{1}$ and $\mu_{2}$ is defined as $$\label{eq:W1def}
W_1(\mu_{1},\mu_{2}):=\inf_{\pi \in \Pi(\mu_1,\mu_2)} {\rm cost}(\pi),$$ where $\pi$ runs over all transport plans in $$\Pi(\mu_1,\mu_2) = \left\{ \pi: V \times V \to [0,1] :
\mu_{1}(x)=\sum_{y\in V}\pi(x,y), \; \mu_{2}(y)=\sum_{x\in
V}\pi(x,y) \right\}.$$ and the *cost* of $\pi$ is defined as $${\rm cost}(\pi) := \sum_{x \in V} \sum_{y \in V} d(x,y)
\pi(x,y).$$
The transportation plan $\pi$ in the above definition moves a mass distribution given by $\mu_1$ into a mass distribution given by $\mu_2$, and $W_1(\mu_1,\mu_2)$ is a measure for the minimal effort which is required for such a transition. If $\pi$ attains the infimum in we call it an [*optimal transport plan*]{} transporting $\mu_{1}$ to $\mu_{2}$. We define the following probability measures $\mu_x^p$ for any $x\in V,\: p\in[0,1]$: $$\mu_x^p(z) = \begin{cases} p & \text{if $z = x$,} \\
\frac{1-p}{d_x} & \text{if $z\sim x$,} \\
0 & \mbox{otherwise.} \end{cases}$$ In this paper we will pay particular attention to $D$-regular graphs and the idleness parameter $p = \frac{1}{D+1}$ and write $\mu_{x}$ for $\mu^{\frac{1}{D+1}}_{x}$, for simplicity.
Let $p\in [0,1]$. The *$p$-Ollivier Ricci curvature* between two different vertices $x,y \in V$ is $$\kappa_{ p}(x,y) = 1 - \frac{W_1(\mu^{ p}_x,\mu^{ p}_y)}{d(x,y)},$$ where $p$ is called the [*idleness*]{}.
If $G$ is $D$-regular then we define the curvature, $\kappa$, as $$\label{kpklly}
\kappa(x,y) = \frac{D+1}{D}\kappa_{\frac{1}{D+1}}(x,y).$$
The motivation for is that $\kappa(x,y)$ agrees with the modified curvature $\kappa_{LLY}(x,y)$ introduced by Lin/Lu/Yau in [@LLY11] for neighbours $x \sim y$, due to [@Idle Theorem 1.1]. That is, we have for neighbours $x \sim y$ in $D$-regular graphs, $$\label{eq:kappaLLY} \kappa(x,y) =
\frac{\kappa_{\frac{1}{D+1}}(x,y)}{1-\frac{1}{D+1}} = \lim_{p \to 1}
\frac{\kappa_p(x,y)}{1-p} =: \kappa_{LLY}(x,y).$$
A straightforward consequence of the fact that Ollivier Ricci curvature is defined via a distance function (Wasserstein distance) is the following: $$\label{eq:infeqinf}
\inf_{x \sim y} \kappa(x,y) \le \inf_{z \neq w} \kappa(z,w),$$ namely that it makes no difference to take the infimum of the curvature over all pairs of different vertices or only over neighbours. Moreover, the Discrete Bonnet-Myers Theorem \[thm:DBM\] follows directly from and the following stronger inequality for any pair of different vertices $z,w \in V$ (see [@Ol09]): $$\label{eq:kapzw_ineq}
\kappa(z,w) \le \frac{2}{d(z,w)},$$ by choosing a pair of vertices $z,w \in V$ of maximal distance, that is, $d(z,w) = {\rm diam}(G)$.
Another fundamental concept in Optimal Transport Theory is Kantorovich duality. First we recall the notion of 1–Lipschitz functions and then state the Kantorovich Duality Theorem.
Let $G=(V,E)$ be a graph and $\phi: V \rightarrow {{\mathbb{R}}}$. We say that $\phi$ is *$1$-Lipschitz* if $$|\phi(x) - \phi(y)| \leq d(x,y) \quad \text{for all $x,y\in V$.}$$ We denote the set of all $1$–Lipschitz functions by $\textrm{\rm{1}--{\rm Lip}}(V)$.
For an arbitrary graph $G = (V,E)$, we denote by $\ell_\infty(V)$ the space of all bounded function $f: V \to {{\mathbb{R}}}$ and by $C_c(V)$ the subspace of all functions with finite support.
\[Kantorovich\] Let $G = (V,E)$ be a graph. Let $\mu_{1},\mu_{2}$ be two probability measures on $V$. Then $$W_1(\mu_{1},\mu_{2}) =
\sup_{\phi \in \textrm{\rm{1}--{\rm Lip}}(V)\, \cap\, \ell_\infty(V)}
\sum_{x\in V}\phi(x)(\mu_{1}(x)-\mu_{2}(x)).$$ If $\phi$ attains the supremum we call it an *optimal Kantorovich potential* transporting $\mu_{1}$ to $\mu_{2}$.
The following fact from [@MW17 Theorem 2.1] is at the heart of all the results presented in Section \[sec:genfacts\]. Moreover, it provides an alternative definition of Ollivier Ricci curvature as a notion induced by a given Laplace operator. This alternative viewpoint allows a more general definition of Ollivier Ricci curvature for various kinds of Laplacians.
Let $G=(V,E)$ be a connected graph and $f: V \to {{\mathbb{R}}}$ be a function on the vertices. We define the *gradient* of $f$ as $$\nabla_{xy} f = \frac{f(x)-f(y)}{d(x,y)} \quad
\text{for all $x,y \in V$, $x \neq y$}.$$ Let $\Delta_G$ be the normalized Laplacian of $G$ defined in . Then, for any pair $x,y \in V$ of different vertices, we have $$\label{eq:kappally_alt}
\kappa_{LLY}(x,y) =
\inf_{\substack{\phi \in \textrm{\rm{1}--{\rm Lip}}(V)\, \cap\, C_c(V)\\ \nabla_{yx}\phi = 1}}
\nabla_{xy} (\Delta_G \phi),$$ where $\kappa_{LLY}$ was defined in .
Transport plans based on triangles and a perfect matching {#sec:transportplanstrianglesmatching}
---------------------------------------------------------
Recall the definition of the combinatorial property $\Lambda(m)$ given in Definition \[def:Lambda\]. The following proposition explains its curvature implication and is useful for our curvature calculations in Section \[sec:examples\]. As mentioned before, it also implies Proposition \[prop:curvcalc\] directly by choosing $m = \frac{2D}{L} - 2$. Besides presenting this proposition and its proof, we also introduce in this subsection the notion of a *transport plan based on triangles and a perfect matching*.
\[prop:curvcalc0\] Let $G=(V,E)$ be a $D$-regular graph and $e = \{x,y\} \in E$ an edge. Then we have $$\label{eq:kappa_est}
\kappa(x,y) \le \frac{2+\#_\Delta(x,y)}{D}$$ Moreover, the following are equivalent:
- $\kappa(x,y) = \frac{2+\#_\Delta(x,y)}{D}$;
- there is a perfect matching between the vertex sets $N_x \backslash (N_{xy} \cup \{y\})$ and $N_y \backslash (N_{xy} \cup \{x\})$.
Assume that $G=(V,E)$ is $D$-regular and $e= \{x,y\} \in E$ is contained in $m = \#_\Delta(x,y)$ triangles. Then we have $$s = |N_{x}\setminus (N_{xy}\cup\{y\})| =
|N_{y}\setminus (N_{xy}\cup\{x\})| = D-1-m.$$ Since all masses are equal to $\frac{1}{D+1}$, we are faced with a Monge Problem and there exists an optimal transport plan $\pi$ transporting $\mu_x$ to $\mu_y$ with $\pi(z,w) \in \{0, \frac{1}{D+1}\}$ for all $z,w \in V$. (That is, $\pi$ is induced by a bijective optimal transport map $T: B_1(x) \to B_1(y)$; for more details see Subsection \[sec:concat\_tramap\].) By [@Idle Lemma 4.1], we can choose $\pi$ to satisfy $\pi(z,z) = \frac{1}{D+1}$ for all $z \in N_{xy} \cup \{x,y\}$, that is there is no mass transport on these vertices, and enumerate the vertices in $N_{x}\setminus (N_{xy}\cup\{y\})$ by $\{x_{i}\}_{i=1}^s$ and in $N_{y}\setminus (N_{xy}\cup\{x\})$ by $\{y_{j}\}_{j=1}^s$ such that $\pi(x_i,y_j) = \frac{1}{D+1} \delta_{ij}$. Explicitly, we have $$\label{eq:traplan}
\pi(u,v) = \begin{cases} \frac{1}{D+1}, & \text{if $(u,v) = (x_i,y_i)$,} \\
\frac{1}{D+1}, & \text{if $u=v \in N_{xy} \cup \{x,y\}$,} \\
0, & \text{otherwise.} \end{cases}$$ Then $$\label{eq:W1calc}
W_{1}(\mu_{x},\mu_{y}) = \sum_{u\in V} \sum_{v\in V} \pi(u,v)d(u,v) \ge
\frac{s}{D+1} = \frac{D-1-m}{D+1},$$ with equality iff $d(x_i,y_i) = 1$, that is, there exists a perfect matching between the vertex sets $N_{x}\setminus (N_{xy}\cup\{y\})$ and $N_{y}\setminus (N_{xy}\cup\{x\})$. Consequently, we have the following curvature estimate with the same matching property in case of equality: $$\label{eq:kappa_calc} \kappa(x,y) = \frac{D+1}{D} \left(1 -
W_1(\mu_x,\mu_y)\right) \le \frac{2+m}{D}.$$
The transport plan $\pi$ chosen in the proof is of a particular structure which will also be important later. Therefore we introduce the following definition:
\[def:plan\_tpm\] Let $e = \{x,y\} \in E$ be an edge of a $D$-regular graph $G=(V,E)$ with the following property: There is a perfect matching between the sets $N_{x}\setminus (N_{xy}\cup\{y\})$ and $N_{y}\setminus (N_{xy}\cup\{x\})$ given by $x_i \sim y_i$, where $x_i$ and $y_i$ are defined as in the above proof. We say that the transport plan $\pi$ defined by is *based on triangles and a perfect matching*.
Cartesian products {#sec:cart}
==================
In this section we show under what conditions Bonnet-Myers sharpness is preserved under taking Cartesian products. First we recall the following result of Lin, Lu and Yau from [@LLY11].
\[LLYcartprods\] Let $G=(V_{G},E_{G})$ be a $d_{G}$-regular graph and $H=(V_{H},E_{H})$ be a $d_{H}$-regular graph. Let $x_{1},x_{2}\in V_{G}$ with $x_{1}\sim x_{2}$ and $y_{1},y_{2}\in V_{H}$ with $y_{1}\sim y_{2}$. Then $$\begin{aligned}
\kappa^{G\times H}((x_{1},y_{1}),(x_{2},y_{1})) & = \frac{d_{G}}{d_{G}+d_{H}} \kappa^{G}(x_{1},x_{2}),
\\
\kappa^{G\times H}((x_{1},y_{1}),(x_{1},y_{2})) & = \frac{d_{H}}{d_{G}+d_{H}} \kappa^{H}(y_{1},y_{2}),
\end{aligned}$$
\[thm:cartprod\] Let $\{G_{i} = (V_{i}, E_{i})\}_{i=1}^{N}$ be a family of regular graphs where $G_{i}$ has valency $D_{i}$ for each $i.$ Let $L_{i}$ be the diameter of $G_{i}.$ Let $G = G_{1}\times\cdots\times G_{N}.$ The following are equivalent:
(i) $G$ is Bonnet-Myers sharp.
(ii) Each $G_{i}$ is Bonnet-Myers sharp and $\frac{D_{1}}{L_{1}}=\cdots=\frac{D_{N}}{L_{N}}.$
Since $G_{1}\times\cdots\times G_{N} = G_{1}\times(G_{2}\times\cdots\times G_{N})$ we may assume that $N=2$ and use induction.
By Theorem \[LLYcartprods\], $$\begin{aligned}
\label{eq:cart_expand}
\inf_{\substack{u,v \in V(G) \\ u\sim v}}\kappa^{G}(u,v)
& = \min\left\{\inf_{\substack{x_{1},x_{2}\in V_{1}\\x_{1}\sim x_{2}\\ y \in V_{2}}} \kappa^{G}((x_{1},y),(x_{2},y)), \inf_{\substack{y_{1},y_{2}\in V_{2}\\y_{1}\sim y_{2}\\ x \in V_{1}}} \kappa^{G}((x,y_{1}),(x,y_{2}))\right\}
\nonumber \\
& = \min\left\{\inf_{\substack{x_{1},x_{2}\in V_{1}\\x_{1}\sim x_{2}}} \frac{D_{1}}{D_{1}+D_{2}}\kappa^{G_{1}}(x_{1},x_{2}), \inf_{\substack{y_{1},y_{2}\in V_{2}\\y_{1}\sim y_{2}}} \frac{D_{2}}{D_{1}+D_{2}}\kappa^{G_{2}}(y_{1},y_{2})\right\}.
\end{aligned}$$
First we prove that (i) implies (ii). Since $G$ is Bonnet-Myers sharp, we have, by Bonnet-Myers theorem applied on $G_1$: $$\frac{2}{L_1+L_2}=\inf_{\substack{u,v \in V(G) \\ u\sim v}}\kappa^{G}(u,v)
\le \inf_{\substack{x_{1},x_{2}\in V_{1}\\x_{1}\sim x_{2}}} \frac{D_{1}}{D_{1}+D_{2}}\kappa^{G_{1}}(x_{1},x_{2}) \le \frac{D_1}{D_1+D_2}\cdot\frac{2}{L_1},$$ which is equivalent to $\frac{D_2}{L_2} \le \frac{D_1}{L_1}$.
On the other hand, Bonnet-Myers on $G_2$ gives $$\frac{2}{L_1+L_2}=\inf_{\substack{u,v \in V(G) \\ u\sim v}}\kappa^{G}(u,v)
\le \inf_{\substack{y_{1},y_{2}\in V_{2}\\y_{1}\sim y_{2}}} \frac{D_{2}}{D_{1}+D_{2}}\kappa^{G_{2}}(y_{1},y_{2}) \le \frac{D_2}{D_1+D_2}\cdot\frac{2}{L_2},$$ which is equivalent to $\frac{D_1}{L_1} \le \frac{D_2}{L_2}$.
Therefore, we can conclude that $\frac{D_1}{L_1} = \frac{D_2}{L_2}$, and all the inequalities above are sharp, that is $G_1$ and $G_2$ are Bonnet-Myers sharp as well.\
\
To prove (ii) implies (i): we simply plug into $$\inf_{\substack{x_{1},x_{2}\in V_{1}\\x_{1}\sim x_{2}}} \kappa^{G_{1}}(x_{1},x_{2})=\frac{2}{L_1} \quad \textup{and} \quad \inf_{\substack{y_{1},y_{2}\in V_{2}\\y_{1}\sim y_{2}}} \kappa^{G_{2}}(y_{1},y_{2})=\frac{2}{L_2}$$ and use the assumption that $\frac{D_1}{L_1}=\frac{D_2}{L_2}$. As a result, we obtain $\inf_{\substack{u,v \in V(G) \\ u\sim v}}\kappa^{G}(u,v)=\frac{2}{L_1+L_2}$.
\[rem:lichcartsharp\] In contrast to the necessary and sufficient condition for Bonnet-Myers sharpness in Theorem \[thm:cartprod\], much less is required for the Cartesian product $G=G_1\times G_2$ to be Lichnerowicz sharp. In fact, $G$ is Lichnerowicz sharp already if $G_1$ is Lichenerowicz sharp and $G_2$ is an arbitrary graph with its curvature lower bound large enough, as explained in the following argument.
Let $\lambda_{1}^{G_{1}}, \lambda_{1}^{G_{2}}, \lambda_{1}^{G}$ be the smallest positive eigenvalues of the Laplacians on $G_{1},G_{2},$ $G$. We have
$$\begin{aligned}
\label{eqn:lich_cart}
\inf_{\substack{u,v \in V(G) \\ u\sim v}}\kappa^{G}(u,v) & = \min\left\{\inf_{\substack{x_{1},x_{2}\in V_{1}\\x_{1}\sim x_{2}\\ y \in V_{2}}} \kappa^{G}((x_{1},y),(x_{2},y)), \inf_{\substack{y_{1},y_{2}\in V_{2}\\y_{1}\sim y_{2}\\ x \in V_{1}}} \kappa^{G}((x,y_{1}),(x,y_{2}))\right\}
\nonumber\\
& = \min\left\{\inf_{\substack{x_{1},x_{2}\in V_{1}\\x_{1}\sim x_{2}}} \frac{D_{1}}{D_{1}+D_{2}}\kappa^{G_{1}}(x_{1},x_{2}), \inf_{\substack{y_{1},y_{2}\in V_{2}\\y_{1}\sim y_{2}}} \frac{D_{2}}{D_{1}+D_{2}}\kappa^{G_{2}}(y_{1},y_{2})\right\}
\nonumber \\
& \le \min\left\{\frac{D_{1}}{D_{1}+D_{2}}\lambda_{1}^{G_{1}}, \frac{D_{2}}{D_{1}+D_{2}}\lambda_{2}^{G_{2}}\right\} = \lambda_{1}^{G}.
\end{aligned}$$
where the inequality comes from Lichnerowicz’ Theorem on each graph $G_i$: $ \inf \kappa^{G_i} \le \lambda_1^{G_i} $. In order to obtain the equality in , a sufficient condition is $ \inf \kappa^{G_1} = \lambda_1^{G_1} $ (i.e. $G_1$ is Lichnerowicz sharp) and $\frac{D_1}{D_2}\inf \kappa^{G_1} \le \inf \kappa^{G_2}$.
Examples of Bonnet-Myers sharp graphs {#sec:examples}
=====================================
Here we present various examples of Bonnet-Myers sharp graphs and study their properties. Interestingly, the $\mu$-graphs in each of the following examples are cocktail party graphs and the $1$-spheres are strongly regular.
Note that a finite simple graph $G=(V,E)$ is called *strongly regular* with parameters $(\nu,k,\lambda,\mu)$ if $G$ is not a complete graph and the following holds true:
- $V$ has cardinality $\nu$,
- every vertex has degree $k$,
- each pair of adjacent vertices has precisely $\lambda$ common neighbours,
- each pair of non-adjacent vertices has precisely $\mu$ common neighbours.
(In contrast to the usual definition, we also consider a set of $n$ isolated points to be a strongly regular graph with parameters $(\nu,k,\lambda,\mu) = (n,0,*,0)$ where $*$ can be any integer.) We say that a strongly regular graph $G$ with these parameters is ${\rm{srg}}(\nu,k,\lambda,\mu)$.
Hypercubes $Q^n$, $n \ge 1$ {#subsec:ex_hypcubes}
---------------------------
The hypercube $Q^n$ can be viewed as the graph whose vertices are elements of $\{0,1\}^n$, and two vertices $x,y \in \{0,1\}^n$ are adjacent if and only if their Hamming distance is one. In [@LLY11] the authors showed that the hypercube $Q^{n}$ has constant curvature $\frac{2}{n}$. Since $Q^{n}$ has diameter $n$, it follows that it is Bonnet-Myers sharp.
It is obvious that every $\mu$-graph of $Q^n$ consists of two isolated points and that every $1$-sphere or $Q^n$ consists of $n$ isolated points. Moreover, hypercubes are self-centered and the antipole of the vertex $(x_1,\dots,x_n) \in \{0,1\}^n$ is given by $(1-x_1,\dots,1-x_n)$.
We will show in Section \[sec:BMextrdiam\] that the hypercubes are the only Bonnet-Myers sharps graph where their valency is equal to their diameter.
Cocktail party graphs $CP(n)$, $n \ge
3$ {#subsec:ex_cocktailparty}
-------------------------------------
The cocktail party graph $CP(n)$ is defined to have vertex set $\{u_{1},\ldots,u_{n},v_{1},\ldots,v_{n}\}$ where all pairs of vertices are adjacent unless they share the same subscript. Note that $CP(n)$ has diameter $L=2$ and is regular with valency $D=2n-2$. $CP(n)$ is self-centered and $u_i$ and $v_i$ are antipoles of each other.
Let $x\sim y$ be an edge in the cocktail party graph $CP(n)$. Then $\kappa(x,y) = 1$, which implies that $CP(n)$ is Bonnet-Myers sharp.
Since $CP(n)$ is edge-transitive it suffices to show that $\kappa(u_{1},u_{2}) = 1$, which is equivalent to showing that $\kappa_{\frac{1}{2n-1}}(u_{1},u_{2}) = \frac{2n-2}{2n-1}$. Note that $\mu_{u_{1}}(v_{1}) = 0$ and $\mu_{u_{1}}$ equals $\frac{1}{2n-1}$ otherwise. Likewise $\mu_{u_{2}}(v_{2}) = 0$ and $\mu_{2}$ equals $\frac{1}{2n-1}$ otherwise. Therefore the only mass that must be transported is a mass of size $\frac{1}{2n-1}$ from $v_{2}$ to $v_{1}$ over a distance of 1. Therefore $$\kappa_{\frac{1}{2n-1}}(u_{1},u_{2}) = 1 - W_{1}(\mu_{u_{1}},\mu_{u_{2}})
= 1 - \frac{1}{2n-1} = \frac{2n-2}{2n-1},$$ as required.
It is easily checked that every $\mu$-graph of $CP(n)$ as well as any induced $1$-sphere is isomorphic to $CP(n-1)$ and therefore ${\rm{srg}}(2n-2,2n-4,2n-6,2n-4)$.
Johnson graphs $J(2n,n)$, $n \ge 3$ {#subsec:ex_johnson}
-----------------------------------
The Johnson graphs are a family of graphs that can be seen as a generalisation of the complete graphs. See [@CLP2018] where their Bakry-Émery curavture is calculated and compared to the curvature of the complete graphs.
The vertices of the Johnson graph $J(n,k)$ are all the subsets of $\{1,\ldots,n\}$ with $1 \le k \le n-1$ elements. Two vertices $u$ and $v$ are connected by an edge if $|u\cap v| = k-1$. Observe that $J(n,1)$ is isomorphic to the complete graph $K_n$ on $n$ vertices.
The Johnson graph $J(n,k)$ is $D=k(n-k)$-regular and has diameter $L=\min\{k,(n-k)\}.$ The smallest non-zero eigenvalue $\lambda_{1}$ of the Laplacian on $J(n,k)$ is $\frac{n}{k(n-k)}$.
Let $n,k\in\mathbb{N}$, $1 \leq k \leq n-1$. Let $x,y$ be to adjacent vertices in $J(n,k)$. Then $$\kappa(x,y) = \frac{n}{k(n-k)}.$$ Therefore $J(n,k)$ is Lichnerowicz sharp. Furthermore, $J(2n,n)$ is Bonnet-Myers sharp.
Without loss of generality, due to edge-transitivity, we may take $x = \{1,\ldots, k\}$ and $y = \{2,\ldots,k+1\}.$ Observe that $$N_{xy} = \left\{\{2,\ldots,k\}\cup\{i\}:i\in\{k+2,\ldots,n\} \right\}
\, \cup \, \left\{ \{1,\ldots,k+1\}\setminus\{i\}: i \in
\{2,\ldots,k\} \right\},$$ and $$\begin{aligned}
N_{x} \setminus \left(N_{xy} \cup\{y\}\right) &=&
\{(\{1,\ldots,k\}\setminus\{i\})\cup\{j\}:i\in\{2,\ldots k\},j\in\{k+2,\ldots,n\}\},
\\
N_{y} \setminus \left(N_{xy}\cup\{x\}\right) &=&
\{(\{2,\ldots,k+1\}\setminus\{i\})\cup\{j\}:i\in\{2,\ldots k\},j\in\{k+2,\ldots,n\}\}.
\end{aligned}$$ Note that $|N_{xy}| = n-2$ and there is an obvious perfect matching between $N_{x} \setminus (N_{xy}\cup \{y\})$ and $N_{y} \setminus (N_{xy}\cup\{x\})$. Therefore, by Proposition \[prop:curvcalc0\], we have $$\kappa(x,y) = \frac{n}{k(n-k)}.$$ In the particular case $J(2n,n)$, we obtain $$\kappa(x,y) = \frac{2n}{n(2n-n)} = \frac{2}{n} = \frac{2}{\min\{n,2n-n\}},$$ showing that $J(2n,n)$ is Bonnet-Myers sharp.
The graphs $J(2n,n)$ are self-centered and the antipole of the vertex $A \subset \{1,\dots,2n\}$ is given by $\{1,\dots,2n\} \backslash A$. Let us also investigate the structures of the $\mu$-graphs and $1$-spheres of $J(2n,n)$. Since Johnson graphs are distance-transitive, it suffices to consider the $\mu$-graph of $x = \{1,\dots,n\}$ and $z = \{3,\dots,n+2\}$. Then we have $$N_{xz} = \{ \{1,3,\dots,n,n+1\}, \{1,3,\dots,n,n+2\},
\{2,3,\dots,n,n+1\}, \{2,3,\dots,n,n+2\} \},$$ and the induced subgraph is a quadrangle, that is the $\mu$-graph of $x$ and $z$ is isomorphic to $CP(2)$. Finally, let us consider $$S_1(x) = \{ y_{ij}:=(\{1,\dots,n\} \backslash \{i\}) \cup \{j\}: i \in
\{1,2,\dots,n\}, j \in \{n+1,n+2,\dots,2n\} \}.$$ There is a natural identification of the vertices in $S_1(x)$ with the elements in the set $\{1,2,\dots,n\} \times \{1,2,\dots,n\}$ via $y_{ij} \mapsto (i,j-n)$. Note that in the induced subgraph $S_1(x)$ we have $y_{ij} \sim y_{kl}$ if and only if ($i=k$ and $j \neq l$) or ($i \neq k$ and $j=l$). This shows that the induced subgraph $S_1(x)$ is isomorphic to the Cartesian product $K_n \times K_n$, where the vertex set of $K_n$ is identified with the set $\{1,2,\dots,n\}$. Note that $K_n \times K_n$ is strongly regular with parameters $(n^2,n,n-2,2)$.
Demi-cubes $Q^{2n}_{(2)}$, $n \ge 3$ {#subsec:ex_demicubes}
------------------------------------
The halved cube graph, $\frac{1}{2}Q^{n}$, is the distance two sub-graph of the hypercube $Q^{n}$, that is the graph with the vertices of $Q^{n}$ formed by connecting any pair of vertices at distance exactly two in the hypercube $Q^n$. The halved cube has two isomorphic connected components and we shall denote either of these components by $Q^{n}_{(2)}.$ The graph $Q^{n}_{(2)}$ is known as the $n$-dimensional demi-cube.
Note that $Q^n_{(2)}$ is a regular graph of valency $D={n \choose
2}=\frac{n(n-1)}{2}$ and diameter $L=\lfloor \frac{n}{2} \rfloor$. Note also that the adjacency matrices of $Q^n$, $Q^n_{(2)}$ and $\frac{1}{2}Q^n$ are related by $$A_{\frac{1}{2}Q^n} = \begin{pmatrix} A_{Q^n_{(2)}} & 0 \\ 0 & A_{Q^n_{(2)}} \end{pmatrix} = \frac{1}{2} \left( A_{Q^n}^2 - n {\rm{Id}} \right).$$ Since the eigenvalues of $Q^n$ are given by $n-2i$, $i=0,\dots,n$, the second largest eigenvalue of $A_{Q^n_{(2)}}$ is $$\theta_1 = \frac{1}{2}\left( (n-2)^2-n \right) = \frac{1}{2}(n-4)(n-1)$$ and the smallest non-zero eigenvalue $\lambda_1$ of the Laplacian $\frac{1}{D} A_{Q^n_{(2)}} - {\rm{Id}}$ is $$\lambda_1 = 1 - \frac{1}{D}\theta_1 = \frac{4}{n}.$$
Let $n\geq 2$ and $x,y$ be two adjacent vertices in $Q^n_{(2)}$. Then $$\kappa(x,y) = \frac{4}{n}.$$ Therefore $Q^n_{(2)}$ is Lichnerowicz sharp and $Q^{2n}_{(2)}$ is Bonnet-Myers sharp.
We may view the vertices of $Q^n_{(2)}$ as elements of $\{0,1\}^n$ that contain an even number of ones. Two vertices are connected by an edge if their Hamming distance is equal to $2$. Let $e_i \in \{0,1\}^n$ be the $i$-th standard vector with precisely one non-zero entry at position $i$. Without loss of generality we may take $x = (0,\dots,0)$ and $y = e_1+e_2$.
Then the common neighbours of $x$ and $y$ are given by $e_1+e_j$ and $e_2+e_j$ with $3 \le j \le n$, that is, we have $|N_{xy}| = 2(n-2)$. Moreover, the vertices in $N_{x} \setminus (N_{xy}\cup \{y\})$ are given by $x_{ij}=e_i+e_j$, $3 \le i < j \le n$ and, similarly, the vertices in $N_{y} \setminus (N_{xy}\cup \{x\})$ are given by $y_{ij}=e_1+e_2+e_i+e_j$, $3 \le i < j \le n$. Obviously, the pairing $x_{ij} \sim y_{ij}$ provides a perfect matching between these sets of vertices. Therefore, by Proposition \[prop:curvcalc0\], we have $$\kappa(x,y) = \frac{4}{n} = \lambda_1.$$ This shows that $Q^n_{(2)}$ is Lichnerowicz sharp. For the graph $Q^{2n}_{(2)}$ we have $$\kappa(x,y) = \frac{4}{2n} = \frac{2}{n} = \frac{2}{L},$$ that is, $Q^{2n}_{(2)}$ is Bonnet-Myers sharp.
To identify the $\mu$-graphs of $Q^{2n}_{(2)}$, we can choose without loss of generality the distance two vertices $x= (0,\dots,0)$ and $z=e_1+e_2+e_3+e_4$. Let $y_{ij} := e_i + e_j$ with $1 \le i < j \le 2n$. Then the common neighbours of $x$ and $z$ are the $6$ vertices $y_{12}$, $y_{13}$, $y_{14}$, $y_{23}$, $y_{24}$, and $y_{34}$. Moreover, the vertex $y_{ij}$ is adjacent to all others in the $\mu$-graph of $x$ and $z$, except for the vertex $y_{kl}$ with $\{k,l\} = \{1,2,3,4\} \backslash \{i,j\}$. This shows that the $\mu$-graph of $x$ and $z$ is isomorphic to $CP(3)$.
Next, let us consider the induced $1$-sphere $S_1(x)$. Its vertices are given by $y_{ij}$, and $y_{ij}$ and $y_{kl}$ are adjacent if and only if $| \{i,j\} \cap \{k,l\} | = 1$. This shows that $S_1(x)$ is isomorphic to the strongly regular graph $J(2n,2)$ with parameters $(n(2n-1),2(2n-2),2n-2,4)$.
Finally, observe that $Q^{2n}_{(2)}$ is self-centered: Identifying the vertices of $Q^{2n}_{(2)}$ with vectors in $\{0,1\}^{2n}$, the antipole of $(x_1,\dots,x_{2n}) \in \{0,1\}^{2n}$ is $(1-x_1,\dots,1-x_{2n})$.
The Gosset graph {#subsec:ex_gosset}
----------------
The Gosset graph is a regular graph of valency $D=27$ with 56 vertices and diameter $L=3$. Its adjaceny matrix can be found in
<http://www.distanceregular.org/graphs/gosset.html>
The Gosset graph can be understood as follows: Take two copies of $K_8$, say $G, G'$ both isomorphic to $K_8$. Denote the vertex sets of $G$ and $G'$ by $\{1,2,\dots,8\}$. The edges in $G$ and $G'$ can be identified with sets $\{i,j\}$, $1 \le i < j \le 8$. However, to distinguish between edges in $G$ and edges in $G'$, we denote them by sets $\{i,j\}$ and $\{i,j\}'$, respectively. There are ${8 \choose 2} = 28$ edges in each of the graphs $G$ and $G'$, and each edge represents a vertex of the Gosset graph. Pairs of edges in the same copy are neighbours (as vertices in the Gosset graph) if and only if they have a vertex in common, that is, $\{i,j\} \sim \{k,l\}$ if and only if $| \{i,j\} \cap \{k,l\} | = 1$. Pairs of edges $\{i,j\}$ and $\{k,l\}'$ are neighbours (as vertices in the Gosset graph if and only if $\{i,j\} \cap \{k,l\} = \emptyset$. With this explicit description, we can prove the following:
Let $x,y$ be two adjacent vertices of the Gosset graph. Then $$\kappa(x,y) = \frac{2}{3},$$ and the Gosset graph is Bonnet-Myers sharp.
Since the Gosset graph is distance-transitive we only need to calculate the curvature of one edge, say, $e=\{ x=\{1,2\},y=\{2,3\} \}$. Note that $e$ lies in precisely $m=16$ triangles: There are $6$ common neighbours of $x$ and $y$ in $G$, namely, $\{1,3\}$, $\{2,4\}$, $\{2,5\}$, $\{2,6\}$, $\{2,7\}$, $\{2,8\}$, and there are $10$ common neighbours of $x$ and $y$ in $G'$, namely, $\{4,5\}'$, $\{4,6\}'$, $\{4,7\}'$, $\{4,8\}'$, $\{5,6\}'$, $\{5,7\}'$, $\{5,8\}'$, $\{6,7\}'$, $\{6,8\}'$, $\{7,8\}'$. Moreover, there is a perfect matching between the $10$ neighbours of $x$ not in $B_1(y)$ and the $10$ neighbours of $y$ not in $B_1(x)$: The $10$ neighbours of $x$ not in $B_1(y)$ are $$z_1=\{1,4\}, z_2=\{1,5\}, z_3=\{1,6\}, z_4=\{1,7\}, z_5=\{1,8\},$$ and $$z_6=\{3,4\}', z_7=\{3,5\}', z_8=\{3,6\}', z_9=\{3,7\}', z_{10}=\{3,8\}'.$$ Similarly, the $10$ neighbours of $y$ not in $B_1(x)$ are $$w_1=\{3,4\}, w_2=\{3,5\}, w_3=\{3,6\}, w_4=\{3,7\}, w_5=\{3,8\},$$ and $$w_6=\{1,4\}', w_7=\{1,5\}', w_8=\{1,6\}', w_9=\{1,7\}', w_{10}=\{1,8\}',$$ and we match $z_j \sim w_j$, $j=1,2,\dots,10$. Applying Proposition \[prop:curvcalc0\] again, we conlude that $$\kappa(x,y) = \frac{2+m}{D} = \frac{18}{27} = \frac{2}{L},$$ finishing the proof.
It is known that the induced $1$-spheres of the Gosset graph are isomorphic to the Schläfli graph which is ${\rm{srg}}(27,16,10,8)$. Moreover, the Gosset graph is self-centered since the vertices $\{i,j\}$ and $\{i,j\}'$ are antipoles of each other. Let us, finally, identify the $\mu$-graphs. Again, by distance-transitivity of the Gosset graph, it suffices to consider the $\mu$-graph of $x=\{1,2\}$ and $z=\{3,4\}$. The common neighbours of $x$ and $y$ in the Gosset graph and corresponding to edges in $G$ are $\{1,3\}$, $\{1,4\}$, $\{2,3\}$ and $\{2,4\}$. The common neighbours of $x$ and $y$ corresponding to edges in $G'$ are $\{5,6\}'$, $\{5,7\}'$, $\{5,8\}'$, $\{6,7\}'$, $\{6,8\}'$, $\{7,8\}'$. Together, these represent $10$ vertices of the Gosset graph and the vertex $\{i,j\} \subset \{1,\dots,4\}$ is adjacent to each of them except to itself and to the vertex $\{1,\dots,4\} \backslash \{i,j\}$. Similarly, the vertex $\{k,l\}'$ with $\{k,l\} \subset \{5,\dots,8\}$ is adjacent to each of these vertices except to itself and to the vertex $\{i,j\}'$ with $\{i,j\} = \{5,\dots,8\} \backslash \{k,l\}$. This shows that the $\mu$-graph of $x$ and $z$ is isomorphic to $CP(5)$.
Revisiting our examples {#sec:revisex}
-----------------------
Let us first mention a few common properties of the $(D,L)$-Bonnet-Myers sharp graphs presented in Subsections \[subsec:ex\_hypcubes\]-\[subsec:ex\_gosset\]. They are all
- self-centered,
- irreducible (with the exception of the hypercubes $Q^n$, $n \ge 2$), and
- distance-regular.
A $D$-regular connected finite graph $G=(V,E)$ of diameter $L$ is called *distance-regular* if there are integers $b_j, c_j$ such that for any two vertices $x,y \in V$ with $d(x,y) = j$ we have $d_x^-(y) = c_j$ and $d_x^+(y) = b_j$. The sequence $$(b_0,\dots,b_{L-1};c_1=1,\dots,c_L\}$$ is called the *intersection array* of the distance regular graph $G$. Moreover, $\mu$ denotes the number of common neighbours of a pair of vertices at distance $2$, i.e., $\mu=c_2$. Distance regular graphs can be also defined as those graphs $G = (V,E)$ with the property that, for any choice of integers $k,l,m \ge 0$, the cardinality of $B_k(x) \cap B_l(y)$ for $x,y \in V$ with $d(x,y) = m$ depends only on the integers $k,l,m$.
Further properties of these examples are listed in Table \[table:BMsharp\_examples\] below. We observe that all the above examples have *symmetric* intersection arrays, that is, we have $$c_j = b_{L-j} \quad \text{for $1 \le j \le L$}.$$ Moreover, we always have $$\label{eq:DL-bc}
b_0 = D, \quad b_1 = D+1-\frac{2D}{L} \quad \text{and} \quad c_2 =
\frac{2(D-L)}{L(L-1)} + 2,$$ and all $\mu$-graphs are isomorphic to the cocktail party graph $CP(c_2/2)$. Furthermore, all $1$-spheres $S_1(x)$ in these graphs are strongly regular with parameters $(\nu,k,\lambda,\mu)$. We say that our examples are *locally* ${\rm{srg}}(\nu,k,\lambda,\mu)$.
The parameters $(\nu,k,\lambda,\mu)$ are already determined by the size of the $\mu$-graphs via the following general result:
\[prop:cocktail-implies-str\] Let $G=(V,E)$ be a self-centered $(D,L)$-Bonnet-Myers sharp graph such that every $\mu$-graph is isomorphic to the cocktail party graph $CP(m)$ with $$m = \frac{D-L}{L(L-1)} + 1.$$ Suppose that the $1$-sphere $S_1(x)$ in $G$ is strongly regular. Then $S_1(x)$ is $${\rm{srg}}\left( D, \frac{2D}{L}-2, \frac{D-1}{L-1}-3,
\frac{2(D-L)}{L(L-1)}
\right).$$ Moreover, the adjacency matrix of $S_1(x)$ has second largest eigenvalue equals $\frac{(D-L)(L-2)}{L(L-1)}$.
The following proof will refer to Theorems \[onespheredegree\] and \[recursionformulas\] which appear later in Section \[sec:genfacts\] but they do not depend on the current section, and hence the following arguments are still applicable.
Assume that the induced subgraph $S_{1}(x)$ is strongly regular with parameters $(\nu,k,\lambda,\mu)$. Clearly $\nu = D$. By Theorem \[onespheredegree\] in the next section, we have $k = \frac{2D}{L}
- 2$, since every vertex is a pole in the case of a self-centered graph.
We now calculate $d_x^-(z)$ for any $z\in S_{2}(x).$ Since the $\mu$-graph of $x,z$ is isomorphic to $CP(m)$ we have $$d_x^-(z) = 2m = \frac{2(D-L)}{L(L-1)} + 2 = av_2^-(x).$$ Let $y\in S_{1}(x).$ By Theorem \[recursionformulas\], we have $$d_x^+(y) = D\left( 1 - \frac{2}{L} \right) + d_x^-(y) =
D\left( 1 - \frac{2}{L} \right) + 1 =
\frac{(L-2)D}{L} + 1 = av_1^+(x).$$ Thus, using $av_1^+(x)|S_{1}(x)| = av_2^-(x)|S_{2}(x)|,$ we obtain $$|S_{2}(x)| = D(L-1)\frac{(L-2)D+L}{2(D+L(L-2))}.$$ Let $\mathcal{T}^-$ be the set of all triangles containing the vertex $x$, $\mathcal{T}^0$ be the set of all triangles in $S_1(x)$ and and $\mathcal{T}^+$ be the set of all triangles containing two vertices in $S_1(x)$ and one vertex in $S_2(x)$. For $z\in S_{2}(x)$ we have that the number of triangles in $\mathcal{T}^+$ containing $z$ is equal to the number of edges in $CP(m),$ which is $2m(m-1)$. Thus $$\label{eq:T+}
|\mathcal{T}^+| = |S_{2}(x)|\cdot 2m(m-1) =
\frac{D}{L(L-1)}((L-2)D+L)\left(\frac{D}{L}-1\right).$$ Let $E(S_{1}(x))$ be the set of edges in $S_{1}(x).$ Since every edge in $G$ is contained in precisely $\frac{2D}{L}-2$ triangles we have $$|E(S_{1}(x))|\cdot \left(\frac{2D}{L} - 2\right) = |\mathcal{T}^-| +
3 |\mathcal{T}^0| + |\mathcal{T}^+|,$$ since every triangle in $\mathcal{T}^- \cup \mathcal{T}^+$ shares precisely one edge with $S_1(x)$ and every triangle in $\mathcal{T}^0$ shares all three edges with $S_1(x)$. Moreover, since $\lambda$ agrees with the number of triangles in ${\mathcal T}^0$ containing a fixed edge in $E(S_1(X))$, we have $$|E(S_{1}(x))|\cdot \lambda = 3 |\mathcal{T}^0|.$$ Thus $$\label{eq:calc_lambda}
|E(S_{1}(x))|\cdot \lambda = |E(S_{1}(x))|\cdot
\left(\frac{2D}{L}-2\right)- |\mathcal{T}^-|-|\mathcal{T}^+|.$$ Note that $|\mathcal{T}^-|$ is equal to the number of edges in $S_1(x)$, that is $$|\mathcal{T}^-| = |E(S_{1}(x))| = \frac{|S_1(x)|}{2} \cdot k =
\frac{D}{2}\left(\frac{2D}{L}-2\right) =
D\left(\frac{D}{L}-1\right).$$ Plugging this into and using leads to $$\lambda = \frac{D-1}{L-1} - 3.$$ Finally, we compute $\mu$ by using $\mu = \frac{k(k-\lambda - 1)}{\nu-k-1}$ (see [@BH p. 116]). This gives $$\mu = 2\frac{D-L}{L(L-1)}.$$
The second largest adjacency eigenvalue of the induced strongly regular subgraph $S_1(x)$ is given by (see [@BH Theorem 9.1.2]) $$\frac{1}{2}\left( \lambda - \mu + \sqrt{(\lambda-\mu)^2+4(k-\mu)} \right).$$ It is straightforward to check that $$(\lambda-\mu)^2 + 4(k-\mu) =
\left( \frac{L^2 + D(L-2)}{L(L-1)} \right)^2,$$ which implies $$\frac{1}{2}\left( \lambda - \mu + \sqrt{(\lambda-\mu)^2+4(k-\mu)} \right)
= \frac{(D-L)(L-2)}{L(L-1)}.$$
Applying Proposition \[prop:cocktail-implies-str\] in our examples and using the relation between the parameters $(D,L)$ and $(b_0,b_1,c_2)$, we conclude that all our distance-regular examples of Bonnet-Myers sharp graphs are locally strongly regular with parameters $${\rm{srg}}\left(b_0,b_0-b_1-1,\frac{b_0(c_2-2)}{b_0-b_1+1}-3,c_2-2 \right).$$ We know from Theorem \[thm:lichn\] that every Bonnet-Myers sharp graph is also Lichnerowicz sharp. Table \[table:BMsharp\_examples\] contains also information about the multiplicity $\dim E_{\lambda_1}$ of the Lichnerowicz eigenvalue $\lambda_1 = \kappa(x,y) = \frac{2}{L}$ for all $x,y \in V$, $x \neq y$. Note that Lichnerowicz sharpness means that the second largest eigenvalue $\theta_1$ of the adjacency matrix agrees with $b_1 -1 = D - \frac{2D}{L}$ and that there is a classification of all distance-regular graphs with second largest adjacency matrix eigenvalue $\theta_1$ equal to $b_1-1$ (see [@BCN89 Theorem 4.4.11]). For the reader’s convenience, we provide here the statement of this theorem.
\[thm:4.4.11\] Let $G=(V,E)$ be a distance-regular graph with second largest eigenvalue $\theta_1=b_1-1$. Then at least one of the following holds:
- $G$ is a strongly regular graph with smallest eigenvalue -2;
- $\mu=1$, i.e., $G$ has numerical girth at least $5$;
- $\mu=2$, and $G$ is a Hamming graph, a Doob graph, or a locally Petersen graph;
- $\mu=4$, and $G$ is a Johnson graph;
- $\mu=6$, and $G$ is a demi-cube;
- $\mu=10$, and $G$ is the Gosset graph.
This classification contains all our examples as well as many others which are not Bonnet-Myers sharp. In Section \[sec:curvandeig\], we will identify all Lichnerowicz sharp graphs in this classification. It can be checked from this classification that all *distance regular* Bonnet-Myers sharp graphs are just the ones given in our examples. In Sections \[sec:transpgeod\] and \[sec:antBMstrspher\], however, we will follow a different route and prove a much stronger result for Bonnet-Myers sharp graphs: a full classification of all *self-centered* Bonnet-Myers sharp graphs.
------------------------- -------------- ---------------- ---------------------- ------------- ------------------ ---------------------
$G$ $(D,L)$ $|V|$ $\dim E_{\lambda_1}$ $\mu$-graph $S_1(x)$ intersection array
\[.1cm\]
\[-.2cm\] $Q^n$ $(n,n)$ $2^n$ $n$ $CP(1)$ $n$ points $c_j=j$
\[.1cm\] $CP(n)$ $(2n-2,2)$ $2n$ $n$ $CP(n-1)$ $CP(n-1)$ $c_2=2n-2$
\[.1cm\] $J(2n,n)$ $(n^2,n)$ $2n \choose n$ $2n-1$ $CP(2)$ $K_n \times K_n$ $c_j = j^2$
\[.1cm\] $Q^{2n}_{(2)}$ $(2n^2-n,n)$ $2^{2n-1}$ $2n$ $CP(3)$ $J(2n,2)$ $c_j=j(2j-1)$
\[.1cm\] Gosset $(27,3)$ $56$ $7$ $CP(5)$ Schläfli $(c_2,c_3)=(10,27)$
------------------------- -------------- ---------------- ---------------------- ------------- ------------------ ---------------------
: $(D,L)$-Bonnet-Myers sharp graphs from Subsections \[subsec:ex\_hypcubes\]–\[subsec:ex\_gosset\][]{data-label="table:BMsharp_examples"}
General facts about Bonnet-Myers sharp graphs {#sec:genfacts}
=============================================
A useful inequality and its applications
----------------------------------------
Henceforth, $\Delta = \Delta_G$ denotes the normalized Laplacian on a graph $G=(V,E)$ defined in . We start with the following lemma.
\[lem:W1\_Delta\] Let $G=(V,E)$ be a finite connected $D$-regular graph and $u,v \in V$, $p \in [0,1]$. Then for any function $f:V\rightarrow {{\mathbb{R}}}$, $$\sum_{z \in V} f(z) \left( \mu_u^p(z)-\mu_v^p(z) \right) = f(u) - f(v) + (1-p)\Delta f(u) -
(1-p)\Delta f(v).$$ In particular, if $f \in \textrm{\rm{1}--{\rm Lip}}(V)$, then $$W_1(\mu_u^p,\mu_v^p) \ge f(u) - f(v) + (1-p)\Delta f(u) -
(1-p)\Delta f(v),$$ with equality iff $f$ is an optimal Kantorovich potential transporting $\mu_u^p$ to $\mu_v^p$.
The first statement is a simple calculation relating $\mu_u^p$ and $\mu_u^p$ to the Laplaction $\Delta$: $$\begin{aligned}
\sum_{z \in V} f(z) \left( \mu_u^p(z)-\mu_v^p(z) \right)
&=& \left[ p f(u) + \frac{1-p}{D} \sum_{z \sim u} f(z) \right] -
\left[ p f(v) + \frac{1-p}{D} \sum_{w \sim v} f(w) \right] \\
&=& f(u) + (1-p)\Delta f(u) - f(v) - (1-p)\Delta f(v).
\end{aligned}$$
In particular, the second statement follows immediately from Theorem \[Kantorovich\] (Kantorovich Duality).
The following result from [@MW17] will prove very useful in our investigations.
\[ineq\] Let $G = (V,E)$ be a finite connected $D$-regular graph with diameter $L$. Let $x,y\in V$ with $d(x,y) = L$, and $z$ be a vertex lying on a geodesic from $x$ to $y$ and $f \in \textrm{\rm{1}--\rm{Lip}}(V)$ satisfy $f(y)-f(x) = L$. Then $$\Delta f(z) \leq 1 -\kappa(x,z)d(x,z).$$
A very similar result was stated in [@MW17 Theorem 4.1] for the specific function $f = d(x,\cdot)$. The proof is a straightforward consequence of the alternative definition of Ollivier Ricci curvature. For the reader’s convenience, we provide a direct proof not making use of .
Observe that the negative function $-f$ lies in $\textrm{{1}--{\rm Lip}}(V)$, too. Choosing $p = \frac{1}{D+1}$, Lemma \[lem:W1\_Delta\] implies $$W_{1}(\mu_{x},\mu_{z}) \ge f(z) - f(x) -\frac{D}{D+1} \Delta f(x) + \frac{D}{D+1} \Delta f(z).$$ Thus $$\Delta f(z) \leq \frac{D+1}{D}W_{1}(\mu_{x},\mu_{z}) +
\frac{D+1}{D}(f(x)-f(z))+\Delta f(x).$$ Note that, since $f \in \textrm{\rm{1}--{\rm Lip}}(V)$ and since $f(y) - f(x) = L = d(y,x)$, $f(z) - f(x) = d(x,z)$ and $\Delta f(x) \leq 1.$ Therefore $$\begin{aligned}
\Delta f(z) & \leq \frac{D+1}{D}(W_{1}(\mu_{x},\mu_{z})-d(x,z))+1
\\
& = 1 - \kappa(x,z)d(x,z).
\end{aligned}$$
\[geosharp\] Let $G = (V,E)$ be a $(D,L)$-Bonnet-Myers sharp graph and let $x,y\in V$ with $d(x,y) = L$. Let $z,w$ be two different vertices lying on a geodesic from $x$ to $y$ with $d(x,z)+d(z,w)+d(w,y) = L$. Then $$\kappa(z,w) = \frac{2}{L}.$$
We have to show that $\kappa_{\frac{1}{D+1}}(z,w) = \frac{D}{D+1}\frac{2}{L}$. The triangle inequality tells us that $$\begin{aligned}
W_{1}(\mu_{x},\mu_{y}) \leq & W_{1}(\mu_{x},\mu_{z})+W_{1}(\mu_{z},\mu_{w})+W_{1}(\mu_{w},\mu_{y})
\\
= & d(x,z)(1-\kappa_{\frac{1}{D+1}}(x,z))+d(z,w)(1-\kappa_{\frac{1}{D+1}}(z,w))+d(w,y)(1-\kappa_{\frac{1}{D+1}}(w,y))
\\
= & L - d(x,z)\kappa_{\frac{1}{D+1}}(x,z)-d(z,w)\kappa_{\frac{1}{D+1}}(z,w)-d(w,y)\kappa_{\frac{1}{D+1}}(w,y).
\end{aligned}$$ Bonnet-Myers sharpness means that $$\kappa_{\frac{1}{D+1}}(u,v) \geq \frac{D}{D+1}\frac{2}{L},$$ for all $u,v\in V, u\neq v$. Assume that $\kappa_{\frac{1}{D+1}}(z,w) > \frac{2D}{D+1}\frac{1}{L}$. Then $$W_{1}(\mu_{x},\mu_{y}) < L -
\frac{D}{D+1}\frac{2}{L}(d(x,z)+d(z,w)+d(w,y)) = L - \frac{2D}{D+1},$$ and so $$\kappa(x,y) > \frac{2}{L},$$ which is a contradiction to . Thus $$\kappa(z,w) = \frac{2}{L}.$$
Note that we can choose $z=x$ or $w = y$ in the statement of the lemma above. This, together with Theorem \[ineq\], leads to the following result:
\[Ftrick\] Let $G = (V,E)$ be a $(D,L)$-Bonnet-Myers sharp graph with diameter $L$. Let $x,y\in V$ with $d(x,y) = L$, and $z$ be a vertex lying on a geodesic from $x$ to $y$, and $f\in \textrm{\rm{1}--{\rm Lip}}$ satisfy $f(y)-f(x) = L$. Then $$\label{eqn:Ftrick}
\Delta f(z) = 1 -\frac{2d(x,z)}{L}.$$
From Lemma \[geosharp\], $\kappa(x,z) = \kappa(y,z) = \frac{2}{L}.$ We obtain, from Theorem \[ineq\] $$\Delta f(z) \leq 1 -\frac{2d(x,z)}{L},$$ and $$\Delta (-f(z)) \leq 1 -\frac{2d(y,z)}{L} = 1 -\frac{2(L-d(x,z))}{L} = \frac{2d(x,z)}{L} - 1 .$$ Thus $$\Delta f(z) = 1 -\frac{2d(x,z)}{L},$$ as required.
\[lieongeos\] Let $G = (V,E)$ be a $(D,L)$-Bonnet-Myers sharp graph and $x,y\in V$ with $d(x,y) = L$. Then any vertex $u\in V$ lies on a geodesic from $x$ to $y$, that is, we have $[x,y] = V$.
Let $f : = d(x,\cdot),\: g:= L - d(\cdot, y)$. We must show that $f = g$. Note that $$f(z)-g(z) = d(x,z) + d(z,y) - L \geq 0.$$ Thus $f\geq g$. Suppose $f \neq g$. Then there exists a vertex $z$ closest to $x$ with $f(z)>g(z)$ with $z \neq x$. Hence there exists a vertex $w\sim z$ with $d(x,w)< d(x,z)$. By our assumptions we have $f(w) = g(w)$. Therefore $w$ is on a geodesic from $x$ to $y$. Thus, by Theorem \[Ftrick\], we have $\Delta f(w) = \Delta g(w)$. However $f(z)>g(z)$ and so $$\begin{aligned}
D \Delta f(w) &=& \sum_{u\sim w} (f(u)-f(w)) \\
&=& f(z) - f(w) + \sum_{\substack{u\sim w \\ u\neq z}}(f(u)-f(w)) \\
&>& g(z) - g(w) + \sum_{\substack{u\sim w \\ u\neq z}}(g(u)-g(w)) \\
&=& D \Delta g(w)
\end{aligned}$$ which is a contradiction. Thus $f = g$, completing the proof.
Let $G = (V,E)$ be Bonnet-Myers sharp. Then every vertex in $V$ has at most one antipole.
Assume $x \in V$ has two different antipoles $y_1$ and $y_2$. Then $y_1$ lies on a geodesic from $x$ to $y_2$, by Theorem \[lieongeos\] and, since $y_1 \neq y_2$, $$d(x,y_1) < d(x,y_2) = {\rm diam}(G),$$ which contradicts to the assumption that $y_1$ is an antipole of $x$.
\[onespheredegree\] Let $G= (V,E)$ be $(D,L)$-Bonnet-Myers sharp graph and $x \in V$ be a pole. Then we have for every edge $\{x,y\} \in E$:
- The edge $\{x,y\}$ lies in precisely $\frac{2D}{L}-2$ triangles, or, in other words, the induced subgraph $S_1(x)$ is $\left( \frac{2D}{L}-2 \right)$-regular;
- There is a perfect matching between the sets $N_x \backslash (N_{xy} \cup \{y\})$ and $N_y \backslash (N_{xy} \cup \{x\})$;
- There is an optimal transport plan $\pi$ transporting $\mu_x$ to $\mu_y$ which is based on triangles and a perfect matching (see Definition \[def:plan\_tpm\]) with the cost $$\textup{cost}(\pi)=\frac{1}{D+1}\left(D+1-\frac{2D}{L}\right).$$
Let $x\in V,$ and define $f(w) = d(x,w)- \frac{L}{2}.$ By , we have $\Delta f +\frac{2}{L} f = 0.$ Let $y\in S_{1}(x).$ We need to show that $d_{x}^{0}(y) = \frac{2D}{L}-2.$ Now since $f(z)-f(y) = 0$ if $z\sim y,$ $z\in S_{1}(x),$ $$\begin{aligned}
0 = \Delta f(y) + \frac{2}{L} f(y) & = \frac{1}{D}\left[ (f(x)-f(y))+ \sum_{z\sim y, z \in S_{2}(x)}(f(z)-f(y))\right]+\frac{2}{L} f(y)
\\
& =\frac{1}{D}(d_{x}^{+}(y)-1)+\frac{2}{L}\left(1-\frac{L}{2}\right)
\\
& = \frac{-1}{D} + \frac{1}{D}d_{x}^{+}(y)- 1 +\frac{2}{L}.
\end{aligned}$$ Rearranging gives $$d_{x}^{+}(y) = D+1 - \frac{2D}{L}$$ and, therefore, $$d_{x}^{0}(y) = D - d^{+}_{x}(y)-d^{-}_{x}(y)= D - (D+1 -\frac{2D}{L}) -1 = \frac{2D}{L} - 2.$$
We now show that there is an optimal transport plan transporting $\mu_{x}$ to $\mu_{y}$ which is based on triangles and a perfect matching of the neighbours of $x$ and $y$. We will prove this indirectly. Suppose that there is no optimal transport plan based on triangles and a perfect matching. Thus, in any optimal transport plan, there must be some mass in $N_{xy}\setminus N_x$ that travels with the distance more than $1$, so the total cost is $$W_{1}(\mu_{x},\mu_{y}) > \frac{1}{D+1}(D+1-\frac{2D}{L}),$$ and thus $$\kappa(x,y) = \frac{D+1}{D}\kappa_{\frac{1}{D+1}}(x,y) < \frac{2}{L},$$ which contradicts to the fact that $G$ is Bonnet-Myers sharp (see Lemma \[geosharp\]).
Therefore, there is an optimal transport plan $\pi$ based on triangles and a perfect matching with the cost: $$\textup{cost}(\pi)=W_{1}(\mu_{x},\mu_{y})= \frac{1}{D+1}(D+1-\frac{2D}{L}).$$
We now give relations between the in, out and spherical degrees of vertices inside Bonnet-Myers sharp graphs.
\[recursionformulas\] Let $G= (V,E)$ be a $(D,L)$-regular Bonnet-Myers sharp graph and $x \in V$ be a pole. Let $y\in S_{k}(x),$ where $k\in \mathbb{N}.$ Then $$\begin{aligned}
d_{x}^{+}(y) - d_{x}^{-}(y) &=& D\left(1-\frac{2k}{L}\right), \label{eq:rec1} \\
2d_{x}^{+}(y) + d_{x}^{0}(y) &=& 2D\left(1-\frac{k}{L}\right), \label{eq:rec2} \\
2d_{x}^{-}(y) + d_{x}^{0}(y) &=& \frac{2kD}{L}. \label{eq:rec3}
\end{aligned}$$
Define $f(w) = d(x,w)- \frac{L}{2}.$ By , we have $\Delta f +\frac{2}{L} f = 0.$ Thus $$\begin{aligned}
\Delta f(y) + \frac{2}{L} f(y) & = \frac{1}{D}\left[ \sum_{z\sim y}(f(z)-f(y))\right]+\frac{2}{L} f(y)
\\
& =\frac{1}{D}\left[ d_{x}^{+}(y)- d_{x}^{-}(y)\right]+\frac{2}{L} \left(k-\frac{L}{2}\right).
\end{aligned}$$ Rearranging gives $$d_{x}^{+}(y) - d_{x}^{-}(y) = D\left(1-\frac{2k}{L}\right).$$ The rest of the formula are obtained by using $D = d_{x}^{+}(y) + d_{x}^{0}(y) + d_{x}^{-}(y) = D $ and algebraic manipulation.
We end this subsection with the proof of Theorem \[thm:DL\_rel\].
Any $(D,L)$-Bonnet-Myers sharp graph satisfies $L \le D$. Moreover $L$ must divide $2D$.
Let $x$ be a pole of $G$. Choosing $k=1$ in Theorem \[recursionformulas\], we conclude from that $L$ must divide $2D$. Therefore, in the case $L > D$, we must have $L=2D$. Choosing $k=L-1$ in would lead to $$2d_{x}^+(y) + d_x^0(y) = 2D \left( 1 - \frac{L-1}{L} \right) = 2D - (L-1) = 1,$$ which would imply $d_{x}^+(y) = 0$ and $d_x^0(y) = 1$, which cannot be since some $y \in S_{L-1}(x)$ must be a neighbour of the antipole of $x$. Therefore, we have ruled out $L=2D$ and we conclude $L \le D$.
Self-centered Bonnet-Myers sharp graphs {#sec:self-centBMsh}
---------------------------------------
This subsection provides two immediate consequences of the results from the previous subsection under the extra assumption of self-centeredness, i.e. every vertex is a pole.
\[constcurv\] Let $G=(V,E)$ be a self-centered $(D,L)$-Bonnet-Myers sharp graph. Then we have for any pair $z,w$ of different vertices $$\kappa(z,w) = \frac{2}{L},$$ that is, $G$ has constant Ollivier-Ricci curvature $\frac{2}{L}$.
Let $z'$ be the antipole of $z$ in $G$. Then $w$ lies on a geodesic from $z$ to $z'$, by Theorem \[lieongeos\] and Lemma \[geosharp\] yields $$\kappa(z,w) = \frac{2}{L}.$$
\[cor:numtriangle\] Let $G=(V,E)$ be a self-centered $(D,L)$-Bonnet-Myers sharp graph. Then we have for every edge $\{x,y\} \in E$:
- The edge $\{x,y\}$ lies in precisely $\frac{2D}{L}-2$ triangles, or, in other words, the induced subgraph $S_1(x)$ is $\left( \frac{2D}{L}-2 \right)$-regular;
- There is a perfect matching between the sets $N_x \backslash (N_{xy} \cup \{y\})$ and $N_y \backslash (N_{xy} \cup \{x\})$;
- There is an optimal transport plan $\pi$ transporting $\mu_x$ to $\mu_y$ which is based on triangles and a perfect matching with the cost $$\textup{cost}(\pi)=\frac{1}{D+1}\left(D+1-\frac{2D}{L}\right).$$
It follows immediately from Theorem \[onespheredegree\], since every vertex of a self-centered graph is a pole.
Curvatures and eigenvalues {#sec:curvandeig}
==========================
The following result agrees with Theorem \[thm:lichn\] and it provides additional information about the choice of a suitable eigenfunction.
Every $(D,L)$-Bonnet-Myers sharp graph $G = (V,E)$ is Lichnerowicz sharp with Laplace eigenfunction $f = d(x,\cdot) - \frac{L}{2}$, where $x$ is a pole of $G$.
Let ${\rm diam}(G) = L$ and $x,y\in V$ satisfy $d(x,y) = L$. Let $f = d(x,\cdot)-\frac{L}{2}$. By Theorem \[lieongeos\] every vertex $z\in V$ lies on a geodesic from $x$ to $y$. Thus, by Lemma \[geosharp\], $\kappa(x,z) = \frac{2}{L}$. Then, by Theorem \[Ftrick\], we have $$\Delta f +\frac{2}{L}f
= 0.$$ Therefore $\lambda_{1}\leq \frac{2}{L}$. By the Discrete Lichnerowicz Theorem \[thm:Lich\] and Bonnet-Myers sharpness, we have $$\lambda_{1}\geq \inf_{\substack{u,v\in V\\ u\neq v}} \kappa(u,v)
= \frac{2}{L}.$$ Thus $\lambda_{1} = \frac{2}{L} = \inf_{u,v\in V, u\neq v} \kappa(u,v)$, completing the proof.
We also have the following general relation between eigenfunctions, curvature and Kantorovich potentials:
Let $G=(V,E)$ be a finite connected $D$-regular graph. Let $f: V \rightarrow {{\mathbb{R}}}$ be a Laplace eigenfunction, which is also an optimal Kantorovich potential transporting $\mu_u^p$ to $\mu_v^p$ for some $u,v \in V$, $u \neq v$, $p \in (\frac{1}{2},1)$. Then $\lambda=\kappa_{LLY}(u,v)$ with $\kappa_{LLY}$ defined in .
Since $f$ is an optimal Kantorovich potential transporting $\mu_u^p$ to $\mu_v^p$ and $\Delta f+\lambda f=0$, Lemma \[lem:W1\_Delta\] yields $$W_1(\mu_u^p,\mu_v^p) = \sum_{z \in V} f(z) \left( \mu_u^p(z)-\mu_v^p(z)
\right) = (1-(1-p)\lambda) ( f(u) - f(v)).$$ Moreover, since $p>\frac{1}{2}$, every optimal transport plan $\pi$ from $\mu_u^p$ to $\mu_v^p$ must satisfy $\pi(u,v)>0$, which then implies by complementary slackness that $f(u)-f(v)=d(u,v)$ (see, e.g., [@Idle Lemma 3.1], which states this fact for neighbours $x,y \in V$, but it is also true for arbitrary pairs of different vertices).
Substituting $f(u)-f(v)=d(u,v)$ in the above equation yields $$\kappa_p(u,v) = 1- \frac{W_1(\mu_u^p,\mu_v^p)}{d(u,v)}
=(1-p)\lambda,$$ which implies $$\frac{\kappa_p(u,v)}{1-p} = \lambda.$$ Since $p \in (\frac{1}{2},1)$, we conclude from [@CK2018 Corollary 3.4] $$\kappa_{LLY}(u,v) = \lim_{p \to 1} \frac{\kappa_p(u,v)}{1-p} =
\frac{\kappa_p(u,v)}{1-p},$$ finishing the proof.
The following result can be derived via similar arguments (but a different logic in its proof):
Let $G=(V,E)$ be Lichnerowicz sharp with a Laplace eigenfunction $f \in \textrm{\rm{1}--{\rm Lip}}(V)$ associated to the eigenvalue $\lambda_1 = \inf_{x \sim y} \kappa(x,y)$. Then, for any pair of different vertices $u,v \in V$ with $f(u)-f(v) = d(u,v)$, $f$ is an optimal Kantorovich potential transporting $\mu_u$ to $\mu_v$.
Assume $f \in \textrm{\rm{1}--{\rm Lip}}(V)$, $\Delta f + \lambda_1 f = 0$ and $p = \frac{1}{D+1}$. Lemma \[lem:W1\_Delta\] yields $$W_1(\mu_u,\mu_v) \ge (1-(1-p)\lambda_1) ( f(u) - f(v)),$$ with equality iff $f$ is an optimal Kantorovich potential transporting $\mu_u$ to $\mu_v$. This is equivalent to $$\kappa_p(u,v) = 1- \frac{W_1(\mu_u,\mu_v)}{d(u,v)}
\le (1-p)\lambda_1 \frac{f(u)-f(v)}{d(u,v)} = (1-p)\lambda_1,$$ using $f(u)-f(v) = d(u,v)$. This, in turn, is equivalent to $$\label{eq:kappalambda1}
\kappa(u,v) = \frac{\kappa_p(u,v)}{1-p} \le \lambda_1.$$ Our assumption $\lambda_1 = \inf_{x \sim y} \kappa(x,y)$ then implies equality in and, therefore, $f$ is an optimal Kantorovich potential transporting $\mu_u$ to $\mu_v$.
Finally, let us identify all Lichnerowicz sharp graphs within an interesting family of distance regular graphs. More precisely, as mentioned in Section \[sec:examples\], there is a classification of all distance-regular graphs with second largest adjacency eigenvalue $\theta_1=b_1-1$ (see Theorem \[thm:4.4.11\]). This class of graphs comprises all strongly regular graphs with smallest adjacency eigenvalue $-2$. In this subclass, we have the following Lichnerowicz sharp graphs.
\[thm:lichstrreg\] The Lichnerowicz sharp strongly regular graphs with smallest adjacency eigenvalue $-2$ are precisely the following ones: The cocktail party graphs $CP(n)$, $n \ge 2$, the lattice graphs $L_2(n) \cong K_n \times K_n$, $n \ge 3$, the triangular graphs $T(n) \cong J(n,2)$, $n \ge 5$, the demi-cube $Q^5_{(2)}$, and the Schläfli graph.
The theorem follows directly from Table \[table:strongly\_reg\_Lichsharp\_examples\] below, where the curvatures $\inf_{x \sim y} \kappa(x,y)$ were determined with the help of the curvature calculator [@CKLLS17] at
<http://www.mas.ncl.ac.uk/graph-curvature/>
Note that Chang stands for any one of the three Chang graphs. For the classification of all strongly regular graphs with smallest adjacency eigenvalue $-2$, see [@BH Theorem 9.2.1].
[l|l|l|l|l]{} $G$ & $(\nu,k,\lambda,\mu)$ & $\theta_1$ & $\lambda_1$ & $\displaystyle{\inf_{x \sim y} \kappa(x,y)}$\
&&&&\
$CP(n)$ & $(2n,2n-2,2n-4,2n-2)$ & $0$ & $1$ & $1$\
$K_n \times K_n$ & $(n^2,2(n-1),N-2,2)$ & $n-2$ & $\frac{n}{2(n-1)}$ & $\frac{n}{2(n-1)}$\
Shrikhande & $(16,6,2,2)$ & $2$ & $\frac{2}{3}$ & $\frac{1}{3}$\
$J(n,2)$ & $\left({n \choose 2},2(n-2),n-2,4\right)$ & $n-4$ & $\frac{n}{2(n-2)}$ & $\frac{n}{2(n-2)}$\
Chang & $(28,12,6,4)$ & $4$ & $\frac{2}{3}$ & $\frac{1}{3}$\
Petersen & $(10,3,0,1)$ & $1$ & $\frac{2}{3}$ & $0$\
$Q^5_{(2)}$ & $(16,10,6,6)$ & $2$ & $\frac{4}{5}$ & $\frac{4}{5}$\
Schläfli & $(27,16,10,8)$ & $4$ & $\frac{3}{4}$ & $\frac{3}{4}$\
In the classification Theorem \[thm:4.4.11\], we can disregard all examples with $\mu=1$ (that is, all vertices at distance $2$ have precisely one neighbour in common), since none of them can be Lichnerowicz sharp due to the following result:
\[thm:mu1\_notlich\] A distance-regular graph with second largest adjacency eigenvalue $\theta_1=b_1-1$ and $\mu=1$ cannot be Lichnerowicz sharp.
Let $G=(V,E)$ be a distance-regular graph of vertex degree $D$ and satisfying $\mu=1$, and $x,z \in V$ with $d(x,z) = 2$. We denote the unique common neighbour of $x$ and $z$ by $y$. Then we have $0 \le d_x^0(z)=:\alpha \le D-1$ and $b_1 = d_x^+(z) =
D-1-\alpha$. The second largest adjacency eigenvalue is then $\theta_1=b_1-1=D-2-\alpha$ and, consequently, the smallest positive Laplace eigenvalue is $$\lambda_1 = 1 - \frac{D-2-\alpha}{D} = \frac{2+\alpha}{D} > 0.$$ Lichnerowicz’ Theorem tells us that $$\inf_{u \sim v} \kappa(u,v) = \inf_{u \neq v} \kappa(u,v) \le \lambda_1 = \frac{2+\alpha}{D}.$$ Let us now estimate $\kappa(x,z)$. We have $$W_1(\mu_x,\mu_z) \ge \frac{1}{D+1}\left( 2 + 2(D-1-\alpha) + \alpha \right)
= \frac{2D-\alpha}{D+1},$$ and, therefore, $$\kappa_{\frac{1}{D+1}}(x,z)
= 1 - \frac{W_1(\mu_x,\mu_z)}{2} \le \frac{1+\frac{\alpha}{2}}{D+1}.$$ This implies that $$\inf_{u \sim v} \kappa(u,v) \le \kappa(x,z) = \frac{D+1}{D} \kappa_{\frac{1}{D+1}}(x,z) \le
\frac{1+\frac{\alpha}{2}}{D} = \frac{\lambda_1}{2} < \lambda_1.$$ This shows that $G$ cannot be Lichnerowicz sharp.
Using the previous two results, the following theorem provides a complete classification of all Lichnerowicz sharp distance-regular graphs with second largest adjacency eigenvalue $\theta_1=b_1-1$:
The Lichnerowicz sharp distance-regular graphs with second largest adjacency eigenvalue $\theta_1=b_1-1$ are precisely the following ones:
1. the cocktail party graphs $CP(n)$ (also Bonnet-Myers sharp);
2. the Hamming graphs $H(n,d) = (K_n)^d$ (only Bonnet-Myers sharp if $n=2$, that is $H(n,d) = Q^d$);
3. the Johnson graphs $J(n,k)$ (only Bonnet-Myers sharp if $n=2k$);
4. the demi-cubes $Q^n_{(2)}$ (only Bonnet-Myers sharp if $n$ is even);
5. the Schläfli graph (not Bonnet-Myers sharp);
6. the Gosset graph (also Bonnet-Myers sharp).
The theorem is an immediate consequence of the classification Theorem \[thm:4.4.11\], Theorems \[thm:lichstrreg\] and \[thm:mu1\_notlich\], and Table \[table:Lichsharp\_examples\] below. As before, the curvatures $\inf_{x \sim y} \kappa(x,y)$ were determined with the help of the curvature calculator [@CKLLS17] at
<http://www.mas.ncl.ac.uk/graph-curvature/>
Note that the Doob graphs are given by ${\rm Doob}^{n,m} = K_4^n \times {\rm Shk}^m$ with $n,m \ge 1$, where ${\rm Shk}$ denotes the Shrikhande graph, and the $(7,2)$-Kneser, Conway-Smith graph and Hall graph are the three locally Petersen graphs.
[l|l|l|l|l|l|l]{} $G$ & $|V|$ & $D$ & $L$ & $\theta_1=b_1-1$ & $\lambda_1$ & $\displaystyle{\inf_{x \sim y} \kappa(x,y)}$\
&&&&&\
$(K_n)^d$ & $n^d$ & $d(n-1)$ & $d$ & $n(d-1)-d$ & $\frac{n}{d(n-1)}$ & $\frac{n}{d(n-1)}$\
${\rm Doob}^{n,m}$ & $4^{n+2m}$ & $3(n+2m)$ & $n+2m$ & $3(n+2m)-4$ & $\frac{4}{3(n+2m)}$ & $\frac{2}{3(n+2m)}$\
$(7,2)$-Kneser & $21$ & $10$ & $2$ & $3$ & $\frac{7}{10}$ & $\frac{1}{2}$\
Conway-Smith & $63$ & $10$ & $4$ & $5$ & $\frac{1}{2}$ & $-\frac{1}{10}$\
Hall & $65$ & $10$ & $3$ & $5$ & $\frac{1}{2}$ & $-\frac{1}{10}$\
$J(n,k)$ & $n \choose k$ & $k(n-k)$ & $\min(k,n-k)$ & $k(n-k)-n$ & $\frac{n}{k(n-k)}$ & $\frac{n}{k(n-k)}$\
$Q^n_{(2)}$ & $2^{n-1}$ & $n \choose 2$ & $\lfloor \frac{n}{2}\rfloor$ & $\frac{(n-4)(n-1)}{2}$ & $\frac{4}{n}$ & $\frac{4}{n}$\
Gosset & $56$ & $27$ & $3$ & $9$ & $\frac{2}{3}$ & $\frac{2}{3}$
Transport geodesics of self-centered Bonnet-Myers sharp graphs {#sec:transpgeod}
==============================================================
This section together with the next one is dedicated to the proof that the examples in Subsections \[subsec:ex\_hypcubes\]-\[subsec:ex\_gosset\] and suitable Cartesian products of them are the only self-centered Bonnet-Myers sharp graphs. Of crucial importance in this proof are transport geodesic techniques. In view of this result, it is natural to ask the following:
Are there any Bonnet-Myers sharp graphs which are not self-centered?
We assume that all Bonnet-Myers sharp graphs are self-centered, but this is currently still an open problem. Let us now start to introduce the relevant tools to achieve the above mentioned goal.
Concatenation of transport maps {#sec:concat_tramap}
-------------------------------
Let $G=(V,E)$ be a simple, connected, $D$-regular graph and $\mu_0,\mu_1$ be probability measures on $V$. A transport plan $\pi \in \Pi(\mu_0,\mu_1)$ is induced by a *transport map* $T: {\rm supp}(\mu_0) \to {\rm supp}(\mu_1)$ if $\mu_1(T(x)) = \mu_0(x)$ for all $x \in V$ and $$\pi(x,y) = \begin{cases} \mu_0(x), & \text{if $x \in {\rm supp}(\mu_0)$ and $y = T(x)$}, \\ 0, & \text{otherwise.} \end{cases}$$ We define the *cost* of a transport map $T$ as the cost of its induced transport plan $\pi: V \times V \to [0,1]$: $${\rm cost}(T) := {\rm cost}(\pi) = \sum_{x \in {\rm supp}(\mu_0)} d(x,T(x)) \mu_0(x).$$ $T$ is called an *optimal transport map* from $\mu_0$ to $\mu_1$ if its induced plan $\pi \in \Pi(\mu_0,\mu_1)$ is an optimal transport plan. The existence of optimal transport maps for given probability measures $\mu_0, \mu_1$ is known as the *Monge Problem*. Let $T_1, T_2$ be transport maps from $\mu_0$ to $\mu_1$ and from $\mu_1$ to $\mu_2$, respectively. These transport maps can be concatenated to a transport map from $\mu_0$ to $\mu_2$ via $$T_2 \circ T_1: {\rm supp}(\mu_0) \to {\rm supp}(\mu_2).$$ The following fact about concatenation will be useful henceforth.
\[prop:transgeod\] Let $G=(V,E)$ be a simple, connected $D$-regular graph and $\mu_0,\mu_1\dots,\mu_k$ be probability measures on $V$. For $1 \le j \le k$, let $T_j$ be a transport map from $\mu_{j-1}$ to $\mu_j$, and $T^j$ be the concatenated map $$T^j := T_j \circ T_{j-1} \circ \cdots \circ T_1: {\rm supp}(\mu_0) \to
{\rm supp}(\mu_j).$$ Then we have $$\label{eq:concat}
{\rm cost}(T^k) \le \sum_{j=1}^k {\rm cost}(T_j).$$
Assume that we have equality in . Then, for each $z \in {\rm supp}(\mu_0)$, the sequence of vertices $$z, T^1(z), T^2(z), \cdots, T^k(z)$$ lies on a geodesic from $z$ to $T^k(z)$, that is, $$\label{eq:eqdisttramap}
d(z,T^k(z)) = d(z,T^1(z)) + d(T^1(z),T^2(z)) + \cdots + d(T^{k-1}(z),T^k(z)).$$
Such sequence of vertices $z, T^1(z), T^2(z), \cdots, T^k(z)$ is hence called a ***transport geodesic***.
Setting $T^0 = {\rm Id}_{{\rm supp}(\mu_0)}$, we have $$\begin{aligned}
{\rm cost}(T^k) &= \sum_{z \in {\rm supp}(\mu_0)} d(z,T^k(z)) \mu_0(z)\\
&\stackrel{\Delta}{\le} \sum_{z \in {\rm supp}(\mu_0)} \left( \sum_{j=1}^k d(T^{j-1}(z),T^j(z)) \right) \mu_0(z)\\
&= \sum_{j=1}^k \left( \sum_{z \in {\rm supp}(\mu_0)} d(T^{j-1}(z),T_j \circ T^{j-1}(z)) \mu_0(z)
\right)\\
&= \sum_{j=1}^k \left( \sum_{x \in {\rm supp}(\mu_{j-1})} d(x,T_j(x)) \mu_{j-1}(x) \right) = \sum_{j=1}^k {\rm cost}(T_j),\end{aligned}$$ with equality if and only if for all $z \in {\rm supp}(\mu_0)$.
Transport geodesics of a Self-centered Bonnet-Myers sharp graph {#subsection: transgeod}
---------------------------------------------------------------
In this subsection and henceforth, we always assume that our $(D,L)$-Bonnet-Myers sharp graph $G=(V,E)$ has the extra condition of self-centeredness.
Let us start with a full-length (i.e. of length $L$) geodesic $g$, and denote the vertices along this geodesic by $$g: \qquad x_0 \sim x_1 \sim x_2 \sim x_3 \sim \cdots \sim x_L.$$
For every $1\le j \le L$, since $x_{j-1}\sim x_j$ consider an optimal transport map $T_j$ from $\mu_{x_{j-1}}$ to $\mu_{x_{j}}$ based on triangles and a perfect matching (see Theorem \[cor:numtriangle\](c)), that is: $$T_j: B_1(x_{j-1}) \rightarrow B_1(x_{j})$$ is a bijective function and satisfies
1. $x=T_j(x)$ if $x\in B_1(x_{j-1})\cap B_1(x_{j})$, and
2. $x\sim T_j(x)$ if $x\in B_1(x_{j-1})\setminus B_1(x_{j})$.
For simplicity, we will write 1. and 2. together as $x\simeq T_j(x)$, where the symbol $\simeq$ means “adjacent or equal to”.
Moreover, for each $z \in B_1(x_0)$, we define $z(0):=z$ and for $1 \le j \le L$, $$z(j) := T^j(z) := T_j \circ \cdots \circ T_1(z) \in B_1(x_j).$$
Note that, in particular, we have $x_0(1) = x_0(0) = x_0$ by condition 1.
\[prop:transgeod\_BMsharp\] Let $G=(V,E)$ be a self-centered $(D,L)$-Bonnet-Myers sharp. Given a full-length geodesic $g$ and maps $T_j$ and $T^j$ (for $1\le j\le L$) defined as above. Then for every $z\in B_1(x_0)$, the sequence of vertices $$z(0) \simeq z(1) \simeq z(2)\simeq \cdots \simeq z(L)$$ is a transport geodesic. Since this transport geodesic follows closely the geodesic $g$, we call it a [**[*transport geodesic along $\boldsymbol{g}$*]{}**]{} and denote it by $g_z$.
\[rem: pm\_not\_uniq\] The definition of a transport geodesic $g_z$ depends on a full-length geodesic $g$ and sets of transport maps $\{T_j\}_{j=1}^L$. Each $T_j$ is a priori not uniquely defined, since the definition of $T_j$ is based on triangles and a perfect matching, the latter of which is not necessarily unique. We will see later (cf. Remark \[rem: pm\_uniq\]) that in fact the maps $\{T_j\}_{j=1}^L$ are already uniquely determined by $g$ in the case of self-centered Bonnet-Myers sharp graphs.
Note that $T^L$ induces a transport plan from $\mu_{x_0}$ to $\mu_{x_L}$, and together with Theorem \[cor:numtriangle\](c), we have $$W_1(\mu_{x_0},\mu_{x_L}) \le {\rm cost}(T^L) \le \sum_{j=1}^L {\rm cost}(T_j) = L \cdot\frac{1}{D+1}\left(D+1-\frac{2D}{L}\right)
= L - \frac{2D}{D+1}.$$ On the other hand, $$\frac{2}{L} \ge \kappa(x_0,x_L) = \frac{D+1}{D} \kappa_{\frac{1}{D+1}}(x_0,x_L) =
\frac{D+1}{D} \left( 1 - \frac{W_1(\mu_{x_0},\mu_{x_L})}{L} \right)$$ implies that $$L - \frac{2D}{D+1} \le W_1(\mu_{x_0},\mu_{x_L}).$$ Bringing these inequalities together, we conclude that $$\label{eq:cost_fullplan}
{\rm cost}(T^L) = \sum_{j=1}^L {\rm cost}(T_j)=L - \frac{2D}{D+1}.$$
Then by Proposition \[prop:transgeod\], for every $z\in B_1(x_0)$, the sequence of vertices $z,T^1(z),T^2(z),\cdots T^L(z)$ is a transport geodesic. This sequence is indeed the same as $$z(0) \simeq z(1) \simeq z(2)\simeq \cdots \simeq z(L)$$ since by definition $z(j)=T^j(z)$ and $z(j-1)\simeq T_j(z(j-1))=z(j)$ for all $j$.
\[prop: ext\_transgeod\] Given the same setup as in Proposition \[prop:transgeod\_BMsharp\]. Then for every $z\in B_1(x_0)$, the corresponding transport geodesic $g_z$, namely $$g_z:\quad z(0) \simeq z(1) \simeq \cdots \simeq z(L)$$ has the length $$\ell(g_z):=d(z(0),z(L))=\begin{cases}
L-1 \quad\textup{, if } z(0)=x_0 \textup{ or } z(L)=x_L\\
L-2 \quad\textup{, otherwise}.
\end{cases}$$ As an immediate consequence, every geodesic $g_z$ can be extended to the geodesic $${\rm ext}(g_z):\quad x_0\simeq z(0) \simeq z(1) \simeq \cdots \simeq z(L)\simeq x_L.$$
Since $z(0)\in B_1(x_0)$ and $z(L)\in B_1(x_L)$, triangle inequality gives $$\begin{aligned}
L=d(x_0,x_L) &\le d(x_0,z(0))+d(z(0),z(L))+d(z(L),x_L) \\
&=\mathbbm{1}_{\{x_0\not=z(0)\}}+\ell(g_z)+\mathbbm{1}_{\{x_L\not=z(L)\}}.\end{aligned}$$
Note also that $z(0)=x_0$ and $z(L)=x_L$ cannot happen simultaneously. Otherwise, it means that $\ell(g_{x_0})=L$ which would imply that the geodesic $g_{x_0}$ contains all distinct vertices $x_0(0)\sim x_0(1) \sim \cdots \sim x_0(L)$, contradicting to the repetition $x_0=x_0(0)=x_0(1)$. Therefore, $$\label{eq:estimate_gz}
\ell(g_z)\ge\begin{cases}
L-1 \quad\textup{, if } z(0)=x_0 \textup{ or } z(L)=x_L\\
L-2 \quad\textup{, otherwise}.
\end{cases}$$ and $$\label{eq:estimate_gz_sum}
\sum\limits_{z \in B_1(x_0)} \ell(g_z) \ge 2(L-1)+(D-1)(L-2).$$
On the other hand, from , $T^L$ has the cost of $L - \frac{2D}{D+1}$. It follows that $$\begin{aligned}
\frac{1}{D+1} \sum_{z \in B_1(x_0)} \ell(g_z)
&=\sum_{z \in B_1(x_0)} d(z,T^L(z)) \frac{1}{D+1}\\
&={\rm cost}(T^L)=L - \frac{2D}{D+1}\\
&=\frac{1}{D+1} \bigg(2(L-1)+(D-1)(L-2)\bigg).\end{aligned}$$ which implies that the equality holds true in , and also in as desired.
Antipoles of intervals in a self-centered Bonnet-Myers sharp graph
------------------------------------------------------------------
We still assume that our graph $G=(V,E)$ is a self-centered $(D,L)$-Bonnet-Myers sharp graph. Henceforth we will use the following notation related to intervals: Given an interval $[x,y] \subset V$ in $G$ and a vertex $z \in [x,y]$, we call a vertex $\overline{z} \in [x,y]$ an *antipole of $z$ w.r.t. $[x,y]$* if $d(x,y) = d(z,\overline{z})$. Note that antipoles were already introduced for graphs and this definition simply means that $z$ and $\overline{z}$ are antipoles of the induced subgraph of $[x,y]$. We now focus on identifying antipoles w.r.t. intervals via the method of transport geodesics.
\[thm:uniq\_ant\] Let $G=(V,E)$ be a self-centered $(D,L)$-Bonnet-Myers sharp, and given a full-length geodesic $g$: $$g: \quad x_0 \sim x_1 \sim \cdots \sim x_L.$$ Then for any $2\le k \le L$, $x_1$ has a unique antipole w.r.t. the interval $[x_0,x_k]$, which we will then denote as $\textup{ant}_{[x_0,x_k]}(x_1)$. In fact, we show that $$\textup{ant}_{[x_0,x_k]}(x_1)=x_0(k)=T^k(x_0) \in B_1(x_k)$$ for any fixed $\{T_j,T^j\}_{j=1}^L$ defined in Subsection \[subsection: transgeod\].
First, fix a set of transport maps $\{T_j,T^j\}_{j=1}^L$ associated to $g$. Suppose that there exists $z\in [x_0,x_k]$ which is an antipole of $x_1$ w.r.t. $[x_0,x_k]$, that is $z\in [x_0,x_k]$ and $d(x_1,z)=d(x_0,x_k)=k$. Since $x_1\sim x_0$ and $d(x_1,z)=k$, we have $d(x_0,z)\ge k-1$. Since $z\in [x_0,x_k]$ and $z\not=x_k$, we must have $d(x_0,z)= k-1$ and $d(z,x_k)=1$. Since $z\in B_1(x_k)$, there is a unique $a\in B_1(x_0)$ such that $a(k)=z$, that is $a=(T^k)^{-1}(z)$ (because $T^k$ is a bijective map). By Proposition \[prop: ext\_transgeod\], $$x_0\simeq a(0) \simeq a(1) \simeq ... \simeq {\underset{\displaystyle\overset{\mkern4mu{\rotatebox{90}{$\,=$}}}{z}}{a(k)}}$$ is part of the geodesic ${\rm ext}(g_a)$, so it is also a geodesic. Therefore it satisfies $$\label{eqn: x0_z}
k-1=d(x_0,z)=d(x_0,a(0))+d(a(0),a(1))+d(a(1),z).$$ On the other hand, since $d(x_1,z)=k$ and $a(1) \in B_1(x_1)$, triangle inequality gives $d(a(1),z)\ge k-1$. Equation (\[eqn: x0\_z\]) then implies $x_0=a(0)=a(1)$, which means $a=x_0$. Thus $z=a(k)=x_0(k)$. So far we have shown that, for every $2\le k \le L$, $x_0(k)$ is the only candidate for an antipole of $x_1$ w.r.t. $[x_0,x_k]$. It remains to show that $x_0(k)$ is in fact the antipole of $x_1$ w.r.t $[x_0,x_k]$.
In particular, when $k=L$, the antipole of $x_1$ w.r.t. $[x_0,x_L]=V$ exists by the assumption that $G$ is self-centered. Denote this antipole by $\overline{x}_1$. By the previous argument, $x_0(L)$ must be $\overline{x}_1$, $d(x_1,x_0(L))=L$.
Consider the transport geodesic $g_{x_0}$: $$g_{x_0}: \quad {\underset{\displaystyle\overset{\mkern4mu{\rotatebox{90}{$\,=$}}}{x_0}}{x_0(0)}}= x_0(1)\simeq x_0(2) \simeq \cdots \simeq {\underset{\displaystyle\overset{\mkern4mu{\rotatebox{90}{$\,=$}}}{\overline{x}_1}}{x_0(L)}}$$
Since $g_{x_0}$ has length $L-1$ (by Proposition \[prop: ext\_transgeod\]), all the “$\simeq$” in $g_{x_0}$ must be strict “$\sim$”. Thus $g_{x_0}$ can be written as $$g_{x_0}: \quad {\underset{\displaystyle\overset{\mkern4mu{\rotatebox{90}{$\,=$}}}{x_0}}{x_0(0)}}= x_0(1)\sim x_0(2) \sim \cdots \sim {\underset{\displaystyle\overset{\mkern4mu{\rotatebox{90}{$\,=$}}}{\overline{x}_1}}{x_0(L)}}.$$ Moreover, since $g_{x_0}$ has length $L-1$ and $d(x_1,\overline{x}_1)=L$, the geodesic $g_{x_0}$ can then be extended (by adding $x_1$ to the left) to another geodesic $g'$: $$g': \quad x_1\sim {\underset{\displaystyle\overset{\mkern4mu{\rotatebox{90}{$\,=$}}}{x_0}}{x_0(0)}}= x_0(1)\sim x_0(2) \sim \cdots \sim {\underset{\displaystyle\overset{\mkern4mu{\rotatebox{90}{$\,=$}}}{\overline{x}_1}}{x_0(L)}}.$$ Consequently, we can read off from the geodesic $g'$ that for every $k\in\{2,...,L\}$
1. $d(x_1,x_0(k))=k$,
2. $x_0(k)\in [x_0,x_k]$, because $k=d(x_0,x_k)\le d(x_0,x_0(k))+d(x_0(k),x_k)\le (k-1)+1$.
Therefore, $x_0(k)$ is the unique antipole of $x_1$ w.r.t. $[x_0,x_k]$ as desired.
Let us first discuss an immediate consequence of Theorem \[thm:uniq\_ant\]. Note that the theorem implies that there is a well-defined antipole map $$\textup{ant}_{[x,y]}: [x,y] \cap B_1(x) \to [x,y] \cap B_1(y).$$ Existence and uniqueness of antipoles for neighbours of $x$ w.r.t. $[x,y]$ implies the following result:
\[cor: ant\_biject\] Let $G = (V,E)$ be a self-centered Bonnet-Myers sharp graph, $x, y\in V$ be two different vertices. Then the antipole map $$\textup{ant}_{[x,y]}: [x,y] \cap B_1(x) \to [x,y] \cap B_1(y)$$ is bijective and, consequently, $$\left| [x,y] \cap B_1(x)\right| = \left| [x,y] \cap B_1(y) \right|.$$
\[rem:ant\_toggle\] Let $x,y \in V$ be two different vertices and $x' \in [x,y] \cap B_1(x)$ with its antipole $y' = \textup{ant}_{[x,y]}(x')$. Observe that then $x,y \in [x',y']$ and $y = \textup{ant}_{[x',y']}(x)$.
Another immediate consequence of Theorem \[thm:uniq\_ant\] is the following corollary.
\[cor: mu\_CP\] Let $G = (V,E)$ be a self-centered Bonnet-Myers sharp graph. Then all $\mu$-graphs of $G$ are cocktail party graphs.
Let $x,y \in V$ with $d(x,y)=2$ and $z \in N_{xy}$. Since $G$ is self-centered Bonnet-Myers sharp, $x=x_0$ has an antipole $x_L \in V$, and we can find a geodesic $g$ from $x_0$ to $x_L$ passing through $z=x_1$ and $y=x_2$ by Theorem \[lieongeos\]: $$g: \quad {\underset{\displaystyle\overset{\mkern4mu{\rotatebox{90}{$\,=$}}}{x}}{x_0}} \sim {\underset{\displaystyle\overset{\mkern4mu{\rotatebox{90}{$\,=$}}}{z}}{x_1}} \sim {\underset{\displaystyle\overset{\mkern4mu{\rotatebox{90}{$\,=$}}}{y}}{x_2}} \sim \cdots \sim x_L.$$ Applying Theorem \[thm:uniq\_ant\] with $k=2$, we conclude that there is a unique $z' \in N_{xy}$ satisfying $d(z,z')=2$. This shows that the $\mu$-graph of $x$ and $y$ is a cocktail party graph.
The fact that all $\mu$-graphs of $G$ are cocktail party graphs allows us to naturally introduce a *switching map*, defined as follows. Consider a pair $x,y\in V$ with $d(x,y)=2$. Then the switching map $\sigma_{xy}:N_{xy}\to N_{xy}$ is defined by $\sigma_{xy}(z):={\rm ant}_{[x,y]}(z)$ and satisfies $\sigma_{xy}^2 = {\rm Id}_{N_{xy}}$.
\[rem: pm\_uniq\] Recall from Remark \[rem: pm\_not\_uniq\] that for general Bonnet-Myers sharp graphs the perfect matchings defining the maps $T_j$ are not necessarily unique. However, under the additional condition of self-centeredness, the fact that all $\mu$-graphs of $G$ are cocktail party graphs implies the uniqueness of these perfect matchings and the associated transport maps $T_j$. Therefore, the definition of a transport geodesic $g_z$ depends only on the geodesic $g$.
In particular, the transport geodesic $g_{x_0}$ containing all antipoles of $x_1$ w.r.t. increasing intervals $[x_0,x_k]$ (see Theorem \[thm:uniq\_ant\]) can be also understood as been generated via the following recursive process of switching maps, as illustrated in Figure \[fig:transport\_geodesic\]: $$\begin{aligned}
x_0(2) &= \sigma_{x_0x_2}(x_1), \\
x_0(3) &= \sigma_{x_0(2)x_3}(x_2), \\
&\vdots \\
x_0(k) &= \sigma_{x_0(k-1)x_k}(x_{k-1}), \\
& \vdots \\
x_0(L) &= \sigma_{x_0(L-1)x_L}(x_{L-1}).
\end{aligned}$$
(0,0)–(12.5,0); (0,0)–(1.5,-1.5)–(11,-1.5); (11,-1.5)–(12.5,0) ; (3,0)–(1.5,-1.5); (4.5,0)–(3,-1.5); (7.5,0)–(6,-1.5); (9,0)–(7.5,-1.5);
(0,0) circle \[radius=0.05\]; at (0,0) [$x_0$]{}; at (0,0) [$\boldsymbol {x_0(0)=x_0(1)}$]{};
(1.5,0) circle \[radius=0.05\]; at (1.5,0) [$x_1$]{};
(3,0) circle \[radius=0.05\]; at (3,0) [$x_2$]{};
(4.5,0) circle \[radius=0.05\]; at (4.5,0) [$x_3$]{};
(7.5,0) circle \[radius=0.05\]; at (7.5,0) [$x_{k-1}$]{};
(9,0) circle \[radius=0.05\]; at (9,0) [$x_{k}$]{};
(12.5,0) circle \[radius=0.05\]; at (12.5,0) [$x_L$]{};
(1.5,-1.5) circle \[radius=0.05\]; at (1.5,-1.5) [$\boldsymbol{x_0(2)}$]{};
(3,-1.5) circle \[radius=0.05\]; at (3,-1.5) [$\boldsymbol{x_0(3)}$]{};
(6,-1.5) circle \[radius=0.05\]; at (5.7,-1.5) [$\boldsymbol{x_0(k-1)}$]{};
(7.5,-1.5) circle \[radius=0.05\]; at (7.5,-1.5) [$\boldsymbol{x_0(k)}$]{};
(11,-1.5) circle \[radius=0.05\]; at (11,-1.5) [$\boldsymbol{x_0(L)}$]{};
Self-centered Bonnet-Myers sharp implies strongly spherical {#sec:antBMstrspher}
===========================================================
The ultimate goal of this section is to prove that all self-centered Bonnet-Myers sharp graphs are strongly spherical (as stated in Theorem \[thm\_spherical\] below). Let us recall the definition of self-centeredness, antipodal, and strongly spherical (introduced in Definition \[def:strspher\] and in Subsection \[sec:graph\_notation\]). For a finite connected graph $G=(V,E)$:\
$\bullet$ $G$ is if for every $x\in V$ there exists $\overline{x}\in V$ such that $d(x,\overline{x})=\textup{diam}(G).$\
$\bullet$ $G$ is if for every $x\in V$ there exists $\overline{x}\in V$ such that $[x,\overline{x}]=V$. The vertex $\overline{x}$ is then called an *antipode* of $x$.\
$\bullet$ $G$ is if $G$ is antipodal, and the induced subgraph of every interval of $G$ is antipodal.
It is important to notice the distinction between the notions “antipole” and “antipode”. Here are basic facts about antipodes:
- Antipodes are also antipoles: Let $\overline{x}$ be an antipode of $x$ in $G$, that is, $[x,\overline{x}]=V$. We choose arbitrary $y,z \in V$ such that $d(y,z) = \textup{diam}(G)$. Then we have by $y,z \in [x,\overline{x}]$ and the triangle inequality $$\begin{gathered}
\textup{diam}(G) \ge d(x,\overline{x}) = \frac{1}{2} (d(x,y) +
d(y,\overline{x})) +
\frac{1}{2} (d(x,z) + d(z,\overline{x})) \\
= \frac{1}{2} (d(y,x) + d(x,z)) + \frac{1}{2} (d(y,\overline{x}) +
d(\overline{x},z)) \ge d(y,z) = \textup{diam}(G).\end{gathered}$$
- Antipodes are necessarily unique: Assume $\overline{x}_1$ and $\overline{x}_2$ are antipodes of $x$. Then $\overline{x}_2$ lies on a geodesic from $x$ to $\overline{x}_1$. Since $d(x,\overline{x}_1)=d(x,\overline{x}_2)$, this implies $\overline{x}_1=\overline{x}_2$.
For the reader’s convenience, let us start with a brief overview of the proof that self-centered Bonnet-Myers sharp graphs are strongly spherical. Note first that every self-centered Bonnet-Myers sharp graph coincides with the interval of any pair of antipoles (by Theorem \[lieongeos\]). Therefore, it suffices to prove that every interval $[x,y]$, $x,y \in V$, of a self-centered Bonnet-Myers sharp graph $G=(V,E)$ is antipodal. This proof is divided into the following four steps:
Step 1:
: Let $x' \in [x,y] \cap S_1(x)$ with antipole $y' = \textup{ant}_{[x,y]}(x')$. We prove for every $z \in [x,y] \cap B_1(x)$ that $z \in [x',y']$ (see Theorem \[thm\_spherical\_step1\]).
Step 2:
: Let $x' \in [x,y] \cap S_1(x)$ with antipole $y' = \textup{ant}_{[x,y]}(x')$. We prove for every $z \in [x,y]$ that $z \in [x',y']$ (see Theorem \[thm\_spherical\_step2\]).
Step 3:
: Let $x' \in [x,y] \cap S_1(x)$ with antipole $y' = \textup{ant}_{[x,y]}(x')$. We prove that $[x,y] = [x',y']$ (see Corollary \[cor\_clockwise\_spherical\]).
Step 4:
: Let $x' \in [x,y]$. We prove that there exists $y' \in [x,y]$ such that $[x,y] = [x',y']$ (see Theorem \[thm\_spherical\]).
Let us now start to prove each of these steps in order. Recall that the existence of antipoles of vertices in $[x,y] \cap B_1(x)$ w.r.t. $[x,y]$ is guaranteed by Corollary \[cor: ant\_biject\].
\[thm\_spherical\_step1\] Let $G=(V,E)$ be self-centered Bonnet-Myers sharp. Let $x,y \in V$ be two different vertices, and consider any $x'\in [x,y]\cap S_1(x)$ with its antipole $y'=\textup{ant}_{[x,y]}(x')$. Then every $z \in [x,y]\cap B_1(y)$ satisfies $z\in[x',y']$.
We start with the set-up and introduce particular sets $A,A_1,A_2,Z,Z_1,Z_2$ and a function $F$ which will be important for the proof of the above theorem.
Let $k=d(x,y)$. We re-label the vertices as $x=x_0$ and $y=x_k$ and $x'=x_1$ and $y'=\overline{x}_1$, as illustrated in Figure \[fig:intervalxy\]. Keep in mind that $\overline{x}_1=\textup{ant}_{[x_0,x_k]}(x_1)$ and $x_0\sim x_1$ and $x_k\sim \overline{x}_1$.
(0,0) to \[out=80,in=180\] (4,2); (0,0) to \[out=-80,in=180\] (4,-2); (4,2) to \[out=0,in=100\] (8,0); (4,-2) to \[out=0,in=-100\] (8,0); (0,0) to \[out=10,in=175\] (6,0.3); (6,0.3) to \[out=-5,in=170\] (8,0);
(0,0) circle \[radius=0.05\]; at (0,0) [$x$]{}; at (0.2,-0.5) [$\boldsymbol{x_0}$]{};
(8,0) circle \[radius=0.05\]; at (8,0) [$y$]{}; at (8,-0.5) [$\boldsymbol{x_k}$]{};
(1.5,1.65) circle \[radius=0.05\]; at (1.5,1.75) [$x'$]{}; at (1.6,1.15) [$\boldsymbol{x_1}$]{};
(6.5,-1.65) circle \[radius=0.05\]; at (6.5,-1.65) [$y'=\textup{ant}_{[x,y]}(x')$]{}; at (6.5,-2.35) [$\boldsymbol{\overline{x}_1=\textup{\bf ant}_{[x_0,x_k]}(x_1)}$]{};
(6,0.3) circle \[radius=0.05\]; at (6,0.3) [$z$]{};
We define the following sets $$\begin{aligned}
& A:=[x_0,x_k]\cap B_1(x_0), && Z:=[x_0,x_k]\cap B_1(x_k),\\
& A_1:=A\cap S_1(x_1)\setminus\{x_0\}, && Z_1:=Z\cap S_1(\overline{x}_1)\setminus\{x_k\},\\
& A_2:=A\cap S_2(x_1), && Z_2:=Z\cap S_2(\overline{x}_1).\end{aligned}$$ Note that the sets $A$ and $Z$ can be partitioned into $$A=\{x_0,x_1\}\sqcup A_1 \sqcup A_2 \qquad \textup{and} \qquad Z=\{x_k,\overline{x}_1\}\sqcup Z_1 \sqcup Z_2.$$
Now fix an arbitrary full-length geodesic $g$ from $x_0$ to $x_L$ (the antipole of $x_0$) which passes through $x_1$ and $x_k$ (this can be done since $x_1\in [x_0,x_k]$ and $x_k\in [x_0,x_L]$ by Theorem \[lieongeos\]), namely $$g: \quad x_0\sim x_1 \sim x_2 \sim \cdots \sim x_k \sim x_{k+1} \sim \cdots \sim x_L.$$ Consider the transport map $T^k:B_1(x_0)\rightarrow B_1(x_k)$ introduced in Subsection \[subsection: transgeod\]. Recall that $T^k$ is bijective. Then define a function $F:Z\rightarrow A$ to be $F(z):=(T^k)^{-1}(z)$ for all $z\in Z\subset B_1(x_k)$. Lemma \[lemma\_FZ\] below guarantees that $F(Z) \subseteq A$, hence $F$ is well-defined.
In order to conclude Theorem \[thm\_spherical\_step1\], we need to prove that $\forall z\in Z:\ z\in [x_1,\overline{x}_1]$, which is divided into Lemma \[lemma\_FZ2\] (dealing with the case $z \in Z_2 \sqcup \{x_k,\overline{x_1}\}$) and Lemma \[lemma\_FZ1\] (dealing with the case $z \in Z_1$).
\[lemma\_FZ\] $F(Z) \subseteq A$ and $F: Z \to A$ is bijective.
\[lemma\_FZ2\] $F(Z_2\sqcup\{x_k,\overline{x}_1\})=A_2\sqcup \{x_0,x_1\}$ and $\forall z\in Z_2\sqcup\{x_k,\overline{x}_1\}:\ z\in [x_1,\overline{x}_1]$.
\[lemma\_FZ1\] $F(Z_1)=A_1$ and $\forall z\in Z_1:\ z\in [x_1,\overline{x}_1]$.
Now we will prove the above three lemmas in order, and then conclude Theorem \[thm\_spherical\_step1\].
First we show that $F(z)\in A$ for all $z\in Z$. Let $a:=F(z)\in B_1(x_0)$, that is $a=a(0)$ and $z=a(k)$. By Proposition \[prop: ext\_transgeod\] we know that $$x_0\simeq a(0) \simeq a(1) \simeq ... \simeq a(k)$$ is a geodesic (as a part of ${\rm ext}(g_a)$). Moreover, since $a(k)=z\in [x_0,x_k]$, this geodesic can be extended to another geodesic $\gamma$, namely $$\label{eq:geodgamma}
\gamma:\quad x_0\simeq {\underset{\displaystyle\overset{\mkern4mu{\rotatebox{90}{$\,=$}}}{a}}{a(0)}} \simeq a(1) \simeq ... \simeq {\underset{\displaystyle\overset{\mkern4mu{\rotatebox{90}{$\,=$}}}{z}}{a(k)}}\simeq x_k.$$ Therefore, $a$ must lie in the interval $[x_0,x_k]$, which means $a\in A$ and we have $F(Z) \subseteq A$.
Next, note that the function $F:Z\rightarrow A$, which is a restriction of $(T^k)^{-1}$, must be injective (because $T^k$ is bijective). Note also that $|A|=|Z|$ because of Corollary \[cor: ant\_biject\]. Therefore, $F$ must be bijective.
A main feature of the following proof is to show $A_2 \sqcup \{x_0,x_1\} \subseteq F(Z_2 \sqcup
\{x_k,\overline{x}_1\})$. For that reason we start with an element $a\in A_2\sqcup \{x_0,x_1\}$. Then there exists a uniqe $z \in Z$ with $F(z)=a$. Consequently, $z=a(k)$ and $z \in [x_0,x_k]$. Consider the following two cases.
From Theorem \[thm:uniq\_ant\], we have $a=x_0=(T^k)^{-1}(\overline {x}_1)=F(\overline {x}_1)$, so $z=\overline {x}_1$ and $z \in [x_1,\overline{x}_1]$.
As in the proof of Lemma \[lemma\_FZ\], we have the following geodesic $\gamma$ of length $k$ (referred to the one in ): $$\gamma: \quad x_0\sim {\underset{\displaystyle\overset{\mkern4mu{\rotatebox{90}{$\,=$}}}{a}}{a(0)}} \simeq a(1) \simeq ... \simeq {\underset{\displaystyle\overset{\mkern4mu{\rotatebox{90}{$\,=$}}}{z}}{a(k)}} \simeq x_k.$$ From this geodesic $\gamma$ and an observation that $$d(x_0,a(1)) = \begin{cases} 1, & \text{if $a=x_1$ (and therefore also $a(1)=x_1$),} \\
2, & \text{if $a \in A_2$ (and therefore $a(1)\neq a(0)$),} \end{cases}$$ we conclude $$\label{eq:da1xk}
d(a(1),x_k) = \begin{cases} k-1, & \text{if $a=x_1$,} \\
k-2, & \text{if $a \in A_2$.} \end{cases}$$ Now we extend the geodesic $$a(1) \simeq ... \simeq {\underset{\displaystyle\overset{\mkern4mu{\rotatebox{90}{$\,=$}}}{z}}{a(k)}} \simeq x_k$$ to $$x_1 \simeq a(1) \simeq ... \simeq {\underset{\displaystyle\overset{\mkern4mu{\rotatebox{90}{$\,=$}}}{z}}{a(k)}} \simeq x_k \sim \overline{x}_1,$$ which is, again a geodesic because of (and recall that $d(x_1,\overline{x}_1) = k$). We can then read off from the above geodesic that $z = x_k$ or $z \in S_2(\overline{x}_1) \cap [x_0,x_k] = Z_2$ and $z \in [x_1,\overline{x}_1]$.
We conclude from both cases that $A_2\sqcup \{x_0,x_1\}\subseteq F(Z_2\sqcup\{x_k,\overline{x}_1\})$. Since $F$ is bijective, it follows that $|A_2| \le |Z_2|$. By switching the roles between $x_0$ and $x_k$ and between the antipoles $x_1$ and $\overline{x}_1$ w.r.t. $[x_0,x_k]$, we obtain the opposite inequality $|Z_2| \le |A_2|$. Therefore, we have $|Z_2| = |A_2|$, and thus $A_2\sqcup \{x_0,x_1\} = F(Z_2\sqcup\{x_k,\overline{x}_1\})$, as desired.
Consequently, if we consider any $z\in Z_2\sqcup\{x_k,\overline{x}_1\}$, then $a\in A_2\sqcup \{x_0,x_1\}$ falls into one of the above cases, in which we have shown $z\in [x_1,\overline{x}_1]$.
Since $F: Z \to A$ is bijective and $F(Z_2\sqcup\{x_k,\overline{x}_1\}) = A_2\sqcup \{x_0,x_1\}$, we conclude $F(Z_1) = A_1$.
Moreover, consider $z\in Z_1$. It follows that $z\sim \overline{x}_1$ and $d(x_1,z)\le k-1$, because $z\not=\overline{x}_1=\textup{ant}_{[x_0,x_k]}(x_1)$. Therefore $$d(x_1,z)+d(z,\overline{x}_1) = d(x_1,z)+1 \le (k-1)+1=k,$$ which means $z\in[x_1,\overline{x}_1]$.
Recalling the original set-up and notation, we only need to show that $z \in [x_1,\overline{x}_1]$. This follows immediately from Lemma $\ref{lemma_FZ2}$ and Lemma \[lemma\_FZ1\].
The next theorem generalized Theorem \[thm\_spherical\_step1\] by removing the restriction $z\in B_1(y)$.
\[thm\_spherical\_step2\] Let $G=(V,E)$ be self-centered Bonnet-Myers sharp. Let $x,y \in V$ be two different vertices, and consider any $x'\in [x,y]\cap S_1(x)$ with its antipole $y'=\textup{ant}_{[x,y]}(x')$. Then every $z \in [x,y]$ satisfies $z\in[x',y']$.
Let $d_1=d(x,y)$ and $d_2=d(z,y)$ (note that $0\le d_2\le d_1$). We will prove the statement of the theorem by induction on $d_1$ and $d_2$.
For any value of $d_1$, the cases $d_2=0,1$ are both covered by Theorem \[thm\_spherical\_step1\].
Assume that the statement is true for $d_1=k-1$ and all $d_2$, and assume that the statement is true for $d_1=k$ and $d_2=j-1$ for some $2\le j\le k-1$. Now consider $d(x,y)=k$ and $z\in [x,y]\cap S_j(y)$. Choose an arbitrary $z_1\in [z,y]\cap S_1(y)$. Hence $x,z,z_1,y$ lies in a geodesic , see Figure \[fig:intervalxy\_induction\]. In particular, $z\in [x,z_1]$.
(0,0) to \[out=80,in=180\] (4,2); (0,0) to \[out=-80,in=180\] (4,-2); (4,2) to \[out=0,in=100\] (8,0); (4,-2) to \[out=0,in=-100\] (8,0); (0,0) to \[out=10,in=175\] (6,0.3); (4,0.4) to \[out=0,in=170\] (8,0);
(0,0) circle \[radius=0.05\]; at (0,0) [$x$]{};
(8,0) circle \[radius=0.05\]; at (8,0) [$y$]{};
(0.8,1.3) circle \[radius=0.05\]; at (0.8,1.3) [$x'$]{};
(7.2,-1.3) circle \[radius=0.05\]; at (7.2,-1.4) [$y'$ $=\textup{ant}_{[x,y]}(x')$]{};
(4,0.4) circle \[radius=0.05\]; at (4,0.4) [$z$]{};
(7,0.18) circle \[radius=0.05\]; at (7,0.18) [$z_1$]{};
Now consider the following three cases whether $d(z_1,y')$ is 0, 1, or 2.
It follows immediately that $z\in [x,z_1]=[x,y'] \subseteq [x',y']$ where the last inclusion is due to $x\in[x',y']$.
Since $z_1 \in [x,y]$, by Theorem \[thm:uniq\_ant\] there is a unique $a_1=\textup{ant}_{[x,y]}(z_1) \in [x,y]$. Since $a_1 \in [x,y] \cap B_1(x)$ and $z_1 \in [x,y] \cap B_1(y)$, by Theorem \[thm\_spherical\_step1\], $z_1, a_1 \in[x',y']$.
The fact that $a_1,z_1\in[x',y']$ and that $d(a_1,z_1)=d(x,y)=d(x',y')$ altogether implies that $a_1$ must be the unique antipole $\textup{ant}_{[x',y']}(z_1)$ by Corollary \[cor: ant\_biject\] since $z_1 \sim y'$. This is illustrated in Figure \[fig:induction\_case2\]. By Remark \[rem:ant\_toggle\], it implies that $y'=\textup{ant}_{[a_1,z_1]}(x')$.
Observe also that $z\in [x,z_1]\subset [a_1,z_1]$ with $d(z,z_1)=j-1$.
We are now in a position to apply the induction hypothesis for the interval $[a_1,z_1]$ (instead of $[x,y]$) and $z \in [a_1,z_1]$ with $d(z,z_1)=j-1$. Note that $d(a_1,z_1) = k$. Note also that $x' \in [a_1,z_1] \cap S_1(a_1)$ and $y' = \textup{ant}_{[a_1,z_1]}(x')$. Then the induction hypothesis implies $z \in [x',y']$, finishing this case.
(0,0) to \[out=80,in=180\] (4,2); (0,0) to \[out=-80,in=180\] (4,-2); (4,2) to \[out=0,in=100\] (8,0); (4,-2) to \[out=0,in=-100\] (8,0); (0,0) to \[out=10,in=175\] (6,0.3); (4,0.4) to \[out=0,in=170\] (8,0); (0,0) to \[out=-30,in=175\] (1, -0.3);
(0.8,1.3) to \[out=-120,in=135\] (1, -0.3) to \[out=-40,in=-160\] (7.2,-1.3) to \[out=60,in=-45\] (7,0.18) to \[out=135,in=20\] (0.8,1.3);
(0,0) circle \[radius=0.05\]; at (0,0) [$x$]{};
(8,0) circle \[radius=0.05\]; at (8,0) [$y$]{};
(0.8,1.3) circle \[radius=0.05\]; at (0.8,1.3) [$x'$]{};
(7.2,-1.3) circle \[radius=0.05\]; at (7.2,-1.4) [$y'$ $=\textup{ant}_{[x,y]}(x')$]{};
(4,0.4) circle \[radius=0.05\]; at (4,0.4) [$z$]{};
(7,0.18) circle \[radius=0.05\]; at (7,0.18) [$z_1$]{};
(1,-0.3) circle \[radius=0.05\]; at (1,-0.3) [$a_1$$=\textup{ant}_{[x,y]}(z_1)$]{};
Since $z_1 \in [x,y] \cap B_1(y)$, by Theorem \[thm\_spherical\_step1\], we have $z_1\in[x',y']$. The condition $d(z_1, y')=2$ then implies that $d(x',z_1)=d(x',y')-2=k-2$. It follows that $$d(x,x')+d(x',z_1)+d(z_1,y)=1+(k-2)+1=k=d(x,y)$$ which means that $x'$ and $z_1$ lie on a geodesic from $x$ to $y$. Let us denote this geodesic by $g^*$: $$g^*: \quad x\sim x'\sim\cdots\sim z_1\sim y.$$ In particular, $x'\in [x,z_1]$. Then $y'':=\textup{ant}_{[x,z_1]}(x')$ exists by Corollary \[cor: ant\_biject\]. The situation is illustrated in Figure \[fig:induction\_case3\].
(0,0) to \[out=80,in=180\] (4,2); (0,0) to \[out=-80,in=180\] (4,-2); (4,2) to \[out=0,in=100\] (8,0); (4,-2) to \[out=0,in=-100\] (8,0); (0,0) to \[out=10,in=175\] (6,0.3); (4,0.4) to \[out=0,in=170\] (8,0);
(0,0) to \[out=80,in=-140\] (0.8,1.3) to \[out=20,in=135\] (7,0.18) to \[out=-90,in=30\] (6.2,-0.8) to \[out=-170,in=-40\] (0,0);
(0,0) circle \[radius=0.05\]; at (0,0) [$x$]{};
(8,0) circle \[radius=0.05\]; at (8,0) [$y$]{};
(0.8,1.3) circle \[radius=0.05\]; at (0.8,1.3) [$x'$]{};
(7.2,-1.3) circle \[radius=0.05\]; at (7.2,-1.5) [$y'=\textup{ant}_{[x,y]}(x')$]{};
(4,0.4) circle \[radius=0.05\]; at (4,0.4) [$z$]{};
(7,0.18) circle \[radius=0.05\]; at (7,0.18) [$z_1$]{};
(6.2,-0.8) circle \[radius=0.05\]; at (6.2,-0.8) [$y''$]{};
Next we apply the induction hypothesis for the interval $[x,z_1]$ (instead of $[x,y]$) and $z \in [x,z_1]$ with $d(z,z_1)=j-1$. Note that $d(x,z_1) = k-1$. Note also that $x' \in [x,z_1] \cap S_1(x)$ and $y'' = \textup{ant}_{[x,z_1]}(x')$. Then the induction hypothesis implies $z \in [x',y'']$.
So far we have that $d(x',z) + d(z,y'') = d(x',y'') = d(x,z_1) = k-1$. It remains to show that $d(y'',y') = 1$ which would imply $$k = d(x',y') \le d(x',z) + d(z,y'') + d(y'',y') = (k-1)+1 = k,$$ that is $z \in [x',y']$, as desired.
To prove $d(y'',y')=1$ we use transport geodesic techniques. Therefore, we relabel the vertices of the geodesic $g^*$ and extend $g^*$ to a full-length geodesic $g$ in $G$ starting from $x=x_0$ as follows: $$g: \quad {\underset{\displaystyle\overset{\mkern4mu{\rotatebox{90}{$\,=$}}}{x_0}}{x}}\sim {\underset{\displaystyle\overset{\mkern4mu{\rotatebox{90}{$\,=$}}}{x_1}}{x'}}\sim \cdots \sim {\underset{\displaystyle\overset{\mkern4mu{\rotatebox{90}{$\,=$}}}{x_{k-1}}}{z_1}}\sim {\underset{\displaystyle\overset{\mkern4mu{\rotatebox{90}{$\,=$}}}{x_k}}{y}} \sim x_{k+1} \sim \cdots \sim x_L,$$ and consider the transport geodesic along $g$ starting at $x_0$.
Theorem \[thm:uniq\_ant\] guarantees that $x_0(m)=\textup{ant}_{[x_0,x_m]}(x_1)$ for all $2\le m\le L$. In particular, we have $y''= \textup{ant}_{[x_0,x_{k-1}]}(x_1) = x_0(k-1)$ and $y' = \textup{ant}_{[x_0,x_k]}(x_1) = x_0(k)$. Therefore, $y'' = x_0(k-1)$ and $y' = x_0(k)$ must be adjacent vertices (as illustrated in Figure \[fig:transport\_geodesic\_g\]), thus completing the proof.
(0,0)–(13.5,0); (0,0)–(1.5,-1.5)–(7.5,-1.5); (7.5,-1.5)–(12,-1.5); (12,-1.5)–(13.5,0) ; (3,0)–(1.5,-1.5); (4.5,0)–(3,-1.5); (7.5,0)–(6,-1.5); (9,0)–(7.5,-1.5);
(0,0) circle \[radius=0.05\]; at (0,0) [$x_0$]{}; (1.5,0) circle \[radius=0.05\]; at (1.5,0) [$x_1$]{};
(3,0) circle \[radius=0.05\]; at (3,0) [$x_2$]{};
(4.5,0) circle \[radius=0.05\]; at (4.5,0) [$x_3$]{};
(7.5,0) circle \[radius=0.05\]; at (7.5,0) [$x_{k-1}$]{};
(9,0) circle \[radius=0.05\]; at (9,0) [$x_{k}$]{};
(13.5,0) circle \[radius=0.05\]; at (13.5,0) [$x_L$]{};
(1.5,-1.5) circle \[radius=0.05\]; at (1.5,-1.5) [$\bf x_0(2)$]{};
(3,-1.5) circle \[radius=0.05\]; at (3,-1.5) [$\bf x_0(3)$]{};
(6,-1.5) circle \[radius=0.05\]; at (5.7,-1.5) [$\boldsymbol{{\underset{\displaystyle\overset{\mkern4mu{\rotatebox{90}{$\,=$}}}{y''}}{x_0(k-1)}}}$]{};
(7.5,-1.5) circle \[radius=0.05\]; at (7.5,-1.5) [$\boldsymbol{{\underset{\displaystyle\overset{\mkern4mu{\rotatebox{90}{$\,=$}}}{y'}}{x_0(k)}}}$]{};
(0,0)–(10.5,0);
(0,0)–(1.5,-1.5)–(7.5,-1.5);
An immediate but important consequence of the above theorem is the following corollary.
\[cor\_clockwise\_spherical\] Let $G=(V,E)$ be self-centered Bonnet-Myers sharp. Let $x,y \in V$ be two different vertices, and consider any $x'\in [x,y]\cap S_1(x)$ with its antipole $y'={\rm ant}_{[x,y]}(x')$. Then $[x',y']=[x,y]$.
Theorem \[thm\_spherical\_step2\] can be rephrased as $[x,y]\subseteq [x',y']$. Since $y=\textup{ant}_{[x',y']}(x)$ by Remark \[rem:ant\_toggle\], we can interchange the roles of $x,y$ and $x',y'$ to obtain the opposite inclusion $[x',y']\subseteq [x,y]$. Therefore $[x',y']=[x,y]$, as desired.
Now, we are ready to conclude the ultimate result of this section by using Corollary \[cor\_clockwise\_spherical\] inductively.
\[thm\_spherical\] Let $G=(V,E)$ be self-centered Bonnet-Myers sharp. Then for two different vertices $x,y \in V$, the induced subgraph of the interval $[x,y]$ is antipodal. Therefore $G$ is strongly spherical.
Let $x' \in [x,y]$. The existence of a vertex $y' \in [x,y]$ satisfying $$[x,y] = [x',y']$$ is proved via induction on $d(x,x')$.
The case $d(x,x')=0$ is trivial and $d(x,x')=1$ is covered by Corollary \[cor\_clockwise\_spherical\].
We assume the statement of the Theorem is true for all $d(x,x') \le m-1$ with $2 \le m \le {\rm{diam}}(G)$. Let $x' \in [x,y]$ with $d(x,x') = m$. We choose a vertex $x_1 \in [x,y] \cap S_1(x)$ such that $x_1,x'$ lie on a geodesic from $x$ to $y$. By the induction hypothesis, there exists $y_1 \in [x,y]$ such that $$[x,y] = [x_1,y_1].$$ Since $d(x_1,x') = m-1$, the induction hypothesis again implies the existence of $y' \in [x_1,y_1] = [x,y]$ such that $$[x_1,y_1] = [x',y'] = [x,y].$$ This finishes the proof.
Classification of Bonnet-Myers sharp graphs with extremal diameters {#sec:BMextrdiam}
===================================================================
In this section we show that there are no Bonnet-Myers sharp graphs of extremal diameter (that is $L=2$ and $L=D$) which are not self-centered. In other words, the only Bonnet-Myers sharp graphs with diameter $L=2$ are cocktail party graphs and the only Bonnet-Myers sharp graphs with diameter $L=D$ are hypercubes.
Characterisation of sharpness for $L = 2$
-----------------------------------------
Let $G=(V,E)$ be a $(D,2)$-Bonnet Myers sharp graph. Then $G$ is isomorphic to a cocktail party graph $CP(n)$ for $n \ge 2$.
Let us first show that all Bonnet-Myers sharp graphs of diameter $L=2$ are necessarily self-centered: if there were a $(D,2)$-Bonnet-Myers sharp graph which is not self-centered, it would have a vertex adjacent to all other vertices and, by $D$-regularity, would have to be the complete graph $K_{D+1}$, which does not have diameter $2$. Now we employ the classification Theorem \[thm:main\] for self-centered Bonnet-Myers sharp graphs and conclude that all Bonnet-Myers sharp graphs of diameter $L=2$ are cocktail party graphs $CP(n)$, $n \ge 2$.
Characterisation of sharpness for $L = D$
-----------------------------------------
We now show that the only Bonnet-Myers sharp graphs with diameter equal to their degree are the hypercubes.
\[l:match\] Let $G=(V,E)$ be a $D$-regular graph. Suppose an edge $\{x,y\} \in E$ is contained in no triangle and satisfies $\kappa(x,y) \geq \frac{2}{D}$. Then, every pair of adjacent edges $w\sim x\sim y$ is contained in a 4-cycle.
This follows immediately from Proposition \[prop:curvcalc0\].
We recall the definitions of the small sphere structure and the non-clustering property from [@LMP2] which have been the key concepts to prove the rigidity result under sharpness of Bonnet-Myers in the Bakry-Émery $\infty$-curvature (see [@LMP2]).
\[def: SSP NCP\] Let $G=(V,E)$ be a $D$-regular graph and let $x \in V$.
1. We say $x$ satisfies the *small sphere property* (SSP) if $$\begin{aligned}
|S_2(x)| \leq {D \choose 2}.
\end{aligned}$$
2. We say $x$ satisfies the *non-clustering property* (NCP) if, whenever $d_x^-(z) = 2$ holds for all $z \in S_2(x)$, one has that for all distinct $y_1,y_2\in S_1(x)$ there is at most one $z \in S_2(x)$ satisfying $y_1\sim z \sim y_2$.
For the arguments below, it is useful to understand structural properties of the hypercube $Q^n$. We view the vertices of $Q^n$ as the elements of $\{0,1\}^n$ which are connected if their Hamming distance is equal to $1$, and assume without loss of generality that $x=(0,0,...,0)$. Then for $1\le k \le n$, $$S_k(x) = \{(a_i)_{i} \in \{0,1\}^n | \ \sum_i a_i=k \},$$ which gives $\#S_k(x) = {D \choose k}$. In particular, $Q^n$ satisfies (SSP).
Moreover, for distinct $y_1,y_2\in S_k(x)$ we always have $d(y_1,y_2) \ge 2$. In the case $d(y_1,y_2)=2$, the entries of $y_1$ and $y_2$ differ in precisely two places and, consequently, $y_1,y_2$ has precisely one common neighbour in $S_{k-1}(x)$ (if $k \ge 1$) and one common neighbour in $S_{k+1}(x)$ (if $k \le n-1$). Therefore, $Q^n$ satisfies also (NCP).
\[l:SSPNCP\] Let $G=(V,E)$ be a $D$-regular graph. If $x \in V$ belongs to no triangle and if $\kappa(x,y) \geq 2/D$ for all $y\sim x$, then $x$ satisfies $(SSP)$ and $(NCP)$.
We first show $|S_2(x)| \leq {D \choose 2}$. Let $z \in S_2(x)$ and let $y\in V$ s.t. $x \sim y \sim z$. Due to Lemma \[l:match\], $z$ is connected to at least two vertices from $S_1(x)$. By double counting, we have $$2 |S_2(x)| \le \sum_{z \in S_2(x)} d_x^-(z) = \sum_{y \in S_1(x)} d_x^+(y)
\le (D-1) |S_1(x)| = (D-1)D,$$ which implies $|S_2(x)| \leq {D \choose 2}$. Therefore $x$ satisfies (SSP).
Next we prove (NCP) at $x$: For all distinct $y_1, y_2 \in S_1(x)$ the pair of adjacent edges $y_1 \sim x \sim y_2$ is contained in a $4$-cycle by Lemma \[l:match\], which means that there is $z \in S_2(x)$ with $y_1 \sim z \sim y_2$. Since $|S_2(x)| \le {D \choose 2}$, there is at most one such $z$ for each such pair $y_1, y_2 \in S_1(x)$.
Now we state the main theorem of this section.
Let $G=(V,E)$ be $(D,L)$-Bonnet-Myers sharp with $L=D$. Then $G$ is the hypercube $Q^D$.
Let $x$ be a pole. We write $B_N:= B_N(x)$ and $S_N:=S_N(x)$.
By Theorem \[onespheredegree\](a), $x$ is not contained in any triangles, therefore, $B_1(x)$ is isomorphic to the $1$-ball in $Q^D$.
Now suppose, the $N$-ball $B_N$ is isomorphic to the $N$-ball of the hypercube with $1 \leq N < D$. We want to show that the $(N+1)$-ball $B_{N+1}$ is then isomorphic to the $(N+1)$-ball of the hypercube, which would prove the theorem by induction.
By in Theorem \[recursionformulas\] and the fact that $d_x^0(z) = 0$ for all $z \in S_N$ (because of the structure of the $N$-ball in the hypercube), we observe $d_x^+(z)= D-N$ for all $z\in S_N$. Let $M:=\{\{v,w\}\subset S_N: d(v,w)=2\}$. Since the $N$-ball $B_N$ is isomorphic to the $N$-ball of the hypercube, we have $$\label{eq:Mest}
|M| \geq {D \choose N-1}{D - N + 1 \choose 2}.$$ Again, due to the structure of the $N$-ball in the hypercube, any pair $\{v,w\} \in M$ cannot have any common neighbours in $S_N$ and can have at most one common neighbour in $S_{N-1}$. Therefore, due to Lemma \[l:match\], since $\kappa \geq \frac{2}{D}$, there exists $p:M \to S_{N+1}$ satisfying $v \sim p(\{v,w\}) \sim w$ for all $\{v,w\} \in M$.
By in Theorem \[recursionformulas\], every $z \in S_{N+1}$ satisfies $d_x^-(z) \le N+1$. We classify the vertices in $S_{N+1}$ by their backwards degree. Let $a_s$ be the number of $z\in S_{N+1}$ with $d_x^-(z)=s$. Remark $a_s = 0$ for $s > N+1$. Therefore, the set $E(S_{N+1},S_N)$ of all edges joining $S_N$ and $S_{N+1}$, satisfies $$|E(S_{N+1},S_N)| = \sum_{s\leq N+1} s a_s.$$ If $d_x^-(z) = s$ for some $z \in S_{N+1}$, then there are at most ${s \choose 2}$ pairs $\{v,w\} \in M$ with $p(\{v,w\})=z$. Thus, $$\label{eq:ME_est}
|M| \leq \sum_{N + 1 \geq s \geq 2} a_s {s\choose 2} \leq \frac{N}2
\sum_{s\leq N+1} sa_s = \frac N 2 |E(S_{N+1},S_N)|.$$ Note that the second inequality in is an equality iff $a_s = 0$ for all $s < N+1$. Therefore, using and , $$\begin{aligned}
|E(S_{N+1},S_N)| \geq \frac 2 N |M| \geq \frac 2 N {D \choose N-1}{D - N + 1 \choose 2} = {D \choose N} (D-N)= |E(S_{N+1},S_N)|
\end{aligned}$$ where the last equality follows since $|S_N| = {D \choose N}$ and since every $z \in S_N$ satisfies $d_x^+(z) = D-N$. Therefore, we have sharpness everywhere which means $a_s=0$ if $s \neq N+1$, i.e., $d_x^-(z)=N+1$ for all $z \in S_{N+1}$. This implies from in Theorem \[recursionformulas\] that $$\label{eq:SN+1-} d_x^0(z)=0 \quad \text{and} \quad
d_x^+(z) = D-N-1 \quad \text{for all $z \in S_{N+1}$}$$ and $|S_{N+1}| = {D \choose N+1}$.
Since $B_N$ is isomorphic to the $N$-ball of the hypercube, any $y \in B_{N-1}$ is not contained in a triangle of $G$. Thus, we can apply Lemma \[l:SSPNCP\] and conclude that $(SSP)$ and $(NCP)$ are satisfied for all $y \in B_{N-1}$. Using this fact and , we can apply [@LMP2 Theorem 6.2] (with $k=N+1$) and conclude that $B_{N+1}$ is isomorphic to the $(N+1)$-ball of the hypercube $Q^D$.
Note the following slight subtlety in this last argument: [@LMP2 Theorem 6.2] requires *bipartiteness*of $G$, which is a priori not known. Instead, we apply this theorem to a modification of $G$. This can be done by the following gluing process of the induced graph $B_{N+1}(x)$ with an $(L-N-1)$-ball of the hypercube $Q^D$: Let $B'_{L-N-1}(x')$ be an $(L-N-1)$-ball of a hypercube $Q^D$ centered at $x'$ and $S'_{L-N-1}(x')$ be the corresponding $(L-N-1)$-sphere. Since $|S_{N+1}(x)| = {D \choose N+1} = |S'_{L-N-1}(x')|$ and $$d_x^+(z) = D-N-1 = d_{x'}^-(z') \quad \text{and} \quad
d_x^0(z)= 0 = d_{x'}^0(z')$$ for all $z \in S_{N+1}(x)$ and $z' \in S'_{L-N-1}(x')$, we can glue these two graphs via a bijective identification of the vertex sets of $S_{N+1}(x)$ and $S'_{L-N-1}(x')$. This guarantees that the new graph is bipartite and $D$-regular.
The proof is now finished by the induction principle.
Bonnet-Myers sharp graphs and Bakry-[É]{}mery curvature {#sec:BM-BE-curvature}
=======================================================
Bakry-Émery curvature {#sec:BE-curvature}
---------------------
Bakry-Émery curvature is a notion based on a fundamental identity in Riemannian Geometry, called *Bochner’s Formula*, involving the Laplace-Beltrami operator. This definition allows to introduce Bakry-Émery curvature also on other spaces with a well-defined Laplacian. The (normalized) Laplacian in our particular discrete setting of a graph $G$ was given in . In this section, we will recall some fundamental properties which will be relevant for relating Bonnet-Myers sharpness in the sense of Ollivier Ricci curvature and Bakry-Émery curvature. More general details about Bakry-Émery curvature can be found in [@CLP2018]. We start with Bakry-Émery’s $\Gamma$-calculus:
\[defn:GammaGamma2\] Let $G=(V,E)$ be a finite simple graph. For any two functions $f,g: V\to \mathbb{R}$, we define $\Gamma(f,g): V\rightarrow \mathbb{R}$ and $\Gamma_2(f,g): V\rightarrow \mathbb{R}$ by $$\begin{aligned}
2\Gamma(f,g)&:=\Delta(fg)-f\Delta g-g\Delta f;\\
2\Gamma_2(f,g)&:=\Delta\Gamma(f,g)-\Gamma(f,\Delta g)-\Gamma(\Delta f,g).
\end{aligned}$$ We write $\Gamma(f):=\Gamma(f,f)$ and $\Gamma_2(f,f):=\Gamma_2(f)$, for short.
\[defn:BEcurvature\] Let $G=(V,E)$ be a finite simple graph. Let $\mathcal{K}\in \mathbb{R}$ and $\mathcal{N}\in (0,\infty)\cup\{\infty\}$. We say that a vertex $x\in V$ satisfies the *curvature-dimension inequality* $CD(\mathcal{K},\mathcal{N})$ if, for any $f:V\to \mathbb{R}$, we have $$\label{eq:CDineq}
\Gamma_2(f)(x)\geq \frac{1}{\mathcal{N}}
(\Delta f(x))^2+\mathcal{K}\Gamma(f)(x).$$ We call $\mathcal{K}$ a lower Ricci curvature bound of $G$ at $x$, and $\mathcal{N}$ a dimension parameter. The graph $G=(V,E)$ satisfies $CD(\mathcal{K},\mathcal{N})$ (globally), if all its vertices satisfy $CD(\mathcal{K},\mathcal{N})$. Let $\mathcal{K}_{G,x}(\infty)$ be the largest real number such that the vertex $x$ satisfies $CD(\mathcal{K}_{G,x}(\infty),\infty)$.
We now recall results from [@CLP2018] which we will need for the rest of this section. Note that the Bakry-Émery curvature $\mathcal{K}_{G,x}$ in [@CLP2018] is based on the non-normalized Laplacian which, in the case of $D$-regular graphs, can be easily translated into the normalized setting presented here. Henceforth, we will denote the Bakry-Émery curvature associated to the normalized Laplacian by $\mathcal{K}^{\rm n}_{G,x}$ (for $D$-regular graphs, we have $\mathcal{K}^{\rm n}_{G,x} = \frac{1}{D} \mathcal{K}_{G,x})$.
Let $G = (V,E)$ be a $D$-regular graph. Theorem 3.1 of [@CLP2018] tells us that $$\label{upperbound}
\mathcal{K}^{\rm n}_{G,x}(\infty) \leq \frac{2}{D} + \frac{\#_{\Delta}(x)}{D^{2}} = \frac{3+D-av_1^+(x)}{2D},$$ for every $x\in V$, where $av_1^+(x)$ was defined in .
We say, as in [@CLP2018], that a $D$-regular graph $G = (V,E)$ is *$\infty$-curvature sharp* at $x \in V$ if holds true with equality.
We now recall a method from [@CLP2018] that allows us to check if a graph $G=(V,E)$ is $\infty$-curvature sharp at a vertex $x$. Let $x \in V$ be an *$S_{1}$-out regular* vertex, that is, $d_{x}^{+}(y)$ is constant for all $y \sim x$. Let $\{ y_1,\dots, y_d \}$ be the vertices of $S_1(x)$. We now define two relevant (weighted) Laplacians $\Delta_{S_1(x)}$ and $\Delta_{S_1'(x)}$ on functions $f: S_1(x) \to {{\mathbb{R}}}$ as follows: $$\Delta_{S_1(x)}f(y_i) = \sum_{y_j: y_j \sim y_i} (f(y_j) - f(y_i)),$$ that is, $\Delta_{S_1(x)}$ be the non-normalized Laplacian of the induced subgraph $S_1(x)$. Let $S_1'(x)$ be the graph with the same vertex set $\{ y_1,\dots, y_d \}$ and an edge between $y_i$ and $y_j$ iff $|\{ z \in S_2(x) \mid y_i \sim z \sim y_j \}| \ge 1$, where $\sim$ describes adjacency in the original graph $G$. We introduce the following weights $w_{y_i y_j}'$ on the edges of $S_1'(x)$: $$w_{y_i y_j}' = \sum_{z \in S_2(x)} \frac{w_{y_i z}w_{z y_j}}{d_x^-(z)}.$$ Where $w_{u v} = 1$ if $u\sim v$ and $0$ otherwise.
The corresponding weighted Laplacian is then given by $$\Delta_{S_1'(x)} f(y_i) = \sum_{j: j \neq i} w_{y_i y_j}' (f(y_j) - f(y_i)).$$ Let $S_1''(x) = S_1(x) \cup S_1'(x)$, i.e., the vertex set of $S_1''(x)$ is $\{ y_1,\dots, y_d \}$ and the edge set is the union of the edge sets of $S_1(x)$ and $S_1'(x)$. Then the sum $\Delta_{S_1(x)} + \Delta_{S_1'(x)}$ can be understood as the weighted Laplacian $\Delta_{S_1''(x)}$ on $S_1''(x)$ with weights $w'' = w + w'$. Note that all our Laplacians $\Delta$ are defined on functions on the vertex set of $S_1(x)$. Let $\lambda_{1}(\Delta_{S_1''(x)})$ denote the smallest non-zero eigenvalue of $\Delta_{S_1''(x)}.$
Theorem 9.1 of [@CLP2018] tells us that an $S_1$-out regular vertex $x$ in a $D$-regular graph $G$ is $\infty-$curvature sharp if and only if $\lambda_{1}(\Delta_{S_1''(x)})\geq \frac{D}{2}$.
On a different note, we also provide the following general result on Cartesian product, which will be useful in the next subsection.
\[lem:BEcart\] Let $G_i = (V_i,E_i)$, $i=1,2$, be two connected, simple $D_i$-regular graphs with diameters $L_i$, respectively. Assume we have $$\label{eq:Kiest}
\mathcal{K}_{G_i,x_i}^{\rm n}(\infty) \le \frac{1}{D_i} + \frac{1}{L_i}$$ at $x_i \in V_i$, $i=1,2$. Then we have $$\label{eq:Kprodest}
\mathcal{K}_{G_1 \times G_2,(x_1,x_2)}^{\rm n}(\infty) \le
\frac{1}{D_1+D_2} + \frac{1}{L_1+L_2}.$$ Moreover, if holds with equality for $i=1,2$ and we have $\frac{D_1}{L_1} = \frac{D_2}{L_2}$, then holds also with equality.
Let $G_i$ be $D_i$-regular with diameter $L_i$, $i = 1,2$ and $x_i
\in V_i$ be the vertices satisfying $${\mathcal K}^{\rm n}_{G_i,x_i}(\infty) \le \frac{1}{D_i} +
\frac{1}{L_i}.$$ Then we have, using [@CLP2018 equation (7.26)], $$\begin{aligned}
{\mathcal K}^{\rm n}_{G_1 \times G_2,(x_1,x_2)}(\infty) &=& \frac{1}{D_1+D_2}
\min_{i=1,2} D_i {\mathcal K}^{\rm n}_{G_i,x_i}(\infty) \\
&\le& \frac{1}{D_1+D_2} \min_{i=1,2} \left( 1+ \frac{D_i}{L_i} \right) \\
&\le& \frac{1}{D_1+D_2} \left(1 + \frac{D_1+D_2}{L_1+L_2} \right) \\
&=& \frac{1}{D_1+D_2} + \frac{1}{L_1+L_2}.
\end{aligned}$$ It is easy to see that in the case of equality in for $i=1,2$, the same calculation leads to equality in .
The Bakry-Émery curvature of Bonnet-Myers sharp graphs {#sec:BEcurvBMsharp}
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As a consequence of Lemma \[lem:BEcart\], the following proposition show that the $\infty$-curvature sharpness is also preserved under taking Cartesian products of Bonnet-Myers sharp graphs (of the same ratios $\frac{D_i}{L_i}$).
\[prop:cartprodBEcurv\] Let $G_i=(V_i,E_i)$, $i=1,2$, be two $(D_i,L_i)$-Bonnet-Myers sharp graphs with ${\mathcal K}^{\rm n}_{G_i,x_i}(\infty) = \frac{1}{D_i}+ \frac{1}{L_i}$ at $x_i \in V_i$. Assume furthermore that $\frac{D_1}{L_1} = \frac{D_2}{L_2}$. Then the Cartesian product $G_1 \times G_2$ is also Bonnet-Myers sharp with $${\mathcal K}^{\rm n}_{G_1\times G_2,(x_1,x_2)}(\infty) = \frac{1}{D_1+D_2}+ \frac{1}{L_1+L_2}.$$
The condition $\frac{D_1}{L_1} = \frac{D_2}{L_2}$ guarantees that the Cartesian product $G_1 \times G_2$ is, again, Bonnet-Myers sharp. The statement about the Bakry-Émery $\infty$-curvature at $(x_1,x_2) \in
V_1 \times V_2$ follows immediately from Lemma \[lem:BEcart\].
For Bonnet-Myers sharp graphs, we have the following $\infty$-curvature estimate at poles.
\[thm:curv\_relation\] Let $G=(V,E)$ be a $(D,L)$-Bonnet-Myers sharp graph. Then we have at every pole $x \in V$: $$\label{eq:be-est} {\mathcal K}^{\rm
n}_{G,x}(\infty) \le \frac{1}{D} + \frac{1}{L}.$$ Moreover, equality in is equivalent to the fact that $G$ is Bakry-Émery $\infty$-curvature sharp at $x$.
Let $x \in V$ be a pole of $G$. Using in Theorem \[recursionformulas\], we have for every $y \in S_1(x)$: $$d_x^+(y) = 1 + D \left( 1-\frac{2}{L} \right).$$ This shows that $x$ is $S_1$-out regular with $av_1^+(x) =d_x^+(y) = 1 + D -\frac{2D}{L} $. We know from that $$\label{eq:BEcurv_est}
{\mathcal K}^{\rm n}_{G,x}(\infty) \le \frac{3+D-av_1^+(x)}{2D} = \frac{1}{2D} \left( 2 + \frac{2D}{L} \right) = \frac{1}{D} + \frac{1}{L}.$$ Equality is equivalent to Bakry-Émery $\infty$-curvature sharpness.
Since every graph has a pole, Theorem \[thm:curv\_relation\] immediately implies Theorem \[thm:BMsharp-BEeq\].
In case of self-centered Bonnet-Myers sharp graphs, Theorem \[thm:curv\_relation\] can be strengthened, where inequality becomes equality at all vertices, resulting in Theorem \[thm:aBM-BEcurv\]:
\[thm:curv\_rel\_strong\] Let $G$ be a self-centered $(D,L)$-Bonnet-Myers sharp graph. Then $G$ is Bakry-Émery $\infty$-curvature sharp at all vertices $x \in V$ and $$\label{eq:curv1D1L}
{\mathcal K}_{G,x}^{\rm n}(\infty) = \frac{1}{D} + \frac{1}{L}.$$
In view of Proposition \[prop:cartprodBEcurv\], it suffices to prove this theorem only for the graphs in the list of Theorem \[thm:main\]. We therefore start with a graph $G= (V,E)$ in the list of Theorem \[thm:main\] and prove for every vertex. Without loss of generality, we can assume $L \ge 2$ since $L=1$ implies $G = K_2$ which follows immediately from ${\mathcal K}_{K_2,x}^{\rm n}(\infty)=2$.
Table \[table:BMsharp\_examples\] in Subsection \[sec:revisex\] confirms that these graphs satisfy all the assumption of Proposition \[prop:cocktail-implies-str\], that is, all $\mu$-graphs of $G$ are cocktail party graphs $CP(m)$ with $$m = \frac{D-L}{L(L-1)} + 1,$$ and all $1$-spheres of $G$ are strongly regular.
Let $x \in V$. Since every vertex of $G$ is a pole, we know from that $${\mathcal K}_{G,x}(\infty) \le \frac{1}{D} + \frac{1}{L}$$ with equality iff $x$ is $\infty$-curvature sharp. We have already seen in the proof of Theorem \[thm:curv\_relation\] that $x$ is $S_1$-out regular. So it only remains to show $\lambda_{1}(\Delta_{S_1''(x)})\geq \frac{D}{2}$.
By Proposition \[prop:cocktail-implies-str\], the strongly regular induced $S_{1}(x)$ has parameters $$(\nu,k,\lambda,\mu) =
(D,\frac{2D}{L}-2,\frac{D-1}{L-1}-3,2\frac{D-L}{L(L-1)}).$$
Let $A$ denote the adjacency matrix of $S_{1}(x).$ Then, by Theorem \[onespheredegree\], $\Delta_{S_{1}(x)} = A - (\frac{2D}{L}-2{\rm )Id}$. Let $y \sim
x$. Then, by in Theorem \[recursionformulas\], $d_{x}^{+}(y) = \frac{(L-2)D}{L}+1$. Since every $\mu$-subgraph of $G$ is $CP(m)$, we have $d_{x}^{-}(z) = 2m = \frac{2}{L-1}(\frac{D}{L}+L-2)$ for all $z\in S_{2}(x)$.
Let us first calculate the adjacency matrix $A'$ of the weighted graph $S_1'(x)$. Recall that the entries $\omega_{yy'}'$ of $A'$ are given by $$\omega_{yy'}' = \sum_{z \in S_2(x), y \sim z \sim y'} \frac{1}{d_x^-(z)} =
\frac{1}{2m} \left| \{ z \in S_2(x) \mid y \sim z \sim y' \} \right|.$$ Assume first that $y$ and $y'$ are not neighbours in the induced $S_1(x)$. There is a unique antipole of $x$ in $\mu(y_1,y_2)$, which is a vertex in $S_2(x)$. Therefore, we have $$y,y' \in S_1(x), y \not\sim y' \quad \Rightarrow \quad \omega_{yy'}' =
\frac{1}{2m}.$$ Now assume that $y \sim y'$. The edge $\{y,y'\}$ lies in precisely $\frac{2D}{L}-2$ triangles, one of them is $\{x,y,y'\}$ and there are precisely $\lambda = \frac{D-1}{L-1}-3$ triangles in the induced $S_1(x)$. The rest of triangles are in $1-1$ correspondence to vertices $z \in S_2(x)$ with $z \sim y$ and $z \sim y'$. Therefore we have $$\left| \{ z \in S_2(x) \mid y \sim z \sim y' \} \right| =
\left(\frac{2D}{L} -2\right) - 1 - \left(\frac{D-1}{L-1} -3 \right)
= \frac{2D}{L} - \frac{D-1}{L-1}.$$ This implies that $$y,y' \in S_1(x), y \sim y' \quad \Rightarrow \quad \omega_{yy'}' =
\left( \frac{2D}{L} - \frac{D-1}{L-1} \right) \frac{1}{2m}.$$ Since the adjacency matrix $A^c$ of the complement of the induced $S_1(x)$ can be written as $A^c = J - {\rm Id} - A$, where $J$ is the all-one matrix, we have for the weighted adjacency matrix $A'$ of $S_1'(x)$ $$A' = \frac{1}{2m} \left( \left( \frac{2D}{L} - \frac{D-1}{L-1} \right) A +
A^c \right) = \frac{1}{2m} \left( \left( \frac{2D}{L} - \frac{D-1}{L-1} -
1 \right) A - {\rm Id} + {\rm J} \right).$$ Note that $$\begin{aligned}
\Delta_{S_1(x)} &=& A - \left( \frac{2D}{L} - 2 \right) {\rm Id}, \\
\Delta_{S_1'(x)} &=& A' - {\rm diag}(v'),
\end{aligned}$$ with $v' = A' {\bf 1}$ where ${\bf 1}$ is the all-one vector. Since $A$ is the adjacency matrix of a $\left(\frac{2D}{L}-2\right)$-regular graph of size $D$, $v'$ is a constant vector with all entries equal to $$\frac{1}{2m} \left( \frac{2D}{L} - \frac{D-1}{L-1} - 1 \right)
\left( \frac{2D}{L} - 2 \right) - \frac{1}{2m} + \frac{D}{2m}.$$ Plugging this information into the formula for $\Delta_{S_1''(x)} = \Delta_{S_1(x)} + \Delta_{S_1'(x)}$ gives, $$\Delta_{S_1''(x)} = \frac{1}{2m}\left(\left(\frac{2D}{L}-\frac{D-1}{L-1}+2m-1\right)\left(A-\left(\frac{2D}{L}-2\right) {\rm Id}\right)-D\cdot {\rm Id}+{\rm J}\right).$$ Observe that the three matrices $A,{\rm Id},{\rm J}$ pairwise commute.
By Proposition \[prop:cocktail-implies-str\], the second largest eigenvalue of $A$ is $\frac{(D-L)(L-2)}{L(L-1)}$. Note that the eigenvector $w$ of the second largest eigenvalue is orthogonal to ${\bf 1}$ and, therefore, ${\rm J}w = 0$. Thus to complete the claim it remains to show that $$\lambda_1(\Delta_{S_1''(x)}) =
\frac{-1}{2m}\left(\left(\frac{2D}{L}-\frac{D-1}{L-1}+2m-1\right)
\underbrace{\left(\frac{(D-L)(L-2)}{L(L-1)}
-\left(\frac{2D}{L}-2\right)\right)}_{= -
\frac{D-L}{L-1}}-D\right)\geq \frac{D}{2}.$$ Multiplying the whole expression by $2m$, we need to show that $$\left( \left(\frac{2D}{L}-\frac{D-1}{L-1}+2m-1\right) \frac{D-L}{L-1}
+D\right) - mD \ge 0,$$ which simplifies, after inserting $m = \frac{D-L}{L(L-1)} + 1$ into the expression, to $$\frac{1}{L(L-1)^2} (L(L-2)+D)(D-L) \ge 0,$$ which is obviously true since $L\leq D$ and $L \geq 2$.
A conjecture about Bakry-Émery curvature
----------------------------------------
In this subsection, let us revisit the following conjecture mentioned in the Introduction:
Let $G=(V,E)$ be a connected, simple $D$-regular graph with diameter $L$. We then have $$\label{eq:conj} \inf_{x \in V }{\mathcal K}^{\rm
n}_{G,x}(\infty) \le \frac{1}{D} + \frac{1}{L}.$$
A simple argument provides the following general estimate. The challenge of the conjecture is thus to remove the final term in .
Let $G=(V,E)$ be a $D$-regular graph of diameter $L$. Then we have $$\label{eq:almostconj}
\inf_{x \in V} {\mathcal K}^{\rm n}_{G,x}(\infty) \le \frac{1}{D} + \frac{1}{L} + \frac{1}{2D^2} \max_{x \in V} \#_\Delta(x).$$
The proof is a combination of the inequalities and .
Here is a list of examples providing supporting evidence for this conjecture:
1. All graphs with $D \le L$: This is an immediate consequence of $$\inf_{x \in V} {\mathcal K}_{G,x}^{\rm n}(\infty) \le \frac{2}{L}$$ proved in [@LMP Corollary 2.2].
2. All Bonnet-Myers sharp graphs: This follows immediately from Theorem \[thm:curv\_relation\].
3. All strongly regular graphs: Note that a strongly regular graph $G=(V,E)$ with parameters $(\nu,D,\lambda,\mu)$ satisfies, as all vertices $x \in V$, $$\#_\Delta(x) = \frac{D \lambda}{2} \le \frac{D (D-2)}{2},$$ since $\lambda \le D-2$ ($G$ cannot be the complete graph). Using , this implies $${\mathcal K}_{G,x}^{\rm n}(\infty) \le \frac{2}{D} + \frac{D-2}{2D} =
\frac{1}{D} + \frac{1}{2}.$$
4. All complete graphs: Note that the complete graph $G=K_n$ has degree $D = n-1$ and Bakry-Émery $\infty$-curvature (see [@CLP2018 Example 5.17]) $${\mathcal K}_{G,x}^{\rm n}(\infty) = \frac{D+3}{2D} \le \frac{1}{D} + 1$$ in all vertices $x$.
5. All demi-cube graphs: The even-dimensional demi-cubes $Q^{2n}_{(2)}$ satisfies ${\mathcal K}_{G,x}^{\rm n}(\infty) = \frac{1}{D}+\frac{1}{L}$ for all vertices $x$ (due to Theorem \[thm:aBM-BEcurv\] as it is self-centered Bonnet-Myers sharp). On the other hand, the odd-dimensional demi-cube $Q^{2n+1}_{(2)}$ has Bakry-Émery $\infty$-curvature $$K_{G,x}^{\rm n}(\infty) \le \frac{3+D-av_1^+(x)}{2D} = \frac{1}{n} = \frac{1}{L} < \frac{1}{D} + \frac{1}{L},$$ where the upper bound $\frac{1}{D} + \frac{1}{L}$ will never be achieved.
6. All Johnson graphs: The Johnson graph $G=J(n,k)$ has the following Bakry-Émery $\infty$-curvature (see [@CLP2018 Example 9.7]) in all vertices $x$ $${\mathcal K}^{\rm n}_{G,x}(\infty) = \frac{n+2}{2k(n-k)}
\le \frac{1}{D} + \frac{1}{L},$$ with vertex degree $D= k(n-k)$ and diameter $L=\min\{ k,n-k\}$.
7. All triangle-free graphs: Since $\#_\Delta(x)=0$ for all $x\in V$, implies that $$\inf_{x \in V} {\mathcal K}^{\rm n}_{G,x}(\infty) \le \frac{1}{D} + \frac{1}{L}.$$
8. Cartesian products: If holds for the graphs $G_i=(V_i,E_i)$, $i=1,2$, then holds also for the Cartesian product $G_1 \times G_2$ due to Lemma \[lem:BEcart\].
[**[Acknowlegdements:]{}**]{} The authors are grateful to David Bourne for many useful discussions and contributions. All authors would also like to thank the University of Science and Technology of China, Hefei, for its hospitality. DC, SL and NP enjoyed the opportunity for further discussions during the 2017 conference “Analysis and Geometry on Graphs and Manifolds” at the University of Potsdam, Germany. DC and FM would also like to thank the Max Planck Institute for Mathematics, Bonn, for the opportunity to participate in the 2017 event “Metric Measure Spaces and Ricci Curvature”. Finally, FM wants to thank the German National Merit Foundation for financial support, and SK wants to thank Thai Institute for the Promotion of Teaching Science and Technology for his scholarship.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Cerium-doped manganite thin films were grown epitaxially by pulsed laser deposition at $720\,^\circ$C and oxygen pressure $p_{O_2}=1-25\,$Pa and were subjected to different annealing steps. According to x-ray diffraction (XRD) data, the formation of CeO$_2$ as a secondary phase could be avoided for $p_{O_2}\ge 8\,$Pa. However, transmission electron microscopy shows the presence of CeO$_2$ nanoclusters, even in those films which appear to be single phase in XRD. With O$_2$ annealing, the metal-to-insulator transition temperature increases, while the saturation magnetization decreases and stays well below the theoretical value for electron-doped La$_{0.7}$Ce$_{0.3}$MnO$_3$ with mixed Mn$^{3+}$/Mn$^{2+}$ valences. The same trend is observed with decreasing film thickness from 100 to 20nm, indicating a higher oxygen content for thinner films. Hall measurements on a film which shows a metal-to-insulator transition clearly reveal holes as dominating charge carriers. Combining data from x-ray photoemission spectroscopy, for determination of the oxygen content, and x-ray absorption spectroscopy (XAS), for determination of the hole concentration and cation valences, we find that with increasing oxygen content the hole concentration increases and Mn valences are shifted from 2+ to 4+. The dominating Mn valences in the films are Mn$^{3+}$ and Mn$^{4+}$, and only a small amount of Mn$^{2+}$ ions can be observed by XAS. Mn$^{2+}$ and Ce$^{4+}$ XAS signals obtained in surface-sensitive total electron yield mode are strongly reduced in the bulk-sensitive fluorescence mode, which indicates hole-doping in the bulk for those films which do show a metal-to-insulator transition.'
author:
- 'R. Werner'
- 'C. Raisch'
- 'V. Leca'
- 'V. Ion'
- 'S. Bals'
- 'G. Van Tendeloo'
- 'T. Chassé'
- 'R. Kleiner'
- 'D. Koelle'
bibliography:
- 'References.bib'
title: 'Transport, magnetic, and structural properties of La$_{0.7}$Ce$_{0.3}$MnO$_3$ thin films. Evidence for hole-doping'
---
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–> 61.05.cj X-ray absorption spectroscopy: EXAFS, NEXAFS, XANES, etc.
61.05.cp X-ray diffraction
–> 68.37.Lp Transmission electron microscopy (TEM)
68.55.J- Morphology of films
–> 71.30.+h Metal–insulator transitions and other electronic transitions
–> 72.60.+g Mixed conductivity and conductivity transitions
72.80.Ga Transition-metal compounds
–> 75.47.Lx Manganites
75.60.Ej Magnetization curves, hysteresis, Barkhausen and related effects
75.70.Ak Magnetic properties of monolayers and thin films
–> 81.15.Fg Laser deposition
Introduction {#Sec:Introduction}
============
Hole-doped manganese perovskite oxides La$_{1-x}A_x$MnO$_3$, where $A$ is a divalent alkaline earth metal, have been intensively studied over the last years due to the interesting interplay between charge, spin, orbital and structural degrees of freedom.[@Imada98; @Coey99; @Salamon01] Without doping, LaMnO$_3$ is an antiferromagnetic insulator due to the super-exchange between the Mn$^{3+}$ ions.[@Millis98] In the hole-doped manganites, the divalent ion introduces holes by changing some Mn valences from Mn$^{3+}$ to Mn$^{4+}$. The properties of the hole-doped manganites are determined by the interplay of Hund´s rule coupling and the Jahn-Teller distortion of the Mn$^{3+}$ ions.[@Millis95] Their behavior can be qualitatively described by the double-exchange model,[@Zener51; @Anderson55] describing the interaction between manganese ions with mixed valences (Mn$^{3+}$ and Mn$^{4+}$). The strong spin-charge coupling via the double-exchange interaction explains the correlation between the metal-to-insulator (MI) and ferromagnet-to-paramagnet (FP) transition. Close to the MI transition temperature $T_{MI}$ an external magnetic field can reduce the spin disorder and therefore enhance the electron hopping between the manganese ions with mixed valences. This results in a large resistivity drop, called colossal magnetoresistance.[@Jonker50]
By substitution of La with a tetravalent ion, like Ce,[@Mandal97; @Gebhardt99; @Ganguly00] Sn,[@Li99a] or Te,[@Tan03a] instead of a divalent one, some of the Mn$^{3+}$ ions become Mn$^{2+}$ with electronic structure t$^3_{2g}$e$^2_g$ (compared to the t$^3_{2g}$e$^1_g$ electronic structure for Mn$^{3+}$). Hence, an extra electron may be induced in the e$_g$-band. Since Mn$^{2+}$ is a non-Jahn-Teller ion, like Mn$^{4+}$, one might expect a similar magnetic interaction between the Mn$^{3+}$ and Mn$^{2+}$ ions as for the well known hole-doped case.[@Mitra03a]
The first attempts to achieve electron-doping by substituting La with Ce were reported by Mandal and Das.[@Mandal97] However, they found hole-doping in their bulk samples. Later on, it was revealed that the bulk samples are a multiphase mixture which leads to the hole-doped behavior.[@Ganguly00; @Philip99] Single phase [La$_{0.7}$Ce$_{0.3}$MnO$_3$]{} (LCeMO) thin films have been prepared without any CeO$_2$ impurities [@Mitra01a; @Raychaudhuri99] regarding x-ray diffraction (XRD) data. The films showed FP and MI transitions similar to the hole-doped manganites. Surface-sensitive X-ray photoemission spectroscopy revealed the existence of Mn$^{2+}$ and Mn$^{3+}$ valences,[@Mitra03a; @Han04] which was interpreted as evidence of electron-doping. However, Hall measurements and thermopower measurements on comparable samples showed a hole-type character. [@Wang06; @Zhao00; @Yanagida04; @Yanagida05] By Ganguly [*et al.*]{} [@Ganguly00] it was further questioned whether LaMnO$_3$ accepts Ce-doping at all. Those authors questioned the reports on single phase LCeMO-films and claimed the presence of multi-phase mixtures, consisting of hole doped La-deficient phases with cerium oxide inclusions. Certainly, the existence of electron-doped manganites could enable new types of spintronic devices, such as $p-n$ junctions based on doped manganites.[@Mitra01] This motivates further research in order to improve understanding of the basic properties of those materials.
In this paper we present the results of studies on transport, magnetic and structural properties of LCeMO thin films grown by pulsed laser deposition (PLD) and their dependence on deposition parameters, annealing procedures and film thickness. We combine a variety of different characterization techniques in order to clarify the nature of the FP and MI transition in our LCeMO thin films.
Experimental Details {#Sec:Experiment}
====================
A commercially available stoichiometric polycrystalline La$_{0.7}$Ce$_{0.3}$MnO$_3$ target was used for thin film growth by PLD on (001) SrTiO$_3$ (STO) substrates (unless stated otherwise). The target was ablated by using a KrF ($\lambda$ = 248 nm) excimer laser at a repetition rate of $2-5\,$Hz. The energy density on the target was $E_d=2\,{\rm J/cm}^2$, while the substrate temperature during deposition was kept at $T_s=720\,^{\circ}$C for all films for which data are presented below, except for sample K with slightly lower $T_s$ and $E_d$ (cf. Tab. \[tab:overview\]). The oxygen pressure $p_{O_2}$ during film growth was varied in the 1–25Pa range with the aim of yielding single phase films with optimum morphology. We used a relatively low deposition pressure as compared to some literature data [@Mitra03a; @Wang06; @Mitra01; @Chang04] in order to avoid over-oxygenation of the films. This is important, as it is known that perovskite rare-earth manganites can accept a large excess of oxygen via the formation of cation vacancies, inducing hole-doping in the parent compound LaMnO$_3$.[@Toepfer97] In-situ high-pressure reflection high energy electron diffraction (RHEED) was used to monitor the growth mode and film thickness. After deposition, most of the films were in-situ annealed for 1h at $T=700\,^{\circ}$C and $p_{O_2}=1\,$bar and then cooled down with 10$^{\circ}$C per minute. In the following, those samples will be called ”in-situ annealed“ films, in contrast to the ”as-deposited“ films which were just cooled down to room temperature under deposition pressure. Some of the samples have been additionally annealed ex-situ at $p_{O_2}=1\,$bar in one or two steps ($1^{\rm st}$ step at $700\,^{\circ}$C; $2^{\rm nd}$ step at $750\,^{\circ}$C; each step for one hour). Table \[tab:overview\] summarizes the fabrication conditions and some characteristics of the LCeMO films described below.
[ccccccc]{}
------------------------------------------------------------------------
\# & $p_{O_2}$ (Pa) & & $d$ (nm) & $c$-axis ($\AA$) & $T_{MI}$ (K)\
& & in-situ & ex-situ & & &\
A & 1 & no & no & 100 & 3.921 & –\
B1 & & & no & & 3.905 & 175\
B2 & \[-1.5ex\]3 & \[-1.5ex\][no]{} & $1\times$ & \[-1.5ex\][90]{} & – & 250\
C & 8 & yes & no & 100 & 3.897 & 190\
D & 25 & no & no & 100 & 3.880 & 180\
E1 & & & no & & 3.894 & 210\
E2 & 8 & yes & $1\times$ & 65 & 3.887 & 216\
E3 & & & $2\times$ & & 3.872 & 230\
F & 8 & yes & no & 40 & 3.879 & 223\
G & 8 & yes & no & 20 & 3.870 & 232\
H & 3 & $(^*)$ & no & 100 & 3.876 & 260\
K & 3$(^{**})$ & no & no & 50 & 3.894 & 180\
$(^*)$ Cooled in 1bar O$_2$ without 1hour in-situ annealing $(^{**})$ deposited at $T_s=700\,^{\circ}$C with $E_d=1.75\,{\rm J/cm}^2$
The surface morphology was checked by atomic force microscopy (AFM) in contact mode. The crystal structure of the films was characterized by XRD and by high-resolution (HR) transmission electron microscopy (TEM). Transport properties were measured with a four probe technique, and a superconducting quantum interference device (SQUID) magnetometer was used to determine the magnetic properties of the samples. Hall measurements were performed in order to obtain information on the dominating type of charge carriers, and x-ray photoemission spectroscopy (XPS) was performed in order to obtain information on the oxygen content of different samples. The valences of the manganese and cerium ions were evaluated by x-ray absorption spectroscopy (XAS). XAS measurements in surface-sensitive total electron yield (TEY) mode and bulk sensitive fluorescence yield (FY) mode were carried out at the WERA dipole beamline (ANKA, Karlsruhe, Germany) with typical energy resolutions set between 100 and 400meV.
Structural Analysis
===================
Figure \[XRDCeO2\] shows the XRD $\Theta-2\Theta$ scans of four LCeMO thin films A, B, C, D (with similar thickness $d=$ 90–100nm) grown under different oxygen pressure $p_{O_2}=$ 1, 3, 8 and 25Pa, respectively. Sample C was in-situ annealed while the other samples were ”as-deposited“ films. According to the XRD data shown in Fig. \[XRDCeO2\], single phase LCeMO films were obtained for $p_{O_2}\ge 8\,$ Pa (samples C and D). For a lower deposition pressure, impurity peaks of CeO$_2$ appear (sample A and B). The substrate temperature $T_s$ also played a crucial role for the phase stability of the LCeMO films. By increasing $T_s$ up to $800\,^{\circ}$C, CeO$_2$ also appears for deposition pressures $p_{O_2}\ge 8\,$Pa. Such a behavior was also observed by Chang [*et al.*]{}.[@Chang04] As shown in the inset of Fig. \[XRDCeO2\], the $c$-axis decreases with increasing deposition pressure $p_{O_2}$. This can be explained by a decreasing concentration of oxygen vacancies with increasing $p_{O_2}$, as it is well known that oxygen vacancies tend to expand the lattice constants.[@Murugavel03]
![(Color online) XRD patterns of samples grown under different deposition pressures: $p_{O_2}=1,\,3,\,8$ and $25\,$Pa for sample A, B, C and D, respectively. CeO$_2$ can be identified in samples A and B. XRD scans are offset for clarity. The inset shows a detailed view around the (001) substrate peak including the (001) film peaks.[]{data-label="XRDCeO2"}](XRDCeO2){width="45.00000%"}
The surface roughness of the films depends strongly on deposition pressure, as shown by AFM and RHEED images on 100nm thick films in Fig. \[AFM-RHEED\] for (a) sample C ($p_{O_2}=8\,$Pa) with an rms roughness of 0.35nm and (b) sample D ($p_{O_2}=25\,$Pa;), with a much larger rms value of 2.15nm. The RHEED images show strong streaky patterns for the film deposited at $p_{O_2}=8\,$Pa \[Fig. \[AFM-RHEED\](a) right\], an indication of an atomically flat surface, while for higher deposition pressure \[here $p_{O_2}=25\,$Pa; Fig. \[AFM-RHEED\](b) right\] an increased surface roughness results in a combination of weaker streaks, together with the formation of a 3D RHEED pattern as a result of island growth. We note that sample C has an extremely smooth surface, showing unit-cell high terrace steps in the AFM image \[c.f. Fig. \[AFM-RHEED\](a) left\], which is quite unusual for such a thick LCeMO film. A similar morphology as for sample C was observed for all films deposited at an oxygen pressure in the range of 1-8Pa. For those conditions the films followed a 2D growth mode, as suggested by the RHEED and AFM data. Increasing the deposition pressure resulted in an increased step density during growth due to lower surface mobility, with the formation of 3D islands. Altogether, we found that $p_{O_2}=8\,$Pa was the optimum pressure for growing films without measurable CeO$_2$ concentration, as detected by XRD, and good surface morphology (rms roughness below 0.4nm).
![(Color online) AFM images (left; frame size $5\times 5\,\mu{\rm m}^2$) and RHEED images (right) of 100nm thick films: (a) sample C, grown at $p_{O_2}=8\,$Pa, and (b) sample D, grown at $p_{O_2}=25\,$Pa.[]{data-label="AFM-RHEED"}](AFM-RHEED){width="43.00000%"}
In order to evaluate the relation between CeO$_2$ formation and the substrate induced strain, 50nm thick LCeMO films were deposited on (001) STO, (110) NdGaO$_3$ and (001) NdGaO$_3$ substrates in the same deposition run.[^1] Here, we used a deposition pressure $p_{O_2}=3\,$Pa, in order to obtain a measurable amount of CeO$_2$. The XRD data showed no discernible difference in the amount of CeO$_2$ for the different substrates.
The growth and phase stability of some complex oxide materials may depend on the type of termination layer of the substrate.[@Huijbregtse01] Therefore, we have grown several LCeMO films on (001) STO substrates with different termination (either SrO or TiO$_2$) in order to determine whether the substrate termination influences the microstructure of the films. The SrO terminated substrates were obtained by annealing at $950\,^{\circ}$C, for 1h in an oxygen flow, while the TiO$_2$-terminated STO substrates were obtained by chemical etching in a BHF solution, following the procedure described in Ref. \[\]. The results showed no correlation between the substrate termination and the CeO$_2$ impurity phase formation. These results suggest that, for the conditions used in this study, the level of strain and the type of substrate termination do not have an important effect on the phase stability in the LCeMO system and that, most probably, the deposition conditions (in particular $T_s$ and $p_{O_2}$) are the determining factors.
Figure \[annealing\] shows the evolution of the $c$-axis with additional ex-situ annealing steps as obtained from XRD data for the (001) peak on sample E. As a result, the $c$-axis decreased from $c=3.894\,$[Å]{} to $c=3.872\,$[Å]{}. As the $a$- and $b$-axis bulk values for LCeMO are smaller than the ones of the STO substrate, the observed shrinking of the $c$-axis cannot be related to strain relaxation effects (which would increase $c$), but most probably to the incorporation of extra oxygen in the film. As another result of the annealing experiments, we did not find a correlation between ex-situ annealing and the CeO$_2$ concentration in our films. This is in contrast to the observations presented by Yanagida [*et al.*]{}[@Yanagida04] and Chang [*et al.*]{};[@Chang04] however, in their work, much longer annealing times (up to 10 hours) have been used. In our case, samples without secondary phase stayed single phase regarding the XRD data. However, while XRD data indicate that films deposited at 8-25Pa O$_2$ are single phase, HRTEM analysis showed evidence for phase separation even in these samples. The results of the microstructural TEM analysis are discussed in the following.
![(Color online) XRD pattern at the (001) peak for sample E, showing the evolution of the $c$-axis with ex-situ annealing steps: after in-situ annealing (1), first (2) and second (3) ex-situ annealing.[]{data-label="annealing"}](XRDrr33annealing){width="35.00000%"}
TEM
===
To obtain a better understanding on the relation between the microstructure and the physical properties of our LCeMO thin films, a few samples grown at different oxygen pressure were selected for TEM analysis. Here, we show results obtained from two films: sample E prepared at $p_{O_2}=8\,$Pa, which appears single phase at XRD, and sample K prepared at $p_{O_2}=3\,$Pa, containing CeO$_2$ as secondary phase. TEM studies were carried out using a JEOL 4000EX microscope operated at 400kV. The instrument has a point-to-point resolution of 0.17nm. Planview TEM specimens were prepared by mechanical polishing of the samples down to a thickness of $30\,\mu$m, followed by Ar ion-milling at grazing incidence to reach electron transparency.
Figure \[TEM\](a) shows a HRTEM plan view image of the LCeMO thin film grown at 8Pa O$_2$ (sample E). Several CeO$_2$ nanoclusters are indicated by arrows. A more detailed HRTEM image of one of the clusters is shown in Fig. \[TEM\](b). Figure \[TEM\](c) shows a TEM plan view image of the LCeMO thin film grown at 3Pa O$_2$ (sample K). In this sample, a higher density of CeO$_2$ nanoclusters in comparison to sample E is observed. Furthermore, the size of the clusters is also larger (although still within the nanometer region). The interface between the CeO$_2$ nanoclusters and the matrix is better defined in comparison to sample E.
![(a) Planview HRTEM images of (a) sample E grown at 8Pa O$_2$; arrows indicate the CeO$_2$ inclusions. An example of an inclusion \[cf. left arrow in (a)\] is shown in more detail in (b). (c) sample K grown at 8Pa O$_2$.[]{data-label="TEM"}](TEM){width="45.00000%"}
HRTEM data for the analyzed samples prove the presence of CeO$_2$ nanoclusters in the perovskite matrix (LCeMO) and show that CeO$_2$ segregation in the 3Pa sample is larger than in the 8Pa sample. In case of the 8Pa sample (and for another 25Pa film not shown) the small total volume of CeO$_2$ clusters made them untraceable by XRD. As an important consequence, our TEM data show that even LCeMO films which appear to be single phase from XRD data contain CeO$_2$ nanoclusters. This observation is important, as it has been shown [@Yanagida04] that the valence state of Mn in LCeMO is sensitive to the degree of Ce segregation, which drives the valences from Mn$^{3+}$ to Mn$^{4+}$, even in the presence of Ce$^{4+}$.
Transport and magnetic properties
=================================
![(Color online) Resistivity vs. temperature for samples A, B and E (with deposition pressure $p_{O_2}$ in parenthesis). The behavior after ex-situ annealing is shown for sample B and E (B1, E1: without ex-situ annealing; B2, E2: after $1^{\rm st}$ ex-situ annealing step; E3: after $2^{\rm nd}$ ex-situ annealing step).[]{data-label="RT"}](RT){width="45.00000%"}
Figure \[RT\] shows resistivity $\rho$ versus temperature $T$ for samples A, B and E. Sample A was ”as-deposited“ at $p_{O_2}=1\,$Pa and shows no metal-to-insulator transition at all. Due to its high resistivity we could not trace out $\rho(T)$ below $T\approx 150\,$K. Sample B, grown at 3Pa (also ”as-deposited“) shows a slight indication of a metal-to-insulator transition, i. e. a maximum in $\rho(T)$ at $T_{MI}= 175\,$K, with a strong increase in resistivity at $T{{\scriptscriptstyle\stackrel{<}{\sim}}}130\,$K, which can be explained by charge localization. Sample E, grown at 8Pa (annealed in-situ) shows a transition at $T_{MI}=210\,$K.
For sample E, the evolution of the $\rho(T)$ curves after two annealing steps (c. f. Sec.\[Sec:Experiment\]) is additionally shown. The $T_{MI}$ transition temperature increases to 230K, which is accompanied by a decreasing resistivity, presumably due to an increasing charge carrier density. This observation is consistent with results obtained by Yanagida [*et al.*]{}[@Yanagida04] and contradicts the picture of an electron-doped manganite: Oxygen annealing should decrease the concentration of Mn$^{2+}$ ions, hence, reduce the density of electrons as charge carriers and therefore lower $T_{MI}$ and increase resistivity.[@Wang06] The annealing steps seem to create more Mn$^{4+}$ in the samples, and the double-exchange between Mn$^{3+}$ and Mn$^{4+}$ gets stronger, which leads to an increase of $T_{MI}$. This interpretation is also supported by the results from measurements of the saturation magnetization ($M_s$) and the spectroscopic analysis, which will be discussed further below.
Figure \[RT\] also shows that $T_{MI}$ of sample B increases more drastically than sample E, even after only a single ex-situ annealing step. This might be due to the higher concentration of a secondary phase (CeO$_2$) in sample B (c. f. Fig. \[XRDCeO2\]), which may favor oxygen diffusion into the film due to crystal defects.
![(Color online) Magnetization vs. applied magnetic field at $T=20\,$K for the as-grown ($p_{O_2}=3\,$Pa) sample B (B1) and after ex-situ annealing (B2).[]{data-label="MvsH1"}](MvsH1){width="35.00000%"}
In Fig. \[MvsH1\] the magnetization $M$ (in units of $\mu_B/$Mn site) vs. applied field $\mu_0 H$ at $T=20\,$K is shown for sample B, measured ”as grown“ (B1) and after ex-situ annealing (B2). The ex-situ annealing step caused a decrease in the saturation magnetization $M_s$, from 2.93 to $2.40\,\mu_B$/Mn-site, while $T_{MI}$ increased from 175 to 250K. With the magnetic moments $m=$5, 4 and $3\,\mu_B$ for Mn$^{2+}$, Mn$^{3+}$ and Mn$^{4+}$, respectively, the theoretical value for the saturation magnetization of electron-doped LCeMO is $M_s=4.3\,\mu_B$/Mn-site.[@Zhang03] Until now, this value has never been achieved. However, for the hole-doped manganites, it is known that excess oxygen increases the valences from Mn$^{3+}$ to Mn$^{4+}$, and therefore decreases the magnetization. Hence, the observed decrease in $M_s$ with oxygen annealing can be explained by the decrease in Mn$^{2+}$ and concomitant increase in Mn$^{4+}$ concentration.
![(Color online) Comparison of samples with different thickness $d$, grown under the same deposition conditions ($p_{O_2}=8\,$Pa; in-situ annealed). (a) XRD $\Theta - 2\Theta$ scans; the inset shows that the $c$-axis value increases with increasing $d$. (b) Resistivity vs. temperature; the inset shows that the transition temperature $T_{MI}$ decreases and the saturation magnetization $M_s$ (from $M(H)$ data; not shown) increases with increasing $d$.[]{data-label="thickness"}](thickness){width="45.00000%"}
In order to study the dependence of structural, transport and magnetic properties on film thickness $d$, four samples (C, E, F, G with $d$=100, 65, 40 and 20nm, respectively) were grown under the same conditions, i.e., at $p_{O_2}=8\,$Pa with in-situ annealing. The $\Theta-2\Theta$ XRD scans of the (001) peak in Fig. \[thickness\](a) show that with decreasing film thickness the $c$-axis shrinks \[see inset\]. Assuming a fixed unit cell volume, this observation might be explained by increasing tensile strain with decreasing $d$, as the bulk in-plane lattice parameters of LCeMO are smaller than those for the STO substrate. However, as oxygen vacancies tend to expand the lattice parameters, an increasing lack of oxygen with increasing $d$ has the same effect. The transport properties shown in Fig. \[thickness\](b) indicate exactly this lack of oxygen with increasing film thickness. Sample C, with largest $d$, shows again charge localization at low $T$, while the thinnest film has the highest $T_{MI}$ \[c. f. inset\] and lowest $\rho$. >From magnetization measurements on samples C, E, F and G we also find that $M_s$ increases with $d$ \[c. f. inset in Fig. \[thickness\](b)\]. The lowest saturation magnetization for the thinnest sample G is another indication for the higher oxygen concentration compared to the others.
Hall measurements
=================
In order to determine the type of majority charge carriers via the Hall effect, we chose one of our films (sample H, $d=100\,$nm) which was deposited at relatively low oxygen pressure ($p_{O_2}=3\,$Pa) and cooled in 1bar, without an annealing step. >From measurements of the longitudinal resistivity $\rho(T)$ of the patterned film we find a clear MI transition with rather high $T_{MI}=260\,$K. The Hall resistivity $\rho_H$ was measured at $T=10$, 50 and 100K in magnetic fields up to 14T. The sign of the Hall voltage was carefully checked by using an $n$-doped silicon reference sample. Figure \[Hall\] shows $\rho_H$ vs. applied magnetic field $H$. The drop of $\rho_H$ in the low-field range reflects the so-called anomalous Hall Effect, $\rho_{aH}=R_{aH}\mu_0M$, which is due to spin orbit interaction.[@Karplus54] Here, $R_{aH}$ is the Hall coefficient for the anomalous Hall effect. With further increasing field, the data show the expected linear behavior of the normal Hall effect $\rho_{nH}=R_{nH}\mu_0H$ with Hall coefficient $R_{nH}=1/ne$ and charge carrier density $n$. The main feature in Fig. \[Hall\] is the positive slope $\partial\rho_H/\partial H$ at high fields, which reveals the majority of the carriers to be holes with $n=1.57$, 1.60 and $1.78\times 10^{22}\,{\rm cm}^{-3}$, for $T=10$, 50 and 100K, respectively. This corresponds to 0.94–1.07 holes/Mn-site. The observation of hole-doping is consistent with the results from transport and magnetization measurements discussed above and also with the spectroscopic analysis, which will be presented in the following section.
![(Color online) Field dependence of the Hall resistivity of sample H. The positive slope at high magnetic field identifies the majority of the carriers to be holes. The solid lines are linear fits to the high-field data.[]{data-label="Hall"}](Hall){width="35.00000%"}
Spectroscopic Analysis
=======================
X-ray Absorption Spectroscopy (XAS) was performed on LCeMO thin films prepared under different conditions, in order to investigate the relation between the manganese valences, the oxygen content and transport and magnetic properties. In total electron yield (TEY) detection mode only the uppermost 5 - 10nm are probed, depending on the electron escape depth, while in fluorescence yield (FY) mode x-ray photons are detected. They have typical attenuation lengths from 100nm (Ce M edge) to 200nm (O K and Mn L edge), thus giving insight into the bulk structure of the samples.
Here we compare two films, D (as-deposited) and G (in-situ annealed), which were deposited at different oxygen pressure $p_{O_2}=25\,$Pa and 8Pa, respectively. >From XPS measurements we find that the oxygen content of G is higher than the one of D. This shows that the higher deposition pressure (for sample D) is not the key to higher oxygen concentration, but that annealing is most relevant. Sample G shows a MI transition at 232K \[c. f. Fig. \[thickness\](b)\], while sample D shows a weak transition at 180K and charge localization at lower temperatures.
A typical spectrum of the O K edge of LCeMO, measured in bulk sensitive fluorescence yield (FY) mode, is seen in Fig. \[XAS-DvsG\] (left). The first structure at about 530eV arises from transitions from the O1s level to states, which are commonly understood to be of mixed Mn3d-O2p character and as being a measure of the hole concentration.[@Abbate92; @Manella05; @Chang05] In fact we found that this prepeak is stronger in sample G, i.e. the more oxidized sample. The second, rather broad and asymmetric feature at 532 to 537eV is attributed to La5d (Ce), La4f (Ce) states hybridized with O2p states. A third set of states (not shown here) is found at about 543eV and is widely believed to derive from hybridization of O2p with higher energy metal-states like Mn 4sp and La 6sp.[@Abbate92]
![(Color online) XA spectra of samples D and G. On the left side the oxygen K edge (FY mode) is shown with the prepeak increasing with higher oxygen content. The right side shows the manganese L$_3$ edge (TEY mode) with different amounts of Mn$^{2+}$ for differently oxidized samples.[]{data-label="XAS-DvsG"}](XAS-DvsG){width="42.00000%"}
The corresponding spectrum at the Mn L edge taken in TEY detection mode is shown in Fig. \[XAS-DvsG\] (right). Both, the L$_3$ edge at 642eV and the L$_2$ edge at 653eV (not shown here) are strongly broadened, indicating the presence of a variety of valence states. The most important feature is the shoulder at 640eV, which is a clear indication of divalent Mn, as can be shown by a comparison with XAS data from MnO.[@Nagel07] In Fig. \[XAS-DvsG\] (right) this shoulder is more pronounced in sample D, i.e., the less oxidized sample. The relative spectral weight of this feature in combination with the relative intensity of Mn3d-O2p states taken from the O K edge is essential to explain the properties of the different samples. A higher degree of oxidation leads to a higher relative spectral weight of the O K prepeak and a lower amount of Mn$^{2+}$. By introducing more oxygen, more holes are created and the manganese valence is increased. This finding is further supported by measurements on three additional samples (not shown here), also showing the effect of film thickness, oxygen pressure during growth and duration of post-growth annealing in oxygen.
The remaining issue is the oxidation state of the Ce ions, which is important for the type of doping. Looking at the Ce absorption M$_5$ edge both in surface-sensitive TEY detection mode and bulk-sensitive FY mode, we found striking differences in the spectral shapes of the measured spectra, as shown in Fig. \[XAS-Ce-Mn\](a). Cerium reference data for CeO$_2$ and CeF$_3$ were taken from Ref. \[\]. In total electron yield detection mode the edge is identical to a pure CeO$_2$ edge, i.e. cerium in a Ce$^{4+}$ state. However, when increasing the information depth by switching to bulk sensitive FY detection,the edge changes drastically. The FY signal contains contributions from Ce$^{4+}$ and Ce$^{3+}$. Note that thermodynamically the reducing power of cerium is not sufficient for the Mn$^{3+}$ - Mn$^{2+}$ transition. The same trends are seen in the FY spectra of the Mn and O edges. Manganese reference data were taken from Ref. \[\]. In case of the Mn edge \[Fig. \[XAS-Ce-Mn\](b)\] a decrease of the Mn$^{2+}$ related feature at 640eV is visible in the FY data, and the edges are broadened towards higher energies than in TEY mode. This indicates an increased amount of Mn$^{3+}$ (642eV) and Mn$^{4+}$ (644eV) species within the film as compared to the near surface region. Finally, at the O K edge (not shown here) the relative prepeak intensity at 530eV increases with growing information depth from TEY to FY mode. As this feature is proportional to the hole concentration, this finding further emphasizes the point that the bulk is more oxidized than the surface and that the majority charge carriers are indeed holes.
![(Color online) XA spectra of sample D in TEY and FY mode (scaled to the TEY intensity) at different absorption edges: (a) Cerium M$_5$ edge; reference spectra of CeO$_2$ and CeF$_3$ were added for comparison. Please note the mixture of Ce$^{3+}$ and Ce$^{4+}$ in FY mode. (b) Manganese L$_3$ edge. The FY data are self-absorption corrected following a procedure by Ref. \[\]. Reference spectra of MnO (blue), Mn$_2$O$_3$ and MnO$_2$ were added for comparison. Please note the missing Mn$^{2+}$ shoulder in FY mode.[]{data-label="XAS-Ce-Mn"}](XAS-Ce-Mn){width="42.00000%"}
Conclusions {#Sec:Conclusions}
===========
We investigated La$_{0.7}$Ce$_{0.3}$MnO$_3$ thin films of variable thickness, grown epitaxially at different oxygen pressure $p_{O_2}$ and subjected to different oxygen annealing procedures. We find that thin film growth at low deposition pressure favors phase separation via the formation of CeO$_2$ inclusions. For higher deposition pressure, still CeO$_2$ nanoclusters are found, as shown by transmission electron microscopy, even for those films which appear to be single phase in x-ray diffraction analysis. Combining electric transport, magnetization and Hall measurements with x-ray photoemission and absorption spectroscopy we obtain a consistent picture in the sense that the appearance of a metal-to-insulator transition in electric transport measurements is always associated with hole doping and the presence of a mixed system of Mn$^{2+}$, Mn$^{3+}$ and Mn$^{4+}$, despite finding Ce$^{4+}$ as a sign of electron doping. The hole-doped behavior of our films may be explained by the presence of cation vacancies (due to CeO$_2$ clustering), which can be occupied by excess oxygen that shifts the valences from Mn$^{2+}$ to Mn$^{3+}$ or Mn$^{4+}$. In particular, oxidation states are well reproduced in the x-ray absorption spectra and fit to the transport properties. Upon oxidizing the samples, the system goes towards Mn$^{3+}$ / Mn$^{4+}$ as expected, while reducing the films forms more Mn$^{2+}$ species. In particular for less oxidized films, we find a reduced layer at the surface with a more oxidized bulk underneath. This explains some of the peculiarities of this system, namely the discrepancy between finding Mn$^{2+}$ and Ce$^{4+}$ and still having holes as majority carriers. Furthermore, this demonstrates that one has to be very careful in relating surface sensitive spectroscopy data to bulk sensitive transport and magnetization data.
We gratefully acknowledge Kathrin Dörr for helpful discussions and Matthias Althammer and Sebastian Gönnenwein for their support with the Hall measurements. Furthermore, we acknowledge the ANKA Angstroemquelle Karlsruhe for the provision of beamtime and we would like to thank P. Nagel, M. Merz and S. Schuppler for the skillful technical assistance using beamline WERA and for valuable discussions. This work was funded by the Deutsche Forschungsgemeinschaft (project no. KO 1303/8-1) and by the European Union under the Framework 6 program for an Integrated Infrastructure Initiative, ref. 026019 ESTEEM. S. B. thanks the Fund for Scientific Research – Flanders.
[^1]: From the bulk values for the LCeMO lattice constants $a=3.821\,\AA$ and $b=3.902\,\AA$ [@Chang04] one obtains an in-plane lattice mismatch ranging from -2% (tensile strain) to +1% (compressive strain) for the different substrates used here.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Using methods from the theory of total positivity, we provide a full classification of attainable term structure shapes in the two-factor Vasicek model. In particular, we show that the shapes normal, inverse, humped, dipped and hump-dip are always attainable. In certain parameter regimes up to four additional shapes can be produced. Our results show that the correlation and the difference in mean-reversion speeds of the two factor processes play a key role in determining the scope of attainable shapes. The mathematical tools from total positivity can likely be applied to higher-dimensional generalizations of the Vasicek model and to other interest rate models as well.'
address: 'Institute for Mathematical Stochastics, TU Dresden'
author:
- 'Martin Keller-Ressel'
bibliography:
- 'references.bib'
title: 'Total positivity and the classification of term structure shapes in the two-factor Vasicek model'
---
Introduction
============
The term structure of interest rates – summarized in the form of the yield or forward curve – is one of the most fundamental economic indicators. Its shape encodes important information on the preferences for short- vs. long-term investments, the desire for liquidity and on expectations of central bank decisions and the general economic outlook. It is therefore a natural question – to be asked of any mathematical model of the term structure – which shapes of yield and forward curves the model is able to (re-)produce. Already in [@vasicek1977equilibrium] a paragraph is dedicated to this question, with Vasicek concluding that [`normal`]{} (increasing), [`inverse`]{} (decreasing) and [`humped`]{} (endowed with a single maximum) shapes can be attained in his single-factor model. The same classification of shapes has been shown to hold in the Cox-Ingersoll-Ross model and furthermore in all one-dimensional affine term structure models (including short-rate models with jumps), see [@cox1985theory Eq. (26)f], [@keller-ressel2008yield; @keller-ressel2018correction].\
It is also well-known, that in the Hull-White extended Vasicek model [@hull1990pricing] *any* initial term structure can be perfectly fitted and therefore that any shape of the term structure can be reproduced at the time of calibration. However, as time progresses, this initial shape will disappear and – due to ergodicity effects – the model will behave closer and closer to a Vasicek model with time-homogeneous coefficients. Therefore, even in view of Hull-White-extended models, the classification of attainable term structure shapes in time-homogeneous short-rate models is a reasonable and important question.\
Here, we provide for the first time a systematic classification of term structure shapes beyond the one-dimensional case. In our main result, Theorem \[thm:main\], we classify all attainable shapes for both the yield and forward curve in the two-dimensional Vasicek model. As expected, several additional shapes, such as a [`dipped`]{} curve, which are not attainable in the one-dimensional case become attainable in the two-factor model. We also give some stronger attainment results, showing for instance that also the locations of humps and dips can typically be chosen without restrictions.\
Our main mathematical tool is the theory of total positivity (see e.g. [@karlin1968total]), a theory linked to the variation-diminishing properties of certain matrices, function systems and integral kernels. Total positivity has broad applications in numerical interpolation, differential equations and stochastic processes. Within mathematical finance, it has been applied to study monotonicity and convexity of options prices [@kijima2002monotonicity] and to the principal-component-analysis of the term structure of interest rates [@salinelli2006correlation; @lord2007level]. Our application to the shape analysis of the term structure is new and fundamentally different from the results in [@salinelli2006correlation; @lord2007level]. While the results in this paper are limited to the two-dimensional Vasicek model, we are confident that the underlying theory can be applied to other multi-factor interest rate models as well.
Preliminaries
=============
Shapes of the term structure
----------------------------
In our terminology *term structure* refers to either the yield curve or the forward curve. The *shape* $\mathsf{S}$ of the term structure is defined by the number and sequence of local maxima or minima of the term structure curve. In common financial market terminology a local maximum is called a ‘hump’ and a local minimum a ‘dip’. As the term structure curves produced by the Vasicek model (or most other models) are smooth, it is clear that the shape of the term structure curve can be conveniently analyzed by considering its derivative: Any sign change of the derivative (from strictly positive to strictly negative or vice versa) corresponds to a local extremum of the term structure; the type of sign change (${\textup{\texttt{+}}}$ to ${\textup{\texttt{-}}}$ or ${\textup{\texttt{-}}}$ to ${\textup{\texttt{+}}}$) determines the type of the extremum (hump or dip). The basic shapes and their conventional names are listed in Table \[tab:shape\]. For ‘higher order’ shapes we use the letters `H` for a hump and `D` for a dip, e.g., the shape ${\texttt{HDH}}{}$ corresponds to a term structure with two local maxima, interlaced by a single local minimum.
\[tab:shape\]
Shape $\mathsf{S}$ of the term structure Description Sign sequence of derivative
------------------------------------------ -------------------------------------------------------- ---------------------------------------------------------------------------------
[`normal`]{} strictly increasing ${\bm{[}{\textup{\texttt{+}}}\bm{]}}$
[`inverse`]{} strictly descreasing ${\bm{[}{\textup{\texttt{-}}}\bm{]}}$
[`humped`]{} single local maximum ${\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}\bm{]}}$
[`dipped`]{} single local minimum ${\bm{[}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}$
[`HD`]{} hump-dip, i.e. local maximum followed by local minimum ${\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}$
[`DH`]{}, [`HDH`]{}, etc. further sequences of multiple ‘dips’ and ‘humps’ ${\bm{[}\dotsc\bm{]}}$
: Shapes of the term structure
Sign sequences
--------------
We introduce several notions associated to a *sign sequence*. The primary purpose of a sign sequence will be to keep track of the number and the directions of sign changes of a numeric sequence or of a continuous function. The notion of a sign sequence appears implicitly in many of the results related to total positivity, however, the terminology introduced here is new.
(i) A **sign sequence** is a non-empty sequence of the symbols ${\textup{\texttt{+}}}$ and ${\textup{\texttt{-}}}$. Only finite sign sequences will be considered here. Also zeroes can be allowed; we comment on this later. We include sign sequences in square brackets and write e.g. $${\bm{[}{\textup{\texttt{+}}}\bm{]}}, \qquad {\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}, \qquad {\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}$$ for some valid sign sequences.
(ii) A sign sequence is called **pure** if the signs ${\textup{\texttt{+}}}$ and ${\textup{\texttt{-}}}$ alternate, e.g., the sequences $${\bm{[}{\textup{\texttt{+}}}\bm{]}}, \qquad {\bm{[}{\textup{\texttt{-}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}\bm{]}}$$ are pure. Any sign sequence can be reduced to a **pure sign sequence** by replacing blocks of ${\textup{\texttt{+}}}$’s by a single ${\textup{\texttt{+}}}$ and blocks of ${\textup{\texttt{-}}}$’s by a single ${\textup{\texttt{-}}}$, e.g. $${\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}} \quad \text{reduces to} \quad {\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}.$$ Note that this reduction preserves the number and direction of sign changes, which is our primary object of interest.
(iii) Two sign sequences are called **equivalent**, if they reduce to the same pure sequence. This defines an equivalence relation $\simeq$, e.g., $${\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}} \simeq {\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}{\textup{\texttt{+}}}{\textup{\texttt{+}}}\bm{]}}.$$
(iv) In a similar way we can define a **subsequence** relation $\subseteq$ where we treat blocks of signs as if they were single signs. E.g. we have $${\bm{[}{\textup{\texttt{-}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}{\textup{\texttt{+}}}{\textup{\texttt{+}}}\bm{]}} \subseteq {\bm{[}{\textup{\texttt{-}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{-}}}\bm{]}}, \qquad {\bm{[}{\textup{\texttt{-}}}\bm{]}} \subseteq {\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}.$$
(v) A subsequence which also preserves the initial sign is called a **head** and a subsequence which preserves the terminal sign is called a **tail**. We write $${\bm{[}{\textup{\texttt{-}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}{\textup{\texttt{+}}}{\textup{\texttt{+}}}\bm{]}}{\overset{\scriptscriptstyle{H}}{\subseteq}}{\bm{[}{\textup{\texttt{-}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{-}}}\bm{]}}, \qquad {\bm{[}{\textup{\texttt{+}}}\bm{]}}{\overset{\scriptscriptstyle{T}}{\subseteq}}{\bm{[}{\textup{\texttt{-}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}$$ for the respective relations.
(vi) Sign sequences should only keep track of ‘strong’ sign changes.[^1] Therefore we add the convention that sign sequences **containing zeroes** can be reduced to a pure sequence by simply omitting all zeroes and then applying the reduction rules described above. E.g. we have $${\bm{[}{\textup{\texttt{+}}}0{\textup{\texttt{+}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}0{\textup{\texttt{+}}}\bm{]}} \simeq {\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}, \qquad {\bm{[}0{\textup{\texttt{-}}}00{\textup{\texttt{-}}}\bm{]}} \simeq {\bm{[}{\textup{\texttt{-}}}\bm{]}}.$$ Note that all strong sign changes (and their direction) are preserved under the described reduction.
(vii) If a variable, say $a$, appears inside a sign sequence, it should be interpreted as ‘sign of $a$’. E.g. the sign sequence ${\bm{[}ab\bm{]}}$ evaluates to ${\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}\bm{]}}$ if $a = 6$ and $b = -1$ and to ${\bm{[}{\textup{\texttt{-}}}\bm{]}}$ if $a = -1$, $b = 0$.
(viii) Let $f$ be a continuous function, defined on a subset $X$ of ${\mathbb{R}}$ and not constantly zero. The **sign sequence of $f$** is the sequence of signs that $f$ takes on between its zeroes. Only functions with finite sign sequences will be considered and we denote the sign sequence of such a function $f$ by ${\mathrm{sseq}}(f)$. For example $$f(x) = x^2 -1\,\text{, defined on $X = [0,\infty)$} \qquad \Longrightarrow \qquad {\mathrm{sseq}}(f) = {\bm{[}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}.$$
In some cases, the first and the last sign in the sign sequence of $f$ will be of particular interest. For them, we use the notation $$S_\text{init}(f)\qquad \text{and} \qquad S_\text{term}(f).$$
Total positivity and Descartes systems {#sec:tot_pos}
--------------------------------------
We introduce some definitions and key results from the theory of total positivity. For background and further details we refer to [@karlin1966tchebycheff; @karlin1968total] and [@borwein1995polynomials].
Let $X, Y \subseteq {\mathbb{R}}$ and let $K$ be a function (‘kernel’) from $X \times Y$ to ${\mathbb{R}}$. If $$\label{eq:K}
\det \begin{pmatrix} K(x_1,y_1) & K(x_1, y_2) & \dotsc & K(x_1, y_m) \\ \vdots & \vdots && \vdots\\ K(x_m, y_1) & K(x_m,y_2) & \dotsc & K(x_m,y_m)\end{pmatrix} \ge 0$$ for any $m \in {\mathbb{N}}$, $x_1 < x_2 < \dotsm < x_m$ in $X$ and $i_1 < i_2 < \dotsm < i_m$ in $Y$, then $K(x,y)$ is called **totally positive**. If strict equality holds in , the kernel is called **strictly totally positive**.
(i) The kernels $K(x,y) = e^{xy}$ and $K(x,y) = {\mathbf{1}_{\left\{y \le x\right\}}}$ are examples of totally positive kernels on ${\mathbb{R}}^2$ (or on any $X \times Y$ with $X, Y \subseteq {\mathbb{R}}$); see [@karlin1968total Ch.1 §2] and [@karlin1968total Ch.3, Eq. (1.13)ff]. The first kernel is even strictly totally positive.
(ii) A totally positive kernel on $X=Y= {\left\{1, \dotsc, n\right\}}$ can be written as a matrix; accordingly such matrices are also called totally positive, cf. [@ando1987totally] or [@hogben2013handbook Ch.29].
A crucial property of totally positive kernels is the following:
\[thm:vardim\_K\] Let $K$ be a totally positive kernel on $X \times Y$, such that $\int_Y K(x,y) dy < \infty$ for all $x \in X$. Let $f: Y \to {\mathbb{R}}$ be a bounded continuous function with finite sign sequence and set $$g(x) := \int_Y K(x,y) f(y) dy.$$ Then $${\mathrm{sseq}}(g) \subseteq {\mathrm{sseq}}(f).$$
This result is a particular case of [@karlin1968total Ch. 5, Thm. 3.1], formulated in the language of sign sequences. It can be extended from integration with respect to Lebesgue measure $dy$ to a large class of $\sigma$-finite measures $d\mu(y)$ on $Y$. These extensions, however, will not be needed here.\
Next, we discuss a closely related definition, which applies to families of functions.
Let $X$ be a subinterval of ${\mathbb{R}}$ and let ${\mathcal{D}}= (\phi_1, \dotsc, \phi_n)$ be a family of continuous functions from $X$ to ${\mathbb{R}}$. If $$\label{eq:Phi}
\det \begin{pmatrix} \phi_{i_1}(x_1) & \phi_{i_2}(x_1) & \dotsc & \phi_{i_m}(x_1)\\ \vdots & \vdots && \vdots\\ \phi_{i_1}(x_m) & \phi_{i_2}(x_m) & \dotsc & \phi_{i_m}(x_m)\end{pmatrix} > 0$$ for any $m \le n$, $x_1 < x_2 < \dotsm < x_m$ in $X$ and $i_1 < i_2 < \dotsm < i_m$ in ${\left\{1, \dotsc, n\right\}}$, then ${\mathcal{D}}$ is called a **Descartes system** on $X$.
\[rem:Descartes\]
(i) The order of the functions $\phi_1, \dotsc, \phi_n$ matters and a permutation of a Descartes system need not be a Descartes system.
(ii) A Descartes system can seen as a strictly totally positive kernel on $X \times {\left\{1, \dotsc, n\right\}}$
(iii) The family of monomials $(1,x,x^2,x^3, \dotsc, x^n)$ is a Descartes system.
(iv) The family of exponential functions $(e^{x\gamma_1}, \dotsc, e^{x \gamma_n})$ is a Descartes system if and only if $\gamma_1 < \gamma_2 < \dotsm < \gamma_n$ \[item:exp\]
Also Descartes systems enjoy variation-diminishing properties:
\[thm:vardim\] Let $(\phi_1, \dots, \phi_n)$ be a Descartes system and let $(a_1, \dotsc, a_n) \subseteq {\mathbb{R}}^n$. Then $$\label{eq:vardim}
{\mathrm{sseq}}\left(\sum_{i=1}^n a_i \phi_i\right) \subseteq {\bm{[}a_1 a_2, \dotsm a_n\bm{]}}.$$
(i) This theorem is [@karlin1966tchebycheff Thm. 3.1, 4.4] (see also [@borwein1995polynomials Thm. 3.2.4]), translated into the language of sign sequences.
(ii) The well-known ‘Descartes’ rule of signs’ for polynomials follows by applying this theorem to the Descartes system $(1, x, x^2, \dotsc, x^n)$; see [@borwein1995polynomials 3.2.E7].
Given a Descartes system ${\mathcal{D}}= (\phi_1, \dots, \phi_n)$, a function of the form $$\phi(x) := \sum_{i=1}^n a_i \phi_i(x)$$ is called a **D-polynomial** in ${\mathcal{D}}$. We call $\phi$ **extremal**, if equality is attained in . The next result concerns the interpolation properties of D-polynomials:
\[thm:extremal\] Let $(\phi_1, \dots, \phi_n)$ be a Descartes system on $X$ and let $r_1 < r_2 < \dotsm < r_{n-1}$ be $n-1$ distinct points in $X$. Then there exists a $D$-polynomial $\phi(x) = \sum_{i =1}^n a_i \phi_i(x)$ with all $a_i$ non-zero, which satisfies:
- $\phi(r_i) = 0$ for all $i \in 1, \dotsc, n-1$;
- $\phi$ has a strong sign change at each $r_i$ in the interior of $X$.
If all $r_i$ are interior points of $X$, then $\phi$ is extremal, i.e.,
- ${\mathrm{sseq}}(\phi) \simeq {\bm{[}a_1\,a_2\dotsm a_n\bm{]}}$.
This result follows from [@karlin1966tchebycheff Ch. I, Thm. 5.1] or [@borwein1995polynomials 3.1.E11], but we provide a self-contained proof and some related results in Sec. \[app:interpolation\], \[app:interpolation2\].
The Vasicek model
=================
The Vasicek model, originally introduced by [@vasicek1977equilibrium] as a single-factor model, has been extended to multiple factors by [@dai2000specification] within the framework of affine term structure models. A detailed study of the two-dimensional case can also be found in [@brigo2007interest].
The Vasicek model with multiple factors
---------------------------------------
The $d$-dimensional Vasicek model is based on a factor process $Z = (Z^1, \dotsc, Z^d)$, with components given under a risk-neutral measure ${\mathbb{Q}}$ by $$\begin{aligned}
\label{eq:vasicek}
dZ^i_t = -\lambda_i ( Z_t^i - \theta_i) \, dt + \sigma_i dB_t^i, \qquad i \in {\left\{1, \dotsc, d\right\}}.\end{aligned}$$ The long-term rates $\theta = (\theta_1, \dotsc, \theta_d)$ are real and the Brownian motions $B^1, \dotsc, B^d$ may be correlated with the covariation matrix of $(\sigma_1 B^1, \dotsc, \sigma_d B^d)$ denoted by $\Sigma$. Moreover, we assume that the mean-reversion speeds are strictly positive and ordered as $$\lambda_1 < \lambda_2 < \dotsm < \lambda_d.$$ Thus, the speed of mean reversion increases with the index $i$ and $Z^1$ is the factor with the slowest mean-reversion. Given a vector $\kappa = (\kappa_1, \dotsc, \kappa_d)$ of strictly positive numbers and $\kappa_0 \in {\mathbb{R}}$, the short rate $r$ of the $d$-dimensional Vasicek model is defined as $$r_t = \kappa_0 + Z_t^\top \kappa = \kappa_0 + \kappa_1 Z_t^1 + \dotsm \kappa_d Z_t^d.$$ From [@dai2000specification], the bond price in this multivariate Vasicek model can be written as $$\label{eq:bond}
P(t,t+x) = {\mathbb{E}^{{\mathbb{Q}}}\left[\left.\exp\left(- \int_t^{t+x} r_s ds\right)\right|{\mathcal{F}}_t\right]} = \exp \left(A(x) + Z_t^\top B(x)\right)$$ where $A$ and $B$ are given as solutions of the ODE system
\[eq:Riccati\] $$\begin{aligned}
A'(x) &= F(B(x)), \qquad A(0) = 0\label{eq:Riccati_A}\\
B'(x) &= R(B(s)), \qquad B(0) = 0\label{eq:Riccati_B}\end{aligned}$$
with
\[eq:FR\_def\] $$\begin{aligned}
F(b) &= \mu^\top b + \frac{1}{2} b^\top \Sigma b- \kappa_0, \qquad \quad \mu^\top = \begin{pmatrix} \lambda_1 \theta_1, \dotsc, \lambda_d \theta_d \end{pmatrix},\label{eq:F_def}\\
R(b) &= -{\mathrm{diag}}(\lambda_1, \dotsc, \lambda_d) b - \kappa.\label{eq:R_def}\end{aligned}$$
The differential equations decouple into scalar linear equations, which can be solved explicitly with solutions given by $$B_i(x) = \frac{\kappa_i}{\lambda_i} \left(e^{-\lambda_i x} - 1\right), \quad i \in {\left\{1,\dotsc, d\right\}}.$$ The explicit form of $A$ can be determined from , but will never be needed here and is therefore omitted.
Yield and forward curves
------------------------
The yield and forward curves in the Vasicek model are easily computed from and as $$\begin{aligned}
f(x;Z_t) &= -\partial_x \log P(t,t+x) = -F(B(x)) - Z_t^\top R(B(x))\label{eq:forward},\\
Y(x;Z_t) &= -\frac{1}{x} \log P(t,t+x) = - \frac{A(x)}{x} - Z_t^\top \frac{B(x)}{x}. \label{eq:yield}\end{aligned}$$ When we want to emphasize the dependency of these curves on the state vector $z$ and on some parameter $p$, we write $f(x;z,p)$ and $Y(x;z,p)$. We use the same notation for all quantities derived from $f$ and $Y$.\
To study the shapes of the yield and forward curve, in particular their local extrema, we need to consider their derivatives.
\[lem:curve\_diff\]The derivatives of the forward and the yield curve in the Vasicek model are given by $$\begin{aligned}
l(x) := \partial_x f(x;z) &= - \mu^\top B'(x) - B(x)^\top \Sigma B'(x) + z ^\top {\mathrm{diag}}(\lambda_1, \dotsc, \lambda_d) B'(x) \label{eq:forward_d}\\
\intertext{and}
m(x) := \partial_x Y(x;z) &= \frac{1}{x^2} \left\{ (A(x) - xF(B(x))) + z^\top (B(x) - xR(B(x))) \right\} = \notag\\
&= \frac{1}{x^2}\int_0^x yl(y) dy.\label{eq:yield_integral}
\end{aligned}$$
First, we calculate the gradient of $F$ and the Jacobian of $R$ as $$\begin{aligned}
\nabla F(b) &= \mu^\top + b^\top \Sigma\\
J_R(b) &= -{\mathrm{diag}}(\lambda_1, \dotsc, \lambda_d).\end{aligned}$$ For the forward curve, differentiation of gives $$l(x) = \partial_x f(x;z) = -\left(\nabla F(B(x)) +z^\top J_R(B(x))\right) B'(x),$$ which is . For the yield curve, differentiation of directly gives the first part of . Taking another derivative of $$x^2 m(x) = \left(A(x) - xF(B(x))\right) + z^\top \left(B(x) - xR(B(x))\right)$$ we obtain after some cancellations that $$\partial_x(x^2 m(x)) = - x \left(\nabla F(B(x)) + z^\top J_R(B(x))\right) B'(x) = x l(x),$$ which yields the integral representation in .
A first application of total positivity
---------------------------------------
The first application of total positivity concerns the relation between yield and forward curves: We show that $m$ is a totally positive transformation of $l$.
\[lem:K\_tot\_pos\] The functions $l$ and $m$ of Lemma \[lem:curve\_diff\] are related by $$m(x) = \int_0^\infty K(x,y) l(y) dy,$$ where the kernel $K(x,y) = \frac{y}{x^2}{\mathbf{1}_{\left\{y \le x\right\}}}$ is totally positive on $(0,\infty) \times (0,\infty)$.
The kernel $K(x,y)$ is of the form $K(x,y) = \phi(x)\psi(y)L(x,y)$, where $\phi(x) = \frac{1}{x^2}$, $\psi(y) = y$ are strictly positive on $(0,\infty)$ and where $L(x,y) = {\mathbf{1}_{\left\{y \le x\right\}}}$. The total positivity of $L(x,y) = {\mathbf{1}_{\left\{y \le x\right\}}}$ is shown in [@karlin1968total Ch. 3, Eq.(1.10)ff]. The total positivity of the composed kernel $K(x,y)$ now follows from [@karlin1968total Ch. 1, Thm. 2.1].
From the variation-diminishing property of $K$ (cf. Theorem \[thm:vardim\_K\]) we can immediately conclude that ${\mathrm{sseq}}(m) \subseteq {\mathrm{sseq}}(l)$ and hence, that the number of local extrema of the yield curve is bounded by the number of local extrema of the forward curve. This result can be slightly strengthened by also considering the initial signs of $l$ and $m$, which are easily obtained from Lemma \[lem:curve\_diff\]. Taking into account that $B(0) = 0$ and $B'(0) = -\kappa$, we obtain from that $$l(0) = \mu^\top \kappa - z^\top {\mathrm{diag}}(\lambda) \kappa.$$ Applying l’Hospital twice to yields $$m(0) = \frac{l(0)}{2},$$ and we conclude that the initial sign of $l$ and $m$ is always the same. Hence, we have shown the following lemma:
The sign sequences of $l$ and $m$ satisfy $${\mathrm{sseq}}(m){\overset{\scriptscriptstyle{H}}{\subseteq}}{\mathrm{sseq}}(l).$$
Reformulating the lemma in terms of term structure shapes, we conclude the following:
\[thm:fw\_yield\] In the multivariate Vasicek model
(a) the initial slope of yield and forward curve has the same sign;
(b) the number of local extrema of the yield curve is less or equal to the number of local extrema of the forward curve;
(c) if the number of local extrema is the same, then also the sequence of types (hump/dip) coincides.
We give an example which demonstrates how concrete restrictions of the yield curve can be derived from this result: Suppose for instance that the forward curve is [`humped`]{}. Then (b) and (c) leave as possible yield curve shapes [`inverse`]{}, [`normal`]{}, and [`humped`]{}. Restriction (a) further eliminates [`inverse`]{}, and the possible forward curve shapes are [`normal`]{}, [`humped`]{}. This is consistent with the analysis of the Vasicek model in the one-dimensional case (which is discussed in more detail below), but – as we have just shown – it also applies in the multivariate case.
The one-dimensional case revisited
----------------------------------
The classification of term structure shapes in the one-dimensional Vasicek model has already been discussed in [@vasicek1977equilibrium] (See also [@keller-ressel2008yield; @keller-ressel2018correction] for the case of general one-dimensional affine short rate models). We revisit this classification problem from the perspective of total positivity. First we calculate $l$, the derivative of the forward curve from Lemma \[lem:curve\_diff\] as $$l(x) = \frac{\sigma^2 \kappa^2}{\lambda} e^{-2\lambda x} + \left\{\kappa \lambda (\theta - z) - \frac{\sigma^2 \kappa^2}{\lambda}\right\} e^{-\lambda x} .$$ From Rem. \[rem:Descartes\] we know that ${\mathcal{D}}= (e^{-2\lambda x}, e^{-\lambda x})$ is a Descartes system. Thus, $l$ is a D-polynomial with coefficients $$\begin{aligned}
u = \frac{\sigma^2 \kappa^2}{\lambda} \qquad \text{and} \qquad w = \kappa \lambda (\theta - z) - \frac{\sigma^2 \kappa^2}{\lambda}.\end{aligned}$$ From the variation-diminishing property of Descartes systems (Thm. \[thm:vardim\]), we conclude that $l$ has at most a single sign change. Thus, only the shapes [`normal`]{}, [`inverse`]{}, [`humped`]{} and [`dipped`]{} are possible. The shape [`dipped`]{} corresponds to a sign sequence ${\bm{[}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}$ of $l$, which is not compatible with the positive sign of $u$. We conclude that the forward curve can only attain the shapes [`normal`]{}, [`inverse`]{}, and [`humped`]{}; the same must be true for the yield curve by Theorem \[thm:fw\_yield\]. This result corresponds to [@keller-ressel2008yield Thm. 3.9] (see also [@keller-ressel2018correction Thm. 2.1]).\
To find the regions of the state space associated to the different shapes it suffices to analyze the initial and terminal sign of $l$ and $m$. First we find $$S_\text{init}(l) = S_\text{init}(m) = {\mathrm{sign}}(\theta - z).$$ For the terminal sign, we find $$\begin{aligned}
S_\text{term}(l) &= {\mathrm{sign}}\left(\lim_{x \to \infty}e^{\lambda x} l(x) \right) = {\mathrm{sign}}\left(\theta - \frac{\sigma^2 \kappa}{\lambda^2} - z\right) \\
S_\text{term}(m) &= {\mathrm{sign}}\left(\lim_{x \to \infty}x^2 m(x) \right) = {\mathrm{sign}}\left(\int_0^x l(x) dx \right) = {\mathrm{sign}}\left(\theta - \frac{3 \sigma^2 \kappa}{4 \lambda^2} - z\right).\end{aligned}$$ Thus we conclude that the forward curve $f(x;Z_t)$ is
- [`normal`]{}, if $Z_t \le \theta - \frac{\sigma^2 \kappa}{\lambda^2}$;
- [`humped`]{}, if $\theta - \frac{\sigma^2 \kappa}{\lambda^2} < Z_t < \theta$; and
- [`inverse`]{}, if $Z_t \ge \theta$.
For the yield curve $Y(x;Z_t)$, we conclude that it is
- [`normal`]{}, if $Z_t \le \theta - \frac{3\sigma^2 \kappa}{4\lambda^2}$;
- [`humped`]{}, if $\theta - \frac{3 \sigma^2 \kappa}{4\lambda^2} < Z_t < \theta$; and
- [`inverse`]{}, if $Z_t \ge \theta$.
These results are consistent with [@vasicek1977equilibrium p. 186f], (see also [@desmettre2018moderne Satz 2.53], [@keller-ressel2018correction Thms. 2.1, 2.3]).
Classification of term structure shapes in the two-dim. case
============================================================
The main result
---------------
Let $\bm{P}$ denote the full parameter space of the two-dimensional Vasicek model, i.e. $$\bm{P} = {\left\{\begin{pmatrix}\theta_1\\\theta_2\end{pmatrix} \in {\mathbb{R}}^2, \kappa_0 \in {\mathbb{R}}, \begin{pmatrix}\kappa_1\\\kappa_2\end{pmatrix} \in (0,\infty)^2, \begin{pmatrix}\sigma_1\\\sigma_2\end{pmatrix} \in [0,\infty)^2, \rho \in [-1,1], 0 < \lambda_1 < \lambda_2\right\}},$$ We start with several definitions related to the attainability of term structure shapes.
\[def:attainable\]
(a) A shape $\mathsf{S}$ of the forward curve is called **attainable**, if we can find a parameter vector $p \in \bm{P}$ and a state vector $z \in {\mathbb{R}}^2$, such that $x \mapsto f(x; z, p)$ has shape $\mathsf{S}$.
(b) The shape $\mathsf{S}$ of the forward curve is called **strictly attainable**, if we can find a parameter vector $p \in \bm{P}$, such that $x \mapsto f(x; Z_t, p)$ attains shape $\mathsf{S}$ with strictly positive probability for all $t > 0$.
(c) A shape $\mathsf{S}$ of the forward curve with $k$ local extrema is called **strongly attainable**, if for any $0 < r_1 < \dotsm < r_k$, we can find a parameter vector $p \in \bm{P}$ and a state vector $z \in {\mathbb{R}}^2$, such that $x \mapsto f(x; z, p)$ has shape $\mathsf{S}$, with its local extrema located at $r_1, \dotsc, r_k$.
The same terminology is applied to the yield curve $x \mapsto Y(x;z,p)$.
We remark that in (b) it makes no difference whether probabilities under the risk-neutral measure ${\mathbb{Q}}$ or probabilities under the statistical measure ${\mathbb{P}}$ are considered, as ${\mathbb{Q}}$ and ${\mathbb{P}}$ are equivalent. It also makes no difference whether ‘all $t > 0$’ or ‘some $t > 0$’ are considered, as in the Vasicek model also the laws of $Z_t$ and $Z_{t'}$ are equivalent for any $t,t' > 0$.
It turns out that stronger attainability results can be obtained, in the sense that not all parameters in $\bm{P}$, but only a subset, need to be varied in order to attain a given shape. To formulate these results, we write $\bm{P}'$ for $\bm{P}$ with the volatility parameters $(\sigma_1, \sigma_2, \rho)$ removed, and introduce the parameter space of covariance matrices $$\bm{\Sigma} := {\left\{\Sigma = \begin{pmatrix} \sigma_1^2 & \rho \sigma_1 \sigma_2 \\ \rho \sigma_1 \sigma_2 & \sigma_2^2 \end{pmatrix}, \sigma_1, \sigma_2 \in [0,\infty), \rho \in [-1,1]\right\}}.$$ Additional restrictions on $\bm{\Sigma}$ are denoted by $\bm{\Sigma}_{\rho <0}$, $\bm{\Sigma}_{\rho = 0}$, etc. We can now introduce the following stronger notion of attainability:
A shape $\mathsf{S}$ of the forward curve is called **$\Sigma$-attainable**, if for any parameter vector $p' \in \bm{P}'$, we can find a covariance matrix $\Sigma \in \bm{\Sigma}$ and a state vector $z \in {\mathbb{R}}^2$, such that $x \mapsto f(x; z, (p',\Sigma))$ has shape $\mathsf{S}$.\
The same terminology is applied to the yield curve $x \mapsto Y(x;z, (p',\Sigma))$.
Combining with Definition \[def:attainable\] we naturally obtain the notions of **strict and strong $\Sigma$-attainability**.\
Our third and final definition concerns the separation of scales in the two-dimensional Vasicek model. Recall that the mean-reversion speeds are ordered as $\lambda_1 < \lambda_2$. Thus, intuitively, $Z_t^1$ controls the long end of the term structure, while $Z_t^2$ controls the short end. The degree of separation between these two effects is captured by the following definition:
The two-dimensional Vasicek model is called
- **scale-separated**, if $2 \lambda_1 < \lambda_2$,
- **scale-proximal**, if $2 \lambda_1 > \lambda_2$, and
- **scale-critical**, if $2 \lambda_1 = \lambda_2$.
We can now formulate our main result on the classification of term structure shapes:
\[thm:main\] Consider the two-dimensional Vasicek model.
(a) In the scale-separated or scale-critical case, the following yield and forward curve shapes are attainable:
> [`normal`]{}, [`inverse`]{}, [`humped`]{}, [`dipped`]{}, [`HD`]{}, [`DH`]{}, [`HDH`]{};
no other shapes are attainable.
(b) In the scale-proximal case, the following yield and forward curve shapes are attainable with $\rho \ge 0$:
> [`normal`]{}, [`inverse`]{}, [`humped`]{}, [`dipped`]{}, [`HD`]{};
no other shapes are attainable with $\rho \ge 0$.
(c) In the scale-proximal case, the following yield and forward curve shapes are attainable with $\rho < 0$:
> [`normal`]{}, [`inverse`]{}, [`humped`]{}, [`dipped`]{}, [`HD`]{}, [`DH`]{}, [`HDH`]{}, [`DHD`]{}, [`HDHD`]{};
no other shapes are attainable with $\rho < 0$.
To attain the listed shapes, it suffices to vary only the correlation and volatility parameters and the state vector:
\[cor:strong\] In all cases of Theorem \[thm:main\], the given shapes are $\Sigma$-attainable. In cases (a) and (b) the shapes are also strongly $\Sigma$- and $\Sigma_{\rho=0}$-attainable. In case (c), all shapes except possibly [`DH`]{}, [`HDH`]{}, [`DHD`]{}, and [`HDHD`]{} are also strongly $\Sigma$- and $\Sigma_{\rho<0}$-attainable.
The second corollary concerns the *strict* attainability in the sense of Def. \[def:attainable\].
\[cor:strict\]
(a) The shapes of Theorem \[thm:main\](a) are strictly $\Sigma_{\rho > 0}$-, $\Sigma_{\rho = 0}$-, and $\Sigma_{\rho < 0}$-attainable.
(b) The shapes of Theorem \[thm:main\](b) are strictly $\Sigma_{\rho > 0}$- and $\Sigma_{\rho = 0}$-attainable
(c) The shapes of Theorem \[thm:main\](c), with possible exception of ${\texttt{DH}}$ and ${\texttt{DHD}}{}$, are strictly $\Sigma_{\rho < 0}$-attainable.
\[rem:sigma\_att\]
(i) In all except the strong attainability results, the sets $\bm{\Sigma}_{\rho = 0}$, etc. can be further restricted to the subsets of *regular* covariance matrices.\[item:regular\]
(ii) In case (c) it remains an open question whether ${\texttt{DH}}{}$ and ${\texttt{DHD}}{}$ are strictly attainable and whether [`DH`]{}, [`HDH`]{}, [`DHD`]{}, and [`HDHD`]{} are strongly attainable.
(iii) We emphasize that these are *theoretical* attainability results. It is for instance not clear whether the more complex shapes can be attained within realistic ranges of parameter values or whether the local extrema that are generated are pronounced enough to be of practical relevance. This is especially true for the cases where a strong attainability result (which allows us to control the locations of extrema) is lacking.
### Role of covariance and correlation
We can immediately make some interesting observations on the role of the correlation parameter:
- In the scale-separated case, the correlation parameter $\rho$ has no effect on the scope of attainable term structure shapes. In fact with $\rho = 0$ the same shapes can be attained as with $\rho > 0$ and $\rho < 0$.
- In comparison, the scope of attainable term structure shapes in the scale-proximal case shrinks for $\rho \ge 0$, but grows for $\rho < 0$. Intuitively, the effects of the long-range factor $Z^1$ and the short-range factor $Z^2$ interact in the scale-proximal case, with positive correlation leading to congruence and negative correlation leading to interference.
Finally, we give a heuristic argument, which supports the conclusion of Cor. \[cor:strong\] and \[cor:strict\], that the variation of the (co-)variance parameters and the state vector is sufficient to attain the listed shapes: For strong attainability of [`HDH`]{}, the most complex shape in case (a), four degrees of freedom are needed: Three for the local extrema and an additional degree of freedom to select between [`HDH`]{} and [`DHD`]{}. The parameter space $\bm{\Sigma}_{\rho = 0}$ has two degrees of freedom and the state space ${\mathbb{R}}^2$ also has two, matching the required four degrees. In case (c) the most complex shape, [`HDHD`]{}, needs five degrees of freedom. The parameter space $\bm{\Sigma}_{\rho < 0}$ provides three of them and the state space ${\mathbb{R}}^2$ provides two. Finally, one could argue that in case (b) the congruent interaction of the two state processes prohibits the full utilization of all five degrees of freedom.\
The proof of Theorem \[thm:main\] and its corollaries rests on the introduction of Descartes systems related to yield and forward curves in the two-dimensional Vasicek model. These Descartes systems are given in Section \[sec:Descartes\] below and allow to apply the results from the theory of total positivity, which were discussed in Section \[sec:tot\_pos\]. The actual proof of Theorem \[thm:main\] is then given in two parts: First, in Section \[sec:admissible\], we show necessity, i.e., that no term structure shapes outside of the lists given in Theorem \[thm:main\] can be attained. Then we show sufficiency, i.e., that all listed shapes are actually attainable. This more difficult part is done in Section \[sec:attainable\].
Descartes systems for the Vasicek model {#sec:Descartes}
---------------------------------------
We introduce several Descartes systems associated to the Vasicek model. As we will show, the derivatives of forward and yield curve, i.e. the functions $l(x)$ and $m(x)$ introduced in Lemma \[lem:curve\_diff\], can be written as D-polynomials in these systems. The next Lemma follows directly from Remark \[rem:Descartes\](\[item:exp\]) and from the ordering of exponents that is implied by the scale-separation properties:
\[lem:Descartes\_f\] The following families of functions are Descartes systems on $[0,\infty)$: $$\begin{aligned}
{\mathcal{D}}_\text{sep} &= (e^{-2 \lambda_2 x}, e^{-(\lambda_1 + \lambda_2)x}, e^{-\lambda_2 x}, e^{-2 \lambda_1 x}, e^{-\lambda_1 x}) \quad &&\text{if} \quad 2 \lambda_1 < \lambda_2\\
{\mathcal{D}}_\text{prox} &= (e^{-2 \lambda_2 x}, e^{-(\lambda_1 + \lambda_2)x}, e^{-2 \lambda_1 x}, e^{-\lambda_2 x}, e^{-\lambda_1 x}) \quad &&\text{if} \quad 2 \lambda_1 > \lambda_2\\
{\mathcal{D}}_\text{crit} &= (e^{-2 \lambda_2 x}, e^{-(\lambda_1 + \lambda_2)x}, e^{-\lambda_2 x}, e^{-\lambda_1 x}) \quad &&\text{if} \quad 2 \lambda_1 = \lambda_2\end{aligned}$$
Note that the only difference between ${\mathcal{D}}_\text{prox}$ and ${\mathcal{D}}_\text{sep}$ are the order of the third and the fourth element. Collapsing these cases yields the boundary case ${\mathcal{D}}_\text{crit}$.\
For the analysis of yield curve shapes a slightly different Descartes system is needed:
\[lem:Descartes\_g\] Set $$g_\alpha(x) = \frac{1}{x^2}\int_0^x y e^{-\alpha y} dy = \tfrac{1}{ \alpha^2 x^2} \left(e^{-\alpha x} - 1 + \alpha x e^{-\alpha x}\right).$$ The following families of functions are Descartes systems on $[0,\infty)$: $$\begin{aligned}
{\mathcal{E}}_\text{sep} &= \left(g_{2 \lambda_2}, g_{\lambda_1 + \lambda_2}, g_{\lambda_2}, g_{2 \lambda _1}, g_{\lambda_1}\right) \quad &&\text{if} \quad 2 \lambda_1 < \lambda_2\\
{\mathcal{E}}_\text{prox} &= \left(g_{2 \lambda_2}, g_{\lambda_1 + \lambda_2}, g_{2 \lambda _1}, g_{\lambda_2}, g_{\lambda_1}\right) \quad &&\text{if} \quad 2 \lambda_1 > \lambda_2\\
{\mathcal{E}}_\text{crit} &= \left(g_{2 \lambda_2}, g_{\lambda_1 + \lambda_2}, g_{\lambda_2}, g_{\lambda_1}\right) \quad &&\text{if} \quad 2 \lambda_1 = \lambda_2\end{aligned}$$
Note that $g_\alpha(x)$ can be written as $$\label{eq:g_rep}
g_\alpha(x) = \int_0^\infty K(x,y) e^{-\alpha y} dy, \quad \text{where} \quad K(x,y) = \frac{y}{x^2}{\mathbf{1}_{\left\{y \le x\right\}}}.$$ Essentially, Lemma \[lem:Descartes\_g\] follows from the total positivity of $K(x,y)$ (see Lemma \[lem:K\_tot\_pos\]); the details are given in Sec. \[app:Descartes\_E\].
Necessary conditions for attainability {#sec:admissible}
--------------------------------------
To derive necessary conditions for attainability of term structure shapes, we write $l$ and $m$ as D-polynomials in the Descartes systems introduced in Lemmas \[lem:Descartes\_f\] and \[lem:Descartes\_g\] and determine their coefficients. Specializing to the case $d=2$, we obtain $$\begin{aligned}
l(x) &= \begin{pmatrix} \kappa_1e^{-\lambda_1 x} \\\kappa_2e^{-\lambda_2 x} \end{pmatrix}^\top
\left\{ - \begin{pmatrix}\lambda_1 \theta_1\\\lambda_2 \theta_2 \end{pmatrix} - \begin{pmatrix}\sigma_1^2 & \rho \sigma_1 \sigma_2\\\rho \sigma_1 \sigma_2 & \sigma_2^2\end{pmatrix} \begin{pmatrix}\frac{\kappa_1}{\lambda_1} (e^{-\lambda_1 x} - 1)\\\frac{\kappa_2}{\lambda_2} (e^{-\lambda_2 x} - 1)\end{pmatrix} + \begin{pmatrix}z_1\\z_2 \end{pmatrix} \right\}.\end{aligned}$$ This can be expanded into $$\label{eq:l_poly}
l(x) = u_2 f_{2\lambda_2}(x) + c f_{\lambda_1 + \lambda_2}(x) + w_2 f_{\lambda_2}(x) + u_1 f_{2\lambda_1}(x) + w_1 f_{\lambda_1}(x),$$ with $f_\alpha(x) = e^{-\alpha x}$ and coefficients given, for $j \in {\left\{1,2\right\}}$, by $$\begin{aligned}
u_j &= \frac{\sigma_j^2 \kappa_j^2}{\lambda_j} \ge 0\\
w_j &= w_j(z_j) = \kappa_j \lambda_j \left(\theta_j - z_j\right) - \frac{\sigma_j^2 \kappa_j^2}{\lambda_j} - \rho \lambda_j \frac{\sigma_1\sigma_2\kappa_1\kappa_2}{\lambda_1\lambda_2}
\intertext{and}
c &= \rho (\lambda_1 + \lambda_2) \frac{\sigma_1\sigma_2\kappa_1\kappa_2}{\lambda_1\lambda_2}.\end{aligned}$$ Using the representation $m(x) = \frac{1}{x^2} \int_0^x y l(y) dy$ from Lemma \[lem:curve\_diff\], it is obvious that also holds for $m(x)$, with $f_\alpha$ replaced by $g_\alpha$. Thus, we have shown the following.
\[lem:coef\]The functions $l(x)$ and $m(x)$ are D-polynomials in the Descartes systems ${\mathcal{D}}$ and ${\mathcal{E}}$ respectively, with coefficients given by
- $(u_2, c, u_1, w_2, w_1)$ in the scale-proximal case,
- $(u_2, c, w_2, u_1, w_1)$ in the scale-separated case,
- $(u_2, c, w_2 + u_1, w_1)$ in the scale-critical case.
We can now use the variation-diminishing property of Descartes systems to derive restrictions on attainable forward and yield curve shapes.
\[thm:sise\_m\]If $\rho \ge 0$, then the sign sequence of $q \in {\left\{l,m\right\}}$, the derivatives of forward and yield curve, satisfies $$\begin{aligned}
&{\mathrm{sseq}}(q)\subseteq {\bm{[}{\textup{\texttt{+}}}w_2 w_1\bm{]}} \quad &&\text{(under scale-proximity)}\\
&{\mathrm{sseq}}(q)\subseteq {\bm{[}{\textup{\texttt{+}}}w_2 {\textup{\texttt{+}}}w_1\bm{]}} \quad &&\text{(under scale-separation)}\\
&{\mathrm{sseq}}(q)\subseteq {\bm{[}{\textup{\texttt{+}}}(u_1 + w_2) w_1\bm{]}} &&\quad \text{(under scale-criticality)}.\end{aligned}$$ If $\rho < 0$ then the sign sequence of $q \in {\left\{l,m\right\}}$ satsifies $$\begin{aligned}
&{\mathrm{sseq}}(q)\subseteq {\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}w_2 w_1\bm{]}} \quad &&\text{(under scale-proximity)}\\
&{\mathrm{sseq}}(q)\subseteq {\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}w_2 {\textup{\texttt{+}}}w_1\bm{]}} \quad &&\text{(under scale-separation)}\\
&{\mathrm{sseq}}(q)\subseteq {\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}(u_1 + w_2) w_1\bm{]}} &&\quad \text{(under scale-criticality)}.\end{aligned}$$
For forward curves this result can be strengthened by using additional information from the terminal sign of $m$.
\[cor:sise\_l\] In Theorem \[thm:sise\_m\] ‘$\subseteq$’ can be replaced by ‘${\overset{\scriptscriptstyle{T}}{\subseteq}}$’ whenever the sign sequence of $l$ is considered.
Theorem \[thm:sise\_m\] follows by applying Theorem \[thm:vardim\] to the coefficients given in Lemma \[lem:coef\]. In doing so, we take into account that $u_j$ has positive sign regardless of the choice of parameters, and apply the reductions of sign sequences described in Sec. 1.1(ii) to arrive at the expressions on the right hand sides.\
For the corollary, the obtained relations can be strengthened from $\subseteq$ to ${\overset{\scriptscriptstyle{T}}{\subseteq}}$ by analyzing the terminal sign of $l$. From we first obtain that $\lim_{x \to \infty} l(x) = 0$, which, however, yields no information on the terminal sign. Rather, the terminal sign of $l$ must be determined by the component with the slowest decay, which is $w_1 f_{\lambda_1}(x) = w_1 e^{-\lambda_1 x}$. Thus, the terminal sign of $l$ is equal to the sign of $w_1$, which is the last sign in all sequences of Lemma \[lem:coef\]. We conclude that ${\mathrm{sseq}}(l)$ is not just a subset, but rather a tail of all the sign sequences that were obtained on the right hand sides.[^2]
Using Theorem \[thm:sise\_m\] we obtain the first part of Therorem \[thm:main\].
Consider the case of the forward curve. The shape of the forward curve is determined by the sign sequence of $l$, and this sign sequence is controlled by the results of Corollary \[cor:sise\_l\]. Hence, restrictions on attainable term structure shapes can be obtained by iterating through all cases of Corollary \[cor:sise\_l\] and through the four possible sign combinations of $w_1$ and $w_2$. Note that we only need to consider the strict signs ${\textup{\texttt{+}}}$ and ${\textup{\texttt{-}}}$, because zeroes can be omitted from sign sequences and do not lead to additional shapes. Instead of listing all possible combinations, we discuss two exemplary cases:
- Suppose that $\rho \ge 0$, $w_1 > 0$ and $w_2 < 0$. In the scale-proximal case we obtain from Corollary \[cor:sise\_l\], that $${\mathrm{sseq}}(l){\overset{\scriptscriptstyle{T}}{\subseteq}}{\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}.$$ The possible tail sequences of ${\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}$ are ${\bm{[}{\textup{\texttt{+}}}\bm{]}}, {\bm{[}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}$ and ${\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}$ itself. These cases correspond to the shapes [`normal`]{}, [`humped`]{} and [`HD`]{}, and we conclude that no other forward curve shapes can be attainable under the given parameter restrictions. Switching to scale-separation, Corollary \[cor:sise\_l\] yields $${\mathrm{sseq}}(l){\overset{\scriptscriptstyle{T}}{\subseteq}}{\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}{\textup{\texttt{+}}}\bm{]}} \simeq {\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}},$$ and the same admissible shapes are obtained as in the scale-separated case.
- Now suppose that $\rho \ge 0$, $w_1 < 0$ and $w_2 < 0$. In the scale-proximal case Corollary \[cor:sise\_l\] yields $${\mathrm{sseq}}(l){\overset{\scriptscriptstyle{T}}{\subseteq}}{\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{-}}}\bm{]}} \simeq {\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}\bm{]}},$$ which leaves the shapes [`inverse`]{}, [`humped`]{} as potentially attainable shapes. In the scale-separated case we obtain $${\mathrm{sseq}}(l){\overset{\scriptscriptstyle{T}}{\subseteq}}{\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}\bm{]}},$$ which, in addition, leaves [`DH`]{} and [`HDH`]{} as potentially attainable.
Applying the same procedure to all other cases produces the lists given in the theorem, in the case of forward curves. The scale-critical case can be treated like the scale-proximal case. For yield curves, we apply Theorem \[thm:sise\_m\] to $m$ in the same manner. Despite the weaker constraint $\subseteq$ instead of ${\overset{\scriptscriptstyle{T}}{\subseteq}}$, it turns out (after iterating through all cases) that the same lists of shapes are obtained.
Sufficient conditions for attainability {#sec:attainable}
---------------------------------------
To complete the proof of Theorem \[thm:main\], we need to show sufficiency, i.e., that all listed shapes are actually attainable. Before going into details, we describe the general strategy of the proof: Let a shape $\mathsf{S}$ of the forward curve with $k$ local extrema be given. Choosing a suitable Descartes-subsystem ${\mathcal{D}}'$ of ${\mathcal{D}}$ with $k+1$ elements, we can apply Theorem \[thm:extremal\] and find a D-polynomial $f$ in ${\mathcal{D}}'$, such that $f$ has a sign sequence with $k$ sign changes, which corresponds to the shape $\mathsf{S}$. Padding the list of coefficients with zeroes, we can write $f$ as a D-polynomial in the full system ${\mathcal{D}}$, i.e. as $$f(x) = a_{2\lambda_2} f_{2\lambda_2}(x) + a_{\lambda_1 + \lambda_2} f_{\lambda_1 + \lambda_2}(x) + a_{\lambda_2} f_{\lambda_2}(x) + a_{2 \lambda_1} f_{2\lambda_1}(x) + a_1 f_{\lambda_1}(x),$$ where we have labeled the coefficients $a$ consistently with the basis functions of ${\mathcal{D}}$. Comparing coefficients with , we can conclude that the shape $\mathsf{S}$ is attainable[^3] in the Vasicek-model, if we can show that the system of equations
\[eq:key\] $$\begin{aligned}
\frac{\sigma_1^2 \kappa_1^2}{\lambda_1} &= a_{2\lambda_1} \label{eq:key_sigma1}\\
\frac{\sigma_2^2 \kappa_2^2}{\lambda_2} &= a_{2\lambda_2} \label{eq:key_sigma2}\\
\rho (\lambda_1 + \lambda_2) \frac{\sigma_1\sigma_2\kappa_1\kappa_2}{\lambda_1\lambda_2} &= a_{\lambda_1 + \lambda_2} \label{eq:rho}\\
\kappa_1 \lambda_1 \left(\theta_1 -z_1\right) - \frac{\sigma_1^2 \kappa_1^2}{\lambda_1} - \rho \lambda_1 \frac{\sigma_1\sigma_2\kappa_1\kappa_2}{\lambda_1\lambda_2} &= a_{\lambda_1}\label{eq:key_z1}\\
\kappa_2 \lambda_2 \left(\theta_2 -z_2\right) - \frac{\sigma_2^2 \kappa_2^2}{\lambda_2} - \rho \lambda_2 \frac{\sigma_1\sigma_2\kappa_1\kappa_2}{\lambda_1\lambda_2} &= a_{\lambda_2}\label{eq:key_z2}\end{aligned}$$
has a solution $(\sigma_1, \sigma_2, \rho, z_1, z_2) \in [0,\infty)^2 \times [-1,1] \times {\mathbb{R}}^2$. The argument for yield curves is analogous, using the appropriate Descartes system ${\mathcal{E}}$ from Lemma \[lem:Descartes\_g\].\
Having reduced the attainability problem to the equation system , we need to discuss its solvability: Clearly, whenever – can be solved for $(\sigma_1, \sigma_2, \rho)$, then also and can be solved for $(z_1, z_2)$. Moreover, the solvability of and only depends on the signs of $a_{2\lambda_1}$ and $a_{2\lambda_2}$. It is therefore only for which solvability is nontrivial, due to the restriction $\rho \in [-1,1]$. These elementary observations are summarized in the following Lemma:
\[lem:key\] Consider the system of equations given in
(a) If $a_{2\lambda_1} < 0$ or $a_{2\lambda_2} < 0$, then has no solution.
(b) If $a_{2\lambda_1} = 0$ and $a_{2\lambda_2} \ge 0$, or if $a_{2\lambda_1} \ge 0$ and $a_{2\lambda_2} = 0$ then has a solution. In this solution $\sigma_1 = \rho = 0$ or $\sigma_2 = \rho = 0$ or both.
(c) If $a_{2\lambda_1} > 0$ and $a_{2\lambda_2} > 0$, then has a solution if and only if $$\rho := \frac{\sqrt{\lambda_1 \lambda_2}}{\lambda_1 + \lambda_2 } \frac{a_{\lambda_1 + \lambda_2}}{\sqrt{a_{2\lambda_1}a_{2\lambda_1}}} \quad \text{is in $[-1,1]$.}$$
To complete the proof of Theorem \[thm:main\] we apply the strategy outlined above on a case-by-case basis to the different shapes:
We partition the proof according to the number $k$ of local extrema of the term structure curve; later we also need to distinguish between the cases (a), (b) and (c) given in Theorem \[thm:main\].
(i) For $k=0$ we use the system ${\mathcal{D}}_1 = (f_{\lambda_1})$. We set $a^\pm_{\lambda_1} = \pm 1$ and all other coefficients to zero. This yields the D-polynomials $f_\pm(x) = \pm f_{\lambda_1}(x) = \pm e^{-\lambda_1 x}$ with sign sequences ${\bm{[}{\textup{\texttt{+}}}\bm{]}}$ and ${\bm{[}{\textup{\texttt{-}}}\bm{]}}$. Setting $z_2 = \sigma_1 = \sigma_2 = \rho = 0$ the system can be solved for $z_1$ in both cases. We conclude that the shapes [`normal`]{} and [`inverse`]{} are attainable.
(ii) For $k=1$ we use the system ${\mathcal{D}}_2 = (f_{\lambda_2}, f_{\lambda_1})$. By Theorem \[thm:extremal\] we can find two extremal D-polynomials $f_+, f_-$ with coefficients $(a^\pm_{\lambda_2}, a^\pm_{\lambda_1})$ and sign sequences ${\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}\bm{]}}$ and ${\bm{[}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}$. Setting $\sigma_1 = \sigma_2 = \rho = 0$ the system can be solved for $(z_2, z_1)$ in both cases. We conclude that the shapes [`dipped`]{} and [`humped`]{} are attainable.
(iii) For $k=2$ we use the system ${\mathcal{D}}_3 = (f_{2 \lambda_2}, f_{\lambda_2}, f_{\lambda_1})$. By Theorem \[thm:extremal\] we can find two extremal D-polynomials $f_+, f_-$ with coefficients $(a^\pm_{2 \lambda_2}, a^\pm_{\lambda_2}, a^\pm_{\lambda_1})$ and sign sequences ${\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}$ and ${\bm{[}{\textup{\texttt{-}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}\bm{]}}$. Setting $\sigma_1 = \rho = 0$ the system can be solved for $(\sigma_2, z_2, z_1)$ in the case of $f_+$. In the case of $f_-$ the system cannot be solved, because $a^-_{2 \lambda_2} < 0$. We conclude that the shape [`HD`]{} is attainable.
At this point we have already covered all attainable shapes in the scale-proximal case with $\rho \ge 0$, i.e., part (b) of the theorem. Next we complete part (a), i.e., the scale-separated case:
(i) For $k=2$ we can alternatively use the system ${\mathcal{D}}_{3,sep} = (f_{\lambda_2}, f_{2 \lambda_1}, f_{\lambda_1})$, which is a subsystem of ${\mathcal{D}}_\text{sep}$.[^4] By Theorem \[thm:extremal\] we can find two extremal D-polynomials $f_+, f_-$ with coefficients $(a^\pm_{\lambda_2}, a^\pm_{2 \lambda_1}, a^\pm_{\lambda_1})$ and sign sequences ${\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}$ and ${\bm{[}{\textup{\texttt{-}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}\bm{]}}$. Setting $\sigma_2 = \rho = 0$ the system can be solved for $(z_2, \sigma_1, z_1)$ in the case of $f_-$. In the case of $f_+$ the system cannot be solved, because $a^-_{2 \lambda_1} < 0$. We conclude that the shape [`DH`]{} is attainable.
(ii) For $k=3$, we use the system ${\mathcal{D}}_\text{4,sep} = (f_{2 \lambda_2} f_{\lambda_2}, f_{2 \lambda_1}, f_{\lambda_1})$, which is a subsystem of ${\mathcal{D}}_\text{sep}$. By Theorem \[thm:extremal\] we can find two extremal D-polynomials $f_+, f_-$ with coefficients $(a^\pm_{2 \lambda_2}, a^\pm_{\lambda_2}, a^\pm_{2 \lambda_1}, a^\pm_{\lambda_1})$ and sign sequences ${\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}\bm{]}}$ and ${\bm{[}{\textup{\texttt{-}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}$. Setting $\rho = 0$ the system can be solved for $(\sigma_2, z_2, \sigma_1, z_1)$ in the case of $f_+$. In the case of $f_-$ the system cannot be solved, because $a^-_{2 \lambda_1} < 0$ and $a^-_{2 \lambda_2} < 0$. We conclude that [`HDH`]{} is attainable.
At this point we have also covered all attainable shapes in the scale-separated case (with arbitrary $\rho$) and thus part (a) is complete. The most difficult case is part (c), i.e., the scale-proximal case with $\rho < 0$. Here, Theorem \[thm:extremal\] is not sufficient to find suitable D-polynomials $f_\pm$ and we have to use the more specialized result Lemma \[lem:special\] instead.
(i) For $k=3$ we use the system ${\mathcal{D}}_{4,prox} = (f_{2 \lambda_2}, f_{\lambda_1 + \lambda_2}, f_{2 \lambda_1}, f_{\lambda_2})$, which is a subsystem of ${\mathcal{D}}_\text{prox}$. By Lemma \[lem:special\] we can find two sets of real numbers $0 < r_1^+ < r_2^+ < r_3^+$ and $0 = r_1^0 < r_2^0 < r_3^0$ as well as D-polynomials $f_+$ and $f_0$ with the following properties:
- The zeroes of $f_+$ and $f_0$ are located exactly at the points $r_1^+, r_2^+, r_3^+$ and $r_1^0 = 0, r_2^0, r_3^0$;
- the sign sequence of $f_+ $ is ${\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}\bm{]}}$ and the sign sequence of $f_0$ is ${\bm{[}{\textup{\texttt{-}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}\bm{]}}$;
- the coefficients of both $f_+$ and $f_0$ have sign sequence ${\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}\bm{]}}$.
Moreover, the coefficients (of both $f_+$ and $f_0$) satisfy $$\left| \frac{a_{\lambda_1 + \lambda_2}}{\sqrt{a_{2 \lambda_1}a_{2 \lambda_2}}} \right| < 2;$$ see . Thus, applying the geometric-arithmetic-mean inequality, we obtain $$\label{eq:rho_solvable}
|\rho| \le \frac{\sqrt{\lambda_1 \lambda_2}}{\lambda_1 + \lambda_2} \left| \frac{a_{\lambda_1 + \lambda_2}}{\sqrt{a_{2 \lambda_1}a_{2 \lambda_2}}} \right| < 1.$$ By Lemma \[lem:key\], this implies that the system of equations is solvable. We conclude that the shapes [`HDH`]{} and [`DH`]{} are attainable.
(ii) For $k=4$ we use the full system ${\mathcal{D}}_\text{prox}$. As in the previous case, we can apply Lemma \[lem:special\] to find two D-polynomials $f_+$ and $f_0$ with prescribed zeroes and with sign sequences ${\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}$ and ${\bm{[}{\textup{\texttt{-}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}$ respectively. The first zero of $f_0$ is located at the boundary point $r_1^0 = 0$. Moreover, the coefficients of both $f_+$ and $f_0$ have sign sequence ${\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}$ and inequality holds. Thus, Lemma \[lem:key\] implies that the system of equations is solvable and we conclude that the shapes [`HDHD`]{} and [`DHD`]{} are attainable.
Having completed part (c), the last case, the proof of Theorem \[thm:main\] is finished.
Strict and strong attainability
-------------------------------
We now discuss how the stronger conclusions of Corollary \[cor:strong\] and \[cor:strict\] can be obtained from the proof of Theorem \[thm:main\] that was given above. First, observe that in all steps (i) - (vii) of the proof, we have shown that the system of equations could be solved by choosing suitable covariance parameters $(\sigma_1, \sigma_2, \rho)$ and state vectors $(z_1, z_2)$ and that it was not necessary to modify any of the remaining parameters in $\bm{P}'$. This shows that attainability can be strengthened to *$\Sigma$-attainability* in all cases.\
Next, observe that that in steps (i) - (v) of the proof we have used Theorem \[thm:extremal\] to find a D-polynomial $f_+$ or $f_-$, which, after solving , equates to $l$, the derivative of the forward curve. Theorem \[thm:extremal\] allows us to predetermine all zeroes $r_1 < \dotsm < r_k$ of $f_\pm$, and hence the locations of the extrema of the forward curve. The same is true for $m$, the derivative of the yield curve. This shows that in cases (i) -(v) we obtain *strong* $\Sigma$-attainability. In addition, note that it was sufficient to choose $\rho = 0$ in all cases (i) - (v). Thus, we even get *strong $\Sigma_{\rho = 0}$-attainability*. This completes the arguments needed for Cor. \[cor:strong\].\
The contents of Cor \[cor:strict\] follow from a perturbation argument. Consider for instance case (iii) in the proof of Thm. \[thm:main\]: There, we have shown that we can find parameters $\sigma_1 = \rho = 0$, $\sigma_2 > 0$ and a state vector $(z_1,z_2) \in {\mathbb{R}}$, which produces the sign sequence ${\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}$ corresponding to shape ${\texttt{HD}}{}$. Suppose that a perturbation $$\sigma_1^\epsilon = \epsilon, \quad \rho^\epsilon = \pm \epsilon \quad \text{and} \quad z_1^\epsilon = z_1 \pm \epsilon, \quad z_2^\epsilon = z_2 \pm \epsilon$$ with $\epsilon$ in some small set $[0,\delta)$ still produces the same sign sequence ${\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}$ and shape ${\texttt{HD}}{}$. Then, we may conclude
- that the shape ${\texttt{HD}}{}$ is strictly $\Sigma$-attainable, as $(Z_t^1, Z_t^2)$ visits any small neighborhood of $(z_1,z_2)$ with strictly positive probability;
- that ${\texttt{HD}}{}$ is also $\Sigma_{\rho > 0}$- and $\Sigma_{\rho < 0}$-attainable, as we have relaxed the condition $\rho = 0$ to $\rho^\epsilon = \pm \epsilon$; and
- that it is sufficient to consider regular matrices $\Sigma$, as we have relaxed the condition $\sigma_1 = 0$ to $\sigma_1^\epsilon = \epsilon$.
The necessary perturbation Lemma is given below. Applying the same argument to each of the cases (i) - (v) in the proof yields part (a) and (b) of Cor. \[cor:strict\]. For cases (vi) and (vii) note that the Lemma can only be applied to the D-polynomials $f_+$, but not to $f_0$, which has a zero at the boundary of $[0,\infty)$ and is not an extremal D-polynomial. This yields part (c) of Cor. \[cor:strict\] and Rem. \[rem:sigma\_att\](i).
Let $\phi = \sum_{i=1}^n a_i \phi_i$ be a non-vanishing D-polynomial in a Descartes system ${\mathcal{D}}= (\phi_1, \dotsc, \phi_n)$ on a subinterval $X \subset {\mathbb{R}}$ which satisfies $${\mathrm{sseq}}\left(\sum_{i=1}^n a_i \phi_i\right) \simeq {\bm{[}a_1 \dotsc a_n\bm{]}}$$ and has no zeroes on the boundary of $X$. Then there exist $(b_i)_{i=1\dotsc n} \in {\left\{-1,+1\right\}}$ and $\delta > 0$, such that $${\mathrm{sseq}}\left(\sum_{i=1}^n a_i^\epsilon \phi_i\right) \simeq {\mathrm{sseq}}\left(\sum_{i=1}^n a_i \phi_i\right)$$ for all $\epsilon \in [0,\delta)$ and with $a_i^\epsilon = a_i + \epsilon b_i$.
First, we show that the sequence $(b_i)$ can be chosen such that $${\bm{[}a_1^\epsilon \dotsc a_n^\epsilon\bm{]}} \simeq {\bm{[}a_1 \dotsc a_n\bm{]}}.$$ To this end define $b_1, \dotsc, b_n$ as follows: $$\begin{aligned}
a_i > 0 \quad &\Longrightarrow &\quad &b_i := +1\\
a_i < 0 \quad &\Longrightarrow &\quad &b_i := -1\\
a_i = 0 \quad &\Longrightarrow &\quad &b_i := \begin{cases}+1 \quad &\text{if the block of zeroes containing $a_i$ borders}\\&\;\text{on at least one $a_j > 0$,} \\-1 \quad &\text{else.}\end{cases}\end{aligned}$$ It is easy to see that the number and direction of strong sign changes in $(a_1^\epsilon, \dotsc, a_n^\epsilon)$ is the same as in $(a_1, \dotsc, a_n)$ for all $\epsilon \ge 0$, i.e., we have $${\bm{[}a_1^\epsilon \dotsc a_n^\epsilon\bm{]}} \simeq {\bm{[}a_1 \dotsc a_n\bm{]}}, \quad \forall\,\epsilon \ge 0.$$ Set $\phi^\epsilon = \sum_{i=1}^n a_i^\epsilon \phi_i$. Then by Theorem \[thm:vardim\] $$\label{eq:sise_sub}
{\mathrm{sseq}}(\phi^\epsilon) \subseteq {\bm{[}a_1^\epsilon \dotsc a_n^\epsilon\bm{]}} \simeq {\bm{[}a_1 \dotsc a_n\bm{]}} \simeq {\mathrm{sseq}}(\phi),$$ for all $\epsilon \ge 0$, and we have shown that $\phi^\epsilon$ cannot have *more* sign changes than $\phi$. It remains to show that equivalence holds for small enough $\epsilon$. Let $k$ be number of strong sign changes of $\phi$. Clearly, we can find $r_0, \dotsc, r_k$ such that the sequence $\phi(r_i)_{i=0, \dotsc, k}$ is of alternating signs. Each interval $(r_i, r_{i+1})$ must contain exactly one zero of $\phi$. Set $$\delta := \frac{\min_{i=0, \dotsc, k} |\phi(r_i)|}{\sum_{j=1}^n \max_{i=0, \dotsc, k} |\phi_j(r_i)|}.$$ Then, $\delta > 0$ and for all $\epsilon \in [0,\delta)$ $$\begin{aligned}
\left|1 - \frac{\phi^\epsilon(r_i)}{\phi(r_i)} \right| &= \left|\frac{\phi(r_i) - \phi^\epsilon(r_i)}{\phi(r_i)} \right| \le \frac{1}{|\phi(r_i)|} \left| \sum_{j=1}^n \epsilon b_j \phi_j(r_i) \right| \le \\ &\le \epsilon \frac{\sum_{j=1}^n |\phi_j(r_i)|}{|\phi(r_i)|} < 1.\end{aligned}$$ This shows that the sequence $\phi^\epsilon(r_i)_{i=0, \dotsc, k}$ has the same alternating signs as $\phi(r_i)_{i=0, \dotsc, k}$ and hence that $\phi^\epsilon$ has at least the same number of zeroes as $\phi$, for all $\epsilon \in [0,\delta)$. Together with , this completes the proof.
Discussion and Outlook
======================
We have shown that the theory of total positivity, in particular the notion of Descartes systems, can be applied to the problem of classifying term structure shapes in the one- and two-dimensional Vasicek model. In principle, this analysis can be extended to Vasicek models with three and more factors, presumably at the expense of even more cases of ‘scale-separation’ and correlation links between the factors that need to be distinguished. In the two-dimensional case, a natural next step that builds on the results given above, is a ‘state-space analysis’ of term structure shapes, i.e., to determine and classify the regions of the state space in which a particular shape of the term structure is produced. Finally, it would be interesting to see, whether the theory of total positivity can also be applied to non-Gaussian affine (or even non-affine) interest rate models, such as those of [@dai2000specification].
Additional results on Descartes systems
=======================================
Let a family $(\phi_1, \dotsc, \phi_k)$ of functions on $X \subseteq {\mathbb{R}}$ be given. We set $\bm{x} = (x_1, \dotsc, x_k) \in X^k$ and $$\Delta_k(X) := {\left\{\bm{x} \in X^k: x_1 < \dotsc < x_k\right\}}.$$ From [@borwein1995polynomials] we adopt the compact notation $$\label{eq:det}
D\begin{pmatrix}\phi_1, \dotsc ,\phi_k\\x_1, \dotsc, x_k\end{pmatrix} := \det \begin{pmatrix} \phi_{1}(x_1) & \phi_{2}(x_1) & \dotsc & \phi_{k}(x_1)\\ \vdots & \vdots && \vdots\\ \phi_{1}(x_k) & \phi_{2}(x_k) & \dotsc & \phi_{k}(x_k)\end{pmatrix}.$$ An important special case is the Vandermonde determinant, which for any real $(\gamma_i)_{i=1, \dotsc, k}$ evaluates as $$\label{eq:Vandermonde}
D\begin{pmatrix}1, x ,x^2, \dotsc, x^{k-1}\\ \gamma_1, \dotsc, \gamma_k\end{pmatrix} = \prod_{j=1}^{k-1} (\gamma_j - \gamma_{j-1}),$$ see e.g. [@hogben2013handbook Ch. 22.4]. For sufficiently differentiable functions $\phi_1, \dotsc, \phi_k$, we also introduce the Wronskian determinant (or simply Wronskian) $$\label{eq:wronskian}
W(\phi_1, \dotsc, \phi_k)(x) = \det \begin{pmatrix} 1 & \phi_1(x) & \phi_1'(x) & \dotsm & \phi_1^{(k)}(x) \\ 1 & \phi_2(x) & \phi_2'(x) & \dotsm & \phi_2^{(k)}(x) \\ \vdots & \vdots & \vdots & & \vdots\\ 1 & \phi_k(x) & \phi_k'(x) & \dotsm & \phi_k^{(k)}(x) \\\end{pmatrix}.$$ In [@karlin1968total Ch. 2, §2] relations between the two determinants in and as well as intermediate notions of ‘derived determinants’ are discussed.
D-polynomials with prescribed zeroes {#app:interpolation}
------------------------------------
Let a Descartes system ${\mathcal{D}}= (\phi_1, \dotsc, \phi_n)$ on $X$ and a set of prescribed zeroes $\bm{r} =(r_1, \dotsc, r_{n-1}) \in \Delta_{n-1}(X)$ be given. We show that the D-polynomial $$\label{eq:interpol}
\phi(x;\bm{r}) = D\begin{pmatrix} \phi_1, &\phi_2, &\dots, &\phi_n\\ x, &r_1, &\dotsc ,&r_{n-1} \end{pmatrix}$$ is the desired interpolation polynomial of Theorem \[thm:extremal\]. First, observe that the determinant vanishes whenever $x = r_i$ for any $i = 1, \dotsc, n-1$, and hence $\phi(x,\bm{r})$ possesses a zero at each $r_i$, which shows (a). Second, as ${\mathcal{D}}$ is a Descartes system, the determinant must be non-zero at all other points in $X$. The point $x$ crossing an interior zero $r_i$ changes the order of two columns in the determinant and hence flips the sign of $\phi(x;\bm{r})$, which shows (b). Claim (c) now follows from Theorem \[thm:vardim\] – because $\phi(x;\bm{r})$ has $n-1$ sign changes, equivalence must hold in .
To prepare for additional results, we remark that the coefficients $a_1, \dotsc, a_n$ of the interpolation D-polynomial $\phi(x;\bm{r})$ can be determined directly from . Expanding the determinant in the first column yields $$\phi(x,\bm{r}) = \sum_{i=1}^n a_i(\bm{r}) \phi_i(x),$$ where $$\label{eq:coefficients}
a_i(\bm{r}) = (-1)^{1+i} D\begin{pmatrix}\phi_1, \dotsc, \phi_{i-1}, \; \phi_{i+1}, \dotsc, \phi_n \\ r_1, \dotsc, r_{n-1}\end{pmatrix}.$$ Because $(\phi_1, \dotsc, \phi_n)$ is a Descartes system, the determinant on the right hand side is strictly positive. This shows that the coefficients of $\phi(x,\bm{r})$ must have alternating signs, starting with ${\textup{\texttt{+}}}$.
The Descartes property of ${\mathcal{E}}$ {#app:Descartes_E}
-----------------------------------------
To show that ${\mathcal{E}}_\text{sep}, {\mathcal{E}}_\text{prox}$ and ${\mathcal{E}}_\text{crit}$ are Descartes systems on $[0,\infty)$, it is sufficient to show that $$D\begin{pmatrix}g_{\alpha_k}, \dotsc, g_{\alpha_1}\\ x_1, \dotsc, x_k\end{pmatrix} > 0$$ for any $\alpha_k > \dotsc > \alpha_1 \ge 0$ and $\bm{x} = (x_1, \dotsc, x_k) \in \Delta_{k-1}[0,\infty)$. Our starting point is the representation of $g_\alpha$ as an integral of $f_\alpha(x) = e^{-\alpha x}$ with respect to the totally positive kernel $$K(x,y) = \frac{y}{x^2}{\mathbf{1}_{\left\{x \le y\right\}}}.$$ From [@karlin1968total Ch. 3, Eq. (1.11)ff] and with $K_i := K(x,y_i)$ we obtain that $$D\begin{pmatrix}K_1, \dotsc, K_k\\ x_1, \dotsc, x_k\end{pmatrix} = \frac{y_1 \dotsm y_{k}}{x_1^2 \dotsm x_k^2} {\mathbf{1}_{\left\{0 \le y_1 \le x_1 \le y_2 \le x_2 \dotsm \le x_k\right\}}}$$ for any $\bm{x}, \bm{y} \in \Delta_k(0,\infty)$. Combining this with the composition formula [@karlin1968total Ch. 3, Eq. (1.2)] we obtain $$\begin{gathered}
D\begin{pmatrix}g_{\alpha_k}, \dotsc, g_{\alpha_1}\\ x_1, \dotsc, x_k\end{pmatrix} = \\ = \int_0^{x_1} \int_{x_1}^{x_2} \dotsm \int_{x_{k-1}}^{x_k} \frac{y_1 \dotsm y_{k}}{x_1^2 \dotsm x_k^2} D\begin{pmatrix}f_{\alpha_k}, \dotsc, f_{\alpha_1}\\ x_1, \dotsc, x_k\end{pmatrix} \,dy_1 dy_2 \dotsm dy_k.\end{gathered}$$ Because $(f_{\alpha_k}, \dotsc, f_{\alpha_1})$ is a Descartes system, the integrand is strictly positive. Moreover, the domain of integration has strictly positive measure. We conclude that the left hand side is strictly positive for any $\bm{x} = (x_1, \dotsc, x_k) \in \Delta_k(0,\infty)$, and hence that ${\mathcal{E}}$ is a Descartes system on $(0,\infty)$. It remains to extend this property to the left-closed interval $[0,\infty)$. By [@karlin1968total Ch. 2, Thm. 2.3] it is sufficient to show the Wronskian $W(g_{\alpha_k}, \dotsc, g_{\alpha_1})(0)$ is strictly positive for any $k$. We first calculate the Taylor expansion $$g_\alpha(x) = \frac{1}{x^2} \int_0^x y e^{-\alpha y} dy = \sum_{k=0}^\infty \frac{(-\alpha)^k}{(k +2)} \frac{x^k}{k!},$$ which follows from the Taylor expansion of the exponential function. We conclude that the $k$-th derivative of $g_\alpha$ at zero is given by $$\label{eq:g_der}
g_\alpha^{(k)}(0) = \frac{(-\alpha)^k}{k+2}.$$ Thus we obtain that the Wronskian at zero is given by $$\label{eq:g_Wronskian}
W(g_{\alpha_k}, \dotsc, g_{\alpha_1})(0) = (k+1)!^{-k} D\begin{pmatrix}1, x, x^2 \dotsc, x^{k-1}\\ -\alpha_k, \dotsc, -\alpha_{1}\end{pmatrix}.$$ The latter is a Vandermonde determinant, which evaluates to $\prod_{j=1}^{k-1}(\alpha_{j+1} - \alpha_j)$ and is therefore strictly positive.
Further results on interpolation polynomials {#app:interpolation2}
--------------------------------------------
\[lem:coef\_ratio\] Let $\alpha_n > \dotsc > \alpha_1 \ge 0$ be given and consider the Descartes system $${\mathcal{D}}= (f_{\alpha_n}, \dotsc, f_{\alpha_1}), \quad \text{where} \quad f_\alpha(x) = e^{-\alpha x}.$$ Let $f(x,\bm{r}) = \sum_{i=1}^n a_i(\bm{r}) f_{\alpha_i}(x)$ be the interpolation D-polynomial of $\bm{r} \in \Delta_{n-1}$. Then its coefficients satisfy, for any $i, j \in {\left\{1, \dotsc, n\right\}}$, $$\label{eq:coef_asymp}
\lim_{\bm{r} \to \bm{0}} \frac{a_i(\bm{r})}{a_j(\bm{r})} = (-1)^{(i-j)}\frac{\alpha_{i+1} - \alpha_{i-1}}{(\alpha_{i+1} - \alpha_{i})(\alpha_{i} - \alpha_{i-1})} \frac{(\alpha_{j+1} - \alpha_{j})(\alpha_{j} - \alpha_{j-1})} {\alpha_{j+1} - \alpha_{j-1}}$$ with the convention that terms containing $\alpha_0$ or $\alpha_{n+1}$ shall be omitted. The same result holds for ${\mathcal{D}}$ replaced with $${\mathcal{E}}= (g_{\alpha_n}, \dotsc, g_{\alpha_1}), \quad \text{where} \quad g_\alpha(x) = \frac{1}{x^2}\int_0^x y e^{-\alpha y} dy.$$
Combining with [@karlin1968total Ch. 6, Eqs.(1.3), (1.4)], we obtain $$\begin{aligned}
\lim_{\bm{r} \to \bm{0}} \frac{a_i(\bm{r})}{a_j(\bm{r})} &= (-1)^{(i-j)} \lim_{\bm{r} \to \bm{0}} \frac{
D\begin{pmatrix}f_n, \dotsc, f_{i+1}, \; f_{i-1}, \dotsc, f_1 \\ r_1, \dotsc, r_{n-1}\end{pmatrix}
}{
D\begin{pmatrix}f_n, \dotsc, f_{j+1}, \; f_{j-1}, \dotsc, f_1 \\ r_1, \dotsc, r_{n-1}\end{pmatrix}
} = \\
&= (-1)^{(i-j)} \frac{
W\left(f_n, \dotsc, f_{i+1}, \; f_{i-1}, \dotsc, f_1 \right)(0)
}{
W\left(f_n, \dotsc, f_{j+1}, \; f_{j-1}, \dotsc, f_1 \right)(0).
} \end{aligned}$$ As $f_\alpha(x) = e^{-\alpha x}$, the Wronskian determinants become Vandermonde determinants, i.e. $$\begin{aligned}
&W\left(f_n, \dotsc, f_{i+1}, \; f_{i-1}, \dotsc, f_1 \right)(0) = D\begin{pmatrix}1, x ,x^2, \dotsc, x^{n-1}\\ -\alpha_n, \dotsc, -\alpha_{i+1}, \; -\alpha_{i-1}, \dotsc, -\alpha_1 \end{pmatrix} = \\ &\qquad \frac{\alpha_{i+1} - \alpha_{i-1}}{(\alpha_{i+1} - \alpha_{i})(\alpha_{i} - \alpha_{i-1})} \prod_{k=1}^{n-1} (\alpha_k - \alpha_{k-1}),\end{aligned}$$ and similarly for $j$. Evaluating their ratio, is obtained. For $g$ the proof is analogous, using to evaluate the Wronskians.
\[lem:special\] Consider the Descartes system ${\mathcal{D}}_{4,\text{prox}} = (f_{2\lambda_2}, f_{\lambda_2 + \lambda_1}, f_{2\lambda_1}, f_{\lambda_2})$ on $[0,\infty)$. There exists a neighborhood $N$ of $\bm{0}$ in $[0,\infty)^3$, such that the coefficients of the interpolation D-polynomial $$f(x;\bm{r}) = a_{2\lambda_2}(\bm{r})f_{2\lambda_2}(x) + a_{\lambda_1 + \lambda_2}(\bm{r})f_{\lambda_1 + \lambda_2}(x) + a_{2\lambda_1}(\bm{r})f_{2\lambda_1}(x) + a_{\lambda_2}(\bm{r})f_{\lambda_2}(x)$$ satisfy $$\label{eq:coef_ineq}
\left| \frac{a_{\lambda_1 + \lambda_2}(\bm{r})}{\sqrt{a_{2 \lambda_1}(\bm{r}) a_{2 \lambda_2}(\bm{r})}}\right| < 2 \qquad \forall \,\bm{r} \in N \cap \Delta_3[0,\infty).$$ The same holds for ${\mathcal{D}}, {\mathcal{E}}_{4,\text{prox}}$ and ${\mathcal{E}}$.
Applying Lemma \[lem:coef\_ratio\] to ${\mathcal{D}}_{4,\text{prox}}$, we calculate the limits $$\begin{aligned}
\lim_{\bm{r} \to \bm{0}} \left|\frac{a_{\lambda_1 + \lambda_2}(\bm{r})}{a_{\lambda_1}(\bm{r})}\right| &= 2 \left(2 - \frac{\lambda_2}{\lambda_1}\right)\\
\lim_{\bm{r} \to \bm{0}} \left|\frac{a_{\lambda_1 + \lambda_2}(\bm{r})}{a_{\lambda_2}(\bm{r})}\right| &= 2\end{aligned}$$ Taking square roots and multiplying, we obtain $$\lim_{\bm{r} \to \bm{0}}\left| \frac{a_{\lambda_1 + \lambda_2}(\bm{r})}{\sqrt{a_{2 \lambda_1}(\bm{r}) a_{2 \lambda_2}(\bm{r})}}\right| = 2 \sqrt{2 - \frac{\lambda_2}{\lambda_1}}.$$ As $\lambda_1 < \lambda_2 < 2 \lambda_1$, the right hand side is contained strictly between $0$ and $2$. Due to , the coefficients of the interpolation D-polynomial $f(x;\bm{r})$ depend continuously on $\bm{r} \in \Delta_3[0,\infty)$, and follows. The proof for ${\mathcal{D}}, {\mathcal{E}}_{4,\text{prox}}$ and ${\mathcal{E}}$ is analogous.
[^1]: A sign change from ${\textup{\texttt{+}}}$ to $0$ and back to ${\textup{\texttt{+}}}$, for example, is not considered a strong sign change, whereas a sign change from ${\textup{\texttt{+}}}$ to $0$ and then to ${\textup{\texttt{-}}}$ is.
[^2]: Note that the same approach does not work for $m$ due to the different asymptotic behaviour as $x$ tends to infinity.
[^3]: In fact even $\Sigma$-attainable.
[^4]: But not a Descartes subsystem of ${\mathcal{D}}_\text{prox}$!
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
Let $k$ be an algebraic closure of finite fields with odd characteristic $p$ and a smooth projective scheme ${\bf X}/W(k)$. Let ${\bf X}^0$ be its generic fiber and $X$ the closed fiber. For ${\bf X}^0$ a curve Faltings conjectured that semistable Higgs bundles of slope zero over ${\bf X}^0_{{{\mathbb C}}_p}$ correspond to genuine representations of the algebraic fundamental group of ${\bf
X}^0_{{{\mathbb C}}_p}$ in his $p$-adic Simpson correspondence [@Fa3]. This paper intends to study the conjecture in the characteristic $p$ setting. Among other results, we show that isomorphism classes of rank two semistable Higgs bundles with trivial chern classes over $X$ are associated to isomorphism classes of two dimensional genuine representations of $\pi_1({\bf X}^0)$ and the image of the association contains all irreducible crystalline representations. We introduce intermediate notions *strongly semistable Higgs bundles* and *quasi-periodic Higgs bundles* between semistable Higgs bundles and representations of algebraic fundamental groups. We show that quasi-periodic Higgs bundles give rise to genuine representations and strongly Higgs semistable are equivalent to quasi-periodic. We conjecture that a Higgs semistable bundle is indeed strongly Higgs semistable.
address:
- 'Institut für Mathematik, Universität Mainz, Mainz, 55099, Germany'
- 'School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, China'
- 'Institut für Mathematik, Universität Mainz, Mainz, 55099, Germany'
author:
- Guitang Lan
- Mao Sheng
- Kang Zuo
title: 'Semistable Higgs bundles and representations of algebraic fundamental groups: Positive characteristic case'
---
\[section\] \[thm\][Theorem]{} \[thm\][Lemma]{} \[thm\][Corollary]{} \[thm\][Proposition]{} \[thm\][Addendum]{} \[thm\][Variant]{} \[thm\][Lemma and Definition]{} \[thm\][Construction]{} \[thm\][Notations]{} \[thm\][Question]{} \[thm\][Problem]{} \[thm\][Remark]{} \[thm\][Remarks]{} \[thm\][Definition]{} \[thm\][Claim]{} \[thm\][Assumption]{} \[thm\][Assumptions]{} \[thm\][Properties]{} \[thm\][Example]{} \[thm\][Conjecture]{} \[thm\][Proposition and Definition]{}
[^1]
Introduction
============
N. Hitchin [@Hitchin] introduced rank two stable Higgs bundles over a compact Riemann surface $X$ and showed that they correspond naturally to irreducible representations of the fundamental group $\pi_1(X)$ by solving a Yang-Mills equation, which generalizes the earlier works by Donaldson, Uhlenbeck-Yau for polystable vector bundles. Later C. Simpson obtained the full correspondence for any polystable Higgs bundles over arbitrary dimensional complex projective manifolds. In [@Fa3] G. Faltings established the correspondence between Higgs bundles and generalized representations of $\pi_1(X)$ over $p$-adic fields. He conjectured that semistable Higgs bundles under his functor shall correspond to usual $p$-adic representations of $\pi_1(X)$. In this paper we intend to study Faltings’s conjecture in the characteristic $p$ setting.\
Let $k$ be the algebraic closure of finite fields of odd characteristic $p$. Let ${\bf X}/W(k)$ be a smooth projective $W:=W(k)$-scheme and $X/k$ its closed fiber. In this paper, if not specified, a Higgs bundle over $X$ means a system of Hodge bundles $$(E=\oplus_{i+j=n}E^{i,j},\theta=\oplus_{i+j=n}\theta^{i,j}),$$ where $E$ is a vector bundle over $X$, $\theta$ is a morphism of ${{\mathcal O}}_X$-modules satisfying $$\theta^{i,j}: E^{i,j}\to
E^{i-1,j+1}\otimes \Omega_{X}, \quad \quad \theta\wedge \theta=0.$$ For simplicity, we assume throughout that $n\leq p-2$. Fix an ample divisor ${\bf H}\subset {\bf X}$ over $W$. The Higgs semistability of $(E,\theta)$ is referred to the $\mu$-semistability with respect to $H\subset X$, the reduction of ${\bf H}$.
There is a functor from the category of quasi-periodic Higgs-de Rham sequences of type $(e,f)$ to the category of crystalline representations of $\pi_1({\bf X'}^0)$ into ${\mathrm{GL}}({{\mathbb F}}_{p^f})$, where ${\bf X'}^0$ is the generic fiber of ${\bf X'}:={\bf
X}\times_{W}{{\mathcal O}}_K$ for a totally ramified extension $\mathrm{Frac}(W)\subset K$ with ramification index $e$. There is also a functor in the opposite direction. These two functors are equivalence of categories in the case $e=0$ and quasi-inverse to each other.
Consequently, we obtain the following
Under the above functors, there is one to one correspondence between the isomorphism classes of irreducible crystalline ${{\mathbb F}}_{p^f}$-representations of $\pi_1({\bf X}^0)$ and the isomorphism classes of periodic Higgs stable bundles of period $f$.
The leading term of a quasi-periodic Higgs-de Rham sequence is a quasi-periodic Higgs bundle. We show that
A quasi-periodic Higgs bundle is strongly Higgs semistable with trivial chern classes. Conversely, A strongly Higgs semistable bundle with trivial chern classes is quasi-periodic.
Strongly semistable vector bundles are strongly semistable Higgs bundles with trivial Higgs fields. As a semistable bundle need not be strongly semistable, the notion of strongly semistability should be replaced by the strongly Higgs semistability. The next result supports our viewpoint.
A rank two semistable Higgs bundle is strongly Higgs semistable.
We would like to make the following
A semistable Higgs bundle is strongly Higgs semistable.
As an application of the above results, we obtain the following
Any isomorphism class of rank two semistable Higgs bundles with trivial chern classes over $X$ is associated to an isomorphism class of crystalline representations of $\pi_1({\bf X}^0)$ into ${\mathrm{GL}}_2(k)$. The image of the association contains all irreducible crystalline representations of $\pi_1({\bf X}^0)$ into ${\mathrm{GL}}_2(k)$.
The plan of our paper is arranged as follows: in Section 2 we introduce the notions *strongly Higgs semistable bundles* which generalizes the notion of strongly semistable vector bundles in the paper [@LS] of Lange-Stuhler and *quasi-periodic Higgs bundles* which generalizes the notion of periodic Higgs subbundles introduced in [@SZ]. We show that a strongly Higgs semistable with trivial chern classes is equivalent to a quasi-periodic Higgs bundle, and a rank two semistable Higgs bundle is strongly Higgs semistable. We conjecture that semistable Higgs bundles of arbitrary rank are strongly Higgs semistable. In Section 3 we show in Theorem \[correspondence in the type (0,f) case\] that there is a one to one correspondence between the strict $p$-torsion category $\mathcal{MF}^{\nabla}_{[0,n],f}({\bf X}/W)$ of Faltings with endomorphism ${{\mathbb F}}_{p^f}$ and the category of periodic Higgs-de Rham sequences of type $(0,f)$. In Section 4, we extend the construction for periodic Higgs bundles to quasi-periodic Higgs bundles. In Section 5, we give some complements and applications of the above theory.\
[**Acknowledgements:**]{} Arthur Ogus has recently pointed to us that the inverse Cartier transform in the paper [@OV] for the nilpotent Higgs bundles coincides with the construction in [@LSZ]. Christopher Deninger has drawn our attention to the work [@Langer], and Adrian Langer has helped us understanding [@Langer]. We thank them heartily.
Strongly semistable Higgs bundles
=================================
In this paper, a vector bundle over $X$ means a torsion free coherent sheaf of ${{\mathcal O}}_X$-module. A Higgs-de Rham sequence over $X$ is a sequence of form $$\xymatrix{
& (H_0,\nabla_0)\ar[dr]^{Gr_{Fil_0}} && (H_1,\nabla_1)\ar[dr]^{Gr_{Fil_1}} \\
(E_0,\theta_0) \ar[ur]^{C_0^{-1}} & & (E_1,\theta_1) \ar[ur]^{C_0^{-1}}&&\ldots }$$ In the sequence, $C_0^{-1}$ is the inverse Cartier transform constructed in [@OV] (see also [@LSZ]). A. Ogus remarked that the exponential twisting of [@LSZ] is equivalent to the more general construction in [@OV] and the equivalence is implicitly implied by Remark 2.10 loc. cit.. $Fil_i$ is a decreasing filtration on $H_i$ with the property $Fil_i^0=H_i$ and $Fil_i^{n+1}=0$ and such that $\nabla_i$ obeys the Griffiths transversality with respect to it.
A Higgs bundle $(E,\theta)$ is called strongly Higgs semistable if it appears in the leading term of a Higgs-de Rham sequence whose Higgs terms $(E_i,\theta_i)$s are all Higgs semistable.
Recall that [@LS] a vector bundle $E$ is said to be strongly semistable if $F_X^{*n}E$ is semistable for all $n\in {{\mathbb N}}$. Clearly, a strongly semistable vector bundle $E$ is strongly Higgs semistable: one takes simply the Higgs-de Rham sequence as $$\xymatrix{
& (F_X^*E,\nabla_{can})\ar[dr]^{Gr_{Fil_{tr}}} && (F_X^{*2}E,\nabla_{can})\ar[dr]^{Gr_{Fil_{tr}}} \\
(E_0,0) \ar[ur]^{C_0^{-1}} & & (F_X^*E,0) \ar[ur]^{C_0^{-1}}&&\ldots }$$ where $\nabla_{can}$ is the canonical connection in the theorem of Cartier descent and $Fil_{tr}$ is the trivial filtration.
A Higgs bundle $(E,\theta)$ is called periodic if it appears in the leading term of a periodic Higgs-de Rham sequence, that is, there exists a natural number $f$ such that there is an isomorphism of Higgs bundles $$(E_{f},\theta_f)\cong (E_0,\theta_0),$$ which via $C_0^{-1}$ induces inductively a filtered isomorphism of de Rham bundles $$(H_{f+i},\nabla_{f+i},Fil_{f+i})\cong
(H_{i},\nabla_{i},Fil_{i}),$$ and hence also an isomorphism of Higgs bundles for all $i\in {{\mathbb N}}$, $$(E_{f+i},\theta_{f+i})\cong (E_i,\theta_i).$$
The minimal number $f\geq 1$ is called the period of the sequence. One understands a periodic Higgs-de Rham sequence of period $f$ through the following diagram: $$\xymatrix{
& (H_0,\nabla_0)\ar[dr]^{Gr_{Fil_0}} && (H_{f-1},\nabla_{f-1})\ar[dr]^{Gr_{Fil_{f-1}}} \\
(E_0,\theta_0) \ar[ur]^{C_0^{-1}} & & \cdots \ar[ur]^{C_0^{-1}}&& (E_f,\theta_f)\ar@/^2pc/[llll]^{\cong}
}$$ In general, we make the following
A Higgs bundle $(E,\theta)$ is called quasi-periodic if it appears in the leading term of a quasi-periodic Higgs-de Rham sequence, i.e., it becomes periodic after a nonnegative integer $e\geq 0$.
We add a simple lemma which follows directly from the construction of $C_0^{-1}$ via the exponential function [@LSZ].
\[degree formula\] Let $(E,\theta)$ be a nilpotent Higgs bundle (not necessary a system of Hodge bundles) with exponent $\leq p-1$. It holds that $\det
C_0^{-1}(E,\theta)=F_{X}^*\det E$. Consequently, $$\deg
C_0^{-1}(E,\theta)=p\deg E.$$
It follows from the fact that in the determinant, the exponential twisting appeared in the construction of $C_0^{-1}(E,\theta)$ is simply the identity.
\[quasiperiodic equivalent to strongly semistable\] A quasi-periodic Higgs bundle is strongly Higgs semistable with trivial chern classes. Conversely, a strongly Higgs semistable bundle with trivial chern classes is quasi-periodic.
One observes that, in a Higgs-de Rham sequence, $c_l(E_{i+1})=p^lc_l(E_{i}), i\geq 0$. This forces the chern classes of a quasi-periodic Higgs bundle to be trivial. By Lemma \[degree formula\], a degree $\lambda$ Higgs subbundle (not necessarily subsystem of Hodge bundles) in $(E_i,\theta_i)$ gives rise to a degree $p\lambda$ Higgs subbundle in $(E_{i+1},\theta_{i+1})$. This implies that, in a Higgs-de Rham sequence of a quasi-periodic Higgs bundle, each Higgs term $(E_i,\theta_i)$ contains no Higgs subbundle of positive degree. So $(E_i,\theta_i)$ is Higgs semistable. Thus we have shown the first statement.\
Assume $X$ has a model over a finite field $k'\subset k$. Let $M_{r,ss}(X)$ be the moduli space of $S$-equivalence classes of rank $r$ semistable Higgs bundles with trivial chern classes over $X$. After A. Langer [@Langer] and C. Simpson [@Si], it is a projective variety over $k'$. For a strongly Higgs semistable bundle $(E,\theta)$ over $X$ with trivial chern classes, we consider the set of $S$-isomorphism classes $\{[(E_i,\theta_i)], i\in {{\mathbb N}}_0\}$, where $(E_i,\theta_i)$s are all Higgs terms in a Higgs-de Rham sequence for $(E,\theta)$. Note that the operators $C_0^{-1}$ and $Gr_{Fil_i}$ do not change the definition field of objects. Thus, if the leading term $(E_0,\theta_0)=(E,\theta)$ is defined over a finite field $k''\supset k'$, all terms in a Higgs-de Rham sequence are defined over $k''$. This implies that the above sequence is a sequence of $k''$-rational points in $M_{r,ss}(X)$ and hence finite. So we find two integers $e$ and $f$ such that $[(E_{e},\theta_e)]=[(E_{e+f},\theta_{e+f})]$. If $(E_e,\theta_e)$ is Higgs stable, then there is a $k''$-isomorphism of Higgs bundles $(E_{e},\theta_e)\cong (E_{e+f},\theta_{e+f})$. If it is only Higgs semistable, we obtain only a $k''$-isomorphism between their gradings. But we do find a $k'''$-isomorphism of Higgs bundles after a certain finite field extension $k''\subset k'''$: there exits a finite field extension $k'''$ of $k''$ such that $(E_e,\theta_e)$ admits a Jordan-Hölder (abbreviated as JH) filtration defined over $k'''$. The operator $Gr_{Fil_e}\circ C_0^{-1}$ transports this JH filtration into a JH filtration on $(E_{e+1},\theta_{e+1})$ defined over the same field $k'''$. Then this holds for any Higgs term $(E_i,\theta_i), i\geq e$. Without loss of generality, we assume that there are only two stable components in the gradings. Then the isomorphism classes of extensions over two stable Higgs bundles are described by a projective space over a finite field. Since there are finitely many $S$-equivalence classes in $\{(E_i,\theta_i),i\geq e\}$ and over each $S$-equivalence class there are only finite many $k'''$-isomorphism classes, there exists a $k'''$-isomorphism $(E_{e},\theta_{e})\cong
(E_{e+f},\theta_{e+f})$ after possibly choosing another $e,f$. It determines via $C_0^{-1}$ an isomorphism of flat bundles between $(H_{e},\nabla_e)$ and $(H_{e+f},\nabla_{e+f})$. This isomorphism defines a filtration $Fil'_{e+f}$ on $H_{e+f}$ from the filtration $Fil_e$ on $H_e$, which may differs from the original one. Put $$(E'_{e+f+1},\theta'_{e+f+1})=Gr_{Fil'_{e+f}}(H_{e+f},\nabla_{e+f}).$$ One has then a tautological isomorphism between $(E_{e+1},\theta_{e+1})$ and $(E'_{e+f+1},\theta'_{e+f+1})$. Continuing the construction, we show that a strongly semistable Higgs bundle with trivial chern classes can be putted into the leading term of a quasi-periodic Higgs-de Rham sequence, hence quasi-periodic. This shows the converse statement.
\[rank two semistable implies strongly semistable\] A rank two semistable Higgs bundle is strongly Higgs semistable.
Let $(E,\theta)$ be a rank two semistable Higgs bundle over $X/k$. Note first that, for the reason of rank, $\theta^2=0$. Hence the operator $C_0^{-1}$ applies. Denote $(H,\nabla)$ for $C_0^{-1}(E,\theta)$, and $HN$ the Harder-Narasimhan filtration on $H$. We need to show that the graded Higgs bundle $Gr_{HN}(H,\nabla)$ is semistable. If $H$ is semistable, there is nothing to prove: in this case, the $HN$ is trivial and hence the induced Higgs field is zero, and $Gr_{HN^\cdot}(H,\nabla)=(H,0)$ is Higgs semistable. Otherwise, the HN filtration is of form $$0\to L_1\to H\to L_2\to 0.$$
$L_1\subset H$ is not $\nabla$-invariant.
We can assume that $\theta\neq 0$. Otherwise, by the Cartier descent, it follows that $L_1\cong F_X^*G_1$ for a rank one sheaf $G_1\subset E$ whose degree is positive, which contradicts with the semistability of $E$. Write $E=E^{1,0}\oplus E^{0,1}$ and $\theta:
E^{1,0}\to E^{0,1}\otimes \Omega_X$ is nonzero. By the local construction of $C_0^{-1}$, the $p$-curvature of $\nabla$ is nilpotent and nonzero. As $L_1$ is of rank one, it follows that the $p$-curvature of $\nabla|_{L_1}$ is zero. Again by the construction of $C_0^{-1}$, $\nabla$ preserves the rank one subsheaf $L'_1:=C_0^{-1}(E^{0,1},0)$ and the restriction $\nabla|_{L'_1}$ has also the $p$-curvature zero property. Let $C\subset X$ be a generic curve. Then the nonzeroness of $\theta$ implies that $E^{0,1}|_C$ has negative degree. So is $L'_1|_{C}$. As $L_1$ has positive degree, they are not the same rank one subsheaf of $H$. Therefore, over a nonempty open subset $U\subset C$, one has $H=L_1\oplus
L'_1$. It contradicts the nonzeroness of the $p$-curvature of $\nabla$.
Then it follows that $$\theta'=Gr_{HN}\nabla: L_1\to L_2\otimes \Omega_X$$ is nonzero. Let $L\subset Gr_{HN}H=L_1\oplus L_2$ be a Higgs sub line bundle. As $\theta'|_{L}=0$, the composite $$L\hookrightarrow L_1\oplus L_2\twoheadrightarrow L_1$$ is zero. Hence the natural map $L\to L_2$ is nonzero and it follows that $$\deg L\leq \deg L_2<0.$$ In this case, $Gr_{HN}(H,\nabla)$ is Higgs stable.
We would like to make the following
A semistable Higgs bundle is strongly Higgs semistable.
A Higgs correspondence
======================
In this section we aim to establish a Higgs correspondence between the category of Higgs-de Rham sequences of periodic Higgs bundles over $X/k$ and the (modified) strict $p$-torsion category $\mathcal{MF}^{\nabla}_{[0,n]}({\bf X}/W), n\leq p-2$ (abbreviated as $\mathcal{MF}$) introduced by Faltings [@Fa1]. Here strict means that each object in the category is annihilated by $p$.\
We introduce first the category $\mathcal{MF}^{\nabla}_{[0,n],f}({\bf X}/W)$, a modification of the Faltings category $\mathcal{MF}^{\nabla}_{[0,n]}({\bf X}/W)$. For each $f\in {{\mathbb N}}$, let ${{\mathbb F}}_{p^f}$ be the unique extension of ${{\mathbb F}}_p$ in $k$ of degree $f$. An object in $\mathcal{MF}^{\nabla}_{[0,n],f}({\bf X}/W)$ (abbreviated as $\mathcal{MF}_{f}$) is a five tuple $(H,\nabla,Fil,\Phi,\iota)$, where $(H,\nabla,Fil,\Phi)$ is object in $\mathcal{MF}^{\nabla}_{[0,n]}({\bf X}/W)$ and $$\iota:
{{\mathbb F}}_{p^f}\hookrightarrow {{\rm End}}_{\mathcal{MF}}(H,\nabla,Fil,\Phi)$$ is an embedding of ${{\mathbb F}}_p$-algebras. A morphism is a morphism in $\mathcal{MF}^{\nabla}_{[0,n]}({\bf X}/W)$ respecting the endomorphism structure. Clearly, the category $\mathcal{MF}^{\nabla}_{[0,n],f}({\bf X}/W)$ for $f=1$ is just the original $\mathcal{MF}^{\nabla}_{[0,n]}({\bf X}/W)$. On the Higgs side, we define the category $\mathcal{HB}_{n,(0,f)}(X/k)$ (abbreviated as $\mathcal{HB}_{(0,f)}$) of the periodic Higgs-de Rham sequences of type $(0,f)$ as follows: an object is a tuple $(E,\theta,Fil_0,\cdots,Fil_{f-1},\phi)$ where $(E,\theta)$ is a Higgs bundle on $X/k$, $Fil_i, 0\leq i\leq f-1$ is a decreasing filtration on $C_0^{-1}(E_i,\theta_i)$ satisfying $Fil_i^0=C_0^{-1}(E_i,\theta_i), Fil_i^{n+1}=0$ and the Griffiths transversality such that $Gr_{Fil_{i}}(H_{i},\nabla_{i})$ is torsion free with $(E_0,\theta_0)=(E,\theta)$ and $(E_i,\theta_i):=Gr_{Fil_{i-1}}(H_{i-1},\nabla_{i-1})$ inductively defined, and $\phi$ is an isomorphism of Higgs bundles $$Gr_{Fil_{f-1}}\circ C_0^{-1}(E_{r-1},\theta_{r-1})\cong (E,\theta).$$ The information of such a tuple is encoded in the following diagram: $$\xymatrix{
& (H_0,\nabla_0)\ar[dr]^{Gr_{Fil_0}} && (H_{f-1},\nabla_{f-1})\ar[dr]^{Gr_{Fil_{f-1}}} \\
(E_0,\theta_0) \ar[ur]^{C_0^{-1}} & & \cdots \ar[ur]^{C_0^{-1}}&& (E_f,\theta_f)\ar@/^2pc/[llll]^{\stackrel{\phi}{\cong} } }$$ Note that $(E,\theta)$ of a tuple in the category is indeed periodic. A morphism between two objects is a morphism of Higgs bundles respecting the additional structures. As an illustration, we explain a morphism in the category $\mathcal{HB}_{(0,1)}$ in detail: let $(E_i,\theta_i,Fil_i,\phi_i), i=1,2$ be two objects and $$f: (E_1,\theta_1,Fil_1,\phi_1)\to (E_2,\theta_2,Fil_2,\phi_2)$$ a morphism. By the functoriality of $C_0^{-1}$, the morphism $f$ of Higgs bundles induces a morphism of flat bundles: $$C_0^{-1}(f): C_0^{-1}(E_1,\theta_1)\rightarrow C_0^{-1}(E_2,\theta_2).$$ It is required to be compatible with the filtrations, and the induced morphism of Higgs bundles is required to be compatible with $\phi$s, that is, there is a commutative diagram $$\label{eq1}
\begin{CD}
Gr_{Fil_1}C_0^{-1}(E_1,\theta_1)@>\phi_1>>(E_1,\theta_1)\\
@VGrC_0^{-1}(f)VV@ VVfV\\
Gr_{Fil_2}C_0^{-1}(E_2,\theta_2)@>\phi_2>>(E_2,\theta_2).
\end{CD}$$
\[correspondence in the type (0,f) case\] There is a one to one correspondence between the category $
\mathcal{MF}^{\nabla}_{[0,n],f}({\bf X}/W)$ and the category $\mathcal{HB}_{n,(0,f)}(X/k)$.
To show the theorem, we choose and fix a small affine covering $\{{\bf U}_i\}$ of ${\bf X}$, together with an absolute Frobenius lifting $F_{{\bf U}_i}$ on each ${\bf U}_i$. By modulo $p$, the covering induces an affine covering $\{U_i\}$ for $X$. We show first a special case of the theorem.
\[correspondence in the type (0,1) case\] There is a one to one correspondence between the Faltings category $\mathcal{MF}^{\nabla}_{[0,n]}({\bf X}/W)$ and the category $\mathcal{HB}_{n,(0,1)}(X/k)$.
Let $(H,\nabla,Fil,\Phi)$ be an object in $\mathcal{MF}$. Put $(E,\theta):=Gr_{Fil}(H,\nabla)$. The following lemma gives a functor $\mathcal{GR}$ from the category $\mathcal{MF}$ to the category $\mathcal{HB}_{(0,1)}$.
\[from Faltings to Higgs the fixed point case\] There is a filtration $Fil_{\exp}$ on $C_0^{-1}(E,\theta)$ together with an isomorphism of Higgs bundles $$\phi_{\exp}: Gr_{Fil_{exp}}(C_0^{-1}(E,\theta))\cong (E,\theta),$$ which is induced by the Hodge filtration $Fil$ and the relative Frobenius $\Phi$.
By Proposition 5 [@LSZ], we showed that the relative Frobenius induces a global isomorphism of flat bundles $$\tilde \Phi: C_0^{-1}(E,\theta)\cong (H,\nabla).$$ So we define $Fil_{exp}$ on $C_0^{-1}(E,\theta)$ to be the inverse image of $Fil$ on $H$ by $\tilde \Phi$. It induces tautologically an isomorphism of Higgs bundles $$\phi_{\exp}=Gr(\tilde \Phi): Gr_{Fil_{exp}}(C_0^{-1}(E,\theta))\cong
(E,\theta).$$
Next, we show that the functor $C_0^{-1}$ induces a functor in the opposite direction. Given an object $(E,\theta,Fil,\phi)\in
\mathcal{HB}_{(0,1)}$, it is clear to define the triple $$(H,\nabla,Fil)=(C_0^{-1}(E,\theta),Fil).$$ What remains is to produce a relative Frobenius $\Phi$ from the $\phi$. Following Faltings [@Fa1] Ch. II. d), it suffices to give for each pair $({\bf U}_i,F_{{\bf U}_i})$ an ${{\mathcal O}}_{U_i}$-morphism $$\Phi_{({\bf
U}_i,F_{{\bf U}_i})}: F_{U_i}^*Gr_{Fil}H|_{U_i}\to H|_{U_i}$$ satisfying
1. strong $p$-divisibility, that is, $\Phi_{({\bf U}_i,F_{{\bf
U}_i})}$ is an isomorphism,
2. horizontal property,
3. over each $U_i\cap U_j$, $\Phi_{({\bf U}_i,F_{{\bf
U}_i})}$ and $\Phi_{({\bf U}_j,F_{{\bf U}_j})}$ are related via the Taylor formula.
Recall [@LSZ] that over each $U_i$ we have the identification (chart) $$\alpha_i:=\alpha_{({\bf U}_i,F_{{\bf U_i}})}:
(F_{U_i}^{*}E|_{U_i},d+\frac{dF_{{\bf
U_i}}}{p}F_{U_i}^*\theta|_{U_i})\cong C_0^{-1}(E,\theta)|_{U_i}.$$ We define $\Phi_{({\bf U}_i,F_{{\bf U}_i})}$ to be the composite $$F_{U_i}^*Gr_{Fil}H|_{U_i}\stackrel{F_{U_i}^*\phi}{\longrightarrow}F_{U_i}^*E|_{U_i}\stackrel{\alpha_i}{\longrightarrow}
C_0^{-1}(E,\theta)|_{U_i}=H|_{U_i}.$$ By construction, $\Phi_{({\bf U}_i,F_{{\bf U}_i})}$ is strongly $p$-divisible. By Proposition 5 loc. cit., the transition function between $\alpha_{i}$ and $\alpha_j$ is given by the Taylor formula. It follows that $\Phi_{({\bf U}_i,F_{{\bf U}_i})}$ and $\Phi_{({\bf
U}_j,F_{{\bf U}_j})}$ are interrelated by the Taylor formula.
\[horizontal property\] Each $\Phi_{({\bf U}_i,F_{{\bf U}_i})}$ is horizontal with respect to $\nabla$.
Put $\tilde H=Gr_{Fil}H$, $\theta'=Gr_{Fil}\nabla$, $\Phi_i=\Phi_{({\bf U}_i,F_{{\bf U}_i})}$ and $F_0$ the absolute Frobenius over $U_i$. Following Faltings [@Fa1] Ch. II. d), it is to show the following commutative diagram $$\CD
F_0^*\tilde H|_{U_i} @>\Phi_i>> H|_{U_i} \\
@V F_{{\bf U}_i}^*\nabla VV @V \nabla VV \\
F_0^{*}\tilde H|_{U_i}\otimes \Omega_{U_i}@>\Phi_i\otimes id>> H|_{U_i}\otimes
\Omega_{U_i}.
\endCD$$ Here $F_{{\bf U}_i}^*\nabla$ is just the composite of $$F_0^*\tilde{H}|_{U_i}\stackrel{F_0^*\theta'}{\longrightarrow}F_0^*\tilde{H}|_{U_i}\otimes
F_0^*\Omega_{U_i}\stackrel{id\otimes\frac{dF_{{\bf
U}_i}}{p}}{\longrightarrow} F_0^*\tilde{H}|_{U_i}\otimes
\Omega_{U_i}.$$ Via the identification $\alpha_i$, it is reduced to show the following diagram commutes: $$\CD
F_0^*\tilde H|_{U_i} @>F_0^{*}\phi>> F_0^{*}E|_{U_i} \\
@V F_{{\bf U}_i}^*\nabla VV @V \frac{dF_{{\bf U}_i}}{p}F_0^{*}\theta VV \\
F_0^{*}\tilde H|_{U_i}\otimes \Omega_{U_i}@>F_0^{*}\phi\otimes id>> F_0^{*}E|_{U_i}\otimes
\Omega_{U_i}.
\endCD$$ As $\phi$ is a morphism of Higgs bundles, one has the following commutative diagram: $$\begin{CD}
\tilde{H}|_{U_ i} @>\phi >> E|_{U_i}\\
@V\theta' VV @VV\theta V\\
\tilde{H}|_{U_i}\otimes\Omega_{U_i}@>\phi\otimes id>> E|_{U_i}\otimes
\Omega_{U_i}.
\end{CD}$$ The pull-back via $F_0^*$ of the above diagram yields the next commutative diagram $$\xymatrix{
F_0^*\tilde{H}|_{U_i}\ar[d]_{F_0^*\theta'} \ar[r]^{F_0^*\phi}
& F_0^*E|_{U_i} \ar[d]_{F_0^*\theta} \ar@/^/[ddr]^{\frac{dF_{{\bf U}_i}}{p}F_0^{*}\theta } \\
F_0^*\tilde{H}|_{U_i}\otimes F_0^*\Omega_{U_i}\ar[r]^{F_0^*\phi\otimes id} \ar@/_/[drr]_{F_0^*\phi\otimes \frac{dF_{{\bf U}_i}}{p}}
& F_0^*E|_{U_i}\otimes F_0^*\Omega_{U_i} \ar@{>}[dr]|-{ id\otimes\frac{dF_{{\bf U}_i}}{p}} \\
& & F_0^*E|_{U_i}\otimes \Omega_{U_i}. }$$ The commutativity of the second diagram follows now from that of the last diagram.
The above lemma provides us with the functor $\mathcal{C}_0^{-1}$ in the opposite direction. Now we can prove Proposition \[correspondence in the type (0,1) case\].
The equivalence of categories follows by providing natural isomorphisms of functors: $$\mathcal{GR}\circ \mathcal{C}_0^{-1}\cong Id, \quad
\mathcal{C}_0^{-1}\circ \mathcal{GR}\cong Id.$$ We define first a natural isomorphism ${{\mathcal A}}$ from $\mathcal{C}_0^{-1}\circ \mathcal{GR}$ to $Id$: for $(H,\nabla,Fil,\Phi)\in \mathcal{MF}$, put $$(E,\theta,
Fil,\phi)=\mathcal{GR}(H,\nabla,Fil,\Phi),\quad (H',\nabla',Fil',
\Phi')=\mathcal{C}_0^{-1}(E,\theta,Fil,\phi).$$ Then one verifies that the map $$\tilde \Phi: (H',\nabla')=C_0^{-1}\circ
Gr_{Fil}(H,\nabla)\cong (H,\nabla)$$ gives an isomorphism from $(H',\nabla',Fil',\Phi')$ to $(H,\nabla,Fil,\Phi)$ in the category $\mathcal{MF}$. We call it ${{\mathcal A}}(H,\nabla,Fil,\Phi)$. It is straightforward to verify that ${{\mathcal A}}$ is indeed a transformation. Conversely, a natural isomorphism $\mathcal{B}$ from $\mathcal{GR}\circ\mathcal{C}_0^{-1}$ to $Id$ is given as follows: for $(E,\theta,Fil,\phi)$, put $$(H,\nabla,Fil,
\Phi)=\mathcal{C}_0^{-1}(E,\theta,Fil,\phi)\quad
(E',\theta',Fil',\phi')=\mathcal{GR}(H,\nabla,Fil,\Phi).$$ Then $\phi: Gr_{Fil}\circ C_0^{-1}(E,\theta)\cong (E,\theta)$ induces an isomorphism from $(E',\theta',Fil',\phi')$ to $(E,\theta,Fil,\phi)$ in $\mathcal{HB}_{(0,1)}$, which we define to be ${{\mathcal B}}(E,\theta,Fil,\phi)$. It is direct to check that ${{\mathcal B}}$ is a natural isomorphism.
Before moving to the proof of Theorem \[correspondence in the type (0,f) case\] in general, we shall introduce an intermediate category, the category of periodic Higgs-de Rham sequences of type $(0,1)$ with endomorphism structure ${{\mathbb F}}_{p^f}$: an object is a five tuple $(E,\theta,Fil,\phi,\iota)$, where $(E,\theta,Fil,\phi)$ is object in $\mathcal{HB}_{(0,1)}$ and $\iota: {{\mathbb F}}_{p^f}\hookrightarrow
{{\rm End}}_{\mathcal{HB}_{(0,1)}}(E,\theta,Fil,\phi)$ is an embedding of ${{\mathbb F}}_p$-algebras. We denote this category by $\mathcal{HB}_f$. A direct consequence of Proposition \[correspondence in the type (0,1) case\] is the following
\[Corresponendence between Faltings catgory with endo and Higgs with endo\] The category $\mathcal{MF}^{\nabla}_{[0,n],f}({\bf X}/W)$ is equivalent to the category $\mathcal{HB}_f$ of Higgs-de Rham sequences of type $(0,1)$ with endomorphism structure ${{\mathbb F}}_{p^f}$.
Corollary \[Corresponendence between Faltings catgory with endo and Higgs with endo\] and the following proposition finish the proof of Theorem \[correspondence in the type (0,f) case\].
\[correspondence from HB\_f and HB\_(0,f)\] There is a one to one correspondence between the category $\mathcal{HB}_{(0,f)}$ of periodic Higgs-de Rham sequences of type $(0,f)$ and the category $\mathcal{HB}_f$ of periodic Higgs-de Rham sequences of type $(0,1)$ with endomorphism structure ${{\mathbb F}}_{p^f}$.
We start with an object $(E,\theta,Fil_0,\cdots,Fil_{f-1},\phi)$ in $\mathcal{HB}_{(0,f)}$. Put $$(G,\eta):=\bigoplus_{i=0}^{f-1}(E_i,\theta_i)$$ with $(E_0,\theta_0)=(E,\theta)$. As the functor $C_0^{-1}$ is compatible with direct sum, one has the identification $$C_0^{-1}(G,\eta)=\bigoplus_{i=0}^{f-1}C_0^{-1}(E_i,\theta_i).$$ We equip the filtration $Fil$ on $C_0^{-1}(G,\eta)$ by $\bigoplus_{i=0}^{f-1}Fil_i$ via the above identification. Also $\phi$ induces a natural isomorphism of Higgs bundles $\tilde \phi:
Gr_{Fil}C_0^{-1}(G,\eta)\cong (G,\eta)$ as follows: as $$Gr_{Fil}C_0^{-1}(G,\eta)=\bigoplus_{i=0}^{r-1}Gr_{Fil_i}C_0^{-1}(E_i,\theta_i),$$ we require that $\tilde \phi$ maps the factor $Gr_{Fil_i}(E_i,\theta_i)$ identically to the factor $(E_{i+1},\theta_{i+1})$ for $0\leq i\leq f-2$ (assume $f\geq 2$ to avoid the trivial case) and the last factor $Gr_{Fil_{f-1}}(E_{f-1},\theta_{f-1})$ isomorphically to $(E_0,\theta_0)$ via $\phi$. Thus the so constructed four tuple $(G,\eta,Fil,\tilde \phi)$ is an object in $\mathcal{HB}_{(0,1)}$.
\[lemma from 0,f to f\] For an object $(E,\theta,Fil_0,\cdots,Fil_{f-1},\phi)$ in $\mathcal{HB}_{(0,f)}$, there is a natural embedding of ${{\mathbb F}}_p$-algebras $$\iota: {{\mathbb F}}_{p^r}\to {{\rm End}}_{\mathcal{HB}_{(0,1)}}(G,\eta,Fil,\tilde
\phi).$$ Thus the extended tuple $(G,\eta,Fil,\tilde \phi,\iota)$ is an object in $\mathcal{HB}_f$.
Without loss of generality, we assume $f=2$. Choose a primitive element $\xi$ in ${{\mathbb F}}_{p^r}|{{\mathbb F}}_{p}$ once and for all. To define the embedding $\iota$, it suffices to specify the image $s:=\iota(\xi)$, which is defined as follows: write $(G,\eta)=(E_0,\theta_0)\oplus
(E_1,\theta_1)$. Then $s=m_{\xi}\oplus m_{\xi^p}$, where $m_{\xi^{p^i}},i=0,1$ is the multiplication map by $\xi^{p^i}$. It defines an endomorphism of $(G,\eta)$ and preserves $Fil$ on $C_0^{-1}(G,\eta)$. Write $(Gr_{Fil}\circ C_0^{-1})(s)$ to be the induced endomorphism of $Gr_{Fil}C_0^{-1}(G,\eta)$. It remains to verify the commutativity $$\tilde \phi\circ s=(Gr_{Fil}\circ
C_0^{-1})(s)\circ \tilde \phi.$$ In terms of a local basis, it boils down to the equation $$\left(
\begin{array}{cc}
0 & 1 \\
\phi & 0 \\
\end{array}
\right)\left(
\begin{array}{cc}
\xi & 0 \\
0 & \xi^p \\
\end{array}
\right)=\left(
\begin{array}{cc}
\xi^p & 0 \\
0 & \xi \\
\end{array}
\right)\left(
\begin{array}{cc}
0 & 1 \\
\phi & 0 \\
\end{array}
\right),$$ which is clear.
Conversely, given an object $(G,\eta,Fil,\phi,\iota)$ in the category $\mathcal{HB}_{f}$, we can associate it an object in $\mathcal{HB}_{(0,f)}$ as follows: the endomorphism $\iota(\xi)$ decomposes $(G,\eta)$ into eigenspaces: $$(G,\eta)=\bigoplus_{i=0}^{f-1}(G_i,\eta_i),$$ where $(G_i,\eta_i)$ is the eigenspace to the eigenvalue $\xi^{p^i}$. The isomorphism $C_0^{-1}(\iota(\xi))$ induces the eigen-decomposition of the de Rham bundle as well: $$(C_0^{-1}(G,\eta),Fil)=\bigoplus_{i=0}^{f-1}(C_0^{-1}
(G_i,\eta_i),Fil_i).$$ Under the decomposition, the isomorphism $\phi:
Gr_{Fil}C_0^{-1}(G,\eta)\cong (G,\eta)$ decomposes into $\oplus_{i=0}^{f-1}\phi_i$ such that $$\phi_i: Gr_{Fil_i}C_0^{-1}(G_i,\eta_i)\cong (G_{i+1\mod
f},\theta_{i+1\mod f}).$$ Put $(E,\theta)=(G_0,\theta_0)$.
\[lemma from f to 0,f\] The filtrations $\{Fil_i\}$s and isomorphisms of Higgs bundles $\{\phi_i\}$s induce inductively the filtration $\widetilde{Fil}_i$ on $C_0^{-1}(E_i,\theta_i), i=0,\cdots,f-1$ and the isomorphism of Higgs bundles $$\tilde \phi: Gr_{\widetilde{Fil}_{f-1}}(E_{f-1},\theta_{f-1})\cong
(E,\theta).$$ Thus the extended tuple $(E,\theta,\widetilde{Fil}_0,\cdots,\widetilde{Fil}_{f-1},\tilde
\phi)$ is an object in $\mathcal{HB}_{(0,f)}$.
Again we shall assume $f=2$. The filtration $\widetilde{Fil}_{0}$ on $C_0^{-1}(E_0,\theta_0)$ is just $Fil_0$. Via the isomorphism $$C_0^{-1}(\phi_0):C_0^{-1}Gr_{Fil_0}C_0^{-1}(G_0,\eta_0)\cong
C_0^{-1}(G_1,\eta_1),$$ we obtain the filtration $\widetilde{Fil}_{1}$ on $C_0^{-1}(E_1,\theta_1)$ from the $Fil_1$. Finally we define $\tilde \phi$ to be the composite: $$Gr_{\widetilde{Fil}_{1}}(E_{1},\theta_{1})=Gr_{\widetilde{Fil}_{1}}C_0^{-1}
Gr_{\widetilde{Fil}_{0}}C_0^{-1}(E,\theta)\stackrel{
Gr_{\widetilde{Fil}_{1}}C_0^{-1}(\phi_0)}{\longrightarrow}
Gr_{\widetilde{Fil}_{1}}C_0^{-1}(G_1,\eta_1) \stackrel{
\phi_1}{\longrightarrow}(E,\theta).$$
We come to the proof of Proposition \[correspondence from HB\_f and HB\_(0,f)\].
Note first that Lemma \[lemma from 0,f to f\] gives us a functor ${{\mathcal E}}$ from $\mathcal{HB}_{(0,f)}$ to $\mathcal{HB}_{f}$, while Lemma \[lemma from f to 0,f\] a functor ${{\mathcal F}}$ in the opposite direction. We show that they give an equivalence of categories. It is direct to see that $${{\mathcal F}}\circ{{\mathcal E}}=Id.$$ So it remains to give a natural isomorphism $\tau$ between $\mathcal{E}\circ\mathcal{F}$ and $Id$. Again we assume that $f=2$ in the following argument. For $(E,\theta,Fil,\phi,\iota)$, put $$\mathcal{F}\{(E,\theta,Fil,\phi,\iota)\}=(G,\eta,Fil_0,
Fil_1,\tilde \phi),\quad \mathcal{E}(G,\eta,Fil_0, Fil_1,\tilde
\phi)=(E',\theta',Fil',\phi',\iota').$$ Notice that $(E',\theta')=(G,\eta)\oplus Gr_{Fil_0}C_0^{-1}(G,\eta)$, we define an isomorphism of Higgs bundles by $$Id\oplus \phi_0: (E',\theta')=(G,\eta)\oplus
Gr_{Fil_0}C_0^{-1}(G,\eta)\cong (E_0,\theta_0)\oplus
(E_1,\theta_1)=(E,\theta).$$ It is easy to check that the above isomorphism gives an isomorphism $\tau(E,\theta,Fil,\phi,\iota)$ in the category $\mathcal{HB}_{f}$. The functorial property of $\tau$ is easily verified.
Faltings showed that the (contravariant) functor ${\bf D}$ [@Fa1] from $\mathcal{MF}^{\nabla}_{[0,n]}({\bf X}/W)$ to the category of continuous ${{\mathbb F}}_p$-representations of $\pi_1({\bf X}^0)$ is fully faithful. The image is closed under subobject and quotient, and its object is called dual crystalline sheaf. In our paper we take the dual of ${\bf D}$ (cf. page 43 loc. cit.) without changing the notation. A crystalline ${{\mathbb F}}_{p^f}$-representation is a crystalline ${{\mathbb F}}_p$-representation ${{\mathbb V}}$ with an embedding of ${{\mathbb F}}_p$-algebras ${{\mathbb F}}_{p^f}\hookrightarrow {{\rm End}}_{\pi_1({\bf
X}^0)}({{\mathbb V}})$.
\[correspondence from crystalline represenations and HB\_(0,f)\] There is an equivalence of categories between the category of crystalline ${{\mathbb F}}_{p^f}$-representations of $\pi_1({\bf X}^0)$ and the category of periodic Higgs-de Rham sequences of type $(0,f)$.
Under the functor ${\bf D}$, an ${{\mathbb F}}_{p^f}$-endomorphism structure on an object of $\mathcal{MF}$ is mapped to an ${{\mathbb F}}_{p^f}$-endomorphism structure on the corresponding ${{\mathbb F}}_p$-representation, and vice versa. The result is then a direct consequence of Theorem \[correspondence in the type (0,f) case\].
Let $\rho$ be a crystalline ${{\mathbb F}}_{p^f}$-representation of $\pi_1({\bf
X}^0)$, and $(E,\theta,Fil_0,\cdots,Fil_{f-1},\phi)$ the corresponding periodic Higgs-de Rham sequence of type $(0,f)$. For $$(E_f,\theta_f)=Gr_{Fil_{f-1}}(H_{f-1},\nabla_{f-1}),$$ $C_0^{-1}(\phi)$ induces the pull-back filtration $C_0^{-1}(\phi)^*Fil_0$ on $C^{-1}_0(E_f,\theta_f)$ and an isomorphism of Higgs bundles $GrC_0^{-1}(\phi)$ on the gradings. It is easy to check that $$(E_1,\theta_1,Fil_1,\cdots,Fil_{f-1},C_0^{-1}(\phi)^*Fil_0,GrC_0^{-1}(\phi))$$ is an object in $\mathcal{HB}_{(0,f)}$, which is called the *shift* of $(E,\theta,Fil_0,\cdots,Fil_{f-1},\phi)$. For any multiple $lf, l\geq 1$, we can lengthen $(E,\theta,Fil_0,\cdots,Fil_{f-1},\phi)$ to an object of $\mathcal{HB}_{(0,lf)}$: as above, we can inductively define the induced filtration on $(H_j,\nabla_j), f\leq j\leq lf-1$ from $Fil_i$s via $\phi$. One has the induced isomorphism of Higgs bundles $(GrC_0^{-1})^{l'f}(\phi):
(E_{(l'+1)f},\theta_{(l'+1)f})\cong (E_{l'f},\theta_{l'f}), 0\leq
l'\leq l-1$. The isomorphism $\phi_{l}: (E_{lf},\theta_{lf})\cong
(E_0,\theta_0)$ is defined to be the composite of them. The obtained object $(E,\theta,Fil_0,\cdots,Fil_{lf-1},\phi_l)$ is called the $l$-th *lengthening* of $(E,\theta,Fil_0,\cdots,Fil_{f-1},\phi)$. The following result is obvious from the construction of the above correspondence.
\[operations on Higgs-de Rham sequences\] Let $\rho$ and $(E,\theta,Fil_0,\cdots,Fil_{f-1},\phi)$ be as above. Then the followings are true:
- The shift of $(E,\theta,Fil_0,\cdots,Fil_{f-1},\phi)$ corresponds to $\rho^{\sigma}=\rho\otimes_{{{\mathbb F}}_{p^f},\sigma}{{\mathbb F}}_{p^f}$, the $\sigma$-conjugation of $\rho$. Here $\sigma\in {\mathrm{Gal}}({{\mathbb F}}_{p^f}|{{\mathbb F}}_p)$ is the Frobenius element.
- For $l\in {{\mathbb N}}$, the $l$-th lengthening of $(E,\theta,Fil_0,\cdots,Fil_{f-1},\phi)$ corresponds to the base extension $\rho\otimes_{{{\mathbb F}}_{p^f}}{{\mathbb F}}_{p^{lf}}$.
We remind also the reader of the following result.
\[locally freeness of periodic bundles\] Periodic Higgs bundles are locally free.
Let $(E,\theta)$ be a periodic Higgs bundle. Then a Higgs-de Rham sequence for it gives an object in the category $\mathcal{HB}_{(0,f)}$ for a certain $f$. Let $(H,\nabla,Fil,\Phi,\iota)$ be the corresponding object in $\mathcal{MF}_f$. The proof of Theorem 2.1 [@Fa1] (cf. page 32 loc. cit.) says that $Fil$ is a filtration of locally free subsheaves of $H$ and the grading $Gr_{Fil}H$ is also locally free. It follows immediately that $(E,\theta)$ is locally free.
Quasi-periodic Higgs bundles
============================
A quasi-periodic Higgs-de Rham sequence of of type $(e,f)$ is a tuple $$(E,\theta,Fil_0,\cdots,Fil_{e+f-1},\phi),$$ where $\phi$ is an isomorphism of Higgs bundles $$\phi: Gr_{Fil_{e+f-1}}(H_{e+f-1},\nabla_{e+f-1})\cong
(E_{e},\theta_e).$$ It follows from Corollary \[locally freeness of periodic bundles\] that the Higgs bundles $(E_i,\theta_i), e\leq i\leq e+f-1$ are locally free. They form the category $\mathcal{HB}_{n,(e,f)}(X/k)$.\
We are going to associate a quasi-periodic Higgs-de Rham sequence of type $(e,f)$ with an object in a Faltings category. We recall first the strict $p$-torsion category $\mathcal{MF}^{\nabla}_{[0,n]}({\bf
X}_V/R_V)$, which is based on the category introduced by Faltings in §3-§4 [@Fa2]. For $V$ a totally ramified extension of $W(k)$, Faltings §2 [@Fa2] introduced the base ring $R_V$ as follows: a uniformizer $\pi$ of $V$ has the minimal polynomial $$f(T)=T^e +
\sum_{0<i<e} a_iT^i\in W[T].$$ It defines the $W$-algebra morphism $W[[T]]\to V, T\mapsto \pi$ and $R_V$ is defined to be the PD-hull of $V$. One has an excellent lifting $X/k$ over $R_V$, that is, one takes ${\bf X}\times_{W}R_V$, the base change of ${\bf X}/W$ to $R_V$. Put ${{\mathcal X}}={\bf X}\times_{W}R_V/p=X\times_{k}R_V/p$. It depends only on the ramification index $e$ of $V$, not on $V$ itself. The sheaf of $k$-algebras ${{\mathcal O}}_{{{\mathcal X}}}$ admits a natural filtration $Fil_{{{\mathcal O}}_{{{\mathcal X}}}}$. The composite of the natural maps $$k=W/p\to
R_V/p\stackrel{T\mapsto 0}{\longrightarrow}k$$ is the identity. It induces the commutative diagram of $k$-schemes $$\xymatrix{
X \ar[dr]_{id} \ar[r]^{\mu}
& {{\mathcal X}}\ar[d]^{\lambda} \\
& X. }$$ An object of the category $\mathcal{MF}^{\nabla}_{[0,n]}({\bf
X}_V/R_V)$ is a four tuple $(H,\nabla,Fil,\Phi)$, where $(H,Fil)$ is a locally filtered-free ${{\mathcal O}}_{{{\mathcal X}}}$-module of finite rank, with a local basis consisting of homogenous elements of degrees between 0 and $n$, $\nabla: H\to H\otimes \Omega_{{{\mathcal X}}/k}$ an integrable connection satisfying the Griffiths transversality, the relative Frobenius $\Phi$ is strongly $p$-divisible (i.e. $\Phi$ locally over ${{\mathcal U}}_i\subset {{\mathcal X}}$ induces an isomorphism $F_{{{\mathcal U}}_{i}}^*Gr_{Fil}^nH\cong H|_{{{\mathcal U}}_i}$) and horizontal with respect to $\nabla$.
The morphism $\lambda$ induces a functor $\lambda^*$ from $\mathcal{HB}_{(e,f)}$ to $\mathcal{MF}^{\nabla}_{[0,n],f}({\bf
X}_V/R_V)$ and the morphism $\mu$ a functor $\mu^*$ from $\mathcal{MF}^{\nabla}_{[0,n],f}({\bf X}_V/R_V)$ to the category $\mathcal{HB}_{(0,f)}$.
For $(E,\theta,Fil_0,\cdots,Fil_{e+f-1},\phi)$, we take $(E',\theta')=\bigoplus_{i=0}^{f-1}(E_i,\theta_i)$. Then $Fil_i$s and $\phi$ induces naturally an object $(E',\theta',Fil'_0,\cdots,Fil'_{e},\phi')$ in $\mathcal{HB}_{(e,1)}$. Thus it suffices to show the above statement for $f=1$.\
Put $H=\lambda^*H_e$, $\nabla=\lambda^*\nabla_e$ and $Fil=Fil_{{{\mathcal O}}_{{{\mathcal X}}}}\otimes \lambda^*Fil_e$. Note that one has a natural isomorphism of ${{\mathcal O}}_{{{\mathcal X}}}$-modules $F_{{{\mathcal U}}_{i}}^*Gr_{Fil}^nH\cong \lambda^*F_{U_i}^*Gr_{Fil_e}H_e$. We define the relative Frobenius $\Phi$ on $H$ via the above isomorphism composed with $\lambda^*\Phi_{({\bf U}_i,F_{{\bf
U}_i})}$, where $\Phi_{({\bf U}_i,F_{{\bf U}_i})}:
F_{U_i}^*Gr_{Fil_e}H_e\to H_e|_{U_i}$ appeared in the paragraph before Lemma \[horizontal property\]. This gives us the functor $\lambda^*$ from $\mathcal{HB}_{(e,1)}$ to $\mathcal{MF}^{\nabla}_{[0,n]}({\bf X}_V/R_V)$. Conversely, given an object $(H,\nabla,Fil,\Phi)\in \mathcal{MF}^{\nabla}_{[0,n]}({\bf
X}_V/R_V)$, the tuple $(\mu^*H,\mu^*\nabla,\mu^*Fil,\mu^*\Phi)$ is naturally an object in $\mathcal{MF}^{\nabla}_{[0,n]}({\bf X}/W)$: over ${{\mathcal U}}_i$, $\Phi$ gives an isomorphism $F_{{{\mathcal U}}_{i}}^*Gr_{Fil}^nH\cong H_{{{\mathcal U}}_i}$. Pulling back the isomorphism via $\mu$, we get $F_{U_i}^*\mu^*Gr^n_{Fil}H\cong
\mu^*H|_{U_i}$. As there is a natural ${{\mathcal O}}_X$-modules isomorphism $Gr_{\mu^*Fil}\mu^*H\cong \mu^*Gr^n_{Fil}H$, we have an isomorphism $F_{U_i}^*Gr_{\mu^*Fil}\mu^*H|_{U_i}\cong \mu^*H|_{U_i}$, which shows that $\mu^*\Phi$ is indeed a relative Frobenius. We define $\mu^*(H,\nabla,Fil,\Phi)\in \mathcal{HB}_{(0,1)}$ to be the object associated to $(\mu^*H,\mu^*\nabla,\mu^*Fil,\mu^*\Phi)$.
\[quasi-periodic corresponds to representation\] There is a functor from the category of quasi-periodic Higgs-de Rham sequences of type $(e,f)$ to the category of crystalline representations of $\pi_1({\bf X'}^0)$ into ${\mathrm{GL}}({{\mathbb F}}_{p^f})$, where ${\bf X'}^0$ is the generic fiber of ${\bf
X'}:={\bf X}\times_{W}{{\mathcal O}}_K$ for a totally ramified extension $\mathrm{Frac}(W)\subset K$ with ramification index $e$. There is also a functor in the converse direction.
The first part follows from the above functor $\lambda^*$ and the proof of Theorem 5. i) [@Fa2]. To provide a functor in the opposite direction, we use the functor $\mu^*$ together with choosing an additional embedding of the category $\mathcal{HB}_{(0,f)}$ into $\mathcal{HB}_{(e,f)}$. This can be done as follows: for an object $(E,\theta,Fil_0,\cdots,Fil_{f-1},\phi)\in
\mathcal{HB}_{(0,f)}$, let $l\in {{\mathbb N}}$ be the minimal number with $e\leq lf$. Then there is a unique object $(E',\theta',Fil'_0,\cdots,Fil'_{e+f-1},\phi')$ in $\mathcal{HB}_{(e,f)}$ obtained from its $l+1$-th lengthening which satisfies the equality $$(E'_{i},\theta'_{i})=(E_{lf-e+i},\theta_{lf-e+i}), 0\leq i\leq e+f.$$
Applications
============
Given a periodic Higgs-de Rham sequence $$\xymatrix{
& (H_0,\nabla_0)\ar[dr]^{Gr_{Fil_0}} && (H_1,\nabla_1)\ar[dr]^{Gr_{Fil_1}} \\
(E_0,\theta_0) \ar[ur]^{C_0^{-1}} & & (E_1,\theta_1) \ar[ur]^{C_0^{-1}}&&\ldots, }$$ we make the following observation:
If $(E,\theta)=(E_0,\theta_0)$ is Higgs stable, then there is a unique periodic Higgs-de Rham sequence for $(E,\theta)$ up to isomorphism.
Let $f\in{{\mathbb N}}$ be the period of the sequence. Thus there is an isomorphism $\phi: (E_f,\theta_f)\cong (E_0,\theta_0)$ such that the tuple $(E,\theta,Fil_0,\cdots,Fil_{f-1},\phi)$ makes an object in $\mathcal{HB}_{(0,f)}$. We show that the datum $Fil_i, 0\leq i\leq
f-1$ and $\phi$ are uniquely determined up to isomorphism. By Theorem \[correspondence in the type (0,f) case\], there is a corresponding object $$(H,Fil,\nabla,\Phi,\iota)\in
\mathcal{MF}_{f}$$ satisfying $Gr_{Fil}(H,\nabla)=\bigoplus_{i=1}^{f}(E_i,\theta_i)$. Because it holds that $$(Gr_{Fil}\circ
C_0^{-1})^{i}(E_f,\theta_f)=(E_i,\theta_i), 1\leq i\leq f-1,$$ each $(E_i,\theta_i)$ is also Higgs stable by Corollary 4.4 [@SZ]. Now we show inductively that $Fil_i$ is unique. This is because of the fact that there is a unique filtration on a flat bundle which satisfies the Griffiths transversality and its grading is Higgs stable. Now we consider $\phi$. For another choice $\varphi$, one notes that $\varphi\circ \phi^{-1}$ is an automorphism of $(E,\theta)$. As it is stable, one must have $\varphi=\lambda\phi$ for a nonzero $\lambda$ in $k$. It is easy to see there is an isomorphism in $\mathcal{HB}_{(0,f)}$: $$(E,\theta,Fil_0,\cdots,Fil_{f-1},\phi)\cong (E,\theta,Fil_0,\cdots,Fil_{f-1},\lambda\phi).$$
Because of the above lemma, the period of a periodic Higgs stable bundle is well defined. We make then the following statement.
\[stable corresponds to irreducible\] Under the equivalence of categories in Corollary \[correspondence from crystalline represenations and HB\_(0,f)\], there is one to one correspondence between the isomorphism classes of irreducible crystalline ${{\mathbb F}}_{p^f}$-representations of $\pi_1({\bf X}^0)$ and the isomorphism classes of periodic Higgs stable bundles of period $f$.
The first examples of periodic Higgs stable bundles are the rank two Higgs subbundles of uniformizing type arising from the study of the Higgs bundle of a universal family of abelian varieties over the good reduction of a Shimura curve of PEL type (see [@SZZ]). In that case, one ’sees’ the corresponding representations because of the existence of extra endomorphisms in the universal family. The above result gives a vast generalization of this primitive example.\
When a periodic Higgs bundle $(E,\theta)$ is only Higgs semistable, the above uniqueness statement is no longer true. We shall make the following
\[assumption on filtration\] For each $0\leq i\leq f-1$, the filtration $Fil_i$ on $H_i$ is preserved by any automorphism of $(H_i,\nabla_i)$.
An isomorphism $\varphi:(E_f,\theta_f)\cong (E_0,\theta_0)$ induces $$(GrC_0^{-1})^{nf}(\varphi):(E_{(n+1)f},\theta_{(n+1)f})\cong
(E_{nf},\theta_{nf}).$$ For $-1\leq i< j$, we define $$\varphi_{j,i}=(GrC_0^{-1})^{(i+1)f}(\varphi)\circ\cdots\circ(GrC_0^{-1})^{jf}(\varphi):(E_{(j+1)f},\theta_{(j+1)f})\cong (E_{(i+1)f},\theta_{(i+1)f}).$$ For $i=-1$ put $\varphi_j=\varphi_{j,-1}$.
\[finiteness implies periodic\] For any two isomorphisms $\varphi, \phi :
(E_f,\theta_f)\cong(E_0,\theta_0)$, there exists a pair $(i,j)$ with $0\leq i<j$ such that $\phi_{j,i}\circ \varphi_{j,i}^{-1}=id$.
If we denote $\tau_s=\phi_{s}\circ \varphi_{s}^{-1}$, then $\tau_s$ is an automorphism of $(E_0,\theta_0)$. Moreover, each element in the set $\{\tau_s\}_{s\in {{\mathbb N}}}$ is defined over the same finite field in $k$. As this is a finite set, there are $j>i\geq 0$ such that $\tau_j=\tau_i$. So the lemma follows.
\[phi plays no role\] Assume \[assumption on filtration\]. Let $(i,j)$ be a pair given by Lemma \[finiteness implies periodic\] for two given isomorphisms $\varphi, \phi : (E_f,\theta_f)\cong(E_0,\theta_0)$. Then there is an isomorphism in $\mathcal{HB}_{(0,(j-i)f)}$: $$(E,\theta,Fil_0,\cdots,Fil_{f-1},\varphi_{j-i-1})\cong (E,\theta,Fil_0,\cdots,Fil_{f-1},\phi_{j-i-1}).$$
Put $\beta=\phi_{i}\circ \varphi_{i}^{-1}: (E_0,\theta_0)\cong
(E_0,\theta_0)$. We shall check that it induces an isomorphism in $\mathcal{HB}_{(0,(j-i)f)}$. By Assumption \[assumption on filtration\], $C_0^{-1}(GrC_0^{-1})^m(\beta)$ for $m\geq 0$ always respects the filtrations. We need only to check that $\beta$ is compatible with $\phi_{j-i-1}$ as well as $\varphi_{j-i-1}$. So it suffices to show that the following diagram is commutative:
E\_[(j-i)f]{}&&E\_0\
& &\
E\_[(j+1)f]{} & & E\_[(i+1)f]{}\
& &\
E\_[(j-i)f]{}& &E\_0\
And it suffices to show that the following diagram is commutative:
E\_[(j-i)f]{}&&E\_0\
& &\
E\_[(j+1)f]{} & & E\_[(i+1)f]{}\
& &\
E\_[(j-i)f]{}& &E\_0\
In the above diagram, the anti-clockwise direction is $$\phi_{j-i-1}\circ\phi_{j,j-i-1}\circ\varphi_{j,j-i-1}^{-1}\circ\varphi_{j-i-1}^{-1}
=\phi_j\circ\varphi_j^{-1}=\phi_i\circ(\phi_{j,i}\circ\varphi_{j,i}^{-1})\circ\varphi_i.$$ By the requirement for $(i,j)$, we have $\phi_{j,i}\circ\varphi_{j,i}^{-1}=id$, so the anti-clockwise direction is $\phi_i\circ\varphi_i$, which is exactly the clockwise direction. So $\beta$ is shown to be compatible with $\phi_{j-i-1}$ and $\varphi_{j-i-1}$.
We deduce some consequences from the above result.
\[rank two semistable bundle corresponds to rep\] Any isomorphism class of rank two semistable Higgs bundles with trivial chern classes over $X$ is associated to an isomorphism class of crystalline representations of $\pi_1({\bf X}^0)$ into ${\mathrm{GL}}_2(k)$. The image of the association contains all irreducible crystalline representations of $\pi_1({\bf X}^0)$ into ${\mathrm{GL}}_2(k)$.
The second statement follows from Theorem \[stable corresponds to irreducible\]. Let $(E,\theta)$ be a rank two semistable Higgs bundle with trivial $c_1$ and $c_2$ over $X$. By Theorems \[rank two semistable implies strongly semistable\] and \[quasiperiodic equivalent to strongly semistable\], it is a quasi-periodic Higgs bundle. Recall that we use the HN-filtration in the proof. Hence we obtain *the* quasi-periodic Higgs-de Rham sequence for $(E,\theta)$. Let $e\in {{\mathbb N}}_0$ be the minimal number such that $(Gr_{HN}\circ C_0^{-1})^e(E,\theta)$ is periodic and say its period is $f\in {{\mathbb N}}$. Thus from $(E,\theta)$ we obtain in the above way an object $$((Gr_{HN}\circ
C_0^{-1})^e(E,\theta),Fil_0=HN,\cdots,Fil_{f-1}=HN,\phi)$$ in $\mathcal{HB}_{(0,f)}$, which is unique up to the choice of $\phi$. Let $\rho$ be the corresponding representation by Theorem \[correspondence from crystalline represenations and HB\_(0,f)\]. As $HN$s clearly satisfy the Assumption \[assumption on filtration\], it follows from Proposition \[phi plays no role\] that the isomorphism class of $\rho\otimes k$ is independent of the choice of $\phi$. It is clear that an isomorphic Higgs bundle to $(E,\theta)$ is associated to the same isomorphism class of crystalline representations. This shows the first statement.
Next, we want to compare the classical construction of Katz and Lange-Stuhler (see §4 [@Katz] and §1 [@LS]) using an Artin-Schreier cover with the one in the current paper. Namely, we consider the isomorphism classes of vector bundles $E$ over $X$ satisfying $F_{X}^{*f}E\cong E$ for an exponent $f\in {{\mathbb N}}$. By Proposition 1.2 and Satz 1.4 in [@LS] (see also §4.1 [@Katz]), they are in bijection with the isomorphism classes of representations $\pi_1(X)\to {\mathrm{GL}}(k)$. Let $[\rho_{KLS}]$ be the isomorphism class of representations $\pi_1(X)\to {\mathrm{GL}}(k)$ corresponding to the isomorphism class of $E$. Let $E$ be such a bundle over $X$ with an isomorphism $\phi: F_X^{*f}E\cong E$. It gives rise to a tuple $(E,0,Fil_{tr},\cdots,Fil_{tr},\phi)$, an object in $\mathcal{HB}_{(0,f)}$. Then by Theorem \[correspondence from crystalline represenations and HB\_(0,f)\], there is a corresponding crystalline representation $\rho:
\pi_1({\bf X}^0)\to {\mathrm{GL}}({{\mathbb F}}_{p^f})$. After Proposition \[phi plays no role\], the isomorphism class of $\rho\otimes_{{{\mathbb F}}_{p^f}}k$ is independent of the choice of $\phi$. The following result follows directly from the construction of the representation due to Faltings [@Fa1].
\[factor through specialization map Part 1\] Let $\tau$ be a crystalline representation of $\pi_1({\bf X}^0)$ into ${\mathrm{GL}}({{\mathbb F}}_p)$ and $(H,\nabla,Fil,\Phi)$ the corresponding object in $\mathcal{MF}^{\nabla}_{[0,n]}({\bf X}/W)$. If the filtration $Fil$ is trivial, namely, $Fil^0H=H,\ Fil^1H=0$, then $\tau$ factors through the specialization map $sp: \pi_1({\bf
X}^0)\twoheadrightarrow \pi_1(X)$.
Let ${\bf U}_i={\mathrm{Spec}}R$ be a small affine subset of ${\bf X}$, and $\Gamma={\mathrm{Gal}}(\bar R|R)$ the Galois group of maximal extension of $R$ étale in characteristic zero (cf. Ch. II. b) [@Fa1]). Let $R^{ur}\subset \bar R$ be the maximal subextension which is étale over $R$ and $\Gamma^{ur}={\mathrm{Gal}}(R^{ur}|R)$. By the local nature of the functor $\bf D$ (cf. Theorem 2.6 loc. cit.), it is to show that the representation ${\bf D}(H_i)$ of $\Gamma$, constructed from the restriction $H_i:=(H,\nabla,Fil,\Phi)|_{{\bf U}_i}\in
\mathcal{MF}^{\nabla}_{[0,n]}(R)$, factors through the natural quotient $\Gamma\twoheadrightarrow \Gamma^{ur}$. To that we have to examine the construction of ${\bf D}(H_i)$ carried in pages 36-39 loc. cit. (see also pages 40-41 for the dual object). First of all, we can choose a basis $f$ of $H_i$ which is $\nabla$-flat. Because $Fil$ is trivial, $\Phi$ is a local isomorphism. So for any basis $e$ of $H_i$, $f=\Phi(e\otimes 1)$ is then a flat basis of $H_i$. The construction of module ${\bf D}(H_i)\subset H_i\otimes \bar R/p$ does not use the connection, but the definition of $\Gamma$-action does (see page 37 loc. cit.). A basis of ${\bf D}(H_i)$ is of form $f\otimes x$, where $x$ is a set of tuples in $\bar R/p$ satisfies the equation $x^p=Ax$, where $A$ is the matrix of $\Phi$ under the basis $f$ (i.e. $\Phi(f\otimes 1)=Af$). Now that $A$ is invertible, the entries of $x$ lie actually in $R^{ur}/p$. Since $f$ is a flat basis, the action of $\Gamma$ on $f\otimes x$ coincides the natural action of $\Gamma$ on the second factor. Thus it factors through the quotient $\Gamma\twoheadrightarrow \Gamma^{ur}$.
By the above lemma, $\rho$ factors as $$\pi_1({\bf X}^0)\stackrel{sp}{\longrightarrow}\pi_1(X)\to
{\mathrm{GL}}({{\mathbb F}}_{p^f}).$$
\[faltings coincide with KLS\] Let $\rho_F: \pi_1(X)\to {\mathrm{GL}}({{\mathbb F}}_{p^f})$ be the induced representation from $\rho$. Then $\rho_F\otimes k$ is in the isomorphism class $[\rho_{KLS}]$.
We can assume that $E$ as well as $\phi$ are defined over $X|k'$ for a finite field $k'$. Then we obtain from Proposition 4.1.1 [@Katz] or Satz 1.4 [@LS] a representation $\rho_{KLS}:
\pi_1(X)\to {\mathrm{GL}}({{\mathbb F}}_{p^f})$. We are going to show that $\rho_F$ and $\rho_{KLS}$ are isomorphic ${{\mathbb F}}_{p^f}$-representations. For $f=1$, this follows directly from their constructions: Katz and Lange-Stuhler construct the representation by solving $\phi$-invariant sections through the equation $x^p=Ax$, which it is exactly what Faltings does in the case of trivial filtration by the above description of his construction. For a general $f$, Katz and Lange-Stuhler solve locally the equation $x^{p^f}=Ax$, which is equivalent to a system of equations of form $$x_0^{p}=x_1,\cdots,x_{f-2}^p=x_{f-1}, x_{f-1}^p=Ax_0.$$ To examine our construction, we take a local basis $e_0=e$ of $E_0=E$ and put $e_i=F_{X}^{*i}e$, a local basis of $E_i$ for $0\leq
i\leq f-1$. Write $\phi(e_{f-1})=Ae_0$. Put $\tilde
e=(e_0,\cdots,e_{f-1})$, and $\tilde x=(x_1,\cdots,x_{f-1})$. Then the $\tilde \phi$ in Lemma \[lemma from f to 0,f\] has the expression $\tilde\phi(\tilde e)=\tilde A\tilde e$ with $$\tilde A= \left(
\begin{array}{cccc}
0 & 1 & \cdots & 0 \\
\vdots & \ddots & \ddots & \vdots \\
0 & \cdots & 0 & 1 \\
\phi & 0 & \cdots &0 \\
\end{array}
\right).$$ One notices that the equation $\tilde x^p=\tilde A\tilde x$ written into components is exactly the above system of equations. Thus one sees that the ${{\mathbb F}}_{p^f}$-representation $\rho_F$ corresponding to $(E,0,Fil_{tr},\cdots,Fil_{tr},\phi)$ by Corollary \[correspondence from crystalline represenations and HB\_(0,f)\] is isomorphic to $\rho_{KLS}$ as ${{\mathbb F}}_{p^f}$-representations.
It may be noteworthy to deduce the following
\[factor through specialization map Part 2\] Let $\tau$ be a crystalline representation of $\pi_1({\bf X}^0)$ with the corresponding object $(H,\nabla,Fil,\Phi)\in
\mathcal{MF}^{\nabla}_{(0,n)}({\bf X}/W)$. Then $\tau$ factors through the specialization map iff the filtration $Fil$ is trivial.
One direction is Lemma \[factor through specialization map Part 1\]. It remains to show the converse direction. Let $\tau_0$ be the induced representation of $\pi_1(X)$ from $\tau$. As it is of finite image, one constructs directly from $\rho_0$ a vector bundle $E$ over $X$ such that $F_{X}^*E\cong E$. Choosing such an isomorphism, we obtain a representation of $\pi_1(X)$ and then a representation $\tau'$ of $\pi_1({\bf X}^0)$ by composing with the specialization map. By Theorem \[faltings coincide with KLS\], $\tau'\otimes
{{\mathbb F}}_{p^f}$ is isomorphic to $\tau\otimes {{\mathbb F}}_{p^f}$ for a certain $f\in {{\mathbb N}}$. It follows from Proposition \[operations on Higgs-de Rham sequences\] (ii) that the filtration $Fil$ is trivial.
We conclude the paper by providing many more examples beyond the rank two semistable Higgs bundles and strongly semistable vector bundles.
Let $(H,\nabla,Fil,\Phi)\in \mathcal{MF}^{\nabla}_{[0,n]}({\bf
X}/W)$. Then any Higgs subbundle $(G,\theta)\subset
Gr_{Fil}(H,\nabla)$ of degree zero is strongly Higgs semistable with trivial chern classes.
Put $(E,\theta)=Gr_{Fil}(H,\nabla)$. Proposition 0.2 [@SXZ] says that $(E,\theta)$ is a semistable Higgs bundle of degree zero. Note that the operator $Gr_{Fil}\circ C_0^{-1}$ does not change the degree, rank and definition field of $(G,\theta)$, and as there are only finitely many Higgs subbundles of $(E,\theta)_0$ with the same degree, rank and definition field as $(G,\theta)$, there exists a pair $(e,f)$ of nonnegative integers with $s>r$ such that $$(Gr_{Fil}\circ C_0^{-1})^s(G,\theta)=(Gr_{Fil}\circ
C_0^{-1})^r(G,\theta)$$ holds. Thus $(G,\theta)$ is quasi-periodic and strongly Higgs semistable with trivial chern classes by Theorem \[quasiperiodic equivalent to strongly semistable\].
[X-X00]{}
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[^1]: This work is supported by the SFB/TR 45 ‘Periods, Moduli Spaces and Arithmetic of Algebraic Varieties’ of the DFG, and partially supported by the University of Science and Technology of China.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The relationship between antiferromagnetic spin fluctuations and superconductivity has become a central topic of research in studies of superconductivity in the iron pnictides. We present unambiguous evidence of the absence of magnetic fluctuations in the non-superconducting collapsed tetragonal phase of CaFe$_2$As$_2$ via inelastic neutron scattering time-of-flight data, which is consistent with the view that spin fluctuations are a necessary ingredient for unconventional superconductivity in the iron pnictides. We demonstrate that the collapsed tetragonal phase of CaFe$_2$As$_2$ is non-magnetic, and discuss this result in light of recent reports of high-temperature superconductivity in the collapsed tetragonal phase of closely related compounds.'
author:
- 'J. H. Soh,$^{1}$ G. S. Tucker,$^{1}$ D. K. Pratt,$^{2}$ D. L. Abernathy,$^{3}$ M. B. Stone,$^{3}$ S. Ran,$^{1}$ S. L. Bud$^{\prime}$ko,$^{1}$ P. C. Canfield,$^{1}$ A. Kreyssig,$^{1}$ R. J. McQueeney$^{1}$ and A. I. Goldman$^{1}$'
title: 'The non-magnetic collapsed tetragonal phase of CaFe$_2$As$_2$ and superconductivity in the iron pnictides'
---
The $A$Fe$_2$As$_2$ ($A$ = Ba, Sr, Ca), or “122”, family of compounds has been one of the most widely studied classes of iron pnictide superconductors [@Johnston_2010; @PandG_2010; @CandB_2010; @Stewart_2011] in recent years, and a great deal of attention has been focused on CaFe$_2$As$_2$ [@Ni_2008; @PC_2009] in particular. At ambient pressure, the substitution of Co or Rh for Fe [@Kumar_2009; @Matusiak_2010; @Harnagea_2011; @Ran_2012; @Danura_2011] results in the suppression of antiferromagnetic (AFM) order and, over some range in substitution, superconductivity (SC) emerges with transition temperatures ($T_{\rm{c}}$) of up to $\approx$ 20 K. Under modest applied pressure Ca(122) manifests fascinating new behavior including a transition to an isostructural volume collapsed tetragonal (cT) phase that is generally believed to be non-magnetic and non-superconducting. The cT phase in Ca(122) is distinguished by a striking 9.5% reduction in the tetragonal **c** lattice parameter, with respect to the high-temperature ambient-pressure tetragonal (T) phase, along with the absence of the stripe-like magnetic order found for the low-temperature ambient-pressure orthorhombic phase [@Goldman_2008].
The first liquid media clamp-cell pressure measurements of Sn-flux solution-grown Ca(122) found traces of SC for applied pressures between roughly 0.25 and 0.9 GPa [@Milton_2008; @Park_2008]. These studies were rapidly followed by transport measurements and neutron diffraction experiments under hydrostatic pressure conditions using He gas pressure cells which showed: (i) no evidence of SC for P $< 0.6$ GPa [@Yu_2009] and; (ii) the existence of the cT structure for $P >$ 0.35 GPa at low temperatures [@Kreyssig_2008; @Goldman_2009]. That work demonstrated that the traces of SC originally found in the frozen liquid clamp-cell measurements probably resulted from significant non-hydrostatic pressure components generated during the transition to the cT phase, although the origin of the SC phase was not identified in these studies. Later experiments, utilizing uniaxial pressure, concluded that the T phase could be stabilized to low temperatures by the presence of non-hydrostatic pressure components and was likely the source of superconductivity in the original liquid clamp-cell measurements [@Prokes_2010].
Recently, superconductivity with $T_{\rm{c}}$ in excess of 45 K has been reported for the substitution of Sr [@Jeffries_2012] or selected rare earths ($R$) [@Lv_2011; @Saha_2012; @Ma_2013] for Ca, or co-doping by La and P [@Kudo_2013], and it has been proposed that these high $T_{\rm{c}}$ values are realized in the cT phase as well [@Saha_2012; @Jeffries_2012]. Since it is generally accepted that there is a close connection between SC in the iron pnictides and the presence of correlated AFM fluctuations in these compounds [@Johnston_2010; @PandG_2010; @CandB_2010; @Stewart_2011; @LandC_2010; @Dai_2012], the possibility of high values of $T_{\rm{c}}$ in the cT phase raises important questions regarding the nature of the cT phase, and the relationship between magnetic fluctuations and unconventional superconductivity in the iron pnictides. It is, therefore, important to clearly establish whether the cT phase of Ca(122) is, in fact, non-magnetic.
There is already evidence that the cT phase of Ca(122) is non-magnetic, consistent with the absence of unconventional superconductivity. First, as noted above, the low-temperature stripe-like AFM order is absent in the cT phase. However, alternative magnetic ground states for the cT phase have been proposed [@Yildirim_2009], and the origin of the suppression of magnetic order, whether it arises from a reduction in the iron moment, changes in the magnetic exchange, or a more subtle change in electronic structure has come under renewed scrutiny [@Jeffries_2012]. Furthermore, the absence of AFM order does not directly speak to the presence or absence of magnetic *fluctuations* in the cT phase. It is well known that strong AFM fluctuations remain after long-range magnetic order is lost in the iron pnictides at optimal doping [@Johnston_2010; @PandG_2010; @CandB_2010; @Stewart_2011; @LandC_2010; @Dai_2012].
Total energy calculations described in Reference predict that the cT phase is non-magnetic and this has been supported by other theoretical studies [@Colonna_2011; @Tomic_2012; @Widom_2013]. Our previous inelastic neutron scattering studies of the T [@Diallo_2010] and cT phases [@Pratt_2009] showed that, at least over a narrow range in momentum transfer (**Q**) close to the AFM wavevector, $\textbf{Q}_{\rm{stripe}}$, and energy transfers ($E$) less than 7 meV, the AFM fluctuations are suppressed, or absent, in the cT phase. Again, this result finds support in other experimental measurements [@Danura_2011; @Ma_2013]. But the narrow scope of the neutron measurements could not exclude the presence of correlated magnetic fluctuations at other positions in reciprocal space [@Yildirim_2009], or simply a change in the energy scale of the fluctuations as has been found, for example, in the well known volume collapse of Ce [@Loong_1987], or very recently in nonsuperconducting Ba(Fe$_{0.85}$Ni$_{0.15}$)$_2$As$_2$ [@Wang_2013]. A much wider view in both **Q** and $E$ must be obtained to clearly establish the presence or absence of magnetic fluctuations in the cT phase of Ca(122).
Here we present unambiguous evidence that the magnetic fluctuations in the non-superconducting cT phase of Ca(122) are absent via inelastic neutron scattering measurements using the ARCS time-of-flight (TOF) instrument [@Abernathy_2012] at the Spallation Neutron Source at Oak Ridge National Laboratory. This result provides clear evidence that the cT phase of Ca(122) is a non-magnetic metal, with no static or dynamic magnetic moment, and supports the view that spin fluctuations are a necessary ingredient for unconventional SC in the iron pnictides. The complete suppression of magnetism in the cT phase also provides a non-magnetic analog for a detailed study of the AFM fluctuation spectrum of the paramagnetic T phase out to energy transfers above 100 meV, and we use this to demonstrate that the dynamical susceptibility, $\chi^{\prime\prime}(\textbf{\rm{Q}},\omega)$, is well described by the model for short-range, over-damped anisotropic spin-correlations introduced in Reference .
The sample used in this study was a co-aligned set of 12 single crystals produced by solution growth using an FeAs flux [@Ran_2011]. The co-alignment provided a total sample mass of $\sim$1.5 grams and a sample mosaic of 1.5$^{\circ}$ full-width-at-half-maximum. As described in Reference , FeAs-flux samples quenched from the melt at 960$^{\circ}$C, or annealed at temperatures above 700$^{\circ}$ C, transform directly from the T phase into the cT structure at low temperature at ambient pressure; the strain field associated with a uniform distribution of fine-sized FeAs precipitates appears to play a key role in the ambient pressure transformation and can be used to systematically tune the behavior of the Ca(122) samples [@Gati_2012]. For the present measurements, the samples were as-grown, quenched from the melt at 960$^{\circ}$ C. Other than a shift in temperature, the transformation from the T phase to the cT phase at ambient pressure is consistent with the T-cT transformation observed for the Sn-flux solution-grown samples under applied pressure [@Ran_2011], eliminating the need for a pressure cell and, therefore, the dominant contribution it makes to the measured background in scattering measurements.
The inelastic neutron scattering experiment was performed using incident beam energies of 75 meV and 250 meV. The sample was attached to the cold-finger of a closed-cycle cryostat and oriented with the tetragonal **c**-axis parallel to the incident beam. In what follows, the neutron scattering data will be described in the tetragonal $I4/mmm$ coordinate system with $\textbf{Q} = \frac{2\pi}{a}(H + K)\hat{\imath} + \frac{2\pi}{a}(H - K)\hat{\jmath} + \frac{2\pi}{c}L\hat{k} = (H + K,H - K,L)$. In this notation, the stripe-like AFM wavevector is $\textbf{Q}_{\rm{stripe}} = (\frac{1}{2},\frac{1}{2},1)$ \[$H = \frac{1}{2}, K = 0$\]. $H$ and $K$ are defined to conveniently describe diagonal cuts in the $I4/mmm$ basal plane as varying $H$ ($K$) corresponds to a longitudinal \[$H,H$\] scan (transverse \[$K,-K$\] scan) through $\textbf{Q}_{\rm{stripe}}$. It can also be shown that $H$ and $K$ are the reciprocal lattice vectors of the Fe square lattice as discussed in Ref. .
\
\
We performed a detailed survey of the spin fluctuations at temperatures above ($T$ = 150 K) and below ($T$ = 10 K) the T-cT transition (at $\approx$ 90 K) and used the MSLICE software [@Coldea_2004] to visualize the data and to take one and two-dimensional cuts through main crystallographic symmetry directions for subsequent data analysis. Figures \[figure1\] and \[figure2\] display the key result of our measurements. Figures \[figure1\] (a) and (b) show the neutron intensity for constant energy slices (integrated over $\Delta$$E$ = $\pm 10$ meV) for $E_{\rm{i}}$ = 75 meV and $E$ = 50 meV \[Fig. \[figure1\] (a)\], and $E_{\rm{i}}$ = 250 meV and $E$ = 80 meV \[Fig. \[figure1\] (b)\] taken at 150 K, above the T-cT transition. The AFM spin fluctuations centered at $\textbf{Q}_{\rm{stripe}}$ and equivalent positions in other Brillouin zones (for $E_{\rm{i}}$ = 250 meV) are clearly observed. Figures \[figure1\] (c) and (d) show the neutron intensity for these same energy slices taken at $T$ = 10 K in the cT phase, demonstrating the absence of magnetic scattering in the vicinity of $\textbf{Q}_{\rm{stripe}}$, and we find no evidence of magnetic intensity at other positions in reciprocal space.
\
\
Figure \[figure2\] shows the energy dependence of the magnetic intensity along the \[$K,-K$\] direction after averaging over the longitudinal \[$H,H$\] direction from 0.45 $< H <$ 0.55 in reciprocal lattice units (r.l.u). Figures \[figure2\] (a) and (b) show the neutron intensity for $E_{\rm{i}}$ = 75 meV and 250 meV, respectively, taken at $T$ = 150 K. In the T phase, the plume of scattering at $\textbf{Q}_{\rm{stripe}}$ extends above 100 meV \[Fig. \[figure2\] (b)\]. The data taken in the cT phase, at $T$ = 10 K, again show no evidence of magnetic scattering in this region (see also Fig. \[figure4\]). Taken together, Figs. \[figure1\] and \[figure2\] clearly demonstrate that AFM fluctuations are absent in the cT structure consistent with the absence of any Fe moment whatsoever.
The full suppression of magnetism and the absence of SC in the cT phase supports current theories of unconventional pairing in the iron pnictides via spin fluctuations, and raises important questions regarding the origin of SC in the cT phase of (Ca$_{1-x}$Sr$_x$)Fe$_2$As$_2$ with $T_{\rm{c}} \simeq 22$ K [@Jeffries_2012], and (Ca$_{1-x}R_x$)Fe$_2$As$_2$ ($R$ = Pr, Nd) with $T_{\rm{c}} > 45$ K [@Saha_2012]. Both References and acknowledge the possibility of the SC originating in a second phase, perhaps within some retained T phase as found for CaFe$_2$As$_2$ under uniaxial pressure [@Prokes_2010]. On the other hand, the values for $T_{\rm{c}}$ in these systems is significantly higher than that found for CaFe$_2$As$_2$ ($\approx$ 10 K), offering the possibility that SC arises from an alternative pairing scenario. Clearly, it would be instructive to study examples of the Sr and $R$-substituted compounds using the TOF methods described here in order to establish whether remnants of magnetic fluctuations persist into the cT phase.
The absence of magnetic scattering in the cT phase provides us with a non-magnetic analog to serve as a background reference for a detailed investigation of spin fluctuations in the paramagnetic T phase. Figure \[figure3\] displays the energy spectrum for the spin fluctuations in the T phase for both incident neutron energies. These plots were obtained from a subtraction of the data obtained at 150 K and 10 K, then folding the resultant difference spectrum across the diagonals of Fig. \[figure1\]. This folding effectively increases the statistics by taking advantage of the fourfold symmetry of the \[$H,H$\]$-$\[$K,\rm{-}K$\] plane. We note that no adjustment of the data to account for the temperature factor was done in the subtraction because it was not possible to assign relative weights to the temperature dependent (e.g. sample, sample holder) and independent (scattering from the cryostat, general background) contributions to the cT data with any certainty. Nevertheless, the absence of a correction for the temperature factor in the subtraction affects only energies below approximately 15 meV and is of no consequence for the analysis described below. The range of integration over **Q** in Fig. \[figure3\] was $\Delta$$H$ = 0.45 to 0.55 r.l.u. and $\Delta$$K$ = -0.06 to +0.06 r.l.u., for consistency with Ref. . The intensity modulation with energy arises from variations in the structure factor along $L$ which are observed as energy-dependent intensity oscillations that are peaked at the AFM zone centers (e.g. $L$ = 1, 3, 5).[@Diallo_2010].
Complementing these data, in Fig. \[figure4\] we show constant-energy cuts through $\textbf{Q}_{\rm{stripe}}$ along the longitudinal \[$H,H$\] and transverse \[$K,-K$\] directions for energy transfers from 20 to 120 meV. Data taken in the paramagnetic T phase at 150 K (blue circles) are contrasted with the corresponding cuts in the cT phase at 10 K (shaded squares), once again demonstrating the absence of any magnetic signal in the cT phase. Furthermore, the background scattering away from $\textbf{Q}_{\rm{stripe}}$ in the T phase is indistinguishable from the scattering in the cT phase indicating that there is no additional incoherent paramagnetic contribution.
Following Ref. , the **Q** and constant-energy cuts in Figs. \[figure3\] and \[figure4\] can be described by a scattering model that includes short-range and anisotropic spin correlations with overdamped dynamics. The dynamic susceptibility can be written as:
$$\label{eqn1}
\chi^{\prime\prime}(\textbf{\rm{Q}},\omega)=\frac{\hbar\omega\gamma\chi_0}{(\hbar\omega)^2 + \gamma^2\{(q^2+\eta q_xq_y)a^2 + (\frac{\xi_T}{a})^{-2} + \eta_c[1 + \cos(\pi L)]\}^2}$$
where $q^2 = q_{x}^{2} + q_{y}^{2}$, $\chi_0$ is the staggered susceptibility, $\gamma$ denotes the damping coefficient originating from the spin decay into particle-hole excitations, and $\xi_T$ and $a$ are the magnetic correlation length at temperature $T$, and the in-plane lattice parameter, respectively. Two dimensionless parameters describe the anisotropy of the in-plane correlation lengths ($\eta$) and the strength of the interlayer spin correlations ($\eta_c = J_c\chi_0$).
The dynamical structure factor, $S(\textbf{\rm{Q}},\omega)$ is related to $\chi^{\prime\prime}(\textbf{\rm{Q}},\omega)$ by the fluctuation-dissipation theorem, so that: $$\label{eqn2}
S(\textbf{\rm{Q}},\omega) = CF(\textbf{\rm{Q}})^2\frac{\chi^{\prime\prime}(\textbf{\rm{Q}},\omega)}{1-e^{-\hbar\omega/kT}}$$ where $F(\textbf{\rm{Q}})$ is the Fe$^{2+}$ magnetic form factor, $C$ is a scaling constant and E = $\hbar\omega$. Fits to the energy spectrum (solid line in Fig. \[figure3\]) and constant-energy **Q**-cuts (solid lines in Fig. \[figure4\]) were performed simultaneously using Eqns. \[eqn1\] and \[eqn2\] and a single scale factor for each incident energy. We obtained values for $\gamma$ = 37$\pm$2 meV, $\xi_T$ = 6.4$\pm$0.2 [Å]{}, $\eta$ = 1.0$\pm$0.2 and $\eta_c$ = 0.16$\pm$0.02 that compare well with those determined for the paramagnetic T phase at 180 K in Ref. : $\gamma$ = 43$\pm$5 meV, $\xi_T$ = 7.9$\pm$0.1 [Å]{}, $\eta$ = 0.55$\pm$0.36 and $\eta_c$ = 0.20$\pm$0.02. We point out here that the present data set extends to much higher energies than previously measured for Ca(122) and, therefore, provides further validation of the nearly AFM spin fluctuation model proposed by Diallo *et al.* [@Diallo_2010] and, in addition, shows that the spin dynamics of the FeAs flux-grown samples and the original Sn flux-grown samples are the same.
In summary, our inelastic neutron scattering data, over an extended range in reciprocal space and energy, demonstrate that the cT phase of Ca(122) is non-magnetic. Based on an accurate background subtraction using the non-magnetic cT phase, we find no evidence for spin fluctuations at other wave vectors, or any incoherent contribution, and conclude that the magnetic fluctuations are exclusive to $\textbf{Q}_{\rm{stripe}}$ for energies below 120 meV. In light of recent reports of high-temperature SC in the cT phase of (Ca$_{1-x}$Sr$_x$)Fe$_2$As$_2$ [@Jeffries_2012] and (Ca$_{1-x}R_x$)Fe$_2$As$_2$ ($R$ = Pr, Nd) [@Saha_2012] the absence of spin fluctuations in the cT phase of Ca(122) clearly calls for further consideration of multiple phases as the source of SC in these systems as well as similar neutron TOF measurements on these compounds.
Work at the Ames Laboratory was supported by the Department of Energy, Basic Energy Sciences under Contract No. DE-AC02-07CH11358. A portion of this research at ORNL’s Spallation Neutron Source was sponsored by the Scientific User Facilities Division, Office of Basic Energy Sciences, U.S. Department of Energy.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Statistical techniques are used in all branches of science to determine the feasibility of quantitative hypotheses. One of the most basic applications of statistical techniques in comparative analysis is the test of equality of two population means, generally performed under the assumption of normality. In medical studies, for example, we often need to compare the effects of two different drugs, treatments or preconditions on the resulting outcome. The most commonly used test in this connection is the two sample $t$-test for the equality of means, performed under the assumption of equality of variances. It is a very useful tool, which is widely used by practitioners of all disciplines and has many optimality properties under the model. However, the test has one major drawback; it is highly sensitive to deviations from the ideal conditions, and may perform miserably under model misspecification and the presence of outliers. In this paper we present a robust test for the two sample hypothesis based on the density power divergence measure [@MR1665873], and show that it can be a great alternative to the ordinary two sample $t$-test. The asymptotic properties of the proposed tests are rigorously established in the paper, and their performances are explored through simulations and real data analysis.'
author:
- 'A. Basu'
- 'A. Mandal'
- 'N. Martin'
- 'L. Pardo'
bibliography:
- 'reference.bib'
date: 'September 28, 2014'
title: '**Robust Tests for the Equality of Two Normal Means based on the Density Power Divergence** '
---
**:** 62F35, 62F03.
: Robustness, Density Power Divergence, Hypothesis Testing.
Introduction: Motivation and Background
=======================================
In many scientific studies, often the main problem of interest is to compare different population groups. In medical studies, for example, the primary research problem could be to test for the difference between the location parameters of two different populations receiving two different drugs, treatments or therapy, or having two different preconditions. The normal distribution often provides the basic setup for statistical analyses in medical studies (as well as in other disciplines). Inference procedures based on the sample mean, the standard deviation and the one and two-sample $t$-tests are often the default techniques for the scenarios where they are applicable. In particular, the two sample $t$-test is the most popular technique in testing for the equality of two means, performed under the assumption of equality of variances. Its applicability in real life situations is, however, tempered by the known lack of robustness of this test against model perturbations. Even a small deviation from the ideal conditions can make the test completely meaningless and lead to nonsensical results. This problem is caused by the fact that the $t$-test is based on the classical estimates of the location and scale parameters (the sample mean and the sample standard deviation). Large outliers tend to distort the mean and inflate the standard deviation. This may lead to false results of both types, i.e. detecting a difference when there isn’t one, and failing to detect a true significance.
In this paper we are going to develop a class of robust tests for the two sample problem which evolves from an appropriate minimum distance technique in a natural way. This class of tests is indexed by two real parameters $%
\beta $ and $\gamma $, and we will constrain each of these parameters to lie within the $[0,1]$ interval. Our general minimum distance approach will allow us to study the likelihood ratio test in an asymptotic sense, as the likelihood ratio test is asymptotically equivalent to the test generated by the parameters $%
\beta =\gamma =0$. Normally we will work with the one parameter family of test statistics corresponding to $%
\beta =\gamma $; the outlier stability of the proposed tests increase with the tuning parameter $\gamma $.
Let $X$ and $Y$ be independent random variables whose distributions are modeled as normals having unknown means $\mu_1$ and $\mu_2$, respectively, with an unknown but common variance $\sigma^2$. We are interested in testing the null hypothesis $$H_{0}:\mu_1=\mu_2\text{ against }H_{1}:\mu_1\neq \mu_2,
\label{EQ:0}$$ under the above set up. It is well known that the exact two sample $t$-test (which is equivalent to the likelihood ratio test) rejects the null hypothesis in (\[EQ:0\]) if and only if $$t=\frac{\left\vert \bar{X}-\bar{Y}\right\vert }{S_{p}\sqrt{\frac{1%
}{n_1}+\frac{1}{n_2}}}>t_{\frac{\alpha }{2}}(n_1+n_2-2),$$ where $\bar{X}$ and $\bar{Y}$ are the sample means corresponding to the random samples $X_{1},X_{2},\ldots ,X_{n_1}$ and $%
Y_{1},Y_{2},\ldots ,Y_{n_2}$ obtained from the two distributions, $$S_{p}^{2}=\frac{(n_1-1)S_{1}^{2}+(n_2-1)S_{2}^{2}}{n_1+n_2-2},$$ $$S_{1}^{2}=\frac{1}{n_1-1}\sum_{i=1}^{n_1}\left( X_i-\bar{X}%
\right) ^{2},\quad S_{2}^{2}=\frac{1}{n_2-1}\sum_{i=1}^{n_2}\left( Y_{i}-%
\bar{Y}\right) ^{2},$$and $t_{\frac{\alpha }{2}}(n_1+n_2-2)$ is the $100(1-\frac{\alpha }{2})$-th quantile of the $t$-distribution with $n_1+n_2-2$ degrees of freedom. The $t$-test is the uniformly most powerful unbiased and invariant test for this hypothesis. Testing the equality of means of independent normal populations with unknown variances which are not necessarily equal, is referred to as the Behrens-Fisher problem.
In this paper we will use the density power divergence (DPD) measure [@MR1665873], which provides a natural robustness option for many standard inference problems. The density power divergence and its variants have been successfully used by many authors in a variety of inference problems; see, eg. [@MR1859416], [@MR2299175; @MR2466551], [@MR3011625; @basu2013], [@MR3117102]. However, the two sample problem requires a non-trivial extension of the currently existing techniques. Our purpose in this paper is to derive the asymptotic properties of the class of two sample tests based on the density power divergence and demonstrate their robust behavior in practical situations.
**Example 1 (Cloth Manufacturing data)**: In order to emphasize the need for applications early, we now present a motivational example. This example illustrates the use of quality control methods practiced in a clothing manufacturing plant. Levi-Strauss manufactures clothing from cloth supplied by several mills. The data used in this example (see Table [TAB:Staudte\_Sheather]{}) are for two of these mills and were obtained from the quality control department of the Levi plant in Albuquerque, New Mexico ([@lambert1987introduction], p. 86). In order to maintain the anonymity of these two mills we have coded them $A$ and $B$. A measure of wastage due to defects in cloth and so on is called *run-up*. It is quoted as percentage of wastage per week and is measured relative to computerized layouts of patterns on the cloth. Since the people working in the plant can often beat the computer in reducing wastage by laying out the patterns by hand, it is possible for run-up to be negative. From the viewpoint of quality control, it is desirable not only that the run-up be small but that the quality from week to week be fairly consistent. There are 22 measurements on run-up for each of the two mills and they are presented in Table \[TAB:Staudte\_Sheather\]. The $t$-test for the equality of the two means against the two-sided alternative has a $p$-value of 0.3428 and fails to reject the null hypothesis; however, when the presumed outliers (presented in bold fonts in Table \[TAB:Staudte\_Sheather\]) are removed from the dataset, the same two-sample $t$-test produces a $p$-value of 0.0308, leading to clear rejection. Choosing $\beta = \gamma$ to be the only parameter, the $p$-values of the DPD tests (to be developed in the next section) for testing the same hypotheses are presented in Figure [fig:Staudte\_Sheather\_book\_p\_val]{} as a function of $\gamma$. It is observed that the $p$-values of the tests with the full data and those with the outlier deleted data are practically identical for $\gamma = 0.2$ or larger, and lead to solid rejection. Thus, while the outliers mask the significance in case of the two sample $t$-test, the more robust DPD tests are able to capture the same.
-------- -------- --------- --------- ------------- --------- --------- ------------- ------------- --------- --------- --------
Mill A $0.12$ $1.01$ $-0.20$ $0.15$ $-0.30$ $-0.07$ $0.32$ $% $-0.32$ $-0.17$ $0.24$
0.27$
$0.03$ $0.35$ $-0.08$ $\bf{2.94}$ $0.28$ $1.30$ $\bf{4.27}$ $0.14$ $% $0.24$ $0.13$
0.30$
Mill B $1.64$ $-0.60$ $-1.16$ $-0.13$ $0.40$ $1.70$ $0.38$ $% $1.04$ $0.42$ $0.85$
0.43$
$0.63$ $0.90$ $0.71$ $0.43$ $1.97$ $0.30$ $0.76$ $\bf{7.02}$ $% $0.60$ $0.29$
0.85$
-------- -------- --------- --------- ------------- --------- --------- ------------- ------------- --------- --------- --------
: Cloth Manufacturing data.[]{data-label="TAB:Staudte_Sheather"}
Our primary motivation for studying the alternatives of the two sample $t$-test has been the need for developing such a test in the context of examples relating to medical data. However, examples abound in practically all scientific disciplines showing that this is a real necessity which is certainly not restricted to the medical field. The example considered above is one such, where the context does not have anything directly to do with a medical problem, but the importance of the problem and the need for a robust solution can immediately be appreciated.
The rest of the paper is organized as follows: In Section \[SEC:MDPDE\] the asymptotic distribution of the minimum DPD estimators in the two sample situation is described. In Section \[SEC:Test\] we introduce our robust two sample test statistic and develop the necessary theory. A large number of real data examples and extensive simulation results are presented in Section \[SEC:numerical\]. Finally Section \[SEC:concluding\] has some concluding remarks.
The Minimum DPD Estimator: Asymptotic Distribution {#SEC:MDPDE}
==================================================
For any two probability density functions $f$ and $g$, the density power divergence measure is defined, as the function of a single tuning parameter $\beta \geq 0$, as $$d_{\beta}(g,f)=\left\{
\begin{array}
[c]{ll}%
\int\left\{ f^{1+\beta}(x)-\left( 1+\frac{1}{\beta}\right) f^{\beta
}(x)g(x)+\frac{1}{\beta}g^{1+\beta}(x)\right\} dx, & \text{for}%
\mathrm{~}\beta>0,\\[2ex]%
\int g(x)\log\left( \displaystyle\frac{g(x)}{f(x)}\right) dx, &
\text{for}\mathrm{~}\beta=0.
\end{array}
\right. \label{EQ:definition_DPD}%$$ Let $X_{1},X_{2},\ldots ,X_n$ be a random sample of size $n$ from a $\mathcal{N}(\mu,\sigma^2)$ distribution, where both parameters are unknown. Let $f_{\mu,\sigma }(x)$ represent the density function of a $\mathcal{N}(\mu,\sigma^2)$ variable. For a given $\beta $, we get the minimum density power divergence estimators (MDPDEs) $\widehat{\mu }%
_{\beta }$ and $\widehat{\sigma }_{\beta}$ of $\mu$ and $\sigma$ by minimizing the following function over $\mu$ and $\sigma$ $$\int_{\mathbb{R}}f_{\mu,\sigma }^{1+\beta }(x)dx-\left( 1+\frac{1}{%
\beta }\right) \frac{1}{n}\sum_{i=1}^{n}f_{\mu,\sigma }^{\beta
}(X_i),\text{\qquad for }\beta >0, \label{1}$$ and $$-\frac{1}{n}\sum_{i=1}^{n}\log f_{\mu,\sigma }(X_i),\text{%
\qquad for }\beta =0. \label{EQ:1.0}$$ For $\beta =0$, the objective function in (\[EQ:1.0\]) is the negative of the usual log likelihood and has the classical maximum likelihood estimator as the minimizer. For a normal density the function in (\[1\]) simplifies to $$h_{n,\beta }(\mu,\sigma )=\frac{1}{\sigma ^{\beta }(2\pi )^{\frac{%
\beta }{2}}}\left\{ \frac{1}{\left( 1+\beta \right) ^{3/2}}-\frac{1}{%
n\beta }\sum_{i=1}^{n}\exp \left( -\frac{1}{2}\left( \frac{X_i-\mu}{{\sigma }}\right) ^{2}\beta \right) \right\} .$$ In order to get $\widehat{\mu }_{\beta }$ and $\widehat{\sigma}_\beta $, we have to solve the estimating equation $$\mathbf{h'}_{n,\beta }(\widehat{\mu }_{\beta },\widehat{\sigma}_\beta ) =%
\begin{pmatrix}
_{1}h'_{n,\beta }(\widehat{\mu }_{\beta },\widehat{\sigma}_\beta ) \\
_{2}h'_{n,\beta }(\widehat{\mu }_{\beta },\widehat{\sigma}_\beta )%
\end{pmatrix} = \boldsymbol{0}_2,
\label{h1}$$ where $$_{1}h_{n,\beta }^{\prime }(\widehat{\mu }_{\beta },\widehat{\sigma}_\beta ) =\left. \frac{%
\partial h_{n,\beta }(\mu,\widehat{\sigma}_\beta )}{\partial \mu}%
\right\vert _{\mu=\widehat{\mu }_{\beta }},\qquad _{2}h_{n,\beta }^{\prime }(\widehat{\mu }_{\beta },\widehat{\sigma}_\beta )
=\left. \frac{\partial h_{n,\beta }(\widehat{\mu }_{\beta },\sigma )}{%
\partial \sigma }\right\vert _{\sigma =\widehat{\sigma}_\beta },$$ and $\boldsymbol{0}_2$ represents a zero vector of length 2. We denote $$\mathbf{H}_{n,\beta }( \mu_0,\sigma_0 ) =\left(
\begin{array}{cc}
_{11}h_{n,\beta }^{\prime \prime }\left( \mu_0,\sigma_0\right) &
_{12}h_{n,\beta }^{\prime \prime }\left( \mu_0,\sigma_0\right)
\\
_{21}h_{n,\beta }^{\prime \prime }\left( \mu_0,\sigma_0\right) &
_{22}h_{n,\beta }^{\prime \prime }\left( \mu_0,\sigma_0\right)
\end{array}%
\right),$$where $$\begin{aligned}
_{11}h_{n,\beta }^{\prime \prime }\left( \mu_0,\sigma_0\right) &
=\left. \dfrac{\partial ^{2}h_{n,\beta }\left( \mu,\sigma
_{0}\right) }{\partial \mu^{2}}\right\vert _{\mu =\mu_0},\qquad
_{12}h_{n,\beta }^{\prime \prime }\left( \mu_0,\sigma_0\right)
=\left. \dfrac{\partial ^{2}h_{n,\beta }\left( \mu,\sigma \right) }{%
\partial \mu\partial \sigma }\right\vert _{\mu =\mu_0,\sigma
=\sigma_0}, \\
_{21}h_{n,\beta }^{\prime \prime }\left( \mu_0,\sigma_0\right) &
=\left. \dfrac{\partial ^{2}h_{n,\beta }\left( \mu,\sigma \right) }{%
\partial \sigma \partial \mu}\right\vert _{\mu =\mu_0,\sigma
=\sigma_0},\qquad _{22}h_{n,\beta }^{\prime \prime }\left( \mu
_{0},\sigma_0\right) =\left. \dfrac{\partial ^{2}h_{n,\beta }\left(
\mu_0,\sigma \right) }{\partial \sigma^2}\right\vert _{\sigma =\sigma
_{0}}.\end{aligned}$$Using a Taylor series expansion of the function in equation (\[h1\]), it is easy to show that $$\begin{aligned}
\sqrt{n}
\begin{pmatrix}
\widehat{\mu }_{\beta } - \mu_0\\
\widehat{\sigma}_\beta - \sigma_0
\end{pmatrix}
&=& \sqrt{n} \mathbf{H}_{n,\beta }^{-1}( \mu_0,\sigma_0 ) \boldsymbol{h}'_{n,\beta }(\mu_0,\sigma_0 ) + o_p(1) \nonumber\\
&=& \sqrt{n} \mathbf{J}_\beta^{-1}( \sigma_0 ) \boldsymbol{h}'_{n,\beta }(\mu_0,\sigma_0 ) + o_p(1),
\label{muSigma}\end{aligned}$$ where $$\boldsymbol{J}_{\beta }(\sigma_0) = \lim_{n \rightarrow \infty }%
\mathbf{H}_{n,\beta } ( \mu_0,\sigma_0 )
= \frac{1}{%
\sqrt{1+\beta }\left( 2\pi \right) ^{\beta /2}\sigma_0 ^{2+\beta }}\left(
\begin{array}{cc}
\frac{1}{1+\beta } & 0 \\
0 & \frac{\beta ^{2}+2}{\left( 1+\beta \right) ^{2}}%
\end{array}%
\right).$$ The joint distribution of $\widehat{\mu }_{\beta }$ and $\widehat{\sigma}_\beta$ then follows (see [@MR3011625]) from the result that $$\sqrt{n}\mathbf{h'}_{n,\beta}(\mu _0,\sigma_0)\underset{%
n\rightarrow \infty }{\overset{\mathcal{L}}{\longrightarrow }}\mathcal{N}%
\left( \boldsymbol{0}_{2},\boldsymbol{K}_{\beta }(\sigma_0)\right) ,
\label{1.1}$$ where $$\begin{aligned}
\boldsymbol{K}_{\beta }(\sigma_0) &=& \left( K_{ij,\beta
}(\sigma_0)\right) _{i,j=1,2} \nonumber\\
&=&
\frac{1}{\sigma_0 ^{2+2\beta
}\left( 2\pi \right) ^{\beta }}\left( \frac{1}{(1+2\beta )^{3/2}}\left(
\begin{array}{cc}
1 & 0 \\
0 & \frac{4\beta ^{2}+2}{1+2\beta }%
\end{array}%
\right) -\left(
\begin{array}{cc}
0 & 0 \\
0 & \frac{\beta ^{2}}{(1+\beta )^{3}}%
\end{array}%
\right) \right) .
\label{2}\end{aligned}$$ We will use the above results to obtain the MDPDEs of the parameters in the two sample setup mentioned below.
Suppose $X_{1},X_{2},\ldots ,X_{n_1}$ is a random sample of size $n_1$ from $X$ which has a $\mathcal{N}(\mu_1,\sigma^2)$ distribution, and $%
Y_{1},Y_{2},\ldots ,Y_{n_2}$ is a random sample of size $n_2$ from $Y$ which has a $\mathcal{N}(\mu_2,\sigma^2)$ distribution; all three parameters are unknown. Let $f_{\mu_1,\sigma }(x)$ and $f_{\mu
_{2},\sigma }(y)$ be the density functions of $X$ and $Y$ respectively. Let us denote the set of unknown parameters by $\boldsymbol{\eta }=(\mu_1,\mu
_{2},\sigma )^{T}$. The MDPDE of $\boldsymbol{\eta }$, denoted by $\widehat{\boldsymbol{\eta }}_{\beta }=(\widehat{\mu}_{1\beta },\widehat{\mu}_{2\beta
},\widehat{\sigma}_{\beta })^{T}$, is obtained by minimizing the following function $$h_{n_1,n_2,\beta }(\boldsymbol{\eta })=\frac{1}{n_1+n_2}%
\left( n_{1\text{ }}h_{n_1,\beta }(\mu_1,\sigma )+n_{2\text{ }%
}h_{n_2,\beta }\left( \mu_2,\sigma \right) \right) .
\label{hn12}$$ It may be noticed that $\hat{\mu}_{1\beta }$ is based only on the first term of the above function, and similarly $\hat{\mu}_{2\beta }$ depends only on the second term. Therefore, the estimating equations are given by $_{1}h_{n_{i},\beta }^{\prime }\left( \mu _{i},\sigma \right) =0$, $i=1,2$, and $_{2}h_{n_1,n_2,\beta }^{\prime }(\boldsymbol{\eta } )=0$, where $$_{2}h_{n_1,n_2,\beta }^{\prime }(\boldsymbol{\eta })=\frac{%
\partial h_{n_1,n_2,\beta }(\boldsymbol{\eta } )}{\partial \sigma }%
=\frac{1}{n_1+n_2}\left( n_1\,\allowbreak _{2}h_{n_1,\beta }^{\prime
}\left( \mu_1,\sigma \right) +n_2\,\allowbreak _{2}h_{n_2,\beta
}^{\prime }\left( \mu_2,\sigma \right) \right) .
\label{h2n12}$$ For $\beta =0$, the above equations can be explicitly solved to get the MDPDEs for this case. It is easily seen that $\widehat{\mu }_{10}=\bar{X%
}$ and $\widehat{\mu }_{20}=\bar{Y}$. Moreover, using equation (\[EQ:1.0\]) we get from (\[hn12\]) $$\begin{aligned}
& h_{n_1,n_2,\beta =0}(\widehat{\boldsymbol{\eta }}_0 ) \\
& =-\frac{1}{n_1+n_2}\left( n_1\frac{1}{n_1}\log
\prod_{i=1}^{n_1}f_{\widehat{\mu }_{10},\widehat{\sigma}_0 }(X_i)+n_2\frac{1}{n_2}\log
\prod_{i=1}^{n_2}f_{\widehat{\mu }_{20},\widehat{\sigma}_0 }(Y_{i})\right) \\
& =\frac{1}{n_1+n_2}\left( (n_1+n_2)\log \widehat{\sigma}_0 +\sum_{i=1}^{n_1}%
\frac{\left( X_i-\bar{X}\right) ^{2}}{2\widehat{\sigma}_0^2}+\sum_{i=1}^{n_2}%
\frac{\left( Y_{i}-\bar{Y}\right) ^{2}}{2\widehat{\sigma}_0^2}+(n_1+n_2)\log
\sqrt{2\pi }\right) .\end{aligned}$$ So, $$_{2}h_{n_1,n_2,\beta }^{\prime }(\widehat{\boldsymbol{\eta }}_0 )
=\frac{1}{\widehat{\sigma}_0 }-\frac{1}{%
\widehat{\sigma}_0 ^{3}(n_1+n_2)}\left\{ (n_1-1)S_{1}^{2} + (n_2-1)S_{2}^{2}\right\} ,$$which leads to the solution $$\widehat{\sigma }_{0}=\left( \frac{(n_1-1)S_{1}^{2}+(n_2-1)S_{2}^{2}}{%
n_1+n_2}\right) ^{\frac{1}{2}}. \label{sig0}$$ Therefore, for $\beta=0$ the MDPDEs turn out to be the MLEs of the corresponding parameters. The following theorem gives the asymptotic distribution of the MDPDE of $\boldsymbol{\eta }$ for a given $\beta$.
\[Th0\]We consider two normal populations with unknown means $\mu_1$ and $\mu_2$ and unknown but common variance $\sigma^2.$ Let $$w=\lim_{n_1,n_2\rightarrow \infty }\frac{n_1}{n_1+n_2}
\label{EQ:w}$$ be the limiting proportion of observations from the first population in the whole sample. We assume that $w \in (0,1)$. Then, the minimum density power divergence estimator $\widehat{\boldsymbol{\eta }}_{\beta }$ of $\boldsymbol{\eta}$ has the asymptotic distribution given by $$\sqrt{\frac{n_1n_2}{n_1+n_2}}(\widehat{\boldsymbol{\eta }}_{\beta }-%
\boldsymbol{\eta }_{0})\underset{n_1,n_2\rightarrow \infty }{\overset{%
\mathcal{L}}{\longrightarrow }}\mathcal{N}\left( \boldsymbol{0}_{3},%
\boldsymbol{\Sigma }_{w,\beta }(\sigma_0)\right) , \label{eqTh0}$$where $\boldsymbol{\eta }_{0}=(\mu _{10},\mu _{20},\sigma_0)^{T}$ is the true value of $\boldsymbol{\eta }$, and $$\boldsymbol{\Sigma }_{w,\beta }(\sigma_0)=\sigma_0^{2}\left(
\begin{array}{ccc}
\left( 1-w\right) \frac{\left( \beta +1\right) ^{3}}{\left( 2\beta +1\right)
^{\frac{3}{2}}} & 0 & 0 \\
0 & w\frac{\left( \beta +1\right) ^{3}}{\left( 2\beta +1\right) ^{\frac{3}{2}%
}} & 0 \\
0 & 0 & w\left( 1-w\right) \frac{\left( \beta +1\right) ^{5}}{\left( \beta
^{2}+2\right) ^{2}}\left( \frac{4\beta ^{2}+2}{(1+2\beta )^{5/2}}-\frac{%
\beta ^{2}}{(1+\beta )^{3}}\right)%
\end{array}%
\right) . \label{8}$$
See Appendix.
The Asymptotic Distribution of the DPD Test Statistic {#SEC:Test}
=====================================================
Let $f_{\mu_1,\sigma _{1}}(x)$ and $f_{\mu_2,\sigma _{2}}(y)$ be the density functions of $X\sim \mathcal{N}(\mu_1,\sigma _{1})$ and $Y\sim
\mathcal{N}(\mu_2,\sigma _{2})$ respectively. The density power divergence measure between the densities of $X$ and $Y$, for $\gamma >0$, is given by $$\begin{aligned}
d_{\gamma }(f_{\mu_1,\sigma _{1}},f_{\mu_2,\sigma _{2}}) =&\frac{1}{%
\sigma _{2}^{\gamma }\sqrt{1+\gamma }\left( 2\pi \right) ^{\gamma /2}}+%
\frac{1}{\gamma\sigma _{1}^{\gamma }\sqrt{1+\gamma }\left( 2\pi \right)
^{\gamma /2}} \\
& -\frac{\gamma +1}{\gamma \sigma _{2}^{\gamma -1}(\gamma \sigma
_{1}^{2}+\sigma _{2}^{2})^{1/2}\left( 2\pi \right) ^{\gamma/2 }} \\
& \times \exp \left\{ \frac{1}{2}\left[ -\left( \tfrac{\mu_2^{2}}{\left(
\frac{\sigma _{2}}{\sqrt{\gamma }}\right) ^{2}}+\tfrac{\mu_1^{2}}{\sigma
_{1}^{2}}\right) +\tfrac{\left( \sigma _{1}^{2}\mu_2+\mu_1\left( \frac{%
\sigma _{2}}{\sqrt{\gamma }}\right) ^{2}\right) ^{2}}{\left( \sigma
_{1}^{2}+\left( \frac{\sigma _{2}}{\sqrt{\gamma }}\right) ^{2}\right) \left(
\frac{\sigma _{2}}{\sqrt{\gamma }}\right) ^{2}\sigma _{1}^{2}}\right]
\right\} ,\end{aligned}$$ and for $\gamma =0$$$d_\gamma(f_{\mu_1,\sigma _{1}},f_{\mu_2,\sigma _{2}})=\log {\frac{\sigma
_{2}}{\sigma _{1}}}-\frac{1}{2}+\frac{\sigma _{1}^{2}}{2\sigma _{2}^{2}}+%
\frac{1}{2\sigma _{2}^{2}}(\mu_1-\mu_2)^{2}.$$To test the null hypothesis given in (\[EQ:0\]), under the assumption that $\sigma _{1}=\sigma _{2}=\sigma $, we will consider the divergence between the two normal populations with the estimated parameters; this yields $$d_{\gamma }(f_{\widehat{\mu }_{1\beta },\widehat{\sigma}_\beta },f_{%
\widehat{\mu }_{2\beta },\widehat{\sigma}_\beta })=\left\{
\begin{array}{ll}
\frac{\sqrt{1+\gamma }}{\gamma \left( \sqrt{2\pi }\widehat{\sigma }_{\beta
}\right) ^{\gamma }}\left[ 1-\exp \left\{ -\frac{\gamma }{2(\gamma
+1)}\left( \frac{\widehat{\mu }_{1\beta }-\widehat{\mu }_{2\beta }}{\widehat{%
\sigma }_{\beta }}\right) ^{2}\right\} \right] , & \text{ for }\gamma >0, \\
\frac{1}{2}\left( \frac{\widehat{\mu }_{1\beta }-\widehat{\mu }_{2\beta }}{%
\widehat{\sigma}_\beta }\right) ^{2}, & \text{ for }\gamma =0.%
\end{array}%
\right. \label{EQ:initial_statistic}$$Naturally, we will reject the null hypothesis for large values of $d_{\gamma }(f_{\widehat{\mu }%
_{1\beta },\widehat{\sigma}_\beta },f_{\widehat{\mu }_{2\beta },\widehat{%
\sigma }_{\beta }})$. To propose the test in a very general setup we have considered two possibly distinct tuning parameters $\gamma$ and $\beta$ in the above expression; the parameter $\gamma$ represents the tuning parameters of the divergence, and the parameter $\beta$ represents the tuning parameter of the MDPDEs. In order to determine the critical region of this test we will find (later in Theorem \[Th2\]) the asymptotic null distribution of the test statistic based on ([EQ:initial\_statistic]{}), standardized with a suitable scaling constant involving $n_1$ and $n_2$.
\[Th1\] For $\gamma >0$, let us define $\boldsymbol{t}_{\gamma }\left( \boldsymbol{\eta }\right) =(t_{\gamma, 1}(%
\boldsymbol{\eta }),t_{\gamma, 2}(\boldsymbol{\eta }),t_{\gamma, 3}(%
\boldsymbol{\eta }))^{T}$, with$$\begin{aligned}
t_{\gamma, 1}(\boldsymbol{\eta })& =\frac{\frac{\mu_1-\mu_2}{\sigma }}{%
\sqrt{1+\gamma }\left( \sqrt{2\pi }\right) ^{\gamma }\sigma ^{\gamma +1}}%
\exp \left\{ -\frac{1}{2}\tfrac{\gamma }{\gamma +1}\left( \tfrac{\mu
_{1}-\mu_2}{\sigma }\right) ^{2}\right\} , \label{t1} \\
%
t_{\gamma, 2}(\boldsymbol{\eta })& =-t_{1}(\boldsymbol{\eta }), \label{t2} \\
%
t_{\gamma, 3}(\boldsymbol{\eta })& = - \tfrac{\sqrt{1+\gamma }}{\left( \sqrt{%
2\pi }\right) ^{\gamma }\sigma ^{\gamma +1}}\left[ 1-\left( 1 - \tfrac{1}{%
1+\gamma }\left( \tfrac{\mu_1-\mu_2}{\sigma }\right) ^{2}\right) \exp
\left\{ -\tfrac{1}{2}\tfrac{\gamma }{\gamma +1}\left( \tfrac{\mu_1-\mu
_{2}}{\sigma }\right) ^{2}\right\} \right] . \label{t3}\end{aligned}$$Then, for $w \in (0,1)$ as defined in (\[EQ:w\]) we have $$\sqrt{\frac{n_1n_2}{n_1+n_2}}\left( d_{\gamma }(f_{\widehat{\mu }%
_{1\beta },\widehat{\sigma}_\beta },f_{\widehat{\mu }_{2\beta },\widehat{%
\sigma }_{\beta }})-d_{\gamma }(f_{\mu _{10},\sigma_0},f_{\mu
_{20},\sigma_0})\right) \underset{n_1,n_2\rightarrow \infty }{\overset%
{\mathcal{L}}{\longrightarrow }}\mathcal{N}\left( 0, \sigma_\gamma^2
\right) , \label{resTh1}$$ where $$\sigma_\gamma^2 = \boldsymbol{t}_{\gamma
}^{T}\left( \boldsymbol{\eta }_{0}\right) \boldsymbol{\Sigma }_{w,\beta
}(\sigma_0)\boldsymbol{t}_{\gamma }\left( \boldsymbol{\eta }_{0}\right),
\label{sigma_gamma}$$ and $\boldsymbol{\Sigma }_{w,\beta }(\sigma_0)$ is given in (\[8\]).
See Appendix.
Notice that $\boldsymbol{t}_{\gamma }^{T}\left( \boldsymbol{\eta }%
_{0}\right) \boldsymbol{\Sigma }_{w,\beta }(\sigma_0)\boldsymbol{t}%
_{\gamma }\left( \boldsymbol{\eta }_{0}\right) \geq 0$. If $\mu _{10}\neq
\mu _{20}$, we observe that $\boldsymbol{t}_{\gamma }\left( \boldsymbol{%
\eta }_{0}\right) \neq \boldsymbol{0}_{3}$, and since $\boldsymbol{\Sigma }%
_{w,\beta }(\sigma_0)$ is positive definite matrix, we have $\boldsymbol{t}%
_{\gamma }^{T}\left( \boldsymbol{\eta }_{0}\right) \boldsymbol{\Sigma }%
_{w,\beta }(\sigma_0)\boldsymbol{t}_{\gamma }\left( \boldsymbol{\eta }%
_{0}\right) >0$. But for $\mu _{10}=\mu _{20}$, $\boldsymbol{t}%
_{\gamma }\left( \boldsymbol{\eta }_{0}\right) =\boldsymbol{0}_{3}$, and hence $\boldsymbol{t}_{\gamma }^{T}\left( \boldsymbol{\eta }_{0}\right)
\boldsymbol{\Sigma }_{w,\beta }(\sigma_0)\boldsymbol{t}_{\gamma }\left(
\boldsymbol{\eta }_{0}\right) =0$. Therefore, to get the asymptotic distribution of the test statistic under the null hypothesis we need a higher order scaling involving $n_1$ and $n_2$ to the quantity given in (\[EQ:initial\_statistic\]).
\[Th2\]Let $w\in (0,1)$ as defined in (\[EQ:w\]) and $\gamma >0$. Then, under the null hypothesis, we have $$S_{\gamma }\left( \widehat{\mu }_{1\beta },\widehat{\mu }_{2\beta },\widehat{%
\sigma }_{\beta }\right) =\frac{2n_1n_2}{n_1+n_2}\frac{d_{\gamma
}(f_{\widehat{\mu }_{1\beta },\widehat{\sigma}_\beta },f_{\widehat{\mu }%
_{2\beta },\widehat{\sigma}_\beta })}{\lambda _{\beta ,\gamma
}\,\allowbreak (\widehat{\sigma}_\beta )}\underset{n_1,n_2\rightarrow
\infty }{\overset{\mathcal{L}}{\longrightarrow }}\chi ^{2}(1),
\label{EQ:test}$$where $$\lambda _{\beta ,\gamma }(\widehat{\sigma}_\beta )=\frac{\left( \beta
+1\right) ^{3}\left( 2\beta +1\right) ^{-\frac{3}{2}}}{\widehat{\sigma }%
_{\beta }^{\gamma }\left( 2\pi \right) ^{\frac{\gamma }{2}}\left( \gamma
+1\right) ^{\frac{1}{2}}}.
\label{lambda1}$$
See Appendix.
The above result indicates that the density power divergence test for the hypothesis in (\[EQ:0\]) can be based on the statistic $S_{\gamma }\left(
\widehat{\mu }_{1\beta },\widehat{\mu }_{2\beta },\widehat{\sigma }_{\beta
}\right) $, where the critical region corresponding to significance level $%
\alpha $ is given by the set of points satisfying $$S_{\gamma }\left( \widehat{\mu }_{1\beta },\widehat{\mu }_{2\beta },\widehat{%
\sigma }_{\beta }\right) >\chi _{\alpha }^{2}(1).$$ Using the result of Theorem \[Th1\] we can get an approximation of the power function of the test statistic. We consider $\mu _{10}\neq \mu _{20}$. In the following we will let $\lambda$ denote the quantity defined in equation (\[lambda1\]) to keep the notation simple. The power function is then given by $$\begin{aligned}
\eta_{\gamma,\beta}(\mu _{10}, \mu _{20}, \sigma_0) &=& P\left( S_{\gamma }\left( \widehat{\mu }_{1\beta },\widehat{\mu }_{2\beta },\widehat{%
\sigma }_{\beta }\right) >\chi _{\alpha }^{2}(1) \right) \\
%
&=& P\left( \frac{2}{\lambda}\frac{n_1n_2}{n_1+n_2} d_{\gamma }(f_{\widehat{\mu }%
_{1\beta },\widehat{\sigma}_\beta },f_{\widehat{\mu }_{2\beta },\widehat{%
\sigma }_{\beta }}) >\chi _{\alpha }^{2}(1) \right) \\
%
&=& P\Bigg( \sqrt{\frac{n_1n_2}{n_1+n_2}}\left( d_{\gamma }(f_{\widehat{\mu }%
_{1\beta },\widehat{\sigma}_\beta },f_{\widehat{\mu }_{2\beta },\widehat{%
\sigma }_{\beta }})-d_{\gamma }(f_{\mu _{10},\sigma_0},f_{\mu
_{20},\sigma_0})\right) \\
%
&& \ \ \ > \frac{\lambda}{2} \sqrt{\frac{n_1 + n_2}{n_1n_2}} \left(\chi _{\alpha }^{2}(1)
- \frac{2 n_1 n_2}{\lambda (n_1 + n_2)} d_{\gamma }(f_{\mu _{10},\sigma_0},f_{\mu
_{20},\sigma_0}) \right)\Bigg) \\
%
&= & 1 - \Phi_n\left( \frac{\lambda}{2\sigma_\gamma}\sqrt{\frac{n_1n_2}{n_1+n_2}} \left(\chi _{\alpha }^{2}(1)
- \frac{2 n_1n_2}{n_1+n_2} d_{\gamma }(f_{\mu _{10},\sigma_0},f_{\mu
_{20},\sigma_0}) \right)\right),\end{aligned}$$ where $\Phi _{n}$ is a sequence of distributions functions tending uniformly to the standard normal distribution function $\Phi$, and $\sigma_\gamma$ is defined in (\[sigma\_gamma\]). We observe that if $\mu _{10}\neq \mu _{20}$ $$\lim_{n_1,n_2\rightarrow \infty }\eta_{\gamma,\beta}(\mu _{10}, \mu _{20}, \sigma_0) =1.$$ Therefore, the test is consistent in the Frasar’s sense [@MR0093863].
\[Cor1\]Let $w\in (0,1)$ as defined in (\[EQ:w\]) and $\gamma =\beta
=0$. Then, under the null hypothesis defined in (\[EQ:0\]), we have $$S_{0}\left( \widehat{\mu }_{10},\widehat{\mu }_{20},\widehat{\sigma }%
_{0}\right) =\frac{n_1n_2}{n_1+n_2}\frac{\left( \bar{X}-%
\bar{Y}\right) ^{2}}{\widehat{\sigma }_{0}^{2}}\underset{%
n_1,n_2\rightarrow \infty }{\overset{\mathcal{L}}{\longrightarrow }}\chi
^{2}(1). \label{eqCor1}$$
The proof of the corollary is straightforward. The test statistic given in the above corollary is closely related to the likelihood ratio test. This correspondence is described in the next corollary.
\[Cor2\]For a given sample the value of the test statistic $S_{0}\left(
\widehat{\mu }_{10},\widehat{\mu }_{20},\widehat{\sigma }_{0}\right) $, defined in (\[eqCor1\]), does not exactly match the value of the likelihood ratio test statistic$$-2\log \Lambda \left( \widehat{\mu }_{10},\widehat{\mu }_{20},\widehat{%
\sigma }_{0}\right) =(n_1+n_2)\log \left( 1+\frac{n_1n_2}{\left(
n_1+n_2\right) ^{2}}\frac{(\bar{X}-\bar{Y})^2}{%
\widehat{\sigma }_{0}^{2}}\right) ,$$where $\widehat{\sigma }_{0}^{2}$ is defined in (\[sig0\]). However, as $%
n_1,n_2\rightarrow \infty $, and $w \in (0,1)$ as defined in (\[EQ:w\]), both test statistics are asymptotically equivalent.
Let us denote $\Theta _{0}=\left\{ \left( \mu ,\mu ,\sigma \right)^T :\mu
\in \mathbb{R},\sigma \in \mathbb{R}^{+}\right\} ,$ $\Theta =\left\{ \left( \mu_1,\mu_2,\sigma \right)^T
:\mu_1,\mu_2\in \mathbb{R},\sigma \in \mathbb{R}^{+}\right\}$. The likelihood function is given by $$\mathcal{L}(\mu_1,\mu_2,\sigma
)=\prod_{i=1}^{n_1}\prod_{j=1}^{n_2}f_{\mu_1,\sigma }(X_i)f_{\mu
_{2},\sigma }(Y_{j}).$$It can be shown that $$\Lambda \left( \widehat{\mu }_{10},\widehat{\mu }_{20},\widehat{\sigma }%
_{0}\right) =\frac{\sup_{\mu_1,\mu_2,\sigma \in \Theta _{0}}\mathcal{L}%
(\mu_1,\mu_2,\sigma )}{\sup_{\mu_1,\mu_2,\sigma \in \Theta }%
\mathcal{L}(\mu_1,\mu_2,\sigma )}=\left( \frac{\sum_{i=1}^{n_1}%
\left( X_i-\widetilde{\mu }\right) ^{2}+\sum_{i=1}^{n_2}\left( Y_{i}-%
\widetilde{\mu }\right) ^{2}}{\sum_{i=1}^{n_1}\left( X_i-\bar{X}%
\right) ^{2}+\sum_{i=1}^{n_2}\left( Y_{i}-\bar{Y}\right) ^{2}}\right)
^{-\frac{n_1+n_2}{2}},$$where $\widetilde{\mu }=\frac{n_1}{n_1+n_2}\bar{X}+\frac{n_2}{%
n_1+n_2}\bar{Y}$. Therefore, asymptotically, the likelihood ratio test rejects the null hypothesis $H_{0}$ if $$-2\log \Lambda \left( \widehat{\mu }_{10},\widehat{\mu }_{20},\widehat{%
\sigma }_{0}\right) =(n_1+n_2)\log \left( \frac{\sum_{i=1}^{n_1}\left(
X_i-\widetilde{\mu }\right) ^{2}+\sum_{i=1}^{n_2}\left( Y_{i}-\widetilde{%
\mu }\right) ^{2}}{\sum_{i=1}^{n_1}\left( X_i-\bar{X}\right)
^{2}+\sum_{i=1}^{n_2}\left( Y_{i}-\bar{Y}\right) ^{2}}\right) >\chi
^{2}(1).$$ Now $$\begin{aligned}
\sum_{i=1}^{n_1}\left( X_i-\widetilde{\mu }\right)
^{2}+\sum\limits_{i=1}^{n_2}\left( Y_{i}-\widetilde{\mu }\right) ^{2}&
=\sum\limits_{i=1}^{n_1}\left( X_i-\bar{X}\right) ^{2}+n_1\left(
\bar{X}-\widetilde{\mu }\right) ^{2}+\sum\limits_{i=1}^{n_2}\left(
Y_{i}-\bar{Y}\right) ^{2}+n_2\left( \bar{Y}-\widetilde{\mu }%
\right) ^{2} \\
& =\sum\limits_{i=1}^{n_1}\left( X_i-\bar{X}\right)
^{2}+\sum\limits_{i=1}^{n_2}\left( Y_{i}-\bar{Y}\right) ^{2}+\frac{%
n_1n_2}{n_1+n_2}(\bar{X}-\bar{Y})^{2}.\end{aligned}$$ So $$\begin{aligned}
-2\log \Lambda \left( \widehat{\mu }_{10},\widehat{\mu }_{20},\widehat{%
\sigma }_{0}\right) & = & (n_1+n_2)\log \left( \frac{\sum_{i=1}^{n_1}\left( X_i-\widetilde{\mu }\right)
^{2}+\sum_{i=1}^{n_2}\left( Y_{i}-\widetilde{\mu }\right) ^{2}}{%
\sum_{i=1}^{n_1}\left( X_i-\bar{X}\right)
^{2}+\sum_{i=1}^{n_2}\left( Y_{i}-\bar{Y}\right) ^{2}}\right)
\\
& =& (n_1+n_2)\log \left( 1+\frac{n_1n_2}{\left( n_1+n_2\right)
^{2}}\frac{(\bar{X}-\bar{Y})^{2}}{\widehat{\sigma }_{0}^{2}}%
\right)
\\
& = &\displaystyle \frac{n_1n_2}{\left( n_1+n_2\right)
}\frac{(\bar{X}-\bar{Y})^{2}}{\widehat{\sigma }_{0}^{2}} + R_{n_1,n_2},\end{aligned}$$where $R_{n_1,n_2} \rightarrow 0$ in probability as $n_1,n_2\rightarrow \infty$ and $w \in (0,1)$. Thus, the test statistics $-2\log \Lambda \left( \widehat{\mu }_{10},\widehat{\mu }%
_{20},\widehat{\sigma }_{0}\right)$ and $S_{0}\left( \widehat{\mu }_{10},%
\widehat{\mu }_{20},\widehat{\sigma }_{0}\right)$ are asymptotically equivalent.
Numerical Studies {#SEC:numerical}
=================
Simulation Study
----------------
In this section we study the performance of our proposed test statistics through simulated data. We have generated two random samples $%
X_{1},X_{2},\ldots ,X_{n_1}$ and $Y_{1},Y_{2},\ldots ,Y_{n_2}$ from $%
\mathcal{N}(\mu_1,\sigma^2)$ and $\mathcal{N}(\mu_2,\sigma^2)$ respectively; thus the total sample size is $n=n_1+n_2$. The value of $w$ in (\[EQ:w\]) is taken to be 0.6, and the sample size from the first population is $n_1=[wn]+1$, where $[x]$ denotes the integer part of $x$. Our aim is to test the null hypothesis given in (\[EQ:0\]). We have taken $%
\sigma^2=1$ in this study. We have compared the results of the ordinary two sample $t$-test and the density power divergence tests with four different values of the tuning parameter $\gamma =\beta =0,0.05, 0.1$ and 0.15; let DPD($\gamma $) represent the DPD test with tuning parameter $%
\gamma $. The nominal level of the tests are 0.05, and all tests are replicated 1,000 times.
In the first case we have taken $\mu_{1}=\mu_{2}=0$. Plot (a) in Figure \[fig:level\_power\] shows the observed levels of the five test statistics for different values of the sample size (obtained as the proportion of test statistics, in the $1000$ replications, that exceed the nominal $\chi ^{2}$ critical value at 5% level of significance). It is seen that the observed levels of the $t$-test are very close to the nominal level. On the other hand, the DPD tests are slightly liberal for very small sample sizes and lead to somewhat inflated observed levels. However, as the sample size increases the levels settle down rapidly around the nominal level.
Next, we have generated data with $\mu_1 = 0$ but $\mu_2 = 1$. The observed power of the tests are presented in plot (b) of Figure \[fig:level\_power\]. There is not much difference among the observed powers in this plot. The DPD tests have slightly higher power than the $t$-test in very small sample sizes. This, however, must be a consequence of the fact that the observed levels of these tests are higher than the nominal level (and higher than the observed level of the $t$-statistic) in small samples.
Now we check the performance of the tests under contaminated data. So, we have generated $n_2$ observations $Y_{1},Y_{2},\ldots ,Y_{n_2}$ from $0.95%
\mathcal{N}(\mu_2,1)+0.05\mathcal{N}(-10,1)$, whereas the $n_1$ observations representing the first population come from the pure $\mathcal{N%
}(\mu_1,1)$ distribution. To evaluate the stability of the level of the tests for testing the hypothesis in (\[EQ:0\]), we have taken $\mu_1=\mu
_{2}=0 $. Figure \[fig:level\_power\] (c) presents the levels for different values of the sample sizes. It may be observed that there is a drastic inflation in the levels for the $t$-test and DPD(0) test statistic, but the levels of the other DPD test statistics remain stable.
Figure \[fig:level\_power\] (d) shows the power of the tests under the contaminated setup considered in the previous paragraph, when $\mu_1 = 0$ and $\mu_2 = 1$. Here, the presence of the outliers lead to a sharp drop in power for the $t$-test and the DPD(0) test. On the other hand, the other tests are clearly more resistant, and hold their power much better as $\gamma$ increases.
On the whole, therefore, it appears that in comparison to the $t$-test, many of our DPD tests are quite competitive in performance when the data come from the pure model. Under contaminated data, however, the robustness properties of the DPD tests appear to be far superior, and they do much better at maintaining the stability of the level and the power in such cases.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![(a) Simulated levels of the DPD tests for pure data; (b) simulated power of the DPD tests for pure data; (c) simulated levels of the DPD tests for contaminated data; (d) simulated power of the DPD tests for contaminated data.[]{data-label="fig:level_power"}](Two_sample_t_nonpara_mu1_0,mu2_0,sigma_1,w_0.6,n_20_1_100,rep_1000_v2.eps "fig:"){height="7.5cm" width="7.5cm"} ![(a) Simulated levels of the DPD tests for pure data; (b) simulated power of the DPD tests for pure data; (c) simulated levels of the DPD tests for contaminated data; (d) simulated power of the DPD tests for contaminated data.[]{data-label="fig:level_power"}](Two_sample_t_nonpara_mu1_0,mu2_1,sigma_1,w_0.6,n_20_1_100,rep_1000_v2.eps "fig:"){height="7.5cm" width="7.5cm"}
![(a) Simulated levels of the DPD tests for pure data; (b) simulated power of the DPD tests for pure data; (c) simulated levels of the DPD tests for contaminated data; (d) simulated power of the DPD tests for contaminated data.[]{data-label="fig:level_power"}](Two_sample_t_nonpara_mu1_0,mu2_0,sigma_1,w_0.6,p_0.05,muC_-10,n_20_1_100,rep_1000_v2.eps "fig:"){height="7.5cm" width="7.5cm"} ![(a) Simulated levels of the DPD tests for pure data; (b) simulated power of the DPD tests for pure data; (c) simulated levels of the DPD tests for contaminated data; (d) simulated power of the DPD tests for contaminated data.[]{data-label="fig:level_power"}](Two_sample_t_nonpara_mu1_0,mu2_1,sigma_1,w_0.6,p_0.05,muC_-10,n_20_1_100,rep_1000_v2.eps "fig:"){height="7.5cm" width="7.5cm"}
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Comparison with Other Robust Tests
----------------------------------
In this section we compare the DPD test with some other popular robust tests. For comparison we have used a parametric test – the two sample trimmed $t$-test proposed by [@yuen1973approximate], as well as two non-parametric tests – the Kolmogorov-Smirnov test (KS-test) and the Wilcoxon two-sample test (which is also known as the Mann-Whitney $U$-test). For the two sample trimmed $t$-test we have trimmed 20% extreme observations from each of the data sets of $X$ and $Y$. The set up, the parameters taken for the simulation and the level of contamination are exactly the same as in the previous section. For comparison we have used only one DPD test in this case, that corresponding to tuning parameter 0.1. To emphasize the robustness properties of these tests we have also included the two sample $t$-test in this investigation. The results are presented in Figure \[fig:level\_power\_v1\].
Figure \[fig:level\_power\_v1\] (a) shows that the observed levels of all the robust tests are very close to the nominal level of 0.05 for the pure normal data. The same result is observed in Figure \[fig:level\_power\_v1\] (c) for the contaminated data. On the other hand, if we consider the observed power of the tests the DPD test is much more powerful than the other tests. Specifically, for the contaminated data, the DPD test does significantly better than the others in holding on to its power. Therefore, on the whole, the DPD tests are not only superior to the two sample $t$-test under contamination, but they also appear to be competitive or better than the other popular robust tests as far as this simulation study is concerned.
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![(a) Simulated levels of different tests for pure data; (b) simulated power of different tests for pure data; (c) simulated levels of different tests for contaminated data; (d) simulated power of different tests for contaminated data.[]{data-label="fig:level_power_v1"}](Two_sample_t_nonpara_mu1_0,mu2_0,sigma_1,w_0.6,n_20_1_100,rep_1000_v1.eps "fig:"){height="7.5cm" width="7.5cm"} ![(a) Simulated levels of different tests for pure data; (b) simulated power of different tests for pure data; (c) simulated levels of different tests for contaminated data; (d) simulated power of different tests for contaminated data.[]{data-label="fig:level_power_v1"}](Two_sample_t_nonpara_mu1_0,mu2_1,sigma_1,w_0.6,n_20_1_100,rep_1000_v1.eps "fig:"){height="7.5cm" width="7.5cm"}
![(a) Simulated levels of different tests for pure data; (b) simulated power of different tests for pure data; (c) simulated levels of different tests for contaminated data; (d) simulated power of different tests for contaminated data.[]{data-label="fig:level_power_v1"}](Two_sample_t_nonpara_mu1_0,mu2_0,sigma_1,w_0.6,p_0.05,muC_-10,n_20_1_100,rep_1000_v1.eps "fig:"){height="7.5cm" width="7.5cm"} ![(a) Simulated levels of different tests for pure data; (b) simulated power of different tests for pure data; (c) simulated levels of different tests for contaminated data; (d) simulated power of different tests for contaminated data.[]{data-label="fig:level_power_v1"}](Two_sample_t_nonpara_mu1_0,mu2_1,sigma_1,w_0.6,p_0.05,muC_-10,n_20_1_100,rep_1000_v1.eps "fig:"){height="7.5cm" width="7.5cm"}
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Real Data Examples
------------------
**Example 2 (Lead Measurement data):** In Table \[TAB:Lake\] the lead measurement data ([@MR922042], p. 280) are presented. The numbers represent the values of $10(x-2)$, $x$ being the level of lead in the water samples from two lakes at randomly chosen locations. To test whether the average pollution levels of the two lakes are equal, we perform tests for equality of the means of the populations represented by the two different samples. The $p$-values of the DPD tests are plotted in Figure \[fig:Lake\_data\_p\_val\]; the solid line represents the $p$-values for the full data, while the dashed line represents for the $p$-values for the outlier deleted data. The less robust tests (corresponding to very small values of $\gamma$) register only borderline significance under full data, and for very small values of $\gamma$ the tests would fail to reject the equality hypothesis at the 1% significance level. However, for all value of $\gamma$, the tests would soundly reject the null hypothesis when the obvious outliers (displayed with bold fonts in Table \[TAB:Lake\]) are removed from the dataset. For higher values of $\gamma$ (0.2 or larger), the $p$-values with or without the outliers are practically identical, demonstrating that the outliers have little effect in such cases. The $p$-values for the two-sample $t$-test with and without the outliers are 0.02397 and 0.0004 respectively. As in Example 1, the presence of the outliers masks the significance of the two-sample $t$-test and the small $\gamma$ DPD tests, but the large $\gamma$ DPD tests successfully discount the effect of the outliers.
------------- --------- -------- --------- --------- --------- --------------- --------- --------------- --------- --------
First Lake $-1.48$ $1.25$ $-0.51$ $0.46$ $0.60$ ${\bf -4.27}$ $0.63$ $-0.14$ $-0.38$ $1.28$
$0.93$ $0.51$ $1.11$ $-0.17$ $-0.79$ $-1.02$ $-0.91$ $0.10$ $0.41$ $1.11$
Second Lake $1.32$ $1.81$ $-0.54$ $2.68$ $2.27$ $2.70$ $0.78$ ${\bf -4.62}$ $1.88$ $0.86$
$2.86$ $0.47$ $-0.42$ $0.16$ $0.69$ $0.78$ $1.72$ $1.57$ $% $1.62$
2.14$
------------- --------- -------- --------- --------- --------- --------------- --------- --------------- --------- --------
: Lead Measurement data.[]{data-label="TAB:Lake"}
**Example 3 (Ozone Control data):** [@MR0443210] report data from a study design to assess the effects of ozone on weight gain in rats. The experimental group consisted of 22 rats, each 70-day old kept in an ozone environment for 7 days. A control group of 23 rats, of the same age, were kept in an ozone-free environment. The weight gains, in grams, are listed in Table \[TAB:Ozone\]. We want to test for the equality of the means of the two groups. The $p$-values of the DPD tests are plotted in Figure \[fig:Ozone\_control\_data\_p\_val\]. The $p$-values of the two-sample $%
t$-test for the full data and the outlier deleted data are $0.0168$ and $%
3.4721\times 10^{-6}$ respectively. The conclusions of this example are similar to those of Examples 1 and 2.
----- ------------- ------------- -------- -------- ------------- --------- ------------- --------- ------------ -------------- -------- ---------
$X$ $\bf{41.0}$ $\bf{38.4}$ $24.4$ $25.9$ $21.9$ $18.3$ $13.1$ $27.3$ $28.5$ $\bf{-16.9}$ $26.0$ $17.4$
$21.8$ $15.4$ $27.4$ $19.2$ $22.4$ $17.7$ $26$ $29.4$ $% $26.6$ $22.7$
21.4 $
$Y$ $10.1$ $6.1$ $20.4$ $7.3$ $14.3$ $15.5$ $-9.9$ $6.8$ $% $17.9$ $-9.0$ $-12.9$
28.2$
$14.0$ $6.6$ $12.1$ $15.7$ $\bf{39.9}$ $-15.9$ $\bf{54.6}$ $-14.7$ $% $-9.0$
\bf{44.1}$
----- ------------- ------------- -------- -------- ------------- --------- ------------- --------- ------------ -------------- -------- ---------
: Ozone Control data[]{data-label="TAB:Ozone"}
**Example 4 (Newcomb’s Light Speed data)**: In 1882 Simon Newcomb, an astronomer and mathematician, measured the time required for a light signal to pass from his laboratory on the Potomac River to a mirror at the base of the Washington Monument and back. The total distance was $%
7443.73 $ meters. Table \[TAB:tNewcomb\] contains these measurements from three samples, as deviations from $24,800$ nanoseconds. For example, for the first observation, $28$, means that the time taken for the light to travel the required $7443.73$ meters is $24,828$ nanoseconds. The data comprises three samples, of sizes $20$, $20$ and $26$, respectively, corresponding to three different days. These data have been analyzed previously by a number of authors including [@MR0455205] and [@Voinov]. The $p$-values of the DPD statistics for the test of the equality of means between Day 1 and Day 2, and Day 1 and Day 3 are plotted in Figure \[fig:Newcomb\_data12\_p\_val\], and \[fig:Newcomb\_data13\_p\_val\] respectively.
The $p$-values for the two-sample $t$-tests for the (Day 1, Day 2) comparison are $0.1058$ for the full data case, and $0.3091$ for the outlier deleted case. The same for the (Day 1, Day 3) comparison are $0.0970$ and $%
0.2895$ respectively. However, for the large $\gamma $, the results from the DPD tests are clearly insignificant with or without the outliers. In this example, therefore, the outliers are forcing the outcome of the two-sample $t$-test (and the DPD tests for small $\gamma $) to the borderline of significance, but the robust tests give insignificant results with or without the outliers, preventing the false significance that is produced by the outliers in the $t$-test; this is unlike the previous three examples where the robust tests overcame a masking effect. These examples demonstrate that the robust DPD tests can give protection against spurious conclusions in both directions.
------- ------ ------ ------ ------ ------ ------------ ------ ------ ------ ----------- ------ ------ ------ ------ ------
day 1 $28$ $26$ $33$ $24$ $34$ $\bf{-44}$ $27$ $16$ $40$ $\bf{-2}$ $29$ $22$ $24$ $21$ $25$
$30$ $23$ $29$ $31$ $19$
day 2 $24$ $20$ $36$ $32$ $36$ $28$ $25$ $21$ $28$ $29$ $37$ $25$ $28$ $26$ $30$
$32$ $36$ $26$ $30$ $22$
day 3 $36$ $23$ $27$ $27$ $28$ $27$ $31$ $27$ $26$ $33$ $26$ $32$ $32$ $24$ $39$
$28$ $24$ $25$ $32$ $25$ $29$ $27$ $28$ $29$ $16$ $23$
------- ------ ------ ------ ------ ------ ------------ ------ ------ ------ ----------- ------ ------ ------ ------ ------
: Newcomb’s Light Speed data.[]{data-label="TAB:tNewcomb"}
**Example 5 (Na Intake data)**: Sodium chloride preference was determined in ten patients with essential hypertension and in 12 normal volunteers. All exhibited normal detection and recognition thresholds for the taste of sodium chloride. All were placed on a constant dry diet containing 9 mEq of Na+ and given, as their only source of fluids, a choice of drinking either distilled water or 0.15 M sodium chloride. Patients with essential hypertension consumed a markedly greater proportion of their total fluid intake as saline (38.2% vs 10.6%, average daily preference over one week) and also showed a greater total fluid intake (1,269 ml vs 668 ml, average daily intake over one week). The hypertensive patients consumed more than four times as much salt as did the normal volunteers. The data are given in Table \[TAB:Na\_intake\]. The $p$-values of the tests for the equality of means are plotted in Figure \[fig:Na\_intake\_data\_p\_val\]. The findings are similar to examples 1, 2 and 3.
----- --------- -------- -------- -------- ------------ -------- -------- -------- -------- ------- -----
$X$ $114.6$ $64.6$ $70.4$ $61.2$ $\bf{297}$ $60.9$ $73.7$ $15.7$ $53.3$
$Y$ $14.2$ $3.2$ $3.7$ $0.0$ $73.6$ $56.6$ $97.2$ $2.4$ $% $4.8$ $0$
0.0$
----- --------- -------- -------- -------- ------------ -------- -------- -------- -------- ------- -----
: Na Intake data.[]{data-label="TAB:Na_intake"}
[**Example 6 (Sri Lanka Zinc Content data)**]{}: The impact of a polluted environment on the health of the residents of an area is a common environmental concern. Large amounts of heavy metals in the body may signal a serious health threat to a community. One study, performed in Sri Lanka, sought to compare rural Sri Lankans with their urban counterparts in terms of the zinc content of their hair. A collection of individuals from rural Sri Lanka was recruited, samples of their hair were taken, and the zinc content in the hair was measured. An independent collection of students from an urban environment was studied, with the zinc content in samples of their hair being measured as well. The data are given in Table \[TAB:SriLanka\]. The $p$-values of the tests for the equality of the means are plotted in Figure \[fig:SriLanka\]. The results again indicate that the presence of outliers can mask the true significance in case of the two sample $t$-test and DPD tests for small values of $\gamma$, but for the large $\gamma$ DPD tests are much more stable in such situations.
------------- ------ ----- ---------------------------------------------------------------------------------------- -- -- -- -- -- -- -- --
Urban ($X$) 1120 230 **[4200]{} & 1200 & 1400 & 750 & 2101 & 430 & 690 & 600 & 834\
Rural ($Y$) & **[3619]{} & 1104 & 243 & 658 & 673 & 598 & 648 & 918 & 133 & 289 & 250\
& 304 & 555 & 640 & 933 & & & & & & &\
****
------------- ------ ----- ---------------------------------------------------------------------------------------- -- -- -- -- -- -- -- --
: Sri Lanka Zinc Content data.[]{data-label="TAB:SriLanka"}
Concluding Remarks {#SEC:concluding}
==================
Without any doubt, the two sample $t$-test is one of the most frequently used tools in the statistics literature. It allows the experimenter to perform tests of the comparative hypotheses, which are the default requirements to be passed before one may declare that a new drug or treatment is an improvement over an existing one. The two sample $t$-test is simple to implement and has several optimality properties. In spite of such desirable attributes, this test is deficient on one count, which is that it does not retain its desired properties under contamination and model misspecification. As few as one, single, large outlier can turn around the decision of the test, and can make the resulting inference meaningless. In this paper we have introduced a test based on the density power divergence; the theoretical properties of the test have been rigorously determined. More importantly, we have demonstrated, through several real data examples, that the DPD test is capable of uncovering both kinds of masking effects caused by outliers – blurring the true difference when one exists, and detecting a difference when there is actually none. The test is simple to use and easy to understand, and we trust that it has the potential to become a powerful tool for the applied statistician.
**Acknowledgments** This work was partially supported by Grants MTM-2012-33740 and ECO-2011-25706. The authors gratefully acknowledge the suggestions of two anonymous referees which led to an improved version of the paper.
Appendix {#appendix .unnumbered}
========
**Proof of Theorem \[Th0\]:** As $\widehat{\mu }_{i\beta }$ is the solution of the estimating equation $_{1}h_{n_{i},\beta }^{\prime }\left( \mu _{i},\sigma \right) =0$, we get from equation (\[muSigma\]) $$\sqrt{n_{i}}(\widehat{\mu }_{i\beta }-\mu _{i0})=\sqrt{n_{i}} \boldsymbol{J}_{11,\beta }^{-1}( \sigma
_{0})\,\allowbreak
_{1}h_{n_{i},\beta }^{\prime }\left( \mu _{i0},\sigma_0\right) +o_{p}(1) ,\quad i=1,2.$$ Hence, using (\[1.1\]) we get $$\sqrt{n_{i}}(\widehat{\mu }_{i\beta }-\mu _{i0})\underset{n_{i}\rightarrow
\infty }{\overset{\mathcal{L}}{\longrightarrow }}\mathcal{N}\left(
0,K_{11,\beta }\boldsymbol{(}\sigma_0)J_{11,\beta }^{-2}( \sigma_0) \right) ,\quad i=1,2,
\label{distmu}$$ where $$K_{11,\beta }\boldsymbol{(}\sigma_0)J_{11,\beta }^{-2}(\sigma
_{0})=\sigma_0^{2}\left( \beta +1\right) ^{3}\left( 2\beta +1\right) ^{-%
\frac{3}{2}}. \label{KJ1}$$ It is clear that $\widehat{\mu }_{1\beta }$ and $\widehat{\mu }_{2\beta }$ are based on two independent set of observations, hence, $Cov(\widehat{\mu }_{1\beta },\widehat{\mu }_{2\beta })=0$. As $_{2}h_{n_1,n_2,\beta }^{\prime }(\widehat{\boldsymbol{\eta }}_\beta )=0$, taking a Taylor series expansion around $\boldsymbol{\eta }_0$ we get $$\begin{aligned}
_{2}h_{n_1,n_2,\beta }^{\prime }(\widehat{\boldsymbol{\eta }}_\beta ) =& \ %\allowbreak
_{2}h_{n_1,n_2,\beta }^{\prime }( \boldsymbol{\eta }_0) + \left. \frac{\partial }{\partial \mu_1}\,\allowbreak
_{2}h_{n_1,n_2,\beta }^{\prime }( \boldsymbol{\eta })\right\vert_{\boldsymbol{\eta }=\boldsymbol{\eta }_0} (\widehat{\mu }_{1\beta }-\mu _{10}) \nonumber\\
& + \left. \frac{\partial }{\partial \mu_2}\,\allowbreak _{2} h_{n_1,n_2,\beta
}^{\prime \prime }( \boldsymbol{\eta }) \right\vert_{\boldsymbol{\eta }=\boldsymbol{\eta }_0} (\widehat{%
\mu }_{2\beta }-\mu _{20}) \nonumber\\
& + \left. \frac{\partial }{\partial \sigma }\,\allowbreak _{2}h_{n_1,n_2,\beta
}^{\prime \prime }( \boldsymbol{\eta })\right\vert_{\boldsymbol{\eta }=\boldsymbol{\eta }_0} \left(
\widehat{\sigma}_\beta -\sigma_0\right) +o_{p}\left(
(n_1+n_2)^{-1/2}\right) \nonumber\\
=& \ 0.
\label{2h'}\end{aligned}$$Notice that $$\begin{aligned}
\lim_{n_1,n_2\rightarrow \infty } \left. \frac{\partial }{\partial \mu_1}\,\allowbreak _{2}h_{n_1,n_2,\beta
}^{\prime }( \boldsymbol{\eta }) \right\vert_{\boldsymbol{\eta }=\boldsymbol{\eta }_0} & = \lim_{n_1,n_2\rightarrow \infty } \frac{\partial }{%
\partial \mu_1}\left( \frac{n_1}{n_1+n_2}\,\allowbreak
_{2}h_{n_1,\beta }^{\prime }\left( \mu _{10},\sigma_0\right) +\frac{%
n_2}{n_1+n_2}\,\allowbreak _{2}h_{n_2}^{\prime }(\mu _{10},\sigma
_{0})\right) \nonumber\\
& =\lim_{n_1,n_2\rightarrow \infty } \frac{n_1}{n_1+n_2} \lim_{n_1,n_2\rightarrow \infty } \left. \frac{\partial }{\partial \mu_1}\,\allowbreak
_{2}h_{n_1,\beta }^{\prime }\left( \mu _{1},\sigma_0\right) \right\vert_{\mu_1 = \mu_{10} }\nonumber\\
&= w \boldsymbol{J}_{12,\beta }\left( \sigma_0\right) = 0.
\label{hmu2}\end{aligned}$$ Similarly we get $$\lim_{n_1,n_2\rightarrow \infty } \left. \frac{\partial }{\partial \mu_2}\,\allowbreak _{2}h_{n_1,n_2,\beta
}^{\prime }( \boldsymbol{\eta })\right\vert_{\boldsymbol{\eta }=\boldsymbol{\eta }_0} =0.
\label{hmu1}$$ Moreover, $$\begin{aligned}
\lim_{n_1,n_2\rightarrow \infty } \left. \frac{\partial }{\partial \sigma }%
\allowbreak _{2}h_{n_1,n_2,\beta }^{\prime }\left( \boldsymbol{\eta } \right)\right\vert_{\boldsymbol{\eta }=\boldsymbol{\eta }_0} &=&\lim_{n_1,n_2\rightarrow \infty }\tfrac{%
n_1}{n_1+n_2}\,\allowbreak _{22}h_{n_1,\beta }^{\prime \prime }(\mu
_{10},\sigma_0)+\lim_{n_1,n_2\rightarrow \infty }\tfrac{n_2}{%
n_1+n_2}\,\allowbreak _{22}h_{n_2,\beta }^{\prime \prime }\left( \mu
_{20},\sigma_0\right) \nonumber\\
&=&w\boldsymbol{J}_{22,\beta }\left( \sigma_0\right) +(1-w)\boldsymbol{J}_{22,\beta }( \sigma_0) = \boldsymbol{J}_{22,\beta } ( \sigma_0).
\label{hsigma}\end{aligned}$$ Therefore, using equations (\[hmu2\]), (\[hmu1\]) and (\[hsigma\]) we get from equation (\[2h’\]) $$\sqrt{n_1+n_2}\left( \widehat{\sigma}_\beta -\sigma_0\right)
=-\boldsymbol{J}_{22,\beta }^{-1}\left( \sigma_0\right) \sqrt{n_1+n_2}%
\,\allowbreak _{2}h_{n_1,n_2,\beta }^{\prime }( \boldsymbol{\eta }_0) +o_{p}(1).
\label{sigmaL}$$ Applying (\[1.1\]) and (\[EQ:w\]) we get $$\begin{aligned}
& \lim_{n_1,n_2\rightarrow \infty } \text{$E$}\left[ \sqrt{n_1+n_2}\,\allowbreak _{2}h_{n_1,n_2,\beta
}^{\prime }(\boldsymbol{\eta }_0) \right] \\
& = \lim_{n_1,n_2\rightarrow \infty } \frac{\sqrt{n_1+n_2}}{n_1+n_2}E\left[ n_1\,\allowbreak
_{2}h_{n_1,\beta }^{\prime }\left( \mu _{10},\sigma_0\right)
+n_2\,\allowbreak _{2}h_{n_2,\beta }^{\prime }\left( \mu _{20},\sigma
_{0}\right) \right] \\
& = \lim_{n_1,n_2\rightarrow \infty }\sqrt{\frac{n_1}{n_1+n_2}} \lim_{n_1,n_2\rightarrow \infty } E\left[ \sqrt{n_1}\,\allowbreak
_{2}h_{n_1,\beta }^{\prime }(\mu _{10},\sigma_0)\right] \\
& \ \ \ + \lim_{n_1,n_2\rightarrow \infty } \sqrt{\frac{%
n_2}{n_1+n_2}} \lim_{n_1,n_2\rightarrow \infty } E\left[ \sqrt{n_2}\,\allowbreak _{2}h_{n_2,\beta
}^{\prime }\left( \mu _{20},\sigma_0\right) \right] \\
&= 0.\end{aligned}$$ Similarly we also have $$\begin{aligned}
& \lim_{n_1,n_2\rightarrow \infty } \text{$Var$}\left[ \sqrt{n_1+n_2}\,\allowbreak
_{2}h_{n_1,n_2,\beta }^{\prime }(\boldsymbol{\eta }_0) \right] \\
& =\lim_{n_1,n_2\rightarrow \infty } (n_1+n_2)\text{$Var$}\left[ \frac{1}{n_1+n_2}\left(
n_1\,\allowbreak _{2}h_{n_1,\beta }^{\prime }(\mu _{10},\sigma
_{0})+n_2\,\allowbreak _{2}h_{n_2,\beta }^{\prime }(\mu _{20},\sigma
_{0}\right) \right] \\
& = \lim_{n_1,n_2\rightarrow \infty } \frac{n_1}{n_1+n_2} \lim_{n_1,n_2\rightarrow \infty } \text{$Var$}\left[ \sqrt{n_1}\,\allowbreak
_{2}h_{n_1,\beta }^{\prime }(\mu _{10},\sigma_0)\right] \\
& \ \ \ + \lim_{n_1,n_2\rightarrow \infty } \frac{n_2}{%
n_1+n_2} \lim_{n_1,n_2\rightarrow \infty } \text{$Var$}\left[ \sqrt{n_2}\,\allowbreak _{2}h_{n_2,\beta
}^{\prime }(\mu _{20},\sigma_0)\right] \\
&= w \boldsymbol{K}_{22,\beta }(\sigma_0) + (1-w) \boldsymbol{K}_{22,\beta }(\sigma_0) \\
&= \boldsymbol{K}_{22,\beta }(\sigma_0) .\end{aligned}$$Hence, $$\sqrt{n_1+n_2}\,\allowbreak _{2}h_{n_1,n_2,\beta }^{\prime }\left(
\boldsymbol{\eta }_0\right) \underset{n_1,n_2\rightarrow
\infty }{\overset{\mathcal{L}}{\longrightarrow }}\mathcal{N}\left(
0,\boldsymbol{K}_{22,\beta }(\sigma_0)\right) .$$Now, from equation (\[sigmaL\]) we get $$\sqrt{n_1+n_2}\left( \widehat{\sigma}_\beta -\sigma_0\right)
\underset{n_1,n_2\rightarrow \infty }{\overset{\mathcal{L}}{%
\longrightarrow }}\mathcal{N}\left( 0,\boldsymbol{K}_{22,\beta }(\sigma_0)\boldsymbol{J}_{22,\beta
}^{-2}(\sigma_0)\right) ,
\label{sigma1}$$where $$\boldsymbol{K}_{22,\beta }(\sigma_0)\boldsymbol{J}_{22,\beta }^{-2}(\sigma_0)=\sigma_0^{2}%
\frac{\left( \beta +1\right) ^{5}}{\left( \beta ^{2}+2\right) ^{2}}\left(
\frac{4\beta ^{2}+2}{(1+2\beta )^{5/2}}-\frac{\beta ^{2}}{(1+\beta )^{3}}%
\right). \label{KJ2}$$As $\boldsymbol{J}_{12,\beta }(\sigma_0)=\boldsymbol{J}_{21,\beta }(\sigma_0)=0$, it is clear that $$\lim_{n_1,n_2\rightarrow \infty } \left. \frac{\partial^2 }{\partial \mu_1 \partial \sigma}\,\allowbreak h_{n_1,n_2,\beta
}( \boldsymbol{\eta })\right\vert_{\boldsymbol{\eta } = \boldsymbol{\eta }_0}
= \lim_{n_1,n_2\rightarrow \infty } \left. \frac{\partial^2 }{\partial \mu_2 \partial \sigma}\,\allowbreak h_{n_1,n_2,\beta
}( \boldsymbol{\eta }) \right\vert_{\boldsymbol{\eta } = \boldsymbol{\eta }_0}
=0.$$ Therefore, $Cov(\widehat{\mu }_{1\beta },\widehat{\sigma}_{\beta })=Cov(\widehat{\mu }_{2\beta },\widehat{\sigma}_{\beta })=0$. Moreover, $Cov(\widehat{\mu }_{1\beta },\widehat{\mu }_{2\beta })=0$. Combining the results in (\[distmu\]) and (\[sigma1\]) we get the variance-covariance matrix of $\sqrt{\frac{n_1n_2}{n_1+n_2}}\widehat{\boldsymbol{\eta }}_{\beta }$ as follows $$\boldsymbol{\Sigma }_{w,\beta }(\sigma_0)=\left(
\begin{array}{ccc}
\left( 1-w\right) \boldsymbol{K}_{11,\beta }\boldsymbol{(}\sigma_0) \boldsymbol{J}_{11,\beta
}^{-2}\left( \sigma_0\right) & 0 & 0 \\
0 & w \boldsymbol{K}_{11,\beta }\boldsymbol{(}\sigma_0) \boldsymbol{J}_{11,\beta }^{-2}(\sigma_0)
& 0 \\
0 & 0 & w\left( 1-w\right) \boldsymbol{K}_{22,\beta }(\sigma_0)\boldsymbol{J}_{22,\beta
}^{-2}\left( \sigma_0\right)
\end{array}%
\right) ,$$ where the values of the diagonal elements are given in (\[KJ1\]) and (\[KJ2\]). Hence, the theorem is proved. ${\blacksquare }$
**Proof of Theorem \[Th1\]**: A Taylor expansion of $%
d_{\gamma }(f_{\widehat{\mu }_{1\beta },\widehat{\sigma}_\beta },f_{%
\widehat{\mu }_{2\beta },\widehat{\sigma}_\beta })$ around $\boldsymbol{\eta }_{0}$ gives$$d_{\gamma }(f_{\widehat{\mu }_{1\beta },\widehat{\sigma}_\beta },f_{%
\widehat{\mu }_{2\beta },\widehat{\sigma}_\beta })=d_{\gamma }(f_{\mu
_{10},\sigma_0},f_{\mu _{20},\sigma_0})+\boldsymbol{t}_{\gamma
}^{T}\left( \boldsymbol{\eta }_{0}\right) (\widehat{\boldsymbol{\eta }}%
_{\beta }-\boldsymbol{\eta }_{0})+o_{p}\left( \left\Vert \widehat{%
\boldsymbol{\eta }}_{\beta }-\boldsymbol{\eta }_{0}\right\Vert \right),$$ where $\boldsymbol{t}_{\gamma }\left( \boldsymbol{\eta }_{0}\right) =\frac{%
\partial }{\partial \boldsymbol{\eta }}\left. d_{\gamma }(f_{\mu_1,\sigma
},f_{\mu_2,\sigma })\right\vert _{\boldsymbol{\eta }=\boldsymbol{\eta }%
_{0}}$; the expressions of the components $t_{\gamma, i}\left( \boldsymbol{\eta }%
_{0}\right) $, $i=1,2,3$, are given in (\[t1\])-(\[t3\]). Hence, the result directly follows from Theorem \[Th0\]. ${\blacksquare }$
**Proof of Theorem \[Th2\]**: If $\mu _{10}=\mu _{20}$, it is obvious that $d_{\gamma }(f_{\mu_{10},\sigma_0},f_{\mu _{20},\sigma_0})=0$, and $\boldsymbol{t}_{\gamma } ( \boldsymbol{\eta }_{0})=0$. Hence, a second order Taylor expansion of $d_{\gamma }(f_{\widehat{\mu }_{1\beta },\widehat{\sigma}_\beta },f_{%
\widehat{\mu }_{2\beta },\widehat{\sigma}_\beta })$ around $\boldsymbol{\eta }_{0}$ gives $$2d_{\gamma }(f_{\widehat{\mu }_{1\beta },\widehat{\sigma}_\beta },f_{%
\widehat{\mu }_{2\beta },\widehat{\sigma}_\beta })=(\widehat{\boldsymbol{%
\eta }}_{\beta }-\boldsymbol{\eta }_{0})^{T}\boldsymbol{A}_{\gamma }\left(
\sigma_0\right) (\widehat{\boldsymbol{\eta }}_{\beta }-\boldsymbol{\eta }%
_{0})+o_p(\left\Vert \widehat{\boldsymbol{\eta }}_{\beta }-\boldsymbol{\eta }%
_{0}\right\Vert ^{2}),
\label{gam}$$where $\boldsymbol{A}_{\gamma }(\sigma_0)$ is the matrix containing the second derivatives of $d_{\gamma }(f_{\mu_1,\sigma },f_{\mu_2,\sigma })\ $evaluated at $\mu_{10}=\mu_{20}$. It can be shown that $$\boldsymbol{A}_{\gamma }\left( \sigma_0\right) \boldsymbol{=}\ell
_{\gamma }(\sigma_0)\left(
\begin{array}{ccc}
1 & -1 & 0 \\
-1 & 1 & 0 \\
0 & 0 & 0%
\end{array}%
\right) ,$$ where $$\ell _{\gamma }(\sigma_0)=\sigma ^{-(\gamma +2)}\left( 2\pi \right) ^{-%
\frac{\gamma }{2}}\left( \gamma +1\right) ^{-\frac{1}{2}}.$$ Therefore, equation (\[gam\]) simplifies to $$2d_{\gamma }(f_{\widehat{\mu }_{1\beta },\widehat{\sigma}_\beta },f_{%
\widehat{\mu }_{2\beta },\widehat{\sigma}_\beta })=\left(
\begin{pmatrix}
\widehat{\mu }_{1\beta } \\
\widehat{\mu }_{2\beta }%
\end{pmatrix}%
-%
\begin{pmatrix}
\mu _{10} \\
\mu _{20}%
\end{pmatrix}%
\right) ^{T}\boldsymbol{A}_{\gamma }^{\ast }\left( \sigma_0\right) \left(
\begin{pmatrix}
\widehat{\mu }_{1\beta } \\
\widehat{\mu }_{2\beta }%
\end{pmatrix}%
-%
\begin{pmatrix}
\mu _{10} \\
\mu _{20}%
\end{pmatrix}%
\right) +o_{p}\left( \left\Vert \widehat{\boldsymbol{\eta }}_{\beta }-%
\boldsymbol{\eta }_{0}\right\Vert ^{2}\right) ,$$ where $$\boldsymbol{A}_{\gamma }^{\ast }\left( \sigma_0\right) =\ell _{\gamma
}(\sigma_0)\left(
\begin{array}{cc}
1 & -1 \\
-1 & 1%
\end{array}%
\right) .$$ From Theorem \[Th0\] we know that $$\sqrt{\frac{n_1n_2}{n_1+n_2}}\left(
\begin{pmatrix}
\widehat{\mu }_{1\beta } \\
\widehat{\mu }_{2\beta }%
\end{pmatrix}%
-%
\begin{pmatrix}
\mu _{10} \\
\mu _{20}%
\end{pmatrix}%
\right) ^{T}\underset{}{\overset{\mathcal{L}}{\longrightarrow }}\mathcal{N}%
\left( \boldsymbol{0}_{2},\boldsymbol{\Sigma }_{w,\beta }^{\ast }(\sigma
_{0})\right),$$ where $$\boldsymbol{\Sigma }_{w,\beta }^{\ast }(\sigma_0)=K_{11,\beta }%
\boldsymbol{(}\sigma_0)J_{11,\beta }^{-2}\left( \sigma_0\right) \left(
\begin{array}{cc}
1-w & 0 \\
0 & w%
\end{array}
\right).$$ Therefore, $\frac{2 n_1n_2}{n_1+n_2} d_{\gamma }(f_{\widehat{\mu }_{1\beta },\widehat{\sigma }_{\beta
}},f_{\widehat{\mu }_{2\beta },\widehat{\sigma}_\beta })$ has the same asymptotic distribution (see [@MR801686]) as the random variable $$\sum\limits_{i=1}^{2}\lambda _{i,\beta ,\gamma }(\sigma_0)Z_{i}^{2},$$ where $Z_{1}$ and $Z_{2}$ are independent standard normal variables, and $$\lambda _{1,\beta ,\gamma }(\sigma_0)=0\text{, and }\lambda _{2,\beta
,\gamma }(\sigma_0)=K_{11,\beta }\boldsymbol{(}\sigma_0)J_{11,\beta
}^{-2}\left( \sigma_0\right) \ell _{\gamma }(\sigma_0)=\lambda _{\beta
,\gamma }(\sigma_0)$$are the eigenvalues of the matrix $\boldsymbol{\Sigma }_{w,\beta }^{\ast
}(\sigma_0)\boldsymbol{A}_{\gamma }^{\ast }\left( \sigma_0\right) $. Hence, $$\frac{2n_1n_2}{n_1+n_2}\frac{d_{\gamma
}(f_{\widehat{\mu }_{1\beta },\widehat{\sigma}_\beta },f_{\widehat{\mu }%
_{2\beta },\widehat{\sigma}_\beta })}{\lambda _{\beta ,\gamma
}\,\allowbreak (\sigma_{0} )} \underset{%
n_1,n_2\rightarrow \infty }{\overset{\mathcal{L}}{\longrightarrow }}\chi
^{2}(1).$$ Finally, since $\widehat{\sigma}_\beta $ is a consistent estimator of $%
\sigma $, replacing $\lambda _{\beta ,\gamma }(\sigma_0)$ by $\lambda
_{\beta ,\gamma }(\widehat{\sigma}_\beta )$ and by following Slutsky’s theorem we obtain the desired result. ${\blacksquare }$
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
We analytically calculate the dominant two-loop electroweak correction, of $\mathcal{O}(G_F^2m_t^4)$, to the partial width of the decay of a Higgs boson, with mass $M_H\ll m_t$, into a bottom-quark pair, and describe the most important conceptual and technical details of our calculation. As a by-product of our analysis, we also recover the $\mathcal{O}(\alpha_sG_Fm_t^{2})$ correction. Relative to the Born result, the $\mathcal{O}(G_F^2m_t^4)$ correction turns out to be approximately $+0.047\%$ and, thus, more than compensates the $\mathcal{O}(\alpha_sG_Fm_t^2)$ one, which amounts to approximately $-0.022\%$.
PACS numbers: 11.10.Gh, 12.15.Ji, 12.15. Lk, 14.80.Bn
author:
- |
Mathias Butenschön, Frank Fugel, Bernd A. Kniehl\
[II. Institut für Theoretische Physik, Universität Hamburg,]{}\
[Luruper Chaussee 149, 22761 Hamburg, Germany]{}
title: |
-3cm
DESY 07-003ISSN 0418-9833
hep-ph/0702215
January 2007
1.5cm $\mathcal{O}(G_F^2m_t^4)$ two-loop electroweak correction to Higgs-boson decay to bottom quarks
---
Introduction
============
The standard model (SM) of elementary particle physics predicts the existence of a last undiscovered particle, the Higgs boson, whose mass $M_H$ is a free parameter of the theory. The direct search for the Higgs boson at the CERN Large Electron-Positron Collider LEP 2 only led to a lower bound of $M_H>114$ GeV at 95% confidence level [@Barate:2003sz]. On the other hand, high-precision measurements, especially at LEP and the SLAC Linear Collider SLC, were sensitive to the Higgs-boson mass via electroweak radiative corrections. These indirect measurements yielded the value $M_H=\left(85^{+39}_{-28}\right)$ GeV and an upper limit of $M_H<166$ GeV at 95% confidence level [@LEPEWWG]. The vacuum-stability and triviality bounds suggest that $130{\rlap{\lower 3.5 pt \hbox{$\mathchar \sim$}} \raise 1pt
\hbox {$<$}}M_H{\rlap{\lower 3.5 pt \hbox{$\mathchar \sim$}} \raise 1pt
\hbox {$<$}}180$ GeV if the SM is valid up to the grand-unification scale (for a review, see Ref. [@Kniehl:2001jy]). For these reasons, one hopes to discover the Higgs boson at the CERN Large Hadron Collider (LHC), which will be capable of producing particles with masses up to 1 TeV. The first question after discovering a new scalar particle will be if it actually is the Higgs boson of the SM, or possibly some particle of an extended Higgs sector. Therefore, it is necessary to know the SM predictions for the production and decay rates of the SM Higgs boson with high precision. Its decay into a bottom-quark pair is of special interest, as it is by far the dominant decay channel for $M_H{\rlap{\lower 3.5 pt \hbox{$\mathchar \sim$}} \raise 1pt
\hbox {$<$}}140$ GeV (see, for instance, Ref. [@Kniehl:1993ay]).
At this point, we wish to summarise the current status of the calculations of radiative corrections to the $H\to b\overline{b}$ decay width in the so-called intermediate mass range, defined by $M_W\le M_H\le 2M_W$. The correction of order ${\cal O}(\alpha_s)$ was first calculated in Ref. [@Braaten:1980yq]. The complete one-loop electroweak correction was found in Ref. [@Kniehl:1991]. As for the ${\cal O}(\alpha_s^2)$ correction, the leading [@Gorishnii:1991zr] and next-to-leading [@Surguladze:1994gc] terms of the expansion in $m_b^2/M_H^2$ of the diagrams without top quarks are known. The diagrams containing a top quark can be divided into two classes. The diagrams containing gluon self-energy insertions were calculated exactly [@Kniehl:1994vq], while for the double-triangle contributions the four leading terms of the expansion in $M_H^2/m_t^2$ are known [@Chetyrkin:1995pd]. In Ref. [@Chetyrkin:1996sr], the ${\cal O}(\alpha_s^3)$ correction without top-quark contributions was calculated in the massless limit. The correction induced by th top quark was subsequently found in Ref. [@Chetyrkin:1997vj] using an appropriate effective field theory. As for the correction of order ${\cal O}(\alpha_s G_F m_t^2)$, the universal part, which appears for any Higgs-boson decay to a fermion pair, was calculated in Ref. [@Kniehl:1994ph] and the non-universal one, using a low-energy theorem, in Ref. [@KniehlSpira]. The latter result was independently found in Ref. [@Kwiatkowski:1994cu]. Apart from the Higgs-boson decay into a $t\overline{t}$ pair, only the one into a $b\overline{b}$ pair has such non-universal top-quark-induced contributions, as bottom is the weak-isospin partner of top. The universal and non-universal corrections of order ${\cal O}(\alpha_s^2 G_F m_t^2)$ were calculated in Refs. [@delu] and [@Chetyrkin:1996ke], respectively. Finally, also a result for the universal correction of order ${\cal O}(G_F^2 m_t^4)$ was published [@Djouadi].
In this paper, we calculate the complete correction of order ${\cal O}(G_F^2 m_t^4)$, including both the universal and non-universal contributions. To this end, we formally assume that $M_H\ll m_t$. This includes the intermediate mass range of the Higgs boson. Our result for the universal contribution in the on-mass-shell scheme agrees with the one found in Ref. [@Djouadi], after correcting an obvious mistake in the latter paper. The key results of our calculation were already presented in a brief communication [@prl]. Here, the full details are exhibited.
Our calculations are performed in ’t Hooft-Feynman gauge. We adopt the on-mass-shell scheme and regularise the ultraviolet divergences by means of dimensional regularisation, with $D=4-2\epsilon$ space-time dimensions and ’t Hooft mass scale $\mu$. We use the anti-commuting definition of $\gamma_5$. As a simplification, we take the Cabibbo-Kobayashi-Maskawa quark mixing matrix to be unity. The Feynman diagrams are generated and drawn using the program `FeynArts` [@Hahn:2000kx] and evaluated using the program `MATAD` [@MATAD], which is written in the programming language `FORM` [@FORM].
In order to check our calculations, we also rederive the correction of order ${\cal O}(\alpha_s G_F m_t^2)$. Our result agrees with Refs. [@Kniehl:1994ph; @KniehlSpira; @Kwiatkowski:1994cu]. Since this calculation follows the lines of the one leading to the ${\cal O}(G_F^2 m_t^4)$ correction, being actually simpler, we refrain from going into details with it.
This paper is organised as follows. In Section \[CapRenSchema\], we describe in detail the renormalisation procedure underlying our analysis. In Section \[CapOurCalc\], we present the details of our diagrammatic calculations. In Section \[CapNieder\], we explain how a part of our calculations can be checked through the application of a low-energy theorem. In Section \[Numerics\], we evaluate the ${\cal O}(G_F^2 m_t^4)$ corrections numerically and compare them with the ${\cal O}(\alpha_s G_F m_t^2)$ ones. We conclude with a summary in Section \[CapZusammenfassung\].
Renormalisation procedure {#CapRenSchema}
=========================
For the reader’s convenience, we present in this section the details of the renormalisation procedure which has to be carried out. We derive general expressions for the mass counterterms and wave-function renormalisation constants in the on-shell scheme, valid for any number of loops. Furthermore, we derive the tadpole renormalisation counterterms and describe the treatment of the corrections due to external legs. In our calculations, we do not need to consider electric-charge renormalisation constants, because, to the orders we consider here, there are no such contributions.
Before going into details, we would like to mention that the expressions for the mass and wave-function renormalisation constants to be derived here are only valid for stable particles. Instable particles do have complex self-energy amplitudes, so that their resummed propagators have complex poles. In that case, the renormalisation conditions are more complicated (see, for instance, Ref. [@Kniehl:1998fn]). Since all self-energy amplitudes appearing in the calculations of this paper are real, we can restrict ourselves to the case of stable particles.
Mass and wave-function renormalisation {#KapMassPar}
--------------------------------------
We write the bare masses in the Lagrangian as sums of the renormalised ones and the mass counterterms. In the on-shell scheme, we fix this splitting by the requirement that the renormalised masses are identical to the poles of the propagators including all radiative corrections. Furthermore, the wave-function renormalisation constants are obtained as the residues of the propagators at their poles.
### Higgs-boson mass and wave-function renormalisation {#sec:higgs}
For the amputated one-particle-irreducible self-energy of the Higgs boson, we write $$\begin{minipage}{112pt}
\begin{picture}(112,32)
\GOval(56,16)(16,16)(0){0.882}
\Text(56,16)[]{1-PI}
\DashLine(40,16)(0,16){4}
\DashLine(72,16)(112,16){4}
\Text(92,19)[b]{$H$}
\Text(20,19)[b]{$H$}
\LongArrow(12,11)(28,11)
\Text(20,9)[t]{$q$}
\end{picture}
\end{minipage}
=i\Sigma_H(q^2).$$ Thus, the dressed propagator, including all radiative corrections, becomes $$\begin{aligned}
S_H^{-1}(q^2)&=&
\begin{minipage}{48pt}
\begin{picture}(48,32)
\DashLine(0,16)(48,16){4}
\end{picture}
\end{minipage}
+
\begin{minipage}{80pt}
\begin{picture}(80,32)
\GOval(40,16)(16,16)(0){0.882}
\Text(40,16)[]{1-PI}
\DashLine(24,16)(0,16){4}
\DashLine(56,16)(80,16){4}
\end{picture}
\end{minipage}
+
\begin{minipage}{112pt}
\begin{picture}(112,32)
\DashLine(0,16)(16,16){4}
\GOval(32,16)(16,16)(0){0.882}
\Text(32,16)[]{1-PI}
\DashLine(48,16)(64,16){4}
\GOval(80,16)(16,16)(0){0.882}
\Text(80,16)[]{1-PI}
\DashLine(96,16)(112,16){4}
\end{picture}
\end{minipage}
+\ldots \nonumber \\
&=& \frac{i}{q^2-M_{H,0}^2}\sum_{n=0}^\infty
\left(i\Sigma_H(q^2)\frac{i}{q^2-M_{H,0}^2}\right)^n
\nonumber \\
&=&\frac{i}{q^2-M_{H,0}^2+\Sigma_H(q^2)}.
\label{HiggsPropSum}\end{aligned}$$
The on-shell renormalisation condition reads $$\label{HiggsMassenBed}
S_H(M_H^2) \stackrel{!}{=} 0.$$ Writing the bare mass of the Higgs boson as the sum of the renormalised mass and a counterterm, $M_{H,0}^2 = M_H^2+\delta M_H^2$, we have $$\label{AusdrDmHq1loop}
\delta M_H^2 = \Sigma_H(M_H^2).$$ Here and in the following, it is understood that, in the expression for a counterterm, all bare quantities have to be replaced by the renormalised ones plus the respective counterterms. In the case of the Higgs-boson mass counterterm, this means that $\Sigma_H(M_H^2)$ has to be expressed in terms of renormalised quantities. For higher-order expressions, this has to be done iteratively.
Expanding Eq. (\[HiggsPropSum\]) about $q^2=M_H^2$ and taking the limit $q^2\to M_H^2$, $$\begin{aligned}
S_H^{-1}(q^2)&=&\frac{i}{q^2-M_H^2}\,
\frac{1}{1+\Sigma_H^\prime\left(M_H^2\right)+{\cal O}\left(q^2-M_H^2\right)}
\nonumber\\
&&{}\xrightarrow{q^2\to M_H^2}\frac{iZ_H}{q^2-M_H^2},\end{aligned}$$ we read off the Higgs-boson wave-function renormalisation constant as $$Z_H = \frac{1}{1+\Sigma_H^\prime(M_H^2)}.
\label{zh}$$ Writing $Z_H= 1+\delta Z_H$ and performing a loop expansion of Eq. (\[zh\]), we have $$\begin{aligned}
\delta Z_H^{(1)} &=& -\Sigma_H^{(1)\prime}(M_H^2),
\label{AusdrDZH1loop}\\
\delta Z_H^{(2)} &=&
-\Sigma_H^{(2)\prime}(M_H^2) +
\left(\Sigma_H^{(1)\prime}(M_H^2)\right)^2.
\label{AusdrDZH2loop}\end{aligned}$$ Here and in the following, numbers placed in parentheses as superscripts specify the loop order of the perturbative expression.
### Fermion mass and wave-function renormalisation {#KapFermMassRen}
The amputated one-particle-irreducible self-energy of fermion $f$ has the form $$\begin{minipage}{112pt}
\begin{picture}(112,32)
\GOval(56,16)(16,16)(0){0.882}
\Text(56,16)[]{1-PI}
\ArrowLine(0,16)(40,16)
\ArrowLine(72,16)(112,16)
\Text(92,20)[b]{$f$}
\Text(20,20)[b]{$f$}
\LongArrow(12,11)(28,11)
\Text(20,9)[t]{$q$}
\end{picture}
\end{minipage}
=i\Sigma_f(q)=i\slashed{q}\omega_- \Sigma_{f,L}(q^2)
+i\slashed{q}\omega_+ \Sigma_{f,R}(q^2)
+ im_{f,0} \Sigma_{f,S}(q^2),
\label{FermSelbstDef}$$ where $m_{f,0}$ is the bare mass of fermion $f$ and $\omega_\pm =(1\pm\gamma_5)/2$ are the projectors onto the helicity eigenstates.
The fermion field $f$ is composed of left- and right-handed components, $l$ and $r$, respectively, as $$f = l + r , \qquad l = \omega_- f , \qquad r = \omega_+ f.$$ In the electroweak theory, $l$ and $r$ interact differently, which has to be accounted for in the renormalisation procedure. In terms of these components, the purely fermionic part of the SM Lagrangian reads: $${\cal L} = \overline{f}(i\slashed{\partial}-m_{f,0})f
= i\overline{l}\slashed{\partial}l
+ i\overline{r}\slashed{\partial}r
- m_{f,0}\overline{r}l - m_{f,0}\overline{l}r.$$ We see that $l$ and $r$ are massless fermion fields with propagators $$\begin{minipage}{48pt}
\begin{picture}(48,32)
\ArrowLine(0,16)(48,16)
\Text(24,20)[b]{$l$}
\LongArrow(16,11)(32,11)
\Text(24,9)[t]{$q$}
\end{picture}
\end{minipage}
=
\begin{minipage}{48pt}
\begin{picture}(48,32)
\ArrowLine(0,16)(48,16)
\Text(24,20)[b]{$r$}
\LongArrow(16,11)(32,11)
\Text(24,9)[t]{$q$}
\end{picture}
\end{minipage}
= \frac{i}{\slashed{q}}.
\label{FermFirst}$$ In addition, we have the following $r$-$l$ transition vertices: $$\begin{minipage}{48pt}
\begin{picture}(48,32)
\ArrowLine(0,16)(24,16)
\Vertex(24,16){2}
\ArrowLine(24,16)(48,16)
\Text(12,20)[b]{$l$}
\Text(36,20)[b]{$r$}
\end{picture}
\end{minipage}
=
\begin{minipage}{48pt}
\begin{picture}(48,32)
\ArrowLine(0,16)(24,16)
\Vertex(24,16){2}
\ArrowLine(24,16)(48,16)
\Text(12,20)[b]{$r$}
\Text(36,20)[b]{$l$}
\end{picture}
\end{minipage}
= -i m_{f,0}.
\label{FermSecond}$$ From Eq. (\[FermSelbstDef\]), we read off the amputated one-particle-irreducible self-energies pertaining to the four different helicity combinations as $$\begin{aligned}
\begin{minipage}{80pt}
\begin{picture}(80,32)
\GOval(40,16)(16,16)(0){0.882}
\Text(40,16)[]{1-PI}
\ArrowLine(0,16)(24,16)
\ArrowLine(56,16)(80,16)
\Text(68,20)[b]{$l$}
\Text(12,20)[b]{$l$}
\LongArrow(4,11)(20,11)
\Text(12,9)[t]{$q$}
\end{picture}
\end{minipage}
&=&i \slashed{q}\Sigma_{f,L}(q^2),
\nonumber\\
\begin{minipage}{80pt}
\begin{picture}(80,32)
\GOval(40,16)(16,16)(0){0.882}
\Text(40,16)[]{1-PI}
\ArrowLine(0,16)(24,16)
\ArrowLine(56,16)(80,16)
\Text(68,20)[b]{$r$}
\Text(12,20)[b]{$r$}
\LongArrow(4,11)(20,11)
\Text(12,9)[t]{$q$}
\end{picture}
\end{minipage}
&=&i \slashed{q}\Sigma_{f,R}(q^2),
\nonumber\\
\begin{minipage}{80pt}
\begin{picture}(80,32)
\GOval(40,16)(16,16)(0){0.882}
\Text(40,16)[]{1-PI}
\ArrowLine(0,16)(24,16)
\ArrowLine(56,16)(80,16)
\Text(68,20)[b]{$r$}
\Text(12,20)[b]{$l$}
\LongArrow(4,11)(20,11)
\Text(12,9)[t]{$q$}
\end{picture}
\end{minipage}
&=&
\begin{minipage}{80pt}
\begin{picture}(80,32)
\GOval(40,16)(16,16)(0){0.882}
\Text(40,16)[]{1-PI}
\ArrowLine(0,16)(24,16)
\ArrowLine(56,16)(80,16)
\Text(68,20)[b]{$l$}
\Text(12,20)[b]{$r$}
\LongArrow(4,11)(20,11)
\Text(12,9)[t]{$q$}
\end{picture}
\end{minipage}
=i m_{f,0}\Sigma_{f,S}(q^2).
\label{FermLast}\end{aligned}$$ Note that above expressions do not yet include the tree-level contributions from Eqs. (\[FermFirst\]) and (\[FermSecond\]). Equations (\[FermFirst\])–(\[FermLast\]) are the ingredients out of which we construct the propagators of the left- and right-handed fields including all radiative corrections. This is done in close analogy to the case of $\gamma$-$Z$-mixing (see, e.g., Ref. [@Hollik:1988ii]). To this end, we introduce the propagator-type symbols $$\begin{aligned}
\begin{minipage}{48pt}
\begin{picture}(48,32)
\ArrowLine(0,16)(48,16)
\Curve{(20,19)(28,19)}
\Curve{(20,13)(28,13)}
\Text(24,22)[b]{$l$}
\end{picture}
\end{minipage}
:&=&
\begin{minipage}{48pt}
\begin{picture}(48,32)
\ArrowLine(0,16)(48,16)
\Text(24,20)[b]{$l$}
\end{picture}
\end{minipage}
+
\begin{minipage}{80pt}
\begin{picture}(80,32)
\GOval(40,16)(16,16)(0){0.882}
\Text(40,16)[]{1-PI}
\ArrowLine(0,16)(24,16)
\ArrowLine(56,16)(80,16)
\Text(68,20)[b]{$l$}
\Text(12,20)[b]{$l$}
\end{picture}
\end{minipage}
+
\begin{minipage}{112pt}
\begin{picture}(112,32)
\ArrowLine(0,16)(16,16)
\Text(8,20)[b]{$l$}
\GOval(32,16)(16,16)(0){0.882}
\Text(32,16)[]{1-PI}
\ArrowLine(48,16)(64,16)
\Text(56,20)[b]{$l$}
\GOval(80,16)(16,16)(0){0.882}
\Text(80,16)[]{1-PI}
\ArrowLine(96,16)(112,16)
\Text(104,20)[b]{$l$}
\end{picture}
\end{minipage}
+\ldots
\nonumber\\
&=& \frac{i}{\slashed{q}} \sum_{n=0}^\infty \left(
i\slashed{q}\Sigma_{f,L}(q^2)\frac{i}{\slashed{q}} \right)^n
= \frac{i}{\slashed{q}\left(1+\Sigma_{f,L}(q^2)\right)},
\nonumber\\
\begin{minipage}{48pt}
\begin{picture}(48,32)
\ArrowLine(0,16)(48,16)
\Curve{(20,19)(28,19)}
\Curve{(20,13)(28,13)}
\Text(24,22)[b]{$r$}
\end{picture}
\end{minipage}
:&=&
\begin{minipage}{48pt}
\begin{picture}(48,32)
\ArrowLine(0,16)(48,16)
\Text(24,20)[b]{$r$}
\end{picture}
\end{minipage}
+
\begin{minipage}{80pt}
\begin{picture}(80,32)
\GOval(40,16)(16,16)(0){0.882}
\Text(40,16)[]{1-PI}
\ArrowLine(0,16)(24,16)
\ArrowLine(56,16)(80,16)
\Text(68,20)[b]{$r$}
\Text(12,20)[b]{$r$}
\end{picture}
\end{minipage}
+
\begin{minipage}{112pt}
\begin{picture}(112,32)
\ArrowLine(0,16)(16,16)
\Text(8,20)[b]{$r$}
\GOval(32,16)(16,16)(0){0.882}
\Text(32,16)[]{1-PI}
\ArrowLine(48,16)(64,16)
\Text(56,20)[b]{$r$}
\GOval(80,16)(16,16)(0){0.882}
\Text(80,16)[]{1-PI}
\ArrowLine(96,16)(112,16)
\Text(104,20)[b]{$r$}
\end{picture}
\end{minipage}
+\ldots
\nonumber\\
&=& \frac{i}{\slashed{q}} \sum_{n=0}^\infty \left(
i\slashed{q}\Sigma_{f,R}(q^2)\frac{i}{\slashed{q}} \right)^n
= \frac{i}{\slashed{q}\left(1+\Sigma_{f,R}(q^2)\right)},\end{aligned}$$ and the vertex-type symbols $$\begin{aligned}
\begin{minipage}{60pt}
\begin{picture}(60,32)
\GOval(30,16)(6,6)(0){0.882}
\Vertex(30,16){2}
\ArrowLine(0,16)(24,16)
\ArrowLine(36,16)(60,16)
\Text(48,20)[b]{$r$}
\Text(12,20)[b]{$l$}
\end{picture}
\end{minipage}
:&=&
\begin{minipage}{48pt}
\begin{picture}(48,32)
\ArrowLine(0,16)(24,16)
\Vertex(24,16){2}
\ArrowLine(24,16)(48,16)
\Text(12,20)[b]{$l$}
\Text(36,20)[b]{$r$}
\end{picture}
\end{minipage}
+
\begin{minipage}{80pt}
\begin{picture}(80,32)
\GOval(40,16)(16,16)(0){0.882}
\Text(40,16)[]{1-PI}
\ArrowLine(0,16)(24,16)
\ArrowLine(56,16)(80,16)
\Text(68,20)[b]{$r$}
\Text(12,20)[b]{$l$}
\end{picture}
\end{minipage}
= im_{f,0}\left(\Sigma_{f,S}(q^2)-1 \right),
\nonumber\\
\begin{minipage}{60pt}
\begin{picture}(60,32)
\GOval(30,16)(6,6)(0){0.882}
\Vertex(30,16){2}
\ArrowLine(0,16)(24,16)
\ArrowLine(36,16)(60,16)
\Text(48,20)[b]{$l$}
\Text(12,20)[b]{$r$}
\end{picture}
\end{minipage}
:&=&
\begin{minipage}{48pt}
\begin{picture}(48,32)
\ArrowLine(0,16)(24,16)
\Vertex(24,16){2}
\ArrowLine(24,16)(48,16)
\Text(12,20)[b]{$r$}
\Text(36,20)[b]{$l$}
\end{picture}
\end{minipage}
+
\begin{minipage}{80pt}
\begin{picture}(80,32)
\GOval(40,16)(16,16)(0){0.882}
\Text(40,16)[]{1-PI}
\ArrowLine(0,16)(24,16)
\ArrowLine(56,16)(80,16)
\Text(68,20)[b]{$l$}
\Text(12,20)[b]{$r$}
\end{picture}
\end{minipage}
= im_{f,0}\left(\Sigma_{f,S}(q^2)-1 \right).\end{aligned}$$ Next, we evaluate the dressed propagator of the left-handed fermion field, including all radiative corrections, as $$\begin{aligned}
S_{ll}^{-1}(q) &=&
\begin{minipage}{48pt}
\begin{picture}(48,32)
\ArrowLine(0,16)(48,16)
\Curve{(20,19)(28,19)}
\Curve{(20,13)(28,13)}
\Text(24,22)[b]{$l$}
\end{picture}
\end{minipage}
+
\begin{minipage}{96pt}
\begin{picture}(96,32)
\ArrowLine(0,16)(24,16)
\Text(12,22)[b]{$l$}
\Curve{(8,19)(16,19)}
\Curve{(8,13)(16,13)}
\GOval(30,16)(6,6)(0){0.882}
\Vertex(30,16){2}
\ArrowLine(36,16)(60,16)
\Text(48,22)[b]{$r$}
\Curve{(42,19)(52,19)}
\Curve{(42,13)(52,13)}
\GOval(66,16)(6,6)(0){0.882}
\Vertex(66,16){2}
\ArrowLine(72,16)(96,16)
\Text(84,22)[b]{$l$}
\Curve{(80,19)(88,19)}
\Curve{(80,13)(88,13)}
\end{picture}
\end{minipage}
+
\begin{minipage}{168pt}
\begin{picture}(168,32)
\ArrowLine(0,16)(24,16)
\Text(12,22)[b]{$l$}
\Curve{(8,19)(16,19)}
\Curve{(8,13)(16,13)}
\GOval(30,16)(6,6)(0){0.882}
\Vertex(30,16){2}
\ArrowLine(36,16)(60,16)
\Text(48,22)[b]{$r$}
\Curve{(42,19)(52,19)}
\Curve{(42,13)(52,13)}
\GOval(66,16)(6,6)(0){0.882}
\Vertex(66,16){2}
\ArrowLine(72,16)(96,16)
\Text(84,22)[b]{$l$}
\Curve{(80,19)(88,19)}
\Curve{(80,13)(88,13)}
\GOval(102,16)(6,6)(0){0.882}
\Vertex(102,16){2}
\ArrowLine(108,16)(132,16)
\Text(120,22)[b]{$r$}
\Curve{(116,19)(124,19)}
\Curve{(116,13)(124,13)}
\GOval(138,16)(6,6)(0){0.882}
\Vertex(138,16){2}
\ArrowLine(144,16)(168,16)
\Text(156,22)[b]{$l$}
\Curve{(152,19)(160,19)}
\Curve{(152,13)(160,13)}
\end{picture}
\end{minipage}
+\ldots
\nonumber\\
&=&\frac{i}{\slashed{q}\left( 1 + \Sigma_{f,L}(q^2)
\right)}\sum_{n=0}^\infty \left[ im_{f,0} \left(
\Sigma_{f,S}(q^2)-1\right) \frac{i}{\slashed{q}\left( 1 + \Sigma_{f,R}(q^2)
\right)} im_{f,0} \left( \Sigma_{f,S}(q^2)-1\right)
\right.
\nonumber\\
&&{}\times\left.
\frac{i}{\slashed{q}\left( 1 + \Sigma_{f,L}(q^2) \right)} \right]^n
\nonumber\\
&=&\frac{i\slashed{q}}{1+\Sigma_{f,L}(q^2)}\,\frac{1}{q^2 -
m_{f,0}^2f(q^2)},
\label{LPropSum}\end{aligned}$$ where $$f(q^2)=\frac{(1-\Sigma_{f,S}(q^2))^2}{(1+\Sigma_{f,L}(q^2))
(1+\Sigma_{f,R}(q^2))}.$$ In a similar way, we find the dressed propagator of the right-handed fermion field, including all radiative corrections, to be $$S_{rr}^{-1}(q) =
\frac{i\slashed{q}}{1+\Sigma_{f,R}(q^2)}\,\frac{1}{q^2 -
m_{f,0}^2 f(q^2)}.
\label{RPropSum}$$ For completeness, we also resum the loop contributions by which a left-handed field converts into a right-handed one and vice versa. Proceeding similarly as in Eq. (\[LPropSum\]), we obtain $$\begin{aligned}
S_{lr}^{-1}(q) &=&
\begin{minipage}{60pt}
\begin{picture}(60,32)
\ArrowLine(0,16)(24,16)
\Text(12,22)[b]{$l$}
\Curve{(8,19)(16,19)}
\Curve{(8,13)(16,13)}
\GOval(30,16)(6,6)(0){0.882}
\Vertex(30,16){2}
\ArrowLine(36,16)(60,16)
\Text(48,22)[b]{$r$}
\Curve{(42,19)(52,19)}
\Curve{(42,13)(52,13)}
\end{picture}
\end{minipage}
+
\begin{minipage}{132pt}
\begin{picture}(132,32)
\ArrowLine(0,16)(24,16)
\Text(12,22)[b]{$l$}
\Curve{(8,19)(16,19)}
\Curve{(8,13)(16,13)}
\GOval(30,16)(6,6)(0){0.882}
\Vertex(30,16){2}
\ArrowLine(36,16)(60,16)
\Text(48,22)[b]{$r$}
\Curve{(42,19)(52,19)}
\Curve{(42,13)(52,13)}
\GOval(66,16)(6,6)(0){0.882}
\Vertex(66,16){2}
\ArrowLine(72,16)(96,16)
\Text(84,22)[b]{$l$}
\Curve{(80,19)(88,19)}
\Curve{(80,13)(88,13)}
\GOval(102,16)(6,6)(0){0.882}
\Vertex(102,16){2}
\ArrowLine(108,16)(132,16)
\Text(120,22)[b]{$r$}
\Curve{(116,19)(124,19)}
\Curve{(116,13)(124,13)}
\end{picture}
\end{minipage}
+\ldots
\nonumber\\
&=&\frac{i}{\slashed{q}\left( 1 + \Sigma_{f,L}(q^2) \right)}
im_{f,0} \left( \Sigma_{f,S}(q^2)-1\right) \frac{i}{\slashed{q}\left( 1 +
\Sigma_{f,R}(q^2) \right)}
\nonumber\\
&&{}\times \sum_{n=0}^\infty \left[ im_{f,0} \left(
\Sigma_{f,S}(q^2)-1\right) \frac{i}{\slashed{q}\left( 1 +
\Sigma_{f,L}(q^2) \right)}
im_{f,0} \left( \Sigma_{f,S}(q^2)-1\right)
\right.
\nonumber\\
&&{}\times\left.\frac{i}{\slashed{q}\left( 1 +
\Sigma_{f,R}(q^2) \right)} \right]^n
\nonumber\\
&=& \frac{im_{f,0}(1-\Sigma_{f,S}(q^2))}
{(1+\Sigma_{f,L}(q^2))(1+\Sigma_{f,R}(q^2))} \,\frac{1}{q^2 -
m_{f,0}^2f(q^2)}.
\label{PropLtoR}\end{aligned}$$ Since Eq. (\[PropLtoR\]) is symmetric under the interchange of the indices $L$ and $R$, we also have $$S_{rl}^{-1}(q)=S_{lr}^{-1}(q).
\label{rightleft}$$
We now derive the fermion mass counterterm. Writing $m_{f,0} = m_f + \delta m_f$, where $m_f$ is the renormalised mass and $\delta m_f$ is the mass counterterm, and imposing the on-shell renormalisation condition, $$\left.S_{ij}(q)u_f(q)\right|_{q^2=m_f^2} \stackrel{!}{=}0,
\label{FermMassA}$$ where $ij=ll,rr,lr,rl$ and $u_f(q)$ is the spinor of the incoming fermion $f$, we obtain $$\frac{\delta m_f}{m_f} =\frac{1}{\sqrt{f\left(m_f^2\right)}}-1.
\label{FermMassB}$$ Expanding Eq. (\[FermMassB\]), we find the explicit one- and two-loop expressions, $$\begin{aligned}
{\frac{\delta m_f^{(1)}}{m_f}}&=& \frac{1}{2}\Sigma_{f,L}^{(1)}(m_f^2)+
\frac{1}{2}\Sigma_{f,R}^{(1)}(m_f^2)+\Sigma_{f,S}^{(1)}(m_f^2),
\label{DeltaMF1Loop}\\
{\frac{\delta m_f^{(2)}}{m_f}}&=&\frac{1}{2}\Sigma_{f,L}^{(2)}(m_f^2)
+\frac{1}{2}\Sigma_{f,R}^{(2)}(m_f^2)+\Sigma_{f,S}^{(2)}(m_f^2)
-\frac{1}{8}\left(\Sigma_{f,L}^{(1)}(m_f^2)-\Sigma_{f,R}^{(1)}(m_f^2)\right)^2
\nonumber\\
&&{}+\Sigma_{f,S}^{(1)}(m_f^2){\frac{\delta m_f^{(1)}}{m_f}}.
\label{DeltaMF2Loop}\end{aligned}$$ The one-loop expression of Eq. (\[DeltaMF1Loop\]) is well known (see, e.g., Ref. [@Kniehl:1991]). The two-loop expression of Eq. (\[DeltaMF2Loop\]) agrees with the one obtained in Ref. [@Faisst] using an alternative procedure.
Finally, we derive the wave-function renormalisation constants for the left-handed and right-handed fields. Expanding Eqs. (\[LPropSum\]) and (\[RPropSum\])–(\[rightleft\]) about $\slashed{q}=m_f$ and taking the limit $\slashed{q}\to m_f$, we have $$\begin{aligned}
S_{ll/rr}^{-1}(q)&=&\frac{i\slashed{q}}{q^2-m_f^2}
\,\frac{1}{\left(1+\Sigma_{f,L/R}(m_f^2)\right)
\left(1-m_f^2\frac{f^\prime(m_f^2)}{f(m_f^2)}\right)
+{\cal O}\left(q^2-m_f^2\right)}
\nonumber\\
&&{}\xrightarrow{q^2\to m_f^2}\frac{i\slashed{q}Z_{f,L/R}}{q^2-m_f^2},
\nonumber\\
S_{lr/rl}^{-1}(q)&=&\frac{im_f}{q^2-m_f^2}
\,\frac{1}{\sqrt{\left(1+\Sigma_{f,L}(m_f^2)\right)
\left(1+\Sigma_{f,R}(m_f^2)\right)}
\left(1-m_f^2\frac{f^\prime(m_f^2)}{f(m_f^2)}\right)
+{\cal O}\left(q^2-m_f^2\right)}
\nonumber\\
&&{}\xrightarrow{q^2\to m_f^2}\frac{im_f\sqrt{Z_{f,L}Z_{f,R}}}{q^2-m_f^2},\end{aligned}$$ where $$Z_{f,L/R} = \frac{1}{
\left(1+\Sigma_{f,L/R}(m_f^2)\right) \left( 1 -
m_f^2\frac{f^\prime(m_f^2)}{f(m_f^2)}\right) }.
\label{FermionZ}$$ Writing $Z_{f,L/R}= 1+\delta Z_{f,L/R}$ and performing a loop expansion of Eq. (\[FermionZ\]), we have $$\begin{aligned}
\delta Z_{f,L}^{(1)} &=& - \Sigma_L^{(1)}
- \Sigma_L^{(1)\prime} - \Sigma_R^{(1)\prime} - 2\Sigma_S^{(1)\prime},
\label{AusdrDZbl1loop}\\
\delta Z_{f,R}^{(1)} &=& - \Sigma_R^{(1)}
- \Sigma_L^{(1)\prime} - \Sigma_R^{(1)\prime} - 2\Sigma_S^{(1)\prime},
\label{AusdrDZbr1loop}\\
\delta Z_{f,L}^{(2)} &=& - \Sigma_L^{(2)} - \Sigma_L^{(2)\prime}
- \Sigma_R^{(2)\prime} - 2\Sigma_S^{(2)\prime}
+ \Sigma_L^{(1)}\left(\Sigma_L^{(1)} + 2\Sigma_L^{(1)\prime}
+ \Sigma_R^{(1)\prime} + 2\Sigma_S^{(1)\prime}\right)
\nonumber\\
&&{}+\Sigma_R^{(1)}\Sigma_R^{(1)\prime}-2\Sigma_S^{(1)}\Sigma_S^{(1)\prime}
+\left(\Sigma_L^{(1)\prime}+\Sigma_R^{(1)\prime}+2\Sigma_S^{(1)\prime}
\right)^2,
\label{AusdrDZbl2loop}\\
\delta Z_{f,R}^{(2)} &=& - \Sigma_R^{(2)} - \Sigma_L^{(2)\prime}
- \Sigma_R^{(2)\prime} - 2\Sigma_S^{(2)\prime}
+ \Sigma_R^{(1)}\left(\Sigma_R^{(1)} + \Sigma_L^{(1)\prime}
+ 2\Sigma_R^{(1)\prime} + 2\Sigma_S^{(1)\prime}\right)
\nonumber\\
&&{}+\Sigma_L^{(1)}\Sigma_L^{(1)\prime}-2\Sigma_S^{(1)}\Sigma_S^{(1)\prime}
+\left(\Sigma_L^{(1)\prime}+\Sigma_R^{(1)\prime}+2\Sigma_S^{(1)\prime}
\right)^2.
\label{AusdrDZbr2loop}\end{aligned}$$ Here, we used the abbreviations $$\begin{aligned}
\Sigma_X^{(n)}&=&\Sigma_{f,X}^{(n)}(m_f^2),
\nonumber\\
\Sigma_X^{(n)\prime} &=& m_f^2\frac{\partial}{\partial
q^2}\Sigma_{f,X}^{(n)}(q^2) \Big|_{q^2=m_f^2},\end{aligned}$$ where $X =L,R,S$. These expressions again agree with Refs. [@Kniehl:1991; @Faisst].
If parity was conserved, we would have $\Sigma_L^f(q^2)=\Sigma_R^f(q^2)$ and thus recover the structure $$S_f^{-1}(q) \xrightarrow{q^2\to m_f^2} \frac{iZ_f}{\slashed{q}-m_f},$$ which is familiar from quantum electrodynamics.
### $W$-boson mass renormalisation
The amputated one-particle-irreducible self-energy of the $W$ boson can be decomposed into a transverse and a longitudinal part as $$\begin{minipage}{112pt}
\begin{picture}(112,32)
\GOval(56,16)(16,16)(0){0.882}
\Text(56,16)[]{1-PI}
\Photon(40,16)(0,16){2}{5}
\Photon(72,16)(112,16){2}{5}
\Text(92,21)[b]{$W_\nu$}
\Text(20,20)[b]{$W_\mu$}
\LongArrow(12,11)(28,11)
\Text(20,9)[t]{$q$}
\end{picture}
\end{minipage}
= -i\Pi_W^{\mu\nu}(q)
= -i \left(\Delta^{\mu\nu}\Sigma_{W,T}(q^2)
+ q^{\mu\nu}\Sigma_{W,L}(q^2)\right),
\label{VecDef}$$ where $$\begin{aligned}
\Delta^{\mu\nu}&=& g^{\mu\nu} - \frac{q^\mu q^\nu}{q^2},
\nonumber\\
q^{\mu\nu}&=& \frac{q^\mu q^\nu}{q^2}.\end{aligned}$$ Owing to the loop-induced mixing of the $W$ boson with the charged Higgs-Kibble ghost $\phi$, we must also take into account the one-particle-irreducible $W\leftrightarrow\phi$ transition amplitudes and the one-particle-irreducible $\phi$-boson self-energy, $$\begin{aligned}
\begin{minipage}{112pt}
\begin{picture}(112,32)
\GOval(56,16)(16,16)(0){0.882}
\Text(56,16)[]{1-PI}
\Photon(40,16)(0,16){2}{5}
\DashLine(72,16)(112,16){4}
\Text(92,21)[b]{$\phi$}
\Text(20,20)[b]{$W_\mu$}
\LongArrow(12,11)(28,11)
\Text(20,9)[t]{$q$}
\end{picture}
\end{minipage}
&=&iq^{\mu}\Sigma_{W\phi}(q^2),
\nonumber\\
\begin{minipage}{112pt}
\begin{picture}(112,32)
\GOval(56,16)(16,16)(0){0.882}
\Text(56,16)[]{1-PI}
\DashLine(40,16)(0,16){4}
\Photon(72,16)(112,16){2}{5}
\Text(92,21)[b]{$W_\mu$}
\Text(20,20)[b]{$\phi$}
\LongArrow(12,11)(28,11)
\Text(20,9)[t]{$q$}
\end{picture}
\end{minipage}
&=&-iq^{\mu}\Sigma_{W\phi}(q^2),
\nonumber\\
\begin{minipage}{112pt}
\begin{picture}(112,32)
\GOval(56,16)(16,16)(0){0.882}
\Text(56,16)[]{1-PI}
\DashLine(40,16)(0,16){4}
\DashLine(72,16)(112,16){4}
\Text(92,21)[b]{$\phi$}
\Text(20,20)[b]{$\phi$}
\LongArrow(12,11)(28,11)
\Text(20,9)[t]{$q$}
\end{picture}
\end{minipage}
&=&i\Sigma_\phi(q^2).\end{aligned}$$ In ’t Hooft-Feynman gauge, the bare propagators of the $W$ and $\phi$ bosons are given by $$\begin{aligned}
G_W^{\mu\nu}(q^2) &=& \frac{-ig^{\mu\nu}}{q^2-M_{W,0}^2},
\label{WProp}\\
G_\phi(q^2)&=&\frac{i}{q^2-M_{W,0}^2},
\label{PhiProp}\end{aligned}$$ with a common bare mass $M_{W,0}$. In order to obtain the dressed $W$-boson propagator, we proceed in two steps. In the first step, we resum the one-particle irreducible self-energies of the $W$ and $\phi$ bosons separately. In the second step, we systematically combine these results by accommodating all possible $W\leftrightarrow\phi$ transitions.
The resummation of the one-particle irreducible $W$-boson self-energy leads to $$\begin{aligned}
\begin{minipage}{48pt}
\begin{picture}(48,32)
\Photon(0,16)(48,16){2}{5}
\Curve{(20,20)(28,20)}
\Curve{(20,12)(28,12)}
\Text(25,22)[b]{$W$}
\end{picture}
\end{minipage}
:&=&
\begin{minipage}{48pt}
\begin{picture}(48,32)
\Photon(0,16)(48,16){2}{5}
\Text(24,20)[b]{$W$}
\end{picture}
\end{minipage}
+
\begin{minipage}{80pt}
\begin{picture}(80,32)
\GOval(40,16)(16,16)(0){0.882}
\Text(40,16)[]{1-PI}
\Photon(0,16)(24,16){2}{3}
\Photon(56,16)(80,16){2}{3}
\Text(68,20)[b]{$W$}
\Text(12,20)[b]{$W$}
\end{picture}
\end{minipage}
+
\begin{minipage}{112pt}
\begin{picture}(112,32)
\Photon(0,16)(16,16){2}{2}
\Text(8,20)[b]{$W$}
\GOval(32,16)(16,16)(0){0.882}
\Text(32,16)[]{1-PI}
\Photon(48,16)(64,16){2}{2}
\Text(56,20)[b]{$W$}
\GOval(80,16)(16,16)(0){0.882}
\Text(80,16)[]{1-PI}
\Photon(96,16)(112,16){2}{2}
\Text(104,20)[b]{$W$}
\end{picture}
\end{minipage}
+\ldots
\nonumber\\
&=& G_{W,\mu\nu}(q^2) + G_{W,\mu\alpha}(q^2)
\left(-i\Pi_W^{\alpha\beta}(q)\right) G_{W,\beta\nu}(q^2)
\nonumber\\
&&{}+G_{W,\mu\alpha}(q^2)
\left(-i\Pi_W^{\alpha\beta}(q)\right) G_{W,\beta\gamma}(q^2)
\left(-i\Pi_W^{\gamma\delta}(q)\right) G_{W,\delta\nu}(q^2)
+\ldots.
\label{WPropSum} \end{aligned}$$ The series in Eq. (\[WPropSum\]) may be resummed by inserting Eqs. (\[VecDef\]) and (\[WProp\]) and exploiting the identities $$\begin{aligned}
{\Delta^\mu}_\nu{\Delta^\nu}_\rho&=&{\Delta^\mu}_\rho,
\nonumber\\
\Delta^{\mu\nu}q_{\nu\rho}&=& 0,
\nonumber\\
q^{\mu\nu}q_{\nu\rho}&=&{q^\mu}_\rho,\end{aligned}$$ as follows $$\begin{aligned}
\begin{minipage}{48pt}
\begin{picture}(48,32)
\Photon(0,16)(48,16){2}{5}
\Curve{(20,20)(28,20)}
\Curve{(20,12)(28,12)}
\Text(25,22)[b]{$W$}
\end{picture}
\end{minipage}
&=&
G_{W,\mu\alpha}(q^2)
\left[{g^\alpha}_\nu -
\frac{{\Delta^\alpha}_\nu\Sigma_{W,T}(q^2)+{q^\alpha}_\nu\Sigma_{W,L}(q^2)}
{q^2-M_{W,0}^2}
\right.
\nonumber\\
&&{}+\left.
\frac{{\Delta^\alpha}_\nu\left(\Sigma_{W,T}(q^2)\right)^2
+{q^\alpha}_\nu\left(\Sigma_{W,L}(q^2)\right)^2}
{(q^2-M_{W,0}^2)^2} - \ldots \right]
\nonumber\\
&=& G_{W,\mu\alpha}(q^2)
\left[{g^\alpha}_\nu + {\Delta^\alpha}_\nu \sum_{n=1}^\infty
\left( \frac{-\Sigma_{W,T}(q^2)}{q^2-M_{W,0}^2} \right)^n +
{q^\alpha}_\nu \sum_{n=1}^\infty \left(
\frac{-\Sigma_{W,L}(q^2)}{q^2-M_{W,0}^2} \right)^n \right]
\nonumber\\
&=&
G_{W,\mu\alpha}(q^2)
\left[ {g^\alpha}_\nu +
{\Delta^\alpha}_\nu\left(\frac{1}{1+\frac{\Sigma_{W,T}(q^2)}{q^2-M_{W,0}^2}}
-1\right)
+{q^\alpha}_\nu\left(\frac{1}{1+\frac{\Sigma_{W,L}(q^2)}{q^2-M_{W,0}^2}}
-1\right)\right]
\nonumber\\
&=&
-i\frac{\Delta_{\mu\nu}}{q^2-M_{W,0}^2+\Sigma_{W,T}(q^2)}
-i\frac{q_{\mu\nu}}{q^2-M_{W,0}^2+\Sigma_{W,L}(q^2)}
\nonumber\\
&=&\left(S_{W,{\rm pure}}^{-1}\right)_{\mu\nu}(q).
\label{eq:Wpure}\end{aligned}$$ The resummation of the one-particle-irreducible $\phi$-boson self-energy proceeds in analogy to the Higgs-boson case discussed in Section \[sec:higgs\] and yields $$\begin{aligned}
\begin{minipage}{48pt}
\begin{picture}(48,32)
\DashLine(0,16)(48,16){4}
\Curve{(20,20)(28,20)}
\Curve{(20,12)(28,12)}
\Text(24,22)[b]{$\phi$}
\end{picture}
\end{minipage}
:&=&
\begin{minipage}{48pt}
\begin{picture}(48,32)
\DashLine(0,16)(48,16){4}
\Text(24,20)[b]{$\phi$}
\end{picture}
\end{minipage}
+
\begin{minipage}{80pt}
\begin{picture}(80,32)
\GOval(40,16)(16,16)(0){0.882}
\Text(40,16)[]{1-PI}
\DashLine(0,16)(24,16){4}
\DashLine(56,16)(80,16){4}
\Text(68,20)[b]{$\phi$}
\Text(12,20)[b]{$\phi$}
\end{picture}
\end{minipage}
+
\begin{minipage}{112pt}
\begin{picture}(112,32)
\DashLine(0,16)(16,16){4}
\Text(8,20)[b]{$\phi$}
\GOval(32,16)(16,16)(0){0.882}
\Text(32,16)[]{1-PI}
\DashLine(48,16)(64,16){4}
\Text(56,20)[b]{$\phi$}
\GOval(80,16)(16,16)(0){0.882}
\Text(80,16)[]{1-PI}
\DashLine(96,16)(112,16){4}
\Text(104,20)[b]{$\phi$}
\end{picture}
\end{minipage}
+\ldots
\nonumber\\
&=&\frac{i}{q^2-M_{W,0}^2+\Sigma_\phi(q^2)}.
\label{eq:Phipure}\end{aligned}$$
The contribution of unmixed $W$-boson propagation in Eq. (\[eq:Wpure\]) needs to be complemented by the contribution that emerges by combining it with the contribution of unmixed $\phi$-boson propagation of Eq. (\[eq:Phipure\]) via the one-particle-irreducible $W\leftrightarrow\phi$ transition amplitudes in all possible ways. This additional contribution is given by $$\begin{aligned}
\left(S_{W,{\rm mix}}^{-1}\right)_{\mu\nu}(q)&=&
\begin{minipage}{96pt}
\begin{picture}(96,32)
\Photon(0,16)(24,16){1.5}{3}
\Text(13,22)[b]{$W$}
\Curve{(8,20)(16,20)}
\Curve{(8,12)(16,12)}
\GOval(30,16)(6,6)(0){0.882}
\Text(30,16)[]{\tiny 1PI}
\DashLine(36,16)(60,16){4}
\Text(47,22)[b]{$\phi$}
\Curve{(42,20)(52,20)}
\Curve{(42,12)(52,12)}
\GOval(66,16)(6,6)(0){0.882}
\Text(66,16)[]{\tiny 1PI}
\Photon(72,16)(96,16){1.5}{3}
\Text(85,22)[b]{$W$}
\Curve{(80,20)(88,20)}
\Curve{(80,12)(88,12)}
\end{picture}
\end{minipage}
+
\begin{minipage}{168pt}
\begin{picture}(168,32)
\Photon(0,16)(24,16){1.5}{3}
\Text(13,22)[b]{$W$}
\Curve{(8,20)(16,20)}
\Curve{(8,12)(16,12)}
\GOval(30,16)(6,6)(0){0.882}
\Text(30,16)[]{\tiny 1PI}
\DashLine(36,16)(60,16){4}
\Text(47,22)[b]{$\phi$}
\Curve{(42,20)(52,20)}
\Curve{(42,12)(52,12)}
\GOval(66,16)(6,6)(0){0.882}
\Text(66,16)[]{\tiny 1PI}
\Photon(72,16)(96,16){1.5}{3}
\Text(85,22)[b]{$W$}
\Curve{(80,20)(88,20)}
\Curve{(80,12)(88,12)}
\GOval(102,16)(6,6)(0){0.882}
\Text(102,16)[]{\tiny 1PI}
\DashLine(108,16)(132,16){4}
\Text(120,22)[b]{$\phi$}
\Curve{(116,20)(124,20)}
\Curve{(116,12)(124,12)}
\GOval(138,16)(6,6)(0){0.882}
\Text(138,16)[]{\tiny 1PI}
\Photon(144,16)(168,16){1.5}{3}
\Text(157,22)[b]{$W$}
\Curve{(152,20)(160,20)}
\Curve{(152,12)(160,12)}
\end{picture}
\end{minipage}
+\ldots
\nonumber\\
&=&\frac{q_{\mu}\Sigma_{W\phi}(q^2)}{q^2-M_{W,0}^2+\Sigma_{W,L}(q^2)} \,
\frac{i}{q^2-M_{W,0}^2+\Sigma_\phi(q^2)}
\nonumber\\
&&{}\times \sum_{n=0}^\infty\left(\frac{q^2
(\Sigma_{W\phi}(q^2))^2}{\left(q^2-M_{W,0}^2+\Sigma_{W,L}(q^2)\right)
\left(q^2-M_{W,0}^2+\Sigma_\phi(q^2)\right)}\right)^n
\nonumber\\
&&{}\times
\frac{-q_{\nu}\Sigma_{W\phi}(q^2)}{q^2-M_{W,0}^2+\Sigma_{W,L}(q^2)}
\nonumber\\
&=&\frac{-iq_{\mu\nu}}{q^2-M_{W,0}^2+\Sigma_{W,L}(q^2)}\,
\frac{-q^2}{q^2 - \frac{\left(q^2-M_{W,0}^2+\Sigma_{W,L}(q^2)\right)
\left(q^2-M_{W,0}^2+\Sigma_\phi(q^2)\right)}
{(\Sigma_{W\phi}(q^2))^2}}.
\label{WPropMixCont}\end{aligned}$$
Adding Eqs. (\[eq:Wpure\]) and (\[WPropMixCont\]), we obtain the fully dressed $W$-boson propagator as $$\left(S_W^{-1}\right)_{\mu\nu}(q)
=\left(S_{W,{\rm pure}}^{-1}\right)_{\mu\nu}(q)
+\left(S_{W,{\rm mix}}^{-1}\right)_{\mu\nu}(q).$$ Its inverse is found to be $$\begin{aligned}
S_W^{\mu\nu}(q)&=&ig^{\mu\nu}(q^2-M_{W,0}^2) +
i\Delta^{\mu\nu}\Sigma_{W,T}(q^2)
+iq^{\mu\nu}\left( \Sigma_{W,L}(q^2)
- \frac{q^2\left(\Sigma_{W\phi}(q^2)\right)^2}
{q^2-M_{W,0}^2+\Sigma_\phi(q^2)} \right).\qquad
\label{InvVPropSumMix}\end{aligned}$$ The on-shell renormalisation condition reads $$\left.S_W^{\mu\nu}(q^2)\epsilon_{W,\nu}(q) \right|_{q^2=M_W^2}
\stackrel{!}{=} 0,
\label{VMassenBed}$$ where $\epsilon_W^\mu(q)$ is the polarisation four-vector of an external $W$ boson. Writing $M_{W,0}^2=M_W^2+\delta M_W^2$ and exploiting the transversality property $q^\mu\epsilon_{W,\mu}(q)=0$, we finally have $$\delta M_W^2 = \Sigma_{W,T}(M_W^2).
\label{AusdrDmwq}$$ We note in passing that Eq. (\[AusdrDmwq\]) is not influenced by $W\leftrightarrow\phi$ mixing.
External-leg corrections {#CapWFR}
------------------------
In this section, we discuss the structure of the amputated matrix element ${\cal A}$ for the decay process $H \to b\overline{b}$ and explain how to obtain from it the transition matrix element ${\cal T}$ by incorporating the wave-function renormalisation constants.
The general form of ${\cal A}$ reads $$\begin{aligned}
\lefteqn{
\begin{minipage}{104.5pt}
\begin{picture}(104.5,62)
\ArrowLine(69.85,39)(104.5,59)
\ArrowLine(104.5,3)(69.85,23)
\GOval(56,31)(16,16)(0){0.882}
\Text(56,31)[]{Amp.}
\DashLine(40,31)(0,31){5}
\Text(20,34)[b]{$H$}
\Text(87.175,16.5)[bl]{$\overline{b}$}
\Text(87.175,45.5)[tl]{$b$}
\LongArrow(10,26)(30,26)
\Text(20,24)[t]{$q_1\!\!+\!q_2$}
\LongArrow(77.7487,49.3301)(91.6051,57.3301)
\Text(85,56)[br]{$q_2$}
\LongArrow(77.7487,12.66987)(91.6051,4.66987)
\Text(86,7)[tr]{$q_1$}
\end{picture}
\end{minipage}
= i {\cal A}}
\nonumber\\
&=& i \left( {\cal A}_1 + \slashed{q}_1 {\cal A}_2 +
\slashed{q}_2 {\cal A}_3 + \slashed{q}_2\slashed{q}_1 {\cal A}_4 +
\gamma_5 {\cal A}_5 + \gamma_5\slashed{q}_1 {\cal A}_6 +
\gamma_5\slashed{q}_2 {\cal A}_7 + \gamma_5\slashed{q}_2
\slashed{q}_1 {\cal A}_8 \right),
\label{HbbStruktur}\end{aligned}$$ where $q_1$ and $q_2$ are the four-momenta of the outgoing $\overline{b}$ and $b$ quarks, respectively, and ${\cal A}_i$ ($i=1,\ldots,8$) are scalar form factors. Projecting onto each of these form factors, we observe that, to the orders we consider in this paper, only two of them are independent. In fact, we have $$\begin{aligned}
{\cal A}_2&=&-{\cal A}_3={\cal A}_6=-{\cal A}_7,
\nonumber\\
{\cal A}_4&=&{\cal A}_5={\cal A}_8=0,\end{aligned}$$ so that ${\cal A}$ collapses to the simple form $${\cal A} = {\cal A}_A + {\cal A}_B \left(
\slashed{q}_2 - \slashed{q}_1 \right)\omega_-,
\label{AAufspaltung}$$ where ${\cal A}_A={\cal A}_1$ and ${\cal A}_B=-2{\cal A}_2$.
Then, ${\cal T}$ is obtained by dressing ${\cal A}$ with the renormalised wave functions of the external legs as $$\begin{aligned}
{\cal T} &=& \sqrt{Z_H} \left( \sqrt{Z_{b,R}} \overline{u}_r(q_2,r_2) +
\sqrt{Z_{b,L}} \overline{u}_l(q_2,r_2) \right) {\cal A} \left(
\sqrt{Z_{b,R}} v_r(q_1,r_1) + \sqrt{Z_{b,L}} v_l(q_1,r_1) \right)
\nonumber\\
&=& \sqrt{Z_H} \overline{u}_b(q_2,r_2) \left(\sqrt{Z_{b,R}}\omega_- +
\sqrt{Z_{b,L}}\omega_+\right) {\cal A} \left(\sqrt{Z_{b,R}}\omega_+ +
\sqrt{Z_{b,L}}\omega_-\right) v_b(q_1,r_1),
\label{TAusAundZVor1}\end{aligned}$$ where $v_b(q_1,r_1)$ and $\overline{u}_b(q_2,r_2)$ denote the spinors of the outgoing $\overline{b}$ and $b$ quarks with spins $r_1$ and $r_2$, respectively. Inserting Eq. (\[AAufspaltung\]) into Eq. (\[TAusAundZVor1\]), we obtain the master formula $${\cal T} = \sqrt{Z_H}\left(\sqrt{Z_{b,L}Z_{b,R}} {\cal A}_A
+m_bZ_{b,L}{\cal A}_B\right)\overline{u}_b(q_2,r_2)v_b(q_1,r_1).
\label{TAusAundZ}$$ Note, that the terms involving $\gamma_5$ vanish upon application of the Dirac equation.
Tadpole renormalisation {#KapTadpoleren}
-----------------------
As is well known (see, for instance, Ref. [@Denner]), one can introduce a so-called tadpole renormalisation in order to avoid the calculation of diagrams containing tadpoles. For the reader’s convenience, in this section, we rederive the counterterm vertices of the tadpole renormalisation along with the counterterm vertices of the Higgs-boson mass renormalisation.
The tadpole renormalisation concerns only the Higgs part of the SM Lagrangian, $${\cal L}_\mathrm{Higgs} = (D_\mu\Phi)^\dagger(D^\mu\Phi)
+\mu^2\Phi^\dagger\Phi-\frac{\lambda}{4}(\Phi^\dagger\Phi)^2,
\label{Lag1}$$ where $\Phi$ is a weak-isospin doublet of two complex scalar fields. The free parameters, $\mu$ and $\lambda$, are chosen in such a way that one stays with a non-vanishing vacuum expectation value $v$, which is defined by $$\frac{v^2}{2} = \left| \langle 0|\Phi(x)|0\rangle \right|^2
=\frac{2\mu^2}{\lambda}.
\label{vac}$$
If we parameterise $$\Phi(x) = {\phi^+(x) \choose \frac{1}{\sqrt{2}}\left(v +
H(x)+i\chi(x)\right)}$$ and substitute $\mu$ and $\lambda$ by $$\begin{aligned}
t&=& v\left(\mu^2-\frac{\lambda v^2}{4}\right),
\nonumber\\
M_H^2&=& - \mu^2 + \frac{3\lambda v^2}{4},\end{aligned}$$ Eq. (\[Lag1\]) takes the form $$\begin{aligned}
{\cal L}_\mathrm{Higgs} &=& \frac{1}{2}(D_\mu H)(D^\mu H) +
\frac{1}{2}(D_\mu\chi)(D^\mu\chi) + (D_\mu\phi^-)(D^\mu\phi^+)
+ t H -\frac{M_H^2}{2}H^2
\nonumber\\
&&{}+ \frac{t}{2v}\left(\chi^2+2 \phi^-\phi^+\right)
-\frac{1}{2v}\left(\frac{t}{v}+M_H^2\right)H
\left(H^2+\chi^2+2 \phi^-\phi^+\right)
\nonumber\\
&&{}-\frac{1}{8v^2}\left(\frac{t}{v}+M_H^2\right)
\left(H^2+\chi^2+2 \phi^-\phi^+\right)^2,
\label{Lag2}\end{aligned}$$ where $\phi^- = \left( \phi^+ \right)^\dagger$. We see that $M_H$ has the physical meaning of the Higgs-boson mass. In this step, we did not exploit Eq. (\[vac\]), which implies that $t=0$, so that we could just have emitted all terms containing $t$. However, as was argued above, it is useful to keep them and to renormalise $t$ along with $M_H^2$ by substituting $$\begin{aligned}
t&\to& t_0 = 0 + \delta t,
\nonumber\\
M_H^2 &\to& M_{H,0}^2 = M_H^2 + \delta M_H^2\end{aligned}$$ in Eq. (\[Lag2\]). Notice that Eq. (\[Lag2\]) represents a bare Lagrangian, so that $v$, $t$, and $M_H$ are actually bare parameters. For consistency, we thus also substitute $v\to v_0$. Then, Eq. (\[Lag2\]) becomes $$\begin{aligned}
{\cal L}_\mathrm{Higgs} &=&\frac{1}{2}(D_\mu H)(D^\mu H) +
\frac{1}{2}(D_\mu\chi)(D^\mu\chi) +
(D_\mu\phi^-)(D^\mu\phi^+) -\frac{M_H^2}{2} H^2
\nonumber\\
&&{}-\frac{M_H^2}{2v_0}H\left(H^2+\chi^2+2\phi^-\phi^+\right)
-\frac{M_H^2}{8v_0^2}\left(H^2+\chi^2+2\phi^-\phi^+\right)^2
\nonumber\\
&&{}+\delta t H-\frac{\delta M_H^2}{2} H^2
+\frac{\delta t}{2v_0}\left(\chi^2 +2\phi^-\phi^+\right)
-\frac{1}{2v_0}\left(\frac{\delta t}{v_0}+\delta M_H^2\right)H
\nonumber\\
&&{}\times\left(H^2+\chi^2+2\phi^-\phi^+\right)
-\frac{1}{8v_0^2}\left(\frac{\delta t}{v_0}+\delta M_H^2\right)
\left(H^2+\chi^2+2\phi^-\phi^+\right)^2.\end{aligned}$$ From the terms proportional to $\delta t$ and $\delta M_H^2$, we can read off the desired counterterm vertices, which we list in Table \[TabCTs\].
The Higgs-boson mass renormalisation condition was already discussed in Section \[sec:higgs\]. As a renormalisation condition for $\delta t$, we set $$\delta t \stackrel{!}{=} -T,
\label{AusdrDt}$$ where $T$ stands for the sum of all amputated one-particle-irreducible tadpole diagrams, $$\begin{minipage}{72pt}
\begin{picture}(72,32)
\GOval(56,16)(16,16)(0){0.882}
\Text(56,16)[]{1-PI}
\DashLine(40,16)(0,16){5}
\Text(20,19)[b]{$H$}
\end{picture}
\end{minipage}
=iT.
\label{DefTadpole}$$ As can be seen from Table \[TabCTs\], there is a one-point Higgs-boson counterterm vertex, $i\delta t$, that forces a cancellation with all diagrams having a tadpole at its place. Therefore, upon tadpole renormalisation, one does not have to consider tadpole diagrams anymore. However, now one has to take into account all the tadpole counterterm vertices in Table \[TabCTs\], except for the one mentioned above.
Results {#CapOurCalc}
=======
In this section, we present the details of our actual calculations. After making some general remarks, we describe in Sections \[SecTree\], \[SecOneLoop\], and \[CapEW2LoopKorr\] the explicit computation of the decay rate at tree level, at the one-loop order ${\cal O}(G_F m_t^2)$, and at the two-loop order ${\cal O}(G_F^2 m_t^4)$, respectively. Section \[SecTree\] also contains the expressions for the renormalisation constants at order ${\cal O}(G_F m_t^2)$, which are needed in the one-loop and two-loop calculations.
In order to compute the leading large-$m_t$ contributions of the various two-loop diagrams, we apply the asymptotic-expansion technique (for a careful introduction, see Ref. [@Smirnov]). However, it turns out that all non-trivial contributions of the self-energy and $Hb\overline{b}$ vertex diagrams (see Figs. \[DiaW2l\], \[DiaHbb2loop\], and \[DiaB2l\]), which are of leading order in $m_t$, cancel among themselves or, in case of the $W$-boson self-energy, in combination with complete counterterm diagrams arising form the Higgs-boson tadpole and mass renormalisations. Specifically, in Fig. \[DiaW2l\], there are non-naive contributions due to the asymptotic expansion of diagrams (i)–(o) that cancel against diagrams (p)–(v); in Fig. \[DiaHbb2loop\], the non-naive contributions of diagrams (a) and (t) cancel; and in Fig. \[DiaB2l\] those of the diagrams (e) and (i) cancel. After these cancellations, only naive contributions due to diagrams involving top-quark propagators remain. Therefore, we can naively expand in all masses and momenta except for the top-quark mass and retain only the leading terms. Obviously, this requires the Higgs-boson mass to be smaller than the top-quark mass, which is compatible with the intermediate-mass range of the Higgs boson, as mentioned in the Introduction.
The ultraviolet divergences which have to disappear in the final expression for the decay rate are cancelled through the application of the renormalisation procedure, which we carry out in the on-mass-shell renormalisation scheme. This provides a non-trivial check for our calculations. As explained in Section \[KapTadpoleren\], we use the counterterm vertices of Table \[TabCTs\] for the Higgs-boson tadpole and mass renormalisations. However, while we renormalise the Higgs-boson mass already at the Lagrangian level, we replace all other bare parameters at the end of the calculations without recourse to any counterterm vertices. This procedure turns out to be most convenient for our purposes.
As a further check on our calculations, we also rederive the correction of order${\cal O}(\alpha_s G_F m_t^2)$. This result is presented in Section \[sec:mixed\]. Finally, we apply a Higgs-boson low-energy theorem [@Kniehl:1995tn], which allows for an independent calculation of the various $Hb\overline{b}$ diagrams at order ${\cal O}(G_F^2m_t^4)$. This is explained in Section \[CapNieder\].
Tree-level result and ${\cal O}(G_F m_t^2)$ renormalisation constants {#SecTree}
---------------------------------------------------------------------
The tree-level diagram is depicted in Fig. \[DiaHbb\](a). Using the notation introduced in Eq. (\[AAufspaltung\]), the corresponding amputated matrix element is in bare form written as $$\label{BornUnren}
{\cal A}^{(0)}_{0} = {\cal A}^{(0)}_{A,0} = -\frac{m_{b,0}}{v_0}.$$ The tree-level transition matrix element is $$\label{ErgT0}
{\cal T}^{(0)}={\cal A}_0^{(0)}\overline{u}_b(q_2,r_2)v_b(q_1,r_1),$$ and the decay rate is $$\label{BornTrans}
\Gamma^{(0)}=\frac{\sqrt{2}N_cG_FM_Hm_b^2}{8\pi}
\left(1-\frac{4m_b^2}{M_H^2}\right)^{3/2},$$ where $N_c=3$ is the number of quark colours. Furthermore, we have introduced Fermi’s constant $G_F$ via the Born relation $$\label{Defv}
\frac{1}{v} = 2^{1/4}G_F^{1/2}.$$
![\[DiaHbb\]Diagrams contributing to $H\to b\overline{b}$ at (a) tree level and (b) order ${\cal O}(G_Fm_t^2)$.](DiaHbb.eps){width="75.00000%"}
In the following, we have to renormalise the vacuum expectation value. Through the order of our calculations, this can be achieved by writing [@Consoli:1989fg] $$\frac{1}{v_0} = 2^{1/4}G_{F,0}^{1/2},$$ with $$G_{F,0} = G_F \frac{M_W^2}{M_{W,0}^2}.$$ Thus, the renormalisation of the vacuum expectation value is reduced to the one of the $W$-boson mass.
In the remainder of this subsection, we list all relevant renormalisation constants of order ${\cal O}(G_F m_t^2)$. They are derived by evaluating the diagrams of Fig. \[Dia1Loop\] and applying Eqs. (\[AusdrDmHq1loop\]), (\[AusdrDZH1loop\]), (\[DeltaMF1Loop\]), (\[AusdrDZbl1loop\]), (\[AusdrDZbr1loop\]), (\[AusdrDmwq\]), and (\[AusdrDt\]). Since we shall compute the correction of order ${\cal O}(G_F^2 m_t^4)$, these renormalisation constants are needed through order ${\cal O}(\epsilon)$ in the expansion in $\epsilon$. The results read $$\begin{aligned}
\delta t^{(1)} &=&C_{\epsilon,0}x_{t,0}m_{t,0}^2v_0N_c
\left[\frac{4}{\epsilon}+4+(4+2\zeta(2))\epsilon+{\cal O}(\epsilon^2)\right],
\label{RC1}\\
\delta M_H^{2(1)} &=&C_{\epsilon,0}x_{t,0}m_{t,0}^2N_c
\left[-\frac{12}{\epsilon}-4+(-4-6\zeta(2))\epsilon+{\cal O}(\epsilon^2)
\right],
\label{RC2}\\
\delta Z_H^{(1)} &=&C_{\epsilon,0}x_{t,0}N_c
\left[-\frac{2}{\epsilon}+\frac{4}{3}-\zeta(2)\epsilon+{\cal O}(\epsilon^2)
\right],
\label{RC7}\\
\frac{\delta m_b^{(1)}}{m_b} &=&C_{\epsilon,0}x_{t,0}
\left[-\frac{3}{2\epsilon}-\frac{5}{4}+\left(-\frac{9}{8}-\frac{3}{4}\zeta(2
)\right)\epsilon+{\cal O}(\epsilon^2)\right],
\label{RC5}\\
\delta Z_{b,L}^{(1)} &=&C_{\epsilon,0}x_{t,0}
\left[-\frac{1}{\epsilon}-\frac{3}{2}+\left(-\frac{7}{4}-\frac{1}{2}\zeta(2
)\right)\epsilon+{\cal O}(\epsilon^2)\right],
\label{RC8}\\
\delta Z_{b,R}^{(1)} &=& 0,
\label{RC9}\\
\frac{\delta m_t^{(1)}}{m_t} &=&C_{\epsilon,0}x_{t,0}
\left[\frac{3}{2\epsilon}+4+\left(9-\frac{5}{4}\zeta(2
)\right)\epsilon+{\cal O}(\epsilon^2)\right],
\label{RC6}\\
\delta M_W^{2(1)} &=&C_{\epsilon,0}x_{t,0}M_{W,0}^2N_c
\left[-\frac{2}{\epsilon}-1+\left(-\frac{1}{2}-\zeta(2)\right)\epsilon
+{\cal O}(\epsilon^2)\right],
\label{RC3}\end{aligned}$$ where we use the abbreviations $$\begin{aligned}
C_\epsilon& =& \left( \frac{4 \pi \mu^2}{m_t^2}e^{-\gamma_E}
\right)^\epsilon,
\nonumber\\
x_t &=& \frac{G_F m_t^2}{8\pi^2\sqrt{2}},\end{aligned}$$ with $\gamma_E$ being Euler’s constant.
![\[Dia1Loop\]One-loop self-energy and tadpole diagrams contributing at order ${\cal O}(G_Fm_t^2)$.](Dia1Loop.eps){width="\textwidth"}
Correction of order ${\cal O}(G_F m_t^2)$ {#SecOneLoop}
-----------------------------------------
At order ${\cal O}(G_F m_t^2)$, only the one diagram depicted in Fig. \[DiaHbb\](b) contributes. Using the notation of Eq. (\[AAufspaltung\]), we obtain for the expansion in $\epsilon$ through order ${\cal O}(\epsilon)$: $$\begin{aligned}
{\cal A}^{(1)}_{A,0} &=&C_{\epsilon,0}x_{t,0}\frac{m_{b,0}}{v_0}
\left[-\frac{2}{\epsilon}+2+(2-\zeta(2))\epsilon+{\cal O}(\epsilon^2)\right]
\nonumber\\
{\cal A}^{(1)}_{B,0} &=&C_{\epsilon,0}x_{t,0}\frac{1}{v_0}
\left(-1-\frac{3}{2}\epsilon+{\cal O}(\epsilon^2)\right).\end{aligned}$$
Expanding Eq. (\[TAusAundZ\]) and replacing the bare masses by the renormalised ones plus their counterterms in Eq. (\[BornUnren\]), we find the transition matrix element to be $${\cal T}^{(1)} = {\cal A}_{A,0}^{(1)} + m_b{\cal A}_{B,0}^{(1)}
+{\cal A}_0^{(0)}
\left(\delta_u^{(1)}+\frac{\delta m_b^{(1)}}{m_b}
+\frac{1}{2} \delta Z_{b,L}^{(1)}+\frac{1}{2} \delta Z_{b,R}^{(1)}\right),
\label{T1Gf}$$ where ${\cal A}^{(0)}$ is the amputated matrix element of Eq. (\[BornUnren\]) and $$\delta_u^{(1)} = \frac{1}{2} \delta Z_H^{(1)}
-\frac{1}{2}\,\frac{\delta M_W^{2(1)}}{M_W^2}$$ is the one-loop contribution to the universal counterterm $\delta_u$, which exhausts the full ${\cal O}(G_F m_t^2)$ corrections for Higgs-boson decays to fermion-antifermion pairs, except for those into $t\overline{t}$ and $b\overline{b}$ pairs. For simplicity, we omitted the spinors on the right-hand side of Eq. (\[T1Gf\]); we shall also do this in the following. $\delta_u^{(1)}$ and ${\cal T}^{(1)}$ are ultraviolet finite and read $$\begin{aligned}
\delta_u^{(1)}&=&x_tN_c\frac{7}{6}
\nonumber\\
&=&x_t\frac{7}{2},
\\
{\cal T}^{(1)} &=& {\cal T}^{(0)} x_t \left(-3+N_c\frac{7}{6} \right).
\label{ErgT1}\end{aligned}$$ The ${\cal O}(G_F m_t^2)$ correction to the decay rate thus becomes $$\begin{aligned}
\frac{\Gamma^{(1)}}{\Gamma^{(0)}}&=&
x_t\left(-6+N_c\frac{7}{3}\right)
\nonumber\\
&=&x_t,\end{aligned}$$ where $\Gamma^{(0)}$ is given in Eq. (\[BornTrans\]). The results of this subsection are in accordance with Ref. [@Kniehl:1991].
Correction of order ${\cal O}(G_F^2 m_t^4)$ {#CapEW2LoopKorr}
-------------------------------------------
Expanding Eq. (\[TAusAundZ\]) up to the two-loop order and replacing all bare masses in the tree-level and one-loop amputated matrix elements by the renormalised masses plus the corresponding counterterms, we find the following master formula for the transition matrix element $$\begin{aligned}
{\cal T}^{(2)} &=&
{\cal A}_{A,0}^{(2)} + m_b{\cal A}_{B,0}^{(2)} + {\cal A}_{A,0}^{(1)}
\left(\frac{\delta m_b^{(1)}}{m_b}+\frac{1}{2} \delta Z_{b,L}^{(1)}
+\frac{1}{2} \delta Z_{b,R}^{(1)}\right)
+m_b{\cal A}_{B,0}^{(1)}\delta Z_{b,L}^{(1)}
\nonumber\\
&&{}+ \left( {\cal A}_{A,0}^{(1)} + m_b {\cal A}_{B,0}^{(1)} \right) \left[
\delta_u^{(1)}
+ 2(1-\epsilon) \frac{\delta m_t^{(1)}} {m_t}
-\frac{\delta M_W^{2(1)}} {M_W^2} \right]
\nonumber\\
&&{}+ {\cal A}_0^{(0)} \left[ \delta_u^{(2)} + \frac{\delta m_b^{(2)}}{m_b}
+ \frac{1}{2} \delta Z_{b,L}^{(2)} + \frac{1}{2} \delta Z_{b,R}^{(2)}
+ \delta_u^{(1)}\left(
\frac{\delta m_b^{(1)}}{m_b}
+ \frac{1}{2} \delta Z_{b,L}^{(1)} + \frac{1}{2} \delta Z_{b,R}^{(1)} \right)
\right.\nonumber\\
&&{}+\left.\frac{1}{2}\,\frac{\delta m_b^{(1)}}{m_b}
\left(\delta Z_{b,L}^{(1)} + \delta Z_{b,R}^{(1)}\right)
- \frac{1}{8} \left(\delta Z_{b,L}^{(1)}-\delta Z_{b,R}^{(1)}\right)^2
\right],
\label{AusdrT2Loop}\end{aligned}$$ where $$\delta_u^{(2)} = \frac{1}{2} \delta Z_H^{(2)}
-\frac{1}{2}\, \frac{\delta M_W^{2(2)}}{M_W^2}
- \frac{1}{8} \left(\delta Z_H^{(1)}\right)^2
- \frac{1}{4} \delta Z_H^{(1)} \frac{\delta M_W^{2(1)}}{M_W^2}
+ \frac{3}{8} \left(\frac{\delta M_W^{2(1)}}{M_W^2} \right)^2
\label{UnivCT}$$ is the universal counterterm.
### Universal counterterm
Let us first calculate the universal counterterm. To this end, we need the two-loop expressions for $\delta Z_H$ and $\delta M_W^2$. The unrenormalised expressions are obtained by evaluating the diagrams in Figs. \[DiaH2l\] and \[DiaW2l\] and applying Eqs. (\[AusdrDZH2loop\]) and (\[AusdrDmwq\]), the results being $$\begin{aligned}
\delta Z_{H,0}^{(2)}&=&C_{\epsilon,0}^2x_{t,0}^2N_c
\left[\frac{3}{\epsilon^2}-\frac{11}{2\epsilon}-\frac{17}{12} +
5\zeta(2)
+N_c\left(\frac{4}{\epsilon^2}-\frac{16}{3\epsilon}+\frac{16}{9} +
4\zeta(2)\right)+{\cal O}(\epsilon)\right],
\nonumber\\
\delta M_{W,0}^{2(2)} &=&C_{\epsilon,0}^2x_{t,0}^2M_{W,0}^2N_c
\left(\frac{3}{\epsilon^2}+\frac{3}{2\epsilon}
-\frac{69}{4}+17\zeta(2)+{\cal O}(\epsilon)\right),\end{aligned}$$ in accordance with Ref. [@Djouadi]. In addition, there are contributions from the renormalisations of the bare parameters in Eqs. (\[RC7\]) and (\[RC3\]), so that $$\begin{aligned}
\delta Z_H^{(2)}&=& \delta Z_{H,0}^{(2)} +
\delta Z_H^{(1)} \left[ 2(1-\epsilon)\frac{\delta m_t^{(1)}}{m_t}
- \frac{\delta M_W^{2(1)}}{M_W^2} \right],
\nonumber\\
\delta M_W^{2(2)}&=& \delta M_{W,0}^{2(2)}+
2(1-\epsilon)\frac{\delta m_t^{(1)}}{m_t}\delta M_W^{2(1)}.\end{aligned}$$ We are now in a position to specify the universal counterterm at order ${\cal O}(G_F^2 m_t^4)$ as defined in Eq. (\[UnivCT\]). The result is $$\begin{aligned}
\delta_u^{(2)}&=&x_t^2N_c\left(\frac{29}{2}-6\zeta(2)
+N_c\frac{49}{24}\right)
\nonumber\\
&=&x_t^2\left(\frac{495}{8}-3\pi^2\right).
\label{ErgUnivCT}\end{aligned}$$
![\[DiaH2l\]Higgs-boson self-energy diagrams contributing at order ${\cal O}(G_F^2 m_t^4)$.](DiaH2l.eps){width="\textwidth"}
![\[DiaW2l\]$W$-boson self-energy diagrams contributing at order ${\cal O}(G_F^2 m_t^4)$. Insertions of $-i\delta M_H^2$ in Higgs-boson lines and of $i\delta t/v_0$ in $\phi$- or $\chi$-boson lines are indicated by crosses.](DiaW2l.eps){width="\textwidth"}
If we convert Eq. (\[ErgUnivCT\]) to a mixed renormalisation scheme which uses on-shell definitions for the particle masses and the definitions of the modified minimal-subtraction ($\overline{\mathrm{MS}}$) scheme for all other basic parameters, then we find agreement with Eq. (15) for $x=0$ in the paper by Djouadi et al. [@Djouadi]. However, the corresponding result for the pure on-shell scheme presented in their Eq. (27) for $x=0$ disagrees with our Eq. (\[ErgUnivCT\]). We can trace this discrepancy to the absence in their Eq. (25) of the additional finite term $\hat{\delta}_{u}^{(1)}\Delta\rho^{(1)}$ which arises from the renormalisation of the one-loop result in their Eq. (7) according to the prescription in their Eq. (18).
### Complete transition matrix element
Having provided $\delta_u^{(2)}$, we now turn to the residual ingredients entering the transition matrix element of Eq. (\[AusdrT2Loop\]). Evaluating the $Hb\overline{b}$ diagrams shown in Fig. \[DiaHbb2loop\], we find the form factors in Eq. (\[AAufspaltung\]) at order ${\cal O}(G_F^2 m_t^4)$ to be $$\begin{aligned}
{\cal A}_{A,0}^{(2)} &=& \frac{m_{b,0}}{v_0}x_{t,0}^2C_{\epsilon,0}^2
\left[\frac{1}{\epsilon^2}-\frac{5}{\epsilon}-5 +7\zeta(2)
+N_c\left(\frac{2}{\epsilon^2}-\frac{2}{\epsilon}-14 -2\zeta(2)\right)
+{\cal O}(\epsilon)\right],
\nonumber\\
{\cal A}_{B,0}^{(2)} &=& \frac{1}{v_0}x_{t,0}^2C_{\epsilon,0}^2
\left[\frac{2}{\epsilon}+1
+N_c\left(\frac{2}{\epsilon}+9\right)
+{\cal O}(\epsilon)\right].\end{aligned}$$
![\[DiaHbb2loop\]Diagrams contributing to $H\to b\overline{b}$ at order ${\cal O}(G_F^2 m_t^4)$. Insertions of $i\delta t/v_0$ in $\phi$-boson lines and of $-i\left(\delta t/v_0+\delta M_H^2\right)/v_0$ in $H\phi\phi$ vertices are indicated by crosses.](DiaHbb2l_4x5.eps){width="\textwidth"}
Evaluating the diagrams depicted in Fig. \[DiaB2l\] and using Eqs. (\[DeltaMF2Loop\]), (\[AusdrDZbl2loop\]), and (\[AusdrDZbr2loop\]), we obtain the bottom-quark mass and wave-function renormalisation constants at order ${\cal O}(G_F^2 m_t^4)$. The renormalisation constants in terms of bare parameters read $$\begin{aligned}
\frac{\delta m_{b,0}^{(2)}}{m_b}&=&C_{\epsilon,0}^2x_{t,0}^2
\left[\frac{27}{8\epsilon^2}+\frac{31}{8\epsilon}+\frac{13}{32}+\frac{59}{8}
\zeta(2)
+N_c\left(\frac{3}{2\epsilon^2}+\frac{15}{4\epsilon} +\frac{55}{8}
-\frac{3}{2}\zeta(2)\right)
+{\cal O}(\epsilon)\right],
\nonumber\\
\delta Z_{b,L,0}^{(2)} &=&C_{\epsilon,0}^2x_{t,0}^2
\left[\frac{2}{\epsilon^2}+\frac{7}{2\epsilon}+1+6\zeta(2)
+N_c\left(\frac{1}{\epsilon^2}+\frac{9}{2\epsilon}+\frac{25}{4}
-\zeta(2)\right)
+{\cal O}(\epsilon)\right],
\nonumber\\
\delta Z_{b,R,0}^{(2)} &=&0.\end{aligned}$$ Additional contributions arise from the replacement of the bare $t$-quark and $W$-boson masses in Eqs. (\[RC5\]), (\[RC8\]), and (\[RC9\]), so that $$\begin{aligned}
\frac{\delta m_b^{(2)}}{m_b} &=&
\frac{\delta m_{b,0}^{(2)}}{m_b}+ \frac{\delta m_b^{(1)}}{m_b}
\left[ 2(1-\epsilon) \frac{\delta m_t^{(1)}}{m_t}
- \frac{\delta M_W^{2(1)}}{M_W^2} \right],
\nonumber\\
\delta Z_{b,L}^{(2)}&=& \delta Z_{b,L,0}^{(2)} +
\delta Z_{b,L}^{(1)} \left[2(1-\epsilon) \frac{\delta m_t^{(1)}}{m_t}
- \frac{\delta M_W^{2(1)}}{M_W^2} \right],
\nonumber\\
\delta Z_{b,R}^{(2)}&=&0.\end{aligned}$$
![\[DiaB2l\]$b$-quark self-energy diagrams contributing at order ${\cal O}(G_F^2 m_t^4)$. Insertions of $i\delta t/v_0$ in $\phi$-boson lines are indicated by crosses.](DiaB2l.eps){width="\textwidth"}
Now all ingredients for the evaluation of the renormalised transition matrix element of order ${\cal O}(G_F^2 m_t^4)$ according to Eq. (\[AusdrT2Loop\]) are available. We find $${\cal T}^{(2)} = {\cal T}^{(0)} x_t^2\left[
-\frac{29}{2} + N_c(18-6\zeta(2)) + N_c^2\frac{49}{24}\right].
\label{ErgT2}$$
Adding Eqs. (\[ErgT0\]), (\[ErgT1\]), and (\[ErgT2\]), squaring, and extracting the ${\cal O}(G_F^2 m_t^4)$ term, we have $$\begin{aligned}
\frac{\Gamma^{(2)}}{\Gamma^{(0)}}&=&
x_t^2\left[-20+N_c(29-12\zeta(2))+N_c^2\frac{49}{9}\right]
\nonumber\\
&=&x_t^2(116-6\pi^2).\end{aligned}$$
Correction of order ${\cal O}(\alpha_s G_F m_t^2)$ {#sec:mixed}
--------------------------------------------------
As a by-product of our analysis, we can also compute the ${\cal O}(\alpha_s G_F m_t^2)$ correction to the $H\to b\overline{b}$ decay width. The comparison of our result with the literature [@Kniehl:1994ph; @KniehlSpira; @Kwiatkowski:1994cu] provides a partial check of our ${\cal O}(G_F^2 m_t^4)$ results. Note, however, that the calculation considerably simplifies as one passes from order ${\cal O}(G_F^2 m_t^4)$ to order ${\cal O}(\alpha_s G_F m_t^2)$. Using our tools, we indeed recover the well-known ${\cal O}(\alpha_s G_F m_t^2)$ results for the universal correction [@Kniehl:1994ph] and the correction to the $H\to b\overline{b}$ decay width [@KniehlSpira; @Kwiatkowski:1994cu], $$\begin{aligned}
\delta_u^{(X_t\alpha_s)}&=&
X_t\frac{\alpha_s}{\pi}C_F N_c\left(-\frac{3}{4}-\frac{\zeta(2)}{2}\right),
\nonumber\\
\frac{\Gamma^{(X_t\alpha_s)}}{\Gamma^{(0)}}&=&
X_t\frac{\alpha_s}{\pi}C_F\left[-12+9\ln \frac{M_H^2}{M_b^2}
+N_c\left(\frac{15}{4}-\zeta(2)-\frac{7}{2}\ln\frac{M_H^2}{M_b^2}\right)
\right],
\label{asxtonshell}\end{aligned}$$ respectively, where $X_t=G_FM_t^2/\left(8\pi^2\sqrt{2}\right)$ and $C_F=(N_c^2-1)/(2N_c)$. In Eq. (\[asxtonshell\]), the bottom- and top-quark masses are denoted with capital letters, $M_b$ and $M_t$, respectively, to indicate that they are pure on-shell masses, i.e. they are defined in the on-shell scheme also with regard to quantum chromodynamics (QCD). The obvious disadvantage of this choice is the appearance of large logarithms of the type $\ln\left(M_H^2/m_b^2\right)$ starting already in order ${\cal O}(\alpha_s)$, which spoil the convergence behaviour of the perturbation expansion. As is well known [@Braaten:1980yq], these logarithms can be resummed into the running bottom-quark mass, if $m_b$ appearing in Eq. (\[BornTrans\]) is QCD-renormalised in the $\overline{\mathrm{MS}}$ scheme at scale $\mu=M_H$, by substituting $m_b=\overline{m}_b(M_H)$. For consistency with the ${\cal O}(G_Fm_t^2)$ and ${\cal O}(G_F^2m_t^4)$ results presented above, which all refer to the electroweak on-shell scheme, we continue our discussion in a mixed renormalisation scheme where the on-shell definition of bottom-quark mass is adopted for electroweak corrections and the $\overline{\mathrm{MS}}$ one for QCD corrections. Since we wish to treat the masses of the top and bottom quarks on the same footing, we adopt this mixed scheme for the top-quark mass as well. Furthermore, the analysis at order ${\cal O}(\alpha_s^2G_Fm_t^2)$ [@delu; @Chetyrkin:1996ke] reveals that Eq. (\[asxtonshell\]) is further improved according to the renormalisation group if $m_t$ and $\alpha_s$ are taken to be $m_t=\overline{m}_t(m_t)$ and $\alpha_s=\alpha_s^{(n_f)}(m_t)$ with $n_f=6$ quark flavours, respectively. In this improved renormalisation scheme, Eq. (\[asxtonshell\]) takes the form $$\begin{aligned}
\delta_u^{(x_t\alpha_s)}&=&x_t\frac{\alpha_s}{\pi}C_FN_c
\left(\frac{19}{12}-\frac{\zeta(2)}{2}\right)
\nonumber\\
&=&x_t\frac{\alpha_s}{\pi}\left(\frac{19}{3}-\frac{\pi^2}{3}\right),
\nonumber\\
\frac{\Gamma^{(x_t\alpha_s)}}{\Gamma^{(0)}}&=&
x_t\frac{\alpha_s}{\pi}C_F
\left[-36+N_c\left(\frac{157}{12}-\zeta(2)\right)\right]
\nonumber\\
&=&x_t\frac{\alpha_s}{\pi}\left(\frac{13}{3}-\frac{2}{3}\pi^2\right).
\label{mixed}\end{aligned}$$ To the order considered here, we have $$m_t=M_t\left(1-\frac{\alpha_s^{(6)}(M_t)}{\pi}C_F\right).$$
Low-energy theorem {#CapNieder}
==================
In this section, we present an alternative way of calculating all but one of the $Hb\overline{b}$ diagrams at order ${\cal O}(G_F^2 m_t^4)$ which is based on the Higgs-boson low-energy theorem [@Kniehl:1995tn]. In fact, the $Hb\overline{b}$ diagrams of Fig. \[DiaHbb2loop\], with the exception of diagram (t), can be generated from the bottom-quark self-energy diagrams of Fig. \[DiaB2l\] by in turn attaching an external Higgs-boson line to each of the top-quark lines. Diagrammatically, this can be represented as follows: $$\begin{aligned}
\begin{minipage}{56pt}
\begin{picture}(56,40)
\ArrowLine(0,10)(56,10)
\Text(28,14)[b]{$t(q)$}
\end{picture}
\end{minipage}
\quad&\longrightarrow&\quad
\begin{minipage}{112pt}
\begin{picture}(112,40)
\ArrowLine(0,10)(56,10)
\Text(28,14)[b]{$t(q)$}
\ArrowLine(56,10)(112,10)
\Text(84,14)[b]{$t(q)$}
\DashLine(56,10)(56,40){5}
\Text(58,40)[lt]{$H$}
\end{picture}
\end{minipage}
\nonumber \\
\frac{i}{\slashed{q}-m_{t,0}} \quad&\longrightarrow&\quad
\frac{i}{\slashed{q}-m_{t,0}}
\,\frac{-im_{t,0}}{v_0} \,\frac{i}{\slashed{q}-m_{t,0}}.\end{aligned}$$ Here, we also made use of the fact that, in the large-$m_t$ approximation, the external Higgs boson does not carry any four-momentum into the respective diagram. Thanks to the identity $$\frac{i}{\slashed{q}-m_{t,0}}\,
\frac{-im_{t,0}}{v_0}\, \frac{i}{\slashed{q}-m_{t,0}} =
\frac{m_{t,0}}{v_0}\,\frac{\partial}{\partial m_{t,0}} \left( \frac{i}{\slashed{q}-m_{t,0}}
\right),$$ the amputated matrix element of $H\to b\overline{b}$ is in the large-$m_t$ limit related to the bottom-quark self-energy as $$\label{FormelNieder}
{\cal A}_0 = \frac{m_{t,0}}{v_0}
\,\frac{\partial}{\partial m_{t,0}} \Sigma_b,$$ where it is understood that the differential operator only acts on masses which stem from propagators, not to those occurring in vertices, and that all quantities in Eq. (\[FormelNieder\]) are taken to be bare. Exploiting the structures underlying Eqs. (\[FermSelbstDef\]) and (\[AAufspaltung\]), Eq. (\[FormelNieder\]) can be decomposed into two scalar equations. Identifying the four-momentum $q$ in Eq. (\[FermSelbstDef\]) with $q_2$ in Eq. (\[AAufspaltung\]) and noticing that $q_2=-q_1$ in the soft-Higgs limit, we have $$\begin{aligned}
{\cal A}_{A,0}&=& m_{b,0}\frac{m_{t,0}}{v_0}
\,\frac{\partial}{\partial m_{t,0}}\Sigma_{b,S},
\nonumber\\
{\cal A}_{B,0}&=&\frac{1}{2}\,\frac{m_{t,0}}{v_0}\,
\frac{\partial}{\partial m_{t,0}}\Sigma^{b,L}.
\label{let}\end{aligned}$$ The fact that the $H\to b\overline{b}$ amplitude does not contain a term proportional to $(\slashed{q}_2-\slashed{q}_1)\omega_+$ is reflected by the fact that the right-handed part of the bottom-quark self-energy, $\Sigma_{b,R}$, vanishes to the orders considered in this paper. The results for ${\cal A}_{A,0}$ and ${\cal A}_{B,0}$ obtained through Eq. (\[let\]) indeed agree with the direct evaluation of the respective diagrams in Fig. \[DiaHbb2loop\].
Numerical results {#Numerics}
=================
Finally, we explore the phenomenological implications of our results. Adopting from Ref. [@PDG] the values $G_F=1.16637\times10^{-5}$ GeV$^{-2}$, $\alpha_s^{(5)}(M_Z)=0.1176$, $M_Z=91.1876$ GeV, and $M_t=174.2$ GeV for our input parameters, so that $\alpha_s^{(6)}(m_t)=0.1076$ and $m_t=166.2$ GeV, we evaluate the relative corrections $\Gamma^{(x)}/\Gamma^{(0)}$ to the $H\to b\overline{b}$ decay width to orders $x=G_F m_t^2$, $G_F^2m_t^4$, and $\alpha_sG_Fm_t^2$. For comparison, we also evaluate the relative corrections to the $H\to l^+l^-$ and $H\to q\overline{q}$ decay widths, where $l=e,\mu,\tau$ and $q=u,d,s,c$, which, to the orders considered here, are given by $$\begin{aligned}
\Delta_l&=&(1+\delta_u)^2-1
\nonumber\\
&=&2\delta_u^{(1)}
+2\delta_u^{(2)}+\left(\delta_u^{(1)}\right)^2
+2\delta_u^{(x_t\alpha_s)},
\nonumber\\
\Delta_q&=&(1+\Delta_\mathrm{QCD})(1+\delta_u)^2-1
\nonumber\\
&=&\Delta_\mathrm{QCD}+2\delta_u^{(1)}
+2\delta_u^{(2)}+\left(\delta_u^{(1)}\right)^2
+2\delta_u^{(x_t\alpha_s)}
+2\Delta_\mathrm{QCD}\delta_u^{(1)},\end{aligned}$$ where [@Braaten:1980yq] $$\Delta_\mathrm{QCD} = \frac{\alpha_s}{\pi}C_F \frac{17}{4}$$ is the ${\cal O}(\alpha_s)$ correction in the limit $m_q\ll M_H$, with $m_q=\overline{m}_q(M_H)$.
The results are listed in Table \[tab:num\]. We observe that the ${\cal O}(G_F^2m_t^4)$ correction to $\Gamma^{(0)}$ increases the enhancement due to the ${\cal O}(G_F m_t^2)$ one by about 16% and has more than twice the magnitude of the negative ${\cal O}(\alpha_sG_Fm_t^2)$ one.
Order $x$ $\Delta_l^{(x)}$ $\Delta_q^{(x)}$ $\Gamma^{(x)}/\Gamma^{(0)}$
--------------------------------- ------------------ ------------------ -----------------------------
$\mathcal{O}(G_Fm_t^2)$ $+2.021\%$ $+2.021\%$ $+0.289\%$
$\mathcal{O}(G_F^2m_t^4)$ $+0.064\%$ $+0.064\%$ $+0.047\%$
$\mathcal{O}(\alpha_sG_Fm_t^2)$ $+0.060\%$ $+0.452\%$ $-0.022\%$
: \[tab:num\]Numerical values of the relative corrections $\Delta_l^{(x)}$, $\Delta_q^{(x)}$, and $\Gamma^{(x)}/\Gamma^{(0)}$ to the $H\to l^+l^-$, $H\to q\overline{q}$, and $H\to b\overline{b}$ decay widths, respectively, at orders $x=G_Fm_t^2$, $G_F^2m_t^4$, and $\alpha_sG_Fm_t^2$.
Conclusions {#CapZusammenfassung}
===========
We analytically calculated the dominant electroweak two-loop correction, of order${\cal O}(G_F^2m_t^4)$, to the $H\to b\overline{b}$ decay width of an intermediate-mass Higgs boson, with $M_H\ll m_t$.
We performed various checks for our analysis. The ultraviolet divergences cancelled through genuine two-loop renormalisation. Our final result is devoid of infrared divergences related to infinitesimal scalar-boson masses. We reproduced those $Hb\overline{b}$ triangle diagrams where the external Higgs boson is coupled to an internal top-quark line, which we had computed directly, through application of a low-energy theorem. After switching to a hybrid renormalisation scheme, our ${\cal O}(G_F^2m_t^4)$ result for the universal correction $\delta_u$ agrees with Ref. [@Djouadi]. Using our techniques, we also recovered the ${\cal O}(\alpha_sG_Fm_t^2)$ correction to the $H\to b\overline{b}$ decay width as well as the universal correction $\delta_u$ in this order.
The ${\cal O}(G_F^2m_t^4)$ correction to the $H\to b\overline{b}$ decay width amplifies the familiar enhancement due to the ${\cal O}(G_Fm_t^2)$ correction by about $+16\%$ and thus more than compensates the screening by about $-8\%$ through QCD effects of order ${\cal O}(\alpha_sG_Fm_t^2)$.
Acknowledgements {#acknowledgements .unnumbered}
================
We like to thank Paolo Gambino, Jan Piclum, Florian Schwennsen, and Matthias Steinhauser for fruitful discussions. This work was supported in part by the German Federal Ministry for Education and Research BMBF through Grant No. 05 HT6GUA and by the German Research Foundation DFG through Graduate School No. GRK 602 [*Future Developments in Particle Physics*]{}.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We present DeepNav, a Convolutional Neural Network (CNN) based algorithm for navigating large cities using locally visible street-view images. The DeepNav agent learns to reach its destination quickly by making the correct navigation decisions at intersections. We collect a large-scale dataset of street-view images organized in a graph where nodes are connected by roads. This dataset contains 10 city graphs and more than 1 million street-view images. We propose 3 supervised learning approaches for the navigation task and show how A\* search in the city graph can be used to generate supervision for the learning. Our annotation process is fully automated using publicly available mapping services and requires no human input. We evaluate the proposed DeepNav models on 4 held-out cities for navigating to 5 different types of destinations. Our algorithms outperform previous work that uses hand-crafted features and Support Vector Regression (SVR) [@mcdonalds].'
author:
- |
Samarth Brahmbhatt\
Georgia Institute of Technology\
Atlanta USA\
[samarth.robo@gatech.edu]{}
- |
James Hays\
Georgia Institute of Technology\
Atlanta USA\
[hays@gatech.edu]{}
title: 'DeepNav: Learning to Navigate Large Cities'
---
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'For a finite cyclic $p$-group $G$ and a discrete valuation domain $R$ of characteristic $0$ with maximal ideal $pR$ the ${R[G]}$-permutation modules are characterized in terms of the vanishing of first degree cohomology on all subgroups (cf. Thm. A). As a consequence any ${R[G]}$-lattice can be presented by ${R[G]}$-permutation modules (cf. Thm. C). The proof of these results is based on a detailed analysis of the category of cohomological $G$-Mackey functors with values in the category of $R$-modules. It is shown that this category has global dimension $3$ (cf. Thm. E). A crucial step in the proof of Theorem E is the fact that a gentle $R$-order category (with parameter $p$) has global dimension less or equal to $2$ (cf. Thm. D).'
address:
- |
B. Torrecillas\
Departamento de Algebra y Análisis Matemático\
Universidad de Almería\
04071 Almería, Spain
- |
Th. Weigel\
Università di Milano-Bicocca\
U5-3067, Via R.Cozzi, 53\
20125 Milano, Italy
author:
- 'B. Torrecillas and Th. Weigel'
bibliography:
- 'gorenstein.bib'
title: 'Lattices and cohomological Mackey functors for finite cyclic p-groups'
---
[^1]
Introduction {#s:intro}
============
For a Dedekind domain $R$ and a finite group $G$ one calls a finitely generated left $R[G]$-module $M$ an [*$R[G]$-lattice*]{}, if $M$ - considered as an $R$-module - is projective. In this paper we focus on the study of $R[G]$-lattice, where $R$ is a discrete valuation domain of characteristic $0$ with maximal ideal $pR$ for some prime number $p$, and $G$ is a finite cyclic $p$-group. The study of such lattices has a long history and was motivated by a promissing result of F.-E. Diederichsen (cf. [@CR:met1 Thm. 34:31], [@died:ham]) who showed that for the finite cyclic group of order $p$ there are precisely three directly indecomposable such lattices up to isomorphism: the trivial $R[G]$-lattice $R$, the free $R[G]$-lattice $R[G]$, and the augmentation ideal $\omega_{{R[G]}}=\operatorname{ker}(R[G]\to R)$. A similar finiteness result holds for cyclic groups of order $p^2$ (cf. [@hr:rep1]). However, for cyclic $p$-groups of order larger than $p^2$ there will be infinitely many isomorphism types of such lattices; even worse, in general this classification problem is “wild” (cf. [@diet:rep1], [@diet:rep2], [@gud:wild]). If the $R[G]$-lattice $M$ is isomorphic to $R[\Omega]$ for some finite left $G$-set $\Omega$, $M$ will be called an [*$R[G]$-permutation lattice*]{}. The main purpose of this paper is to establish the following characterization of $R[G]$-permutation lattices for finite cyclic $p$-groups (cf. Cor. \[cor:hil90lat1\], Prop. \[prop:elequi\]).
Let $R$ be a discrete valuation domain of characteristic $0$ with maximal ideal $pR$ for some prime number $p$, let $G$ be a finite cyclic $p$-group, and let $M$ be an $R[G]$-lattice. Then the following are equivalent.
- $M$ is an $R[G]$-permutation lattice,
- $H^1(U,\operatorname{res}^G_U(M))=0$ for all subgroups $U$ of $G$,
- $M_U$ is $R$-torsion free for all subgroups $U$ of $G$,
where $M_U=M/\omega_{R[U]}M$ denotes the $U$-coinvariants of $M$.
By a result of I. Reiner (cf. [@CR:met1 Thm. 34.31], [@rei:intcyc]), one knows that there are ${\mathbb{Z}}[C_p]$-lattices satisfying (ii), where $C_p$ is the cyclic group of order $p$, which are not ${\mathbb{Z}}[C_p]$-permutation lattices. Hence the conclusion of Theorem A does not hold for the ring $R={\mathbb{Z}}$.
Theorem A has a number of interesting consequences which we would like to explain in more detail. For a finite $p$-group $G$ it is in general quite difficult to decide whether a given ${R[G]}$-lattice $M$ is indeed an ${R[G]}$-permutation lattice. A sufficient criterion to the just mentioned problem was given by A. Weiss in [@weiss:rig] for an arbitrary finite $p$-group $G$ and the ring of $p$-adic integers $R={\mathbb{Z}}_p$. He showed that if for a normal subgroup $N$ of $G$ the ${\mathbb{Z}}_p[G/N]$-module $M^N$ of $N$-invariants is a ${\mathbb{Z}}_p[G/N]$-permutation module, and $\operatorname{res}^G_N(M)$ is a free ${\mathbb{Z}}_p[N]$-module, then $M$ is a ${\mathbb{Z}}_p[G]$-permutation module (cf. [@karp:ind Chap. 8, Thm. 2.6]). Theorem A extends A. Weiss’ result for cyclic $p$-groups in the following way (cf. Prop. \[prop:exweiss\]).
Let $R$ be a discrete valuation domain of characteristic $0$ with maximal ideal $pR$ for some prime number $p$, let $G$ be a finite cyclic $p$-group, and let $N$ be a normal subgroup of $G$. Suppose that the $R[G]$-lattice $M$ is satisfying the following two hypothesis.
- $\operatorname{res}^G_N(M)$ is an $R[N]$-permutation module, and
- $M^N$ is an $R[G/N]$-permutation module.
Then $M$ is an ${R[G]}$-permutation module.
Although it seems impossible to describe all isomorphism types of directly indecomposable $R[G]$-lattices, where $R$ is a discrete valuation domain of characteristic $0$ with maximal ideal $pR$ and $G$ is a finite cyclic $p$-group, one can (re)present such lattices in a very natural way (cf. Thm. \[thm:preslat\]).
Let $R$ be a discrete valuation domain of characteristic $0$ with maximal ideal $pR$ for some prime number $p$, let $G$ be a finite cyclic $p$-group, and let $M$ be an $R[G]$-lattice. Then there exist finite $G$-sets $\Omega_0$ and $\Omega_1$, and a short exact sequence $$\label{eq:preslat}
\xymatrix{
0\ar[r]& R[\Omega_1]\ar[r]&R[\Omega_0]\ar[r]&M\ar[r]&0
}$$ of $R[G]$-lattices.
The proof of Theorem A and Theorem C is based on the theory of [*cohomological Mackey functors*]{} for a finite group $G$. Mackey functors were first introduced by A.W.M. Dress in [@dress:mac]. Cohomological Mackey functors satisfy an additional identity (cf. [@pw:user]). The category of cohomological $G$-Mackey functors ${\mathfrak{cMF}}_G({{}_R{\mathbf{mod}}})$ with values in the category of $R$-modules coincides with the category of contravariant functors of an $R^\circledast$-order category ${{\mathcal{M}}}_R(G)$ (cf. §\[ss:maccat\]). In case that $G$ is a cyclic $p$-group or order $p^n$, one has a [*unitary projection functor*]{} (cf. §\[ss:funRcat\]) $$\label{eq:unipro}
\pi\colon{\mathfrak{cMF}}_R(G)\longrightarrow{{\mathcal{G}}}_R(n,p)$$ which can be used to analyze the category ${\mathfrak{cMF}}_G({{}_R{\mathbf{mod}}})$. Here ${{\mathcal{G}}}_R(n,p)$ denotes the [*gentle $R$-order category*]{} supported on $n+1$ vertices and parameter $p$ (cf. §\[ss:gentordcat\]) which can be seen as an $R$-order version of the gentle algebra ${{\mathcal{G}}}_{\mathbb{F}}(n)$ defined over a field ${\mathbb{F}}$. The gentle algebra has been subject to intensive investigations (cf. [@gere:gent]), e.g., it is well known that ${{\mathcal{G}}}_{\mathbb{F}}(n)$ is $1$-Gorenstein (resp. $0$-Gorenstein for $n=0$ or $n=1$), but for $n\geq 1$ it is not of finite global dimension. Hence the following property of the gentle $R$-order category is somehow surprising (cf. Thm. \[thm:gldimgent\]).
Let $p$ be a prime number, and let $R$ be a principal ideal domain of characteristic $0$ such that $p.1\in R$ is a prime element. Then $\operatorname{gldim}({{\mathcal{G}}}_R(n,p))= 2$ for $n\geq 2$, and $\operatorname{gldim}({{\mathcal{G}}}_R(n,p))= 1$ for $n=1$ or $0$.
In §\[s:secmac\] we study [*section cohomology groups*]{} which can be associated to any cohomological Mackey functor and any normal section of a finite group. This allows us to introduce the notion of cohomological Mackey functors with the [*Hilbert$^{90}$-property*]{} (cf. §\[ss:h0\]). Theorem A and Theorem C are a direct consequence of a more general result which states that for a discrete valuation domain $R$ of characteristic $0$ and maximal ideal $pR$ every cohomological $G$-Mackey functor with values in the category of $R$-lattices and with the Hilbert$^{90}$ property is projective (cf. Thm. \[thm:defcat\]). The proof of this more general result is achieved in two steps. The first step is to show that the deflation functor associated to $\pi$ (cf. ) maps Hilbert$^{90}$ $R$-lattice functors to projective functors of the gentle $R$-order category. The second step is to establish injectivity and surjectivity criteria which ensure that a given natural transformation $\phi\colon{\mathbf{X}}\to{\mathbf{Y}}$ envolving Hilbert$^{90}$ $R$-lattice functors is indeed an isomorphism (cf. Prop. \[prop:inj\], Prop. \[prop:surmac\]).
The first step is based on a sufficient criterion (cf. Thm. \[thm:whitehead\]) which guarantees that the deflation functor associated to a unitary projection $\pi$ is mapping $\circledast$-acyclic $R$-lattice functors to projective $R$-lattice functors. Here $\circledast$ denotes the [*Yoneda dual*]{} (cf. §\[ss:yondual\]) which can be seen as the standard dualizing procedure for $R^\circledast$-categories. Although this criterion is based on what is usually called “abstract nonsense”, it will turn out to be quite useful: two of the three hypothesis one has to claim can be verified easily for the unitary projection $\pi$ and involve the Hilbert$^{90}$ property, while the third is a direct consequence of Theorem D.
The two main results known to authors concerning the cohomology of cohomological Mackey functors are due to S. Bouc (cf. [@bouc:com]) and D. Tambara (cf. [@tamb:hom]), but concern cohomological Mackey functors with values in a field of positive characteristic. Although the just-mentioned results indicate that for cyclic groups the theory of cohomological Mackey functors should be significantly easier (and different) than in the general case, the following consequence is nevertheless surprising.
Let $R$ be a discrete valuation domain of characteristic $0$ with maximal ideal $pR$ for some prime number $p$, and let $G$ be a non-trivial finite cyclic $p$-group. Then $\operatorname{gldim}_R({{\mathcal{M}}}_R(G))=3$.
$R^\circledast$-categories {#s:rord}
==========================
Let $R$ be a commutative ring with $1$, and let ${{}_R{\mathbf{mod}}}$ denote the abelian category of $R$-modules. An $R$-module $M$ will be called an [*$R$-lattice*]{}, if $M$ is a finitely generated projective $R$-module. We denote by ${{}_R{\mathbf{lat}}}$ the full subcategory of ${{}_R{\mathbf{mod}}}$ the objects of which are $R$-lattices, and by ${{}_R{\mathbf{mod}}}^{\operatorname{f.g.}}$ the full subcategory of ${{}_R{\mathbf{mod}}}$ the objects of which are finitely generated $R$-modules. For certain applications we have to restrict our considerations to Dedekind domains. For such a ring $R$ one has the following property: If $\phi\colon M\to Q$ is a surjective homomorphism of $R$-lattices, then $\ker(\phi)$ is an $R$-lattice and the canonical map $\ker(\phi)\to M$ is split-injective.
Following [@bass:kth Chap. 2. §2] one calls a category ${{\mathcal{C}}}$ an [*$R$-category*]{}, if $\operatorname{Hom}_{{\mathcal{C}}}(A,B)$ is an $R$-module for any pair of objects $A,B\in\operatorname{ob}({{\mathcal{C}}})$, and composition $$\label{eq:bilin}
{\underline{\phantom{x}}}\circ{\underline{\phantom{x}}}\colon\operatorname{Hom}_{{\mathcal{C}}}(B,C)\times\operatorname{Hom}_{{\mathcal{C}}}(A,B)\longrightarrow\operatorname{Hom}_{{\mathcal{C}}}(A,C)$$ is $R$-bilinear for any three objects $A,B,C\in\operatorname{ob}({{\mathcal{C}}})$. E.g., ${{}_R{\mathbf{mod}}}$ is an $R$-category. Note that ${{\mathcal{C}}}^{\operatorname{op}}$ is an $R$-category for every $R$-category ${{\mathcal{C}}}$. A (covariant) functor $\phi\colon {{\mathcal{C}}}\to{{\mathcal{D}}}$ between $R$-categories ${{\mathcal{C}}}$ and ${{\mathcal{D}}}$ is called [*$R$-linear*]{}, if $$\label{eq:Rlin}
\phi_{A,B}\colon\operatorname{Hom}_{{{\mathcal{C}}}}(A,B)\longrightarrow\operatorname{Hom}_{{{\mathcal{D}}}}(\phi(A),\phi(B))$$ is a homomorphism of $R$-modules for every pair of objects $A,B\in\operatorname{ob}({{\mathcal{C}}})$.
$R^\circledast$-order categories {#ss:Rcat}
--------------------------------
An $R$-category ${{\mathcal{C}}}$ will be called an [*$R$-order category*]{}, if $\operatorname{ob}({{\mathcal{C}}})$ is a finite set and $\operatorname{Hom}_{{{\mathcal{C}}}}(A,B)$ is an $R$-lattice for all $A,B\in \operatorname{ob}({{\mathcal{C}}})$. E.g., if $\mu$ is an $R$-order, then $\mu\bullet$, the category with one object $\bullet$ and $\operatorname{Hom}_{\mu\bullet}(\bullet,\bullet)=\mu$, is an $R$-order category. An $R$-category ${{\mathcal{C}}}$ together with an $R$-linear functor $\sigma\colon{{\mathcal{C}}}\to{{\mathcal{C}}}^{\operatorname{op}}$ satisfying $\sigma(A)=A$ for all $A\in\operatorname{ob}({{\mathcal{C}}})$ and $\sigma\circ\sigma=\operatorname{id}_{{{\mathcal{C}}}}$ will be called an [*$R^\circledast$-category*]{}. E.g., if $\mu$ is an $R$-algebra with an $R$-linear antipode $\sigma_\mu\colon \mu\to\mu^{\operatorname{op}}$ of order $2$, i.e., $\sigma\circ\sigma=\operatorname{id}_{\mu}$, then $\mu\bullet$ is an $R^\circledast$-category. An $R^\circledast$-category $({{\mathcal{C}}},\sigma)$, where ${{\mathcal{C}}}$ is an $R$-order category, will be called an [*$R^\circledast$-order category*]{}.
Additive functors {#ss:add}
-----------------
Let ${{\mathcal{C}}}$ be an $R$-category. By ${\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}})$ we denote the category of $R$-linear functors from ${{\mathcal{C}}}^{\operatorname{op}}$ to ${{}_R{\mathbf{mod}}}$, i.e., ${\mathbf{F}}\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}}))$ is a contravariant $R$-linear functor from ${{\mathcal{C}}}$ to ${{}_R{\mathbf{mod}}}$. Morphisms in ${\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}})$ are given by the [*$R$-linear natural transformations*]{}, i.e., $\eta\in\operatorname{nat}_R({\mathbf{F}},{\mathbf{G}})$ is called [*$R$-linear*]{}, if $\eta_A\colon{\mathbf{F}}(A)\to{\mathbf{G}}(A)$ is $R$-linear for every $A\in\operatorname{ob}({{\mathcal{C}}})$. It is well known that ${\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}})$ is an abelian category (cf. [@mcl:hom Chap. IX, Prop. 3.1]).
A functor ${\mathbf{F}}\in \operatorname{ob}({\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}}))$ will be called an [*$R$-lattice functor*]{} if ${\mathbf{F}}(A)$ is an $R$-lattice for every object $A\in\operatorname{ob}({{\mathcal{C}}})$. By ${\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{lat}}})\subseteq{\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}})$ we denote the full subcategory of $R$-lattice functors.
Let $({{\mathcal{C}}},\sigma)$ be an $R^\circledast$-category, and let ${\underline{\phantom{x}}}^\ast=\operatorname{Hom}_R({\underline{\phantom{x}}},R)\colon{{}_R{\mathbf{lat}}}\longrightarrow{{}_R{\mathbf{lat}}}^{\operatorname{op}}$ denote the dualizing functor in ${{}_R{\mathbf{lat}}}$. Composition of ${\underline{\phantom{x}}}^\ast$ with $\sigma$ yields a dualizing functor $$\label{eq:equiv}
{\underline{\phantom{x}}}^\ast\colon
{\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{lat}}})\longrightarrow{\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{lat}}})^{\operatorname{op}},$$ where ${\mathbf{F}}^\ast(A)={\mathbf{F}}(A)^\ast$ and ${\mathbf{F}}^\ast(\phi)={\mathbf{F}}(\sigma(\phi))^\ast$ for ${\mathbf{F}}\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{lat}}}))$ and $\phi\colon A\to B\in\operatorname{Hom}_{{\mathcal{C}}}(A,B)$.
Projectives {#ss:proj}
-----------
Let ${{\mathcal{C}}}$ be an $R$-category, and let $A\in\operatorname{ob}({{\mathcal{C}}})$. Then $$\label{eq:defP}
{\mathbf{P}}^A=\operatorname{Hom}_{{\mathcal{C}}}({\underline{\phantom{x}}},A)\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}}))$$ is an $R$-linear functor from ${{\mathcal{C}}}^{\operatorname{op}}$ to ${{}_R{\mathbf{mod}}}$. Moreover, if ${{\mathcal{C}}}$ is an $R$-order category, then ${\mathbf{P}}^A$ is an $R$-lattice functor. One has the following property (cf. [@sten:roq Prop. IV.7.3]).
\[fact:Pfirst\] Let ${{\mathcal{C}}}$ be an $R$-category, let $A\in\operatorname{ob}({{\mathcal{C}}})$ and ${\mathbf{F}}\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}}))$. Then one has a canonical isomorphism $$\label{eq:nattrans}
\theta_{A,{\mathbf{F}}}\colon\operatorname{nat}_R({\mathbf{P}}^A,{\mathbf{F}})\longrightarrow{\mathbf{F}}(A)$$ given by $\theta_{A,{\mathbf{F}}}(\xi)=\xi_A(\operatorname{id}_A)$, $\xi\in\operatorname{nat}_R({\mathbf{P}}^A,{\mathbf{F}})$.
The inverse of $\theta_{A,{\mathbf{F}}}$ can be given explicit. For $f\in{\mathbf{F}}(A)$ and $B\in\operatorname{ob}({{\mathcal{C}}})$ one has $$\label{eq:yonnat}
\theta_{A,{\mathbf{F}}}^{-1}(f)_B\colon{\mathbf{P}}^A(B)\to{\mathbf{F}}(B),\quad
\theta_{A,{\mathbf{F}}}^{-1}(f)_B(\phi)={\mathbf{F}}(\phi)(f),\quad
\phi\in\operatorname{Hom}_{{\mathcal{C}}}(B,A).$$ It is straightforward to verify that $\theta_{A,{\mathbf{F}}}^{-1}(f)\in\operatorname{nat}_R({\mathbf{P}}^A,{\mathbf{F}})$. Let $\chi\colon{\mathbf{F}}\to{\mathbf{G}}$ be an $R$-linear natural transformation. Then one has a commutative diagram $$\label{eq:yonnat2}
\xymatrix{
{\mathbf{F}}(A)\ar[d]_{\chi_A}\ar[r]^-{\theta_{A,{\mathbf{F}}}^{-1}}&\operatorname{nat}_R({\mathbf{P}}^A,{\mathbf{F}})\ar[d]^{\chi\circ-}\\
{\mathbf{G}}(A)\ar[r]^-{\theta_{A,{\mathbf{G}}}^{-1}}&\operatorname{nat}_R({\mathbf{P}}^A,{\mathbf{G}}).
}$$ From this fact one concludes the following well known property (see [@sten:roq Cor. 7.5]).
\[fact:yon\] Let ${{\mathcal{C}}}$ be an $R$-category, let ${\mathbf{F}},{\mathbf{G}}\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}}))$, and let $\phi\in\operatorname{nat}_R({\mathbf{F}},{\mathbf{G}})$ be a natural transformation such that $\phi_A\colon {\mathbf{F}}(A)\to{\mathbf{G}}(A)$ is surjective. Then for every natural transformation $\chi\in\operatorname{nat}_R({\mathbf{P}}^A,{\mathbf{G}})$ there exists ${\tilde{\chi}}\in\operatorname{nat}_R({\mathbf{P}}^A,{\mathbf{F}})$ making the diagram $$\label{eq:diayon}
\xymatrix{
&{\mathbf{P}}^A\ar[d]^{\chi}\ar@{-->}[dl]_{{\tilde{\chi}}}&\\
{\mathbf{F}}\ar[r]^{\phi}&{\mathbf{G}}\ar[r]&0
}$$ commute. In particular, ${\mathbf{P}}^A\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}}))$ is projective.
As a consequence one has the following.
\[fact:enproj\] Let ${{\mathcal{C}}}$ be an $R$-category such that $\operatorname{ob}({{\mathcal{C}}})$ is a set. Then ${\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}})$ is an abelian category with enough projectives.
If ${{\mathcal{C}}}$ is an $R$-category and $\operatorname{ob}({{\mathcal{C}}})$ is a set, we denote for ${\mathbf{F}}\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}}))$ the right derived functors of $\operatorname{nat}_R({\underline{\phantom{x}}},{\mathbf{F}})$ by $\operatorname{ext}_R^k({\underline{\phantom{x}}},{\mathbf{F}})$, $k\geq 0$.
Let ${{\mathcal{C}}}$ be an $R$-order category. Then by definition ${\mathbf{P}}^A$ is an $R$-lattice functor. A projective object ${\mathbf{P}}\in\operatorname{ob}({\mathfrak{F}}({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}}))$ which is a lattice functor will be a called a [*projective $R$-lattice functor*]{}. For these functors one concludes the following:
\[fact:rlatproj\] Let $R$ be a Dedekind domain, and let ${{\mathcal{C}}}$ be an $R$-order category. Then every $R$-lattice functor ${\mathbf{F}}\in\operatorname{ob}({\mathfrak{F}}({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}}))$ has a projective resolution $({\mathbf{P}}_k,\partial_k^{\mathbf{P}},{\varepsilon}_{\mathbf{F}})$, where ${\mathbf{P}}_k$ is a projective $R$-lattice functor for every $k\geq 0$.
\[rem:rep\] Let ${{\mathcal{C}}}$ be an $R$-order category such that for all $A,B\in\operatorname{ob}({{\mathcal{C}}})$, $A\not= B$, one has $A\not\simeq B$. Let $\mu_{{\mathcal{C}}}$ be the $R$-order given by $\textstyle{\mu_{{\mathcal{C}}}=\bigoplus_{A,B\in\operatorname{ob}({{\mathcal{C}}})}\operatorname{Hom}_{{\mathcal{C}}}(A,B)}$, where the product is given by $$\label{eq:defmuC}
\alpha\cdot\beta=
\begin{cases}
\alpha\circ\beta&\ \text{for $B_1=B_2$,}\\
\hfil 0\hfil &\ \text{for $B_1\not=B_2$.}
\end{cases}$$ for $\alpha\in\operatorname{Hom}_{{\mathcal{C}}}(B_2,C)$, $\beta\in\operatorname{Hom}_{{\mathcal{C}}}(A,B_1)$. Then one has a canonical $R$-linear functor $\rho_{{\mathcal{C}}}\colon{{\mathcal{C}}}\to\mu_{{\mathcal{C}}}\bullet$ (cf. §\[ss:Rcat\]) induced by the identity on morphisms. Moreover, the category ${\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}})$ is naturally equivalent to the category of right $\mu_{{{\mathcal{C}}}}$-modules ${\mathbf{mod}}_{\mu_{{{\mathcal{C}}}}}$. This equivalence is achieved by assigning a right $\mu_{{\mathcal{C}}}$-module $M$ the functor ${\mathbf{F}}_M\in \operatorname{ob}({\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}}))$ given by ${\mathbf{F}}_M(A)=M\cdot\operatorname{id}_A$ for $A\in\operatorname{ob}({{\mathcal{C}}})$. For $\phi\in\operatorname{Hom}_{{\mathcal{C}}}(A,B)$ the mapping ${\mathbf{F}}_M(\phi)\colon{\mathbf{F}}_M(B)\to{\mathbf{F}}_M(A)$ is given by right multiplication with $\phi$. A functor ${\mathbf{F}}\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}}))$ can be made into a right $\mu_{{\mathcal{C}}}$-module $M_{\mathbf{F}}$, where $M_{\mathbf{F}}=\bigoplus_{A\in\operatorname{ob}({{\mathcal{C}}})}{\mathbf{F}}(A)$. For $f\in{\mathbf{F}}(B)$ and $\phi\in\operatorname{Hom}_{{{\mathcal{C}}}}(A,B)$ one has $f\cdot\phi={\mathbf{F}}(\phi)(f)$.
For $A\in\operatorname{ob}({{\mathcal{C}}})$, $\operatorname{id}_A$ is an idempotent in $\mu_{{{\mathcal{C}}}}$. Moreover, under the identification mentioned above ${\mathbf{P}}^A$ corresponds to the right $\mu_{{{\mathcal{C}}}}$-module $\operatorname{id}_A\cdot\mu_{{{\mathcal{C}}}}$.
Dimensions {#ss:Ldim}
----------
Let $R$ be a Dedekind domain, let ${{\mathcal{C}}}$ be an $R$-order category, and let ${\mathbf{F}}\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}}))$. Then ${\mathbf{F}}$ has [*projective $R$-dimension less or equal to $d$*]{} if it has a projective resolution $({\mathbf{P}}_k,\partial_k^{\mathbf{P}},{\varepsilon}_{\mathbf{F}})$ with ${\mathbf{P}}_k=0$ for $k>d$. The minimal such number $d\in{\mathbb{N}}_0\cup\{\infty\}$ is called the [*projection $R$-dimension*]{} of ${\mathbf{F}}$ and will be denoted by $\operatorname{proj.dim}({\mathbf{F}})$. The numbers $$\label{eq:gldim}
\begin{aligned}
\operatorname{gldim}_R({{\mathcal{C}}})&=\sup\{\,\operatorname{proj.dim}_R({\mathbf{F}})\mid {\mathbf{F}}\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}}))\,\},\\
\operatorname{Ldim}_R({{\mathcal{C}}})&=\sup\{\,\operatorname{proj.dim}_R({\mathbf{F}})\mid {\mathbf{F}}\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{lat}}}))\,\},
\end{aligned}$$ will be called the [*global $R$-dimension*]{} and the [*global $R$-lattice dimension*]{} of ${{\mathcal{C}}}$, respectively. By a result of M. Auslander, one has $$\label{eq:gldim2}
\operatorname{gldim}_R({{\mathcal{C}}})=\sup\{\,\operatorname{proj.dim}_R({\mathbf{F}})\mid {\mathbf{F}}\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}}^{\operatorname{f.g.}}))\,\}$$ (cf. [@rotman:hom Thm. 9.12]). In particular, $$\label{eq:dims}
\operatorname{Ldim}_R({{\mathcal{C}}})\leq\operatorname{gldim}_R({{\mathcal{C}}})\leq\operatorname{Ldim}_R({{\mathcal{C}}})+1.$$ E.g., $\operatorname{Ldim}_R({{\mathcal{C}}})=0$ if, and only if, every $R$-lattice functor is projective. An $R$-order category satisfying $\operatorname{Ldim}_R({{\mathcal{C}}})\leq 1$ will be called [*pseudo-hereditary*]{}. Such a category has the following property: Any subfunctor ${\mathbf{F}}$ of a projective $R$-lattice functor ${\mathbf{P}}$ such that ${\mathbf{P}}/{\mathbf{F}}$ is an $R$-lattice functor is projective.
The Yoneda dual {#ss:yondual}
---------------
Let $({{\mathcal{C}}},\sigma)$ be an $R^\circledast$-category. For $\phi\in\operatorname{Hom}_{{{\mathcal{C}}}}(A,B)$ one has an $R$-linear natural transformation ${\mathbf{P}}(\phi)\colon{\mathbf{P}}^A\to{\mathbf{P}}^B$ given by composition with $\phi$. Hence one has a functor $$\label{eq:cdash}
{\underline{\phantom{x}}}^\circledast\colon
{\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}})\longrightarrow {\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}})^{\operatorname{op}},$$ given by ${\mathbf{F}}^\circledast(A)=\operatorname{nat}_R({\mathbf{F}},{\mathbf{P}}^A)$ for ${\mathbf{F}}\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}}))$ and $A\in\operatorname{ob}({{\mathcal{C}}})$, and ${\mathbf{F}}^\circledast(\phi)={\mathbf{P}}(\sigma(\phi))\circ{\underline{\phantom{x}}}\colon{\mathbf{F}}^\circledast(B)\to{\mathbf{F}}^\circledast(A)$ for $\phi\colon A\to B\in\operatorname{Hom}_{{{\mathcal{C}}}}(A,B)$. We call the functor ${\underline{\phantom{x}}}^\circledast$ the [*Yoneda dual*]{}.
\[rem:yondual\] Let $\mu$ be an $R$-algebra with $R$-linear antipode $\sigma\colon\mu\to\mu^{\operatorname{op}}$. Then ${\mathfrak{F}}_R(\mu\bullet^{\operatorname{op}},{{}_R{\mathbf{mod}}})$ can be identified with the category of right $\mu$-modules (cf. Rem. \[rem:rep\]). Under this identification, the Yoneda dual satisfies ${\underline{\phantom{x}}}^\circledast=\operatorname{Hom}_\mu({\underline{\phantom{x}}},\mu)^\times$. Here we used the symbol ${}^\times$ to express that for a right $\mu$-module $M$, the left $\mu$-module $\operatorname{Hom}_\mu(M,\mu)$ is considered as right $\mu$-module via the map $\sigma$.
The Yoneda dual has the following property:
\[prop:yonproj\] Let $({{\mathcal{C}}},\sigma)$ be an $R^\circledast$-category, and let $A\in\operatorname{ob}({{\mathcal{C}}})$. Then one has a canonical natural isomorphism $j_A\colon ({\mathbf{P}}^A)^\circledast\to{\mathbf{P}}^A$ which is natural in $A$, i.e., for $\psi\colon A\to D\in\operatorname{Hom}_{{\mathcal{C}}}(A,D)$ one has a commutative diagram $$\label{eq:isoyon0}
\xymatrix{
({\mathbf{P}}^D)^\circledast\ar[d]_{{\mathbf{P}}(\psi)^\circledast}\ar[r]^{j_D}&
{\mathbf{P}}^D\ar[d]^{{\mathbf{P}}(\sigma(\psi))}\\
({\mathbf{P}}^A)^\circledast\ar[r]^{j_A}&{\mathbf{P}}^A.
}$$ In particular, if $R$ is a Dedekind domain and $({{\mathcal{C}}},\sigma)$ is an $R^\circledast$-order category, then ${\underline{\phantom{x}}}^\circledast$ maps projective $R$-lattice functors to projective $R$-lattice functors, and $R$-lattice functors to $R$-lattice functors.
Let $\phi\colon B\to C$ be a morphism in ${{\mathcal{C}}}$. By the definition of ${\mathbf{P}}^{{\underline{\phantom{x}}}}$ and Fact \[fact:Pfirst\], one has canonical isomorphisms $$\label{eq:isoyon1}
\xymatrix@R=.3truecm{
({\mathbf{P}}^A)^\circledast(B)\ar@{=}[d]\ar[0,3]^{j_A(B)}&&&{\mathbf{P}}^A(B)\ar@{=}[d]\\
\operatorname{nat}_R({\mathbf{P}}^A,{\mathbf{P}}^B)\ar[r]^-{\theta_{A,{\mathbf{P}}^B}}&{\mathbf{P}}^B(A)\ar@{=}[r]&\operatorname{Hom}_{{\mathcal{C}}}(A,B)\ar[r]^{\sigma}&\operatorname{Hom}_{{{\mathcal{C}}}}(B,A).}$$ and the diagram $$\label{eq:isoyon2}
\xymatrix{
\operatorname{nat}_R({\mathbf{P}}^A,{\mathbf{P}}^C)\ar[r]\ar[d]_{{\mathbf{P}}(\sigma(\phi))\circ-}
&\operatorname{Hom}_{{\mathcal{C}}}(A,C)\ar[r]^{\sigma}\ar[d]^{\sigma(\phi)\circ-}
&\operatorname{Hom}_{{{\mathcal{C}}}}(C,A)\ar[d]^{-\circ\phi}\\
\operatorname{nat}_R({\mathbf{P}}^A,{\mathbf{P}}^B)\ar[r]&\operatorname{Hom}_{{\mathcal{C}}}(A,B)\ar[r]^{\sigma}&\operatorname{Hom}_{{{\mathcal{C}}}}(B,A)\\
}$$ commutes. This shows that $j_A$ is a natural isomorphism. The commutativity of the diagram $$\label{eq:isoyon3}
\xymatrix{
\operatorname{nat}_R({\mathbf{P}}^D,{\mathbf{P}}^B)\ar[r]\ar[d]_{-\circ{\mathbf{P}}(\psi)}
&\operatorname{Hom}_{{\mathcal{C}}}(D,B)\ar[r]^{\sigma}\ar[d]^{-\circ\psi}
&\operatorname{Hom}_{{{\mathcal{C}}}}(B,D)\ar[d]^{\sigma(\psi)\circ-}\\
\operatorname{nat}_R({\mathbf{P}}^A,{\mathbf{P}}^B)\ar[r]&\operatorname{Hom}_{{\mathcal{C}}}(A,B)\ar[r]^{\sigma}&\operatorname{Hom}_{{{\mathcal{C}}}}(B,A)\\
}$$ shows the commutativity of . The final remark is straightforward.
Let $A,B\in\operatorname{ob}({{\mathcal{C}}})$ and let ${\mathbf{F}}\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}}))$. For any $\chi\in\operatorname{nat}_R({\mathbf{F}},{\mathbf{P}}^B)$ one has an $R$-linear map $$\label{eq:eta0}
\sigma\circ\chi_A\colon\xymatrix{
{\mathbf{F}}(A)\ar[r]^{\chi_A}&{\mathbf{P}}^B(A)\ar[r]^{\sigma_{A,B}}&{\mathbf{P}}^A(B)}.$$ Let $f\in{\mathbf{F}}(A)$, and let $\eta^{f,B}_{{\mathbf{F}},A}\colon\operatorname{nat}_R({\mathbf{F}},{\mathbf{P}}^B)\longrightarrow{\mathbf{P}}^A(B)$ be given by $$\label{eq:eta1}
\eta^{f,B}_{{\mathbf{F}},A}(\chi)=\sigma(\chi_A(f)).$$ For $\phi\colon B\to C\in\operatorname{Hom}_{{\mathcal{C}}}(B,C)$ one has a commutative diagram $$\label{eq:eta2}
\xymatrix{
\operatorname{nat}_R({\mathbf{F}},{\mathbf{P}}^C)\ar[r]^-{\eta^{f,C}_{{\mathbf{F}},A}}\ar[d]_{{\mathbf{P}}(\sigma(\phi))\circ-}
&{\mathbf{P}}^A(C)\ar[d]^{-\circ\phi}\\
\operatorname{nat}_R({\mathbf{F}},{\mathbf{P}}^B)\ar[r]^-{\eta^{f,B}_{{\mathbf{F}},A}}&{\mathbf{P}}^A(B).\\
}$$ Hence $\eta^{f,-}_{{\mathbf{F}},A}\colon{\mathbf{F}}^\circledast\to{\mathbf{P}}^A$ is an $R$-linear natural transformation. The mapping $$\label{eq:eta3}
\eta_{{\mathbf{F}},A}\colon {\mathbf{F}}(A)\longrightarrow\operatorname{nat}_R({\mathbf{F}}^\circledast,{\mathbf{P}}^A)$$ is $R$-linear, and for $\psi\colon D\to A\in\operatorname{Hom}_{{\mathcal{C}}}(D,A)$, the diagram $$\label{eq:eta4}
\xymatrix{
{\mathbf{F}}(A)\ar[d]_{{\mathbf{F}}(\psi)}\ar[r]^-{\eta_{{\mathbf{F}},A}}&
\operatorname{nat}_R({\mathbf{F}}^\circledast,{\mathbf{P}}^A)\ar[d]^{{\mathbf{P}}(\sigma(\psi))\circ-}\\
{\mathbf{F}}(D)\ar[r]^-{\eta_{{\mathbf{F}},D}}&\operatorname{nat}_R({\mathbf{F}}^\circledast,{\mathbf{P}}^D)
}$$ commutes. Thus it defines an $R$-linear, natural transformation $\eta_{\mathbf{F}}\colon{\mathbf{F}}\to{\mathbf{F}}^{\circledast\circledast}$. Let $\alpha\in\operatorname{nat}_R({\mathbf{F}},{\mathbf{G}})$. For all $A\in\operatorname{ob}({{\mathcal{C}}})$ one has a commutative diagram $$\label{eq:eta5}
\xymatrix{
{\mathbf{F}}(A)\ar[d]_{\alpha_A}
\ar[r]^-{\eta_{{\mathbf{F}},A}}&\operatorname{nat}_R({\mathbf{F}}^\circledast,{\mathbf{P}}^A)\ar[d]^{-\circ\alpha^\circledast}\\
{\mathbf{G}}(A)\ar[r]^-{\eta_{{\mathbf{G}},A}}&\operatorname{nat}_R({\mathbf{G}}^\circledast,{\mathbf{P}}^A).
}$$ Hence one has the following.
\[prop:isoeta\] Let $({{\mathcal{C}}},\sigma)$ be an $R^\circledast$-category. Then $$\label{eq:nateta}
\eta\colon\operatorname{id}_{{\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}})}\longrightarrow{\underline{\phantom{x}}}^{\circledast\circledast}$$ is a natural transformation. For every $E\in\operatorname{ob}({{\mathcal{C}}})$, $\eta_{{\mathbf{P}}^E}\colon{\mathbf{P}}^E\to({\mathbf{P}}^E)^{\circledast\circledast}$ is a natural isomorphism. In particular, if $R$ is a Dedekind domain and $({{\mathcal{C}}},\sigma)$ is an $R^\circledast$-order category, then $\eta_{{\mathbf{P}}}\colon {\mathbf{P}}\to{\mathbf{P}}^{\circledast\circledast}$ is an isomorphism for every projective $R$-lattice functor ${\mathbf{P}}\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}}))$.
It suffices to show that $\eta_{{\mathbf{P}}^E}\colon{\mathbf{P}}^E\to({\mathbf{P}}^E)^{\circledast\circledast}$ is a natural isomorphism for every $E\in\operatorname{ob}({{\mathcal{C}}})$. For $A\in\operatorname{ob}({{\mathcal{C}}})$ one has a commutative diagram $$\label{eq:eta6}
\xymatrix@C=2.0truecm{
{\mathbf{P}}^A(E)\ar[r]^-{\theta^{-1}_{E,{\mathbf{P}}^A}}\ar[d]_{\sigma}&\operatorname{nat}_R({\mathbf{P}}^E,{\mathbf{P}}^A)\ar[d]^{-\circ j_E}\\
{\mathbf{P}}^E(A)\ar[r]^-{\eta_{{\mathbf{P}}^E,A}}&\operatorname{nat}_R(({\mathbf{P}}^E)^\circledast,{\mathbf{P}}^A),
}$$ and all maps apart from $\eta_{{\mathbf{P}}^E,A}$ are isomorphisms (cf. , ). Hence $\eta_{{\mathbf{P}}^E,A}$ is an isomorphism, and this yields the claim.
Derived functors of the Yoneda dual {#ss:deryon}
-----------------------------------
Let $({{\mathcal{C}}},\sigma)$ be an $R^\circledast$-category such that $\operatorname{ob}({{\mathcal{C}}})$ is a set. Then ${\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}})$ is an abelian category with enough projectives (cf. Fact \[fact:enproj\]).
The Yoneda dual ${\underline{\phantom{x}}}^\circledast\colon {\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}})\to
{\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}})^{\operatorname{op}}$ is additive and left-exact. Let ${{\mathcal{R}}}^k({\underline{\phantom{x}}})^\circledast$, $k\geq 1$, denote its right-derived functors, i.e., one has that $$\label{eq:rightyo}
{{\mathcal{R}}}^k({\mathbf{F}})^\circledast(A)=\operatorname{ext}_R^k({\mathbf{F}},{\mathbf{P}}^A),\qquad
\text{for ${\mathbf{F}}\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}}))$,}$$ and ${{\mathcal{R}}}^k({\mathbf{F}})^\circledast(\phi)=\operatorname{ext}_R^k({\mathbf{F}},{\mathbf{P}}(\sigma(\phi)))$ for $\phi\in\operatorname{Hom}_{{\mathcal{C}}}(A,B)$. A functor ${\mathbf{F}}$ will be called [*$\circledast$-acyclic*]{}, if ${{\mathcal{R}}}^k({\mathbf{F}})^\circledast=0$ for all $k>0$. E.g., every projective functor is $\circledast$-acyclic.
Let $R$ be a Dedekind domain, and let $({{\mathcal{C}}},\sigma)$ be an $R^\circledast$-order category. An $R$-lattice functor ${\mathbf{F}}\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}}))$ will be called [*$\circledast$-bi-acyclic*]{}, if ${\mathbf{F}}$ and ${\mathbf{F}}^\circledast$ are $\circledast$-acyclic. The $R^\circledast$-order category $({{\mathcal{C}}},\sigma)$ will be called [*$\circledast$-symmetric*]{}, if every $\circledast$-acyclic $R$-lattice functor is $\circledast$-bi-acyclic.
Gorenstein projective functors {#ss:compproj}
------------------------------
Let $R$ be a Dedekind domain, let $({{\mathcal{C}}},\sigma)$ be an $R^\circledast$-order category, and let ${\mathbf{F}}\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{lat}}}))$. A chain complex $({\mathbf{P}}_\bullet,\partial_\bullet^{\mathbf{P}})$ together with a natural transformation ${\varepsilon}\colon {\mathbf{P}}_0\to{\mathbf{F}}$ will be called a [*complete projective $R$-lattice functor resolution*]{} of ${\mathbf{F}}$, if
- ${\mathbf{P}}_k$ is a projective $R$-lattice functor for all $k\in{\mathbb{Z}}$;
- $({\mathbf{P}}_\bullet,\partial_\bullet^{\mathbf{P}})$ is exact;
- ${\varepsilon}\circ\partial_1=0$ and ${\varepsilon}$ induces a natural isomorphism $\tilde{{\varepsilon}}\colon\operatorname{coker}(\partial_1)\to{\mathbf{F}}$.
For an exact chain complex of projective $R$-lattice functors $({\mathbf{P}}_\bullet,\partial_\bullet^{\mathbf{P}})$ we denote by $({\mathbf{Q}}_\bullet,\partial_\bullet^{\mathbf{Q}})=({\mathbf{P}}_\bullet,\partial_\bullet^{\mathbf{P}})^\circledast$ the chain complex of projective $R$-lattice functors given by ${\mathbf{Q}}_k={\mathbf{P}}^\circledast_{-k-1}$ and $\partial_k^{\mathbf{Q}}=(\partial_{-k}^{\mathbf{P}})^\circledast$. Note that by Proposition \[prop:isoeta\], $({\mathbf{P}}_\bullet,\partial_\bullet^{\mathbf{P}})^{\circledast\circledast}$ is canonically isomorphic to $({\mathbf{P}}_\bullet,\partial_\bullet^{\mathbf{P}})$. The complete projective $R$-lattice functor resolution $({\mathbf{P}}_\bullet,\partial_\bullet^{\mathbf{P}},{\varepsilon}_{\mathbf{F}})$ of ${\mathbf{F}}$ will be called [*$\circledast$-exact*]{} if
- $({\mathbf{P}}_\bullet,\partial_\bullet^{\mathbf{P}})^\circledast$ is exact.
An $R$-lattice functor with a $\circledast$-exact complete projective $R$-lattice functor resolution is also called a [*Gorenstein projective functor*]{}. One has the following property.
\[prop:explores\] Let $R$ be a Dedekind domain, let $({{\mathcal{C}}},\sigma)$ be an $R^\circledast$-order category, and let ${\mathbf{F}}\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{lat}}}))$ be an $R$-lattice functor. Then ${\mathbf{F}}$ is Gorenstein projective if, and only if, ${\mathbf{F}}$ is $\circledast$-bi-acyclic.
Suppose that ${\mathbf{F}}$ is $\circledast$-bi-acyclic. By Fact \[fact:rlatproj\], ${\mathbf{F}}$ has a projective resolution $({\mathbf{P}}_\bullet,\partial_\bullet^{\mathbf{P}},{\varepsilon}_{\mathbf{F}})$ by projective $R$-lattice functors. Let ${\mathbf{Q}}_{k}={\mathbf{P}}^\circledast_{-k}$, $\partial^{\mathbf{Q}}_{k}=(\partial_{1-k}^{\mathbf{P}})^\circledast$. Then $({\mathbf{Q}}_\bullet,\partial^{\mathbf{Q}}_\bullet)$ is a chain complex of projective $R$-lattice functors concentrated in non-positive degrees (cf. Prop. \[prop:yonproj\]). As ${\mathbf{F}}$ is $\circledast$-acyclic, one has $$\label{eq:comres1}
H_k({\mathbf{Q}}_\bullet,\partial^{\mathbf{Q}}_\bullet)\simeq
\begin{cases}
{\mathbf{F}}^\circledast&\ \text{for $k=0$;}\\
\hfil 0\hfil&\ \text{for $k\not=0$.}
\end{cases}$$ As ${\mathbf{F}}^\circledast$ is an $R$-lattice functor, it has a projective resolution $({\mathbf{R}}_\bullet,\partial_\bullet^{\mathbf{R}},\mu_{{\mathbf{F}}^\circledast})$ by projective $R$-lattice functors. Let $({\mathbf{S}}_\bullet,\partial_\bullet^{\mathbf{S}})$ be the chain complex given by ${\mathbf{S}}_k={\mathbf{R}}_k$ for $k\geq 0$ and ${\mathbf{S}}_k={\mathbf{Q}}_{k+1}$ for $k<0$, and mappings $\partial_k^{\mathbf{S}}=\partial_k^{\mathbf{R}}$ for $k\geq 1$, $\partial_k^{\mathbf{S}}=\partial_{k+1}^{\mathbf{Q}}$ for $k\leq -1$, and $\partial_0={\varepsilon}_{{\mathbf{F}}}^\circledast\circ\mu_{{\mathbf{F}}^\circledast}$. Then $({\mathbf{S}}_\bullet,\partial_\bullet^{\mathbf{S}},\mu_{{\mathbf{F}}^\circledast})$ is a complete projective $R$-lattice functor resolution of ${\mathbf{F}}^\circledast$.
Let $({\mathbf{T}}_\bullet,\partial_\bullet^{\mathbf{T}})=({\mathbf{S}}_\bullet,\partial_\bullet^{\mathbf{S}})^\circledast$. Then one has ${\mathbf{T}}_0={\mathbf{P}}_0^{\circledast\circledast}$, $\rho={\varepsilon}\circ\eta_{{\mathbf{P}}_0}^{-1}\colon{\mathbf{T}}_0\to{\mathbf{F}}$ (cf. Prop. \[prop:isoeta\]) is satisfying $\rho\circ\partial_1^{\mathbf{T}}=0$, and the induced map $\tilde{\rho}\colon\operatorname{coker}(\partial_1^{\mathbf{T}})\to{\mathbf{F}}$ is a natural isomorphism. By construction and Proposition \[prop:isoeta\], one has $H_k({\mathbf{T}}_\bullet,\partial_\bullet^{\mathbf{T}})=0$ for $k>0$. As ${\mathbf{F}}^\circledast$ is $\circledast$-acyclic, one has also $H_k({\mathbf{T}}_\bullet,\partial_\bullet^{\mathbf{T}})=0$ for $k<-1$.
Let ${\mathbf{T}}^{<0}_\bullet$ and ${\mathbf{T}}^{\geq0}_\bullet$ denote the truncated chain complexes, respectively, and consider the short exact sequence of chain complexes $0\to{\mathbf{T}}^{<0}_\bullet\to{\mathbf{T}}_\bullet\to{\mathbf{T}}^{\geq0}_\bullet\to 0$. By construction, the connecting homomorphism $H_0(\delta)\colon H_0({\mathbf{T}}^{\geq0}_\bullet)\to H_{-1}({\mathbf{T}}^{<0}_\bullet)$ is an isomorphism. The long exact sequence in homology implies that the chain complex $({\mathbf{T}}_\bullet,\partial_\bullet^{\mathbf{T}})$ has trivial homology, and hence is exact. Thus by Proposition \[prop:yonproj\], $({\mathbf{T}}_\bullet,\partial_\bullet^{\mathbf{T}},\rho)$ is a complete projective $R$-lattice functor resolution of ${\mathbf{F}}$. As $({\mathbf{T}}_\bullet,\partial_\bullet^{\mathbf{T}})^\circledast$ is canonically isomorphic to $({\mathbf{S}}_\bullet,\partial_\bullet^{\mathbf{S}})$, $({\mathbf{T}}_\bullet,\partial_\bullet^{\mathbf{T}},\rho)$ is also $\circledast$-exact.
Let $({\mathbf{T}}_\bullet,\partial_\bullet^{\mathbf{T}},{\varepsilon}_{{\mathbf{F}}})$ be a $\circledast$-exact complete projective $R$-lattice functor resolution of ${\mathbf{F}}$. Then $$\label{eq:yonres1}
{{\mathcal{R}}}^k({\mathbf{F}})^\circledast=H_{-k-1}(({\mathbf{T}}_\bullet,\partial_\bullet^{\mathbf{T}})^\circledast)=0,\ \ k>0,$$ i.e., ${\mathbf{F}}$ is $\circledast$-acyclic. Replacing the chain complex $({\mathbf{T}}_\bullet,\partial_\bullet^{\mathbf{T}})$ by the chain complex $({\mathbf{T}}_\bullet,\partial_\bullet^{\mathbf{T}})^\circledast$ shows that ${\mathbf{F}}^\circledast$ is also $\circledast$-acyclic.
Gorenstein $R^\circledast$-order categories {#ss:gor}
-------------------------------------------
Let $R$ be a Dedekind[^2] domain, and let $({{\mathcal{C}}},\sigma)$ be an $R^\circledast$-order category. For $A\in\operatorname{ob}({{\mathcal{C}}})$ the functors $$\label{eq:defJ}
{\mathbf{J}}^A=({\mathbf{P}}^A)^\ast\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}}))$$ are $R$-lattice functors which are [*relative injective*]{} in ${\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{lat}}})$ in the following sense: Let $\alpha\colon{\mathbf{F}}\to{\mathbf{G}}$ be a split-injective, $R$-linear transformation of $R$-lattice functors, and let $\beta\colon {\mathbf{F}}\to{\mathbf{J}}^A$ be any $R$-linear natural transformation. Then there exists an $R$-linear natural transformation ${\tilde{\beta}}\colon{\mathbf{G}}\to{\mathbf{J}}^A$ such that the diagram $$\label{eq:relinj}
\xymatrix{
0\ar[r]&{\mathbf{F}}\ar[r]^\alpha\ar[d]_{\beta}&{\mathbf{G}}\ar@{-->}[dl]^{{\tilde{\beta}}}\\
&{\mathbf{J}}^A&
}$$ commutes. Here we called a natural transformation $\alpha\colon{\mathbf{F}}\to{\mathbf{G}}$ of $R$-lattice functors [*split injective*]{}, if it is injective and $\operatorname{coker}(\alpha_B)$ is an $R$-lattice for every $B\in\operatorname{ob}({{\mathcal{C}}})$. The $R^\circledast$-order category $({{\mathcal{C}}},\sigma)$ is called [*$m$-Gorenstein*]{}, $m\geq 0$, if $\operatorname{proj.dim}({\mathbf{J}}^A)\leq m$ for all $A\in\operatorname{ob}({{\mathcal{C}}})$. A $0$-Gorenstein $R^\circledast$-order category is also called [*Frobenius*]{}. For $m$-Gorenstein $R^\circledast$-order categories one has the following:
\[prop:gor\] Let $R$ be a Dedekind domain, and let $({{\mathcal{C}}},\sigma)$ be an $m$-Gorenstein $R^\circledast$-order category. Then for any ${\mathbf{F}}\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{lat}}}))$ and $k> m$ one has ${{\mathcal{R}}}^k({\mathbf{F}})^\circledast=0$.
By Fact \[fact:rlatproj\], ${\mathbf{F}}$ has a projective resolution $({\mathbf{P}}_i,\partial_i^{\mathbf{P}},{\varepsilon}_{\mathbf{F}})$ by projective $R$-lattice functors. Moreover, by hypothesis, for $A\in\operatorname{ob}({{\mathcal{C}}})$ the functor ${\mathbf{J}}^A$ has a finite projective resolution $({\mathbf{Q}}_j,\partial_j^{\mathbf{Q}},{\varepsilon}_A)$ by projective $R$-lattice functors and ${\mathbf{Q}}_j=0$ for $j>m$. Thus ${\mathbf{P}}^A\simeq({\mathbf{J}}^A)^\ast$ has a finite, relative injective resolution $({\mathbf{I}}^j,\delta^j,\mu_A)$, ${\mathbf{I}}^j={\mathbf{Q}}_j^\ast$, $\delta^{j-1}=(\partial_j^{\mathbf{Q}})^\ast$, $\mu_A={\varepsilon}_A^\ast$ with ${\mathbf{I}}^j=0$ for $j>m$. Consider the double complex $(E_0^{s,t},\partial_v,\partial_h)$, where $E_0^{s,t}=\operatorname{nat}_R({\mathbf{P}}_s,{\mathbf{I}}^t)$ and $\partial_v$ and $\partial_h$ are the vertical and horizontal differential induced by $\partial_\bullet^{\mathbf{P}}$ and $\delta^\bullet$, respectively. The cohomology of the total complex $(\operatorname{Tot}^\bullet(E_0^{s,t}),\partial_v+(-1)^\bullet\partial_h)$ can be calculated in two ways.
Applying first the vertical and then the horizontal differential yields a spectral sequence with $E_2$-term $$\label{eq:Ev}
{}^vE_2^{s,t}=
\begin{cases}
\operatorname{ext}_R^s({\mathbf{F}},{\mathbf{P}}^A)&\ \ \text{for $t=0$},\\
\hfil 0\hfil&\ \ \text{for $t\not=0$,}
\end{cases}$$ concentrated on the $(t=0)$-line. By definition, $\operatorname{nat}_R({\underline{\phantom{x}}},{\mathbf{I}}^j)$ is exact for every short exact sequence of $R$-lattice functors. Since $R$ is a Dedekind domain, $0\to\operatorname{ker}(\partial_s^{\mathbf{P}})\to{\mathbf{P}}_s\to\operatorname{im}(\partial_s^{\mathbf{P}})\to 0$ is a short exact sequence of $R$-lattice functors for every $s\geq 0$. Hence applying first the horizontal and then the vertical differential yields a spectral sequence with $E_1$-term concentrated on the $(s=0)$-line, and ${}^hE_1^{0,t}=0$ for $t>m$. The claim then follows from the fact that both spectral sequences converge to the cohomology of the total complex.
$R^\circledast$-order categories with the Whitehead property {#ss:beauty}
------------------------------------------------------------
Let $R$ be a Dedekind domain. The Gorenstein property of an $R^\circledast$-order category is a quantitative measurement for the failure of being Frobenius. However, for our main purpose another property plays a more important role. We say that an $R^\circledast$-order category $({{\mathcal{C}}},\sigma)$ has the [*Whitehead property*]{}[^3], if any $\circledast$-acyclic $R$-lattice functor is projective. The following property is well known (cf. [@brown:coh Prop. VIII.6.7]).
\[fact:glwh\] Let $R$ be a Dedekind domain, and let $({{\mathcal{C}}},\sigma)$ be an $R^\circledast$-order category of finite global $R$-lattice dimension. Then $({{\mathcal{C}}},\sigma)$ is Gorenstein and has the Whitehead property. Moreover, $$\label{eq:glwh}
\operatorname{Ldim}_R({{\mathcal{C}}})=\max\{\,k\geq 0\mid{{\mathcal{R}}}^k({\underline{\phantom{x}}})^\circledast\not=0\,\}.$$
\[rem:glwh\] Let $R$ be a Dedekind domain, and let $({{\mathcal{C}}},\sigma)$ be an $R^\circledast$-order category. For $m\geq 0$ one has the implications $$\label{eq:glwh2}
\operatorname{gldim}_R({{\mathcal{C}}})\leq m\ \Longrightarrow\
{{\mathcal{C}}}\ \text{$m$-Gorenstein \& Whitehead}\ \Longrightarrow\
{{\mathcal{C}}}\ \text{$m$-Gorenstein}.$$ If $G$ is a finite group, then $({\mathbb{Z}}[G]\bullet,\sigma)$, where $\sigma(g)=g^{-1}$ for $g\in G$, is $0$-Gorenstein. But $({\mathbb{Z}}[G]\bullet,\sigma)$ has the Whitehead property if, and only if, $G$ is the trivial group. Hence the second implication cannot be reversed. For certain values of $m$ one can reverse the first implication. E.g., if $({{\mathcal{C}}},\sigma)$ is $0$-Gorenstein, then it has the Whitehead property if, and only if, every $R$-lattice functor is projective, i.e., $\operatorname{Ldim}_R({{\mathcal{C}}})=0$. This is also the case for $m=1$.
\[fact:gldim1\] Let $R$ be a Dedekind domain, and let $({{\mathcal{C}}},\sigma)$ be a $1$-Gorenstein $R^\circledast$-order category. Then $({{\mathcal{C}}},\sigma)$ has the Whitehead property if, and only if, ${{\mathcal{C}}}$ is pseudo-hereditary, i.e., $\operatorname{Ldim}_R({{\mathcal{C}}})\leq 1$.
By Fact \[fact:glwh\], it suffice to show the reverse direction of the first implication of . Suppose that $({{\mathcal{C}}},\sigma)$ is $1$-Gorenstein and has the Whitehead property. Let ${\mathbf{F}}\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{lat}}}))$. Then there exists a surjective natural transformation $\pi\colon {\mathbf{P}}\to{\mathbf{F}}$ for some projective $R$-lattice functor ${\mathbf{P}}$, and ${\mathbf{Q}}=\operatorname{ker}(\pi)$ is an $R$-lattice functor. By Proposition \[prop:gor\] and the long exact sequence, the sequence $$\label{eq:semiher2}
{{\mathcal{R}}}^1({\mathbf{F}})^\circledast\longrightarrow
{{\mathcal{R}}}^1({\mathbf{P}})^\circledast\longrightarrow
{{\mathcal{R}}}^1({\mathbf{Q}})^\circledast\longrightarrow 0$$ is exact. As ${{\mathcal{R}}}^1({\mathbf{P}})^\circledast=0$, ${\mathbf{Q}}$ is $\circledast$-acyclic and thus, by hypothesis, projective.
Functors between $R^\circledast$-categories {#ss:funRcat}
-------------------------------------------
Let $({{\mathcal{C}}},\sigma_{{\mathcal{C}}})$ and $({{\mathcal{D}}},\sigma_{{\mathcal{D}}})$ be $R^\circledast$-categories. An $R$-linear functor $\phi\colon{{\mathcal{C}}}\to{{\mathcal{D}}}$ will be called [*unitary*]{}, if $$\label{eq:Runi}
\sigma_{{\mathcal{D}}}\circ\phi=\phi\circ\sigma_{{{\mathcal{C}}}}.$$ If $\operatorname{ob}({{\mathcal{C}}})$ and $\operatorname{ob}({{\mathcal{D}}})$ are sets the unitary functor $\pi\colon({{\mathcal{C}}},\sigma_{{\mathcal{C}}})\to({{\mathcal{D}}},\sigma_{{\mathcal{D}}})$ will be called a [*unitary projection*]{}, if $\pi\colon\operatorname{ob}({{\mathcal{C}}})\longrightarrow\operatorname{ob}({{\mathcal{D}}})$ is a bijection, and $$\label{eq:Runi2}
\pi_{A,B}\colon\operatorname{Hom}_{{\mathcal{C}}}(A,B)\longrightarrow\operatorname{Hom}_{{\mathcal{D}}}(\pi(A),\pi(B))$$ is surjective for any pair of objects $A,B\in\operatorname{ob}({{\mathcal{C}}})$. For such a functor composition with $\pi$ induces an exact inflation functor $$\label{eq:inf}
\operatorname{inf}^\pi({\underline{\phantom{x}}})={\underline{\phantom{x}}}\circ\pi\colon{\mathfrak{F}}_R({{\mathcal{D}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}})\longrightarrow{\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}}).$$ Let $\pi\colon ({{\mathcal{C}}},\sigma_{{\mathcal{C}}})\to({{\mathcal{D}}},\sigma_{{\mathcal{D}}})$ be a unitary projection of $R^\circledast$-order categories. Then $\pi$ induces a surjective homomorphisms of $R$-orders $\mu(\pi)\colon\mu_{{{\mathcal{C}}}}\to\mu_{{{\mathcal{D}}}}$ (cf. Rem. \[rem:rep\]). Moreover, the inflation functor $\operatorname{inf}_{\mu_{{{\mathcal{D}}}}}^{\mu_{{{\mathcal{C}}}}}({\underline{\phantom{x}}})\colon{\mathbf{mod}}_{\mu_{{{\mathcal{D}}}}}\to
{\mathbf{mod}}_{\mu_{{{\mathcal{C}}}}}$ has a left-adjoint $$\label{eq:dfl1}
\operatorname{def}_{\mu_{{{\mathcal{D}}}}}^{\mu_{{{\mathcal{C}}}}}({\underline{\phantom{x}}})={\underline{\phantom{x}}}\otimes_{\mu_{{{\mathcal{C}}}}}\mu_{{{\mathcal{D}}}}\colon{\mathbf{mod}}_{\mu_{{{\mathcal{C}}}}}\to
{\mathbf{mod}}_{\mu_{{{\mathcal{D}}}}}.$$ From this fact one concludes the following.
\[fact:dfl\] Let $\pi\colon ({{\mathcal{C}}},\sigma_{{\mathcal{C}}})\to({{\mathcal{D}}},\sigma_{{\mathcal{D}}})$ be a unitary projection of $R^\circledast$-order categories.
- There exists a functor $$\label{eq:dfl2}
\operatorname{def}^\pi({\underline{\phantom{x}}})\colon {\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}})\longrightarrow{\mathfrak{F}}_R({{\mathcal{D}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}})$$ which is left-adjoint to $\operatorname{inf}^\pi({\underline{\phantom{x}}})$.
- The unit of the adjunction $\eta\colon\operatorname{id}_{{\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}})}\longrightarrow \operatorname{inf}^\pi\circ\operatorname{def}^\pi$ is a natural surjection, and the the co-unit $\varepsilon\colon \operatorname{def}^\pi\circ\operatorname{inf}^\pi\longrightarrow\operatorname{id}_{{\mathfrak{F}}_R({{\mathcal{D}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}})}$ is a natural isomorphism.
- For all $A\in\operatorname{ob}({{\mathcal{C}}})$ there exists an isomorphism $\xi_A\colon \operatorname{def}^\pi({\mathbf{P}}^A)\to{\mathbf{P}}^{\pi(A)}$ making the diagram $$\label{eq:dlf3}
\xymatrix{
\operatorname{def}^\pi({\mathbf{P}}^A)\ar[r]^-{\xi_A}\ar[d]_{\operatorname{def}^\pi({\mathbf{P}}(\phi))}&{\mathbf{P}}^{\pi(A)}\ar[d]^{{\mathbf{P}}(\pi(\phi))}\\
\operatorname{def}^\pi({\mathbf{P}}^B)\ar[r]^-{\xi_B}&{\mathbf{P}}^{\pi(B)}
}$$ commute for all $\phi\colon A\to B\in\operatorname{Hom}_{{\mathcal{C}}}(A,B)$.
For $A\in\operatorname{ob}({{\mathcal{C}}})$ and $\phi\in\operatorname{Hom}_{{\mathcal{C}}}(A,B)$ put ${\mathbf{Q}}^A=\operatorname{inf}^\pi({\mathbf{P}}^{\pi(A)})$ and ${\mathbf{Q}}(\phi)=\operatorname{inf}^\pi({\mathbf{P}}(\pi(\phi)))$. Then $$\label{eq:dfl4}
\xymatrix@C=2truecm{
\tau_A\colon {\mathbf{P}}^A\ar[r]^-{\eta_{{\mathbf{P}}^A}} &\operatorname{inf}^\pi(\operatorname{def}^\pi({\mathbf{P}}^A))\ar[r]^-{\operatorname{inf}^\pi(\xi_A)}&{\mathbf{Q}}^A
}$$ is a surjection satisfying $\tau_B\circ{\mathbf{P}}(\phi)={\mathbf{Q}}(\phi)\circ\tau_A$ for all $\phi\in\operatorname{Hom}_{{\mathcal{C}}}(A,B)$.
For ${\mathbf{F}}\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}}))$ let ${\mathbf{F}}^\boxtimes\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}}))$ be the functor given by ${\mathbf{F}}^\boxtimes(A)=\operatorname{nat}_R({\mathbf{F}},{\mathbf{Q}}^A)$ and ${\mathbf{F}}^\boxtimes(\phi)={\mathbf{Q}}(\sigma_{{\mathcal{C}}}(\phi))\circ{\underline{\phantom{x}}}$ for $\phi\in\operatorname{Hom}_{{{\mathcal{C}}}}(A,B)$. Then ${\underline{\phantom{x}}}^\boxtimes\colon{\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}})\to{\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}})^{\operatorname{op}}$ is a functor. By , one has $$\label{eq:dfl5}
{\mathbf{P}}(\pi(\sigma_{{\mathcal{C}}}(\phi)))={\mathbf{P}}(\sigma_{{\mathcal{D}}}(\pi(\phi)))\colon {\mathbf{P}}^{\pi(B)}\to{\mathbf{P}}^{\pi(A)}$$ for all $\phi\in\operatorname{Hom}_{{\mathcal{C}}}(A,B)$. Hence the mapping ${\tilde{\tau}}\colon{\underline{\phantom{x}}}^\circledast\longrightarrow{\underline{\phantom{x}}}^\boxtimes$ induced by $\tau$ is a natural transformation. Since $$\label{eq:dfl6}
{\mathbf{F}}^\boxtimes(A)=\operatorname{nat}_R({\mathbf{F}},{\mathbf{Q}}^A)\simeq\operatorname{nat}_R(\operatorname{def}^\pi({\mathbf{F}}),{\mathbf{P}}^{\pi(A)})=
\operatorname{inf}^\pi(\operatorname{def}^\pi({\mathbf{F}})^\circledast),$$ ${\underline{\phantom{x}}}^\boxtimes$ can be identified with $\operatorname{inf}^\pi(\operatorname{def}^\pi({\underline{\phantom{x}}})^\circledast)$. Thus by the left-adjointness of $\operatorname{def}^\pi({\underline{\phantom{x}}})$, ${\tilde{\tau}}$ induces a natural transformation $$\label{eq:dfl7}
{\hat{\tau}}\colon \operatorname{def}^\pi({\underline{\phantom{x}}}^\circledast)\longrightarrow\operatorname{def}^\pi({\underline{\phantom{x}}})^\circledast
\colon{\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}})\longrightarrow{\mathfrak{F}}_R({{\mathcal{D}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}})^{\operatorname{op}}.$$ For $A\in\operatorname{ob}({{\mathcal{C}}})$ the mapping ${\hat{\tau}}_{{\mathbf{P}}^A}\colon \operatorname{def}^\pi(({\mathbf{P}}^A)^\circledast)\to\operatorname{def}({\mathbf{P}}_A)^\circledast$ coincides with the isomorphism $$\label{eq:dfl8}
{\hat{\tau}}_{{\mathbf{P}}^A}\colon \operatorname{Hom}_{\mu_{{{\mathcal{C}}}}}(\operatorname{id}_A\cdot\mu_{{{\mathcal{C}}}},\mu_{{{\mathcal{C}}}})^\times
\otimes_{\mu_{{{\mathcal{C}}}}}\mu_{{{\mathcal{D}}}}\longrightarrow
\operatorname{Hom}_{\mu_{{{\mathcal{D}}}}}(\operatorname{id}_{\pi(A)}\cdot\mu_{{{\mathcal{D}}}},\mu_{{{\mathcal{D}}}})^\times.$$ From this fact one concludes the following.
\[fact:uni\] Let $\pi\colon ({{\mathcal{C}}},\sigma_{{\mathcal{C}}})\to({{\mathcal{D}}},\sigma_{{\mathcal{D}}})$ be a unitary projection of $R^\circledast$-order categories. Then ${\hat{\tau}}_{{\mathbf{P}}}\colon\operatorname{def}^\pi({\mathbf{P}}^\circledast)\to\operatorname{def}^\pi({\mathbf{P}})^\circledast$ is an isomorphism for every projective $R$-lattice functor ${\mathbf{P}}\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}}))$.
If $\pi\colon ({{\mathcal{C}}},\sigma_{{\mathcal{C}}})\to({{\mathcal{D}}},\sigma_{{\mathcal{D}}})$ is a unitary projection of $R^\ast$-order categories, its deflation functor $\operatorname{def}^\pi({\underline{\phantom{x}}})\colon{\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}})\to{\mathfrak{F}}_R({{\mathcal{D}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}})$ is right exact and maps projectives to projectives (cf. [@weib:hom Prop. 2.3.10]). We denote by ${{\mathcal{L}}}_k\operatorname{def}^\pi({\underline{\phantom{x}}})$ its left derived functors. Functors ${\mathbf{F}}\in\operatorname{ob}({\mathfrak{F}}({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{mod}}}))$ satisfying ${{\mathcal{L}}}_k\operatorname{def}^\pi({\mathbf{F}})=0$ for all $k>0$ will be called [*$\pi$-acyclic*]{}.
\[thm:whitehead\] Let $R$ be a Dedekind domain, and let $\pi\colon({{\mathcal{C}}},\sigma_{{\mathcal{C}}})\to({{\mathcal{D}}},\sigma_{{\mathcal{D}}})$ be a unitary projection of $R^\ast$-order categories. Assume further that
- $({{\mathcal{C}}},\sigma_{{\mathcal{C}}})$ is $\circledast$-symmetric ;
- $({{\mathcal{D}}},\sigma_{{\mathcal{D}}})$ has the Whitehead property ;
- if ${\mathbf{F}}^\circledast\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{lat}}}))$ is $\circledast$-acyclic, ${\mathbf{F}}$ is also $\pi$-acyclic.
Then $\operatorname{def}^\pi({\mathbf{G}})$ is a projective $R$-lattice functor for any $\circledast$-acyclic $R$-lattice functor ${\mathbf{G}}\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{lat}}}))$.
Suppose that ${\mathbf{G}}\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{C}}}^{\operatorname{op}},{{}_R{\mathbf{lat}}}))$ is $\circledast$-acyclic. By hypothesis (i), ${\mathbf{G}}$ is $\circledast$-bi-acyclic and thus Gorenstein projective (cf. Prop. \[prop:explores\]), i.e., ${\mathbf{G}}$ admits a $\circledast$-exact complete projective $R$-lattice functor resolution $({\mathbf{P}}_\bullet,\partial_\bullet^{\mathbf{P}},{\varepsilon}_{\mathbf{G}})$. Shifting the chain complex $({\mathbf{P}}_\bullet,\partial_\bullet^{\mathbf{P}})$ appropriately, one concludes that every functor ${\mathbf{C}}_k=\operatorname{coker}(\partial_{k+1})$ admits a $\circledast$-exact complete projective $R$-lattice functor resolution for all $k\in{\mathbb{Z}}$, and hence is Gorenstein projective. Thus by Proposition \[prop:explores\], ${\mathbf{C}}_k$ is $\circledast$-bi-acyclic, i.e., ${\mathbf{C}}_k$ and ${\mathbf{C}}_k^\circledast$ are $\pi$-acyclic for all $k\in{\mathbb{Z}}$.
Let $({\mathbf{Q}}_\bullet,\partial_\bullet^{\mathbf{Q}})=(\operatorname{def}^\pi({\mathbf{P}}_\bullet),\operatorname{def}^\pi(\partial_\bullet^{\mathbf{P}}))$. As $H_k({\mathbf{Q}}_\bullet,\partial_\bullet^{\mathbf{Q}})\simeq {{\mathcal{L}}}_1\operatorname{def}^\pi({\mathbf{C}}_{k-1})=0$, $({\mathbf{Q}}_\bullet,\partial_\bullet^{\mathbf{Q}})$ is exact. By Fact \[fact:uni\], one has an isomorphism of chain complexes $$\label{eq:uni1}
({\mathbf{Q}}_\bullet,\partial_\bullet^{\mathbf{Q}})^\circledast\simeq (\operatorname{def}^\pi(({\mathbf{P}}_\bullet,\partial_\bullet^{\mathbf{P}})^\circledast).$$ Moreover, as $H_k(({\mathbf{Q}}_\bullet,\partial_\bullet^{\mathbf{Q}})^\circledast)\simeq
{{\mathcal{L}}}_1\operatorname{def}^\pi({\mathbf{C}}^\circledast_{1-k})=0$ (cf. §\[ss:compproj\]), $({\mathbf{Q}}_\bullet,\partial_\bullet^{\mathbf{Q}})^\circledast$ is also exact. Hence $({\mathbf{Q}}_\bullet,\partial_\bullet^{\mathbf{Q}},\operatorname{def}^\pi({\varepsilon}_{\mathbf{G}}))$ is a $\circledast$-exact complete projective $R$-lattice functor resolution of $\operatorname{def}^\pi({\mathbf{G}})$. In particular, $\operatorname{def}^\pi({\mathbf{G}})$ is Gorenstein projective and thus $\circledast$-bi-acyclic (cf. Prop. \[prop:explores\]). Hence by hypothesis (ii), $\operatorname{def}^\pi({\mathbf{G}})$ is projective.
Cohomological Mackey functors {#ss:mack}
=============================
Throughout this section $G$ will denote a finite group, and - if not stated otherwise - $R$ will denote a commutative ring with unit $1_R\in R$.
Cohomological $G$-Mackey functors {#ss:Gmac1}
---------------------------------
A [*cohomological $G$-Mackey functor ${\mathbf{X}}$*]{} with values in the category of $R$-modules is a family of $R$-modules $({\mathbf{X}}_U)_{U\subseteq G}$ together with homomorphisms of $R$-modules $$\label{eq:mac1}
\begin{aligned}
i_{U,V}^{\mathbf{X}}\colon&{\mathbf{X}}_U\longrightarrow {\mathbf{X}}_V,\\
t_{V,U}^{\mathbf{X}}\colon&{\mathbf{X}}_V\longrightarrow {\mathbf{X}}_U,\\
c_{g,U}^{\mathbf{X}}\colon& {\mathbf{X}}_U\longrightarrow{\mathbf{X}}_{{}^gU},
\end{aligned}$$ for $U,V\subseteq G$, $V\subseteq U$, $g\in G$, which satisfy the identities:
- $i_{U,U}^{\mathbf{X}}=t_{U,U}^{\mathbf{X}}=c_{u,U}^{\mathbf{X}}=\operatorname{id}_{{\mathbf{X}}_U}$ for all $U\subseteq G$ and all $u\in U$;
- $i_{V,W}^{\mathbf{X}}\circ i_{U,V}^{\mathbf{X}}=i_{U,W}^{\mathbf{X}}$ and $t_{V,U}^{\mathbf{X}}\circ t_{W,V}^{\mathbf{X}}=t_{W,U}^{\mathbf{X}}$ for all $U,V,W\subseteq G$ and $W\subseteq V\subseteq U$;
- $c_{h,{}^gU}^{\mathbf{X}}\circ c_{g,U}^{\mathbf{X}}=c_{hg,U}^{\mathbf{X}}$ for all $U\subseteq G$ and $g,h\in G$;
- $i_{{}^gU,{}^gV}^{\mathbf{X}}\circ c_{g,U}^{\mathbf{X}}=c_{g,V}^{\mathbf{X}}\circ i_{U,V}^{\mathbf{X}}$ for all $U,V\subseteq G$, $V\subseteq U$, and $g\in G$;
- $t_{{}^gV,{}^gU}^{\mathbf{X}}\circ c_{g,V}^{\mathbf{X}}=c_{g,U}^{\mathbf{X}}\circ t_{V,U}^{\mathbf{X}}$ for all $U,V\subseteq G$, $V\subseteq U$, and $g\in G$;
- $i_{U,W}^{\mathbf{X}}\circ t_{V,U}^{\mathbf{X}}=\sum_{g\in W\setminus U/V}
t_{ {}^gV\cap W,W}^{\mathbf{X}}\circ c_{g,V\cap W^g}^{\mathbf{X}}\circ i_{V,V\cap W^g}^{\mathbf{X}}$, where $W^g=g^{-1}Wg$ for all subgroups $U, V, W\subseteq G$ and $V,W\subseteq U$;
- $t_{V,U}^{\mathbf{X}}\circ i_{U,V}^{\mathbf{X}}=|U:V|.\operatorname{id}_{{\mathbf{X}}_U}$ for all subgroups $U,V\subseteq G$, $V\subseteq U$.
A homomorphism of cohomological Mackey functors $\phi\colon{\mathbf{X}}\to{\mathbf{Y}}$ is a family of $R$-module homomorphisms $\phi_U\colon{\mathbf{X}}_U\to{\mathbf{Y}}_U$, $U\subseteq G$ which commute with all the mappings $i_{.,.}$, $t_{.,.}$ and $c_{g,.}$, $g\in G$. By ${\mathfrak{cMF}}_G({{}_R{\mathbf{mod}}})$ we denote the abelian category of all cohomological $G$-Mackey functors with values in the category of $R$-modules. For ${\mathbf{X}}$, ${\mathbf{Y}}\in\operatorname{ob}({\mathfrak{cMF}}_G({{}_R{\mathbf{mod}}}))$ we denote by $\operatorname{nat}_G({\mathbf{X}},{\mathbf{Y}})$ the morphisms in the category ${\mathfrak{cMF}}_G({{}_R{\mathbf{mod}}})$. For further details on Mackey functors see [@dress:mac], [@pj:simp], [@pw:user].
The Mackey category {#ss:maccat}
-------------------
Let ${{\mathcal{M}}}(G)$ be the category the objects of which are subgroups of $G$ with morphisms given by $$\label{eq:mormf}
\operatorname{Hom}_{{{\mathcal{M}}}(G)}(U,V)=\operatorname{Hom}_G({\mathbb{Z}}[G/U],{\mathbb{Z}}[G/V]).$$ Then ${{\mathcal{M}}}(G)$ is a ${\mathbb{Z}}$-order category which is generated by the morphisms $$\begin{aligned}
\rho^U_g\colon&{\mathbb{Z}}[G/{}^gU]\longrightarrow{\mathbb{Z}}[G/U],&
\rho_g^U(xgUg^{-1})&=xgU;\label{eq:mormf11}\\
{\mathfrak{i}}_{V,U}\colon&{\mathbb{Z}}[G/V]\longrightarrow{\mathbb{Z}}[G/U],&
{\mathfrak{i}}_{V,U}(xV)&=xU;\label{eq:mormf12}\\
{\mathfrak{t}}_{U,V}\colon&{\mathbb{Z}}[G/U]\longrightarrow{\mathbb{Z}}[G/V],&
{\mathfrak{t}}_{U,V}(xU)&=\textstyle{\sum_{r\in{{\mathcal{R}}}}} xrV; \label{eq:mormf13}\end{aligned}$$ $g\in G$, $U,V\subseteq G$, $V\subseteq U$, where ${{\mathcal{R}}}\subseteq U$ is a set of right $V$-coset representatives. The assignment $$\label{eq:sigmamac}
\sigma(U)=U,\
\sigma(\rho_g^U)=\rho_{g^{-1}}^{{}^gU} ,\
\sigma({\mathfrak{i}}_{V,U})={\mathfrak{t}}_{U,V},\
\sigma({\mathfrak{t}}_{U,V})={\mathfrak{i}}_{V,U},$$ for $U,V\subseteq G$, $V\subseteq U$, $g\in G$, defines an antipode $\sigma\colon{{\mathcal{M}}}(G)\to{{\mathcal{M}}}(G)^{op}$. Let ${{\mathcal{M}}}_R(G)$ denote the $R$-order category obtained from ${{\mathcal{M}}}(G)$ by tensoring with $R$. Assigning to every cohomological $G$-Mackey functor ${\mathbf{X}}$ with values in ${{}_R{\mathbf{mod}}}$ the contravariant functor ${\tilde{{\mathbf{X}}}}$ given by $$\label{eq:maccon1}
{\tilde{{\mathbf{X}}}}(U)={\mathbf{X}}_U,\
{\tilde{{\mathbf{X}}}}(\rho^U_g)=c_{g,U}^{\mathbf{X}},\
{\tilde{{\mathbf{X}}}}({\mathfrak{i}}_{V,U})=i_{U,V}^{\mathbf{X}},\
{\tilde{{\mathbf{X}}}}({\mathfrak{t}}_{U,V})=t_{V,U}^{\mathbf{X}},$$ yields an identification between ${\mathfrak{cMF}}_G({{}_R{\mathbf{mod}}})$ and ${\mathfrak{F}}_R({{\mathcal{M}}}_R(G)^{\operatorname{op}},{{}_R{\mathbf{mod}}})$. Note that some authors prefer to identify the category of cohomological Mackey functors with the category of covariant functors of ${{\mathcal{M}}}_R(G)$. The existence of the antipode $\sigma\colon{{\mathcal{M}}}_G(R)\to{{\mathcal{M}}}_G(R)^{\operatorname{op}}$ showes that both approaches are equivalent.
The cohomological Mackey functors ${\boldsymbol{\Upsilon}}$ and ${\mathbf{T}}$ {#ss:ST}
------------------------------------------------------------------------------
Let $G$ be a finite group. There are two particular cohomological $G$-Mackey functors based on the $R$-module $R$. Let ${\boldsymbol{\Upsilon}}\in\operatorname{ob}({\mathfrak{cMF}}_G({{}_R{\mathbf{mod}}}))$ be given by $$\label{eq:defbuT1}
{\boldsymbol{\Upsilon}}_U=R,\ \ i_{U,V}^{{\boldsymbol{\Upsilon}}}=|U:V|\operatorname{id}_R,\ \ t_{V,U}^{{\boldsymbol{\Upsilon}}}=\operatorname{id}_R,\ \
c_{g,U}^{{\boldsymbol{\Upsilon}}}=\operatorname{id}_R,$$ and ${\mathbf{T}}\in\operatorname{ob}({\mathfrak{cMF}}_G({{}_R{\mathbf{mod}}}))$ be given by $$\label{eq:defboT}
{\mathbf{T}}_U=R,\ \ i_{U,V}^{\mathbf{T}}=\operatorname{id}_R,\ \ t_{V,U}^{\mathbf{T}}=|U:V|\operatorname{id}_R,\ \
c_{g,U}^{\mathbf{T}}=\operatorname{id}_R,$$ for $U,V\subseteq G$, $V\subseteq U$. Then ${\boldsymbol{\Upsilon}}$ and ${\mathbf{T}}$ are $R$-lattice functors, and one has ${\mathbf{T}}\simeq{\boldsymbol{\Upsilon}}^\ast$.
Let $R$ be an integral domain of characteristic $0$. For such a ring the subfunctor ${\boldsymbol{\Sigma}}\subseteq{\mathbf{T}}$ given by ${\boldsymbol{\Sigma}}_U= |U|\cdot{\mathbf{T}}_U$ is canonically isomorphic to ${\boldsymbol{\Upsilon}}$, i.e., there exists a canonical injective natural transformation $j\colon{\boldsymbol{\Upsilon}}\to{\mathbf{T}}$. We denote by ${\mathbf{B}}=\operatorname{coker}(j)$ the cokernel of this canonical map.
Let ${\mathbf{X}}\in\operatorname{ob}({\mathfrak{cMF}}_G({{}_R{\mathbf{mod}}}))$, and let $\phi\colon{\mathbf{T}}\to{\mathbf{X}}$ be a natural transformation. Then $\phi$ is uniquely determined by $\phi_G\colon{\mathbf{T}}_G\to{\mathbf{X}}_G$, and every such morphism defines a unique natural transformation $\phi\colon{\mathbf{T}}\to{\mathbf{X}}$. Hence one has a canonical isomorphism $$\label{eq:homT}
\operatorname{nat}_G({\mathbf{T}},{\mathbf{X}})\simeq{\mathbf{X}}_G.$$ In a similar fashion one shows that $$\label{eq:homuT}
\operatorname{nat}_G({\boldsymbol{\Upsilon}},{\mathbf{X}})\simeq{\mathbf{X}}_{\{1\}}^G.$$
Invariants and coinvariants {#ss:invcoinv}
---------------------------
There are two standard procedures which turn a left ${R[G]}$-module $M$ into a cohomological $G$-Mackey functor with values in the category of $R$-modules. By ${\mathbf{h}}^0(M)$ we denote what is called the [*fixed-point-functor*]{} in [@pj:simp]. In more detail, one has ${\mathbf{h}}^0(M)_U=M^U$, for $U,V\subseteq G$, $V\subseteq U$, $i_{U,V}^{{\mathbf{h}}^0(M)}\colon M^U\to M^V$ is the canonical map, $t_{V,U}^{{\mathbf{h}}^0(M)}\colon M^V\to M^U$ is given by the transfer, i.e., if ${{\mathcal{R}}}\subseteq U$ denotes a system of coset representative of $U/V$ then $t_{V,U}^{{\mathbf{h}}^0(M)}$ is given by multiplication with $\sum_{r\in{{\mathcal{R}}}}r$, and $c_{g,U}^{{\mathbf{h}}^0(M)}\colon M^U\to M^{{}^gU}$ is left-multiplication by $g\in G$.
By ${\mathbf{h}}_0(M)$ we denote the cohomological $G$-Mackey functors of [*coinvariants*]{}. Thus ${\mathbf{h}}_0(M)_U=M/\omega_{R[U]}M$, where $\omega_{R[U]}=\operatorname{ker}(R[U]\to R)$ is the augmentation ideal in $R[U]$, and for $U,V\subseteq G$, $V\subseteq U$, $t_{V,U}^{{\mathbf{h}}_0(M)}\colon M_V\to M_U$ is the canonical map, the map $i_{U,V}^{{\mathbf{h}}_0(M)}\colon M_U\to M_V$ is induced by multiplication with $\sum_{r\in{{\mathcal{R}}}} r^{-1}$, and the map $c_{g,U}^{{\mathbf{h}}^0(M)}\colon M_U\to M_{{}^gU}$ is induced by multiplication with $g\in G$. E.g., one has canonical isomorphisms of cohomological $G$-Mackey functors ${\boldsymbol{\Upsilon}}\simeq{\mathbf{h}}_0(R)$ and ${\mathbf{T}}\simeq{\mathbf{h}}^0(R)$, where $R$ denotes the trivial left ${R[G]}$-module.
Standard projective cohomological Mackey functors {#ss:projmac}
-------------------------------------------------
By §\[ss:proj\], one knows that for $W\subseteq G$ the functor $$\label{eq:standproj}
{\mathbf{P}}^W=\operatorname{Hom}_G(R[G/{\underline{\phantom{x}}}],R[G/W])\in\operatorname{ob}({\mathfrak{cMF}}_G({{}_R{\mathbf{mod}}})),\ $$ where ${\mathbf{P}}^W_U=\operatorname{Hom}_G(R[G/U],R[G/W])=R[G/W]^U$, is projective in ${\mathfrak{cMF}}_G({{}_R{\mathbf{mod}}})$. These functors can be described as follows.
\[fact:projcohmac\] Let $G$ be a finite group, and let $W\subseteq G$. Then one has canonical isomorphisms $$\label{eq:standproj2}
{\mathbf{P}}^W\simeq{\mathbf{h}}^0(R[G/W])\simeq{\mathbf{h}}^0({\mathbf{P}}^W_{\{1\}})\simeq\operatorname{\mathbf{ind}}_W^G({\mathbf{T}}^W),$$ where $\operatorname{\mathbf{ind}}_W^G({\underline{\phantom{x}}})$ denotes the induction functor in the category of Mackey functors , and ${\mathbf{T}}^W\in\operatorname{ob}({\mathfrak{cMF}}_W({{}_R{\mathbf{mod}}}))$ is the cohomological $W$-Mackey functor described in subsection \[ss:ST\].
Both descriptions of the standard projective cohomological $G$-Mackey functors will be useful for our purpose. Note that one has canonical isomorphisms ${\mathbf{P}}^G\simeq{\mathbf{T}}$, i.e., ${\mathbf{T}}$ is projective. We also put ${\mathbf{Q}}={\mathbf{P}}^{\{1\}}={\mathbf{h}}^0({R[G]})$.
\[rem:minproj\] Let $G$ be a finite $p$-group, and let $R$ be a discrete valuation domain of characteristic $0$ with maximal ideal $pR$. For $W\subseteq G$ there exists a simple cohomological $G$-Mackey functor ${\mathbf{S}}^W$ with values in the category of $R$-modules given by $$\label{eq:simpmacfun}
{\mathbf{S}}^W_U=
\begin{cases}
{\mathbb{F}}&\ \text{for $U= {}^gW$;}\\
0&\ \text{for $U\not= {}^gW$.}
\end{cases}$$ In particular, for $U,V\subseteq G$, $V\subsetneq U$, one has $i_{U,V}^{{\mathbf{S}}^W}=0$ and $t_{V,U}^{{\mathbf{S}}^W}=0$. Moreover, any simple cohomological $G$-Mackey functor is isomorphic to some ${\mathbf{S}}^W$, $W\subseteq G$ (cf. [@pj:simp]). The Nakayama relations and show that for $V\subseteq G$ and $V\not={}^gW$ one has $$\label{eq:simp1}
\operatorname{nat}_G({\mathbf{P}}^V,{\mathbf{S}}^W)=\operatorname{nat}_V({\mathbf{T}},\operatorname{res}^G_V({\mathbf{S}}^W))\simeq {\mathbf{S}}^W_V=0.$$ On the other hand for $V={}^gW$ one has $$\label{eq:simp2}
\operatorname{nat}_G({\mathbf{P}}^V,{\mathbf{S}}^W)=\operatorname{nat}_V({\mathbf{T}},\operatorname{res}^G_V({\mathbf{S}}^W))\simeq {\mathbf{S}}^W_V={\mathbb{F}}.$$ Hence ${\mathbf{P}}^W$ is the (minimal) projective cover of ${\mathbf{S}}^W$ for all $W\subseteq G$.
Standard relative injective cohomological Mackey functors {#ss:standinj}
---------------------------------------------------------
Let $G$ be a finite group, and $W\subseteq G$. The functor $\operatorname{\mathbf{ind}}_W^G({\underline{\phantom{x}}})$ commutes with the functor ${\underline{\phantom{x}}}^\ast$ on $R$-lattice functors, i.e., one has a natural isomorphism $$\label{eq:natisoast}
\operatorname{\mathbf{ind}}_W^G({\underline{\phantom{x}}}^\ast)\simeq\operatorname{\mathbf{ind}}_W^G({\underline{\phantom{x}}})^\ast\colon
{\mathfrak{cMF}}_W({{}_R{\mathbf{lat}}})\longrightarrow{\mathfrak{cMF}}_G({{}_R{\mathbf{lat}}})^{\operatorname{op}}.$$ Thus ${\mathbf{J}}^W=({\mathbf{P}}^W)^\ast\simeq\operatorname{\mathbf{ind}}_W^G({\boldsymbol{\Upsilon}})\in\operatorname{ob}({\mathfrak{cMF}}_G({{}_R{\mathbf{lat}}}))$.
The Yoneda dual {#ss:yonmac}
---------------
Let ${\mathbf{X}}\in\operatorname{ob}({\mathfrak{cMF}}_G({{}_R{\mathbf{lat}}}))$. As ${\mathbf{P}}^W$, $W\subseteq G$, takes values in the category of $R$-lattices, the Nakayama relations and yield canonical isomorphisms $$\label{eq:yondumac}
\begin{aligned}
\operatorname{nat}_G({\mathbf{X}},{\mathbf{P}}^W)&\simeq\operatorname{nat}_G({\mathbf{J}}^W,{\mathbf{X}}^\ast)
\simeq\operatorname{nat}_G(\operatorname{\mathbf{ind}}_W^G({\boldsymbol{\Upsilon}}),{\mathbf{X}}^\ast)\\
&\simeq\operatorname{nat}_W({\boldsymbol{\Upsilon}},\operatorname{res}^G_W({\mathbf{X}}^\ast))
\simeq({\mathbf{X}}_{\{1\}}^\ast)^W
\end{aligned}$$ From this one concludes the following property (cf. Fact \[fact:projcohmac\]).
\[fact:yondumac\] Let ${\mathbf{X}}\in\operatorname{ob}({\mathfrak{cMF}}_G({{}_R{\mathbf{lat}}}))$. Then ${\mathbf{X}}^\circledast\simeq{\mathbf{h}}^0({\mathbf{X}}^\ast_{\{1\}})$.
Section cohomology of cohomological Mackey functors {#s:secmac}
===================================================
If not stated otherwise $R$ will denote a commutative ring with unit $1_R\in R$. Let $G$ be a finite group, and let ${\mathbf{X}}\in\operatorname{ob}({\mathfrak{cMF}}_G({{}_R{\mathbf{mod}}}))$. For $U,V\subseteq G$, $V\triangleleft U$, one defines the [*section cohomology groups*]{} of ${\mathbf{X}}$ by $$\label{eq:seccoh}
\begin{aligned}
{\mathbf{k}}^0(U/V,{\mathbf{X}})&=\operatorname{ker}(i_{U,V}^{\mathbf{X}}),&{\mathbf{k}}^1(U/V,{\mathbf{X}})&={\mathbf{X}}_V^U,\\
{\mathbf{c}}_0(U/V,{\mathbf{X}})&=\operatorname{coker}(t_{V,U}^{\mathbf{X}}),&{\mathbf{c}}_1(U/V,{\mathbf{X}})&=\operatorname{ker}(t_{V,U}^{\mathbf{X}})/\omega_{U/V}{\mathbf{X}}_V.
\end{aligned}$$ The following properties were established in [@tw:stand §2.4].
\[prop:sec\] Let $G$ be a finite group, let $U$ and $V$ be subgroups of $G$ such that $V$ is normal in $U$, and let ${\mathbf{X}}$ be a cohomological $G$-Mackey functor with values in ${{}_R{\mathbf{mod}}}$.
- The canonical maps yield an exact sequence of $R$-modules $$\label{eq:6term}
\xymatrix{
0\ar[r]&{\mathbf{c}}_1(U/V,{\mathbf{X}})\ar[r]&
{\hat{H}}^{-1}(U/V,{\mathbf{X}}_V)\ar[r]&
{\mathbf{k}}^0(U/V,{\mathbf{X}})\ar[d]\\
0&{\mathbf{k}}^1(U/V,{\mathbf{X}})\ar[l]&{\hat{H}}^{0}(U/V,{\mathbf{X}}_V)\ar[l]&{\mathbf{c}}_0(U/V,{\mathbf{X}})\ar[l]\\
}$$ where ${\hat{H}}^\bullet(U/V,{\underline{\phantom{x}}})$ denotes the Tate cohomology groups.
- Let $\xymatrix{0\ar[r]&{\mathbf{X}}\ar[r]^\phi&{\mathbf{Y}}\ar[r]^\psi&{\mathbf{Z}}\ar[r]&0}$ be a short exact sequence of cohomological $G$-Mackey functors with values in ${{}_R{\mathbf{mod}}}$. Then one has exact sequences $$\label{eq:longk}
\xymatrix@R=3pt{
0\ar[r]&{\mathbf{k}}^0(U/V,{\mathbf{X}})\ar[r]^{{\mathbf{k}}^0(\phi)}&{\mathbf{k}}^0(U/V,{\mathbf{Y}})\ar[r]^{{\mathbf{k}}^0(\psi)}&
{\mathbf{k}}^0(U/V,{\mathbf{Z}})\ar[r]&\ldots\\
\ldots\ar[r]&{\mathbf{k}}^1(U/V,{\mathbf{X}})\ar[r]^{{\mathbf{k}}^1(\phi)}&{\mathbf{k}}^1(U/V,{\mathbf{Y}})\ar[r]^{{\mathbf{k}}^1(\psi)}&
{\mathbf{k}}^1(U/V,{\mathbf{Z}})
}$$ and $$\label{eq:longc}
\xymatrix@R=3pt{
&{\mathbf{c}}_1(U/V,{\mathbf{X}})\ar[r]^{{\mathbf{c}}_1(\phi)}&{\mathbf{c}}_1(U/V,{\mathbf{Y}})\ar[r]^{{\mathbf{c}}_1(\psi)}&
{\mathbf{c}}_1(U/V,{\mathbf{Z}})\ar[r]&\ldots\\
\ldots\ar[r]&{\mathbf{c}}_0(U/V,{\mathbf{X}})\ar[r]^{{\mathbf{c}}_0(\phi)}&{\mathbf{c}}_0(U/V,{\mathbf{Y}})\ar[r]^{{\mathbf{c}}_0(\psi)}&
{\mathbf{c}}_0(U/V,{\mathbf{Z}})\ar[r]&0.
}$$
Section cohomology for cyclic subgroups {#ss:cycsec}
---------------------------------------
Let $W$ be a non-trivial cyclic subgroup of the finite group $G$ generated by the element $w\in W$. Taking coinvariants of the chain complex of $R[W]$-modules $R[W]\overset{w-1}{\longrightarrow} R[W]$ yields an exact sequence $$\label{eq:seccyc2}
\xymatrix@C=1.3truecm{
0\ar[r]&{\mathbf{T}}^W\ar[r]^{{\mathbf{P}}({\mathfrak{t}}_{W,\{1\}})}&{\mathbf{Q}}^W\ar[r]^{{\mathbf{h}}^0(w-1)}&{\mathbf{Q}}^W\ar[r]&{\boldsymbol{\Upsilon}}^W\ar[r]&0},$$ (cf. §\[ss:ST\], §\[ss:projmac\]). If $R$ is an integral domain of characteristic $0$, one has additionally a short exact sequence $$\label{eq:seccyc1}
\xymatrix{
0\ar[r]&{\boldsymbol{\Upsilon}}^W\ar[r]&{\mathbf{T}}^W\ar[r]&{\mathbf{B}}^W\ar[r]&0}.$$ Splicing together the short exact sequences and yields a projective resolution of the cohomological $W$-Mackey functor ${\mathbf{B}}^W$. Using this projective resolution, Fact \[fact:Pfirst\], and one concludes the following.
\[fact:seccyc\] Let $R$ be a integral domain of characteristic $0$, and let $W\subseteq G$ be cyclic subgroup of the finite group $G$. Then for $k\in\{0,1\}$ one has canonical isomorphisms $$\label{eq:seccyc3}
\begin{aligned}
\operatorname{ext}_G^k(\operatorname{\mathbf{ind}}_W^G({\mathbf{B}}^W),{\mathbf{X}})&\simeq {\mathbf{k}}^k(W/\{1\},{\mathbf{X}}),\\
\operatorname{ext}_G^{3-k}(\operatorname{\mathbf{ind}}_W^G({\mathbf{B}}^W),{\mathbf{X}})&\simeq {\mathbf{c}}_k(W/\{1\},{\mathbf{X}}).
\end{aligned}$$
Note that Fact \[fact:seccyc\] shows also that for a cyclic subgroup $W\subseteq G$ one can consider the groups ${\mathbf{k}}^\bullet(W/\{1\},{\underline{\phantom{x}}})$, ${\mathbf{c}}_{3-\bullet}(W/\{1\},{\underline{\phantom{x}}})$ together with the respective connecting homomorphisms as a cohomological functor (cf. [@mcl:hom §XII.8]).
Cohomological Mackey functors of type $H^0$ and $H_0$ {#ss:h0}
-----------------------------------------------------
Let $G$ be a finite group, and let ${\mathbf{X}}\in\operatorname{ob}({\mathfrak{cMF}}_G({{}_R{\mathbf{mod}}}))$. Then ${\mathbf{X}}$ will be called [*$i$-injective*]{}, if for all $U,V\subseteq G$, $V\subseteq U$, the map $i^{\mathbf{X}}_{U,V}$ is injective; i.e., ${\mathbf{X}}$ is $i$-injective if, and only if, for all $U,V\subseteq G$, $V\triangleleft U$, one has ${\mathbf{k}}^0(U/V,{\mathbf{X}})=0$. Moreover, ${\mathbf{X}}$ will be called [*of type $H^0$*]{} (or to satisfy [*Galois descent*]{}), if ${\mathbf{X}}$ is $i$-injective and ${\mathbf{k}}^1(U/V,{\mathbf{X}})=0$ for all $U,V\subseteq G$, $V\triangleleft U$, i.e., ${\mathbf{X}}$ is of type $H^0$ if, and only if, one has a canonical isomorphism (induced by $i$) $$\label{eq:hHo0}
{\mathbf{X}}\simeq{\mathbf{h}}^0({\mathbf{X}}_{\{1\}}).$$ The cohomological $G$-Mackey functor ${\mathbf{X}}$ will be called to be [*Hilbert$^{90}$*]{}, if it is of type $H^0$ and $H^1(U,{\mathbf{X}}_{\{1\}})=0$ for every subgroup $U$ of $G$. One has the following property.
\[prop:H90\] Let $G$ be a finite group, and let ${\mathbf{X}}\in\operatorname{ob}({\mathfrak{cMF}}_G({{}_R{\mathbf{mod}}}))$ be Hilbert$^{90}$. Then for all $U,V\subset G$, $V\triangleleft U$, one has $H^1(U/V,{\mathbf{X}}_V)=0$.
By the 5-term exact sequence, inflation $H^1(U/V,{\mathbf{X}}_{\{1\}}^V)\to H^1(U,{\mathbf{X}}_{\{1\}})$ is injective. Hence $H^1(U/V,{\mathbf{X}}_{\{1\}}^V)=0$. As ${\mathbf{X}}$ is of type $H^0$, ${\mathbf{X}}_{\{1\}}^V$ and ${\mathbf{X}}_V$ are isomorphic $R[U/V]$-modules. This yields the claim.
In a similar fashion one calls ${\mathbf{X}}$ to be [*$t$-surjective*]{}, if for all $U,V\subseteq G$, $V\subseteq U$, the map $t^{\mathbf{X}}_{V,U}$ is surjective; i.e., ${\mathbf{X}}$ is $t$-surjective if, and only if, for all $U,V\subseteq G$, $V\triangleleft U$, one has ${\mathbf{c}}_0(U/V,{\mathbf{X}})=0$. The cohomological $G$-Mackey functor ${\mathbf{X}}$ will be called [*of type $H_0$*]{} (or to satisfy [*Galois co-descent*]{}), if ${\mathbf{X}}$ is $t$-surjective and ${\mathbf{c}}_1(U/V,{\mathbf{X}})=0$ for all $U,V\subseteq G$, $V\triangleleft U$, i.e., ${\mathbf{X}}$ is of type $H_0$ if, and only if, one has a canonical isomorphism (induced by $t$) $$\label{eq:hHu0}
{\mathbf{h}}_0({\mathbf{X}}_{\{1\}})\simeq{\mathbf{X}}.$$ Furthermore, ${\mathbf{X}}$ will be called to be [*co-Hilbert$^{90}$*]{}, if it is of type $H_0$ and for every subgroup $U$ of $G$ one has ${\hat{H}}^{-1}(U,{\mathbf{X}}_{\{1\}})=0$.
\[rem:projH0\] Every projective cohomological $G$-Mackey functor ${\mathbf{P}}$ with values in the category of $R$-modules is a direct summand of a coproduct of standard projective cohomological $G$-Mackey functors. Hence by Fact \[fact:projcohmac\] every projective cohomological $G$-Mackey functor ${\mathbf{P}}$ is of type $H^0$. However, if $R$ is an integral domain of characteristic $0$, the Nakayama relations imply that $H^1(G,R[\Omega])=0$ for any $G$-set $\Omega$. In particular, ${\mathbf{P}}$ is even Hilbert$^{90}$.
The periodicity of Tate cohomology for finite cyclic groups has the following consequence.
\[prop:H90c\] Let $G$ be a finite group, and let ${\mathbf{X}}\in\operatorname{ob}({\mathfrak{cMF}}_G({{}_R{\mathbf{mod}}}))$ be Hilbert$^{90}$. Let $U,V\subseteq G$, $V\triangleleft U$, be such that $U/V$ is cyclic. Then ${\mathbf{c}}_1(U/V,{\mathbf{X}})=0$.
By Proposition \[prop:H90\] and the periodicity of Tate cohomology (of period $2$), one has ${\hat{H}}^{-1}(U/V,{\mathbf{X}}_V)\simeq H^1(U/V,{\mathbf{X}}_V)=0$. Hence yields the claim.
Tate duality {#ss:tatedual}
------------
Let $R$ be a principal ideal domain of characteristic $0$, and let $K=\operatorname{quot}(R)$ denote its quotient field. Then ${\mathbb{I}}=K/R$ is an injective $R$-module[^4] Let $G$ be a finite group, and let $M$ be a left ${R[G]}$-lattice. Then one has an exact sequence of left ${R[G]}$-modules $$\label{eq:tatedual1}
\xymatrix{
0\ar[r]&M^\ast\ar[r]&\operatorname{Hom}_R(M,K)\ar[r]&\operatorname{Hom}_R(M,{\mathbb{I}})\ar[r]&0,
}.$$ where $M^\ast=\operatorname{Hom}_R(M,R)$. The following property is also known as [*Tate duality*]{}.
\[prop:tatedual\] Let $R$ be a principal ideal domain of characteristic $0$, let $K=\operatorname{quot}(R)$ be the quotient field of $R$, and let ${\mathbb{I}}=K/R$. Let $G$ be a finite group, and let $M$ be an ${R[G]}$-lattice. Then for all $k\in{\mathbb{Z}}$ one has natural isomorphisms $$\label{eq:tatedual2}
{\hat{H}}^k(G,M^\ast)\simeq\operatorname{Hom}_R({\hat{H}}^{-k}(G,M),{\mathbb{I}}).$$
It is well known that one has natural isomorphisms $$\label{eq:tatedual3}
{\hat{H}}^{k-1}(G,\operatorname{Hom}_R(M,{\mathbb{I}}))\simeq\operatorname{Hom}_R({\hat{H}}^{-k}(G,M),{\mathbb{I}})$$ for all $k\in{\mathbb{Z}}$ (cf. [@brown:coh p. 148, Ex. VI.7.4]). Moreover, as ${\hat{H}}^k(G,\operatorname{Hom}_R(M,K))=0$ for all $k\in{\mathbb{Z}}$, one has also natural isomorphisms $$\label{eq:tatedual4}
{\hat{H}}^{k-1}(G,\operatorname{Hom}_R(M,{\mathbb{I}}))\simeq {\hat{H}}^k(G,M^\ast).$$ This yields the claim.
Section cohomology of $R$-lattice functors {#ss:seclat}
------------------------------------------
Let $R$ be an integral domain of characteristic $0$, let $G$ be a finite group, and let ${\mathbf{X}}\in\operatorname{ob}({\mathfrak{cMF}}_G({{}_R{\mathbf{lat}}}))$ be an $R$-lattice functor. For $U,V\subseteq G$, $V\subseteq U$, the axiom (cMF$_7$) (cf. §\[ss:Gmac1\]) implies that ${\mathbf{X}}$ is $i$-injective. Hence by one has an isomorphism $$\label{eq:isolat}
{\mathbf{c}}_1(U/V,{\mathbf{X}})\simeq{\hat{H}}^{-1}(U/V,{\mathbf{X}}_V)$$ and a short exact sequence $$\label{eq:23term}
\xymatrix{
0\ar[r]&{\mathbf{c}}_0(U/V,{\mathbf{X}})\ar[r]&{\hat{H}}^0(U/V,{\mathbf{X}}_V)\ar[r]&{\mathbf{k}}^1(U/V,{\mathbf{X}})\ar[r]&0.
}$$ Let $R$ be a principal ideal domain, and let $\phi\colon A\to B$ be a homomorphism of $R$-lattices. Then $\phi$ is split injective if, and only if, $\phi^\ast\colon B^\ast\to A^\ast$ is surjective. From this fact one concludes the following properties.
\[prop:latsec\] Let $R$ be a principal ideal domain of characteristic $0$, let $G$ be a finite group, and let ${\mathbf{X}}\in\operatorname{ob}({\mathfrak{cMF}}_G({{}_R{\mathbf{lat}}}))$.
- ${\mathbf{X}}$ is of type $H^0$ if, and only if, ${\mathbf{X}}^\ast$ is $t$-surjective.
- The following are equivalent:
- ${\mathbf{X}}$ is Hilbert$^{90}$;
- ${\mathbf{X}}^\ast$ is of type $H_0$;
- ${\mathbf{X}}^\ast$ is co-Hilbert$^{90}$.
Let $U,V\subseteq G$, $V\subseteq U$.
\(a) The map $i_{U,V}\colon {\mathbf{X}}_U\to{\mathbf{X}}_V$ is split-injective if, and only if, ${\mathbf{k}}^1(U/V,{\mathbf{X}})=0$. Hence the previously mentioned remark yields the claim.
\(b) Suppose that ${\mathbf{X}}$ is Hilbert$^{90}$. Then $H^1(U/V,{\mathbf{X}}_V)=0$ for all $U,V\subseteq G$, $V\triangleleft U$ (cf. Prop. \[prop:H90\]). By Tate duality (cf. Prop. \[prop:tatedual\]), one has $$\label{eq:tatedual5}
{\hat{H}}^{-1}(U/V,{\mathbf{X}}_V^\ast)\simeq\operatorname{Hom}_R(H^1(U/V,{\mathbf{X}}_V),{\mathbb{I}}_R)=0$$ whenever $V$ is normal in $U$. Hence ${\mathbf{c}}_1(U/V,{\mathbf{X}}^\ast)=0$ for all $U,V\subseteq G$, $V\triangleleft U$ (cf. ). Thus by (a), ${\mathbf{X}}^\ast$ is of type $H_0$. If ${\mathbf{X}}^\ast$ is of type $H_0$, implies that ${\hat{H}}^{-1}(U/V,{\mathbf{X}}_V^\ast)=0$ for all $U,V\subseteq G$, $V\triangleleft U$, i.e., ${\mathbf{X}}^\ast$ is co-Hilbert$^{90}$. If ${\mathbf{X}}^\ast$ is co-Hilbert$^{90}$, then (a) implies that ${\mathbf{X}}$ is of type $H^0$. By Tate duality (cf. Prop. \[prop:tatedual\]), one has $$\label{eq:tatedual6}
H^{1}(U/V,{\mathbf{X}}_V)\simeq\operatorname{Hom}_R({\hat{H}}^{-1}(U/V,{\mathbf{X}}_V^\ast),{\mathbb{I}}_R)=0$$ This yields the claim.
Finite cyclic groups {#ss:fincyc}
--------------------
If $G$ is a finite group and $R$ is any commutative ring with unit $1$, one has ${\mathbf{P}}^{\{1\}}\simeq ({\mathbf{P}}^{\{1\}})^\ast$, i.e., ${\mathbf{P}}^{\{1\}}$ is projective and relative injective. If $G$ is a finite cyclic group, and $W\subseteq G$ is a non-trivial subgroup of $G$, applying $\operatorname{\mathbf{ind}}_W^G({\underline{\phantom{x}}})$ to the exact sequence yields an exact sequence $$\label{eq:exact2}
\xymatrix@C=1.6truecm{
0\ar[r]&{\mathbf{P}}^W\ar[r]^{{\mathbf{P}}({\mathfrak{t}}_{W,\{1\}})}
&{\mathbf{P}}^{\{1\}}\ar[r]^{\operatorname{\mathbf{ind}}^G_W(w-1)}&{\mathbf{P}}^{\{1\}}\ar[r]&{\mathbf{J}}^W\ar[r]&0,
}$$ where $w\in W$ is a generating element of $W$. In particular, $$\label{eq:projinjdim}
\operatorname{proj.dim}({\mathbf{J}}^W)\leq 2,\qquad W\subseteq G,\ W\not=\{1\},$$ and $\operatorname{proj.dim}({\mathbf{J}}^{\{1\}})=0$. Thus one has (cf. §\[ss:gor\]).
\[prop:gorcm\] Let $R$ be a Dedekind domain, and let $G$ be a finite cyclic group. Then ${{\mathcal{M}}}_R(G)$ is $2$-Gorenstein.
For $\operatorname{ext}_G^k({\mathbf{J}}^W,{\mathbf{X}})={{\mathcal{R}}}^k\!\operatorname{nat}_G({\mathbf{J}}^W,{\mathbf{X}})$, ${\mathbf{X}}\in\operatorname{ob}({\mathfrak{cMF}}_G({{}_R{\mathbf{mod}}}))$, $W\subseteq G$, one obtains the following.
\[prop:extJ\] Let $R$ be a Dedekind domain, let $G$ be a finite cyclic group, and let ${\mathbf{X}}\in\operatorname{ob}({\mathfrak{cMF}}_G({{}_R{\mathbf{mod}}}))$. Then for $W\subseteq G$ one has
- $\operatorname{ext}^0_G({\mathbf{J}}^W,{\mathbf{X}})=\operatorname{nat}_G({\mathbf{J}}^W,{\mathbf{X}})\simeq{\mathbf{X}}_{\{1\}}^W$;
- $\operatorname{ext}^1_G({\mathbf{J}}^W,{\mathbf{X}})\simeq{\mathbf{c}}_1(W/\{1\},{\mathbf{X}})$;
- $\operatorname{ext}^2_G({\mathbf{J}}^W,{\mathbf{X}})\simeq {\mathbf{c}}_0(W/\{1\},{\mathbf{X}})$;
and $\operatorname{ext}^k_G({\mathbf{J}}^W,{\mathbf{X}})=0$ for $k\geq 3$.
For $W=\{1\}$, one has $\operatorname{ext}^1_G({\mathbf{J}}^W,{\mathbf{X}})=\operatorname{ext}^2_G({\mathbf{J}}^W,{\mathbf{X}})=0$, and $\operatorname{ext}^0_G({\mathbf{J}}^W,{\mathbf{X}})=\operatorname{nat}_G({\mathbf{J}}^W,{\mathbf{X}})={\mathbf{X}}_{\{1\}}$ (cf. Fact \[fact:Pfirst\]). Hence the claim holds in this case, and we may assume that $W\not=\{1\}$. From the Nakayama relations and one concludes that $\operatorname{ext}^k_G({\mathbf{J}}^W,{\mathbf{X}})$ coincides with the $k^{th}$-cohomology of the cochain complex $$\label{eq:exact3}
\xymatrix{
&0\ar[r]&{\mathbf{X}}_{\{1\}}\ar[r]^{w-1}&{\mathbf{X}}_{\{1\}}\ar[r]^{t^{\mathbf{X}}_{\{1\},W}}&{\mathbf{X}}_W\ar[r]&0}$$ concentrated in degrees $0$, $1$ and $2$. This yields the claim in case that $W\not=\{1\}$.
From Proposition \[prop:extJ\] one obtains the following description of the higher derived functors of the Yoneda dual.
\[prop:Ryon\] Let $R$ be a principal ideal domain of characteristic $0$, let $G$ be a finite cyclic group, and let ${\mathbf{X}}\in\operatorname{ob}({\mathfrak{cMF}}_G({{}_R{\mathbf{lat}}}))$ be a cohomological $G$-Mackey functor with values in the category of $R$-lattices. Then the following are equivalent.
- ${\mathbf{X}}$ is Hilbert$^{90}$;
- ${\mathbf{X}}^\ast$ is co-Hilbert$^{90}$;
- ${\mathbf{X}}$ is $\circledast$-acyclic;
- ${\mathbf{X}}^\circledast$ is Hilbert$^{90}$;
- ${\mathbf{X}}^\circledast$ is $\circledast$-acyclic.
In particular, $({{\mathcal{M}}}_R(G),\sigma)$ is a $\circledast$-symmetric $R^\circledast$-order category.
By Proposition \[prop:latsec\](b), (i) and (ii) are equivalent. For $W\subseteq G$ one has $$\label{eq:acyc1}
{{\mathcal{R}}}^k({\mathbf{X}})^\circledast_W=\operatorname{ext}_G^k({\mathbf{X}},{\mathbf{P}}^W)\simeq\operatorname{ext}_G^k({\mathbf{J}}^W,{\mathbf{X}}^\ast).$$ Hence Proposition \[prop:extJ\] implies that (ii) and (iii) are equivalent, and thus also (iv) and (v) are equivalent. By Fact \[fact:yondumac\], ${\mathbf{X}}^\circledast\simeq{\mathbf{h}}^0({\mathbf{X}}^\ast_{\{1\}})$. Let $W\subseteq G$. The periodicity of Tate cohomology (or period 2) and Tate duality (cf. ) imply that $$\label{eq:tatedual7}
H^1(W,{\mathbf{X}}_{\{1\}}^\ast)\simeq
{\hat{H}}^{-1}(W,{\mathbf{X}}_{\{1\}}^\ast)\simeq\operatorname{Hom}_R(H^1(W,{\mathbf{X}}_{\{1\}}),{\mathbb{I}}_R).$$ Hence (i) implies (iv). Replacing ${\mathbf{X}}$ by ${\mathbf{X}}^\circledast$ shows that (iv) implies (i). This yields the claim.
The following property will allow us to analyze the projective dimensions of cohomological Mackey functors for finite cyclic groups.
\[prop:cycint\] Let $R$ be a Dedekind domain of characteristic $0$, let $G$ be a finite cyclic group, and let $\phi\colon{\mathbf{P}}\to{\mathbf{X}}$ be a surjective natural transformation in ${\mathfrak{cMF}}_G({{}_R{\mathbf{mod}}})$, where ${\mathbf{P}}$ is a projective $R$-lattice functor. Then
- $\operatorname{ker}(\phi)$ is an $R$-lattice functor;
- if ${\mathbf{X}}$ is $i$-injective, $\operatorname{ker}(\phi)$ is of type $H^0$;
- if ${\mathbf{X}}$ is of type $H^0$, $\operatorname{ker}(\phi)$ is Hilbert$^{90}$.
For (a) there is nothing to prove. Put ${\mathbf{K}}=\operatorname{ker}(\phi)$, and let $U,V\subseteq G$, $V\subseteq U$. By Remark \[rem:projH0\] and Fact \[fact:seccyc\], one has an exact sequence $$\label{eq:seccyc4}
\xymatrix{
{\mathbf{c}}_1(U/V,{\mathbf{X}})\ar[d]&0\ar[l]&\\
{\mathbf{c}}_0(U/V,{\mathbf{K}})\ar[r]&
{\mathbf{c}}_0(U/V,{\mathbf{P}})\ar[r]&
{\mathbf{c}}_0(U/V,{\mathbf{X}})\ar[r]&0
}$$ and isomorphisms $$\begin{aligned}
{\mathbf{k}}^0(U/V,{\mathbf{X}})&\simeq{\mathbf{k}}^1(U/V,{\mathbf{K}}),\label{eq:seccyc5}\\
{\mathbf{k}}^1(U/V,{\mathbf{X}})&\simeq{\mathbf{c}}_1(U/V,{\mathbf{K}}).\label{eq:seccyc6}\end{aligned}$$ Hence implies (b). If ${\mathbf{X}}$ is of type $H^0$, yields that ${\mathbf{c}}_1(U/V,{\mathbf{K}})=0$. Thus by and the periodicity of Tate cohomology (of period $2$), one has $$\label{eq:seccyc7}
H^1(U/V,{\mathbf{K}}_V)\simeq {\hat{H}}^{-1}(U/V,{\mathbf{K}}_V)\simeq{\mathbf{c}}_1(U/V,{\mathbf{K}})=0.$$ This yields the claim.
The following property will turn out to be useful for our purpose.
\[prop:torhil\] Let $R$ be an integral domain of characteristic $0$, and let $p\in R$. Assume further that $G$ is a finite cyclic group, and that ${\mathbf{X}}\in\operatorname{ob}({\mathfrak{cMF}}_G({{}_R{\mathbf{lat}}}))$ is an $R$-lattice functor with the Hilbert$^{90}$ property. Then ${\mathbf{Y}}={\mathbf{X}}/p{\mathbf{X}}$ is of type $H^0$.
We may suppose that $p\not=0$. Then $p{\mathbf{X}}\simeq{\mathbf{X}}$. Let $U,V\in G$, $V\subseteq U$. By Fact \[fact:seccyc\], one has a long exact sequence $$\label{eq:seccyc8}
\xymatrix{
{\mathbf{k}}^0(U/V,{\mathbf{Y}})\ar[d]&{\mathbf{k}}^0(U/V,{\mathbf{X}})\ar[l]&{\mathbf{k}}^0(U/V,{\mathbf{X}})\ar[l]_{p}&0\ar[l]\\
{\mathbf{k}}^1(U/V,{\mathbf{X}})\ar[r]^p&
{\mathbf{k}}^1(U/V,{\mathbf{X}})\ar[r]&
{\mathbf{k}}^1(U/V,{\mathbf{Y}})\ar[d]&\\
{\mathbf{c}}_1(U/V,{\mathbf{Y}})\ar[d]&{\mathbf{c}}_1(U/V,{\mathbf{X}})\ar[l]&{\mathbf{c}}_1(U/V,{\mathbf{X}})\ar[l]_{p}\\
{\mathbf{c}}_0(U/V,{\mathbf{X}})\ar[r]^p&
{\mathbf{c}}_0(U/V,{\mathbf{X}})\ar[r]&
{\mathbf{c}}_0(U/V,{\mathbf{Y}})\ar[r]&0
}$$ As ${\mathbf{k}}^0(U/V,{\mathbf{X}})={\mathbf{k}}^1(U/V,{\mathbf{X}})={\mathbf{c}}_1(U/V,{\mathbf{X}})=0$, one concludes that ${\mathbf{k}}^0(U/V,{\mathbf{Y}})={\mathbf{k}}^1(U/V,{\mathbf{Y}})=0$. This yields the claim.
Injectivity criteria {#ss:injcrit}
--------------------
For a finite $p$-group $G$ there are useful criteria ensuring that a homomorphism $\phi\colon{\mathbf{X}}\to{\mathbf{Y}}$ of cohomological $G$-Mackey functors is injective. These criteria are based on the following fact.
\[fact:soctriv\] Let $G$ be a finite $p$-group, let ${\mathbb{F}}$ be a field of characteristic $p$, and let $M$ be a non-trivial, finitely generated left ${\mathbb{F}}[G]$-module. Let $B\subseteq M$ be an ${\mathbb{F}}[G]$-submodule satisfying $B\cap M^G=0$. Then $B=0$.
The ${\mathbb{F}}$-algebra ${\mathbb{F}}[G]$ is artinian. Moreover, as every irreducible left ${\mathbb{F}}[G]$-module is isomorphic to the trivial left ${\mathbb{F}}[G]$-module, for any finitely generated left ${\mathbb{F}}[G]$-module $B$ one has $\operatorname{soc}(B)=B^G$. Hence the hypothesis implies $\operatorname{soc}_G(B)=0$. Thus $B=0$.
From Fact \[fact:soctriv\] one concludes the following injectivity criterion.
\[lem:injmod\] Let $G$ be a finite $p$-group, and let ${\mathbb{F}}$ be a field of characteristic $p$. Suppose that for $\phi\colon{\mathbf{X}}\to{\mathbf{Y}}\in\operatorname{mor}({\mathfrak{cMF}}_G({{}_{{\mathbb{F}}}{\mathbf{mod}}}))$ one has that
- $\phi_G\colon {\mathbf{X}}_G\to{\mathbf{Y}}_G$ is injective, and
- ${\mathbf{X}}$ is of type $H^0$, and ${\mathbf{Y}}$ is $i$-injective.
Then $\phi$ is injective.
By hypothesis (i), $i^{{\mathbf{Y}}}_{G,U}\circ\phi_G\colon{\mathbf{X}}_G\to{\mathbf{Y}}_U$ is injective for all $U\subseteq G$. If $V\subseteq G$ is normal in $G$, one has a commutative diagram $$\label{dia:inj1}
\xymatrix{
{\mathbf{X}}_G\ar[r]^{\phi_G}\ar[d]_{i^{{\mathbf{X}}}_{G,V}}&{\mathbf{Y}}_G\ar[d]^{i^{{\mathbf{Y}}}_{G,V}}\\
{\mathbf{X}}_V\ar[r]^{\phi_V}&{\mathbf{Y}}_V
}$$ As ${\mathbf{X}}$ is of type $H^0$, $\operatorname{im}(i_{G,V}^{{\mathbf{X}}})={\mathbf{X}}_V^{G/V}=\operatorname{soc}_{G/V}({\mathbf{X}}_V)$ (cf. ). Moreover, since $\phi_V\circ i^{{\mathbf{X}}}_{G,V}=i^{{\mathbf{Y}}}_{G,V}\circ\phi_G$ is injective, $\phi_V\vert_{\operatorname{soc}_{G/V}({\mathbf{X}}_V)}\colon\operatorname{soc}_{G/V}({\mathbf{X}}_V)\to{\mathbf{Y}}_V$ is injective, i.e., $\operatorname{ker}(\phi_V)\cap\operatorname{soc}_{G/V}({\mathbf{X}}_V)=0$. Thus by Fact \[fact:soctriv\], $\operatorname{ker}(\phi_V)=0$ and $\phi_V$ is injective.
Let $U$ be any subgroup of $G$, and let $V\subseteq U$ be a subgroup of $U$ which is normal in $G$. By the previously mentioned remark one has a commutative diagram $$\label{dia:inj2}
\xymatrix{
{\mathbf{X}}_U\ar[r]^{\phi_U}\ar[d]_{i^{{\mathbf{X}}}_{U,V}}&{\mathbf{Y}}_U\ar[d]^{i^{{\mathbf{Y}}}_{U,V}}\\
{\mathbf{X}}_V\ar[r]^{\phi_V}&{\mathbf{Y}}_V
}$$ with $\phi_V$ is injective. By hypothesis (ii), $i_{U,V}^{{\mathbf{X}}}$ is injective. Hence $\phi_U$ is injective, and this yields the claim.
Let $R$ be a discrete valuation domain of characteristic $0$ with prime ideal $pR$ for some prime number $p$, i.e., ${\mathbb{F}}=R/pR$ is a field of characteristic $p$. For a finitely generated $R$-module $A$ let $\operatorname{gr}_\bullet(A)$ denote the graded ${\mathbb{F}}[t]$-module associated to the p-adic filtration $(p^k.A)_{k\geq 0}$. Then every homogeneous component $\operatorname{gr}_k(A)$ is a finite-dimensional ${\mathbb{F}}$-vector space. Moreover, $A$ is a free $R$-module if, and only if, $\operatorname{gr}_\bullet(A)$ is a free ${\mathbb{F}}[t]$-module. Let $\phi\colon A\to B$ be a homomorphism of finitely generated $R$-modules. Then $\phi$ induces a homomorphism of ${\mathbb{F}}[t]$-modules $\operatorname{gr}_\bullet(\phi)\colon\operatorname{gr}_\bullet(A)\to\operatorname{gr}_\bullet(B)$. Moreover, one has the following.
\[fact:disgr\] Let $R$ be a discrete valuation domain of characteristic $0$ with prime ideal $pR$ for some prime number $p$, and let $\phi\colon A\to B$ be a homomorphism of $R$-lattices. Then the following are equivalent:
- $\phi$ is split-injective;
- $\operatorname{gr}_\bullet(\alpha)\colon\operatorname{gr}_\bullet(A)\to\operatorname{gr}_\bullet(B)$ is injective;
- $\operatorname{gr}_0(\alpha)\colon\operatorname{gr}_0(A)\to\operatorname{gr}_0(B)$ is injective.
Lemma \[lem:injmod\] and Fact \[fact:disgr\] imply the following criterion for split-injectivity.
\[prop:inj\] Let $G$ be a finite $p$ group, let $R$ be a discrete valuation domain of characteristic $0$ with prime ideal $pR$, and let $\phi\colon {\mathbf{X}}\to{\mathbf{Y}}\in \operatorname{mor}({\mathfrak{cMF}}_G({{}_R{\mathbf{lat}}}))$ be a natural transformation of cohomological $R$-lattice functors with the following properties:
- $\operatorname{gr}_0(\phi_G)\colon \operatorname{gr}_0({\mathbf{X}}_G)\to\operatorname{gr}_0({\mathbf{Y}}_G)$ is injective;
- $\operatorname{gr}_0({\mathbf{X}})$ is of type $H^0$ and $\operatorname{gr}_0({\mathbf{Y}})$ is $i$-injective.
Then $\phi\colon{\mathbf{X}}\to{\mathbf{Y}}$ is split-injective.
Gentle $R^\circledast$-order categories {#ss:gent}
=======================================
Throughout this section we fix a prime number $p$ and assume further that $R$ is a principal ideal domain of characteristic $0$ such that $pR$ is a prime ideal, i.e., ${\mathbb{F}}=R/pR$ is a field, the [*residue field*]{} of $R$ at $pR$. By $K=\operatorname{quot}(R)$ we denote the quotient field of $R$.
Gentle $R^\circledast$-order categories {#ss:gentordcat}
---------------------------------------
By ${{\mathcal{G}}}_R(n,p)$, $n\geq 0$, we denote the $R$-order category with objects $\operatorname{ob}({{\mathcal{G}}}_R(n,p))=\{0,\ldots,n\}$ and morphisms given by $$\label{eq:morgent}
\operatorname{Hom}_{{{\mathcal{G}}}_R(n,p)}(j,k)=
\begin{cases}
R.t_{j,k}&\ \text{for $j<k$,}\\
R.\operatorname{id}_k&\ \text{for $j=k$,}\\
R.i_{j,k}&\ \text{for $j>k$,}
\end{cases}$$ for $0\leq j,k\leq n$ subject to the relations
- $i_{l,j}=i_{k,j}\circ i_{l,k} $ for $j\leq k\leq l$;
- $t_{j,l}=t_{k,l}\circ t_{j,k} $ for $j\leq k\leq l$;
- $i_{j+1,j}\circ t_{j,j+1}=p.\operatorname{id}_j$ for $j\in \{0,\ldots,n-1\}$;
- $ t_{k-1,k}\circ i_{k,k-1}=p.\operatorname{id}_k$ for $k\in \{1,\ldots,n\}$;
where we put $t_{k,k}=i_{k,k}=\operatorname{id}_k$ for $k\in\{0,\ldots,n\}$. It comes equipped with the natural equivalence $\sigma\colon{{\mathcal{G}}}_R(n,p)\to{{\mathcal{G}}}_R(n,p)^{\operatorname{op}}$ of order $2$, i.e., $\sigma\circ\sigma=\operatorname{id}_{{{\mathcal{G}}}_R(n,p)}$, given by $$\label{eq:antiG}
\sigma(k)=k,\ \ \sigma(t_{j,k})=i_{k,j},\ \ \sigma(i_{k,j})=t_{j,k},\ \ 0\leq j\leq k\leq n;$$ and thus forms an $R^\circledast$-order category.
\[rem:gentle\] Let $\mu=\mu_{{{\mathcal{G}}}_R(n,p)}$ be the $R$-order representing ${{\mathcal{G}}}_R(n,p)$ (cf. Remark \[rem:rep\]). Then $\mu\otimes_R{\mathbb{F}}$ is a gentle ${\mathbb{F}}$-algebra. It is well known that these algebras are $1$-Gorenstein (cf. [@gere:gent]). However, for $n\geq 1$ they are not of finite global dimension, and, therefore, they do not have the Whitehead property (cf. Fact \[fact:glwh\] and \[fact:gldim1\]).
The unitary projection {#ss:unipro1}
----------------------
Let $C_{p^n}$ be the cyclic group of order $p^n$. Then $$\label{eq:unitar1}
\pi\colon{{\mathcal{M}}}_R(C_{p^n})\longrightarrow{{\mathcal{G}}}_R(n,p),$$ given by $\pi(U)=\log_p(|G:U|)$, $\pi({\mathfrak{i}}_{V,U})=i_{j,i}$, $\pi({\mathfrak{t}}_{U,V})=t_{i,j}$, $\rho^U_g=\operatorname{id}_i$, for $U,V\subseteq G$, $|U|=p^{n-i}$, $|V|=p^{n-j}$, $j\geq i$, is a unitary projection. Applying $\operatorname{inf}^\pi({\underline{\phantom{x}}})$ shows that every functor ${\mathbf{F}}\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{G}}}_R(n,p)))$ can also be considered as a cohomological Mackey functor for the finite group $C_{p^n}$. The deflation functor $\operatorname{def}^\pi({\underline{\phantom{x}}})$ can be described explicitly using the functor of $C_{p^n}$-coinvariants ${\underline{\phantom{x}}}_{C_{p^n}}$, i.e., for ${\mathbf{X}}\in\operatorname{ob}({\mathfrak{cMF}}_{C_{p^n}}({{}_R{\mathbf{mod}}}))$ one has $$\label{eq:unitar2}
\operatorname{def}^\pi({\mathbf{X}})(k)=({\mathbf{X}}_U)_{C_{p^n}},\ \ \text{$|U|=p^{n-k}$,}$$ and $\operatorname{def}^\pi(\alpha)(k)=(\alpha_U)_{C_{p^n}}\colon({\mathbf{X}}_U)_{C_{p^n}}\to({\mathbf{Y}}_U)_{C_{p^n}}$ for $\alpha\in\operatorname{Hom}_{{{\mathcal{M}}}_R(C_{p^n})}({\mathbf{X}},{\mathbf{Y}})$. Furthermore, by Fact \[fact:dfl\](c), one has for $W\subseteq G$, $|W|=p^{n-k}$, that $$\label{eq:dflps}
\operatorname{def}^\pi({\mathbf{P}}^W)\simeq {\mathbf{P}}^k.$$
Simple functors {#ss:simpgent}
---------------
As every functor ${\mathbf{F}}\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{G}}}_R(n,p)^{\operatorname{op}},{{}_R{\mathbf{mod}}}))$ is in particular a cohomological $C_{p^n}$-Mackey functor, one can use the description given in [@pj:simp] in order to determine all simple functors in $\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{G}}}_R(n,p)^{\operatorname{op}},{{}_R{\mathbf{mod}}}^{\operatorname{f.g.}}))$. For every $\ell\in\{0,\ldots,n\}$ there exists a simple functor ${\mathbf{S}}^\ell$ given by $$\label{eq:simp}
{\mathbf{S}}^\ell(k)=\begin{cases}
{\mathbb{F}},&\ \text{for $k=\ell$,}\\
0,&\ \text{for $k\not=\ell$;}
\end{cases}
\ \ {\mathbf{S}}^\ell(t_{j,k})=0,\ \ {\mathbf{S}}^\ell(i_{k,j})=0,\ \
0\leq j\leq k\leq n.$$ From Remark \[rem:minproj\] one concludes that if $R$ is discrete valuation ring of characteristic $0$ with maximal ideal $pR$, then every simple functor ${\mathbf{S}}\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{G}}}_R(n,p)^{\operatorname{op}},{{}_R{\mathbf{mod}}}^{\operatorname{f.g.}}))$ must be naturally isomorphic to some ${\mathbf{S}}^\ell$, $0\leq \ell\leq n$.
$R$-lattice functors of rank $1$ {#ss:rank1}
--------------------------------
Let ${\mathbf{F}}\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{G}}}_R(n,p)^{\operatorname{op}},{{}_R{\mathbf{lat}}}))$ be an $R$-lattice functor. Then ${\mathbf{F}}(i_{0,k})\otimes_R K\colon {\mathbf{F}}(k)\otimes_RK\to{\mathbf{F}}(0)\otimes_RK$ is an isomorphism of finite-dimensional $K$-vector spaces, i.e., $\operatorname{rk}({\mathbf{F}}(k))=\operatorname{rk}({\mathbf{F}}(0))$ for all $k\in\{0,\ldots,n\}$, where $\operatorname{rk}({\mathbf{F}}(0))$ denotes the rank of the free $R$-module ${\mathbf{F}}(0)$. We define the [*rank of ${\mathbf{F}}$*]{} by $\operatorname{rk}({\mathbf{F}})=rk({\mathbf{F}}(0))$.
If $M$ is an $R$-lattice and $B\subseteq M$ is an $R$-submodule of $M$, we denote by $$\label{eq:defsat}
\operatorname{sat}_M(B)=\{\,b\in M\mid\exists r\in R\setminus\{0\}\colon\ r.b\in B\,\}$$ the [*saturation*]{} of $B$ in $M$. It is again an $R$-submodule of $M$. Let ${\mathbf{G}}$ be a subfunctor of the $R$-lattice functor ${\mathbf{F}}\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{G}}}_R(n,p)^{\operatorname{op}},{{}_R{\mathbf{lat}}}))$. Then $\operatorname{sat}_{{\mathbf{F}}}({\mathbf{G}})$ given by $$\label{eq:defsat2}
\operatorname{sat}_{{\mathbf{F}}}({\mathbf{G}})(k)=\operatorname{sat}_{{\mathbf{F}}(k)}({\mathbf{G}}(k)),$$ $0\leq k\leq n$, is a subfunctor of ${\mathbf{F}}$ containing ${\mathbf{G}}$. The subfunctor ${\mathbf{G}}$ will be called [*saturated*]{}, if $\operatorname{sat}_{\mathbf{F}}({\mathbf{G}})={\mathbf{G}}$. The following fact allows us to reduce some considerations to $R$-lattice functors of rank $1$.
\[fact:rank11\] Let ${\mathbf{F}}\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{G}}}_R(n,p)^{\operatorname{op}},{{}_R{\mathbf{lat}}}))$, $\operatorname{rk}({\mathbf{F}})> 0$. Then ${\mathbf{F}}$ contains a saturated subfunctor of rank $1$. In particular, there exists an ascending chain $({\mathbf{F}}_j)_{0\leq j\leq \operatorname{rk}({\mathbf{F}})}$ of subfunctors of ${\mathbf{F}}$ satisfying ${\mathbf{F}}_0=0$, ${\mathbf{F}}_{j-1}\subseteq{\mathbf{F}}_j$, ${\mathbf{F}}_{\operatorname{rk}({\mathbf{F}})}={\mathbf{F}}$ and ${\mathbf{F}}_j/{\mathbf{F}}_{j-1}$ is an $R$-lattice functor of rank $1$.
Let $a\in{\mathbf{F}}(0)$, $a\not=0$. Then ${\mathbf{R}}\subseteq{\mathbf{F}}$ given by ${\mathbf{R}}(k)=\operatorname{sat}_{{\mathbf{F}}(k)}(R{\mathbf{F}}(i_{k,0})(a))$ together with the canonical maps is a saturated [subfunctor]{} of ${\mathbf{F}}$. The final remark follows by induction.
Let ${\mathbf{F}}$ be an $R$-lattice functor of rank $1$. By (iii) and (iv) of the definition, for $k\in\{0,\ldots n-1\}$ either ${\mathbf{F}}(t_{k,k+1})$ is an isomorphism, or ${\mathbf{F}}(i_{k+1,k})$ is an isomorphism. Thus we can represent ${\mathbf{F}}$ by a diagram $\Delta_{{\mathbf{F}}}$, where we draw an arrow from $k+1$ to $k$ if ${\mathbf{F}}(t_{k,k+1})\colon{\mathbf{F}}(k+1)\to{\mathbf{F}}(k)$ is an isomorphism, and an arrow from $k$ to $k+1$ if ${\mathbf{F}}(i_{k+1,k})\colon{\mathbf{F}}(k)\to{\mathbf{F}}(k+1)$ is an isomorphism. It is straightforward to verify that the isomorphism type of ${\mathbf{F}}$ is uniquely determined by $\Delta_{{\mathbf{F}}}$, and that for every arrow diagram $\Delta$ there exists an $R$-lattice functor ${\mathbf{F}}_\Delta$ which is represented by this diagram.
\[rem:diafun\] (a) For $\ell\in\{0,\ldots,n\}$ let ${\mathbf{P}}^\ell=\operatorname{Hom}_{{{\mathcal{G}}}}({\underline{\phantom{x}}},\ell)$ be the standard projective $R$-lattice functor associated to $\ell$ (cf. §\[ss:proj\]). Then ${\mathbf{P}}^\ell$ has rank $1$ and is represented by the arrow diagram $$\label{eq:arrP}
\xymatrix@C.5truecm{
0&1\ar[l]&\cdots\ar[l]&\ell-1\ar[l]&\ell\ar[r]\ar[l]&\ell+1\ar[r]&\cdots\ar[r]&n-1\ar[r]&n
}.$$
\(b) If ${\mathbf{F}}$ is represented by the diagram $\Delta_{{\mathbf{F}}}$, then ${\mathbf{F}}^\ast$ is represented by the diagram $\Delta_{{\mathbf{F}}^\ast}=\bar{\Delta}_{{\mathbf{F}}}$, where all arrows are reversed.
\(c) Let ${\mathbf{J}}^\ell=({\mathbf{P}}^\ell)^\ast$, $\ell\in\{0,\ldots,n\}$. Then ${\mathbf{J}}^\ell$ is relative injective and, by (a) and (b), ${\mathbf{J}}^\ell$ is represented by the diagram $$\label{eq:arrJ}
\xymatrix@C.5truecm{
0\ar[r]&1\ar[r]&\cdots\ar[r]&\ell-1\ar[r]&\ell&\ell+1\ar[l]&\cdots\ar[l]&n-1\ar[l]&n\ar[l]
}.$$ In particular, ${\mathbf{P}}^0\simeq{\mathbf{J}}^n$ and ${\mathbf{P}}^n\simeq{\mathbf{J}}^0$ are relative injective.
Let ${\mathbf{F}}\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{G}}}_R(n,p)^{\operatorname{op}},{{}_R{\mathbf{lat}}}))$ be a an $R$-lattice functor of rank $1$. Then $\Delta_{{\mathbf{F}}}$ defines a connected graph $\Gamma_{{\mathbf{F}}}$ in the plane ${\mathbb{R}}^2={\mathbb{R}}e_1\oplus{\mathbb{R}}e_2$, where all arrows are diagonal and point in negative $e_2$-direction, e.g., for ${\mathbf{F}}\in
\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{G}}}_R(8,p)^{\operatorname{op}},{{}_R{\mathbf{lat}}}))$ with $\Delta_{\mathbf{F}}$ given by $$\label{eq:arrex1}
\xymatrix@C.5truecm{
0&1\ar[l]\ar[r]&2\ar[r]&3&4\ar[r]\ar[l]&5\ar[r]&6\ar[r]&7&8\ar[l]
}$$ one obtains the graph $\Gamma_{\mathbf{F}}$ $$\label{eq:arrex2}
\xymatrix@C1.1truecm @M=0pt @W=0pt @R=.5cm{
\ar@{-}[r]\ar@{-}[d]&\ar@{-}[r]&\ar@{-}[r]&\ar@{-}[r]&\ar@{-}[r]&\ar@{-}[r]&\ar@{-}[r]&\ar@{-}[r]&\ar@{-}[d]\\
\ar@{-}[d]&\ar[dl]\ar[dr]{\scriptscriptstyle{\blacksquare}}&&&&&&&\ar@{-}[d]\\
\bullet\ar@{-}[d]&&\bullet\ar[dr]&&{\scriptscriptstyle{\blacksquare}}\ar[dl]\ar[dr]&&&&\ar@{-}[d]\\
\ar@{-}[d]&&&\circ\ar@{-->}[dr]\ar@{..>}[dl]&&\bullet\ar[dr]&&&\ar@{-}[d]\\
\ar@{-}[d]&&\ar@{..>}[dl]&&\ar@{-->}[dr]&&\bullet\ar[dr]&&{\scriptscriptstyle{\blacksquare}}\ar[dl]\ar@{-}[d]\\
\ar@{-}[d]&\ar@{..>}[dl]&&&&\ar@{-->}[dr]&&\circ&\ar@{-}[d]\\
\ar@{-}[d]&&&&&&\ar@{-->}[dr]&&\ar@{-}[d]\\
\ar@{-}[d]&&&&&&&\ar@{-->}[dr]&\ar@{-}[d]\\
&&&&&&&&\ar@{-}[d]\\
\ar@{->}[u]^{e_2}\ar@{->}[r]_{e_1}&\ar@{-}[r]&\ar@{-}[r]&\ar@{-}[r]&\ar@{-}[r]&\ar@{-}[r]&\ar@{-}[r]&\ar@{-}[r]&\\
0&1&2&3&4&5&6&7&8
}.$$ Let $\max({\mathbf{F}})\subset\operatorname{ob}({{\mathcal{G}}}_R(n,p))$ be the set of objects corresponding to local maxima in the graph $\Gamma_{{\mathbf{F}}}$, i.e., $k\not\in\{0,n\}$ is contained in $\max({\mathbf{F}})$ if, and only if, $\Delta_{\mathbf{F}}$ contains a subdiagram of the form $(\xymatrix{k-1&k\ar[r]\ar[l]&k+1})$. Moreover, $0\in\max({\mathbf{F}})$ if $(\xymatrix{0\ar[r]&1})$ is a subdiagram of $\Delta_{{\mathbf{F}}}$, while $n\in \max({\mathbf{F}})$ if $(\xymatrix{n-1&\ar[l]n})$ is a subdiagram of $\Delta_{{\mathbf{F}}}$. E.g., for ${\mathbf{F}}\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{G}}}_R(8,p)^{\operatorname{op}},{{}_R{\mathbf{lat}}}))$ as in one has that $\max({\mathbf{F}})=\{1,4,8\}$. By $\min({\mathbf{F}})$ we denote the subset of $\{1,\ldots,n-1\}$ corresponding to local minima in the graph $\Delta_{{\mathbf{F}}}$, i.e., $\ell\not\in\{0,n\}$ is contained in $\min({\mathbf{F}})$ if, and only if, $\Delta_{\mathbf{F}}$ contains a subdiagram of the form $(\xymatrix{k-1\ar[r]&k&k+1\ar[l]})$. E.g., for the functor ${\mathbf{F}}\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{G}}}_R(8,p)^{\operatorname{op}},{{}_R{\mathbf{lat}}}))$ as in one has $\min({\mathbf{F}})=\{3, 7\}$. Thus by construction, one has $|\max({\mathbf{F}})|=|\min({\mathbf{F}})|+1$. The following fact is straightforward.
\[fact:hdrk1\] Let ${\mathbf{F}}\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{G}}}_R(n,p)^{\operatorname{op}},{{}_R{\mathbf{lat}}}))$ be an $R$-lattice functor of rank $1$. Then $\operatorname{nat}_R({\mathbf{F}},{\mathbf{S}}^\ell)\simeq {\mathbb{F}}$ if $\ell\in\max({\mathbf{F}})$, and $\operatorname{nat}_R({\mathbf{F}},{\mathbf{S}}^\ell)=0$ if $\ell\not\in\max({\mathbf{F}})$. Moreover, ${\mathbf{F}}$ is projective if, and only if, $\min({\mathbf{F}})=\emptyset$.
Let ${\mathbf{F}}\in \operatorname{ob}({\mathfrak{F}}_R({{\mathcal{G}}}_R(n,p)^{\operatorname{op}},{{}_R{\mathbf{lat}}}))$ be an $R$-lattice functor of rank $1$ which is not projective. Let $s({\mathbf{F}})\in\max({\mathbf{F}})$ be the smallest element in $\max({\mathbf{F}})$, and let $t({\mathbf{F}})$ be the smallest element in $\min({\mathbf{F}})$. The projective $R$-lattice functor ${\mathbf{P}}^{s({\mathbf{F}})}$ corresponds to the diagram obtained from the diagram $\Delta_{{\mathbf{F}}}$ by changing all arrows between vertices $\alpha$ and $\alpha+1$, $\alpha \geq t({\mathbf{F}})$ to $\xymatrix{(\alpha\ar[r]&\alpha+1)}$. Let ${\mathbf{F}}^\wedge \in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{G}}}_R(n)^{\operatorname{op}},{{}_R{\mathbf{lat}}}))$ be the $R$-lattice functor of rank $1$ corresponding to the diagram obtained from the diagram $\Delta_{{\mathbf{F}}}$ by changing all arrows between vertices $\alpha-1$ and $\alpha$, $\alpha \leq t({\mathbf{F}})$ to $\xymatrix{(\alpha-1&\alpha\ar[l])}$. E.g., for ${\mathbf{F}}\in \operatorname{ob}({\mathfrak{F}}_R({{\mathcal{G}}}_R(8,p)^{\operatorname{op}},{{}_R{\mathbf{lat}}}))$ as in , $\Gamma_{{\mathbf{F}}^\wedge}$ is given by replacing the first segment by the path $\xymatrix{\ar@{..>}[r]&}$ in ; and $\Delta_{{\mathbf{F}}^\wedge}$ is given by $$\label{eq:arrex3}
\xymatrix@C.5truecm{
0&1\ar[l]&2\ar[l]&3\ar[l]&4\ar[r]\ar[l]&5\ar[r]&6\ar[r]&7&8\ar[l]
}$$
The global dimension of ${{\mathcal{G}}}_R(n,p)$ {#ss:globgent}
------------------------------------------------
The following property will be essential for the subsequent analysis.
\[lem:rank1seq\] Let ${\mathbf{F}}\in \operatorname{ob}({\mathfrak{F}}_R({{\mathcal{G}}}_R(n,p)^{\operatorname{op}},{{}_R{\mathbf{lat}}}))$ be an $R$-lattice functor of rank $1$ which is not projective. Then one has a short exact sequence of $R$-lattice functors $$\label{eq:rank1seq}
\xymatrix{
0\ar[r]&{\mathbf{P}}^{t({\mathbf{F}})}\ar[r]^-{\psi}&{\mathbf{P}}^{s({\mathbf{F}})}\oplus{\mathbf{F}}^{\wedge}\ar[r]^-{\phi}&{\mathbf{F}}\ar[r]&0.
}$$
One can identify ${\mathbf{P}}^{s({\mathbf{F}})}$ and ${\mathbf{F}}^\wedge$ as subfunctors of ${\mathbf{F}}$ by putting $$\label{eq:subPs}
\begin{aligned}
{\mathbf{P}}^{s({\mathbf{F}})}(k)&=
\begin{cases}
\hfil{\mathbf{F}}(k)\hfil&\ \text{for $k\leq t({\mathbf{F}})$},\\
\operatorname{im}({\mathbf{F}}(i_{k,t({\mathbf{F}})}))&\ \text{for $k> t({\mathbf{F}})$};
\end{cases}\\
{\mathbf{F}}^\wedge(k)&=
\begin{cases}
\operatorname{im}({\mathbf{F}}(t_{k,t({\mathbf{F}})}))&\ \text{for $k\leq t({\mathbf{F}})$},\\
\hfil{\mathbf{F}}(k)\hfil&\ \text{for $k> t({\mathbf{F}})$}.
\end{cases}
\end{aligned}$$ for $k\in\{0,\ldots,n\}$. Let $\phi_1\colon {\mathbf{P}}^{s({\mathbf{F}})}\to {\mathbf{F}}$ and $\phi_2\colon {\mathbf{F}}^\wedge\to{\mathbf{F}}$ denote the canonical inclusions. By construction, $\phi=\phi_1\oplus\phi_2\colon{\mathbf{P}}^{s({\mathbf{F}})}\oplus{\mathbf{F}}^\wedge\to{\mathbf{F}}$ is surjective with kernel $\operatorname{ker}(\phi)\subseteq {\mathbf{P}}^{s({\mathbf{F}})}\oplus{\mathbf{F}}^\wedge$ given by $$\label{eq:kerphi}
\operatorname{ker}(\phi)(k)=\{\,(x,-x)\in{\mathbf{P}}^{s({\mathbf{F}})}(k)\oplus{\mathbf{F}}^\wedge(k)
\mid x\in {\mathbf{P}}^{s({\mathbf{F}})}(k)\cap{\mathbf{F}}^\wedge(k)\,\},$$ i.e., $\operatorname{ker}(\phi)\simeq{\mathbf{P}}^{s({\mathbf{F}})}\cap{\mathbf{F}}^\wedge$. By construction, ${\mathbf{X}}={\mathbf{P}}^{s({\mathbf{F}})}\cap{\mathbf{F}}^\wedge$ is an $R$-lattice functor of rank $1$ with all maps ${\mathbf{X}}(t_{j,t({\mathbf{F}})})$ and ${\mathbf{X}}(i_{k,t({\mathbf{F}})})$ surjective for $0\leq j<t({\mathbf{F}})<k\leq n$. Hence all maps ${\mathbf{X}}(t_{j,t({\mathbf{F}})})$, ${\mathbf{X}}(i_{k,t({\mathbf{F}})})$, $0\leq j<t({\mathbf{F}})<k\leq n$, are isomorphisms. Thus $\Delta_{{\mathbf{X}}}=\Delta_{{\mathbf{P}}^{t({\mathbf{F}})}}$, and this yields the claim.
The equality $|\max({\mathbf{F}}^\wedge)|=|\min({\mathbf{F}})|+1$ has the following consequence.
\[prop:projrank1\] Let ${\mathbf{F}}\in \operatorname{ob}({\mathfrak{F}}_R({{\mathcal{G}}}_R(n,p)^{\operatorname{op}},{{}_R{\mathbf{lat}}}))$ be an $R$-lattice functor of rank $1$. Then one has a short exact sequence of $R$-lattice functors $$\label{eq:rank1seq2}
\xymatrix{
0\ar[r]&
\textstyle{\bigoplus_{j\in\min({\mathbf{F}})}}{\mathbf{P}}^{j}\ar[r]^-{\alpha}&
\textstyle{\bigoplus_{k\in\max({\mathbf{F}})}}{\mathbf{P}}^{k}\ar[r]^-{\beta}&{\mathbf{F}}\ar[r]&0.
}$$ In particular, $\operatorname{proj.dim}_R({\mathbf{F}})\leq 1$.
We proceed by induction on $m=|\max({\mathbf{F}})|$. If $|\max({\mathbf{F}})|=1$, one has $\min({\mathbf{F}})=\emptyset$, and hence ${\mathbf{F}}$ is projective. Therefore we may assume that $m>1$, and that the assertion is true for all $R$-lattice functors ${\mathbf{G}}$ of rank $1$ satisfying $|\max({\mathbf{G}})|<m$. Let ${\mathbf{F}}\in \operatorname{ob}({\mathfrak{F}}_R({{\mathcal{G}}}_R(n,p)^{\operatorname{op}},{{}_R{\mathbf{lat}}}))$ with $|\max({\mathbf{F}})|=m>1$. Hence $|\max({\mathbf{F}}^\wedge)|=m-1$, and, by induction, one has a short exact sequence $$\label{eq:rank1seq3}
\xymatrix{
0\ar[r]&
\textstyle{\bigoplus_{j\in\min({\mathbf{F}}^\wedge)}}{\mathbf{P}}^{j}\ar[r]^-{\alpha^\wedge}&
\textstyle{\bigoplus_{k\in\max({\mathbf{F}}^\wedge)}}{\mathbf{P}}^{k}\ar[r]^-{\beta^\wedge}&{\mathbf{F}}^\wedge\ar[r]&0.
}$$ For $s=s({\mathbf{F}})$, $t=t({\mathbf{F}})$ and $\beta=(\operatorname{id}_{{\mathbf{P}}^s}\oplus\beta^\wedge)\circ\phi$ one has a commutative and exact diagram $$\label{eq:rank1seq4}
\xymatrix{
&0\ar@{..>}[d]&0\ar[d]&&\\
0\ar@{..>}[r]&\operatorname{ker}(\zeta)\ar@{..>}[d]\ar@{..>}[r]&\textstyle{\bigoplus_{j\in\min({\mathbf{F}}^\wedge)}}{\mathbf{P}}^{j}
\ar[d]_{\alpha^\wedge}\ar[r]&0\ar[d]&\\
0\ar[r]&\ker(\beta)\ar[r]\ar@{-->}[d]_{\zeta}
&{\mathbf{P}}^s\oplus\textstyle{\bigoplus_{k\in\max({\mathbf{F}}^\wedge)}}{\mathbf{P}}^{k}
\ar[d]_{\operatorname{id}_{{\mathbf{P}}^s}\oplus\beta^\wedge}\ar[r]^-\beta
&{\mathbf{F}}\ar@{=}[d]\ar[r]&0\\
0\ar[r]&{\mathbf{P}}^t\ar@{..>}[d]\ar[r]^\psi&{\mathbf{P}}^s\oplus{\mathbf{F}}^\wedge\ar[r]^\phi\ar[d]&{\mathbf{F}}\ar[r]\ar[d]&0\\
&0&0&0&
}$$ where $\psi$ and $\phi$ are as in Lemma \[lem:rank1seq\] and $\zeta$ is the induced map. By the snake lemma, one may extend this diagram by the arrows “$\xymatrix@C.3truecm{\ar@{..>}[r]&}$”. Hence $\ker(\beta)\simeq\bigoplus_{j\in\min({\mathbf{F}})}{\mathbf{P}}^j$, and this yields the claim.
\[rem:gentn1\] By Remark \[rem:diafun\], every functor ${\mathbf{F}}\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{G}}}_R(1,p)^{\operatorname{op}},{{}_R{\mathbf{lat}}}))$ of rank $1$ is projective and relative injective.
Finally, one concludes the following theorem which is somehow counterintuitive in view of Remark \[rem:gentle\].
\[thm:gldimgent\] Let $p$ be a prime number, and let $R$ be a principal ideal domain of characteristic $0$ such that $pR$ is a prime ideal. Then
- $\operatorname{Ldim}_R({{\mathcal{G}}}_R(1,p))=0$ and $\operatorname{gldim}_R({{\mathcal{G}}}_R(1,p))=1$; and
- $\operatorname{Ldim}_R({{\mathcal{G}}}_R(n,p))=1$ and $\operatorname{gldim}_R({{\mathcal{G}}}_R(n,p))=2$ for $n\geq 2$.
In particular, $({{\mathcal{G}}}_R(n,p),\sigma)$ has the Whitehead property.
Suppose that $n\geq 2$. By Proposition \[prop:projrank1\], the projective dimension of any $R$-lattice functor of rank 1 is less or equal to $1$. Hence by Fact \[fact:rank11\], induction on the rank and the Horseshoe lemma [@ben:coh1 Lemma 2.5.1], ${{\mathcal{G}}}_R(n,p)$ is of global $R$-lattice dimension less or equal to $1$. Since there are $R$-lattice functors of rank 1 which are not projective, one concludes that $\operatorname{Ldim}_R({{\mathcal{G}}}_R(n,p))=1$. For any simple functor ${\mathbf{S}}^\ell$, $0\leq \ell\leq n$, one has $\operatorname{proj.dim}({\mathbf{S}}^\ell)=2$. Thus $\operatorname{gldim}_R({{\mathcal{G}}}_R(n,p))=2$.
By Remark \[rem:gentn1\], any $R$-lattice functor of rank 1 of ${{\mathcal{G}}}_R(1,p)$ is projective and relative injective. Hence by Fact \[fact:rank11\], induction on the rank and the Horseshoe lemma, any $R$-lattice functor is projective, i.e., $\operatorname{Ldim}_R({{\mathcal{G}}}_R(1,p))=0$. For the simple functors ${\mathbf{S}}^\ell$, $\ell\in\{0,1\}$, one has $\operatorname{proj.dim}({\mathbf{S}}^\ell)=1$. Thus $\operatorname{gldim}_R({{\mathcal{G}}}_R(n,p))=1$.
The final remark is a direct consequence of Fact \[fact:glwh\].
Projective $R$-lattice functors {#ss:projgent}
-------------------------------
Let $R$ be a discrete valuation domain of characteristic $0$ with maximal ideal $pR$. Then $R$ is a noetherian ring, and every proper subfunctor ${\mathbf{G}}\subsetneq{\mathbf{F}}$ of a functor ${\mathbf{F}}\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{G}}}_R(n,p)^{\operatorname{op}},{{}_R{\mathbf{mod}}}^{\operatorname{f.g.}}))$ must be contained in a maximal subfunctor ${\mathbf{M}}\subsetneq{\mathbf{F}}$. Moreover, from the discussion in subsection \[ss:simpgent\] one concludes that ${\mathbf{F}}/{\mathbf{M}}\simeq{\mathbf{S}}^\ell$ for some $\ell\in\{0,\ldots,n\}$. We define the [*radical*]{} of ${\mathbf{F}}\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{G}}}_R(n,p)^{\operatorname{op}},{{}_R{\mathbf{mod}}}^{\operatorname{f.g.}}))$ by $$\label{eq:rad}
\operatorname{rad}({\mathbf{F}})=\bigcap_{\substack{{\mathbf{M}}\subsetneq{\mathbf{F}}\\ {\mathbf{M}}\ \text{maximal}}}{\mathbf{M}}.$$ and the [*head*]{} of ${\mathbf{F}}$ by $\operatorname{hd}({\mathbf{F}})={\mathbf{F}}/\operatorname{rad}({\mathbf{F}})$. In particular, there exist non-negative integers $f_0,\ldots,f_n\in{\mathbb{N}}_0$ such that $$\label{eq:hdgent}
\operatorname{hd}({\mathbf{F}})\simeq f_0{\mathbf{S}}^0\oplus\cdots\oplus f_n{\mathbf{S}}^n.$$ Here we used the abbreviation $m{\mathbf{Z}}=\oplus_{1\leq j\leq m} {\mathbf{Z}}$. Moreover, $\operatorname{hd}({\mathbf{F}})=0$ if, and only if, ${\mathbf{F}}=0$. Furthermore, the following property holds.
\[fact:sur\] Let $R$ be a discrete valuation domain of characteristic $0$ with maximal ideal $pR$, and let $\phi\colon{\mathbf{G}}\to{\mathbf{F}}\in
\operatorname{mor}({\mathfrak{F}}_R({{\mathcal{G}}}_R(n,p)^{\operatorname{op}},{{}_R{\mathbf{mod}}}^{\operatorname{f.g.}}))$ be a natural transformation of functors with values in the category of finitely generated $R$-modules. Then $\phi$ is surjective if, and only if, the induced map $\operatorname{hd}(\phi)\colon\operatorname{hd}({\mathbf{G}})\to\operatorname{hd}({\mathbf{F}})$ is surjective.
From this one concludes the following property.
\[fact:rankgent\] Let $R$ be a discrete valuation domain of characteristic $0$ with maximal ideal $pR$, and let ${\mathbf{F}}\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{G}}}_R(n,p)^{\operatorname{op}},{{}_R{\mathbf{lat}}}))$. Then $\operatorname{rk}({\mathbf{F}})\leq\dim_{{\mathbb{F}}}(\operatorname{hd}({\mathbf{F}}))$, and equality holds if, and only if, ${\mathbf{F}}$ is projective.
Suppose that $\operatorname{hd}({\mathbf{F}})\simeq f_0{\mathbf{S}}^0\oplus\cdots\oplus f_n{\mathbf{S}}^n$. Put ${\mathbf{P}}=f_0{\mathbf{P}}^0\oplus\cdots\oplus f_n{\mathbf{P}}^n$. Since ${\mathbf{P}}$ is projective, there exists a natural transformation $\phi\colon{\mathbf{P}}\to{\mathbf{F}}$ such that $\operatorname{hd}(\phi)\colon\operatorname{hd}({\mathbf{P}})\to\operatorname{hd}({\mathbf{F}})$ is an isomorphism. By Fact \[fact:sur\], $\phi$ is surjective, and thus $$\label{eq:rankgent}
\dim_{{\mathbb{F}}}(\operatorname{hd}({\mathbf{F}}))=\dim_{{\mathbb{F}}}(\operatorname{hd}({\mathbf{P}}))=\operatorname{rk}({\mathbf{P}})\geq\operatorname{rk}({\mathbf{F}}).$$ If $\operatorname{rk}({\mathbf{F}})=\dim_{{\mathbb{F}}}(\operatorname{hd}({\mathbf{F}}))$, then $\phi$ must be an isomorphism. Assume that ${\mathbf{F}}$ is projective. Then $\phi$ is split-surjective, i.e., there exists a projective $R$-lattice functor ${\mathbf{Q}}\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{G}}}_R(n,p)^{\operatorname{op}},{{}_R{\mathbf{lat}}}))$ such that ${\mathbf{P}}\simeq{\mathbf{F}}\oplus{\mathbf{Q}}$. As $\operatorname{hd}(\phi)$ is an isomorphism, this yields $\operatorname{hd}({\mathbf{Q}})=0$. Hence ${\mathbf{Q}}=0$, and ${\mathbf{F}}$ is isomorphic to ${\mathbf{P}}$.
The proof of Fact \[fact:rankgent\] has shown also the following.
\[cor:projgent\] Let $R$ be a discrete valuation domain of characteristic $0$ with maximal ideal $pR$, and let ${\mathbf{P}}\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{G}}}_R(n,p)^{\operatorname{op}},{{}_R{\mathbf{lat}}}))$ be a projective $R$-lattice functor satisfying $\operatorname{hd}({\mathbf{P}})\simeq f_0{\mathbf{S}}^0\oplus\cdots\oplus f_n{\mathbf{S}}^n$. Then ${\mathbf{P}}\simeq f_0{\mathbf{P}}^0\oplus\cdots\oplus f_n{\mathbf{P}}^n$.
Cohomological Mackey functors for cyclic p-groups {#s:cmcp}
=================================================
Throughout this section we assume that $R$ is a discrete valuation domain of characteristic $0$ with maximal ideal $pR$ and that $G$ is a finite cyclic $p$-group of order $p^n$.
The deflation functor {#ss:unipro2}
---------------------
Let $\pi\colon {{\mathcal{M}}}_R(G)\longrightarrow{{\mathcal{G}}}_R(n,p)$ denote the unitary projection (cf. §\[ss:unipro1\]), let ${\underline{\phantom{x}}}^\pi=\operatorname{inf}^\pi(\operatorname{def}^\pi({\underline{\phantom{x}}}))$, and let $\eta\colon\operatorname{id}_{{\mathfrak{cMF}}_G({{}_R{\mathbf{mod}}})}\to{\underline{\phantom{x}}}^\pi$ denote the unit of the adjunction. In particular, $\eta_{\mathbf{X}}\colon{\mathbf{X}}\to{\mathbf{X}}^\pi$ is surjective, and $\eta_{{\mathbf{X}},G}\colon{\mathbf{X}}_G\to{\mathbf{X}}^\pi_G$ is an isomorphism for all ${\mathbf{X}}\in\operatorname{ob}({\mathfrak{cMF}}_G({{}_R{\mathbf{mod}}}))$.
\[fact:defcat1\] Let $G$ be a finite cyclic group of order $p^n$, let $R$ be an integral domain of characteristic $0$, and let ${\mathbf{X}}\in\operatorname{ob}({\mathfrak{cMF}}_G({{}_R{\mathbf{lat}}}))$ be a cohomological $R$-lattice functor which is Hilbert$^{90}$. Then one has a canonical isomorphism $$\label{eq:defcat1}
\operatorname{def}^\pi({\mathbf{X}})(k)\simeq \operatorname{im}(t_{U,G}^{\mathbf{X}})\subseteq {\mathbf{X}}_G,$$ for $U\subseteq G$, $|U|=p^{n-k}$.
Let $g\in G$ be a generator of $G$, i.e., $\operatorname{def}^\pi({\mathbf{X}})(k)={\mathbf{X}}_U/(1-g){\mathbf{X}}_U$. Periodicity of Tate cohomology implies that ${\hat{H}}^{-1}(G/U,{\mathbf{X}}_U)=H^1(G/U,{\mathbf{X}}_U)=0$. Thus by , $\operatorname{ker}(t^{{\mathbf{X}}}_{U,G})=(1-g){\mathbf{X}}_U$. Hence the induced map $$\label{eq:defcat2}
\xymatrix{
{\mathbf{X}}_U/(1-g){\mathbf{X}}_U\ar[r]^-{\tilde{t}^{\mathbf{X}}_{U,G}}&{\mathbf{X}}_G
}$$ is injective. This yields the claim.
From Fact \[fact:defcat1\] one concludes the following.
\[cor:defcat\] Let $G$ be a finite cyclic group of order $p^n$, let $R$ be an integral domain of characteristic $0$, and let $\phi\colon{\mathbf{X}}\to{\mathbf{Y}}\in\operatorname{mor}({\mathfrak{cMF}}_G({{}_R{\mathbf{lat}}}))$ be a natural transformation of cohomological $R$-lattice functors with the following properties:
- ${\mathbf{X}}$ and ${\mathbf{Y}}$ are Hilbert$^{90}$;
- $\phi_G\colon{\mathbf{X}}_G\to{\mathbf{Y}}_G$ is injective.
Then $\operatorname{def}^\pi(\phi)\colon\operatorname{def}^\pi({\mathbf{X}})\to\operatorname{def}^\pi({\mathbf{Y}})$ is injective. In particular, if $$\label{eq:defcat3}
\xymatrix{
0\ar[r]&
{\mathbf{X}}\ar[r]^{\alpha}&
{\mathbf{Y}}\ar[r]^{\beta}&
{\mathbf{Z}}\ar[r]& 0}$$ is a short exact sequence of $R$-lattice functors all of which are Hilbert$^{90}$, then $$\label{eq:defcat4}
\xymatrix{
0\ar[r]&
\operatorname{def}^\pi({\mathbf{X}})\ar[r]^{\operatorname{def}^\pi(\alpha)}&
\operatorname{def}^\pi({\mathbf{Y}})\ar[r]^{\operatorname{def}^\pi(\beta)}&
\operatorname{def}^\pi({\mathbf{Z}})\ar[r]& 0}$$ is exact.
\[prop:defcat\] Let $G$ be a finite cyclic group of order $p^n$, let $R$ be an Dedekind domain of characteristic $0$, and let ${\mathbf{X}}\in\operatorname{ob}({\mathfrak{cMF}}_G({{}_R{\mathbf{lat}}}))$ be a cohomological $R$-lattice functor which is Hilbert$^{90}$. Then ${\mathbf{X}}$ is $\pi$-acyclic.
Let $({\mathbf{P}}_\bullet,\partial^{\mathbf{P}}_\bullet,{\varepsilon}_{\mathbf{X}})$ be a projective resolution of ${\mathbf{X}}$ in ${\mathfrak{cMF}}_G({{}_R{\mathbf{mod}}})$ with ${\mathbf{P}}_k$ projective $R$-lattice functors. In particular, ${\mathbf{Q}}_k=\operatorname{im}(\partial_k^P)$, $k\geq 1$, is an $R$-lattice functor. By construction, one has the short exact sequences $$\label{eq:defcat5}
\xymatrix@R=3pt{
0\ar[r]&{\mathbf{Q}}_1\ar[r]&{\mathbf{P}}_0\ar[r]&{\mathbf{X}}\ar[r]&0,\\
0\ar[r]&{\mathbf{Q}}_{k+1}\ar[r]&{\mathbf{P}}_k\ar[r]&{\mathbf{Q}}_k\ar[r]&0,
}$$ for $k\geq 1$. Thus by induction and Proposition \[prop:cycint\], ${\mathbf{Q}}_k$ is a Hilbert$^{90}$ $R$-lattice functors for all $k\geq 1$. Hence by Corollary \[cor:defcat\], one has short exact sequences $$\label{eq:defcat6}
\xymatrix@R=3pt{
0\ar[r]&\operatorname{def}^\pi({\mathbf{Q}}_1)\ar[r]&\operatorname{def}^\pi({\mathbf{P}}_0)\ar[r]&\operatorname{def}^\pi({\mathbf{X}})\ar[r]&0,\\
0\ar[r]&\operatorname{def}^\pi({\mathbf{Q}}_{k+1})\ar[r]&\operatorname{def}^\pi({\mathbf{P}}_k)\ar[r]&\operatorname{def}^\pi({\mathbf{Q}}_k)\ar[r]&0,
}$$ for $k\geq 1$. This implies that ${{\mathcal{L}}}_k\operatorname{def}^\pi({\mathbf{X}})=0$ for all $k\geq 1$.
Let $R$ be a discrete valuation domain of characteristic $0$ with maximal ideal $pR$, and let $G$ be a cyclic $p$-group. As in subsection \[ss:projgent\] one concludes that every proper subfunctor ${\mathbf{Y}}\subsetneq{\mathbf{X}}$ of a cohomological $G$-Mackey functor ${\mathbf{X}}\in\operatorname{ob}({\mathfrak{cMF}}_G({{}_R{\mathbf{mod}}}^{\operatorname{f.g.}}))$ must be contained is a maximal subfunctor ${\mathbf{M}}\subsetneq{\mathbf{X}}$. Therefore we define the [*radical*]{} of ${\mathbf{X}}\in\operatorname{ob}({\mathfrak{cMF}}_G({{}_R{\mathbf{mod}}}^{\operatorname{f.g.}}))$ by $$\label{eq:radmac}
\operatorname{rad}({\mathbf{X}})=\bigcap_{\substack{{\mathbf{M}}\subsetneq{\mathbf{X}}\\ {\mathbf{M}}\ \text{maximal}}}{\mathbf{M}}.$$ and the [*head*]{} of ${\mathbf{X}}$ by $\operatorname{hd}({\mathbf{X}})={\mathbf{X}}/\operatorname{rad}({\mathbf{X}})$. By Remark \[rem:minproj\], there exist non-negative integers $f_U\in{\mathbb{N}}_0$, $U\subseteq G$, such that $$\label{eq:hdmac}
\operatorname{hd}({\mathbf{X}})\simeq\textstyle{\bigoplus_{U\subseteq G} f_U{\mathbf{S}}^U}.$$ Since every simple cohomological $G$-Mackey functor ${\mathbf{S}}\in\operatorname{ob}({\mathfrak{cMF}}_G({{}_R{\mathbf{mod}}}^{\operatorname{f.g.}}))$ is isomorphic to $\operatorname{inf}^\pi({\boldsymbol{\Sigma}})$ for some simple functor ${\boldsymbol{\Sigma}}\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{G}}}_R(n,p)^{\operatorname{op}},{{}_R{\mathbf{mod}}}^{\operatorname{f.g.}}))$, one has $\operatorname{ker}(\eta_{\mathbf{X}})\subseteq \operatorname{rad}({\mathbf{X}})$. This inclusion has the following consequence.
\[prop:surmac\] Let $R$ be a discrete valuation domain of characteristic $0$ with maximal ideal $pR$, and let $G$ be a finite cyclic $p$-group. Let $\phi\colon{\mathbf{X}}\to{\mathbf{Y}}$ be a natural transformation of cohomological $G$-Mackey functors with values in the category ${{}_R{\mathbf{mod}}}^{\operatorname{f.g.}}$. Then the following are equivalent.
- $\phi$ is surjective;
- $\phi^\pi\colon{\mathbf{X}}^\pi\to{\mathbf{Y}}^\pi$ is surjective;
- $\operatorname{hd}(\phi)\colon\operatorname{hd}({\mathbf{X}})\to\operatorname{hd}({\mathbf{Y}})$ is surjective.
The natural surjection $\tau\colon\operatorname{id}\to\operatorname{hd}({\underline{\phantom{x}}})$ factors through the natural surjection $\eta\colon\operatorname{id}\to{\underline{\phantom{x}}}^\pi$, i.e., there exists a natural surjection $\psi\colon{\underline{\phantom{x}}}^\pi\to\operatorname{hd}({\underline{\phantom{x}}})$ such that $\tau=\psi\circ\eta$. This yields the implications (i) $\Rightarrow$ (ii) $\Rightarrow$ (iii). Suppose that (iii) holds and that $\phi$ is not surjective. Then $\operatorname{im}(\phi)$ is contained in a maximal subfunctor ${\mathbf{M}}\subsetneq{\mathbf{Y}}$. Thus for ${\mathbf{S}}={\mathbf{Y}}/{\mathbf{M}}$, the kernel of the map $\phi_\ast\colon\operatorname{nat}_G({\mathbf{Y}},{\mathbf{S}})\to\operatorname{nat}_G({\mathbf{X}},{\mathbf{S}})$ is non-trivial. However, in the commutative diagram $$\label{eq:diasur}
\xymatrix{
\operatorname{nat}_G(\operatorname{hd}({\mathbf{Y}}),{\mathbf{S}})\ar[r]^{\operatorname{hd}(\phi)_\ast}\ar[d]_{(\tau_{\mathbf{Y}})_\ast}&
\operatorname{nat}_G(\operatorname{hd}({\mathbf{X}}),{\mathbf{S}})\ar[d]^{(\tau_{\mathbf{X}})_\ast}\\
\operatorname{nat}_G({\mathbf{Y}},{\mathbf{S}})\ar[r]^{\phi_\ast}&\operatorname{nat}_G({\mathbf{X}},{\mathbf{S}})
}$$ the vertical maps are isomorphisms, and $\operatorname{hd}(\phi)_\ast$ is injective forcing $\operatorname{ker}(\phi_\ast)=0$, a contradiction. This yields the claim.
We are now ready to prove the following theorem which is one of the key results in this paper.
\[thm:defcat\] Let $R$ be a discrete valuation domain of characteristic $0$ with maximal ideal $pR$, let $G$ be a finite cyclic $p$-group, and let ${\mathbf{X}}\in\operatorname{ob}({\mathfrak{cMF}}_G({{}_R{\mathbf{lat}}}))$ be a cohomological $G$-Mackey functor with values in the category of $R$-lattices which is Hilbert$^{90}$. Then there exists a finite $G$-set $\Omega$ such that ${\mathbf{X}}\simeq{\mathbf{h}}^0(R[\Omega])$. In particular, ${\mathbf{X}}$ is projective.
The deflation functor $\operatorname{def}^\pi\colon{\mathfrak{cMF}}_G({{}_R{\mathbf{mod}}})\longrightarrow{\mathfrak{F}}_R({{\mathcal{G}}}_R(p,n),{{}_R{\mathbf{mod}}})$ associated to the unitary projection $\pi\colon{{\mathcal{M}}}_R(G)\longrightarrow
{{\mathcal{G}}}_R(p,n)$ has the following properties:
- $({{\mathcal{M}}}_R(G)),\sigma)$ is $\circledast$-symmetric (cf. Prop. \[prop:Ryon\]).
- ${{\mathcal{G}}}_R(n,p)$ has global $R$-lattice dimension less or equal to $1$ (cf. Thm. \[thm:gldimgent\]), and thus has the Whitehead property (cf. Fact \[fact:glwh\] and Fact \[fact:gldim1\]).
- An $R$-lattice functor ${\mathbf{Y}}\in\operatorname{ob}({\mathfrak{cMF}}_G({{}_R{\mathbf{lat}}}))$ is $\circledast$-acyclic if, and only if, it has the Hilbert$^{90}$ property (cf. Prop. \[prop:Ryon\]). By Proposition \[prop:defcat\], such a functor is $\pi$-acyclic.
In particular, the hypothesis of Theorem \[thm:whitehead\] are satisfied, and one concludes that ${\mathbf{Z}}=\operatorname{def}^\pi({\mathbf{X}})\in\operatorname{ob}({\mathfrak{F}}_R({{\mathcal{G}}}_R(n,p),{{}_R{\mathbf{mod}}}))$ is projective. Hence there exist non-negative integers $f_0,\ldots, f_n$ such that ${\mathbf{Z}}\simeq f_0{\mathbf{P}}^0\oplus\cdots f_n{\mathbf{P}}^n$ (cf. Cor. \[cor:projgent\]).
Let $\eta_{{\mathbf{X}}}\colon{\mathbf{X}}\to{\mathbf{X}}^\pi$ be the canonical map (cf. §\[ss:unipro2\]), i.e., ${\mathbf{X}}^\pi\simeq\bigoplus_{0\leq k\leq n}\operatorname{inf}^\pi(f_k{\mathbf{P}}^k)$. Let $U_k\subseteq G$ denote the unique subgroup of $G$ of index $p^k$, and let $\Omega$ be the $G$-set $\Omega=\bigsqcup_{0\leq k\leq n} f_k(G/U_k)$. Put ${\mathbf{P}}={\mathbf{h}}^0(R[\Omega])$. Then ${\mathbf{P}}\in\operatorname{ob}({\mathfrak{cMF}}_G({{}_R{\mathbf{lat}}}))$ is projective (cf. Fact \[fact:projcohmac\]). Since $\operatorname{def}^\pi({\mathbf{h}}^0(R[G/U_k]))\simeq {\mathbf{P}}^k$ for all $k\in\{0,\ldots,n\}$, one has an isomorphism $\phi^\pi\colon{\mathbf{P}}^\pi\to{\mathbf{X}}^\pi$. Since ${\mathbf{P}}$ is projective, there exists a homomorphism of cohomological $G$-Mackey functors such that the diagram $$\xymatrix{
{\mathbf{P}}\ar[r]^{\phi}\ar[d]_{\eta_{{\mathbf{P}}}}&{\mathbf{X}}\ar[d]^{\eta_{\mathbf{X}}}\\
{\mathbf{P}}^\pi\ar[r]^{\phi^\pi}&{\mathbf{X}}^{\pi}
}$$ commutes. By construction, $\phi^\pi_G$ is an isomorphism, and $\eta_{{\mathbf{P}},G}$ and $\eta_{{\mathbf{X}},G}$ are isomorphisms (cf. §\[ss:unipro2\]). Thus $\phi_G$ is an isomorphism. In particular, with the same notations as used in subsection \[ss:injcrit\], the map $\operatorname{gr}_0(\phi_G)\colon\operatorname{gr}_0({\mathbf{P}})\to\operatorname{gr}_0({\mathbf{X}})$ is an isomorphism. By hypothesis, ${\mathbf{X}}$ is an $R$-lattice functor with the Hilbert$^{90}$ property, and the same is true for ${\mathbf{P}}$ (cf. Remark \[rem:projH0\]). Hence $\operatorname{gr}_0({\mathbf{X}})$ and $\operatorname{gr}_0({\mathbf{P}})$ are of type $H^0$ (cf. Prop. \[prop:torhil\]), and $\phi$ is split-injective (cf. Prop. \[prop:inj\]). Moreover, by Proposition \[prop:surmac\], $\phi$ must be surjective. This yields the claim.
As an immediate consequence of Remark \[rem:projH0\] we obtain the following.
\[cor:profperm\] Let $R$ be a discrete valuation domain of characteristic $0$ with maximal ideal $pR$, let $G$ be a finite cyclic $p$-group of order $p^n$, and let ${\mathbf{P}}\in\operatorname{ob}({\mathfrak{cMF}}_G({{}_R{\mathbf{lat}}}))$ be a projective $R$-lattice functor. Then there exist non-negative integers $f_W\in{\mathbb{N}}_0$, $W\subseteq G$, such that ${\mathbf{P}}\simeq\bigoplus_{U\subseteq G} f_W{\mathbf{P}}^W$, i.e., $$\label{eq:K0}
K_0({{\mathcal{M}}}_R(G))\simeq\operatorname{B}(G)\simeq {\mathbb{Z}}^n,$$ where $\operatorname{B}(G)$ denotes the Burnside ring of $G$.
In case that the ${R[G]}$-lattice $M$ satisfies a Hilbert$^{90}$ property, one obtains the following.
\[cor:hil90lat1\] Let $R$ be a discrete valuation domain of characteristic $0$ with maximal ideal $pR$, let $G$ be a finite cyclic $p$-group, and let $M$ be an ${R[G]}$-lattice such that $H^1(U,\operatorname{res}^G_U(M))=0$ for every subgroup $U$ of $G$. Then there exists a finite $G$-set $\Omega$ such that $M\simeq R[\Omega]$.
By hypothesis, ${\mathbf{X}}={\mathbf{h}}^0(M)$ is a cohomological $G$-Mackey functor with values in the category of $R$-lattices satisfying the Hilbert$^{90}$ property. Thus by Theorem \[thm:defcat\], there exists a finite $G$-set $\Omega$ such that ${\mathbf{X}}\simeq{\mathbf{h}}^0(R[\Omega])$. Hence evaluating the functors ${\mathbf{X}}$ and ${\mathbf{h}}^0(R[\Omega])$ on the subgroup $\{1\}$ yields the claim.
The following property is a direct consequence of Tate duality (cf. Prop. \[prop:tatedual\]) and completes the proof of Theorem A.
\[prop:elequi\] Let $R$ be a principal ideal domain of characteristic $0$, let $G$ be a finite group, let $U$ be a subgroup of $G$, and let $M$ be an ${R[G]}$-lattice. Then the following are equivalent.
- $H^1(U,\operatorname{res}^G_U(M^\ast))=0$;
- ${\hat{H}}^{-1}(U,\operatorname{res}^G_U(M))=0$;
- $M/\omega_{R[U]}M$ is torsion free.
By , (i) and (ii) are equivalent. Let $N_U\colon M\to M^U$ be the $U$-norm map, i.e., for $m\in M$ one has $N_U(m)=\sum_{u\in U}u\cdot m$. As $M$ is an $R[U]$-lattice, $M^U$ is an $R$-lattice. Hence $$\label{eq:torid}
\operatorname{tor}_R(M/\omega_{R[U]}M)=\operatorname{ker}(N_U)/\omega_{R[U]}M={\hat{H}}^{-1}(U,\operatorname{res}^G_U(M)),$$ where $\operatorname{tor}_R({\underline{\phantom{x}}})$ denotes the $R$-submodule of $R$-torsion elements. Thus (ii) and (iii) are equivalent.
Projective dimensions {#ss:projdim}
---------------------
In conjunction with Proposition \[prop:cycint\], Theorem \[thm:defcat\] has strong implications on the projective dimension of a cohomological Mackey functor of a cyclic $p$-group.
\[thm:projdimmac\] Let $R$ be a discrete valuation domain of characteristic $0$ with maximal ideal $pR$, let $G$ be a finite cyclic $p$-group, and let ${\mathbf{X}}\in\operatorname{ob}({\mathfrak{cMF}}_G({{}_R{\mathbf{mod}}}^{\operatorname{f.g.}}))$. Let $$\label{eq:parproj}
\xymatrix{
{\mathbf{P}}_2\ar[r]^{{\partial}_2}&{\mathbf{P}}_1\ar[r]^{{\partial}_1}&
{\mathbf{P}}_0\ar[r]^{{\varepsilon}_{{\mathbf{X}}}}&{\mathbf{X}}\ar[r]&0}$$ be a partial projective resolution of ${\mathbf{X}}$ by projective $R$-lattice functors. Then
- $\operatorname{ker}({\partial}_2)$ is a projective $R$-lattice functor, i.e., $\operatorname{proj.dim}({\mathbf{X}})\leq 3$.
- If ${\mathbf{X}}$ is $i$-injective, then $\operatorname{ker}({\partial}_1)$ is a projective $R$-lattice functor, i.e., one has $\operatorname{proj.dim}({\mathbf{X}})\leq 2$.
- If ${\mathbf{X}}$ is of type $H^0$, then $\operatorname{ker}({\partial}_0)$ is a projective $R$-lattice functor, i.e., one has $\operatorname{proj.dim}({\mathbf{X}})\leq 1$.
In particular, if $G$ is non-trivial then $\operatorname{Ldim}({{\mathcal{M}}}_R(G))= 2$, and $\operatorname{gldim}({{\mathcal{M}}}_R(G))= 3$.
\(a) By Proposition \[prop:cycint\](a), (b) and (c), $\operatorname{ker}({\partial}_0)$ is an $R$-lattice functor and thus $i$-injective, $\operatorname{ker}({\partial}_1)$ is of type $H^0$, and $\operatorname{ker}({\partial}_2)$ is Hilbert$^{90}$. Hence Theorem \[thm:defcat\] yields the claim in this case. (b) and (c) follow by a similar argument. From (a) one concludes that $\operatorname{gldim}({{\mathcal{M}}}_R(G))\leq 3$, and (b) implies $\operatorname{Ldim}({{\mathcal{M}}}_R(G))\leq 2$. If $G$ is non-trivial, the discussion in subsection \[ss:cycsec\] shows that $\operatorname{proj.dim}({\mathbf{B}}^G)=3$. This yields the final remark (cf. ).
\[rem:gorfin\] Let ${\mathbb{F}}$ be a field of characteristic $p$, and let $G$ be a non-trivial, finite cyclic $p$-group. Then ${{\mathcal{M}}}_{{\mathbb{F}}}(G)$ is not of finite global dimension, but ${{\mathcal{M}}}_{{\mathbb{F}}}(G)$ is $2$-Gorenstein (cf. Prop. \[prop:gorcm\]). This phenomenon occured already for the gentle $R$-order categories (in dimension $1$) (cf. Rem. \[rem:gentle\]).
Lattices {#ss:latt}
--------
From Theorem \[thm:defcat\] and Theorem \[thm:projdimmac\](b), one concludes the following.
\[thm:preslat\] Let $R$ be a discrete valuation domain of characteristic $0$ with maximal ideal $pR$, let $G$ be a finite cyclic $p$-group, and let $M$ be an ${R[G]}$-lattice. Then there exist finite $G$-sets $\Omega_0$ and $\Omega_1$, and a short exact sequence $$\label{eq:shexseqlat}
\xymatrix{
0\ar[r]&R[\Omega_1]\ar[r]&R[\Omega_0]\ar[r]&M\ar[r]&0.
}$$
Let ${\mathbf{X}}={\mathbf{h}}^0(M)$. As ${\mathbf{X}}$ is of type $H^0$, Theorem \[thm:projdimmac\](b) implies that ${\mathbf{X}}$ has a projective resolution $$\label{eq:profreslat}
\xymatrix{
0\ar[r]&{\mathbf{P}}_1\ar[r]^{{\partial}_1}&{\mathbf{P}}_0\ar[r]^{{\varepsilon}_{\mathbf{X}}}&{\mathbf{X}}\ar[r]&0,
}$$ where ${\mathbf{P}}_0$ and ${\mathbf{P}}_1$ are projective $R$-lattice functors. As ${\mathbf{P}}_0$ and ${\mathbf{P}}_1$ have the Hilbert$^{90}$ property (cf. Remark \[rem:projH0\]), Theorem \[thm:defcat\] implies that there exist finite $G$-sets $\Omega_0$ and $\Omega_1$ such that ${\mathbf{P}}_i={\mathbf{h}}^0(R[\Omega_i])$, $i\in\{0,1\}$. Thus evaluating the functors on $\{1\}$ yields the claim.
Extending A. Weiss’ theorem {#ss:aweiss}
---------------------------
The following property can be seen as an extension of A. Weiss’ theorem for finite cyclic $p$-groups.
\[prop:exweiss\] Let $R$ be a discrete valuation domain of characteristic $0$ with maximal ideal $pR$ for some prime number $p$, let $G$ be a finite cyclic $p$-group, and let $M$ be an ${R[G]}$-lattice. Suppose that for some subgroup $N$ of $G$ one has
- $\operatorname{res}^G_N(M)$ is an $R[N]$-permutation module;
- $M^N$ is an $R[G/N]$-permutation module.
Then $M$ is isomorphic to an ${R[G]}$-permutation module, i.e., there exists some finite $G$-set $\Omega$ such that $M\simeq R[\Omega]$.
Let $U$ be a subgroup of $G$. If $U\subseteq N$, then by (i), $\operatorname{res}^G_U(M)$ is an $R[U]$-permutation module. Thus one has $H^1(U,\operatorname{res}^G_U(M))=0$. Suppose that $N\subsetneq U$. Since $N$ is a normal subgroup of $U$, the $5$-term exact sequence in cohomology yields an exact sequence $$\label{eq:5term}
\xymatrix{0\ar[r]&H^1(U/N,M^N)\ar[r]&H^1(U,\operatorname{res}^G_U(M))\ar[r]&H^1(N,\operatorname{res}^G_N(M))^{G/N}
}$$ Hence by (i), one has $H^1(N,\operatorname{res}^G_N(M))=0$. From (ii) one concludes that $M^N$ is an $R[U/N]$-permutation module, and therefore $H^1(U/N,M^N)=0$. Thus by one has $H^1(U,\operatorname{res}^G_U(M))=0$. The assertion then follows from Theorem A.
[^1]: The first author is supported by the grants MTM2011-27090 from the “Ministerio de Ciencia e Innovacion” and P07-FQM-0312 from the “Junta de Andaluca” (FEDER).
[^2]: For an arbitrary commutative ring $R$ with $1$ the kernel of a surjective homomorphism $\phi\colon M\to Q$ of $R$-lattices is not necessarily an $R$-lattice. This is the reason why we restrict all subsequent considerations to $R$-order categories over a Dedekind domain $R$.
[^3]: The famous [*Whitehead problem*]{}, stated by J. H. Whitehead around 1950, is the question whether every abelian group $A$ satisfying $\operatorname{Ext}^1_{\mathbb{Z}}(A,{\mathbb{Z}})=0$ must be a free abelian group. For finitely generated abelian groups this is easily seen to be true, and K. Stein showed (cf. [@stein:ab]) that the statement remains valid for countable abelian groups. However, by the extra-ordinary work of S. Shelah (cf. [@shel:wh1], [@shel:wh2], [@shel:wh3]) one knows now that this problem is in general undecidable.
[^4]: This follows by an argument similar to the proof of [@mcl:hom Cor. III.7.3].
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'C. E. Ekuma'
- 'M. Jarrell'
- 'J. Moreno'
- 'D. Bagayoko'
title: |
\[Supplementary Information\]\
Re-examining the electronic structure of germanium: A first-principle study
---
DFT and Progress in band gap in materials
=========================================
Despite the great progress made possible by density functional theory (DFT), from 1964 to present, problems associated with obtaining theoretically the measured energy or band gaps, for finite and crystalline semiconductors, respectively, have persisted. Specifically, most DFT calculations, with emphasis on those utilizing local density approximation (LDA) and semi-local potentials, have led to semiconductor band gaps that are 30 – 50% smaller than their corresponding, measured values. Much effort has been deployed to find explanations of and remedies to this recalcitrant band gap problem. Perdew and Zunger [@PhysRevB.23.5048] introduced the self interaction correction (SIC) to local spin density (LSD) approximation calculations. While the exact functional for the ground state is self interaction free, these authors discussed corrections that appear to be needed for the description of finite systems, beginning with atoms, and of localized states in solids. This self interaction is argued to contribute to the underestimation of the band gaps of insulators by DFT calculations [@PhysRevB.23.5048]. Consequently, self interaction corrections (SIC) are expected to improve the agreement between calculated band gaps and measured ones, in addition to improving binding energies and bringing orbital energies closer to removal energies [@PhysRevB.23.5048; @PhysRevB.26.5445]. While self interaction corrections have led to some improvements in band gap calculations, they have not totally resolved the problem. Applications of SIC have mostly overestimated the band gap of semiconductors [@PhysRevB.52.R14316; @PhysRevB.54.5495]. According to Cohen [@Cohen2008], self interaction is well-defined only for one-electron systems.
According to the literature, a major source of the theoretical underestimation of band gaps consists of the derivative discontinuity of the exchange correlation energy, Exc [@PhysRevLett.51.1888; @PhysRevLett.49.1691; @PhysRevLett.51.1884; @PhysRevB.32.3883]. Perdew [@PhysRevLett.49.1691], following a thought experiment on a diatomic molecule, established the existence of a derivative discontinuity of the exchange correlation energy, i.e., a discontinuity in the exchange correlation potential, V$_{xc}$. Perdew and Levy [@PhysRevLett.51.1884] generalized this discontinuity to the case of semiconductors. They showed that the exchange correlation potential may jump by the discontinuity, $\Delta_{xc}$, when the number of electrons in the system under study increases by one. Band gaps calculated with a local density approximation (LDA) potentials, according to their findings, are to be augmented by this discontinuity in order to reproduce the corresponding, measured values. The authors suggested, without claiming to have a proof of it, that this discontinuity is a non zero (and positive) in real semiconductors and insulators. Sham and Schlüter [@PhysRevLett.51.1888] also found a derivative discontinuity of E$_{xc}$ in insulators. These authors, however, asserted that their work does not show whether or not this discontinuity is non zero in real insulators. Subsequent work by Sham and Schlüter [@PhysRevB.32.3883] derived the discontinuity of the functional derivative of E$_{xc}$ in insulators by considering an increase of the number of electrons by one. Cautiously, these authors concluded that the discrepancy between calculated and measured band gap is a measure of the discontinuity $\Delta_{xc}$ - given the results from several calculations – if the employed LDA potentials are assumed to be good approximations. The description of our method below indicates the strong possibility that some current LDA and GGA potentials may be very good approximations.
Despite its popular use to explain the disagreement between calculated and measured band gaps, the above discontinuity has not yet been established to be non zero in real semiconductors or insulators. Further, Sham and Schlüter [@PhysRevB.32.3883] underscored the fact that, in principle, DFT and Kohn Sham LDA hold only if the number of particle is kept constant. The question could arise whether or not the discontinuity, derived by considering a change of the number of particle, is strictly applicable to DFT or LDA calculations. From the preceding, it has not yet been established that DFT or LDA calculations cannot obtain the correct band gaps, despite the fact that presently known LDA potentials do not have a discontinuity and that most of the numerous, previous ab-initio DFT and LDA calculations did not.
Another presumed contributor to the band gap underestimation by theory stems from the use of local (LDA) and semi-local (GGA) potentials. The question naturally arises as to what extent the local and semi-local potentials fail to capture key feature of the exact one. We are aware of no definitive answer, given that the exact one is not known. We would have had to delve into this matter further if we were dealing with molecules or their dissociation. The solid state systems of interest to us, to judge by previous results obtained with our method [@Bagayoko2004; @Bagayoko2008], possible errors due to the use of local and semi-local potentials appear to be very small.
There exist several approaches that have been introduced to address the band gap problem. Review articles and books are the best sources for discussing these approaches and for examples of the many DFT calculations that led to band gaps much smaller than their corresponding, experimental counterparts. In contrast, a summary of results from BZW LDA calculations for over 10 materials show agreement between theory and experiment. Illustrative examples of discrepancies between theory and experiment follow. The case of Ge is summarized in this article. Some previous LDA, GGA, and GW calculations did not yield the measured band gap, from first principle. A table provided by Ekuma and Bagayoko [@Ekumab2011] shows a multitude of DFT calculations with vastly different band gaps for titanium dioxide. With the computational method described here, Ekuma and Bagayoko obtained the measured, direct gap and predicted an indirect one. For elemental silicon, Grüning [@Myrta2006] reported an LDA band gap of 0.7 eV, much smaller than the 1.25 eV they reported as the measured value. These authors also performed calculations with the exact exchange (EXX), EXX plus LDA, EXX plus the random phase approximation (RPA). The last approach or scheme yielded 0.6 eV, a gap smaller than the above LDA gap, while the first two led to 1.5 and 1.6 eV respectively, values much larger than the experimental one. With the original version of our method, Zhao et al. utilized an LDA potential to obtain a gap of 1.02 for Si, much closer to the experimental one. Generalized gradient approximation (GGA) calculations have led to improvements of calculated properties of materials, including lattice parameters. Specifically, Hao [@PhysRevB.85.014111] reported revised Tao-Perdew-Staroverov-Scuseria (revTPSS) meta-GGA calculated lattice parameters that are in agreement with experiment following a zero-point phonon correction, for over 50 materials. Despite this very significant success, most GGA and meta-GGA calculations, including the previous ones discussed here for Ge, have not produced band gaps in agreement with experiment.
From the above summary, historical overview of the band gap problem, it appears that the scientific community believes that the derivative discontinuity of the exchange correlation energy is the main source of the disagreement between DFT calculated energy and band gaps and their corresponding, measured ones. This belief led to the development of several schemes aimed at resolving the band gap problem. Except for the few, most of these schemes are ad hoc as they include adjustable parameters that vary with the material under study. The continuing growth in the number of these schemes seems be a problem in itself, the ad hoc nature of most of them does not lend itself to predictive capabilities from first principle, the aim of theory to inform and to guide experiment. The only exception to the above trend consists of the work of our group. This work has not yet gotten the attention of the community at large, presumably due to the strength of the above belief, on the one hand, and the preponderance of results that are explained with the discontinuity, on the other hand. As we previously noted [@Bagayoko2005], the situation resembles that of the Ptolemaic model of the solar system where epicycles were continually introduced to explain its disagreement with observations. The quintessential point in support of the our method, described below, is the following: For all DFT calculations of energy bands, the *minima of the occupied energies, which add up to yield the ground state energy of the electron system, are obtained from the theory if the “correct” ground state charge density is utilized, subject to the constraint that the number of particle is kept constant* [@PhysRev.136.B864; @PhysRev.140.A1133].
Most of the previous DFT calculations, including those with GGA and LDA potentials, have consisted of judicious selecting large basis set and of performing iterations to obtain self consistent eigenvalues of the Kohn-Sham type equation. It is assumed that the single basis set in question leads to the correct representation of the electronic cloud in the system under study, a system that can be drastically different from an atomic or ionic one. *In particular, as we recently pointed out, polarization ($p$, $d$, and $f$ orbitals) has primacy over spherical symmetry ($s$ orbital) for systems varying from diatomic molecules to solids.* Hence, utilizing basis sets derived from calculations of properties of atoms for the study of solids is potentially problematic. Indeed, the angular symmetries in these systems are vastly different from those for atoms and ions. The need for the method described below becomes apparent with the realization that, irrespective of the degree of convergence of the iterations, a single trial basis set that has a major symmetry inadequacy for the description of the system is not going to lead to physically valid DFT eigenvalues as the implacable condition of using the “correct” ground state density will not be met.
Description of the Bagayoko-Zhao-Williams-Ekuma-Franklin Method
===============================================================
The original version of our method, named after Bagayoko, Zhao, and Williams (BZW) was introduced in 1998 [@Bagayoko1998] and further explained in 1999 [@Zhao1999]. The method consists of implementing the linear combination of atomic orbitals (LCAO) formalism by methodically searching for the smallest basis set, called the optimal basis set, that leads to the minima of the occupied energies. This search begins with a deliberately small basis set that is not smaller than the minimum basis set, i.e., the smallest one needed to account for all the electrons in the system. The self consistent calculation with this basis set is followed by another whose basis set uses the previous one plus one additional orbital. The occupied energies from the two calculations are compared numerically and graphically. In the more than 20 systems we have studied, these occupied energies from these first two calculations have been different, with those of calculation II generally lower than the corresponding one from Calculation I. Calculation III is then carried out, using the basis set in Calculation II plus an additional orbital. The occupied, self consistent energies from Calculations II and III are also compared. This process of augmenting a basis set, performing new self-consistent calculations, and comparing its results with those of the one immediately preceding it, continues until a calculation is found, say N, to have exactly the same occupied energies as the one immediately following it. This perfect superposition establishes that the minima of the occupied energies have been reached and that the corresponding basis set give the best representation of the ground state charge density of the system.
Before elaborating further on the physical content of the method, we note that adding an orbital means increasing the size of the basis set (and hence the dimension of the Hamiltonian) by 2, 6, 10, or 14 depending on the $s$, $p$, $d$, or $f$ character, respectively, of the orbital in question. In the original BZW, we added orbitals in the order of their energies resulting from the atomic calculations, i.e., the orbitals corresponding to the lowest laying excited, atomic state were successively added. As apparent from our earlier results, the BZW method practically led to the measured band gap of semiconductors we studied. Further, our predictions of the band gaps and other properties for cubic Si$_\mathrm{3}$N$_\mathrm{4}$ [@Bagayoko2001] and cubic InN [@Bagayoko2004] were totally confirmed by experiment [@PhysRevB.65.161202; @Egdell2003; @Sch2006]. Following the works of Ekuma and Franklin [@Ekuma2011; @ekuma:012189; @Ekumab2011; @ibid2011], we realized that valence electrons in multi-atomic systems simply do not follow the symmetry landscape that prevail for isolated atoms or ions. The aim is to obtain a better representation of the electronic cloud (ground state charge density) of the system under study. Hence, in most of our subsequent calculations, for a given principal quantum number, we add $p$, $d$, and $f$ (if applicable) orbitals before the $s$ orbital for that principal quantum number. This counter-intuitive ordering, for isolated atoms, is simply needed for ‘some’ multi-atomic systems. The initials of Ekuma and Franklin (EF) are added to the name of the enhanced method \[BZW-EF\] in recognition of their extensive calculations whose results led Bagayoko to see the necessity for this new order. While the BZW method led to band gaps that were sometimes smaller by 0.1 – 0.3 eV, insignificant differences between BZW-EF calculated gaps and corresponding experiment ones are in the second decimal place for the systems studied to date. For ZnO [@Franklin2013729], the BZW-EF method led to an upper valence band width more than 1.0 eV larger than was obtained with the BZW, with a significant improvement in agreement with experiment.
The origin of the changes in the band structure and the band gap when the basis set increases toward the optimal one consists of the progressively better representation of the ground state charge density. As per the derivation of DFT, the minima of the occupied energies are obtained if the “correct” charge density for the ground state is employed. These changes are due to physical interactions, given that the charge density, the potential and hence the Hamiltonian change from one calculation to the next. We should underscore here that while our focus is on occupied energies (i.e., DFT is a fundamentally ground state theory), when these energies reach their minima, so do the low laying unoccupied energies, up to 9 – 10 eV for the materials studied to date with the BZW-EF method. For the many systems with the BZW, most low laying unoccupied energies also converged up to 5 – 6 eV, as was the case for wurtzite indium nitride [@Jin2007]. For a few materials, this convergence of the lowest unoccupied energies was not achieved with the original BZW method. For metals, as shown by Zhao [@Zhao1999], the low-laying unoccupied energies converge when the occupied ones do, due to the fact that at least one band crosses the Fermi level. This fact may partly explain the early successes of DFT in describing metals as compared to semiconductors.
The description of what occurs when basis sets much larger than the optimal ones are employed will complete the description of the BZW-EF method. We first recall that the basis set immediately following the optimal one leads to the occupied energies obtained with the optimal one and to the same unoccupied energies up to 9 – 10 eV. So, by much larger basis sets, we mean the ones that are larger than the basis sets immediately following the optimal ones. Earlier works by Bagayoko and Co-workers [@Zhao1999; @Bagayoko1998] verified that basis sets larger than the optimal one do not change the charge density, the potential, and the Hamiltonian, nor do they change the occupied energies. In the absence of changes in the Hamiltonian, i.e., the physics of the study, the additional lowering of unoccupied energies with these much larger basis set cannot be ascribed to DFT. However, the Rayleigh theorem provides an explanation of the unphysical lowering of these energies. The theorem states that when the same eigenvalue equation is solved with two basis sets of different sizes, such that the larger one includes the smaller one, then the eigenvalue obtained with the larger basis set are lower than or equal to their corresponding one obtained with the smaller basis set.
Clearly, after the optimal basis set is reached, and that the occupied energies are no longer changed from their values obtained with the optimal one, the lowering of unoccupied energies can be ascribed to a mathematical artifact that is the manifestation of the above theorem. We therefore contend that this extra lowering, a variational basis set effect [@Zhao1999; @Bagayoko1998], is a major source of discrepancies between many previous DFT calculations and between these calculations and experiment, as far as band gaps are concerned. *The lowest laying conduction bands, with full physical meaning when they result from the optimal basis set, partly lose their physical content due to the above effect.* It is important to note that this is the case for any basis set that is not the optimal one, whether it is smaller or larger than the optimal, or simply lacks orbital or angular features of the optimal one. For basis sets that do not totally include their corresponding optimal ones, even the occupied energies are not totally DFT results. The preceding lines in this paragraph point to the great difficulty in obtaining physically meaning DFT occupied and low laying unoccupied energies with a single trial basis set, irrespective of the degree of convergence of the applicable iterations for self consistency.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'High-frequency (up to $\omega = 6 \,10^4 \un{rad/s}$) rheological measurements combined with light scattering investigations show that an isotropic and multiconnected phase of surfactant micelles exhibits a terminal relaxation time of a few $\un{\mu s}$, much smaller than in solutions of entangled wormlike micelles. This result is explained in terms of the local hexagonal order of the microscopic structure and we discuss its relevance for the understanding of dynamic behaviour in related systems, such as wormlike micelles and sponge phases.'
author:
- 'D. Constantin[^1]'
- 'J.-F. Palierne'
- 'É. Freyssingeas'
- 'P. Oswald'
title: 'High-frequency rheological behaviour of a multiconnected lyotropic phase'
---
In recent years, experimental evidence was presented as to the existence of isotropic phases consisting of connected surfactant micelles [@danino1; @kato1; @kato2]. It has been proposed that they provide an intermediate structure between entangled wormlike micelles and sponge phases [@porte1; @drye1]. Indeed, experimental results [@porte1; @appell1; @khatory1] show that, in some ionic wormlike micellar systems, a dramatic decrease in both viscosity and relaxation time is induced by increasing the counterion concentration, feature that could be explained by the appearance of connections in the micellar network. On the theoretical side, models for the flow behaviour of these connected phases have been developed [@drye1; @lequeux1], and rheology data has been interpreted according to these models in order to characterize the appearance of connections, qualitatively [@narayanan1; @hassan1; @aitali1] or quantitatively [@in1]. Throughout this body of work, however, only the relaxation modes specific to polymer systems have been considered. This approach is certainly valid in dilute phases with not too many connections, but it must fail when the density of connections becomes important and in concentrated systems, where the micelles begin to interact (sterically or otherwise). How does the system behave then and which are the relevant concepts ?
In this Letter, we try to answer these questions by investigating a concentrated and highly connected isotropic phase of a nonionic surfactant/water mixture. We argue that, in the absence of reptation (suppressed by the connections), it can be short-range order (for a concentrated system) that dominates the rheological behaviour.
We employ high-frequency rheology and dynamical light scattering (DLS) to study the isotropic phase in the [ ]{}lyotropic mixture, where [ ]{}is the non-ionic surfactant hexa-ethylene glycol mono-n-dodecyl-ether, or (for the phase diagram see [@mitchell]). Its dynamic behaviour has already been investigated by measuring the shear viscosity [@strey1; @darrigo1], sound velocity and ultrasonic absorption [@darrigo1] as well as NMR relaxation rates [@burnell1], all pointing to the presence of wormlike micelles (at least above 10 % surfactant concentration by weight [@darrigo1]). In previous experiments [@sallen1; @constantin] we have shown that, for 50 % wt surfactant concentration, above the hexagonal mesophase, the isotropic phase has a structure consisting of surfactant cylinders that locally preserve the hexagonal order over a distance $d$ that varies from about $40 \, \un{nm}$ at $40 { {\,}^{\circ} \mbox{C}}$ to $25 \, \un{nm}$ at $60 { {\,}^{\circ} \mbox{C}}$. Between the cylinders there is a large number of thermally activated connections (with an estimated density $n \sim 10^{6} \, \un{\mu m^{-3}}$) [@constantin].
We prepared the [ ]{}mixture with 50.0 % [ ]{}weight concentration. The surfactant was purchased from Nikko Chemicals Ltd. and used without further purification. We used ultrapure water from Fluka Chemie AG. The mixture was carefully homogenized by repeatedly heating, stirring and centrifuging and then allowed to equilibrate at room temperature over a few days.
Rheology measurements were performed in a piezorheometer, the principle of which has been described in reference [@cagnon] : the liquid sample of thickness $60 \un{\mu m}$ is contained between two glass plates mounted on piezoelectric ceramics. One of the plates is made to oscillate vertically with an amplitude of about 1 nm by applying a sine wave to the ceramic. This movement induces a squeezing flow in the sample and the stress transmitted to the second plate is measured by the other piezoelectric element. The shear is extremely small : $\gamma \leq 10^{-4}$, so the sample structure is not altered by the flow. The setup allows us to measure the storage ($G'$) and loss ($G''$) shear moduli for frequencies ranging from $1$ to $6 \, 10^{4} \un{rad/s}$ with five points per frequency decade. The entire setup is temperature regulated within $0.05 { {\,}^{\circ} \mbox{C}}$ and hermetically sealed to avoid evaporation.
Ten temperature points in the isotropic phase have been investigated, from $38.85 { {\,}^{\circ} \mbox{C}}$ (transition temperature from the hexagonal phase) up to $48 { {\,}^{\circ} \mbox{C}}$. The results are displayed in figure \[fig1\]. For clarity, only curves corresponding to 40, 42, 44, 46, and $48 { {\,}^{\circ} \mbox{C}}$ are plotted. Values below $1 \un{Pa}$ (solid horizontal line) are not reliable, as the signal/noise ratio becomes poor. At low frequencies, the response is purely viscous; it is only above $\omega = 10^{3} \un{rad/s}$ that there is a noticeable increase in the value of the storage modulus $G'$.
On general grounds, the low-frequency behaviour of the storage and loss moduli in a fluid is [@ferry] : $G' \propto \omega ^2$ and $G'' \propto \omega$. The slope of $G'$ vs. $\omega$ yields the “zero-shear viscosity” $\eta _0$ and the two curves cross at a frequency $\omega=1/\tau$, where $\tau$ is the terminal relaxation time. The ratio $\eta _0 / \tau$ defines a shear modulus. If $\tau$ is the only relevant time scale in the system, the complex modulus $G^*(\omega) = G' + i G''$ has a simple analytical expression, known as the Maxwell model [@ferry] : $$\label{maxwell}
G^*(\omega) = \frac{i \omega \eta _0}{1+i \omega \tau} \, .$$ The relaxation time $\tau$ separates two regimes : for $\omega \tau \ll 1$, the system can be considered as a viscous fluid with viscosity $\eta _0$, while for $\omega \tau \gg 1$ it exhibits elasticity, with a shear modulus $G_{\infty} = \eta _0 / \tau$.
As shown in \[fig2\], we obtain robust results for the static viscosity $\eta _0$ and for the relaxation time $\tau$ (plotted vs. temperature in figure \[fig3\]).
The temperature variation of the parameters $\eta _0$ and $\tau$ can be described by Arrhenius laws; for the viscosity : $$\label{arrhenius}
\eta _0 (T) = \eta _0 (T^*) \exp \left [ \frac{E_{\eta}}{k_B} \left (\frac{1}{T}-\frac{1}{T^*}\right ) \right ] \, ,$$ yielding an activation energy $E_{\eta} = 35 \pm 1 \, k_B T$ (solid curve in figure \[fig3\]). For comparison, continuous shear measurements in a Couette rheometer (Haake, model RS100), give an activation energy $E_{\eta} = 31 \, k_B T$ [@sallen3]. The relaxation time has an activation energy $E_{\tau} = 38 \pm 6 \, k_B T$ (solid curve in figure \[fig3\]). Within experimental precision, $E_{\eta} = E_{\tau}$. The high-frequency elastic modulus is therefore constant in temperature : $$\label{eq:ginf}
G_{\infty} = \eta _0 / \tau = 44 \pm 6 \, 10^3 \, \un{Pa} \, .$$
The DLS setup uses an Ar laser ($\lambda = 514 \, \un{nm}$), delivering up to $1.5 \un{W}$, a thermostated bath of an index matching liquid (decahydronaphthalene, $n = 1.48$), a photomultiplier and a PC-controlled 256 channel Malvern correlator with sample times as fast as $0.1 \un{\mu s}$. The scattering vector $q$ varies in the range $4 \, 10^{6}$ – $3 \, 10^{7} \un{m^{-1}}$. The signal is monoexponential over the whole range. In figure \[fig5\] we show the relaxation rate $\Omega (q)$ vs. $q^2$ for temperatures between $40$ and $49 { {\,}^{\circ} \mbox{C}}$ . The data fit well to a diffusion law (although there is a slight indication of super-diffusive behaviour). Since the scattered intensity is related to the variations in refractive index produced by concentration fluctuations, we obtain the collective diffusion constant for the concentration field; its temperature variation can be described by an Arrhenius fit (solid curve in figure \[fig5\] – inset) with an activation energy $E_D \simeq 4 \, k_B T$. The average value : $$D = 1.65 \, 10^{-10} \un{m^2/s}
\label{eq:diff}$$ is in good agreement with the one previously obtained from directional-growth experiments [@sallen2] : $D = 1.2 \, 10^{-10} \un{m^2/s}$ at the transition temperature ($38.85 { {\,}^{\circ} \mbox{C}}$).
In unconnected wormlike micellar systems [@drye1; @cates1; @cates2], the relevant relaxation process is reptation, the micelle gradually disengaging from its initial deformed environment and adopting a stress-free configuration. The typical reptation time is given by : $\tau _{\rm{rep}} \simeq L_{\rm{m}} ^2 / D_{\rm{c}}$, with $L_{\rm{m}}$ the average length of a micelle and $D_{\rm{c}}$ the curvilinear diffusion constant. However, if the micelles can break up (with a lifetime $\tau _{\rm{br}}$) this provides an additional pathway for disengagement, the two resulting ends being free to recombine in a different environment. For $ \tau _{\rm{br}} \ll \tau _{\rm{rep}}$, the terminal relaxation time is given by : $\tau = (\tau _{\rm{br}} \tau _{\rm{rep}})^{1/2}$ [@cates1]. As an illustration, in the CTAB/H$_2$0/KBr system the typical micelle length is $L_{\rm{m}} \simeq 1 \mu\rm{m}$, while $\tau$ varies between 0.1 and 1 s depending on the surfactant concentration [@candau1].
Let us now consider the effect of connections; following Drye and Cates [@drye1], we will introduce a typical micelle length between cross-links $L_{\rm{c}}$. The effect of the connections is that reptation occurs on distances of the order of $L_{\rm{c}}$, instead of the much larger $L_{\rm{m}}$ [@cates2]. This explains the fact (counterintuitive at first sight) that connecting the network does in fact reduce the viscosity. If $L_{\rm{c}}$ is small enough, the network is saturated, and the concept of entanglement is no longer applicable; neither is the reptation mechanism. The system we investigate is well in the saturated case, since the typical distance between connections on a micelle is only four times the mean distance between micelles [@constantin].
What is then the origin of viscoelasticity ? We begin the discussion of our results with the very general observation that, when a system is dynamically correlated over a typical distance $L$, one can only observe elastic behaviour by probing the system on scales smaller than the correlation distance [@dimension]. The time $\tau$ needed to relax the stress can then be estimated as : $$\label{tau}
\tau \sim L^2 / (2 \delta D) \, ,$$ where $\delta$ is the space dimension and $D$ is the diffusion constant associated to the relaxation process (a classical example is provided by the Nabarro–Herring creep in solids [@quere]). The system under investigation is very concentrated so, in contrast with the semi-dilute wormlike micellar solutions usually studied, the micelle-micelle interaction plays an important role in the dynamics of the phase. This interaction locally induces hexagonal order as mentioned above; the relevant correlation length is the distance $d$ over which the micelles preserve local order. A pictorial representation is given in figure \[fig6\] : consider a material with short-range order confined between two plates. The system can be seen as consisting of elasticity-endowed units of typical size $d$, the correlation distance. After applying an instantaneous shear $\gamma$ by moving the upper plate to the left, one such unit (represented in thick line) has been advected from point 1 to point 2. At time $t=0^+$ after the deformation, the stress on the upper plate is $\sigma = G_{\infty} \gamma$. Since there is no long-range restoring force, once the particles equilibrate their internal configuration (over a distance $d$), the elastic stress is completely relaxed; thus, after a time $\tau$ given by eq. \[tau\], $\sigma = 0$.
Does this mechanism account for the observed behaviour ? In light of the previous discussion, let us estimate the relaxation time for our system. With the value of $d$ obtained from X-ray scattering and the DLS collective diffusion coefficient (eq. \[eq:diff\]), one has : $$\tau \simeq d^2 / (6D) \sim 10^{-6} \un{s} \, ,
\label{tau2}$$ in good agreement with the experimental results (figure \[fig3\]).
A rough estimate of $G_{\infty}$ can be obtained by noticing that at short range (less than $d$), the structure of the phase resembles that of the hexagonal one, so it should exhibit a similar shear modulus when probed on very short scales. The shear modulus of the hexagonal phase can be estimated as $G_{\rm{hex}} = k_B T / a^3 \simeq 2 \, 10^4 \un{Pa}$ (with $a=6 \un{nm}$ the lattice parameter), in agreement with our result (eq. \[eq:ginf\]). This value can also be compared with preliminary measurements of the shear modulus in the hexagonal phase of [ ]{}[@pieranski1] yielding : $$G_{\rm{hex}} \simeq 2 \, 10^{5} \un{Pa}$$ at room temperature, of the same order of magnitude as our result. The shear modulus of the hexagonal phase should vary very little with temperature, in agreement with our experimental findings.
However, our very simple model does not accurately describe the temperature variation of the physical parameters in equation \[tau2\]. An Arrhenius fit of $d(T)$ (from the X-ray data of reference [@constantin]) yields an activation energy $E_d = 7 \pm 1 \, k_B T$. From equation (7) we would expect that : $$E_{\tau} = 38 \pm 6 \, k_B T \sim 2 E_d + E_D = 18 \pm 2 \, k_B T$$ which is clearly off by a factor of two.
A tentative explanation involves the possible anisotropy of the correlated domains; in this case, the value obtained from the X-ray diffractogram is an average between a transverse correlation length $d=d_{\bot}$ (which is the one relevant for the relaxation) and a $d_{\|}$ (which need not exhibit the same temperature variation). The same observation applies for $D$ : we measure an average value, but at small scale the structure is anisotropic.
A more detailed comparison with theory requires additional data on unsaturated structures. We are currently investigating the same isotropic phase at lower surfactant concentration, where preliminary experiments show rather complicated rheological behaviour.
Finally, we suggest that this approach can also be applied to sponge phases, the characteristic distance being $\xi$, the correlation length. These phases are equally very fluid and, at low frequency (up to at least $10^2 \un{s^{-1}}$), display pure Newtonian behaviour [@snabre1; @vinches1]. Within the framework of the same highly simplified model (equation \[tau\]), we predict a relaxation time of order $\tau \sim \xi ^2 / (6D)$. For instance, in the /hexanol/water system at 5.3 % volume fraction of membrane, where $\xi \simeq 0.1 \un{\mu m}$ and $D \simeq 2 \, 10^{-12} \un{m^2/s}$ [@freyssingeas1], we expect $\tau \sim 10^{-3} \un{s}$.
In conclusion, we study the dynamics of the isotropic (micellar) phase in the [ ]{}mixture at high concentration, where it is highly connected. We show that the observed viscoelastic behaviour can be related to the local hexagonal order of the system.
We would like to thank P. Pieranski for communicating experimental results prior to publication and R. Strey for providing a reprint.
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. From the softening of the structure the authors infer that, even at 50 % surfactant concentration, the isotropic phase is still composed of wormlike micelles (they assume that a connected structure would be more rigid). In fact, as discussed in references [@porte1; @drye1; @appell1; @khatory1; @lequeux1] and in the present work, connecting the micelles can render the structure more fluid.
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Provided that the local order is 2D or 3D. If the order is lamellar (1D), the associated shear modulus is zero.
experiments in progress.
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[^1]: E-mail address :
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'Yuta Tsuchimoto, Patrick Knüppel, Aymeric Delteil, Zhe Sun, Martin Kroner,'
- Ataç Imamoğlu
title: |
SUPPLEMENTAL MATERIAL\
Quantum interface between photonic and superconducting qubits
---
Quantum Monte Carlo method
--------------------------
We consider optical to microwave conversion. In the quantum trajectory formalism, the time evolution of the system is given by the Schrödinger equation: $$\frac{d}{dt}\ket{\psi(t)} = \frac{1}{i\hbar} H_{\mathrm{eff}} \ket{\psi(t)},$$ where $\psi(t)$ is a stochastic wavefunction and $H_{\mathrm{eff}}$ is the non-Hermitian Hamiltonian given by $$\label{hamiltonian}
H_{\mathrm{eff}} = H_{\mathrm{s}} + H_{\mathrm{t}} + H_{\mathrm{st}} - \frac{i \hbar}{2} \sum_k \hat{C}^\dagger_k \hat{C}_k.$$ where $H_{\mathrm{s}}$ ($H_{\mathrm{t}}$) is the source (target) Hamiltonian and $H_{\mathrm{st}}$ the interaction Hamiltonian. The collapse operators $\hat{C}_k$ of this system are $$\begin{aligned}
\hat{C}_1 &= \sqrt{\Gamma_{\mathrm{FE}}^{({\mathrm{s}})}} \sigma_{\mathrm{EF}}^{({\mathrm{s}})} + \sqrt{\Gamma_{\mathrm{FG}}^{({\mathrm{t}})} \eta} \sigma_{\mathrm{GF}}^{({\mathrm{t}})}, \\
\hat{C}_2 &= \sqrt{\Gamma_{\mathrm{FG}}^{({\mathrm{t}})} (1-\eta)} \sigma_{\mathrm{GF}}^{({\mathrm{t}})}, \\
\hat{C}_3 &= \sqrt{\Gamma_{\mathrm{EG}}^{({\mathrm{t}})}} \sigma_{\mathrm{GE}}^{({\mathrm{t}})}, \\
\hat{C}_4 &= \sqrt{\kappa_{\mathrm{c}}} \hat{a}_{\mathrm{c}},
\end{aligned}$$ where $\sigma_{ij} = \ket{i}\bra{j}$ express the projection ($i=j$) and lowering or rising operator ($i \neq j$) respectively. (s) and (t) stand for source and target (interface). $\hat{C}_1$ corresponds to a detection event where photons are emitted either in the $\ket{F}_{\mathrm{s}} \rightarrow \ket{E}_{\mathrm{s}}$ or $\ket{F,0_{\mathrm{c}}}_{\mathrm{t}} \rightarrow \ket{G,0_{\mathrm{c}}}_{\mathrm{t}}$ transitions. The latter transition is coupled to the incident light with a coupling efficiency $\eta$. The operator $\hat{C}_2$ denotes an event originating from the $\ket{F,0_{\mathrm{c}}}_{\mathrm{t}} \rightarrow \ket{G,0_{\mathrm{c}}}_{\mathrm{t}}$ transition which does not couple to the incident mode. $\hat{C}_3$ describes an event associated with the $\ket{E,1_{\mathrm{c}}}_{\mathrm{t}} \rightarrow \ket{G,1_{\mathrm{c}}}_{\mathrm{t}}$ transition. Finally, $\hat{C}_4$ accounts for the cavity decay event. By substituting these collapse operators, equation \[hamiltonian\] becomes: $$\begin{aligned}
H_{\mathrm{eff}} = && \hbar \Omega_{\mathrm{L}} (\sigma_{\mathrm{GF}}^{({\mathrm{s}})} + \sigma_{\mathrm{FG}}^{({\mathrm{s}})}) + \hbar g_{\mathrm{c}} (\hat{a}_{\mathrm{c}} \sigma_{\mathrm{FE}}^{({\mathrm{t}})} + \hat{a}_{\mathrm{c}}^\dagger \sigma_{\mathrm{EF}}^{({\mathrm{t}})})\nonumber\\ &&- i\hbar \frac{\Gamma_{\mathrm{FE}}^{({\mathrm{s}})}}{2}\sigma_{\mathrm{FF}}^{({\mathrm{s}})} - i\hbar \frac{\Gamma_{\mathrm{FG}}^{({\mathrm{t}})}}{2}\sigma_{\mathrm{FF}}^{({\mathrm{t}})} - i \hbar \frac{\Gamma_{\mathrm{EG}}^{({\mathrm{t}})}}{2}\sigma_{\mathrm{EE}}^{({\mathrm{t}})}\nonumber\\ && -i\hbar \sqrt{\Gamma_{\mathrm{FG}}^{({\mathrm{t}})}\Gamma_{\mathrm{FE}}^{({\mathrm{s}})}\eta} \sigma_{\mathrm{EF}}^{({\mathrm{s}})} \sigma_{\mathrm{FG}}^{({\mathrm{t}})} -i \hbar \frac{\kappa_c}{2}\hat{a}_{\mathrm{c}}^\dagger \hat{a}_{\mathrm{c}}.\end{aligned}$$ Here, we neglected the term $\sigma_{\mathrm{FE}}^{({\mathrm{s}})}\sigma_{\mathrm{GF}}^{({\mathrm{t}})}$ describing the reverse process where an emitted photon from the $\ket{F,0_{\mathrm{c}}}_{\mathrm{t}} \rightarrow \ket{G,0_{\mathrm{c}}}_{\mathrm{t}}$ transition drives the $\ket{E}_{\mathrm{s}} \rightarrow \ket{F}_{\mathrm{s}}$ transition because we assume unidirectional coupling realized by a Faraday rotator or a chiral waveguide. We calculated the time evolution of the stochastic wave function using the quantum Monte Carlo wave function approach. We set the initial state as the ground states for both the source and interface and assumed that the cavity does not have microwave photons, i.e. $\ket{\psi_{\mathrm{initial}}} = \ket{G}_{\mathrm{s}}\ket{G,0_{\mathrm{c}}}_{\mathrm{t}}$. As the excitation laser pulse, we chose a Gaussian pulse with a peak Rabi frequency $\Omega_0$ which is of the same order as $\Gamma_{\mathrm{FG}}^{({\mathrm{t}})}$. The bandwidth of the pulse was set to be smaller than $\Gamma_{\mathrm{FG}}^{({\mathrm{t}})}$. The generated single-photon pulse shape from the $\ket{F}_{\mathrm{s}} \rightarrow \ket{E}_{\mathrm{s}}$ transition was ensured to be Gaussian by keeping $\Omega_0/\Gamma_{\mathrm{FE}}^{({\mathrm{s}})}$ small. $\eta$ was assumed to be 1.0. By assuming a realistic CQD and SC cavity, we set each parameter as follows: $\Gamma_{\mathrm{FG}}^{({\mathrm{t}})}/2\pi = 300{\mskip3mu}{\mathrm{MHz}}$, $\kappa_c/2\pi = 3 {\mskip3mu}{\mathrm{MHz}}$, $g_c/2\pi = 50-400{\mskip3mu}{\mathrm{MHz}}$, $\Gamma_{\mathrm{EG}}^{({\mathrm{t}})}/2\pi = 0-750 {\mskip3mu}{\mathrm{MHz}}$. Since the $\ket{E,1_{\mathrm{c}}}_{\mathrm{t}} \rightarrow \ket{G,1_{\mathrm{c}}}_{\mathrm{t}}$ transition heralds a successful optical-to-microwave photon conversion, we counted this event to estimate the conversion efficiency and rate.
Next, we consider microwave to optical conversion. Here, the collapse operators of this scheme are as follows: $$\begin{aligned}
\hat{C'}_1 &= \sqrt{\Gamma_{\mathrm{FG}}^{({\mathrm{t}})}} \sigma_{\mathrm{GF}}^{({\mathrm{t}})}, \\
\hat{C'}_2 &= \sqrt{\Gamma_{\mathrm{EG}}^{({\mathrm{t}})}} \sigma_{\mathrm{GE}}^{({\mathrm{t}})}, \\
\hat{C'}_3 &= \sqrt{\kappa_{\mathrm{c}}} \hat{a}_{\mathrm{c}}.
\end{aligned}$$ The effective Hamiltonian is then given by $$\begin{aligned}
H_{\mathrm{eff}} =&& \hbar \Omega_{0} (\sigma_{\mathrm{GE}}^{({\mathrm{t}})} + \sigma_{\mathrm{EG}}^{({\mathrm{t}})}) + \hbar g_{\mathrm{c}} (\hat{a}_{\mathrm{c}} \sigma_{\mathrm{FE}}^{({\mathrm{t}})} + \hat{a}_{\mathrm{c}}^\dagger \sigma_{\mathrm{EF}}^{({\mathrm{t}})})\nonumber\\ && - i\hbar \frac{\Gamma_{\mathrm{FG}}^{({\mathrm{t}})}}{2}\sigma_{\mathrm{FF}}^{({\mathrm{t}})}
- i \hbar \frac{\Gamma_{\mathrm{EG}}^{({\mathrm{t}})}}{2}\sigma_{\mathrm{EE}}^{({\mathrm{t}})} -i \hbar \frac{\kappa_{\mathrm{c}}}{2} \hat{a}_{\mathrm{c}}^\dagger \hat{a}_{\mathrm{c}}\end{aligned}$$ The initial state is $\ket{\psi_{\mathrm{initial}}} = \ket{G,1_{\mathrm{c}}}_{\mathrm{t}}$. The parameters used for this simulation are the same as those of the optical to microwave conversion except for $\Omega_0 = \Gamma_{\mathrm{FG}}^{({\mathrm{t}})}/3$. We counted the decay event $\hat{C'}_1$ to estimate the conversion efficiency and rate.
Analytical formula for the optical-to-microwave conversion efficiency
---------------------------------------------------------------------
We consider a simple case where weak coherent field resonantly couples the target CQD with perfect mode matching $\eta = 1.0$ based on van Enk [@vanEnk]. Here, the coherent field satisfies $$b_{\mathrm{in}}(t)\ket{\beta} = \beta {\mathrm{exp}}(-i\omega_{\mathrm{in}}t)\ket{\beta},$$ where $b_{\mathrm{in}}$ is the input field operator, $\omega_{\mathrm{in}}$ is the incident photon frequency, and $\beta$ is the amplitude of the input field. Based on the input-output formalism, we write the mean output field $\braket{b_{\mathrm{out}}}$ generated from the $\ket{F,0_{\mathrm{c}}}_{\mathrm{t}} \rightarrow \ket{G,0_{\mathrm{c}}}_{\mathrm{t}}$ transition as follows: $$\braket{b_{\mathrm{out}}} = \beta+\sqrt{\Gamma_{\mathrm{FG}}^{({\mathrm{t}})}}\braket{\sigma_{\mathrm{FG}}^{({\mathrm{t}})}}.$$ Assuming that $\kappa_{\mathrm{c}}$ is sufficiently smaller than $g_{\mathrm{c}}$ and all the other decay rates, we find that an analytical formula of the mean field is $$\braket{b_{\mathrm{out}}} = \beta-\frac{2\beta\Gamma_{\mathrm{EG}}^{({\mathrm{t}})}}{4g_{\mathrm{c}}^2/\Gamma_{\mathrm{FG}}^{({\mathrm{t}})}+\Gamma_{\mathrm{EG}}^{({\mathrm{t}})}-ig_{\mathrm{c}}}.$$ Here, we define a normalized mean field $\braket{b_{\mathrm{out}}^{\mathrm{n}}} = \braket{b_{\mathrm{out}}}/\beta$. The efficiency of the $\ket{F,0_{\mathrm{c}}}_{\mathrm{t}} \rightarrow \ket{G,0_{\mathrm{c}}}_{\mathrm{t}}$ transition is given by the conjugate product $\braket{b_{\mathrm{out}}^{\mathrm{n\dagger}} b_{\mathrm{out}}^{\mathrm{n}}}$. One can calculate the conversion efficiency $\zeta$ as the complement of this decay efficiency as shown in the main text.
{width="75.00000%"}
Erasure of time-bin information of the herald photons
-----------------------------------------------------
Figure \[S1\] shows a proposed optical setup which compensates for the delay between the two time-bin components of photons emitted from the $\ket{E,1_{\mathrm{c}}}_{\mathrm{t}}\rightarrow \ket{G,0_{\mathrm{c}}}_{\mathrm{t}}$ transition, which allows to remove entanglement between the transferred qubits and the herald photons. The polarization of these photons is defined by the QD selection rules. Here, we note $\ket{H}$ ($\ket{V}$) the horizontally (vertically) polarized state, and we assume that the photons are initially $H$-polarized. A Pockels-cell driven synchronously with the state transfer protocol selectively rotates the early component from $\ket{H}$ to $\ket{V}$. This allows to use a polarized beam splitter to channel the two time components into two separate optical paths, such that the early component is delayed by $t_2 - t_1$ and rotated back to $\ket{H}$. Therefore the two components arrive simultaneously and in the same polarization state onto a second (non-polarized) beam splitter. A single-photon detector placed at one of the output port heralds a successful operation. This process is inherently probabilistic but does not require additional local operation on the qubit. On the other hand, the scheme can be in principle rendered deterministic by using two single-photon detectors of high ($\sim 100\%$) efficiency at the two output ports of the beam splitter. In this case, an additional qubit rotation would have to be performed on the transmon qubit state depending on which detector has clicked, due to the $\pi$ phase difference between the final qubit states left after a photon detection in either of the two output ports of the beam splitter.
Coupling strength
-----------------
We show that the large dipole of the CQD interacting with enhanced cavity vacuum electric field leads to large $g_{\mathrm{c}}$ necessary for the fast and high-efficient conversion. The dipole moment of a CQD is given by $p \sim a \cdot e/2$ where $e$ is a charge of an electron and $a$ is the distance between the two quantum dot layers. We assume a typical distance $a \sim 10{\mskip3mu}\text{nm}$, resulting in $p \sim 8 \times 10^{-28} {\mskip3mu}\text{C$\cdot$m}$. This large dipole moment interacts with the cavity vacuum electric field given by $$E_{\mathrm{rms}} = \sqrt{\frac{\hbar\omega_{\mathrm{c}}}{2\epsilon_{\mathrm{0}}\epsilon_{\mathrm{eff}}V_{\mathrm{eff}}}}
\label{eq:Vrms}$$ where $\omega_{\mathrm{c}}$ is the resonant frequency of the SC cavity. $\epsilon_{\mathrm{0}}$ and $\epsilon_{\mathrm{eff}}$ are the permittivity of vacuum and the effective dielectric constant of the SC cavity, respectively. This vacuum field can also be expressed as a function of the cavity impedance as follows: $$E_{\mathrm{rms}} = \sqrt{\frac{Z_{\mathrm{cav}}\hbar\omega_{\mathrm{c}}^2}{\pi d^2}}.
\label{eq:Evac}$$ where $d$ is the the gap between the center and ground conductor. $Z_{\mathrm{cav}}$ is the characteristic impedance of the SC cavity. For a typical SC cavity, $Z_{\mathrm{cav}}$ is about $50 {\mskip3mu}{\Omega}$. A high impedance cavity can be used to enhance the vacuum field [@Stoc17]. If we assume $Z_{\mathrm{cav}} \sim 2000 {\mskip3mu}\Omega$, $d \sim 7 {\mskip3mu}\mu\text{m}$ and $\omega_{\mathrm{c}}/2\pi \sim11{\mskip3mu}\text{GHz}$, the vacuum field is $E_\text{rms} \sim 3{\mskip3mu}\text{V/m}$. Furthermore, $E_\text{rms}$ is enhanced between the top gate and the ground plane by factor of $\sqrt{\epsilon_{\mathrm{eff}}/\epsilon_{\mathrm{GaAs}}} \cdot d/d'$ (see Fig. 1 in the main text). $\epsilon_{\mathrm{GaAs}}$ is the dielectric constant of GaAs, which is about 13. $\epsilon_{\mathrm{eff}}$ is given by $(\pi c/l\omega_{\mathrm{c}})^2$ where $c$ is the speed of light in vacuum and $l$ is the length of the cavity. We assumed $l \sim3{\mskip3mu}\text{mm}$ and $d' \sim 200 {\mskip3mu}\text{nm}$ as realistic values. The coupling strength is therefore $$g_{\mathrm{c}}/2\pi = \frac{p \cdot \sqrt{\epsilon_{\mathrm{eff}}/\epsilon_{\mathrm{GaAs}}} \cdot d/d' \cdot E_{\mathrm{rms}}}{2\pi\hbar} \sim 200{\mskip3mu}\text{MHz}.
\label{eq:g}$$ This large coupling strength satisfies the conditions for the high conversion efficiency and rate discussed in the paper.
S. J. van Enk, Phys. Rev. A **69**, 043813 (2004).
A. Stockklauser, P. Scarlino, J. V. Koski, S. Gasparinetti, C. K. Andersen, C. Reichl, W. Wegscheider, T. Ihn, K. Ensslin, and A. Wallraff, Phys. Rev. X **7**, 011030 (2017).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
We have used the event generator LUCIAE to analyse NA35 data of the $\bar p$ and $\bar{\Lambda}$ yields, the ratio $\bar{\Lambda}$/$\bar p$, and the transverse mass distributions of $\bar p$ and $\bar{\Lambda}$ in pp and central sulphur-nucleus collisions at 200A GeV. The NA35 data could be reproduced reasonably if one assumes that the s quark pair suppression factor and concerns in nucleus-nucleus collisions are larger than the nucleon-nucleon collision. It seems to indicate that NA35 data might imply the reduction of strangeness suppression in ultrarelativistic nucleus-nucleus collisions comparing to the nucleon-nucleon collision at the same energy. However, the ratio $\bar{\Lambda}$/$\bar p$ approaching to unity in AA collisions comparing to the pp collision does not necessarily mean a flavor symmetry since hadronic rescattering plays a role as well.
PACS number: 25.75.+r
---
**[Strange Antibaryon Production Data and Reduction of Strangeness Suppression in Sulphur-Nucleus Collisions at 200A GeV]{}\
Sa Ben-Hao$^{1,2,3,5}$ and Tai An$^4$\
**
ttttt = tt = 1. CCAST (World Lab.), P. O. Box 8730 Beijing, China.\
2. China Institute of Atomic Energy, P. O. Box 275 (18),\
Beijing, 102413 China.\
3. Department of Physics, University of Hiroshima,\
Higashi-Hiroshima, 739 Japan.\
4. Institute of High Energy Physics, Academia Sinica,\
P. O. Box 918, Beijing, 100039 China.\
5. Institute of Theoretical Physics, Academia Sinica,\
Beijing China.\
0.8cm =0.3cm =0.3cm Strangeness production is expected to be a powerful probe for the mechanism of nucleus-nucleus collisions. Strangeness enhancement, the increased strangeness particle production in nucleus-nucleus collisions comparing to the nucleon-nucleon collision, is predicted to be a sensitive signature of the QGP formation in the ultrarelativistic nucleus-nucleus collisions \[1\].
The first experimental results of the enhanced production of strange particles in nucleus-nucleus collisions at 200A GeV incident energy was reported six years ago \[2\]. Later on this enhancement was confirmed by more and more experiments \[3-7\].
Strange antibaryon production, in particular, might bring messages of the equilibrium and flavour symmetry of quarks in ultrarelativistic nucleus-nucleus collisions. It has been estimated \[8\] that if there is QGP formed in the ultrarelativistic nucleus-nucleus collisions, strange (antistrange) quark pair might be copiously reproduced, resulting in an approximate flavour symmetry among u, d and s quarks \[9\]. Thus the ratio of $\bar{\Lambda}$/$\bar p$ should approach unity, since $\bar{\Lambda}$ is composed of $\bar u\bar d\bar s$ and $\bar p$ is $\bar u\bar d\bar d$.
Recently published NA35 data of antibaryon production in sulphur-nucleus collisions at 200A GeV \[7\] did really observe that the ratios $\bar{\Lambda}$ /$\bar p$ are all approaching, even over, unity and far exceeding the corresponding value of $\sim$0.25 in the nucleon-nucleon collision at the same energy. Although the NA35 data of the $\bar{\Lambda}$/$\bar p$ has been fairly reproduced by RQMD 1.08 \[10,7\] with fusion of overlpping strings into a color rope and with hadronic rescattering, the data of transverse mass distributions of $\bar p$ and $\bar{\Lambda}$ have not been explained yet, the physics behind the data have not been exposed especially.
Based on the idea that the strangeness enhancement in ultrarelativistic nucleus-nucleus collision compared to the nucleon-nucleon collision at the same energy could be investigated via the reduction of s quark suppression \[11-14\], in this letter the event generator LUCIAE \[15\] is used to analyse the NA35 data and to explore the physics behind the data.
LUCIAE was updated based on FRITIOF 7.02 \[16\] by taking into account the collective string interaction \[17-18\] during the emission of gluon bremsstrahlung and the rescattering of produced particles \[19\]. One knows that in FRITIOF 7.02 \[16\], two colliding hadrons are excited due to the longitudinal momentum transfer and/or a Rutherford parton scattering. The highly excited states will emit bremsstrahlung gluons according to the soft radiation model. The deexcited states are then treated as Lund strings allowing to decay into the final hadronic state due to the Lund fragmentation scheme. However, in the ultrarelativistic nucleus-nucleus collisions there are generally many excited strings formed close by each other. These strings would behave like vortex lines in a color superconducting QCD vacuum and will interact with each other (as the repulsive interaction suffered by the “ordinary” vortex lines in a type II superconductor). This kind of collective interaction among strings, not included in FRITIOF 7.02, is depicted by the firecracker model \[15,18\] dealing with the large p$_t$ gluon (firecracker gluon) production from the collective interaction among strings. The firecracker gluon will work as a gluon-kink excitation on a string, which tends to shorten the longitudinal size of the string and then brings about the increasing of the string tension (thus it should be regarded as an effective string tension). In firecracker model it is assumed that the groups of neighboring strings might form interacting quantum states, the large common energy density (corresponding to the collective interaction among strings) might then affect the emission of gluonic bremsstrahlung \[17-18\].
A rescattering model has been developed to describe the reinteraction of produced particles, from FRITIOF event generator, with each other and with the participant and the spectator nucleons \[19\]. In the model, the produced particles and the participant (wounded) nucleons, from FRITIOF event generator, are distributed randomly in the geometrical overlapping region between the projectile and the target nuclei under a given impact parameter. The target (projectile) spectator nucleons are distributed randomly outside the overlapping region and inside the target (projectile) sphere. A rescattering cascade process has evolved since then, cf. Ref. \[15,19\] for the detail. The considered inelastic reactions, concerning strangeness, are here cataloged into:
tttttttttttttttttttttttttttttttttttttttttttttttttttt= $\pi\pi \rightleftharpoons k\bar{k}$;\
$\pi N \rightleftharpoons kY$, $\pi\bar{N} \rightleftharpoons \bar{k}\bar{Y}$;\
$\pi Y \rightleftharpoons k\Xi$, $\pi\bar{Y} \rightleftharpoons \bar{k}\bar{\Xi}$;\
$\bar{k}N \rightleftharpoons \pi Y$ , $k\bar{N} \rightleftharpoons \pi\bar{Y}$;\
$\bar{k}Y \rightleftharpoons \pi\Xi$, $k\bar{Y} \rightleftharpoons \pi\bar{\Xi}$;\
$\bar{k}N \rightleftharpoons k\Xi$, $k\bar{N} \rightleftharpoons \bar{k}\bar{\Xi}$;\
$\pi\Xi \rightleftharpoons k\Omega^- $, $\pi\bar{\Xi} \rightleftharpoons \bar{k}\overline{\Omega^-}$;\
$k\bar{\Xi} \rightleftharpoons \pi\overline{\Omega^-}$, $\bar{k}\Xi \rightleftharpoons \pi\Omega^-$;\
$\bar{N}N$ annihilation;\
$\bar{Y}N$ annihilation;\
where $Y$ refers to the $\Lambda$ or $\Sigma$ and $\Xi$ refers to the $\Xi^-$ or $\Xi^0$. There are 299 inelastic reactions involved altogether. As the reactions introduced above do not make up the full inelastic cross section, the remainder is again treated as elastic scattering \[19\]. The cross section of $\pi\pi \rightarrow k\bar{k}$ is taken to be 2.0 mb as usual \[10\]. The isospin averaged parametrization of Ref. \[9\] is adopted for the cross sections of the reactions $\pi N \rightarrow kY$ and for the other strange quark production reactions. Of course, the difference in threshold energy among reactions is taken into account. Following Ref. \[9\], the cross section of strange quark exchange reaction, $\bar{k}N\rightarrow\pi Y
$ for instance, is assumed to be equal to ten times the value of the cross section of the strangeness production reaction. As for the cross section of the reverse reaction, the detailed balance assumption \[20\] is required.
The cross sections of the inelastic reactions given by the isospin averaged parameterization formulas of \[9\] for the $\pi N \rightarrow kY$ decrease exponentially with the CMS energy of the two colliding particles. But we know that the total inelastic cross section of the $\pi N$ is approximately energy -independent, which means that more inelastic channels would occur like $\pi^-$ + p $\rightarrow K^{*0}$ + $\Lambda \rightarrow$ K + $\pi + \Lambda$ and $\pi^-$ + p $\rightarrow K^{*0}$ + $\Sigma^0 \rightarrow$ K + $\pi + \Sigma^0$ etc., when the CMS energy of the two colliding particles is increasing. In some sense the reactions we listed above should be looked upon as ’representative channels’ of certain types of reactions. Therefore instead of applying energy-dependent inelastic cross section, we could alternatively give a constant ’effective cross section’ to the reaction $\pi N \rightarrow kY$ \[15,19\].
In JETSET routine, which runs together with LUCIAE event generator, there are model parameters parj(2) (or ’s’) and parj(3) which are responsible for the s quark suppression and related to the effective string tension. ’s’ refers to the suppression of s quark pair production in the color field compared to u or d pair production. parj(3) is the extra suppression of strange diquark production compared to the normal suppression of strange quark pair. Besides ’s’ and parj(3) there is parj(1), which stands for the suppression of diquark- antidiquark pair production in the color field compared to the quark-antiquark pair production and is related to the effective string tension as well. Originally, in the LUND fragmentation scheme ’s’ was assumed to be a ’constant’ and this assumption was confirmed by the e$^+$e$^-$ physics from the low energies to Z$^0$ energy. However, there are experimental facts that this parameter is energy dependent when the fragmentation scheme is applied to the Deep Inelastic Scattering (DIS) and hh collisions. In DIS experiments, ’s’ is seen to be rising up with the increase of energy from 0.15 at $\sqrt{S}$=5 GeV to 0.35 at $\sqrt{S}$=20 GeV \[21\]. An energy dependent ’s’ for hh collisions has been known for many years. It varies from 0.2 at the ISR energy to about 0.4 at $\sqrt{S}$= 1.8 TeV \[21\]. In addition, this parameter is also observed strongly phase space dependence in DIS and hh collisions. The corresponding value of ’s’ runs from 0.15 to 0.55 with a mean value close to 0.3 \[22\]. Thus the idea of the reduction of s quark suppression in nucleus-nucleus collisions comparing to the nucleon-nucleon collision is possible to be executed via changing the ’s’ and concerning parameters in JETSET routine.
From LUND string model point of view s quark suppression factor could be related to the effective string tension. It is reasonable to expected that the more violent collision, the stronger collective interaction among strings, the larger effective string tension and then the larger s quark suppression factor. Generally speaking, a nucleus-nucleus collision is more violent than the nucleon-nucleon collision at the same interaction energy. That might be the reason of the reduction of s quark suppression in nucleus- nucleus collisions comparing to the nucleon-nucleon collision at the same energy.
The purpose of this letter is to explore the physics behind the NA35 data and not to fit the data as good as possible. We fix a set of somewhat larger parameters (relative to the defaults: ’s’=0.3, parj(3)=0.4 and parj(1)=0.1) of ’s’=0.4, parj(3)=0.5333 and parj(1)=0.1333 (referred to as parameter set 1, later on) in the calculations of nucleus-nucleus collisions and a set of somewhat smaller parameters of ’s’=0.2, parj(3)=0.2666 and parj(1)=0.06666 ( referred to as parameter set 2) in the calculation of the nucleon-nucleon collision . These results are given in table 1 and figure 1 and obtained from the average over 10$^5$ generated events for pp, 2000 events for $\bar{\Lambda}$ and $\bar{p}$ yield in AA, and 3000 for m$_t$ distribution in AA. The y and p$_t$ acceptances are set to be the same as the experiment \[7\], correspondingly. Of course, the two sets of parameters above imply that one makes an assumption there that those three parameters are linearly proportional to the effective string tension in their responsibilities to the final hadronic production.
Tab. 1 gives the rapidity densities of $\bar{p}$, $\bar{\Lambda}$, and $h^-$ and the ratio $\bar{\Lambda}$/$\bar{p}$ in p + p and S + S, S + Ag, and S + Au central collisions at 200A GeV. The rapidity acceptance is 3 $\leq$ y $\leq$ 4. The results of LUCIAE are comparable with the corresponding NA35 data. The LUCIAE results of $\bar{\Lambda}$ /$\bar{p}$ seem going down monotonously from S + S to S + Au, which might attribute to the fact that one did not consider the possible difference of the reduction of strangeness suppression among them. The results of LUCIAE and RQMD are comparable with each other, since both of them do have contained the collective interaction among strings and the hadronic rescattering \[23\]. The hadronic rescattering, in LUCIAE and RQMD, is expected to play a similar role in the final state distributions though details of the hadronic rescattering are not the same in these two generators. In RQMD the effect of highly dense strings in ultrarelativistic nucleus-nucleus collisions is considered via the fusion of strings into a color rope, which brings the enhanced production of strange antibaryon. In LUCIAE the highly dense strings are considered by the firecracker model \[15,18\], where the effect of highly dense strings is depicted analytically only in the emission of gluon bremsstrahlung. It leaves the possibility to enlarge the s quark pair suppression factor and concerns in JETSET routine to bring the enhanced production of strange antibaryon.
Fig. 1 gives $\bar{p}$ and $\bar{\Lambda}$ transverse mass distributions in central S + S (upper frame), S + Ag (middle frame), and S + Au (lower frame) reactions at 200A GeV, respectively. In this figure, the open squares and open circles are the NA35 data of $\bar{p}$ and $\bar{\Lambda}$, respectively and the corresponding results of LUCIAE are given by full squares and full circles. The solid lines are the exponential fits of the NA35 data, cf. Ref. 7 for details. One sees from this figure that the agreement between the NA35 data and the results of LUCIAE is fair for $\bar{p}$ and reasonably good for $\bar{\Lambda}$.
Table 2 gives the average yield (in full phase space) of $\Lambda$, $\bar{\Lambda}$, K$^+$ and K$^0_s$, in p+p and central S + S and S + Ag collisions at 200A GeV. In this table the data were taken from \[5\] besides p+p, which was taken from \[24\]. One sees from this table that although we are here aiming at analysing strange antibaryon production data, the strange particle production data are also reproduced reasonably good at the same time.
In order to distinguish the role of the s quark pair suppression factor and concerns from the hadronic rescattering and the role of ’s’ and parj(3) from parj(1) one calculates the results of table 3 and figure 2. In table 3 one compares the results of the rapidity densities of $\bar{p}$, $\bar{\Lambda}$, and $h^-$ and the ratio $\bar{\Lambda}$/$\bar{p}$ in S + S central collisions at 200A GeV calculated by using different parameters and with or without rescattering: ’LUCIAE 1’: parameter set 2 and without rescattering; ’LUCIAE 2’: parameter set 2 and with rescattering; ’LUCIAE 3’: parameter set 1 but the value of parj(1) is changed to 0.06666 and with rescattering; and ’LUCIAE 4’: parameter set 1 and with rescattering. One sees from this table that the ’LUCIAE 1’ result of $\bar{\Lambda}$/$\bar{p}$ is close to the corresponding result in the pp collision, as it should be. For results of ’LUCIAE 2’, $\bar{\Lambda}$ yield is close to the corresponding result of ’LUCIAE 1’ but $\bar{p}$ yield is lower than ’LUCIAE 1’, which brings about the larger ratio $\bar{\Lambda}$/$\bar{p}$ in ’LUCIAE 2’ than in ’LUCIAE 1’. Here one knows that the hadronic rescattering seems to play nearly null role for ${\bar\Lambda}$ yield, since ${\bar\Lambda}$ production in rescattering is mainly via $\pi\bar{N} \rightleftharpoons\bar{k}\bar{Y}$ and $k\bar{N} \rightleftharpoons \pi\bar{Y}$, that is nearly canceled by the corresponding inverse reactions and $\bar{Y}N$ annihilations especially. On the contrary, the rescattering tends to reduce $\bar{p}$ multiplicity through the $\bar{p}p$ annihilation. However, relying on hadronic rescattering only is not possible to have the ratio $\bar{\Lambda}$/$\bar{p}$ approaching to unity. Furthermore, comparing the results of ’LUCIAE 3’ with ’LUCIAE 4’ one knows that although by increasing parj(1) the yield of $\bar{p}$ and $\bar{\Lambda}$ are increased the ratio $\bar{\Lambda}$/$\bar{p}$ is hardly affected.
Figure 2 gives the transverse mass distributions of ${\bar\Lambda}$ (upper frame) and $\bar{p}$ (lower frame) in S + S reaction at 200A GeV, respectively. The full triangles, the open squares, and the full circles are calculated individually for the case 1: parameter set 2 and with rescattering; the case 2: parameter set 1 and without rescattering; and the case 3: parameter set 1 and with rescattering. The rapidity acceptance are all set to be 1 $\leq$ y $\leq$ 3. One knows from this figure again that the hadronic rescattering seems to play nearly null role for ${\bar\Lambda}$ production, but hadronic rescattering tends to reduce $\bar{p}$ multiplicity, at lower m$_t$ region especially, through the $\bar{p}p$ annihilation.
In summary, we have roughly reproduced the NA35 data of $\bar p$, $\bar{\Lambda}$ yields , the corresponding ratio $\bar{\Lambda}$/$\bar p$, and the transverse mass distributions of $\bar p$ and $\bar{\Lambda}$ in pp and central S + S, S + Ag, and S + Au collisions at 200A GeV, using event generator LUCIAE and via increasing s quark pair suppression factor and concerns in AA collisions. It seems to be true that the NA35 data \[5,7\] imply the reduction of strangeness suppression in ultrarelativistic nucleus-nucleus collisions comparing to the pp collision at the same energy. Although the ratio $\bar{\Lambda}$/$\bar p$ approaching to unity is the result of reduction of s quark suppression, it does not necessarily mean a flavor symmetry, since reproducing the NA35 data does not require the ’s’ value should be equal to one and the hadronic rescattering also plays a role in enlarging the ratio $\bar{\Lambda}$/$\bar p$. However, it is absolutely needed to have a further study for the microscopic mechanism of the reduction of strangeness suppression before making a conclusion using strangeness enhancement as a signal of QGP.
Acknowledgment
We are grateful to J. Eschke and D. R$\ddot{o}$hrich for providing the NA35 data. Thanks go to O. Miyamura, K. Kumagai and T. Sasaki for discussions and helps. SBH thanks Department of Physics, University of Hiroshima for hospitality and JSPS for financial support staying in Japan to finish most of the calculations. This work is supported by the national Natural Science Foundation of China as a cooperation program between NSFC of China and JSPS of Japan.
References
1. M. Jacob and J. Tran Van., Phys. Rep., $\bf{88}$, 321(1982);\
J. Rafelski, Phys. Rep., $\bf{88}$, 331(1982);\
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J. Ellis and U. Heinz, Phys. Lett., $\bf{B233}$, 223(1989).
2. J. Bartke, et al., NA35 Colla., Z. Phys., $\bf{C48}$, 191(1990).
3. E. Andersen, et al., NA36 Colla., Phys. Lett., $\bf{B316}$, 603(1993).
4. S. Abatzis, et al., WA85 Colla., Phys. Lett., $\bf{B244}$, 127(1990).
5. T. Alber, et al., NA35 Colla., Z. Phys., $\bf{C64}$, 195(1994).
6. E. Andersen, et al., NA36 Colla., Nucl. Phys., $\bf{A590}$, 291c(1995).
7. T. Alber, et al., NA35 Colla., Phys. Lett., $\bf{B366}$, 56(1996).
8. E. V. Shuryak, Phys. Rep., $\bf{61}$, 71(1980); $\bf{115}$, 151(1984).
9. P. Koch, B. M$\ddot{u}$ller, and J. Rafelski, Phys. Rep., $\bf{142}$, 167 (1986).
10. H. Sorge, Phys. Rev., $\bf{C52}$, 3291(1995).
11. Sa Ben-Hao and Tai An, Phys. Rev., $\bf{C55}$, 2010 (1997)
12. A. K. Wr$\acute{o}$blewski, Acta Phys. Pol., $\bf{B16}$, 379(1985).
13. H. Bialkowska, M. Ga$\acute{z}$dzicki, W. Retyk, and E. Skrzypczak, Z. Phys., $\bf{C55}$, 491(1992).
14. M. Ga$\acute{z}$dzicki, and U. Heinz, Phys. Rev., $\bf{C54}$, 1496(1996).
15. Sa Ben-Hao and Tai An, Comp. Phys. Commu., $\bf{90}$, 121(1995).
16. B. Andersson, G. Gustafson, and Hong Pi, Z. Phys., $\bf{C57}$, 485(1993).
17. B. Andersson, Phys. Lett., $\bf{B256}$, 337(1991).
18. B. Andersson and An Tai, Z. Phys., $\bf{C71}$, 155(1996).
19. Sa Ben-Hao, Wang Zhong-Qi, Zhang Xiao-Ze, Song Guang, Lu Zhong-Dao, and Zheng Yu-Ming, Phys. Rev., $\bf{C48}$, 2995(1993);\
Sa Ben-Hao, Tai An, and Lu Zhong-Dao, Phys. Rev., $\bf{C52}$, 2069(1995);\
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20. G. Bertsch, and S. Das Gupta, Phys. Rep., $\bf{160}$, 189(1988).
21. A. K. Wr$\acute{o}$blewski, Proceedings of the 25th International conference on HEP, p. 125, Singapore, 1990.
22. ZEUS Colla., Z. Phys., $\bf{C68}$, 29(1995);\
Fermilab E665 Colla., Z. Phys., $\bf{C61}$, 539(1994).
23. H. Sorge, Z. Phys., $\bf{C67}$, 479(1995).
24. M. Ga$\acute{z}$dzicki, and O. Hansen, Nucl. Phys., $\bf{A528}$, 754 (1991).
Figure Captions
> Fig. 1 Transverse mass distributions of $\bar p$ (3 $\leq$ y $\leq$ 4) and $\bar\Lambda$ produced in central S+S (upper frame, 1 $\leq$ y $\leq$ 3), S+Ag (middle frame, 1 $\leq$ y $\leq$ 3), and S+Au (lower frame, 3 $\leq$ y $\leq$ 5) collisions at 200A GeV. The open squares and open circles are the NA35 data of $\bar p$ and $\bar\Lambda$, respectively and the full squares and full circles are the corresponding results of LUCIAE.
>
> Fig. 2 Transverse mass distributions (1 $\leq$ y $\leq$ 3) of $\bar\Lambda$ (upper frame) and $\bar p$ (lower frame) in S+S reaction at 200A GeV. The full triangles, the open squares, and the full circles are calculated individually for the case 1: parameter set 2 (’s’=0.2, parj(3)=0.2666 and parj(1)=0.06666) and with rescattering; the case 2: parameter set 1 (’s’=0.4, parj(3)=0.5333, parj(1)=0.1333) and without rescattering; and the case 3: parameter set 1 and with rescattering.
[cccccc]{}\
\
reaction & &$\bar p$ & $\bar\Lambda$ & $\bar\Lambda$/$\bar p$ &h$^-$\
p + p& data & 0.02$\pm$0.02 & 0.005$\pm$0.002 & 0.25$\pm$0.1 & 0.74$\pm$0.04\
&LUCIAE & 0.017 & 0.0035 & 0.21 & 0.60\
&RQMD & 0.015 &0.005 & 0.3 & $-$\
S + S& data & 0.4$\pm$0.1 & 0.76$\pm$0.16 & 1.9$^{+0.7}_{-0.6}$ &25$\pm$1\
central &LUCIAE & 0.65 & 0.66 & 1.03 & 23.7\
&RQMD & 0.7 & 0.75 & 1.1 & $-$\
S + Ag& data & 0.6$\pm$0.2 & 0.75$\pm$0.19 & 1.3$^{+0.7}_{-0.5}$ & 40$\pm$2\
central&LUCIAE & 1.08 & 0.99 & 0.91 & 39.3\
&RQMD & 1.0 & 0.9 & 0.9 & $-$\
S + Au& data & 0.7$\pm$0.2 & 0.75$\pm$0.1 & 1.1$^{+0.4}_{-0.3}$ & 47$\pm$5\
central&LUCIAE & 1.09 & 0.91 & 0.84 & 42.7\
&RQMD & 1.4 & 1.2 & 0.9 & $-$\
[cccccc]{}\
\
reaction & & $\Lambda$ & $\bar\Lambda$ & K$^+$ & K$^0_s$\
p + p& data & 0.096$\pm$0.015 & 0.013$\pm$0.005 & $-$ & 0.17$\pm$0.01\
& LUCIAE & 0.10 & 0.011 & 0.22 & 0.16\
S + S& data & 9.4$\pm$1.0 & 2.2$\pm$0.4 & $-$ & 10.5$\pm$1.7\
central & LUCIAE & 8.0 & 1.9 & 11.9 & 9.83\
S + Ag& data & 15.2$\pm$1.2 & 2.6$\pm$0.3 & $-$ & 15.5$\pm$1.5\
central & LUCIAE & 13.9 & 3.1 & 21.3 & 17.6\
[cccccc]{}\
\
reaction & &$\bar p$ & $\bar\Lambda$ & $\bar\Lambda$/$\bar p$ &h$^-$\
p + p& data & 0.02$\pm$0.02 & 0.005$\pm$0.002 & 0.25$\pm$0.1 & 0.74$\pm$0.04\
&LUCIAE & 0.017 & 0.0035 & 0.21 & 0.60\
&RQMD & 0.015 &0.005 & 0.3 & $-$\
S + S& data & 0.4$\pm$0.1 & 0.76$\pm$0.16 & 1.9$^{+0.7}_{-0.6}$ &25$\pm$1\
central &LUCIAE 1$^a$ & 0.62 & 0.17 & 0.28 & 25.2\
&LUCIAE 2$^b$ & 0.42 & 0.17 & 0.40 & 24.8\
&LUCIAE 3$^c$ & 0.36 & 0.38 & 1.05 & 24.7\
&LUCIAE 4$^d$ & 0.65 & 0.66 & 1.03 & 23.7\
\
\
\
\
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'Alexander Grigor’yan and Meng Yang'
title: 'Local and Non-Local Dirichlet Forms on the Sierpiński Carpet'
---
[^1] [^2] [^3] [^4]
Introduction
============
Sierpiński carpet (SC) is a typical example of non p.c.f. (post critically finite) self-similar sets. It was first introduced by Wacław Sierpiński in 1916 which is a generalization of Cantor set in two dimensions, see Figure \[fig\_SC\].
![Sierpiński Carpet[]{data-label="fig_SC"}](carpet){width="50.00000%"}
SC can be obtained as follows. Divide the unit square into nine congruent small squares, each with sides of length $1/3$, remove the central one. Divide each of the eight remaining small squares into nine congruent squares, each with sides of length $1/9$, remove the central ones, see Figure \[fig\_construction\]. Repeat above procedure infinitely many times, SC is the compact connected set $K$ that remains.
(0,0)–(9,0)–(9,9)–(0,9)–cycle; (3,3)–(6,3)–(6,6)–(3,6)–cycle;
(11,0)–(20,0)–(20,9)–(11,9)–cycle; (14,3)–(17,3)–(17,6)–(14,6)–cycle; (12,1)–(13,1)–(13,2)–(12,2)–cycle; (15,1)–(16,1)–(16,2)–(15,2)–cycle; (18,1)–(19,1)–(19,2)–(18,2)–cycle; (12,4)–(13,4)–(13,5)–(12,5)–cycle; (18,4)–(19,4)–(19,5)–(18,5)–cycle; (12,7)–(13,7)–(13,8)–(12,8)–cycle; (15,7)–(16,7)–(16,8)–(15,8)–cycle; (18,7)–(19,7)–(19,8)–(18,8)–cycle;
In recent decades, self-similar sets have been regarded as underlying spaces for analysis and probability. Apart from classical Hausdorff measures, this approach requires the introduction of Dirichlet forms. Local regular Dirichlet forms or associated diffusions (also called Brownian motion (BM)) have been constructed in many fractals, see [@BP88; @BB89; @Lin90; @KZ92; @Kig93; @Bar98; @Kig01]. In p.c.f. self-similar sets including Sierpiński gasket, this construction is relatively transparent, while similar construction on SC is much more involved.
For the first time, BM on SC was constructed by Barlow and Bass [@BB89] using *extrinsic* approximation domains in ${\mathbb{R}}^2$ (see black domains in Figure \[fig\_construction\]) and time-changed reflected BMs in those domains. Technically, [@BB89] is based on the following two ingredients in approximation domains:
(a) \[enum\_a\] Certain resistance estimates.
(b) \[enum\_b\] Uniform Harnack inequality for harmonic functions with Neumann boundary condition.
For the proof of the uniform Harnack inequality, Barlow and Bass used certain probabilistic techniques based on Knight move argument (this argument was generalized later in [@BB99a] to deal also with similar problems in higher dimensions).
Subsequently, Kusuoka and Zhou [@KZ92] gave an alternative construction of BM on SC using *intrinsic* approximation graphs and Markov chains in those graphs. However, in order to prove the convergence of Markov chains to a diffusion, they used the two aforementioned ingredients of [@BB89], reformulated in terms of approximation graphs.
However, the problem of a purely analytic construction of a local regular Dirichlet form on SC (similar to that on p.c.f. self-similar sets) has been open until now and was explicitly raised by Hu [@Hu13]. The main result of this paper is a direct purely *analytic* construction of a local regular Dirichlet form on SC.
The most essential ingredient of our construction is a certain resistance estimate in approximation graphs which is similar to the ingredient (\[enum\_a\]). We obtain the second ingredient—the uniform Harnack inequality in approximation graphs as a consequence of (\[enum\_a\]). A possibility of such an approach was mentioned in [@BCK05]. In fact, in order to prove a uniform Harnack inequality in approximation graphs, we extend resistance estimates from finite graphs to the infinite graphical SC (see Figure \[fig\_graphSC\]) and then deduce from them a uniform Harnack inequality-first on the infinite graph and then also on finite graphs. By this argument, we avoid the most difficult part of the proof in [@BB89].
in [0,1,...,27]{} (,0)–(,28);
in [0,1,...,27]{} (0,)–(28,);
in [0,1,2]{} in [0,1,2]{} (9\*+3,9\*+3)–(9\*+6,9\*+3)–(9\*+6,9\*+6)–(9\*+3,9\*+6)–cycle;
(9,9)–(18,9)–(18,18)–(9,18)–cycle;
in [0,1,...,27]{} in [0,0.5,1,...,27.5]{} (,) circle (0.08);
in [0,1,...,27]{} in [0,0.5,1,...,27.5]{} (,) circle (0.08);
(9.25,9.25)–(17.75,9.25)–(17.75,17.75)–(9.25,17.75)–cycle;
in [0,1,2]{} in [0,1,2]{} (9\*+3.25,9\*+3.25)–(9\*+5.75,9\*+3.25)–(9\*+5.75,9\*+5.75)–(9\*+3.25,9\*+5.75)–cycle;
The self-similar local regular Dirichlet form ${\mathcal{E}}_{{\mathrm{loc}}}$ on SC has the following self-similarity property. Let $f_0,\ldots,f_7$ be the contraction mappings generating SC. For all function $u$ in the domain ${\mathcal{F}}_{{\mathrm{loc}}}$ of ${\mathcal{E}}_{{\mathrm{loc}}}$ and for all $i=0,\ldots,7$, we have $u\circ f_i\in{\mathcal{F}}_{{\mathrm{loc}}}$ and $${\mathcal{E}}_{{\mathrm{loc}}}(u,u)=\rho\sum_{i=0}^7{\mathcal{E}}_{{\mathrm{loc}}}(u\circ f_i,u\circ f_i).$$ Here $\rho>1$ is a parameter from the aforementioned resistance estimates, whose exact value remains still unknown. Barlow, Bass and Sherwood [@BB90; @BBS90] gave two bounds as follows:
- $\rho\in[7/6,3/2]$ based on shorting and cutting technique.
- $\rho\in[1.25147,1.25149]$ based on numerical calculation.
McGillivray [@McG02] generalized above estimates to higher dimensions.
The heat semigroup associated with ${\mathcal{E}}_{{\mathrm{loc}}}$ has a heat kernel $p_t(x,y)$ satisfying the following estimates: for all $x,y\in K,t\in(0,1)$ $$\label{eqn_hk}
p_t(x,y)\asymp\frac{C}{t^{\alpha/\beta^*}}\exp\left(-c\left(\frac{|x-y|}{t^{1/\beta^*}}\right)^{\frac{\beta^*}{\beta^*-1}}\right),$$ where $\alpha=\log8/\log3$ is the Hausdorff dimension of SC and $$\label{eqn_beta_up}
\beta^*:=\frac{\log(8\rho)}{\log3}.$$ The parameter $\beta^*$ is called the *walk dimension of BM* and is frequently denoted also by $d_w$. The estimates (\[eqn\_hk\]) were obtained by Barlow and Bass [@BB92; @BB99a] and by Hambly, Kumagai, Kusuoka and Zhou [@HKKZ00]. Equivalent conditions of sub-Gaussian heat kernel estimates for local regular Dirichlet forms on metric measure spaces were explored by many authors, see Andres and Barlow [@AB15], Grigor’yan and Hu [@GH14a; @GH14b], Grigor’yan, Hu and Lau [@GHL10; @GHL15], Grigor’yan and Telcs [@GT12]. We give an alternative proof of the estimates (\[eqn\_hk\]) based on the approach developed by the first author and others.
Consider the following stable-like non-local quadratic form $$\begin{aligned}
&{\mathcal{E}}_\beta(u,u)=\int_K\int_K\frac{(u(x)-u(y))^2}{|x-y|^{\alpha+\beta}}\nu({\mathrm{d}}x)\nu({\mathrm{d}}y),\\
&{\mathcal{F}}_\beta={\left\{u\in L^2(K;\nu):{\mathcal{E}}_\beta(u,u)<+\infty\right\}},
\end{aligned}$$ where $\alpha=\mathrm{dim}_{\mathcal{H}}K$ as above, $\nu$ is the normalized Hausdorff measure on $K$ of dimension $\alpha$, and $\beta>0$ is so far arbitrary. Then the *walk dimension of SC* is defined as $$\label{eqn_beta_low}
\beta_*:=\sup{\left\{\beta>0:({\mathcal{E}}_\beta,{\mathcal{F}}_\beta)\text{ is a regular Dirichlet form on }L^2(K;\nu)\right\}}.$$ Using the estimates (\[eqn\_hk\]) and subordination technique, it was proved in [@Pie00; @GHL03] that $({\mathcal{E}}_\beta,{\mathcal{F}}_\beta)$ is a regular Dirichlet form on $L^2(K;\nu)$ if $\beta\in(0,\beta^*)$ and that ${\mathcal{F}}_\beta$ consists only of constant functions if $\beta>\beta^*$, which implies the identity $$\beta_*=\beta^*.$$ In this paper, we give another proof of this identity without using the estimates (\[eqn\_hk\]), but using directly the definitions (\[eqn\_beta\_up\]) and (\[eqn\_beta\_low\]) of $\beta^*$ and $\beta_*$.
Barlow raised in [@Bar13] a problem of obtaining bounds of the walk dimension $\beta^*$ of BM without using directly ${\mathcal{E}}_{{\mathrm{loc}}}$. We partially answer this problem by showing that $$\beta_*\in\left[\frac{\log\left(8\cdot\frac{7}{6}\right)}{\log3},\frac{\log\left(8\cdot\frac{3}{2}\right)}{\log3}\right],$$ which gives then the same bound for $\beta^*$. However, the same bound for $\beta^*$ follows also from the estimate $\rho\in[7/6,3/2]$ mentioned above. We hope to be able to improve this approach in order to get better estimates of $\beta_*$ in the future.
Using the estimates (\[eqn\_hk\]) and subordination technique, it was proved in [@Pie08] that $$\label{eqn_approximation}
\varliminf_{\beta\uparrow\beta^*}(\beta^*-\beta){\mathcal{E}}_\beta(u,u)\asymp{\mathcal{E}}_{{\mathrm{loc}}}(u,u)\asymp\varlimsup_{\beta\uparrow\beta^*}(\beta^*-\beta){\mathcal{E}}_\beta(u,u)$$ for all $u\in{\mathcal{F}}_{{\mathrm{loc}}}$. This is similar to the following classical result $$\lim_{\beta\uparrow2}(2-\beta)\int_{{\mathbb{R}}^n}\int_{{\mathbb{R}}^n}\frac{(u(x)-u(y))^2}{|x-y|^{n+\beta}}{\mathrm{d}}x{\mathrm{d}}y=C(n)\int_{{\mathbb{R}}^n}|\nabla u(x)|^2{\mathrm{d}}x,$$ for all $u\in W^{1,2}({\mathbb{R}}^n)$, where $C(n)$ is some positive constant (see [@FOT11 Example 1.4.1]). We reprove (\[eqn\_approximation\]) as a direct corollary of our construction without using the estimates (\[eqn\_hk\]).
The idea of our construction of ${\mathcal{E}}_{{\mathrm{loc}}}$ is as follows. In the first step, we construct another quadratic form $E_\beta$ equivalent to ${\mathcal{E}}_\beta$ and use it to prove the identity $$\label{eqn_walk}
\beta_*=\beta^*:=\frac{\log(8\rho)}{\log3}.$$ It follows that ${\mathcal{E}}_\beta$ is a regular Dirichlet form for all $\beta\in(\alpha,\beta^*)$. Then, we use another quadratic form ${\mathfrak{E}}_\beta$, also equivalent to ${\mathcal{E}}_\beta$, and define ${\mathcal{E}}$ as a $\Gamma$-limit of a sequence ${\left\{(\beta^*-\beta_n){\mathfrak{E}}_{\beta_n}\right\}}$ with $\beta_n\uparrow\beta^*$. We prove that ${\mathcal{E}}$ is a regular closed form, where the main difficulty lies in the proof of the uniform density of the domain ${\mathcal{F}}$ of ${\mathcal{E}}$ in $C(K)$. However, ${\mathcal{E}}$ is not necessarily Markovian, local or self-similar. In the last step, ${\mathcal{E}}_{{\mathrm{loc}}}$ is constructed from ${\mathcal{E}}$ by means of an argument from [@KZ92]. Then ${\mathcal{E}}_{{\mathrm{loc}}}$ is a self-similar local regular Dirichlet form with a Kigami’s like representation (\[eqn\_Kigami\]) which is similar to the representations in Kigami’s construction on p.c.f. self-similar sets, see [@Kig01]. We use the latter in order to obtain certain resistance estimates for ${\mathcal{E}}_{{\mathrm{loc}}}$, which imply the estimates (\[eqn\_hk\]) by [@GHL14; @GH14a].
Let us emphasize that the resistance estimates in approximation graphs and their consequence—the uniform Harnack inequality, are mainly used in order to construct one *good* function on $K$ with certain energy property and separation property, which is then used to prove the identity (\[eqn\_walk\]) and to ensure the non-triviality of ${\mathcal{F}}$.
An important fact about the local regular Dirichlet form ${\mathcal{E}}_{{\mathrm{loc}}}$ is that this Dirichlet form is a resistance form in the sense of Kigami whose existence gives many important corollaries, see [@Kig01; @Kig03; @Kig12].
Statement of the Main Results
=============================
Consider the following points in ${\mathbb{R}}^2$: $$p_0=(0,0),p_1=(\frac{1}{2},0),p_2=(1,0),p_3=(1,\frac{1}{2}),$$ $$p_4=(1,1),p_5=(\frac{1}{2},1),p_6=(0,1),p_7=(0,\frac{1}{2}).$$ Let $f_i(x)=(x+2p_i)/3$, $x\in{\mathbb{R}}^2$, $i=0,\ldots,7$. Then the Sierpiński carpet (SC) is the unique non-empty compact set $K$ in ${\mathbb{R}}^2$ satisfying $K=\cup_{i=0}^7f_i(K)$.
Let $\nu$ be the normalized Hausdorff measure on $K$. Let $({\mathcal{E}}_\beta,{\mathcal{F}}_\beta)$ be given by $$\begin{aligned}
&{\mathcal{E}}_\beta(u,u)=\int_K\int_K\frac{(u(x)-u(y))^2}{|x-y|^{\alpha+\beta}}\nu({\mathrm{d}}x)\nu({\mathrm{d}}y),\\
&{\mathcal{F}}_\beta={\left\{u\in L^2(K;\nu):{\mathcal{E}}_\beta(u,u)<+\infty\right\}},
\end{aligned}$$ where $\alpha=\log8/\log3$ is Hausdorff dimension of SC, $\beta>0$ is so far arbitrary. Then $({\mathcal{E}}_\beta,{\mathcal{F}}_\beta)$ is a quadratic form on $L^2(K;\nu)$ for all $\beta\in(0,+\infty)$. Note that $({\mathcal{E}}_\beta,{\mathcal{F}}_\beta)$ is not necessary to be a regular Dirichlet form on $L^2(K;\nu)$ related to a stale-like jump process. The *walk dimension* of SC is defined as $$\beta_*:=\sup{\left\{\beta>0:({\mathcal{E}}_\beta,{\mathcal{F}}_\beta)\text{ is a regular Dirichlet form on }L^2(K;\nu)\right\}}.$$
Let $$V_0={\left\{p_0,\ldots,p_7\right\}},V_{n+1}=\cup_{i=0}^7f_i(V_n)\text{ for all }n\ge0.$$ Then ${\left\{V_n\right\}}$ is an increasing sequence of finite sets and $K$ is the closure of $\cup_{n=0}^\infty V_n$. Let $W_0={\left\{\emptyset\right\}}$ and $$W_n={\left\{w=w_1\ldots w_n:w_i=0,\ldots,7,i=1,\ldots,n\right\}}\text{ for all }n\ge1.$$ For all $w^{(1)}=w^{(1)}_1\ldots w^{(1)}_m\in W_m,w^{(2)}=w^{(2)}_1\ldots w^{(2)}_n\in W_n$, denote $w^{(1)}w^{(2)}$ as $w=w_1\ldots w_{m+n}\in W_{m+n}$ with $w_i=w^{(1)}_i$ for all $i=1,\ldots,m$ and $w_{m+i}=w^{(2)}_i$ for all $i=1,\ldots n$. For all $i=0,\ldots,7$, denote $i^n$ as $w=w_1\ldots w_n\in W_n$ with $w_k=i$ for all $k=1,\ldots,n$.
For all $w=w_1\ldots w_n\in W_n$, let $$\begin{aligned}
f_w&=f_{w_1}\circ\ldots\circ f_{w_n},\\
V_w&=f_{w_1}\circ\ldots\circ f_{w_n}(V_0),\\
K_w&=f_{w_1}\circ\ldots\circ f_{w_n}(K),\\
P_w&=f_{w_1}\circ\ldots\circ f_{w_{n-1}}(p_{w_n}),
\end{aligned}$$ where $f_\emptyset=\mathrm{id}$ is the identity map.
Our semi-norm $E_\beta$ is given as follows. $$E_\beta(u,u):=\sum_{n=1}^\infty3^{(\beta-\alpha)n}\sum_{w\in W_n}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(u(p)-u(q))^2.$$
Our first result is as follows.
\[lem\_equiv\] For all $\beta\in(\alpha,+\infty),u\in C(K)$, we have $$E_\beta(u,u)\asymp{\mathcal{E}}_\beta(u,u).$$
The second author has established similar equivalence on Sierpiński gasket (SG), see [@MY17 Theorem 1.1].
We use Lemma \[lem\_equiv\] to give bound of walk dimension as follows.
\[thm\_bound\] $$\label{eqn_bound_beta}
\beta_*\in\left[\frac{\log\left(8\cdot\frac{7}{6}\right)}{\log3},\frac{\log\left(8\cdot\frac{3}{2}\right)}{\log3}\right].$$
This estimate follows also from the results of [@BB90] and [@BBS90] where the same bound for $\beta^*$ was obtained by means of shorting and cutting techniques, while the identity $\beta_{*}=\beta^{*}$ follows from the sub-Gaussian heat kernel estimates by means of subordination technique. Here we prove the estimate (\[eqn\_bound\_beta\]) of $\beta_*$ directly, without using heat kernel or subordination technique.
We give a direct proof of the following result.
\[thm\_walk\] $$\beta_*=\beta^*:=\frac{\log(8\rho)}{\log3},$$ where $\rho$ is some parameter in resistance estimates.
Hino and Kumagai [@HK06] established other equivalent semi-norms as follows. For all $n\ge1,u\in L^2(K;\nu)$, let $$P_nu(w)=\frac{1}{\nu(K_w)}\int_{K_w}u(x)\nu({\mathrm{d}}x),w\in W_n.$$ For all $w^{(1)},w^{(2)}\in W_n$, denote $w^{(1)}\sim_nw^{(2)}$ if $\mathrm{dim}_{\mathcal{H}}(K_{w^{(1)}}\cap K_{w^{(2)}})=1$. Let $${\mathfrak{E}}_\beta(u,u):=\sum_{n=1}^\infty3^{(\beta-\alpha)n}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
w^{(1)}\sim_nw^{(2)}
\end{subarray}
$
}}}
\left(P_nu(w^{(1)})-P_nu(w^{(2)})\right)^2.$$
\[lem\_equivHK\]([@HK06 Lemma 3.1]) For all $\beta\in(0,+\infty),u\in L^2(K;\nu)$, we have $${\mathfrak{E}}_\beta(u,u)\asymp{\mathcal{E}}_\beta(u,u).$$
We combine $E_\beta$ and ${\mathfrak{E}}_\beta$ to construct a local regular Dirichlet form on $K$ using $\Gamma$-convergence technique as follows.
\[thm\_BM\] There exists a self-similar strongly local regular Dirichlet form $({\mathcal{E}}_{{{\mathrm{loc}}}},{\mathcal{F}}_{{{\mathrm{loc}}}})$ on $L^2(K;\nu)$ satisfying $$\begin{aligned}
&{\mathcal{E}}_{{{\mathrm{loc}}}}(u,u)\asymp\sup_{n\ge1}3^{(\beta^*-\alpha)n}\sum_{w\in W_n}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(u(p)-u(q))^2,\label{eqn_Kigami}\\
&{\mathcal{F}}_{{{\mathrm{loc}}}}={\left\{u\in C(K):\sup_{n\ge1}3^{(\beta^*-\alpha)n}\sum_{w\in W_n}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(u(p)-u(q))^2<+\infty\right\}}.\nonumber\end{aligned}$$
By uniqueness result in [@BBKT10], we have above local regular Dirichlet form coincides with that given by [@BB89] and [@KZ92].
We have a direct corollary that non-local Dirichlet forms can approximate local Dirichlet form as follows.
\[cor\_approx\] There exists some positive constant $C$ such that for all $u\in{\mathcal{F}}_{{\mathrm{loc}}}$ $$\frac{1}{C}{\mathcal{E}}_{{\mathrm{loc}}}(u,u)\le\varliminf_{\beta\uparrow\beta^*}(\beta^*-\beta){\mathcal{E}}_\beta(u,u)\le\varlimsup_{\beta\uparrow\beta^*}(\beta^*-\beta){\mathcal{E}}_{\beta}(u,u)\le C{\mathcal{E}}_{{\mathrm{loc}}}(u,u).$$
Let us introduce the notion of Besov spaces. Let $(M,d,\mu)$ be a metric measure space and $\alpha,\beta>0$ two parameters. Let $$\begin{aligned}
&\left[u\right]_{B^{2,2}_{\alpha,\beta}(M)}&=\sum_{n=1}^\infty3^{(\alpha+\beta)n}\int\limits_M\int\limits_{d(x,y)<3^{-n}}(u(x)-u(y))^2\mu({\mathrm{d}}y)\mu({\mathrm{d}}x),\\
&\left[u\right]_{B^{2,\infty}_{\alpha,\beta}(M)}&=\sup_{n\ge1}3^{(\alpha+\beta)n}\int\limits_M\int\limits_{d(x,y)<3^{-n}}(u(x)-u(y))^2\mu({\mathrm{d}}y)\mu({\mathrm{d}}x),\\
\end{aligned}$$ and $$\begin{aligned}
B_{\alpha,\beta}^{2,2}(M)&={\left\{u\in L^2(M;\mu):[u]_{B^{2,2}_{\alpha,\beta}(M)}<+\infty\right\}},\\
B_{\alpha,\beta}^{2,\infty}(M)&={\left\{u\in L^2(M;\mu):[u]_{B^{2,\infty}_{\alpha,\beta}(M)}<+\infty\right\}}.\\
\end{aligned}$$ By the following Lemma \[lem\_equiv1\] and Lemma \[lem\_holder\], we have ${\mathcal{F}}_\beta=B^{2,2}_{\alpha,\beta}(K)$ for all $\beta\in(\alpha,+\infty)$.
We characterize $({\mathcal{E}}_{{\mathrm{loc}}},{\mathcal{F}}_{{\mathrm{loc}}})$ on $L^2(K;\nu)$ as follows.
\[thm\_Besov\] ${\mathcal{F}}_{{\mathrm{loc}}}=B_{\alpha,\beta^*}^{2,\infty}(K)$ and ${\mathcal{E}}_{{\mathrm{loc}}}(u,u)\asymp[u]_{B^{2,\infty}_{\alpha,\beta^*}(K)}$ for all $u\in{\mathcal{F}}_{{\mathrm{loc}}}$.
We give a direct proof of this theorem using (\[eqn\_Kigami\]) and thus avoiding heat kernel estimates, while using some geometric properties of SC. Similar characterization of the domains of local regular Dirichlet forms was obtained in [@Jon96] for SG, [@Pie99] for simple nested fractals and [@HW06] for p.c.f. self-similar sets. In [@Pie00; @GHL03; @KS05], the characterization of the domains of local regular Dirichlet forms was obtained in the setting of metric measure spaces assuming heat kernel estimates.
Finally, using (\[eqn\_Kigami\]) of Theorem \[thm\_BM\], we give an alternative proof of sub-Gaussian heat kernel estimates as follows.
\[thm\_hk\] $({\mathcal{E}}_{{\mathrm{loc}}},{\mathcal{F}}_{{\mathrm{loc}}})$ on $L^2(K;\nu)$ has a heat kernel $p_t(x,y)$ satisfying $$p_t(x,y)\asymp\frac{C}{t^{\alpha/\beta^*}}\exp\left(-c\left(\frac{|x-y|}{t^{1/\beta^*}}\right)^{\frac{\beta^*}{\beta^*-1}}\right),$$ for all $x,y\in K,t\in(0,1)$.
This paper is organized as follows. In Section \[sec\_equiv\], we prove Lemma \[lem\_equiv\]. In Section \[sec\_bound\], we prove Theorem \[thm\_bound\]. In Section \[sec\_resistance\], we give resistance estimates. In Section \[sec\_harnack\], we give uniform Harnack inequality. In Section \[sec\_monotone\], we give two weak monotonicity results. In Section \[sec\_good\], we construct one good function. In Section \[sec\_walk\], we prove Theorem \[thm\_walk\]. In Section \[sec\_BM\], we prove Theorem \[thm\_BM\]. In Section \[sec\_Besov\], we prove Theorem \[thm\_Besov\]. In Section \[sec\_hk\], we prove Theorem \[thm\_hk\].
NOTATION. The letters $c,C$ will always refer to some positive constants and may change at each occurrence. The sign $\asymp$ means that the ratio of the two sides is bounded from above and below by positive constants. The sign $\lesssim$ ($\gtrsim$) means that the LHS is bounded by positive constant times the RHS from above (below).
Proof of Lemma \[lem\_equiv\] {#sec_equiv}
=============================
We need some preparation as follows.
\[lem\_equiv1\]([@MY17 Lemma 2.1]) For all $u\in L^2(K;\nu)$, we have $$\int_K\int_K\frac{(u(x)-u(y))^2}{|x-y|^{\alpha+\beta}}\nu({\mathrm{d}}x)\nu({\mathrm{d}}y)\asymp\sum_{n=0}^\infty3^{(\alpha+\beta)n}\int_K\int_{B(x,3^{-n})}(u(x)-u(y))^2\nu({\mathrm{d}}y)\nu({\mathrm{d}}x).$$
\[cor\_arbi\]([@MY17 Corollary 2.2]) Fix arbitrary integer $N\ge0$ and real number $c>0$. For all $u\in L^2(K;\nu)$, we have $$\int_K\int_K\frac{(u(x)-u(y))^2}{|x-y|^{\alpha+\beta}}\nu({\mathrm{d}}x)\nu({\mathrm{d}}y)\asymp\sum_{n=N}^\infty3^{(\alpha+\beta)n}\int_K\int_{B(x,c3^{-n})}(u(x)-u(y))^2\nu({\mathrm{d}}y)\nu({\mathrm{d}}x).$$
The proofs of above results are essentially the same as those in [@MY17] except that contraction ratio $1/2$ is replaced by $1/3$. We also need the fact that SC satisfies the chain condition, see [@GHL03 Definition 3.4].
The following result states that a Besov space can be embedded in some Hölder space.
\[lem\_holder\]([@GHL03 Theorem 4.11 ([3]{})]) Let $u\in L^2(K;\nu)$ and $$E(u):=\int_K\int_K\frac{(u(x)-u(y))^2}{|x-y|^{\alpha+\beta}}\nu({\mathrm{d}}x)\nu({\mathrm{d}}y),$$ then $$|u(x)-u(y)|^2\le cE(u)|x-y|^{\beta-\alpha}\text{ for }\nu\text{-almost every }x,y\in K,$$ where $c$ is some positive constant.
If $E(u)<+\infty$, then $u\in C^{\frac{\beta-\alpha}{2}}(K)$.
Note that the proof of above lemma does not rely on heat kernel.
We divide Lemma \[lem\_equiv\] into the following Theorem \[thm\_equiv1\] and Theorem \[thm\_equiv2\]. The idea of the proofs of these theorems comes form [@Jon96]. But we do need to pay special attention to the difficulty brought by non p.c.f. property.
\[thm\_equiv1\] For all $u\in C(K)$, we have $$\sum_{n=1}^\infty3^{(\beta-\alpha)n}\sum_{w\in W_n}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(u(p)-u(q))^2\lesssim\int_K\int_K\frac{(u(x)-u(y))^2}{|x-y|^{\alpha+\beta}}\nu({\mathrm{d}}x)\nu({\mathrm{d}}y).$$
First fix $n\ge1,w=w_1\ldots w_n\in W_n$, consider $${\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(u(p)-u(q))^2.$$ For all $x\in K_w$, we have $$(u(p)-u(q))^2\le2(u(p)-u(x))^2+2(u(x)-u(q))^2.$$ Integrating with respect to $x\in K_w$ and dividing by $\nu(K_w)$, we have $$(u(p)-u(q))^2\le\frac{2}{\nu(K_w)}\int_{K_w}(u(p)-u(x))^2\nu({\mathrm{d}}x)+\frac{2}{\nu(K_w)}\int_{K_w}(u(x)-u(q))^2\nu({\mathrm{d}}x),$$ hence $${\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(u(p)-u(q))^2\le2\cdot2\cdot2\sum_{p\in V_w}\frac{1}{\nu(K_w)}\int_{K_w}(u(p)-u(x))^2\nu({\mathrm{d}}x).$$ Consider $(u(p)-u(x))^2$, $p\in V_w$, $x\in K_w$. There exists $w_{n+1}\in{\left\{0,\ldots,7\right\}}$ such that $p=f_{w_1}\circ\ldots\circ f_{w_n}(p_{w_{n+1}})$. Let $k,l\ge1$ be integers to be determined, let $$w^{(i)}=w_1\ldots w_nw_{n+1}\ldots w_{n+1}$$ with $ki$ terms of $w_{n+1}$, $i=0,\ldots,l$. For all $x^{(i)}\in K_{w^{(i)}}$, $i=0,\ldots,l$, we have $$\begin{aligned}
(u(p)-u(x^{(0)}))^2&\le2(u(p)-u(x^{(l)}))^2+2(u(x^{(0)})-u(x^{(l)}))^2\\
&\le2(u(p)-u(x^{(l)}))^2+2\left[2(u(x^{(0)})-u(x^{(1)}))^2+2(u(x^{(1)})-u(x^{(l)}))^2\right]\\
&=2(u(p)-u(x^{(l)}))^2+2^2(u(x^{(0)})-u(x^{(1)}))^2+2^2(u(x^{(1)})-u(x^{(l)}))^2\\
&\le\ldots\le2(u(p)-u(x^{(l)}))^2+2^2\sum_{i=0}^{l-1}2^i(u(x^{(i)})-u(x^{(i+1)}))^2.
\end{aligned}$$ Integrating with respect to $x^{(0)}\in K_{w^{(0)}}$, …, $x^{(l)}\in K_{w^{(l)}}$ and dividing by $\nu(K_{w^{(0)}})$, …, $\nu(K_{w^{(l)}})$, we have $$\begin{aligned}
&\frac{1}{\nu(K_{w^{(0)}})}\int_{K_{w^{(0)}}}(u(p)-u(x^{(0)}))^2\nu({\mathrm{d}}x^{(0)})\\
&\le\frac{2}{\nu(K_{w^{(l)}})}\int_{K_{w^{(l)}}}(u(p)-u(x^{(l)}))^2\nu({\mathrm{d}}x^{(l)})\\
&+2^2\sum_{i=0}^{l-1}\frac{2^i}{\nu(K_{w^{(i)}})\nu(K_{w^{(i+1)}})}\int_{K_{w^{(i)}}}\int_{K_{w^{(i+1)}}}(u(x^{(i)})-u(x^{(i+1)}))^2\nu({\mathrm{d}}x^{(i)})\nu({\mathrm{d}}x^{(i+1)}).
\end{aligned}$$ Now let us use $\nu(K_{w^{(i)}})=(1/8)^{n+ki}=3^{-\alpha(n+ki)}$. For the first term, by Lemma \[lem\_holder\], we have $$\begin{aligned}
\frac{1}{\nu(K_{w^{(l)}})}\int_{K_{w^{(l)}}}(u(p)-u(x^{(l)}))^2\nu({\mathrm{d}}x^{(l)})&\le \frac{cE(u)}{\nu(K_{w^{(l)}})}\int_{K_{w^{(l)}}}|p-x^{(l)}|^{\beta-\alpha}\nu({\mathrm{d}}x^{(l)})\\
&\le{2}^{(\beta-\alpha)/2}cE(u){3}^{-(\beta-\alpha)(n+kl)}.
\end{aligned}$$ For the second term, for all $x^{(i)}\in K_{w^{(i)}},x^{(i+1)}\in K_{w^{(i+1)}}$, we have $$|x^{(i)}-x^{(i+1)}|\le\sqrt{2}\cdot3^{-(n+ki)},$$ hence $$\begin{aligned}
&\sum_{i=0}^{l-1}\frac{2^i}{\nu(K_{w^{(i)}})\nu(K_{w^{(i+1)}})}\int_{K_{w^{(i)}}}\int_{K_{w^{(i+1)}}}(u(x^{(i)})-u(x^{(i+1)}))^2\nu({\mathrm{d}}x^{(i)})\nu({\mathrm{d}}x^{(i+1)})\\
&\le\sum_{i=0}^{l-1}{2^{i}\cdot3^{\alpha k+2\alpha(n+ki)}}\int\limits_{K_{w^{(i)}}}\int\limits_{|x^{(i+1)}-x^{(i)}|\le\sqrt{2}\cdot3^{-(n+ki)}}(u(x^{(i)})-u(x^{(i+1)}))^2\nu({\mathrm{d}}x^{(i)})\nu({\mathrm{d}}x^{(i+1)}),
\end{aligned}$$ and $$\begin{aligned}
&\frac{1}{\nu(K_w)}\int_{K_w}(u(p)-u(x))^2\nu({\mathrm{d}}x)=\frac{1}{\nu(K_{w^{(0)}})}\int_{K_{w^{(0)}}}(u(p)-u(x^{(0)}))^2\nu({\mathrm{d}}x^{(0)})\\
&\le 2\cdot{2}^{(\beta-\alpha)/2}cE(u)3^{-(\beta-\alpha)(n+kl)}\\
&+4\sum_{i=0}^{l-1}{2^{i}\cdot3^{\alpha k+2\alpha(n+ki)}}\int\limits_{K_{w^{(i)}}}\int\limits_{|x-y|\le\sqrt{2}\cdot3^{-(n+ki)}}(u(x)-u(y))^2\nu({\mathrm{d}}x)\nu({\mathrm{d}}y).
\end{aligned}$$ Hence $$\begin{aligned}
&\sum_{w\in W_n}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(u(p)-u(q))^2\\
&\le8\sum_{w\in W_n}\sum_{p\in V_w}\frac{1}{\nu(K_w)}\int_{K_w}(u(p)-u(x))^2\nu({\mathrm{d}}x)\\
&\le8\sum_{w\in W_n}\sum_{p\in V_w}\left(2\cdot{2}^{(\beta-\alpha)/2}cE(u)3^{-(\beta-\alpha)(n+kl)}\right.\\
&\left.+4\sum_{i=0}^{l-1}{2^{i}\cdot3^{\alpha k+2\alpha(n+ki)}}\int\limits_{K_{w^{(i)}}}\int\limits_{|x-y|\le\sqrt{2}\cdot3^{-(n+ki)}}(u(x)-u(y))^2\nu({\mathrm{d}}x)\nu({\mathrm{d}}y)\right).
\end{aligned}$$ For the first term, we have $$\sum_{w\in W_n}\sum_{p\in V_w}3^{-(\beta-\alpha)(n+kl)}=8\cdot8^n\cdot3^{-(\beta-\alpha)(n+kl)}=8\cdot3^{\alpha n-(\beta-\alpha)(n+kl)}.$$ For the second term, fix $i=0,\ldots,l-1$, different $p\in V_w$, $w\in W_n$ correspond to different $K_{w^{(i)}}$, hence $$\begin{aligned}
&\sum_{i=0}^{l-1}\sum_{w\in W_n}\sum_{p\in V_w}2^{i}\cdot3^{\alpha k+2\alpha(n+ki)}\int\limits_{K_{w^{(i)}}}\int\limits_{|x-y|\le\sqrt{2}\cdot3^{-(n+ki)}}(u(x)-u(y))^2\nu({\mathrm{d}}x)\nu({\mathrm{d}}y)\\
&\le\sum_{i=0}^{l-1}2^{i}\cdot3^{\alpha k+2\alpha(n+ki)}\int\limits_{K}\int\limits_{|x-y|\le\sqrt{2}\cdot3^{-(n+ki)}}(u(x)-u(y))^2\nu({\mathrm{d}}x)\nu({\mathrm{d}}y)\\
&=3^{\alpha k}\sum_{i=0}^{l-1}2^{i}\cdot3^{-(\beta-\alpha)(n+ki)}\left(3^{(\alpha+\beta)(n+ki)}\int\limits_{K}\int\limits_{|x-y|\le\sqrt{2}\cdot3^{-(n+ki)}}(u(x)-u(y))^2\nu({\mathrm{d}}x)\nu({\mathrm{d}}y)\right).
\end{aligned}$$ For simplicity, denote $$E_{n}(u)=3^{(\alpha+\beta)n}\int_{K}\int_{|x-y|\le\sqrt{2}\cdot3^{-n}}(u(x)-u(y))^2\nu({\mathrm{d}}x)\nu({\mathrm{d}}y).$$ We have $$\label{eqn_equiv1_1}
\begin{aligned}
&\sum_{w\in W_n}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(u(p)-u(q))^2\\
&\le128\cdot{2}^{(\beta-\alpha)/2}cE(u)3^{\alpha n-(\beta-\alpha)(n+kl)}+32\cdot3^{\alpha k}\sum_{i=0}^{l-1}2^{i}\cdot3^{-(\beta-\alpha)(n+ki)}E_{n+ki}(u).
\end{aligned}$$ Hence $$\begin{aligned}
&\sum_{n=1}^\infty3^{(\beta-\alpha)n}\sum_{w\in W_n}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(u(p)-u(q))^2\\
&\le128\cdot{2}^{(\beta-\alpha)/2}cE(u)\sum_{n=1}^\infty3^{\beta n-(\beta-\alpha)(n+kl)}+32\cdot3^{\alpha k}\sum_{n=1}^\infty\sum_{i=0}^{l-1}2^{i}\cdot3^{-(\beta-\alpha)ki}E_{n+ki}(u).
\end{aligned}$$ Take $l=n$, then $$\begin{aligned}
&\sum_{n=1}^\infty3^{(\beta-\alpha)n}\sum_{w\in W_n}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(u(p)-u(q))^2\\
&\le128\cdot{2}^{(\beta-\alpha)/2}cE(u)\sum_{n=1}^\infty3^{\left[\beta-(\beta-\alpha)(k+1)\right]n}+32\cdot3^{\alpha k}\sum_{n=1}^\infty\sum_{i=0}^{n-1}2^{i}\cdot3^{-(\beta-\alpha)ki}E_{n+ki}(u)\\
&=128\cdot{2}^{(\beta-\alpha)/2}cE(u)\sum_{n=1}^\infty3^{\left[\beta-(\beta-\alpha)(k+1)\right]n}+32\cdot3^{\alpha k}\sum_{i=0}^\infty2^{i}\cdot3^{-(\beta-\alpha)ki}\sum_{n=i+1}^{\infty}E_{n+ki}(u)\\
&\le128\cdot{2}^{(\beta-\alpha)/2}cE(u)\sum_{n=1}^\infty3^{\left[\beta-(\beta-\alpha)(k+1)\right]n}+32\cdot3^{\alpha k}\sum_{i=0}^\infty3^{\left[1-(\beta-\alpha)k\right]i}C_1E(u),
\end{aligned}$$ where $C_1$ is some positive constant from Corollary \[cor\_arbi\]. Take $k\ge1$ sufficiently large such that $\beta-(\beta-\alpha)(k+1)<0$ and $1-(\beta-\alpha)k<0$, then above two series converge, hence $$\sum_{n=1}^\infty3^{(\beta-\alpha)n}\sum_{w\in W_n}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(u(p)-u(q))^2\lesssim\int_K\int_K\frac{(u(x)-u(y))^2}{|x-y|^{\alpha+\beta}}\nu({\mathrm{d}}x)\nu({\mathrm{d}}y).$$
\[thm\_equiv2\] For all $u\in C(K)$, we have $$\label{eqn_equiv2_1}
\int_K\int_K\frac{(u(x)-u(y))^2}{|x-y|^{\alpha+\beta}}\nu({\mathrm{d}}x)\nu({\mathrm{d}}y)\lesssim\sum_{n=1}^\infty3^{(\beta-\alpha)n}\sum_{w\in W_n}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(u(p)-u(q))^2,$$ or equivalently for all $c\in(0,1)$ $$\label{eqn_equiv2_2}
\begin{aligned}
&\sum_{n=2}^\infty3^{(\alpha+\beta)n}\int\limits_K\int\limits_{B(x,c3^{-n})}(u(x)-u(y))^2\nu({\mathrm{d}}y)\nu({\mathrm{d}}x)\\
&\lesssim\sum_{n=1}^\infty3^{(\beta-\alpha)n}\sum_{w\in W_n}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(u(p)-u(q))^2.
\end{aligned}$$
Note $V_n=\cup_{w\in W_n}V_w$, it is obvious that its cardinal $\#V_n\asymp8^n=3^{\alpha n}$. Let $\nu_n$ be the measure on $V_n$ which assigns $1/\#V_n$ on each point of $V_n$, then $\nu_n$ converges weakly to $\nu$.
First, for $n\ge2,m>n$, we estimate $$3^{(\alpha+\beta)n}\int_K\int_{B(x,c3^{-n})}(u(x)-u(y))^2\nu_m({\mathrm{d}}y)\nu_m({\mathrm{d}}x).$$ Note that $$\begin{aligned}
\int\limits_K\int\limits_{B(x,c3^{-n})}(u(x)-u(y))^2\nu_m({\mathrm{d}}y)\nu_m({\mathrm{d}}x)=\sum\limits_{w\in W_n}\int\limits_{K_w}\int\limits_{B(x,c3^{-n})}(u(x)-u(y))^2\nu_m({\mathrm{d}}y)\nu_m({\mathrm{d}}x).
\end{aligned}$$ Fix $w\in W_n$, there exist at most nine $\tilde{w}\in W_n$ such that $K_{\tilde{w}}\cap K_w\ne\emptyset$, see Figure \[fig\_Kw\].
(0,0)–(6,0)–(6,6)–(0,6)–cycle; (2,0)–(2,6); (4,0)–(4,6); (0,2)–(6,2); (0,4)–(6,4); (2,2)–(4,2)–(4,4)–(2,4)–cycle;
(3,3) node [$K_w$]{};
Let $$K_w^*=
{\bigcup_{\mbox{\tiny
$
\begin{subarray}{c}
\tilde{w}\in W_n\\
K_{\tilde{w}}\cap K_w\ne\emptyset
\end{subarray}
$
}}}
K_{\tilde{w}}.$$ For all $x\in K_w$, $y\in B(x,c3^{-n})$, we have $y\in K_w^*$, hence $$\begin{aligned}
&\int_{K_w}\int_{B(x,c3^{-n})}(u(x)-u(y))^2\nu_m({\mathrm{d}}y)\nu_m({\mathrm{d}}x)\le\int_{K_w}\int_{K_w^*}(u(x)-u(y))^2\nu_m({\mathrm{d}}y)\nu_m({\mathrm{d}}x)\\
&=
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
\tilde{w}\in W_n\\
K_{\tilde{w}}\cap K_w\ne\emptyset
\end{subarray}
$
}}}
\int_{K_w}\int_{K_{\tilde{w}}}(u(x)-u(y))^2\nu_m({\mathrm{d}}y)\nu_m({\mathrm{d}}x).
\end{aligned}$$ Note ${\left\{P_w\right\}}=K_w\cap V_{n-1}$ for all $w\in W_n$. Fix $\tilde{w},w\in W_n$ with $K_{\tilde{w}}\cap K_w\ne\emptyset$. If $P_{\tilde{w}}\ne P_w$, then $|P_{\tilde{w}}-P_w|=2^{-1}\cdot3^{-(n-1)}$ or there exists a unique $z\in V_{n-1}$ such that
$$\label{eqn_med}
\lvert P_{\tilde{w}}-z\rvert=\lvert P_w-z\rvert=2^{-1}\cdot3^{-(n-1)}.$$
Let $z_1=P_{\tilde{w}}$, $z_3=P_w$ and $$z_2=
\begin{cases}
P_{\tilde{w}}=P_w,&\text{if }P_{\tilde{w}}=P_w,\\
P_{\tilde{w}},&\text{if }|P_{\tilde{w}}-P_w|=2^{-1}\cdot3^{-(n-1)},\\
z,&\text{if }P_{\tilde{w}}\ne P_w\text{ and }z \text{ is given by Equation (\ref{eqn_med})}.
\end{cases}$$ Then for all $x\in K_w$, $y\in K_{\tilde{w}}$, we have $$\begin{aligned}
&(u(x)-u(y))^2\\
&\le4\left[(u(y)-u(z_1))^2+(u(z_1)-u(z_2))^2+(u(z_2)-u(z_3))^2+(u(z_3)-u(x))^2\right].
\end{aligned}$$ For $i=1,2$, we have $$\begin{aligned}
&\int_{K_w}\int_{K_{\tilde{w}}}(u(z_i)-u(z_{i+1}))^2\nu_m({\mathrm{d}}y)\nu_m({\mathrm{d}}x)=(u(z_i)-u(z_{i+1}))^2\left(\frac{\#(K_w\cap V_m)}{\#V_m}\right)^2\\
&\asymp(u(z_i)-u(z_{i+1}))^2\left(\frac{8^{m-n}}{8^m}\right)^2=3^{-2\alpha n}(u(z_i)-u(z_{i+1}))^2.
\end{aligned}$$ Hence $$\begin{aligned}
&\sum_{w\in W_n}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
\tilde{w}\in W_n\\
K_{\tilde{w}}\cap K_w\ne\emptyset
\end{subarray}
$
}}}
\int_{K_w}\int_{K_{\tilde{w}}}(u(x)-u(y))^2\nu_m({\mathrm{d}}y)\nu_m({\mathrm{d}}x)\\
&\lesssim3^{-\alpha n}\sum\limits_{w\in W_n}\int\limits_{K_w}(u(x)-u(P_w))^2\nu_m({\mathrm{d}}x)+3^{-2\alpha n}\sum_{w\in W_{n-1}}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-(n-1)}
\end{subarray}
$
}}}
(u(p)-u(q))^2\\
&\asymp3^{-\alpha(m+n)}\sum\limits_{w\in W_n}\sum\limits_{x\in K_w\cap V_m}(u(x)-u(P_w))^2\\
&+3^{-2\alpha n}\sum_{w\in W_{n-1}}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-(n-1)}
\end{subarray}
$
}}}
(u(p)-u(q))^2.
\end{aligned}$$ Let us estimate $(u(x)-u(P_w))^2$ for $x\in K_w\cap V_m$. We construct a finite sequence $$p_1,\ldots,p_{4(m-n+1)},p_{4(m-n+1)+1}$$ such that $p_1=P_w$, $p_{4(m-n+1)+1}=x$ and for all $k=0,\ldots,m-n$, we have $$p_{4k+1},p_{4k+2},p_{4k+3},p_{4k+4},p_{4(k+1)+1}\in V_{n+k},$$ and for all $i=1,2,3,4$, we have $$\lvert p_{4k+i}-p_{4k+i+1}\rvert=0\text{ or }2^{-1}\cdot3^{-(n+k)}.$$ Then $$\begin{aligned}
\left(u(x)-u(P_w)\right)^2\lesssim\sum_{k=0}^{m-n}4^{k}&\left[(u(p_{4k+1})-u(p_{4k+2}))^2+(u(p_{4k+2})-u(p_{4k+3}))^2\right.\\
&\left.+(u(p_{4k+3})-u(p_{4k+4}))^2+(u(p_{4k+4})-u(p_{4(k+1)+1}))^2\right].
\end{aligned}$$ For all $k=n,\ldots,m$, for all $p,q\in V_k\cap K_w$ with $|p-q|=2^{-1}\cdot 3^{-k}$, the term $(u(p)-u(q))^2$ occurs in the sum with times of the order $8^{m-k}=3^{\alpha(m-k)}$, hence $$\begin{aligned}
&3^{-\alpha(m+n)}\sum\limits_{w\in W_n}\sum\limits_{x\in K_w\cap V_m}(u(x)-u(P_w))^2\\
&\lesssim3^{-\alpha(m+n)}\sum_{k=n}^{m}4^{k-n}\cdot3^{\alpha(m-k)}\sum_{w\in W_k}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-k}
\end{subarray}
$
}}}
(u(p)-u(q))^2\\
&=\sum_{k=n}^{m}4^{k-n}\cdot3^{-\alpha(n+k)}\sum_{w\in W_k}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-k}
\end{subarray}
$
}}}
(u(p)-u(q))^2.
\end{aligned}$$ Hence $$\begin{aligned}
&\int_K\int_{B(x,c3^{-n})}(u(x)-u(y))^2\nu_m({\mathrm{d}}y)\nu_m({\mathrm{d}}x)\\
&\lesssim\sum_{k=n}^{m}4^{k-n}\cdot3^{-\alpha(n+k)}\sum_{w\in W_k}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-k}
\end{subarray}
$
}}}
(u(p)-u(q))^2\\
&+3^{-2\alpha n}\sum_{w\in W_{n-1}}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-(n-1)}
\end{subarray}
$
}}}
(u(p)-u(q))^2.
\end{aligned}$$ Letting $m\to+\infty$, we have $$\label{eqn_equiv2_3}
\begin{aligned}
&\int_K\int_{B(x,c3^{-n})}(u(x)-u(y))^2\nu({\mathrm{d}}y)\nu({\mathrm{d}}x)\\
&\lesssim\sum_{k=n}^\infty4^{k-n}\cdot3^{-\alpha(n+k)}\sum_{w\in W_k}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-k}
\end{subarray}
$
}}}
(u(p)-u(q))^2\\
&+3^{-2\alpha n}\sum_{w\in W_{n-1}}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-(n-1)}
\end{subarray}
$
}}}
(u(p)-u(q))^2.
\end{aligned}$$ Hence $$\begin{aligned}
&\sum_{n=2}^\infty3^{(\alpha+\beta)n}\int_K\int_{B(x,c3^{-n})}(u(x)-u(y))^2\nu({\mathrm{d}}y)\nu({\mathrm{d}}x)\\
&\lesssim\sum_{n=2}^\infty\sum_{k=n}^\infty4^{k-n}\cdot3^{\beta n-\alpha k}\sum_{w\in W_k}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-k}
\end{subarray}
$
}}}
(u(p)-u(q))^2\\
&+\sum_{n=2}^\infty3^{(\beta-\alpha)n}\sum_{w\in W_{n-1}}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-(n-1)}
\end{subarray}
$
}}}
(u(p)-u(q))^2\\
&\lesssim\sum_{k=2}^\infty\sum_{n=2}^k4^{k-n}\cdot3^{\beta n-\alpha k}\sum_{w\in W_k}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-k}
\end{subarray}
$
}}}
(u(p)-u(q))^2\\
&+\sum_{n=1}^\infty3^{(\beta-\alpha)n}\sum_{w\in W_{n}}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(u(p)-u(q))^2\\
&\lesssim\sum_{n=1}^\infty3^{(\beta-\alpha)n}\sum_{w\in W_{n}}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(u(p)-u(q))^2.
\end{aligned}$$
Proof of Theorem \[thm\_bound\] {#sec_bound}
===============================
First, we consider lower bound. We need some preparation.
\[prop\_lower\] Assume that $\beta\in(\alpha,+\infty)$. Let $f:[0,1]\to{\mathbb{R}}$ be a strictly increasing continuous function. Assume that the function $U(x,y)=f(x)$, $(x,y)\in K$ satisfies ${\mathcal{E}}_{\beta}(U,U)<+\infty$. Then $({\mathcal{E}}_\beta,{\mathcal{F}}_\beta)$ is a regular Dirichlet form on $L^2(K;\nu)$.
Above proposition means that only *one* good enough function contained in the domain can ensure that the domain is large enough.
We only need to show that ${\mathcal{F}}_\beta$ is uniformly dense in $C(K)$. Then ${\mathcal{F}}_\beta$ is dense in $L^2(K;\nu)$. Using Fatou’s lemma, we have ${\mathcal{F}}_\beta$ is complete under $({\mathcal{E}}_\beta)_1$ metric. It is obvious that ${\mathcal{E}}_\beta$ has Markovian property. Hence $({\mathcal{E}}_\beta,{\mathcal{F}}_\beta)$ is a Dirichlet form on $L^2(K;\nu)$. Moreover, ${\mathcal{F}}_\beta\cap C(K)={\mathcal{F}}_\beta$ is trivially $({\mathcal{E}}_\beta)_1$ dense in ${\mathcal{F}}_\beta$ and uniformly dense in $C(K)$. Hence $({\mathcal{E}}_\beta,{\mathcal{F}}_\beta)$ on $L^2(K;\nu)$ is regular.
Indeed, by assumption, $U\in{\mathcal{F}}_\beta$, ${\mathcal{F}}_\beta\ne\emptyset$. It is obvious that ${\mathcal{F}}_\beta$ is a sub-algebra of $C(K)$, that is, for all $u,v\in{\mathcal{F}}_\beta$, $c\in{\mathbb{R}}$, we have $u+v,cu,uv\in{\mathcal{F}}_\beta$. We show that ${\mathcal{F}}_\beta$ separates points. For all distinct $(x^{(1)},y^{(1)}),(x^{(2)},y^{(2)})\in K$, we have $x^{(1)}\ne x^{(2)}$ or $y^{(1)}\ne y^{(2)}$.
If $x^{(1)}\ne x^{(2)}$, then since $f$ is strictly increasing, we have $$U(x^{(1)},y^{(1)})=f(x^{(1)})\ne f(x^{(2)})=U(x^{(2)},y^{(2)}).$$ If $y^{(1)}\ne y^{(2)}$, then let $V(x,y)=f(y)$, $(x,y)\in K$, we have $V\in{\mathcal{F}}_\beta$ and $$V(x^{(1)},y^{(1)})=f(y^{(1)})\ne f(y^{(2)})=V(x^{(2)},y^{(2)}).$$ By Stone-Weierstrass theorem, ${\mathcal{F}}_\beta$ is uniformly dense in $C(K)$.
Now, we give lower bound.
The point is to construct an explicit function. We define $f:[0,1]\to{\mathbb{R}}$ as follows. Let $f(0)=0$ and $f(1)=1$. First, we determine the values of $f$ at $1/3$ and $2/3$. We consider the minimum of the following function $${\varphi}(x,y)=3x^2+2(x-y)^2+3(1-y)^2,x,y\in{\mathbb{R}}.$$ By elementary calculation, ${\varphi}$ attains minimum $6/7$ at $(x,y)=(2/7,5/7)$. Assume that we have defined $f$ on $i/3^n$, $i=0,1,\ldots,3^n$. Then, for $n+1$, for all $i=0,1,\ldots,3^{n}-1$, we define $$f(\frac{3i+1}{3^{n+1}})=\frac{5}{7}f(\frac{i}{3^n})+\frac{2}{7}f(\frac{i+1}{3^n}),f(\frac{3i+2}{3^{n+1}})=\frac{2}{7}f(\frac{i}{3^n})+\frac{5}{7}f(\frac{i+1}{3^n}).$$ By induction principle, we have the definition of $f$ on all triadic points. It is obvious that $f$ is uniformly continuous on the set of all triadic points. We extend $f$ to be continuous on $[0,1]$. It is obvious that $f$ is increasing. For all $x,y\in[0,1]$ with $x<y$, there exist triadic points $i/3^n,(i+1)/3^n\in(x,y)$, then $f(x)\le f(i/3^n)<f((i+1)/3^n)\le f(y)$, hence $f$ is strictly increasing.
Let $U(x,y)=f(x)$, $(x,y)\in K$. By induction, we have $$\sum_{w\in W_{n+1}}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-(n+1)}
\end{subarray}
$
}}}
(U(p)-U(q))^2=\frac{6}{7}\sum_{w\in W_{n}}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(U(p)-U(q))^2\text{ for all }n\ge1.$$ Hence $$\label{eqn_lower}
\sum_{w\in W_{n}}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(U(p)-U(q))^2=\left(\frac{6}{7}\right)^n\text{ for all }n\ge1.$$ For all $\beta\in(\log8/\log3,\log(8\cdot7/6)/\log3)$, we have $3^{\beta-\alpha}<7/6$. By Equation (\[eqn\_lower\]), we have $$\sum_{n=1}^\infty3^{(\beta-\alpha)n}\sum_{w\in W_n}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(U(p)-U(q))^2<+\infty.$$ By Lemma \[lem\_equiv\], ${\mathcal{E}}_\beta(U,U)<+\infty$. By Proposition \[prop\_lower\], $({\mathcal{E}}_\beta,{\mathcal{F}}_\beta)$ is a regular Dirichlet form on $L^2(K;\nu)$ for all $\beta\in(\log8/\log3,\log(8\cdot7/6)/\log3)$. Hence $$\beta_*\ge\frac{\log(8\cdot\frac{7}{6})}{\log3}.$$
The construction of above function is similar to that given in the proof of [@Bar13 Theorem 2.6]. Indeed, above function is constructed in a self-similar way. Let $f_n:[0,1]\to{\mathbb{R}}$ be given by $f_0(x)=x$, $x\in[0,1]$ and for all $n\ge0$ $$f_{n+1}(x)=
\begin{cases}
\frac{2}{7}f_n(3x),&\text{if }0\le x\le\frac{1}{3},\\
\frac{3}{7}f_n(3x-1)+\frac{2}{7},&\text{if }\frac{1}{3}<x\le\frac{2}{3},\\
\frac{2}{7}f_n(3x-2)+\frac{5}{7},&\text{if }\frac{2}{3}<x\le1.
\end{cases}$$ It is obvious that $$f_n(\frac{i}{3^n})=f(\frac{i}{3^n})\text{ for all }i=0,\ldots,3^n,n\ge0,$$ and $$\max_{x\in[0,1]}\lvert f_{n+1}(x)-f_n(x)\rvert\le\frac{3}{7}\max_{x\in[0,1]}\lvert f_{n}(x)-f_{n-1}(x)\rvert\text{ for all }n\ge1,$$ hence $f_n$ converges uniformly to $f$ on $[0,1]$. Let $g_1,g_2,g_3:{\mathbb{R}}^2\to{\mathbb{R}}^2$ be given by $$g_1(x,y)=\left(\frac{1}{3}x,\frac{2}{7}y\right),g_2(x,y)=\left(\frac{1}{3}x+\frac{1}{3},\frac{3}{7}y+\frac{2}{7}\right),g_3(x,y)=\left(\frac{1}{3}x+\frac{2}{3},\frac{2}{7}y+\frac{5}{7}\right).$$ Then ${\left\{(x,f(x)):x\in[0,1]\right\}}$ is the unique non-empty compact set $G$ in ${\mathbb{R}}^2$ satisfying $$G=g_1(G)\cup g_2(G)\cup g_3(G).$$
Second, we consider upper bound. We shrink SC to another fractal. Denote $\mathcal{C}$ as Cantor ternary set in $[0,1]$. Then $[0,1]\times\mathcal{C}$ is the unique non-empty compact set $\tilde{K}$ in ${\mathbb{R}}^2$ satisfying $$\tilde{K}=\cup_{i=0,1,2,4,5,6}f_i(\tilde{K}).$$ Let $$\tilde{V}_0={\left\{p_0,p_1,p_2,p_4,p_5,p_6\right\}},\tilde{V}_{n+1}=\cup_{i=0,1,2,4,5,6}f_i(\tilde{V}_n)\text{ for all }n\ge0.$$ Then ${\left\{\tilde{V}_n\right\}}$ is an increasing sequence of finite sets and $[0,1]\times\mathcal{C}$ is the closure of $\cup_{n=0}^\infty\tilde{V}_n$. Let $\tilde{W}_0={\left\{\emptyset\right\}}$ and $$\tilde{W}_n={\left\{w=w_1\ldots w_n:w_i=0,1,2,4,5,6,i=1,\ldots,n\right\}}\text{ for all }n\ge1.$$ For all $w=w_1\ldots w_n\in\tilde{W}_n$, let $$\tilde{V}_w=f_{w_1}\circ\ldots\circ f_{w_n}(\tilde{V}_0).$$
Assume that $({\mathcal{E}}_\beta,{\mathcal{F}}_\beta)$ is a regular Dirichlet form on $L^2(K;\nu)$, then there exists $u\in{\mathcal{F}}_\beta$ such that $u|_{{\left\{0\right\}}\times[0,1]}=0$ and $u|_{{\left\{1\right\}}\times[0,1]}=1$. By Lemma \[lem\_equiv\], we have $$\label{eqn_upper}
\begin{aligned}
+\infty&>\sum_{n=1}^\infty3^{(\beta-\alpha)n}\sum_{w\in W_n}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(u(p)-u(q))^2\\
&\ge\sum_{n=1}^\infty3^{(\beta-\alpha)n}\sum_{w\in\tilde{W}_n}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in\tilde{V}_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(u(p)-u(q))^2\\
&=\sum_{n=1}^\infty3^{(\beta-\alpha)n}\sum_{w\in\tilde{W}_n}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in\tilde{V}_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
((u|_{[0,1]\times\mathcal{C}})(p)-(u|_{[0,1]\times\mathcal{C}})(q))^2\\
&\ge\sum_{n=1}^\infty3^{(\beta-\alpha)n}\sum_{w\in\tilde{W}_n}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in\tilde{V}_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(\tilde{u}(p)-\tilde{u}(q))^2,
\end{aligned}$$ where $\tilde{u}$ is the function on $[0,1]\times\mathcal{C}$ that is the minimizer of $$\sum_{n=1}^\infty3^{(\beta-\alpha)n}\sum_{w\in\tilde{W}_n}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in\tilde{V}_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(\tilde{u}(p)-\tilde{u}(q))^2:\tilde{u}|_{{\left\{0\right\}}\times\mathcal{C}}=0,\tilde{u}|_{{\left\{1\right\}}\times\mathcal{C}}=1,\tilde{u}\in C([0,1]\times\mathcal{C}).$$ By symmetry of $[0,1]\times\mathcal{C}$, $\tilde{u}(x,y)=x,(x,y)\in [0,1]\times\mathcal{C}$. By induction, we have $$\sum_{w\in\tilde{W}_{n+1}}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in\tilde{V}_w\\
|p-q|=2^{-1}\cdot3^{-(n+1)}
\end{subarray}
$
}}}
(\tilde{u}(p)-\tilde{u}(q))^2=\frac{2}{3}\sum_{w\in\tilde{W}_n}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in\tilde{V}_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(\tilde{u}(p)-\tilde{u}(q))^2\text{ for all }n\ge1,$$ hence $$\sum_{w\in\tilde{W}_{n}}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in\tilde{V}_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(\tilde{u}(p)-\tilde{u}(q))^2=\left(\frac{2}{3}\right)^n\text{ for all }n\ge1.$$ By Equation (\[eqn\_upper\]), we have $$\sum_{n=1}^\infty3^{(\beta-\alpha)n}\left(\frac{2}{3}\right)^n<+\infty,$$ hence, $\beta<\log(8\cdot3/2)/\log3$. Hence $$\beta_*\le\frac{\log(8\cdot\frac{3}{2})}{\log3}.$$
Resistance Estimates {#sec_resistance}
====================
In this section, we give resistance estimates using electrical network techniques.
We consider two sequences of finite graphs related to $V_n$ and $W_n$, respectively.
For all $n\ge1$. Let ${\mathcal{V}}_n$ be the graph with vertex set $V_n$ and edge set given by $${\left\{(p,q):p,q\in V_n,|p-q|=2^{-1}\cdot3^{-n}\right\}}.$$ For example, we have the figure of ${\mathcal{V}}_2$ in Figure \[fig\_V2\].
in [0,1,...,9]{} (,0)–(,9);
in [0,1,...,9]{} (0,)–(9,);
(3,3)–(6,3)–(6,6)–(3,6)–cycle;
in [0,1,...,9]{} in [0,0.5,1,...,9]{} (,) circle (0.06);
in [0,1,...,9]{} in [0,0.5,1,...,9]{} (,) circle (0.06);
(3.25,3.25)–(5.75,3.25)–(5.75,5.75)–(3.25,5.75)–cycle;
Let ${\mathcal{W}}_n$ be the graph with vertex set $W_n$ and edge set given by $${\left\{(w^{(1)},w^{(2)}):w^{(1)},w^{(2)}\in W_n,\mathrm{dim}_{\mathcal{H}}\left(K_{w^{(1)}}\cap K_{w^{(2)}}\right)=1\right\}}.$$ For example, we have the figure of ${\mathcal{W}}_2$ in Figure \[fig\_W2\].
(0,0)–(8,0)–(8,8)–(0,8)–cycle; (2,0)–(2,8); (6,0)–(6,8); (0,2)–(8,2); (0,6)–(8,6); (3,0)–(3,2); (5,0)–(5,2); (0,3)–(2,3); (0,5)–(2,5); (6,3)–(8,3); (6,5)–(8,5); (3,6)–(3,8); (5,6)–(5,8); (2,1)–(3,1); (5,1)–(6,1); (1,2)–(1,3); (7,2)–(7,3); (1,5)–(1,6); (7,5)–(7,6); (2,7)–(3,7); (5,7)–(6,7);
(0,0) circle (0.06); (1,0) circle (0.06); (2,0) circle (0.06); (3,0) circle (0.06); (4,0) circle (0.06); (5,0) circle (0.06); (6,0) circle (0.06); (7,0) circle (0.06); (8,0) circle (0.06);
(0,1) circle (0.06); (2,1) circle (0.06); (3,1) circle (0.06); (5,1) circle (0.06); (6,1) circle (0.06); (8,1) circle (0.06);
(0,2) circle (0.06); (1,2) circle (0.06); (2,2) circle (0.06); (3,2) circle (0.06); (4,2) circle (0.06); (5,2) circle (0.06); (6,2) circle (0.06); (7,2) circle (0.06); (8,2) circle (0.06);
(0,3) circle (0.06); (1,3) circle (0.06); (2,3) circle (0.06); (6,3) circle (0.06); (7,3) circle (0.06); (8,3) circle (0.06);
(0,4) circle (0.06); (2,4) circle (0.06); (6,4) circle (0.06); (8,4) circle (0.06);
(0,5) circle (0.06); (1,5) circle (0.06); (2,5) circle (0.06); (6,5) circle (0.06); (7,5) circle (0.06); (8,5) circle (0.06);
(0,6) circle (0.06); (1,6) circle (0.06); (2,6) circle (0.06); (3,6) circle (0.06); (4,6) circle (0.06); (5,6) circle (0.06); (6,6) circle (0.06); (7,6) circle (0.06); (8,6) circle (0.06);
(0,7) circle (0.06); (2,7) circle (0.06); (3,7) circle (0.06); (5,7) circle (0.06); (6,7) circle (0.06); (8,7) circle (0.06);
(0,8) circle (0.06); (1,8) circle (0.06); (2,8) circle (0.06); (3,8) circle (0.06); (4,8) circle (0.06); (5,8) circle (0.06); (6,8) circle (0.06); (7,8) circle (0.06); (8,8) circle (0.06);
On ${\mathcal{V}}_n$, the energy $${\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_n\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(u(p)-u(q))^2,u\in l(V_n),$$ is related to a weighted graph with the conductances of all edges equal to $1$. While the energy $$\sum_{w\in W_n}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(u(p)-u(q))^2,u\in l(V_n),$$ is related to a weighted graph with the conductances of some edges equal to $1$ and the conductances of other edges equal to $2$, since the term $(u(p)-u(q))^2$ is added either once or twice.
Since $$\begin{aligned}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_n\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(u(p)-u(q))^2&\le\sum_{w\in W_n}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(u(p)-u(q))^2\\
&\le 2
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_n\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(u(p)-u(q))^2,
\end{aligned}$$ we use $$D_n(u,u):=\sum_{w\in W_n}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(u(p)-u(q))^2,u\in l(V_n),$$ as the energy on ${\mathcal{V}}_n$. Assume that $A,B$ are two disjoint subsets of $V_n$. Let $$R_n(A,B)=\inf{\left\{D_n(u,u):u|_A=0,u|_B=1,u\in l(V_n)\right\}}^{-1}.$$ Denote $$R_n^V=R_n(V_n\cap{\left\{0\right\}}\times[0,1],V_n\cap{\left\{1\right\}}\times[0,1]),$$ $$R_n(x,y)=R_n({\left\{x\right\}},{\left\{y\right\}}),x,y\in V_n.$$ It is obvious that $R_n$ is a metric on $V_n$, hence $$R_n(x,y)\le R_n(x,z)+R_n(z,y)\text{ for all }x,y,z\in V_n.$$
On ${\mathcal{W}}_n$, the energy $${\mathfrak{D}}_n(u,u):=\sum_{w^{(1)}\sim_nw^{(2)}}(u(w^{(1)})-u(w^{(2)}))^2,u\in l(W_n),$$ is related to a weighted graph with the conductances of all edges equal to $1$. Assume that $A,B$ are two disjoint subsets of $W_n$. Let $${\mathfrak{R}}_n(A,B)=\inf{\left\{{\mathfrak{D}}_n(u,u):u|_A=0,u|_B=1,u\in l(W_n)\right\}}^{-1}.$$ Denote $${\mathfrak{R}}_n(w^{(1)},w^{(2)})={\mathfrak{R}}_n({\left\{w^{(1)}\right\}},{\left\{w^{(2)}\right\}}),w^{(1)},w^{(2)}\in W_n.$$ It is obvious that ${\mathfrak{R}}_n$ is a metric on $W_n$, hence $${\mathfrak{R}}_n(w^{(1)},w^{(2)})\le{\mathfrak{R}}_n(w^{(1)},w^{(3)})+{\mathfrak{R}}_n(w^{(3)},w^{(2)})\text{ for all }w^{(1)},w^{(2)},w^{(3)}\in W_n.$$
The main result of this section is as follows.
\[thm\_resist\] There exists some positive constant $\rho\in\left[7/6,3/2\right]$ such that for all $n\ge1$ $$R_n^V\asymp\rho^n,$$ $$R_n(p_0,p_1)=\ldots=R_n(p_6,p_7)=R_n(p_7,p_0)\asymp\rho^n,$$ $${\mathfrak{R}}_n(0^n,1^n)=\ldots={\mathfrak{R}}_n(6^n,7^n)={\mathfrak{R}}_n(7^n,0^n)\asymp\rho^n.$$
By triangle inequality, for all $i,j=0,\ldots,7,n\ge1$ $$R_{n}(p_i,p_j)\lesssim\rho^n,$$ $${\mathfrak{R}}_{n}(i^n,j^n)\lesssim\rho^n.$$
We have a direct corollary as follows.
\[cor\_resist\_upper\] For all $n\ge1,p,q\in V_n,w^{(1)},w^{(2)}\in W_n$ $$R_n(p,q)\lesssim\rho^n,$$ $${\mathfrak{R}}_n(w^{(1)},w^{(2)})\lesssim\rho^n.$$
We only need to show that ${\mathfrak{R}}_n(w,0^n)\lesssim\rho^n$ for all $w\in W_n,n\ge1$. Then for all $w^{(1)},w^{(2)}\in W_n$ $${\mathfrak{R}}_n(w^{(1)},w^{(2)})\le{\mathfrak{R}}_n(w^{(1)},0^n)+{\mathfrak{R}}_n(w^{(2)},0^n)\lesssim\rho^n.$$ Similarly, we have the proof of $R_n(p,q)\lesssim\rho^n$ for all $p,q\in V_n,n\ge1$.
Indeed, for all $n\ge1,w=w_1\ldots w_n\in W_n$, we construct a finite sequence as follows. $$\begin{aligned}
w^{(1)}&=w_1\ldots w_{n-2}w_{n-1}w_n=w,\\
w^{(2)}&=w_1\ldots w_{n-2}w_{n-1}w_{n-1},\\
w^{(3)}&=w_1\ldots w_{n-2}w_{n-2}w_{n-2},\\
&\ldots\\
w^{(n)}&=w_1\ldots w_1w_1w_1,\\
w^{(n+1)}&=0\ldots 000=0^n.
\end{aligned}$$ For all $i=1,\ldots,n-1$, by cutting technique $$\begin{aligned}
&{\mathfrak{R}}_n(w^{(i)},w^{(i+1)})={\mathfrak{R}}_n(w_1\ldots w_{n-i}w_{n-i+1}\ldots w_{n-i+1},w_1\ldots w_{n-i}w_{n-i}\ldots w_{n-i})\\
&\le{\mathfrak{R}}_i(w_{n-i+1}\ldots w_{n-i+1},w_{n-i}\ldots w_{n-i})={\mathfrak{R}}_i(w_{n-i+1}^i,w_{n-i}^i)\lesssim\rho^i.
\end{aligned}$$ Since ${\mathfrak{R}}_n(w^{(n)},w^{(n+1)})={\mathfrak{R}}_n(w_1^n,0^n)\lesssim\rho^n$, we have $${\mathfrak{R}}_n(w,0^n)={\mathfrak{R}}_n(w^{(1)},w^{(n+1)})\le\sum_{i=1}^n{\mathfrak{R}}_n(w^{(i)},w^{(i+1)})\lesssim\sum_{i=1}^n\rho^i\lesssim\rho^n.$$
We need the following results for preparation.
First, we have resistance estimates for some symmetric cases.
\[thm\_resist1\] There exists some positive constant $\rho\in[7/6,3/2]$ such that for all $n\ge1$ $$R_n^V\asymp\rho^n,$$ $$R_n(p_1,p_5)=R_n(p_3,p_7)\asymp\rho^n,$$ $$R_n(p_0,p_4)=R_n(p_2,p_6)\asymp\rho^n.$$
The proof is similar to [@BB90 Theorem 5.1] and [@McG02 Theorem 6.1] where flow technique and potential technique are used. We need discrete version instead of continuous version.
Hence there exists some positive constant $C$ such that $$\frac{1}{C}x_nx_m\le x_{n+m}\le Cx_nx_m\text{ for all }n,m\ge1,$$ where $x$ is any of above resistances. Since above resistances share the same complexity, there exists *one* positive constant $\rho$ such that they are equivalent to $\rho^n$ for all $n\ge1$.
By shorting and cutting technique, we have $\rho\in[7/6,3/2]$, see [@Bar13 Equation (2.6)] or [@BB99a Remarks 5.4].
Second, by symmetry and shorting technique, we have the following relations.
\[prop\_resist2\] For all $n\ge1$ $$R_n(p_0,p_1)\le{\mathfrak{R}}_n(0^n,1^n),$$ $$R_n^V\le R_n(p_1,p_5)=R_n(p_3,p_7)\le{\mathfrak{R}}_n(1^n,5^n)={\mathfrak{R}}_n(3^n,7^n),$$ $$R_n^V\le R_n(p_0,p_4)=R_n(p_2,p_6)\le{\mathfrak{R}}_n(0^n,4^n)={\mathfrak{R}}_n(2^n,6^n).$$
Third, we have the following relations.
\[prop\_resist3\] For all $n\ge1$ $${\mathfrak{R}}_n(0^n,1^n)\lesssim R_n(p_0,p_1),$$ $${\mathfrak{R}}_n(1^n,5^n)={\mathfrak{R}}_n(3^n,7^n)\lesssim R_n(p_1,p_5)=R_n(p_3,p_7),$$ $${\mathfrak{R}}_n(0^n,4^n)={\mathfrak{R}}_n(2^n,6^n)\lesssim R_n(p_0,p_4)=R_n(p_2,p_6).$$
The idea is to use electrical network transformations to *increase* resistances to transform weighted graph ${\mathcal{W}}_n$ to weighted graph ${\mathcal{V}}_{n-1}$.
First, we do the transformation in Figure \[fig\_trans1\] where the resistances of the resistors in the new network only depend on the shape of the networks in Figure \[fig\_trans1\] such that we obtain the weighted graph in Figure \[fig\_trans2\] where the resistances between any two points are larger than those in the weighted graph ${\mathcal{W}}_n$. For ${\mathfrak{R}}_n(i^n,j^n)$, we have the equivalent weighted graph in Figure \[fig\_trans3\].
(0,0)–(2,0)–(2,1)–(0,1)–cycle; (1,0)–(1,1);
(0,0) circle (0.06); (1,0) circle (0.06); (2,0) circle (0.06); (0,1) circle (0.06); (1,1) circle (0.06); (2,1) circle (0.06);
(3,0)–(3,1); (4,0)–(4,1); (5,0)–(5,1); (3,0.5)–(5,0.5);
(3,0) circle (0.06); (4,0) circle (0.06); (5,0) circle (0.06); (3,1) circle (0.06); (4,1) circle (0.06); (5,1) circle (0.06);
(2.5,0.5) node [$\Rightarrow$]{};
(0,0) circle (0.06); (0.5,0) circle (0.06); (1,0) circle (0.06); (1.5,0) circle (0.06); (2,0) circle (0.06); (2.5,0) circle (0.06); (3,0) circle (0.06); (3.5,0) circle (0.06); (4,0) circle (0.06); (4.5,0) circle (0.06); (5,0) circle (0.06); (5.5,0) circle (0.06); (6,0) circle (0.06); (6.5,0) circle (0.06); (7,0) circle (0.06); (7.5,0) circle (0.06); (8,0) circle (0.06); (8.5,0) circle (0.06);
(0,0.5) circle (0.06); (1,0.5) circle (0.06); (1.5,0.5) circle (0.06); (2.5,0.5) circle (0.06); (3,0.5) circle (0.06); (4,0.5) circle (0.06); (4.5,0.5) circle (0.06); (5.5,0.5) circle (0.06); (6,0.5) circle (0.06); (7,0.5) circle (0.06); (7.5,0.5) circle (0.06); (8.5,0.5) circle (0.06);
(0,1) circle (0.06); (0.5,1) circle (0.06); (1,1) circle (0.06); (1.5,1) circle (0.06); (2,1) circle (0.06); (2.5,1) circle (0.06); (3,1) circle (0.06); (3.5,1) circle (0.06); (4,1) circle (0.06); (4.5,1) circle (0.06); (5,1) circle (0.06); (5.5,1) circle (0.06); (6,1) circle (0.06); (6.5,1) circle (0.06); (7,1) circle (0.06); (7.5,1) circle (0.06); (8,1) circle (0.06); (8.5,1) circle (0.06);
(0,1.5) circle (0.06); (0.5,1.5) circle (0.06); (1,1.5) circle (0.06); (3,1.5) circle (0.06); (3.5,1.5) circle (0.06); (4,1.5) circle (0.06); (4.5,1.5) circle (0.06); (5,1.5) circle (0.06); (5.5,1.5) circle (0.06); (7.5,1.5) circle (0.06); (8,1.5) circle (0.06); (8.5,1.5) circle (0.06);
(0,2) circle (0.06); (1,2) circle (0.06); (3,2) circle (0.06); (4,2) circle (0.06); (4.5,2) circle (0.06); (5.5,2) circle (0.06); (7.5,2) circle (0.06); (8.5,2) circle (0.06);
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(0,3) circle (0.06); (0.5,3) circle (0.06); (1,3) circle (0.06); (1.5,3) circle (0.06); (2,3) circle (0.06); (2.5,3) circle (0.06); (3,3) circle (0.06); (3.5,3) circle (0.06); (4,3) circle (0.06); (4.5,3) circle (0.06); (5,3) circle (0.06); (5.5,3) circle (0.06); (6,3) circle (0.06); (6.5,3) circle (0.06); (7,3) circle (0.06); (7.5,3) circle (0.06); (8,3) circle (0.06); (8.5,3) circle (0.06);
(0,3.5) circle (0.06); (1,3.5) circle (0.06); (1.5,3.5) circle (0.06); (2.5,3.5) circle (0.06); (3,3.5) circle (0.06); (4,3.5) circle (0.06); (4.5,3.5) circle (0.06); (5.5,3.5) circle (0.06); (6,3.5) circle (0.06); (7,3.5) circle (0.06); (7.5,3.5) circle (0.06); (8.5,3.5) circle (0.06);
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(0,5) circle (0.06); (1,5) circle (0.06); (1.5,5) circle (0.06); (2.5,5) circle (0.06); (3,5) circle (0.06); (4,5) circle (0.06);
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(0,6.5) circle (0.06); (1,6.5) circle (0.06); (3,6.5) circle (0.06); (4,6.5) circle (0.06);
(0,7) circle (0.06); (0.5,7) circle (0.06); (1,7) circle (0.06); (3,7) circle (0.06); (3.5,7) circle (0.06); (4,7) circle (0.06);
(0,7.5) circle (0.06); (0.5,7.5) circle (0.06); (1,7.5) circle (0.06); (1.5,7.5) circle (0.06); (2,7.5) circle (0.06); (2.5,7.5) circle (0.06); (3,7.5) circle (0.06); (3.5,7.5) circle (0.06); (4,7.5) circle (0.06);
(0,8) circle (0.06); (1,8) circle (0.06); (1.5,8) circle (0.06); (2.5,8) circle (0.06); (3,8) circle (0.06); (4,8) circle (0.06);
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(0,0)–(8.5,0); (0,0)–(0,8.5);
(1,1)–(1.5,1); (1,1)–(1,1.5); (1.25,0)–(1.25,1); (0,1.25)–(1,1.25); (1,0.5)–(1.5,0.5); (0.5,1)–(0.5,1.5);
(1.5,1)–(3,1); (2.75,1)–(2.75,0); (2.5,0.5)–(3,0.5);
(3,1)–(3,1.5); (4,1)–(4,1.5); (3,1.25)–(4,1.25); (3.5,1)–(3.5,1.5); (4,1)–(4.5,1); (4,0.5)–(4.5,0.5); (4.25,0)–(4.25,1);
(4.5,1)–(4.5,1.5); (5,1)–(5,1.5); (5.5,1)–(5.5,1.5); (4.5,1.25)–(5.5,1.25); (5.5,1)–(6,1); (5.5,0.5)–(6,0.5); (5.75,0)–(5.75,1); (6,1)–(7,1); (7,1)–(7.5,1); (7,0.5)–(7.5,0.5); (7.25,0)–(7.25,1); (7.5,1)–(7.5,1.5); (8,1)–(8,1.5); (8.5,1)–(8.5,1.5); (7.5,1.25)–(8.5,1.25);
(8.5,0)–(9,0); (8.5,0.5)–(9,0.5); (8.5,1)–(9,1); (8.75,0)–(8.75,1);
(1,1.5)–(1,3); (0.5,2.5)–(0.5,3); (0,2.75)–(1,2.75);
(1,3)–(3,3); (3,3)–(3,1);
(3.5,2.5)–(3.5,3); (4,2.5)–(4,3); (3,2.75)–(4,2.75); (4,1.5)–(4.5,1.5); (4,2)–(4.5,2); (4,2.5)–(4.5,2.5); (4.25,1.5)–(4.25,2.5);
(4.5,2.5)–(4.5,3); (5,2.5)–(5,3); (5.5,2.5)–(5.5,3); (4.5,2.75)–(5.5,2.75); (5.5,1.5)–(5.5,2.5); (5.5,3)–(7.5,3); (7.5,3)–(7.5,1.5);
(8,2.5)–(8,3); (8.5,2.5)–(8.5,3); (7.5,2.75)–(8.5,2.75);
(8.5,1.5)–(9,1.5); (8.5,2)–(9,2); (8.5,2.5)–(9,2.5); (8.75,1.5)–(8.75,2.5);
(8.5,3)–(9,3); (8.5,3.5)–(9,3.5); (8.5,4)–(9,4); (8.75,3)–(8.75,4);
(7,3.5)–(7.5,3.5); (7,4)–(7.5,4); (7.25,3)–(7.25,4);
(5.5,3.5)–(6,3.5); (5.5,4)–(6,4); (5.75,3)–(5.75,4);
(6,4)–(7,4); (4.5,4)–(5.5,4);
(4,3)–(4.5,3); (4,3.5)–(4.5,3.5); (4,4)–(4.5,4); (4.25,3)–(4.25,4);
(4,4)–(4,4.5); (3.5,4)–(3.5,4.5); (3,4)–(3,4.5); (3,4.25)–(4,4.25);
(2.5,4)–(3,4); (2.5,3.5)–(3,3.5); (2.75,3)–(2.75,4);
(1,4)–(1.5,4); (1,3.5)–(1.5,3.5); (1.25,3)–(1.25,4);
(7.5,4)–(8.5,4);
(0.5,4)–(0.5,4.5); (1,4)–(1,4.5); (0,4.25)–(1,4.25);
(1.5,4)–(1.5,4.5); (2,4)–(2,4.5); (2.5,4)–(2.5,4.5); (1.5,4.25)–(2.5,4.25);
(1,4.5)–(1.5,4.5); (1,5)–(1.5,5); (1,5.5)–(1.5,5.5); (1.25,4.5)–(1.25,5.5);
(2.5,4.5)–(3,4.5); (2.5,5)–(3,5); (2.5,5.5)–(3,5.5); (2.75,4.5)–(2.75,5.5);
(4,4.5)–(4,5.5); (1.5,5.5)–(2.5,5.5);
(0.5,5.5)–(0.5,6); (1,5.5)–(1,6); (0,5.75)–(1,5.75);
(3,5.5)–(3,6); (3.5,5.5)–(3.5,6); (4,5.5)–(4,6); (3,5.75)–(4,5.75);
(1,6)–(1,7); (3,6)–(3,7); (4,6)–(4,7);
(0.5,7)–(0.5,7.5); (1,7)–(1,7.5); (0,7.25)–(1,7.25);
(3,7)–(3,7.5); (3.5,7)–(3.5,7.5); (4,7)–(4,7.5); (3,7.25)–(4,7.25);
(1,7.5)–(3,7.5);
(1,8)–(1.5,8); (1,8.5)–(1.5,8.5); (1.25,7.5)–(1.25,8.5);
(2.5,8)–(3,8); (2.5,8.5)–(3,8.5); (2.75,7.5)–(2.75,8.5);
(0,8.5)–(0,9); (0.5,8.5)–(0.5,9); (1,8.5)–(1,9); (1.5,8.5)–(1.5,9); (2,8.5)–(2,9); (2.5,8.5)–(2.5,9); (3,8.5)–(3,9); (3.5,8.5)–(3.5,9); (4,8.5)–(4,9);
(0,8.75)–(1,8.75); (1.5,8.75)–(2.5,8.75); (3,8.75)–(4,8.75);
(4,8.5)–(4,7.5);
(2,9.5) node […]{}; (9.5,2) node ;
(0,0) circle (0.06); (0.5,0) circle (0.06); (1,0) circle (0.06); (1.5,0) circle (0.06); (2,0) circle (0.06); (2.5,0) circle (0.06); (3,0) circle (0.06); (3.5,0) circle (0.06); (4,0) circle (0.06); (4.5,0) circle (0.06); (5,0) circle (0.06); (5.5,0) circle (0.06); (6,0) circle (0.06); (6.5,0) circle (0.06); (7,0) circle (0.06); (7.5,0) circle (0.06); (8,0) circle (0.06); (8.5,0) circle (0.06);
(0,0.5) circle (0.06); (1,0.5) circle (0.06); (1.5,0.5) circle (0.06); (2.5,0.5) circle (0.06); (3,0.5) circle (0.06); (4,0.5) circle (0.06); (4.5,0.5) circle (0.06); (5.5,0.5) circle (0.06); (6,0.5) circle (0.06); (7,0.5) circle (0.06); (7.5,0.5) circle (0.06); (8.5,0.5) circle (0.06);
(0,1) circle (0.06); (0.5,1) circle (0.06); (1,1) circle (0.06); (1.5,1) circle (0.06); (2,1) circle (0.06); (2.5,1) circle (0.06); (3,1) circle (0.06); (3.5,1) circle (0.06); (4,1) circle (0.06); (4.5,1) circle (0.06); (5,1) circle (0.06); (5.5,1) circle (0.06); (6,1) circle (0.06); (6.5,1) circle (0.06); (7,1) circle (0.06); (7.5,1) circle (0.06); (8,1) circle (0.06); (8.5,1) circle (0.06);
(0,1.5) circle (0.06); (0.5,1.5) circle (0.06); (1,1.5) circle (0.06); (3,1.5) circle (0.06); (3.5,1.5) circle (0.06); (4,1.5) circle (0.06); (4.5,1.5) circle (0.06); (5,1.5) circle (0.06); (5.5,1.5) circle (0.06); (7.5,1.5) circle (0.06); (8,1.5) circle (0.06); (8.5,1.5) circle (0.06);
(0,2) circle (0.06); (1,2) circle (0.06); (3,2) circle (0.06); (4,2) circle (0.06); (4.5,2) circle (0.06); (5.5,2) circle (0.06); (7.5,2) circle (0.06); (8.5,2) circle (0.06);
(0,2.5) circle (0.06); (0.5,2.5) circle (0.06); (1,2.5) circle (0.06); (3,2.5) circle (0.06); (3.5,2.5) circle (0.06); (4,2.5) circle (0.06); (4.5,2.5) circle (0.06); (5,2.5) circle (0.06); (5.5,2.5) circle (0.06); (7.5,2.5) circle (0.06); (8,2.5) circle (0.06); (8.5,2.5) circle (0.06);
(0,3) circle (0.06); (0.5,3) circle (0.06); (1,3) circle (0.06); (1.5,3) circle (0.06); (2,3) circle (0.06); (2.5,3) circle (0.06); (3,3) circle (0.06); (3.5,3) circle (0.06); (4,3) circle (0.06); (4.5,3) circle (0.06); (5,3) circle (0.06); (5.5,3) circle (0.06); (6,3) circle (0.06); (6.5,3) circle (0.06); (7,3) circle (0.06); (7.5,3) circle (0.06); (8,3) circle (0.06); (8.5,3) circle (0.06);
(0,3.5) circle (0.06); (1,3.5) circle (0.06); (1.5,3.5) circle (0.06); (2.5,3.5) circle (0.06); (3,3.5) circle (0.06); (4,3.5) circle (0.06); (4.5,3.5) circle (0.06); (5.5,3.5) circle (0.06); (6,3.5) circle (0.06); (7,3.5) circle (0.06); (7.5,3.5) circle (0.06); (8.5,3.5) circle (0.06);
(0,4) circle (0.06); (0.5,4) circle (0.06); (1,4) circle (0.06); (1.5,4) circle (0.06); (2,4) circle (0.06); (2.5,4) circle (0.06); (3,4) circle (0.06); (3.5,4) circle (0.06); (4,4) circle (0.06); (4.5,4) circle (0.06); (5,4) circle (0.06); (5.5,4) circle (0.06); (6,4) circle (0.06); (6.5,4) circle (0.06); (7,4) circle (0.06); (7.5,4) circle (0.06); (8,4) circle (0.06); (8.5,4) circle (0.06);
(0,4.5) circle (0.06); (0.5,4.5) circle (0.06); (1,4.5) circle (0.06); (1.5,4.5) circle (0.06); (2,4.5) circle (0.06); (2.5,4.5) circle (0.06); (3,4.5) circle (0.06); (3.5,4.5) circle (0.06); (4,4.5) circle (0.06);
(0,5) circle (0.06); (1,5) circle (0.06); (1.5,5) circle (0.06); (2.5,5) circle (0.06); (3,5) circle (0.06); (4,5) circle (0.06);
(0,5.5) circle (0.06); (0.5,5.5) circle (0.06); (1,5.5) circle (0.06); (1.5,5.5) circle (0.06); (2,5.5) circle (0.06); (2.5,5.5) circle (0.06); (3,5.5) circle (0.06); (3.5,5.5) circle (0.06); (4,5.5) circle (0.06);
(0,6) circle (0.06); (0.5,6) circle (0.06); (1,6) circle (0.06); (3,6) circle (0.06); (3.5,6) circle (0.06); (4,6) circle (0.06);
(0,6.5) circle (0.06); (1,6.5) circle (0.06); (3,6.5) circle (0.06); (4,6.5) circle (0.06);
(0,7) circle (0.06); (0.5,7) circle (0.06); (1,7) circle (0.06); (3,7) circle (0.06); (3.5,7) circle (0.06); (4,7) circle (0.06);
(0,7.5) circle (0.06); (0.5,7.5) circle (0.06); (1,7.5) circle (0.06); (1.5,7.5) circle (0.06); (2,7.5) circle (0.06); (2.5,7.5) circle (0.06); (3,7.5) circle (0.06); (3.5,7.5) circle (0.06); (4,7.5) circle (0.06);
(0,8) circle (0.06); (1,8) circle (0.06); (1.5,8) circle (0.06); (2.5,8) circle (0.06); (3,8) circle (0.06); (4,8) circle (0.06);
(0,8.5) circle (0.06); (0.5,8.5) circle (0.06); (1,8.5) circle (0.06); (1.5,8.5) circle (0.06); (2,8.5) circle (0.06); (2.5,8.5) circle (0.06); (3,8.5) circle (0.06); (3.5,8.5) circle (0.06); (4,8.5) circle (0.06);
(0,0)–(8.5,0); (0,0)–(0,8.5);
(1,1)–(1.5,1); (1,1)–(1,1.5); (1.25,0)–(1.25,1); (0,1.25)–(1,1.25);
(1.5,1)–(3,1); (2.75,1)–(2.75,0);
(3,1)–(3,1.5); (4,1)–(4,1.5); (3,1.25)–(4,1.25); (4,1)–(4.5,1); (4.25,0)–(4.25,1);
(4.5,1)–(4.5,1.5); (5.5,1)–(5.5,1.5); (4.5,1.25)–(5.5,1.25); (5.5,1)–(6,1); (5.75,0)–(5.75,1); (6,1)–(7,1); (7,1)–(7.5,1); (7.25,0)–(7.25,1); (7.5,1)–(7.5,1.5); (8.5,1)–(8.5,1.5); (7.5,1.25)–(8.5,1.25);
(8.5,0)–(9,0); (8.5,1)–(9,1); (8.75,0)–(8.75,1);
(1,1.5)–(1,3); (0,2.75)–(1,2.75);
(1,3)–(3,3); (3,3)–(3,1);
(4,2.5)–(4,3); (3,2.75)–(4,2.75); (4,1.5)–(4.5,1.5); (4,2.5)–(4.5,2.5); (4.25,1.5)–(4.25,2.5);
(4.5,2.5)–(4.5,3); (5.5,2.5)–(5.5,3); (4.5,2.75)–(5.5,2.75); (5.5,1.5)–(5.5,2.5); (5.5,3)–(7.5,3); (7.5,3)–(7.5,1.5);
(8.5,2.5)–(8.5,3); (7.5,2.75)–(8.5,2.75);
(8.5,1.5)–(9,1.5); (8.5,2.5)–(9,2.5); (8.75,1.5)–(8.75,2.5);
(8.5,3)–(9,3); (8.5,4)–(9,4); (8.75,3)–(8.75,4);
(7,4)–(7.5,4); (7.25,3)–(7.25,4);
(5.5,4)–(6,4); (5.75,3)–(5.75,4);
(6,4)–(7,4); (4.5,4)–(5.5,4);
(4,3)–(4.5,3); (4,4)–(4.5,4); (4.25,3)–(4.25,4);
(4,4)–(4,4.5); (3,4)–(3,4.5); (3,4.25)–(4,4.25);
(2.5,4)–(3,4); (2.75,3)–(2.75,4);
(1,4)–(1.5,4); (1.25,3)–(1.25,4);
(7.5,4)–(8.5,4);
(1,4)–(1,4.5); (0,4.25)–(1,4.25);
(1.5,4)–(1.5,4.5); (2.5,4)–(2.5,4.5); (1.5,4.25)–(2.5,4.25);
(1,4.5)–(1.5,4.5); (1,5.5)–(1.5,5.5); (1.25,4.5)–(1.25,5.5);
(2.5,4.5)–(3,4.5); (2.5,5.5)–(3,5.5); (2.75,4.5)–(2.75,5.5);
(4,4.5)–(4,5.5); (1.5,5.5)–(2.5,5.5);
(1,5.5)–(1,6); (0,5.75)–(1,5.75);
(3,5.5)–(3,6); (4,5.5)–(4,6); (3,5.75)–(4,5.75);
(1,6)–(1,7); (3,6)–(3,7); (4,6)–(4,7);
(1,7)–(1,7.5); (0,7.25)–(1,7.25);
(3,7)–(3,7.5); (4,7)–(4,7.5); (3,7.25)–(4,7.25);
(1,7.5)–(3,7.5);
(1,8.5)–(1.5,8.5); (1.25,7.5)–(1.25,8.5);
(2.5,8.5)–(3,8.5); (2.75,7.5)–(2.75,8.5);
(0,8.5)–(0,9); (1,8.5)–(1,9); (1.5,8.5)–(1.5,9); (2.5,8.5)–(2.5,9); (3,8.5)–(3,9); (4,8.5)–(4,9);
(0,8.75)–(1,8.75); (1.5,8.75)–(2.5,8.75); (3,8.75)–(4,8.75);
(4,8.5)–(4,7.5);
(2,9.5) node […]{}; (9.5,2) node ;
(0.25-1,0.25)–(0.25-1,-0.25)–(-0.25-1,-0.25)–(-0.25-1,0.25)–cycle; (0-1,0.25)–(0-1,0.75); (0-1,-0.25)–(0-1,-0.75); (0.25-1,0)–(0.75-1,0); (-0.25-1,0)–(-0.75-1,0);
(0.25-1,0.25) circle (0.06); (0.25-1,-0.25) circle (0.06); (-0.25-1,-0.25) circle (0.06); (-0.25-1,0.25) circle (0.06); (-1,0) circle (0.06);
(0,0) node [$\Rightarrow$]{};
(0.25,0)–(1.75,0); (1,-0.75)–(1,0.75);
(0.25+1,0.25) circle (0.06); (0.25+1,-0.25) circle (0.06); (-0.25+1,-0.25) circle (0.06); (-0.25+1,0.25) circle (0.06); (1,0) circle (0.06);
(-5,-1.5)–(-5,-2)–(-4.5,-2); (-5,-1.5) circle (0.06); (-5,-2) circle (0.06); (-4.5,-2) circle (0.06); (-5,-1.75) circle (0.06); (-4.75,-2) circle (0.06); (-4.75,-1.75) circle (0.06);
(-4,-2) node [$\Rightarrow$]{};
(-3,-1.75)–(-2.5,-1.75)–(-2.5,-2); (-2.75,-2)–(-2.75,-1.5)–(-3,-1.5); (-5+2,-1.5) circle (0.06); (-5+2,-2) circle (0.06); (-4.5+2,-2) circle (0.06); (-5+2,-1.75) circle (0.06); (-4.75+2,-2) circle (0.06); (-4.75+2,-1.75) circle (0.06);
(3,-2)–(3,-1.5)–(3.5,-1.5); (3,-2) circle (0.06); (3,-1.5) circle (0.06); (3.5,-1.5) circle (0.06); (3,-1.75) circle (0.06); (3.25,-1.5) circle (0.06); (3.25,-1.75) circle (0.06);
(4,-2) node [$\Rightarrow$]{};
(5,-1.75)–(5.5,-1.75)–(5.5,-1.5); (5.25,-1.5)–(5.25,-2)–(5,-2); (5,-2) circle (0.06); (5,-1.5) circle (0.06); (5.5,-1.5) circle (0.06); (5,-1.75) circle (0.06); (5.25,-1.5) circle (0.06); (5.25,-1.75) circle (0.06);
(-5,-2.5)–(-4.5,-2.5)–(-4.5,-3); (-5,-2.5) circle (0.06); (-4.5,-2.5) circle (0.06); (-4.5,-3) circle (0.06); (-4.75,-2.5) circle (0.06); (-4.5,-2.75) circle (0.06); (-4.75,-2.75) circle (0.06);
(-4,-3) node [$\Rightarrow$]{};
(-2.75,-2.5)–(-2.75,-3)–(-2.5,-3); (-3,-2.5)–(-3,-2.75)–(-2.5,-2.75); (-5+2,-2.5) circle (0.06); (-4.5+2,-2.5) circle (0.06); (-4.5+2,-3) circle (0.06); (-4.75+2,-2.5) circle (0.06); (-4.5+2,-2.75) circle (0.06); (-4.75+2,-2.75) circle (0.06);
(3,-3)–(3.5,-3)–(3.5,-2.5); (3,-3) circle (0.06); (3.5,-3) circle (0.06); (3.5,-2.5) circle (0.06); (3.25,-3) circle (0.06); (3.5,-2.75) circle (0.06); (3.25,-2.75) circle (0.06);
(4,-3) node [$\Rightarrow$]{};
(5,-3)–(5,-2.75)–(5.5,-2.75); (5.25,-3)–(5.25,-2.5)–(5.5,-2.5); (3+2,-3) circle (0.06); (3.5+2,-3) circle (0.06); (3.5+2,-2.5) circle (0.06); (3.25+2,-3) circle (0.06); (3.5+2,-2.75) circle (0.06); (3.25+2,-2.75) circle (0.06);
Second, we do the transformations in Figure \[fig\_trans4\] where the resistances of the resistors in the new networks only depend on the shape of the networks in Figure \[fig\_trans4\] such that we obtain a weighted graph with vertex set $V_{n-1}$ and all conductances equivalent to $1$. Moreover, the resistances between any two points are larger than those in the weighted graph ${\mathcal{W}}_n$, hence we obtain the desired result.
Now we estimate $R_n(p_0,p_1)$ and ${\mathfrak{R}}_n(0^n,1^n)$ as follows.
The idea is that replacing one point by one network should increase resistances by multiplying the resistance of an individual network.
By Proposition \[prop\_resist2\] and Proposition \[prop\_resist3\], we have for all $n\ge1$ $$R_n(p_0,p_1)\asymp{\mathfrak{R}}_n(0^n,1^n).$$ By Theorem \[thm\_resist1\] and Proposition \[prop\_resist2\], we have for all $n\ge1$ $${\mathfrak{R}}_n(0^n,1^n)\ge R_n(p_0,p_1)\ge\frac{1}{4}R_n(p_1,p_5)\asymp\rho^n.$$ We only need to show that for all $n\ge1$ $${\mathfrak{R}}_n(0^n,1^n)\lesssim\rho^n.$$
First, we estimate ${\mathfrak{R}}_{n+1}(0^{n+1},12^n)$. Cutting certain edges in ${\mathcal{W}}_{n+1}$, we obtain the electrical network in Figure \[fig\_resist1\] which is equivalent to the electrical networks in Figure \[fig\_resist2\].
(0,0)–(2,0)–(2,2)–(0,2)–cycle; (3,0)–(5,0)–(5,2)–(3,2)–cycle; (6,0)–(8,0)–(8,2)–(6,2)–cycle; (0,3)–(2,3)–(2,5)–(0,5)–cycle; (6,3)–(8,3)–(8,5)–(6,5)–cycle; (0,6)–(2,6)–(2,8)–(0,8)–cycle; (3,6)–(5,6)–(5,8)–(3,8)–cycle; (6,6)–(8,6)–(8,8)–(6,8)–cycle;
(2,2)–(2,3); (2,2)–(3,2); (0,5)–(0,6); (2,8)–(3,8); (5,6)–(6,6); (6,6)–(6,5); (8,3)–(8,2); (5,0)–(6,0);
(0,0) circle (0.06); (5,0) circle (0.06);
(0,-0.5) node [$0^{n+1}$]{}; (5,-0.5) node [$12^n$]{};
(1,1) node [$0W_n$]{}; (4,1) node [$1W_n$]{}; (7,1) node [$2W_n$]{}; (7,4) node [$3W_n$]{}; (7,7) node [$4W_n$]{}; (4,7) node [$5W_n$]{}; (1,7) node [$6W_n$]{}; (1,4) node [$7W_n$]{};
Hence $$\begin{aligned}
{\mathfrak{R}}_{n+1}(0^{n+1},12^n)&\le{\mathfrak{R}}_n(0^n,4^n)+\frac{\left(5{\mathfrak{R}}_n(0^n,4^n)+7\right)\left({\mathfrak{R}}_n(0^n,4^n)+1\right)}{\left(5{\mathfrak{R}}_n(0^n,4^n)+7\right)+\left({\mathfrak{R}}_n(0^n,4^n)+1\right)}\\
&\lesssim{\mathfrak{R}}_n(0^n,4^n)+\frac{5}{6}{\mathfrak{R}}_n(0^n,4^n)=\frac{11}{6}{\mathfrak{R}}_n(0^n,4^n)\lesssim\rho^{n+1}.
\end{aligned}$$ Second, from $0^{n+1}$ to $1^{n+1}$, we construct a finite sequence as follows. For $i=1,\ldots,n+2$, $$w^{(i)}=
\begin{cases}
1^{i-1}0^{n+2-i},\text{ if }i\text{ is an odd number},\\
1^{i-1}2^{n+2-i},\text{ if }i\text{ is an even number}.\\
\end{cases}$$ By cutting technique, if $i$ is an odd number, then $$\begin{aligned}
&{\mathfrak{R}}_{n+1}(w^{(i)},w^{(i+1)})={\mathfrak{R}}_{n+1}(1^{i-1}0^{n+2-i},1^{i}2^{n+1-i})\\
&\le{\mathfrak{R}}_{n+2-i}(0^{n+2-i},12^{n+1-i})\lesssim\rho^{n+2-i}.
\end{aligned}$$ If $i$ is an even number, then $$\begin{aligned}
&{\mathfrak{R}}_{n+1}(w^{(i)},w^{(i+1)})={\mathfrak{R}}_{n+1}(1^{i-1}2^{n+2-i},1^{i}0^{n+1-i})\\
&\le{\mathfrak{R}}_{n+2-i}(2^{n+2-i},10^{n+1-i})={\mathfrak{R}}_{n+2-i}(0^{n+2-i},12^{n+1-i})\lesssim\rho^{n+2-i}.
\end{aligned}$$ Hence $$\begin{aligned}
&{\mathfrak{R}}_{n+1}(0^{n+1},1^{n+1})={\mathfrak{R}}_{n+1}(w^{(1)},w^{(n+2)})\\
&\le\sum_{i=1}^{n+1}{\mathfrak{R}}_{n+1}(w^{(i)},w^{(i+1)})\lesssim\sum_{i=1}^{n+1}\rho^{n+2-i}=\sum_{i=1}^{n+1}\rho^{i}\lesssim\rho^{n+1}.
\end{aligned}$$
Uniform Harnack Inequality {#sec_harnack}
==========================
In this section, we give uniform Harnack inequality as follows.
\[thm\_harnack\] There exist some constants $C\in(0,+\infty),\delta\in(0,1)$ such that for all $n\ge1,x\in K,r>0$, for all nonnegative harmonic function $u$ on $V_n\cap B(x,r)$, we have $$\max_{V_n\cap B(x,\delta r)}u\le C\min_{V_n\cap B(x,\delta r)}u.$$
The point of above theorem is that the constant $C$ is *uniform* in $n$.
The idea is as follows. First, we use resistance estimates in finite graphs $V_n$ to obtain resistance estimates in an infinite graph $V_\infty$. Second, we obtain Green function estimates in $V_\infty$. Third, we obtain elliptic Harnack inequality in $V_\infty$. Finally, we transfer elliptic Harnack inequality in $V_\infty$ to uniform Harnack inequality in $V_n$.
Let ${\mathcal{V}}_\infty$ be the graph with vertex set $V_\infty=\cup_{n=0}^\infty3^nV_n$ and edge set given by $${\left\{(p,q):p,q\in V_\infty,|p-q|=2^{-1}\right\}}.$$ We have the figure of ${\mathcal{V}}_\infty$ in Figure \[fig\_graphSC\].
Locally, ${\mathcal{V}}_\infty$ is like ${\mathcal{V}}_n$. Let the conductances of all edges be $1$. Let $d$ be the graph distance, that is, $d(p,q)$ is the minimum of the lengths of all paths connecting $p$ and $q$. It is obvious that $$d(p,q)\asymp|p-q|\text{ for all }p,q\in V_\infty.$$
By shorting and cutting technique, we reduce ${\mathcal{V}}_\infty$ to ${\mathcal{V}}_n$ to obtain resistance estimates as follows. $$R(x,y)\asymp\rho^{\frac{\log d(x,y)}{\log 3}}=d(x,y)^{\frac{\log\rho}{\log3}}=d(x,y)^\gamma\text{ for all }x,y\in V_\infty,$$ where $\gamma=\log\rho/\log3$.
Let $g_B$ be the Green function in a ball $B$. We have Green function estimates as follows.
([@GHL14 Proposition 6.11])\[thm\_green\] There exist some constants $C\in(0,+\infty),\eta\in(0,1)$ such that for all $z\in V_\infty,r>0$, we have $$g_{B(z,r)}(x,y)\le Cr^\gamma\text{ for all }x,y\in B(z,r),$$ $$g_{B(z,r)}(z,y)\ge\frac{1}{C}r^\gamma\text{ for all }y\in B(z,\eta r).$$
We obtain elliptic Harnack inequality in $V_\infty$ as follows.
([@GT01 Lemma 10.2],[@GH14a Theorem 3.12])\[thm\_harnack\_infinite\] There exist some constants $C\in(0,+\infty)$, $\delta\in(0,1)$ such that for all $z\in V_\infty,r>0$, for all nonnegative harmonic function $u$ on $V_\infty\cap B(z,r)$, we have $$\max_{B(z,\delta r)}u\le C\min_{B(z,\delta r)}u.$$
We give an alternative approach as follows. It was proved in [@BCK05] that sub-Gaussian heat kernel estimates are equivalent to resistance estimates for random walks on fractal graph under strongly recurrent condition. Hence we obtain sub-Gaussian heat kernel estimates, see [@BCK05 Example 4]. It was proved in [@GT02 Theorem 3.1] that sub-Gaussian heat kernel estimates imply elliptic Harnack inequality. Hence we obtain elliptic Harnack inequality in $V_\infty$.
Now we obtain Theorem \[thm\_harnack\] directly.
Weak Monotonicity Results {#sec_monotone}
=========================
In this section, we give two weak monotonicity results.
For all $n\ge1$, let $$a_n(u)=\rho^n\sum_{w\in W_n}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(u(p)-u(q))^2,u\in l(V_n).$$
We have one weak monotonicity result as follows.
\[thm\_monotone1\] There exists some positive constant $C$ such that for all $n,m\ge1,u\in l(V_{n+m})$, we have $$a_n(u)\le Ca_{n+m}(u).$$
For all $w\in W_n,p,q\in V_w$ with $|p-q|=2^{-1}\cdot3^{-n}$, by cutting technique and Corollary \[cor\_resist\_upper\] $$\begin{aligned}
\left(u(p)-u(q)\right)^2&\le R_m(f_w^{-1}(p),f_w^{-1}(q))
\sum_{v\in W_m}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
x,y\in V_{wv}\\
|x-y|=2^{-1}\cdot3^{-(n+m)}
\end{subarray}
$
}}}
(u(x)-u(y))^2\\
&\le C\rho^m\sum_{v\in W_m}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
x,y\in V_{wv}\\
|x-y|=2^{-1}\cdot3^{-(n+m)}
\end{subarray}
$
}}}
(u(x)-u(y))^2.
\end{aligned}$$ Hence $$\begin{aligned}
a_n(u)&=\rho^n\sum_{w\in W_n}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(u(p)-u(q))^2\\
&\le\rho^n\sum_{w\in W_n}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
\left(C\rho^m\sum_{v\in W_m}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
x,y\in V_{wv}\\
|x-y|=2^{-1}\cdot3^{-(n+m)}
\end{subarray}
$
}}}
(u(x)-u(y))^2\right)\\
&=C\rho^{n+m}\sum_{w\in W_{n+m}}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-(n+m)}
\end{subarray}
$
}}}
(u(p)-u(q))^2=Ca_{n+m}(u).
\end{aligned}$$
For all $n\ge1$, let $$b_n(u)=\rho^n
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
w^{(1)}\sim_nw^{(2)}
\end{subarray}
$
}}}
(P_nu(w^{(1)})-P_nu(w^{(2)}))^2,u\in L^2(K;\nu).$$
We have another weak monotonicity result as follows.
\[thm\_monotone2\] There exists some positive constant $C$ such that for all $n,m\ge1,u\in L^2(K;\nu)$, we have $$b_n(u)\le Cb_{n+m}(u).$$
This result was also obtained in [@KZ92 Proposition 5.2]. Here we give a direct proof using resistance estimates.
This result can be reduced as follows.
For all $n\ge1$, let $$B_n(u)=\rho^n\sum_{w^{(1)}\sim_nw^{(2)}}(u(w^{(1)})-u(w^{(2)}))^2,u\in l(W_n).$$
For all $n,m\ge1$, let $M_{n,m}:l(W_{n+m})\to l(W_n)$ be a mean value operator given by $$(M_{n,m}u)(w)=\frac{1}{8^m}\sum_{v\in W_m}u(wv),w\in W_n,u\in l(W_{n+m}).$$
\[thm\_monotonicity2\] There exists some positive constant $C$ such that for all $n,m\ge1,u\in l(W_{n+m})$, we have $$B_n(M_{n,m}u)\le CB_{n+m}(u).$$
For all $u\in L^2(K;\nu)$, note that $$P_nu=M_{n,m}(P_{n+m}u),$$ hence $$\begin{aligned}
b_n(u)&=\rho^n\sum_{w^{(1)}\sim_nw^{(2)}}(P_nu(w^{(1)})-P_nu(w^{(2)}))^2=B_n(P_nu)\\
&=B_n(M_{n,m}(P_{n+m}u))\le CB_{n+m}(P_{n+m}u)\\
&=C\rho^{n+m}\sum_{w^{(1)}\sim_{n+m}w^{(2)}}(P_{n+m}u(w^{(1)})-P_{n+m}u(w^{(2)}))^2=Cb_{n+m}(u).
\end{aligned}$$
Fix $n\ge1$. Assume that $W\subseteq W_n$ is connected, that is, for all $w^{(1)},w^{(2)}\in W$, there exists a finite sequence ${\left\{v^{(1)},\ldots,v^{(k)}\right\}}\subseteq W$ such that $v^{(1)}=w^{(1)},v^{(k)}=w^{(2)}$ and $v^{(i)}\sim_nv^{(i+1)}$ for all $i=1,\ldots,k-1$. Let $${\mathfrak{D}}_W(u,u):=
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
w^{(1)},w^{(2)}\in W\\
w^{(1)}\sim_nw^{(2)}
\end{subarray}
$
}}}
(u(w^{(1)})-u(w^{(2)}))^2,u\in l(W).$$ For all $w^{(1)},w^{(2)}\in W$, let $$\begin{aligned}
{\mathfrak{R}}_W(w^{(1)},w^{(2)})&=\inf{\left\{{\mathfrak{D}}_W(u,u):u(w^{(1)})=0,u(w^{(2)})=1,u\in l(W)\right\}}^{-1}\\
&=\sup{\left\{\frac{(u(w^{(1)})-u(w^{(2)}))^2}{{\mathfrak{D}}_W(u,u)}:{\mathfrak{D}}_W(u,u)\ne0,u\in l(W)\right\}}.
\end{aligned}$$ It is obvious that $$(u(w^{(1)})-u(w^{(2)}))^2\le{\mathfrak{R}}_W(w^{(1)},w^{(2)}){\mathfrak{D}}_W(u,u)\text{ for all }w^{(1)},w^{(2)}\in W,u\in l(W),$$ and ${\mathfrak{R}}_W$ is a metric on $W$, hence $${\mathfrak{R}}_W(w^{(1)},w^{(2)})\le{\mathfrak{R}}_W(w^{(1)},w^{(3)})+{\mathfrak{R}}_W(w^{(3)},w^{(2)})\text{ for all }w^{(1)},w^{(2)},w^{(3)}\in W.$$
Fix $w^{(1)}\sim_nw^{(2)}$, there exist $i,j=0,\ldots,7$ such that $w^{(1)}i^m\sim_{n+m}w^{(2)}j^m$, see Figure \[fig\_monotonicity\].
(0,0)–(3,0)–(3,3)–(0,3)–cycle; (4,0)–(7,0)–(7,3)–(4,3)–cycle; (3,0)–(4,0); (3,3)–(4,3);
(3.5,1.5) node [$\vdots$]{};
(3,0) circle (0.06); (4,0) circle (0.06);
(1.5,1.5) node [$w^{(1)}W_m$]{}; (5.5,1.5) node [$w^{(2)}W_m$]{}; (2.5,-0.5) node [$w^{(1)}i^m$]{}; (4.5,-0.5) node [$w^{(2)}j^m$]{};
(1,2.5) node [$w^{(1)}v$]{}; (1,2.2) circle (0.06); (5,2.5) node [$w^{(2)}v$]{}; (5,2.2) circle (0.06);
Fix $v\in W_m$ $$(u(w^{(1)}v)-u(w^{(2)}v))^2\le{\mathfrak{R}}_{w^{(1)}W_m\cup w^{(2)}W_m}(w^{(1)}v,w^{(2)}v){\mathfrak{D}}_{w^{(1)}W_m\cup w^{(2)}W_m}(u,u).$$ By cutting technique and Corollary \[cor\_resist\_upper\] $$\begin{aligned}
&{\mathfrak{R}}_{w^{(1)}W_m\cup w^{(2)}W_m}(w^{(1)}v,w^{(2)}v)\\
&\le{\mathfrak{R}}_{w^{(1)}W_m\cup w^{(2)}W_m}(w^{(1)}v,w^{(1)}i^m)+{\mathfrak{R}}_{w^{(1)}W_m\cup w^{(2)}W_m}(w^{(1)}i^m,w^{(2)}j^m)\\
&+{\mathfrak{R}}_{w^{(1)}W_m\cup w^{(2)}W_m}(w^{(2)}j^m,w^{(2)}v)\\
&\le{\mathfrak{R}}_m(v,i^m)+1+{\mathfrak{R}}_m(v,j^m)\lesssim\rho^m.
\end{aligned}$$ Hence $$\begin{aligned}
&(u(w^{(1)}v)-u(w^{(2)}v))^2\lesssim\rho^m{\mathfrak{D}}_{w^{(1)}W_m\cup w^{(2)}W_m}(u,u)\\
&=\rho^m\left({\mathfrak{D}}_{w^{(1)}W_m}(u,u)+{\mathfrak{D}}_{w^{(2)}W_m}(u,u)\right.\\
&\left.+
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
v^{(1)},v^{(2)}\in W_m\\
w^{(1)}v^{(1)}\sim_{n+m}w^{(2)}v^{(2)}
\end{subarray}
$
}}}
(u(w^{(1)}v^{(1)})-u(w^{(2)}v^{(2)}))^2\right).
\end{aligned}$$ Hence $$\begin{aligned}
&\left(M_{n,m}u(w^{(1)})-M_{n,m}u(w^{(2)})\right)^2=\left(\frac{1}{8^m}\sum_{v\in W_m}\left(u(w^{(1)}v)-u(w^{(2)}v)\right)\right)^2\\
&\le\frac{1}{8^m}\sum_{v\in W_m}\left(u(w^{(1)}v)-u(w^{(2)}v)\right)^2\\
&\lesssim\rho^m\left({\mathfrak{D}}_{w^{(1)}W_m}(u,u)+{\mathfrak{D}}_{w^{(2)}W_m}(u,u)\right.\\
&\left.+
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
v^{(1)},v^{(2)}\in W_m\\
w^{(1)}v^{(1)}\sim_{n+m}w^{(2)}v^{(2)}
\end{subarray}
$
}}}
(u(w^{(1)}v^{(1)})-u(w^{(2)}v^{(2)}))^2\right).
\end{aligned}$$ In the summation with respect to $w^{(1)}\sim_nw^{(2)}$, the terms ${\mathfrak{D}}_{w^{(1)}W_m}(u,u),{\mathfrak{D}}_{w^{(2)}W_m}(u,u)$ are summed at most $8$ times, hence $$\begin{aligned}
B_n(M_{n,m}u)&=\rho^n\sum_{w^{(1)}\sim_nw^{(2)}}\left(M_{n,m}u(w^{(1)})-M_{n,m}u(w^{(2)})\right)^2\\
&\lesssim\rho^n\sum_{w^{(1)}\sim_nw^{(2)}}\rho^m\left({\mathfrak{D}}_{w^{(1)}W_m}(u,u)+{\mathfrak{D}}_{w^{(2)}W_m}(u,u)\right.\\
&\left.+
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
v^{(1)},v^{(2)}\in W_m\\
w^{(1)}v^{(1)}\sim_{n+m}w^{(2)}v^{(2)}
\end{subarray}
$
}}}
(u(w^{(1)}v^{(1)})-u(w^{(2)}v^{(2)}))^2\right)\\
&\le8\rho^{n+m}\sum_{w^{(1)}\sim_{n+m}w^{(2)}}\left(u(w^{(1)})-u(w^{(2)})\right)^2=8B_{n+m}(u).
\end{aligned}$$
One Good Function {#sec_good}
=================
In this section, we construct *one* good function with energy property and separation property.
By standard argument, we have Hölder continuity from Harnack inequality as follows.
\[thm\_holder\] For all $0\le\delta_1<{\varepsilon}_1<{\varepsilon}_2<\delta_2\le1$, there exist some positive constants $\theta=\theta(\delta_1,\delta_2,{\varepsilon}_1,{\varepsilon}_2)$, $C=C(\delta_1,\delta_2,{\varepsilon}_1,{\varepsilon}_2)$ such that for all $n\ge1$, for all bounded harmonic function $u$ on $V_n\cap(\delta_1,\delta_2)\times[0,1]$, we have $$|u(x)-u(y)|\le C|x-y|^\theta\left(\max_{V_n\cap[\delta_1,\delta_2]\times[0,1]}|u|\right)\text{ for all }x,y\in V_n\cap[{\varepsilon}_1,{\varepsilon}_2]\times[0,1].$$
The proof is similar to [@BB89 Theorem 3.9].
For all $n\ge1$. Let $u_n\in l(V_n)$ satisfy $u_n|_{V_n\cap{\left\{0\right\}}\times[0,1]}=0,u_n|_{V_n\cap{\left\{1\right\}}\times[0,1]}=1$ and $$D_n(u_n,u_n)=\sum_{w\in W_n}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(u_n(p)-u_n(q))^2=(R_n^V)^{-1}.$$ Then $u_n$ is harmonic on $V_n\cap(0,1)\times[0,1]$, $u_n(x,y)=1-u_n(1-x,y)=u_n(x,1-y)$ for all $(x,y)\in V_n$ and $$u_n|_{V_n\cap{\left\{\frac{1}{2}\right\}}\times[0,1]}=\frac{1}{2},u_n|_{V_n\cap[0,\frac{1}{2})\times[0,1]}<\frac{1}{2},u_n|_{V_n\cap(\frac{1}{2},1]\times[0,1]}>\frac{1}{2}.$$ By Arzelà-Ascoli theorem, Theorem \[thm\_holder\] and diagonal argument, there exist some subsequence still denoted by ${\left\{u_n\right\}}$ and some function $u$ on $K$ with $u|_{{\left\{0\right\}}\times[0,1]}=0$ and $u|_{{\left\{1\right\}}\times[0,1]}=1$ such that $u_n$ converges uniformly to $u$ on $K\cap[{\varepsilon}_1,{\varepsilon}_2]\times[0,1]$ for all $0<{\varepsilon}_1<{\varepsilon}_2<1$. Hence $u$ is continuous on $K\cap(0,1)\times[0,1]$, $u_n(x)\to u(x)$ for all $x\in K$ and $u(x,y)=1-u(1-x,y)=u(x,1-y)$ for all $(x,y)\in K$.
\[prop\_u\] The function $u$ given above has the following properties.
(1) There exists some positive constant $C$ such that $$a_n(u)\le C\text{ for all }n\ge1.$$
(2) For all $\beta\in(\alpha,\log(8\rho)/\log3)$, we have $$E_{\beta}(u,u)<+\infty.$$ Hence $u\in C^{\frac{\beta-\alpha}{2}}(K)$.
(3) $$u|_{K\cap{\left\{\frac{1}{2}\right\}}\times[0,1]}=\frac{1}{2},u|_{K\cap[0,\frac{1}{2})\times[0,1]}<\frac{1}{2},u|_{K\cap(\frac{1}{2},1]\times[0,1]}>\frac{1}{2}.$$
\(1) By Theorem \[thm\_resist1\] and Theorem \[thm\_monotone1\], for all $n\ge1$, we have $$\begin{aligned}
&a_n(u)=\lim_{m\to+\infty}a_{n}(u_{n+m})\le C\varliminf_{m\to+\infty}a_{n+m}(u_{n+m})\\
&=C\varliminf_{m\to+\infty}\rho^{n+m}D_{n+m}(u_{n+m},u_{n+m})=C\varliminf_{m\to+\infty}\rho^{n+m}\left(R_{n+m}^V\right)^{-1}\le C.
\end{aligned}$$
\(2) By (1), for all $\beta\in(\alpha,\log(8\rho)/\log3)$, we have $$E_{\beta}(u,u)=\sum_{n=1}^\infty\left(3^{\beta-\alpha}\rho^{-1}\right)^na_n(u)\le C\sum_{n=1}^\infty\left(3^{\beta-\alpha}\rho^{-1}\right)^n<+\infty.$$ By Lemma \[lem\_equiv\] and Lemma \[lem\_holder\], we have $u\in C^{\frac{\beta-\alpha}{2}}(K)$.
\(3) It is obvious that $$u|_{K\cap{\left\{\frac{1}{2}\right\}}\times[0,1]}=\frac{1}{2},u|_{K\cap[0,\frac{1}{2})\times[0,1]}\le\frac{1}{2},u|_{K\cap(\frac{1}{2},1]\times[0,1]}\ge\frac{1}{2}.$$ By symmetry, we only need to show that $$u|_{K\cap(\frac{1}{2},1]\times[0,1]}>\frac{1}{2}.$$ Suppose there exists $(x,y)\in K\cap(1/2,1)\times[0,1]$ such that $u(x,y)=1/2$. Since $u_n-\frac{1}{2}$ is a nonnegative harmonic function on $V_n\cap(\frac{1}{2},1)\times[0,1]$, by Theorem \[thm\_harnack\], for all $1/2<{\varepsilon}_1<x<{\varepsilon}_2<1$, there exists some positive constant $C=C({\varepsilon}_1,{\varepsilon}_2)$ such that for all $n\ge1$ $$\max_{V_n\cap[{\varepsilon}_1,{\varepsilon}_2]\times[0,1]}\left(u_n-\frac{1}{2}\right)\le C\min_{V_n\cap[{\varepsilon}_1,{\varepsilon}_2]\times[0,1]}\left(u_n-\frac{1}{2}\right).$$ Since $u_n$ converges uniformly to $u$ on $K\cap[{\varepsilon}_1,{\varepsilon}_2]\times[0,1]$, we have $$\sup_{K\cap[{\varepsilon}_1,{\varepsilon}_2]\times[0,1]}\left(u-\frac{1}{2}\right)\le C\inf_{K\cap[{\varepsilon}_1,{\varepsilon}_2]\times[0,1]}\left(u-\frac{1}{2}\right)=0.$$ Hence $$u-\frac{1}{2}=0\text{ on }K\cap[{\varepsilon}_1,{\varepsilon}_2]\times[0,1]\text{ for all }\frac{1}{2}<{\varepsilon}_1<x<{\varepsilon}_2<1.$$ Hence $$u=\frac{1}{2}\text{ on }K\cap(\frac{1}{2},1)\times[0,1].$$ By continuity, we have $$u=\frac{1}{2}\text{ on }K\cap[\frac{1}{2},1]\times[0,1],$$ contradiction!
Proof of Theorem \[thm\_walk\] {#sec_walk}
==============================
First, we consider upper bound. Assume that $({\mathcal{E}}_\beta,{\mathcal{F}}_\beta)$ is a regular Dirichlet form on $L^2(K;\nu)$, then there exists $u\in{\mathcal{F}}_\beta$ such that $u|_{{\left\{0\right\}}\times[0,1]}=0$ and $u|_{{\left\{1\right\}}\times[0,1]}=1$. Hence $$\begin{aligned}
+\infty&>E_\beta(u,u)=\sum_{n=1}^\infty3^{(\beta-\alpha)n}D_n(u,u)\ge\sum_{n=1}^\infty3^{(\beta-\alpha)n}D_n(u_n,u_n)\\
&=\sum_{n=1}^\infty3^{(\beta-\alpha)n}\left(R_n^V\right)^{-1}\ge C\sum_{n=1}^\infty\left(3^{\beta-\alpha}\rho^{-1}\right)^n.
\end{aligned}$$ Hence $3^{\beta-\alpha}\rho^{-1}<1$, that is, $\beta<{\log\left(8\rho\right)}/{\log3}=\beta^*$. Hence $\beta_*\le\beta^*$.
Second, we consider lower bound. Similar to the proof of Proposition \[prop\_lower\], to show that $({\mathcal{E}}_\beta,{\mathcal{F}}_\beta)$ is a regular Dirichlet form on $L^2(K;\nu)$ for all $\beta\in(\alpha,\beta^*)$, we only need to show that ${\mathcal{F}}_\beta$ separates points.
Let $u\in C(K)$ be the function in Proposition \[prop\_u\]. By Proposition \[prop\_u\] (2), we have $E_{\beta}(u,u)<+\infty$, hence $u\in{\mathcal{F}}_\beta$.
For all distinct $z_1=(x_1,y_1),z_2=(x_2,y_2)\in K$, without lose of generality, we may assume that $x_1<x_2$. Replacing $z_i$ by $f_w^{-1}(z_i)$ with some $w\in W_n$ and some $n\ge1$, we only have the following cases.
(1) $x_1\in[0,\frac{1}{2}),x_2\in[\frac{1}{2},1]$.
(2) $x_1\in[0,\frac{1}{2}],x_2\in(\frac{1}{2},1]$.
(3) $x_1,x_2\in[0,\frac{1}{2})$, there exist distinct $w_1,w_2\in{\left\{0,1,5,6,7\right\}}$ such that $$z_1\in K_{w_1}\backslash K_{w_2}\text{ and }z_2\in K_{w_2}\backslash K_{w_1}.$$
(4) $x_1,x_2\in(\frac{1}{2},1]$, there exist distinct $w_1,w_2\in{\left\{1,2,3,4,5\right\}}$ such that $$z_1\in K_{w_1}\backslash K_{w_2}\text{ and }z_2\in K_{w_2}\backslash K_{w_1}.$$
For the first case, $u(z_1)<{1}/{2}\le u(z_2)$. For the second case, $u(z_1)\le{1}/{2}<u(z_2)$.
(0,0)–(6,0)–(6,6)–(0,6)–cycle; (2,0)–(2,6); (4,0)–(4,6); (0,2)–(6,2); (0,4)–(6,4);
(1,1) node [$K_0$]{}; (3,1) node [$K_1$]{}; (5,1) node [$K_2$]{}; (5,3) node [$K_3$]{}; (5,5) node [$K_4$]{}; (3,5) node [$K_5$]{}; (1,5) node [$K_6$]{}; (1,3) node [$K_7$]{};
For the third case. If $w_1,w_2$ do *not* belong to the same one of the following sets $${\left\{0,1\right\}},{\left\{7\right\}},{\left\{5,6\right\}},$$ then we construct a function $w$ as follows. Let $v(x,y)=u(y,x)$ for all $(x,y)\in K$, then $$v|_{[0,1]\times{\left\{0\right\}}}=0,v|_{[0,1]\times{\left\{1\right\}}}=1,$$ $$v(x,y)=v(1-x,y)=1-v(x,1-y)\text{ for all }(x,y)\in K,$$ $$E_\beta(v,v)=E_\beta(u,u)<+\infty.$$ Let $$w=
\begin{cases}
v\circ f_i^{-1}-1,&\text{on }K_i,i=0,1,2,\\
v\circ f_i^{-1},&\text{on }K_i,i=3,7,\\
v\circ f_i^{-1}+1,&\text{on }K_i,i=4,5,6,\\
\end{cases}$$ then $w\in C(K)$ is well-defined and $E_\beta(w,w)<+\infty$, hence $w\in{\mathcal{F}}_\beta$. Moreover, $w(z_1)\ne w(z_2)$, $w|_{[0,1]\times{\left\{0\right\}}}=-1,w|_{[0,1]\times{\left\{1\right\}}}=2,w(x,y)=w(1-x,y)=1-w(x,1-y)$ for all $(x,y)\in K$.
If $w_1,w_2$ *do* belong to the same one of the following sets $${\left\{0,1\right\}},{\left\{7\right\}},{\left\{5,6\right\}},$$ then it can only happen that $w_1,w_2\in{\left\{0,1\right\}}$ or $w_1,w_2\in{\left\{5,6\right\}}$, without lose of generality, we may assume that $w_1=0$ and $w_2=1$, then $z_1\in K_0\backslash K_1$ and $z_2\in K_1\backslash K_0$.
Let $$w=
\begin{cases}
u\circ f_i^{-1}-1,&\text{on }K_i,i=0,6,7,\\
u\circ f_i^{-1},&\text{on }K_i,i=1,5,\\
u\circ f_i^{-1}+1,&\text{on }K_i,i=2,3,4,\\
\end{cases}$$ then $w\in C(K)$ is well-defined and $E_{\beta}(w,w)<+\infty$, hence $w\in{\mathcal{F}}_\beta$. Moreover $w(z_1)\ne w(z_2)$, $w|_{{\left\{0\right\}}\times[0,1]}=-1,w|_{{\left\{1\right\}}\times[0,1]}=2,w(x,y)=w(x,1-y)=1-w(1-x,y)$ for all $(x,y)\in K$.
For the forth case, by reflection about ${\left\{\frac{1}{2}\right\}}\times[0,1]$, we reduce to the third case.
Hence ${\mathcal{F}}_\beta$ separates points, hence $({\mathcal{E}}_\beta,{\mathcal{F}}_\beta)$ is a regular Dirichlet form on $L^2(K;\nu)$ for all $\beta\in(\alpha,\beta^*)$, hence $\beta_*\ge\beta^*$.
In conclusion, $\beta_*=\beta^*$.
Proof of Theorem \[thm\_BM\] {#sec_BM}
============================
In this section, we use $\Gamma$-convergence technique to construct a local regular Dirichlet form on $L^2(K;\nu)$ which corresponds to the BM. The idea of this construction is from [@KS05].
The construction of local Dirichlet forms on p.c.f. self-similar sets relies heavily on some monotonicity result which is ensured by some compatibility condition, see [@Kig93; @Kig01]. Our key observation is that even with some weak monotonicity results, we still apply $\Gamma$-convergence technique to obtain some limit.
We need some preparation about $\Gamma$-convergence.
In what follows, $K$ is a locally compact separable metric space and $\nu$ is a Radon measure on $K$ with full support. We say that $({\mathcal{E}},{\mathcal{F}})$ is a *closed form on $L^2(K;\nu)$ in the wide sense* if $\mathcal{F}$ is complete under the inner product ${\mathcal{E}}_1$ but ${\mathcal{F}}$ is not necessary to be dense in $L^2(K;\nu)$. If $({\mathcal{E}},{\mathcal{F}})$ is a closed form on $L^2(K;\nu)$ in the wide sense, we extend ${\mathcal{E}}$ to be $+\infty$ outside ${\mathcal{F}}$, hence the information of ${\mathcal{F}}$ is encoded in ${\mathcal{E}}$.
\[def\_gamma\] Let ${\mathcal{E}}^n,{\mathcal{E}}$ be closed forms on $L^2(K;\nu)$ in the wide sense. We say that ${\mathcal{E}}^n$ is $\Gamma$-convergent to ${\mathcal{E}}$ if the following conditions are satisfied.
(1) For all ${\left\{u_n\right\}}\subseteq L^2(K;\nu)$ that converges strongly to $u\in L^2(K;\nu)$, we have $$\varliminf_{n\to+\infty}{\mathcal{E}}^n(u_n,u_n)\ge{\mathcal{E}}(u,u).$$
(2) For all $u\in L^2(K;\nu)$, there exists a sequence ${\left\{u_n\right\}}\subseteq L^2(K;\nu)$ converging strongly to $u$ in $L^2(K;\nu)$ such that $$\varlimsup_{n\to+\infty}{\mathcal{E}}^n(u_n,u_n)\le{\mathcal{E}}(u,u).$$
We have the following result about $\Gamma$-convergence.
\[prop\_gamma\]([@Dal93 Proposition 6.8, Theorem 8.5, Theorem 11.10, Proposition 12.16]) Let ${\left\{({\mathcal{E}}^n,{\mathcal{F}}^n)\right\}}$ be a sequence of closed forms on $L^2(K;\nu)$ in the wide sense, then there exist some subsequence ${\left\{({\mathcal{E}}^{n_k},{\mathcal{F}}^{n_k})\right\}}$ and some closed form $({\mathcal{E}},{\mathcal{F}})$ on $L^2(K;\nu)$ in the wide sense such that ${\mathcal{E}}^{n_k}$ is $\Gamma$-convergent to ${\mathcal{E}}$.
In what follows, $K$ is SC and $\nu$ is Hausdorff measure.
We need an elementary result as follows.
\[prop\_ele\] Let ${\left\{x_n\right\}}$ be a sequence of nonnegative real numbers.
(1) $$\varliminf_{n\to+\infty}x_n\le\varliminf_{\lambda\uparrow1}(1-\lambda)\sum_{n=1}^\infty\lambda^nx_n\le\varlimsup_{\lambda\uparrow1}(1-\lambda)\sum_{n=1}^\infty\lambda^nx_n\le\varlimsup_{n\to+\infty}x_n\le\sup_{n\ge1}x_n.$$
(2) If there exists some positive constant $C$ such that $$x_n\le Cx_{n+m}\text{ for all }n,m\ge1,$$ then $$\sup_{n\ge1}x_n\le C\varliminf_{n\to+\infty}x_n.$$
The proof is elementary using ${\varepsilon}$-$N$ argument.
Take ${\left\{\beta_n\right\}}\subseteq(\alpha,\beta^*)$ with $\beta_n\uparrow\beta^*$. By Proposition \[prop\_gamma\], there exist some subsequence still denoted by ${\left\{\beta_n\right\}}$ and some closed form $({\mathcal{E}},{\mathcal{F}})$ on $L^2(K;\nu)$ in the wide sense such that $(\beta^*-\beta_n){\mathfrak{E}}_{\beta_n}$ is $\Gamma$-convergent to ${\mathcal{E}}$. Without lose of generality, we may assume that $$0<\beta^*-\beta_n<\frac{1}{n+1}\text{ for all }n\ge1.$$
We have the characterization of $({\mathcal{E}},{\mathcal{F}})$ on $L^2(K;\nu)$ as follows.
\[thm\_gamma\] $$\begin{aligned}
&{\mathcal{E}}(u,u)\asymp\sup_{n\ge1}3^{(\beta^*-\alpha)n}\sum_{w\in W_n}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(u(p)-u(q))^2,\\
&{\mathcal{F}}={\left\{u\in C(K):\sup_{n\ge1}3^{(\beta^*-\alpha)n}\sum_{w\in W_n}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(u(p)-u(q))^2<+\infty\right\}}.
\end{aligned}$$ Moreover, $({\mathcal{E}},{\mathcal{F}})$ is a regular closed form on $L^2(K;\nu)$.
Recall that $\rho=3^{\beta^*-\alpha}$, then $$\begin{aligned}
E_{\beta}(u,u)&=\sum_{n=1}^\infty3^{(\beta-\alpha)n}\sum_{w\in W_n}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(u(p)-u(q))^2=\sum_{n=1}^\infty3^{(\beta-\beta^*)n}a_n(u),\\
{\mathfrak{E}}_\beta(u,u)&=\sum_{n=1}^\infty3^{(\beta-\alpha)n}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
w^{(1)}\sim_nw^{(2)}
\end{subarray}
$
}}}
\left(P_nu(w^{(1)})-P_nu(w^{(2)})\right)^2=\sum_{n=1}^\infty3^{(\beta-\beta^*)n}b_n(u).
\end{aligned}$$
We use weak monotonicity results Theorem \[thm\_monotone1\], Theorem \[thm\_monotone2\] and elementary result Proposition \[prop\_ele\].
For all $u\in L^2(K;\nu)$, there exists ${\left\{u_n\right\}}\subseteq L^2(K;\nu)$ converging strongly to $u$ in $L^2(K;\nu)$ such that $$\begin{aligned}
&{\mathcal{E}}(u,u)\ge\varlimsup_{n\to+\infty}(\beta^*-\beta_n){\mathfrak{E}}_{\beta_n}(u_n,u_n)=\varlimsup_{n\to+\infty}(\beta^*-\beta_n)\sum_{k=1}^\infty3^{(\beta_n-\beta^*)k}b_k(u_n)\\
&\ge\varlimsup_{n\to+\infty}(\beta^*-\beta_n)\sum_{k=n+1}^\infty3^{(\beta_n-\beta^*)k}b_k(u_n)\ge C\varlimsup_{n\to+\infty}(\beta^*-\beta_n)\sum_{k=n+1}^\infty3^{(\beta_n-\beta^*)k}b_n(u_n)\\
&=C\varlimsup_{n\to+\infty}\left\{b_n(u_n)\left[(\beta^*-\beta_n)\frac{3^{(\beta_n-\beta^*)(n+1)}}{1-3^{\beta_n-\beta^*}}\right]\right\}.
\end{aligned}$$ Since $0<\beta^*-\beta_n<1/(n+1)$, we have $3^{(\beta_n-\beta^*)(n+1)}>1/3$. Since $$\lim_{n\to+\infty}\frac{\beta^*-\beta_n}{1-3^{\beta_n-\beta^*}}=\frac{1}{\log3},$$ there exists some positive constant $C$ such that $$(\beta^*-\beta_n)\frac{3^{(\beta_n-\beta^*)(n+1)}}{1-3^{\beta_n-\beta^*}}\ge C\text{ for all }n\ge1.$$ Hence $${\mathcal{E}}(u,u)\ge C\varlimsup_{n\to+\infty}b_n(u_n).$$ Since $u_n\to u$ in $L^2(K;\nu)$, for all $k\ge1$, we have $$b_k(u)=\lim_{n\to+\infty}b_k(u_n)=\lim_{k\le n\to+\infty}b_k(u_n)\le C\varliminf_{n\to+\infty}b_n(u_n).$$ For all $m\ge1$, we have $$\begin{aligned}
(\beta^*-\beta_m)\sum_{k=1}^\infty3^{(\beta_m-\beta^*)k}b_k(u)&\le C(\beta^*-\beta_m)\sum_{k=1}^\infty3^{(\beta_m-\beta^*)k}\varliminf_{n\to+\infty}b_n(u_n)\\
&=C(\beta^*-\beta_m)\frac{3^{\beta_m-\beta^*}}{1-3^{\beta_m-\beta^*}}\varliminf_{n\to+\infty}b_n(u_n).
\end{aligned}$$ Hence ${\mathcal{E}}(u,u)<+\infty$ implies ${\mathfrak{E}}_{\beta_m}(u,u)<+\infty$, by Lemma \[lem\_holder\], we have ${\mathcal{F}}\subseteq C(K)$. Hence $$\varliminf_{m\to+\infty}(\beta^*-\beta_m)\sum_{k=1}^\infty3^{(\beta_m-\beta^*)k}b_k(u)\le C\varliminf_{n\to+\infty}b_n(u_n).$$ Hence for all $u\in{\mathcal{F}}\subseteq C(K)$, we have $$\begin{aligned}
{\mathcal{E}}(u,u)&\ge C\varlimsup_{n\to+\infty}b_n(u_n)\ge C\varliminf_{n\to+\infty}b_n(u_n)\ge C\varliminf_{m\to+\infty}(\beta^*-\beta_m)\sum_{k=1}^\infty3^{(\beta_m-\beta^*)k}b_k(u)\\
&\ge C\varliminf_{m\to+\infty}(\beta^*-\beta_m)\sum_{k=1}^\infty3^{(\beta_m-\beta^*)k}a_k(u)\ge C\sup_{n\ge1}a_n(u).
\end{aligned}$$
On the other hand, for all $u\in{\mathcal{F}}\subseteq C(K)$, we have $$\begin{aligned}
&{\mathcal{E}}(u,u)\le\varliminf_{n\to+\infty}(\beta^*-\beta_n){\mathfrak{E}}_{\beta_n}(u,u)\\
&\le C\varliminf_{n\to+\infty}(\beta^*-\beta_n)E_{\beta_n}(u,u)=C\varliminf_{n\to+\infty}(\beta^*-\beta_n)\sum_{k=1}^\infty3^{(\beta_n-\beta^*)k}a_k(u)\\
&=C\varliminf_{n\to+\infty}\frac{\beta^*-\beta_n}{1-3^{\beta_n-\beta^*}}(1-3^{\beta_n-\beta^*})\sum_{k=1}^\infty3^{(\beta_n-\beta^*)k}a_k(u)\le C\sup_{n\ge1}a_n(u).
\end{aligned}$$ Therefore, for all $u\in{\mathcal{F}}\subseteq C(K)$, we have $${\mathcal{E}}(u,u)\asymp\sup_{n\ge1}a_n(u)=\sup_{n\ge1}3^{(\beta^*-\alpha)n}\sum_{w\in W_n}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(u(p)-u(q))^2,$$ and $${\mathcal{F}}={\left\{u\in C(K):\sup_{n\ge1}3^{(\beta^*-\alpha)n}\sum_{w\in W_n}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(u(p)-u(q))^2<+\infty\right\}}.$$
It is obvious that the function $u\in C(K)$ in Proposition \[prop\_u\] is in ${\mathcal{F}}$. Similar to the proof of Theorem \[thm\_walk\], we have ${\mathcal{F}}$ is uniformly dense in $C(K)$. Hence $({\mathcal{E}},{\mathcal{F}})$ is a regular closed form on $L^2(K;\nu)$.
Now we prove Theorem \[thm\_BM\] as follows.
For all $n\ge1,u\in l(V_{n+1})$, we have $$\begin{aligned}
&\rho\sum_{i=0}^7a_n(u\circ f_i)=\rho\sum_{i=0}^7\rho^n\sum_{w\in W_n}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(u\circ f_i(p)-u\circ f_i(q))^2\\
&=\rho^{n+1}\sum_{w\in W_{n+1}}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-(n+1)}
\end{subarray}
$
}}}
(u(p)-u(q))^2=a_{n+1}(u).
\end{aligned}$$ Hence for all $n,m\ge1,u\in l(V_{n+m})$, we have $$\rho^m\sum_{w\in W_m}a_n(u\circ f_w)=a_{n+m}(u).$$ For all $u\in{\mathcal{F}},n\ge1,w\in W_n$, we have $$\sup_{k\ge1}a_k(u\circ f_w)\le\sup_{k\ge1}\sum_{w\in W_n}a_k(u\circ f_w)=\rho^{-n}\sup_{k\ge1}a_{n+k}(u)\le\rho^{-n}\sup_{k\ge1}a_{k}(u)<+\infty,$$ hence $u\circ f_w\in{\mathcal{F}}$.
Let $${\overline{{\mathcal{E}}}}^{(n)}(u,u)=\rho^n\sum_{w\in W_n}{\mathcal{E}}(u\circ f_w,u\circ f_w),u\in{\mathcal{F}},n\ge1.$$ Then $$\begin{aligned}
{\overline{{\mathcal{E}}}}^{(n)}(u,u)&\ge C\rho^n\sum_{w\in W_n}\varlimsup_{k\to+\infty}a_k(u\circ f_w)\ge C\rho^n\varlimsup_{k\to+\infty}\sum_{w\in W_n}a_k(u\circ f_w)\\
&=C\varlimsup_{k\to+\infty}a_{n+k}(u)\ge C\sup_{k\ge1}a_k(u).
\end{aligned}$$ Similarly $$\begin{aligned}
{\overline{{\mathcal{E}}}}^{(n)}(u,u)&\le C\rho^n\sum_{w\in W_n}\varliminf_{k\to+\infty}a_k(u\circ f_w)\le C\rho^n\varliminf_{k\to+\infty}\sum_{w\in W_n}a_k(u\circ f_w)\\
&=C\varliminf_{k\to+\infty}a_{n+k}(u)\le C\sup_{k\ge1}a_k(u).
\end{aligned}$$ Hence $${\overline{{\mathcal{E}}}}^{(n)}(u,u)\asymp\sup_{k\ge1}a_k(u)\text{ for all }u\in{\mathcal{F}},n\ge1.$$ Moreover, for all $u\in{\mathcal{F}}$, $n\ge1$, we have $$\begin{aligned}
&{\overline{{\mathcal{E}}}}^{(n+1)}(u,u)=\rho^{n+1}\sum_{w\in W_{n+1}}{\mathcal{E}}(u\circ f_w,u\circ f_w)=\rho^{n+1}\sum_{i=0}^7\sum_{w\in W_n}{\mathcal{E}}(u\circ f_i\circ f_w,u\circ f_i\circ f_w)\\
&=\rho\sum_{i=0}^7\left(\rho^n\sum_{w\in W_n}{\mathcal{E}}((u\circ f_i)\circ f_w,(u\circ f_i)\circ f_w)\right)=\rho\sum_{i=0}^7{\overline{{\mathcal{E}}}}^{(n)}(u\circ f_i,u\circ f_i).
\end{aligned}$$ Let $$\tilde{{\mathcal{E}}}^{(n)}(u,u)=\frac{1}{n}\sum_{l=1}^n{\overline{{\mathcal{E}}}}^{(l)}(u,u),u\in{\mathcal{F}},n\ge1.$$ It is obvious that $$\tilde{{\mathcal{E}}}^{(n)}(u,u)\asymp\sup_{k\ge1}a_k(u)\text{ for all }u\in{\mathcal{F}},n\ge1.$$ Since $({\mathcal{E}},{\mathcal{F}})$ is a regular closed form on $L^2(K;\nu)$, by [@CF12 Definition 1.3.8, Remark 1.3.9, Definition 1.3.10, Remark 1.3.11], we have $({\mathcal{F}},{\mathcal{E}}_1)$ is a separable Hilbert space. Let ${\left\{u_i\right\}}_{i\ge1}$ be a dense subset of $({\mathcal{F}},{\mathcal{E}}_1)$. For all $i\ge1$, ${\left\{\tilde{{\mathcal{E}}}^{(n)}(u_i,u_i)\right\}}_{n\ge1}$ is a bounded sequence. By diagonal argument, there exists a subsequence ${\left\{n_k\right\}}_{k\ge1}$ such that ${\left\{\tilde{{\mathcal{E}}}^{(n_k)}(u_i,u_i)\right\}}_{k\ge1}$ converges for all $i\ge1$. Since $$\tilde{{\mathcal{E}}}^{(n)}(u,u)\asymp\sup_{k\ge1}a_k(u)\asymp{\mathcal{E}}(u,u)\text{ for all }u\in{\mathcal{F}},n\ge1,$$ we have ${\left\{\tilde{{\mathcal{E}}}^{(n_k)}(u,u)\right\}}_{k\ge1}$ converges for all $u\in{\mathcal{F}}$. Let $${\mathcal{E}}_{{{\mathrm{loc}}}}(u,u)=\lim_{k\to+\infty}\tilde{{\mathcal{E}}}^{(n_k)}(u,u)\text{ for all }u\in{\mathcal{F}}_{{{\mathrm{loc}}}}:={\mathcal{F}}.$$ Then $${\mathcal{E}}_{{{\mathrm{loc}}}}(u,u)\asymp\sup_{k\ge1}a_k(u)\asymp{\mathcal{E}}(u,u)\text{ for all }u\in{\mathcal{F}}_{{\mathrm{loc}}}={\mathcal{F}}.$$ Hence $({\mathcal{E}}_{{\mathrm{loc}}},{\mathcal{F}}_{{\mathrm{loc}}})$ is a regular closed form on $L^2(K;\nu)$. It is obvious that $1\in{\mathcal{F}}_{{\mathrm{loc}}}$ and ${\mathcal{E}}_{{\mathrm{loc}}}(1,1)=0$, by [@FOT11 Lemma 1.6.5, Theorem 1.6.3], we have $({\mathcal{E}}_{{\mathrm{loc}}},{\mathcal{F}}_{{\mathrm{loc}}})$ on $L^2(K;\nu)$ is conservative.
For all $u\in{\mathcal{F}}_{{\mathrm{loc}}}={\mathcal{F}}$, we have $u\circ f_i\in{\mathcal{F}}={\mathcal{F}}_{{\mathrm{loc}}}$ for all $i=0,\ldots,7$ and $$\begin{aligned}
&\rho\sum_{i=0}^7{\mathcal{E}}_{{\mathrm{loc}}}(u\circ f_i,u\circ f_i)=\rho\sum_{i=0}^7\lim_{k\to+\infty}\tilde{{\mathcal{E}}}^{(n_k)}(u\circ f_i,u\circ f_i)\\
&=\rho\sum_{i=0}^7\lim_{k\to+\infty}\frac{1}{n_k}\sum_{l=1}^{n_k}{\overline{{\mathcal{E}}}}^{(l)}(u\circ f_i,u\circ f_i)=\lim_{k\to+\infty}\frac{1}{n_k}\sum_{l=1}^{n_k}\left[\rho\sum_{i=0}^7{\overline{{\mathcal{E}}}}^{(l)}(u\circ f_i,u\circ f_i)\right]\\
&=\lim_{k\to+\infty}\frac{1}{n_k}\sum_{l=1}^{n_k}{\overline{{\mathcal{E}}}}^{(l+1)}(u,u)=\lim_{k\to+\infty}\frac{1}{n_k}\sum_{l=2}^{n_k+1}{\overline{{\mathcal{E}}}}^{(l)}(u,u)\\
&=\lim_{k\to+\infty}\left[\frac{1}{n_k}\sum_{l=1}^{n_k}{\overline{{\mathcal{E}}}}^{(l)}(u,u)+\frac{1}{n_k}{\overline{{\mathcal{E}}}}^{(n_k+1)}(u,u)-\frac{1}{n_k}{\overline{{\mathcal{E}}}}^{(1)}(u,u)\right]\\
&=\lim_{k\to+\infty}\tilde{{\mathcal{E}}}^{(n_k)}(u,u)={\mathcal{E}}_{{\mathrm{loc}}}(u,u).
\end{aligned}$$ Hence $({\mathcal{E}}_{{\mathrm{loc}}},{\mathcal{F}}_{{\mathrm{loc}}})$ on $L^2(K;\nu)$ is self-similar.
For all $u,v\in{\mathcal{F}}_{{\mathrm{loc}}}$ satisfying $\mathrm{supp}(u),\mathrm{supp}(v)$ are compact and $v$ is constant in an open neighborhood $U$ of $\mathrm{supp}(u)$, we have $K\backslash U$ is compact and $\mathrm{supp}(u)\cap(K\backslash U)=\emptyset$, hence $\delta=\mathrm{dist}(\mathrm{supp}(u),K\backslash U)>0$. Taking sufficiently large $n\ge1$ such that $3^{1-n}<\delta$, by self-similarity, we have $${\mathcal{E}}_{{\mathrm{loc}}}(u,v)=\rho^n\sum_{w\in W_n}{\mathcal{E}}_{{\mathrm{loc}}}(u\circ f_w,v\circ f_w).$$ For all $w\in W_n$, we have $u\circ f_w=0$ or $v\circ f_w$ is constant, hence ${\mathcal{E}}_{{\mathrm{loc}}}(u\circ f_w,v\circ f_w)=0$, hence ${\mathcal{E}}_{{\mathrm{loc}}}(u,v)=0$, that is, $({\mathcal{E}}_{{\mathrm{loc}}},{\mathcal{F}}_{{\mathrm{loc}}})$ on $L^2(K;\nu)$ is strongly local.
For all $u\in{\mathcal{F}}_{{\mathrm{loc}}}$, it is obvious that $u^+,u^-,1-u,{\overline{u}}=(0\vee u)\wedge1\in{\mathcal{F}}_{{\mathrm{loc}}}$ and $${\mathcal{E}}_{{\mathrm{loc}}}(u,u)={\mathcal{E}}_{{\mathrm{loc}}}(1-u,1-u).$$ Since $u^+u^-=0$ and $({\mathcal{E}}_{{\mathrm{loc}}},{\mathcal{F}}_{{\mathrm{loc}}})$ on $L^2(K;\nu)$ is strongly local, we have ${\mathcal{E}}_{{\mathrm{loc}}}(u^+,u^-)=0$. Hence $$\begin{aligned}
{\mathcal{E}}_{{\mathrm{loc}}}(u,u)&={\mathcal{E}}_{{\mathrm{loc}}}(u^+-u^-,u^+-u^-)={\mathcal{E}}_{{\mathrm{loc}}}(u^+,u^+)+{\mathcal{E}}_{{\mathrm{loc}}}(u^-,u^-)-2{\mathcal{E}}_{{\mathrm{loc}}}(u^+,u^-)\\
&={\mathcal{E}}_{{\mathrm{loc}}}(u^+,u^+)+{\mathcal{E}}_{{\mathrm{loc}}}(u^-,u^-)\ge{\mathcal{E}}_{{\mathrm{loc}}}(u^+,u^+)={\mathcal{E}}_{{\mathrm{loc}}}(1-u^+,1-u^+)\\
&\ge{\mathcal{E}}_{{\mathrm{loc}}}((1-u^+)^+,(1-u^+)^+)={\mathcal{E}}_{{\mathrm{loc}}}(1-(1-u^+)^+,1-(1-u^+)^+)={\mathcal{E}}_{{\mathrm{loc}}}({\overline{u}},{\overline{u}}),
\end{aligned}$$ that is, $({\mathcal{E}}_{{\mathrm{loc}}},{\mathcal{F}}_{{\mathrm{loc}}})$ on $L^2(K;\nu)$ is Markovian. Hence $({\mathcal{E}}_{{\mathrm{loc}}},{\mathcal{F}}_{{\mathrm{loc}}})$ is a self-similar strongly local regular Dirichlet form on $L^2(K;\nu)$.
The idea of the construction of ${\overline{{\mathcal{E}}}}^{(n)},\tilde{{\mathcal{E}}}^{(n)}$ is from [@KZ92 Section 6]. The proof of Markovain property is from the proof of [@BBKT10 Theorem 2.1].
Proof of Theorem \[thm\_Besov\] {#sec_Besov}
===============================
Theorem \[thm\_Besov\] is a special case of the following result.
\[prop\_equiv\_local\] For all $\beta\in(\alpha,+\infty),u\in C(K)$, we have $$\sup_{n\ge1}3^{(\beta-\alpha)n}\sum_{w\in W_n}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(u(p)-u(q))^2\asymp[u]_{B^{2,\infty}_{\alpha,\beta}(K)}.$$
Similar to non-local case, we need the following preparation.
([@GHL03 Theorem 4.11 ([3]{})])\[lem\_holder\_local\] Let $u\in L^2(K;\nu)$ and $$F(u):=\sup_{n\ge1}3^{(\alpha+\beta)n}\int_K\int_{B(x,3^{-n})}(u(x)-u(y))^2\nu({\mathrm{d}}y)\nu({\mathrm{d}}x),$$ then $$|u(x)-u(y)|^2\le cF(u)|x-y|^{\beta-\alpha}\text{ for }\nu\text{-almost every }x,y\in K,$$ where $c$ is some positive constant.
If $F(u)<+\infty$, then $u\in C^{\frac{\beta-\alpha}{2}}(K)$.
The proof is very similar to that of Lemma \[lem\_equiv\]. We only point out the differences. To show that LHS$\lesssim$RHS, by the proof of Theorem \[thm\_equiv1\], we still have Equation (\[eqn\_equiv1\_1\]) where $E(u)$ is replaced by $F(u)$. Then $$\begin{aligned}
&3^{(\beta-\alpha)n}\sum_{w\in W_n}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(u(p)-u(q))^2\\
&\le128\cdot2^{(\beta-\alpha)/2}cF(u)3^{\beta n-(\beta-\alpha)(n+kl)}+32\cdot3^{\alpha k}\sum_{i=0}^{l-1}2^i\cdot3^{-(\beta-\alpha)ki}E_{n+ki}(u).
\end{aligned}$$ Take $l=n$, then $$\begin{aligned}
&3^{(\beta-\alpha)n}\sum_{w\in W_n}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(u(p)-u(q))^2\\
&\le128\cdot2^{(\beta-\alpha)/2}cF(u)3^{[\beta-(\beta-\alpha)(k+1)]n}+32\cdot3^{\alpha k}\sum_{i=0}^{n-1}2^i\cdot3^{-(\beta-\alpha)ki}E_{n+ki}(u)\\
&\le128\cdot2^{(\beta-\alpha)/2}cF(u)3^{[\beta-(\beta-\alpha)(k+1)]n}+32\cdot3^{\alpha k}\sum_{i=0}^{\infty}3^{[1-(\beta-\alpha)k]i}\left(\sup_{n\ge1}E_{n}(u)\right).
\end{aligned}$$ Take $k\ge1$ sufficiently large such that $\beta-(\beta-\alpha)(k+1)<0$ and $1-(\beta-\alpha)k<0$, then $$\begin{aligned}
&\sup_{n\ge1}3^{(\beta-\alpha)n}\sum_{w\in W_n}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(u(p)-u(q))^2\\
&\lesssim\sup_{n\ge1}3^{(\alpha+\beta)n}\int_K\int_{B(x,3^{-n})}(u(x)-u(y))^2\nu({\mathrm{d}}y)\nu({\mathrm{d}}x).
\end{aligned}$$
To show that LHS$\gtrsim$RHS, by the proof of Theorem \[thm\_equiv2\], we still have Equation (\[eqn\_equiv2\_3\]). Then $$\begin{aligned}
&\sup_{n\ge2}3^{(\alpha+\beta)n}\int_K\int_{B(x,c3^{-n})}(u(x)-u(y))^2\nu({\mathrm{d}}y)\nu({\mathrm{d}}x)\\
&\lesssim\sup_{n\ge2}\sum_{k=n}^\infty4^{k-n}\cdot3^{\beta n-\alpha k}\sum_{w\in W_k}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-k}
\end{subarray}
$
}}}
(u(p)-u(q))^2\\
&+\sup_{n\ge2}3^{(\beta-\alpha)n}\sum_{w\in W_{n-1}}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-(n-1)}
\end{subarray}
$
}}}
(u(p)-u(q))^2\\
&\lesssim\sup_{n\ge2}\sum_{k=n}^\infty4^{k-n}\cdot3^{\beta(n-k)}\left(\sup_{k\ge1}3^{(\beta-\alpha)k}\sum_{w\in W_k}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-k}
\end{subarray}
$
}}}
(u(p)-u(q))^2\right)\\
&+\sup_{n\ge1}3^{(\beta-\alpha)n}\sum_{w\in W_n}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(u(p)-u(q))^2\\
&\lesssim\sup_{n\ge1}3^{(\beta-\alpha)n}\sum_{w\in W_n}
{\sum_{\mbox{\tiny
$
\begin{subarray}{c}
p,q\in V_w\\
|p-q|=2^{-1}\cdot3^{-n}
\end{subarray}
$
}}}
(u(p)-u(q))^2.
\end{aligned}$$
We have the following properties of Besov spaces for large exponents.
\[cor\_chara\] $B^{2,2}_{\alpha,\beta^*}(K)={\left\{\text{constant functions}\right\}}$, $B^{2,\infty}_{\alpha,\beta^*}(K)$ is uniformly dense in $C(K)$. $B^{2,2}_{\alpha,\beta}(K)=B^{2,\infty}_{\alpha,\beta}(K)={\left\{\text{constant functions}\right\}}$ for all $\beta\in(\beta^*,+\infty)$.
By Theorem \[thm\_BM\] and Theorem \[thm\_Besov\], we have $B^{2,\infty}_{\alpha,\beta^*}(K)$ is uniformly dense in $C(K)$. Assume that $u\in C(K)$ is non-constant, then there exists $N\ge1$ such that $a_N(u)>0$. By Theorem \[thm\_monotone1\], for all $\beta\in[\beta^*,+\infty)$, we have $$\sum_{n=1}^\infty3^{(\beta-\beta^*)n}a_n(u)\ge\sum_{n=N+1}^\infty3^{(\beta-\beta^*)n}a_n(u)\ge C\sum_{n=N+1}^\infty3^{(\beta-\beta^*)n}a_N(u)=+\infty,$$ for all $\beta\in(\beta^*,+\infty)$, we have $$\sup_{n\ge1}3^{(\beta-\beta^*)n}a_n(u)\ge\sup_{n\ge N+1}3^{(\beta-\beta^*)n}a_n(u)\ge C\sup_{n\ge N+1}3^{(\beta-\beta^*)n}a_N(u)=+\infty.$$
By Lemma \[lem\_equiv\] and Proposition \[prop\_equiv\_local\], we have $B^{2,2}_{\alpha,\beta}(K)={\left\{\text{constant functions}\right\}}$ for all $\beta\in[\beta^*,+\infty)$ and $B^{2,\infty}_{\alpha,\beta}(K)={\left\{\text{constant functions}\right\}}$ for all $\beta\in(\beta^*,+\infty)$.
Proof of Theorem \[thm\_hk\] {#sec_hk}
============================
We use effective resistance as follows.
Let $(M,d,\mu)$ be a metric measure space and $({\mathcal{E}},{\mathcal{F}})$ a regular Dirichlet form on $L^2(M;\mu)$. Assume that $A,B$ are two disjoint subsets of $M$. Define *effective resistance* as $$R(A,B)=\inf{\left\{{\mathcal{E}}(u,u):u|_A=0,u|_B=1,u\in{\mathcal{F}}\cap C_0(M)\right\}}^{-1}.$$ Denote $$R(x,B)=R({\left\{x\right\}},B),R(x,y)=R({\left\{x\right\}},{\left\{y\right\}}),x,y\in M.$$ It is obvious that if $A_1\subseteq A_2$, $B_1\subseteq B_2$, then $$R(A_1,B_1)\ge R(A_2,B_2).$$
First, we show that $$R(x,y)\asymp|x-y|^{\beta^*-\alpha}\text{ for all }x,y\in K.$$ By Lemma \[lem\_holder\_local\], we have $$(u(x)-u(y))^2\le c{\mathcal{E}}_{{\mathrm{loc}}}(u,u)|x-y|^{\beta^*-\alpha}\text{ for all }x,y\in K,u\in{\mathcal{F}}_{{\mathrm{loc}}},$$ hence $$R(x,y)\lesssim|x-y|^{\beta^*-\alpha}\text{ for all }x,y\in K.$$ On the other hand, we claim $$R(x,B(x,r)^c)\asymp r^{\beta^*-\alpha}\text{ for all }x\in K,r>0\text{ with }B(x,r)^c\ne\emptyset.$$ Indeed, fix $C>0$. If $u\in{\mathcal{F}}_{{\mathrm{loc}}}$ satisfies $u(x)=1$, $u|_{B(x,r)^c}=0$, then $\tilde{u}:y\mapsto u(x+C(y-x))$ satisfies $\tilde{u}\in{\mathcal{F}}_{{\mathrm{loc}}}$, $\tilde{u}(x)=1$, $\tilde{u}|_{B(x,Cr)^c}=0$. By Theorem \[thm\_BM\], it is obvious that $${\mathcal{E}}_{{\mathrm{loc}}}(\tilde{u},\tilde{u})\asymp C^{-(\beta^*-\alpha)}{\mathcal{E}}_{{\mathrm{loc}}}(u,u),$$ hence $$R(x,B(x,Cr)^c)\asymp C^{\beta^*-\alpha}R(x,B(x,r)^c).$$ Hence $$R(x,B(x,r)^c)\asymp r^{\beta^*-\alpha}.$$ For all $x,y\in K$, we have $$R(x,y)\ge R(x,B(x,|x-y|)^c)\asymp|x-y|^{\beta^*-\alpha}.$$
Then, we follow a standard analytic approach as follows. First, we obtain Green function estimates as in [@GHL14 Proposition 6.11]. Then, we obtain heat kernel estimates as in [@GH14a Theorem 3.14]. Note that we are dealing with compact set, the final estimates only hold for some finite time $t\in(0,1)$.
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[^1]: *Date*:
[^2]: *MSC2010*: 28A80
[^3]: *Keywords*: Sierpiński carpet, non-local quadratic form, walk dimension, $\Gamma$-convergence, Brownian motion, effective resistance, heat kernel
[^4]: The authors were supported by SFB701 and SFB1283 of the German Research Council (DFG). The second author is very grateful to Dr. Qingsong Gu for very helpful discussions. Part of the work was carried out while the second author was visiting the Chinese University of Hong Kong, he is very grateful to Prof. Ka-Sing Lau for the arrangement of the visit.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
A new measure of the crystal-field strength, complementary to the conventional one, is defined. It is based on the rotational invariants $\left|B_{k0}\right|_{\rm av}$ or $\left|\sum_{k}B_{k0}\right|_{\rm av}$, $k=2,4,6$, of the crystal-(ligand)-field (CF) Hamiltonian ${\cal H}_{\rm CF}$ parametrizations, i.e. on the axial CF parameters modules averaged over all reference frame orientations. They turn out to be equal to $\left|{\cal
H}_{\rm CF}^{(k)}\right|_{\rm av}$ and $\left|{\cal H}_{\rm CF}\right|_{\rm av}$, respectively. While the traditional measure is established on the parametrization modules or on the second moment of the CF energy levels, the introduced scale employs rather the first moment of the energy modules and has better resolving power. The new scale is able to differentiate the strength of various iso-modular parametrizations according to the classes of rotationally equivalent parametrizations. Using both the compatible CF strength measures one may draw more accurate conclusions about the Stark levels arrays and particularly their total splitting magnitudes.
author:
- '**J. Mulak$^{1}$ and M. Mulak$^{2}$**'
date: |
[*$^{1}$ W Trzebiatowski Institute of Low Temperature and Structure Research,\
Polish Academy of Sciences, 50–950, PO Box 1410, Wroclaw, Poland\
$^{2}$ Institute of Physics, Wroclaw University of Technology,\
Wyb. Wyspianskiego 27, 50–370 Wroclaw, Poland*]{}
title: '**On a complementary scale of crystal-field strength**'
---
[*PACS*]{}: 71.70.Ch\
[*Key words*]{}: crystal-field strength, crystal-field splitting\
1. Introduction {#introduction .unnumbered}
===============
Solid state experimentalists, especially spectroscopists, still need a reliable scale quantitatively characterizing the effect of crystal-field interaction, i.e. defining the so-called crystal-field strength. Such a parameter could directly verify and compare various parametrizations of the crystal-field Hamiltonian ${\cal
H}_{\rm CF}$, which may come from different fittings experimental data when the orientations of reference frames associated with these parametrizations are unknown in the majority of cases.
Although such a conventional scale for measuring the strength of the crystal-field has been already introduced over twenty years ago \[1,2\], in some cases it seems to be insufficiently precise. It employs the basic rotational invariants of the ${\cal H}_{\rm CF}$, i.e. the modules of its $2^{k}$-pole components ${\cal H}_{\rm
CF}^{(k)}$, defined as $M_{k}=\left(\sum_{q}|B_{kq}|^{2}\right)^{1/2}$, as well as uses the global ${\cal
H}_{\rm CF}$ modulus $M=\left(\sum_{k}\sum_{q}|B_{kq}|^{2}\right)^{1/2}$. In the first case the partial crystal-field strength is defined as $S_{k}\!=\!\left(\frac{1}{2k+1}\right)^{1/2}M_{k}$, while in the second case the global crystal-field strength is given by $S=\left(\sum_{k}S_{k}^{2}\right)^{1/2}$. Throughout the paper the tensor (Wybourne) notation for the crystal-field Hamiltonian and the crystal-field parameters (CFPs), ${\cal H}_{\rm CF}=\sum_{k}\sum_{q}B_{kq}C_{q}^{(k)}$, is consistently used \[3\]. The summations over $k$ and $q$ indices run, in each individual case, over strictly specified values according to the kind of central ion and its point symmetry.
Both the parameters $S_{k}$ and $S$ themselves are not a direct measure of the real magnitude of the initial state splitting, since the crystal-field effect depends also on the properties of an object (a paramagnetic ion) upon which the ${\cal H}_{\rm CF}$ acts. Namely, the response of the system to the ${\cal H}_{\rm CF}$ perturbation reflects the symmetry of the electron density distribution of the central ion open-shell. For instance, an $S$-type ion like Gd$^{3+}$ feels no crystal field (in the first order of perturbation) no matter how strong is the surrounding field.
The effect of splitting can be most simply expressed by the so-called second moments $\sigma_{k}^{2}$ or $\sigma^{2}$ of the CF sublevels within the initial state upon switching on the ${\cal H}^{(k)}_{\rm CF}$ (or ${\cal H}_{\rm CF}$) perturbation. In fact, the second moment is easily represented by the scalar crystal-field strength parameters, either $S_{k}$ or $S$ (section 2). However, although the effective ${\cal H}^{(k)}_{\rm
CF}$ multipoles (for $k=2,4,6$) contribute to the energy of individual Stark levels independently (as an algebraic sum), the simple linear relations between $\sigma_{k}^{2}$ (or $\sigma^{2}$), and $S_{k}^{2}$ (or $S^{2}$) are always fulfilled. As it is proved these relations strongly confine both the maximal $(\Delta {\cal
E}_{\rm max})$ and minimal $(\Delta {\cal E}_{\rm min})$ nominally allowed splittings of the initial state (section 3). Moreover the actual crystal field splittings $\Delta E$ can be additionally restricted (section 5). Naturally, all the iso-modular ${\cal H}_{\rm CF}$ parametrizations correspond to the same crystal-field strengths $S_{k}$ and $S$. However, apart from the modules $M_{k}$ and $M$, there exist also other rotational invariants of the ${\cal H}^{(k)}_{\rm CF}$ or ${\cal H}_{\rm CF}$ which distinguish the whole classes of the rotationally equivalent ${\cal H}_{\rm CF}$ parameterizations, in other words the parameterizations referring to the same real crystal-field potential, but expressed in variously oriented reference frame. Interestingly, the new invariants turn out to be the average values of the axial parameter modulus $|B_{k0}|_{\rm av}$, $k=2,4,6$, in the case of ${\cal H}^{(k)}_{\rm CF}$, or $|\sum_{k}B_{k0}|_{\rm av}$ for the global ${\cal H}_{\rm CF}$ obtained after the averaging over all orientations of the reference frame, i.e. over the solid angle $4\pi$. As it is shown in the paper the average value of the axial parameter modulus or the average of the modulus of their sum are just equal to $|{\cal H}^{(k)}_{\rm CF}|_{\rm av}$ and $|{\cal H}_{\rm CF}|_{\rm av}$, respectively (section 4).
The new scale of the crystal-field strength based on the above invariants is in principle consistent with the conventional one but it reveals more resolving power. Applying the new measure to the iso-modular parametrizations may lead to different strength parameters what is exemplified below for several cases (section 5). The introduced more subtle strength gradation established rather on the first moment of the sublevel energy modules gives, comparing to the second moment, additional information about the Stark levels array for various iso-modular ${\cal H}_{\rm CF}$s, including the magnitude of the total splitting gap of the states. In this paper we confine ourselves to the pure model states of the zero-order approximation with a well defined angular momentum quantum number and the corresponding degeneration. These could be for instance Russell-Saunders coupled states $|\alpha L S J \rangle$ coming from the $^{2S+1}$L terms, where $\alpha$ stands for the remaining quantum numbers needed for their complete determination. Such states have a well defined quantum number $J$ and the degeneration $2J+1$. The derivation of the analogical expressions including $J$-mixing effects \[4\] or a transformation to other functional bases of the zero-order approximation can be accomplished by using standard angular momentum re-coupling techniques \[4-8\]. In section 5 we analyse by way of example the crystal-field splitting of $p^{1}$, $d^{1}$ and $f^{1}$ one-electron configurations and a typical complex state $^{3}H_{4}$ for various iso-modular ${\cal H}_{\rm CF}^{(k)}$, $k=2,4,6$. In the first three cases we avoid complex states re-coupling procedure which is a side issue to the problem under consideration. Since we study the differentiation of the effects due to various iso-modular Hamiltonians ${\cal H}_{\rm CF}^{(k)}$, all CFPs values along with the Stark levels energies are given in $M_{k}$ units.
2. Conventional definition of the crystal-field strength parameter {#conventional-definition-of-the-crystal-field-strength-parameter .unnumbered}
------------------------------------------------------------------
The comparison and scaling of the crystal-field impact can be based upon the two types of scalar quantities, $M_{k}$ and/or $M$, since both of them are rotationally invariant. A scalar crystal-field strength parameter of this kind was given firstly by Auzel and Malta \[1,2\] as (in original notation): $$\begin{aligned}
N_{v} &=& \left[\sum\limits_{k,q}|B_{q}^{k}|^2\left(\frac{2\pi}{2k+1}\right)\right]^{1/2},\end{aligned}$$ which is nothing more but $M$ in the space spanned by spherical harmonics $Y_{q}^{k}$. In other words, $N_{v}$ is a norm representing a distance in the space. Currently there are two definitions widely used in the literature \[9-12\]: $$\begin{aligned}
S_{k} &=& \left(\frac{1}{2k+1}\sum\limits_{q}|B_{kq}|^2\right)^{1/2}=
\left\{\frac{1}{2k+1}\left[B_{k0}^{2}+2\sum\limits_{q>0}({\rm Re}B_{kq})^2+({\rm
Im}B_{kq})^2\right]\right\}^{1/2},\end{aligned}$$ for $k=2,4$ and $6$ in the case of $2^{k}$-pole ${\cal H}_{\rm CF}$ component and $$\begin{aligned}
S &=& \left(S_{2}^{2}+S_{4}^{2}+S_{6}^{2}\right)^{1/2} \qquad {\rm or} \qquad
S=\left[\frac{1}{3}\left(S_{2}^{2}+S_{4}^{2}+S_{6}^{2}\right)\right]^{1/2},\end{aligned}$$ for the global ${\cal H}_{\rm CF}=\sum_{k}{\cal H}_{\rm CF}^{(k)}$ \[4\]. A word of caution seems to be worthy at this point. Namely, the values of $S_{k}$ or $S$ can differ according to the type of the ${\cal H}_{\rm CF}$ parametrization (operators) applied. They can be compared with each other only after proper recalculation. Since both these quantities are independent of the assumed axis system they allow to check whether the original CFP data sets and the transformed ones are compatible. The strengths $S_{k}$ or $S$ enable also a broad comparison of CFP data sets when the axis systems have not been explicitly defined, and undoubtedly they play a central role in the CF theory. What is also important and useful they are linked to the second moment of the Stark levels within a particular initial state $|\alpha S L J\rangle$ \[4,13\].
The second moment of the sublevels $|n\rangle$ within $|\alpha S L J\rangle$ state upon introduction of a ${\cal
H}_{\rm CF}$ perturbation is defined by $$\begin{aligned}
\sigma^{2}(|\alpha S L J\rangle) &=& \frac{1}{2J+1}\sum\limits_{n}\left[E_{n}-\bar{E}(|\alpha S L J\rangle)\right]^{2},\end{aligned}$$ where the center of gravity of the Stark levels belonging to the state $|\alpha S L J\rangle$ is given by $$\begin{aligned}
\bar{E}(|\alpha S L J\rangle)&=& \frac{1}{2J+1}\sum\limits_{n}E_{n},\end{aligned}$$ and $E_{n}$ is the $|n\rangle$ sublevel energy. Since ${\cal H}_{\rm CF}$ is diagonal in the $|n\rangle$ basis and the second order effect of ${\cal H}_{\rm CF}$ interaction is neglected \[4\] $$\begin{aligned}
\sigma^{2}(|\alpha S L J\rangle) &=& \frac{1}{2J+1}Tr\left\{{\cal H}_{\rm CF}^{2}\right\}.\end{aligned}$$ Hence $$\begin{aligned}
\sigma^{2}(|\alpha S L J\rangle) &=& \frac{1}{2J+1}\sum\limits_{k}S_{k}^{2}
\left(\langle\alpha SLJ||C^{(k)}||\alpha SLJ\rangle\right)^{2},\end{aligned}$$ what implies from the orthogonality of 3-$j$ symbols \[2,5,13\]. The symbols $\langle\alpha SLJ||C^{(k)}||\alpha
SLJ\rangle$ are the double-bar or reduced matrix elements of the spherical tensor operators. According to Wigner-Eckart theorem \[14\] they are independent of the reference frame orientation. Their origin and physical meaning stem from the following relationships \[5-8,15\]: $$\begin{aligned}
\langle\alpha SLJM_{J}|C_{q}^{(k)}|\alpha SL^{\prime}J^{\prime}M_{J}^{\prime}\rangle &=&
(-1)^{J-M_{J}}\left(\begin{array}{ccc}
J & k & J^{\prime} \\
-M_{J} & q & M_{J}^{\prime} \\
\end{array}\right)
\langle\alpha SLJM_{J}||C^{(k)}||\alpha SL^{\prime}J^{\prime}\rangle,\end{aligned}$$ where the reduced matrix element follows the 3-$j$ factor. Further use of tensor formalism yields $$\begin{aligned}
\langle\alpha SLJ||C^{(k)}||\alpha SL^{\prime}J^{\prime}\rangle &=&
(-1)^{S+L^{\prime}+J+k}
\left[(2J+1)(2J^{\prime}+1)\right]^{1/2}\left\{\begin{array}{ccc}
J & J^{\prime} & k \\
L^{\prime} & L & S \\
\end{array}\right\}
\langle\alpha SL||C^{(k)}||\alpha SL^{\prime}\rangle,\end{aligned}$$ where the double-reduced matrix element follows now the 6-$j$ symbol. We can also pass to the matrix elements of the unit operators $U^{(k)}$ \[5-8,15\], i.e. normalized equivalents of $C^{(k)}$, since $$\begin{aligned}
\langle\alpha SL||C^{(k)}||\alpha SL^{\prime}\rangle &=& \langle\alpha SL||U^{(k)}||\alpha SL^{\prime}\rangle
\langle\l||C^{(k)}||l\rangle,\end{aligned}$$ and $l$ is the angular momentum quantum number of the open-shell electrons. The reduced matrix elements of the $U^{(k)}$ operators have been compiled by Nielson and Koster \[16\], whereas the 3-$j$ and 6-$j$ symbols can be found in the tables by Rotenberg et al \[7\].
The simple relation between the $\sigma_{k}^{2}$ and $S_{k}^{2}$ (Eq.4) can be also proved employing Vieta’s formulas for roots of the ${\cal H}_{\rm CF}^{(k)}$ matrix characteristic polynomial $$\begin{aligned}
E^{n}+a_{1}E^{n-1}+a_{2}E^{n-2}+\ldots a_{n}&=& 0,\end{aligned}$$ which is here of order of $n=2J+1$. All its coefficients and roots must be real what follows obviously from the ${\cal H}_{\rm CF}$ hermiticity. Interestingly, some characteristics of the sublevels spectrum may be described in terms of the elementary algebra. Firstly, as the energy center of gravity of the initial state is conserved, i.e. $\left(\sum_{i=1}^{n}E_{i}=0\right)^2$, the $a_{1}$ coefficient standing at $E^{n-1}$ must vanish. Next, since $0=\left(\sum_{i}^{n}E_{i}=0\right)^2=\sum_{i=1}^{n}E_{i}^{2}+ 2\sum_{i>j}E_{i}E_{j}$, the second moment, i.e. the sum of the root squares (divided by $2J+1$) is equal to $\frac{-2a_{2}}{2J+1}$. It can be also shown that $$\begin{aligned}
-2a_{2} &=& \frac{1}{2k+1}\;M_{k}^{2}\;\langle J||C^{(k)}||J\rangle^{2},\end{aligned}$$ where the simplified notation for the reduced matrix element representing only the last quantum number has been introduced. Hence, between $\sigma_{k}^{2}$ and $S_{k}^{2}$ a simple formula holds (Eq.4) $$\begin{aligned}
\sigma_{k}^{2} &=& \frac{1}{2J+1}\;S_{k}^{2}\;
\langle J||C^{(k)}||J\rangle^{2}.\end{aligned}$$ In other words, $\sigma_{k}$ is proportional to $S_{k}$. Finally, a free term of the characteristic polynomial is given as $a_{n}=(-1)^{n}E_{1}E_{2}\ldots E_{n}$, what may be helpful analyzing the solutions. For instance, if one root equals zero then a free term vanishes.
The problem becomes more complex for the global crystal-field strength $S$ (Eq.2), since then the components $S_{k}^{2}$ contribute to the sum with their weights $\langle J||C^{(k)}||J\rangle^{2}$ (Eq.4). This is why there is no straightforward proportionality between $\sigma^{2}$ and $S^{2}$ in this case. Nevertheless $\sigma$ is a positively defined quadratic form of $S_{k}$ and, in consequence, the inputs of particular ${\cal H}_{\rm
CF}$ $2^{k}$-poles into $\sigma^{2}$ cannot compensate themselves. The condition that $\sigma^{2}$ is constant for various iso-modular ${\cal H}_{\rm CF}$ does not exclude, however, a possible differentiation of the CF sublevels sequence and structure, as well as the initial state total splitting. In fact, $\sigma$ and $S$ could be correlated similarly as $\sigma_{k}$ and $S_{k}$ in the previous case, only if the elements $\langle
J||C^{(k)}||J\rangle$ were equal for all $k$. Nevertheless, the second moment of the Stark levels within a particular state $|\alpha S L J\rangle$ is simply given in terms of $S_{2}$, $S_{4}$ and $S_{6}$. Auzel and Malta \[2\] made an attempt to average the $\sigma^{2}$ quadratic form by bringing down its respective ellipsoid $\sum_{k}S_{k}^{2}\langle J||C^{(k)}||J\rangle^{2}$ in the $k$-space to a sphere of the same volume $\sum_{k}S_{k}^{2}\left[\Pi_{k}\langle J||C^{(k)}||J\rangle^{2}\right]^{1/3}$ and having a radius equal to the geometric mean of the three ellipsoid axes. In practice, unfortunately, this elegant approach does not always lead to acceptable results. In the literature the overall effect of the crystal-field interaction is often characterized by a quantitative comparison of the crystal-field strength \[17-20\]. Additionally, a systematic correlation between the free ion parameters and the CF strength is observed, namely increase of the crystal-field interaction results in the reduction of the free-ion parameters \[17\]. The CF strength increases in the RE series with decreasing ionic radius of the RE$^{3+}$ host cation \[19\]. The physical meaning of the CF strength scalar parameter is also supported by the fact that it rises with pressure applied to a sample \[21,22\]. The CF strength parameter has also been used to compare the root mean square error obtained for crystal fields of different strength. However, its use in such a case is restricted only to comparisons of the identical site symmetries \[10,23\]. Furthermore, within the approximation to the second order in the crystal-field, the shift in the center of gravity of a particular $^{2S+1}L_{J}$ state due to $J$-mixing effects is a simple linear function of the $S_{k}^{2}$ \[4,17\]. The concept of the $S_{k}^{2}$ or $S$ can be extended to define the quantities $C_{k}$ and $C_{G}$ \[13\] as normalized “scalar products” of any two compared parametrizations. These quantities represent the “angles” between the two considered parametrizations and are a convenient measure of the closeness, i.e. the correlation of any two CFPs sets.
3. The correspondence of the Stark levels second moment of $|J\rangle$ state to its nominally allowed splittings {#the-correspondence-of-the-stark-levels-second-moment-of-jrangle-state-to-its-nominally-allowed-splittings .unnumbered}
----------------------------------------------------------------------------------------------------------------
The second moment of CF levels, $\sigma^{2}$, essentially limits a formally allowed range of the initial state $|J\rangle$ total splittings $\Delta {\cal E}$ for different but iso-modular ${\cal H}_{\rm CF}$s. Such energy splitting confinement differs for non-Kramers and Kramers ions what is specified in details below.
Let us firstly study the case of any integer $J$, i.e. non-Kramers ions. Having to keep a constant $\sigma^{2}$ the minimal hypothetical splitting $\Delta {\cal E}_{\rm min}$ of the $(2J+1)$-fold degenerate state takes place when $J$ levels assume identical energy of $\frac{J+1}{2J+1}\Delta{\cal E}_{\rm min}$, and the remaining $J+1$ levels take the energy $\frac{-J}{2J+1}\Delta {\cal E}_{\rm min}$, or vice versa. Further this is referred as Type I splitting. Then $$\begin{aligned}
\sigma^{2} &=& \frac{J(J+1)(\Delta {\cal E}_{\rm min})^{2}}{(2J+1)^{2}}, \qquad \qquad
\Delta {\cal E}_{\rm min}=\sigma\frac{2J+1}{\sqrt{J(J+1)}}.\end{aligned}$$ In turn, the maximal hypothetical splitting $\Delta {\cal E}_{\rm max}$ occurs for one level of $\Delta {\cal
E}_{\rm max}/2$ energy, one of $-\Delta {\cal E}_{\rm max}/2$, and the rest $(2J-1)$ levels with zero energy. Further this is referred as Type II splitting. Then $$\begin{aligned}
\sigma^{2} &=& \frac{2\left(\Delta {\cal E}_{\rm max}/2\right)^2}{2J+1}, \qquad \qquad
\Delta {\cal E}_{\rm max}=\sigma \sqrt{2(2J+1)},\end{aligned}$$ and hence $$\begin{aligned}
\frac{\Delta {\cal E}_{\rm max}}{\Delta {\cal E}_{\rm min}} &=& \sqrt{\frac{2J(J+1)}{2J+1}}.\end{aligned}$$ Let us also consider, following Auzel and Malta \[2\], the case of the homogenous splitting $\Delta {\cal E}_{\rm
hom}$, when $$\begin{aligned}
\sigma^{2} &=& \frac{2(1+4+\ldots+J^{2})\left(\Delta {\cal E}_{\rm hom}/2J\right)^2}{2J+1} \qquad {\rm and} \qquad
\Delta {\cal E}_{\rm hom}=2\sigma \sqrt{\frac{3J}{J+1}}.\end{aligned}$$ Below it will be referred as Type III splitting. For example, if $J=4$ then $\Delta {\cal E}_{\rm
min}=\sigma\frac{9}{2\sqrt{5}}=2.01\sigma$, $\Delta {\cal E}_{\rm max}=\sigma3\sqrt{2}\sigma=4.24\sigma$, $\Delta {\cal E}_{\rm hom}=4\sigma\sqrt{3/5}=3.10\sigma$, and finally the ratio $\frac{\Delta {\cal E}_{\rm
max}}{\Delta {\cal E}_{\rm min}} = 2.11$.
For $J=1$, $\Delta {\cal E}_{\rm min}=\sigma\frac{3}{\sqrt{2}}$, $\Delta {\cal E}_{\rm max}=\Delta {\cal E}_{\rm
hom}=\sigma\sqrt{6}$. The ratio $\frac{\Delta {\cal E}_{\rm max}}{\Delta {\cal E}_{\rm min}} = 2/\sqrt{3}=1.16$ and this narrow interval strictly limits the $\Delta {\cal E}$ variation. It has a simple graphical interpretation. As is known, three real roots of a third order equation must fulfill the conditions (Cardan’s formulas) presented in Fig.1a, where the angle $\varphi$ is a function of the equation coefficients. The maximal and minimal splittings $\Delta {\cal E}$ correspond to the solutions shown in Figs 1b and 1c, respectively.
Let us now pass to the Kramers ions with a half integer $J$. Here, two cases should be analyzed. Firstly, if an even number of doublets $(2J+1)/2$ occurs the minimal $|J\rangle$ state splitting takes place when $(2J+1)/4$ doublets have the energy $\Delta {\cal E}_{\rm min}/2$, and the next $(2J+1)/4$ doublets the energy $-\Delta
{\cal E}_{\rm min}/2$. Then, $$\begin{aligned}
\sigma^{2} &=& \frac{4[(2J+1)/4]\left(\Delta {\cal E}_{\rm min}/2\right)^2}{2J+1}, \qquad \qquad
\Delta {\cal E}_{\rm min}=2\sigma.\end{aligned}$$ In turn, the maximal splitting, $\Delta {\cal E}_{\rm max}$, will appear if one of the doublets will be of energy $\Delta {\cal E}_{\rm max}/2$, and the second one of energy $-\Delta {\cal E}_{\rm max}/2$ with all the rest of levels with zero energy. This time $$\begin{aligned}
\sigma^{2} &=& \frac{4\left(\Delta {\cal E}_{\rm max}/2\right)^2}{2J+1}, \qquad \qquad
\Delta {\cal E}_{\rm max}=\sigma \sqrt{2J+1},\end{aligned}$$ and therefore now $$\begin{aligned}
\frac{\Delta {\cal E}_{\rm max}}{\Delta{\cal E}_{\rm min}} &=& \frac{\sqrt{2J+1}}{2}.\end{aligned}$$
Secondly, for an odd number of Kramers doublets $\Delta {\cal E}_{\rm
min}=\frac{2\sigma(2J+1)}{\sqrt{(2J+3)(2J-1)}}\;$ and $\;\Delta {\cal E}_{\rm max}=\sigma\sqrt{2J+1}$, with $\frac{\Delta {\cal E}_{\rm max}}{\Delta {\cal E}_{\rm min}} = \frac{1}{2}\sqrt{\frac{(2J+3)(2J-1)}{2J+1}}$.
The homogenous splitting $\Delta {\cal E}_{\rm hom}$ for an even number of doublets $(J=(4k+3)/2)$ and for an odd number of doublets $(J=(4k+1)/2)$, where $k=0,1,\ldots$, amounts correspondingly to $\sigma\sqrt{\frac{3(2J-1)}{J}}$ and $2\sigma\sqrt{\frac{3(2J-1)}{2J+3}}$. By way of example, if $J=9/2$ (i.e. for five doublets), then $\Delta {\cal E}_{\rm min}=\sigma\frac{5}{\sqrt{6}}=2.04\sigma$, $\Delta {\cal E}_{\rm
max}=\sigma\sqrt{10}=3.16\sigma$, and $\Delta {\cal E}_{\rm hom}=2.83\sigma$. In the case of Kramers ions the $\Delta {\cal E}$ variation range turns out to be smaller than that for non-Kramers ions, which is seen comparing the $\Delta {\cal E}_{\rm min}$ and $\Delta {\cal E}_{\rm max}$ for $J=4$ and $J=9/2$. Finally, taking the most extreme case of $J=15/2$ for $f$-electron configurations (e.g. for Dy$^{3+}$, Er$^{3+}$) with eight doublets, we would obtain $\Delta {\cal E}_{\rm min}=\sigma\frac{16}{3\sqrt{7}}=2.02\sigma$, $\Delta {\cal
E}_{\rm max}=4\sigma$, and $\Delta {\cal E}_{\rm hom}=2.37\sigma$.
4. The new scale of the crystal-field strength. Comparison of both the scales $S_{k}=\frac{1}{2k+1}M_{k}$ and $S_{k}^{\prime}=|{\cal H}_{\rm CF}^{(k)}|_{\rm av}$ {#the-new-scale-of-the-crystal-field-strength.-comparison-of-both-the-scales-s_kfrac12k1m_k-and-s_kprimecal-h_rm-cfk_rm-av .unnumbered}
=================================================================================================================================================================
4.1. Average values of the axial parameter modules $|B_{k0}^{\prime}|_{\rm av}$ and $|\sum_{k}B_{k0}^{\prime}|_{\rm av}$, where $k=2,4,6$ – the rotational invariants of the equivalent ${\cal
H}_{\rm CF}$ parametrizations {#average-values-of-the-axial-parameter-modules-b_k0prime_rm-av-and-sum_kb_k0prime_rm-av-where-k246-the-rotational-invariants-of-the-equivalent-calh_rm-cf-parametrizations .unnumbered}
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Rotating the reference frame by the two Euler angles $\alpha$ and $\beta$ we obtain all the equivalent ${\cal
H}_{\rm CF}$ parametrizations (with the accuracy to the third Euler angle $\gamma$ about the $z$ axis) \[5,24\]. Their axial parameters for a $2^{k}$-pole component are given as: $$\begin{aligned}
B_{k0}^{\prime} &=& \sum\limits_{q=-k}^{k}{\cal D}_{0q}^{(k)} (\alpha,\beta,0)B_{kq}=
\sum\limits_{q=-k}^{k}C_{q}^{(k)}(\beta,\alpha)B_{kq} \nonumber\\
&=& C_{0}^{(k)}(\beta)B_{k0} + 2\sum\limits_{q=1}^{k}C_{q}^{(k)}(\beta)\cos q(\alpha+\varphi_{q})
|B_{kq}|,\end{aligned}$$ where ${\cal D}_{0q}^{(k)} (\alpha,\beta,\gamma)$ are the middle row rotation matrix elements, $C_{q}^{(k)}(\beta,\alpha)=\left(\frac{4\pi}{2k+1}\right)^{1/2}Y_{q}^{k}(\beta,\alpha)$ are the spherical tensors, whereas $C_{q}^{(k)}(\beta)=(-1)^{q}\left[\frac{(k-q)!}{(k+q)!}\right]^{1/2}P_{k}^{q}(\cos\beta)$, and $P_{k}^{q}(\cos\beta)$ are the associated Legendre functions, $B_{kq}=|B_{kq}|e^{{\rm i}q\varphi_{q}}$, and $B_{k-q}=(-1)^{q}|B_{kq}|e^{-{\rm i}q\varphi_{q}}$. The primed parameters correspond to the transformed parametrization while the unprimed to the initial one. It can be directly proved that $\left(B_{k0}^{\prime}\right)_{\rm av}=0$ and $\left(\sum_{k}B_{k0}^{\prime}\right)_{\rm av}=0$, while the average absolute values $$\begin{aligned}
\left|B_{k0}^{\prime}\right|_{\rm av}&=& \frac{1}{4\pi}\int\limits_{0}^{2\pi}\int\limits_{0}^{\pi}
\left|B_{k0}^{\prime}|(\alpha,\beta)\right|\sin\beta d\beta d\alpha \;,\nonumber\\
\left|\sum_{k}B_{k0}^{\prime}\right|_{\rm av}&=& \frac{1}{4\pi}\int\limits_{0}^{2\pi}\int\limits_{0}^{\pi}
\left|\sum_{k}B_{k0}^{\prime}(\alpha,\beta)\right|\sin\beta d\beta d\alpha \;,\end{aligned}$$ as the rotational group invariants are discriminants of the equivalent parametrizations classes \[24\]. By the mean value we understand the magnitude averaged over all possible orientations of the reference frame, i.e. over the solid angle $4\pi$. Interestingly, they can be used to estimate the CF strength independently of the parametrization modulus.
4.2. Average values of the modules $\left|{\cal H}_{\rm CF}^{(k)}\right|_{\rm av}$ and $\left|{\cal H}_{\rm CF}\right|_{\rm av}$ {#average-values-of-the-modules-leftcal-h_rm-cfkright_rm-av-and-leftcal-h_rm-cfright_rm-av .unnumbered}
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Since the expression $$\begin{aligned}
{\cal H}_{\rm CF}^{(k)} &=& \sum\limits_{q=-k}^{k}B_{kq}C_{q}^{(k)}(\beta,\alpha),\end{aligned}$$ where $\beta$ and $\alpha$ are the spherical angle coordinates in the central-ion reference system, is identical with that for $B_{k0}^{\prime}$ (Eq.10), the following important identity holds $$\begin{aligned}
\left|{\cal H}_{\rm CF}^{(k)}\right|_{\rm av}&=& \frac{1}{4\pi}\int\limits_{0}^{2\pi}\int\limits_{0}^{\pi}
\left|{\cal H}_{\rm CF}^{(k)}|(\alpha,\beta)\right|\sin\beta d\beta d\alpha=
\left|B_{k0}^{\prime}\right|_{\rm av}.\end{aligned}$$ The average value of the modulus of the $2^{k}$-pole ${\cal H}_{\rm CF}^{(k)}$ component turns out to be equal to the average value of the modulus of the relevant axial parameter $B_{k0}$. This identity, Eq.12, obvious when we properly interpret the rotation angles in both cases of averaging, associates $\left|B_{k0}^{\prime}\right|_{\rm av}$ with the complementary measure of the CF strength $S_{k}^{\prime}$: $$\begin{aligned}
S_{k}^{\prime} &=& \left|{\cal H}_{\rm CF}^{(k)}\right|_{\rm av}= \left|B_{k0}^{\prime}\right|_{\rm av}\end{aligned}$$ Although the expression for $S_{k}^{\prime}$ in the above form is limited to a given $2^{k}$-pole ${\cal H}_{\rm
CF}^{(k)}$ component, it may be generalized for the global ${\cal H}_{\rm CF}$ $$\begin{aligned}
S^{\prime} &=& \left|\sum\limits_{k}{\cal H}_{\rm CF}^{(k)}\right|_{\rm av}=
\left|\sum\limits_{k}B_{k0}^{\prime}\right|_{\rm av}.\end{aligned}$$ Contrary to the conventional CF strengths $S_{k}$ and $S$ (Eqs 1,2), which are constant for all the iso-modular parametrizations, the new strengths $S_{k}^{\prime}$ and $S^{\prime}$ calculated for the constant modulus (and modules) change their magnitudes within certain ranges discussed in the next section. To compare both the measures it is convenient to express $S_{k}^{\prime}$ in the product form $f_{k}\cdot M_{k}$, where $f_{k}$ is a specified factor. Now, these two measures will be compatible if the factor $f_{k}$ is close to $\sqrt{\frac{1}{2k+1}}$, Eq.1, i.e. to $0.447$, $0.333$ and $0.277$ for $k=2,4$ and $6$, respectively. This compatibility is demonstrated in the next section, where a thorough discussion of the relation between both the CF strength scales is provided, by way of example of the CF splitting of $p^{1}$, $d^{1}$ and $f^{1}$ electron configurations with the spin-orbit coupling deliberately neglected , and the $^{3}H_{4}$ state for various iso-modular ${\cal H}_{\rm CF}^{(k)}$s.
5. Computational results and discussion {#computational-results-and-discussion .unnumbered}
---------------------------------------
5.1. Crystal-field splitting of $p^{1}$, $d^{1}$ and $f^{1}$ electron configurations for various iso-modular ${\cal H}_{\rm CF}^{(k)}$s, $k=2,4,6$. {#crystal-field-splitting-of-p1-d1-and-f1-electron-configurations-for-various-iso-modular-cal-h_rm-cfks-k246. .unnumbered}
---------------------------------------------------------------------------------------------------------------------------------------------------
We consider the model results of interaction of any iso-modular ${\cal H}_{\rm CF}^{(k)}$ ($k=2,4,6$) with $M_{k}=1$, on the initial states with well defined angular momentum quantum numbers. The magnitudes of all quantities under discussion, i.e. the new CF strength parameters $S_{k}^{\prime}$, the total splittings $\Delta
E^{(k)}$, the second moments $\sigma_{k}^{2}$ of CF levels and the averages of the absolute values of the Stark level energies $|E_{n}^{(k)}|_{\rm av}$ are given in $M_{k}$ units. Tables 1, 2 and 3 present a comprehensive review of $S_{k}^{\prime}$ values for various iso-modular ${\cal H}_{\rm CF}^{(k)}$s, with $k=2,4,6$, respectively. Correspondingly, these five, ten and eleven ${\cal H}_{\rm CF}^{(k)}$s compiled in Tables are the representative ones including those with the highest and lowest $S_{k}^{\prime}$ values found during the survey. No other ${\cal H}_{\rm CF}^{(k)}$s seem to yield $S_{k}^{\prime}$ out of these ranges. The strength parameters $S_{k}^{\prime}$ change themselves within the rather narrow intervals: $0.368 - 0.385$, $0.251 - 0.287$ and $0.195 - 0.239$, while the relevant $S_{k}$ are constant and equal to $0.447$, $0.333$ and $0.277$ for $k=2,4$ and $6$, respectively. The maximal $S_{k}^{\prime}$ parameters refer to the purely axial ${\cal H}_{\rm
CF}^{(k)}$s when $B_{k0}$s achieve 1. For other parametrizations this maximal value of 1 is not achieved in any reference frame.
As implies from Tables 4, 5 and 6 there is a certain mapping between the above $S_{k}^{\prime}$ ranges and the referring to them intervals of $\Delta E^{(k)}$ and $|E_{n}^{(k)}|_{\rm av}$. As is shown in the paper this quantitative mapping is determined by the roots of the ${\cal H}_{\rm CF}^{(k)}$ matrix characteristic polynomial, and the key part of the matrix elements is the product $(-1)^{M_{J}}B_{kq}\left(\begin{array}{ccc}
J & k & J^{\prime} \\
-M_{J} & q & M_{J}^{\prime} \\
\end{array}\right)$. The remaining factors coming into the matrix elements are common and play the role of a scaling factor. In the below examples concerning the CF splitting of one-electron states with $J=J^{\prime}=l$ for $l=1,2$ and $3$, the role of such a scaling factor play the double-bar matrix elements $\langle l||C^{(k)}||l \rangle$.
It should be pointed out, however, that the mappings $S_{k}^{\prime}\leftarrow\!\!\rightarrow \Delta E^{(k)}$, $S_{k}^{\prime}\leftarrow\!\!\rightarrow |E_{n}^{(k)}|_{\rm av}$, $\Delta E^{(k)}\leftarrow\!\!\rightarrow
|E_{n}^{(k)}|_{\rm av}$ are neither straightforward nor explicit. With the increase of the initial state degeneration $2J+1$ they become less clear due to a big variety of possible splitting schemes. Nevertheless, one may presume a dominant tendency: the greater $S_{k}^{\prime}$ the greater $|E_{n}^{(k)}|_{\rm av}$ and the lesser $\Delta E^{(k)}$ (Type I splittings). In the reverse case, i.e. for a small $S_{k}^{\prime}$, Type II splittings are expected. However, such reasoning does not take into account the unique characteristics of the Hamiltonian averages $\left|{\cal H}_{\rm CF}^{(k)}\right|_{\rm av}$, and the space density distribution of unpaired electrons in the states of various $J$. From this point of view the analysis of Tables 4, 5 and 6 seems to be instructive, indeed.
On the other hand, the allowed spans of the $\Delta {\cal E}^{(k)}$ for a fix $M_{k}$, i.e. $\sigma_{k}$ are known. In the light of the above mapping it turns out that not all of these values $\Delta {\cal E}^{(k)}$, and corresponding to them splitting schemes, can actually occur. Namely, depending on the initial state quantum number $J$ and the multipole’s rank $k$ some specified limitations of the $\Delta E^{(k)}$ are observed (Tables 4, 5 and 6). They are listed briefly below.
For $l=1$ ($p$-electron) and $k=2$ the full nominal range of the $\Delta {\cal E}^{(2)}$ and all splittings of Types I, II and III are admitted. More particularly, $\Delta E^{(2)}$ can vary from $0.600\;M_{2}$ to $0.693\;M_{2}$ (Table 4).
For $l=2$ ($d$-electron) and $k=2$ the magnitude of $\Delta E^{(2)}$ is constant and equals $0.572\;M_{2}$ for each iso-modular ${\cal H}_{\rm CF}^{(2)}$ what corresponds to splittings similar to those of Type I. Other splittings, including e.g. $\Delta {\cal E}^{(2)}_{\rm hom}$ are impossible in this case (Table 5).
For $l=2$ ($p$-electron) and $k=4$ again the full nominal range of the $\Delta {\cal E}^{(4)}$ is allowed beginning from the smallest $0.363M_{4}$ for the cubic ${\cal H}_{\rm CF}^{(4)}$, up to the biggest $0.564
M_{4}$ for ${\cal H}_{\rm CF}^{(4)}=\frac{1}{\sqrt{2}}B_{44}C_{4}^{(4)}+\frac{1}{\sqrt{2}}B_{4-4}C_{-4}^{(4)}$ (Table 5).
For $l=3$ ($f$-electron) and the value of $k=2$ $\Delta E^{(2)}$ weakly depends on $S_{2}^{\prime}$, varying in all its range merely from $0.600M_{2}$ to $0.608M_{2}$, i.e. somewhat below the $\Delta {\cal E}^{(2)}_{\rm
hom}$ (Type III splittings) (Table 6).
Next, for $l=3$ and $k=4$, the possible $\Delta E^{(4)}$ varies within the range from $0.358M_{4}$ to $0.482M_{4}$, i.e. around the $\Delta {\cal E}^{(4)}_{\rm hom}$ (Table 6).
Finally, for $l=3$ and $k=6$ the allowed $\Delta E^{(6)}$ varies from $0.326M_{6}$ to $0.501M_{6}$ covering the majority of the nominal range together with its upper limit, but excluding the smallest splittings (Table 6).
The obtained results may be generalized for states with $J$ or $L$ equal to 1, 2 or 3, multiplying $\Delta
E^{(k)}$ and $|E_{n}^{(k)}|_{\rm av}$ by the scaling factors $\langle J||C^{(k)}||J\rangle$ or $\langle
L||C^{(k)}||L\rangle$.
5.2. Crystal-field splitting of $^{3}H_{4}$ state in various iso-modular ${\cal H}_{\rm CF}^{(k)}$s, $k=2,4,6$. {#crystal-field-splitting-of-3h_4-state-in-various-iso-modular-cal-h_rm-cfks-k246. .unnumbered}
---------------------------------------------------------------------------------------------------------------
Let us end up with the analysis of splitting of nine-fold degenerate $^{3}H_{4}$ state subjected to the iso-modular ${\cal H}_{\rm CF}^{(k)}$s enclosed in Tables 1, 2 and 3. Table 7 shows the correlation between $S_{k}^{\prime}$, $\Delta E^{(k)}$ and $|E_{n}^{(k)}|_{\rm av}$. The scaling factors $\langle
J\!=\!4||C^{(k)}||J\!=\!4\rangle=\langle J\!=\!4||U^{(k)}||J\!=\!4\rangle\langle f||C^{(k)}||f\rangle$, required here due to the coupled initial state $(L=5, S=1, J=4)$, are equal to $-1.2365$, $-0.7389$ and $0.7706$ for $k=2,4,6$, respectively. Hence $\sigma_{2}=0.184M_{2}$, $\sigma_{4}=0.082M_{4}$ and $\sigma_{6}=0.071M_{6}$, while the global second moment of the Stark levels takes the form $$\begin{aligned}
\sigma^{2}&=& \frac{1}{9}\left[\frac{1}{5}(-1.2365)^{2}M_{2}^{2}+\frac{1}{9}(-0.7389)^{2}M_{4}^{2}
+\frac{1}{13}(0.7706)^{2}M_{6}^{2}\right].\end{aligned}$$ The ranges of the formally allowed $\Delta {\cal E}^{(k)}$ corresponding to the above second moments $\sigma_{2},\sigma_{4},\sigma_{6}$ are marked in Fig.2 by the solid lines.
We can see in Fig.2 that from the set of all potentially allowed total splittings $\Delta {\cal E}^{(k)}$ only certain $\Delta E^{(k)}$ may be realized (those between the dashed lines), and consequently, only certain splitting schemes (roughly between Types I and III) may occur. For instance, in the case of all the three effective multipoles neither $\Delta {\cal E}^{(k)}_{\rm max}$ nor $\Delta {\cal E}^{(k)}_{\rm min}$ are possible, while $\Delta {\cal E}^{(k)}_{\rm hom}$ can appear solely in the case of $2^{6}$-pole. Based on Table 7 it is seen also that for all the three effective ${\cal H}_{\rm CF}^{(k)}$s the biggest $\Delta E^{(k)}$ are achieved for intermediate $S_{k}^{\prime}$ values.
6. Conclusions {#conclusions .unnumbered}
--------------
The conventional scales $S_{k}$ or $S$ with the associated second moments of the CF levels, $\sigma_{k}$ or $\sigma$, do not distinguish the iso-modular ${\cal H}_{\rm CF}^{(k)}$ or ${\cal H}_{\rm CF}$ parametrizations, which, however, can be differentiated by an another scale – the spherically averaged $S_{k}^{\prime}=\left|{\cal H}_{\rm CF}^{(k)}\right|_{\rm av}$ and $S^{\prime}=\left|{\cal H}_{\rm
CF}\right|_{\rm av}$. It is proved that the $S_{k}^{\prime}$ variation ranges for all the iso-modular parametrizations are limited and lie slightly below the relevant $S_{k}$ magnitudes. The span of these ranges amounts to 5, 10 and 20% of their values for $k=2,4$ and $6$, respectively. There exists a direct mapping of $S_{k}^{\prime}$ ranges into the total splitting $\Delta E^{(k)}$ ranges and $|E_{n}^{(k)}|_{\rm av}$ intervals, which may be interpreted more clearly for the initial states with low degeneration. Such mapping allows to estimate the total splittings $\Delta E^{(k)}$ or $\Delta E$ to be expected and characterize their spectrum. It is shown that not all the nominally admitted total $\Delta {\cal E}^{(k)}$ or $\Delta {\cal E}$ splittings determined by the modules $M_{k}$ or $M$, i.e. the second moments $\sigma_{k}$ or $\sigma$, can actually occur. This essentially confines the set of the allowed splitting schemes.
[10]{} Auzel F 1979 [*Matt. Res. Bull.*]{} [**14**]{} 223 Auzel F and Malta O L 1983 [*J. Physique*]{} [**44**]{} 201 Wybourne B G 1965 [*Spectroscopic Properties of Rare Earths*]{} (New York: John Wiley) Leavitt R P 1982 [*J. Chem Phys.*]{} [**77**]{} 1661 Edmonds A R 1960 [*Angular Momentum in Quantum Mechanics*]{} (Princeton, New York: Princeton University Press) Judd B R 1963 [*Operator Techniques in Atomic Spectroscopy*]{} (New York: Mc Graw-Hill) Rotenberg M, Bivins R, Metropolis N and Wooten J K Jr. 1963 [*The 3-j and 6-j Symbols*]{} (Cambridge, Ma: MIT Press) Newman D J and Ng B K C (ed) 2000 [*Crystal Field Handbook*]{} (Cambridge, Ma: MIT Press) Chapter 3 Burdick G W and Reid M F 2004 [*Molecular Physics*]{} [**102**]{} 1141 Chang N C, Gruber J B, Leavitt R P and Morrison C A 1982 [*J. Chem. Phys.*]{} [**76**]{} 3877 Yeung Y Y and Newman D J 1985 [*J Chem. Phys.*]{} [**82**]{} 3747 Rudowicz C and Qin J 2004 [*J. Lumin.*]{} [**110**]{} 39 Newman D J and Ng B K C (ed) 2000 [*Crystal Field Handbook*]{} (Cambridge, Ma: MIT Press) Chapter 8 Kaplan I G 1975 [*Symmetry of Many Electron Systems*]{} (New York: Academic Press) Mulak J and Gajek Z 2000 [*The Effective Crystal-field Potential*]{} (Amsterdam: Elsevier) Nielson C W and Koster G F 1963 [*Spectroscopic Coefficients for $p^{n}$, $d^{n}$ and $f^{n}$ Configurations*]{} (Cambridge, Ma: MIT Press) Liu G K 2005 [*J. Sol. State Chem.*]{} [**178**]{} 489 Malta O L, Antic-Fidancev E, Lemaitre-Blaise M, Milicic-Tang A and Taibi M 1995 [*J. Alloys Compd.*]{} [**228**]{} 41 Antic-Fidancev E, H[ö]{}lsa J and Lastusaari M 2002 [*J. Alloys Compd.*]{} [**341**]{} 82 Lavin V, Babu P, Jayasankar C K, Martin I R and Rodriguez V D 2001 [*J. Chem. Phys.*]{} [**115**]{} 10935
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----- -- ---------------------- ------------------------------------------------ ----------------------------------------------------- -- -------
No. $S_{2}^{\prime}=|{\cal H}_{\rm CF}^{(2)}|_{\rm av}$
$B_{20}$ $B_{21}$ $B_{22}$
1 1 0 0 0.385
2 $\frac{1}{\sqrt{5}}$ $\frac{1}{\sqrt{5}}$ $-\frac{1}{\sqrt{5}}$ 0.381
3 $\frac{1}{\sqrt{5}}$ $\frac{1}{\sqrt{5}}$ $-\frac{1}{\sqrt{5}}$ 0.374
4 $\frac{1}{\sqrt{5}}$ $\frac{1}{\sqrt{5}}\;\;{\rm e}^{{\rm i}\pi/4}$ $\frac{1}{\sqrt{5}}$ 0.369
5 0 0 $\frac{1}{\sqrt{2}}$ 0.368
----- -- ---------------------- ------------------------------------------------ ----------------------------------------------------- -- -------
: The spherical averages of five representative iso-modular ${\cal H}_{\rm CF}^{(2)}$s, $\;S_{2}^{\prime}=|{\cal H}_{\rm CF}^{(2)}|_{av}$, acc. to Eqs 11-13, expressed in $M_{2}$ units. Only $B_{2q}$ CFPs are given, $B_{2-q}=(-1)^{q}B_{2q}^{\ast}$[]{data-label="tab"}
----- -- --------------------------------- --------------------------------------- ----------------------------------------------------- ---------------------------------------- --------------------------------- -- -------
No. $S_{4}^{\prime}=|{\cal H}_{\rm CF}^{(4)}|_{\rm av}$
$B_{40}$ $B_{41}$ $B_{42}$ $B_{43}$ $B_{44}$
1 1 0 0 0 0 0.287
2 $\frac{1}{2}\sqrt{\frac{7}{3}}$ 0 0 0 $\frac{1}{2}\sqrt{\frac{5}{6}}$ 0.280
3 $\frac{1}{3}$ $\frac{1}{3}$ $\frac{1}{3}\;{\rm e}^{{\rm i}\pi/2}$ $\frac{1}{3}$ $\frac{1}{3}$ 0.277
4 $\frac{1}{3}$ $-\frac{1}{3}$ $\frac{1}{3}$ $\frac{1}{3}$ $\frac{1}{3}$ 0.276
5 $\frac{1}{3}$ $\frac{1}{3}$ $\frac{1}{3}$ $-\frac{1}{3}\;{\rm e}^{{\rm i}\pi/2}$ $\frac{1}{3}$ 0.273
6 0 0 $\frac{1}{\sqrt{2}}$ 0 0 0.269
7 0 0 0 $\frac{1}{\sqrt{2}}$ 0 0.266
8 $\frac{1}{3}$ $\frac{1}{3}$ $\frac{1}{3}$ $\frac{1}{3}$ $\frac{1}{3}$ 0.265
9 $\frac{1}{3}$ $\frac{1}{3}\;{\rm e}^{{\rm i}\pi/4}$ $\frac{1}{3}$ $\frac{1}{3}$ $\frac{1}{3}$ 0.261
10 0 0 0 0 $\frac{1}{\sqrt{2}}$ 0.251
----- -- --------------------------------- --------------------------------------- ----------------------------------------------------- ---------------------------------------- --------------------------------- -- -------
: The spherical averages of ten representative iso-modular ${\cal H}_{\rm CF}^{(4)}$s, $\;S_{4}^{\prime}=|{\cal H}_{\rm CF}^{(4)}|_{av}$, acc. to Eqs 11-13, expressed in $M_{4}$ units. Only $B_{4q}$ CFPs are given, $B_{4-q}=(-1)^{q}B_{4q}^{\ast}$[]{data-label="tab"}
----- -- ------------------------ ------------------------ ----------------------------------------------------- ----------------------- ------------------------- ----------------------- ------------------------ -- -------
No. $S_{6}^{\prime}=|{\cal H}_{\rm CF}^{(6)}|_{\rm av}$
$B_{60}$ $B_{61}$ $B_{62}$ $B_{63}$ $B_{64}$ $B_{65}$ $B_{66}$
1 1 0 0 0 0 0 0 0.239
2 $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $-\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ 0.231
3 $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $-\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ 0.228
4 $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $-\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $-\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ 0.227
5 $-\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ 0.225
6 $\frac{1}{\sqrt{13}}$ $-\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ 0.223
7 $\frac{1}{2\sqrt{2}}$ 0 0 0 $\pm\frac{\sqrt{7}}{4}$ 0 0 0.223
8 $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $-\frac{1}{\sqrt{13}}$ 0.222
9 0 $\frac{1}{\sqrt{2}}$ 0 0 0 0 0 0.219
10 $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ 0.213
11 0 0 0 0 0 0 $\frac{1}{\sqrt{2}}$ 0.195
----- -- ------------------------ ------------------------ ----------------------------------------------------- ----------------------- ------------------------- ----------------------- ------------------------ -- -------
: The spherical averages of eleven representative iso-modular ${\cal H}_{\rm CF}^{(6)}$s, $\;S_{6}^{\prime}=|{\cal H}_{\rm CF}^{(6)}|_{av}$, acc. to Eqs 11-13, expressed in $M_{6}$ units. Only $B_{6q}$ CFPs are given, $B_{6-q}=(-1)^{q}B_{6q}^{\ast}$[]{data-label="tab"}
[lcccc]{} No.& &$\left|{\cal H}_{\rm CF}^{(2)}\right|_{\rm av}$&$\Delta E^{(2)}$& $\left|E_{n}^{(2)}\right|_{\rm av}$\
1& &0.385& 0.600 &0.267\
2& &0.381& 0.656 &0.262\
3& &0.374& 0.683 &0.250\
4& &0.369& 0.692 &0.239\
5& &0.368& 0.693 &0.231\
[lcccc]{} No.& &$\left|{\cal H}_{\rm CF}^{(2)}\right|_{\rm av}$&$\Delta E^{(2)}$& $\left|E_{n}^{(2)}\right|_{\rm av}$\
1& &0.385& 0.572 &0.229\
2& &0.381& 0.572 &0.227\
3& &0.374& 0.572 &0.221\
4& &0.369& 0.572 &0.217\
5& &0.368& 0.572 &0.213\
\
\
& &$\left|{\cal H}_{\rm CF}^{(4)}\right|_{\rm av}$&$\Delta E^{(4)}$& $\left|E_{n}^{(4)}\right|_{\rm av}$\
6& &0.287& 0.476 &0.152\
7& &0.280& 0.363 &0.174\
8& &0.277& 0.449 &0.169\
9& &0.276& 0.437 &0.169\
10& &0.273& 0.463 &0.164\
11& &0.269& 0.426 &0.159\
12& &0.266& 0.398 &0.159\
13& &0.265& 0.549 &0.137\
14& &0.261& 0.555 &0.132\
15& &0.251& 0.564 &0.113\
[lcccc]{} No.& &$\left|{\cal H}_{\rm CF}^{(2)}\right|_{\rm av}$&$\Delta E^{(2)}$& $\left|E_{n}^{(2)}\right|_{\rm av}$\
1& &0.385& 0.600 &0.190\
2& &0.381& 0.603 &0.191\
3& &0.374& 0.607 &0.192\
4& &0.369& 0.608 &0.193\
5& &0.368& 0.608 &0.193\
\
\
& &$\left|{\cal H}_{\rm CF}^{(4)}\right|_{\rm av}$&$\Delta E^{(4)}$& $\left|E_{n}^{(4)}\right|_{\rm av}$\
6& &0.287& 0.394 &0.121\
7& &0.280& 0.417 &0.119\
8& &0.277& 0.449 &0.116\
9& &0.276& 0.458 &0.117\
10& &0.273& 0.464 &0.115\
11& &0.269& 0.478 &0.113\
12& &0.266& 0.482 &0.115\
13& &0.265& 0.399 &0.129\
14& &0.261& 0.363 &0.129\
15& &0.251& 0.358 &0.131\
\
\
& &$\left|{\cal H}_{\rm CF}^{(6)}\right|_{\rm av}$&$\Delta E^{(6)}$& $\left|E_{n}^{(6)}\right|_{\rm av}$\
16& &0.239& 0.408 &0.107\
17& &0.231& 0.326 &0.130\
18& &0.228& 0.365 &0.120\
19& &0.227& 0.379 &0.123\
20& &0.225& 0.445 &0.106\
21& &0.223& 0.420 &0.112\
22& &0.223& 0.346 &0.127\
23& &0.222& 0.422 &0.112\
24& &0.219& 0.468 &0.097\
25& &0.213& 0.481 &0.097\
26& &0.195& 0.501 &0.072\
[lcccc]{} No.& &$\left|{\cal H}_{\rm CF}^{(2)}\right|_{\rm av}$&$\Delta E^{(2)}$& $\left|E_{n}^{(2)}\right|_{\rm av}$\
1& &0.385& 0.504 &0.163\
2& &0.381& 0.543 &0.170\
3& &0.374& 0.562 &0.164\
4& &0.369& 0.560 &0.164\
5& &0.368& 0.524 &0.155\
\
\
& &$\left|{\cal H}_{\rm CF}^{(4)}\right|_{\rm av}$&$\Delta E^{(4)}$& $\left|E_{n}^{(4)}\right|_{\rm av}$\
6& &0.287& 0.215 &0.079\
7& &0.280& 0.227 &0.073\
8& &0.277& 0.249 &0.070\
9& &0.276& 0.241 &0.071\
10& &0.273& 0.232 &0.073\
11& &0.269& 0.230 &0.064\
12& &0.266& 0.232 &0.067\
13& &0.265& 0.231 &0.077\
14& &0.261& 0.229 &0.076\
15& &0.251& 0.196 &0.077\
\
\
& &$\left|{\cal H}_{\rm CF}^{(6)}\right|_{\rm av}$&$\Delta E^{(6)}$& $\left|E_{n}^{(6)}\right|_{\rm av}$\
16& &0.239& 0.202 &0.058\
17& &0.231& 0.192 &0.069\
18& &0.228& 0.249 &0.051\
19& &0.227& 0.212 &0.060\
20& &0.225& 0.224 &0.062\
21& &0.223& 0.233 &0.059\
22& &0.223& 0.245 &0.053\
23& &0.222& 0.233 &0.061\
24& &0.219& 0.258 &0.052\
25& &0.213& 0.208 &0.059\
26& &0.195& 0.206 &0.058\
FIGURE CAPTIONS: Fig.1. Crystal-field splitting of $|J\!=\!1\rangle$ state – geometrical interpretation ($x$ is the energy) a) general case: $\Delta {\cal E}^{(2)}=x_{1}-x_{2}$, b) $\Delta {\cal E}^{(2)}_{\rm min}=x_{1}-x_{2}$, and c) $\Delta {\cal E}^{(2)}_{\rm hom}=\Delta {\cal E}^{(2)}_{\rm max}=x_{2}-x_{3}$.
Fig.2. Nominally allowed $\Delta {\cal E}^{(k)}$ (bold solid borders) and the actual $\Delta E^{(k)}$ (dashed borders) ranges of the total splittings of the $^{3}H_{4}$ state subjected to the iso-modular ${\cal
H}_{CF}^{(k)}$. The $\Delta {\cal E}^{(k)}_{\rm hom}$ are also given (thin solid lines).
{width="17cm"}
Fig.1
{width="15cm"}
Fig.2
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
Clinical applicability of automated decision support systems depends on a robust, well-understood classification interpretation. Artificial neural networks while achieving class-leading scores fall short in this regard. Therefore, numerous approaches have been proposed that map a salient region of an image to a diagnostic classification. Utilizing heuristic methodology, like blurring and noise, they tend to produce diffuse, sometimes misleading results, hindering their general adoption.
In this work we overcome these issues by presenting a model agnostic saliency mapping framework tailored to medical imaging. We replace heuristic techniques with a strong neighborhood conditioned inpainting approach, which avoids anatomically implausible artefacts. We formulate saliency attribution as a map-quality optimization task, enforcing constrained and focused attributions. Experiments on public mammography data show quantitatively and qualitatively more precise localization and clearer conveying results than existing state-of-the-art methods.
address: 'VRVis Zentrum für Virtual Reality und Visualisierung Forschungs-GmbH, Vienna, Austria'
bibliography:
- 'refs.bib'
title: Interpreting medical image classifiers by optimization based counterfactual impact analysis
---
at (current page.north) ;
Classifier Decision Visualization, Image Inpainting, Mammography, Explainable AI
Introduction {#sec:intro}
============
Consulting radiologists will routinely back their findings by pinpointing and describing a specific region on a radiograph. Contrary, acting as a highly efficient black box, artificial neural networks (ANN) [@litjens2017] fall short of this form of explanation for their predictions. ANNs’ high dimensional, nonlinear nature, does not induce a canonical map between derived prediction and input image. Understandably, a plethora of approaches have been presented that try to derive a so called saliency map, that is, a robust mapping between pixel space and prediction class [@zintgraf2017; @fong2017; @dabkowski2017; @simonyan2013; @zhou2016; @shrikumar17; @chang2018; @petsiuk2018; @uzunova2019].
![Our saliency mapping framework: (i) classification score of an image is obtained, (ii) a hole mask is generated, inpainted and classified, (iii) saliency loss is computed based on the score difference of original/inpainted images and map quality, (iv) optimization is continued for a fixed number of steps and a result map/mask is derived.[]{data-label="fig:overview"}](architecture_2.pdf){width=".48\textwidth"}
Most frequently this form of reasoning is based on *local explanations* (LE), i.e. on concrete maps for image-prediction pairs [@fong2017; @lipton2016]. A clinically applicable LE needs to be *informative* for the radiologists, that is, focusing on regions coinciding with medical knowledge [@lombrozo2006]. Moreover, a methodologically sound LE is *faithful* to the classifier, i.e. dependent on architecture, parametrization, and its preconditions like training-set distribution [@adebayo2018].
*Direct approaches* efficiently utilize the assumed analytic nature or the layered architecture of an ANN classifier to derive the desired saliency map for a LE [@simonyan2013; @zhou2016]. While frequently applied, the obtained results of this class are possibly incomplete, diffuse, hard to interpret, and as recent work shows misleading [@zintgraf2017; @fong2017; @dabkowski2017; @shrikumar17; @adebayo2018]. Thereby they violate both criteria, informativeness and faithfulness, hindering their general application in medical imaging.
Contrary, *reference based* LE approaches [@fong2017] try to mitigate these issues by studying how the given classifier reacts to perturbations of the input image. Using the original input as a reference and marginalizing a dedicated image region’s contribution, they estimate this region’s effect on the classification score. Solutions mainly vary in the ways this marginalization is achieved. They range from heuristic approaches, e.g. blurring, noise, or graying out [@dabkowski2017; @fong2017], over local neighbourhood conditioning [@zintgraf2017], to utilizing strong conditional generative models [@chang2018; @uzunova2019]. These methods address *informativeness*, however, applied to medical images, they introduce noise, possibly pathological indications, anatomical implausible tissue or other adversarial artefacts. By this, they amplify the out-of-distribution problem, similar to an adversarial attack: they expect a meaningful classification result for an image, that is not within the training-set distribution. Hence, they fall short of *faithfulness* for clinical applications.
Marginalization for medical imaging, i.e. the replacement of pathological regions with counterfactual healthy tissue, is being actively explored and addressed by generative adversarial network setups (GAN). Besides promising results, authors report resolution limitations, and the same underlying out-of-distribution issue[@bermudez2018; @baumgartner2017; @becker2019; @andermatt2019].
**Contribution:** We address the open challenge of *faithful and informative* medical black-box classifier interpretation by expanding natural image classifier visualization approaches [@zintgraf2017; @dabkowski2017; @fong2017]. We propose a reference based optimization framework tailored to medical images, focusing on the interactions between original and marginalized image classification-scores, and map quality. To tackle anatomical correctness of marginalization in medical images, partial convolution inpainting [@liu2018] is adapted. Hence, instead of a globally acting GAN, we utilize local per-pixel reconstruction without sacrificing global image composition. We validate our approach on publicly available mammography data, and show quantitatively and qualitatively more precise localization, and clearer conveying results than existing state-of-the-art methods.
Methods {#sec:methods}
=======
Our goal is to estimate a *faithful* and *informative* saliency map between a medical image and its classification score: given an image, we search for and visually attribute the *specific* pixel-set that contributes towards a confident classification for a fixed class (see Fig. \[fig:overview\]). Following [@dabkowski2017; @zintgraf2017] we formulate the general problem as finding the *smallest deletion region* (SDR) of a class $c$, i.e. the pixel-set whose marginalization w.r.t. the classifier lowers the classification score for $c$.
**Image-wise Saliency Mapping**: Informally, we search for the smallest smooth map, that indicates the regions we need to change (inpaint) such that we get a *sufficiently healthy* image able to *fool the classifier*. We formalize the problem as follows: Let $I$ denote an image of a domain $\mathcal{I}$ with pixels on a discrete grid $m_1 \times m_2$, $c$ a fixed class, and $f$ a classifier capable of estimating $p(c|I)$, the probability of $c$ for $I$. Also let $M$ denote the saliency mask for image $I$ and class $c$, hence $M \in M^{m_1 \times m_2}(\{0,1\})$. We use *total variation* $tv(M)$ [@dabkowski2017], and *size* $ar(M)$, to measure the mask’s shape. Note that *size* here is ambiguous. Experimentally we found dice overlap with regions-of-interest like organ masks to be favourable over the map’s average pixel value[@dabkowski2017]. With $\odot$ denoting elementwise multiplication, and $ \pi(M)$ the inpainting result of a hole image $I \odot M$, we can define $ \phi(M) := -1 \cdot \log ( p(c| \pi(M)) ) $ and $\psi(M) := \log (\text{odds}(I)) - \log (\text{odds}(\pi(M))) $, where $\text{odds}(I) = \frac{ p(c|I) }{ 1 - p(c|I)}$. Both, $\phi$ and $\psi$, weigh the new probability of the inpainted image. If we assume class $c$ to denote *pathological*, then healthy images, and large score differences will be favoured. With this preparation we define our desired optimization function as $$\mathcal{L}(M) := \lambda_1 \cdot (\phi(M) + \psi(M)) + \lambda_2 \cdot tv(M) + \lambda_3 \cdot ar(M)$$ where $\lambda_i \in {\rm I\!R}$ are regularization parameters, and search for $\arg\min_{M} \; \mathcal{L}(M)$. There are two collaborating parts in $\mathcal{L}$. The first term enforces the class probability to drop, the latter two emphasize an informative mask. Focusing on medical images, $\mathcal{L}$ directly solves the SDR task, thereby minimizing medically implausible and adversarial artefacts caused by inpainting of large classifier-neutral image regions, as observable in [@liu2018; @dabkowski2017; @fong2017]. The optimization problem is solved by local search through stochastic gradient descent, starting from a regular grid initialization. By design, no restrictions are applied on the classifier $f$. For optimization we relax the mask’s domain to $M^{m_1 \times m_2}([0,1])$, and threshold at $\theta \in (0,1)$.
**Image Inpainting with Partial Convolutions**: For marginalization, we want to emphasize local context, while still considering global joint region interaction, and thereby favor a globally sound anatomy. Therefore, we adapt the U-Net like architecture of [@liu2018], capable of handling masks with irregular shapes, fitting our optimization requirements for pathological regions of different sizes and shapes. The chosen architecture consists of eight partial convolution layers on both encoding and decoding parts. It takes an image with holes $I \odot M$ and the hole mask $M$ as an input, and outputs the inpainted image $\pi(M)$. The partial convolution layers insert only the convolution result of the current sliding convolution-window when image information is present. The convolution filter $W$ is applied on the features $X$ using the binary mask $M$ and yields new features $x'$ the following way:
![Left: Comparison of classifier performance without inpainting (green), with inpainting in healthy tissue (blue) and in mass tissue (red) over 10 random runs (shadowed). Right: Original image with mass (top), inpainted with replaced healthy texture (bottom).[]{data-label="fig:inpainter_performance"}](inpainter_roc_comparison.png){width="6.7cm" height="6.7cm"}
![Left: Comparison of classifier performance without inpainting (green), with inpainting in healthy tissue (blue) and in mass tissue (red) over 10 random runs (shadowed). Right: Original image with mass (top), inpainted with replaced healthy texture (bottom).[]{data-label="fig:inpainter_performance"}](inp_orig_zoom.png){width="2.5cm" height="2.5cm"}
![Left: Comparison of classifier performance without inpainting (green), with inpainting in healthy tissue (blue) and in mass tissue (red) over 10 random runs (shadowed). Right: Original image with mass (top), inpainted with replaced healthy texture (bottom).[]{data-label="fig:inpainter_performance"}](inp_result_zoom.png){width="2.5cm" height="2.5cm"}
$$x' = \begin{cases}
W^{T}(X\odot M)\frac{1}{sum(M)}+b, & \text{if} \;sum(M)>0 \\
0, & \text{otherwise}
\end{cases}$$ where $b$ is the bias term. The convolution operation is scaled by $\frac{1}{sum(M)}$ according to the amount of information available in the current sliding window. Moreover a new mask is passed to the next layer which is updated by setting its values to $1$ in the sliding window if $sum(M)>0$. The authors of [@liu2018] propose to train the network with a loss function concentrating on both per-pixel reconstruction performance of the hole/non-hole regions and on overall appearance of the image. To improve the overall appearance a perceptual loss and a style loss are applied which match images in a mapped feature space. Total variation is used as a last loss component to ensure a smooth transition between hole regions and present image regions.
P *$D_{ours}$* *$D_{cam}$* *$D_{sal}$*
---------- -------------------- ----------------- ----------------
50 **137.1$\pm$69.5** 200.1$\pm$65.3 164.2$\pm$36.0
75 **137.1$\pm$69.5** 182.1$\pm$78.9 162.9$\pm$36.9
90 **137.1$\pm$69.5** 144.1$\pm$89.3 166.3$\pm$44.0
Ablation 163.3$\pm$35.9 - -
P *$H_{ours}$* *$H_{cam}$* *$H_{sal}$*
50 **227.1$\pm$82.1** 359.3$\pm$83.9 371.3$\pm$61.5
75 **227.1$\pm$82.1** 308.5$\pm$103.4 350.3$\pm$63.1
90 **227.1$\pm$82.1** 232.2$\pm$121 328.9$\pm$64.5
Ablation 328.1$\pm$69.5 - -
: Results of the saliency mapping on test data with masses for different percentiles $P$: $D...$ average Euclidean, $H...$ average Hausdorff distances, between result mask connected components and ground-truth annotations (in pixels).[]{data-label="tab:results_optimization"}
P *$A_{ours}$* *$A_{cam}$* *$A_{sal}$* *$O_{cam}$* *$O_{sal}$*
---- ----------------- ------------- ------------- -------------- ------------- -- -- --
50 **.02$\pm$.02** .43$\pm$.10 .50$\pm$.0 .60$\pm$ .33 .86$\pm$.14
75 **.02$\pm$.02** .24$\pm$.03 .25$\pm$.0 .44$\pm$ .32 .64$\pm$.22
90 **.02$\pm$.02** .10$\pm$.0 .10$\pm$.0 .23$\pm$ .25 .39$\pm$.23
: For three threshold-levels, chosen by percentiles of the output-distribution of *CAM* resp. *SAL*: $A$: Average overlaps between attributions and breast area; $O$: Overlap between our attribution and *CAM* resp. *SAL*.[]{data-label="tab:results_overlap"}
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Experimental Setup {#sec:methods}
==================
**Dataset:** In this work the Database for Screening Mammography (DDSM) [@heath2000] and the Curated Breast Imaging Subset of DDSM (CBIS-DDSM) [@lee2016] were used, downsampled to a resolution of 576x448 pixels. Data was split into 1231 scans containing masses and 2000 healthy samples for training, and into 334 mass and 778 healthy scans for testing. Scans with masses contain pixel-wise ground-truth annotation (GT).
**Image Classifier:** The basis of our saliency detection framework is a MobileNet [@howard2017] binary classifier to categorize images either as healthy or as a sample with masses. The network was trained on all training scans with batch size of 4 using the Adam optimizer with a learning rate ($lr$) of $1\text{e-}5$ for 250 epochs using early stopping. Rotation, zoom, horizontal and vertical flips were used for data augmentation. It was pretrained by 50k 224x224 pixel patches from the training data with the task of classifying background vs. masses.
**Inpainting:** The inpainter was trained on the healthy training samples with a batch size of 1 in two phases [@liu2018]. The first phase was set up with batch normalization (BN) and $lr=1\text{e-}5$ for 100 epochs, the second without BN in the encoder part and with $lr=1\text{e-}6$ for 50 epochs. For each image up to 400 8x8 pixel holes were generated at random positions, where both single small holes and larger clusters were simulated to mimic configurations during optimization.
The inpainter has the task to change the classification score of an image towards healthy when replacing mass tissue, no considerable change should happen otherwise. To demonstrate that, we computed (i) a ROC curve using the classifier on all test samples without any inpainting, (ii) ROC curves for inpainting only in healthy tissue over 10 runs with randomly sampled holes and (iii) ROC curves for inpainting of mass tissue in unhealthy scans over 10 runs (Fig. \[fig:inpainter\_performance\] left).
**Saliency Mapping:** Parametrization was experimentally chosen based on grid-search, restricted by $\lambda_i \in [0,1]$, for $i=1,2,3$. Similar to [@chang2018; @fong2017] we found the resulting masks to be especially sensitive to $\lambda_2$. This smoothness controlling term, balances between noisy result-maps and compression induced information-loss. We exemplify this behaviour with an ablation study, where contributions of smoothing and sizing are set to zero (cf. Table \[tab:results\_optimization\]). The final optimization results were derived in 100 steps per image, with $lr=2\text{e-}3$, $\theta = 0.5$ and setting $\lambda_1 = 1.0$, $\lambda_{2,3} = 0.1$.
We compared our approach against two established methods based on widespread adaptation in medical imaging [@rajpurkar2017; @baumgartner2017], and inherent validity [@adebayo2018]. We chose the gradient based *Saliency Map* [@simonyan2013] (SAL) and the network-derived *Cam* [@zhou2016] (CAM) visualizations. As our domain prohibits the utilization of blurring, noise, etc. we could not meaningfully test against reference based methods [@zintgraf2017; @fong2017; @dabkowski2017]. For evaluation, we derived four measures, that (i) relate the result masks (RM) to the mass annotations (GT), and (ii) compare the result masks to each other (cf. Table \[tab:results\_optimization\], \[tab:results\_overlap\]). Particularly for (i) we studied $D$ the average of Euclidean distances between the centres of GT masks and RMs’ connected components, and $H$ the average Hausdorff distances between GT masks and RM. Here, lower values indicate better localization, i.e. a better vicinity to the pathology. Conversely, for (ii) we calculated $A$ the ratio between derived RMs and the organ masks (the breast area), and $O$ the overlap coefficients between our RMs and those of CAM and SAL. Here lower values of $A$ indicate a more compact map, i.e. better visual distinction. $O$ scores describe the correlation level between result masks. Statistically significant difference between all resulting findings was formalized using Wilcoxon signed-rank tests, for $\alpha < 0.01$.
All measurements were performed on binary masks, for which CAM and SAL had to be thresholded as those maps’ noise covers the complete image. We therefore chose the $50/75/90$th percentiles, i.e. $50/25/10$ percent of the map-points. Where multiple masses, or mapping results occurred we used their median for a robust estimation per image.
Results and Conclusion {#sec:results}
======================
**Inpainting Evaluation:** The ROC curves in Fig. \[fig:inpainter\_performance\] represent an AUC of $0.89$ for original images (green), average AUCs of $0.88$ for inpainting tissue in healthy cases (blue) and $0.83$ for inpainting only in masses for pathological cases (red). Besides the AUCs, the visual separability of the green/blue curves from the red one indicates that the inpainter behaves correctly and introduces significant changes only when replacing mass tissue w.r.t. the classifier. The inpainting quality of replacing mass with healthy tissue is visible in Fig. \[fig:inpainter\_performance\] right.
**Saliency Evaluation:** *Quantitatively*, our framework yields saliency masks significantly closer to GT masks based on Euclidean distances $D$ and Hausdorff distances $H$, both substantiated by p-values below $2\text{e-}12$ for all tested percentiles (cf. Table \[tab:results\_optimization\]). Considering the map sizes $A$ (Table \[tab:results\_overlap\]), we report overall significantly smaller masks, and again p-values below $2\text{e-}12$ for all percentiles. This behavior changes when the shape-specific regularization parameters $\lambda_{2,3}$ are relaxed, as exemplified by the ablation study. As shown in the last row of both parts of Table \[tab:results\_optimization\], our feature attributions become scattered and noisy. Close inspection of the overlap-values, esp. $O$ in Table \[tab:results\_overlap\], reveals that on average our method’s attributions have a higher overlap with SAL than CAM. This indicates that our results tend to adhere to the dense localization spots of SAL, but alleviate the latter’s noise and interpretation issue described in [@fong2017; @zintgraf2017].
*Qualitatively*, as depicted in Fig. \[fig:results\] (b), our salient regions appear at the circumference of masses which is reasonable w.r.t. the fact that this is the discriminative region for the presence of masses. This is in line with [@becker2019], which reports on injection of poorly *circumscribed*, malignant looking masses while transforming healthy cases into pathological ones using a GAN variant. In addition, our method yields more accurate visualizations than CAM and SAL (Fig. \[fig:results\] first row), i.e. it has a smaller, more precise and more informative feature attribution than these standard visualization methods (Fig. \[fig:results\] (b)-(d)).
**Conclusion:** We presented a novel, model agnostic, optimization-based, accurate saliency visualization approach tailored to medical images. Image region marginalization was solved by partial convolution based inpainting, treating medical images correctly, thus overcoming the out-of-distribution problem. Our method showed informative and faithful results on public mammography data, and is therefore suitable as a classification interpretation tool in radiological workflows.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this paper, we consider finitely many interval maps simultaneously acting on the unit interval $I = [0,\, 1]$ in the real line $\mathbb{R}$; each with utmost finitely many jump discontinuities and study certain important statistical properties. Even though we use the symbolic space on $N$ letters to reduce the case of simultaneous dynamics to maps on an appropriate space, our aim in this paper remains to resolve ergodicity, rates of recurrence, decay of correlations and invariance principles leading upto the central limit theorem for the dynamics that evolves through simultaneous action. In order to achieve our ends, we define various Ruelle operators, normalise them by various means and exploit their spectra.'
author:
- |
Aswin Gopakumar\
[aswin15@iisertvm.ac.in]{}\
Kirthana Rajasekar\
[kirthanarajasekar15@iisertvm.ac.in]{}\
Shrihari Sridharan\
[shrihari@iisertvm.ac.in]{}\
[*Indian Institute of Science Education and Research*]{}\
[*Thiruvananthapuram (IISER-TVM), India.*]{}
title: '<span style="font-variant:small-caps;">Simultaneous Action of Finitely Many Interval Maps: Some Dynamical and Statistical Properties</span>'
---
--------------------- --- -----------------------------------------------------
**Keywords** : Growth of typical trajectories;
Invariance principles;
Ruelle operator and the pressure function;
Simultaneous action of finitely many interval maps.
**AMS Subject** : 37E05, 37C35, 37D35, 37B10.
**Classifications**
--------------------- --- -----------------------------------------------------
Introduction
============
Various dynamical properties and statistical properties help us understand the behaviour of dynamics caused by the action of a transformation $T$ on some phase space $X$. Important among such properties include the Birkhoff’s pointwise ergodic theorem, asymptotic estimates on rates of recurrence of typical orbits, decay of correlations, invariance principles, central limit theorem, law of iterated logarithms [*etc*]{}. Each of these theorems provide us a deeper glimpse into the structure, the dynamics builds in its phase space of action or an invariant subset, thereof.
Birkhoff’s pointwise ergodic theorem observes a considered dynamical system through a real-valued continuous function and states that the sequence of local time averages along the orbit of any typical point converges to the global space average, whenever the transformation $T$ acting on $X$ is ergodic. Though the result is extremely strong, it does allow some points (though negligible, meaning with collective measure zero) to fluctuate from this mean behaviour. Thus, an interesting study in the dynamics of such ergodic systems is to obtain a good understanding of the set of points that violate the Birkhoff’s ergodic conditions. An easy way to approach this subject locally is to work out the ergodic sums of the observables and consider the cardinality of the set of points whose ergodic sum calculated at various times remain inside some chosen interval $[a,\, b] \subset \mathbb{R}$. However, on a global scale, an alternate way to understand the deviation from the average behaviour of typical orbits is achieved by formalising the central limit theorem.
The central limit theorem is an important tool in mathematics that distributes the random variables along a bell-curve (normal distribution), as more and more independent random variables are appropriately included under the ambit of study. This is a central object of investigation in understanding deterministic dynamical systems, owing to its natural appeal, when we consider the various orbits in the phase space. However, as important as the central limit theorem is, we see that they are subsumed by more general invariance principles. A sequence of random variables $\big\{ X_{n} \big\}_{n\, \ge\, 1}$ is said to satisfy an almost sure invariance principle if the sequence can be approximated almost surely by another sequence, preferably with certain desired properties and with a relatively small margin of error.
Several mathematicians have studied these properties in many deterministic dynamical systems, where the phase space is a compact interval of the real line, [@md:86; @ci:96; @lsv:99; @ps:02; @cr:07; @cm:15], the Julia set of some rational map that occurs as a compact subset of the Riemann sphere, [@du:91; @dpu:96; @ss:07; @ss:09], [*etc.*]{}, however, with a single transformation acting on the appropriate space, based on which the dynamics evolves. Examples of continuous time dynamical systems that has remained in the focus of the dynamics community include expanding flows restricted on a compact subset of the Riemannian manifold, [@dp:84; @spl:89; @mp:91] or some mixing Axiom $A$ diffeomorphism restricted on a basic set, [@mr:73; @ks:90; @rs:93; @na:00; @ps:01]. A particularly desirable feature of all the above-mentioned maps restricted on their respective sets is that they can be studied through an associated symbolic model [@rb:73; @mr:73]. There are also various studies carried out by several mathematicians that analyse statistical results in various settings of dynamical systems. Prominent among them include [@ps:75; @hh:80; @cp:90; @pp:90; @ps:94; @lsy:99; @si:99; @fmt:03; @mtk:05; @mn:05; @hntv:17].
What we intend to investigate in this paper is slightly richer in dynamics, than what is explained so far. In this paper, we consider the compact unit interval $[0,\, 1] \subset \mathbb{R}$; however with finitely many maps acting on the space simultaneously. Thus, the dynamics evolves along the multiple branches provided by each of these maps. In fact, we work with finitely many interval maps defined on $[0,\, 1]$; each of which has a discrete set of utmost finitely many discontinuities. As an expert reader may realise, these results are readily transferable to various settings including the simultaneous action of finitely many rational maps restricted on the appropriate Julia set, as defined in [@hs:00] or to the action of a holomorphic correspondence restricted on the support of its Dinh-Sibony measure, as defined in [@bs:16]. We shall explain our claim of transferability of the main theorems of this paper, as written above, in the final section, .
This paper is structured as follows: In the next two introductory sections and , we narrate the basic settings of this paper and develop certain notations and dynamical notions, however only as skeletal to enable us to state the main theorems of this paper in section . The main results of this paper describe the ergodicity of the system in theorems and , the rates of recurrence in theorems and , the exponential decay of correlations in theorems and , almost sure invariance principles in theorems and and a few more statistical properties such as the central limit theorems and the laws of iterated logarithms in theorems and . The reason why each theorem appears twice in the list above will be clear, by the time we reach section . In section , we recall the setting of symbolic dynamics that comes in handy as a book-keeping mechanism in our study. In sections , and , we define three kinds of Ruelle operators on the appropriate Banach space of continuous functions and Hölder continuous functions defined on the phase spaces that interest us, compare their spectra and normalise them in different ways in order that they help us in proving our main theorems. Having achieved these, we embark on writing the proofs of the main theorems in sections , , , and . We conclude the paper with a few remarks in section .
Preliminaries and the pressure function {#prelims}
=======================================
In this section, we explain the setting of our paper and define certain basic terminologies that help us in constructing the necessary notions to state our main results.
Let $I$ denote the unit interval on the real line, [*i.e.*]{}, $I = [0,\, 1]$. We are interested in studying the dynamics of finitely many interval maps acting simultaneously on $I$, *i.e.*, given $N \in \mathbb{N}$ and $1 \leq d \leq N$, we consider the interval maps $T_{d} : I \longrightarrow I$ of degree $(d + 1)$ given by $$T_{d}\, (x)\ \ :=\ \ (d + 1)\; x \pmod 1.$$ The simultaneous action is explained as follows: For any $x_{0} \in I$, its forward orbit at times $t = 0,\, 1,\, 2,\, \cdots,\, n,\, \cdots$ is defined as $$\label{1storbit}
\left\{ x_{0},\, x_{1} \in \bigcup_{d\, =\, 1}^{N} T_{d} (x_{0}),\, x_{2} \in \bigcup_{d\, =\, 1}^{N} T_{d} (x_{1}),\, \cdots,\, x_{n} \in \bigcup_{d\, =\, 1}^{N} T_{d} (x_{n - 1}), \cdots \right\}.$$ Thus, at every stage, we have $N$ many maps to choose from to move forward and the totality of all these branches describe the forward orbit. Observe that the dynamics that arises out of such a process can also be described by the action of a semigroup generated by the same interval maps, $\mathscr{S} = \big\langle T_{1},\, T_{2},\, \cdots,\, T_{N} \big\rangle$, or as a correspondence on $I \times I$, appropriately defined.
Suppose $\mathcal{C} (V, \mathbb{F})$ denotes the space of all continuous functions defined on the space $V$ that takes values on the field $\mathbb{F}$. Then, for any $f \in \mathcal{C}(I, \mathbb{C})$, the set of all complex-valued continuous functions defined on $I$, we define a *composition operator*, $\mathscr{O}$ by $$\mathscr{O} (f) \in \bigcup\limits_{d\, =\, 1}^{N} \big( f \circ T_{d} \big).$$ Along every orbit of the point $x_{0}$, as described in , this composition operator chooses the map $T_{d}$ every time in such a fashion that $T_{d} (x_{k - 1})\, =\, x_{k}$. Hence, even though $\mathscr{O} (f)$ is not single-valued, we have by definition that $\left( \mathscr{O} (f) \right) (x_{0})$ to be single-valued along every chosen orbit of $x_{0}$. We further describe this idea in section .
To assist us in this study, we will make use of the space consisting of infinitely long words on $N$ symbols, [*i.e.*]{}, suppose $S = \big\{ 1,\, 2,\, \cdots,\, N \big\}$, we consider $$\Sigma_{N}^{+}\ \ :=\ \ S^{\mathbb{Z}_{+}}\ \ =\ \ \Big\{ w = \left( w_{1}\, w_{2}\, \cdots\, w_{n}\, \cdots \right) : w_{i} \in S \Big\}.$$ As we shall explain in section , $\Sigma_{N}^{+}$ is a compact measurable metric space equipped with the Bernoulli measure, where the shift map $\sigma$ defined by $\left( \sigma (w) \right)_{n} = w_{n\, +\, 1}$ is continuous and non-invertible, but a local homeomorphism.
From now on, we denote by $X$ the product phase space given by $X := \Sigma_{N}^{+} \times I$, where we define a skew-product map $T$ as $$\label{spm}
T (w,\, x)\ \ :=\ \ ( \sigma w,\, T_{w_{1}} x),\ \ \ \ \text{where}\ \ w = (w_{1}\, w_{2}\, \cdots).$$
By a standard argument due to Tychonoff, as may be found in [@mj:00], we consider the natural product topology on $X$ that gives rise to the metric $d_{X} (\cdot,\, \cdot)$ on $X$. Further, we also have the product sigma-algebra and the product measure defined on $X$. Let $\mu$ denote some $T$-invariant measure supported on $X$. For example, the appropriate product measure of the Bernoulli measure on cylinder sets of $\Sigma_{N}^{+}$ and the Lebesgue measure on open intervals of $I$ is a $T$-invariant probability measure on $X$.
The definition of the skew-product map $T$ entails that the forward orbit of $(w, x)$ at times $t = 0,\, 1,\, 2,\, \cdots,\, n,\, \cdots$ under $T$ is given by $$\Big\{ (w,\, x),\; (\sigma w,\, T_{w_{1}} x),\; \left( \sigma^{2} w,\, \left( T_{w_{2}} \circ T_{w_{1}} \right) x \right),\; \cdots,\; \left( \sigma^{n} w,\, \left( T_{w_{n}} \circ T_{w_{n - 1}} \cdots \circ T_{w_{1}} \right) x \right),\; \cdots \Big\}.$$ Thus, for a chosen $w$ in $\Sigma_{N}^{+}$, the natural projection on the second co-ordinate $\mathbf{Proj}_{2} : X \longrightarrow I$ captures the sectional idea behind the orbit of $x \in I$ as described in . Taking the union over all possible $w \in \Sigma_{N}^{+}$ captures the idea in its entirety.
For ease of notations, we fix little letters like $f,\, g,\, h,$ [*etc*]{}. to represent functions defined on the interval, $I$ and use big letters like $F,\, G,\, H$ [*etc*]{}. to represent functions defined on the product space $X = \Sigma_{N}^{+} \times I$. Although one may think of $f$ as being a restriction of $F$ on the interval, [*i.e.*]{}, $F = f \circ \mathbf{Proj}_{2}$ that yields $F((w,\, x)) = f(x)$, it need not be the case always.
The space $X$ now facilitates us to redefine the composition operator, in this setting denoted by $\mathscr{Q}$ defined on $\mathcal{C} (X, \mathbb{C})$ given by $$\mathscr{Q} (F)\ \ :=\ \ F \circ T.$$
Let $\mathscr{F}_{\alpha}(X, \mathbb{C})$ denote the space of all complex-valued $\alpha$-Hölder continuous functions defined on $X$. For $F \in \mathscr{F}_{\alpha}(X, \mathbb{C})$, we define the following norm, $$\big\Vert F \big\Vert_{\alpha}\ \ :=\ \ \big\vert F \big\vert_{\alpha} + \big\Vert F \big\Vert_{\infty},$$ where $$\big\vert F \big\vert_{\alpha}\ \ :=\ \ \sup\limits_{(w,\, x)\; \neq\; (v,\, y)} \left\{ \frac{\big\vert F((w,\, x))\; -\; F((v,\, y)) \big\vert}{\big( d_{X}((w,\, x),\; (v,\, y)) \big)^{\alpha}}\ :\ (w,\, x),\; (v,\, y) \in X \right\}$$ denotes the $\alpha$-Hölder semi-norm and $\big\Vert F \big\Vert_{\infty}$ denotes the usual supremum norm. Then, $\mathscr{F}_{\alpha}(X, \mathbb{C})$ is a Banach space under the norm $\big\Vert \cdot \big\Vert_{\alpha}$.
Given any function $F \in \mathcal{C}(X, \mathbb{R})$, its *pressure* is defined as $$\label{pressure}
\mathfrak{P} (F)\ \ :=\ \ \sup \left\{ h_{\mu} (T) + \int\! F d \mu \right\},$$ where the supremum is taken over all $T$-invariant probability measures supported on $X$. Further, $h_{\mu} (T)$ is the measure theoretic entropy of $T$ with respect to $\mu$. An *equilibrium measure* for the function $F$ denoted by $\mu_{F}$ is defined as that measure for which the supremum is attained in the definition of pressure, as stated in . The unique existence of $\mu_{F}$ for every $F \in \mathscr{F}_{\alpha} (X, \mathbb{R})$ is assured by Denker and Urbanski in [@du:91] and Sumi and Urbanski in [@su:09], for an analogous setting.
The pressure function and the equilibrium measure have respective analogues for dynamics under simultaneous actions of the interval maps. We will establish the same later in section . However, for the sake of stating the results, we mention the following properties. Given $f \in \mathscr{F}_{\alpha}(I, \mathbb{R})$, its *pressure* under simultaneous dynamics that we denote by $\mathbb{P}(f)$ coincides with the quantity $\mathfrak{P}(f \circ \mathbf{Proj}_{2})$. Similarly, the measure $\mathfrak{m}_{f}$ on $I$, whose relation with $\mu_{f \circ \mathbf{Proj}_{2}}$, which will be an easy observation once defined, is given by, $$\int\! g\, d \mathfrak{m}_{f}\ \ =\ \ \int\! ( g \circ \mathbf{Proj}_{2})\, d \mu_{f\, \circ\, \mathbf{Proj}_{2}},\ \ \ \forall g \in \mathscr{F}_{\alpha}(I, \mathbb{R}).$$
Periodic points and other dynamical notions {#peri}
===========================================
A point $(w,\, x) \in X$ is said to be a *periodic point* of period $p$ for $T$ iff $\sigma^{p} w\; =\; w$ and $\left( T_{w_{p}} \circ T_{w_{p - 1}} \circ \cdots \circ T_{w_{1}} \right)\, x\; =\; x$. We denote the set of all $p$-periodic points by ${\rm Fix}_{p} (T)$. Once we determine the periodic points of $T$, it becomes easier for us to identify the periodic orbits of simultaneous action of the $N$ many interval maps.
We first introduce a few notations here. For any $x_{1} \in I$, let $\mathscr{R}_{n} (x_{1})$ denote the set of all rays starting from $x_{1}$ that describe the initial $n$-long itinerary of the trajectory of $x_{1}$ in the order that the point visits, [*i.e.*]{}, $$\mathscr{R}_{n} (x_{1})\ \ :=\ \ \Big\{ \left( x_{1},\, x_{2},\, \cdots,\, x_{n} \right) \in I^{n}\ :\ \forall 2 \le k \le n,\ \exists 1 \le d \le N\ \text{such that}\ T_{d} (x_{k - 1}) = x_{k} \Big\}.$$ Further, we denote by $\mathscr{R} (x_{1}) = \mathscr{R}_{\infty} (x_{1})$ the set of all infinite rays starting from $x_{1}$, where we produce each point in $\mathscr{R}_{n} (x_{1})$ to an infinitely long sequence, as allowed by the dynamics, [*i.e.*]{}, $$\mathscr{R} (x_{1})\ \ :=\ \ \Big\{ \left( x_{1},\, x_{2},\, \cdots\, \right) \in I^{\mathbb{Z}_{+}}\ :\ \forall k \ge 2,\ \exists 1 \le d \le N\ \text{for which}\ T_{d} (x_{k - 1}) = x_{k} \Big\}.$$
For any $m \le n < \infty$, we define the following projection operators on the set $\mathscr{R}_{n} (x_{1})$ as follows: $$\begin{array}{c c c c l c r c l}
\Pi_{m} & : & \mathscr{R}_{n} (x_{1}) & \longrightarrow & \mathscr{R}_{m} (x_{1}) & \text{defined by} & \Pi_{m} \left( ( x_{1},\, x_{2},\, \cdots,\, x_{n} ) \right) & = & \left( x_{1},\, x_{2},\, \cdots,\, x_{m} \right); \\
\pi_{m} & : & \mathscr{R}_{n} (x_{1}) & \longrightarrow & I & \text{defined by} & \pi_{m} \left( ( x_{1},\, x_{2},\, \cdots,\, x_{n} ) \right) & = & x_{m}.
\end{array}$$ Allowing a slight abuse of notations, for any $m < \infty$, analogous definitions can be written for the projection operators $\Pi_{m}$ and $\pi_{m}$ defined on $\mathscr{R} (x_{1})$. $$\begin{array}{c c c c l c r c l}
\Pi_{m} & : & \mathscr{R} (x_{1}) & \longrightarrow & \mathscr{R}_{m} (x_{1}) & \text{defined by} & \Pi_{m} \left( ( x_{1},\, x_{2},\, \cdots ) \right) & = & \left( x_{1},\, x_{2},\, \cdots,\, x_{m} \right); \\
\pi_{m} & : & \mathscr{R} (x_{1}) & \longrightarrow & I & \text{defined by} & \pi_{m} \left( ( x_{1},\, x_{2},\, \cdots ) \right) & = & x_{m}.
\end{array}$$
We say $x_{1} \in I$ is a *periodic point* of period $p$ with *periodic orbit* $(x_{1},\, x_{2},\, \cdots,\, x_{p}) \in \mathscr{R}_{p} (x_{1})$ pertaining to the combinatorial data given by some $p$-lettered word $w = (w_{1}\, w_{2}\, \cdots\, w_{p})$ on $N$ letters (the length of $w$ denoted by $|w| = p$), if
1. $\pi_{1} \left( ( x_{1},\, x_{2},\, \cdots ) \right)\ \ =\ \ \pi_{p + 1} \left( ( x_{1},\, x_{2},\, \cdots ) \right)$;
2. $p$ is the least such positive integer for which the first condition is true, [*i.e.*]{},\
$\pi_{1} \left( ( x_{1},\, x_{2},\, \cdots ) \right)\ \ \ne\ \ \pi_{q} \left( ( x_{1},\, x_{2},\, \cdots ) \right)\ \forall q \le p$; and
3. there does not exist any distinct $1 \le q, r \le p$ for which\
$\pi_{q} \left( ( x_{1},\, x_{2},\, \cdots,\, x_{p} ) \right)\ \ =\ \ \pi_{r} \left( ( x_{1},\, x_{2},\, \cdots,\, x_{p} ) \right)$.
We identify such a periodic point $x_{1} \in I$ with period $p$ and periodic orbit $(x_{1},\, x_{2},\, \cdots,\, x_{p})$ by looking for periodic blocks of points in $\mathscr{R} (x_{1})$ that satisfy, $$\Pi_{p} \left( ( x_{1},\, x_{2},\, \cdots ) \right)\ \ =\ \ \Pi_{p} \left( ( x_{mp\, +\, 1},\, x_{mp\, +\, 2},\, \cdots ) \right)\ \ \ \forall m \in \mathbb{Z}_{+}.$$
It is a simple observation that corresponding to any $p$-periodic point $x \in I$, there exists a $p$-lettered word $w = (w_{1}\, w_{2}\, \cdots\, w_{p})$ on $N$ letters such that $$T_{w} x\ \ :=\ \ \big( T_{w_{p}} \circ \cdots \circ T_{w_{1}} \big) x\ \ =\ \ x.$$ For any $n$-lettered word $w = ( w_{1}\, w_{2}\, \cdots\, w_{n} )$, we collect the points satisfying $T_{w} x = x$ in the set ${\rm Fix} (T_{w})$.
For any $T$-invariant probability measure $\mu$ supported on $X$, let $F, G$ be any two complex-valued integrable functions defined on $X$, the appropriate space denoted by $L^{1} (\mu)$. We say the functions $F$ and $G$ are *cohomologous* to each other if there exists a function $H \in L^{1} (\mu)$ such that $F\, -\, G\ =\ \mathscr{Q} (H)\, -\, H$. If $F$ is cohomologous to the constant function $\mathbf{0}$, then $F$ is called a *coboundary*. For any $F \in \mathcal{C} (X, \mathbb{C})$, we denote and define its $n$-th ergodic sum by $$\label{nthergodicsumF}
F^{n}\ \ :=\ \ F + \mathscr{Q} (F) + \mathscr{Q}^{2} (F) + \cdots + \mathscr{Q}^{n - 1} (F).$$ Hence, for any two cohomologous functions $F$ and $G$, it is obvious that their $n$-th ergodic sums evaluated at a periodic point of period $n$ must be the same, [*i.e.*]{}, $$F^{n}((w,\, x))\ \ =\ \ G^{n}((w,\, x))\ \ \forall (w,\, x) \in {\rm Fix}_{n} (T).$$
Let $F,\, G \in \mathscr{F}_{\alpha}(X, \mathbb{R})$ with $F$ not being cohomologous to any constant function. Then, by a result due to Ruelle in [@dr:78], the function $t \longmapsto \mathfrak{P}(F + t G)$, where $t \in \mathbb{R}$, is convex and real analytic. Further, from [@cp:90] and [@pp:90], we have $$\begin{aligned}
\label{firstderivative}
\left. \frac{d}{dt} \Big( \mathfrak{P}(F + t G) \Big) \right|_{t\, =\, 0} & = & \int\! G\, d \mu_{F} \\
\label{secondderivative}
\left. \frac{d^{2}}{d t^{2}} \Big( \mathfrak{P}(F + t G) \Big) \right|_{t\, =\, 0} & = & \lim_{n\, \rightarrow\, \infty} \frac{1}{n} \int\! \left( G^{n} - n \int\! G\, d \mu_{F} \right)^{2}\, d \mu_{F}\ >\ 0.\end{aligned}$$
For simultaneous action of $N$ interval maps at a point $x \in I$, the $n$-th order ergodic sum of any $f \in \mathcal{C} (I, \mathbb{C})$ must be calculated over its appropriate orbit, [*i.e.*]{}, given a $n$-lettered word $w = (w_{1}\, w_{2}\, \cdots\, w_{n})$ on $N$ letters, we define the composition operator $\mathscr{O}_{w}$ by $$\mathscr{O}_{w} (f)\ \ :=\ \ f \circ T_{w},\ \ \ \ \text{where}\ \ T_{w}\ :=\ T_{w_{n}} \circ \cdots \circ T_{w_{1}}.$$ Then, the $n$-th order ergodic sum of the function $f$ with respect to the given $n$-lettered word $w$, or (by a slight abuse of notations) the $n$-th order ergodic sum of the function $f$ with respect to any given infinite-lettered word $w = (w_{1}\, w_{2}\, \cdots\, w_{n}\, \cdots)$ on $N$ letters that agree with our $n$-lettered word on the initial $n$ positions is given by $$\begin{aligned}
f^{n}_{w} (x) & := & \left( f + \mathscr{O}_{(w_{1})} (f) + \cdots + \mathscr{O}_{(w_{1}\, w_{2}\, \cdots\, w_{n - 1})} (f) \right) (x) \\
& = & \left( f + f ( T_{w_{1}} ) + \cdots + f \left( T_{w_{n - 1}} \circ \cdots \circ T_{w_{2}} \circ T_{w_{1}} \right) \right) (x). \end{aligned}$$
For a given $n$-lettered word $w = (w_{1}\, w_{2}\, \cdots\, w_{n})$ on $N$ letters, we say that two Lebesgue integrable functions $f, g \in L^{1} (\lambda)$ defined on $I$ are *$w$-cohomologous* to each other if there exists an integrable function $h \in L^{1} (\lambda)$, also defined on $I$ such that $f\, -\, g\ =\ \mathscr{O}_{w} h\, -\, h$. Hence, for any two $w$-cohomologous functions $f$ and $g$, we observe that the values of the function evaluated at a periodic point $x_{1}$ of period $n$ with periodic orbit $(x_{1},\, x_{2},\, \cdots,\, x_{n})$ pertaining to the combinatorial data given by the $n$-lettered word $w$, necessarily agree; and so do their $n$-th order ergodic sums. Further, if $f$ is $w$-cohomologous to the constant function $\mathbf{0}$, then $f$ is called a *$w$-coboundary*.
Statements of results {#main}
=====================
In this section, we state the main theorems of this paper. The first five results concern the setting of the dynamics of the skew-product map $T$ defined on $X = \Sigma_{N}^{+} \times I$, while the next five results concern the setting of simultaneous dynamics of the concerned interval maps on $I$; thus generalising the situation to maps that evolve with multiple branches.
\[erg1\] The action of $T$ on the product space $X$ is necessarily ergodic with respect to the product measure $\mu$. In other words, the measure of any subset $B \subseteq X$, in the product sigma-algebra of $X$ that satisfies $T^{-1} B = B$, is necessarily $0$ or $1$.
\[ror1\] Consider $F \in \mathscr{F}_{\alpha}(X, \mathbb{R})$ that satisfies the approximability condition, [*i.e.*]{}, there exists distinct points $(w_{1},\, x_{1}) \in {\rm Fix}_{p_{1}} (T),\ (w_{2},\, x_{2}) \in {\rm Fix}_{p_{2}} (T)$ and $(w_{3},\, x_{3}) \in {\rm Fix}_{p_{3}} (T)$ with $p_{i} \neq p_{j}$ for $i \neq j$ for which $$\label{dioF}
\frac{F^{p_{2}}((w_{2},\, x_{2}))\; -\; F^{p_{1}}((w_{1},\, x_{1}))}{F^{p_{3}}((w_{3},\, x_{3}))\; -\; F^{p_{1}}((w_{1},\, x_{1}))}\ \ =:\ \ \mathfrak{d}_{1}$$ is a Diophantine number, [*i.e.*]{}, there exists $l > 2$ and $m > 0$ such that we have $$\label{dioph}
\left| \mathfrak{d}_{1} - \frac{p}{q} \right|\ \ \geq\ \ \frac{m}{q^{l}},\ \ \forall p, q \in \mathbb{Z}_{+}.$$ Further, suppose that there exists a unique real number $\kappa$ such that $$\int\! F d \mu_{\kappa F}\ \ =\ \ 0.$$ Then, for every $n \in \mathbb{Z}_{+},\ a, b \in \mathbb{R}$ with $a < b$, there exists a positive real constant $C_{1} > 0$ such that $$\label{ror1eq}
\# \Big\{ (w,\, x) \in {\rm Fix}_{n} (T) : a \leq F^{n}((w,\, x)) \leq b \Big\}\ \ \sim\ \ C_{1}\ \frac{e^{n \mathfrak{P} (\kappa F)}}{\sqrt{n}}\ \int_{a}^{b}\!\! e^{- \kappa t}\, dt.$$
Here, in equation , by $a_{n} \sim b_{n}$, we mean that $\displaystyle{\lim\limits_{n\, \to\, \infty} \frac{a_{n}}{b_{n}} = 1}$.
\[doc1\] For any $F \in \mathscr{F}_{\alpha} (X, \mathbb{R})$ whose equilibrium measure is denoted by $\mu_{F}$, there exists a constant $\vartheta \in (0, 1)$ such that for all $G,\ H \in \mathscr{F}_{\alpha} (X, \mathbb{R})$, we have $C_{2} > 0$ (depending on $G$ and $H$) that satisfies $$\label{doce1}
\left\vert \int\! \mathscr{Q}^{n} (G) H\, d \mu_{F}\; -\; \int\! G\, d \mu_{F}\, \int\! H\, d \mu_{F} \right\vert\ \ \le\ \ C_{2} \vartheta^{n};\ \ \forall n \geq 1.$$
The preceding theorem defines the exponential decay of correlations of the distributions $\mathscr{Q}^{n} (G)$ and $H$ with respect to the measure $\mu_{F}$ as $n \rightarrow \infty$. The exponential nature of the decay is evident in the statement of the theorem. The next theorem relates the ergodic sum of $G \in \mathscr{F}_{\alpha} (X, \mathbb{R})$ to what is known as the Brownian motion on some richer probability space.
\[asip1\] For any $F \in \mathscr{F}_{\alpha} (X, \mathbb{R})$ whose equilibrium measure is denoted by $\mu_{F}$, consider $G \in \mathscr{F}_{\alpha}(X, \mathbb{R})$ satisfying $$\int G\, d \mu_{F}\ \ =\ \ 0.$$ Suppose the variance of $H \in \mathscr{F}_{\alpha} (X, \mathbb{R})$ is defined as $$\left( \varsigma (H) \right)^{2}\ \ :=\ \ \lim_{n \rightarrow \infty} \frac{1}{n} \int\! \left( H^{n}\; -\; n\, \int\! H\, d \mu_{F} \right)^{2}\, d \mu_{F}.$$ Then, there exists a Hölder continuous function $\Phi \in \mathscr{F}_{\alpha}(X, \mathbb{R})$ cohomologous to $G$, a one-dimensional Brownian motion $\Big\{ \mathfrak{B}(t) \Big\}_{t\, \ge\, 0}$ with variance $t \left( \varsigma (G) \right)^{2}$ and a sequence of random variables $\big\{ \mathfrak{Y}_{n} : \Omega \longrightarrow \mathbb{R} \big\}_{n\, \ge\, 0}$ such that $\big\{ \mathfrak{Y}_{n} \big\}_{n\, \ge\, 0}$ and $\big\{ \Phi^{n} \big\}_{n\, \ge\, 0}$ are equal in distribution and given any $\delta > 0$, $$\mathfrak{Y}_{\lfloor t \rfloor}(\omega)\ \ =\ \ \mathfrak{B} (t) (\omega)\; +\; O(t^{\frac{1}{4}\, +\, \delta}),\ \ \forall t \geq 0,\ \ \mu_{F} \text{-a.e.},$$ provided $G$ is not a coboundary.
The almost sure invariance principle leads to a few important corollaries such as the central limit theorem and the law of iterated logarithms.
\[osp1\] For any $F \in \mathscr{F}_{\alpha} (X, \mathbb{R})$ whose equilibrium measure is denoted by $\mu_{F}$, consider $G \in \mathscr{F}_{\alpha}(X, \mathbb{R})$ satisfying $$\int\! G\, d \mu_{F}\ \ =\ \ 0.$$ Suppose $G$ is not a coboundary. Then,
1. $G$ satisfies the central limit theorem, [*i.e.*]{}, $\frac{1}{\sqrt{n}} G^{n}$ converges in distribution to a normal distribution with mean zero and variance $\left( \varsigma (G) \right)^{2}$ as $n \rightarrow \infty$.
2. $G$ satisfies the law of iterated logarithms, [*i.e.*]{}, $$\limsup_{n\, \rightarrow\, \infty} \frac{G^{n} ((w,\, x))}{\varsigma (G) \sqrt{2n \log \log n}}\ \ =\ \ 1\ \ \mu_{F}\text{-a.e.}$$
In the next five theorems, we state theorems captioned under the same titles, however, by suppressing the first co-ordinate of $X$ and looking at a genuine simultaneous action of the finitely many interval maps under consideration.
\[erg2\] Consider the interval maps $T_{1},\, T_{2},\, \cdots,\, T_{N}$ that act simultaneously on the interval $I$. Let $\lambda$ denote the Lebesgue measure on $\mathbb{R}$. Then, for any real-valued Lebesgue integrable function $f \in L^{1} (\lambda)$, for $\lambda$-a.e. $x \in I$, we have $$\lim_{n\, \to\, \infty}\, \frac{1}{n}\, \frac{1}{N^{n}}\, \sum_{w\; :\; |w|\, =\, n} \Big[ f\, +\, \mathscr{O}_{(w_{1})} (f)\, +\, \cdots\, +\, \mathscr{O}_{(w_{1}\, w_{2}\, \cdots\, w_{n - 1})} (f) \Big] (x)\ \ =\ \ \int_{0}^{1}\! f\, d \lambda.$$
\[ror2\] Consider $f \in \mathscr{F}_{\alpha}(I, \mathbb{R})$ that satisfies the approximability condition, [*i.e.*]{}, there exists distinct periodic points $x,\ y$ and $z$ in $I$ with distinct periods $p_{x},\ p_{y}$ and $p_{z}$, pertaining to the combinatorial data given by $w_{x},\ w_{y}$ and $w_{z}$ such that $$\label{dioph2}
\frac{f^{p_{y}}_{w_{y}} (y)\ -\ f^{p_{x}}_{w_{x}} (x)}{f^{p_{z}}_{w_{z}} (z)\ -\ f^{p_{x}}_{w_{x}} (x)}\ \ =\ \ \mathfrak{d}_{2}$$ is a Diophantine number. Further, suppose there exists unique $\kappa > 0$ such that $$\int\! f d\mathfrak{m}_{\kappa f}\ \ =\ \ 0.$$ Then, for every $n \in \mathbb{Z}_{+},\ a, b \in \mathbb{R}$ with $a < b$, there exists a positive real constant $C_{3} > 0$ such that $$\sum\limits_{w\; :\; |w|\, =\, n} \# \big\{ x \in {\rm Fix} (T_{w})\ :\ a \leq f^{n}_{w}(x) \leq b \big\}\ \ \sim\ \ C_{3} \frac{e^{n \mathbb{P}(\kappa f)}}{\sqrt{n}}\ \int_{a}^{b}\! e^{-\kappa t}\, d t.$$
\[doc2\] Let $\lambda$ denote the Lebesgue measure on $I$. Then, there exist a constant $\vartheta \in (0, 1)$ such that for all $\alpha$-Hölder continuous functions $g,\ h \in \mathscr{F}_{\alpha}(I, \mathbb{R})$, we have $C_{4} > 0$ (depending on $g,\ h$ and some $n$-lettered word $w$) that satisfies $$\label{doce2}
\left\vert \int\! \mathscr{O}_{w} (g) h\, d \lambda\; -\; \int\! g\, d \lambda \, \int\! h\, d \lambda \right\vert\ \ \le\ \ C_{4} \vartheta^{n};\ \ \forall n \geq 1.$$
\[asip2\] Let $\lambda$ denote the Lebesgue measure on $I$. For any $g \in \mathscr{F}_{\alpha} (I, \mathbb{R})$ with $$\int\! g\, d \lambda\ \ =\ \ 0,$$ and $w = (w_{1}\, w_{2}\, \cdots) \in \Sigma_{N}^{+} $, assume that the variance of $g$ with respect to the word $w$ denoted by $\big( \varsigma_{w} (g) \big)^{2}$ and defined by $$\big( \varsigma_{w} (g) \big)^{2}\ \ :=\ \ \lim\limits_{n\, \to\, \infty} \frac{1}{n} \int\! \left( g_{w}^{n} \right)^{2}\, d \lambda\ \ >\ \ 0.$$ Then, there exists a probability space $( \Omega,\; \mathscr{A},\; \nu )$, a sequence of random variables $\big\{ Y_{w}^{n} \big\}_{n\, \ge\, 0}$ and a standard Brownian motion $\big\{ \mathfrak{B}^{*} (t) \big\}_{t\, \ge\, 0}$ such that $g_{w}^{n}$ and $Y_{w}^{n}$ are equal in distribution and given any $\delta > 0$, $$\label{varsigmawng}
Y_{w}^{n} (\omega)\ -\ \mathfrak{B}^{*} \left( \left( \varsigma_{w}^{(n)} (g) \right)^{2} \right) (\omega)\ \ =\ \ O \big( n^{\frac{1}{4}\, +\, \delta} \big),\ \ \ \ \nu\text{-a.e.}\ \ \ \text{where}\ \ \left( \varsigma_{w}^{(n)} (g) \right)^{2}\ \ =\ \ \int\! \left( g_{w}^{n} \right)^{2}\, d \lambda,$$ provided $g_{\Pi_{n} (w)}$ is not a $\Pi_{n} (w)$-coboundary for any $n \ge 1$.
\[osp2\] For a function $g \in \mathscr{F}_{\alpha} (I, \mathbb{R})$ that satisfies $$\int\! g\, d \lambda\ \ =\ \ 0,$$ suppose $w \in \Sigma_{N}^{+} $ is such that the variance $\big( \varsigma_{w} (g) \big)^{2} > 0$ and $g_{\Pi_{n} (w)}$ is not a $\Pi_{n} (w)$-coboundary for any $n \ge 1$. Then,
1. $g$ satisfies the central limit theorem *i.e.,* $\frac{1}{\sqrt{n}} g_{w}^{n} $ converges in distribution to a normal distribution with mean zero and variance $\big( \varsigma_{w} (g) \big)^{2}$ as $n \to \infty$.
2. $g$ satisfies the law of iterated logarithms, [*i.e.*]{}, $$\limsup_{n\, \rightarrow\, \infty} \frac{g_{w}^{n} (x)}{\varsigma_{w} (g) \sqrt{2n \log \log n}}\ \ =\ \ 1\ \ \lambda\text{-a.e.}$$
A book-keeping mechanism, $\Sigma_{N}^{+}$ {#SigmaNplus}
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In this section, we define the space $\Sigma_{N}^{+}$ and discuss certain properties that will be useful in the sequel. Interested readers may refer to [@bk:98], for more details on this space. Recall the definition of $\Sigma _{N}^{+}$ from section , $$\Sigma_{N}^{+}\ \ :=\ \ S^{\mathbb{Z}_{+}}\ \ =\ \ \big\{ 1,\, 2,\, \cdots,\, N \big\}^{\mathbb{Z}_{+}}\ \ =\ \ \Big\{ w = \left( w_{1}\, w_{2}\, \cdots\, w_{n}\, \cdots \right)\ :\ w_{i} \in \left\{ 1,\, 2,\, \cdots,\, N \right\} \Big\}.$$ Observe that one can define the maps $\Pi_{m}$ and $\pi_{m}$ on the symbolic space $S^{n}$ as well as $\Sigma_{N}^{+}$, analogous to its definitions on $I^{n}$ and $I^{\mathbb{Z}_{+}}$. We make use of the same to define a metric between any words $v, w \in \Sigma_{N}^{+}$. Fix any $\theta \in (0, 1)$, and define $$d_{\Sigma_{N}^{+}} (v,\, w)\ \ :=\ \ \theta^{n (v,\, w)},\ \ \text{where}\ \ n (v,\, w)\ :=\ \sup \Big\{ k \in \mathbb{Z}_{+}\ :\ \Pi_{k} (v)\ =\ \Pi_{k} (w) \Big\}.$$ Here, we define $n ( v, v ) := \infty$, thereby $d_{\Sigma_{N}^{+}} ( v, v ) = 0$. Thus, it is clear that we have a family of metrics on the space $\Sigma_{N}^{+}$. The discrete topology that separates any two distinct symbols on the set $\{ 1, 2, \cdots, N \}$ accords a product topology on $\Sigma_{N}^{+}$ with which the above described family of metrics is compatible. We shall fix a value of $\theta$, according to our need in a later section. In this topology, the cylinder sets given by fixing a finite set of co-ordinates, are the sets that are both closed and open. For ease of explanations, we shall always consider cylinder sets whose co-ordinates are fixed from the first co-ordinate onwards, for example, a cylinder set of length $m$ looks like $$\big[ v_{1}\, v_{2}\, \cdots\, v_{m} \big]\ \ =\ \ \Big\{ w \in \Sigma_{N}^{+}\ :\ \Pi_{m} (w)\ =\ (v_{1}\, v_{2}\, \cdots\, v_{m}) \Big\}.$$ These cylinder sets form a basis for the $\sigma$-algebra on $\Sigma_{N}^{+}$ on which one could define a measure for $\Sigma_{N}^{+}$. An easily describable measure on the space, $\Sigma_{N}^{+}$ is the Bernoulli measure defined thus. For any fixed probability vector $p = \big( p_{1},\, p_{2},\, \cdots,\, p_{N} \big)$, the measure is defined as $$\mu \Big( \big[ v_{1}\, v_{2}\, \cdots\, v_{m} \big] \Big)\ \ =\ \ p_{v_{1}}\; p_{v_{2}}\; \cdots\; p_{v_{m}}.$$ Observe that the shift map $\sigma$ defined on $\Sigma_{N}^{+}$ satisfies the properties asked for with respect to the topology defined on $\Sigma_{N}^{+}$. Further, $\Sigma_{N}^{+}$ is a compact metric space with topological dimension $0$.
We now appeal to Tychonoff and accord some structure on $X$. For any two points $(w,\, x)$ and $(v,\, y)$ in $X$, we define a metric $$d_{X} ((w,\, x),\, (v,\, y))\ \ :=\ \ \max \Big\{ d_{\Sigma_{N}^{+}} (w,\, v),\; |x - y| \Big\}.$$ Thus, we work with the appropriate product topology and the product $\sigma$-algebra and the product measure, while we work with $X$.
Various Ruelle operators {#vro}
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Making use of the Ruelle operator, as given in [@pp:90] for every $1 \le d \le N$, we define a Ruelle operator for the skew-product map in this section. Later, we consider each of these Ruelle operators to define a collective Ruelle operator for the case of simultaneous action of all these maps.
For every $1 \le d \le N$, fix $f \in \mathcal{C} (I, \mathbb{C})$, consider $\mathcal{L}_{f}^{(d)} : \mathcal{C} (I, \mathbb{C}) \longrightarrow \mathcal{C} (I, \mathbb{C})$ given by $$\label{ruelled}
\left( \mathcal{L}_{f}^{(d)} g \right) (x)\ \ :=\ \ \sum_{T_{d} y\, =\, x} e^{f(y)} g(y).$$ Observe that this definition entails the following iterative formula given by, $$\label{iterue}
\left( \left( \mathcal{L}_{f}^{(d)} \right)^{\!\circ n} g \right) (x)\ \ :=\ \ \sum_{T_{d}^{n} y\, =\, x} e^{f^{n}_{(d\, d\, \cdots\, d)} (y)} g(y).$$ This is the usual Ruelle operator, as defined in [@pp:90]. For such an operator $\mathcal{L}_{f}^{(d)}$, we have the Ruelle operator theorem, as stated in [@pp:90].
\[rotd\] Suppose $f \in \mathscr{F}_{\alpha} (I, \mathbb{R})$. Then, the Ruelle operator $\mathcal{L}_{f}^{(d)}$ has a simple maximal positive eigenvalue, $\rho^{(d)}$. The remainder of the spectrum lies in a disc of radius strictly smaller than $\rho^{(d)}$. The eigenfunction $\phi_{d}$ corresponding to the maximal eigenvalue is strictly positive. Further, there exists an eigenmeasure corresponding to the maximal eigenvalue, in the space of all $T_{d}$-invariant probability measures supported on $I$, for the dual operator $\left( \mathcal{L}_{f}^{(d)} \right)^{\!*}$.
Appealing to variational principles studied by several authors including Bowen in [@rb:73], Ruelle in [@dr:78] and Parry and Pollicott in [@pp:90], we know that the maximal eigenvalue for the Ruelle operator $\mathcal{L}_{f}^{(d)}$, for $f \in \mathscr{F}_{\alpha} (I, \mathbb{R})$ can also be described using the $d$-pressure function, [*i.e.*]{}, $\rho^{(d)} = e^{\mathcal{P}^{(d)} (f)}$, where $$\mathcal{P}^{(d)} (f)\ \ :=\ \ \sup \left\{ h_{m^{(d)}} (T_{d}) + \int\! f\, d m^{(d)} \right\}.$$ The supremum in the above definition is taken over all $T_{d}$-invariant probability measures supported on $I$ and $h_{m^{(d)}} (T_{d})$ is the measure theoretic entropy of $T_{d}$ with respect to the measure $m^{(d)}$. Then, the existence of the unique equilibrium measure, denoted by $m_{f}^{(d)}$, that realises the supremum in the definition of $d$-pressure is assured by Denker and Urbanskii in [@du:91]. Further, the variational principle states that this unique equilibrium measure, $m_{f}^{(d)}$ is equivalent to the eigenmeasure corresponding to the maximal eigenvalue $\rho^{(d)}$ for the dual operator $\left( \mathcal{L}_{f}^{(d)} \right)^{\!*}$, as given in theorem .
Taking cue from definition , we define the Ruelle operator for the skew-product setting as follows: Fix $F \in \mathcal{C} (X, \mathbb{C})$ and consider $\mathfrak{L}_{F} : \mathcal{C} (X, \mathbb{C}) \longrightarrow \mathcal{C} (X, \mathbb{C})$ given by $$\label{ruelleT}
\left( \mathfrak{L}_{F} G \right) ((w,\, x))\ \ :=\ \ \sum_{T ((v,\, y))\, =\, (w,\, x)} e^{F((v,\, y))} G((v,\, y)).$$ It is a simple observation that the iterates of the Ruelle operator $\mathfrak{L}_{F}$ agrees with the appropriate iterative formula given in . We merely state the same here. $$\left( \left( \mathfrak{L}_{F} \right)^{\!\circ n} G \right) ((w,\, x))\ \ :=\ \ \sum_{T^{n} ((v,\, y))\, =\, (w,\, x)} e^{F^{n} ((v,\, y))} G((v,\, y)).$$ Further, the operator $\mathfrak{L}_{F}$ satisfies the properties mentioned in the Ruelle operator theorem, as mentioned in theorem , whenever $F \in \mathscr{F}_{\alpha} (X, \mathbb{R})$. We include the statement of the theorem, in the context of the skew-product map, for readers’ convenience.
\[rotskew\] Suppose $F \in \mathscr{F}_{\alpha} (X, \mathbb{R})$. Then, the Ruelle operator $\mathfrak{L}_{F}$ has a simple maximal eigenvalue at $\varrho = e^{\mathfrak{P} (F)}$. The remainder of the spectrum lies in a disc of radius strictly smaller than $e^{\mathfrak{P} (F)}$. The eigenfunction $\Phi$ corresponding to the maximal eigenvalue is strictly positive. Further, the eigenmeasure corresponding to the maximal eigenvalue for the dual operator $\left( \mathfrak{L}_{F} \right)^{\!*}$ is equivalent to the equilibrium measure $\mu_{F}$, that realises the supremum in the definition of pressure, as stated in equation .
Fixing $f \in \mathcal{C} (I, \mathbb{C})$, we now define a third Ruelle operator, denoted by $$\mathbb{L}_{f}\ :\ \mathcal{C} (I, \mathbb{C})\ \longrightarrow\ \mathcal{C} (I, \mathbb{C}),$$ that captures the idea of simultaneous action of interval maps. Making sense of the $n$-th iterate of the Ruelle operators $\mathcal{L}_{f}^{(d)}$ and $\mathfrak{L}_{F}$ that captures the set of all $n$-th order pre-images of the point where the operator acts and taking the $n$-th ergodic sum over each of those orbits, we are inclined to define the $n$-th iterate of the Ruelle operator $\mathbb{L}_{f}$ acting on a point $x$ by considering those points that would reach $x$ in $n$ steps, by the action of a combination of $n$ many maps from the collection $\{ T_{1},\, T_{2},\, \cdots,\, T_{N} \}$ and taking the $n$-th ergodic sum dictated by all such $n$-lettered words. The same can be expressed as $$\begin{aligned}
\left( \left( \mathbb{L}_{f} \right)^{\!\circ n} g \right) (x) & = & \sum_{w\, :\, |w|\, =\, n} \sum_{\left( T_{w_{n}} \circ T_{w_{n - 1}} \circ \cdots \circ T_{w_{1}} \right) y\, =\, x} e^{f^{n}_{w} (y)} g(y) \\
& = & \sum_{w_{n}\, =\, 1}^{N} \cdots \sum_{w_{1}\, =\, 1}^{N} \sum_{\left( T_{w_{n}} \circ T_{w_{n - 1}} \circ \cdots \circ T_{w_{1}} \right) y\, =\, x} e^{f (y)\, +\, f (T_{w_{1}} y)\, +\, \cdots\, +\, f ( T_{w_{n - 1}} \circ \cdots \circ T_{w_{1}} y)} g(y). \end{aligned}$$
This understanding paves the way for us to define the Ruelle operator, in this case as $$\label{ruellesimul}
\mathbb{L}_{f} g (x)\ \ :=\ \ \sum_{d\, =\, 1}^{N} \sum_{T_{d} y\, =\, x} e^{f(y)} g(y)\ \ =\ \ \sum_{d\, =\, 1}^{N} \mathcal{L}_{f}^{(d)} g (x).$$
Spectrum of the operators $\mathfrak{L}_{F}$ and $\mathbb{L}_{f}$ {#spectrum}
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In this section, we establish a relationship between the Ruelle operators $\mathfrak{L}_{F}$ and $\mathbb{L}_{f}$, as defined in equations and .
Let $$\label{qoff}
Q\ :\ \mathscr{F}_{\alpha} (I, \mathbb{C}) \longrightarrow \mathscr{F}_{\alpha} (X, \mathbb{C})\ \ \text{be defined as}\ \ \left( Q(f) \right) (w,\, x)\ \ :=\ \ f (x).$$ Then, for any Hölder continuous function $f \in \mathscr{F}_\alpha (I, \mathbb{C})$, $$\begin{aligned}
\big| Q(f) ((w,\, x))\ -\ Q(f)((v,\, y)) \big| & = & \big| f(x) - f(y) \big| \\
& \le & M_{f} \left| x - y \right|^{\alpha} \\
& \le & M_{f}\, \big[ d_{X} ((w,\, x),\ (v,\, y)) \big]^{\alpha}, \end{aligned}$$ for some $M_{f} > 0$ and for any $0 < \theta < 1$, on which the metric on $\Sigma_{N}^{+}$ depends. Thus, the map $Q$ is well defined. Further, the above inequality also proves that the Hölder constant $M_{f}$ remains unperturbed for the function $Q(f)$ in the product space as well, [*i.e.*]{}, $M_{f} \equiv M_{Q(f)}$. Moreover, it is clear from the definition of the various Ruelle operators that $$Q \left( \mathbb{L}_{f} g \right)\ \ =\ \ \mathfrak{L}_{Q(f)} Q(g),\ \ \forall f,\, g \in \mathscr{F}_{\alpha} (I, \mathbb{C}).$$ Thus, the action of $\mathbb{L}_{f}$ is similar to that of the action of $\mathfrak{L}_{Q(f)}$ restricted to the subspace, ${\rm Image} (Q) \subseteq \mathscr{F}_{\alpha} (X, \mathbb{C})$. As a consequence, we can relate the spectrum of $\mathbb{L}_{f}$ and $\mathfrak{L}_{Q(f)}$. The following lemma narrates the same.
\[rotsa\] For some fixed $f \in \mathscr{F}_{\alpha}(I, \mathbb{C})$, let $\Phi \in \mathscr{F}_{\alpha} (X, \mathbb{C})$ be an eigenfunction of $\mathfrak{L}_{Q(f)}$ with corresponding eigenvalue $\varrho$, [*i.e.*]{}, $\mathfrak{L}_{Q(f)} \Phi = \varrho\, \Phi$. Then, $\Phi \in {\rm Image}(Q) \subset \mathscr{F}_{\alpha} (X, \mathbb{C})$.
In order to prove this lemma, it is sufficient to prove that the eigenfunction $\Phi$ is independent of the first co-ordinate, [*i.e.*]{}, $\Phi ((v,\, x)) = \Phi ((w,\, x)),\ \forall v,\, w \in \Sigma_{N}^{+}$ and $x \in [0,\, 1]$. In fact, we prove that given any $\epsilon > 0$, there exists $M_{\epsilon} \in \mathbb{Z}_{+}$ such that $$\Big\vert \left( \mathfrak{L}_{Q(f)} \right)^{\!\circ n} \Phi ((v,\, x))\ -\ \left( \mathfrak{L}_{Q(f)} \right)^{\!\circ n} \Phi ((w,\, x)) \Big\vert\ \ \le\ \ \varrho^{n} \epsilon,\ \ \forall n \ge M_{\epsilon}.$$ A mere application of the eigenfunction equation to the above inequality, then yields $$\varrho^{n} \Big| \Phi ((v,\, x))\ -\ \Phi ((w,\, x)) \Big|\ \ \le\ \ \varrho^{n} \epsilon,\ \ \forall n \ge M_{\epsilon},$$ implying $$\Big| \Phi ((v,\, x))\ -\ \Phi ((w,\, x)) \Big|\ \ \le\ \ \epsilon.$$ The proof is then complete, appealing to the arbitrary choice that we can make for $\epsilon$.
Using the definition of the Ruelle operator, we have $$\begin{aligned}
& & \Big| \left( \mathfrak{L}_{Q(f)} \right)^{\!\circ n} \Phi ((v,\, x)) - \left( \mathfrak{L}_{Q(f)} \right)^{\!\circ n} \Phi ((w,\, x)) \Big| \\
& = & \left| \sum_{u\; :\; |u|\, =\, n} \left( \sum_{T^{n} ((u v,\, y))\, =\, (v,\, x)} e^{(Q(f))^{n} (u v,\, y)} \Phi ((u v,\, y)) \right. \right. \\
& & \hspace{+2in} \left. \left. -\ \sum_{T^{n} ((u w,\, y))\, =\, (w,\, x)} e^{(Q(f))^{n} (u w,\, y)} \Phi ((u w,\, y)) \right) \right| \\
& \le & \sum_{u\; :\; |u|\, =\, n} \sum_{T_{u_{n}} \circ \cdots \circ T_{u_{1}} y\, =\, x} \Big| e^{f_{u}^{n} (y)} \Big| \Big| \Phi ((u v,\, y)) - \Phi ((u w,\, y)) \Big| \\
& \leq & e^{n \| f \|_{\infty}}\, M_{\Phi}\, N^{n}\, \theta^{n \alpha} \\
& = & \varrho^{n}\, M_{\Phi}\, \left( \frac{e^{\| f \|_{\infty}}\, N\, \theta^{\alpha}}{\varrho} \right)^{n}. \end{aligned}$$
Once we choose the functions $f$ and $\Phi$, observe that the quantities $\| f \|_{\infty}$ and $M_{\Phi}$ are determined. Thus, we can only rely on our choice of $\theta$ to make the above estimate, as small as necessary. We therefore fix $\theta$ in such fashion that $$\frac{e^{\| f \|_{\infty}}\, N\, \theta^{\alpha}}{\varrho}\ \ <\ \ 1.$$ Further, we remark that we shall use the same $\theta$, so fixed to suit our purpose in the above inequality, in the remainder of this paper. Now, it is clear that we have some threshold, say $M_{\epsilon}$ such that $$\Big| \left( \mathfrak{L}_{Q(f)} \right)^{\!\circ n} \Phi ((v,\, x))\ -\ \left( \mathfrak{L}_{Q(f)} \right)^{\!\circ n} \Phi ((w,\, x)) \Big|\ \ \le\ \ \varrho^{n} \epsilon,\ \ \forall n \ge M_{\epsilon}.$$
A careful reader may observe that lemma is merely the statement of the Ruelle operator theorem stated for the simultaneous dynamics of finitely many interval maps. In particular, $\Phi \in {\rm Image}(Q)$ implies that there exists $\phi \in \mathscr{F}_{\alpha} (I, \mathbb{C})$ such that $Q(\phi) = \Phi$ and $\mathbb{L}_{f} \phi = \varrho \phi$. Thus, the set of eigenvalues of $\mathfrak{L}_{Q(f)}$ and $\mathbb{L}_{f}$ remains equal. In particular, the simple maximal eigenvalue $\varrho$ of $\mathfrak{L}_{Q(f)}$ is also the simple maximal eigenvalue of $\mathbb{L}_{f}$.
Normalising the operators $\mathcal{L}_{f}^{(d)},\ \mathfrak{L}_{F}$ and $\mathbb{L}_{f}$ {#Normalising}
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For technical convenience, we normalise the Ruelle operators $\mathcal{L}_{f}^{(d)},\ \mathfrak{L}_{Q(f)}$ and $\mathbb{L}_{f}$, in different ways that would suit our purposes to prove the main theorems.
For any $f \in \mathscr{F}_{\alpha} (I, \mathbb{R})$, we define the normalised Ruelle operator by considering the Ruelle operator, as defined in , corresponding to the function $$\widetilde{f}_{d}\ \ :=\ \ f\; +\; \log \phi_{d}\; -\; \log \phi_{d} \circ T_{d}\; -\; \mathcal{P}^{(d)} ( f ),$$ where $\phi_{d}$ is the eigenfunction corresponding to the maximal eigenvalue of the operator $\mathcal{L}_{f}^{(d)}$. Thus, $$\widetilde{\mathcal{L}}_{f}^{(d)}\, \mathbf{1} (x)\ \ :=\ \ \mathcal{L}_{\widetilde{f}_{d}}^{(d)}\, \mathbf{1} (x)\ \ =\ \ \mathbf{1} (x).$$
For any $f \in \mathscr{F}_{\alpha} (I, \mathbb{R})$, we define the normalised Ruelle operator in the skew-product setting by considering the Ruelle operator, as defined in , corresponding to the function $$\widetilde{Q (f)}\ \ :=\ \ Q(f)\; +\; \log \Phi\; -\; \log \Phi \circ T\, ,$$ where $\Phi$ is the eigenfunction corresponding to the maximal eigenvalue $\varrho = e^{\mathfrak{P} ( Q(f))}$ of the operator $\mathfrak{L}_{Q(f)}$. Thus, this normalisation effects the eigenfunction of the normalised operator to be equal to the constant function $\mathbf{1}$, however, leaves the the maximal eigenvalue $\varrho$ unchanged, [*i.e.*]{}, $$\widetilde{\mathfrak{L}}_{Q(f)}\, \mathbf{1} ((w,\, x))\ \ :=\ \ \mathfrak{L}_{\widetilde{Q(f)}}\, \mathbf{1} ((w,\, x))\ \ =\ \ \varrho\, \mathbf{1} ((w,\, x)).$$ In particular, the operator $\varrho^{-1} \widetilde{\mathfrak{L}}_{Q(f)}$ has $1$ as its maximal eigenvalue with corresponding eigenfunction $\mathbf{1}$. Further, since by definition, $Q(f)$ and $\widetilde{Q(f)}$ are cohomologous to each other, their ergodic sums are preserved.
Further, once we normalise the operators $\mathcal{L}_{f}^{(d)}$ and $\mathfrak{L}_{Q(f)}$ as prescribed, we also observe that the equilibrium measure with respect to the Hölder continuous functions $f \in \mathscr{F}_{\alpha} (I, \mathbb{R})$ for the map $T_{d}$ and $Q(f) \in \mathscr{F}_{\alpha} (X, \mathbb{R})$ for the skew-product map $T$, namely $m_{f}^{(d)}$ and $\mu_{Q(f)}$ are nothing but the eigenmeasures corresponding to the maximal eigenvalues of the normalised operators $\widetilde{\mathcal{L}}_{f}^{(d)}$ and $\varrho^{-1} \widetilde{\mathfrak{L}}_{Q(f)}$ respectively. Thus, we use the notations $m_{f}^{(d)}$ and $\mu_{Q(f)}$ for the equilibrium measure, as well as the eigenmeasure corresponding to the maximal eigenvalue of the normalised operators $\widetilde{\mathcal{L}}_{f}^{(d)}$ as well as $\varrho^{-1} \widetilde{\mathfrak{L}}_{Q(f)}$, respectively.
Moreover, we define the normalised operator in the setting of simultaneous dynamics thus: For any $f \in \mathscr{F}_{\alpha} (I, \mathbb{R})$, let $$\widetilde{\mathbb{L}}_{f} g (x)\ \ :=\ \ \frac{1}{\phi (x)}\, \mathbb{L}_{f + \log \phi} g (x)\ \ =\ \ \frac{1}{\phi (x)}\, \sum_{d\, =\, 1}^{N} \sum_{T_{d} y\, =\, x} e^{f(y)\, +\, \log \phi (y)} g(y),$$ where $\phi$ is the strictly positive eigenfunction corresponding to the maximal eigenvalue of the operator $\mathbb{L}_{f}$, that we denote by $e^{\mathbb{P} (f)}$, where $\mathbb{P} (f)$ is the *pressure* of the function $f$, in this setting. It must be noted that this normalisation does not change the maximal eigenvalue, but has an effect on the corresponding eigenfunction, making it to be $\mathbf{1}$. Further, this normalisation also takes a toll in this setting; $\widetilde{\mathbb{L}}_{f}$ is no longer a Ruelle operator, but merely a bounded linear operator.
The operator $Q : \mathcal{C} (I, \mathbb{C}) \longrightarrow \mathcal{C} (X, \mathbb{C})$, as defined in , has a natural transpose $$Q^{*}\ :\ \mathcal{C}^{*} (X, \mathbb{C}) \longrightarrow \mathcal{C}^{*} (I, \mathbb{C}).$$ On a restricted space, this transpose gives us the map, $$Q^{*}\ :\ \big\{ \mu\ :\ \mu\ \text{is a probability measure on}\ X \big\} \longrightarrow \big\{ \mathfrak{m}\ :\ \mathfrak{m}\ \text{is a probability measure on}\ I \big\},$$ defined by $$\int\! f\, d \left( Q^{*} \mu \right)\ \ :=\ \ \int\! Q(f)\, d \mu,\ \ \forall f \in \mathcal{C} (I, \mathbb{C}).$$ For $f \in \mathscr{F}_{\alpha}(X, \mathbb{R})$, we define $\mathfrak{m}_{f} := Q^{*}(\mu_{Q(f)})$, thereby, $$\int_{I}\! g\, d \mathfrak{m}_{f}\ \ =\ \ \int_{X}\! Q(g)\, d \mu_{Q(f)},\ \ \forall g \in \mathscr{F}_{\alpha} (I, \mathbb{R}).$$ Thus, it is an easy observation that as a consequence of the definitions of $\mathfrak{m}_{f}$ and $\widetilde{\mathbb{L}}_{f},\ \mathfrak{m}_{f}$ coincides with the eigenmeasure corresponding to maximal eigenvalue of the operator $e^{-\mathbb{P}(f)} \left( \widetilde{\mathbb{L}}_{f} \right)^{*}$.
In the sequel, we will be interested in a complex perturbation of the operators $\widetilde{\mathfrak{L}}_{Q(f)}$ and $\widetilde{\mathbb{L}}_{f}$ for $f \in \mathscr{F}_{\alpha} (I, \mathbb{R})$, [*i.e.*]{}, for $\zeta = \kappa + i \xi \in \mathbb{C}$, we consider the operators $\mathfrak{L}_{\kappa Q(f)}$ and $\mathbb{L}_{\kappa f}$, normalise the same as explained above respectively and then perturb the operator. We denote them respectively by $\widetilde{\mathfrak{L}}_{\zeta Q(f)}$ and $\widetilde{\mathbb{L}}_{\zeta f}$. We conclude this section with a few lemmas that provide a bound on the operator norm for the iterates of the normalised, yet perturbed operators.
For any $F \in \mathscr{F}_{\alpha} (X, \mathbb{R})$, there exists a positive constant $C_{5} > 0$, such that for every $n \ge 0$, and any $G \in \mathscr{F}_{\alpha} (X, \mathbb{C})$, we have $$\bigg\Vert \left( \varrho^{-1} \widetilde{\mathfrak{L}}_{\zeta F} \right)^{\!\circ n}\, G \bigg\Vert_{\alpha}\ \ \ge\ \ C_{5}\, | \xi |\, \big\Vert G \big\Vert_{\infty}\; +\; \alpha^{n}\, \big\vert G \big\vert_{\alpha}.$$
\[zeropointtwo\] Let $F \in \mathscr{F}_{\alpha} (X, \mathbb{R})$ satisfy the approximability condition, as mentioned in equations and of theorem . Then, there exists positive constants $C_{6},\ C_{7}$ and $C_{8}$ such that $$\bigg\Vert \left( \varrho^{-1} \widetilde{\mathfrak{L}}_{\zeta F} \right)^{\! \circ (2 n R)} \bigg\Vert\ \ \le\ \ C_{6}\, | \xi |\, \left( 1 - \frac{1}{| \xi |^{C_{7}}} \right)^{n - 1},\ \ \ \forall n \ge 1,$$ where $\left| \xi \right|$ is sufficiently large and $R$ is the greatest integer contained in $C_{8} \log \left| \xi \right|$.
We conclude this section with a lemma that provides a bound on the operator norm for the iterates of the normalised, but perturbed operator $\widetilde{\mathbb{L}}_{\zeta f}$, for some $f \in \mathscr{F}_{\alpha} (I, \mathbb{R})$. We also include a short proof of the same, for readers’ convenience.
\[eightpointthree\] Let $f \in \mathscr{F}_{\alpha}(I, \mathbb{R})$ satisfy the approximability condition, as mentioned in equation of theorem . Let $\zeta = \kappa + i \xi \in \mathbb{C}$. Then, there exists positive constants $C_{9},\ C_{10}$ and $C_{11}$ such that $$\bigg\Vert \left( e^{-\mathbb{P}(\kappa f)} \widetilde{\mathbb{L}}_{\zeta f} \right)^{\!\circ (2 n R)} \bigg\Vert\ \ \le\ \ C_{9}\ \left\vert \xi \right\vert\ \Bigg( 1 - \frac{1}{\left\vert \xi \right\vert^{C_{10}}} \Bigg)^{n - 1},\ \ \ \forall n \ge 1,$$ where $\left\vert \xi \right\vert$ is sufficiently large and $R$ is the greatest integer contained in $C_{11} \log \left\vert \xi \right\vert$.
Given $f \in \mathscr{F}_{\alpha} (I, \mathbb{R})$ that satisfies the approximability condition, it is an easy observation that $Q(f) \in \mathscr{F}_{\alpha} (X, \mathbb{R})$ satisfies the appropriate approximability condition. Thus, lemma comes to our rescue. Further, from the definitions of the operators as stated in and , we obtain that $$\label{landl}
\bigg\Vert \left( \widetilde{\mathbb{L}}_{\zeta f} \right)^{\!\circ n} \bigg\Vert\ \ \le\ \ \bigg\Vert \left( \widetilde{\mathfrak{L}}_{\zeta Q(f)} \right)^{\!\circ n} \bigg\Vert\ \ \ \ \forall n \ge 1.$$
Proof of the ergodicity theorems {#ergproof}
================================
In this section, we prove the ergodicity theorems as stated in for the skew-product setting and for the simultaneous dynamics setting.
Let $B \subseteq X$ be a completely $T$-invariant set in the $\sigma$-algebra of $X$ with strictly positive measure, [*i.e.*]{}, $\mu (B) > 0$. As said earlier in section , a natural candidate for $\mu$ is the product measure of the Bernoulli measure on cylinder sets of $\Sigma_{N}^{+}$ and the Lebesgue measure on open intervals of $I$. Since we have assumed that the set $B$ is completely $T$-invariant, [*i.e.*]{}, $T^{-1} B\, =\, B$ with strictly positive measure, it is sufficient for us to show that $\mu (B) = 1$, to prove the theorem.
It is easy to observe that, in its most general form, the set $B$ can be expressed as $$B\ \ =\ \ \bigcup_{j\, \ge\, 1} \big( U_{j} \times V_{j} \big),$$ where $U_{j}$’s are cylinder sets in $\Sigma_{N}^{+}$ and $V_{j}$’s are open subsets of $I$. Let $U \times V \subseteq B$ be a product set in this collection where $U = \big[ v_{1}\, v_{2}\, \cdots\, v_{n} \big]$ and $V \subseteq I$ is an open set. Since $B$ is completely $T$-invariant, we have $U \times V \subseteq T^{-1} B$. Thus, there exists $B'$ in the $\sigma$-algebra of $X$ such that $B' \subseteq B$ and $U \times V \subseteq T^{-1} (B')$. For the smallest such subset $B'$, the countably many possibilities for the form of $B'$ are given by
1. $\big[ v_{2}\, v_{3}\, \cdots\, v_{n} \big] \times V'$;
2. $\bigcup\limits_{d\, =\, 1}^{N} \Big( \big[ v_{2}\, v_{3}\, \cdots\, v_{n}\, d \big] \times V_{d}' \Big)$;
3. $\bigcup\limits_{d_{1}\, =\, 1}^{N} \bigcup\limits_{d_{2}\, =\, 1}^{N} \Big( \big[ v_{2}\, v_{3}\, \cdots\, v_{n}\, d_{1}\, d_{2} \big] \times V_{(d_{1}\, d_{2})}' \Big)$;
4. $\cdots$.
It is clear that in the above enumeration of possibilities, the cylinder sets in all cases starting from (1) onwards are subsumed by the cylinder set in case (0). Suppose we also prove that the appropriate open subsets of $I$ in each of the possibilities starting from case (1) are subsumed by the open subset of $I$ in case (0), it is sufficient to merely work with case (0). For the same purpose, we consider the general case (m), as listed in the above possibilities, namely,
1. $\bigcup\limits_{d_{1}\, =\, 1}^{N} \bigcup\limits_{d_{2}\, =\, 1}^{N} \cdots \bigcup\limits_{d_{m}\, =\, 1}^{N}\Big( \big[ v_{2}\, v_{3}\, \cdots\, v_{n}\, d_{1}\, d_{2}\, \cdots\, d_{m} \big] \times V_{(d_{1}\, d_{2}\, \cdots\, d_{m})}' \Big)$.
Observe that $$\begin{aligned}
\label{UtimesV}
U \times V & \subseteq & T^{-1} B' \nonumber \\
& = & T^{-1} \Bigg( \bigcup\limits_{d_{1}\, =\, 1}^{N} \bigcup\limits_{d_{2}\, =\, 1}^{N} \cdots \bigcup\limits_{d_{m}\, =\, 1}^{N}\Big( \big[ v_{2}\, v_{3}\, \cdots\, v_{n}\, d_{1}\, d_{2}\, \cdots\, d_{m} \big] \times V_{(d_{1}\, d_{2}\, \cdots\, d_{m})}' \Big) \Bigg) \nonumber \\
& = & \bigcup\limits_{d_{1}\, =\, 1}^{N} \bigcup\limits_{d_{2}\, =\, 1}^{N} \cdots \bigcup\limits_{d_{m}\, =\, 1}^{N}\Bigg( T^{-1} \Big( \big[ v_{2}\, v_{3}\, \cdots\, v_{n}\, d_{1}\, d_{2}\, \cdots\, d_{m} \big] \times V_{(d_{1}\, d_{2}\, \cdots\, d_{m})}' \Big) \Bigg) \nonumber \\
& = & \bigcup\limits_{d_{1}\, =\, 1}^{N} \bigcup\limits_{d_{2}\, =\, 1}^{N} \cdots \bigcup\limits_{d_{m}\, =\, 1}^{N} \Bigg( \bigcup_{d\, =\, 1}^{N} \Big( \big[ d\, v_{2}\, v_{3}\, \cdots\, v_{n}\, d_{1}\, d_{2}\, \cdots\, d_{m} \big] \times T_{d}^{-1} V_{(d_{1}\, d_{2}\, \cdots\, d_{m})}' \Big) \Bigg). \nonumber \\
& & \end{aligned}$$
However, since $U = \big[ v_{1}\, v_{2}\, \cdots\, v_{n} \big]$, the only possibility in the right hand side of the countable union in equation where $U \times V$ can be a subset reduces to $$U \times V\ \ \subseteq\ \ \bigcup\limits_{d_{1}\, =\, 1}^{N} \bigcup\limits_{d_{2}\, =\, 1}^{N} \cdots \bigcup\limits_{d_{m}\, =\, 1}^{N} \Big( \big[ v_{1}\, v_{2}\, v_{3}\, \cdots\, v_{n}\, d_{1}\, d_{2}\, \cdots\, d_{m} \big] \times T_{v_{1}}^{-1} V_{(d_{1}\, d_{2}\, \cdots\, d_{m})}' \Big).$$
The above union is over sets that are disjoint with respect to their first component and thus $V \subseteq T_{v_{1}}^{-1} V_{(d_{1}\, d_{2}\, \cdots\, d_{m})}'$, for any $m$-lettered word $(d_{1}\, d_{2}\, \cdots\, d_{m})$. Now using minimality of the set $B'$, we observe that the set $V_{(d_{1}\, d_{2}\, \cdots\, d_{m})}'$ remains independent of the $m$-lettered word $(d_{1}\, d_{2}\, \cdots\, d_{m})$. Hence, $B' = \big[ v_{2}\, v_{3}\, \cdots\, v_{n} \big] \times V'$, as mentioned in case (0).
Based on our observation that for every $U \times V = U_{0} \times V_{0} \subseteq B$, there exists a $B_{1} = U_{1} \times V_{1} \subseteq B$ that satisfies $U_{0} \times V_{0} \subseteq T^{-1} \left( U_{1} \times V_{1} \right)$, we obtain a sequence of sets $\left( U_{k} \times V_{k} \right)$ that satisfy, $\left( U_{k - 1} \times V_{k - 1} \right) \subseteq T^{-1} \left( U_{k} \times V_{k} \right)$. However, this process must end in a finite number of steps, precisely $n$, since $U_{0}$ is a cylinder set that fixes only $n$ many positions. Thus, after those finitely many steps, we obtain $\Sigma_{N}^{+} \times V_{n} \subseteq B$ for some open subset $V_{n} \subseteq I$ with strictly positive Lebesgue measure, [*i.e.*]{}, $\lambda (V_{n}) > 0$.
Choose the maximal subset $V \subseteq I$ such that $\Sigma_{N}^{+} \times V \subseteq B$, [*i.e.*]{}, suppose there exists a subset $V' \subseteq I$ such that $\Sigma_{N}^{+} \times V' \subseteq B$ then, $V' \subseteq V$. For such a maximal subset $V \subseteq I$, consider $\big[ w \big] \times V$ for some choice of $w \in \big\{ 1,\, 2,\, \cdots,\, N \big\}$. It is obvious that $\big[ w \big] \times V \subseteq \Sigma_{N}^{+} \times V$. Then, there exists a $V' \subseteq I$ such that $$\label{wtimesV}
\big[ w \big] \times V\ \ \subseteq\ \ T^{-1} \big( \Sigma_{N}^{+} \times V' \big)\ \ =\ \ \bigcup_{d\, =\, 1}^{N} \Big( \big[ d \big] \times T_{d}^{-1} V' \Big).$$ Arguing as earlier, we reduce the countable union in the right hand side of equation to $$\big[ w \big] \times V\ \ \subseteq\ \ \big[ w \big] \times T_{w}^{-1} V'.$$ However, owing to the maximality of $V$, we have $V' \subseteq V$ that implies $T_{w}^{-1} V' \subseteq T_{w}^{-1} V$. Thus, $$\big[ w \big] \times V\ \ \subseteq\ \ \big[ w \big] \times T_{w}^{-1} V'\ \ \subseteq \big[ w \big] \times T_{w}^{-1} V.$$ This implies $V \subseteq T_{w}^{-1} V$ for any choice of $w \in \big\{ 1,\, 2,\, \cdots,\, N \big\}$. However, each of these interval maps preserves the Lebesgue measure, [*i.e.*]{}, $\lambda (V) = \lambda (T_{w}^{-1} V)$ for all $w \in \big\{ 1,\, 2,\, \cdots,\, N \big\}$. Thus, by eliminating an appropriate set of Lebesgue measure zero from $V$, we obtain $\widetilde{V}$ that satisfies $\widetilde{V} = T_{w}^{-1} \widetilde{V}$ for all $w \in \big\{ 1,\, 2,\, \cdots,\, N \big\}$. Finally, we appeal to the ergodicity of each of these interval maps $T_{w}$ to conclude that the completely $T_{w}$-invariant set $\widetilde{V}$ for all $w$ of strictly positive measure must be of Lebesgue measure $1$.
Thus, the completely $T$-invariant set $B$ that satisfies $\mu (B) > 0$ has measure $1$, thereby completing the proof of theorem .
\[erg1rem\] The above proof, in fact provides a stronger result than is stated in theorem , namely, for any set $B$ in the $\sigma$-algebra of $X$ that is completely $T$-invariant, meaning $T^{-1} B = B$, we have either $\mu (B) = 0$ or $B$ can be expressed as $\Sigma_{N}^{+} \times \widetilde{V}$ where $\widetilde{V}$ is a set of full Lebesgue measure in $I$, [*i.e.*]{}, $\lambda \left( \widetilde{V} \right) = 1$.
We now prove the ergodic theorem for simultaneous action of finitely many interval maps, as stated in theorem .
For a Lebesgue integrable real-valued function $f$ defined on $I$, define $F := Q(f) \in L^{1} (\mu)$, where $\mu$ is the product measure of the Bernoulli measure on the cylinder sets of $\Sigma_{N}^{+}$ and the Lebesgue measure on the open intervals of $I$. Define $$E\ :=\ \Bigg\{ (w,\, x) \in X\; :\; \liminf_{n\, \rightarrow\, \infty} \frac{1}{n} \sum\limits_{j\, =\, 0}^{n - 1} \mathscr{Q}^{j} (F) (w,\, x)\ =\ \limsup_{n\, \rightarrow\, \infty} \frac{1}{n} \sum\limits_{j\, =\, 0}^{n - 1} \mathscr{Q}^{j} (F) (w,\, x)\ =\ \int\! F\, d \mu \Bigg\}.$$ One can easily observe that $E$ is a $T$-invariant subset of $X$. Thus, $\mu (E)$ is either zero or one, by theorem . The set of points collected in $E$ is the set of all points in $X$ that satisfies the Birkhoff’s pointwise ergodic theorem to the dynamical system $T$ acting on $X$. Hence, $\mu (E) = 1$. Further, from remark , we have that $E = \Sigma_{N}^{+} \times E'$, where $\lambda(E') = 1$. Hence, we now have for $\lambda$-almost every $x \in I$ and for every $w \in \Sigma_{N}^{+}$, $$\lim_{n\, \rightarrow\, \infty} \frac{1}{n} \sum\limits_{j\, =\, 0}^{n - 1} \mathscr{Q}^{j} (F) (w,\, x)\ \ =\ \ \int\! F\, d \mu.$$
\[thereshold\] For a fixed $x_{0} \in E'$, given $\epsilon > 0$, there exists $M_{\epsilon} \in \mathbb{N}$, independent of $w$, such that for all $n \geq M_{\epsilon}$, we have $$\left| \lim_{n\, \rightarrow\, \infty} \frac{1}{n} \sum\limits_{j\, =\, 0}^{n - 1} \mathscr{Q}^{j} (F) (w,\, x_{0})\ -\ \int\! F\, d \mu \right|\ \ <\ \ \epsilon,\ \ \ \forall w \in \Sigma_{N}^{+}.$$
We initially prove the theorem, assuming claim to be true. We shall prove the claim immediately thereafter. From the definition of the composition operators, we have for any arbitrary $\epsilon > 0$, there exists $M_{\epsilon} \in \mathbb{N}$ such that for every $n \geq M_{\epsilon}$, $$\left| \frac{1}{n} f_{w}^{n} (x)\ -\ \int_{0}^{1}\! f\, d \lambda \right|\ \ \le\ \ \epsilon,\ \ \text{for}\ \lambda\text{-a.e.}\ x \in I\ \text{and}\ \forall w = (w_{1}\, w_{2}\, \cdots\, w_{n}).$$ This implies $$\left| \sum\limits_{w\ :\ |w|\, =\, n} \left( \frac{1}{n} f_{w}^{n} (x)\ -\ \int_{0}^{1}\! f\, d \lambda \right) \right|\ \ \le\ \ N^{n} \epsilon.$$ Thus, $$\left| \frac{1}{n} \frac{1}{N^{n}} \sum\limits_{w\ :\ |w|\, =\, n} f_{w}^{n} (x)\ -\ \int_{0}^{1}\! f\, d \lambda \right|\ \ \le\ \ \epsilon,\ \ \text{for}\ \ \lambda\text{-a.e.}\ x \in I,$$ proving the theorem.
We now complete this section, by proving the claim in .
For a fixed $x_{0} \in E'$, consider the sequence $\displaystyle{\left\{ \frac{1}{n} \sum\limits_{j\, =\, 0}^{n - 1} \mathscr{Q}^{j} (F) (w,\, x_{0}) \right\}_{n\, \ge\, 1}}$ of functions defined on the compact space $\Sigma_{N}^{+}$, converging to the constant $\int\! F\, d \mu$. This convergence is uniform over $w \in \Sigma_{N}^{+}$, [*i.e.*]{}, given any $\epsilon > 0$, there exists $M_{1} = M_{1} (\epsilon)$ such that for all $n > M_{1}$, we have $$\label{kirone}
\left| \frac{1}{n} \sum_{j\, =\, 0}^{n - 1} \mathscr{Q}^{j} (F) (w,\, x_{0})\ -\ \frac{1}{n} \sum_{j\, =\, 0}^{n - 1} \mathscr{Q}^{j} (F) (v,\, x_{0}) \right|\ \ \le\ \ \frac{\epsilon}{2},\ \ \forall w, v \in \Sigma_{N}^{+}.$$ We already know that for some fixed $v \in \Sigma_{N}^{+}$, there exists $M_{2} = M_{2}(\epsilon, v)$ such that for all $n > M_{2}$, we have $$\label{kirtwo}
\left| \frac{1}{n} \sum\limits_{j\, =\, 0}^{n - 1} \mathscr{Q}^{j} (F) (v,\, x_{0})\ -\ \int\! F\, d \mu \right|\ \ \le\ \ \frac{\epsilon}{2}.$$ Taking $M_{3} := \max\{ M_{1},\, M_{2} \}$ where $M_{1}$ and $M_{2}$ are the quantities prescribed by equations and , we get for all $n > M_{3}$ and $w \in \Sigma_{N}^{+}$, $$\begin{aligned}
& & \left| \frac{1}{n} \sum_{j\, =\, 0}^{n - 1} \mathscr{Q}^{j} (F) (w,\, x_{0})\ -\ \int\! F\, d \mu \right| \\
& \le & \left| \frac{1}{n} \sum_{j\, =\, 0}^{n - 1} \mathscr{Q}^{j} (F) (w,\, x_{0})\ -\ \frac{1}{n} \sum_{j\, =\, 0}^{n - 1} \mathscr{Q}^{j} (F) (v,\, x_{0}) \right|\ +\ \left| \frac{1}{n} \sum_{j\, =\, 0}^{n - 1} \mathscr{Q}^{j} (F) (v,\, x_{0})\ -\ \int\! F\, d \mu \right| \\
& \le & \epsilon.\end{aligned}$$ It is clear from the definition that $M_{3}$ is independent of the word $w$ and only dependent on $\epsilon$ and $x_{0}$.
Rates of recurrence {#rorproof}
===================
In this section, we write the proofs of theorems and . Throughout this section, we work with a fixed $F \in \mathscr{F}_{\alpha} (X, \mathbb{R})$, along with the corresponding normalised operator $\varrho^{-1} \widetilde{\mathfrak{L}}_{F}$ with equilibrium measure $\mu_{F}$. We know that the iterates of this operator obey the result given in lemma . Further, for some $\zeta = \kappa + i \xi \in \mathbb{C}$, we note that the pressure function associated with the operator $\mathfrak{L}_{\zeta F}$ and its normalised version $\widetilde{\mathfrak{L}}_{\zeta F}$ are one and the same, owing to the definition of pressure and the method of normalisation.
The following lemma gives an approximation for the eigenvalue of the normalised yet perturbed operator, $\widetilde{\mathfrak{L}}_{\zeta F}$. We urge the reader to observe that the statement of the lemma and hence, its proof, are merely mentioned for a change of variables along the imaginary variable, even though more is true, as one may obtain from, say [@ps:94].
\[morse\] For $\zeta = \kappa + i \xi$, there exists a change of variables $\Upsilon = \Upsilon(\xi)$ such that for $| \xi | < \delta$, we can expand $$e^{\mathfrak{P} \left( \zeta F \right)}\ \ =\ \ e^{\mathfrak{P} \left( \kappa F \right)}\; \Big( 1\; -\; \Upsilon^{2}\; +\; i\, \Theta \left( \Upsilon \right) \Big),$$ where $\Theta$ is a real-valued function that satisfies $\Theta (\Upsilon) = O ( | \Upsilon |^{3} )$.
By perturbation theory, we know that there exists $\delta > 0$ such that, for $G \in \mathscr{F}_{\alpha}(X, \mathbb{C})$ satisfying $\| G - \kappa F \|_{\alpha} < \delta$, $$\label{analytic}
G\ \ \longmapsto\ \ \mathfrak{P}(G)\ \ \ \ \text{is an analytic map}.$$
The analyticity of the above map from the Banach space $\mathscr{F}_{\alpha} (X, \mathbb{C})$ to $\mathbb{C}$, in a neighbourhood of $\kappa F$ assures the existence of a linear map, say $\mathfrak{D} : \mathscr{F}_{\alpha} (X, \mathbb{C}) \longrightarrow \mathbb{C}$ such that $$\lim_{G\, \rightarrow\, \kappa F} \frac{\mathfrak{P} \big( G \big)\ -\ \mathfrak{P} \big( \kappa F \big)\ -\ \mathfrak{D} \big( G - \kappa F \big)}{\| G\ -\ \kappa F \|_{\alpha}}\ \ =\ \ 0,$$ where $\mathfrak{D}$ is the differential of $\mathfrak{P}$ at $\kappa F$. From equation , we have for $\xi \in \mathbb{R}$, $$\lim_{\xi \rightarrow 0} \frac{\mathfrak{P} \big( ( \kappa + \xi) F \big)\ -\ \mathfrak{P} \big( \kappa F \big)\ -\ \mathfrak{D} \big( \xi F \big)}{\| \xi F \|_{\alpha}}\ \ =\ \ 0,$$ for the choice $$\frac{\mathfrak{D} \big( F \big)}{\| F \|_{\alpha}}\ \ =\ \ \left. \frac{d}{d\xi} \mathfrak{P} \big( ( \kappa + \xi ) F \big) \right|_{\xi\, =\, 0}\ \ =\ \ \int\! F\, d \mu_{\kappa F}.$$ Since $\mathfrak{D}$ is linear, we get $$\left. \frac{d}{d \xi} \mathfrak{P} \big( ( \kappa + i \xi ) F \big) \right|_{\xi\, =\, 0}\ \ =\ \ \lim_{\xi\, \rightarrow\, 0} \frac{\mathfrak{D} \big( i \xi F \big)}{\| i \xi F \|_{\alpha}}\ \ =\ \ i \lim_{\xi\, \rightarrow\, 0} \frac{\mathfrak{D} \big( \xi F \big)}{\| \xi F \|_{\alpha}}\ \ =\ \ \int\! i F\, d \mu_{\kappa F}\ \ =\ \ 0.$$
Similarly from equation we have for $\xi \in \mathbb{R}$, $$\left. \frac{d^{2}}{d \xi^{2}} \mathfrak{P} \big( ( \kappa + \xi ) F \big) \right|_{\xi\, =\, 0}\ \ =\ \ \lim_{n\, \rightarrow\, \infty} \frac{1}{n} \int\! \Big( F^{n} ((w,\, x)) \Big)^{2}\, d \mu_{\kappa F}.$$ This implies $$\left. \frac{d^{2}}{d \xi^{2}} \mathfrak{P} \big( ( \kappa + i \xi ) F \big) \right|_{\xi\, =\, 0}\ \ =\ \ \lim_{n\, \rightarrow\, \infty} \frac{1}{n} \int\! \Big( i F^{n} \Big)^{2}\, d \mu_{\kappa F}\ \ =\ \ \lim_{n\, \rightarrow\, \infty} \frac{-1}{n} \int\! \Big( F^{n} \Big)^{2}\, d \mu_{\kappa F}\ \ <\ \ 0.$$
Since $G \longmapsto \mathfrak{P} (G)$ is an analytic map, as mentioned in , we have the map $\zeta \longmapsto \mathfrak{P} ( \zeta F )$ to be analytic too in a neighbourhood of $\kappa$, where $\zeta = \kappa + i \xi \in \mathbb{C}$. This implies that ${\rm Im} \big( \mathfrak{P} ( \zeta F ) \big)$ is a harmonic function around $\kappa$, [*i.e.*]{}, for $\xi \in \mathbb{R}$, $$\left. \frac{d^{2}}{d \xi^{2}} {\rm Im} \big( \mathfrak{P} ( ( \kappa + \xi ) F ) \big) \right|_{\xi\, =\, 0}\ \ =\ \ \left. \frac{d^{2}}{d \xi^{2}} {\rm Im} \big( \mathfrak{P} ( ( \kappa + i \xi ) F ) \big) \right|_{\xi\, =\, 0}.$$ We know that for $\xi \in \mathbb{R},\ \mathfrak{P} \big( ( \kappa + \xi ) F \big) \in \mathbb{R}$, [*i.e.*]{}, $$\left. \frac{d^{2}}{d \xi^{2}} {\rm Im} \big( \mathfrak{P} ( ( \kappa + \xi ) F ) \big) \right|_{\xi\, =\, 0}\ \ =\ \ 0\ \ \ \ \Longrightarrow\ \ \ \ \left. \frac{d^{2}}{d \xi^{2}} {\rm Im} \big( \mathfrak{P} ( ( \kappa + i \xi ) F ) \big) \right|_{\xi\, =\, 0}\ \ =\ \ 0.$$
Thus we have $$\begin{aligned}
\left. \frac{d}{d \xi} \mathfrak{P} \big( ( \kappa + i \xi ) F \big) \right|_{\xi\, =\, 0} & = & 0; \\
\left. \frac{d^{2}}{d \xi^{2}} {\rm Re} \left( \mathfrak{P} \big( ( \kappa + i \xi ) F \big) \right) \right|_{\xi\, =\, 0} & < & 0; \\
\left. \frac{d^{2}}{d \xi^{2}} {\rm Im} \left( \mathfrak{P} \big( ( \kappa + i \xi ) F \big) \right) \right|_{\xi\, =\, 0} & = & 0. \end{aligned}$$
Thus, the map $\xi \longmapsto {\rm Re} \left( \mathfrak{P} \big( ( \kappa + i \xi ) F \big) \right)$ satisfies the hypothesis of Morse lemma for non degenerate critical points. Thus, similar to lemma (4) in [@ps:94] that has been derived in an analogous context as lemma (6) in [@ss:07], we obtain a change of variables $\Upsilon = \Upsilon(\xi)$ for $- \delta < \xi < \delta$ such that $$e^{\mathfrak{P} \big( ( \kappa + i \xi ) F \big)}\ \ =\ \ e^{\mathfrak{P} ( \kappa F )} \big( 1\ -\ \Upsilon^{2}\ +\ i \Theta \left( \Upsilon \right) \big),$$ where $\Theta \left( \Upsilon ( \xi ) \right) = e^{- \mathfrak{P} ( \kappa F )} {\rm Im} \big( e^{\mathfrak{P} \big( ( \kappa + i \xi ) F \big)} \big)$ and $\Theta \left( \Upsilon \right) = O \big( \left| \Upsilon \right|^{3} \big)$, thus proving the lemma.
We now start by considering the left hand side of equation , as mentioned in theorem , that measures the cardinality of the set $$\Big\{ (w,\, x)\, \in\, {\rm Fix}_{n} (T)\ :\ a\, \leq\, F^{n}((w,\, x))\, \leq\, b \Big\}.$$ It is clear that it can be expressed in terms of the indicator function, $\chi_{[a,\, b]}$, [*i.e.*]{}, $$\# \Big\{ (w,\, x)\, \in\, {\rm Fix}_{n} (T)\ :\ a\, \leq\, F^{n}((w,\, x))\, \leq\, b \Big\}\ \ =\ \ \sum_{(w,\, x)\, \in\, {\rm Fix}_{n} (T)} \chi_{[a,\, b]} \Big( F^{n} ((w,\, x)) \Big).$$ Since we know that any characteristic function can be approximated by a sequence of smooth functions with compact support under the integral norm, we initially prove a slightly modified result, as stated in proposition , where the indicator function $\chi_{[a,\, b]}$ is replaced by some smooth function with compact support, say $\tau : \mathbb{R} \longrightarrow \mathbb{R} $.
\[prop 1\] Suppose $F \in \mathscr{F}_{\alpha} (X, \mathbb{R})$ satisfies the hypothesis in theorem , [*i.e.*]{}, $F$ satisfies the approximability condition and there exists a unique real number $\kappa$ such that $\int\! F\, d \mu_{\kappa F} = 0$. Then, there exists a positive constant $C_{12} > 0$ such that $$\digamma_{\!\! \tau} (n)\ \ :=\ \ \sum_{(w,\, x)\; \in\; {\rm Fix}_{n} (T)} \tau \Big( F^{n} ((w,\, x)) \Big)\ \ \sim\ \ C_{12}\; \frac{e^{n\, \mathfrak{P}(\kappa F)}}{\sqrt{n}}\; \int_{\mathbb{R}}\! \tau (t)\, e^{- \kappa t}\; d t.$$
For $y \in \mathbb{R}$ and $\kappa$ as in the hypothesis, define $$\label{tausubkappa}
\tau_{\kappa} (y)\ \ :=\ \ \tau (y) e^{-\, \kappa y},$$ in order that $\digamma_{\!\! \tau} (n)$ can now be expressed as $$\label{taukappa}
\digamma_{\!\! \tau} (n)\ \ =\ \ \sum_{(w,\, x)\; \in\; {\rm Fix}_{n} (T)} \tau_{\kappa}\; \Big( F^{n} ((w,\, x)) \Big)\; e^{\kappa F^{n} ((w,\, x))}.$$ Using inverse Fourier transform and Fubini’s theorem, we rewrite equation as $$\begin{aligned}
\digamma_{\!\! \tau} (n) & = & \sum_{(w,\, x)\; \in\; {\rm Fix}_{n} (T)} \int_{\mathbb{R}}\! \widehat{\tau_{\kappa}} (\xi)\; e^{(\kappa\, +\, i \xi)\; F^{n} ((w,\, x))}\; d \xi \\
& = & \int_{\mathbb{R}}\! \widehat{\tau_{\kappa}} (\xi) \sum_{(w,\, x)\; \in\; {\rm Fix}_{n} (T)} e^{(\kappa\, +\, i \xi)\; F^{n} ((w,\, x))}\; d \xi. \end{aligned}$$
By definition, $\tau_{\kappa}$ lives inside a compact support. Hence, by an application of the Paley-Wiener theorem, we deduce that it is sufficient to estimate $$\sum\limits_{(w,\, x)\; \in\; {\rm Fix}_{n} (T)} e^{(\kappa\, +\, i \xi)\; F^{n}((w,\, x))},\ \ \text{for some}\ \xi\; \in\; \mathbb{R}.$$ The following lemma, from [@dr:73] helps us approximate this sum in terms of the iterates of the appropriate normalised operator such that $$\widetilde{\mathfrak{L}}_{\kappa F} \mathbf{1}\ \ =\ \ \varrho\, \mathbf{1}\ \ \ \ \text{where we recall}\ \ \varrho\ \ =\ \ e^{\mathfrak{P}(\kappa F)}.$$
\[ror1lemma1\] Let $\kappa \in \mathbb{R}$ be the unique real number that satisfies $\int\! F\, d \mu_{\kappa F} = 0$. Then, there exists $0 < \eta < \varrho^{-1}$ such that for every point $(w, x)\; \in\; {\rm Fix}_{n} (T)$, we have $$\sum_{(v,\, y)\; \in\; {\rm Fix}_{n} (T)} e^{(\kappa\, +\, i \xi) F^{n}((v,\, y))}\ \ =\ \ \left( \left( \widetilde{\mathfrak{L}}_{(\kappa\, +\, i \xi) F} \right)^{\!\circ n} \mathbf{1} \right) ((w,\, x))\; \Big( 1\; +\; O \left( n \eta^{n}\, \max\big\{ 1,\; |\xi| \big\} \right) \Big).$$
Using the above lemma, we write $\digamma_{\!\! \tau} (n)$ as $$\label{split1}
\digamma_{\!\! \tau} (n)\ \ =\ \ \int_{\mathbb{R}}\! \widehat{\tau_{\kappa}} (\xi) \left( \left( \widetilde{\mathfrak{L}}_{(\kappa\, +\, i \xi) F} \right)^{\!\circ n} \mathbf{1} \right) ((w,\, x))\ \Big( 1\, +\, O \left( n \eta^{n}\, \max\big\{ 1,\; |\xi| \big\} \right) \Big)\; d \xi.$$
The second term in the above equation is dominated by $(\varrho \eta)^{n}$, which converges to zero faster than any polynomial of $n$. In the remainder of the proof, we estimate the first term of equation .
If $\Xi_{\zeta F} : \mathscr{F}_{\alpha}(X, \mathbb{C}) \longrightarrow \mathscr{F}_{\alpha}(X, \mathbb{C})$ is the one-dimensional eigenprojection associated with $\widetilde{\mathfrak{L}}_{\zeta F}$, for $- \delta < \xi < \delta$, we know by perturbation theory that $\Xi_{\zeta F} (\mathbf{1}) = \mathbf{1} + O \big( \left| \Upsilon \right| \big)$. Thus, by perturbation theory and lemma , for $- \delta < \xi < \delta$ and some $0 < \vartheta < 1$, we obtain $$\begin{aligned}
\left( \widetilde{\mathfrak{L}}_{\zeta F} \right)^{\!\circ n} \mathbf{1} & = & e^{n \mathfrak{P} ( \zeta F )} \Big( 1 + O \left( \big| \Upsilon \big| \right) \Big) + O ( \vartheta^{n} ) \\
& = & e^{n \mathfrak{P} \big( \kappa F \big)} \Big( 1\; -\; \Upsilon^{2}\; +\; i\, \Theta \left( \Upsilon \right) \Big)^{n} \Big( 1 + O \left( \big| \Upsilon \big| \right) \Big) + O ( \vartheta^{n} ). \end{aligned}$$
The above equation facilitates the splitting of the integral in equation into two integrals given by, $$\begin{aligned}
\label{split2}
\int_{\mathbb{R}}\! \widehat{\tau_{\kappa}} (\xi) \left( \left( \widetilde{\mathfrak{L}}_{(\kappa\, +\, i \xi) F} \right)^{\!\circ n} \mathbf{1} \right) ((w,\, x))\; d \xi & = & \int_{| \xi |\, <\, \delta}\! \widehat{\tau_{\kappa}} (\xi) \left( \left( \widetilde{\mathfrak{L}}_{(\kappa\, +\, i \xi) F} \right)^{\!\circ n} \mathbf{1} \right) ((w,\, x))\; d \xi \nonumber \\
& & + \int_{| \xi |\, \ge\, \delta}\! \widehat{\tau_{\kappa}} (\xi) \left( \left( \widetilde{\mathfrak{L}}_{(\kappa\, +\, i \xi) F} \right)^{\!\circ n} \mathbf{1} \right) ((w,\, x))\; d \xi. \nonumber \\
& & \end{aligned}$$
We first estimate the first integral in equation , using the change in variables from lemma . $$\begin{aligned}
\label{split3}
& & \int_{| \xi |\, <\, \delta}\! \widehat{\tau_{\kappa}} (\xi) \left( \left( \widetilde{\mathfrak{L}}_{(\kappa\, +\, i \xi) F} \right)^{\!\circ n} \mathbf{1} \right) ((w,\, x))\, d \xi \nonumber \\
& = & \int_{| \xi |\, <\, \delta} e^{n \mathfrak{P} ( \kappa F )} \Big( 1 - \Upsilon^{2} + i \Theta \left( \Upsilon \right) \Big)^{n} \Big( 1 + O \left( \big| \Upsilon \big| \right) \Big) \Big( \widehat{\tau_{\kappa}} \big( \xi ( \Upsilon ) \big) \Big) \frac{d \xi}{d \Upsilon} d \Upsilon + O ( \vartheta^{n} ) \nonumber \\
& = & C_{13}\; \widehat{\tau_{\kappa}} (0)\; e^{n \mathfrak{P} ( \kappa F )} \int_{| \xi |\, <\, \delta} \Big( 1 - \Upsilon^{2} + i \Theta \left( \Upsilon \right) \Big)^{n} \Big( 1 + O \left( \big| \Upsilon \big| \right) \Big)\; d \Upsilon\ +\ O(n^{-1})\ +\ O( \vartheta^{n} ), \nonumber \\
& & \end{aligned}$$ where $C_{13} > 0$ is a constant dependent on $\tau_{\kappa}$ and the Jacobian of change of variables. We now apply binomial expansion to the expression in the integral of equation , thus splitting it into three parts and define them as follows: $$\begin{aligned}
I_{0} (n) & := & \int_{- \delta}^{\delta}\! \big( 1\ -\ \Upsilon^{2} \big)^{n}\, d \Upsilon; \\
\sum_{j\, =\, 1}^{n} I_{j} (n) & := & \left| \sum_{j\, =\, 1}^{n} \binom{n}{j} \int_{- \delta}^{\delta} \big( 1\ -\ \Upsilon^{2} \big)^{n - j}\; \big( i \Theta (\Upsilon) \big)^{j}\; \left( 1\ +\ O \big( |\Upsilon| \big) \right)\, d \Upsilon \right|; \\
J(n) & := & \int_{- \delta}^{\delta}\! \big( 1\ -\ \Upsilon^{2} \big)^{n}\; O \big( |\Upsilon| \big)\, d \Upsilon. \end{aligned}$$ Using techniques from [@ps:94], we estimate the above integrals to get the following inequalities. $$\begin{aligned}
I_{0} (n) & = & C_{14} \frac{\Gamma \left( n + 1 \right)}{\Gamma \left( n + 1 + \frac{1}{2} \right)} + O ( ( 1 - \delta^{2} )^{n} ) \\
\sum_{j\, =\, 1}^{n} I_{j} (n) & \le & C_{15} \frac{\Gamma \left( n + 1 \right)}{\Gamma \left( n + 1 + \frac{1}{2} \right)}\;\frac{1}{\sqrt{n}} \\
| J(n) | & \le & C_{16} \frac{\Gamma(n + 1)}{\Gamma(n + 2)}, \end{aligned}$$ for some positive constants $C_{14},\ C_{15}$ and $C_{16}$. We now use the identity, $$\lim_{n\, \rightarrow\, \infty} \frac{\Gamma(n + \beta)}{\Gamma(n) n^{\beta}}\ \ =\ \ 1,$$ to conclude that $I_{0} (n)\ \sim\ \frac{1}{\sqrt{n}} C_{14}$ and that the rest of the terms inside the integral in equation converge to zero faster than $\frac{1}{\sqrt{n}}$.
What remains now is the integral in equation over $| \xi | \geq \delta$. For that, we make use of the following lemma, which is a standard result in the theory of Fourier transforms.
\[lemmafourier\] Let $\chi : \mathbb{R} \longrightarrow \mathbb{R}$ be a compactly supported $\mathcal{C}^{r}$ function. Then the Fourier transform $\widehat{\chi} (\xi) = O ( | \xi |^{- r} )$ as $\xi \rightarrow \infty$.
Since $\tau_{\kappa}$ is smooth, we may suppose that $\tau_{\kappa}$ is a compactly supported $\mathcal{C}^{r}$ function, for any arbitrary $r \in \mathbb{N}$. Now using lemma and lemma , we obtain the following expression. $$\int_{| \xi |\, \ge\, \delta} \widehat{\tau_{\kappa}} ( \xi )\; \Big( \left( \widetilde{\mathfrak{L}}_{(\kappa + i \xi ) F} \right)^{\!\circ n} \mathbf{1} \Big)((w, x))\, d \xi\ \ =\ \ O \left( e^{n \mathfrak{P} \big( \kappa F \big)} \int_\delta^{\infty}\! \left( 1 - \frac{1}{\xi^{C_{7}}} \right)^{\frac{n}{2R}} \xi^{- r}\, d \xi \right),$$ where $R = [C_{8} \log \xi]$. We now prove that the integral in the right hand side of the above quantity approaches zero faster than $\frac{1}{\sqrt{n}}$ as $n$ goes to $\infty$, using techniques from [@ps:94]. In order to achieve the same, we further split the integral into two parts, as $$\int_{\delta}^{\infty} \left( 1\ -\ \frac{1}{\xi^{C_{7}}} \right)^{\frac{n}{2R}} \xi^{- r}\, d \xi\ \ =\ \ \int_{\delta}^{n^{\delta'}} \left( 1\ -\ \frac{1}{\xi^{C_{7}}} \right)^{\frac{n}{2R}} \xi^{- r}\, d \xi\ +\ \int_{n^{\delta'}}^{\infty} \left( 1\ -\ \frac{1}{\xi^{C_{7}}} \right)^{\frac{n}{2R}} \xi^{- r}\, d \xi,$$ where $\delta < \delta' < \frac{1}{C_{7}}$. Thus, by our choice of $\delta'$, we have $C_{7} \delta' < 1$. Hence, we get the following estimates. The convergence rate of the first term to zero is faster than any polynomial while the second part converges to zero at a polynomial rate dependent on $r$. $$\begin{aligned}
\label{split5.1}
\int_{\delta}^{n^{\delta'}} \left( 1\ -\ \frac{1}{\xi^{C_{7}}} \right)^{\frac{n}{2R}} \xi^{- r}\, d \xi & = & O \left( n^{\delta'} \left( 1\ -\ \frac{1}{n^{C_{7} \delta'}} \right)^{\frac{n}{2 \delta' \beta \log n}} \right) \\
\label{split5.2}
\int_{n^{\delta '}}^{\infty} \left( 1\ -\ \frac{1}{\xi^{C_{7}}} \right)^{\frac{n}{2R}} \xi^{- r}\, d \xi & = & O \left( n^{( 1 - r ) \delta'} \right) \end{aligned}$$ Using the bounds in equations and , we now estimate the remaining part of equation to be the following. $$\int_{| \xi |\, \ge\, \delta} \widehat{\tau_{\kappa}} (\xi) \Big( \widetilde{\mathfrak{L}}_{(\kappa + i \xi) F} \Big)^{\! \circ n}\mathbf{1} ((w, x))\, d \xi\ \ =\ \ O \Big( e^{n \mathfrak{P} ( \kappa F )} n^{(1 - r) \delta'} \Big).$$ Since $\widehat{\tau_{\kappa}}$ is smooth, we prove that the rate of growth of the second term in equation is smaller than $\frac{1}{\sqrt{n}} e^{n \mathfrak{P} ( \kappa F )}$, by considering $r > 1 + \frac{1}{2 \delta'}$. Thus, we obtain the following asymptotic relation. $$\digamma_{\!\! \tau}(n)\ \ \sim\ \ C_{12} \frac{e^{n\mathfrak{P} (\kappa F)}}{\sqrt{n}} \widehat{\tau_{\kappa}} (0).$$
The proof of theorem now follows from proposition where we replace the function $F$ by the characteristic function $\chi_{[a,\, b]}$.
We now proceed to prove theorem , using techniques similar to those used in the above proof. Hence, we only highlight the important steps to complete this proof.
As earlier, we begin with a generalisation of the left hand side of the assertion of theorem for a compactly supported function and define $\widetilde{\digamma}_{\!\! \tau} (n)$ as $$\widetilde{\digamma}_{\!\! \tau} (n)\ \ :=\ \ \sum\limits_{w\; :\, |w|\, =\, n} \sum\limits_{x\, \in\, {\rm Fix} T_{w}} \tau ( f^{n}_{w} (x) ).$$ Clearly, $\widetilde{\digamma}_{\chi_{[a,\, b]}} (n)$ coincides with the expression we need to estimate. By replacing $\tau$ by $\tau_{\kappa}$, as defined in equation and applying Fourier transforms we get the following: $$\widetilde{\digamma}_{\tau} (n)\ \ =\ \ \int_{\mathbb{R}}\! \widehat{\tau_{\kappa}} (\xi) \sum\limits_{w\; :\; |w|\, =\, n} \sum\limits_{x\, \in\, {\rm Fix} (T_{w})} e^{((\kappa + i \xi ) f^{n}_{w} (x))}\, d \xi.$$ Thus, in order to estimate $\widetilde{\digamma}_{\tau} (n)$, we first estimate the sum inside the integral, namely $$\sum\limits_{w\; :\; |w|\, =\, n} \sum\limits_{x\, \in\, {\rm Fix} (T_{w})} e^{((\kappa + i \xi) f^{n}_{w} (x))}\ \ \text{for}\ \ \xi \in \mathbb{R}.$$ The following lemma gives a relation between the sum to be estimated and the iterates of the normalised Ruelle operator, $\widetilde{\mathbb{L}}_{(\kappa + i \xi) f}$. By observing the relation between $(\widetilde{\mathbb{L}}_{(\kappa + i \xi) f})^{\circ n}$ and $(\widetilde{\mathfrak{L}}_{(\kappa + i \xi) Q(f)})^{\circ n}$, as prescribed in equation , we write the following lemma which is nothing but an easy corollary of lemma .
Let $\kappa \in \mathbb{R}$ be the unique real number that satisfies $\int\! f\, d \mathfrak{m}_{\kappa f} = 0$. Then, there exists $0 < \eta < e^{- \mathbb{P}(\kappa f)}$ such that for every point $x$ that satisfies $T_{w}^{n} x = x$ for some $n$-lettered word $w$, we have $$\sum_{v\; :\; |v|\, =\, n} \sum\limits_{y\, \in\, {\rm Fix} (T_{v})} e^{(\kappa\, +\, i \xi) f^{n}_{v} (y)}\ \ =\ \ \left( \left( \widetilde{\mathbb{L}}_{(\kappa\, +\, i \xi) f} \right)^{\!\circ n} \mathbf{1} \right) (x)\; \Big( 1\; +\; O \left( n \eta^{n}\, \max\big\{ 1,\; |\xi| \big\} \right) \Big).$$
Using the above lemma, we rewrite $\widetilde{\digamma}_{\tau}(n)$ as, $$\label{integralform}
\widetilde{\digamma}_{\tau}(n) = \int_{\mathbb{R}} \widehat{\tau}_{\kappa}(\xi) \left( \left((\widetilde{\mathbb{L}}_{(\kappa+i\xi)f})^{\circ n} \mathbf{1} \right)(x)\ \left(1 + O(n \eta^{n} \max \{1, |\xi|\})\right) \right) d\xi.$$ The second term in the above integral is dominated by $(\eta e^{\mathbb{P} (\kappa f)})^{n}$ and thus goes to zero, faster than polynomial of $n$, as $n$ tends to infinity. Using the definition of $\mathbb{P} ( ( \kappa + i \xi ) f )$, theorem and lemma , we conclude that the pressure functions $\mathbb{P} ( ( \kappa + i \xi ) f )$ and $\mathfrak{P} ( ( \kappa + i \xi ) Q(f) )$ coincide. Further, since $\int\! Q(f)\, d \mu_{\kappa Q(f)} = \int\! f\, d \mathfrak{m}_{\kappa f} = 0$, we obtain the following lemma as a corollary of lemma .
For $\zeta = \kappa + i \xi$, there exists a change of variables $\Upsilon = \Upsilon(\xi)$ such that for $| \xi | < \delta$, we can expand $$e^{\mathbb{P} \left( \zeta f \right)}\ \ =\ \ e^{\mathbb{P} \left( \kappa f \right)}\; \Big( 1\; -\; \Upsilon^{2}\; +\; i\, \Theta \left( \Upsilon \right) \Big),$$ where $\Theta$ is a real-valued function that satisfies $\Theta (\Upsilon) = O ( | \Upsilon |^{3} )$.
By perturbation theory, the one dimensional eigenprojection associated with $\widetilde{\mathbb{L}}_{( \kappa + i \xi ) f}$ is of the form $$\Xi_{( \kappa + i \xi ) f} \mathbf{1}\ \ =\ \ \mathbf{1} + O ( | \Upsilon | ).$$ Thus for $- \delta < \xi < \delta$ and for some $0 < \vartheta < 1$, we have $$\left( \widetilde{\mathbb{L}}_{( \kappa + i \xi ) f} \right)^{\!\circ n} \mathbf{1}\ \ =\ \ e^{n \mathbb{P} \big( ( \kappa + i \xi ) f \big)} \Big( \mathbf{1} + O \left( \big| \Upsilon \big| \right) \Big) + O ( \vartheta^{n} ).$$ The integral in the equation can now be split accordingly to get $$\begin{aligned}
\label{split integral}
\int_{\mathbb{R}}\! \widehat{\tau_{\kappa}} (\xi) \left( \left( \widetilde{\mathbb{L}}_{(\kappa\, +\, i \xi) f} \right)^{\!\circ n} \mathbf{1} \right) (x)\; d \xi & = & \int_{| \xi |\, <\, \delta}\! \widehat{\tau_{\kappa}} (\xi) \left( \left( \widetilde{\mathbb{L}}_{(\kappa\, +\, i \xi) f} \right)^{\!\circ n} \mathbf{1} \right) (x)\; d \xi \nonumber \\
& & + \int_{| \xi |\, \ge\, \delta}\! \widehat{\tau_{\kappa}} (\xi) \left( \left( \widetilde{\mathbb{L}}_{(\kappa\, +\, i \xi) f} \right)^{\!\circ n} \mathbf{1} \right) (x)\; d \xi. \nonumber \\
& & \end{aligned}$$ We then apply change of variables to the first integral in the above equation to get $$\begin{aligned}
& & \int_{| \xi |\, <\, \delta}\! \widehat{\tau_{\kappa}} (\xi) \left( \left( \widetilde{\mathbb{L}}_{(\kappa\, +\, i \xi) f} \right)^{\!\circ n} \mathbf{1} \right) (x)\, d \xi \\
& = & C_{17}\; \widehat{\tau_{\kappa}} (0)\; e^{n \mathfrak{P} ( \kappa f )} \int_{| \xi |\, <\, \delta} \Big( 1 - \Upsilon^{2} + i \Theta \left( \Upsilon \right) \Big)^{n} \Big( 1 + O \left( \big| \Upsilon \big| \right) \Big)\; d \Upsilon\ +\ O(n^{-1})\ +\ O( \vartheta^{n} ), \\ \end{aligned}$$ where the expression inside the integral is the same as in equation . Thus, the techniques used in the proof of theorem can be used again to obtain $$\int_{| \xi |\, <\, \delta}\! \widehat{\tau_{\kappa}} (\xi) \left( \left( \widetilde{\mathbb{L}}_{(\kappa\, +\, i \xi) f} \right)^{\!\circ n} \mathbf{1} \right) (x)\, d \xi\ \sim\ \frac{1}{\sqrt{n}} C_{18} e^{n \mathbb{P} ( \kappa f)} \widehat{\tau_{\kappa}}(0).$$
Now what remains is to prove that the second integral in equation grows at a rate strictly smaller than $\frac{1}{\sqrt{n}} e^{n \mathbb{P} ( \kappa f )}$, as $n \to \infty$, to complete the proof. This can be achieved again using lemma , as earlier.
Decay of correlations {#docsec}
=====================
In this section, we prove theorems pertaining to the decay of correlations in the skew-product setting and in the setting of simultaneous action of finitely many interval maps, namely theorems and .
Fix $F \in \mathscr{F}_{\alpha} (X, \mathbb{R})$ and consider the normalised Ruelle operator $\widetilde{\mathfrak{L}}_{F}$ along with its corresponding equilibrium measure $\mu_{F}$. Denote by $\mathscr{F}^{F}_{\alpha} (X, \mathbb{R})$, the set of all $\alpha$-Hölder continuous functions whose integral with respect to $\mu_{F}$ is zero, [*i.e.*]{}, $$\mathscr{F}^{F}_{\alpha} (X, \mathbb{R})\ \ :=\ \ \Bigg\{ G \in \mathscr{F}_{\alpha} (X, \mathbb{R})\ :\ \int\! G\, d \mu_{F} = 0 \Bigg\}.$$ It is easily verifiable that $\mathscr{F}^{F}_{\alpha} (X, \mathbb{R})$ is a subspace of $\mathscr{F}_{\alpha} (X, \mathbb{R})$. Further, the space $\mathscr{F}^{F}_{\alpha} (X, \mathbb{R})$ is preserved by the action of the operator $\widetilde{\mathfrak{L}}_{F}$, [*i.e.*]{}, $\widetilde{\mathfrak{L}}_{F} : \mathscr{F}^{F}_{\alpha} (X, \mathbb{R}) \longrightarrow \mathscr{F}^{F}_{\alpha} (X, \mathbb{R})$. The Ruelle operator theorem states that the action of $\widetilde{\mathfrak{L}}_{F}$ on $\mathscr{F}^{F}_{\alpha} (X, \mathbb{R})$ has a spectral radius strictly smaller than $\varrho = e^{\mathfrak{P} (F)}$. Equivalently one may consider the operator $\varrho^{-1} \widetilde{\mathfrak{L}}_{F}$ on $\mathscr{F}_{\alpha}^{F} (X, \mathbb{R})$ that has a spectral radius, say $\varrho_{F} < 1$.
We first state and prove a lemma, that will be useful to prove our main results in this section.
\[lem4.1\] For any $\vartheta \in (\varrho_{F},\, 1)$, there exists a positive constant $C_{19} > 0$ such that $$\bigg\Vert \left( \varrho^{-1} \widetilde{\mathfrak{L}}_{F} \right)^{\!\circ n} G \bigg\Vert_{\alpha}\ \ \le\ \ C_{19}\, \vartheta^{n}\, \big\Vert G \big\Vert_{\alpha}\ \ \forall n \ge 0\ \ \text{and}\ \ \forall G \in \mathscr{F}^{F}_{\alpha} (X, \mathbb{R}).$$
Fix a number $\vartheta \in (\varrho_{F},\, 1)$. Choose $\epsilon > 0$ such that $\varrho_{F} + \epsilon < \vartheta$. For this $\epsilon$, there exists $M_{\epsilon} \in \mathbb{Z}_{+}$ such that $$\bigg\Vert \left( \varrho^{-1} \widetilde{\mathfrak{L}}_{F} \right)^{\!\circ n} \bigg\Vert^{\frac{1}{n}}\ \ <\ \ \varrho_{F} + \epsilon,\ \ \forall n \ge M_{\epsilon},\ \ \ \text{since}\ \ \varrho_{F}\ \ = \inf\limits_{n\, \ge\, 1} \bigg\Vert \left( \widetilde{\mathfrak{L}}_{F} \right)^{\!\circ n} \bigg\Vert^{\frac{1}{n}},$$ where we only consider the action of $\varrho^{-1} \widetilde{\mathfrak{L}}_{F}$ on $\mathscr{F}_{\alpha}^{F} (X, \mathbb{R})$. This implies $$\bigg\Vert \left( \varrho^{-1} \widetilde{\mathfrak{L}}_{F} \right)^{\!\circ n} G \bigg\Vert_{\alpha}\ \ <\ \ \left( \varrho_{F} + \epsilon \right)^{n}\, \big\Vert G \big\Vert_{\alpha}\ \ <\ \ \vartheta^{n}\, \big\Vert G \big\Vert_{\alpha},\ \ \forall n \ge M_{\epsilon},\ \ \text{and}\ \ \forall G \in \mathscr{F}^{F}_{\alpha} (X, \mathbb{R}).$$ Further, since $\widetilde{\mathfrak{L}}_{F}$ is a bounded operator, there exists a positive constant $D_{0} > 0$ such that $$\bigg\Vert \left( \varrho^{-1} \widetilde{\mathfrak{L}}_{F} \right)^{\!\circ n} G \bigg\Vert_{\alpha}\ \ \le\ \ D_{0}\, \big\Vert G \big\Vert_{\alpha},\ \ \forall n \ge 1.$$ Now, an application of the Archimedean property of $\mathbb{R}$ results in finitely many finite constants $D_{1},\, D_{2},\, \cdots,\, D_{M_{\epsilon}}$ that satisfy $D_{n} \vartheta^{n} > D_{0}$ for $1 \le n \le M_{\epsilon}$. Hence, $$\bigg\Vert \left( \varrho^{-1} \widetilde{\mathfrak{L}}_{F} \right)^{\!\circ n} G \bigg\Vert_{\alpha}\ \ \le\ \ D_{n}\, \vartheta^{n}\, \big\Vert G \big\Vert_{\alpha},\ \ \text{for}\ 1 \le n \le M_{\epsilon}.$$ The result now follows by taking $C_{19} = \max \big\{ 1,\, D_{1},\, D_{2},\, \cdots,\, D_{M_{\epsilon}} \big\}$.
We are now ready to prove theorem that states the decay of correlations result for the skew-product map $T$.
For any $G \in \mathscr{F}_{\alpha}^{F} (X, \mathbb{R})$, the left hand side of equation in theorem can be written as $$\begin{aligned}
\int\! \mathscr{Q}^{n} (G) H\, d \mu_{F}\ -\ \int\! G\, d \mu_{F} \int\! H\, d \mu_{F} & = & \int\! \mathscr{Q}^{n} (G) H\, d \mu_{F}\ -\ \int\! \mathscr{Q}^{n} (G)\, d \mu_{F} \int\! H\, d \mu_{F} \\
& = & \int\! \mathscr{Q}^{n} (G) \left( H - \int\! H\, d \mu_{F} \right)\, d \mu_{F}. \end{aligned}$$ Suppose we denote $\displaystyle{\widetilde{H} := \left( H - \int\! H\, d \mu_{F} \right)}$, then it is easy to see that $\widetilde{H} \in \mathscr{F}^{F}_{\alpha} (X, \mathbb{R})$. Further, on the space of $\mu_{F}$-square integrable real-valued functions defined on $X$ denoted by $L^{2} (\mu_{F})$, the operator $\widetilde{\mathfrak{L}}_{F}$ has a natural extension, with its adjoint given by the operator $\mathscr{Q}$, [*i.e.*]{}, $$\big\langle \mathscr{Q} \Phi,\, \Psi \big\rangle\ \ =\ \ \big\langle \Phi,\, \widetilde{\mathfrak{L}}_{F} \Psi \big\rangle,\ \ \forall \Phi, \Psi \in L^{2} (\mu_{F}).$$ Hence, $$\int\! \mathscr{Q}^{n} (G) \widetilde{H}\, d \mu_{F}\ \ =\ \ \int\! G \left( \varrho^{-1} \widetilde{\mathfrak{L}}_{F} \right)^{\!\circ n} \widetilde{H}\, d \mu_{F}.$$ Therefore, $$\left\vert \int\! G \left( \varrho^{-1} \widetilde{ \mathfrak{L}}_{F} \right)^{\!\circ n} \widetilde{H}\, d \mu_{F} \right\vert\ \ \le\ \ \int\! \left\vert G \left( \varrho^{-1} \widetilde{\mathfrak{L}}_{F} \right)^{\!\circ n} \widetilde{H} \right\vert\, d \mu_{F}\ \ \le\ \ \left\Vert G \right\Vert_{2}\, \left\Vert \left( \varrho^{-1} \widetilde{\mathfrak{L}}_{F} \right)^{\!\circ n} \widetilde{H} \right\Vert_{2}.$$ Further, $$\begin{aligned}
\bigg\Vert \left( \varrho^{-1} \widetilde{\mathfrak{L}}_{F} \right)^{\!\circ n} \widetilde{H} \bigg\Vert_{2} & \le & \bigg\Vert \left( \varrho^{-1} \widetilde{\mathfrak{L}}_{F} \right)^{\!\circ n} \widetilde{H} \bigg\Vert_{\alpha} \\
& \le & C_{19}\, \vartheta^{n}\, \left\Vert \widetilde{H} \right\Vert_{\alpha}\ \hspace{+6cm} (\text{using lemma \eqref{lem4.1}}) \\
& \le & C_{19}\, \vartheta^{n}\, \left( \big\Vert H \big\Vert_{\alpha}\ +\ \left\vert \int H\, d \mu_{F} \right\vert \right)\ \hspace{+2.7cm} (\text{using definition of}\ \widetilde{H}) \\
& \le & 2 C_{19}\, \vartheta^{n} \big\Vert H \big\Vert_{\alpha}. \end{aligned}$$ Thus, we obtain the result with the constant $C_{2}\; =\; 2 C_{19}\, \left\Vert G \right\Vert_{2}\, \left\Vert H \right\Vert_{\alpha}$ to complete the proof of theorem .
To prove theorem , we start by considering the functions $f_{d} = - \log | T_{d}' | \in \mathscr{F}_{\alpha} (I, \mathbb{R})$, for $1 \le d \le N$. By the definition of the Ruelle operator $\mathcal{L}_{f_{d}}^{(d)}$, as stated in equation , we know that $$\Big( \mathcal{L}_{f_{d}}^{(d)} g \Big) (x)\ \ =\ \ \sum_{T_{d} y\, =\, x} \frac{g(y)}{| T_{d}' (y) |}.$$ Observe that $\mathcal{L}_{f_{d}}^{(d)} = \widetilde{\mathcal{L}}_{f_{d}}^{(d)}$, [*i.e.*]{}, the operator $\mathcal{L}_{f_{d}}^{(d)}$ has eigenvalue $1$, with corresponding eigenfunction $\mathbf{1}$. Further, it is evident from Boyarsky and Góra ([@bg:97], section (4.3)) that the dual operator $\left( \mathcal{L}_{f_{d}}^{(d)} \right)^{\!*}$ fixes the Lebesgue measure $\lambda$ [*i.e.*]{}, $\left( \mathcal{L}_{f_{d}}^{(d)} \right)^{\!*} \lambda = \lambda$. Moreover, for every $1 \le d \le N$, the operator $\mathscr{O}_{d}$ defined by $\mathscr{O}_{d} g = g \circ T_{d}$ satisfies $\mathcal{L}_{f_{d}}^{(d)} \mathscr{O}_{d} = {\rm id}$, the identity operator in $\mathcal{C} (I, \mathbb{R})$.
Denoting by $\mathscr{F}_{\alpha}^{\lambda} (I, \mathbb{R})$, the set of all real-valued $\alpha$-Hölder continuous functions on $I$ whose Lebesgue integral is equal to $0$, [*i.e.*]{}, $$\label{Falphalambda}
\mathscr{F}_{\alpha}^{\lambda} (I, \mathbb{R})\ \ :=\ \ \left\{ f \in \mathscr{F}_{\alpha} (I, \mathbb{R})\ :\ \int\! f \, d \lambda = 0 \right\},$$ and observing that $\mathcal{L}_{f_{d}}^{(d)}$ preserves the space $\mathscr{F}_{\alpha}^{\lambda} (I, \mathbb{R})$ for all $1 \le d \le N$, we now state a lemma whose proof runs [*mutatis mutandis*]{} as the proof of lemma . We know that the action of $\mathcal{L}_{f_{d}}^{(d)}$ on $\mathscr{F}_{\alpha}^{\lambda} (I, \mathbb{R})$ has a spectral radius, say $\rho_{\lambda}^{(d)} < 1$, owing to theorem .
\[elevenpointtwo\] For any $\vartheta^{(d)} \in (\rho_{\lambda}^{(d)},\, 1)$, there exists a constant $C_{20}^{(d)} > 0$ such that $$\bigg\Vert \left( \mathcal{L}_{f_{d}}^{(d)} \right)^{\!\circ n} g \bigg\Vert_{\alpha}\ \ \le\ \ C_{20}^{(d)}\, \left( \vartheta^{(d)} \right)^{n}\, \big\Vert g \big\Vert_{\alpha}\ \ \forall n \ge 1\ \ \text{and}\ \ \forall g \in \mathscr{F}^{\lambda}_{\alpha} (I, \mathbb{R}),\ \ \text{for}\ \ 1 \le d \le N.$$
We are now thoroughly equipped to prove theorem .
For any $g \in \mathscr{F}_{\alpha}^{\lambda} (I, \mathbb{R})$, the left hand side of equation in theorem can be written as $$\begin{aligned}
\int\! \mathscr{O}_{w} (g) h\, d \lambda\ -\ \int\! g\, d \lambda \int\! h\, d \lambda & = & \int\! \mathscr{O}_{w} (g) h\, d \lambda\ -\ \int\! \mathscr{O}_{w} (g)\, d \lambda \int\! h\, d \lambda \\
& = & \int\! \mathscr{O}_{w} (g) \left( h - \int\! h\, d \lambda \right)\, d \lambda. \end{aligned}$$ Let $\displaystyle{\widetilde{h} = h - \int\! h\, d \lambda} \in \mathscr{F}_{\alpha}^{\lambda} (I, \mathbb{R})$. Then, since $\mathcal{L}_{f_{d}}^{(d)}$ is the adjoint of $\mathscr{O}_{d}$ in the space of Lebesgue square integrable real-valued functions defined on $I,\ L^2(I,\mathbb{R})$, we have $$\begin{aligned}
\left\vert \int\! \left( \mathscr{O}_{w} g \right) \widetilde{h}\, d \lambda \right\vert & = & \left\vert \int\! g \left( \mathcal{L}_{f_{w_{n}}}^{(w_{n})} \mathcal{L}_{f_{w_{n - 1}}}^{(w_{n - 1})} \cdots \mathcal{L}_{f_{w_{1}}}^{(w_{1})} \right) \widetilde{h}\, d \lambda \right\vert \\
& \le & \left\Vert g \right\Vert_{2} \left\Vert \mathcal{L}_{f_{w_{n}}}^{(w_{n})} \mathcal{L}_{f_{w_{n - 1}}}^{(w_{n - 1})} \cdots \mathcal{L}_{f_{w_{1}}}^{(w_{1})} \widetilde{h} \right\Vert_{2}. \end{aligned}$$ Now making use of the inequality $$\left\Vert \left( \mathcal{L}_{f_{w_{n}}}^{(w_{n})} \mathcal{L}_{f_{w_{n - 1}}}^{(w_{n - 1})} \cdots \mathcal{L}_{f_{w_{1}}}^{(w_{1})} \right) \widetilde{h} \right\Vert_{2}\ \ \le\ \ \left\Vert \left( \mathcal{L}_{f_{w_{n}}}^{(w_{n})} \mathcal{L}_{f_{w_{n - 1}}}^{(w_{n - 1})} \cdots \mathcal{L}_{f_{w_{1}}}^{(w_{1})} \right) \widetilde{h} \right\Vert_{\alpha},$$ and redistributing the operators for $1 \le d \le N$, we obtain $$\begin{aligned}
\left\Vert \left( \mathcal{L}_{f_{w_{n}}}^{(w_{n})} \mathcal{L}_{f_{w_{n - 1}}}^{(w_{n - 1})} \cdots \mathcal{L}_{f_{w_{1}}}^{(w_{1})} \right) \widetilde{h} \right\Vert_{\alpha} & = & \left\Vert \left( \left( \mathcal{L}_{f_{1}}^{(1)} \right)^{\gamma_{1}} \left( \mathcal{L}_{f_{2}}^{(2)} \right)^{\gamma_{2}} \cdots \left( \mathcal{L}_{f_{N}}^{(N)} \right)^{\gamma_{N}} \right) \widetilde{h} \right\Vert_{\alpha} \\
& \le & C_{20}^{(1)}\, C_{20}^{(2)}\, \cdots\, C_{20}^{(N)}\, \left( \vartheta^{(1)} \right)^{\gamma_{1}} \left( \vartheta^{(2)} \right)^{\gamma_{2}} \cdots \left( \vartheta^{(N)} \right)^{\gamma_{N}} \left\Vert \widetilde{h} \right\Vert_{\alpha}, \end{aligned}$$ appealing to lemma .
Finally, defining $C_{20} := C_{20}^{(1)}\, C_{20}^{(2)}\, \cdots\, C_{20}^{(N)}$ and $\vartheta := \max \left\{ \vartheta^{(1)},\, \vartheta^{(2)},\, \cdots,\, \vartheta^{(N)} \right\}$, we obtain $$\begin{aligned}
\left\vert \int\! \mathscr{O}_{w} (g) h\, d \lambda\ -\ \int\! g\, d \lambda \int\! h\, d \lambda \right\vert & \le & C_{20} \vartheta^{n} \left\Vert g \right\Vert_{2} \left\Vert \widetilde{h} \right\Vert_{\alpha} \\
& \le & 2 C_{20} \vartheta^{n} \left\Vert g \right\Vert_{2} \left\Vert h \right\Vert_{\alpha}, \end{aligned}$$ thus completing the proof.
Almost sure invariance principles {#asipsec}
=================================
In this section, we prove the almost sure invariance principles as stated in theorems and for both the settings, that we focus in this paper. As in section , we begin by fixing a real-valued Hölder continuous function $F \in \mathscr{F}_{\alpha} (X, \mathbb{R})$ and considering the corresponding normalised Ruelle operator $\varrho^{-1} \widetilde{\mathfrak{L}}_{F}$ along with its equilibrium measure $\mu_{F}$ and the subspace $\mathscr{F}_{\alpha}^{F} (X, \mathbb{R}) \subseteq \mathscr{F}_{\alpha} (X, \mathbb{R})$. The proof closely follows the method of proof given by Pollicott and Sharp in [@ps:02] and Sridharan in [@ss:09].
For any function $G \in \mathscr{F}_{\alpha}^{F} (X, \mathbb{R})$, define $$H\ \ :=\ \ \sum\limits_{n\, \ge \, 1} \left( \varrho^{-1} \widetilde{\mathfrak{L}}_{F} \right)^{\!\circ n} G.$$ Observe that the infinite series that defines $H$ converges, owing to lemma . Then, $$\widetilde{\mathfrak{L}}_{F} \Big( G\, +\, H\, -\, \mathscr{Q} (H) \Big)\ \ =\ \ \widetilde{\mathfrak{L}}_{F} G\, +\, \widetilde{\mathfrak{L}}_{F} H\, -\, \varrho H\ \ =\ \ \mathbf{0}.$$ Thus, defining $\Phi\; :=\; G + H - \mathscr{Q} (H)$, we observe that $$\begin{aligned}
\Big\vert G^{n} ((w,\, x))\; -\; \Phi^{n} ((w,\, x)) \Big\vert & = & \Big\vert \mathscr{Q}^{n} H ((w,\, x))\; -\; H ((w,\, x)) \Big\vert \\
& \le & \Big\vert \mathscr{Q}^{n} H ((w,\, x)) \Big\vert\; +\; \Big\vert H ((w,\, x)) \Big\vert \\
& \le & 2 \big\Vert H \big\Vert_{\alpha}. \end{aligned}$$
Thus, we have proved:
(c.f.[@ps:02], Lemma 2) For any function $G \in \mathscr{F}_{\alpha}^{F} (X, \mathbb{R})$, there exists a function $H \in \mathscr{F}_{\alpha}^{F} (X, \mathbb{R})$ such that $\Phi = G + \big( H - \mathscr{Q} (H) \big)$ satisfies $$\widetilde{\mathfrak{L}}_{F} \Phi\ \ =\ \ 0\ \ \ \ \text{and}\ \ \ \ G^{n} ((w,\, x))\ \ =\ \ \Phi^{n} ((w,\, x)) + O(1).$$
Given that $\Phi$ and $G$ are cohomologous to each other, we have $$\label{eqnvar}
\varsigma(G)^{2}\ \ =\ \ \int\! \Phi ((w,\, x))^{2}\, d \mu_{F}\; +\; 2 \sum_{n\, \ge\, 0} \int\! \Phi ((w,\, x)) \Phi (T^{n} ((w,\, x)))\, d \mu_{F}.$$ Since $\widetilde{\mathfrak{L}}_{f} \Phi = 0$, we obtain $$\begin{aligned}
\int\! \Phi((w,\, x)) \Phi(T^{n} ((w,\, x)))\, d \mu_{F} & = & \int\! \left( \varrho^{-1} \widetilde{\mathfrak{L}}_{F} \right)^{\!\circ n} \left( \Phi ((w,\, x)) \Phi(T^{n} ((w,\, x))) \right)\, d \mu_{F} \\
& = & \int\! \left( \varrho^{-1} \widetilde{\mathfrak{L}}_{F}\right)^{\!\circ n} \Phi((w,\, x)) \Phi((w,\, x))\, d \mu_{F} \\
& = & 0. \end{aligned}$$ Therefore, equation becomes $$\varsigma(G)^{2}\ \ =\ \ \int\! \Phi ((w,\, x))^{2}\, d \mu_{F}.$$
Let $$\widehat{X}\ \ :=\ \ \Big\{ (w_{n},\, x_{n})_{n\, \le\, 0} \in X^{- \mathbb{N}}\ :\ T((w_{n - 1},\, x_{n - 1})) = (w_{n},\, x_{n}) \Big\}.$$ For the purposes of proofs in this section, we fix the following notations. Elements in $\widehat{X}$ will be represented as $\overline{(w,\, x)} = (w_{n},\, x_{n})_{n\, \le\, 0}$. Making use of the canonical projection $${\rm Pr} : \widehat{X} \longrightarrow X\ \ \ \text{defined by}\ \ \ {\rm Pr} \left( \overline{(w,\, x)} \right)\ =\ (w_{0},\, x_{0}),$$ we denote and define the natural extension of the map $T : X \longrightarrow X$ on $\widehat{X}$ by $$\widehat{T}\ :\ \widehat{X} \longrightarrow \widehat{X}\ \ \ \text{such that}\ \ \ {\rm Pr} \left( \widehat{T} \left( \overline{(w,\, x)} \right) \right)\ =\ T((w_{0},\, x_{0})).$$ Given a function $\Phi \in \mathcal{C} (X, \mathbb{R})$, let $\widehat{\Phi}$ be its natural extension on $\widehat{X}$ given by $$\label{widehatPhi}
\widehat{\Phi} \left( \overline{(w,\, x)} \right)\ \ =\ \ \Phi \left( {\rm Pr} \left (\overline{(w,\, x)} \right) \right)\ \ =\ \ \Phi( (w_{0},\, x_{0}) ).$$ Thus, the function $F \in \mathscr{F}_{\alpha} (X, \mathbb{R})$ that we fixed in the beginning of this section along with its equilibrium measure $\mu_{F}$ are written as $\widehat{F}$ and $\widehat{\mu_{F}}$ on the space $\widehat{X}$. Since $\mu_{F}$ is a $T$-invariant probability measure on $X$, it is clear that $\widehat{\mu_{F}}$ is a $\widehat{T}$-invariant probability measure on $\widehat{X}$. Suppose $\mathscr{B}$ is a $\sigma$-algebra on $X$, define a sequence of $\sigma$-algebras on $\widehat{X}$ by $$\mathscr{B}_{0}\ \ :=\ \ {\rm Pr}^{- 1} \mathscr{B}\ \ \ \ \text{and}\ \ \ \ \mathscr{B}_{n}\ \ :=\ \ \left( \widehat{T} \right)^{n} \left( \mathscr{B}_{0} \right)\ \ \text{for}\ \ n \in \mathbb{N}.$$
On a probability space $(\Omega, \nu)$, let $\big\{ \mathscr{B}_{n} \big\}_{n\, \ge\, 0}$ be an increasing sequence of $\sigma$-algebras and $\big\{ \Psi_{n} : \Omega \longrightarrow \mathbb{R} \big\}_{n\, \ge\, 0}$ be a collection of functions. Then, $\big\{ \Psi_{n}, \mathscr{B}_{n} \big\}_{n\, \ge\, 0}$ is called an *increasing martingale* if $\Psi_{n}$ is $\mathscr{B}_{n}$-measurable and ${\rm E} \left[ \Psi_{n + 1} \mid \mathscr{B}_{n} \right] = \Psi_{n}$ for $n\geq 0$.
Thus, defining $$\begin{aligned}
\left( \widehat{\Phi} \right)^{n} \left( \overline{(w,\, x)} \right) & := & \widehat{\Phi} \left( \left( \widehat{T} \right)^{- 1} \left( \overline{(w,\, x)} \right) \right)\; +\; \widehat{\Phi} \left( \left( \widehat{T} \right)^{- 2} \left( \overline{(w,\, x)} \right) \right) \\
& & \hspace{+4cm} +\; \cdots +\; \widehat{\Phi} \left( \left( \widehat{T} \right)^{- n} \left( \overline{(w,\, x)} \right) \right), \end{aligned}$$ that captures the $n$-th ergodic sum $\Phi^{n}((w,\, x))$, as defined in equation , on the base space, helps us form an increasing martingale on $\widehat{X}$, related to $\Phi^{n}$.
[@ps:02] The sequence $\left\{ \left( \widehat{\Phi} \right)^{n}, \mathscr{B}_{n} \right\}_{n\, \ge\, 1}$ forms an increasing martingale on $\widehat{X}$.
Before we embark on the proof of theorem , we state the Skorokhod embedding theorem, as in Appendix I of [@hh:80]. The statement of this theorem will come in handy, in writing the proof.
Let $\left\{ \widehat{\Psi}_{n}, \mathscr{B}_{n} \right\}_{n\, \ge\, 0}$ be a zero mean and square integrable martingale on $\widehat{X}$. Then, there exists a probability space $( \Omega, \mathcal{A}, \nu)$ that supports a Brownian motion $\mathfrak{B}$ such that $\mathfrak{B}(t)$ has variance $t$, an increasing sequence of $\sigma$-algebras $\big\{ \mathcal{F}_{n} \big\}_{n\, \ge\, 0}$ and a sequence of non negative random variables $\big\{ \mathfrak{X}_{n} \big\}_{n\, \ge\, 1}$ such that if $\mathcal{S}_{0} = 0$ and $\mathcal{S}_{n} = \sum\limits_{j\, =\, 1}^{n} \mathfrak{X}_{j}$ for $n \geq 1$, then
1. $\mathfrak{Y}_{n}\ \ :=\ \ \mathfrak{B} \left( \mathcal{S}_{n} \right)\ \ \stackrel{{\rm d}}{=}\ \ \widehat{\Psi}_{n}$,\
where $\stackrel{{\rm d}}{=}$ represents equality in distribution, [*i.e.*]{}, for any Borel measurable set $V$ in $\mathbb{R}$, $$\widehat{\mu_{F}} \left( \left\{ \overline{(w,\, x)} \in \widehat{X}\ :\ \widehat{\Psi}_{n} \left( \overline{(w,\, x)} \right) \in V \right\} \right)\ \ =\ \ \nu \left( \big\{ \omega \in \Omega\ :\ \mathfrak{Y}_{n} (\omega) \in V \big\} \right);$$
2. $\mathfrak{Y}_{n}$ and $\mathcal{S}_{n}$ are $\mathcal{F}_{n}$-measurable;
3. ${\rm E}\left[ \mathfrak{X}_{n} \mid \mathcal{F}_{n - 1}) \right]\ \ =\ \ {\rm E}\left[ \left( \mathfrak{Y}_{n} - \mathfrak{Y}_{n - 1} \right)^{2} \mid \mathcal{F}_{n - 1} \right],\ \nu$-a.e. for $n \ge 1$.
We now make use of the Skorokhod embedding theorem and prove theorem .
Since $\left( \widehat{\Phi} \right)^{n}$ is a square integrable function with mean zero, we can apply the Skorokhod embedding thoerem. Further, making use of the definition of $\widehat{\Phi}$, as given in equation , we obtain $$\mathfrak{Y}_{n}\ \ \stackrel{{\rm d}}{=}\ \ \left( \widehat{\Phi} \right)^{n}\ \ \stackrel{{\rm d}}{=}\ \ \Phi^{n}.$$
Thus, in order to complete the proof of theorem , we make the following claim.
\[claim2\] Given any $\delta > 0$, $$\mathfrak{Y}_{n} (\omega)\ \ =\ \ \mathfrak{B} (n) (\omega)\; +\; O \left( n^{\frac{1}{4}\, +\, \delta} \right)\ \ \forall n \ge 0,\ \ \ \nu\text{-a.e.}$$
Pending proof of claim , it follows from the properties of Brownian motion that $$\mathfrak{Y}_{\lfloor t \rfloor}\ \ =\ \ \mathfrak{B} (t)\; +\; O \left( t^{\frac{1}{4}\, +\, \delta} \right)\ \ \forall t \ge 0,\ \ \ \nu\text{-a.e.}$$ This proves the theorem.
We now prove our claim .
Since $\mathfrak{Y}_{n} = \mathfrak{B} ( \mathcal{S}_{n} )$, we approximate $\mathcal{S}_{n}$ by $n \varsigma(G)^{2}$, as follows.
$$\begin{aligned}
\label{eqn12}
\mathcal{S}_{n}\; -\; n \varsigma(G)^{2} & = & \sum_{j\, =\, 1}^{n} \Big( \mathfrak{X}_{j}\, -\, {\rm E} \big[ \mathfrak{X}_{j} \mid \mathcal{F}_{j - 1} \big] \Big) \nonumber \\
& & +\; \sum_{j\, =1\, }^{n} \Big( {\rm E} \big[ \left( \mathfrak{Y}_{j}\, -\, \mathfrak{Y}_{j - 1} \right)^{2} \mid \mathcal{F}_{j - 1} \big]\; -\; \left( \mathfrak{Y}_{j}\, -\, \mathfrak{Y}_{j - 1} \right)^{2} \Big) \nonumber \\
& & +\; \sum_{j\, =\, 1}^{n} \Big( \mathfrak{Y}_{j}\, -\, \mathfrak{Y}_{j - 1} \Big)^{2}\; -\; n \varsigma(G)^{2}. \end{aligned}$$
Given any sequence $\left\{ \widehat{\Psi}_{n} \right\}_{n\, \ge\, 0}$ of functions and an increasing sequence of $\sigma$-algebras $\big\{ \mathcal{F}_{n} \big\}_{n\, \ge\, 0}$ such that $\widehat{\Psi}_{n}$ is $\mathcal{F}_{n}$ measurable for all $n \ge 0$, the sequence defined by $$\left\{ \mathbf{\widehat{\Psi}}_{n}\ \ :=\ \ \sum\limits_{j\, =\, 1}^{n} \left( \widehat{\Psi}_{j}\, -\, {\rm E} \big[ \widehat{\Psi}_{j} \mid \mathcal{F}_{j - 1} \big] \right),\; \mathcal{F}_{n} \right\}_{n\, \ge\, 1}$$ forms a martingale. Hence, the first and the second terms on the right hand side of equation are martingales. By the strong law of large numbers for martingales, as can be found in [@wf:71], we can see that for every $\delta > 0$ $$\begin{aligned}
\sum_{j\, =\, 1}^{n} \Big( \mathfrak{X}_{j}\, -\, {\rm E} \big[ \mathfrak{X}_{j} \mid \mathcal{F}_{j - 1} \big] \Big) & = & O \left( n^{\frac{1}{2}\, +\, \delta} \right); \\
\sum_{j\, =1\, }^{n} \Big( {\rm E} \big[ \left( \mathfrak{Y}_{j}\, -\, \mathfrak{Y}_{j - 1} \right)^{2} \mid \mathcal{F}_{j - 1} \big]\; -\; \left( \mathfrak{Y}_{j}\, -\, \mathfrak{Y}_{j - 1} \right)^{2} \Big) & = & O \left( n^{\frac{1}{2}\, +\, \delta} \right). \end{aligned}$$ We can therefore write equation as $$\label{eqn13}
\mathcal{S}_{n}\; -\; n \varsigma(G)^{2}\ \ =\ \ \sum_{j\, =\, 1}^{n} \Big( \mathfrak{Y}_{j}\, -\, \mathfrak{Y}_{j - 1} \Big)^{2}\; -\; n \varsigma(G)^{2}\; +\; O \left( n^{\frac{1}{2}\, +\, \delta} \right).$$
We estimate the sum on the right hand side of equation with the help of the following series. $$\sum_{j\, \ge\, 1} \frac{1}{j^{\frac{1}{2}\, +\, \delta}} \left[ \Big( \mathfrak{Y}_{j}\, -\, \mathfrak{Y}_{j - 1} \Big)^{2}\; -\; \varsigma(G)^{2} \right].$$
The following integrals are equal. $$\begin{aligned}
\mathfrak{I} (\delta) & := & \int\! \left( \sum_{j\, \ge\, 1} \frac{1}{j^{\frac{1}{2}\, +\, \delta}} \left[ \Big( \mathfrak{Y}_{j}\, -\, \mathfrak{Y}_{j - 1} \Big)^{2}\; -\; \varsigma(G)^{2} \right] \right)^{2}\, d \nu \\
& = & \int\! \left( \sum_{j\, \ge\, 1} \frac{1}{j^{\frac{1}{2}\, +\, \delta}} \left[ \Phi \left( T^{j} ((w, x)) \right)^{2}\, -\, \int\! \Phi^{2}\, d \mu_{F} \right] \right)^{2}\, d \mu_{F}. \end{aligned}$$
We already have that $\mathfrak{Y}_{n} \stackrel{{\rm d}}{=} \Phi^{n}$. Thus, from a proposition of Brieman, L. as in [@lb:68], we deduce that for any measurable function $\Theta : \mathbb{R}^{\mathbb{N}} \longrightarrow \mathbb{R}$, $$\int\! \Theta \Big( \big( \mathfrak{Y}_{j} (\omega) \big)_{j\, =\, 0}^{\infty} \Big)\, d \nu\ \ =\ \ \int\! \Theta \Big( \big( \Phi^{j} ((w, x)) \big)_{j\, =\, 0}^{\infty} \Big)\, d \mu_{F}.$$ The result follows from an appropriate choice of the function $\Theta$, say $$\Theta \Big( \big( y_{j} \big)_{j\, \ge\, 0} \Big)\ \ =\ \ \left( \sum_{j\, \ge\, 1} \frac{1}{j^{\frac{1}{2}\, +\, \delta}} \left[ \left[y_{j + 1}\, -\, y_{j} \right]^{2}\; -\; \int\! \Phi^{2}\, d \mu \right] \right)^{2}.$$
A simple calculation now yields that for any $\delta > 0,\ \mathfrak{I} (\delta) < \infty$. Hence, $$\sum_{j\, \ge\, 1} \frac{1}{j^{\frac{1}{2}\, +\, \delta}} \left[ \Big( \mathfrak{Y}_{j}\, -\, \mathfrak{Y}_{j - 1} \Big)^{2}\; -\; \varsigma(G)^{2} \right]\ \ <\ \ \infty,\ \ \nu\text{-a.e.}$$ Applying the Kronecker lemma as in [@hh:80], we deduce that $$\label{eqn18}
\sum_{j\, =\, 1}^{n} \left[ \Big( \mathfrak{Y}_{j}\, -\, \mathfrak{Y}_{j - 1} \Big)^{2}\; -\; n \varsigma(G)^{2} \right]\ \ =\ \ O \left( n^{\frac{1}{2}\, +\, \delta} \right).$$ Thus, from equations ) and , we have $\mathcal{S}_{n}\; -\; n \varsigma(G)^{2}\ =\ O \left( n^{\frac{1}{2}\, +\, \delta} \right),\ \nu$-a.e.
Finally, defining $\widetilde{\mathfrak{B}} (t) := \mathfrak{B} ( t \varsigma(G)^{2} )$, we have for $n \ge 0$, $$\mathfrak{B} ( \mathcal{S}_{n} )\ \ =\ \ \mathfrak{B} \left( n \varsigma(G)^{2} \right)\; +\; O \left( n^{\frac{1}{4}\, +\, \delta} \right)\ \ =\ \ \widetilde{\mathfrak{B}} (n)\; +\; O \left( n^{\frac{1}{4}\, +\, \delta} \right),\ \ \nu\text{-a.e.}$$ This proves the equation in claim , namely, $$\mathfrak{Y}_{n}\ \ =\ \ \widetilde{\mathfrak{B}} (n)\; +\; O \left( n^{\frac{1}{4}\, +\, \delta} \right),\ \ \nu\text{-a.e.}$$
We now proceed to prove the next theorem on almost sure invariance principles for simultaneous action of the interval maps, as stated in theorem . We draw motivation from the proof of a similar result in an article by Haydn, Nicol, Török and Vaienti in [@hntv:17] and achieve a better bound. We first state a theorem due to Cuny and Merlévede as in [@cm:15], that would be helpful in our proof.
[@cm:15] \[CM 2.3\] Let $\big\{ U_{n} \big\}_{n\, \ge\, 0}$ be a sequence of square integrable random variables adapted to some non-increasing sequence of $\sigma$-algebras $\big\{ \mathscr{A}_{n} \big\}_{n\, \ge\, 0}$ on $\mathbb{R}$. Assume that $${\rm E} \big[ U_{n} \mid \mathscr{A}_{n + 1} \big]\ \ =\ \ 0\ \text{a.s.};\ \ \ \ \varsigma_{n}^{2}\ \ =\ \ \sum_{k\, =\, 0}^{n - 1} {\rm E} \big[ U_{k}^{2} \big]\ \ \to\ \ \infty;\ \ \ \ \sup\limits_{n\, \ge\, 0} {\rm E} \big[ U_{n}^{2} \big]\ \ <\ \ \infty.$$ Let $\big\{ a_{n} \big\}_{n\, \ge\, 0}$ be a non-decreasing sequence of positive numbers such that $$\left\{ \frac{a_{n}}{\varsigma_{n}} \right\}_{n\, \ge\, 0}\ \ \text{is non-decreasing}\ \ \ \text{and}\ \ \ \left\{ \frac{a_{n}}{\varsigma_{n}^{2}} \right\}_{n\, \ge\, 0}\ \ \text{is non-increasing}.$$ Further, assume that
1. $\sum\limits_{k\, =\, 0}^{n - 1} \Big( {\rm E} \big[ U_{k}^{2} \mid \mathscr{A}_{k + 1} \big]\; -\; {\rm E} \big[ U_{k}^{2} \big] \Big)\ \ =\ \ o(a_{n}),\ \lambda$-a.s.;
2. $\sum\limits_{n\, \ge\, 0} a_{n}^{- r} {\rm E} \big[ |U_{n}|^{2r} \big]\ \ <\ \ \infty$ for some $1 \le r \le 2$.
Then enlarging our probability space, if necessary, it is possible to find a sequence $\big\{ \mathcal{U}_{n} \big\}_{n\, \ge\, 0}$ of independent centered Gaussian variables with ${\rm E} \big[ \mathcal{U}_{n}^{2} \big] = {\rm E} \big[ U_{n}^{2} \big]$ such that $$\sup_{0\, \le\, k\, \le\, n - 1} \left| \sum_{j\, =\, 0}^{k} U_{j}\; -\; \sum_{j\, =\, 0}^{k} \mathcal{U}_{j} \right|\ \ =\ \ o \left( \left[ a_{n} \left( \left| \log \left( \frac{\varsigma_{n}^{2}}{a_{n}} \right) \right|\; +\; \log \log a_{n} \right) \right]^{\frac{1}{2}} \right),\ \ \ \lambda\text{-a.s.}$$
Note that the assertion of theorem can be rewritten by considering another probability space $(\Omega, \mathscr{A}, \nu)$ and a sequence of random variables, say $\big\{ \mathcal{V}_{n} \big\}_{n\, \ge\, 0}$ such that $U_{n} \stackrel{{\rm d}}{=} \mathcal{V}_{n}$. Then, $$\sup_{0\, \le\, k\, \le\, n - 1} \left| \sum_{j\, =\, 0}^{k} \mathcal{V}_{j}\; -\; \sum_{j\, =\, 0}^{k} \mathcal{U}_{j} \right|\ \ =\ \ o \left( \left[ a_{n} \left( \left| \log \left( \frac{\varsigma_{n}^{2}}{a_{n}} \right) \right|\; +\; \log \log a_{n} \right) \right]^{\frac{1}{2}} \right),\ \ \ \nu\text{-a.s.}$$
We will now prove theorem .
Recall the definition of the space $\mathscr{F}_{\alpha}^{\lambda} (I, \mathbb{R})$ from equation , $$\mathscr{F}_{\alpha}^{\lambda} (I, \mathbb{R})\ \ :=\ \ \left\{ f \in \mathscr{F}_{\alpha} (I, \mathbb{R})\ :\ \int\! f \, d \lambda = 0 \right\},$$ and the property that for $f_{d} = - \log | T_{d}' |$, the operator $\mathcal{L}_{f_{d}}^{(d)}$ preserves the space $\mathscr{F}_{\alpha}^{\lambda} (I, \mathbb{R})$ for all $1 \le d \le N$.
Let $g \in \mathscr{F}_{\alpha}^{\lambda} (I, \mathbb{R})$ and $w \in \Sigma_{N}^{+}$. Then, define a sequence of $\sigma$-algebras $$\mathscr{B}_{w}^{n}\ \ :=\ \ \left( \mathscr{O}_{(w_{1}\, w_{2}\, \cdots\, w_{n})} \right)^{- 1} \mathscr{B}\ \ \ \ \text{for}\ \ n \ge 0,$$ where $\mathscr{B}$ is the Borel $\sigma$-algebra on $I$.
Suppose for all $n \ge 1$, we denote by $\mathfrak{g}_{w}^{n}$, the sum $$\mathfrak{g}_{w}^{n}\ \ :=\ \ \Big( \mathcal{L}_{f_{w_{n}}}^{(w_{n})} \Big) g\; +\; \Big( \mathcal{L}_{f_{w_{n}}}^{(w_{n})} \mathcal{L}_{f_{w_{n - 1}}}^{(w_{n - 1})} \Big) g\; +\; \cdots\; +\; \Big( \mathcal{L}_{f_{w_{n}}}^{(w_{n})} \mathcal{L}_{f_{w_{n - 1}}}^{(w_{n - 1})} \cdots \mathcal{L}_{f_{w_{1}}}^{(w_{1})} \Big) g$$ and $\mathfrak{g}_{w}^{0} = 0$ for all $w \in \Sigma_{N}^{+}$. It is easy to see that $$\mathcal{L}_{f_{w_{n + 1}}}^{(w_{n + 1})} \mathbb{g}_{w}^{n}\ \ =\ \ 0,\ \ \ \ \text{where}\ \ \ \ \mathbb{g}_{w}^{n}\ \ =\ \ g\; +\; \mathfrak{g}_{w}^{n}\; -\; T_{w_{n + 1}} \mathfrak{g}_{w}^{n + 1}.$$ Defining $\mathbb{h}_{w}^{n} = \mathscr{O}_{(w_{1}\, w_{2}\, \cdots\, w_{n})} \left( \mathbb{g}_{w}^{n} \right)$, one can observe that $\mathbb{h}_{w}^{n}$ agrees with the definition of a reverse martingale difference sequence for the sequence of $\sigma$-algebras $\mathscr{B}_{w}^{n}$, as defined in Conze and Raugi [@cr:07], as given below.
Given a sequence of random variables $\big\{ X_{n} \big\}_{n\, \in\, \mathbb{N}}$ adapted to a non-increasing sequence of $\sigma$- algebras $\big\{ \mathscr{A}_{n} \big\}_{n\, \in\, \mathbb{N}},\ \big\{ X_{n},\, \mathscr{A}_{n} \big\}_{n\, \in\, \mathbb{N}}$ is a *reverse martingale* or equivalently, $\big\{ X_{n} \big\}_{n\, \in\, \mathbb{N}}$ is a reverse martingale adapted to $\big\{ \mathscr{A}_{n} \big\}_{n\, \in\, \mathbb{N}}$ if $\bigg\{ \widetilde{X}_{n},\, \widetilde{\mathscr{A}}_{n} \bigg\}_{n\, \le\, -1}$ forms a martingale, where $\widetilde{X}_{n} = X _{- n}$ and $\widetilde{\mathscr{A}}_{n} = \mathscr{A}_{- n}$ for each $n \in - \mathbb{N}$.
Now, $$\begin{aligned}
\sum_{k\, =\, 0}^{n - 1} \mathbb{h}_{w}^{k} & = & \sum_{k\, =\, 0}^{n - 1} \left( \mathscr{O}_{(w_{1}\, w_{2}\, \cdots\, w_{k})} g\, +\, \mathscr{O}_{(w_{1}\, w_{2}\, \cdots\, w_{k})} \mathfrak{g}_{w}^{k}\, -\, \mathscr{O}_{(w_{1}\, w_{2}\, \cdots\, w_{k + 1})} \mathfrak{g}_{w}^{k + 1} \right) \\
& = & \sum_{k\, =\, 0}^{n - 1} \mathscr{O}_{(w_{1}\, w_{2}\, \cdots\, w_{k})} g\, -\, \mathscr{O}_{(w_{1}\, w_{2}\, \cdots\, w_{n})} \mathfrak{g}_{w}^{n}. \end{aligned}$$ Further, $\left\Vert \mathfrak{g}_{w}^{n} \right\Vert_{\alpha}$ is uniformly bounded. Hence, $$\begin{aligned}
{\rm E} \left[ \left( \sum_{k\, =\, 0}^{n - 1} \mathbb{h}_{w}^{k} \right)^{2} \right] & = & {\rm E} \left[ \left( \sum_{k\, =\, 0}^{n - 1} \mathscr{O}_{(w_{1}\, w_{2}\, \cdots\, w_{k})} g \right)^{2} \right]\; +\; {\rm E} \big[ \left( \mathscr{O}_{(w_{1}\, w_{2}\, \cdots\, w_{n})} \mathfrak{g}_{w}^{n} \right)^{2} \big] \\
& & \hspace{+2cm} -\; 2 {\rm E} \left[ \left( \sum_{k\, =\, 0}^{n - 1} \mathscr{O}_{(w_{1}\, w_{2}\, \cdots\, w_{k})} g \right) \big( \mathscr{O}_{(w_{1}\, w_{2}\, \cdots\, w_{n})} \mathfrak{g}_{w}^{n} \big) \right] \\
& = & \left( \varsigma_{w}^{(n)} (g) \right)^{2}\; +\; o \left( \left( \varsigma_{w}^{(n)} (g) \right) \right), \end{aligned}$$ where we recall the definition of $\varsigma_{w}^{(n)} (g)$ from equation as $$\left( \varsigma_{w}^{(n)} (g) \right)^{2}\ \ =\ \ \int\! \left( g_{w}^{n} \right)^{2}\, d \lambda.$$
Haydn, Nicol, Török and Vaienti in [@hntv:17] show us that ${\rm E} \big[ \mathbb{h}_{w}^{j} \mathbb{h}_{w}^{k} \big] = 0$, for $j \ne k$ and therefore $$\sum_{k\, =\, 0}^{n - 1} {\rm E} \big[ \left( \mathbb{h}_{w}^{k} \right)^{2} \big]\ \ =\ \ {\rm E} \left[ \left( \sum_{k\, =\, 0}^{n - 1} \mathbb{h}_{w}^{k} \right)^{2} \right]\ \ =\ \ \left( \varsigma_{w}^{(n)} (g) \right)^{2}\; +\; o \left( \left( \varsigma_{w}^{(n)} (g) \right) \right),$$ which implies, $\sum\limits_{k\, =\, 0}^{n - 1} {\rm E} \big[\left( \mathbb{h}_{w}^{k} \right)^{2} \big] \to \infty$.
Thus, we have constructed a sequence of square integrable random variables $\left\{ \mathbb{h}_{w}^{n} \right\}_{n\, \ge\, 0}$ adapted to a non-increasing sequence of $\sigma$-algebras $\big\{ \mathscr{B}_{w}^{n} \big\}_{n\, \ge\, 0}$ that satisfies $${\rm E} \big[ \mathbb{h}_{w}^{n} \mid \mathscr{B}_{w}^{n + 1} \big]\ \ =\ \ 0\ \text{a.s.};\ \ \ \ \sum_{k\, =\, 0}^{n - 1} {\rm E} \big[ \left( \mathbb{h}_{w}^{k} \right)^{2} \big]\ \ \to\ \ \infty;\ \ \ \ \sup\limits_{n\, \ge\, 0} {\rm E} \big[ \left( \mathbb{h}_{w}^{n} \right)^{2} \big]\ \ <\ \ \infty.$$ Further, defining a sequence $\left\{ a_{n} := \left( \varsigma_{w}^{(n)} (g) \right)^{1\, +\, \epsilon} \right\}_{n\, \ge\, 0}$ for some sufficiently small $\epsilon > 0$, we observe that the sequences satisfy $$\left\{ \frac{a_{n}}{\left( \varsigma_{w}^{(n)} (g) \right)} \right\}_{n\, \ge\, 0}\ \ \text{is non-decreasing}\ \ \ \text{and}\ \ \ \left\{ \frac{a_{n}}{\left( \varsigma_{w}^{(n)} (g) \right)^{2}} \right\}_{n\, \ge\, 0}\ \ \text{is non-increasing}.$$
Thus, in order to appeal to theorem and exploit the assertions there, we only need to verify the two enumerated assumptions in the statement. We will, for now take the relevant assumptions to be true and proceed to complete the proof of theorem . Once the proof is complete, we will complete the verifications of the enumerated statements.
By theorem , we have that there exist sequences $\big\{ \mathcal{Y}_{w}^{n} \big\}_{n\, \ge\, 0}$ and $\big\{ Z_{w}^{n} \big\}_{n\, \ge\, 0}$ such that $$\begin{aligned}
\sup_{0\, \le\, k\, \le\, n - 1} \left| \sum_{j\, =\, 0}^{k} \mathcal{Y}_{w}^{j}\; -\; \sum_{j\, =\, 0}^{k} Z_{w}^{j} \right| & = & o \left( \left[ a_{n} \left( \left| \log \left( \frac{\varsigma_{n}^{2}}{a_{n}} \right) \right|\; +\; \log \log a_{n} \right) \right]^{\frac{1}{2}} \right) \\
& = & o \left( \left( n^{\frac{1}{2}\, +\, \epsilon} \left( \left| \log \left( n^{\frac{1}{2}\, -\, \epsilon} \right) \right|\; +\; \log \log n^{\frac{1}{2}\, +\, \epsilon} \right) \right)^{\frac{1}{2}} \right) \\
& = & O \left( n^{\frac{1}{4}\, +\, \delta} \right),\ \ \ \ \nu\text{-a.s., for some}\ \delta > 0. \end{aligned}$$
Further, by the result due to Cuny and Merlevede [@cm:15], we know that $$\sum_{j\, =\, 0}^{n - 1} {\rm E} \big[ \left( Z_{w}^{j} \right)^{2} \big]\ \ =\ \ \left( \varsigma_{w}^{(n)} (g) \right)^{2}\; +\; O \left( \varsigma_{w}^{(n)} (g) \right)\ \ =\ \ \left( \varsigma_{w}^{(n)} (g) \right)^{2}\; +\; o \left( n^{\frac{1}{2}\; +\; \delta'} \right),$$ for some $\delta' > 0$. Hence, we can replace the random variables with a standard Brownian motion $\big\{ \mathfrak{B}^{*} (t) \big\}_{t\, \ge\, 0}$ such that $$\sup\limits_{0\, \le\, k\, \le\, n - 1} \left| \sum_{j\, =\, 0}^{k} Z_{w}^{j}\; -\; \mathfrak{B}^{*} \left( \left( \varsigma_{w}^{(k)} (g) \right)^{2}\; +\; o \left( k^{\frac{1}{2}\, +\, \delta'} \right) \right) \right|\ \ =\ \ 0,\ \ \ \nu\text{-a.s.}$$ which implies that $$\sup\limits_{0\, \le\, k\, \le\, n - 1} \left| \sum_{j\, =\, 0}^{k} Z_{w}^{j}\; -\; \mathfrak{B}^{*} \left( \left( \varsigma_{w}^{(k)} (g) \right)^{2} \right) \right|\ \ =\ \ o \left( n^{\frac{1}{4}\, +\, \delta'} \right),\ \ \ \nu\text{-a.s.}$$ Therefore, if we replace the independent centered Gaussian variables with the standard Brownian motion, we get $$\sum_{j\, =\, 0}^{n - 1} \mathcal{Y}_{w}^{j}\; -\; \mathfrak{B}^{*} \left( \left( \varsigma_{w}^{(n)} (g) \right)^{2} \right)\ \ =\ \ O \left( n^{\frac{1}{4}\, +\, \delta} \right),\ \ \ \ \nu\text{-a.s.}$$ Further, it is easy to see that there exist a sequence of random variables $\big\{ Y_{w}^{n} \big\}_{n\, \ge\, 0}$ such that $$\left\vert Y_{w}^{n}\; -\; \sum_{j\, =\, 0}^{n} \mathcal{Y}_{w}^{j} \right\vert\ \ =\ \ O(1),$$ and $g_{w}^{n}$ and $Y_{w}^{n}$ are equal in distribution, thus proving theorem .
We now complete the verifications of the enumerated conditions in theorem .
\[cond1\] The first of the enumerated condition in theorem looks like $$\sum_{k\, =\, 0}^{n - 1} \left( {\rm E} \left[ \left( \mathbb{h}_{w}^{k} \right)^{2} \mid \mathscr{B}_{w}^{k + 1} \right]\; -\; {\rm E} \left[ \left( \mathbb{h}_{w}^{k} \right)^{2} \right] \right)\ \ =\ \ o(a_{n}).$$
\[cond2\] The second of the enumerated conditions in theorem looks like $$\sum\limits_{n\, \ge\, 0} a_{n}^{- r} {\rm E} \big[ \left| \mathbb{h}_{w}^{n} \right|^{2r} \big]\ \ <\ \ \infty,\ \ \text{for some}\ 1 \le r \le 2.$$
From Conze and Raugi [@cr:07], we get that $${\rm E} \big[ \left( \mathbb{h}_{w}^{n} \right)^{2} \mid \mathscr{B}_{w}^{n + 1} \big]\ \ =\ \ \mathscr{O}_{(w_{1}\, w_{2}\, \cdots\, w_{n})} \left( \mathcal{L}_{f_{w_{n + 1}}}^{(w_{n + 1})} \left( \mathbb{g}_{w}^{n} \right)^{2} \right)\ \ \text{and}$$ $$\int\! \left| \sum_{k\, =\, 0}^{n - 1} \left( {\rm E} \left[ \left( \mathbb{h}_{w}^{k} \right)^{2} \mid \mathscr{B}_{w}^{k + 1} \right]\; -\; {\rm E} \left[ \left( \mathbb{h}_{w}^{k} \right)^{2} \right] \right) \right|^{2}\, d \lambda\ \ \le\ \ C_{21} \sum_{k\, =\, 0}^{n - 1} {\rm E}\big[ \left( \mathbb{h}_{w}^{k} \right)^{2} \big]\ \ \le\ \ C_{22} \left( \varsigma_{w}^{(n)} (g) \right)^{2},$$ for some positive constants $C_{21}$ and $C_{22}$. Hence, by Gal Koksma Theorem as in [@zl:14; @sw:60], we have $$\begin{aligned}
\sum_{k\, =\, 0}^{n - 1} \left( {\rm E} \left[ \left( \mathbb{h}_{w}^{k} \right)^{2} \mid \mathscr{B}_{w}^{k + 1} \right]\; -\; {\rm E} \left[ \left( \mathbb{h}_{w}^{k}\right)^{2} \right] \right) & = & O \left( \left( \varsigma_{w}^{(n)} (g) \right)\; +\; \log^{\frac{3}{2}\, +\, \epsilon} \left( \left( \varsigma_{w}^{(n)} (g) \right)^{2} \right) \right) \\
& = & o \left( \left( \varsigma_{w}^{(n)} (g) \right)^{1\, +\, \epsilon'} \right) \\
& = & o(a_{n}), \end{aligned}$$ where $\epsilon'$ is some small positive quantity, possibly less than or equal to $\epsilon$.
Here, we begin with an easy observation that $\left( \varsigma_{w}^{(n)} (g) \right)^{2} = O(n)$. Thus given $\delta > 0$, there exists a threshold $M_{\delta} \in \mathbb{N}$ and a positive real number $C_{23} > 0$ such that $$\left| \frac{\varsigma_{w}^{(n)} (g)}{\sqrt{n}}\; -\; C_{23} \right|\ \ \le\ \ \delta,\ \ \ \forall n \ge M_{\delta}.$$ Suppose $m < M_{\delta}$. Then, by the Archimedean property of the reals, we have $$\varsigma_{w}^{(m)} (g)\ \ \ge\ \ \sqrt{m} D_{m}.$$ Choosing $C_{24} = \min \big\{ C_{23} - \delta,\, D_{1},\, D_{2},\, \cdots,\, D_{N_{\delta}} \big\}$, we have $$\varsigma_{w}^{(n)} (g)\ \ \ge\ \ \sqrt{n} C_{24},\ \ \ \ \forall n \in \mathbb{N}.$$
For $r = 2$ in condition (2) in the enumerated statement of theorem , we have $$\begin{aligned}
\sum_{n\, \ge\, 0} a_{n}^{- 2} {\rm E} \left[ \left\vert \mathbb{h}_{w}^{n} \right\vert^{4} \right] & = & \sum_{n\, \ge\, 0} \left( \varsigma_{w}^{(n)} (g) \right)^{- (2\, +\, \epsilon)} {\rm E} \left[ \left\vert \mathbb{h}_{w}^{n} \right\vert^{4} \right] \\
& \le & C_{25}\; +\; \sum_{n\, \ge\, 1} \frac{1}{C_{24}^{2\, +\, \epsilon} n^{1\, +\, 2 \epsilon}} {\rm E} \left[ \left\vert \mathbb{h}_{w}^{n} \right\vert^{4} \right] \\
& \le & C_{25}\; +\; C_{26} \sum_{n\, \ge\, 1} \frac{1}{n^{1\, +\, 2 \epsilon}} \\
& < & \infty, \end{aligned}$$ since $\sup\limits_{n\, \ge\, 0} {\rm E} \left[ \left\vert \mathbb{h}_{w}^{n} \right\vert^{4} \right] < \infty$.
Finally, when we replace the standard Brownian motion with a Brownian motion $\big\{ \widetilde{\mathfrak{B}^{*}} (t) \big\}_{t\, \ge\, 0}$ such that $\widetilde{\mathfrak{B}^{*}} (t)$ has variance $t \left( \varsigma_{w} (g) \right)^{2}$, we get $$Y_{w}^{n}\; -\; \widetilde{\mathfrak{B}^{*}} \left( \left( n \right) \right)\ \ =\ \ O \left( n^{\frac{1}{2}\, -\, \gamma} \right),\ \ \ \ \nu\text{-a.s., for some}\ \gamma > 0.$$
Proofs of other statistical properties {#seccor}
======================================
In this section, we write the proofs of the other statistical properties such as the central limit theorem, weak invariance principles and the law of iterated logarithms, as mentioned in theorems and .
1. [*Proof of the central limit theorem*]{}: Recall from the proof of theorem that
1. $\mathfrak{Y}_{n}\ \ \stackrel{\text{d}}{=}\ \ \Phi^{n}\ \ \forall n \ge 1$ and
2. $\mathfrak{Y}_{n}\ \ =\ \ \mathfrak{B} (n)\; +\; O \left( n^{\frac{1}{4}\, +\, \delta} \right),\ \ \nu$-a.e.
Further, owing to condition (b) above, we have that for some $\epsilon > 0$, $$\frac{1}{\sqrt{n}} \mathfrak{Y}_{n}\ \ =\ \ \frac{1}{\sqrt{n}} \mathfrak{B} (n)\; +\; O\left( n^{- \epsilon} \right),\ \ \nu\text{-a.e.}$$ and therefore, $\frac{1}{\sqrt{n}} \big( \mathfrak{Y}_{n} - \mathfrak{B} (n) \big) \stackrel{{\rm p}}{\longrightarrow} 0$, [*i.e.*]{}, converges in probability to $0$ as $n \rightarrow \infty$. But $\frac{1}{\sqrt{n}} \mathfrak{B} (n)$ is a normal distribution with mean zero and variance $\varsigma(G)^{2}$ for all $n \geq 1$. Further, owing to condition (a), we have that $$\frac{1}{\sqrt{n}} \mathfrak{Y}_{n}\ \ \stackrel{\text{d}}{=}\ \ \frac{1}{\sqrt{n}} \Phi^{n}\ \ \forall n \ge 1.$$ Making use of both the conditions, we have $$\frac{1}{\sqrt{n}} \Phi^{n}\ \ \stackrel{\text{d}}{\longrightarrow}\ \ \mathcal{N} \left( 0,\, \varsigma(G)^{2} \right),$$ where $\mathcal{N} \left( 0,\, \varsigma^{2} \right)$ denotes the normal distribution with mean $0$ and variance $\varsigma^{2}$. The result follows since $G^{n}((w, x)) = \Phi^{n} ((w, x)) + O(1)$.
2. [*Proof of the law of iterated logarithms*]{}: If $\Phi$ in theorem satisfies the law of iterated logarithms, then so does $G$, since $$\limsup_{n\, \rightarrow\, \infty} \left[ \frac{G^{n} ((w, x)) - \Phi^{n} ((w, x))}{\varsigma(G) \sqrt{2n \log \log n}} \right]\ \ \stackrel{\text{p}}{\longrightarrow}\ \ 0\ \ \ \ \text{as}\ n \rightarrow \infty.$$ The following lemma is the key to proving the law of iterated logarithms for the given function $\Phi$.
Any Brownian motion with variance $\varsigma^{2}$ satisfies the law of iterated logarithms [*i.e.*]{}, $$\limsup_{t\, \rightarrow\, \infty} \frac{\mathfrak{B} (t) (\omega)}{\varsigma \sqrt{2t \log \log t}}\ \ =\ \ 1,\ \ \ \nu\text{-a.e.}$$
Since we have a Brownian motion which by condition (b) in the proof of the central limit theorem is equal to $\mathfrak{Y}_{n} + O \left( n^{\frac{1}{4}\, +\, \delta} \right)$, we have by theorem that $$\limsup_{n\, \rightarrow\, \infty} \frac{\mathfrak{Y}_{n} (\omega)}{\varsigma \sqrt{2n \log \log n}}\ \ =\ \ 1,\ \ \ \nu\text{-a.e.}$$ Since $\mathfrak{Y}_{n}$ and $\Phi^{n}$ have the same distribution with the latter having variance $\big( \varsigma (G) \big)^{2}$, we conclude that $$\limsup_{n\, \rightarrow\, \infty} \frac{\Phi^{n} ((w, x))}{\varsigma(G) \sqrt{2n \log \log n}}\ \ =\ \ 1,\ \ \ \mu_{F}\text{-a.e.}$$
We now prove theorem to conclude the proofs of all the theorems in this paper.
The proofs of both the statements run [*mutatis mutandis*]{} as the proofs of their analogous statements in theorem . Hence, we merely highlight the following for readers’ convenience.
1. $Y_{w}^{n}\ \ \stackrel{\text{d}}{=}\ \ g_{w}^{n}\ \ \ \forall n \geq 1$.
2. $Y_{w}^{n}\ \ =\ \ \widetilde{\mathfrak{B}}^{*} (n) + O \left( n^{\frac{1}{2}\, -\, \gamma} \right),\ \ \nu$-a.e.
Further, $$\limsup_{n\, \rightarrow\, \infty} \frac{g_{w}^{n}(x)}{\varsigma_{w}(g) \sqrt{2n \log \log n}}\ \ =\ \ 1,\ \ \ \lambda\text{-a.e.}$$
Concluding remarks {#concl}
==================
As stated in the introductory section, the theorems proved in this paper are easily transferable to several other analogous settings of dynamical systems. We conclude this paper by merely pointing to some of those.
- Instead of working with the specified interval maps $T_{d} : I \longrightarrow I$ of degree $(d + 1)$ for $1 \le d \le N$ given by $T_{d} (x) = (d + 1) x \pmod 1$, one may well consider the action of any $N$ (piecewise) linear interval maps, $S_{d} : I \longrightarrow I;\ 1 \le d \le N$, each with integer degree at least $2$. The measure, in this case still remains Lebesgue.
- One might as well consider the simple monomial maps $P_{d};\ 1 \le d \le N$, defined on the Riemann sphere $\overline{\mathbb{C}} = \mathbb{C} \cup \{ \infty \}$ and given by $P_{d} (z) = z^{k_{d}}$ where $k_{d} \ge 2$ for all $1 \le d \le N$. We know from, say [@afb:91], that the Julia set $\mathbb{J} (P_{d})$ of the monomial map $P_{d}$ is the unit circle $\mathbb{S}^{1}$ in $\mathbb{C}$. In such a case, the Julia set $\mathbb{J} (P)$ of the skew-product map $P$ appropriately defined analogous to equation is also the unit circle, as one may find from [@hs:00]. Owing to the Julia set $\mathbb{J} (P)$ being completely $P$-invariant, one may undertake an analogous study, as done in this paper, to the dynamics generated by the monomial maps $P_{d};\ 1 \le d \le N$ restricted on the Julia set $\mathbb{J} (P) = \mathbb{S}^{1} \subset \mathbb{C}$ and obtain analogous results, employing the Haar measure on $\mathbb{S}^{1}$.
- Let $R_{d};\ 1 \le d \le N$ be a collection of rational maps acting on the Riemann sphere $\overline{\mathbb{C}}$; each with degree $k_{d} \ge 2$. We suppose that the rational maps are so chosen that the Julia set $\mathbb{J} (R_{d})$ of the map $R_{d}$ is topologically connected. Then, defining the skew-product map $R$ appropriately and restricting its action on the $R$-invariant Julia set $\mathbb{J} (R)$, as one may obtain from [@hs:00], it is possible to investigate analogous results for Boyd’s measure, as defined in [@db:99]. It must be borne in mind that the Boyd’s measure is a generalisation of the Lyubich’s measure, as in [@myl:86].
- Finally, we consider the dynamical system obtained by iterating certain relations on $\mathbb{C}$. The relation can be explained as the zero set of a polynomial, say $Q \in \mathbb{C}[\zeta,\, \omega]$ of a certain form such that:
- $Q \big( \cdot,\, \omega \big)$ and $Q \big( \zeta,\, \cdot \big)$ are generically multiple-valued;
- if $\mathscr{G}_{Q}$ denotes the biprojective completion of $\big\{ Q\, =\, 0 \big\}$ in $\overline{\mathbb{C}}\, \times\, \overline{\mathbb{C}}$, then no irreducible component of $\mathscr{G}_{Q}$ is of the form $\big\{ a \big\}\, \times\, \overline{\mathbb{C}}$ or $\overline{\mathbb{C}}\, \times\, \big\{ a \big\}$, where $a \in \overline{\mathbb{C}}$.
Such dynamical systems have been studied by Dinh and Sibony in [@ds:06] and Bharali and Sridharan in [@bs:16]. Upon satisfying certain technical conditions, one may study analogous dynamical and statistical properties with respect to the Dinh-Sibony measure, when the action of the holomorphic correspondence $Q \big( \zeta,\, \omega \big)$ is restricted on the support of the Dinh-Sibony measure, as defined in [@ds:06; @bs:16].
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
We consider the problem of viscosity solution of integro-partial differential equation(IPDE in short) with one obstacle via the solution of reflected backward stochastic differential equations(RBSDE in short) with jumps. We show existence and uniqueness of a continuous viscosity solution of equation with non local terms, in case the generator is not monotonous and Levy’s measure is infinite.\
author:
- '<span style="font-variant:small-caps;">L. SYLLA</span>'
date: 'December, 21, 2017'
title: ': '
---
**Keywords**: Integro-partial differential equation; Reflected stochastic differential equations with jumps; Viscosity solution; Non-local operator.\
\
**MSC 2010 subject classifications**: 35D40, 35R09, 60H30.
Introduction
============
We consider the following system of integro-partial differential equation with one-obstacle $\ell$, which is a function of $(t,x)$: $\forall i\in\{1,\ldots,m\}$, $$\label{eq1}
\left
\{\begin{array}{ll}
\min\Big\{ u^{i}(t,x)-\ell(t,x);-\partial_{t}u^{i}(t,x)-b(t,x)^{\top}\mathrm{D}_{x}u^{i}(t,x)-\frac{1}{2}\mathrm{Tr}(\sigma\sigma^{\top}(t,x)\mathrm{D}^{2}_{xx}u^{i}(t,x))\\
\quad\quad-\mathrm{K}_{i}u^{i}(t,x)-\mathit{h}^{(i)}(t,x,u^{i}(t,x),(\sigma^{\top}\mathrm{D}_{x}u^{i})(t,x),\mathrm{B}_{i}u^{i}(t,x))\Big\}=0,\quad (t,x)\in\left[ 0,T\right] \times\mathbb{R}^{k};\\
u^{i}(T,x)=g^{i}(x);
\end{array}
\right.$$ where the operators $\mathrm{B}_{i}$ and $\mathrm{K}_{i}$ are defined as follows: $$\begin{aligned}
\mathrm{B}_{i}u^{i}(t,x) & = & \displaystyle\int_{\mathrm{E}}\gamma^{i}(t,x,e)(u^{i}(t,x+\beta(t,x,e))-u^{i}(t,x))\lambda(\mathrm{d} e);\label{2.2}\\
\mathrm{K}_{i}u^{i}(t,x) & = & \displaystyle\int_{\mathrm{E}}(u^{i}(t,x+\beta(t,x,e))-u^{i}(t,x)-\beta(t,x,e)^{\top}\mathrm{D}_{x}u^{i}(t,x))\lambda(de).\nonumber\end{aligned}$$ The resolution of (\[eq1\]) is in connection with the following system of backward stochastic differential equations with jumps and one-obstacle $\ell$:
$$\label{eq2}
\left
\{\begin{array}{ll}
(i)~dY^{i;t,x}_{s}=-f^{(i)}(s,X^{t,x}_{s},(Y^{i;t,x}_{s})_{i=1,m},Z^{i;t,x}_{s},U^{i;t,x}_{s})ds-
\mathrm{d}\mathrm{K}^{i;t,x}_{s}\\
\quad\quad\quad\quad\quad\quad
\quad\quad+Z^{i;t,x}_{s}\mathrm{d}
\mathrm{B}_{s}+\displaystyle\int_{\mathrm{E}}\mathrm{U}^{i;t,x}
_{s}(e)\tilde{\mu}(\mathrm{d}s,\mathrm{d}e),\quad s\leq T;\\
(ii)~Y^{i;t,x}_{s}\geq \ell(s,X^{t,x}_{s})~\textrm{and}~ \displaystyle\int^{T}_{0}(Y^{i;t,x}_{s}- \ell(s,X^{t,x}_{s}))\mathrm{d}\mathrm{K}^{i;t,x}_{s}=0;
\end{array}
\right.$$
and\
the following standard stochastic differential equation of diffusion-jump type: $$\label{2.4}
X^{t,x}_{s}=x+\displaystyle\int^{s}_{t}b(r,X^{t,x}_{r})\, \mathrm{d}r+\displaystyle\int^{s}_{t}\sigma(r,X^{t,x}_{r})\, \mathrm{d}B_{r}+\displaystyle\int^{s}_{t}
\displaystyle\int_{E}\beta(r,X^{t,x}_{r-},e)\tilde{\mu}(dr,de),$$ for $s\in[t,T]$ and $X^{t,x}_{s}=x$ if $s\leq t$.\
It is recalled that pioneering work was done for the resolution of (\[eq1\]), among these works we can mention those of Barles and al. [@bar] in case without obstacle, Harraj and al. [@har] in the case with two obstacles; with as common point the hypothesis of monotony on the generator and $\gamma \geq 0$. But recently Hamadène and Morlais relaxed these conditions with $\lambda(.)$ finite [@hamaMor].\
In this work we propose to solve (\[eq1\]) by relaxing the monotonicity of the generator and the positivity of $\gamma$ and assuming that $\lambda=\infty$.\
Our paper is organized as follows: in the next section we give the notations and the assumptions of our objects; in section $3$ we recall a number of existing results; in section $4$ we build estimates and properties for a good resolution of our problem; section $5$ is reserved to give our main result and the section $6$ for doing an extension of our result.\
And in the end, classical definition of the concept of viscosity solution is put in appendix.
Notations and assumptions
=========================
Let $\left(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\leq T},\mathbb{P}\right)$ be a stochastic basis such that $\mathcal{F}_{0}$ contains all $\mathbb{P}-$null sets of $\mathcal{F}$, and $\mathcal{F}_{t}=\mathcal{F}_{t+}:=\bigcap_{\epsilon>0}
\mathcal{F}_{t+\epsilon},~t\geq 0$, and we suppose that the filtration is generated by the two mutually independents processes:\
(i) $B:=(B_{t})_{t\geq 0}$ a $d$-dimensional Brownian motion and,\
(ii) a Poisson random measure $\mu$ on $\mathbb{R}^{+}\times\mathrm{E}$ where $\mathrm{E}:=\mathbb{R}^{\ell}-\{0\}$ is equipped with its Borel field $\mathcal{E}$ $(\ell\geq 1)$. The compensator $\nu(\mathrm{d}t,\mathrm{d}e)=\mathrm{d}t\lambda(\mathrm{d}e)$ is such that $\{\tilde{\mu}(\left[0,t\right]\times A)=(\mu-\lambda)(\left[ 0,t\right]\times A)\}_{t\geq 0}$ is a martingale for all $A\in\mathcal{E}$ satisfying $\lambda(A)<\infty$. We also assume that $\lambda$ is a $\sigma$-finite measure on $(E,\mathcal{E})$, integrates the function $(1\wedge\mid e\mid ^{2})$ and $\lambda(E)=\infty$.\
Let’s now introduce the following spaces:\
(iii) $\mathcal{P}~(resp.~\mathbf{P})$ the field on $\left[0,T\right]\times \Omega$ of $\mathcal{F}_{t\leq T}$-progressively measurable (resp. predictable) sets.\
(iv) For $\kappa\geq 1$, $\mathbb{L}^{2}_{\kappa}(\lambda)$ the space of Borel measurable functions $\varphi:=(\varphi(e))_{e\in E}$ from $E$ into $\mathbb{R}^{\kappa}$ such that $\|\varphi\|^{2}_{\mathbb{L}^{2}_{\kappa}(\lambda)}=\displaystyle\int_{E}\left|\varphi(e)\right|^{2}_{\kappa}\lambda(\mathrm{d}e)<\infty$; $\mathbb{L}^{2}_{1}(\lambda)$ will be simply denoted by $\mathbb{L}^{2}(\lambda)$;\
(v) $\mathcal{S}^{2}(\mathbb{R}^{\kappa})$ the space of RCLL (for right continuous with left limits) $\mathcal{P}$-measurable and $\mathbb{R}^{\kappa}$-valued processes such that $\mathbb{E}[\sup_{s\leq T} \left|Y_{s}\right|^{2}]<\infty$; $\mathcal{A}^{2}_{c}$ is its subspace of continuous non-decreasing processes $(\mathrm{K}_{t})_{t\leq T}$ such that $\mathrm{K}_{0}=0$ and $\mathbb{E}\left[(\mathrm{K}_{T})^{2} \right]<\infty$;\
(vi) $\mathbb{H}^{2}(\mathbb{R}^{\kappa\times d})$ the space of processes $Z:=(Z_{s})_{s\leq T}$ which are $\mathcal{P}$-measurable, $\mathbb{R}^{\kappa\times d}$-valued and satisfying $\mathbb{E}\left[\displaystyle\int^{T}_{0}\left|Z_{s}\right|^{2}\, \mathrm{d} s\right]<\infty$;\
(vii) $\mathbb{H}^{2}(\mathbb{L}^{2}_{\kappa}(\lambda))$ the space of processes $U:=(U_{s})_{s\leq T}$ which are $\mathbf{P}$-measurable, $\mathbb{L}^{2}_{\kappa}(\lambda)$-valued and satisfying $\mathbb{E}\left[\displaystyle\int^{T}_{0}\|U_{s}(\omega)\|^{2}_{\mathbb{L}^{2}_{\kappa}(\lambda)}\, \mathrm{d} s\right]<\infty$;\
(viii) $\Pi_{g}$ the set of deterministics functions\
$\varpi:~(t,x)\in [0,T]\times \mathbb{R}^{\kappa}\mapsto\varpi(t,x)\in\mathbb{R}$ of polynomial growth, i.e., for which there exists two non-negative constants $C$ and $p$ such that for any $(t,x)\in [0,T]\times \mathbb{R}^{\kappa}$, $$\left|\varpi(t,x)\right|\leq C(1+\left|x\right|^{p}).$$ The subspace of $\Pi_{g}$ of continuous functions will be denoted by $\Pi^{c}_{g}$;\
(ix) $\mathcal{U}$ the subclass of $\Pi^{c}_{g}$ which consists of functions\
$\Phi:~(t,x)\in [0,T]\times \mathbb{R}^{\kappa}\mapsto\mathbb{R}$ such that for some non-negative constants $C$ and $p$ we have $$\left|\Phi(t,x)-\Phi(t,x')\right|\leq C(1+\left|x\right|^{p}+\left|x'\right|^{p})\left|x-x'\right|,~\textrm{for any}~t,~x,~x'.$$ (x) For any process $\theta:=(\theta_{s})_{s\leq T}$ and $t\in(0,T],~\theta_{t-}=\lim_{s\nearrow t}\theta_{s}$ and\
$$\Delta_{t}\theta=\theta_{t}-\theta_{t-}.$$ Now let $b$ and $\sigma$ be the following functions: $$b:(t,x)\in [0,T]\times \mathbb{R}^{k}\mapsto b(t,x)\in\mathbb{R}^{k};$$ $$\sigma:(t,x)\in [0,T]\times \mathbb{R}^{k}\mapsto\sigma(t,x)\in\mathbb{R}^{k\times d}.$$ We assume that they are jointly continuous in $(t,x)$ and Lipschitz continuous w.r.t. $x$ uniformly in $t$, i.e., there exists a constant $C$ such that, $$\label{2.5}
\forall (t,x,x')\in[0,T]\times \mathbb{R}^{k+k},~\left|b(t,x)-b(t,x')\right|+\left|\sigma(t,x)-\sigma(t,x')\right|\leq C\left|x-x'\right|.$$ Let us notice that by (\[2.5\]) and continuity, the functions $b$ and $\sigma$ are of linear growth, i.e., there exists a constant $C$ such that $$\label{2.6}
\forall (t,x,x')\in[0,T]\times \mathbb{R}^{k+k},~\left|b(t,x)\right|+\left|\sigma(t,x)\right|\leq C\left|1+x\right|.$$ Let $\beta:(t,x,e)\in [0,T]\times \mathbb{R}^{k}\times E\mapsto \beta(t,x,e)\in\mathbb{R}^{k}$ be a measurable function such that for some real constant $C$, and for all $e\in E$, $$\begin{aligned}
\label{2.7}
& (i) & \left|\beta(t,x,e)\right| \leq C(1\wedge\left|e\right|); \\ \nonumber
& (ii) & \left|\beta(t,x,e)-\beta(t,x',e)\right|\leq C\left|x-x'\right|(1\wedge\left|e\right|);\\
& (iii) & \textrm{the mapping}~(t,x)\in[0,T]\times \mathbb{R}^{k}\mapsto \beta(t,x,e)\in\mathbb{R}^{k}~\textrm{is continuous for any}~\mathrm{e}\in\mathrm{E}\nonumber.\end{aligned}$$ We are now going to introduce the objects which are specifically connected to the RBSDE with jumps we will deal with. Let $\ell$ the barrier of (\[eq2\]); $(g^{i})_{i=1,m}$ and $(h^{(i)})_{i=1,m}$ be two functions defined as follows: for $i=1,\ldots,m$, $$\begin{aligned}
g^{i}:\mathbb{R}^{k} & \longrightarrow & \mathbb{R}^{m}\nonumber \\
{}{}x & \longmapsto & g^{i}(x)\nonumber\end{aligned}$$ and $$\begin{aligned}
h^{(i)}:[0,T]\times\mathbb{R}^{k+m+d+1} & \longrightarrow & \mathbb{R}\nonumber \\
{}{}(t,x,y,z,q) & \longmapsto & h^{(i)}(t,x,y,z,q).\nonumber\end{aligned}$$ Moreover we assume they satisfy:\
(**H1**)\[2.10\]: The reflecting barrier $\ell$ is real valued and $\mathcal{P}$-measurable process satisfying, $\ell\in\mathcal{U}$ i.e., it is continuous and there exists constants $C$ and $p$ such that,\
$\left|\ell(t,x)-\ell(t,x')\right|\leq C(1+\left|x\right|^{p}+\left|x'\right|^{p})\left|x-x'\right|$, for any $t\geq 0$, $x$, $x'$.\
\
(**H2**): For any $i\in\left\lbrace 1,\ldots,m\right\rbrace$, the function $g^{i}$ belongs to $\mathcal{U}$.\
\
(**H3**): For any $i\in\left\lbrace 1,\ldots,m\right\rbrace$, $$\begin{aligned}
& (i) &~\textrm{the function}~h^{(i)}~\textrm{is Lipschitz in}~ (y,z,q)~\textrm{uniformly in}~(t,x),~\textrm{i.e., there exists a real constant}\nonumber\\
& {}{} & \textrm{C such that for any}~ (t,x)\in[0,T]\times\mathbb{R}^{k}, (y,z,q)~\textrm{and}~(y',z',q')~\textrm{elements of}~\mathbb{R}^{m+d+1},\nonumber\\
& {}{} &\left|h^{(i)}(t,x,y,z,q)-h^{(i)}(t,x,y',z',q')\right|\leq C(\left|y-y'\right|+\left|z-z'\right|+\left|q-q'\right|);\label{2.11}\\
& (ii) & ~\textrm{the}~(t,x)\mapsto h^{(i)}(t,x,y,z,q), ~\textrm{for fixed}~ (y,z,q)\in\mathbb{R}^{m+d+1},~\textrm{belongs uniformly to}~\mathcal{U},~\textrm{i.e., it}\nonumber\\
& {}{} &\textrm{is continuous and there exists constants C and p (which do not depend on}~(y,z,q))~\textrm{such that},\nonumber\\
& {} & \left|h^{(i)}(t,x,y,z,q)-h^{(i)}(t,x',y,z,q)\right|\leq C(1+\left|x\right|^{p}+\left|x'\right|^{p})\left|x-x'\right|,~\textrm{for any}~t\geq 0,~x,~x'.\end{aligned}$$ Next let $\gamma^{i},~i=1,\ldots,m$ be Borel measurable functions defined from $[0,T]\times\mathbb{R}^{k}\times E$ into $\mathbb{R}$ and satisfying: $$\begin{aligned}
\label{2.12}
& (i) &\left|\gamma^{i}(t,x,e)\right|\leq C(1\wedge\left|e\right|);\nonumber\\
& (ii) & \left|\gamma^{i}(t,x,e)-\gamma^{i}(t,x',e)\right|\leq C(1\wedge\left|e\right|)\left|x-x'\right|(1+\left|x\right|^{p}+\left|x'\right|^{p});\\
& (iii) & \textrm{the mapping}~t\in[0,T]\mapsto \gamma^{i}(t,x,e)\in\mathbb{R}~ \textrm{is continuous for any}~(x,e)\in\mathbb{R}^{k}\times E.\nonumber\end{aligned}$$ Finally we introduce the following functions $(f^{(i)})_{i=1,m}$ defined by: $$\label{2.13}
\forall (t,x,y,z,\zeta)\in[0,T]\times\mathbb{R}^{k+m+d}\times \mathbb{L}^{2}(\lambda),~f^{(i)}(t,x,y,z,\zeta):=h^{(i)}\left(t,x,y,z,\displaystyle\int_{E}\gamma^{i}(t,x,e)\zeta(e)\lambda(de)\right).$$ The functions $(f^{(i)})_{i=1,m}$, enjoy the two following properties:
$$\begin{aligned}
\label{2.15}
& (a) &~\textrm{The function}~f^{(i)}~\textrm{is Lipschitz in}~ (y,z,\zeta)~\textrm{uniformly in}~(t,x),~\textrm{i.e., there exists a real constant}\nonumber\\
& {}{} & \textrm{C such that}\nonumber\\
& {}{} &\left|f^{(i)}(t,x,y,z,\zeta)-f^{(i)}(t,x,y',z',\zeta')\right|\leq C(\left|y-y'\right|+\left|z-z'\right|+\|\zeta-\zeta'\|_{\mathbb{L}^{2}(\lambda)});\\
\nonumber
& {}{} & \textrm{since}~h^{(i)}~\textrm{is uniformly Lipschitz in}~(y,z,q)~\textrm{and}~ \gamma^{i}~\textrm{verifies (\ref{2.11})-(i)};\nonumber\\
& (b) &~\textrm{The function}~(t,x) \in[0,T]\times\mathbb{R}^{k}\mapsto f^{(i)}(t,x,0,0,0)~\textrm{belongs}~to~\Pi^{c}_{g};\nonumber\\
\nonumber
& {}{} & \textrm{and then}~ \mathbb{E}\left[\displaystyle\int^{T}_{0}\left|f^{(i)}(r,X^{t,x}_{r},0,0,0)\right|^{2}\,dr \right]<\infty.\end{aligned}$$
Having defined our data and put our assumptions, we can look at the state of the art.
Preliminaires
=============
A class of diffusion processes with jumps
-----------------------------------------
Let $(t,x)\in[0,T]\times\mathbb{R}^{d}$ and $(X^{t,x}_{s})_{s\leq T}$ be the stochastic process solution of (\[2.4\]). Under assumptions (\[2.5\])-(\[2.7\]) the solution of Equation (\[2.4\]) exists and is unique (see [@fuj] for more details). We state some properties of the process $\{(X^{t,x}_{s}),~s\in[0,T]\}$ which can found in [@fuj].
For each $t\geq 0$, there exists a version of $\{(X^{t,x}_{s}),~s\in[t,T]\}$ such that $s\rightarrow X^{t}_{s}$ is a $C^{2}(\mathbb{R}^{d})$-valued rcll process. Moreover it satisfies the following estimates: $\forall p\geq 2,~x,x'\in\mathbb{R}^{d}$ and $s\geq t$, $$\begin{aligned}
\mathbb{E}[\sup_{t\leq r\leq s} \left|X^{t,x}_{r}-x\right|^{p}] & \leq & M_{p}(s-t)(1+\left|x\right|^{p});\nonumber\\
\mathbb{E}[\sup_{t\leq r\leq s} \left|X^{t,x}_{r}-X^{t,x'}_{r}-(x-x')^{p}\right|^{p}] & \leq & M_{p}(s-t)(\left|x-x'\right|^{p});\label{3.16}\end{aligned}$$ for some constant $M_{p}$.
Existence and uniqueness for a RBSDE with jumps
-----------------------------------------------
Let $(t,x)\in[0,T]\times\mathbb{R}^{d}$ and we consider the following m-dimensional RBSDE with jumps: $$\label{3.17}
\left
\{\begin{array}{ll}
(i)~\vec{Y}^{t,x}:=(Y^{i,t,x})_{i=1,m}\in\mathcal{S}^{2}(\mathbb{R}^{m}),~Z^{t,x}:=(Z^{i,t,x})_{i=1,m}\in\mathbb{H}^{2}(\mathbb{R}^{m\times d}),\\
K^{t,x}:=(K^{i,t,x})_{i=1,m}\in\mathcal{A}^{2}_{c},~ U^{t,x}:=(U^{i,t,x})_{i=1,m}\in\mathbb{H}^{2}(\mathbb{L}^{2}_{m}(\lambda));\\
(ii)~dY^{i;t,x}_{s}=-f^{(i)}(s,X^{t,x}_{s},(Y^{i;t,x}_{s})_{i=1,m},Z^{i;t,x}_{s},U^{i;t,x}_{s})ds-
\mathrm{d}\mathrm{K}^{i;t,x}_{s}\\
\quad\quad\quad\quad\quad\quad
\quad\quad+Z^{i;t,x}_{s}\mathrm{d}
\mathrm{B}_{s}+\displaystyle\int_{\mathrm{E}}\mathrm{U}^{i;t,x}
_{s}(e)\tilde{\mu}(\mathrm{d}s,\mathrm{d}e),\quad s\leq T;\\
(iii)~Y^{i;t,x}_{s}\geq \ell(s,X^{t,x}_{s})~\textrm{and}~ \displaystyle\int^{T}_{0}(Y^{i;t,x}_{s}- \ell(s,X^{t,x}_{s}))\mathrm{d}\mathrm{K}^{i;t,x}_{s}=0;
\end{array}
\right.$$ where for any $i\in\{1,\ldots,m\}$, $Z^{i;t,x}_{s}$ is the ith row of $Z^{t,x}_{s}$, $K^{i;t,x}_{s}$ is the ith component of $K^{t,x}_{s}$ and $U^{i;t,x}_{s}$ is the ith component of $U^{t,x}_{s}$.\
The following result is related to existence and uniqueness of a solution for the RBSDE with jumps (\[3.17\]).\
Its proof is given in [@hamaOuk] by using the penalization method (see p .5-12) and the Snell envelope method (see p. 14-16).
Assume that assumptions $(\mathbf{H1}),~(\mathbf{H2})~\text{and }(\mathbf{H3})$ hold. Then for any $(t,x)\in[0,T]\times\mathbb{R}^{d}$, the RBSDE (\[3.17\]) has an unique solution $(\vec{Y}^{t,x},Z^{t,x},U^{t,x},K^{t,x})$.
The solution of this RBSDE with jumps exist and is unique since: $$\begin{aligned}
& (i) &\mathbb{E}\left[\left|g(X^{t,x}_{T})\right|^{2}\right]<\infty,~\textrm{due to polynomial growth of}~g~\textrm{and estimate}~(\ref{3.16})~on~X^{t,x};\nonumber\\
& (ii) & \textrm{for any}~i=1,\ldots,m,~f^{(i)}~\textrm{verifies the properties (a)-(b) related to uniform}\\
& {}{} & \textrm{Lipschitz w.r.t.}~(y,z,\zeta)~\textrm{and}~ds\otimes d\mathbb{P}-\textrm{square integrability of the process }\\
& {}{} & (f^{(i)}(s,X^{t,x}_{s},0,0,0))_{s\leq T}.\end{aligned}$$
Viscosity solutions of integro-differential partial equation with one obstacle
------------------------------------------------------------------------------
The following result on one obstacle is proved with two obstacles in Harraj and al. (see. [@har], Theorem 4.6. p. 47 by using (4.1) p. 44), establishes the relationship between the solution of (\[3.17\]) and the one of system (\[eq1\]).
Assume that (**H1**), (**H2**), (**H3**) are fulfilled. Then there exists deterministic continuous functions $(u^{i}(t,x))_{i=1,m}$ which belong to $\Pi_{g}$ such that for any $(t,x)\in[0,T]\times\mathbb{R}^{k}$, the solution of the RBSDE (\[3.17\]) verifies: $$\label{3.18}
\forall i\in\{1,\ldots,m\},~\forall s\in[t,T],~Y^{i;t,x}_{s}=u^{i}(s,X^{t,x}_{s}).$$ Moreover if for any $i\in\{1,\ldots,m\}$, $$\begin{aligned}
& (i) & \gamma^{i}\geq 0;\\
& (ii) & \textrm{for any fixed}~(t,x,\vec{y},z)\in[0,T]\times \mathbb{R}^{k+m+d},~\textrm{the mapping}\\
& {}{} & (q\in\mathbb{R})\longmapsto h^{(i)}(t,x,\vec{y},z,q)\in\mathbb{R}~\textrm{is non-decreasing}. \end{aligned}$$
The function $(u^{i})_{i=1,m}$ is a continuous viscosity solution (in Barles and al. ’s sense, see Definition $6.1$ in the Appendix) of (\[eq1\]).\
For the proof see [@har] for the same way.\
Finally, the solution $(u^{i})_{i=1,m}$ of (\[eq1\]) is unique in the class $\Pi^{c}_{g}$.
(see [@har]) Under the assumptions (**H1**), (**H2**), (**H3**), there exists a unique viscosity solution of (\[eq1\]) in the class of functions satisfying $$\lim\limits_{\left|x\right| \to +\infty}\left|u(t,x)\right| e^{-\tilde{A}[\log(\left|x\right|)]^{2}}=0$$ uniformly for $t\in[0,T]$, for some $\tilde{A}>0$.
Estimates and properties
========================
In this section we provide estimates for the functions $(u^{i})_{i=1,m}$ defined in (\[3.18\]). Recall that,\
$(\vec{Y}^{t,x},Z^{t,x},U^{t,x},K^{t,x}):=((Y^{i;t,x})_{i=1,m},
(Z^{i;t,x})_{i=1,m},(U^{i;t,x})_{i=1,m},(K^{i;t,x})_{i=1,m})$ is the unique solution of the RBSDE with jumps (\[3.17\]).
Under assumption (**H1**), (**H2**), (**H3**), for any $p\geq 2$ there exists two non-negative constants $C$ and $\rho$ such that, $$\label{4.22}
\mathbb{E}\left[\left\lbrace \displaystyle\int^{T}_{0} ds \left(\displaystyle\int_{E}\left|U^{t,x}_{s}(e)\right|^{2}\lambda(de)\right)\right\rbrace^{\frac{p}{2}}\right]=\mathbb{E}\left[\left\lbrace \displaystyle\int^{T}_{0} ds\|U^{t,x}_{s}\|^{2}_{\mathbb{L}^{2}_{m}(\lambda)}\right\rbrace^{\frac{p}{2}}\right]\leq C\left(1+\left|x\right|^{\rho}\right).$$
First let us point out that since $X^{t,x}_{s}=x$ for $s\in[0,t]$ then, the uniqueness of the solution of RBSDE of (\[3.17\]) implies that, $$\label{4.23}
Z^{t,x}_{s}=U^{t,x}_{s}=K^{t,x}_{s}=0,~ds\otimes d\mathbf{P}-\textrm{a.e. on}~[0,t]\times\Omega.$$ Next let $p\geq 2$ be fixed. Using the representation (\[3.18\]), for any $i\in\{1,\ldots,m\}$ and $s\in[t,T]$ we have, $$\begin{aligned}
\label{4.24}
Y^{i;t,x}_{s} & = & g^{i}(X^{t,x}_{T})+\displaystyle\int^{T}_{s}f^{(i)}(r,X^{t,x}_{r},(u^{j}(X^{t,x}_{r}))_{j=1,m},Z^{i;t,x}_{r},U^{i;t,x}_{s})dr+
\mathrm{K}^{i;t,x}_{T}-\mathrm{K}^{i;t,x}_{s}\nonumber\\
&{}{}&-\displaystyle\int^{T}_{s}Z^{i;t,x}_{r}\mathrm{d}
\mathrm{B}_{r}-\displaystyle\int^{T}_{s}\displaystyle\int_{\mathrm{E}}\mathrm{U}^{i;t,x}
_{r}(e)\tilde{\mu}(\mathrm{d}r,\mathrm{d}e).\end{aligned}$$ This implies that the system of RBSDEs with jumps (\[3.17\]) turns into a decoupled one since the equations in (\[4.24\]) are not related each other.\
Next for any $i=1,\ldots,m$, the functions $u^{i}$, $g^{i}$ and $(t,x)\mapsto f^{(i)}(t,x,0,0,0)$ are of polynomial growth and finally $y\mapsto f^{(i)}(t,x,y,0,0)$ is Lipschitz uniformly w.r.t. $(t,x)$. Then for some $C$ and $\rho\geq 0$, $$\label{4.25}
\mathbb{E}\left[\left|g^{i}(X^{t,x}_{T})\right|^{p}+\left( \displaystyle\int^{T}_{s}\left|f^{(i)}(r,X^{t,x}_{r},(u^{j}(X^{t,x}_{r}))_{j=1,m},0,0)\right|^{2}dr\right)^{\frac{p}{2}}\right]\leq C\left(1+\left|x\right|^{\rho}\right).$$ Let us now fix $i_{0}\in\{1,\ldots,m\}$, $\forall s\in[t,T]$, $$\label{4.26}
\left
\{\begin{array}{ll}
(i)~Y^{i_{0},t,x}_{s}\in\mathcal{S}^{2}(\mathbb{R}),~Z^{i_{0},t,x}_{s}\in\mathbb{H}^{2}(\mathbb{R}^{d}),~
K^{i_{0},t,x}_{s}\in\mathcal{A}^{2}_{c},~U^{i_{0},t,x}_{s}\in\mathbb{H}^{2}(\mathbb{L}^{2}(\lambda));\\
(ii)~Y^{i_{0},t,x}_{s}= g^{i_{0}}(X^{t,x}_{T})+\displaystyle\int^{T}_{s}f^{(i_{0})}(r,X^{t,x}_{r},(u^{j}(X^{t,x}_{r}))_{j=1,m},Z^{i_{0},t,x}_{r},U^{i_{0},t,x}_{r})dr+
K^{i_{0},t,x}_{T}-K^{i_{0},t,x}_{s}\nonumber\\
\qquad\qquad-\displaystyle\int^{T}_{s}Z^{i_{0},t,x}_{r}\mathrm{d}
\mathrm{B}_{r}-\displaystyle\int^{T}_{s}
\displaystyle\int_{\mathrm{E}}U^{i_{0},t,x}_{r}(e)\tilde{\mu}(\mathrm{d}r,\mathrm{d}e).\\
(iii)~Y^{i_{0},t,x}_{s}\geq \ell(s,X^{t,x}_{s})~\textrm{and}~ \displaystyle\int^{T}_{0}(Y^{i_{0},t,x}_{s}- \ell(s,X^{t,x}_{s}))\mathrm{d}K^{i_{0},t,x}_{s}=0.
\end{array}
\right.$$ Applying Itô formula to $\left|Y^{i_{0},t,x}_{s}\right|^{2}$ between $s$ and $T$, we have $$\begin{aligned}
\label{4.24bis}
& {}{} &\left|Y^{i_{0},t,x}_{s}\right|^{2}+\displaystyle\int^{T}_{s}\left|Z^{i_{0},t,x}_{r}\right|^{2}\,dr+\sum_{s\leq r\leq T}(\Delta Y^{i_{0},t,x}_{r})^{2}\\
& {}{} &=\left|g^{i_{0}}(X^{t,x}_{T})\right|^{2}+2\displaystyle\int^{T}_{s}
Y^{i_{0},t,x}_{r}f^{(i_{0})}(r,X^{t,x}_{r},(u^{j}(X^{t,x}_{r}))_{j=1,m},Z^{i_{0},t,x}_{r},U^{i_{0},t,x}_{r})\,dr+2\displaystyle\int^{T}_{s}Y^{i_{0},t,x}_{r}\,dK^{i_{0},t,x}_{r}\nonumber \\
& {}{} &-2\displaystyle\int^{T}_{s}
\displaystyle\int_{\mathrm{E}}Y^{i_{0},t,x}_{r}U^{i_{0},t,x}_{r}\tilde{\mu}(\mathrm{d}r,\mathrm{d}e)-2\displaystyle\int^{T}_{s}Y^{i_{0},t,x}_{r}Z^{i_{0},t,x}_{r}\,dB_{r}.\nonumber\end{aligned}$$ Notice that $Y^{i_{0},t,x}_{r}=u^{i_{0}}(r,X^{t,x}_{r})$ and we have that $|u^{i_{0}}(r,X^{t,x}_{r})|\leq C(1+|X^{t,x}_{r}|^q)$. Next let us set $\Sigma=1+\sup_{s\leq T}|X^{t,x}_{s}|$. Therefore $|Y^{i_{0},t,x}_{r}|\leq C_{q}\Sigma^{q}$ and $\left|g^{i_{0}}(X^{t,x}_{T})\right|\leq C_{q}\Sigma^{q}$.\
By raising to the power $\frac{p}{2}$ and then taking expectation, it follows from (\[4.24bis\]) and the fact of, there exists $C>0$ such that,\
$\mathbb{E}\left[\left(\displaystyle\int^{T}_{s}\|U^{i_{0},t,x}_{r}\|^{2}_{\mathbb{L}^{2}(\lambda)}\,dr\right)^{\frac{p}{2}}\right]\leq C\mathbb{E}\left[\left(\sum_{s\leq r\leq T}(\Delta Y^{i_{0},t,x}_{r})^{2}\right)^{\frac{p}{2}}\right]$, (see [@len] p. 28-45), $$\begin{aligned}
\label{4.25ori}
& {}{} &\mathbb{E}\left[\left|Y^{i_{0},t,x}_{s}\right|^{p}\right]+\mathbb{E}\left[\left( \displaystyle\int^{T}_{s}\left|Z^{i_{0},t,x}_{r}\right|^{2}\,dr\right)^{\frac{p}{2}}\right] +\mathbb{E}\left[\left(\displaystyle\int^{T}_{s}\|U^{i_{0},t,x}_{r}\|^{2}_{\mathbb{L}^{2}(\lambda)}\,dr\right)^{\frac{p}{2}}\right]\nonumber\\
& {}{} &\leq 5^{\frac{p}{2}-1}\mathbb{E}\left[\left|g^{i_{0}}(X^{t,x}_{T})\right|^{p}\right]+5^{\frac{p}{2}-1}\mathbb{E}\left[\left(\displaystyle\int^{T}_{s}2\left|
Y^{i_{0},t,x}_{r}f^{(i_{0})}(r,X^{t,x}_{r},(u^{j}(X^{t,x}_{r}))_{j=1,m},Z^{i_{0},t,x}_{r},U^{i_{0},t,x}_{r})\right|\,dr\right)^{\frac{p}{2}}\right]\nonumber\\
& {}{} &+5^{\frac{p}{2}-1}\mathbb{E}\left[\left|\displaystyle\int^{T}_{s}Y^{i_{0},t,x}_{r}\,dK^{i_{0},t,x}_{r}\right|^{\frac{p}{2}}\right]\nonumber \\
& {}{} &+5^{\frac{p}{2}-1}\mathbb{E}\left[\left|\displaystyle\int^{T}_{s}
\displaystyle\int_{\mathrm{E}}2Y^{i_{0},t,x}_{r}U^{i_{0},t,x}_{r}\tilde{\mu}(\mathrm{d}r,\mathrm{d}e)\right|^{\frac{p}{2}}\right]+5^{\frac{p}{2}-1}\mathbb{E}\left[\left|\displaystyle\int^{T}_{s}2Y^{i_{0},t,x}_{r}Z^{i_{0},t,x}_{r}\,dB_{r}\right|^{\frac{p}{2}}\right].\end{aligned}$$ For more comprehension, we adopt the following scripture for inequality (\[4.25ori\]);\
$$\begin{aligned}
\label{4.26bis}
& {}{} &\mathbb{E}\left[\left|Y^{i_{0},t,x}_{s}\right|^{p}\right]+\mathbb{E}\left[\left( \displaystyle\int^{T}_{s}\left|Z^{i_{0},t,x}_{r}\right|^{2}\,dr\right)^{\frac{p}{2}}\right] +\mathbb{E}\left[\left(\displaystyle\int^{T}_{s}\|U^{i_{0},t,x}_{r}\|^{2}_{\mathbb{L}^{2}(\lambda)}\,dr\right)^{\frac{p}{2}}\right]\leq 5^{\frac{p}{2}-1}\mathbb{E}\left[\left|g^{i_{0}}(X^{t,x}_{T})\right|^{p}\right]\nonumber\\
& {}{} & +5^{\frac{p}{2}-1}T_{1}(s)+5^{\frac{p}{2}-1}T_{2}(s)+5^{\frac{p}{2}-1}T_{3}(s)+5^{\frac{p}{2}-1}T_{4}(s)\end{aligned}$$ where,\
$$\begin{aligned}
T_{1}(s) & = & \mathbb{E}\left[\left(\displaystyle\int^{T}_{s}2\left|
Y^{i_{0},t,x}_{r}f^{(i_{0})}(r,X^{t,x}_{r},(u^{j}(X^{t,x}_{r}))_{j=1,m},Z^{i_{0},t,x}_{r},U^{i_{0},t,x}_{r})\right|\,dr\right)^{\frac{p}{2}}\right];\\
T_{2}(s) & = & \mathbb{E}\left[\left|\displaystyle\int^{T}_{s}Y^{i_{0},t,x}_{r}\,dK^{i_{0},t,x}_{r}\right|^{\frac{p}{2}}\right];\\
T_{3}(s) & = & \mathbb{E}\left[\left|\displaystyle\int^{T}_{s}
\displaystyle\int_{\mathrm{E}}2Y^{i_{0},t,x}_{r}U^{i_{0},t,x}_{r}\tilde{\mu}(\mathrm{d}r,\mathrm{d}e)\right|^{\frac{p}{2}}\right];\\
T_{4}(s) & = & \mathbb{E}\left[\left|\displaystyle\int^{T}_{s}2Y^{i_{0},t,x}_{r}Z^{i_{0},t,x}_{r}\,dB_{r}\right|^{\frac{p}{2}}\right].\end{aligned}$$ We will estimate $T_{1}(s)$, $T_{2}(s)$, $T_{3}(s)$ and $T_{4}(s)$, $\forall s\in[t,T]$.\
(a) Before starting our estimations, let’s linearize $f$ with respect to $(u^{j}(X^{t,x}_{r}))_{j=1,m}$ and $Z^{i_{0},t,x}_{r}$ i.e.\
$f^{(i_{0})}(r,X^{t,x}_{r},(u^{j}(X^{t,x}_{r}))_{j=1,m},Z^{i_{0},t,x}_{r},U^{i_{0},t,x}_{r})=a^{t,x}_{r}Z^{i_{0},t,x}_{r}+b^{t,x}_{r}(u^{j}(X^{t,x}_{r}))_{j=1,m}+f^{(i_{0})}(r,X^{t,x}_{r},0,0,U^{i_{0},t,x}_{r})$;\
where $a^{t,x}_{r}$ and $b^{t,x}_{r}$ are progressively measurable processes respectively bounded by the Lipschitz constants of $f$ in $Z^{t,x}_{r}$ and $(u^{j}(X^{t,x}_{r}))_{j=1,m}$ i.e $|a^{t,x}_{r}|\leq C_{Z}$ and $|b^{t,x}_{r}|\leq\lambda_{1}$.\
(b) We also take the fact that $f$ is Lipschitz in $U^{i_{0},t,x}_{r}$ i.e there exists a constant Lipschitz $\lambda_{2}$ such that $|f^{(i_{0})}(r,X^{t,x}_{r},0,0,U^{i_{0},t,x}_{r})|\leq|f^{(i_{0})}(r,X^{t,x}_{r},0,0,0)|+\lambda_{2}\|U^{i_{0},t,x}_{r}\|_{\mathbb{L}^{2}_{m}(\lambda)}$.\
By combining (a) and (b) we have $$\begin{aligned}
\label{linea}
|f^{(i_{0})}(r,X^{t,x}_{r},(u^{j}(X^{t,x}_{r}))_{j=1,m},Z^{i_{0},t,x}_{r},U^{i_{0},t,x}_{r})|& \leq & |a^{t,x}_{r}Z^{i_{0},t,x}_{r}|+|b^{t,x}_{r}(u^{j}(X^{t,x}_{r}))_{j=1,m}|+|f^{(i_{0})}(r,X^{t,x}_{r},0,0,0)|\nonumber\\
& {}{} &+\lambda_{2}\|U^{i_{0},t,x}_{r}\|_{\mathbb{L}^{2}_{m}(\lambda)}.\end{aligned}$$ Let’s start our estimates.\
\
\
By using (\[linea\]), it follows that; $$\begin{aligned}
\label{4.25bis}
& {}{} &\displaystyle\int^{T}_{s}2\left|
Y^{i_{0},t,x}_{r}f^{(i_{0})}(r,X^{t,x}_{r},(u^{j}(X^{t,x}_{r}))_{j=1,m},Z^{i_{0},t,x}_{r},U^{i_{0},t,x}_{r})\right|\,dr\nonumber\\
& {}{} &\leq \displaystyle\int^{T}_{s}2\left|
Y^{i_{0},t,x}_{r}\left(a^{t,x}_{r}Z^{i_{0},t,x}_{r}\right)\right|\,dr +\displaystyle\int^{T}_{s}2\left|
Y^{i_{0},t,x}_{r}\left(b^{t,x}_{r}(u^{j}(X^{t,x}_{r}))_{j=1,m}\right)\right|\,dr+\lambda_{2}\displaystyle\int^{T}_{s}2\left|
Y^{i_{0},t,x}_{r}\left(U^{i_{0},t,x}_{r}\right)\right|dr\nonumber\\
& {}{} &+\displaystyle\int^{T}_{s}2\left|
Y^{i_{0},t,x}_{r}f^{(i_{0})}(r,X^{t,x}_{r},0,0,0)\right|\,dr\nonumber\\
& {}{} &\leq C^{2}_{q}C_{Z}T\epsilon^{-1}_{1}\Sigma^{2q}+\epsilon_{1}C_{Z}\displaystyle\int^{T}_{s}\left|Z^{i_{0},t,x}_{r}\right|^{2}\,dr+C^{2}_{q}\lambda_{1}T\epsilon^{-1}_{2}\Sigma^{2q}+\epsilon_{2}\lambda_{1}C^{2}_{q}T\Sigma^{2q}+C^{2}_{q}T\lambda_{2}\epsilon^{-1}_{3}\Sigma^{2q}\nonumber\\
& {}{} &+\lambda_{2}\epsilon_{3}\displaystyle\int^{T}_{s}\|U^{i_{0},t,x}_{r}\|^{2}_{\mathbb{L}^{2}_{m}(\lambda)}\,dr+C^{2}_{q}T\epsilon^{-1}_{4}\Sigma^{2q}+C^{2}_{q}T\epsilon_{4}\Sigma^{2q}.\nonumber\\end{aligned}$$ By raising to the power $\frac{p}{2}$ and then taking expectation it follows that,\
$$\begin{aligned}
\label{T1}T_{1}(s) & \leq & 8^{\frac{p}{2}-1}CC^{p}_{q}\left\lbrace (C_{Z}T\epsilon^{-1}_{1})^{\frac{p}{2}}+(\lambda_{1}T\epsilon^{-1}_{2})^{\frac{p}{2}}+C^{p}_{q}(\epsilon_{2}\lambda_{1}T)^{\frac{p}{2}}+(T\lambda_{2}\epsilon^{-1}_{3})^{\frac{p}{2}}+(T\epsilon^{-1}_{4})^{\frac{p}{2}}\right.\nonumber\\
& {}{} & \left.+(T\epsilon_{4})^{\frac{p}{2}}\right\rbrace\left|1+|x|^{pq}\right|+8^{\frac{p}{2}-1}(\epsilon_{1}C_{Z})^{\frac{p}{2}}\mathbb{E}\left[\left(\displaystyle\int^{T}_{s}\left|Z^{i_{0},t,x}_{r}\right|^{2}\,dr\right)^{\frac{p}{2}}\right]\nonumber\\
& {}{} & +8^{\frac{p}{2}-1}(\lambda_{2}\epsilon_{3})^{\frac{p}{2}}\mathbb{E}\left[\left(\displaystyle\int^{T}_{s}\|U^{i_{0},t,x}_{r}\|^{2}_{\mathbb{L}^{2}_{m}(\lambda)}\,dr\right)^{\frac{p}{2}}\right].\end{aligned}$$
Before estimating $T_{2}(s)$, let us first give an estimate of $\mathbb{E}\left[|K^{i_{0},t,x}_{T}-K^{i_{0},t,x}_{s}|^{p}\right]$ which will serve us in that of $T_{2}(s)$.\
$$\begin{aligned}
K^{i_{0},t,x}_{T}-K^{i_{0},t,x}_{s} &=& Y^{i_{0},t,x}_{s}-g^{i_{0}}(X^{t,x}_{T})-\displaystyle\int^{T}_{s}f^{(i_{0})}(r,X^{t,x}_{r},(u^{j}(X^{t,x}_{r}))_{j=1,m},Z^{i_{0},t,x}_{r},U^{i_{0},t,x}_{r})dr
\nonumber\\
& {}{} & +\displaystyle\int^{T}_{s}Z^{i_{0},t,x}_{r}\mathrm{d}
\mathrm{B}_{r}+\displaystyle\int^{T}_{s}
\displaystyle\int_{\mathrm{E}}U^{i_{0},t,x}_{r}(e)\tilde{\mu}(\mathrm{d}r,\mathrm{d}e).\nonumber\end{aligned}$$ By (\[linea\]) and Cauchy-Schwartz inequality; it follows that, $$\begin{aligned}
|K^{i_{0},t,x}_{T}-K^{i_{0},t,x}_{s}| & \leq & 2C_{q}\Sigma^{q}+C_{q}\lambda_{1}T\Sigma^{q}+C_{q}T\Sigma^{q}+C_{Z}\left(\displaystyle\int^{T}_{s}|Z^{i_{0},t,x}_{r}|^{2}\,dr\right)^{\frac{1}{2}}\nonumber\\
& {}{} &+\lambda_{2}\left(\displaystyle\int^{T}_{s}\|U^{i_{0},t,x}_{r}\|^{2}\,dr\right)^{\frac{1}{2}}+\displaystyle\int^{T}_{s}Z^{i_{0},t,x}_{r}\mathrm{d}
\mathrm{B}_{r}+\displaystyle\int^{T}_{s}
\displaystyle\int_{\mathrm{E}}U^{i_{0},t,x}_{r}(e)\tilde{\mu}(\mathrm{d}r,\mathrm{d}e).\nonumber\end{aligned}$$ By raising to the power $p$, taking expectation and BDG inequality we have, $$\begin{aligned}
\label{KT}
\mathbb{E}\left[|K^{i_{0},t,x}_{T}-K^{i_{0},t,x}_{s}|^{p}\right]& \leq & 5^{p-1}CC^{p}_{q}\left\lbrace 2^{p}+(T\lambda_{1})^{p}+T^{p}\right\rbrace|1+|x|^{pq}|+5^{p-1}(C^{p}_{Z}+C_{p})\mathbb{E}\left(\displaystyle\int^{T}_{0}|Z^{i_{0},t,x}_{s}|^{2}\,ds\right)^{\frac{p}{2}}\nonumber\\
& {}{} &+5^{p-1}(\lambda^{p}_{2}+C_{p})\mathbb{E}\left\lbrace \displaystyle\int^{T}_{0} ds\|U^{i_{0},t,x}_{s}\|^{2}_{\mathbb{L}^{2}_{m}(\lambda)}\right\rbrace^{\frac{p}{2}}.\end{aligned}$$
$$\begin{aligned}
\label{KT2}
\displaystyle\int^{T}_{s}|Y^{i_{0},t,x}_{s}|\mathrm{d}K^{i_{0},t,x}_{s} & \leq & \displaystyle\int^{T}_{s}|(Y^{i_{0},t,x}_{s}-\ell(s,X^{t,x}_{s}))|\mathrm{d}K^{i_{0},t,x}_{s}+\displaystyle\int^{T}_{s}|\ell(s,X^{t,x}_{s})|\mathrm{d}K^{i_{0},t,x}_{s}\nonumber\\
{} & \leq & \sup_{s\leq T}|\ell(s,X^{t,x}_{s})|K^{i_{0},t,x}_{T}\nonumber\\
{} & \leq & \epsilon^{-1}_{5}\sup_{s\leq T}|\ell(s,X^{t,x}_{s})|^{2}+\epsilon_{5}(K^{i_{0},t,x}_{T})^{2}\nonumber\\
{} & \leq & \epsilon^{-1}_{5}C^{2}_{q}\Sigma^{2q}+\epsilon_{5}(K^{i_{0},t,x}_{T})^{2}.\end{aligned}$$ By using (\[KT\]) and (\[KT2\]); it follows that $$\begin{aligned}
T_{2}(s) & \leq & 2^{\frac{p}{2}-1}CC^{p}_{q}(\epsilon^{-1}_{5})^{\frac{p}{2}}|1+|x|^{pq}|+2^{\frac{p}{2}-1}(\epsilon_{5})^{\frac{p}{2}}\mathbb{E}\left[(K^{i_{0},t,x}_{T})^{p}\right]\nonumber\\
{} & \leq & 2^{\frac{p}{2}-1}CC^{p}_{q}(\epsilon^{-1}_{5})^{\frac{p}{2}}|1+|x|^{pq}|+(\epsilon_{5})^{\frac{p}{2}}2^{\frac{p}{2}-1}7^{p-1}CC^{p}_{q}\left\lbrace 2^{p}+(T\lambda_{1})^{p}+T^{p}\right\rbrace|1+|x|^{pq}|+\nonumber\\
& {}{} &(\epsilon_{5})^{\frac{p}{2}}2^{\frac{p}{2}-1}7^{p-1}(C^{p}_{Z}+C_{p})\mathbb{E}\left(\displaystyle\int^{T}_{0}|Z^{i_{0},t,x}_{s}|^{2}\,ds\right)^{\frac{p}{2}}\nonumber\\
& {}{} &+(\epsilon_{5})^{\frac{p}{2}}2^{\frac{p}{2}-1}7^{p-1}(\lambda^{p}_{2}+C_{p})\mathbb{E}\left\lbrace \displaystyle\int^{T}_{0} ds\|U^{i_{0},t,x}_{s}\|^{2}_{\mathbb{L}^{2}_{m}(\lambda)}\right\rbrace^{\frac{p}{2}}\nonumber\\
T_{2}(s) & \leq & \left\lbrace 2^{\frac{p}{2}-1}CC^{p}_{q}(\epsilon^{-1}_{5})^{\frac{p}{2}}+(\epsilon_{5})^{\frac{p}{2}}2^{\frac{p}{2}-1}7^{p-1}CC^{p}_{q}(2^{p}+(T\lambda_{1})^{p}+T^{p})\right\rbrace \left(1+|x|^{pq}\right)\nonumber\\
& {}{} &+(\epsilon_{5})^{\frac{p}{2}}2^{\frac{p}{2}-1}7^{p-1}(C^{p}_{Z}+C_{p})\mathbb{E}\left(\displaystyle\int^{T}_{0}|Z^{i_{0},t,x}_{s}|^{2}\,ds\right)^{\frac{p}{2}}\nonumber\\
& {}{} &+(\epsilon_{5})^{\frac{p}{2}}2^{\frac{p}{2}-1}7^{p-1}(\lambda^{p}_{2}+C_{p})\mathbb{E}\left\lbrace \displaystyle\int^{T}_{0} ds\|U^{i_{0},t,x}_{s}\|^{2}_{\mathbb{L}^{2}_{m}(\lambda)}\right\rbrace^{\frac{p}{2}}.\nonumber\end{aligned}$$\
By BDG inequality, $$\begin{aligned}
T_{3}(s) & \leq & C_{p}\mathbb{E}\left(\displaystyle\int^{T}_{0}|Y^{i_{0},t,x}_{s}|^{2}|Z^{i_{0},t,x}_{s}|^{2}\,ds\right)^{\frac{p}{4}}\nonumber\\
{} & \leq & C_{p}\mathbb{E}\left(\sup_{s\leq T}|Y^{i_{0},t,x}_{s}|^{2}\displaystyle\int^{T}_{0}|Z^{i_{0},t,x}_{s}|^{2}\,ds\right)^{\frac{p}{4}}\nonumber\\
{} & \leq & C_{p}C^{p}_{q}\epsilon^{-1}_{6}(1+|x|^{pq})+C_{p}\epsilon_{6}\mathbb{E}\left(\displaystyle\int^{T}_{0}|Z^{i_{0},t,x}_{s}|^{2}\,ds\right)^{\frac{p}{2}}.\end{aligned}$$\
By BDG inequality, $$\begin{aligned}
T_{4}(s) & \leq & C_{p}\mathbb{E}\left(\displaystyle\int^{T}_{0}|Y^{i_{0},t,x}_{s}|^{2}\|U^{i_{0},t,x}_{s}\|^{2}_{\mathbb{L}^{2}_{m}(\lambda)}\,ds\right)^{\frac{p}{4}}\nonumber\\
{} & \leq & C_{p}\mathbb{E}\left(\sup_{s\leq T}|Y^{i_{0},t,x}_{s}|^{2}\displaystyle\int^{T}_{0}\|U^{i_{0},t,x}_{s}\|^{2}_{\mathbb{L}^{2}_{m}(\lambda)}\,ds\right)^{\frac{p}{4}}\nonumber\\
{} & \leq & C_{p}C^{p}_{q}\epsilon^{-1}_{7}(1+|x|^{pq})+C_{p}\epsilon_{7}\mathbb{E}\left(\displaystyle\int^{T}_{0}\|U^{i_{0},t,x}_{s}\|^{2}_{\mathbb{L}^{2}_{m}(\lambda)}\,ds\right)^{\frac{p}{2}}.\end{aligned}$$ Finally by taking estimation of $T_{1}(s)$, $T_{2}(s)$, $T_{3}(s)$, $T_{4}(s)$ and choosing $\epsilon_{1}$, $\epsilon_{2}$, $\epsilon_{3}$, $\epsilon_{4}$, $\epsilon_{5}$, $\epsilon_{6}$, $\epsilon_{7}$ such that;\
$\{(\epsilon_{5})^{\frac{p}{2}}2^{\frac{p}{2}-1}7^{p-1}(C^{p}_{Z}+C_{p})+C_{p}\epsilon_{6}\}<1$,\
$\{(\epsilon_{5})^{\frac{p}{2}}2^{\frac{p}{2}-1}7^{p-1}(\lambda^{p}_{2}+C_{p})+C_{p}\epsilon_{7}\}<1$,\
and the sum of all coefficients of $(1+|x|^{pq})$ was small than $1$.\
It follows then $$\mathbb{E}\left[\left\lbrace \displaystyle\int^{T}_{0} ds\|U^{i_{0},t,x}_{s}\|^{2}_{\mathbb{L}^{2}_{m}(\lambda)}\right\rbrace^{\frac{p}{2}}\right]\leq C\left(1+\left|x\right|^{\rho}\right).$$ Where $\rho=pq$.\
Finally since $i_{0}\in\{1,\ldots,m\}$ is arbitrary we then obtain the estimate (\[4.22\]).
For any $i=1,\ldots,m$, $u^{i}$ belongs to $\mathcal{U}$.
Let $x$ and $x'$ be elements of $\mathbb{R}^{k}$. Let $(\vec{Y}^{t,x},Z^{t,x},U^{t,x},K^{t,x})$ $(\textrm{resp. }(\vec{Y}^{t,x'},Z^{t,x'},U^{t,x'},K^{t,x'}))$ be the solution of the RBSDE with jumps (\[3.17\]) associated with $f(s,X^{t,x}_{s},y,\eta,\zeta,g(X^{t,x}_{T}))$\
$(\textrm{resp. }f(s,X^{t,x'}_{s},y,\eta,\zeta,g(X^{t,x'}_{T})))$. Applying Itô formula to $\left|\vec{Y}^{t,x}-\vec{Y}^{t,x'}\right|^{2}$ between $s$ and $T$, we have $$\begin{aligned}
\label{4.31}
& {}{} &\left|\vec{Y}^{t,x}_{s}-\vec{Y}^{t,x'}_{s}\right|^{2}+\displaystyle\int^{T}_{s}\left|\Delta Z_{r}\right|^{2}\,dr+\sum_{s\leq r\leq T}(\Delta_{r}\vec{Y}^{t,x}_{r})^{2}\\
& {}{} &=\left|g(X^{t,x}_{T})-g(X^{t,x'}_{T})\right|^{2}+2\displaystyle\int^{T}_{s}
<\left(\vec{Y}^{t,x}_{s}-\vec{Y}^{t,x'}_{s}\right),\Delta f(r)>\,dr+2\displaystyle\int^{T}_{s}\left(\vec{Y}^{t,x}_{r}
-\vec{Y}^{t,x'}_{r}\right)\,
d\left(\Delta K_{r}\right)\nonumber \\
& {}{} &-2\displaystyle\int^{T}_{s}
\displaystyle\int_{\mathrm{E}}\left(\vec{Y}^{t,x}_{r}
-\vec{Y}^{t,x'}_{r}\right)\left(\Delta U_{r}(e)\right)\tilde{\mu}(\mathrm{d}r,\mathrm{d}e)-2\displaystyle\int^{T}_{s} \left(\vec{Y}^{t,x}_{r}-\vec{Y}^{t,x'}_{r}\right)\left(\Delta Z_{r}\right)\,dB_{r};\nonumber\end{aligned}$$ and taking expectation we obtain: $\forall s\in[t,T]$, $$\begin{aligned}
\label{4.33}
& {}{} &\mathbb{E}\left[\left|\vec{Y}^{t,x}_{s}-\vec{Y}^{t,x'}_{s}\right|^{2}+\displaystyle\int^{T}_{s}\left|\Delta Z_{r}\right|^{2}\,dr+\displaystyle\int^{T}_{s}\|\Delta U_{r}\|^{2}_{\mathbb{L}^{2}(\lambda)}\,dr\right]\\
& {}{} &\leq\mathbb{E}\left[\left|g(X^{t,x}_{T})-g(X^{t,x'}_{T})\right|^{2}+2\displaystyle\int^{T}_{s}
<\left(\vec{Y}^{t,x}_{s}-\vec{Y}^{t,x'}_{s}\right),\Delta f(r)>\,dr\right]\nonumber\\
& {}{} &\qquad\qquad+\mathbb{E}\left[2\displaystyle\int^{T}_{s}\left(\vec{Y}^{t,x}_{r}
-\vec{Y}^{t,x'}_{r}\right)\,
d\left(\Delta K_{r}\right)\right],\nonumber \end{aligned}$$ where the processes $\Delta X_{r}$, $\Delta Y_{r}$, $\Delta f(r)$, $\Delta K_{r}$, $\Delta Z_{r}$, $\Delta U_{r}$ and $\Delta \ell_{r}$ are defined as follows: $\forall r\in[t,T]$,\
$\Delta f(r):=((\Delta f^{(i)}(r))_{i=1,m}=(f^{(i)}(r,X^{i;t,x}_{r},\vec{Y}^{t,x}_{r},Z^{i;t,x}_{r},U^{i;t,x}_{r})-f^{(i)}(r,X^{i;t,x'}_{r},\vec{Y}^{i;t,x'}_{r},Z^{i;t,x'}_{r},U^{i;t,x'}_{r}))_{i=1,m}$, $\Delta X_{r}=X^{t,x}_{r}-X^{t,x'}_{r}$, $\Delta Y(r)=\vec{Y}^{t,x}_{r}-\vec{Y}^{t,x'}_{r}=(Y^{j;t,x}_{r}-Y^{j;t,x'}_{r})_{j=1,m}$,\
$\Delta K_{r}=K^{t,x}_{r}-K^{t,x'}_{r}$, $\Delta Z_{r}=Z^{t,x}_{r}-Z^{t,x'}_{r}$, $\Delta U_{r}=U^{t,x}_{r}-U^{t,x'}_{r}$ and $\Delta\ell_{r}=\left(\ell(r,X^{t,x}_{r}
)-\ell(r,X^{t,x'}_{r})\right)$ ($<\cdot,\cdot>$ is the usual scalar product on $\mathbb{R}^{m}$). Now we will give an estimation of each three terms of the second member of inequality (\[4.33\]).\
$\bullet$ As for any $i\in\{1,\ldots,m\}$ $g^{i}$ belongs to $\mathcal{U}$; therefore\
$$\begin{aligned}
\mathbb{E}\left[\left|g(X^{t,x}_{T})-g(X^{t,x'}_{T})\right|^{2}\right] & \leq & C\left|X^{t,x}_{T}-X^{t,x'}_{T}\right|^{2}(1+\left|X^{t,x}_{T}\right|^{2p}+\left|X^{t,x'}_{T}\right|^{2p})\nonumber\\
& \leq & \mathbb{E}\left[\left|x-x'\right|^{2}(1+\left|(X^{t,x}_{T}-x)+x\right|^{2p}+\left|(X^{t,x'}_{T}-x')+x'\right|^{2p}\right],\nonumber\end{aligned}$$ and by subsequently using the triangle inequality, the relation of proposition $3.1$ and the fact that $$(a+b){^p}\leq 2^{p-1}(a^{p}+b^{p}).$$ $$\label{4.34}
\mathbb{E}\left[\left|g(X^{t,x}_{T})-g(X^{t,x'}_{T})\right|^{2}\right]\leq C\left|x-x'\right|^{2}(1+\left|x\right|^{2p}+\left|x'\right|^{2p}),$$ $\bullet$ using (iii) of (\[3.17\]): $\mathbb{E}\left[2\displaystyle\int^{T}_{s}\left(\vec{Y}^{t,x}_{r}
-\vec{Y}^{t,x'}_{r}\right)\,
d\left(\Delta K_{r}\right)\right]$ can be replaced by\
$$\mathbb{E}\left[2\displaystyle\int^{T}_{s}\left(\ell(r,X^{t,x}_{r}
)-\ell(r,X^{t,x'}_{r})\right)\,
d\left(\Delta K_{r}\right)\right].$$\
Now by (**H1**) and Cauchy-Schwartz inequality we obtain: $$\label{4.35}
\mathbb{E}\left[\sup_{0\leq t\leq T}(\Delta\ell_{t})^{2}\right]\times\mathbb{E}\left[\left(\Delta K_{T}\right)^{2}\right]\leq 2CC'\left|x-x'\right|^{2}(1+\left|x\right|^{2p}+\left|x'\right|^{2p});$$ where $C'=\mathbb{E}\left[\left(\Delta K_{T}\right)^{2}\right]$.\
$\bullet$ To complete our estimation of (\[4.33\]) we need to deal with $\mathbb{E}\left[2\displaystyle\int^{T}_{s}
<\left(\vec{Y}^{t,x}_{s}-\vec{Y}^{t,x'}_{s}\right),\Delta f(r)>\,dr\right].$ Taking into account the expression of $f^{(i)}$ given by (\[2.13\]) we then split $\Delta f(r)$ in the follows way: for $r\leq T$, $$\Delta f(r)=(\Delta f(r))_{i=1,m}=\Delta_{1}(r)+\Delta_{2}(r)+\Delta_{3}(r)+\Delta_{4}(r)=(\Delta^{i}_{1}(r)+\Delta^{i}_{2}(r)+\Delta^{i}_{3}(r)+\Delta^{i}_{4}(r))_{i=1,m},$$ where for any $i=1,\ldots,m$, $$\begin{aligned}
\Delta^{i}_{1}(r) & = & h^{(i)}\left(r,X^{t,x}_{r},\vec{Y}^{t,x}_{r},Z^{i;t,x}_{r},\displaystyle\int_{E}\gamma^{i}(r,X^{t,x}_{r},e)U^{i;t,x}_{r}(e)\lambda(de)\right)\\
&{}{}& -h^{(i)}\left(r,X^{t,x'}_{r},\vec{Y}^{t,x}_{r},Z^{i;t,x}_{r},\displaystyle\int_{E}\gamma^{i}(r,X^{t,x}_{r},e)U^{i;t,x}_{r}(e)\lambda(de)\right);\\
\Delta^{i}_{2}(r) & = & h^{(i)}\left(r,X^{t,x'}_{r},\vec{Y}^{t,x}_{r},Z^{i;t,x}_{r},\displaystyle\int_{E}\gamma^{i}(r,X^{t,x}_{r},e)U^{i;t,x}_{r}(e)\lambda(de)\right)\\
&{}{}& -h^{(i)}\left(r,X^{t,x'}_{r},\vec{Y}^{t,x'}_{r},Z^{i;t,x}_{r},\displaystyle\int_{E}\gamma^{i}(r,X^{t,x}_{r},e)U^{i;t,x}_{r}(e)\lambda(de)\right);\\
\Delta^{i}_{3}(r) & = & h^{(i)}\left(r,X^{t,x'}_{r},\vec{Y}^{t,x'}_{r},Z^{i;t,x}_{r},\displaystyle\int_{E}\gamma^{i}(r,X^{t,x}_{r},e)U^{i;t,x}_{r}(e)\lambda(de)\right)\\
&{}{}& -h^{(i)}\left(r,X^{t,x'}_{r},\vec{Y}^{t,x'}_{r},Z^{i;t,x'}_{r},\displaystyle\int_{E}\gamma^{i}(r,X^{t,x}_{r},e)U^{i;t,x}_{r}(e)\lambda(de)\right);\\
\Delta^{i}_{4}(r) & = & h^{(i)}\left(r,X^{t,x'}_{r},\vec{Y}^{t,x'}_{r},Z^{i;t,x'}_{r},\displaystyle\int_{E}\gamma^{i}(r,X^{t,x}_{r},e)U^{i;t,x}_{r}(e)\lambda(de)\right)\\
&{}{}& -h^{(i)}\left(r,X^{t,x'}_{r},\vec{Y}^{t,x'}_{r},Z^{i;t,x'}_{r},\displaystyle\int_{E}\gamma^{i}(r,X^{t,x'}_{r},e)U^{i;t,x'}_{r}(e)\lambda(de)\right).\end{aligned}$$ By Cauchy-Schwartz inequality, the inequality $2ab\leq\epsilon a^{2}+\frac{1}{\epsilon}b^{2}$, the relation (2.11) and the estimate (\[3.16\]) we have: $$\begin{aligned}
\label{4.36}
\mathbb{E}\left[2\displaystyle\int^{T}_{s}
\scriptstyle<\Delta Y(r),\Delta_{1}(r)>\,dr\right] & \leq & \mathbb{E}\left[\frac{1}{\epsilon}\int^{T}_{s}\scriptstyle{|\Delta Y(r)|^{2}\,dr+C^{2}\epsilon}\displaystyle\int^{T}_{s}\scriptstyle{|X^{t,x}_{r}-X^{t,x'}_{r}|^{2}(1+|X^{t,x}_{r}|^{p}+|X^{t,x'}_{r}|^{p})^{2}\,dr}\right]\nonumber\\
& \leq & \mathbb{E}\left[\frac{1}{\epsilon}\int^{T}_{s}|\Delta Y(r)|^{2}\,dr\right]+C^{2}\epsilon|x-x'|^{2}(1+|x|^{p}+|x'|^{p})^{2}.\end{aligned}$$ Besides since $h^{(i)}$ is Lipschitz w.r.t. $(y,z,q)$ then, $$\label{4.37}
\mathbb{E}\left[2\displaystyle\int^{T}_{s}<\Delta Y(r),\Delta_{2}(r)>\,dr\right]\leq 2C\mathbb{E}\left[\int^{T}_{s}|\Delta Y(r)|^{2}\,dr\right],$$ and $$\label{4.38}
\mathbb{E}\left[2\displaystyle\int^{T}_{s}<\Delta Y(r),\Delta_{3}(r)>\,dr\right]\leq\mathbb{E}\left[\frac{1}{\epsilon}\int^{T}_{s}|\Delta Y(r)|^{2}\,dr+C^{2}\epsilon\int^{T}_{s}|\Delta Z(r)|^{2}\,dr\right].$$ It remains to obtain a control of the last term. But for any $s\in[t,T]$ we have, $$\begin{aligned}
\label{4.39}
& {}{}& \mathbb{E}\left[2\displaystyle\int^{T}_{s}<\Delta Y(r),\Delta_{4}(r)>\,dr\right]\\
& \leq & 2C\mathbb{E}\left[\int^{T}_{s}|\Delta Y(r)|\,dr\times \left|\int_{E}\left(\gamma(r,X^{t,x}_{r},e)U^{t,x}_{r}(e)-\gamma(r,X^{t,x'}_{r},e)U^{t,x'}_{r}(e)\right)\,\lambda(de)\right|\right]\nonumber.\end{aligned}$$ Next by splitting the crossing terms as follows $\gamma(r,X^{t,x}_{r},e)U^{t,x}_{r}(e)-\gamma(r,X^{t,x'}_{r},e)U^{t,x'}_{r}(e)=\Delta U_{s}(e)\gamma(s,X^{t,x}_{s},e)+U^{t,x'}_{s}\left(\gamma(s,X^{t,x}_{s},e)-\gamma(s,X^{t,x'}_{s},e)\right)$\
and setting $\Delta \gamma_{s}(e):=\left(\gamma(s,X^{t,x}_{s},e)-\gamma(s,X^{t,x'}_{s},e)\right)$,\
we obtain, $$\begin{aligned}
\label{4.40}
\mathbb{E}\left[2\displaystyle\int^{T}_{s}<\Delta Y(r),\Delta_{4}(r)>\,dr\right]& \leq & 2C\mathbb{E}\left[\int^{T}_{s}|\scriptstyle\Delta Y(r)|\times\left(\displaystyle\int_{E}\scriptstyle(|U^{t,x'}_{r}(e)\Delta\gamma_{r}(e)|+|\Delta U_{r}(e)\gamma(r,X^{t,x}_{r},e)|)\,\lambda(de)\right)\,dr\right]\nonumber\\
& \leq & \frac{2}{\epsilon}\mathbb{E}\left[\int^{T}_{s}|\Delta Y(r)|^{2}\,dr\right]+C^{2}\epsilon\mathbb{E}\left[\int^{T}_{s}\left(\int_{E}(|U^{t,x'}_{r}(e)\Delta\gamma_{r}(e)|\lambda(de)\right)^{2}\,dr\right]\nonumber\\
& {}{} &+C^{2}\epsilon\mathbb{E}\left[\int^{T}_{s}\left(\int_{E}(|\Delta U_{r}(e)\gamma(r,X^{t,x}_{r},e)|\lambda(de)\right)^{2}\,dr\right].\end{aligned}$$ By Cauchy-Schwartz inequality, (\[2.12\]) and (\[3.16\]), and the result of Lemma 4.1 it holds: $$\begin{aligned}
\label{4.41}
\mathbb{E}\left[\int^{T}_{s}\left(\int_{E}(|U^{t,x'}_{r}(e)\Delta\gamma_{r}(e)|\lambda(de)\right)^{2}\,dr\right] & \leq & \mathbb{E}\left[\int^{T}_{s}\,dr\left(\int_{E}|U^{t,x'}_{r}(e)|^{2}\lambda(de)\right)\left(\int_{E}|\Delta\gamma_{r}(e)|^{2}\lambda(de)\right)\right]\nonumber\\
&\leq & C\mathbb{E}\left[\{\scriptstyle\sup_{r\in[t,T]}|X^{t,x}_{r}-X^{t,x'}_{r}|^{2}(1+\sup_{r\in[t,T]}|X^{t,x}_{r}|^{p}+|X^{t,x'}_{r}|^{p})^{2}\,dr\}\right]\nonumber\\
& {}{} &\times \mathbb{E}\left[\int^{T}_{s}\,dr\left(\int_{E}|U^{t,x'}_{r}(e)|^{2}\lambda(de)\right)\right]\nonumber\\
& \leq & C\sqrt{\mathbb{E}\left[\{\scriptstyle\sup_{r\in[t,T]}|X^{t,x}_{r}-X^{t,x'}_{r}|^{4}(1+\sup_{r\in[t,T]}|X^{t,x}_{r}|^{p}+|X^{t,x'}_{r}|^{p})^{4}\}\right]}\nonumber\\
& {}{} &\times\sqrt{\mathbb{E}\left[\left\lbrace \int^{T}_{s}\,dr\left(\int_{E}|U^{t,x'}_{r}(e)|^{2}\lambda(de)\right)\right\rbrace^{2}\right]}\nonumber\\
& \leq & C\left|x-x'\right|^{2}(1+\left|x\right|^{2p}+\left|x'\right|^{2p}).\end{aligned}$$ For some exponent $p$. On the other hand using once more Cauchy-Schwartz inequality and (\[2.12\])-(i) we get $$\begin{aligned}
\label{4.42}
\mathbb{E}\left[\int^{T}_{s}\left(\int_{E}(\scriptstyle|\Delta U_{r}(e)\gamma(r,X^{t,x}_{r},e)|\lambda(de)\right)^{2}\,dr\right] & \leq & \mathbb{E}\left[\int^{T}_{s}\,dr\left(\int_{E}(\scriptstyle|\Delta U_{r}(e)|^{2}\lambda(de)\right)\left(\int_{E}|\gamma(r,X^{t,x}_{r},e)|^{2}\lambda(de)\right)\right]\nonumber\\
& \leq & C\mathbb{E}\left[\int^{T}_{s}\,dr\left(\int_{E}(|\Delta U_{r}(e)|^{2}\lambda(de)\right)\right].\end{aligned}$$ Taking now into account inequalities (\[4.36\])-(\[4.42\]) we obtain: $$\begin{aligned}
\label{4.43}
& {}{} &\mathbb{E}\left[\left|\vec{Y}^{t,x}_{s}-\vec{Y}^{t,x'}_{s}\right|^{2}+\displaystyle\int^{T}_{s}\left|\Delta Z_{r}\right|^{2}\,dr+\displaystyle\int^{T}_{s}\|\Delta U_{r}\|^{2}_{\mathbb{L}^{2}(\lambda)}\,dr\right]\\
& {}{} &\leq\mathbb{E}\left[\left|g(X^{t,x}_{T})-g(X^{t,x'}_{T})\right|^{2}+2\displaystyle\int^{T}_{s}
<\left(\vec{Y}^{t,x}_{s}-\vec{Y}^{t,x'}_{s}\right),\Delta f(r)>\,dr\right]\\
& {}{} &\qquad\qquad+\mathbb{E}\left[2\displaystyle\int^{T}_{s}\left(\vec{Y}^{t,x}_{r}
-\vec{Y}^{t,x'}_{r}\right)\,
d\left(\Delta K_{r}\right)\right]\\
& \leq & \left|x-x'\right|^{2}(1+\left|x\right|^{2p}+\left|x'\right|^{2p})(C+2CC'+C^{2}\epsilon+C^{3}\epsilon)+\left(\frac{3}{\epsilon}+2C\right)\mathbb{E}\left[\int^{T}_{s}|\Delta Y(r)|^{2}\,dr\right]\\
&{}{}&+C^{2}\epsilon\mathbb{E}\left[\int^{T}_{s}|\Delta Z(r)|^{2}\,dr\right]+C^{3}\epsilon\mathbb{E}\left[\int^{T}_{s}\,dr\left(\int_{E}(|\Delta U_{r}(e)|^{2}\lambda(de)\right)\right].\end{aligned}$$ Choosing now $\epsilon$ small enough we deduce the existence of a constant $C\geq 0$ such that for any $s\in[t,T]$,\
$\mathbb{E}\left[|\Delta Y(s)|^{2}\right]\leq C\left|x-x'\right|^{2}(1+\left|x\right|^{2p}+\left|x'\right|^{2p})+\mathbb{E}\left[\displaystyle\int^{T}_{s}|\Delta Y(r)|^{2}\,dr\right]$\
and by Gronwall lemma this implies that for any $s\in[t,T]$,\
$$\mathbb{E}\left[|\Delta Y(s)|^{2}\right]\leq C\left|x-x'\right|^{2}(1+\left|x\right|^{2p}+\left|x'\right|^{2p}).$$ Finally in taking $s=t$ and considering (\[3.18\]) we obtain the desired result.
This result give also estimate of $U$ where we use the function $h^{(i)}$ $\forall~i=1,\ldots,m$ contrary in estimate (\[4.22\]).
For $u^{i}\in\mathcal{U}$ $\forall i=1,\ldots,m$ $B_{i}u^{i}$ defined in (\[2.2\]) is well posed since the functions $\beta$ and $(\gamma_{i})_{i=1,m}$ verify (\[2.7\]) and (\[2.12\]) respectively.
The main point to notice is that $\lambda$ integrates $(1\wedge |e|^{p})$ $\forall p\geq 2$.\
We have that $$\begin{aligned}
|\mathrm{B}_{i}u^{i}(t,x)| & \leq & \displaystyle\int_{\mathrm{E}}|\gamma^{i}(t,x,e)|\times|(u^{i}(t,x+\beta(t,x,e))-u^{i}(t,x))|\,\lambda(de)\\
&{}{}&\leq\displaystyle\int_{\mathrm{E}}C(1\wedge|e|)|\beta(t,x,e)|(1+|x+\beta(t,x,e)|^{p}+|x|^{p})\,\lambda(de)\\
&{}{}&\leq C^{2}(1+|x|^{p}(1+2^{p-1}))\displaystyle\int_{\mathrm{E}}C(1\wedge|e|^{2})\,\lambda(de)+(2^{\frac{p-1}{p}}C^{\frac{p+2}{p}})^{p}\displaystyle\int_{\mathrm{E}}C(1\wedge|e|^{p})\,\lambda(de).\end{aligned}$$ Which finish the proof.
Now by remark 3, the last estimate of $U$ confirm the following result;
For any $i=1,\ldots,m$, $(t,x)\in[0,T]\times\mathbb{R}^{k}$, $$U^{i;t,x}_{s}(e)=u^{i}(s,X^{t,x}_{s-}+\beta(s,X^{t,x}_{s-},e))-u^{i}(s,X^{t,x}_{s-}),~~d\mathbb{P}\otimes ds\otimes d\lambda-\text{a.e. on}~\Omega\times[t,T]\times E.$$
First note that since the measure $\lambda$ is note finite, then we cannot use the same technique as in [@hamaMor] where the authors use the jumps of processes and (\[3.18\]).\
In our case $U^{i;t,x}$ is only square integrable and not necessarily integrable w.r.t. $d\mathbb{P}\otimes ds\otimes d\lambda$. Therefore we first begin by truncating the Lévy measure as the same way in [@hama].\
**Step 1: Truncation of the Lévy measure**\
For any $k\geq 1$, let us first introduce a new Poisson random measure $\mu_{k}$ (obtained from the truncation of $\mu$) and its associated compensator $\nu_{k}$ as follows: $$\mu_{k}(ds,de)=1_{\{|e|\geq\frac{1}{k}\}}\mu(ds,de)~~\text{and }\nu_{k}(ds,de)=1_{\{|e|\geq\frac{1}{k}\}}\nu(ds,de).$$ Which means that, as usual, $\tilde{\mu_{k}}(ds,de):=(\mu_{k}-\nu_{k})(ds,de)$, is the associated random martingale measure.\
The main point to notice is that $$\begin{aligned}
\lambda_{k}(E)=\displaystyle\int_{E}\,\lambda_{k}(de)& = &\displaystyle\int_{E}1_{\{|e|\geq\frac{1}{k}\}}\,\lambda(de)\nonumber\\
{}{}&=&\displaystyle\int_{\{|e|\geq\frac{1}{k}\}}\,\lambda(de)\nonumber\\
{}{}&=&\lambda(\{|e|\geq\frac{1}{k}\})<\infty.\end{aligned}$$ As in [@hama], let us introduce the process $^{k}X^{t,x}$ solving the following standard SDE of jump-diffusion type: $$\begin{aligned}
& {}{} & ^{k}X^{t,x}_{s}=x+\displaystyle\int^{s}_{t}b(r,^{k}X^{t,x}_{r})\,dr+\displaystyle\int^{s}_{t}\sigma(r,^{k}X^{t,x}_{r})\,dB_{r}\nonumber\\
& {}{} &\qquad\qquad+\displaystyle\int^{s}_{t}\displaystyle\int_{\mathrm{E}}\beta(r,^{k}X^{t,x}_{r-},e)\tilde{\mu}_{k}\,(dr,de),~~~t\leq s\leq T;~^{k}X^{t,x}_{r}=x~\text{if }s\leq t.\nonumber\\\end{aligned}$$ Note that thanks to the assumptions on $b$, $\sigma$, $\beta$ the process $^{k}X^{t,x}$ exists and is unique. Moreover it satisfies the same estimates as in (\[3.16\]) since $\lambda_{k}$ is just a truncation at the origin of $\lambda$ which integrates $(1\wedge|e|^{2})_{e\in E}$.\
On the other hand let us consider the following Markovian RBSDE with jumps $$\label{4.46}
\left
\{\begin{array}{ll}
(i)~\mathbb{E}\left[\sup_{s\leq T}\left|^{k}Y^{t,x}_{s}\right|^{2}+\displaystyle\int^{T}_{s}\left|^{k}Z^{t,x}_{r}\right|^{2}\,dr+\displaystyle\int^{T}_{s}\|^{k}U^{t,x}_{r}\|^{2}_{\mathbb{L}^{2}(\lambda_{k})}\,dr\right]<\infty\\
(ii)~^{k}{Y}^{t,x}:=(^{k}Y^{i,t,x})_{i=1,m}\in\mathcal{S}^{2}(\mathbb{R}^{m}),~^{k}Z^{t,x}:=(^{k}Z^{i,t,x})_{i=1,m}\in\mathbb{H}^{2}(\mathbb{R}^{m\times d}),\\
^{k}K^{t,x}:=(^{k}K^{i,t,x})_{i=1,m}\in\mathcal{A}^{2}_{c},~ ^{k}U^{t,x}:=(^{k}U^{i,t,x})_{i=1,m}\in\mathbb{H}^{2}(\mathbb{L}^{2}_{m}(\lambda_{k}));\\
(iii)~^{k}Y^{t,x}_{s}=g(^{k}X^{t,x}_{T})+\displaystyle\int^{T}_{s}f_{\mu_{k}}(r,^{k}X^{t,x}_{r},^{k}Y^{t,x}_{r},^{k}Z^{t,x}_{r},^{k}U^{t,x}_{r})\,dr+
^{k}K^{t,x}_{T}-^{k}K^{t,x}_{s}\\
\quad\quad\quad\quad\quad\quad
\quad\quad-\displaystyle\int^{T}_{s}\left\lbrace ^{k}Z^{t,x}_{r}\,d
B_{r}+\displaystyle\int_{\mathrm{E}}^{k}U^{t,x}_{r}(e)\tilde{\mu}_{k}(dr,de)\right\rbrace ,\quad s\leq T;\\
(iv)~^{k}Y^{i;t,x}_{s}\geq \ell(s,^{k}X^{t,x}_{s})~\textrm{and}~ \displaystyle\int^{T}_{0}(^{k}Y^{i;t,x}_{s}- \ell(s,^{k}X^{t,x}_{s}))\,d(^{k}K^{i;t,x}_{s})=0.
\end{array}
\right.$$ Finally let us introduce the following functions $(f^{(i)})_{i=1,m}$ defined by: $\forall (t,x,y,z,\zeta)\in[0,T]\times\mathbb{R}^{k}\times\mathbb{R}^{m}\times\mathbb{R}^{m\times d}\times\mathbb{L}^{2}_{m}(\lambda_{k}),\\f_{\mu_{k}}(t,x,y,z,\zeta)=(f^{(i)}_{\mu_{k}}(t,x,y,z_{i},\zeta_{i}))_{i=1,m}:=\left( h^{(i)}\left(t,x,y,z,\displaystyle\int_{E}\gamma^{i}(t,x,e)\zeta_{i}(e)\lambda_{k}(de)\right)\right)_{i=1,m}$. First let us emphasize that this latter RBSDE is related to the filtration $(\mathcal{F}^{k}_{s})_{s\leq T}$ generated by the Brownian motion and the independent random measure $\mu_{k}$. However this point does not raise major issues since for any $s\leq T$, $\mathcal{F}^{k}_{s}\subset \mathcal{F}_{s}$ and thanks to the relationship between $\mu$ and $\mu_{k}$.\
Next by the properties of the functions $b$, $\sigma$, $\beta$ and by the same opinions of proposition $3.2$ and proposition $3.3$, there exists an unique quadriple $(^{k}Y^{t,x},^{k}K^{t,x},^{k}Z^{t,x},^{k}U^{t,x})$ solving (\[4.46\]) and there also exists a function $u^{k}$ from $[0,T]\times \mathbb{R}^{k}$ into $\mathbb{R}^{m}$ of $\Pi^{c}_{g}$ such that $$\label{4.47}
\forall s\in[t,T],~~^{k}Y^{t,x}:=u^{k}(s,^{k}X^{t,x}),~\mathbb{P}-a.s.$$ Moreover as in proposition $4.2$, there exists positive constants $C$ and $p$ wich do not depend on $k$ such that: $$\label{4.48}
\forall t,x,x',~~|u^{k}(t,x)-u^{k}(t,x')|\leq C\left|x-x'\right|(1+\left|x\right|^{p}+\left|x'\right|^{p}).$$ Finally as $\lambda_{k}$ is finite then we have the following relationship between the process $^{k}U^{t,x}:=(^{k}U^{i;t,x})_{i=1,m}$ and the deterministics functions $u^{k}:=(u^{k}_{i})_{i=1,m}$ (see [@hamaMor]): $\forall i=1,\ldots,m$; $$^{k}U^{i;t,x}_{s}(e)=u^{k}_{i}(s,^{k}X^{t,x}_{s-}+\beta(s,^{k}X^{t,x}_{s-},e))-u^{k}_{i}(s,^{k}X^{t,x}_{s-}),~~d\mathbb{P}\otimes ds\otimes d\lambda_{k}-a.e.~\text{on }\Omega\times[t,T]\times E.$$ This is mainly due to the fact that $^{k}U^{t,x}$ belongs to $\mathbb{L}^{1}\cap\mathbb{L}^{2}(ds\otimes d\mathbb{P}\otimes d\lambda_{k})$ since $\lambda_{k}(E)<\infty$ and then we can split the stochastic integral w.r.t. $\tilde{\mu}_{k}$ in (\[4.46\]). Therefore for all $i=1,\ldots,m$, $$\label{4.49}
^{k}U^{i;t,x}_{s}(e)1_{\{|e|\geq \frac{1}{k}\}}=(u^{k}_{i}(s,^{k}X^{t,x}_{s-}+\beta(s,^{k}X^{t,x}_{s-},e))-u^{k}_{i}(s,^{k}X^{t,x}_{s-}))1_{\{|e|\geq \frac{1}{k}\}},~~d\mathbb{P}\otimes ds\otimes d\lambda_{k}-a.e.~\text{on }\Omega\times[t,T]\times E.$$
**Step 2: Convergence of the auxiliary processes**\
Let’s now prove the following convergence result; $$\begin{aligned}
\label{4.51}
&{}{}&\mathbb{E}\left[\sup_{s\leq T}\left|Y^{t,x}_{s}-^{k}Y^{t,x}_{s}\right|^{2}+(K^{t,x}_{T}-^{k}K^{t,x}_{T})^{2}+\displaystyle\int^{T}_{0}\left|Z^{t,x}_{s}-^{k}Z^{t,x}_{s}\right|^{2}\,ds
\right.\nonumber\\
&{}{}&\left.
+\displaystyle\int^{T}_{0}\,ds\displaystyle\int_{E}\lambda(de)\left|U^{t,x}_{s}(e)-^{k}U^{t,x}_{s}(e)1_{\{|e|\geq \frac{1}{k}\}}\right|^{2}\right]\substack{\longrightarrow\\ k\longrightarrow+\infty}0;\end{aligned}$$ where $(Y^{t,x},K^{t,x},Z^{t,x},U^{t,x})$ is solution of the RBSDE with jumps (\[3.17\]).\
First note that the following convergence result was established in [@hama] $$\label{4.50}
\mathbb{E}\left[\sup_{s\leq T}\left|X^{t,x}_{s}-^{k}X^{t,x}_{s}\right|^{2}\right]\substack{\longrightarrow\\ k\longrightarrow+\infty}0.$$ We now focus on (\[4.51\]). Note that we can apply Ito’s formula, even if the RBSDEs are related to filtrations and Poisson random measures which are not the same, since:\
(i) $\mathcal{F}^{k}_{s}\subset\mathcal{F}_{s}$, $\forall s\leq T$;\
(ii) for any $s\leq T$, $\displaystyle\int^{s}_{0}\displaystyle\int_{\mathrm{E}}^{k}U^{i;t,x}(e)\tilde{\mu}_{k}\,(dr,de)=\displaystyle\int^{s}_{0}\displaystyle\int_{\mathrm{E}}^{k}U^{i;t,x}(e)1_{\{|e|\geq\frac{1}{k}\}}\tilde{\mu}\,(dr,de)$ and then the first $(\mathcal{F}^{k}_{s})_{s\leq T}-$martingale is also an $(\mathcal{F}_{s})_{s\leq T}-$martingale. $\forall s\in[0,T]$, $$\begin{aligned}
& {}{} &\left|\vec{Y}^{t,x}_{s}-^{k}Y^{t,x}_{s}\right|^{2}+\displaystyle\int^{T}_{0}\left|Z^{t,x}_{s}-^{k}Z^{t,x}_{s}\right|^{2},ds+\sum_{s\leq r\leq T}(^{k}\Delta_{r}\vec{Y}^{t,x}_{r})^{2}\nonumber\\
& {}{} &=\left|g(X^{t,x}_{T})-g(^{k}X^{t,x}_{T})\right|^{2}+2\displaystyle\int^{T}_{s}
\left(\vec{Y}^{t,x}_{r}-^{k}Y^{t,x}_{r}\right)\times ^{k}\Delta f(r)\,dr+2\displaystyle\int^{T}_{s}\left(\vec{Y}^{t,x}_{r}
-^{k}Y^{t,x}_{r}\right)\,
d\left(^{k}\Delta K_{r}\right)\nonumber \\
& {}{} &-2\displaystyle\int^{T}_{s}
\displaystyle\int_{\mathrm{E}}\left(\vec{Y}^{t,x}_{r}
-^{k}Y^{t,x}_{r}\right)\left(^{k}\Delta U_{r}(e)\right)\tilde{\mu}(\mathrm{d}r,\mathrm{d}e)-2\displaystyle\int^{T}_{s} \left(\vec{Y}^{t,x}_{r}-^{k}\vec{Y}^{t,x}_{r}\right)\left(^{k}\Delta Z_{r}\right)\,dB_{r};\nonumber\end{aligned}$$ and taking expectation we obtain: $\forall s\in[t,T]$,
$$\begin{aligned}
\label{4.53}
& {}{} &\mathbb{E}\left[\left|\vec{Y}^{t,x}_{s}-^{k}Y^{t,x}_{s}\right|^{2}+\left|^{k}\Delta K_{T}\right|^{2}+\displaystyle\int^{T}_{0}\left\lbrace\left|Z^{t,x}_{s}-^{k}Z^{t,x}_{s}\right|^{2}+\displaystyle\int_{E}\left|U^{t,x}_{s}-^{k}U^{t,x}_{s}1_{\{|e|\geq\frac{1}{k}\}}\right|^{2}\,\lambda(de)\right\rbrace\,ds\right]\nonumber\\
& {}{} &\leq\mathbb{E}\left[\left|g(X^{t,x}_{T})-g(^{k}X^{t,x}_{T})\right|^{2}+2\displaystyle\int^{T}_{s}
\left(\vec{Y}^{t,x}_{r}-^{k}Y^{t,x}_{r}\right)\times ^{k}\Delta f(r)\,dr\right]+\mathbb{E}\left[\sup_{s\leq T}\left|^{k}\Delta\ell_{s}\right|^{2}\right];\nonumber\\\end{aligned}$$
where the processes $^{k}\Delta X_{r}$, $^{k}\Delta Y_{r}$, $^{k}\Delta f(r)$, $^{k}\Delta K_{r}$, $^{k}\Delta Z_{r}$, $^{k}\Delta U_{r}$ and $^{k}\Delta \ell_{r}$ are defined as follows: $\forall r\in[0,T]$,\
$^{k}\Delta f(r):=((^{k}\Delta f^{(i)}(r))_{i=1,m}=(f^{(i)}(r,X^{t,x}_{r},\vec{Y}^{t,x}_{r},Z^{i;t,x}_{r},U^{i;t,x}_{r})-f^{(i)}_{k}(r,^{k}X^{t,x}_{r},^{k}Y^{t,x}_{r},^{k}Z^{t,x}_{r},^{k}U^{t,x}_{r}))_{i=1,m}$, $^{k}\Delta X_{r}=X^{t,x}_{r}-^{k}X^{t,x}_{r}$, $^{k}\Delta Y(r)=\vec{Y}^{t,x}_{r}-^{k}Y^{t,x}_{r}=(Y^{j;t,x}_{r}-^{k}Y^{j;t,x}_{r})_{j=1,m}$,\
$^{k}\Delta K_{r}=K^{t,x}_{r}-^{k}K^{t,x}_{r}$, $^{k}\Delta Z_{r}=Z^{t,x}_{r}-^{k}Z^{t,x}_{r}$, $^{k}\Delta U_{r}=U^{t,x}_{r}-^{k}U^{t,x}_{s}1_{\{|e|\geq\frac{1}{k}\}}$ and $^{k}\Delta\ell_{r}=\left(\ell(r,X^{t,x}_{r}
)-\ell(r,^{k}X^{t,x}_{r})\right)$.\
Next let us set for $r\leq T$, $$^{k}\Delta f(r)=(f(r,X^{t,x}_{r},\vec{Y}^{t,x}_{r},Z^{t,x}_{r},U^{t,x}_{r})-f_{k}(r,^{k}X^{t,x}_{r},^{k}Y^{t,x}_{r},^{k}Z^{t,x}_{r},^{k}U^{t,x}_{r}))=A(r)+B(r)+C(r)+D(r);$$ where for any $i=1,\ldots,m$, $$\begin{aligned}
A(r) & = & \left(h^{(i)}\left(r,X^{t,x}_{r},\vec{Y}^{t,x}_{r},Z^{i;t,x}_{r},\displaystyle\int_{E}\gamma^{i}(r,X^{t,x}_{r},e)U^{i;t,x}_{r}(e)\lambda(de)\right)
\right.\\
&{}{}&\left.
-h^{(i)}\left(r,^{k}X^{t,x}_{r},\vec{Y}^{t,x}_{r},Z^{i;t,x}_{r},\displaystyle\int_{E}\gamma^{i}(r,X^{t,x}_{r},e)U^{i;t,x}_{r}(e)\lambda(de)\right)\right)_{i=1,m};\\
B(r) & = & \left(h^{(i)}\left(r,^{k}X^{t,x}_{r},\vec{Y}^{t,x}_{r},Z^{i;t,x}_{r},\displaystyle\int_{E}\gamma^{i}(r,X^{t,x}_{r},e)U^{i;t,x}_{r}(e)\lambda(de)\right)
\right.\\
&{}{}&\left.
-h^{(i)}\left(r,^{k}X^{t,x}_{r},^{k}Y^{t,x}_{r},Z^{i;t,x}_{r},\displaystyle\int_{E}\gamma^{i}(r,X^{t,x}_{r},e)U^{i;t,x}_{r}(e)\lambda(de)\right)\right)_{i=1,m};\\
C(r) & = & \left(h^{(i)}\left(r,^{k}X^{t,x}_{r},^{k}Y^{t,x}_{r},Z^{i;t,x}_{r},\displaystyle\int_{E}\gamma^{i}(r,X^{t,x}_{r},e)U^{i;t,x}_{r}(e)\lambda(de)\right)
\right.\\
&{}{}&\left.
-h^{(i)}\left(r,^{k}X^{t,x}_{r},^{k}Y^{t,x}_{r},^{k}Z^{i;t,x}_{r},\displaystyle\int_{E}\gamma^{i}(r,X^{t,x}_{r},e)U^{i;t,x}_{r}(e)\lambda(de)\right)\right)_{i=1,m};\\
D(r) & = & \left( h^{(i)}\left(r,^{k}X^{t,x}_{r},^{k}Y^{t,x}_{r},^{k}Z^{i;t,x}_{r},\displaystyle\int_{E}\gamma^{i}(r,X^{t,x}_{r},e)U^{i;t,x}_{r}(e)\lambda(de)\right)
\right.\\
&{}{}&\left.
-h^{(i)}\left(r,^{k}X^{t,x}_{r},^{k}Y^{t,x}_{r},^{k}Z^{i;t,x}_{r},\displaystyle\int_{E}\gamma^{i}(r,^{k}X^{t,x}_{r},e)^{k}U^{i;t,x}_{r}(e)\lambda_{k}(de)\right)\right)_{i=1,m}.\end{aligned}$$ By (\[4.50\]) and the of $g\in\mathcal{U}$ and $\ell\in\mathcal{U}$ we have, $$\label{4.54}
\mathbb{E}\left[\left|g(X^{t,x}_{T})-g(^{k}X^{t,x}_{T})\right|^{2}\right]\substack{\displaystyle\longrightarrow 0\\ k\rightarrow+\infty}$$ and $$\label{4.55}
\mathbb{E}\left[\sup_{s\leq T}\left|\ell(X^{t,x}_{s})-\ell(^{k}X^{t,x}_{s})\right|^{2}\right]\substack{\displaystyle\longrightarrow 0\\ k\rightarrow+\infty}.$$ Now we will interest to $\mathbb{E}\left[\displaystyle\int^{T}_{s}
\left(\vec{Y}^{t,x}_{r}-^{k}Y^{t,x}_{r}\right)\times ^{k}\Delta f(r)\,dr\right]$ for found (\[4.51\]).\
By (\[2.10\]) and (\[2.11\]), we have: $\forall r\in[0,T]$ $$\begin{aligned}
\label{4.56}
\left|A(r)\right| &\leq & C\left|X^{t,x}_{r}-^{k}X^{t,x}_{r}\right|(1+\left|X^{t,x}_{r}\right|^{p}+\left|^{k}X^{t,x}_{r}\right|^{p});\\
\left|B(r)\right| &\leq & C\left|\vec{Y}^{t,x}_{r}-^{k}Y^{t,x}_{r}\right|~\text{and } \left|C(r)\right|\leq \left|Z^{t,x}_{r}-^{k}Z^{t,x}_{r}\right|;\nonumber\end{aligned}$$ where $C$ is a constant. Finally let us deal with $D(r)$ which is more involved. First note that $D(r)=(D_{i}(r))_{i=1,m}$ where $$\begin{aligned}
D_{i}(r)& = & h^{(i)}\left(r,^{k}X^{t,x}_{r},^{k}Y^{t,x}_{r},^{k}Z^{i;t,x}_{r},\displaystyle\int_{E}\gamma^{i}(r,X^{t,x}_{r},e)U^{i;t,x}_{r}(e)\lambda(de)\right)\\
&{}{}&-h^{(i)}\left(r,^{k}X^{t,x}_{r},^{k}Y^{t,x}_{r},^{k}Z^{i;t,x}_{r},\displaystyle\int_{E}\gamma^{i}(r,^{k}X^{t,x}_{r},e)^{k}U^{i;t,x}_{r}(e)\lambda_{k}(de)\right).\end{aligned}$$ But as $h^{(i)}$ is Lipschitz w.r.t to the last component $q$ then, $$\begin{aligned}
\label{4.57}
\left|D(r)\right|^{2} & \leq & C\left\lbrace \displaystyle\int_{E}\left|\gamma^{i}(r,X^{t,x}_{r},e)U^{i;t,x}_{r}(e)-\gamma^{i}(r,^{k}X^{t,x}_{r},e)^{k}U^{i;t,x}_{r}(e)1_{\{|e|\geq\frac{1}{k}\}}\right|^{2}\lambda(de)\right\rbrace\nonumber\\
{}&\leq & C\left\lbrace \left\lbrace \displaystyle\int_{E}\left|\gamma^{i}(r,X^{t,x}_{r},e)-\gamma^{i}(r,^{k}X^{t,x}_{r},e)\right|\left|U^{i;t,x}_{r}(e)\right|\,\lambda(de)\right\rbrace^{2}
\right.\nonumber\\
&{}{}&\left.
+\left\lbrace \displaystyle\int_{E}\left|\gamma^{i}(r,X^{t,x}_{r},e)\right|\left| U^{i;t,x}_{r}(e)-^{k}U^{i;t,x}_{r}(e)1_{\{|e|\geq\frac{1}{k}\}}\right|\lambda(de)\right\rbrace^{2}\right\rbrace\nonumber\\
{}&\leq & C\left\lbrace \left|X^{t,x}_{r}-^{k}X^{t,x}_{r}\right|(1+\left|X^{t,x}_{r}\right|^{p}+\left|^{k}X^{t,x}_{r}\right|^{p})\left|\displaystyle\int_{E}U^{i;t,x}_{r}(e)\right|\lambda(de)\right\rbrace^{2}\nonumber\\
&{}{}&+C\displaystyle\int_{E}(1\wedge|e|)\left| U^{i;t,x}_{r}(e)-^{k}U^{i;t,x}_{r}(e)1_{\{|e|\geq\frac{1}{k}\}}\right|^{2}\lambda(de),\end{aligned}$$ and (\[4.53\]) become by using the majorations obtain in (\[4.56\]) and in (\[4.57\]); $$\begin{aligned}
\label{4.58}
& {}{} &\mathbb{E}\left[\left|\vec{Y}^{t,x}_{s}-^{k}Y^{t,x}_{s}\right|^{2}+\left|^{k}\Delta K_{T}\right|^{2}+\displaystyle\int^{T}_{0}\left\lbrace\left|Z^{t,x}_{s}-^{k}Z^{t,x}_{s}\right|^{2}+\displaystyle\int_{E}\left|U^{t,x}_{s}-^{k}U^{t,x}_{s}1_{\{|e|\geq\frac{1}{k}\}}\right|^{2}\,\lambda(de)\right\rbrace\,ds\right]\nonumber\\
& {}{} &\leq\mathbb{E}\left[\left|g(X^{t,x}_{T})-g(^{k}X^{t,x}_{T})\right|^{2}\right]+\mathbb{E}\left[\sup_{s\leq T}\left|\ell(X^{t,x}_{s})-\ell(^{k}X^{t,x}_{s})\right|^{2}\right] +C\mathbb{E}\left[\displaystyle\int^{T}_{s}
\left|\vec{Y}^{t,x}_{s}-^{k}Y^{t,x}_{s}\right|^{2}\right]\nonumber\\
& {}{} &+C\mathbb{E}\left[\displaystyle\int^{T}_{0}\left|X^{t,x}_{r}-^{k}X^{t,x}_{r}\right|^{2}(1+\left|X^{t,x}_{r}\right|^{p}+\left|^{k}X^{t,x}_{r}\right|^{p})^{2}\,dr\right]\nonumber\\
& {}{} &+C\mathbb{E}\left[\displaystyle\int^{T}_{0}\,dr\left\lbrace \left|X^{t,x}_{r}-^{k}X^{t,x}_{r}\right|(1+\left|X^{t,x}_{r}\right|^{p}+\left|^{k}X^{t,x}_{r}\right|^{p})\displaystyle\int_{E}U^{i;t,x}_{r}(e)\lambda(de)\right\rbrace^{2}\right].\nonumber\\\end{aligned}$$ The two first terms converge to $0$ by (\[4.54\]) and (\[4.55\]).\
For the fourth term we have: $$\begin{aligned}
&{}{}&\mathbb{E}\left[\displaystyle\int^{T}_{0}\left|X^{t,x}_{r}-^{k}X^{t,x}_{r}\right|^{2}(1+\left|X^{t,x}_{r}\right|^{p}+\left|^{k}X^{t,x}_{r}\right|^{p})^{2}\,dr\right]\\
{}&\leq & \mathbb{E}\left[\sup_{r\leq T}\left|X^{t,x}_{r}-^{k}X^{t,x}_{r}\right|^{2}\displaystyle\int^{T}_{0}(1+\left|X^{t,x}_{r}\right|^{p}+\left|^{k}X^{t,x}_{r}\right|^{p})^{2}\,dr\right]\nonumber\\
{}&\leq & \left\lbrace\mathbb{E}\left[\sup_{r\leq T}\left|X^{t,x}_{r}-^{k}X^{t,x}_{r}\right|^{2}\right]\right\rbrace^{\frac{1}{2}}\left\lbrace\mathbb{E}\left[\left(\displaystyle\int^{T}_{0}(1+\left|X^{t,x}_{r}\right|^{p}+\left|^{k}X^{t,x}_{r}\right|^{p})^{2}\left|X^{t,x}_{r}-^{k}X^{t,x}_{r}\right|\right)^{2}\,dr\right]\right\rbrace^{\frac{1}{2}}.\end{aligned}$$ The first factor in the right-hand side of this inequality goes to $0$ when $k\rightarrow \infty$ due to (\[4.50\]) and the second factor is uniformly bounded by the uniform estimates (\[3.16\]) of $X^{t,x}$ and $^{k}X^{t,x}$.\
Note also the last term converge to $0$ when $k\rightarrow \infty$, it is a consequence of (\[4.50\]), the fact that $^{k}X^{t,x}$ verifies estimates (\[3.16\]) uniformly, the Cauchy-Schwartz inequality (used twice) and finally (\[4.22\]) of lemma $4.1$. Then by Gronwall’s lemma we deduce first that for any $s\leq T$, $$\label{4.59}
\mathbb{E}\left[\left|\vec{Y}^{t,x}_{s}-^{k}Y^{t,x}_{s}\right|^{2}\right]\substack{\displaystyle\longrightarrow 0\\ k\rightarrow+\infty}$$ and in taking $s=t$ we obtain $u^{k}(t,x)\substack{\displaystyle\longrightarrow u(t,x)\\ k\rightarrow+\infty}$. As $(t,x)\in[0,T]\times \mathbb{R}^{k}$ is arbitrary then u$^{k}\substack{\displaystyle\longrightarrow u\\ k\rightarrow+\infty}$ pointwisely.\
Next going back to (\[4.58\]) take the limit w.r.t $k$ and using the uniform polynomial growth of $u^{k}$ and the Lebesgue dominated convergence theorem as well, to obtain: $$\label{4.60}
\mathbb{E}\left[\displaystyle\int^{T}_{t}\displaystyle\int_{E}\left|U^{t,x}_{s}-^{k}U^{t,x}_{s}1_{\{|e|\geq\frac{1}{k}\}}\right|^{2}\,\lambda(de)\,ds\right]\substack{\displaystyle\longrightarrow 0\\ k\rightarrow+\infty}.$$ **Step 3: Conclusion**\
First note that by (\[4.48\]) and the pointwise convergence of $(u^{k})_{k}$ to $u$, if $(x_{k})_{k}$ is a sequence of $\mathbb{R}^{k}$ which converge to $x$ then $((u^{k}(t,x_{k}))_{k})$ converge to $u(t,x)$.\
Now let us consider a subsequence which we still denote by $\{k\}$ such that $\sup_{s\leq T}\left|X^{t,x}_{s}-^{k}X^{t,x}_{s}\right|^{2}\substack{\displaystyle\longrightarrow 0\\ k\rightarrow+\infty}$, $\mathbb{P}$-a.s. (and then $\left|X^{t,x}_{s-}-^{k}X^{t,x}_{s-}\right|\substack{\displaystyle\longrightarrow 0\\ k\rightarrow+\infty}$ since $\left|X^{t,x}_{s-}-^{k}X^{t,x}_{s-}\right|\leq \sup_{s\leq T}\left|X^{t,x}_{s}-^{k}X^{t,x}_{s}\right|^{2}$). By (\[4.50\]), this subsequence exists. As the mapping $x\mapsto \beta(t,x,e)$ is Lipschitz then the sequence $$\begin{aligned}
\label{4.61}
&{}{}&\left(^{k}U^{t,x}_{s}(e)1_{\{|e|\geq\frac{1}{k}\}}\right)_{k}=\left((u^{k}_{i}(s,^{k}X^{t,x}_{s-}+\beta(s,^{k}X^{t,x}_{s-},e))-u^{k}_{i}(s,^{k}X^{t,x}_{s-}))1_{\{|e|\geq\frac{1}{k}\}}\right)_{k\geq 1}\substack{\displaystyle\longrightarrow {}\\ k\rightarrow+\infty}\nonumber\\
&{}{}& (u_{i}(s,X^{t,x}_{s-}+\beta(s,X^{t,x}_{s-},e))-u_{i}(s,X^{t,x}_{s-})),\quad d\mathbb{P}\otimes ds\otimes d\lambda-a.e.\quad \text{on }\Omega\times[t,T]\times E\quad\end{aligned}$$ for any $i=1,\ldots,m$. Finally from (\[4.60\]) we deduce that $$\label{4.62}
U^{t,x}_{s}(e)=(u_{i}(s,X^{t,x}_{s-}+\beta(s,X^{t,x}_{s-},e))-u_{i}(s,X^{t,x}_{s-})),\quad\text{on }\Omega\times[t,T]\times E$$ which is the desired result.
The main result
===============
First we give the definition of viscosity solution of IPDEs as given in [@hama] and [@hamaMor]. Our main result deal with this definition.
We say that a family of deterministics functions $u=(u^{i})_{i=1,m}$ which belongs to $\mathcal{U}\quad\forall i\in\{1,\ldots,m\}$ is a viscosity sub-solution (resp. super-solution) of the IPDE (\[eq1\]) if:\
$(i)\quad \forall x\in\mathbb{R}^{k}$, $u^{i}(x,T)\leq g^{i}(x)$ (resp. $u^{i}(x,T)\geq g^{i}(x)$);\
$(ii)\quad\text{For any } (t,x)\in[0,T]\times\mathbb{R}^{k}$ and any function $\phi$ of class $C^{1,2}([0,T]\times\mathbb{R}^{k})$ such that $(t,x)$ is a global maximum point of $u^{i}-\phi$ (resp. global minimum point of $u^{i}-\phi$) and $(u^{i}-\phi)(t,x)=0$ one has $$\label{5.63}
\min\left\lbrace u^{i}(t,x)-\ell(t,x);-\partial_{t}\phi(t,x)-\mathcal{L}^{X}\phi(t,x)-h^{i}(t,x,(u^{j}(t,x))_{j=1,m},\sigma^{\top}(t,x))D_{x}\phi(t,x),B_{i}u^{i}(t,x))\right\rbrace\leq 0$$ $\left(resp.
\right.$ $$\label{5.64}
\left.
\min\left\lbrace u^{i}(t,x)-\ell(t,x);-\partial_{t}\phi(t,x)-\mathcal{L}^{X}\phi(t,x)-h^{i}(t,x,(u^{j}(t,x))_{j=1,m},\sigma^{\top}(t,x))D_{x}\phi(t,x),B_{i}u^{i}(t,x))\right\rbrace\geq 0\right).$$ The family $u=(u^{i})_{i=1,m}$ is a viscosity solution of (\[eq1\]) if it is both a viscosity sub-solution and viscosity super-solution.\
Note that $\mathcal{L}^{X}\phi(t,x)=b(t,x)^{\top}\mathrm{D}_{x}\phi(t,x)+\frac{1}{2}\mathrm{Tr}(\sigma\sigma^{\top}(t,x)\mathrm{D}^{2}_{xx}\phi(t,x))+\mathrm{K}\phi(t,x)$;\
where $\mathrm{K}\phi(t,x)=\displaystyle\int_{\mathrm{E}}(\phi(t,x+\beta(t,x,e))-\phi(t,x)-\beta(t,x,e)^{\top}\mathrm{D}_{x}\phi(t,x))\lambda(de).$
Under assumptions (**H1**), (**H2**) and (**H3**), the IPDE (\[eq1\]) has unique solution which is the $m$-tuple of functions $(u^{i})_{i=1,m}$ defined in proposition $3.3$ by (\[3.18\]).
*:* *Existence*\
Assume that assumptions (**H1**), (**H2**) and (**H3**) are fulfilled, then the following multi-dimensional RBSDEs with jumps $$\label{5.65}
\left
\{\begin{array}{ll}
(i)~\underline{\vec{Y}}^{t,x}:=(\underline{Y}^{i;t,x})_{i=1,m}\in\mathcal{S}^{2}(\mathbb{R}^{m}),~\underline{Z}^{t,x}:=(\underline{Z}^{i;t,x})_{i=1,m}\in\mathbb{H}^{2}(\mathbb{R}^{m\times d}),~
\underline{K}^{t,x}:=(\underline{K}^{i;t,x})_{i=1,m}\in\mathcal{A}^{2}_{c},\\\underline{U}^{t,x}:=(\underline{U}^{i;t,x})_{i=1,m}\in\mathbb{H}^{2}(\mathbb{L}^{2}_{m}(\lambda));\\
(ii)~\underline{Y}^{i;t,x}_{s}= g^{i}(X^{t,x}_{T})+
\underline{K}^{i;t,x}_{T}-\underline{K}^{i;t,x}_{s}-\displaystyle\int^{T}_{s}\underline{Z}^{i;t,x}\mathrm{d}
\mathrm{B}_{r}-\displaystyle\int^{T}_{s}
\displaystyle\int_{\mathrm{E}}\underline{U}^{i;t,x}_{r}(e)\tilde{\mu}(\mathrm{d}r,\mathrm{d}e).\\
\quad\quad\quad+\displaystyle\int^{T}_{s}h^{(i)}(r,X^{t,x}_{r},\underline{Y}^{i;t,x}_{r},\underline{Z}^{i;t,x}_{r},\displaystyle\int_{\mathrm{E}}\gamma^{i}(t,X^{t,x}_{r},e)\{(u^{i}(t,X^{t,x}_{r-}+\beta(t,X^{t,x}_{r-},e))-u^{i}(t,X^{t,x}_{r-}))\}\,\lambda(de))dr\\
(iii)~\underline{Y}^{i;t,x}_{s}\geq \ell(s,X^{t,x}_{s})~\textrm{and}~ \displaystyle\int^{T}_{0}(\underline{Y}^{i;t,x}_{s}- \ell(s,X^{t,x}_{s}))\mathrm{d}\underline{K}^{i;t,x}_{s}=0;
\end{array}
\right.$$ has unique solution $(\underline{Y},\underline{Z},\underline{K},\underline{U})$. Next as for any $i=1,\ldots,m$, $u^{i}$ belongs to $\mathcal{U}$, then by proposition $3.3$ the (\[3.18\]), there exists a family of deterministics continuous functions of polynomial growth $(\underline{u}^{i})_{i=1,m}$ that fact for any $(t,x)\in[0,T]\times\mathbb{R}^{k}$, $$\forall s\in[t,T],\qquad \underline{Y}^{i;t,x}_{s}=\underline{u}^{i}(s,X^{t,x}_{s}).$$ Such that by the same proposition, the family $(\underline{u}^{i})_{i=1,m}$ is a viscosity solution of the following system: $$\label{5.66}
\left
\{\begin{array}{ll}
\min\Big\{\underline{u}^{i}(t,x)-\ell(t,x);-\partial_{t}\underline{u}^{i}(t,x)-b(t,x)^{\top}\mathrm{D}_{x}\underline{u}^{i}(t,x)-\frac{1}{2}\mathrm{Tr}(\sigma\sigma^{\top}(t,x)\mathrm{D}^{2}_{xx}\underline{u}^{i}(t,x))\\
\quad\quad-\mathrm{K}_{i}\underline{u}^{i}(t,x)-\mathit{h}^{(i)}(t,x,(\underline{u}^{j}(t,x))_{j=1,m},(\sigma^{\top}\mathrm{D}_{x}\underline{u}^{i})(t,x),\mathrm{B}_{i}u^{i}(t,x))\Big\}=0,\quad (t,x)\in\left[ 0,T\right] \times\mathbb{R}^{k};\\
u^{i}(T,x)=g^{i}(x).
\end{array}
\right.$$ Now we have the family $(\underline{u}^{i})_{i=1,m}$ is a viscosity solution, our main objective is to found relation between $(\underline{u}^{i})_{i=1,m}$ and $(u^{i})_{i=1,m}$ which is defined in (\[3.18\]).\
For this, let us consider the system of RBSDE with jumps $$\label{5.67}
\left
\{\begin{array}{ll}
(i)~\vec{Y}^{t,x}:=(Y^{i;t,x})_{i=1,m}\in\mathcal{S}^{2}(\mathbb{R}^{m}),~Z^{t,x}:=(Z^{i;t,x})_{i=1,m}\in\mathbb{H}^{2}(\mathbb{R}^{m\times d}),~K^{t,x}:=(K^{i;t,x})_{i=1,m}\in\mathcal{A}^{2}_{c},\\U^{t,x}:=(U^{i;t,x})_{i=1,m}\in\mathbb{H}^{2}(\mathbb{L}^{2}_{m}(\lambda));\\
(ii)~Y^{i;t,x}_{s}= g^{i}(X^{t,x}_{T})+
K^{i;t,x}_{T}-K^{i;t,x}_{s}-\displaystyle\int^{T}_{s}Z^{i;t,x}\mathrm{d}
\mathrm{B}_{r}-\displaystyle\int^{T}_{s}
\displaystyle\int_{\mathrm{E}}U^{i;t,x}_{r}(e)\tilde{\mu}(\mathrm{d}r,\mathrm{d}e).\\
\quad\quad\quad+\displaystyle\int^{T}_{s}h^{(i)}(r,X^{t,x}_{r},Y^{i;t,x}_{r},\underline{Z}^{i;t,x}_{r},\displaystyle\int_{\mathrm{E}}\gamma^{i}(t,X^{t,x}_{r},e)U^{i;t,x}_{r}(e)\,\lambda(de))dr;\\
(iii)~Y^{i;t,x}_{s}\geq \ell(s,X^{t,x}_{s})~\textrm{and}~ \displaystyle\int^{T}_{0}(Y^{i;t,x}_{s}- \ell(s,X^{t,x}_{s}))\mathrm{d}K^{i;t,x}_{s}=0.
\end{array}
\right.$$ By uniqueness of the solution of the RBSDEs with jumps (\[5.64\]), that for any $s\in[t,T]$ and $\forall i\in\{1\ldots,m\}$, $\underline{Y}^{i;t,x}_{s}=Y^{i;t,x}_{s}$.\
Therefore $\underline{u}^{i}=u^{i}$, such that by (\[4.60\]) we obtain $U^{t,x}_{s}(e)=(u_{i}(s,X^{t,x}_{s-}+\beta(s,X^{t,x}_{s-},e))-u_{i}(s,X^{t,x}_{s-})),\quad\text{on }\Omega\times[t,T]\times E$, which give the viscosity solution in the sense of definition $5.1$ (see [@hama]) by pluging (\[4.61\]) in $h^{(i)}$ of (\[5.66\]).
*:* *Uniqueness*\
For uniqueness, let $(\overline{u}^{i})_{i=1,m}$ be another family of $\mathcal{U}$ which is solution viscosity of the system (\[eq1\]) in the sense of definition $5.1$ and we consider RBSDE with jumps defined with $\overline{u}^{i}$. $$\label{5.68}
\left
\{\begin{array}{ll}
(i)~\vec{\overline{Y}}^{t,x}:=(\overline{Y}^{i;t,x})_{i=1,m}\in\mathcal{S}^{2}(\mathbb{R}^{m}),~\overline{Z}^{t,x}:=(\overline{Z}^{i;t,x})_{i=1,m}\in\mathbb{H}^{2}(\mathbb{R}^{m\times d}),~\overline{K}^{t,x}:=(\overline{K}^{i;t,x})_{i=1,m}\in\mathcal{A}^{2}_{c},\\\overline{U}^{t,x}:=(\overline{U}^{i;t,x})_{i=1,m}\in\mathbb{H}^{2}(\mathbb{L}^{2}_{m}(\lambda));\\
(ii)~\overline{Y}^{i;t,x}_{s}= g^{i}(X^{t,x}_{T})+
\overline{K}^{i;t,x}_{T}-\overline{K}^{i;t,x}_{s}-\displaystyle\int^{T}_{s}\overline{Z}^{i;t,x}\mathrm{d}
\mathrm{B}_{r}-\displaystyle\int^{T}_{s}
\displaystyle\int_{\mathrm{E}}\overline{U}^{i;t,x}_{r}(e)\tilde{\mu}(\mathrm{d}r,\mathrm{d}e).\\
\quad\quad\quad+\displaystyle\int^{T}_{s}h^{(i)}(r,X^{t,x}_{r},\overline{Y}^{i;t,x}_{r},\overline{Z}^{i;t,x}_{r},\displaystyle\int_{\mathrm{E}}\gamma^{i}(t,X^{t,x}_{r},e)(\overline{u}_{i}(s,X^{t,x}_{s-}+\beta(s,X^{t,x}_{s-},e))-\overline{u}_{i}(s,X^{t,x}_{s-}))\,\lambda(de))dr;\\
(iii)~\overline{Y}^{i;t,x}_{s}\geq \ell(s,X^{t,x}_{s})~\textrm{and}~ \displaystyle\int^{T}_{0}(\overline{Y}^{i;t,x}_{s}- \ell(s,X^{t,x}_{s}))\mathrm{d}\overline{K}^{i;t,x}_{s}=0.
\end{array}
\right.$$ By Feynman Kac formula $\overline{u}^{i}(s,X^{t,x}_{s})=Y^{i;t,x}_{s}$ where $Y^{i;t,x}_{s}$ satisfies the RBSDE with jumps (\[eq2\]) associated to IPDE (\[eq1\]).\
Since that the RBSDE with jumps (\[5.66\]) has solution and it is unique by assumed that (**H1**), (**H2**) and (**H3**) are verified. By proposition $3.3$ the (\[3.18\]), there exists a family of deterministic continuous functions of polynomial growth $(v^{i})_{i=1,m}$ that fact for any $(t,x)\in[0,T]\times\mathbb{R}^{k}$, $$\forall s\in[t,T],\qquad \overline{Y}^{i;t,x}_{s}=v^{i}(s,X^{t,x}_{s}).$$ Such that by the same proposition, the family $(v^{i})_{i=1,m}$ is a viscosity solution of the following system: $$\label{5.69}
\left
\{\begin{array}{ll}
\min\Big\{v^{i}(t,x)-\ell(t,x);-\partial_{t}v^{i}(t,x)-b(t,x)^{\top}\mathrm{D}_{x}v^{i}(t,x)-\frac{1}{2}\mathrm{Tr}(\sigma\sigma^{\top}(t,x)\mathrm{D}^{2}_{xx}v^{i}(t,x))\\
\quad\quad-\mathrm{K}_{i}v^{i}(t,x)-\mathit{h}^{(i)}(t,x,(v^{j}(t,x))_{j=1,m},(\sigma^{\top}\mathrm{D}_{x}v^{i})(t,x),\mathrm{B}_{i}\overline{u}^{i}(t,x))\Big\}=0,\quad (t,x)\in\left[ 0,T\right] \times\mathbb{R}^{k};\\
u^{i}(T,x)=g^{i}(x).
\end{array}
\right.$$ By uniqueness of solution of (\[5.67\]) $\overline{u}^{i}$ is viscosity solution of (\[5.68\]); and by proposition $3.3$ $v^{i}=\overline{u}^{i}$ $\forall i\in\{1,\ldots,m\}$.\
Now for completing our proof we show that on $\Omega\times[t,T]\times E$, $ds\otimes d\mathbb{P}\otimes d\lambda-\text{a.e.}\quad\forall i\in\{1,\ldots, m\}$; $$\begin{aligned}
\label{5.71}
\overline{U}^{i;t,x}_{s}(e) & = & (v^{i}(s,X^{t,x}_{s-}+\beta(s,X^{t,x}_{s-},e))-v^{i}(s,X^{t,x}_{s-}))\nonumber\\
{} & = & (\overline{u}_{i}(s,X^{t,x}_{s-}+\beta(s,X^{t,x}_{s-},e))-\overline{u}_{i}(s,X^{t,x}_{s-})).\end{aligned}$$ By Remark $3.4$ in [@hama]; let us considere $(x_{k})_{k\geq 1}$ a sequence of $\mathbb{R}^{k}$ which converges to $x\in\mathbb{R}^{k}$ and the two following RBSDE with jumps (adaptation is w.r.t. $\mathcal{F}^{k}$): $$\label{5.72}
\left
\{\begin{array}{ll}
(i)~\vec{\overline{Y}}^{k,t,x}:=(\overline{Y}^{i;k,t,x})_{i=1,m}\in\mathcal{S}^{2}(\mathbb{R}^{m}),~\overline{Z}^{k,t,x}:=(\overline{Z}^{i;k,t,x})_{i=1,m}\in\mathbb{H}^{2}(\mathbb{R}^{m\times d}),\\\overline{K}^{k,t,x}:=(\overline{K}^{i;k,t,x})_{i=1,m}\in\mathcal{A}^{2}_{c},~\overline{U}^{k,t,x}:=(\overline{U}^{i;k,t,x})_{i=1,m}\in\mathbb{H}^{2}(\mathbb{L}^{2}_{m}(\lambda));\\
(ii)~\overline{Y}^{i;k,t,x}_{s}= g^{i}(X^{k,t,x}_{T})+
\overline{K}^{i;k,t,x}_{T}-\overline{K}^{i;k,t,x}_{s}-\displaystyle\int^{T}_{s}\overline{Z}^{i;k,t,x}\mathrm{d}
\mathrm{B}_{r}-\displaystyle\int^{T}_{s}
\displaystyle\int_{\mathrm{E}}\overline{U}^{i;k,t,x}_{r}(e)\tilde{\mu}(\mathrm{d}r,\mathrm{d}e)\\
\quad\quad\quad+\displaystyle\int^{T}_{s}h^{(i)}\left(r,X^{k,t,x}_{r},\overline{Y}^{i;k,t,x}_{r},\overline{Z}^{i;k,t,x}_{r},
\right.\\
\left.
\qquad\qquad\qquad\qquad\qquad\displaystyle\int_{\mathrm{E}}\gamma^{i}(t,X^{k,t,x_{k}}_{r},e)(\overline{u}_{i}(s,X^{k,t,x_{k}}_{s-}+\beta(s,X^{k,t,x_{k}}_{s-},e))-\overline{u}_{i}(s,X^{k,t,x_{k}}_{s-}))\,\lambda(de)\right)dr;\\
(iii)~\overline{Y}^{i;k,t,x_{k}}_{s}\geq \ell(s,X^{k,t,x_{k}}_{s})~\textrm{and}~ \displaystyle\int^{T}_{0}(\overline{Y}^{i;k,t,x_{k}}_{s}- \ell(s,X^{k,t,x_{k}}_{s}))\mathrm{d}\overline{K}^{i;k,t,x_{k}}_{s}=0;
\end{array}
\right.$$ and $$\label{5.73}
\left
\{\begin{array}{ll}
(i)~\vec{\overline{Y}}^{k,t,x_{k}}:=(\overline{Y}^{i;k,t,x_{k}})_{i=1,m}\in\mathcal{S}^{2}(\mathbb{R}^{m}),~\overline{Z}^{k,t,x_{k}}:=(\overline{Z}^{i;k,t,x_{k}})_{i=1,m}\in\mathbb{H}^{2}(\mathbb{R}^{m\times d}),\\\overline{K}^{k,t,x_{k}}:=(\overline{K}^{i;k,t,x_{k}})_{i=1,m}\in\mathcal{A}^{2}_{c},~\overline{U}^{k,t,x_{k}}:=(\overline{U}^{i;k,t,x_{k}})_{i=1,m}\in\mathbb{H}^{2}(\mathbb{L}^{2}_{m}(\lambda));\\
(ii)~\overline{Y}^{i;k,t,x_{k}}_{s}= g^{i}(X^{k,t,x_{k}}_{T})+
\overline{K}^{i;k,t,x_{k}}_{T}-\overline{K}^{i;k,t,x_{k}}_{s}-\displaystyle\int^{T}_{s}\overline{Z}^{i;k,t,x_{k}}\mathrm{d}
\mathrm{B}_{r}-\displaystyle\int^{T}_{s}
\displaystyle\int_{\mathrm{E}}\overline{U}^{i;k,t,x_{k}}_{r}(e)\tilde{\mu}(\mathrm{d}r,\mathrm{d}e)\\
\quad\quad\quad+\displaystyle\int^{T}_{s}h^{(i)}\left(r,X^{k,t,x_{k}}_{r},\overline{Y}^{i;k,t,x_{k}}_{r},\overline{Z}^{i;k,t,x_{k}}_{r},
\right.\\
\left.
\qquad\qquad\qquad\qquad\qquad\displaystyle\int_{\mathrm{E}}\gamma^{i}(t,X^{k,t,x_{k}}_{r},e)(\overline{u}_{i}(s,X^{k,t,x_{k}}_{s-}+\beta(s,X^{k,t,x_{k}}_{s-},e))-\overline{u}_{i}(s,X^{k,t,x_{k}}_{s-}))\,\lambda(de)\right)dr;\\
(iii)~\overline{Y}^{i;k,t,x_{k}}_{s}\geq \ell(s,X^{k,t,x_{k}}_{s})~\textrm{and}~ \displaystyle\int^{T}_{0}(\overline{Y}^{i;k,t,x_{k}}_{s}- \ell(s,X^{k,t,x_{k}}_{s}))\mathrm{d}\overline{K}^{i;k,t,x_{k}}_{s}=0.
\end{array}
\right.$$ By proof of step $2$ of proposition $4.4$, $(\overline{Y}^{i;k,t,x}, \overline{K}^{i;k,t,x},\overline{Z}^{i;k,t,x},\overline{U}^{i;k,t,x}1_{\{|e|\geq \frac{1}{k}\}})_{k}$ converge to $(\overline{Y}^{i;t,x}, \overline{K}^{i;t,x},\overline{Z}^{i;t,x},\\\overline{U}^{i;t,x})$ in $\mathcal{S}^{2}(\mathbb{R})\times\mathcal{A}^{2}_{c}\times\mathbb{H}^{2}(\mathbb{R}^{\kappa\times d})\times\mathbb{H}^{2}(\mathbb{L}^{2}(\lambda)) $.\
Let $((v^{k}_{i=1,m}))_{k\geq 1}$ be the sequence of continuous deterministics functions such that for any $t\leq T$ and $s\in[t,T]$,\
$$\overline{Y}^{i;k,t,x}_{s}=\overline{v}^{k}_{i}(s,^{k}X^{t,x}_{s})~\text{and } \overline{Y}^{i;k,t,x_{k}}_{s}=\overline{v}^{k}_{i}(s,^{k}X^{t,x_{k}}_{s})~~\forall i=1,\ldots,m.$$ Such that we have respectively by proof of proposition $4.4$ in step $1$ and step $2$:\
$(i)~\overline{U}^{i;k,t,x}_{s}(e)=(v^{i}(s,^{k}X^{t,x}_{s-}+\beta(s,^{k}X^{t,x}_{s-},e))-v^{i}(s,^{k}X^{t,x}_{s-}))$, $ds\otimes d\mathbb{P}\otimes d\lambda_{k}$-a.e on $[t,T]\times\Omega\times E$;\
$(ii)~\text{the sequence } ((v^{k}_{i=1,m}))_{k\geq 1}$ converge to $v^{i}(t,x)$ by using (\[4.59\]).\
So that $x_{k}\longrightarrow_{k} x$ we take the following estimation which is obtaining by Ito’s formula and by the properties of $h^{(i)}$. $$\begin{aligned}
\label{5.74}
& {}{} &\mathbb{E}\left[\left|\vec{Y}^{k,t,x_{k}}_{s}-Y^{k,t,x}_{s}\right|^{2}+\left|K^{k,t,x_{k}}_{T}-K^{k,t,x}_{T}\right|^{2}+\displaystyle\int^{T}_{0}\left\lbrace\left|Z^{k,t,x_{k}}_{s}-Z^{k,t,x}_{s}\right|^{2}
\right.
\right.
\nonumber\\
&{}{}&\qquad\qquad\qquad\left.\left.+\displaystyle\int_{E}\left|U^{k,t,x_{k}}_{s}-U^{k,t,x}_{s}\right|^{2}\,\lambda_{k}(de)\right\rbrace\,ds\right]\nonumber\\
& {}{} &\leq\mathbb{E}\left[\left|g(^{k}X^{t,x_{k}}_{T})-g(^{k}X^{t,x}_{T})\right|^{2}\right]+\mathbb{E}\left[\sup_{s\leq T}\left|\ell(^{k}X^{t,x_{k}}_{s})-\ell(^{k}X^{t,x}_{s})\right|^{2}\right] +C\mathbb{E}\left[\displaystyle\int^{T}_{s}
\left|\vec{Y}^{k,t,x_{k}}_{r}-\vec{Y}^{k,t,x}_{r}\right|^{2}\,dr\right]\nonumber\\
& {}{} &+C\mathbb{E}\left[\displaystyle\int^{T}_{0}\left|^{k}X^{t,x_{k}}_{r}-^{k}X^{t,x}_{r}\right|^{2}(1+\left|^{k}X^{t,x_{k}}_{r}\right|^{p}+\left|^{k}X^{t,x}_{r}\right|^{p})^{2}\,dr\right]\nonumber\\
& {}{} &+C\displaystyle\sum_{i=1,m}\mathbb{E}\left[\displaystyle\int^{T}_{s}\left|\mathrm{B}_{i}\overline{u}^{i}(r,^{k}X^{t,x_{k}}_{r})-\mathrm{B}_{i}\overline{u}^{i}(r,^{k}X^{t,x}_{r})\right|^{2}\,dr\right].\nonumber\\\end{aligned}$$ Next using (\[4.54\]) and (\[4.55\]), the continuty of the function $(t,x)\mapsto \mathrm{B}_{i}\overline{u}^{i}(t,x)$ and the fact of it belong to $\Pi_{g}$ and in the other hand the majoration of the fourth term of (\[4.58\]); we can use Gronwall’s lemma for $s=t$ $\forall i=1,\ldots,m$,\
$$v^{k}_{i}(t,x_{k})\longrightarrow_{k}v^{k}_{i}(t,x).$$ Therefore by (i)-(ii) we have, for any $i=1,\ldots,m$,\
$$\label{5.75}
\overline{U}^{i;t,x}_{s}(e)=(v^{i}(s,X^{t,x}_{s-}+\beta(s,X^{t,x}_{s-},e))-v^{i}(s,X^{t,x}_{s-}))\quad ds\otimes d\mathbb{P}\otimes d\lambda-\text{a.e. in }[t,T]\times\Omega\times E, \quad\forall i\in\{1,\ldots, m\}.$$ By this result we can replace $(\overline{u}_{i}(s,X^{t,x}_{s-}+\beta(s,X^{t,x}_{s-},e))-\overline{u}_{i}(s,X^{t,x}_{s-}))$ by $\overline{U}^{i;t,x}_{s}(e)$ in (\[5.71\]), we deduce that the quadriple $(\overline{Y}^{t,x}, \overline{K}^{t,x},\overline{Z}^{t,x},\overline{U}^{t,x})$ verifies: $\forall i\in\{1,\ldots,m\}$ $$\label{5.76}
\left
\{\begin{array}{ll}
(i)~\vec{\overline{Y}}^{t,x}:=(\overline{Y}^{i;t,x})_{i=1,m}\in\mathcal{S}^{2}(\mathbb{R}^{m}),~\overline{Z}^{t,x}:=(\overline{Z}^{i;t,x})_{i=1,m}\in\mathbb{H}^{2}(\mathbb{R}^{m\times d}),~\overline{K}^{t,x}:=(\overline{K}^{i;t,x})_{i=1,m}\in\mathcal{A}^{2}_{c},\\\overline{U}^{t,x}:=(\overline{U}^{i;t,x})_{i=1,m}\in\mathbb{H}^{2}(\mathbb{L}^{2}_{m}(\lambda));\\
(ii)~\overline{Y}^{i;t,x}_{s}= g^{i}(X^{t,x}_{T})+
\overline{K}^{i;t,x}_{T}-\overline{K}^{i;t,x}_{s}-\displaystyle\int^{T}_{s}\overline{Z}^{i;t,x}_{r}\,\mathrm{d}
\mathrm{B}_{r}-\displaystyle\int^{T}_{s}
\displaystyle\int_{\mathrm{E}}\overline{U}^{i;t,x}_{r}(e)\,\tilde{\mu}(\mathrm{d}r,\mathrm{d}e).\\
\quad\quad\quad+\displaystyle\int^{T}_{s}h^{(i)}(r,X^{t,x}_{r},\overline{Y}^{i;t,x}_{r},\overline{Z}^{i;t,x}_{r},\displaystyle\int_{\mathrm{E}}\gamma^{i}(r,X^{t,x}_{r},e)\overline{U}^{i;t,x}_{r}\,\lambda(de))dr;\\
(iii)~\overline{Y}^{i;t,x}_{s}\geq \ell(s,X^{t,x}_{s})~\textrm{and}~ \displaystyle\int^{T}_{0}(\overline{Y}^{i;t,x}_{s}- \ell(s,X^{t,x}_{s}))\mathrm{d}\overline{K}^{i;t,x}_{s}=0.
\end{array}
\right.$$ It follows that $$\forall i\in\{1,\ldots,m\},\quad \overline{Y}^{i;t,x}={Y}^{i;t,x}.$$ With the uniqueness of solution (\[5.68\]), we have $u^{i}=\overline{u}^{i}=v^{i}$ which means that the solution of (\[eq1\]) in the sense of Definition $5.1$ is unique inside the class $\mathcal{U}$.
Extension
=========
In this section, we will redefine the function $h^{(i)}$ as a function of $\|U^{i,t,x}\|_{\mathbb{L}^{2}(\lambda)}$ $\forall i\in\{1,\ldots,m\}$.\
And to show that the results of the previous section remain valid.\
Let us consider for any $i\in\{1,\ldots,m\}$ the functions $f^{(i)}$, defined by $$\forall (t,x,y,z,\zeta)\in[0,T]\times\mathbb{R}^{k}\times\mathbb{R}^{m+d}\times\mathbb{L}^{2}(\lambda);\quad\quad f^{(i)}(t,x,y,z,\zeta)=h^{(i)}(t,x,y,z,\|\zeta\|_{\mathbb{L}^{2}(\lambda)});$$ where the functions $(h^{(i)})_{i=1,m}$ are the sames defined in section $2$.\
We recall that the result of Theorem $5.2$ is obtained by having mainly $U^{t,x}_{s}(e)=(u^{i}(s,X^{t,x}_{s-}+\beta(s,X^{t,x}_{s-},e))-u^{i}(s,X^{t,x}_{s-}))$; this makes it possible to have the definition 4.1 by passing through a modification of the expression of $B_{i}u^{i}$ $\forall i\in\{1,\ldots,m\}$.\
We show that $\|U^{i,t,x}_{s}(e)\|^{2}_{\mathbb{L}^{2}(\lambda)}=\|(u^{i}(s,X^{t,x}_{s-}+\beta(s,X^{t,x}_{s-},e))-u^{i}(s,X^{t,x}_{s-}))|\|^{2}_{\mathbb{L}^{2}(\lambda)}$ and that in this case $B_{i}u^{i}$ is well $\forall i\in\{1,\ldots,m\}$.\
Let now $(t,x)\in[0,T]\times\mathbb{R}^{d}$ and let us consider the following m-dimensional RBSDE with jumps: $$\label{6.76}
\left
\{\begin{array}{ll}
(i)~\vec{Y}^{t,x}:=(Y^{i,t,x})_{i=1,m}\in\mathcal{S}^{2}(\mathbb{R}^{m}),~Z^{t,x}:=(Z^{i,t,x})_{i=1,m}\in\mathbb{H}^{2}(\mathbb{R}^{m\times d}),\\
K^{t,x}:=(K^{i,t,x})_{i=1,m}\in\mathcal{A}^{2}_{c},~ U^{t,x}:=(U^{i,t,x})_{i=1,m}\in\mathbb{H}^{2}(\mathbb{L}^{2}_{m}(\lambda));\\
\forall i\in\{1,\ldots, m\}~ Y^{i;t,x}_{T}= g^{i}(X^{t,x}_{T})~\text{and};\\
(ii)~dY^{i;t,x}_{s}=-f^{(i)}(s,X^{t,x}_{s},(Y^{i;t,x}_{s})_{i=1,m},Z^{i;t,x}_{s},\|U^{t,x}_{s}(e)\|_{\mathbb{L}^{2}(\lambda)})ds-
\mathrm{d}\mathrm{K}^{i;t,x}_{s}\\
\quad\quad\quad\quad\quad\quad
\quad\quad+Z^{i;t,x}_{s}\mathrm{d}
\mathrm{B}_{s}+\displaystyle\int_{\mathrm{E}}\mathrm{U}^{i;t,x}
_{s}(e)\tilde{\mu}(\mathrm{d}s,\mathrm{d}e),\quad s\leq T;\\
(iii)~Y^{i;t,x}_{s}\geq \ell(s,X^{t,x}_{s})~\textrm{and}~ \displaystyle\int^{T}_{0}(Y^{i;t,x}_{s}- \ell(s,X^{t,x}_{s}))\mathrm{d}\mathrm{K}^{i;t,x}_{s}=0.
\end{array}
\right.$$ By assumed that (**H1**), (**H2**) and (**H3**) are verified and by proposition $3.3$ the (\[3.18\]), there exists a family of deterministics continuous functions of polynomial growth $(w^{i})_{i=1,m}$ that fact for any $(t,x)\in[0,T]\times\mathbb{R}^{k}$, $$\forall s\in[t,T],\qquad Y^{i;t,x}_{s}=w^{i}(s,X^{t,x}_{s}).$$ Such that by the same proposition, the family $(w^{i})_{i=1,m}$ is a viscosity solution of the following system: $$\label{6.77}
\left
\{\begin{array}{ll}
\min\Big\{w^{i}(t,x)-\ell(t,x);-\partial_{t}w^{i}(t,x)-b(t,x)^{\top}\mathrm{D}_{x}w^{i}(t,x)-\frac{1}{2}\mathrm{Tr}(\sigma\sigma^{\top}(t,x)\mathrm{D}^{2}_{xx}w^{i}(t,x))\\
\quad\quad-\mathrm{K}_{i}w^{i}(t,x)-\mathit{h}^{(i)}(t,x,(w^{j}(t,x))_{j=1,m},(\sigma^{\top}\mathrm{D}_{x}w^{i})(t,x),\mathrm{B}_{i}w^{i}(t,x))\Big\}=0,\quad (t,x)\in\left[ 0,T\right] \times\mathbb{R}^{k};\\
w^{i}(T,x)=g^{i}(x).
\end{array}
\right.$$ Indeed, using Lemma $4.1$ and the fact that $U^{i;t,x_{k}}$ converges to $U^{i;t,x}$ $\forall i\in\{1,\ldots,m\}$ when $x_{k}\longrightarrow_{k} x$, we deduce that $\|U^{i,t,x_{k}}_{s}(e)\|_{\mathbb{L}^{2}(\lambda)}\longrightarrow_{k} \|U^{i,t,x}_{s}(e)\|_{\mathbb{L}^{2}(\lambda)}$.\
Moreover, from property of $h^{(i)}$ and the proof of theorem $5.2$ step $2$ (viscosity solution uniqueness), $\|U^{i,t,x}_{s}(e)\|^{2}_{\mathbb{L}^{2}(\lambda)}=\|(w^{i}(s,X^{t,x}_{s-}+\beta(s,X^{t,x}_{s-},e))-w^{i}(s,X^{t,x}_{s-}))\|^{2}_{\mathbb{L}^{2}(\lambda)}$; from where\
$B_{i}w^{i}=\left\lbrace\displaystyle\int_{E}|(w^{i}(s,X^{t,x}_{s-}+\beta(s,X^{t,x}_{s-},e))-w^{i}(s,X^{t,x}_{s-}))|^{2}\,\lambda(de)\right\rbrace^{\frac{1}{2}}$.\
Thanks to corollary $4.3$, we deduce that $B_{i}w^{i}$ is well defined $\forall i\in\{1,\ldots,m\}$.
Appendix. Barles et al.’s definition for viscosity solution of IPDE (\[eq1\]) {#appendix.-barles-et-al.s-definition-for-viscosity-solution-of-ipde-eq1 .unnumbered}
=============================================================================
In the paper by Barles et al. [@bar], the definition of the viscosity solution of the system (\[eq1\]) is given as follows.
We say that a family of deterministics functions $u=(u^{i})_{i=1,m}$ which is continuous $\forall i\in\{1,\ldots,m\}$, is a viscosity sub-solution (resp. super-solution) of the IPDE (\[eq1\]) if:\
$(i)\quad \forall x\in\mathbb{R}^{k}$, $u^{i}(x,T)\leq g^{i}(x)$ (resp. $u^{i}(x,T)\geq g^{i}(x)$);\
$(ii)\quad\text{For any } (t,x)\in[0,T]\times\mathbb{R}^{k}$ and any function $\phi$ of class $C^{1,2}([0,T]\times\mathbb{R}^{k})$ such that $(t,x)$ is a global maximum point of $u^{i}-\phi$ (resp. global minimum point of $u^{i}-\phi$) and $(u^{i}-\phi)(t,x)=0$, one has $$\min\left\lbrace u^{i}(t,x)-\ell(t,x);-\partial_{t}\phi(t,x)-\mathcal{L}^{X}\phi(t,x)-h^{i}(t,x,(u^{j}(t,x))_{j=1,m},\sigma^{\top}(t,x))D_{x}\phi(t,x),B_{i}\phi(t,x))\right\rbrace\leq 0$$ $\left(resp.
\right.$ $$\left.
\min\left\lbrace u^{i}(t,x)-\ell(t,x);-\partial_{t}\phi(t,x)-\mathcal{L}^{X}\phi(t,x)-h^{i}(t,x,(u^{j}(t,x))_{j=1,m},\sigma^{\top}(t,x))D_{x}\phi(t,x),B_{i}\phi(t,x)(t,x))\right\rbrace\scriptstyle\geq 0\right).$$ The family $u=(u^{i})_{i=1,m}$ is a viscosity solution of (\[eq1\]) if it is both a viscosity sub-solution and viscosity super-solution.\
Note that $\mathcal{L}^{X}\phi(t,x)=b(t,x)^{\top}\mathrm{D}_{x}\phi(t,x)+\frac{1}{2}\mathrm{Tr}(\sigma\sigma^{\top}(t,x)\mathrm{D}^{2}_{xx}\phi(t,x))+\mathrm{K}\phi(t,x)$;\
where $\mathrm{K}\phi(t,x)=\displaystyle\int_{\mathrm{E}}(\phi(t,x+\beta(t,x,e))-\phi(t,x)-\beta(t,x,e)^{\top}\mathrm{D}_{x}\phi(t,x))\lambda(de)$.
[16]{}
Barles G., Buckdahn R. and Pardoux E., *Backward stochastic differential equations and integral-partial differential equations. Stochastics: An International Journal of Probability and Stochastic Processes*, **60**, pp. 57-83, 1997.
Fujiwara T., Kunita H.: Stochastic differential equations of jump type and Lévy processes in differomorphism group, J. Math. Kyoto Univ. 25, 1, 71-106, 1985.
Hamadène S., *Viscosity solutions of second order integral-partial differential equations without monotonicity condition: A new result, Nonlinear Analysis 147 (2016) 213-235*.
Hamadène S., M.-A. Morlais, *Viscosity solutions for second order integro-differential equations without monotonicity condition: The probabilistic Approach, Stochastics 88 (4) (2016)*.
Hamadène S., Ouknine Y., *Reflected backward stochastic differential equation with jumps and random obstacle, Electron. J. Probab.* **8** *(2003), no. 2, 1-20*.
Harraj N., Ouknine Y., Turpin I., *Double barriers Reflected BSDEs with jumps and viscosity solutions of parabolic Integro-differential PDEs, J. Appl. Math. Stoch. Anal. 1 (2005) 37-53*.
Lenglart E., Lépingle D., Pratelli M., *Présentation unifiée de certaines inégalités de la théorie des martingales, Séminaire de probabilités (Strasbourg), tome 14 (1980), p. 26-48*.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We construct a fundamental region for the action on the $2d+1$-dimensional affine space of some free, discrete, properly discontinuous groups of affine transformations preserving a quadratic form of signature $(d+1, d)$, where $d$ is any odd positive integer.'
author:
- Ilia Smilga
date: 'Received: date / Accepted: date'
title: Fundamental domains for properly discontinuous affine groups
---
Introduction {#sec:intro}
============
Background and motivation {#sec:background}
-------------------------
The present paper is part of a larger effort to understand discrete groups $\Gamma$ of affine transformations (subgroups of the affine group $GL_n(\mathbb{R}) \rtimes \mathbb{R}^n$) acting properly discontinuously on the affine space $\mathbb{R}^n$. The case where $\Gamma$ consists of isometries (in other words, $\Gamma \subset O_n(\mathbb{R}) \rtimes \mathbb{R}^n$) is well-understood: a classical theorem by Bieberbach says that such a group always has an abelian subgroup of finite index.
Define a *crystallographic* group to be a discrete group $\Gamma \subset GL_n(\mathbb{R}) \rtimes \mathbb{R}^n$ acting properly discontinuously and such that the quotient space $\mathbb{R}^n / \Gamma$ is compact. In [@Aus64], Auslander conjectured that any crystallographic group is virtually solvable, that is, contains a solvable subgroup of finite index. Later, Milnor [@Mil77] asked whether this statement is actually true for any affine group acting properly discontinuously. The answer turned out to be negative: Margulis [@Mar83; @Mar87] gave a counterexample in dimension 3. On the other hand, Fried and Goldman [@FG83] proved the Auslander conjecture in dimension 3 (the cases $n=1$ and $2$ are easy). Later, Abels, Margulis and Soifer proved it in dimension $n \leq 6$. See [@AbSur] for a survey of already known results.
In his PhD thesis and subsequent papers [@Dru92; @Dru93], Drumm elaborated on Margulis’s result by explicitly describing fundamental domains for the groups $\Gamma$ introduced by Margulis, which allowed him in particular to deduce the topology of the quotient $\mathbb{R}^3 / \Gamma$. On the other hand, Abels, Margulis and Soifer [@AMS02] constructed a family of counterexamples to Milnor’s conjecture in dimension $4n+3$, preserving a quadratic form of signature $(2n+2,2n+1)$. The purpose of this paper is to adapt Drumm’s construction to Abels-Margulis-Soifer groups: describe a fundamental domain and deduce the topology of the quotient space. Here is the main result:
Let $d$ be an odd positive integer. Then any generalized Schottky subgroup of $SO(d+1, d)$ with sufficiently contracting generators has a nonempty open set of affine deformations $\Gamma$ that act properly discontinuously on $\mathbb{R}^{d+1, d}$, with the quotient $\mathbb{R}^{d+1, d}/\Gamma$ homeomorphic to a solid $(2d+1)$-dimensional handlebody.
To do this, we use mainly two sources of inspiration. The first one is of course [@AMS02], the original work of Abels, Margulis and Soifer. The second one is an article by Charette and Goldman [@CG00] presenting Drumm’s results.
Plan of the paper {#sec:plan}
-----------------
We start, in section \[sec:basic\], by giving some elementary geometrical properties of a space equipped with a form of signature $(d+1, d)$ where $d$ is odd. We describe, in subsection \[sec:MTIS’es\], its maximal totally isotropic subspaces; in subsection \[sec:frames\], its pseudohyperbolic maps (roughly maps whose space of fixed points has the smallest possible dimension); in subsection \[sec:orientation\], an orientation trick (taken from [@AMS02]) that allows to extend any two transversal maximal totally isotropic subspaces into half-$d+1$-dimensional spaces that still have zero intersection. Finally, in subsection \[sec:metric\], we introduce metrics on various spaces (in particular projective spaces) we need to work with, and we define the strength of contraction of a pseudohyperbolic map.
In the next two sections, we consider subgroups of $SO(d+1,d)$ generated by pseudohyperbolic maps. In section \[sec:group\], we study their action on $\mathbb{P}(\Lambda^d \mathbb{R}^{d+1,d})$. We show that, provided the generators are sufficiently contracting, such a group is free and every element is pseudohyperbolic. We also control the geometry and strength of contraction of all cyclically reduced words on the generators. This result is very similar to Lemma 5.24 from [@AMS02], and we follow closely its proof. (For a more concise proof of a similar result, see also section 6 of [@Ben96].)
In section \[sec:tennis\_ball\], we study the action of these subgroups directly on $\mathbb{P}(\mathbb{R}^{d+1,d})$. We show that, supposing again that the generators are sufficiently contracting, this action is similar to the action of a Schottky group (which shows again that such a group is free). The way we construct the fundamental domain was partly inspired by Drumm’s ideas, but his “crooked planes” do not directly generalize to higher dimensions. Instead, we have used “angular” neighborhoods of some half-spaces (namely of the “positive wings” defined in section \[sec:orientation\]).
Finally, in section \[sec:affine\], we study affine groups $\Gamma$ whose linear parts satisfy the conditions of the previous two sections. We prove the Main Theorem (after stating it more precisely: see Theorem \[main\_theorem\]). Here we closely follow section 4 of [@CG00]. First, we describe a set $\mathcal{H}^0$ as the complement to $2n$ “sources” and “sinks” corresponding to the $n$ generators of $\Gamma$. We show (Proposition \[fundamental\_region\]) that under some conditions, $\mathcal{H}^0$ is a fundamental domain for $\Gamma$. Indeed, we see immediately that its images under elements of the group “fit together nicely”. To prove that they cover the whole space, by contradiction, we turn our attention to a hypothetical point not covered by any “tile”. We include it in a nested sequence of domains, then show (by methods adapted from [@CG00]) that these domains must, in a sense, run away to infinity.
Conventions, definitions and basic properties {#sec:basic}
=============================================
Let $p$ and $q$ be two positive integers. We write $\mathbb{R}^{p, q}$ as shorthand for the space $\mathbb{R}^{p+q}$ equipped with a quadratic form $Q$ of signature $(p, q)$. The group of automorphisms of $\mathbb{R}^{p, q}$ (that is, automorphisms of $\mathbb{R}^{p+q}$ that preserve the quadratic form) is $O(p, q)$. This group has four connected components; we call $SO^+(p,q)$ the connected component of the identity.
We equip $\mathbb{R}^{p, q}$ with some additional structure. We choose a maximal positive definite subspace $S$ of $\mathbb{R}^{p, q}$, and we set $T = S^\perp$ the corresponding maximal negative definite subspace. We may then define orthogonal projections $\pi_S: \mathbb{R}^{p, q} \to S$ and $\pi_T: \mathbb{R}^{p, q} \to T$, and positive definite forms $N_S := {{\left. Q \right|}_{S}}$ and $N_T := -{{\left. Q \right|}_{T}}$, so that $$\label{eq:form_decomposition}
\forall x \in \mathbb{R}^{p,q},\quad Q(x) = N_S(\pi_S(x)) - N_T(\pi_T(x)).$$
Maximal totally isotropic subspaces {#sec:MTIS'es}
-----------------------------------
From now on, the acronym MTIS stands for a maximal totally isotropic subspace. If $V$ is a MTIS of $\mathbb{R}^{p, q}$, then (supposing that $p \geq q$) we have $\dim V = q$, $V \subset V^\perp$ and $\dim V^\perp = p$. We write $\mathscr{L}$ the set of all MTIS’es.
A very useful tool for the study of MTIS’es is the following bijection between $\mathscr{L}$ and the space $O(T, S)$ of orthogonal linear maps from $T$ to $S$ (seen as Euclidean spaces via the forms $N_S$ and $N_T$): $$\label{eq:MTIS_bijection}
\xymatrix@R=3pt{
\mathscr{L} \ar@{<->}[r]^{\sim} & O(T, S) \\
V \ar@{|->}[r] & f_V := \pi_S \circ \left({{\left. \pi_T \right|}_{V}}\right)^{-1} \\
V_f := {\left\{ t + f(t) \; \middle| \; t \in T \right\}} & f \ar@{|->}[l]
}$$
It is straightforward to check that both of these maps are well-defined and reciprocal to each other. Indeed, for any $V \in \mathscr{L}$ and $f \in O(T, S)$, we have:
- ${{\left. \pi_T \right|}_{V}}$ is bijective. Indeed, since $V \cap T^\perp = V \cap S = \emptyset$, this map is injective, and the spaces $T$ and $V$ have equal dimension.
- $f_V \in O(T, S)$. Indeed, let $t \in T$; we set $v := ({{\left. \pi_T \right|}_{V}})^{-1}(t)$. Then $v \in V$, and we have $0 = Q(v) = N_S(\pi_S(v)) - N_T(\pi_T(v)) = N_S(f_V(t)) - N_T(t)$.
- $V_f$ is a MTIS. Indeed, this space has dimension $q$, and for all $t \in T$, we have $Q(t + f(t)) = N_S(f(t)) - N_T(t) = 0$.
- $V_{f_V} = V$. Indeed, let $v \in V$; then we have $v = \pi_T(v) + \pi_S(v) = \pi_T(v) + f_V(\pi_T(v))$, hence $v \in V_{f_V}$; and we know that $V$ and $V_{f_V}$ have the same dimension.
- $f_{V_f} = f$. Indeed, let $t \in T$; then we have $f_{V_f}(t) = \pi_S(t + f(t)) = f(t)$.
Here is a first application of this bijection. Later in the paper we shall prove some facts about families of $2n$ pairwise transversal MTIS’es. It would be wise to check that these statements are not vacuous, [i.e. ]{}that such families do indeed exist. This might seem obvious, but it turns out that, while it works for the particular values of $p$ and $q$ we deal with, it is false in general:
\[pairwise\_transversal\] Let $p$, $q$ be two integers, $p \geq q \geq 0$. Then it is possible to find infinitely many pairwise transversal MTIS’es in $\mathbb{R}^{p, q}$, [**unless**]{} $p = q$ and $p$ is odd, in which case it is impossible to find more than two of them.
Let $V_1$ and $V_2$ be two MTIS’es, and $f_i := f_{V_i}$ their images under the bijection . We claim that $V_1$ and $V_2$ are transversal iff $f_1 - f_2$ is injective. Indeed, we have $$x \in V_1 \cap V_2 \iff \exists t \in T,\quad x = t + f_1(t) = t + f_2(t),$$ hence $V_1 \cap V_2 = 0 \iff \ker (f_1 - f_2) = 0$.
The question now becomes: how many orthogonal maps from $T$ to $S$ — or, equivalently, from $\mathbb{R}^q$ to $\mathbb{R}^p$ — can we find such that their differences are pairwise injective, [i.e. ]{}such that the images of any nonzero vector under these maps are pairwise different?
Suppose first that we may find an even integer $r$ such that $q \leq r \leq p$. Let $f_0: \mathbb{R}^q \to \mathbb{R}^p$ be any orthogonal (hence injective) map, and let $E$ be any $r$-dimensional linear space such that $f_0(\mathbb{R}^q) \subset E \subset \mathbb{R}^p$. Then we may find in $O(E)$ an infinite subgroup whose nontrivial elements have no fixed points: for example, the group $G$ formed by matrices $$\begin{pmatrix}
R_\theta & 0 & 0 \\
0 & \ddots & 0 \\
0 & 0 & R_\theta
\end{pmatrix}$$ (where $R_\theta = \begin{pmatrix}\cos \theta & - \sin \theta \\ \sin \theta & \cos \theta\end{pmatrix}$), with $\theta$ running in $\mathbb{R}$. Now consider the set of all maps $g \circ f_0$ with $g \in G$. Let $x \in \mathbb{R}^q \setminus \{0\}$: then $f_0(x) \neq 0$, and the images of $f_0(x)$ under the elements of $G$ are pairwise different. It follows that these maps have indeed pairwise injective differences.
Otherwise, we have $p = q$ and $p$ is odd. The identity and the map $x \mapsto -x$ are two maps of $O(p)$ with injective difference. Now take any three maps in $O(p)$. Then at least two of them, let us call them $f_1$ and $f_2$, have the same determinant: in other terms $f_1 \circ f_2^{-1} \in SO(p)$. But for odd $p$, any map of $SO(p)$ has a fixed point. It follows that $f_1 - f_2$ is not injective.
Pseudohyperbolic maps and frames {#sec:frames}
--------------------------------
From now on, we fix a positive integer $d$ and we set $(p, q) = (d+1, d)$. Take any map $g \in GL(\mathbb{R}^{d+1, d})$. Then we may decompose $\mathbb{R}^{d+1, d}$ into a direct sum of three spaces $\mathbb{R}^{d+1, d} = V_{{\mathsmaller{<}}}(g) \oplus V_{{\mathsmaller{=}}}(g) \oplus V_{{\mathsmaller{>}}}(g)$ stable by $g$ and such that all eigenvalues $\lambda$ of ${{\left. g \right|}_{V_{{\mathsmaller{<}}}(g)}}$ (resp. $V_{{\mathsmaller{=}}}$, $V_{{\mathsmaller{>}}}$) satisfy $|\lambda| < 1$ (resp. $=1$, $>1$).
We shall say that $g$ is *pseudohyperbolic* if $g \in O(d+1, d)$, $\dim V_{{\mathsmaller{=}}}(g) = 1$ and the eigenvalue of $g$ lying in $V_{{\mathsmaller{=}}}(g)$ is $1$ (not $-1$). (As we will soon show, all pseudohyperbolic maps actually lie in $SO(d+1, d)$). In this case, we define the *frame* of $g$ to be the ordered pair $\mathcal{V}(g) := (V_{{\mathsmaller{<}}}(g), V_{{\mathsmaller{>}}}(g))$, and the *dynamical part* of $g$ (as opposed to the frame, which is the “geometrical part”) to be the map $g_{{\mathsmaller{<}}} := {{\left. g \right|}_{V_{{\mathsmaller{<}}}(g)}}$.
Then a pseudohyperbolic map is uniquely defined by its frame and dynamical part. However, these must satisfy some conditions. To state them, we shall need the following notation: for any linear map $g$, we denote by $\rho(g)$ its *spectral radius*, that is, the largest modulus of any eigenvalue of $g$.
\[frame\_and\_dynamical\_part\] Pseudohyperbolic maps are in one-to-one correspondence (via the previous definition) with (ordered) triples $(V_{{\mathsmaller{<}}}, V_{{\mathsmaller{>}}}, g_{{\mathsmaller{<}}})$ such that $V_{{\mathsmaller{<}}}$ and $V_{{\mathsmaller{>}}}$ are two transversal MTIS’es and $g_{{\mathsmaller{<}}}$ is an automorphism of $V_{{\mathsmaller{<}}}$ with $\rho(g_{{\mathsmaller{<}}}) < 1$.
First, let us check that the frame and dynamical part of any pseudohyperbolic map $g$ do satisfy the required conditions. Indeed:
- The fact that $g_{{\mathsmaller{<}}}$ is an automorphism of $V_{{\mathsmaller{<}}}$ and the limitation on its spectral radius follow immediately from the definition of $V_{{\mathsmaller{<}}}$.
- Also by definition, $V_{{\mathsmaller{<}}}(g) \cap V_{{\mathsmaller{>}}}(g) = 0$.
- Let $x_{{\mathsmaller{<}}} \in V_{{\mathsmaller{<}}}(g)$. Then we have $$Q(x_{{\mathsmaller{<}}})
= \lim_{n \to +\infty} Q(g^n(x_{{\mathsmaller{<}}}))
= Q \left( \lim_{n \to +\infty} g_{{\mathsmaller{<}}}^n(x_{{\mathsmaller{<}}}) \right)
= 0,$$ since $\rho(g_{{\mathsmaller{<}}}) < 1$. This shows that $V_{{\mathsmaller{<}}}$ is a totally isotropic subspace.
- Similarly, by using $g^{-1}$ instead of $g$, we can show that $V_{{\mathsmaller{>}}}$ is totally isotropic. Now since $\mathbb{R}^{d+1,d} = V_{{\mathsmaller{<}}} \oplus V_{{\mathsmaller{=}}} \oplus V_{{\mathsmaller{>}}}$, we have $$2d+1 = \dim V_{{\mathsmaller{<}}} + \dim V_{{\mathsmaller{=}}} + \dim V_{{\mathsmaller{>}}} \leq d + 1 + d = 2d+1,$$ hence the inequality must be an equality, that is, $V_{\mathsmaller{<}}$ and $V_{{\mathsmaller{>}}}$ have maximal dimension.
Now let $V_{{\mathsmaller{<}}}$ and $V_{{\mathsmaller{>}}}$ be any pair of transversal MTIS’es and $g_{{\mathsmaller{<}}}$ any automorphism of $V_{{\mathsmaller{<}}}$ with $\rho(g_{{\mathsmaller{<}}}) < 1$. Let us show that there is at most one pseudohyperbolic map with frame $(V_{{\mathsmaller{<}}}, V_{{\mathsmaller{>}}})$ and dynamical part $g_{{\mathsmaller{<}}}$. Indeed, let $g$ be such a map. Then we may calculate $V_{{\mathsmaller{<}}}(g)$, $V_{{\mathsmaller{=}}}(g)$, $V_{{\mathsmaller{>}}}(g)$ and the restrictions of $g$ onto these subspaces, which determines $g$ uniquely. Indeed:
- By definition, $V_{{\mathsmaller{<}}}(g) = V_{{\mathsmaller{<}}}$ and $V_{{\mathsmaller{>}}}(g) = V_{{\mathsmaller{>}}}$.
- Let $x_{{\mathsmaller{<}}} \in V_{{\mathsmaller{<}}}$, $x_{{\mathsmaller{=}}} \in V_{{\mathsmaller{=}}}(g)$. Then we have (denoting by $\langle \bullet, \bullet \rangle$ the bilinear form corresponding to the quadratic form $Q$): $$\langle x_{{\mathsmaller{<}}}, x_{{\mathsmaller{=}}} \rangle
= \lim_{n \to +\infty} \langle g^n(x_{{\mathsmaller{<}}}), g^n(x_{{\mathsmaller{=}}}) \rangle
= \left\langle \lim_{n \to +\infty} g_{{\mathsmaller{<}}}^n(x_{{\mathsmaller{<}}}),\; x_{{\mathsmaller{=}}} \right\rangle
= 0.$$ This shows that $V_{{\mathsmaller{=}}}(g) \perp V_{{\mathsmaller{<}}}$. In the same way, we get $V_{{\mathsmaller{=}}}(g) \perp V_{{\mathsmaller{>}}}$; hence $V_{{\mathsmaller{=}}}(g) \subset V_{{\mathsmaller{<}}}^\perp \cap V_{{\mathsmaller{>}}}^\perp$. But clearly, the right-hand side is a space of dimension at most 1; hence $V_{{\mathsmaller{=}}}(g) = V_{{\mathsmaller{<}}}^\perp \cap V_{{\mathsmaller{>}}}^\perp$.
- By definition, ${{\left. g \right|}_{V_{{\mathsmaller{<}}}}} = g_{{\mathsmaller{<}}}$ and ${{\left. g \right|}_{V_{{\mathsmaller{=}}}}}$ is the identity.
- For $x \in \mathbb{R}^{d+1,d}$, we define $x_{{\mathsmaller{<}}}, x_{{\mathsmaller{=}}}, x_{{\mathsmaller{>}}}$ to be the components of $x$ lying in $V_{{\mathsmaller{<}}}(g)$, $V_{{\mathsmaller{=}}}(g)$, $V_{{\mathsmaller{>}}}(g)$ (so that $x = x_{{\mathsmaller{<}}} + x_{{\mathsmaller{=}}} + x_{{\mathsmaller{>}}}$). For every $x$, since $Q(x_{{\mathsmaller{<}}}) = Q(x_{{\mathsmaller{>}}}) = \langle x_{{\mathsmaller{<}}}, x_{{\mathsmaller{=}}} \rangle = \langle x_{{\mathsmaller{>}}}, x_{{\mathsmaller{=}}} \rangle = 0$, we have $$Q(x) = 2\langle x_{{\mathsmaller{<}}}, x_{{\mathsmaller{>}}} \rangle + Q(x_{{\mathsmaller{=}}}).$$ Now if we apply $g$, we get: $$Q(g(x)) = 2\langle g_{{\mathsmaller{<}}}(x_{{\mathsmaller{<}}}), g_{{\mathsmaller{>}}}(x_{{\mathsmaller{>}}}) \rangle + Q(x_{{\mathsmaller{=}}}),$$ hence for every $x_{{\mathsmaller{<}}} \in V_{{\mathsmaller{<}}}$ and $x_{{\mathsmaller{>}}} \in V_{{\mathsmaller{>}}}$, we have $\langle x_{{\mathsmaller{<}}}, x_{{\mathsmaller{>}}} \rangle = \langle g_{{\mathsmaller{<}}}(x_{{\mathsmaller{<}}}), g_{{\mathsmaller{>}}}(x_{{\mathsmaller{>}}}) \rangle$. It follows that $g_{{\mathsmaller{>}}}$ is adjoint to $g_{{\mathsmaller{<}}}^{-1}$. More rigorously, we have $$\label{eq:duality}
g_{{\mathsmaller{>}}} = \Phi_\mathcal{V}^{-1} \circ (g_{{\mathsmaller{<}}}^{-1})^* \circ \Phi_\mathcal{V},$$ where $\Phi_\mathcal{V}: V_{{\mathsmaller{>}}} \to V_{{\mathsmaller{<}}}^*$ is the appropriate restriction and factoring of the canonical isomorphism $\Phi_Q: \mathbb{R}^{d+1,d} \to (\mathbb{R}^{d+1,d})^*$ defined by $\Phi_Q(x) \cdot y = \langle x, y \rangle$. This determines $g_{{\mathsmaller{>}}}$ uniquely.
Finally, let $V_{{\mathsmaller{=}}} := V_{{\mathsmaller{<}}}^\perp \cap V_{{\mathsmaller{>}}}^\perp$. Then $\dim V_{{\mathsmaller{=}}} = 1$ and $\mathbb{R}^{d+1,d} = V_{{\mathsmaller{<}}} \oplus V_{{\mathsmaller{=}}} \oplus V_{{\mathsmaller{>}}}$. Consider the map $g := g_{{\mathsmaller{<}}} \oplus \operatorname{Id}_{V_{{\mathsmaller{=}}}} \oplus g_{{\mathsmaller{>}}}$, with $g_{{\mathsmaller{>}}}$ defined by . Then it is straightforward to check that $g$ is a pseudohyperbolic map with frame $(V_{{\mathsmaller{<}}}, V_{{\mathsmaller{>}}})$ and dynamical part $g_{{\mathsmaller{<}}}$. (Note that it follows from that the eigenvalues of $g_{{\mathsmaller{>}}}$ are reciprocal to the eigenvalues of $g_{{\mathsmaller{<}}}$).
Incidentally, we can now prove — as announced earlier — that all pseudohyperbolic maps $g$ lie in $SO(d+1, d)$. Indeed, for all such $g$, we have $\det g_{{\mathsmaller{>}}} = (\det g_{{\mathsmaller{<}}})^{-1}$, hence $\det g = (\det g_{{\mathsmaller{<}}})(\det \operatorname{Id})(\det g_{{\mathsmaller{>}}}) = 1$.
We define a *frame* in general to be an ordered pair of transversal MTIS’es. If $\mathcal{V}$ is a frame, we write:
- $V_{{\mathsmaller{<}}}$ its first component and $V_{{\mathsmaller{>}}}$ its second component;
- $V_{{\mathsmaller{=}}}$ the line $V_{{\mathsmaller{<}}}^\perp \cap V_{{\mathsmaller{>}}}^\perp$;
- $V_{{\mathsmaller{\leq}}} := V_{{\mathsmaller{<}}}^\perp = V_{{\mathsmaller{<}}} \oplus V_{{\mathsmaller{=}}}$ and $V_{{\mathsmaller{\geq}}} := V_{{\mathsmaller{>}}}^\perp = V_{{\mathsmaller{>}}} \oplus V_{{\mathsmaller{=}}}$.
Orientation {#sec:orientation}
-----------
\[orientation\] It is possible to choose an orientation on all the MTIS’es $V$ (resp. on their orthogonal subspaces $V^\perp$), such that every $f \in SO^+(d+1,d)$ induces a direct isomorphism from $V$ to $f(V)$ (resp. from $V^\perp$ to $f(V^\perp) = f(V)^\perp$).
We first treat the case of the spaces orthogonal to the MTIS’es. We fix some orientations on $S$ and $T$ (recall that these are two mutually orthogonal maximal definite spaces, one positive and one negative). Then, for any MTIS $V$, $\pi_S$ induces an isomorphism from $V^\perp$ to $S$. Indeed, both spaces have dimension $d+1$, and $$\ker {{\left. \pi_S \right|}_{V^\perp}} = V^\perp \cap \ker \pi_S = V^\perp \cap T = \{0\},$$ since $V^\perp$ is a positive and $T$ a negative definite subspace. We then choose the orientation of $V^\perp$ that makes ${{\left. \pi_S \right|}_{V^\perp}}$ a direct isomorphism.
Now consider the map from $S$ to itself given by the composition of $$S \overset{\pi_S^{-1}}{{\xrightarrow{\hspace*{1cm}}}} V^\perp
\overset{f}{{\xrightarrow{\hspace*{1cm}}}} f(V^\perp)
\overset{\pi_S}{{\xrightarrow{\hspace*{1cm}}}} S.$$ It is easy to see that its determinant depends continuously on $f$ and never vanishes for $f \in SO^+(d+1,d)$. Since $SO^+(d+1,d)$ is connected, the determinant must have constant sign, hence the result.
Replacing $S$ by $T$, the same argument adapts for the MTIS’es themselves.
From now on, let us fix such a family of orientations.
The *positive wing* supported by a MTIS $V$ is the half-space $$V^{{\mathsf{L}}} := {\left\{ v + xe \; \middle| \; v \in V, \; x \geq 0 \right\}},$$ where $e \in V^\perp$ is any vector such that whenever $(e_1, \ldots, e_d)$ is a direct basis of $V$, $(e_1, \ldots, e_d, e)$ is a direct basis of $V^\perp$. (The symbol ${\mathsf{L}}$ should be read as “half-perp”; it is intended to represent half the symbol $\perp$.)
\[wings\_disjoint\] If $d$ is odd, the positive wings supported by any two transversal MTIS’es $V_{{\mathsmaller{<}}}$ and $V_{{\mathsmaller{>}}}$ have a trivial intersection: $$V_{{\mathsmaller{<}}}^{{\mathsf{L}}} \cap V_{{\mathsmaller{>}}}^{{\mathsf{L}}} = \{0\}.$$
Let $\mathcal{B}_{{\mathsmaller{>}}} = (e_{{\mathsmaller{>}}}^1, \ldots, e_{{\mathsmaller{>}}}^d)$ be any direct basis of $V_{{\mathsmaller{>}}}$. We set $$\mathcal{B}_{{\mathsmaller{<}}} = (e_{{\mathsmaller{<}}}^1, \ldots, e_{{\mathsmaller{<}}}^d) := \Phi_\mathcal{V}^{-1}(-\mathcal{B}_{{\mathsmaller{>}}}^*)$$ to be the basis of $V_{{\mathsmaller{>}}}$ dual to the basis $-\mathcal{B}_{{\mathsmaller{>}}} = (-e_{{\mathsmaller{>}}}^1, \ldots, -e_{{\mathsmaller{>}}}^d)$ (see the proof of Proposition \[frame\_and\_dynamical\_part\] for the definition of $\Phi_\mathcal{V}$). Let also $e_{{\mathsmaller{=}}}$ be the vector of unit norm lying in $V_{{\mathsmaller{=}}}$ such that $(\mathcal{B}_{{\mathsmaller{>}}}, e_{{\mathsmaller{=}}})$ is a direct basis of $V_{{\mathsmaller{>}}}^\perp = V_{{\mathsmaller{>}}} \oplus V_{{\mathsmaller{=}}}$. We now define a basis of $\mathbb{R}^{d+1,d}$ by joining together these bases of $V_{{\mathsmaller{<}}}$, $V_{{\mathsmaller{>}}}$ and $V_{{\mathsmaller{=}}}$: $$\mathcal{B} := (e_{{\mathsmaller{<}}}^1, \ldots, e_{{\mathsmaller{<}}}^d, e_{{\mathsmaller{>}}}^1, \ldots, e_{{\mathsmaller{>}}}^d, e_{{\mathsmaller{=}}}).$$ In this basis, the quadratic form $Q$ is then given by the matrix $$\begin{pmatrix}
0 & -I_d & 0 \\
-I_d & 0 & 0 \\
0 & 0 & 1
\end{pmatrix}.$$ Now consider the automorphism $f$ given, in basis $\mathcal{B}$, by the matrix $$\begin{pmatrix}
0 & I_d & 0 \\
I_d & 0 & 0 \\
0 & 0 & (-1)^d
\end{pmatrix}.$$ It is easy to show that $f \in SO^+(d+1,d)$ (for details, see [@AMS02], proof of Lemma 3.1 — they call this map $h_\pi$). But $f$ maps $\mathcal{B}_{{\mathsmaller{>}}}$ onto $\mathcal{B}_{{\mathsmaller{<}}}$ and $(\mathcal{B}_{{\mathsmaller{>}}}, e_{{\mathsmaller{=}}})$ onto $(\mathcal{B}_{{\mathsmaller{<}}}, (-1)^de_{{\mathsmaller{=}}})$. Hence by Proposition \[orientation\], the latter are direct bases of $V_{{\mathsmaller{>}}}$ and $V_{{\mathsmaller{>}}}^\perp$. This implies that $$\begin{cases}
V_{{\mathsmaller{<}}}^{{\mathsf{L}}} = V_{{\mathsmaller{<}}} + (-1)^d\mathbb{R}^{\geq 0}e_{{\mathsmaller{=}}} \\
V_{{\mathsmaller{>}}}^{{\mathsf{L}}} = V_{{\mathsmaller{>}}} + \mathbb{R}^{\geq 0}e_{{\mathsmaller{=}}}.
\end{cases}$$ Hence $$V_{{\mathsmaller{<}}}^{{\mathsf{L}}} \cap V_{{\mathsmaller{>}}}^{{\mathsf{L}}} = \left((-1)^d\mathbb{R}^{\geq 0} \cap \mathbb{R}^{\geq 0}\right)e_{{\mathsmaller{=}}};$$ since $d$ is odd, the conclusion follows.
Suppose now that $d$ is even. Then the same argument shows that two positive wings *always* have a nontrivial intersection. Thus with our methods, there is no hope to construct a non-abelian free properly discontinuous subgroup in $SO(d+1, d) \rtimes \mathbb{R}^{2d+1}$ for even $d$, since a crucial point is the existence of $2n$ pairwise disjoint wings (for $n > 1$). Indeed, it was shown in [@AMS02] (Theorem A), using a very similar orientation argument, that such subgroups do not exist.
\[direction\] For every frame $\mathcal{V}$ (or $\mathcal{V}'$, $\mathcal{V}_i$, and so on), we denote by $e_{{\mathsmaller{=}}}$ (resp. $e'_{{\mathsmaller{=}}}$, $e_{i,{\mathsmaller{=}}}$, and so on) the vector of unit norm contained in the half-line $V_{{\mathsmaller{=}}} \cap V_{{\mathsmaller{>}}}^{{\mathsf{L}}}$. If $d$ is odd, we then have:$$\begin{cases}
V_{{\mathsmaller{<}}}^{{\mathsf{L}}} = V_{{\mathsmaller{<}}} - \mathbb{R}^{\geq 0}e_{{\mathsmaller{=}}} \\
V_{{\mathsmaller{>}}}^{{\mathsf{L}}} = V_{{\mathsmaller{>}}} + \mathbb{R}^{\geq 0}e_{{\mathsmaller{=}}}.
\end{cases}$$
Strength of contraction and other metric considerations {#sec:metric}
-------------------------------------------------------
From now on, we assume $d$ to be odd. We introduce on $\mathbb{R}^{d+1,d}$, in addition to its structural quadratic form $Q$, several positive definite quadratic forms. Every such form $N$ gives us an inner product (written $\langle x, y \rangle_N$), a Euclidean norm (written $\|x\|_N := N(x)^{\frac{1}{2}}$; hence also a metric on $\mathbb{R}^{d+1,d}$), and an operator norm (written also $\|g\|_N := \sup \frac{\|g(x)\|_N}{\|x\|_N}$).
First, we need a “global” norm, that we shall use most of the time: it will enable us to take measurements that do not depend on a particular frame. Insofar as all norms on a finite-dimensional space are equivalent, its choice does not really matter; however, the following particular expression will simplify some of the proofs. We define the form $N_0$ by $$\label{eq:norm_definition}
\forall x \in \mathbb{R}^{d+1,d},\quad N_0(x) = N_S(\pi_S(x)) + N_T(\pi_T(x))$$ (compare this with ).
However, for every frame $\mathcal{V}$, we also need a “local” norm, that will make calculations involving this frame easier. We define $N_\mathcal{V}$ to be the (positive definite) quadratic form on $\mathbb{R}^{d+1,d}$ that makes the spaces $V_{{\mathsmaller{<}}}$, $V_{{\mathsmaller{=}}}$ and $V_{{\mathsmaller{>}}}$ pairwise orthogonal, but whose restriction to any of these spaces coincides with $N_0$.
Consider a vector space $E$ (for the moment, the reader may suppose that $E = \mathbb{R}^{d+1,d}$; later we will also need the case $E = \Lambda^d \mathbb{R}^{d+1,d}$). We define $$\begin{cases}
\pi_\mathbb{S}: E \setminus \{0\} \to \mathbb{S}(E) \\
\pi_\mathbb{P}: E \setminus \{0\} \to \mathbb{P}(E)
\end{cases}$$ to be, respectively, the canonical projections onto the sphere $\mathbb{S}(E) := (E \setminus \{0\})/\mathbb{R}^{> 0}$ and the projective space $\mathbb{P}(E) := (E \setminus \{0\})/\mathbb{R}^*$. (Readers who think of the sphere as a subset of $E$ might get confused when we change the norm; this is why we define $\mathbb{S}(E)$ as an abstract quotient space.) For every linear map $g: E \to E$, we define the corresponding maps $g_\mathbb{S}: \mathbb{S}(E) \to \mathbb{S}(E)$ and $g_\mathbb{P}: \mathbb{P}(E) \to \mathbb{P}(E)$ (written simply $g$ when no confusion is possible.)
Consider a metric space $(X, \delta)$; let $A$ and $B$ be two subsets of $X$. We shall denote the ordinary, minimum distance between $A$ and $B$ by $$\delta(A, B) := \inf_{a \in A} \inf_{b \in B} \delta(a, b),$$ as opposed to the Hausdorff distance, which we shall denote by $$\delta^\mathrm{Haus}(A, B) := \max\left( \sup_{a \in A} \delta(a, B),\; \sup_{b \in B} \delta(b, A) \right).$$
For every positive definite quadratic form $N$ on $E$, for every $\overline{x}, \overline{y} \in \mathbb{S}(E)$, we define the distance $$\alpha_N (\overline{x}, \overline{y}) := \arccos \frac{\langle x, y \rangle_N}{\|x\|_N \|y\|_N},$$ where $x$ and $y$ are any vectors representing respectively $\overline{x}$ and $\overline{y}$ (obviously, the value does not depend on the choice of $x$ and $y$). This measures the angle between the half-lines $\overline{x}$ and $\overline{y}$. For shortness’ sake, we will usually simply write $\alpha_N(x, y)$ with $x, y \in E \setminus \{0\}$, to mean $\alpha_N (\pi_\mathbb{S}(x), \pi_\mathbb{S}(y))$.
In a similar way, we equip $\mathbb{P}(E)$ with the distance $$\alpha_N^\mathrm{Proj} (x, y) := \alpha_N (\mathbb{R}x, \mathbb{R}y) = \min(\alpha_N(x, y),\; \alpha_N(x, -y)).$$ Note that for sets $X$ and $Y$ symmetric about the origin (such as vector spaces), we have $\alpha_N^\mathrm{Proj}(X, Y) = \alpha_N(X, Y)$: in this situation, we may ignore the distinction between the spherical and projective cases.
For any set $X \subset \mathbb{S}(E)$ and any radius ${\varepsilon}> 0$, we shall denote the ${\varepsilon}$-neighborhood of $X$ with respect to the distance $\alpha_N$ by: $$B_N(X, {\varepsilon}) := {\left\{ x \in \mathbb{S}(E) \; \middle| \; \alpha_N(x,X) < {\varepsilon}\right\}}.$$ When $X$ is symmetric, we shall sometimes treat $B_N(X, {\varepsilon})$ as a subset of $\mathbb{P}(E)$.
For the sake of briefness, we shall often specify a “default” form at the beginning of some sections or paragraphs. In the rest of that section or paragraph, *every* mention of *any* of the metric-dependent values or functions defined above without explicit mention of the metric itself (such as $\langle x, y \rangle$, $\alpha(x, y)$, $B(X, {\varepsilon})$ and so on) is understood to refer to the current “default” metric.
Finally, we introduce the following notation. Let $A$ and $B$ be two positive quantities, and $p_1, \ldots, p_k$ some parameters. Whenever we write $$A \ll_{p_1, \ldots, p_k} B,$$ we mean that there is a constant $C$, depending on nothing but $p_1, \ldots, p_k$, such that $A \leq CB$. (If we do not write any parameters, this means of course that $C$ is an absolute constant.) Whenever we write $$A \asymp_{p_1, \ldots, p_k} B,$$ we mean that $A \ll_{p_1, \ldots, p_k} B$ and $B \ll_{p_1, \ldots, p_k} A$ at the same time.
Let $g$ be a pseudohyperbolic map, $\mathcal{V}$ its frame, $g_{{\mathsmaller{<}}} = {{\left. g \right|}_{V_{{\mathsmaller{<}}}(g)}}$ its dynamical part, $g_{{\mathsmaller{>}}} = {{\left. g \right|}_{V_{{\mathsmaller{>}}}(g)}}$. Since $V_{{\mathsmaller{<}}}$ and $V_{{\mathsmaller{>}}}$ are transversal, by Proposition \[wings\_disjoint\], $V_{{\mathsmaller{<}}}^{{\mathsf{L}}}$ and $V_{{\mathsmaller{>}}}^{{\mathsf{L}}}$ have zero intersection. Their projections onto the sphere are then disjoint; being compact, they are always separated by a positive distance. We define the *separation* of $\mathcal{V}$ (or, by abuse of terminology, of $g$) to be $${\varepsilon}(g) = {\varepsilon}(\mathcal{V}) := \alpha_{N_0}(V_{{\mathsmaller{<}}}^{{\mathsf{L}}}, V_{{\mathsmaller{>}}}^{{\mathsf{L}}}),$$ the distance between these projections in global metric. (The distance in local metric, $\alpha_{N_\mathcal{V}}(V_{{\mathsmaller{<}}}^{{\mathsf{L}}}, V_{{\mathsmaller{>}}}^{{\mathsf{L}}})$, is by definition always equal to $\frac{\pi}{2}$). For any constant ${\varepsilon}> 0$, we say that $\mathcal{V}$ (or $g$) is *${\varepsilon}$-separated* if ${\varepsilon}(\mathcal{V}) \geq {\varepsilon}$.
The *strength of contraction* of $g$ is the quantity $$s(g) := \max\left( \|g_{{\mathsmaller{<}}}\|, \|g_{{\mathsmaller{>}}}^{-1}\| \right)$$ (with the metric given indifferently by $N_0$ or $N_{\mathcal{V}(g)}$: both coincide on $V_{{\mathsmaller{<}}}(g)$ and $V_{{\mathsmaller{>}}}(g)$.) For $s > 0$, we say that $g$ is *$s$-contracting* if $s(g) \leq s$. In this case, for all $x_{{\mathsmaller{<}}} \in V_{{\mathsmaller{<}}}(g)$ and $x_{{\mathsmaller{>}}} \in V_{{\mathsmaller{>}}}(g)$, we have $$\frac{\|g(x_{{\mathsmaller{<}}})\|}{\|x_{{\mathsmaller{<}}}\|} \leq s \text{ and } \frac{\|g(x_{{\mathsmaller{>}}})\|}{\|x_{{\mathsmaller{>}}}\|} \geq s^{-1}.$$
Note that if $d > 1$, there is no constant $C$ such that all pseudohyperbolic maps would be $C$-contracting, as the norm may be much larger than the spectral radius. However, for any pseudohyperbolic map $g$, we have $$s(g^n) = \underset{n \to \infty}{O} \left( \rho(g_{{\mathsmaller{<}}})^n \right) \underset{n \to \infty}{\to} 0.$$
Now we need to formulate an essential property of the metrics defined above, that we shall very often use subsequently. All of the norms $\| \bullet \|_{N_\mathcal{V}}$ and the associated distances $\alpha_{N_\mathcal{V}}$ are Lipschitz-equivalent, with a common Lipschitz constant that depends only on the separation of $\mathcal{V}$. More precisely:
\[uniformly\_equivalent\] For every ${\varepsilon}> 0$ and every ${\varepsilon}$-separated frame $\mathcal{V}$, we have: $$\forall x \in \mathbb{R}^{d+1,d},\quad
\|x\|_{N_\mathcal{V}} \asymp_{{\varepsilon}} \|x\|_{N_0};$$ $$\forall x, y \in \mathbb{S}(\mathbb{R}^{d+1,d}),\quad
\alpha_{N_\mathcal{V}}(x, y) \asymp_{{\varepsilon}} \alpha_{N_0}(x, y).$$
For any frame $\mathcal{V}$, let $C(\mathcal{V})$ be the Lipschitz constant between the norms given by $N_0$ and $N_\mathcal{V}$, [i.e. ]{}the smallest constant satisfying the first inequality above. Then $C(\mathcal{V})$ is always finite, and may be expressed as the operator norm of the identity map subordinated to the norms given by $N_0$ and $N_\mathcal{V}$: hence it depends continuously on $\mathcal{V}$. Since for any fixed ${\varepsilon}> 0$, the set of all ${\varepsilon}$-separated frames is compact, the first claim follows.
Now if two norms given by $N$ and $N'$ are $C$-Lipschitz-equivalent, then the corresponding distances $\alpha_N$ and $\alpha_{N'}$ are always $C^2$-Lipschitz-equivalent. Indeed, in dimension 2, this follows from a straightforward calculation; in the general case, we may simply fix two vectors $x$ and $y$ and restrict our attention to the subspace they span. Hence the second estimation follows from the first.
Pseudohyperbolicity of products {#sec:group}
===============================
The goal of this section is to prove Proposition \[product\_pseudohyperbolic\], which essentially states that under some conditions, the product of several pseudohyperbolic maps is still pseudohyperbolic.
Proximal case {#sec:proximal}
-------------
Let $E$ be a vector space. We fix a default quadratic form $\hat{N}_0$ on $E$. (In practice, we will apply the results of this subsection to $E = \Lambda^d \mathbb{R}^{d+1, d}$.)
Our first goal is to show Lemma \[product\_proximal\], which is analogous to Proposition \[product\_pseudohyperbolic\] (and will be used to prove it), but with proximal maps instead of pseudohyperbolic ones. We begin by a few definitions.
Let $f \in GL(E)$. Let $\lambda$ be an eigenvalue of $f$ with maximal modulus. We say that $f$ is *proximal* if $\lambda$ is unique and has multiplicity 1. We may then decompose $E$ into a direct sum of a line $V_s(f)$, called its *attracting space*, and a hyperplane $V_u(f)$, called its *repulsing space*, both stable by $f$ and such that: $$\begin{cases}
{{\left. f \right|}_{V_s}} = \pm \lambda \operatorname{Id}\\
\text{for every eigenvalue } \mu \text{ of } {{\left. f \right|}_{V_u}},\; |\mu| < |\lambda|.
\end{cases}$$ We define the *separation* of $f$ to be $\eta(f) := \alpha (V_s(f), V_u(f))$. For any constant $\eta > 0$, we say that $f$ is *$\eta$-separated* if $\eta(f) \geq \eta$. For any quadratic form $N$ on $E$, we define the *strength of contraction* of $f$ with respect to $N$ by $$\hat{s}_N(f) := \frac{\|{{\left. f \right|}_{V_u}}\|_N}{|\lambda|}$$ (we remind that writing simply $\hat{s}$ means $\hat{s}_{\hat{N}_0}$.) Note that these definitions are different from the ones we used in the context of pseudohyperbolic maps (hence the new notation $\hat{s}$).
An *independent proximal system* is a tuple $F = (f_1, \ldots, f_n)$ of maps $f_i \in GL(E)$ such that:
(i) every $f_i$ and every $f_i^{-1}$ is proximal;
(ii) for every indices $i$, $i'$ and signs $\sigma$, $\sigma'$ such that $(i', \sigma') \neq (i, -\sigma)$, we have $$\alpha(V_s(f_i^\sigma), V_u(f_{i'}^{\sigma'})) > 0.$$
In this case, we define the *separation* of $F$ to be $$\eta(F) := \min_{(i', \sigma') \neq (i, -\sigma)} \alpha(V_s(f_i^\sigma), V_u(f_{i'}^{\sigma'})),$$ and the *contraction strength* of $F$ to be $$\hat{s}(F) := \max_{i, \sigma} \hat{s}(f_i^\sigma).$$
\[cyclically\_reduced\] Take a nonnegative integer $k$, and take $k$ couples $(i_1, \sigma_1), \ldots, (i_k, \sigma_k)$ such that for every $l$, $1 \leq i_l \leq n$ and $\sigma_l = \pm 1$. Consider the word $f = f_{i_1}^{\sigma_1} \ldots f_{i_k}^{\sigma_k}$.
We say that $f$ is *reduced* if for every $l$ such that $1 \leq l \leq k-1$, we have $(i_{l+1}, \sigma_{l+1}) \neq (i_l, -\sigma_l)$. We say that $f$ is *cyclically reduced* if it is reduced and also satisfies $(i_1, \sigma_1) \neq (i_k, -\sigma_k)$.
Now we prove an analog of Proposition \[product\_pseudohyperbolic\] in the proximal case:
\[product\_proximal\] For every $\eta > 0$, there is a constant $\hat{s}(\eta) > 0$ with the following property. Let $F = (f_1, \ldots, f_n)$ be any $\eta$-separated, $\hat{s}(\eta)$-contracting independent proximal system. Let $f = f_{i_1}^{\sigma_1} \ldots f_{i_k}^{\sigma_k}$ (with $\sigma_l = \pm 1$) any nonempty cyclically reduced word. Then $f$ is proximal, $\hat{s}(f) \ll_{\eta} \hat{s}(F)$ and $$\alpha(V_s(f),\; V_s(f_{i_1}^{\sigma_1})) \;\ll_{\eta}\; \hat{s}(F).$$
Before proceeding, we need a technical lemma that relates the abstract strength of contraction $\hat{s}(f)$ and some actual Lipschitz constants of $f$ acting on the projective space $\mathbb{P}(E)$. For any set $X \subset \mathbb{P}(E)$, we introduce the following notation for the Lipschitz constant of $f$ restricted to $X$ in metric given by $N$: $$\mathcal{L}_N(f, X) :=
\sup_{\substack{(x, y) \in X^2 \\ x \neq y}}
\frac{\alpha_N^\mathrm{proj}(f(x), f(y))}{\alpha_N^\mathrm{proj}(x, y)}$$
\[Lipschitz\] For any $\eta > 0$, $\zeta > 0$, for any proximal $\eta$-separated map $f$, we have :
\[eq:Lipschitz\] $$\label{eq:Lipschitz1}
\mathcal{L} \left( f,\; \mathbb{P}(E) \setminus B (V_u(f), \zeta) \right) \ll_{\eta, \zeta} \hat{s}(f)$$ $$\label{eq:Lipschitz2}
\hat{s}(f) \ll_{\eta, \zeta} \mathcal{L} \left( f,\; B (V_s(f), \zeta) \right)$$
(using of course the metric given by $\hat{N}_0$.)
Let $\eta > 0$, $\zeta > 0$. For every proximal $f$, we define on $E$ a quadratic form $\hat{N}_f$ that makes $V_s(f)$ and $V_u(f)$ orthogonal but coincides with $\hat{N}_0$ on these spaces. By an obvious generalization of Lemma \[uniformly\_equivalent\], for every proximal $\eta$-separated map $f$, we have $$\label{eq:N0_to_Nf}
\alpha^\mathrm{proj}_{\hat{N}_f} \asymp_{\eta} \alpha^\mathrm{proj}_{\hat{N}_0}.$$ (Lemma \[uniformly\_equivalent\] referred to $\alpha$ rather than $\alpha^\mathrm{proj}$, but since both distances are locally equal, this makes little difference.) Now consider a proximal map $f$, and note the following facts:
- From , it follows $$\begin{aligned}
\mathbb{P}(E) \setminus B (V_u(f),\; \zeta)\; &\subset\; \mathbb{P}(E) \setminus B_{\hat{N}_f} (V_s(f),\; \zeta'),\\
B (V_s(f),\; \zeta)\; &\supset\; B_{\hat{N}_f} (V_s(f),\; \zeta'),\end{aligned}$$ where $\zeta' = C\zeta$ for some constant $C$ depending only on $\eta$. Moreover, it is clear that $X \subset Y$ implies $\mathcal{L}(f, X) \leq \mathcal{L}(f, Y)$.
- For all $X$, we have $$\mathcal{L}_{\hat{N}_0}(f, X) \asymp_{\eta} \mathcal{L}_{\hat{N}_f}(f, X)$$
- For any $\zeta' > 0$, we have $$\mathcal{L}_{\hat{N}_f} \left( f,\; B_{\hat{N}_f} (V_s(f), \zeta') \right)
\asymp_{\zeta'} \hat{s}_{\hat{N}_f}(f).$$ Indeed, consider the projection $\pi_u: \mathbb{P}(E) \setminus V_u(f) \to V_u(f)$ parallel to $V_s(f)$, defined by $\pi_u(x_u:1) = x_u$ (with obvious notations). It induces a homeomorphism from $B_{\hat{N}_f} (V_s(f), \zeta')$ to the ball ${\left\{ x \in V_u(f) \; \middle| \; \|x\|_{\hat{N}_f} \leq \frac{1}{\tan \zeta'} \right\}}$. A straightforward calculation shows that the said homeomorphism is bilipschitz (with respect to the metrics $\alpha^\mathrm{proj}_{\hat{N}_f}$ and $\|\bullet\|_{\hat{N}_f}$), with a Lipschitz constant $C(\zeta')$ that does not at all depend on $f$ or $\eta$. On the other hand, the Lipschitz constant of the conjugate function $\pi_u \circ f \circ \pi_u^{-1}$ is nothing other than $\hat{s}_{\hat{N}_f}(f)$. Hence $f$ is Lipschitz-continuous with constant $C(\zeta')^2\hat{s}_{\hat{N}_f}(f)$, hence the conclusion.
- Since $\hat{N}_f$ and $\hat{N}_0$ coincide on $V_u(f)$ and $V_s(f)$, we have $\hat{s}_{\hat{N}_f}(f) = \hat{s}_{\hat{N}_0}(f)$.
Now to show , we simply apply all these steps in succession, keeping in mind that $$\mathbb{P}(E) \setminus B_{\hat{N}_f} (V_u(f), \zeta') = B_{\hat{N}_f} (V_s(f), \frac{\pi}{2} - \zeta').$$ To show , we apply the same steps in the reverse order.
Let $\eta > 0$, and let $F = (f_1, \ldots, f_n)$ be an $\eta$-separated, $\hat{s}(\eta)$-contracting independent proximal system (for a value $\hat{s}(\eta)$ to be specified later).
An immediate corollary of Lemma \[Lipschitz\] is that for every $\eta$-separated proximal map $\phi$ and every $\zeta \leq \eta$, we have $$\label{eq:eps6}
\phi \left( \mathbb{P}(E) \setminus B(V_u(\phi), \zeta) \right) \subset B\left( V_s(\phi),\; C\left( \eta, \zeta \right)\hat{s}(\phi) \right)$$ for some constant $C(\eta, \zeta)$. Indeed, $V_s(\phi) \in \mathbb{P}(E) \setminus B(V_u(\phi), \zeta)$ is a fixed point of $\phi$ and $\mathrm{diam}(\mathbb{P}(E) \setminus B(V_u(\phi), \zeta)) \leq \frac{\pi}{2} \ll 1$.
Let $\eta' = C(\eta, \frac{\eta}{3})\hat{s}(F)$. For every $l$ in the range from $1$ to $k$, we set $$\begin{cases}
X_l^- := B(V_u(f_{i_l}^{\sigma_l}), \frac{\eta}{3}) \\
X_l^+ := B(V_s(f_{i_l}^{\sigma_l}), \eta').
\end{cases}$$ Then by , for every $l$ we have $f_{i_l}^{\sigma_l}(\mathbb{P}(E) \setminus X_l^-) \subset X_l^+$. Since $\hat{s}(F) \leq \hat{s}(\eta)$, if we choose $\hat{s}(\eta)$ small enough, we may suppose that $\eta' \leq \frac{\eta}{3}$. Then for every $l$ we also have $X_l^+ \subset \mathbb{P}(E) \setminus X_{l-1}^-$ (since the word $f$ is reduced). By induction, it follows that $$f(\mathbb{P}(E) \setminus X_k^-) \subset X_1^+.$$
Now by , we know that for every $l$
$$\label{eq:lip_less_s}
\mathcal{L} \left( f_{i_l}^{\sigma_l},\; \mathbb{P}(E) \setminus X_l^- \right) \ll_{\eta} \hat{s}(F) \leq \hat{s}(\eta).$$
Once again, choosing $\hat{s}(\eta)$ small enough, we may actually suppose that $$\label{eq:lip_less_1}
\mathcal{L} \left( f_{i_l}^{\sigma_l},\; \mathbb{P}(E) \setminus X_l^- \right) < 1.$$
Since $f$ is cyclically reduced, we have $X_1^+ \subset \mathbb{P}(E) \setminus X_k^-$; hence $X_1^+$ is stable by $f$ and, by induction, we get $$\mathcal{L} \left( f,\; X_1^+ \right) < 1.$$
It follows that $f$ is proximal and $V_s(f) \in X_1^+$ (see [@Tits72], Lemma 3.8 for a proof), which settles the first and third statement of the conclusion. On the other hand, it is easy to see that $V_u(f) \subset X_k^-$ (indeed, consider any point $x \in \mathbb{P}(E)$ belonging to $V_u(f)$ but not to $X_k^-$: then we would have $\lim_{n \to \infty} f^n(x) = V_s(f)$, which contradicts the fact that $V_u(f)$ is a stable subspace). But we know that $$\begin{aligned}
\alpha(X_1^+, X_k^-)
&\geq \textstyle
\alpha(V_s(f_{i_1}^{\sigma_1}), V_u(f_{i_k}^{\sigma_k})) - \eta' - \frac{\eta}{3}\\
&\geq \textstyle
\eta - \frac{\eta}{3} - \frac{\eta}{3}\\
&= \textstyle \frac{\eta}{3},\end{aligned}$$ hence $f$ is $\frac{\eta}{3}$-separated.
This allows us to apply to $f$: $$\hat{s}(f) \ll_{\eta} \mathcal{L} \left( f,\; B(V_s(f), \frac{\eta}{3}) \right).$$ We know that $B(V_s(f), \frac{\eta}{3}) \subset B(V_s(f_{i_1}^{\sigma_1}), \frac{2\eta}{3}) \subset \mathbb{P}(E) \setminus X_k^-$, hence $$\mathcal{L} \left( f,\; B(V_s(f), \frac{\eta}{3}) \right)
\leq \mathcal{L} \left( f,\; \mathbb{P}(E) \setminus X_k^- \right).$$ On the other hand, using in combination with , we get that $$\mathcal{L} \left( f,\; \mathbb{P}(E) \setminus X_k^- \right)
\ll_{\eta} \hat{s}(F).$$ Stringing together these inequalities, we get $$\hat{s}(f) \ll_{\eta} \hat{s}(F),$$ which settles the second statement of the conclusion.
Pseudohyperbolic case {#sec:pseudohyperbolic}
---------------------
Throughout this section, we work by default in metric given by $N_0$.
We define a *frameset* $\mathcal{W}$ to be a set of $n$ frames $\mathcal{V}_1, \ldots, \mathcal{V}_n$ whose $2n$ components $V_{1, {\mathsmaller{<}}}, V_{1, {\mathsmaller{>}}}, \ldots, V_{n, {\mathsmaller{<}}}, V_{n, {\mathsmaller{>}}}$ are pairwise transversal. We define the *separation* ${\varepsilon}(\mathcal{W})$ of the frameset to be the minimal separation between any two MTIS’es forming the frameset.
Let $\mathcal{W} = (\mathcal{V}_1, \ldots, \mathcal{V}_n)$ be a frameset. A *group based on $\mathcal{W}$* is a group $G$ generated by pseudohyperbolic maps $g_1, \ldots, g_n$ with respective frames $\mathcal{V}_1, \ldots, \mathcal{V}_n$. For $s > 0$, we say that $G$ is *$s$-contracting* if all of its generators are $s$-contracting; the *contraction strength* of $G$ is the number $$s(G) := \max_i s(g_i).$$
- By the “separation between $V$ and $V'$”, we mean here the separation of the frame $(V, V')$. Take care that we take the minimum over all of the $\binom{2n}{2}$ possible pairings, not just the frames $\mathcal{V}_1, \ldots, \mathcal{V}_n$.
- Lemma \[pairwise\_transversal\] guarantees that framesets with an arbitrarily large number of frames exist.
\[product\_pseudohyperbolic\] For every ${\varepsilon}> 0$, there is a constant $s_1({\varepsilon}) > 0$ with the following property. Let $\mathcal{W}$ be any ${\varepsilon}$-separated frameset, $G =\,<g_1, \ldots, g_n>$ any $s_1({\varepsilon})$-contracting group based on $\mathcal{W}$, $g = g_{i_1}^{\sigma_1} \ldots g_{i_k}^{\sigma_k}$ (with $\sigma_l = \pm 1$) any nonempty cyclically reduced word.
Then $g$ is pseudohyperbolic, $\frac{{\varepsilon}}{3}$-separated, 1-contracting, and $$\alpha_{N_0}^\mathrm{Haus}(V_{{\mathsmaller{>}}}(g), V_{{\mathsmaller{>}}}(g_{i_1}^{\sigma_1}))\; \ll_{{\varepsilon}}\; s(G).$$
Such a group will be called a *pseudohyperbolic group*.
A pseudohyperbolic group is always free. Indeed, take any reduced word formed on its generators. We may find a cyclically reduced word conjugate to it, and we then know that it is a pseudohyperbolic map. Hence it is not equal to the identity.
The Proposition follows from Lemma \[product\_proximal\] applied to the space $E := \Lambda^d \mathbb{R}^{d+1, d}$. Indeed, there is a correspondence between pseudohyperbolic maps in $\mathbb{R}^{d+1, d}$ and proximal maps in $E$, as will be shown below.
For every map $g \in L(\mathbb{R}^{d+1, d})$, we define the corresponding map $\Lambda^d g \in L(E)$, and for every quadratic form $N$ on $\mathbb{R}^{d+1, d}$, we define the corresponding quadratic form $\Lambda^d N$ on $E$ by $$\langle x_1 \wedge \ldots \wedge x_d,\; y_1 \wedge \ldots \wedge y_d \rangle_{\Lambda^d N}
:= \sum_{\sigma \in \mathcal{S}_d} \epsilon_\sigma \prod_{i=1}^d \langle x_i, y_{\sigma(i)} \rangle_N$$ (where $\mathcal{S}_d$ is the set of permutations of $\{1, \ldots, d\}$ and $\epsilon_\sigma$ stands for the signature of $\sigma$). We set the default form on $E$ to be $\hat{N}_0 = \Lambda^d N_0$. Let us now formulate the desired correspondence:
\[pseudoh-to-prox\]
(i) For $g \in SO(d+1,d)$, $\Lambda^d g$ is proximal iff $g$ is pseudohyperbolic. Moreover, the attracting (resp. repulsing) space of $\Lambda^d g$ depends on nothing but $V_{{\mathsmaller{>}}}(g)$ (resp. $V_{{\mathsmaller{<}}}(g)$): $$\label{eq:frame_transformation}
\begin{cases}
V_s(\Lambda^d g) = \Lambda^d V_{{\mathsmaller{>}}}(g) \\
V_u(\Lambda^d g) = {\left\{ x \in E \; \middle| \; x \wedge \Lambda^{d+1} V_{{\mathsmaller{\leq}}}(g) = 0 \right\}}.
\end{cases}$$
(ii) For every ${\varepsilon}> 0$, there is a constant $\eta({\varepsilon}) > 0$ such that for every ${\varepsilon}$-separated frame $\mathcal{V}$, we have $$\alpha (V_s, V_u) \geq \eta({\varepsilon})$$ (with $V_s$ and $V_u$ defined as in ).
(iii) For every ${\varepsilon}> 0$, for every ${\varepsilon}$-separated pseudohyperbolic map $g \in SO(d+1,d)$, we have $$s(g) \ll_{{\varepsilon}} \hat{s}(\Lambda^d g).$$ If in addition $s(g) < 1$, we have $$s(g) \asymp_{{\varepsilon}} \hat{s}(\Lambda^d g).$$
(iv) For any two $d$-dimensional subspaces $V_1$ and $V_2$ of $\mathbb{R}^{d+1, d}$, we have $$\alpha_{N_0}^\mathrm{Haus}(V_1, V_2)
\;\asymp\; \alpha_{\Lambda^d N_0}(\Lambda^d V_1, \Lambda^d V_2).$$
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(i) Let $g \in SO(d+1,d)$. Let $\lambda_{1}, \ldots \lambda_{2d+1}$ be the eigenvalues of $g$ counted with multiplicity and ordered by increasing absolute value. Then we know that the eigenvalues of $\Lambda^d g$ counted with multiplicity are exactly the products of the form $\lambda_{i_1}\ldots\lambda_{i_d}$, where $1 \leq i_1 < \ldots < i_d \leq 2d+1$. As the two largest of them are $\lambda_{d+2} \ldots \lambda_{2d+1}$ and $\lambda_{d+1}\lambda_{d+3} \ldots \lambda_{2d+1}$, it follows that $\Lambda^d g$ is proximal iff $|\lambda_{d+1}| < |\lambda_{d+2}|$.
Suppose that this is the case. Being isotropic spaces, $V_{{\mathsmaller{<}}}(g)$ and $V_{{\mathsmaller{>}}}(g)$ have dimension at most $d$; it follows that $|\lambda_{d+1}| = 1$. We then have $|\lambda_{d+2}| > 1$, hence $\dim V_{{\mathsmaller{>}}}(g) = d$. Since $V_{{\mathsmaller{=}}}(g) \subset V_{{\mathsmaller{>}}}(g)^\perp$ and $V_{{\mathsmaller{=}}}(g)$ is transversal to $V_{{\mathsmaller{>}}}(g)$, we get that $\dim V_{{\mathsmaller{=}}}(g) = 1$. Having all this, it is easy to show that the identity holds, hence $\lambda_{d+1} = \frac{\det g}{(\det g_{{\mathsmaller{<}}})(\det g_{{\mathsmaller{>}}})} = 1$. We conclude that $g$ is pseudohyperbolic. The converse is obvious.
As for the expression of $V_s$ and $V_u$, it follows immediately by considering a basis that trigonalises $g$.
(ii) Let ${\varepsilon}> 0$. Clearly, $\alpha(V_s, V_u)$ depends continuously on $V_{{\mathsmaller{<}}}$ and $V_{{\mathsmaller{>}}}$, and never vanishes when $V_{{\mathsmaller{<}}}$ and $V_{{\mathsmaller{>}}}$ are transversal. Since the set of all ${\varepsilon}$-separated frames is compact, this expression must have a positive lower bound.
(iii) Let ${\varepsilon}> 0$; let $g \in SO(d+1, d)$ be an ${\varepsilon}$-separated pseudohyperbolic map with frame $\mathcal{V}$. We proceed in three steps.
- First, note that, by , we have $\|g_{{\mathsmaller{>}}}^{-1}\| \asymp_{{\varepsilon}} \|g_{{\mathsmaller{<}}}\|$, hence $$\label{eq:g_<_and_g_>}
s(g) \asymp_{{\varepsilon}} \|g_{{\mathsmaller{>}}}^{-1}\|.$$
- Second, let us show that for any proximal map $f$, we have $$\label{eq:exterior_uniformly_equivalent}
\hat{s}_{\Lambda^d N_0}(f) \asymp_{{\varepsilon}} \hat{s}_{\Lambda^d N_\mathcal{V}}(f).$$ (Caution: $\Lambda^d N_\mathcal{V}$ is in general not the same as $\hat{N}_f$.) Indeed, note that if some norms given by $N$ and $N'$ are $C$-Lipschitz-equivalent, then the norms given by $\Lambda^d N$ and $\Lambda^d N'$ are $C^d$-Lipschitz-equivalent. The above inequalities then follow from Lemma \[uniformly\_equivalent\].
- The last step is to prove the result in metric given by $N_{\mathcal{V}(g)}$. Let $s_{1} \leq \ldots \leq s_{d}$ (resp. $s'_{1} \geq \ldots \geq s'_{d}$) be the singular values of $g_{{\mathsmaller{>}}}$ (resp. $g_{{\mathsmaller{<}}}$), so that $\|g_{{\mathsmaller{<}}}\|_{N_\mathcal{V}} = \|g_{{\mathsmaller{<}}}\|_{N_0} = s'_{1}$ and $\|g_{{\mathsmaller{>}}}^{-1}\| = s_{1}^{-1}$. Since the spaces $V_{{\mathsmaller{<}}}$, $V_{{\mathsmaller{=}}}$ and $V_{{\mathsmaller{>}}}$ are stable by $g$ and pairwise $N_\mathcal{V}$-orthogonal, we get that the singular values of $g$ in metric given by $N_\mathcal{V}$ are $$s'_{d}, \ldots, s'_{1}, 1, s_{1}, \ldots, s_{d}$$ (note however that if we do not suppose $s(g) < 1$, this list might not be sorted in increasing order.) On the other hand, we know that the singular values of $\Lambda^d g$ in metric given by $\Lambda^d N_\mathcal{V}$ are products of $d$ distinct singular values of $g$ in metric given by $N_\mathcal{V}$. Since $V_s(\Lambda^d g)$ is $\Lambda^d N_\mathcal{V}$-orthogonal to $V_u(\Lambda^d g)$, we may once again analyze the singular values separately for each subspace. We know that the singular value corresponding to $V_s$ is equal to $s_{1} \ldots s_{d}$; we deduce that $\left\| {{\left. \Lambda^d g \right|}_{V_u}} \right\|_{\Lambda^d N_\mathcal{V}}$ is equal to the maximum of the remaining singular values. In particular it is larger than $1 \cdot s_{2} \ldots s_{d}$. On the other hand, if $\lambda$ is the largest eigenvalue of $\Lambda^d g$, then we have $$|\lambda| = |\lambda_1 \ldots \lambda_d| = |\det g_{{\mathsmaller{>}}}| = s_{1} \ldots s_{d}$$ (where $\lambda_1, \ldots, \lambda_d$ are the eigenvalues of $g_{{\mathsmaller{>}}}$). It follows that: $$\label{eq:s'_lower_bound}
\hat{s}_{\Lambda^d N_\mathcal{V}}(\Lambda^d g)
= \frac{\left\| {{\left. \Lambda^d g \right|}_{V_u}} \right\|_{\Lambda^d N_\mathcal{V}}}{|\lambda|}
\geq \frac{1 \cdot s_{2} \ldots s_{d}}{s_{1} \ldots s_{d}}
= s_{1}^{-1}
= \|g_{{\mathsmaller{>}}}^{-1}\|.$$ By combining , and , we get the first estimation.
Now suppose that $s(g) < 1$. Then we have $s'_{1} \leq s(g) < 1$ and $1 < s(g)^{-1} \leq s_{1}$, which means that the singular values of $\Lambda^d g$ are indeed sorted in the “correct” order. Hence $1 \cdot s_{2} \ldots s_{d}$ is actually the largest singular value of ${{\left. \Lambda^d g \right|}_{V_u}}$, and the inequality becomes an equality: $\hat{s}_{\Lambda^d N_\mathcal{V}}(\Lambda^d g) = \|g_{{\mathsmaller{>}}}^{-1}\|$. The second estimation follows.
(iv) Let $V_1$ and $V_2$ be two $d$-dimensional spaces. We introduce the notations: $$\alpha_1 := \alpha_{N_0}^\mathrm{Haus}(V_1, V_2);$$ $$\alpha_2 := \alpha_{\Lambda^d N_0}(\Lambda^d V_1, \Lambda^d V_2).$$ We may find an $N_0$-orthonormal basis $(e_1, \ldots, e_{2d+1})$ of $\mathbb{R}^{d+1,d}$ such that $V_1$ has basis $(e_1, \ldots, e_d)$ and $V_2$ has basis $$\left( (\cos \theta_i) e_i + (\sin \theta_i) e_{d+i} \right)_{1 \leq i \leq d},$$ for some angles $0 \leq \theta_1 \leq \ldots \leq \theta_d \leq \frac{\pi}{2}$. In this case, we have: $$\alpha_1 = \theta_d$$ and $$\cos \alpha_2 = \prod_{i=1}^d \cos \theta_i,$$ hence $$(\cos \alpha_1)^d \leq \cos \alpha_2 \leq \cos \alpha_1.$$ On the other hand, from the concavity of the function $y \mapsto (\arccos \exp y)^2$, it follows that for every $\theta \in [0, \frac{\pi}{2}]$, we have $$\arccos((\cos \theta)^d) \leq \sqrt{d}\theta.$$ Finally we get $$\alpha_1 \leq \alpha_2 \leq \sqrt{d}\alpha_1,$$ QED.
We may now prove the main Proposition.
Let ${\varepsilon}> 0$; let $\mathcal{W} = (\mathcal{V}_1, \ldots, \mathcal{V}_n)$ be an ${\varepsilon}$-separated frameset and $G =\,<g_1, \ldots, g_n>$ be an $s_1({\varepsilon})$-contracting group based on $\mathcal{W}$, for some constant $s_1({\varepsilon})$ to be specified later. Let $g = g_{i_1}^{\sigma_1} \ldots g_{i_k}^{\sigma_k}$ be a nonempty cyclically reduced word.
For every $i$, take $f_i = \Lambda^d g_i$. Let us check that we may apply Lemma \[product\_proximal\]. Indeed:
- By Lemma \[pseudoh-to-prox\] (i), $F = (f_1, \ldots, f_n)$ is an independent proximal system. (Conditions (i) and (ii) follow, respectively, from the first and second part of Lemma \[pseudoh-to-prox\] (i).)
- By Lemma \[pseudoh-to-prox\] (ii), we have $\eta(F) \leq \eta({\varepsilon})$; in other words, $F$ is $\eta({\varepsilon})$-separated. We set $\eta = \eta({\varepsilon})$: then “$\ll_\eta$” always implies “$\ll_{\varepsilon}$”.
- Without loss of generality, we may suppose $s(G) < 1$. Then by Lemma \[pseudoh-to-prox\] (iii), we have $\hat{s}(F) \ll_{\varepsilon}s(G)$, which is in turn no greater than $s_1({\varepsilon})$. If we choose $s_1({\varepsilon})$ sufficiently small (since $\eta$ is entirely determined by ${\varepsilon}$), we then have $$\hat{s}(F) \leq \hat{s}(\eta).$$
Now let us deduce the conclusions of the Proposition \[product\_pseudohyperbolic\] from the conclusions of Lemma \[product\_proximal\], applied to the word $\Lambda^d g = f_{i_1}^{\sigma_1} \ldots f_{i_k}^{\sigma_k}$:
- That $g$ is pseudohyperbolic follows from Lemma \[pseudoh-to-prox\] (i).
- Let us show that $$\alpha^\mathrm{Haus}(V_{{\mathsmaller{>}}}(g),\; V_{{\mathsmaller{>}}}(g_{i_1}^{\sigma_1})) \ll_{\varepsilon}s(G).$$ Indeed, we have: $$\begin{aligned}
\alpha^\mathrm{Haus}(V_{{\mathsmaller{>}}}(g),\; V_{{\mathsmaller{>}}}(g_{i_1}^{\sigma_1}))
&\ll \alpha(\Lambda^d V_{{\mathsmaller{>}}}(g),\; \Lambda^d V_{{\mathsmaller{>}}}(g_{i_1}^{\sigma_1})) &&\text{by Lemma \ref{pseudoh-to-prox} (iv)}\\
&= \alpha(V_s(\Lambda^d g),\; V_s(f_{i_1}^{\sigma_1})) &&\text{by Lemma \ref{pseudoh-to-prox} (i)}\\
&\ll_\eta \hat{s}(F) &&\text{by Lemma \ref{product_proximal}}\\
&\ll_{\varepsilon}s(G) &&\text{by Lemma \ref{pseudoh-to-prox} (iii);}\\\end{aligned}$$ and we know that “$\ll_\eta$” implies “$\ll_{\varepsilon}$”.
- Let us show that $g$ is $\frac{{\varepsilon}}{3}$-separated. Since $s(G) \leq s_1({\varepsilon})$, we may choose $s_1({\varepsilon})$ sufficiently small to deduce, from the previous point, the following inequality: $$\alpha^\mathrm{Haus}(V_{{\mathsmaller{>}}}(g),\; V_{{\mathsmaller{>}}}(g_{i_1}^{\sigma_1})) \leq \frac{{\varepsilon}}{3}.$$ Replacing $g$ by $g^{-1}$, we get similarly $$\alpha^\mathrm{Haus}(V_{{\mathsmaller{<}}}(g),\; V_{{\mathsmaller{<}}}(g_{i_k}^{\sigma_k})) \leq \frac{{\varepsilon}}{3}.$$ Finally, since $g$ is cyclically reduced and $\mathcal{W}$ is ${\varepsilon}$-separated, we know that $$\alpha(V_{{\mathsmaller{>}}}(g_{i_1}^{\sigma_1}),\; V_{{\mathsmaller{<}}}(g_{i_k}^{\sigma_k})) \geq {\varepsilon}.$$ From these three inequalities, it follows that $$\alpha(V_{{\mathsmaller{<}}}(g), V_{{\mathsmaller{>}}}(g)) \geq \frac{{\varepsilon}}{3}.$$
- Let us show that $g$ is 1-contracting. Using Lemma \[product\_proximal\] and Lemma \[pseudoh-to-prox\] (iii), we get $$\begin{aligned}
s(g) &\ll_{{\varepsilon}(g)} \hat{s}(\Lambda^d g) &&\text{by Lemma \ref{pseudoh-to-prox} (iii)}\\
&\ll_\eta \hat{s}(F) &&\text{by Lemma \ref{product_proximal}}\\
&\ll_{\varepsilon}s(G) &&\text{by Lemma \ref{pseudoh-to-prox} (iii) (since $s(G) < 1$.}\\\end{aligned}$$ Since ${\varepsilon}(g) \geq \frac{{\varepsilon}}{3}$ and $\eta = \eta({\varepsilon})$, we get $s(g) \ll_{\varepsilon}s(G) \leq s_1({\varepsilon})$. If we take $s_1({\varepsilon})$ sufficiently small, we deduce that $$s(g) < 1.$$
The “tennis ball” and generalized Schottky groups {#sec:tennis_ball}
=================================================
Let ${\varepsilon}> 0$, and let $\mathcal{V}$ be a frame.
\[tennis\_ball\_domains\] We define, on the sphere $\mathbb{S}(\mathbb{R}^{d+1,d})$ (from now on simply referred to as $\mathbb{S}$), the following domains: $$\begin{cases}
\mathcal{H}_\mathbb{S}^- := B_{N_\mathcal{V}}(\pi_\mathbb{S}(V_{{\mathsmaller{<}}}^{{\mathsf{L}}}), {\varepsilon}) \\
\mathcal{H}_\mathbb{S}^+ := B_{N_\mathcal{V}}(\pi_\mathbb{S}(V_{{\mathsmaller{>}}}^{{\mathsf{L}}}), {\varepsilon}).
\end{cases}$$ (Of course, they depend on $\mathcal{V}$ and ${\varepsilon}$, but to simplify the notations, we shall leave this dependence implicit.) We call them *tennis ball domains* (to understand why, draw them for $d=1$).
In the following Proposition and its proof, we work in metric given by $N_{\mathcal{V}(g)}$.
\[tennis\_ball\] For every ${\varepsilon}> 0$, there is a constant $s_2({\varepsilon})$ such that for any $s_2({\varepsilon})$-contracting pseudohyperbolic map $g$ (with frame $\mathcal{V}$), we have
\[eq:tennis\_ball\] $$\label{eq:tb1}
g_\mathbb{S} \left( \mathbb{S} \setminus
\overline{\mathcal{H}_\mathbb{S}^-} \right)
\subset \mathcal{H}_\mathbb{S}^+$$ $$\label{eq:tb2}
g^{-1}_\mathbb{S} \left( \mathbb{S} \setminus
\overline{\mathcal{H}_\mathbb{S}^+} \right)
\subset \mathcal{H}_\mathbb{S}^-.$$
Since we work here in metric given by $N_\mathcal{V}$, the separation of $g$ does not matter and ${\varepsilon}$ has nothing to do with it. Instead ${\varepsilon}$ defines the “aperture” of the tennis ball domains $\mathcal{H}_\mathbb{S}^\pm$.
\[tb12\] As $g_\mathbb{S}$ is a homeomorphism and the domains under consideration are regular, these two relations are actually equivalent. Also, since $V_{{\mathsmaller{<}}}(g^{-1}) = V_{{\mathsmaller{>}}}$, $V_{{\mathsmaller{>}}}(g^{-1}) = V_{{\mathsmaller{<}}}$ and $s(g^{-1}) = s(g)$, is nothing else than applied to $g^{-1}$.
Let ${\varepsilon}> 0$ and $g$ be a pseudohyperbolic map with frame $\mathcal{V}$. As previously done, for $x \in \mathbb{R}^{d+1,d}$, we define the triple $(x_{{\mathsmaller{<}}}, x_{{\mathsmaller{=}}}, x_{{\mathsmaller{>}}}) \in V_{{\mathsmaller{<}}} \times V_{{\mathsmaller{=}}} \times V_{{\mathsmaller{>}}}$ such that $x_{{\mathsmaller{<}}} + x_{{\mathsmaller{=}}} + x_{{\mathsmaller{>}}} = x$; these are the $N_\mathcal{V}$-orthogonal projections of $x$ on the corresponding spaces. The vector $e_{{\mathsmaller{=}}}$ (see Definition \[direction\]) gives an orientation on $V_{{\mathsmaller{=}}}$, which allows us to define an order on this 1-dimensional space: we say that $x_{{\mathsmaller{=}}} \geq y_{{\mathsmaller{=}}}$ iff $\langle x_{{\mathsmaller{=}}}, e_{{\mathsmaller{=}}} \rangle \geq \langle y_{{\mathsmaller{=}}}, e_{{\mathsmaller{=}}} \rangle$.
\[banana\_description\] Let $x \in \mathbb{R}^{d+1,d} \setminus \{0\}$. Then we have:
\[eq:banana\_description\] $$\label{eq:bd1}
\pi_\mathbb{S}(x) \in B (\pi_\mathbb{S}(V_{{\mathsmaller{<}}}^{{\mathsf{L}}}), {\varepsilon})
\iff \begin{cases}
x_{{\mathsmaller{=}}} \leq 0 \text{ \rm and } \frac{\|x_{{\mathsmaller{>}}}\|}{\|x_{{\mathsmaller{<}}} + x_{{\mathsmaller{=}}}\|} < \tan {\varepsilon}\\
\text{\rm or} \\
x_{{\mathsmaller{=}}} \geq 0 \text{ \rm and } \frac{\|x_{{\mathsmaller{>}}} + x_{{\mathsmaller{=}}}\|}{\|x_{{\mathsmaller{<}}}\|} < \tan {\varepsilon}\end{cases}$$ and $$\label{eq:bd2}
\pi_\mathbb{S}(x) \in B (\pi_\mathbb{S}(V_{{\mathsmaller{>}}}^{{\mathsf{L}}}), {\varepsilon})
\iff \begin{cases}
x_{{\mathsmaller{=}}} \leq 0 \text{ \rm and } \frac{\|x_{{\mathsmaller{<}}} + x_{{\mathsmaller{=}}}\|}{\|x_{{\mathsmaller{>}}}\|} < \tan {\varepsilon}\\
\text{\rm or} \\
x_{{\mathsmaller{=}}} \geq 0 \text{ \rm and } \frac{\|x_{{\mathsmaller{<}}}\|}{\|x_{{\mathsmaller{>}}} + x_{{\mathsmaller{=}}}\|} < \tan {\varepsilon}\end{cases}$$
(and by replacing everywhere “$< \tan {\varepsilon}$” by “$\leq \tan {\varepsilon}$”, we may characterize in a similar way the closures of these domains.)
Without loss of generality, let us concentrate on (the other statement follows simply by interchanging $V_{{\mathsmaller{<}}}$ and $V_{{\mathsmaller{>}}}$ and swapping the orientation of $V_{{\mathsmaller{=}}}$.) Remember that $x \in V_{{\mathsmaller{>}}}^{{\mathsf{L}}}$ iff $x_{{\mathsmaller{<}}} = 0$ and $x_{{\mathsmaller{=}}} > 0$.
- Suppose $x_{{\mathsmaller{=}}} \geq 0$. As $V_{{\mathsmaller{>}}}^{{\mathsf{L}}} \subset V_{{\mathsmaller{\geq}}}$, we have $\alpha(x, V_{{\mathsmaller{>}}}^{{\mathsf{L}}}) \geq \alpha(x, V_{{\mathsmaller{\geq}}})$. On the other hand, we have $\alpha(x, V_{{\mathsmaller{\geq}}}) = \alpha(x, x_{{\mathsmaller{>}}} + x_{{\mathsmaller{=}}})$ and $x_{{\mathsmaller{>}}} + x_{{\mathsmaller{=}}} \in V_{{\mathsmaller{>}}}^{{\mathsf{L}}}$, which shows the opposite inequality. Hence $\alpha(x, V_{{\mathsmaller{>}}}^{{\mathsf{L}}}) = \alpha(x, V_{{\mathsmaller{\geq}}})$.
- Suppose $x_{{\mathsmaller{=}}} \leq 0$; without loss of generality, we may assume that $\|x\| = 1$. Since $V_{{\mathsmaller{>}}} \subset V_{{\mathsmaller{>}}}^{{\mathsf{L}}}$, obviously $\alpha(x, V_{{\mathsmaller{>}}}^{{\mathsf{L}}}) \leq \alpha(x, V_{{\mathsmaller{>}}})$. Now let $y \in V_{{\mathsmaller{>}}}^{{\mathsf{L}}}$; then we have: $$\begin{aligned}
\cos \alpha(x, y) &= \frac{\langle x, y \rangle}{\|y\|} \\
&= \frac{\langle x_{{\mathsmaller{<}}} + x_{{\mathsmaller{>}}} + x_{{\mathsmaller{=}}},\; y_{{\mathsmaller{>}}} + y_{{\mathsmaller{=}}} \rangle}{\|y_{{\mathsmaller{>}}} + y_{{\mathsmaller{=}}}\|} \\
&= \frac{\langle x_{{\mathsmaller{>}}}, y_{{\mathsmaller{>}}} \rangle + \langle x_{{\mathsmaller{=}}}, y_{{\mathsmaller{=}}} \rangle}{\|y_{{\mathsmaller{>}}} + y_{{\mathsmaller{=}}}\|} \\
&\leq \frac{\langle x_{{\mathsmaller{>}}}, y_{{\mathsmaller{>}}} \rangle}{\|y_{{\mathsmaller{>}}}\|} \\
&= \cos \alpha(x, y_{{\mathsmaller{>}}}),\end{aligned}$$ since $\langle x_{{\mathsmaller{=}}}, y_{{\mathsmaller{=}}} \rangle \leq 0$ and $\|y_{{\mathsmaller{>}}} + y_{{\mathsmaller{=}}}\| \geq \|y_{{\mathsmaller{>}}}\|$. Hence $\alpha(x, y) \geq \alpha(x, y_{{\mathsmaller{>}}})$, with $y_{{\mathsmaller{>}}} \in V_{{\mathsmaller{>}}}$. This shows the opposite inequality. Hence $\alpha(x, V_{{\mathsmaller{>}}}^{{\mathsf{L}}}) = \alpha(x, V_{{\mathsmaller{>}}})$.
The result now follows from the fact that for any vector subspace $E \subset \mathbb{R}^{d+1,d}$, we have $$\alpha(x, E) = \alpha(x, x_E) = \arccos \frac{\|x_E\|}{\|x\|} = \arctan \frac{\|x - x_E\|}{\|x_E\|},$$ where $x_E$ is the $N_\mathcal{V}$-orthogonal projection of $x$ onto $E$.
By virtue of Remark \[tb12\], it is enough to show . Let $x \in \mathbb{R}^{d+1,d} \setminus \{0\}$ such that $\alpha(x, V_{{\mathsmaller{<}}}^{{\mathsf{L}}}) > {\varepsilon}$; it is enough to prove that if $s(g) \leq s_2({\varepsilon})$ (for a value of $s_2({\varepsilon})$ to be specified later), we have $\alpha(g(x), V_{{\mathsmaller{>}}}^{{\mathsf{L}}}) < {\varepsilon}$.
Suppose that $x_{{\mathsmaller{=}}} \leq 0$. Then we have, by Lemma \[banana\_description\], $$\frac{\|x_{{\mathsmaller{>}}}\|}{\|x_{{\mathsmaller{<}}} + x_{{\mathsmaller{=}}}\|} > \tan {\varepsilon}.$$ We deduce that $$\begin{aligned}
\frac{\|g(x)_{{\mathsmaller{<}}} + g(x)_{{\mathsmaller{=}}}\|^2}{\|g(x)_{{\mathsmaller{>}}}\|^2} &= \frac{\|g(x_{{\mathsmaller{<}}})\|^2 + \|g(x_{{\mathsmaller{=}}})\|^2}{\|g(x_{{\mathsmaller{>}}})\|^2} \\
&\leq \frac{s(g)^2\|x_{{\mathsmaller{<}}}\|^2 + \|x_{{\mathsmaller{=}}}\|^2}{s(g)^{-2}\|x_{{\mathsmaller{>}}}\|^2} \\
&\leq s(g)^2\frac{\|x_{{\mathsmaller{<}}}\|^2 + \|x_{{\mathsmaller{=}}}\|^2}{\|x_{{\mathsmaller{>}}}\|^2} \\
&\leq s(g)^2(\tan {\varepsilon})^{-2} \\
&< (\tan {\varepsilon})^2,\end{aligned}$$ provided that $s(g) < (\tan {\varepsilon})^4$, which is true if we take $s_2({\varepsilon})$ to be smaller than this value. Hence $\alpha(g(x), V_{{\mathsmaller{>}}}) < {\varepsilon}$. On the other hand, we have $g(x)_{{\mathsmaller{=}}} = g(x_{{\mathsmaller{=}}}) = x_{{\mathsmaller{=}}} \leq 0$. It follows that $\alpha(g(x), V_{{\mathsmaller{>}}}^{{\mathsf{L}}}) < {\varepsilon}$. In the case where $x_{{\mathsmaller{=}}} \geq 0$, a completely analogous calculation yields the same result.
Now consider a frameset $\mathcal{W} = (\mathcal{V}_1, \ldots, \mathcal{V}_n)$ and a set of radii ${\varepsilon}_1, \ldots, {\varepsilon}_n$.
\[tennis\_ball\_sets\] Just as in Definition \[tennis\_ball\_domains\], we define for every index $i$ the domains $$\begin{cases}
\mathcal{H}_{\mathbb{S}, i}^- := B_{N_{\mathcal{V}_i}}(\pi_\mathbb{S}(V_{i, {\mathsmaller{<}}}^{{\mathsf{L}}}), {\varepsilon}_i) \\
\mathcal{H}_{\mathbb{S}, i}^+ := B_{N_{\mathcal{V}_i}}(\pi_\mathbb{S}(V_{i, {\mathsmaller{>}}}^{{\mathsf{L}}}), {\varepsilon}_i).
\end{cases}$$ (Once again, they depend on $\mathcal{W}$ and the ${\varepsilon}_i$, but to simplify the notations, we keep this dependence implicit.) Let $G =\,<g_1, \ldots, g_n>$ be any group based on $\mathcal{W}$. If the sets $\overline{\mathcal{H}_{\mathbb{S}, i}^\pm}$ are pairwise disjoint and for every $i$, $s(g_i)$ is small enough to apply Proposition \[tennis\_ball\], we say that $G$ is *$({\varepsilon}_1, \ldots, {\varepsilon}_n)$-Schottky*.
In this case, it follows from Proposition \[tennis\_ball\] that $G$ is free. Indeed, we have for every $i$: $$\label{eq:sph_ping_pong}
\begin{cases}
g_i \left( \mathbb{S} \setminus \overline{\mathcal{H}_{\mathbb{S}, i}^-} \right)
\subset \mathcal{H}_{\mathbb{S}, i}^+ \\
g_i^{-1} \left( \mathbb{S} \setminus \overline{\mathcal{H}_{\mathbb{S}, i}^+} \right)
\subset \mathcal{H}_{\mathbb{S}, i}^-,
\end{cases}$$ and we may apply the ping-pong lemma (see for example [@Tits72], Proposition 1.1).
Affine deformations {#sec:affine}
===================
Let $G \subset SO(d+1, d)$ be any linear group. An *affine deformation* of $G$ is any group $\Gamma \subset \mathbb{R}^{d+1,d} \rtimes SO(d+1, d)$ such that the canonical projection $L: \mathbb{R}^{d+1,d} \rtimes SO(d+1, d) \to SO(d+1, d)$ induces an isomorphism from $\Gamma$ to $G$. In other terms, it is a group of affine transformations that does not contain pure translations and whose linear parts form the group $G$.
Now suppose $G =\,<g_1, \ldots, g_n>$ is a free group; let $\Gamma$ be any affine deformation of $G$. Then it is generated by the elements $\gamma_1, \ldots, \gamma_n$ whose linear parts are $g_1, \ldots, g_n$, respectively. This means that, $G$ being fixed, $\Gamma$ is entirely determined by the translational parts of its generators, namely the vectors $\gamma_1(0), \ldots, \gamma_n(0)$. Reciprocally, for any family of vectors ${\bf t} = (t_1, \ldots, t_n) \in \left(\mathbb{R}^{d+1, d}\right)^n$, we may define $\gamma_1, \ldots, \gamma_n$ by $\gamma_i(x) = g_i(x) + t_i$ for all $i$. Since $G$ is free, the group generated by these elements is then an affine deformation of $G$, that we shall call $G({\bf t})$. This defines a bijection between the set of all affine deformations of $G$ and $\left(\mathbb{R}^{d+1, d}\right)^n$.
We may now state the Main Theorem more precisely:
\[main\_theorem\] For every ${\varepsilon}> 0$, there is a constant $s_3({\varepsilon})$ with the following property. Let $\mathcal{W}$ be any ${\varepsilon}$-separated frameset, $G$ any $s_3({\varepsilon})$-contracting group based on $\mathcal{W}$. Then we can say that:
(i) The group $G$ is free;
(ii) There is a nonempty open set ${\bf T} \subset \left(\mathbb{R}^{d+1, d}\right)^n$ (depending on $G$) such that for every ${\bf t} \in {\bf T}$, the affine deformation $G({\bf t})$ acts properly discontinuously on $\mathbb{R}^{d+1, d}$;
(iii) For ${\bf t} \in {\bf T}$, the quotient space $\mathbb{R}^{d+1, d}/G({\bf t})$ is homeomorphic to a solid $(2d+1)$-dimensional handlebody with $n$ handles.
We begin by giving a few definitions and notations, to be fixed for the remainder of this section.
- We fix ${\varepsilon}> 0$, $\mathcal{W}$ an ${\varepsilon}$-separated frameset, $G$ an $s_3({\varepsilon})$-contracting group based on $\mathcal{W}$. We will determine the value of $s_3({\varepsilon})$ in the course of the proof.
- For every $i$, we choose a constant ${\varepsilon}_i > 0$ such that for any set $X \subset \mathbb{S}$, we have $$\label{eq:eps_i_and_eps}
B_{N_{\mathcal{V}_i}}(X, {\varepsilon}_i) \subset B_{N_0}(X, \frac{{\varepsilon}}{3}).$$ We define the tennis-ball domains $\mathcal{H}_{\mathbb{S}, i}^\sigma$ accordingly (see definition \[tennis\_ball\_sets\]).
- For any ${\bf t} \in \left(\mathbb{R}^{d+1, d}\right)^n$, for any $i$ such that $1 \leq i \leq n$ and $\sigma = \pm 1$, we introduce the following domain. It is a subset of $\mathbb{R}^{d+1, d}$ constructed as a cone, whose apex depends on the translational part and whose base is the corresponding tennis-ball domain: $$\mathcal{H}_i^\sigma({\bf t}) := \pi_\mathbb{S}^{-1}(\mathcal{H}_{\mathbb{S}, i}^\sigma) + \sigma u_i,$$ where $u_i$ is the solution to the equation $u_i + g_i(u_i) = t_i$. (Since $g_i$ is pseudohyperbolic, it does not have $-1$ as an eigenvalue, so that the equation has indeed a unique solution.)
- For any such ${\bf t}$, we also introduce, for every index $i$, the domains $$\begin{cases}
\mathcal{\tilde{H}}_i^-({\bf t}) := \mathcal{H}_i^-({\bf t}) \\
\mathcal{\tilde{H}}_i^+({\bf t}) := \gamma_i \left(\mathbb{R}^{d+1,d} \setminus \overline{\mathcal{H}_i^-({\bf t})}\right)
\end{cases}$$ (where $\gamma_i: x \mapsto g_i(x) + t_i$ is the $i$-th generator of the affine deformation $G({\bf t})$: it depends implicitly on ${\bf t}$.) We also introduce the domain $$\mathcal{H}^0 := \mathbb{R}^{d+1,d} \setminus \bigcup_{i=1}^n \bigcup_{\sigma = \pm} \overline{\mathcal{\tilde{H}}_i^\sigma}.$$
- We define ${\bf T}$ to be the set of all ${\bf t} \in \left(\mathbb{R}^{d+1, d}\right)^n$ such that the $2n$ sets $\overline{\mathcal{H}_i^\pm({\bf t})}$ are pairwise disjoint.
Now (i) follows immediately from either Proposition \[product\_pseudohyperbolic\], or Proposition \[tennis\_ball\] combined with . The claim (ii) follows from Lemma \[regions\_disjoint\_nonempty\], Lemma \[regions\_disjoint\_open\] and Proposition \[fundamental\_region\] below (since the existence of a fundamental domain is equivalent to proper discontinuity). The latter Proposition is interesting in its own right, as it describes the exact shape of the fundamental domain. It also allows us to prove (iii). Indeed, if the fundamental domain is $\mathcal{H}^0$, then the quotient space $\mathbb{R}^{d+1, d}/G({\bf t})$ is homeomorphic to the space obtained from $\overline{\mathcal{H}^0}$ by identifying for every $i$ the border of $\mathcal{\tilde{H}}_i^-$ with the border of $\mathcal{\tilde{H}}_i^+$. But clearly, the borders of $\mathcal{\tilde{H}}_i^-$ and $\mathcal{\tilde{H}}_i^+$ are homeomorphic to $\mathbb{R}^{2d}$ (or, if you wish, to $2d$-dimensional open “disks”), and $\mathcal{H}^0$ is homeomorphic to $\mathbb{R}^{2d+1}$ (or a $2d+1$-dimensional open ball).
\[regions\_disjoint\_nonempty\] The set ${\bf T}$ is nonempty.
Recall that $e_{i, {\mathsmaller{=}}}$ is the vector of unit norm $N_0$ fixed by $g_i$ with a suitably chosen sign (see Definition \[direction\]). We set $${\bf t}_0 := (2e_{1, {\mathsmaller{=}}}, \ldots, 2e_{n, {\mathsmaller{=}}}).$$ Then we have, for every $i$, $u_i = e_{i, {\mathsmaller{=}}}$, since by definition $g_i(e_{i, {\mathsmaller{=}}}) = e_{i, {\mathsmaller{=}}}$.
Let us show that ${\bf t}_0 \in {\bf T}$, that is, that the sets $\overline{\mathcal{H}_i^\pm({\bf t}_0)}$ are pairwise disjoint. To do this, we include them in the sets $\pi_\mathbb{S}^{-1}(\mathcal{H}_{\mathbb{S}, i}^\pm)$, that we already know to be pairwise disjoint.
Indeed, let us fix an index $i$ and a sign $\sigma$. In this proof, we work in metric given by $N_{\mathcal{V}_i}$. We need to show that: $$\overline{\mathcal{H}_i^\sigma({\bf t}_0)}
\; := \; \overline{\pi_\mathbb{S}^{-1}(\mathcal{H}_{\mathbb{S}, i}^\sigma)} + \sigma e_{i, {\mathsmaller{=}}}
\; \subset \; \pi_\mathbb{S}^{-1}(\mathcal{H}_{\mathbb{S}, i}^\sigma).$$ Suppose, without loss of generality, that $\sigma = +1$; let $x \in \overline{\pi_\mathbb{S}^{-1}(\mathcal{H}_{\mathbb{S}, i}^+)}$. If $x = 0$, clearly, we have $$x + e_{i, {\mathsmaller{=}}} = e_{i, {\mathsmaller{=}}} \in V_{i, {\mathsmaller{>}}}^{{\mathsf{L}}} \subset \pi_\mathbb{S}^{-1}(\mathcal{H}_{\mathbb{S}, i}^+).$$ Otherwise, it is easy to see that $\pi_\mathbb{S}(x) \in \overline{\mathcal{H}_{\mathbb{S}, i}^+}$. Clearly, we may apply Lemma \[banana\_description\], provided we replace strict inequalities with non-strict ones. We reuse the notation $x = x_{{\mathsmaller{<}}} + x_{{\mathsmaller{=}}} + x_{{\mathsmaller{>}}}$ from that lemma. Let us distinguish three cases:
- If $x_{{\mathsmaller{=}}} \geq 0$, then we have $$\frac{\|x_{{\mathsmaller{<}}}\|}{\|x_{{\mathsmaller{>}}} + x_{{\mathsmaller{=}}}\|} \leq \tan {\varepsilon}_i.$$ We still have $x_{{\mathsmaller{=}}} + e_{{\mathsmaller{=}}} \geq 0$ and $\|x_{{\mathsmaller{=}}} + e_{{\mathsmaller{=}}}\| > \|x_{{\mathsmaller{=}}}\|$, hence $$\frac{\|x_{{\mathsmaller{<}}}\|}{\|x_{{\mathsmaller{>}}} + x_{{\mathsmaller{=}}} + e_{{\mathsmaller{=}}}\|} < \frac{\|x_{{\mathsmaller{<}}}\|}{\|x_{{\mathsmaller{>}}} + x_{{\mathsmaller{=}}}\|} \leq \tan {\varepsilon}_i,$$ and we conclude that $x + e_{{\mathsmaller{=}}} \in \pi_\mathbb{S}^{-1}(B_{N}(\pi_\mathbb{S}(V_{{\mathsmaller{>}}}^{{\mathsf{L}}}), {\varepsilon}_i))$.
- If $x_{{\mathsmaller{=}}} \leq -e_{{\mathsmaller{=}}}$, then, similarly, $$\frac{\|x_{{\mathsmaller{<}}} + x_{{\mathsmaller{=}}} + e_{{\mathsmaller{=}}}\|}{\|x_{{\mathsmaller{>}}}\|} < \frac{\|x_{{\mathsmaller{<}}} + x_{{\mathsmaller{=}}}\|}{\|x_{{\mathsmaller{>}}}\|} \leq \tan {\varepsilon}_i,$$ and we reach the same conclusion.
- If $-e_{{\mathsmaller{=}}} < x_{{\mathsmaller{=}}} < 0$, then we have $$\frac{\|x_{{\mathsmaller{<}}} + x_{{\mathsmaller{=}}}\|}{\|x_{{\mathsmaller{>}}}\|} \leq \tan {\varepsilon}_i,$$ from which we deduce $$\begin{aligned}
\|x_{{\mathsmaller{<}}}\| &< \|x_{{\mathsmaller{<}}} + x_{{\mathsmaller{=}}}\| \\
&\leq (\tan {\varepsilon}_i) \|x_{{\mathsmaller{>}}}\| \\
&< (\tan {\varepsilon}_i) \|x_{{\mathsmaller{>}}} + x_{{\mathsmaller{=}}} + e_{{\mathsmaller{=}}}\|;\end{aligned}$$ since $x_{{\mathsmaller{=}}} + e_{{\mathsmaller{=}}} \geq 0$, we reach again the same conclusion.
\[regions\_disjoint\_open\] The set ${\bf T}$ is open.
Let ${\bf t}_0 = (t_{0,1}, \ldots, t_{0,n})$ be any element of ${\bf T}$. We know that any two of the sets $\overline{\mathcal{H}_i^\pm({\bf t}_0)}$ are disjoint; we claim that they are separated by a positive distance. Indeed, take any ball $B$ whose radius is large compared to ${\bf t}_0$. Then the parts that fall inside $B$ are compact and disjoint, hence separated by a positive distance. As for the parts that fall outside $B$, they are separated because asymptotically, their projections onto $\mathbb{S}$ — namely $\overline{\mathcal{H}_{\mathbb{S}, i}^\pm}$ — are also compact and disjoint.
Let $d_{\min}$ be the smallest of these distances. Consider the set of all ${\bf t} = (t_1, \ldots, t_n)$ such that for every index $i$, $\|u_i - u_{0,i}\| < \frac{d_{\min}}{2}$. Then clearly this set is a neighborhood of ${\bf t}_0$, and is included in ${\bf T}$.
\[fundamental\_region\] For any ${\bf t} \in {\bf T}$, the action of the affine deformation $\Gamma := G({\bf t})$ on the affine space $\mathbb{R}^{d+1,d}$ has fundamental domain $\mathcal{H}^0$. More precisely:
(i) The images of $\mathcal{H}^0$ under the elements of $\Gamma$ are pairwise disjoint;
(ii) The images of its closure cover the whole space: $$\bigcup_{\gamma \in \Gamma} \overline{\gamma(\mathcal{H}^0)} = \mathbb{R}^{d+1,d}.$$
Let us fix a value ${\bf t} = (t_1, \ldots, t_n) \in {\bf T}$; we call $\Gamma =\;<\gamma_1, \ldots, \gamma_n>\;:= G({\bf t})$ the corresponding affine deformation. The first thing to understand is that the domains $\mathcal{H}_i^\pm$ (from now on, we shall no longer mention the dependence on ${\bf t}$) satisfy “ping-pong identities” similar to . Namely, it follows from that $$\begin{aligned}
\gamma_i \left( \mathbb{R}^{d+1,d} \setminus \overline{\mathcal{H}_i^-} \right)
&= \gamma_i \left( \mathbb{R}^{d+1,d} \setminus \overline{\pi_\mathbb{S}^{-1}(\mathcal{H}_{\mathbb{S}, i}^-) - u_i} \right) \\
&= g_i \left( \pi_\mathbb{S}^{-1} \left( \mathbb{S} \setminus
\overline{\mathcal{H}_{\mathbb{S}, i}^-} \right)\right) - g_i(u_i) + t_i \\
&\subset \pi_\mathbb{S}^{-1}(\mathcal{H}_{\mathbb{S}, i}^+) + u_i \\
&= \mathcal{H}_i^+,\end{aligned}$$ and the same holds for $\gamma_i^{-1}$. Thus we get: $$\label{eq:aff_ping_pong}
\begin{cases}
\gamma_i \left( \mathbb{R}^{d+1,d} \setminus \overline{\mathcal{H}_i^-} \right)
\subset \mathcal{H}_i^+ \\
\gamma_i^{-1}\left( \mathbb{R}^{d+1,d} \setminus \overline{\mathcal{H}_i^+} \right)
\subset \mathcal{H}_i^-.
\end{cases}$$ If we replace $\mathcal{H}_i^\pm$ by $\mathcal{\tilde{H}}_i^\pm$, the inclusions become sharp equalities:
\[eq:sharp\_ping\_pong\] $$\label{eq:spp1}
\gamma_i \left(\mathbb{R}^{d+1,d} \setminus \overline{\mathcal{\tilde{H}}_i^-}\right) = \mathcal{\tilde{H}}_i^+;$$ $$\label{eq:spp2}
\gamma_i^{-1} \left(\mathbb{R}^{d+1,d} \setminus \overline{\mathcal{\tilde{H}}_i^+}\right) = \mathcal{\tilde{H}}_i^-.$$
Indeed is true by definition of $\mathcal{\tilde{H}}_i^+$, and follows from because $\gamma_i$ is continuous and $\mathcal{H}_i^-$ is a regular domain.
In order to have a real “ping-pong configuration”, we also need to check that the sets $\mathcal{\tilde{H}}_i^\pm$ are pairwise disjoint. But using their definition and , we know that they are included in the sets $\mathcal{\tilde{H}}_i^\pm$, which are pairwise disjoint by hypothesis (${\bf t} \in {\bf T}$).
(i) is now trivial. Indeed, we see by induction that for every $\gamma = \gamma_{i_1}^{\sigma_1} \ldots \gamma_{i_k}^{\sigma_k} \in \Gamma$, the image $\gamma(\mathcal{H}^0)$ lies in $\mathcal{\tilde{H}}_{i_1}^{\sigma_1}$, which is by definition disjoint from $\mathcal{H}^0$.
(ii) is the hard part. The particular case $d=1$ was done by Drumm in [@Dru93] (proof of Theorem 4); see also [@CG00], section 4. Our proof is closely analogous.
From now on, we work in metric given by $N_0$.
Before proceeding, we need a small geometric lemma.
\[angle\_control\]
(i) Let $V$ and $W$ be two MTIS’es. Then we have $$\alpha(V_Q^\perp, W_Q^\perp) = \alpha(V, W)$$ (where $V_Q^\perp$ means the space $Q$-orthogonal to $V$).
(ii) Let $V'$ be a space $Q$-orthogonal to some MTIS $V$ and $x \in S$. Then we have $$\sin \alpha(x,\; V' \cap S) = \sqrt{2} \sin \alpha(x,\; V').$$
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(i) It is obvious that $\alpha(V_{N_0}^\perp, W_{N_0}^\perp) = \alpha(V, W)$ (where $V_{N_0}^\perp$ means, similarly, the space $N_0$-orthogonal to $V$). On the other hand, for every MTIS $V$, we have $V_Q^\perp = \varsigma(V_{N_0}^\perp)$, where the map $\varsigma := \operatorname{Id}_S \oplus (-\operatorname{Id}_T)$ is an $N_0$-isometry. The required equality follows.
(ii) First note that any plane contained in $V'$ intersects $S$ at an angle of $\frac{\pi}{4}$. Indeed, such a plane contains vectors $x$ such that $Q(x, x) = 0$ (because it intersects $V$), but no vectors $x$ such that $Q(x, x) < 0$. On the other hand, it follows from the definition of $N_0$ and $Q$ that $\alpha(x, S) <$ (resp. $=$, $>$) $\frac{\pi}{4}$ iff $Q(x, x) >$ (resp. $=$, $<$) $0$.
Now let $X$ be the 3-space spanned by $x$, $\pi_{V'}(x)$ and the line $V' \cap S$ (here $\pi_{V'}$ stands for the $N_0$-orthogonal projection onto $V'$). We know that the planes $V' \cap X$ and $S \cap X$ intersect at an angle of $\frac{\pi}{4}$, and that the plane spanned by $x$ and $\pi_{V'}(x)$ is perpendicular to $V' \cap X$. The spherical version of the law of sines yields the identity $\sin \alpha(x, V' \cap S) = \sqrt{2} \sin \alpha(x, \pi_{V'}(x))$, QED.
We proceed by contradiction: let $x_0 \in \mathbb{R}^{d+1,d}$ such that $$\label{eq:x_condition}
\forall \gamma \in \Gamma,\quad \gamma(x_0) \not\in \overline{\mathcal{H}^0}.$$ Then there is a (unique) sequence $(i_k, \sigma_k)$ (indexed by $k \geq 1$) of elements of $\{1, \ldots, n\} \times \{-, +\}$, such that for all $k \geq 0$, we have $$\gamma^{-[k]}(x_0)
\in \mathcal{\tilde{H}}_{i_{k+1}}^{\sigma_{k+1}},$$ where $\gamma^{[k]} := \gamma_{i_1}^{\sigma_1} \ldots \gamma_{i_k}^{\sigma_k}$, and $\gamma^{-[k]}$ is shorthand for $(\gamma^{[k]})^{-1}$. Indeed, by induction, suppose that we have constructed the first $k$ terms (for some $k \geq 0$); then we have, by hypothesis $$\gamma^{-[k]}(x_0)
\;\in\; \mathbb{R}^{d+1,d} \setminus \overline{\mathcal{H}^0}
\;=\; \bigcup_{i=1}^n \bigcup_{\sigma = \pm} \mathcal{\tilde{H}}_i^\sigma,$$ which allows us to pick an appropriate pair $(i_{k+1}, \sigma_{k+1})$ (which is actually unique, since the $\mathcal{\tilde{H}}_i^\pm$ are disjoint).
Note also that the word $\gamma^{[k]}$ is always reduced, [i.e. ]{}for all $k \geq 1$, we have $(i_{k+1}, \sigma_{k+1}) \neq (i_k, -\sigma_k)$. Indeed, we have $\gamma^{-[k-1]}(x_0) \in \mathcal{\tilde{H}}_{i_k}^{\sigma_k}$; since $\gamma_{i_k}^{\sigma_k}$ is bijective, applying (assuming $\sigma_k = +1$; otherwise ), we get $$\begin{aligned}
\gamma^{-[k]}(x_0) &= \gamma_{i_k}^{-\sigma_k} \left( \gamma^{-[k-1]}(x_0) \right) \\
&\in \mathbb{R}^{d+1,d} \setminus \overline{\mathcal{\tilde{H}}_{i_k}^{-\sigma_k}}.\end{aligned}$$
We may also suppose that for infinitely many values of $k$, the word $\gamma^{[k]}$ is cyclically reduced (in other terms, $(i_k, \sigma_k) \neq (i_1, -\sigma_1)$.) Indeed, otherwise, we may replace $x_0$ by $\gamma_i^\sigma(x_0)$, where $(i, \sigma)$ is a pair such that the set of indices $k$ such that $(i_k, \sigma_k) \neq (i, -\sigma)$ is infinite and also contains 1 (such a pair always exists). Then the new value still satisfies , and the sequence $(i_k, \sigma_k)$ changes by appending $(i, \sigma)$ at the beginning.
Without loss of generality, let us suppose that $(i_1, \sigma_1) = (1, +)$.
Now an easy induction shows that the following domains form a decreasing sequence that concentrates on $x_0$ : $$\label{eq:decreasing_sequence}
\mathcal{\tilde{H}}_1^+
\supset \gamma^{[1]}(\mathcal{\tilde{H}}_{i_2}^{\sigma_2})
\supset \gamma^{[2]}(\mathcal{\tilde{H}}_{i_3}^{\sigma_3})
\supset \ldots
\ni x_0.$$
Next, we define $$\Delta := S \cap V_{1, {\mathsmaller{>}}}^{{\mathsf{L}}}$$ (recall that $S$ is a maximal positive definite space). We know that $S$ is a $(d+1)$-dimensional space, $V_{1, {\mathsmaller{>}}}^{{\mathsf{L}}}$ is half a $(d+1)$-dimensional space and $S \cap V_{1, {\mathsmaller{>}}} = 0$ (since $S$ is positive definite and $V_{1, {\mathsmaller{>}}}$ is isotropic). Thus $\Delta$ is a half-line. We also define $P$ to be the ($d$-dimensional) hyperplane of $S$ that is $N_S$-orthogonal (or, equivalently, $Q$-orthogonal, or also $N_0$-orthogonal) to $\Delta$. To avoid cumbersome periphrases, in the following, we shall often use terms such as “above” and “below”, having in mind that “up” is the direction where $\Delta$ points.
Now consider the set $\mathcal{\tilde{H}}_1^+ \cap (x_0 + S)$ (here $(x_0 + S)$ stands for the affine space passing through $x_0$ and parallel to $S$). It is contained in an affine half-space of $(x_0 + S)$ lying above a hyperplane parallel to $P$. Indeed, from it follows that $$\mathcal{H}_{\mathbb{S},1}^+ \subset B_{N_0} \left( \pi_\mathbb{S}(V_{1, {\mathsmaller{>}}}^{\mathsf{L}}), \frac{{\varepsilon}}{3} \right).$$ Without loss of generality, we may suppose that ${\varepsilon}\leq \operatorname{diam}\mathbb{P}(\mathbb{R}^{d+1, d}) = \frac{\pi}{2}$ (indeed the separation of no frame or frameset may exceed that value). Then the radius of the right-hand-side neighborhood is no larger than $\frac{\pi}{6}$. It follows from Lemma \[angle\_control\] that $$B_{N_0} \left( \pi_\mathbb{S}(V_{1, {\mathsmaller{>}}}^{\mathsf{L}}), \frac{\pi}{6} \right) \cap S
\;\;\subset\; B_{N_0}\left( \pi_\mathbb{S}(V_{1, {\mathsmaller{>}}}^{\mathsf{L}}\cap S), \frac{\pi}{4} \right)$$ (the angle $\frac{\pi}{4}$ is the solution to $\sin x = \sqrt{2} \sin \frac{\pi}{6}$). The desired property may be deduced from here.
Applying , we see that for every $k \geq 0$, the domain $$\gamma^{[k]}(\mathcal{\tilde{H}}_{i_{k+1}}^{\sigma_{k+1}}) \cap (x_0 + S)$$ is included in a half-space of $(x_0 + S)$ lying above a hyperplane parallel to $P$. We define $P_k$ to be the uppermost such hyperplane; we call $a_k$ the intersection of $P_k$ with the line containing $(x_0 + \Delta)$, and we set, for $k \geq 1$, $\delta_k = a_k - a_{k-1}$.
The result now follows from:
\[gap\_bounded\_below\] There is a constant $\delta_{\min} > 0$ such that for every $k \geq 1$, whenever $(i_k, \sigma_k) \neq (1, -)$, $\|\delta_k\|_{N_0} \geq \delta_{\min}$.
Indeed, from , it follows that the sequence $(a_k)$ is increasing and bounded above by $x_0$. However, we have chosen $x_0$ in such a way that the condition of Lemma \[gap\_bounded\_below\] occurs infinitely often. It follows that $(a_k)$ is unbounded, which is a contradiction.
We still work in metric given by $N_0$.
Let $k \geq 1$ be an index such that $(i_k, \sigma_k) \neq (1, -)$, so that $g^{[k]}$ is cyclically reduced. We know that the group $G$ is pseudohyperbolic; by Proposition \[product\_pseudohyperbolic\], we have $$\alpha(V_{{\mathsmaller{>}}}(g^{[k]}), V_{1, {\mathsmaller{>}}}) \ll_{\varepsilon}s(G).$$ As $s(G) \leq s_3({\varepsilon})$, by choosing $s_3({\varepsilon})$ small enough, we may suppose that this angle is no larger than $\frac{\pi}{6}$. By Lemma \[angle\_control\], it follows that $$\alpha(V_{{\mathsmaller{\geq}}}(g^{[k]}) \cap S,\; V_{1, {\mathsmaller{\geq}}} \cap S) \leq \frac{\pi}{4}$$ (remember that $V_{1, {\mathsmaller{\geq}}} \cap S$ is the line containing $\Delta$).
Now let $\eta_k$ be the projection of $\delta_k$ onto $V_{{\mathsmaller{\geq}}}(g^{[k]}) \cap S$ parallel to $P$. Then we have $$\label{eq:eta_delta}
\|\delta_k\| \geq \left(\cos \frac{\pi}{4}\right) \|\eta_k\| = \frac{\sqrt{2}}{2}\|\eta_k\|.$$
Next, still by Proposition \[product\_pseudohyperbolic\], we know that $g^{[k]}$ is pseudohyperbolic, $\frac{{\varepsilon}}{3}$-separated and 1-contracting; let $\mathcal{V}^{[k]}$ be its frame. By definition, the norm of $g^{[k]}$ restricted to $V_{{\mathsmaller{>}}}(g^{[k]})$ (resp. $V_{{\mathsmaller{=}}}(g^{[k]})$) is equal to $s(g^{[k]})$ (resp. $1$). It follows that $$\begin{aligned}
\left\| {{\left. g^{-[k]} \right|}_{V_{{\mathsmaller{\geq}}}(g^{[k]})}} \right\|_{N_{\mathcal{V}^{[k]}}}
&= \max \left(\left\| {{\left. g^{-[k]} \right|}_{V_{{\mathsmaller{>}}}(g^{[k]})}} \right\|,\;
\left\| {{\left. g^{-[k]} \right|}_{V_{{\mathsmaller{=}}}(g^{[k]})}} \right\|\right) \\
&= \max \left(\left\| (g^{[k]}_{{\mathsmaller{>}}})^{-1} \right\|,\; 1 \right) \\
&= 1.\end{aligned}$$ From Lemma \[uniformly\_equivalent\], we deduce $$\left\| {{\left. g^{-[k]} \right|}_{V_{{\mathsmaller{\geq}}}(g^{[k]})}} \right\|_{N_0} \ll_{\varepsilon}1;$$ given that, by construction, $\eta_k \in V_{{\mathsmaller{\geq}}}(g^{[k]})$, we get $$\label{eq:eta_eta}
\|\eta_k\| \gg_{\varepsilon}\|g^{-[k]}(\eta_k)\|.$$
Finally, let $x_k$ be any point that lies both in $P_k$ and $\overline{\gamma^{[k]}(\mathcal{\tilde{H}}_{i_{k+1}}^{\sigma_{k+1}})}$ (the intersection is nonempty by definition of $P_k$). Set $y_{k-1} := x_k - \eta_k$. Since the orthogonal projection of $\eta_k$ onto $\Delta$ is equal to $\delta_k$, it follows that $y_{k-1} \in P_{k-1}$, and in particular $y_{k-1} \not\in \gamma^{[k-1]}(\mathcal{\tilde{H}}_{i_k}^{\sigma_k})$. Applying $\gamma^{-[k]}$, we get $$\begin{cases}
\gamma^{-[k]}(x_k) \in \overline{\mathcal{\tilde{H}}_{i_{k+1}}^{\sigma_{k+1}}} \\
\gamma^{-[k]}(y_{k-1}) \in \overline{\mathcal{\tilde{H}}_{i_k}^{-\sigma_k}}.
\end{cases}$$ Since $(i_{k+1}, \sigma_{k+1}) \neq (i_k, -\sigma_k)$, we have $$\label{eq:eta_bound}
\|g^{-[k]}(\eta_k)\| = \|\gamma^{-[k]}(x_k) - \gamma^{-[k]}(y_{k-1})\| \geq d_{\min},$$ where $d_{\min}$ is the smallest distance between any of the $\mathcal{\tilde{H}}_i^\sigma$ (which is nonzero as shown in the proof of Lemma \[regions\_disjoint\_open\]).
Joining , and together, we get indeed a lower bound for $\|\delta_k\|$ that does not depend on $k$.
I would like to thank my PhD advisor, Mr. Yves Benoist, whose help while I was working on this paper has been invaluable to me.
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"pile_set_name": "ArXiv"
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---
abstract: 'LOFT-P is a mission concept for a NASA Astrophysics Probe-Class ($<$\$1B) X-ray timing mission, based on the LOFT M-class concept originally proposed to ESA’s M3 and M4 calls. LOFT-P requires very large collecting area, high time resolution, good spectral resolution, broad-band spectral coverage (2–30 keV), highly flexible scheduling, and an ability to detect and respond promptly to time-critical targets of opportunity. It addresses science questions such as: What is the equation of state of ultra dense matter? What are the effects of strong gravity on matter spiraling into black holes? It would be optimized for sub-millisecond timing of bright Galactic X-ray sources including X-ray bursters, black hole binaries, and magnetars to study phenomena at the natural timescales of neutron star surfaces and black hole event horizons and to measure mass and spin of black holes. These measurements are synergistic to imaging and high-resolution spectroscopy instruments, addressing much smaller distance scales than are possible without very long baseline X-ray interferometry, and using complementary techniques to address the geometry and dynamics of emission regions. LOFT-P would have an effective area of $>$6 m$^2$, $>10\times$ that of the highly successful Rossi X-ray Timing Explorer (RXTE). A sky monitor (2–50 keV) acts as a trigger for pointed observations, providing high duty cycle, high time resolution monitoring of the X-ray sky with $\sim$20 times the sensitivity of the RXTE All-Sky Monitor, enabling multi-wavelength and multi-messenger studies. A probe-class mission concept would employ lightweight collimator technology and large-area solid-state detectors, segmented into pixels or strips, technologies which have been recently greatly advanced during the ESA M3 Phase A study of LOFT. Given the large community interested in LOFT ($>$800 supporters[^1], the scientific productivity of this mission is expected to be very high, similar to or greater than RXTE ($\sim 2000$ refereed publications). We describe the results of a study, recently completed by the MSFC Advanced Concepts Office, that demonstrates that such a mission is feasible within a NASA probe-class mission budget.'
author:
- 'Colleen A. Wilson-Hodge'
- 'Paul S. Ray'
- Deepto Chakrabarty
- Marco Feroci
- Laura Alvarez
- Michael Baysinger
- Chris Becker
- Enrico Bozzo
- Soren Brandt
- Billy Carson
- Jack Chapman
- Alexandra Dominguez
- Leo Fabisinski
- Bert Gangl
- Jay Garcia
- Christopher Griffith
- Margarita Hernanz
- Robert Hickman
- Randall Hopkins
- Michelle Hui
- Luster Ingram
- Peter Jenke
- Seppo Korpela
- Tom Maccarone
- Malgorzata Michalska
- Martin Pohl
- Andrea Santangelo
- Stephane Schanne
- Andrew Schnell
- Luigi Stella
- Michiel van der Klis
- Anna Watts
- Berend Winter
- Silvia Zane
- 'on behalf of the LOFT Consortium, the US-LOFT SWG, and the LOFT-P collaboration'
bibliography:
- 'report.bib'
title: 'Large Observatory for x-ray Timing (LOFT-P): A Probe-class Mission Concept Study'
---
INTRODUCTION {#sec:intro}
============
*LOFT-P* is a probe-class X-ray observatory designed to work in the 2–30 keV band with huge collecting area ($>10 \times$ NASA’s highly successful *Rossi X-ray Timing Explorer (RXTE)*) and good spectral resolution ($<$260 eV). It is optimized for the study of matter in the most extreme conditions found in the Universe and addresses several key science areas including:
- [Probing the behavior of matter spiraling into black holes (BHs) to explore the effects of strong gravity and measure the masses and spins of BHs.]{}
- [Using multiple neutron stars (NSs) to measure the ultradense matter equation of state over an extended range.]{}
- [Continuously surveying the dynamic X-ray sky with a large duty cycle and high time-resolution to characterize the behavior of X-ray sources over a vast range of time scales.]{}
- [Enabling multiwavelength and multi-messenger study of the dynamic sky through cross-correlation with high-cadence time-domain surveys in the optical and radio (LSST, LOFAR, SKA pathfinders) and with gravitational wave interferometers like LIGO and VIRGO.]{}
Detailed simulations[@yellowbook; @Watts2016] have demonstrated that an order of magnitude larger collecting area than *RXTE* (i.e., $>$6 m$^2$) is required to meet these BH and NS objectives, and a previous engineering study[@Ray2011] has shown that such an instrument is too large for the Explorer (EX) class and requires a probe-class mission.
The *LOFT-P* mission concept, which has been under study in both the Europe and the US since 2010[@yellowbook; @Feroci2012; @Feroci2014; @Feroci2016], comprises two instruments. The Large Area Detector (LAD) consists of collimated arrays of silicon drift detectors (SDDs) with a 1-degree field of view and a baseline peak effective area of 10 m$^2$ at 8 keV (Fig. \[fig:loft-p\_effarea\]), optimized for submillisecond timing and spectroscopy of NSs and BHs. The sensitive Wide Field Monitor (WFM) is a 2–50 keV coded-mask imager (also using SDDs) that acts as a trigger for pointed LAD observations of X-ray transients and also provides nearly continuous imaging of the X-ray sky with a large instantaneous field of view.
\[ht\]
![ \[fig:loft-p\_effarea\] Effective area as a function of area shown for the LOFT-P LAD baseline concept. Several existing and planned missions are shown for comparison.](LOFT-P_effarea.pdf){height="5cm"}
We first presented LOFT-P as a concept, based on the ESA M3 studies of LOFT[@yellowbook], at the American Astronomical Society (AAS) High Energy Astrophysics Division (HEAD): High-Energy Large- and Medium-class Space Missions in the 2020s meeting in 2015[^2], where it was well received. It was later presented as an example probe-class mission in the NASA Physics of the Cosmos Program Analysis Group (PhysPAG) final presentation to the head of NASA’s Astrophysics Division, to demonstrate the strong community support for creation of a “probe class,” for NASA astrophysics missions that cost between \$500M and \$1B. We submitted a white paper[@Wilson2016] describing LOFT-P science and this simple assessment to NASA’s PhysPAG’s Call for White Papers: Probe-class Mission Concepts, for which 14 white papers were received[^3]. At the April 2016 HEAD meeting, NASA’s PhysPAG endorsed the option that NASA issue a ROSES solicitation for Astrophysics Probe mission concept study proposals for input to the 2020 Astrophysics Decadal Survey[^4]. In May 2016 the Advanced Concepts Office at NASA MSFC performed a preliminary study (Fig. \[fig:loft-p\_pic\]) to verify the cost of *LOFT-P* as a US-led probe-class mission and to investigate a US-led design on a US launcher, in preparation.
\[ht\]
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![ \[fig:loft-p\_pic\] LOFT-P spacecraft configuration with 122 LAD modules and 10 WFM cameras (left). This configuration fits within the volume of a Falcon 9 fairing (right). A Falcon Heavy is required to deliver LOFT-P to a 0 $\deg$ orbit from Cape Canaveral. An astronaut is added to both figures to give a sense of scale.](loft-p_pic_v2.pdf "fig:"){height="7cm"} ![ \[fig:loft-p\_pic\] LOFT-P spacecraft configuration with 122 LAD modules and 10 WFM cameras (left). This configuration fits within the volume of a Falcon 9 fairing (right). A Falcon Heavy is required to deliver LOFT-P to a 0 $\deg$ orbit from Cape Canaveral. An astronaut is added to both figures to give a sense of scale.](LOFT-P_in_Falcon9_v2.png "fig:"){height="7cm"}
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
SCIENCE GOALS AND MISSION REQUIREMENTS
======================================
Science Goals
-------------
[**Strong gravity and black hole spin.**]{} Unlike the small perturbations of Newtonian gravity found in the weak-field regime of general relativity (GR), strong-field gravity results in gross deviations from Newtonian physics and qualitatively new behavior for motion near compact objects, including the existence of event horizons and an innermost stable circular orbit (ISCO). *LOFT-P* observations will probe strong gravitational fields of NSs and BHs in a way that is complementary to gravitational wave interferometers like LIGO and VIRGO. Accretion flows and the X-ray photons they emit are “test particles” that probe the stationary spacetimes of compact objects, whereas gravitational waves carry information about the dynamical evolution of these spacetimes. As a result, *LOFT-P* observations will allow mapping the stationary spacetimes of black holes and testing the no-hair theorem. In GR, only two parameters (mass and spin) are required to completely describe an astrophysical BH, and the X-rays originating in the strong gravity regions necessarily encode information about these fundamental parameters.
*LOFT-P* observations of accreting stellar-mass BHs will be unique in providing three independent measurements of each BH spin from high-frequency quasi-periodic oscillations (HFQPOs), relativistic reflection modelling of Fe (and other) lines, and disk continuum spectra, each using techniques with differing systematic uncertainties. In those systems in which HFQPOs have already been detected with $\sim
5$% rms amplitude by *RXTE*, deeper observations with *LOFT-P* will allow detections of the 5–10 additional QPO peaks predicted by theory. This will identify their frequencies with particular linear or resonant accretion disk modes; this will be possible once a spectrum of modes is observed, instead of just a pair. *LOFT-P*’s timing capabilities can also test whether the correct spins have been obtained by reverberation mapping of the X-ray reflection in X-ray binaries and AGN (for which it will provide significantly better S/N than *Athena*).
[**Properties of ultradense matter.**]{} How does matter behave at the very highest densities? This seemingly simple question has profound consequences for quantum chromodynamics and for compact object astrophysics. The equation of state (EOS) of ultradense matter (which relates density and pressure) is still poorly known, and exotic new states of matter such as deconfined quarks or color superconducting phases may emerge at the very high densities that occur in NS interiors. This regime of supranuclear density but low temperature is inaccessible to laboratory experiments (where high densities can only be reached in very energetic heavy ion collisions), but its properties are reflected in the mass-radius ($M$-$R$) relation of NSs. Consequently, measurement of NS $M$ and $R$ is the crucial ingredient for determining the ultradense matter EOS.
*LOFT-P* will obtain $M$ and $R$ measurements by fitting energy-resolved oscillation models to the millisecond X-ray pulsations arising in a hot spot from rotating, accreting NSs. The detailed pulse shape is distorted by gravitational self-lensing, relativistic Doppler shifts, and beaming in a manner which encodes $M$ and $R$. Detailed modeling of the pulse profile can extract $M$ and $R$ separately. Measurements of both $M$ and $R$ for three or more NSs, made with $\approx$5% precision, would definitively determine the EOS of ultradense matter, while measurement of a larger number of NSs with $<$10% precision would still place strong constraints. The recently approved *NICER* mission will apply this same technique to faint rotation-powered pulsars, a different class of NSs. This is complementary to *LOFT-P* rather than duplicative. A key difference between the NSs targeted by *NICER* and *LOFT-P* is that the *NICER* targets generally rotate more slowly ($<$300 Hz) than the *LOFT-P* targets ($>$600 Hz). As a result, *NICER* observations cannot fully exploit Doppler effects to break degeneracies between $M$ and $R$, making precise and uncorrelated measurements more difficult. *NICER* will obtain precise ($<$5%) determinations of $R$ for only 1–2 sources; this is unlikely to be sufficient to solve the EOS problem, since multiple measurements are required to measure the slope of the $M$-$R$ curve and, hence, of the pressure-density relation that describes the EOS. Combining $M$-$R$ measurements from *NICER* and *LOFT-P* will triple the sample size.
The $M$-$R$ relation of neutron stars can also be probed with magnetar oscillations. Like with the HFQPOs, by dramatically improving the collecting area, enough frequencies should be found to allow mode identification. Given the relative precision of timing calibration to response matrix calibration, timing-based models should eventually allow the highest precision measurements possible.
[**Observatory science.**]{} As a flexible observatory with superb spectral-timing capabilities with wide field coverage of the sky over a broad range of timescales, *LOFT-P* will serve a large user community and make significant scientific impact on many topics in astrophysics. The LAD will study accretion physics, jet dynamics (especially in conjunction with timing studies in the infrared that will be possible on medium-sized telescopes), and disk winds (taking advantage of the high throughput and high spectral resolution which will allow very rapid detection of the turn-on of a disk wind).
The WFM’s combination of angular resolution, sensitivity, spectral resolution and [*instantaneous*]{} wide-field coverage will enable studies of black hole transients, tidal disruption events, and gamma-ray bursts too faint for current instrumentation. It will also enable instant spectroscopic follow-up of these events, as the positional accuracy will be smaller than the fields of view of modern integral field units. The WFM’s mission-long survey of the sky in Fe K$\alpha$ will be more sensitive to Compton thick AGN than *eROSITA*.
The WFM will be unique as a discovery machine for the earliest stages of supernova shock breakouts by working in the X-rays, and having the sensitivity and instantaneous field of view to have an expected detection rate of a few breakouts per year within 20 Mpc. This will allow much more rapid spectroscopic follow-up than other means of discovering supernovae, allowing crucial studies of the early stages of the explosions that can be used to probe details of the explosion mechanisms and the binarity of supernova progenitors. *LOFT-P* will be ideal for detecting and localizing X-ray counterparts to gravitational wave sources, fast radio bursts, and optical transients in the era of LSST.
Mission Requirements
--------------------
For purposes of this LOFT-P study, the science requirements were assumed to be identical to those for LOFT M3[@yellowbook]. The large effective area and good spectral resolution were driven by the need to reduce Poisson noise for relatively bright sources to access weak timing features or to gather high-quality spectra for phenomena occurring on very short timescales. Examples include, simultaneously measuring both mass and radius for several neutron star systems to 3-5%, directly observing millisecond orbital motion close to stellar mass black holes, and Fe-line tomography in AGNs to constrain the spin of the supermassive black hole.
Because many of the target sources are highly variable, and because the desired observations can only occur in particular states, the Wide Field Monitor is required. Furthermore, the observatory needs to be relatively agile, able to respond to targets-of-opportunity in order to observe sources in the desired states and to respond to outbursts of new and interesting sources relevant to the LOFT-P science.
[ll]{} Parameter & Baseline\
\
Effective Area & 9.5 m$^2$ @ 8 keV\
Energy Range & 2–30 keV\
Spectral Resolution & $<240$ eV @ 6 keV\
Deadtime & $<$ 0.1% @ 1 Crab\
Time Resolution & 10 $\mu$s\
Collimated field-of-view & 1$^\circ$ FWHM\
Sensitivity (5 $\sigma$) & 0.1 mCrab in 100 s\
\
Source Localization & 1 arcmin\
Angular Resolution & 5 arcmin\
Energy Range & 2-50 keV\
Spectral Resolution & 300 eV @ 6 keV\
Effective Area & 170 cm$^2$ (peak)\
Field of view & 4.1 steradian\
Sky Coverage & 50% of LAD accessible sky\
Sensitivity (5 $\sigma$) & 3 mCrab (50 ks)\
\
Low Earth Orbit & 550 km, $< 5 ^\circ$ inclination\
Sky visibility (Field-of-Regard) & $>$ 35% ($>$ 50% extended)\
Pointing Accuracy & 1 arcmin on 3 axis\
Pointing knowledge & 5 arcsec\
Telemetry Rate & 100 Gbit/day\
Slew Rate & 4$^\circ$/min\
SCIENCE INSTRUMENTS
===================
For purposes of the LOFT-P study, the science instruments were assumed to be identical to those described in the LOFT Yellowbook[@yellowbook]. They are described briefly below. Parameters of these instruments are listed in Table \[tab:instr\_miss\_req\].
Large Area Detector (LAD)
-------------------------
The LAD provides the capability to revolutionize the study of X-ray variability on millisecond timescales. This instrument has previously been studied and described in detail[@Zane2014]. To provide that capability, two advances are needed over past instruments: dramatically larger area and improved spectral resolution. A modular design based on Silicon Drift Detectors (SDDs) is used to achieve the large area. Each LAD module has an array of $4 \times 4$ detectors and $4 \times 4$ collimators, the module back end electronics, and the ASICs, that control the detectors and read out the digitized events. For purposes of the LOFT-P study, the design unit is a LAD module, enabling the number of modules to be a study parameter. To meet the effective area requirement, 120 modules are needed. The field-of-view of the LAD is limited to 1 $\deg$ by lead glass micro-channel plate collimators. On the back side of each module, there will be a radiator for passive cooling and a shield to reduce the background. For every 25–30 LAD modules, there is a single panel back end electronics unit. Mass and power assumptions for the LAD modules for purposes of the LOFT-P study are listed in Table \[tab:assumptions\]. In the LOFT-P configuration, there are 122 LAD modules, vs. 126 in the LOFT M3 configuration[@yellowbook].
Wide Field Monitor (WFM)
------------------------
The WFM provides broad sky coverage to monitor potential LAD targets for transitions into desired observational states and provides considerable science in its own right. The WFM has previously been studeid and described in detail[@Brandt2014]. The WFM images the sky using coded mask cameras with solid-state class energy resolution, through the use of SDD detectors, the same detectors as are used for the LAD. Since the SDDs provide accurate positions in only one direction, pairs of orthogonal cameras are used to provide accurate source positions. The cameras have a Tungsten mask with a 25% open area to optimize sensitivity for weaker sources. Like the LOFT M3 design[@yellowbook], LOFT-P also includes 5 pairs of WFM cameras. Each camera pair has an effective field of view of $70^\circ \times 70^\circ$. Mass and power assumptions for the WFM are listed in Table \[tab:assumptions\].
MISSION DESIGN
==============
In this section, we describe the results of the LOFT-P mission concept study, a one month study performed by NASA MSFC’s Advanced Concepts Office (ACO). MSFC’s ACO is an engineering design facility for conceptual and preliminary design and analysis of launch vehicles, in-space vehicles and satellites, surface systems, human systems, and overall mission architecture concepts. The ACO is unique among the NASA preliminary design facilities because they have participated in development of every type of spacecraft flown by NASA. The team has the expertise to perform end-to-end analysis of new and innovative missions and vehicle systems. Details of ACO’s capabilities, history, and people are provided here[^5]. Past and current studies of astrophysics missions include Hubble, Chandra, and X-ray Surveyor. The goal of this study was to take a preliminary look at whether or not a US-led LOFT-P mission would fit within the \$500M-\$1B Probe class (excluding launch vehicle).
Assumptions and Requirements
----------------------------
A new spacecraft was designed to meet the requirements listed in Tables \[tab:instr\_miss\_req\] and \[tab:assumptions\]. Table \[tab:assumptions\] also lists key information about the science instruments based on the LOFT M3 study[@yellowbook].
[ll]{} Parameter & Required Value (Goal)\
\
Estimated Launch Year & 2027-2030\
Mission duration & 4 (5) years\
Science data downlink & 6.7 Gbits (14 Gbits) per orbit\
Orbit & LEO, Minimizing time in SAA,\
& 600 km upper limit,\
& $<5^\circ$ inclination\
\
Basic Mass & 6.05 kg\
Power & 8.25 W\
Thermal requirement & EoL LAD detector temperature\
& requirement of $-10^\circ$C over\
& nominal FoR; Up to $+11^\circ$ C for\
& extended FoR\
Alignment & co-aligned within 3 arcmin\
Quantity & 120 modules (minimum)\
\
Basic Mass & 9.29 kg\
Power & 7.56 W\
Quantity & 10 cameras\
Mission Analysis
----------------
The orbit selection was driven by minimizing three factors: passage through the South Atlantic Anomaly (SAA), radiation at higher altitudes, and atmospheric drag. Based on the ESA M3 study[@yellowbook], an orbital inclination of $<5^\circ$ was required. To avoid higher radiation exposures, the altitude must be no greater than 600 km. Station-keeping requirements, driven mostly by atmospheric drag, determined the minimum altitude. The NASA Debris Assessment Software[@Opiela2007] (DAS) v2.0.2 was used to estimate the orbital altitude decay rate. Imposing a limit of 10 km degradation in altitude before raising the orbit back to the initial value resulted in a minimum recommended initial altitude of 550 km. Lower altitudes are possible, but station-keeping requirements increase substantially below 500 km, and do not offer any increase in payload mass capability to the launch vehicle. Analytical Graphics Systems Tool Kit (STK)[^6] was used to estimate ground station contact times, and determine the number of stations needed. Results indicated that in order to meet the daily science data download requirements, two ground stations are required. Details are provided in the Communications subsection below.
Launch Vehicle
--------------
Based on the ESA M3 study[@yellowbook], a launch mass of 4070 kg was required. According to the NASA Launch Services Program (LSP) website[^7] assuming launch from Cape Canaveral Air Force Station, the maximum launch mass that can be delivered by a Falcon 9 to a 500–600 km orbit with an inclination of 5$^\circ$ is 3705 kg, meaning that a Falcon 9 has insufficient performance to deliver LOFT-P to the required orbital altitude and inclination. NASA LSP stated that it is reasonable to assume that a Falcon Heavy or a similar vehicle will be on contract and available by the late 2020s. Since NASA LSP was not able to provide performance estimates, ACO estimated expected performance using the performance degradation from an SLS Block 1B going to a 28.5$^\circ$ vs a 0$^\circ$ orbit, and from the Falcon 9 to those same orbits. Applying this performance degradation to the advertised Falcon Heavy capability to 28.5$^\circ$ resulted in an estimated capability of 12,200 kg to an equatorial orbit. Applying the Falcon 9 performance degradation ratio resulted in an estimated worst case Falcon Heavy payload capability of 5630 kg to an equatorial orbit. Since launch vehicles with boosters lose less performance when going to lower inclinations, the study team feels that the 5630 kg estimate is much too conservative, with the actual capability being closer to the 12,200 kg estimate. Therefore, analysts used the average of the two values, and estimate a capability of 8900 kg to an equatorial orbit, easily placing LOFT-P into the desired location.
Spacecraft Configuration
------------------------
The large payload dynamic envelope in the Falcon 9 fairing[^8] enabled a monolithic design for LOFT-P rather than the deployable design adopted for ESA M3 and M4 LOFT[@Feroci2012; @Feroci2014]. This design accommodated 122 LAD modules, only slightly fewer than the 126 modules on LOFT M3, and the full 10-camera WFM, identical to LOFT M3. Additional WFM cameras could be added to the configuration if cost and telemetry allow it. The overall configuration is conservative and does allow room for component growth and for extra subsystem components to be added that were not analyzed in this study. Table \[tab:mel\] gives the master equipment list for this design. Masses include a 20% mass growth allowance for structures, power, communication, command and data handling, guidance, navigation, and control, and for the science instruments. For the thermal control system, a 30% mass growth allowance is included. For the propulsion system (excluding propellant) mass growth allowances of 5-25% are used, depending on TRL and knowledge of specific components to be used. Mass growth allowances are based on AIAA standards[^9]. The total wet mass is the combined total of the spacecraft dry mass, science instruments, and propellant.
Equipment Mass (kg)
---------------------------------------------- -----------
Structures (incl. LAD frame panel) 2160
Thermal Control 300
Power 190
Avionics, Control, Comm. 380
Propulsion 150
Mass growth allowance[^10] 530
Spacecraft DRY MASS 3710
LAD (122 Modules) 740
WFM (10 cameras, ICU, & harness) 100
Mass growth allowance (20%) 170
Total Science Instrument Mass[^11] 1010
Total Dry Mass 4720
Propellant (incl. 20% mass growth allowance) 940
Total Wet Mass 5660
: Master equipment list and mass budget for LOFT-P\[tab:mel\]
### Spacecraft Structure
A finite element model was used to size the LOFT-P spacecraft and bus. MSC Patran was used to pre- and post-process the finite element model. MSC Nastran was used as the finite element model solver. Collier Research Hypersizer was used for the model optimization and sizing checks. Structural assessment includes strength, stability, and stiffness checks. Falcon Heavy envelope loads (launch/ascent) of 6 g axial and 2 g lateral were assesses. A constraint was applied at the LOFT-P Bus to payload adapter interface. The frame will be manufactured using Quasi-Isotropic IM-7 8552 composite laminates. This monolithic structure sizing is driven by stiffness. Structural deflections are well within the dynamic payload volume during launch and ascent. LAD mis-alignment due to non-uniform thermal loading can exceed the requirement of 3 arcmin if the thermal gradient is larger than $\sim 17 ^{\circ}$ C. The LOFT-P normal modes are low with first torsion at approximately 8 Hz.
### Communications System
An X-band system with a fixed omnidirectional antenna is used for the downlink data system. A fixed antenna is more reliable and reduces mission risk as compared to a gimballed antenna. A ground link analysis based on link times and daily accesses was performed to determine the best selection and number of required ground stations at 0, 5, and 10 degree inclinations. South Point Hawaii, Kourou, Guam, and Malindi were considered. South Point had no capability for a 0$^\circ$ orbit. Downlink averages were 3.8–5.6, 3.3–5.7, and 4.5–5.3 Gbits/orbit for 0, 5, and 10 degree, respectively, assuming a maximum X-band downlink rate of 10 Mbps, indicating that no single ground station gave sufficient time to download the required 6.7 Gbits/orbit of science data, and that 3–4 ground stations were required for the desired goal of 14 Gbits/orbit of science data. Initial investigations were started into using TDRSS, which allows downlink rates up to 300 Mbps, but requires a much higher power transmitter than is incorporated into the current LOFT-P design. Using TDRSS during launch and start-up operations is desirable, but further investigation into using TDRSS for normal operations is needed.
The communications system also includes a secondary VHF LOFT burst alert system with components based upon Orion EVA system heritage. This system will provide rapid alerts of transient events, e.g., gamma-ray bursts and X-ray bursts, to ground-based VHF receivers.
### Power Systems
The overall power demand, including the spacecraft, science instrumentation and 30% mass growth allowance, is 2068W for the LOFT-P Falcon Heavy configuration. The power system supplies all of this demand. Power is generated by two conventional, folding, rigid panel solar arrays, 7.2 m$^2$ each, with a conversion efficiency of 25% (beginning of life) and a total end of life power output of 3670W. The solar arrays were sized as folding rigid panel arrays using physics-based sizing relations based on manufacturers cell data. The power electronics sizing is based on flight heritage boards integrated into existing space qualified enclosures. Cabling is estimated using spacecraft dimensions and physics-based sizing tools. Cables are sized for a 2% loss. Power requirements are aggregated from all other subsystems with a 30% design margin per AIAA requirements. Energy storage is provided by six primary batteries. The power system mass (excluding mass growth allowance) is 191 kg.
### Avionics and GN&C
Two fully redundant Proton2x-Box flight computers from Space Micro are the core of the avionics system. These computers combine a commercial product set of building blocks, including a Proton400k processor, a power supply, DIO flash, up to 250 Gbit data storage, and 150 Mbs data rate transmission.
Attitude knowledge is achieved using a redundant pair of Ball Aerospace star trackers and Northrop Grumman inertial measurement units (IMUs). The star trackers provide 4“ of accuracy, meeting the 5” mission requirement. Both the star trackers and IMUs are at or above TRL 8.
Pointing requirements for this spacecraft are modest, with a required pointing accuracy of 1 arcmin (3 $\sigma$) on 3-axis. Pointing stability is frequency dependent. The spacecraft will be normally inertially pointed, with uninterrupted observation times of about 1 ks to 100 ks (hours to days). Slew speeds will be about 2$^\circ$/min for normal slewing, with faster slews of 4$^\circ$/min for target-of-opportunity observations. For this study, an operational mode of slewing about the Y or Z axis was assumed. Because of the large mass and surface area of the system, damping launch tip-off rates is challenging, driving actuator sizing to unreasonably large sizes. Therefore, use of thrusters is recommended to damp tip-off rates. In our analysis, actuator sizing did not use tip-off rates.
Three axis drives were needed for 3-axis control, plus an additional one for single-fault tolerance. Control moment gyroscopes (CMGs) were selected because no reaction wheels were found that provided the required torque. The current design includes a pyramid configuration of 4 Ball Aerospace CMGs, with 129 Nms momentum storage, 2.64 Nm torque (up to 6.1 Nm as a set) to allow slew rates up to 4 $\deg$/min. A set of 3 Cayuga Astronautics L-series Magnetic torquers are used for continuous momentum unloading with 100% margin, excluding tip-off rates.
### Thermal Control System
Thermal control of the LOFT-P spacecraft will utilize passive high-TRL components such as MLI, white paint, passive radiators, and heaters to maintain spacecraft subsystem components within acceptable temperature ranges. A simplified model was developed in Thermal Desktop. The model was based on the LAD panel frame and spacecraft bus structures. A simplified thermal model of the LAD modules and front end electronics, based on the ESA M3 study[@yellowbook], was incorporated into the LOFT-P thermal model. The analysis estimated the average temperature of the structural panel frame across the Field of Regard (FoR) and was used to size the thermal control components for the spacecraft. Hot and cold cases were studied with Sun beta angles for 0, 5, and 10 deg inclinations and 600 km orbits. Sun avoidance angles of 0 deg to 90 deg were also analyzed to evaluate the feasibility of meeting the LAD temperature requirements. A LAD detector temperature requirement of -10C, over the nominal FoR, is the driving requirement that influences thermal control. The FoR of the LAD constrains the solar flux seen by the LAD modules. The LOFT-P concept uses a local radiator design to lower the overall panel temperature without recourse to shading from sunlight. Analysis shows that the LAD structural panel average temperature is $<-10$C at a sun aspect angle of 30 deg, which compares well to the previous ESA designs. The LOFT-P concept provides additional conservatism due to the ability to shade the LAD modules with the primary LAD panel structure as well as mass margin for local sun shading if necessary. However, further analysis of the LAD modules and electronics needs to be performed to verify the overall thermal control approach. The WFM is protected from direct sunlight with a sun shield (as shown in Fig. \[fig:loft-p\_pic\]) to avoid deformations of the coded mask[@Brandt2014]. The model was used to estimate the mass of a conceptual thermal control system for the spacecraft, propulsion system, and instruments. The estimated mass was 298 kg, not including 30% mass growth allowance. Total estimated power of the thermal control system is 50 W.
### Propulsion System
The propulsion system includes TRL 9+ hardware components and heritage derived hardware. The propulsion system’s primary purpose is to de-orbit the spacecraft at the end of the mission, including 5 reentry maneuvers, and to perform orbit maintenance maneuvers. Secondary purposes include launch vehicle insertion error corrections, tip-off damping, collision avoidance, and momentum unloading. A simple monopropellant blowdown system with maximum off-the-shelf components, is selected for this task. The system consists of four PSI-ATK 80514-1 tanks that are loaded with hydrazine and nitrogen pressurant. The thruster configuration comprises 4 pods, each containing three Aerojet MR-104 attitude control system thrusters (2N). One pod also includes an orbit adjust thruster (440 N), an Aerojet MR-111E. The system is single fault tolerant at the component level, two fault tolerant to failure at the system level. The system provides a total delta-V (with margin) of 298 m/s. Margins are 25% for launch vehicle insertion errors, orbit maintenance, collision avoidance, and momentum unloading. For reentry, for which the delta-V is well determined, a 10% margin is assumed. Tank sizing allows up to 378 m/s delta-V. The predicted dry mass of the system without contigency is 154 kg. Margins are low (5%) for the high-TRL off-the-shelf components such as the hydrazine tanks, the thrusters, and the isolation latch valve. Propellant dominates the mass of the system, with 894 kg of hydrazine and 46 kg of nitrogen pressurant including 20% mass growth allowance.
### Preliminary Cost Estimate
Costs for the LOFT-P mission were estimated using the following parametric models PCEC (Project Cost Estimating Capability), SEER-H, NICM (NASA Instrument Cost Model), and MOCET (Mission Operations Cost Estimating Tool) for ground data systems/mission operations systems costs. Two cost estimates were performed during the study. The first was based on the ESA M3 study of LOFT[@yellowbook]. The second was based on the MSFC Advanced Concepts Office study of LOFT-P. Both were assumed to be NASA-led for cost assumptions. The NASA Standard Level WBS for space flight projects was assumed, based on NPR 71020.5E: NASA Space Flight Program and Project Management Requirements, Appendix H. Costs were estimated in FY2016 dollars, with a fee of 12.5%, and cost reserves at 35%. Launch vehicle costs were excluded from both estimates. Mass with contingencies was used. MOCET was used to calculate all phase E costs, based on the Fermi mission,
The costs for both concepts assumed the following for the LAD: 125 detector modules, 5 Panel Back end Electronics, 2 ICUs. Costs were based on one development and production of 125 modules. For the WFM, the model assumed 10 cameras, 2 WFM ICUs. Costs were based on one development and production of 10 cameras. Average modification on the electronic components of both instruments was assumed, given the considerable development that has already taken place in Europe. For both concepts, major modification for the spacecraft structure, average modification for the C&DH system, and minor modification for the electrical, thermal, propulsion, and communication systems were assumed. The same phase A-D schedules taken from[@yellowbook] were used. For consistency with ESA estimates, a 3 year mission (5 year goal) was assumed for LOFT M3, while for the LOFT-P mission was assumed to have a duration of 4 years (5 year goal). Using these cost models, our preliminary cost estimates show a 15-25% margin with respect to the \$1B Probe-class cost cap, including 35% cost reserve. Both cost estimates include full life cycle costs, including labor, instruments, spacecraft, mission operations, and ground data systems. Both cost estimates compare well with other astrophysics missions in the ONCE database, including Fermi.
FUTURE WORK & CONCLUSIONS
=========================
The LOFT-P concept complements existing LOFT designs and bounds options. A single panel was chosen for LOFT-P to reduce complexity, but requires increased mass to meet stiffness and stability requirements. The large panel manufacturing and mass may offset the reduced complexity. Future studies need to trade a single LAD panel vs a multipanel deployed configuration, including analysis for low frequency vibrations, to verify impacts from LAD module assembly and alignment, and to assess the impacts on overall spacecraft maneuverability and stability. The large moment of the monolithic design drives the need for thrusters to control tip-off and the need for CMGs, which limits fast slew rates and would likely be a major driver in a future trade study of a single panel vs multipanel design. Further studies are also needed for the fast slew rate, including considering feasibility of using thrusters for fast slews, which would likely allow for the use of reaction wheels instead of the more-expensive CMGs for attitude control. Cost fidelity can also be improved by refining the mass basis and investigating instrument/component modeling, including definition of heritage/high TRL components for model inputs and conducting a sensitivity analysis.
The LOFT-P study has shown that a LOFT-like mission is feasible as a probe-class mission. The estimated cost of the monolithic LOFT-P design is similar to the multipanel LOFT M3 design. This study has positioned LOFT-P well for a more detailed concept study in preparation for the 2020 Astrophysics Decadal Survey. LOFT-P science is timely. With its highly capable LAD and WFM, LOFT-P will address fundamental physics, and time-domain science.
NRL‘s work on X-ray astrophysics is funded by the Chief of Naval Research (CNR). The LOFT-P study was funded internally by NASA MSFC. The work of the MSSL-UCL and Leichester SRC on the LOFT-LAD project has been supported by the UK Space Agency. The work of the ICE (CSIC-IEEC) on the LOFT-WFM project has been supported by funds from the Spanish MINECO.
[^1]: [http:// www.isdc.unige.ch/loft/index.php/loft-team/community-members](http:// www.isdc.unige.ch/loft/index.php/loft-team/community-members)
[^2]: <https://files.aas.org/head2015_workshop/HEAD_2015_Colleen_Wilson-Hodge.pdf>
[^3]: <http://pcos.gsfc.nasa.gov/physpag/whitepapers.php>
[^4]: <http://pcos.gsfc.nasa.gov/physpag/meetings/head-apr2016/TH02_Bautz_HEAD_PCOS_update_Apr2016_v2.pdf>
[^5]: <http://www.nasa.gov/centers/marshall/capabilities/adv_capabilities.html>
[^6]: <http://www.agi.com>
[^7]: <http://www.nasa.gov/centers/kennedy/launchingrockets/index.html>
[^8]: <http://www.spacex.com/sites/spacex/files/falcon_9_users_guide_rev_2.0.pdf>
[^9]: <https://www.aiaa.org/StandardsDetail.aspx?id=3918>
[^10]: Per AIAA standards: 30% for thermal control system; 7% for\
propulsion system (very high TRL); 20% for all other systems
[^11]: Per ESA M3 Study
| {
"pile_set_name": "ArXiv"
} |
Cuprate high-temperature superconductors (HTS) are layered anisotropic materials. Therefore the electrodynamic problem of the magnetic field penetration depth in HTS in the low-field limit is characterized by two length parameters, namely, $\lambda_{ab}$ controlled by screening currents running in the CuO$_2$ planes (in-plane penetration depth) and $\lambda_c$ due to currents running in the direction perpendicular to these planes (out-of-plane or $c$-axis penetration depth). The temperature dependence of the penetration depth in HTS is largely determined by the superconductivity mechanism. It is known (see, e.g., Ref. [@Tru1] and references therein) that $\Delta\lambda_{ab}(T)\propto T$ in the range $T<T_c/3$ in high-quality HTS samples at the optimal level of doping, and this observation has found the most simple interpretation in the $d$-wave model of the high-frequency response in HTS [@Scal]. Measurements of $\lambda_c(T)$ are quoted less frequently than those of $\lambda_{ab}(T)$. Most of such data published by far were derived from microwave measurements of the surface impedance of HTS crystals [@Shib1; @Mao; @Kit1; @Bon1; @Jac1; @Shib2; @Srik; @Kit2; @Hos]. There is no consensus in literature about $\Delta\lambda_c(T)$ at low temperatures. Even in reports on low-temperature properties of high-quality YBCO crystals, which are the most studied objects, one can find both linear, $\Delta\lambda_c(T)\propto T$ [@Mao; @Srik], and quadratic dependences [@Hos] in the range $T<T_c/3$. In BSCCO materials, the shape of $\Delta\lambda_c(T)$ depends on the level of oxygen doping: in samples with maximal $T_c\simeq 90$ K $\Delta\lambda_c(T)\propto T$ at low temperatures [@Jac1; @Shib2]; at higher oxygen contents (overdoped samples) $T_c$ is lower and the linear function $\Delta\lambda_c(T)$ transforms to a quadratic one [@Shib2]. The common feature of all microwave experiments is that the change in the ratio $\Delta\lambda_c(T)/\lambda_c(0)$ is smaller than in $\Delta\lambda_{ab}(T)/\lambda_{ab}(0)$ because in all HTS $\lambda_c(0)\gg\lambda_{ab}(0)$. The length $\lambda_c(0)$ is especially large in BSCCO crystals, $\lambda_c(0)>10$ $\mu$m and, according to some estimates, it ranges up to $\sim 500$ $\mu$m. The large spread of $\lambda_c(0)$ is caused by two factors, namely, the poor accuracy of the techniques used in determination of $\lambda_c(0)$ and effects of local and extended defects in tested samples, whose range is of order of 1 mm and comparable to both $\lambda_c$ and total sample dimensions.
Recently we suggested [@Nic] a new technique for determination of $\lambda_c(0)$ based on the measurements of the surface barrier field $H_J(T)\propto 1/\lambda_c(T)$ at which Josephson vortices penetrate into the sample. The field $H_J$ corresponds to the onset of microwave absorption in the locked state of BSCCO single crystals. This paper suggests an alternative technique based on comparison between microwave measurements of BSCCO crystals aligned differently with respect to ac magnetic field and a numerical solution of the electrodynamic problem of the magnetic field distribution in an anisotropic plate at an arbitrary temperature. Moreover, since $\lambda_c(0)$ in BSCCO single crystals is relatively large, we managed to determine $\lambda_c(T)$ from the temperature dependences of ac-susceptibility and compare these measurements to results of microwave experiments.
Single crystals of BSCCO were grown by the floating-zone method [@Tam] and shaped as rectangular platelets. This paper presents measurements of two BSCCO samples with various levels of oxygen doping. The first sample (\#1), characterized by a higher critical temperature, $T_c\approx 90$ K (optimally doped), has dimensions $a\times b\times c\simeq 1.5\times 1.5\times 0.1$ mm$^3$ ($a\approx b$). The second (\#2, $a\times b\times c\simeq 0.8\times 1.8\times 0.03$ mm$^3$) is slightly overdoped ($T_c\approx 84$ K).
When measuring the ac-susceptibility $\chi=\chi'-i\chi''$, we placed a sample inside one of two identical induction coils. The coils were connected to one another, and the out-of-phase and in-phase components of the imbalance signal were measured at a frequency of $10^5$ Hz. These components are proportional to the real and imaginary parts of the sample magnetic moment $M$, respectively: $M=\chi vH_0$, where $v$ is the sample volume and $H_0$ is the ac magnetic field amplitude, which was within 0.1 Oe in our experiments.
Figure 1 shows temperature dependences $\chi'(T)/|\chi'(0)|$ in
sample \#1 for three different sample alignments with respect to the ac magnetic field: the transverse (T) orientation, ${\bf H}_{\omega}\parallel {\bf c}$, (the inset on the left of Fig. 1), when the screening current flows in the $ab$-plane (full circles); in the longitudinal (L) orientation, ${\bf H}_{\omega}\perp {\bf c}$, (the inset on the right of Fig. 1, ${\bf H}_{\omega}$ is parallel to the $b$-edge of the crystal), when currents running in the directions of both CuO$_2$ planes and the $c$-axis are present (up triangles); in the L-orientation, ${\bf H}_{\omega}\perp {\bf c}$, whose difference from the previous configuration is that the sample is turned around the $c$-axis through $90^\circ$ (down triangles). Fig. 1 clearly shows that at $T<T_c$ $\chi'_{ab}(T)$ is notably smaller in the T-orientation than $\chi'_{ab+c}(T)$ in the L-orientation (the subscripts of $\chi'$ denote the direction of the screening current). The coincidence of $\chi'_{ab+c}(T)$ curves at ${\bf H}_{\omega}\perp {\bf c}$ and the small width of the superconducting transition at ${\bf H}_{\omega}\parallel {\bf c}$ ($\Delta T_c< 1$ K) indicate that the quality of the tested sample \#1 is fairly high. This is supported by precision measurements of surface impedance $Z_s(T)=R_s(T)+iX_s(T)$ of sample \#1 at frequency $f=9.4$ GHz in the T-orientation, which are plotted in Fig. 2. The measurement technique was described in detail elsewhere [@Tru1]. It applies to both surface impedance components $R_s(T)$ and $X_s(T)$:
$$R_s=\Gamma_s\,\Delta(1/Q),\qquad X_s=-2\,\Gamma_s\,\delta f/f,\label{RX}$$
Here $\Gamma_s=\omega \mu_0\int_V H^2_{\omega}dV /[\int_S
H_s^2 dS]$ is the sample geometrical factor ($\omega=2\pi f$, $\mu_0=4\pi\cdot10^{-7}$ H/m, $V$ is the volume of the cavity, $H_{\omega}$ is the magnetic field generated in the cavity, $S$ is the total sample surface area, and $H_s$ is the tangential component of the microwave magnetic field on the sample surface); $\Delta (1/Q)$ is the difference between the values 1/Q of the cavity with the sample inside and empty cavity; $\delta f$ is the frequency shift relative to that which would be measured for a sample with perfect screening, i.e., no penetration of the microwave fields. In the experiment we measure the difference $\Delta f(T)$ between resonant frequency shifts with temperature of the loaded and empty cavity, which is equal to $\Delta f(T)=\delta f(T)+f_0$, where $f_0$ is a constant [@exp]. The constant $f_0$ includes both the perfect-conductor shift and the uncontrolled contribution caused by opening and closing the cavity. In HTS single crystals, the constant $f_0$ can be directly derived from measurements of the surface impedance in the normal state; in particular, in the T-orientation $f_0$ can be derived from the condition that the real and imaginary parts of the impedance should be equal above $T_c$ (normal skin-effect). In Fig. 2 $R_s(T)=X_s(T)$ at $T\ge T_c$, and its temperature dependence is adequately described by the expression $2R_s^2(T)/\omega
\mu_0=\rho(T)=\rho_0+bT$ with $\rho_0\approx 13$ $\mu\Omega\cdot$cm and $b\approx 0.3$ $\mu\Omega\cdot$cm/K. Given $R_s(T_{\rm c})=\sqrt{\omega
\mu_0 \rho(T_{\rm c})/2}\approx 0.12$ $\Omega$, we obtain the resistivity $\rho(T_{\rm c})\approx 40$ $\mu\Omega\cdot$cm. The insets to Fig. 2 show $R_s(T)$ and $\lambda (T)=X_s(T)/\omega\mu_0$ for $T<0.7\,T_{\rm c}$ plotted on a linear scale. The extrapolation of the low-temperature sections of these curves to $T=0$ yields estimates of $\lambda_{ab}(0)\approx 2600$ Å and the residual surface resistance $R_{\rm res}\approx 0.5$ m$\Omega$. $R_{\rm res}$ is due to various defects in the surface layer of the superconductor and it is generally accepted that the lower the $R_{{\rm res}}$, the better the sample quality. The above mentioned parameters of sample \#1 indicate that its quality is fairly high. In the T-orientation, linear functions $\Delta R_s(T)$ and $\Delta\lambda_{ab}(T)$ in the low-temperature range were previously observed in optimally doped BSCCO crystals at a frequency of about 10 GHz [@Jac1; @Shib2; @Lee]. In the slightly overdoped sample \#2 we also observed $\Delta\lambda_{ab}(T)$, $\Delta R_s(T)\propto T$ at low temperature, moreover, the measurement $R_{\rm {res}}\approx 120$ $\mu\Omega$ is, to the best of our knowledge, the lowest value ever obtained in BSCCO single crystals.
In both superconducting and normal states of HTS, the relation between electric field and current density is local: $j=\hat\sigma E$, where the conductivity $\hat\sigma$ is a tensor characterized by components $\sigma_{ab}$ and $\sigma_c$. In the normal state, ac field penetrates in the direction of the $c$-axis through the skin depth $\delta_{ab}=\sqrt{2/\omega\mu_0\sigma_{ab}}$ and in the CuO$_2$ plane through $\delta_c=\sqrt{2/\omega\mu_0\sigma_c}$. In the superconducting state all parameters $\delta_{ab}$, $\delta_c$, $\sigma_{ab}=\sigma'_{ab}-i\sigma''_{ab}$, and $\sigma_c=\sigma'_c
-i\sigma''_c$ are complex. In the temperature range $T<T_c$, if $\sigma'\ll \sigma''$, the field penetration depths are given by the formulas $\lambda_{ab}=\sqrt{1/\omega\mu_0\sigma''_{ab}}$, $\lambda_c=\sqrt{1/\omega\mu_0\sigma''_c}$. In the close neighborhood of $T_c$, if $\sigma'\agt \sigma''$, the decay of magnetic field in the superconductor is characterized by the functions $\rm {Re}\,(\delta_{ab})$ and $\rm {Re}\,(\delta_c)$, which turn to $\delta_{ab}$ and $\delta_c$, respectively, at $T\ge T_c$.
In the L-orientation of BSCCO single crystals at $T<0.9\,T_c$ the penetration depth is smaller than characteristic sample dimensions. If we neglect the anisotropy in the $ab$-plane and the contribution from $ac$-faces (see the inset to Fig. 1), which is a factor $\sim c/b$ smaller than that of the $ab$-surfaces, the effective impedance $Z_s^{ab+c}$ in the L-orientation can be expressed in terms of $Z_s^{ab}$ and $Z_s^c$ averaged over the surface area [@Tru1; @Kit1] (the superscripts of $Z_s$ denote the direction of the screening current). Thus, given measurements of $\Delta\lambda_{ab}(T)=\Delta X_s^{ab}(T)/\omega\mu_0$ in the T-orientation and of the effective value $\Delta\lambda_{ab+c}(T)=\Delta
X_s^{ab+c}(T)/\omega\mu_0$ in the L-orientation, we obtain $$\Delta\lambda_c=\left[(a+c)\,\Delta\lambda_{ab+c}-
a\,\Delta\lambda_{ab}\right]/c~.
\label{DL}$$
This technique for determination of $\Delta\lambda_c(T)$ was used in microwave experiments [@Shib1; @Mao; @Kit1; @Bon1; @Jac1; @Shib2; @Srik] at low temperatures, $T<T_c$. Even so, it cannot be applied to the range of higher temperatures because the size effect plays an important role. Really, at $T>0.9\,T_{\rm c}$ the lengths $\lambda_c$ and $\delta_c$ are comparable to the sample dimensions. In order to analyze our measurements in both superconducting and normal states of BSCCO, we used formulas [@Gou] for field distributions in an anisotropic long strip ($b\gg{a,c}$) in the L-orientation. These formulas neglect the effect of $ac$-faces of the crystal, but take account of the size effect. In addition, in a sample shaped as a long strip, there is a simple relation between its surface impedance components and ac-susceptibility, which is expressed in terms of parameter $\mu$ introduced in Ref. [@Gou]: $$\Delta\,(1/Q)-2i\,\delta f/f=i \gamma \mu v/V,\qquad \chi=-1+\mu,
\label{QF}$$ where $\gamma=VH_0^2/[\int_V H^2_{\omega}dV]=10.6$ is a constant characterizing our cavity [@Tru1]. At an arbitrary temperature, the complex parameter $\mu=\mu'-i\mu''$ is controlled by the components $\sigma_{ab}(T)$ and $\sigma_c(T)$ of the conductivity tensor: $$\mu={8 \over \pi^2}\sum_n {1\over n^2}\left\{
{\tan(\alpha_n) \over \alpha_n}+
{\tan(\beta_n) \over \beta_n}
\right\},
\label{MU}$$ where the sum is performed over odd integers $n>0$, and $$\begin{aligned}
\alpha_n^2=-{a^2 \over \delta_c^2}
\left({i\over 2}+{\pi^2 \over 4}{\delta_{ab}^2 \over c^2}n^2 \right),\,
\beta_n^2=-{c^2 \over \delta_{ab}^2}
\left({i\over 2}+{\pi^2 \over 4}{\delta_c^2 \over a^2}n^2 \right).
\nonumber\end{aligned}$$
In the superconducting state at $T<0.9\,T_c$ we find that $\lambda_{ab}\ll c$ and $\lambda_c\ll a$. In this case, we derive from Eq. (\[MU\]) a simple expression for the real part of $\mu$: $$\mu'=1+\chi'={2\lambda_c \over a}+ {2\lambda_{ab} \over c}~.
\label{CH}$$
One can easily check up that in the range of low temperatures the change in $\Delta\lambda_{\rm c}(T)$ prescribed by Eq. (5) is identical to Eq. (2). Figure 3 shows measurements of $\Delta\lambda_c(T)$ in sample \#1 (circles) and sample \#2 (squares) at $T<0.9\,T_c$. The
open symbols plot low-frequency measurements obtained in accordance with Eq. (5), the full symbols plot microwave measurements processed by Eq. (2). Agreement between measurements of sample \#2 (lower curve) is fairly good, but in fitting together experimental data from sample \#1 (upper curve) we had to divide by a factor of 1.8 all $\Delta\lambda_c(T)$ derived from measurements of ac-susceptibility using Eq. (5). The cause of the difference between $\Delta\lambda_c(T)$ measured in sample \#1 at different frequencies is not quite clear. We rule out a systematic experimental error that could be caused by misalignment of the sample with respect to the ac magnetic field in the coil because (i) curves of $\Delta\lambda_c(T)$ were accurately reproduced when square sample \#1 was turned through an angle of $90^\circ$ in the L-orientation, and (ii) a small tilt of this sample with respect to the magnetic field generated by the coil would lead to a larger difference (more than a factor of 1.8) between the two sets of experimental data. It seems more plausible that Eqs. (2) and (5), which neglect the contribution of $ac$-faces, yield inaccurate results concerning sample \#1: its $ac$-faces, which have a notable area (sample \#1 is thick), can host a lot of defects (for example, those of the capacitive type), and the latter can affect the character of field penetration as a function of frequency.
The curves of $\Delta\lambda_c(T)$ at $T<0.5\,T_c$ plotted in Fig. 3 are almost linear: $\Delta\lambda_c(T)\propto T$. The inset to Fig. 3 shows the low-temperature section of the curve of $\Delta\lambda_c(T)$ in sample \#1. Its slope is 0.3 $\mu$m/K and equals that from Ref. [@Shib2]. Note also that changes in $\Delta\lambda_c(T)$ are smaller in the oxygen-overdoped sample \#2 than in sample \#1.
We can estimate $\lambda_c(0)$ by substituting in Eq. (1) $\delta f(0)$ obtained by comparing of $\Delta(1/Q)$ and $\Delta f=\delta f-f_0$ measurements taken in the T- and L-orientations to numerical calculations by Eqs. (3) and (4), which take account of the size effect in the high-frequency response of an anisotropic crystal. The procedure of comparison for sample \#1 is illustrated by Fig. 4. Unlike the case of the T-orientation, the measured temperature dependence of $\Delta(1/Q)$
in the L-orientation deviates from $(-2\Delta f/f)$ owing to the size effect. Using the measurements of $R_s=\sqrt{\omega\mu_0/2\sigma_{ab}}$ at $T>T_c$ in the T-orientation (Fig. 2) for determination of $\sigma_{ab}(T)$, alongside the data on $\Delta(1/Q)$ in the L-orientation (open squares in Fig. 4), from Eqs. (3) and (4) we obtain the curve of $\rho_c(T)=1/\sigma_c(T)$ shown in the right-hand inset to Fig. 4. Further, using the functions $\sigma_c(T)$ and $\sigma_{ab}(T)$, we calculate $(-2\delta f/f)$ versus temperature for $T>T_c$, which is plotted by the solid line in Fig. 4. This line is approximately parallel to the experimental curve of $-2\Delta f/f$ in the L-orientation (open circles in Fig. 4). The difference $-2(\delta f-\Delta f)/f$ yields the additive constant $f_0$. Given $f_0$ and $\Delta f(T)$ measured in the range $T<T_c$, we also obtain $\delta f(T)$ in the superconducting state in the L-orientation. As a result, with due account of $\lambda_{ab}(T)$ (the inset to Fig. 2), we derive from Eqs. (3) and (5) $\lambda_c(0)$, which equals approximately 150 $\mu$m in sample \#1. A similar procedure performed with sample \#2 yields $\lambda_c(0)\approx 50$ $\mu$m, which is in agreement with our measurements of overdoped BSCCO obtained using a different technique [@Nic]. We also estimated $\lambda_c(0)$ on the base of absolute measurements of the susceptibility $\chi'_c(0)$ from Eq. (5), and we obtained $\lambda_c(0)\approx 210$ $\mu$m for sample \#1 and $\lambda_c(0)\approx 70$ $\mu$m for sample \#2. These results are in reasonable agreement with our microwave measurements if we take into consideration the fact that the accuracy of $\lambda_c$ measurements is rather poor and the error can be up to 30%.
In conclusion, we have used the ac-susceptibility and cavity perturbation techniques in studying anisotropic high frequency properties of BSCCO single crystals. We have observed almost linear dependences $\Delta\lambda_c(T)\propto T$, which are in fair agreement with both experimental [@Jac1; @Shib2; @Nic] and theoretical [@Klem] results by other researchers. We have also investigated a new technique for determination of $\lambda_c(0)$, which is a factor of three higher in the optimally doped BSCCO sample than in the overdoped crystal. The ratio between the slopes of curves of $\Delta\lambda_c(T)$ in the range $T\ll T_c$ is the same. These facts could be put down to dependences of $\lambda_c(0)$ and $\Delta\lambda_c(T)$ on the oxygen content in these samples. At the same time, we cannot rule out influence of defects in the samples on $\lambda_c(0)$, even though their quality in the $ab$-plane is fairly high, according to our experiments. In order to draw ultimate conclusions concerning the nature of the transport properties along the $c$-axis in BSCCO single crystals, studies of more samples with various oxygen contents are needed.
We would like to thank V. F. Gantmakher for helpful discussions. This research was supported by grant No. 4985 of CNRS-RAS cooperation. The work at ISSP was also supported by the Russian Fund for Basic Research (grants 97-02-16836 and 98-02-16636) and Scientific Council on Superconductivity (project 96060).
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We computed the power spectrum of weak cosmic shear in models with non-Gaussian primordial density fluctuations. Cosmological initial conditions deviating from Gaussianity have recently attracted much attention in the literature, especially with respect to their effect on the formation of non-linear structures and because of the bounds that they can put on the inflationary epoch. The fully non-linear matter power spectrum was evaluated with the use of the physically motivated, semi-analytic halo model, where the mass function and linear halo bias were suitably corrected for non-Gaussian cosmologies. In agreement with previous work, we found that a level of non-Gaussianity compatible with CMB bounds and with positive skewness produces an increase in power of the order of a few percent at intermediate scales. We then used the matter power spectrum, together with observationally motivated background source redshift distributions in order to compute the cosmological weak lensing power spectrum. We found that the degree of deviation from the power spectrum of the reference Gaussian model is small compared to the statistical error expected from even future weak lensing surveys. However, summing the signal over a large range of multipoles can beat down the noise, bringing to a significant detection of non-Gaussianity at the level of $|f_\mathrm{NL}| \simeq $ few tens, when all other cosmological parameters are held fixed. Finally, we have shown that the constraints on the level of non-Gaussianity can be improved by $\sim 20\%$ with the use of weak lensing tomography.'
author:
- |
C. Fedeli$^{1,2,3}$ and L.Moscardini$^{1,2,3}$\
$^1$ Dipartimento di Astronomia, Università di Bologna, Via Ranzani 1, I-40127 Bologna, Italy (cosimo.fedeli@unibo.it)\
$^2$ INAF-Osservatorio Astronomico di Bologna, Via Ranzani 1, I-40127 Bologna, Italy\
$^3$ INFN, Sezione di Bologna, Viale Berti Pichat 6/2, I-40127 Bologna, Italy\
bibliography:
- './master.bib'
title: 'Cosmic shear statistics in cosmologies with non-Gaussian initial conditions'
---
Introduction {#sct:introduction}
============
One of the major success of the inflationary scenario for the early Universe is that it explains the formation of the seed fluctuations in the dark matter density field that, due to gravitational instability, eventually formed non-linear structures such as galaxies, galaxy clusters and voids (@GU81.1 [@BR84.1; @KO87.1]). In the simplest model of inflation, the early accelerated expansion phase of the Universe was driven by a single, minimally coupled scalar field. In this case, density fluctuations are predicted to follow an almost Gaussian probability distribution. Significant deviations from Gaussianity are however predicted in many of the more elaborated models of inflation that have been developed up to date (@MA86.1 [@AL87.1]).
The most recent analysis of the Cosmic Microwave Background radiation power spectrum of temperature fluctuations (CMB henceforth, @DU09.1 [@KO09.1], see also @SM09.1) is consistent with Gaussian primordial density perturbations, although a significant level of non-Gaussianity is still allowed. Inflationary models exist predicting a scale dependent behavior for the non-Gaussian amplitude [@LO08.1], implying that the amount of deviation from Gaussianity might be different between the large scales probed by the CMB and the small scales probed by galaxies and galaxy clusters.
In light of this, it is important to understand the effect on structure formation in the Universe of non-Gaussianity levels compatible with CMB bounds and/or with a significant scale evolution. This kind of problem has recently attracted much attention in the literature, with efforts directed towards the abundance of nonlinear structures (@MA00.2; @VE00.1; @MA04.1 [@KA07.1]; @GR07.1 [@GR09.1; @MA09.2]), halo biasing (@DA08.1 [@MC08.1]; @FE09.1), galaxy bispectrum [@SE07.2; @JE09.1], mass density distribution (@GR08.2) and topology [@MA03.2; @HI08.2], integrated Sachs-Wolfe effect [@AF08.1; @CA08.1], Ly-$\alpha$ flux from low-density intergalactic medium [@VI09.1], $21$ cm fluctuations [@CO06.2; @PI07.1] and reionization (@CR09.1).
One particular observable quantity that should be affected in a non-trivial way by non-Gaussianity is the fully non-linear power spectrum of the large-scale dark matter distribution. Studies of the effect of non-Gaussian initial conditions on this observable have been recently put forward with numerical $n$-body simulations [@GR08.2], with renormalized perturbation theory [@TA08.1] and by using both [@GI09.1]. Although differences exist between different works, they all agree in setting the effect of non-Gaussianity to a few percent at most on mildly non-linear scales.
Observationally, the matter power spectrum on scales smaller than CMB scales is usually measured by looking at the distribution of pairs of galaxies, that are known to be biased tracers of the underlying matter density field. More recently however the gravitational deflection of light has also been shown to be usable in order to map the large scale distribution of dark matter, having the additional advantage of being insensitive to the problems related with the bias of luminous tracers. The tradeoff for this advantage is that cosmic shear can measure only a projected version of the matter power spectrum, that depends on the assumed redshift distribution of the background source galaxies.
In this paper we focused on this approach, namely we aimed at understanding what kind of constraints can be put on deviations from primordial Gaussianity using the weak lensing power spectrum. As an example, attention was devoted to planned wide field optical surveys, such as the ESA Cosmic Vision project EUCLID [@LA09.1]. The rest of this work is organized as follows. In Section \[sct:ng\] we describe the non-Gaussian cosmologies considered in this work and how deviations from Gaussianity alter the mass function and the halo bias, both required for computing the non-linear power spectrum. In Section \[sct:modeling\] we discuss in detail the way in which we modeled the fully non-linear matter power spectrum, with particular attention to the assumptions, advantages and drawbacks underlying the method. In Section \[sct:results\] we describe our results concerning the weak lensing power spectrum, and in Section \[sct:discussion\] we draw our conclusions.
For the relevant calculations we adopted as a reference cosmology the one resulting from the best fit WMAP-$5$ parameters together with type-Ia supernovae and the observed Baryon Acoustic Oscillation (BAO). The present values of the density parameters for matter, dark energy and baryons are $\Omega_{\mathrm{m},0} = 0.279$, $\Omega_{\Lambda,0} = 0.721$ and $\Omega_{\mathrm{b},0} = 0.046$, respectively. The Hubble constant reads $H_0 = h$100 km s$^{-1}$ Mpc$^{-1}$, with $h = 0.701$. The slope of the primordial power spectrum of density fluctuations is $n = 0.96$, while the normalization is set by the *rms* of the density field on a comoving scale of $8 h^{-1}$ Mpc, $\sigma_8 = 0.817$. To construct the linear power spectrum we used the matter transfer function of @BA86.1, modified according to the shape factor of @SU95.1. The more sophisticated prescription of @EI98.1 is almost coincident with the former, except for the presence of the BAO, that is not of interest here.
Non-Gaussian cosmologies {#sct:ng}
========================
Simple generalizations of the most standard model of inflation give rise to seed primordial density fluctuations that follow a non-Gaussian probability distribution. A particularly simple way to parametrize the deviation of this distribution from a Gaussian consists of writing the Bardeen’s gauge invariant potential $\Phi$ as the sum of a linear Gaussian term and a quadratic correction [@SA90.1; @GA94.1; @VE00.1; @KO01.1],
$$\label{eqn:ng}
\Phi = \Phi_\mathrm{G} + f_\mathrm{NL}*\left( \Phi_\mathrm{G}^2 - \langle \Phi_\mathrm{G}^2 \rangle \right).$$
In Eq. (\[eqn:ng\]) the symbol $*$ denotes convolution between functions, and reduces to simple multiplication only in the particular case in which $f_\mathrm{NL}$ is a constant, while in general it is a function of the scale. Note that on scales smaller than the Hubble radius $\Phi$ equals minus the Newtonian peculiar gravitational potential.
In the following, we adopted the Large Scale Structure convention (as opposed to the CMB convention, see @AF08.1 [@PI09.1; @CA08.1]; @GR09.1) for defining the fundamental parameter $f_\mathrm{NL}$. According to this, the primordial value of $\Phi$ has to be linearly extrapolated at $z = 0$, and as a consequence the constraints given on $f_\mathrm{NL}$ by the CMB have to be raised of $\sim 30\%$ to comply with this paper’s convention (see also @FE09.1 for a concise explanation).
If the distribution of the primordial density (and potential) perturbations is not Gaussian, then it cannot be fully described by the power spectrum $P_\Phi(\bf{k})$ only, but we also need higher-order moments such as the bispectrum $B_\mathrm{\Phi}({\bf k}_1,{\bf k}_2,{\bf k}_3)$. In particular, different models of inflation give rise to different shapes of the bispectrum. In the following we shall adopt two particularly popular bispectrum shapes. The first one is dubbed the *local* shape. In this case, the bispectrum is maximized for configurations in which one of the three momenta is much smaller than the other two (“squeezed” configurations). The parameter $f_\mathrm{NL}$ is a dimensionless constant, and the bispectrum can be written as [@CR07.1; @LO08.1]
$$B_\Phi({\bf k}_1,{\bf k}_2,{\bf k}_3) = 2f_\mathrm{NL} B^2 \left[ k_1^{n-4}k_2^{n-4} + k_1^{n-4}k_3^{n-4} + k_2^{n-4}k_3^{n-4} \right],$$
where $k_i = \|{\bf k}_i\|$. The constant $B$ is the amplitude of the spectrum $P_\Phi(k)$, related to the amplitude $A$ of the power spectrum of density fluctuations by the relation $B = 9AH_0^4\Omega_{\mathrm{m},0}^2/4$. The second bispectrum shape is the *equilateral* shape, where the bispectrum is maximized by configurations where the three arguments have approximately the same magnitude. In the latter case, the primordial bispectrum takes the cumbersome form
$$\begin{aligned}
B_\Phi({\bf k}_1,{\bf k}_2,{\bf k}_3) &=& 6f_\mathrm{NL} B^2 \left[ k_1^{(n-4)/3}k_2^{2(n-4)/3}k_3^{n-4} \right. +
\nonumber\\
&+& k_3^{(n-4)/3}k_1^{2(n-4)/3}k_2^{n-4} + k_2^{(n-4)/3}k_3^{2(n-4)/3}k_1^{n-4} +
\nonumber\\
&+& k_1^{(n-4)/3}k_3^{2(n-4)/3}k_2^{n-4} + k_2^{(n-4)/3}k_1^{2(n-4)/3}k_3^{n-4} +
\nonumber\\
&+& k_3^{(n-4)/3}k_2^{2(n-4)/3}k_1^{n-4} -k_1^{n-4}k_2^{n-4} - k_1^{n-4}k_3^{n-4} -
\nonumber\\
&-& \left. k_2^{n-4}k_3^{n-4} - 2k_1^{2(n-4)/3}k_2^{2(n-4)/3}k_3^{2(n-4)/3}\right]. \end{aligned}$$
Most importantly, in inflationary models that predict an equilateral primordial bispectrum, the parameter $f_\mathrm{NL}$ is in general dependent on the scales. We adopt here the functional form suggested by @LO08.1, according to which
$$\label{eqn:kcmb}
f_\mathrm{NL}({\bf k}_1,{\bf k}_2,{\bf k}_3) = f_{\mathrm{NL},0} \left( \frac{k_1+k_2+k_3}{k_\mathrm{CMB}} \right)^{-2\kappa}.$$
The functional form of Eq. (\[eqn:kcmb\]) is chosen in order to avoid violating the WMAP constraints. Specifically, $f_{\mathrm{NL},0}$ represents the non-linear parameter evaluated at the scale $k_\mathrm{CMB} = 0.086 h$ Mpc$^{-1}$ roughly corresponding to the largest multipole used by [@KO09.1] to estimate non-Gaussianity in the WMAP data, $l = 700$. The constant free parameter $\kappa$ is assumed to be $|\kappa| \ll 1$ between CMB and cluster scales [@LO08.1; @CR09.1], in order to enhance non-Gaussianity on scales smaller than CMB. In previous work (@FE09.1), we adopted the values $\kappa = 0, -0.1, -0.2$. Here, for simplicity we limit ourselves to the case $\kappa = -0.2$, that is expected to give the largest effect. Also, for ease of notation, in the equilateral case we shall henceforth write $f_\mathrm{NL}$ meaning $f_{\mathrm{NL},0}$, since no ambiguity will arise.
At least two of the ingredients that make up the non-linear matter power spectrum (Section \[sct:modeling\]) are modified in cosmologies with non-Gaussian initial conditions: the halo mass function and the linear bias. The mass function $n(M,z)$ is the number of structures within the unit mass around $M$ that at redshift $z$ is contained in the unit comoving volume. An often used prescription for the mass function in Gaussian cosmologies is the one of [@PR74.1], that however tends to overpredict halo abundance at low masses with respect to the results of numerical $n$-body simulations. Other prescriptions exist, whose parameters are fitted against $n$-body simulations [@JE01.1; @WA06.1; @TI08.1] or determined based on more realistic models for the collapse of density perturbations [@SH01.1; @SH02.1]. We used the latter prescription in the remainder of this work. The @SH02.1 mass function in non-Gaussian models can then be written as
$$n(M,z) = n^{\mathrm{(G)}}(M,z) \frac{n_\mathrm{PS}(M,z)}{n_\mathrm{PS}^\mathrm{(G)}(M,z)},$$
where in our case $n^{\mathrm{(G)}}(M,z)$ is the mass function in the Gaussian cosmology computed according to the [@SH02.1] prescription and $n_\mathrm{PS}(M,z)$ and $n_\mathrm{PS}^\mathrm{(G)}(M,z)$ represent the [@PR74.1] mass functions in the non-Gaussian and reference Gaussian models respectively. Following [@LO08.1], we write the mass function $n_\mathrm{PS}(M,z)$ as
$$\begin{aligned}
\label{eqn:mfps}
n_\mathrm{PS}(M,z) &=& - \sqrt{\frac{2}{\pi}} \frac{\rho_\mathrm{m}}{M} \exp\left[ -\frac{\delta_\mathrm{c}^2(z)}{2\sigma_M^2} \right] \left[ \frac{d\ln \sigma_M}{dM} \left( \frac{\delta_\mathrm{c}(z)}{\sigma_M} + \right.\right.
\nonumber\\
&+& \left. \left. \frac{S_3\sigma_M}{6} \left( \frac{\delta_\mathrm{c}^4(z)}{\sigma^4_M} -2\frac{\delta^2_\mathrm{c}(z)}{\sigma^2_M} -1\right) \right) + \right.
\nonumber\\
&+& \left. \frac{1}{6} \frac{dS_3}{dM}\sigma_M \left( \frac{\delta^2_\mathrm{c}(z)}{\sigma^2_M} -1\right) \right].\end{aligned}$$
Eq. (\[eqn:mfps\]) has been obtained by Edgeworth expanding the probability distribution for smoothed density fluctuations and $\delta_\mathrm{c}(z) \equiv \Delta_\mathrm{c}/D_+(z)$, where the quantity $\Delta_\mathrm{c}$ is the linear density threshold for spherical collapse, that is constant in an Einstein-de Sitter model and only mildly dependent on redshift in models with a cosmological constant. We included this redshift dependence in our calculations but do not indicate it explicitely, since it is practically irrelevant. The function $D_+(z)$ is the linear growth factor for density fluctuations, while $\sigma_M$ is the *rms* of density perturbations smoothed on a scale corresponding to mass $M$. The function $S_3(M) \equiv f_{\mathrm{NL},0} \mu_3(M)/\sigma_M^4$ is the normalized skewness. Note that Eq. (\[eqn:mfps\]) reduces to the standard $n_\mathrm{PS}^\mathrm{(G)}(M,z)$ in the case in which $S_3(M)$ vanishes identically. The third-order moment $\mu_3(M)$ can be written as
$$\mu_3(M) = \int_{\mathbb{R}^9} \mathcal{M}_R(k_1) \mathcal{M}_R(k_2) \mathcal{M}_R(k_3) B_\Phi({\bf k}_1,{\bf k}_2,{\bf k}_3) \frac{d{\bf k}_1d{\bf k}_2d{\bf k}_3}{(2\pi)^9}.$$
The function $\mathcal{M}_R(k)$ relates the Fourier transform of density fluctuations smoothed on some scale $R$ to the relative peculiar potential, and it is defined as
$$\mathcal{M}_R(k) \equiv \frac{2}{3} \frac{T(k)k^2}{H_0^2 \Omega_{\mathrm{m},0}} W_R(k),$$
where $T(k)$ is the matter transfer function and $W_R(k)$ is the Fourier transform of the top-hat window function. For an alternative derivation of $n_\mathrm{PS}(M,z)$, see @MA00.2.
The linear bias describes how well dark matter halos trace the underlying large scale matter distribution, and is needed in order to account for the correlation between different halos. For it we adopted the @SH01.1 modification of the original @MO96.1 formula, obtained with [@PR74.1]-like considerations in Gaussian cosmologies, that reads
$$\begin{aligned}
\label{eqn:bias}
b^\mathrm{(G)}(M,z) &=& 1 + a\frac{\Delta_\mathrm{c}}{D_+^2(z)\sigma^2_M} - \frac{1}{\Delta_\mathrm{c}} +
\nonumber\\
&+& \frac{2p}{\Delta_\mathrm{c}} \left[ \frac{[D_+(z)\sigma_M]^{2p}}{[D_+(z)\sigma_M]^{2p} + [\sqrt{a} \Delta_\mathrm{c}]^{2p}} \right].\end{aligned}$$
The original @MO96.1 formula is obtained by setting $a = 1$ and $p = 0$ in Eq. (\[eqn:bias\]), while the @SH01.1 revision is obtained with the values $a = 0.75$ and $p = 0.3$.
In non-Gaussian models, the bias acquires an extra scale dependence that can be written as [@MA08.1]
$$b(M,z,k) = b^\mathrm{(G)}(M,z) + \Delta b(M,z,k),$$
where
$$\Delta b(M,z,k) = \left[ b^\mathrm{(G)}(M,z)-1 \right] \delta_\mathrm{c}(z) \Gamma_R(k).$$
The term $\Gamma_R(k)$ encapsulates all the dependence on the scale, and can be written as
$$\begin{aligned}
\Gamma_R(k) &=& \frac{1}{8\pi^2\sigma_M^2\mathcal{M}_R(k)} \int_0^{+\infty} \zeta^2\mathcal{M}_R(\zeta) \times
\nonumber\\
&\times& \left[ \int_{-1}^1 \mathcal{M}_R\left(\sqrt{\alpha}\right) \frac{B_\Phi\left( \zeta,\sqrt{\alpha},k \right)}{P_\Phi(k)} d\mu \right] d\zeta,\end{aligned}$$
where $\alpha \equiv k^2 + \zeta^2 + 2k\zeta\mu$ and $R$ is the top-hat radius corresponding to the mass $M$. In the particular case of a primordial bispectrum of local shape, the previous equation simplifies to
$$\begin{aligned}
\Gamma_R(k) &=& \frac{2f_\mathrm{NL}}{8\pi^2\sigma_M^2\mathcal{M}_R(k)} \int_0^{+\infty} \zeta^2\mathcal{M}_R(\zeta) P_\Phi(\zeta)\times
\nonumber\\
&\times& \left[ \int_{-1}^1 \mathcal{M}_R\left(\sqrt{\alpha}\right) \frac{P_\Phi\left(\sqrt{\alpha}\right)}{P_\Phi(k)} d\mu \right] d\zeta.\end{aligned}$$
In the next section we show how the mass function and halo bias enter the non-linear matter power spectrum, and what is the subsequent effect of primordial non-Gaussianity.
Modeling the non-linear power spectrum {#sct:modeling}
======================================
We computed the fully non-linear matter power spectrum by using the halo model developed by @MA00.3 and @SE00.1. In this model, the power spectrum is set by the sum of two terms. The first one, dominating on large scales, is given by dark-matter particle pairs residing in different halos, hence it depends on the correlations of individual halos. The second term, dominating on the smallest scales, takes into account particle pairs that are included in the same halo, hence it is extremely sensitive to the inner structure of halos themselves. This kind of decomposition of the matter power spectrum probably has a deeper rooting than simple power spectrum modeling, since it also arises in renormalized perturbation theory [@CR06.2; @CR06.1].
The main ingredients entering in this model are the mass function, the halo bias and the halo internal structure. The first two were discussed in Section \[sct:ng\], including their modifications due to non-Gaussian initial conditions. The only additional point that we make here is that one of the hypotheses underlying the halo model for the power spectrum is that all the matter in the Universe is included inside halos of some mass. This imposes the constraint
$$\label{eqn:c1}
\int_0^{+\infty} n(M,z) \frac{M}{\rho_\mathrm{m}} dM = 1,$$
where $\rho_\mathrm{m}$ is the average comoving matter density. As can be easily verified, this constraint is fulfilled in Gaussian cosmologies by the [@PR74.1] mass function, and also the [@SH02.1] mass function is normalized such to satisfy Eq. (\[eqn:c1\]). Numerically however it is not possible to push the lower integration bound down to zero, rather the integration will be stopped to some $M_\mathrm{inf} > 0$. As noted by [@RE02.2], decreasing $M_\mathrm{inf}$ makes the integral on the left-hand side of Eq. (\[eqn:c1\]) approach unity, however this happens very slowly in CDM models, so that effectively, even if $M_\mathrm{inf}$ is very small, the integral will still be significantly different from unity. We solved this following [@RE02.2], namely by enforcing the constraint in Eq. (\[eqn:c1\]) by adding a constant to the mass function in the smallest mass bin that is considered when performing the integration numerically. In the present work, we assumed $M_\mathrm{inf} = 10^6 M_\odot h^{-1}$.
Halo density profile
--------------------
We assumed that dark matter halos in both Gaussian and non-Gaussian cosmological models are on average well described by a generalized @NA96.1 (NFW henceforth, see also @NA95.1 [@NA97.1]) density profile, written as
$$\rho(r) = \frac{\rho_\mathrm{s}}{(r/r_\mathrm{s})^\alpha(1+r/r_\mathrm{s})^{3-\alpha}},$$
which reduces to the standard NFW model for $\alpha = 1$ [@AM04.1]. For full generality we developed the following calculations for any value of $\alpha$, however when evaluating the effect of non-Gaussianity on the power spectrum we specialized mostly to the case $\alpha = 1$, with only a minor discussion on the role of the inner slope. The total mass of the halo that is included inside some radius $r$ can be computed as [@TA03.2]
$$\begin{aligned}
\label{eqn:mass}
M(r) &=& 4\pi \int_0^r \rho(x) x^2 dx =
\nonumber\\
&=&\frac{4\pi \rho_\mathrm{s}r_\mathrm{s}^3}{3-\alpha} \left._2F_1 \right. \left( 3-\alpha,3-\alpha;4-\alpha;-\frac{r}{r_\mathrm{s}} \right) \left( \frac{r}{r_\mathrm{s}} \right)^{3-\alpha} \equiv
\nonumber\\
&\equiv& 4\pi \rho_\mathrm{s}r_\mathrm{s}^3G_\alpha(r/r_\mathrm{s}).\end{aligned}$$
In Eq. (\[eqn:mass\]), the function $\left._2F_1 \right.$ is the Gauss hypergeometric function, and in the particular case $\alpha = 1$, the function $G_\alpha$ reduces to the well known form
$$G_1(x) = \ln(1+x) - \frac{x}{1+x}.$$
The two parameters $r_\mathrm{s}$ and $\rho_\mathrm{s}$ completely specify the density profile and can be expressed in terms of the virial mass $M$ and concentration $c$. In the remainder of this work we defined the virial mass as the mass contained in the sphere whose mean density equals $\Delta_\mathrm{v} = 200$ times the *average* density of the Universe, that is
$$\label{eqn:massv}
M = \frac{4}{3}\pi R_\mathrm{v}^3 \Delta_\mathrm{v} \rho_\mathrm{m}.$$
The virial radius $R_\mathrm{v}$ is the radius of this sphere, and the concentration is defined as the ratio between the virial radius and the scale radius of the profile, $c \equiv R_\mathrm{v}/r_\mathrm{s}$. It is important to note that different authors use different definitions for the viral radius. In some cases the overdensity is not referred to the mean matter density, rather to the critical density $\rho_\mathrm{c}(z) = 3H^2(z)/8\pi G$. Yet others use different values of the overdensity, usually the value computed for the collapse of a spherical overdensity, that in an Einstein-de Sitter universe is constant and equals $\Delta_\mathrm{v} \simeq 178$. It is expected that different choices assign different values of the concentration to the same virial mass, hence effectively modifying the power spectrum on small scales. We adopted the overdensity with respect to the average density for consistency with the [@SH02.1] mass function, that was calibrated against numerical simulations where dark matter lumps were detected via spherical overdensity methods [@TO98.1]. They actually used $\Delta_\mathrm{v} = 178$, but we checked that this does not make a significant difference for the resultant power spectrum.
Given the previous considerations, the comoving scale density can then be written as
$$\rho_\mathrm{s} = \frac{\Delta_\mathrm{v}}{3} \rho_\mathrm{m} \frac{c^3}{G_\alpha(c)},$$
and the comoving scale radius as
$$r_\mathrm{s} = \left( \frac{3M}{4\pi c^3 \Delta_\mathrm{v}\rho_\mathrm{m}} \right)^{1/3}.$$
Please note that the scale radius is independent on the slope $\alpha$, while the scale density is not.
The concentration of a dark matter halo is actually linked to the virial mass according to the hierarchical paradigm for structure formation, since it is expected that small structures collapse earlier and hence are more compact at a given redshift. We discuss the exact nature of this relationship further below. Thus, once the inner slope of the profile $\alpha$ is specified, the dark matter distribution depends effectively only on mass and redshift. Hence, for a dark matter halo of mass $M$ at redshift $z$ from this moment on we write the density profile as $\rho(r,M,z)$. We indicate with $\hat{\rho}(k,M,z)$ the Fourier transform of $\rho(r,M,z)$ with respect to the radius, which can be written as
$$\label{eqn:ft}
\hat{\rho}(k,M,z) = 4\pi \int_0^{R_\mathrm{v}} \rho(r,M,z) \frac{\sin(kr)}{kr}r^2dr.$$
This definition conveniently implies $\hat{\rho}(0,M,z) = M$, although it neglects matter outside the virial radius (see the discussion below). When $\alpha = 1$ the Fourier transform of the density profile can be expressed analitically, according to
{width="0.45\hsize"} {width="0.45\hsize"}
$$\begin{aligned}
\hat{\rho}(k,M,z) &=& 4\pi \rho_\mathrm{s}r_\mathrm{s}^3 \left[\frac{}{}\sin(kr_\mathrm{s}) \left(\mathrm{Si}[(1+c)kr_\mathrm{s}] - \mathrm{Si}(kr_\mathrm{s}) \right) \right. +
\nonumber\\
&+&\cos(kr_\mathrm{s}) \left(\mathrm{Ci}[(1+c)kr_\mathrm{s}] - \mathrm{Ci}(kr_\mathrm{s}) \right) -
\nonumber\\
&-&\left.\frac{\sin(kr_\mathrm{s}c)}{(1+c)kr_\mathrm{s}} \right]\end{aligned}$$
(@SC01.1; @RU08.2), where $\mathrm{Si}(x)$ and $\mathrm{Ci}(x)$ are the sine and cosine integrals respectively. When $\alpha$ is different from unity, we have instead to resort to numerical integration. Please note that, for $x \rightarrow 0$, $\mathrm{Si}(x) \simeq 0$ while $\mathrm{Ci}(x) \simeq \ln(x)$, thus we have that $\hat{\rho}(0,M,z) = 4\pi \rho_\mathrm{s}r_\mathrm{s}^3 G_1(c)$, that, according to Eq. (\[eqn:mass\]), correctly equals the virial mass of the halo.
In Figure \[fig:profile\] we show the density profiles of dark matter halos with different masses and redshifts and their Fourier transforms. We plot results for three different values of the inner slope $\alpha$, and we adopted the @EK01.1 prescription in order to relate the mass to the concentration of dark matter halos. We discuss this choice further below. Due to the sine function present in the integral in Eq. (\[eqn:ft\]), the Fourier transform of the density profile presents wiggles at small scales, whose strength increases with decreasing $\alpha$. This can be naively understood since profiles with smaller $\alpha$ are flatter, and hence it is expected that their Fourier transforms have more fluctuations at large wavenumbers.
Power spectrum
--------------
Let now $P_\mathrm{L}(k,z)$ be the linear power spectrum of density fluctuations extrapolated at redshift $z$. We assumed it to be the same both in the reference Gaussian model and in non-Gaussian cosmologies. According to the halo model, the fully non-linear spectrum $P(k,z)$ can be written as the sum of two terms that read as follows.
$$\label{eqn:p1}
P_1(k,z) = \int_0^{+\infty} n(M,z) \left[\frac{\hat{\rho}(M,z,k)}{\rho_\mathrm{m}}\right]^2 dM$$
and $$\label{eqn:p2}
P_2(k,z) = \left[ \int_0^{+\infty} n(M,z)b(M,z,k)\frac{\hat{\rho}(M,z,k)}{\rho_\mathrm{m}}dM \right]^2 P_\mathrm{L}(k,z).$$ In the two previous equations, $n(M,z)$ is the standard mass function while $b(M,z,k)$ is the linear bias, both introduced in Section \[sct:ng\] above.
The full non-linear power spectrum on large scales is dominated by the second term , $P(k,z) \simeq P_2(k,z)$. As noted by [@SE00.1], for self-consistency it is necessary that this term approaches the linear power spectrum in the limit $k \ll 1 h$ Mpc$^{-1}$, which imposes the nontrivial constraint
$$\label{eqn:c2}
\int_0^{+\infty} n(M,z)b(M,z,k)\frac{M}{\rho_\mathrm{m}}dM = 1$$
in that limit. Analogously to the previous mass function constraint in Eq. (\[eqn:c1\]), we practically enforced the constraint in Eq. (\[eqn:c2\]) by adding a constant to the bias in the smallest mass bin adopted in the numerical integration. There is one point worth of discussion about Eq. (\[eqn:c2\]). While in the Gaussian model this constraint can be computed only once, given the redshift, in a non-Gaussian cosmology this computation has to be performed at each scale at which we are interested in computing the power spectrum. Strictly speaking, the condition needs to be enforced only in the limit $k \ll 1 h$ Mpc$^{-1}$, therefore we would have the freedom to relax it for $k \gtrsim 1 h$ Mpc$^{-1}$. However, it is not clear at which scale the transition between corrected and uncorrected bias should happen, neither how fast this transition should be. Therefore, we chose not to use this freedom, and to enforce the condition in Eq. (\[eqn:c2\]) for all $k$ in non-Gaussian models.
In the remainder of this paper, unless explicitly noted, instead of the power spectrum we will refer to the dimensionless power $\Delta^2(k,z)$, defined as
$$\Delta^2(k,z) \equiv \frac{4\pi k^3 P(k,z)}{(2\pi)^3}.$$
The last thing that remains to be defined in order to fully specify the halo model is the relation between the virial mass and the concentration of dark matter halos. This is of fundamental importance at very small scales, where the power spectrum is expected to be dominated by the correlations of dark matter particle pairs that are inside the same halo. In order to do this, there exist prescriptions based on the study of samples of dark matter halos extracted from high-resolution Gaussian cosmological simulations (@NA96.1; @BU01.1; @EK01.1; @DO04.1 [@GA08.1]). Many authors however [@CO00.2; @RE02.2], prefer to adopt a concentration-mass relation for which the halo model matter power spectrum is a good fit to the spectrum measured in $n$-body simulations of Gaussian cosmologies. The latter is often assumed to be well represented by the prescriptions of [@PE94.1] and of @SM03.1, both of which are implemented by the publicly available `halofit` code.
In a perfectly consistent picture of structure formation, the two described approaches should give equivalent results. In practice however, this is not the case. As noted by [@SE03.2], in order for the first approach to reproduce numerically simulated spectra, the amplitude of the concentration-mass relation adopted should drop with redshift more steeply than predicted by both the @BU01.1 and @EK01.1 prescriptions. [@SE03.2] argue that the halo model with a concentration-mass relation derived directly by simulated dark matter halos should be adopted as physically motivated, while the fits of [@PE94.1] and [@SM03.1] to numerically simulated power spectra should not be trusted beyond the range where they have been tested, that is $k \lesssim 40 h$ Mpc$^{-1}$ at $z = 0$ and even $k \lesssim 10 h$ Mpc$^{-1}$ at high redshift. We obtained results that are somewhat consistent with this interpretation. In fact, we found that by using the halo model with the concentration-mass relation given by @EK01.1 produces a $z = 0$ power spectrum that is in broad agreement with the `halofit` result. Moving at high redshift we find instead that the halo model power spectrum is higher than the `halofit` results for $k \gtrsim 10 - 20 h$ Mpc$^{-1}$, with the difference growing with redshift. If we require the concentration to drop with redshift more steeply than with the @EK01.1 recipe, as suggested by [@SE03.2], we would reduce the power at small scales and high redshift, hence reducing the discrepancy. As a matter of fact, multiplying the @EK01.1 concentrations by the extra factor $(1+z)^{-1/2}$ we obtained a fair agreement, as shown in Figure \[fig:spectraZ\].
![The dimensionless power computed according to the halo model in the reference Gaussian cosmology for three different redshifts, $z = 0$, $z = 1$ and $z = 2$, from top to bottom respectively. Results are compared with the recipes of @PE94.1 and @SM03.1 computed with `halofit`, as labeled. The concentration-mass relation is the one prescribed by @EK01.1, with concentrations multiplied by the additional factor $(1+z)^{-1/2}$.[]{data-label="fig:spectraZ"}](Figures/spectraZ){width="\hsize"}
We also tried to use the concentration-mass relation that [@CO00.2] found to give a good fit to numerically simulated power spectra (see also @RE02.2). We found that the resulting power at small scales is higher than the `halofit` results. This is likely a consequence of the fact that the fit of [@CO00.2] (as the authors themselves note) is valid only for the specific cosmological model they used, that in particular has a higher normalization $\sigma_8$ than our. This implies larger halo concentrations for fixed mass and redshift, and hence more power at large wavenumbers. Approaches similar in spirit to those of [@CO00.2] and [@RE02.2] have recently been followed by @BE09.1 in order to find a suitable version of the mass-concentration relation of dark matter halos.
A distinct possibility is that the fits to numerically simulated power spectra are indeed correct even beyond their range of applicability, but the halo profile to be inserted in the halo model is not the true profile of dark matter halos, since other effects such as halo substructure and triaxiality can affect the power spectrum on small scales [@CO00.2]. However, [@SE00.1] and [@SE03.2] argue that at least the effect of substructures is not significant.
![The fully non-linear dark matter dimensionless power in the reference Gaussian cosmology at $z = 0$ for three different values of the inner slope of dark matter halos, as labelled in the plot. Also, the long-dashed and dot-dashed curves represent the 1-halo and 2-halo contributions to the non-linear power spectrum for $\alpha = 1$. In all cases, the mass-concentration relation from @EK01.1 has been adopted, with the redshift correction needed to match the `halofit` results.[]{data-label="fig:spectra"}](Figures/spectra){width="\hsize"}
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Finally, as a last point we note that some degree of disagreement is also present in the intermediate regime in Figure \[fig:spectraZ\]. This was also found and discussed by [@SE03.2], who state that the halo model is not expected to be perfect at intermediate scales because of several factors, for instance the fact that we ignore the matter outside the virial radius in the Fourier transform in Eq. (\[eqn:ft\]) and the contribution from the single halo term of the power spectrum is certainly overestimated for $k \ll 1/R_\mathrm{v}$. It is quite clear that with the precision of upcoming experiments for the measurement of the non-linear power spectrum, it is going to be necessary to find a unique prescription for the power spectrum, valid on all scales and in agreement with the mean structure of individual halos. However, for the time being we are interested in relative deviations with respect to the fiducial Gaussian $\Lambda$CDM cosmology, hence we stick to our choice.
In Figure \[fig:spectra\] we show the fully non-linear power spectrum of the dark matter in the Gaussian cosmological model for three different values of the inner slope of dark matter halos. The concentrations are computed again according to the prescription of @EK01.1 with the correction factor $(1+z)^{-1/2}$, and for the case $\alpha = 1$ the separate contributions from the $1$-halo and $2$-halo terms are also shown. As it is expected, the inner slope mainly affects the power spectrum at large wavenumbers, with the dimensionless power increasing with increasing $\alpha$. The effect of the inner slope of halo density profiles on the power spectrum for the rather extreme values shown in Figure \[fig:spectra\] can be quite large at small scales, and in fact we show below that for large enough wavenumbers it does overcome the effect of primordial non-Gaussianity captured by the halo model.
For realistic models of non-Gaussianity, it turns out that the shift of the matter power spectrum due to the corrections of the bias and of the mass function is very small, such that it is almost not visible on the scale of e.g., Figure \[fig:spectra\] (see for instance @GR08.2). Hence, in Figure \[fig:ratio\], we plot the ratio of the matter power spectrum for non-Gaussian cosmologies to the Gaussian one, with all quantities computed adopting $\alpha=1$. Shown are the results for primordial bispectra of the local and equilateral shapes (with $\kappa = -0.2$), with $f_\mathrm{NL}$ matching the largest possible positive values allowed by CMB constraints [@KO09.1].
As can be seen, due to non-Gaussianity with positive $f_\mathrm{NL}$, the power spectrum is increased at intermediate scales. At large scales the ratios get closer to unity, because the power spectra are dominated by the two-halo term, which is normalized such to reproduce the linear power spectrum, which is the same in Gaussian and non-Gaussian cosmologies. We show results for three different redshifts, indicating that the effect of non-Gaussianity is larger for higher $z$, as one might expect. The absolute increase in power at intermediate scales is quite moderate, being at most of $\sim 4\%$ at high-$z$ for the local shape and $\sim 6\%$ for the equilateral shape. The general qualitative behavior of the ratio of the non-Gaussian power spectra to the Gaussian one is quite independent of the shape of the primordial bispectrum, and also resembles the one reported by [@AM04.1], that adopted completely different non-Gaussian models. Thus, this trend seems to be a quite general property of non-Gaussian distributions with positive skewness. The fact that non-Gaussianities of the equilateral shape give a larger deviation in terms of the matter power spectrum might seem counter intuitive, since the corrections to the bias are larger in the local than in the equilateral case (@MA08.1; @FE09.1). However, it should be recalled that the bounds on $f_\mathrm{NL}$ given by CMB are looser for the equilateral as compared to the local shape. As a matter of fact, we checked that considering a local non-Gaussian model with $f_\mathrm{NL} = 330$, as is the case for the equilateral model, the ratio between the non-Gaussian and the Gaussian power spectra is in fact larger than for the equilateral case, reaching up to $\sim 8\%$ for the high redshift curve.
In Figure \[fig:ratio\] we have shown results only for positive values of $f_\mathrm{NL}$. This is because the CMB constraints on the level of non-Gaussinity, that we chose to follow in this part of the work, are highly asymmetric. For instance, the largest negative value of $f_\mathrm{NL}$ that is allowed by CMB constraints for non-Gaussianity of the local type is only $f_\mathrm{NL} = -12$, for which we expect almost no effect. As we verified, negative values of $f_\mathrm{NL}$ give rise to a symmetric behavior of the matter power spectrum around $P_\mathrm{G}(k,z)$ with respect to positive values. This is also clear from other works [@GI09.1] and from the following discussion in Section \[sct:results\].
Numerical simulations of non-Gaussian matter density fields, and computations based on renormalized perturbation theory (@GR08.2; @DE09.1 [@PI09.1; @TA08.1; @GI09.1]) also find an increment in the matter power spectrum due to local non-Gaussianity with positive skewness at the level of few percent. Also, they verify the qualitative behavior according to which at large scales (the only that can be probed by contemporary simulations and renormalized perturbation theory with one or two loop corrections), the effect of non-Gaussianity is milder for higher redshift.
There are factors that can affect the matter power spectrum differently in a non-Gaussian and Gaussian cosmology that are not captured by the simple halo model. For instance, due to the different structure formation process, it is possible that the mean halo density profile, ellipticity and substructure content are different, as well as the amount of matter outside the virial radius, all of which is neglected by the halo model. In order to exemplify one of these effects, in Figure \[fig:ratio\_alpha\] we show the effect of variations in the inner slope of average dark matter halo density profiles in non-Gaussian models, and compare it with the effect of primordial non-Gaussianity itself on the mass function and halo bias, that is instead automatically included in the halo model. As can be seen, the effect of a $\sim 10\%$ shift of the inner slope overcomes the effect of non-Gaussianity included in the halo model at scales $k \lesssim 1h$ Mpc. Larger deviations from the fiducial slope $\alpha=1$ produce even starker modifications, shifting the transition scale up to $k \sim 0.6 h$ Mpc$^{-1}$. The fact that the effect of halo density profile on the matter power spectrum can become larger than the effect of non-Gaussianity captured by the halo model was already pointed out by @AM04.1.
![The ratio between the matter power spectrum in the labeled cosmology with non-Gaussian initial conditions computed assuming an inner slope for the mean dark matter halo density profile different from unity (again as labeled in the plot) to the same quantity computed instead with $\alpha=1$. For reference, we also report the ratio of the power spectrum in the non-Gaussian model to the same quantity in the Gaussian one for $\alpha=1$ (as in Figure \[fig:ratio\]). All quantities refer to $z=0$.[]{data-label="fig:ratio_alpha"}](Figures/ratio_alpha){width="\hsize"}
Another fact that is to be taken into account is that, while no secure prediction on the inner slope of dark matter halos in non-Gaussian cosmologies has been produced (with the exception of @AV03.1, that however used very different non-Gaussian models from our own), some uncertainty on the value of $\alpha$ is also present in Gaussian cosmologies [@MO98.2; @DI05.1]. As we verified, if the fluctuations on $\alpha$ in Gaussian and non-Gaussian cosmologies are similar, the two effects cancel, so that the ratio between the two power spectra is almost unchanged on the scales of Figure \[fig:ratio\]. This, together with the very fact that no sharp increase (or decrease) of the matter power spectrum for non-Gaussian cosmologies with respect to the Gaussian case is detected at the scales probed by numerical simulations (e.g., @GR08.2) indicates that there should be no large differences in the value of $\alpha$ in the two kinds of cosmological models.
In order to properly gauge the effect of primordial non-Gaussianity on the matter power spectrum that is not automatically captured by the simple halo model, a more detailed understanding of the mean dark matter halo profile, especially in non-Gaussian cosmologies, should therefore be achieved. This, as well as the assessment of the other uncertainties mentioned above, would require larger ensembles of non-Gaussian simulations and a more careful analysis thereof, which is clearly not the point in our study. Here we limit ourselves to point out that, although the general effect of non-Gaussianity is fairly captured, one should expect some higher order difference between the halo model, numerical simulations, and other ways to estimate the matter power spectrum. With this cautionary remark in mind, we proceed with the computation of the weak lensing power spectrum by setting $\alpha=1$, deferring further discussion of the effects that are not considered here to Section \[sct:discussion\].
Results {#sct:results}
=======
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Weak lensing power spectrum
---------------------------
According to @BA00.1 and @BA01.1, the power spectrum of the lensing convergence can be written as
$$\label{eqn:wl}
C_l = \frac{9H_0^4\Omega_{\mathrm{m},0}^2}{4 c^4} \int_0^{\chi_\mathrm{H}} P\left(\frac{l}{f_K(\chi)},\chi \right) \frac{W^2(\chi)}{a^2(\chi)} d\chi,$$
where $\chi = \chi(z)$ is the comoving distance out to redshift $z$, $a(\chi)$ is the scale factor normalized to unity today and $f_K(\chi)$ is the comoving angular-diameter distance corresponding to the comoving distance $\chi$, which depends on the spatial curvature $K$ of the Universe. The integral in Eq. (\[eqn:wl\]) extends formally out to the horizon size $\chi_\mathrm{H}$, however the integrand becomes zero well before this limit is reached, due to the absence of sources at $z \gtrsim 4$ (see below). The redshift distribution of sources $n(z)$ has a fundamental role in the evaluation of the weak lensing power spectrum, as it defines the integration kernel
$$W(\chi) = \int_\chi^{\chi_\mathrm{H}} n(\chi') \frac{f_K(\chi-\chi')}{f_K(\chi')} d\chi'.$$
The Eq. (\[eqn:wl\]) for the convergence power spectrum was obtained using Fourier expansion and the Limber’s approximation [@BA01.1], while the exact expression would make use of spherical harmonic expansion. However, it was recently shown by @JE09.1 that, at least when considering only the convergence power spectrum, the accuracy of the Limber’s approximation is very good, better than $1\%$ at $l>10$, corresponding to $2\pi/l \lesssim 2 \times 10^3$ arcmin.
Several choices for the redshift distribution of background sources to be adopted for cosmic shear studies are available in the literature. One of the most recent ones is presented in the work of [@BE07.2], where a detailed analysis of the photometric redshift distribution in four different fields is reported. The four fields considered were the Canada-France-Hawaii Telescope Legacy Survey (CFHTLS) wide survey [@VA02.1; @HO06.1], the GaBoDS field [@HE07.1], the VIRMOS-DESCART project [@VA01.1; @MC03.1; @LE04.1], and the RCS survey [@HO02.1]. [@BE07.2] fitted the photometric redshift distribution in the four fields using the three-parameter formula
$$\label{eqn:z1}
n(z) = \frac{\beta}{z_0\Gamma\left[(1+\alpha)/\beta\right]} \left( \frac{z}{z_0} \right)^\alpha \exp \left[ -\left( \frac{z}{z_0} \right)^\beta \right].$$
The same fitting formula has also been used by @VA05.2 and [@SE06.1] to fit the photometric redshift distribution of the Hubble Deep Field (HDF). As noted by [@BE07.2] and suggested by @VA06.1 however, the HDF suffers of sample variance, and maybe it is also subject to a selection bias. [@BE07.2] noted that the formula in Eq. (\[eqn:z1\]) does not fit very well their photometric redshift distribution if all galaxies are included, and proposed a different functional form that performed a better fit. However, when considering only their high-confidence redshift interval (outside which the fraction of catastrophic errors reaches $40-70\%$), Eq. (\[eqn:z1\]) becomes a good fit. We chose to stick to this choice, and for each of the three parameters in Eq. (\[eqn:z1\]) we adopted the mean of the values for the four fields analyzed by [@BE07.2].
In order to show the effect of different choices for the background source redshift distribution, in Figure \[fig:zDist\] we report the redshift distributions of @VA05.2, @SE06.1, and @BE07.2 with the respective integration kernels. As expected, when $z \rightarrow 0$ the kernel tends to the integral over the source redshift distribution, that is correctly normalized to unity. Independently of the assumed source redshift distribution, the kernel already vanishes at $\chi \simeq 1.5 c/H_0$, corresponding to $z \simeq 3$. In Figure \[fig:wl\] we show the weak lensing power spectra that we computed in the $\Lambda$CDM model for the three distributions, using the matter power spectrum evaluated with the halo model described above in Section \[sct:modeling\]. As can be seen, the difference between different power spectra can be quite significant, especially at intermediate scales, implying that the choice of the redshift distribution must be carefully addressed, given the precision level reached by future surveys.
![The fully non-linear weak lensing power spectra computed for the three different choices of source redshift distribution detailed in the text, as labelled in the plot. The gray shaded area shows statistical uncertainty on the power spectrum computed with the redshift distribution of @VA05.2. The background cosmology is the reference Gaussian one.[]{data-label="fig:wl"}](Figures/weakLensing){width="\hsize"}
The three power spectra tend to coincide in the linear regime at very large scales. This is because on these scales the integral in Eq. (\[eqn:wl\]) is dominated by the low-redshift contribution, where the differences between the three integration kernels are minimal. Moving in the non-linear regime, at small scales, differences between different $n(z)$ become apparent, in particular the power spectrum for the redshift distribution of @BE07.2 deviates more significantly from the power spectra for the redshift distributions of @VA05.2 and @SE06.1. This is consistent with the larger deviations that are apparent for the former distribution in the weight function $W(\chi)$.
Also shown in the same figure is the Gaussian statistical error for the power spectrum computed for the redshift distribution of @VA05.2. According to [@KA92.1; @KA98.1; @SE98.1; @HU02.1] this has been evaluated with the prescription
$$\label{eqn:er}
\Delta C_l = \sqrt{\frac{2}{(2l+1)\Delta lf_\mathrm{sky}}} \left( C_l + \frac{\gamma^2}{\bar{n}} \right).$$
In Eq. (\[eqn:er\]), $\bar{n}$ is the average number density of galaxies in the survey at hand, that we assumed equal to $\bar{n} = 40$ arcmin$^{-2}$, $f_\mathrm{sky}$ is the survey area in units of the sky area, that we posed equal to $0.5$ and $\gamma$ is the *rms* intrinsic shape noise for each galaxy, that we set equal to $\gamma = 0.22$ [@ZH09.1]. Such numerical values are the goals for the proposed ESA space mission EUCLID, and will be used for the rest of this paper. The parameter $\Delta l$ in Eq. (\[eqn:er\]) represents the width of the multipole bin within which power is measured. For simplicity, we set $\Delta l = 1$ here and in the remainder of this paper. In [@TA07.1; @TA09.1] it was shown that the multipole bin width $\Delta l$ does not significantly affect the likelihood values and the parameter estimation as long as the multipole binning is not too coarse and the weak lensing power spectrum does not show sharp variations within each bin. Finally, it has been shown that the simple Gaussian prescription in Eq. (\[eqn:er\]) is in agreement with more elaborated error definitions on the scales where non-Gaussian errors are negligible [@FO08.1].
We computed the weak lensing power spectra for the non-Gaussian models with local and equilateral shapes of the primordial bispectrum, using the three source redshift distributions described above and computing the matter power spectrum as described in the previous Sections. In Figure \[fig:wlNG\] we show the ratio of the power spectra computed in the non-Gaussian models to the reference Gaussian case. As before, the models have the maximal values of the parameter $f_\mathrm{NL}$ that are allowed by CMB constraints (see @KO09.1).
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The deviations from Gaussianity are very small, at the level of a few percent at most. Interestingly, if we exclude the small scale part of the plot, the deviation from the Gaussian model would be maximal at scales included between $\sim 10$ arcmin and $\sim 100$ arcmin, that is where the statistical error on the ratio is minimal (of the order of $\sim 10\%$ according to standard error propagation), hence we expect the bulk of the cosmological signal to come from this region. We note that the deviations from the Gaussian weak lensing power spectrum are consistent with the deviations on the three-dimensional matter power spectrum, which, e.g., in the local shape case, grow above $4\%$ only at $z>2$, where very few sources are present. The trend presented in Figure \[fig:wlNG\], namely of non-Gaussian power spectra being larger than the reference Gaussian one at intermediate scales and lower at very small scales is in qualitative agreement with the results of [@RE02.2], implying that this might be a generic feature of all non-Gaussian models with a positive skewness. We also note that virtually no difference in this result is due to the choice of the background source redshift distribution. Therefore from this moment on, unless noted otherwise, we focused uniquely on one distribution, namely the one of @BE07.2.
Weak lensing tomography
-----------------------
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It is possible to increase the amount of information that is obtainable from weak lensing surveys by employing weak lensing tomography [@HU99.1; @TA04.1]. This consists of subdividing the redshift distribution in several bins, compute the power spectra considering those sources that are in each bin only, and then combine the information from different redshift bins. More practically, Eq. (\[eqn:wl\]) can be generalized to consider the cross correlation power spectra for different redshift bins, as
$$\label{eqn:wlt}
C_l^{ij} = \frac{9H_0^4\Omega_{\mathrm{m},0}^2}{4 c^4} \int_0^{\chi_\mathrm{H}} P\left(\frac{l}{f_K(\chi)},\chi \right) \frac{W_i(\chi)W_j(\chi)}{a^2(\chi)} d\chi,$$
where the kernels now read
$$W_i(\chi) = \int_\chi^{\chi_\mathrm{H}} n_i(\chi') \frac{f_K(\chi-\chi')}{f_K(\chi')} d\chi'.$$
In the previous equation, the redshift distribution $n_i$ refers to the $i-$th redshift bin, and must be normalized such that
$$\int_0^{\chi_\mathrm{H}}n_i(\chi)d\chi = 1$$
for all $i$. Here, we considered three redshift bins, each one of which contains one third of the total amount of sources, adopting the distribution of @BE07.2. The two redshifts that separate the three bins are $z_1 = 0.60$ and $z_2 = 0.96$. In our case hence $i$ and $j$ run between $1$ and $n_z=3$. As noted in @MA06.1 and @SU09.1, the discriminating power of cosmic shear increases with increasing $n_z$ until $n_z = 5$, however in order to be conservative we decided not to make the redshift binning too fine, limiting ourselves to three redshift bins only.
In Figure \[fig:weakLensingZ\] we show the ratios of the non-Gaussian power spectra computed for each of the three bins considered here to the corresponding quantities evaluated in the Gaussian cosmology. As can be seen, for higher redshift bins the peak of the deviation tends to shift toward smaller angular scales. This could be naively expected, since higher source redshift bins include higher redshift matter power spectrum information, and according to Figure \[fig:ratio\] the peak of deviation between the Gaussian and non-Gaussian spectra shifts at smaller scales with increasing redshift. Nevertheless, no significant change in the maximum deviation is seen for the three source redshift bins considered here but, as we show below, combining information of different bins does improve the constraining power of cosmic shear. We additionally tried to recompute Figure \[fig:weakLensingZ\] by adopting the two other source redshift distributions detailed above. We found no practical change in the effect of primordial non-Gaussianity, implying that the three $n(z)$ are too similar to each other in order for one to be appreciably preferred over the others.
We attempted a statistical analysis in order to understand what kind of constraints can be put on the value of $f_\mathrm{NL}$ by using weak lensing only, and how these improve upon inclusion of the tomographic information. We stress here that this statistical analysis is simplified, since it does not include, e.g., a proper treatment of weak lensing shear systematics and the effect of marginalization over other cosmological parameters. For this analysis, we considered multipoles included in the range between $l_1 = 50$ and $l_2 = 3000$, since little cosmological information can be extracted outside this range. Additionally, at $l>l_2$ the effect of baryon physics, that we ignored, starts to be important (@WH04.1 [@ZH04.1; @JI06.1]), and non-Gaussian errors due to the coupling of different models caused by non-linear clustering, that we did not include, begin to be significant with respect to Gaussian errors (@WH00.1 [@CO01.1]).
Given all the above, the covariance matrix for weak lensing tomography has $n_z(n_z+1)n_l/2$ independent elements, where $n_l = l_2-l_1 = 2950$, and it can be written as [@HU06.1; @MA06.1; @SU09.1]
$$\begin{aligned}
\label{eqn:cov}
\Gamma\left(C_l^{ij},C_{l'}^{km}\right) &=& \frac{\delta_{ll'}}{(2l+1)\Delta lf_\mathrm{sky}} \left[ \left( C_l^{ik} + \delta_{ik}\frac{\gamma^2}{\bar{n}_i} \right)\left( C_l^{jm} + \delta_{jm}\frac{\gamma^2}{\bar{n}_j} \right) \right. +
\nonumber\\
&+& \left.\left( C_l^{im} + \delta_{im}\frac{\gamma^2}{\bar{n}_i} \right)\left( C_l^{jk} + \delta_{jk}\frac{\gamma^2}{\bar{n}_j} \right)\right]\equiv \Gamma_{\alpha\beta},\end{aligned}$$
where $\alpha$ and $\beta$ run from $1$ through $n_z(n_z+1)n_l/2$. We note that the presence of $\delta_{ll'}$ implies that no correlation is considered between different multipoles, and this is a consequence of the fact that we ignored the non-Gaussian part of the signal covariance. This means that, if we let the matrix indices run over all the redshift bin pairs for a fixed $l$ and then change $l$ and repeat the operation, the covariance matrix (and hence its inverse) is a block diagonal matrix, where individual blocks correspond to covariance matrices between different redshift bins for a fixed multipole. Here, as before, we adopted $\Delta l = 1$ and values of the three parameters $f_\mathrm{sky}$, $\gamma$, and $\bar{n}$ corresponding to the EUCLID goals.
According to this discussion, we can define a $\chi^2 \left(f_\mathrm{NL}\right)$ function as
$$\chi^2(f_\mathrm{NL}) = \sum_l \sum_{\alpha\beta}\left[\frac{}{}C_l^{\alpha} - y_l^{\alpha}(f_\mathrm{NL})\right]\Gamma^{-1}_{\alpha\beta}(l)\left[\frac{}{}C_l^{\beta} - y_l^{\beta}(f_\mathrm{NL})\right],$$
where $\alpha$ and $\beta$ now do not run from $1$ to $n_z(n_z+1)n_l/2$, but only from $1$ through $n_z(n_z+1)/2 = 6$, i.e., the number of redshift bin independent pairs. This kind of procedure is correct as long as we can neglect the non-Gaussian part of the covariance matrix (see @TA07.1 [@TA09.1] for details). We assumed the measured data $C_l^{ij}$ to be the weak lensing power and cross spectra computed in the Gaussian cosmology, and the models $y_l^{ij}(f_\mathrm{NL})$ to be the spectra computed in a given non-Gaussian model with a fixed value of $f_\mathrm{NL}$. To account for the fact that the measured power and cross spectra values would not be the exact theoretical values, at each multipole we randomly perturbed the values of the spectra around the fiducial theoretical values according to a Gaussian distribution with variance given by $\Gamma\left(C_l^{ij},C_{l'}^{ij}\right)$. We repeated this procedure $128$ times, each time changing the seed for the generation of random numbers, and used the average $\chi^2(f_\mathrm{NL})$ values for the subsequent analysis.
When no tomography is applied, i.e., we have only one redshift bin, then the covariance matrix takes the form
$$\Gamma\left( C_l,C_{l'} \right) = \frac{\delta_{ll'}}{(2l+1)\Delta lf_\mathrm{sky}}2\left( C_l + \frac{\gamma}{\bar{n}}\right)^2 = \delta_{ll'}\Delta C_l^2,$$
and hence the $\chi^2(f_\mathrm{NL})$ function reads $$\chi^2(f_\mathrm{NL}) = \sum_l \frac{\left[ C_l - y_l(f_\mathrm{NL})\right]^2}{\Delta C_l^2},$$ which is the standard $\chi^2(f_\mathrm{NL})$ definition.
Probe $68.3\%$ CL $90\%$ CL $95.4\%$ CL $99\%$ CL
----------------------------- ------------------------------- ------------------------------- ------------------------------- -------------------------------
Weak lensing power spectrum $|f_\mathrm{NL}| \lesssim 17$ $|f_\mathrm{NL}| \lesssim 28$ $|f_\mathrm{NL}| \lesssim 34$ $|f_\mathrm{NL}| \lesssim 43$
Weak lensing tomography $|f_\mathrm{NL}| \lesssim 14$ $|f_\mathrm{NL}| \lesssim 23$ $|f_\mathrm{NL}| \lesssim 28$ $|f_\mathrm{NL}| \lesssim 36$
Given the assumptions above, it follows that the minimum value of $\chi^2(f_\mathrm{NL})$ is reached for $f_\mathrm{NL} = 0$, and it is approximately equal to the number of degrees of freedom $\nu$. Hence, we can set $\chi^2_\mathrm{min} \equiv \chi^2(0) \simeq \nu$ and define a $\Delta \chi^2(f_\mathrm{NL})$ function as $\Delta \chi^2(f_\mathrm{NL}) \equiv \chi^2(f_\mathrm{NL})-\chi^2_\mathrm{min}$. In the simple weak lensing case we have $\nu = n_l - 1 = 2949$, while for weak lensing tomography $\nu = n_z(n_z+1)n_l/2 - 1 = 17699$. Were the non-Gaussian power and cross spectra identical to the Gaussian ones, then we would have had $\Delta \chi^2 = 0$. In any other case, $\Delta \chi^2 > 0$. The bigger $\Delta \chi^2$ is, the better the Gaussian and non-Gaussian models can be distinguished or, in other words, at the highest confidence the non-Gaussian model can be excluded provided we measure the Gaussian power and cross spectra. For the highest value of $f_\mathrm{NL}$ consistent with CMB data in the local case, $f_\mathrm{NL} = 145$, if no tomography is applied we found $\Delta\chi^2 \simeq 73$. For the equilateral shape, for which we assumed $f_{\mathrm{NL}} = 330$, we obtained $\Delta \chi^2 \simeq 150$. These very high values of $\Delta \chi^2(f_\mathrm{NL})$ mean that these non-Gaussian models would be excluded at a very high confidence level, which would seem at odds with the results presented in Figure \[fig:wlNG\]. As a matter of fact, we have shown in that Figure that the degree of deviation from the weak lensing power spectrum of the $\Lambda$CDM model due to primordial non-Gaussianity is much smaller than the statistical error, the former being of the order of a few percent while the latter being at least of $\sim 10\%$. However we have to remind that we are summing the signal over thousands of multipoles, hence even a very modest signal for a fixed multipole can bring to a significant integrated discriminative power. Introducing tomography and computing the $\Delta \chi^2(f_\mathrm{NL})$ as described above, the two aforementioned values raised to $\Delta \chi^2 = 104$ and $\Delta \chi^2 = 215$ respectively. The raise is expected, since adding tomographic information increases the discriminative power of the method.
![The $\Delta \chi^2(f_\mathrm{NL})$ function for the weak lensing power spectrum (black empty circles) and for the weak lensing tomography (red filled squares), computed assuming a non-Gaussian cosmology with local shape of the primordial bispectrum. The black solid and red dot-dashed lines represent the best fit parabolae, as detailed in the text, while the green dotted horizontal line represents the $\Delta \chi^2$ value corresponding to a $99\%$ Confidence Level detection. Note that in order to perform the fit, we also used points at $|f_\mathrm{NL}| > 100$, that are not visible in this plot.[]{data-label="fig:fit"}](Figures/fit_local){width="\hsize"}
Because of the computational cost in evaluating weak lensing and cross power spectra and in order to better relate $\chi^2$ intervals into constraints on $f_\mathrm{NL}$, we computed the value of $\Delta \chi^2 (f_\mathrm{NL})$ for several different values of $f_\mathrm{NL}$, and then fitted the obtained points with some analytic expression. The result of this procedure is reported in Figure \[fig:fit\], where we limited ourselves to the local shape of the primordial bispectrum since it is expected to give the larger effect for a fixed $f_\mathrm{NL}$ value. We found that a parabola is an excellent fit to the $\Delta \chi^2(f_\mathrm{NL})$ function, implying the interesting fact that the power and cross spectra change linearly with the level of non-Gaussianity $f_\mathrm{NL}$. This remains true despite the slight asymmetry seen in the computed points between positive and negative $f_\mathrm{NL}$ values, which is due to the limited number of Monte-Carlo $\chi^2$ realizations that we could compute. The symmetry in the $\Delta \chi^2(f_\mathrm{NL})$ function reflects the symmetry in the behavior of the matter power spectrum with $f_\mathrm{NL}$ that we discussed in Section \[sct:modeling\]. As one could naively expect, the $\Delta \chi^2(f_\mathrm{NL})$ function in the case of weak lensing tomography is narrower than when we consider the power spectrum only, implying that stronger constraints can be put in the former case.
Since we are operating with one single parameter and under the assumption of normally distributed errors, we can directly translate $\chi^2$ variations into Confidence Levels (CLs) for the amount of non-Gaussianity $f_\mathrm{NL}$. The kind of constraints that can be expected by a EUCLID-like survey by using weak lensing only are $|f_\mathrm{NL}| \lesssim$ few tens, with a $\sim 20\%$ improvement given by weak lensing tomography. In Table \[tab:par\] we summarize the outcome of our statistical analysis, reporting the constraints on $f_\mathrm{NL}$ at different CLs by using both the weak lensing power spectrum only and the weak lensing tomography. As can be seen, the constraints at $99\%$ CL are already a factor of $\sim 3$ better than current WMAP constraints on positive $f_\mathrm{NL}$ values [@KO09.1]. At $68\%$ CL, the constraints coming from cosmic shear are expected to be competitive with other future experiments based, e.g., on the Integrated Sachs-Wolfe effect [@SA67.1; @CA08.1]. Currently, the only constraints coming from probes alternative to, and claimed to be more powerful than the CMB constraints are detailed in @SL08.1. There, the authors make use of Large Scale Structure data, as probed by a combination of different tracers, getting $f_\mathrm{NL}<70$ at $95\%$ CL. According to Table \[tab:par\] our results show that the cosmic shear measurements by EUCLID should improve upon this of about a factor of $2$. In the future however, the leading constraints on the level of non-Gaussianity should still be mostly given by the CMB, with *Planck* predicted to detect deviations from Gaussianity at the level of $f_\mathrm{NL} \simeq$ few [@SE09.1].
Discussion and conclusions {#sct:discussion}
==========================
In this work we computed the weak lensing power spectrum in various cosmological models with non-Gaussian initial conditions, and the relative statistical uncertainties that are expected for future large area optical surveys. The underlying non-linear matter power spectrum was evaluated using the semi-analytic halo model, that by construction depends on the internal structure of dark matter halos. As mentioned in Section \[sct:modeling\], it is possible that this internal structure is different in non-Gaussian with respect to Gaussian models. For instance, there has been some indication in the literature that the concentration and/or inner logarithmic slope of dark matter halos in cosmological models with primordial non-Gaussianity and positive skewness is larger than in the standard $\Lambda$CDM cosmology [@AV03.1]. This is intuitively in agreement with the fact that in such models it is easier to have high-density peaks, that should cross earlier the threshold for collapse, with the corresponding structures having more time to relax and compactify. A larger halo concentration would bring more power at very large wavenumbers, modifying the weak lensing power spectrum at small angular scales. On the other hand, it is not straightforward if and to what extent this expectation is fulfilled in arbitrary non-Gaussian models, and this should be verified for the models at hand with cosmological $n$-body simulations. We plan to explore this issue in the future with the numerical simulations presented in [@GR07.1] (see also @GR08.2). In Any case, the effect of a different inner halo structure should show up at scales where the statistical error is very large, thus affecting our conclusions only in a minor way.
As partly discussed in the main paper body, the matter power spectrum has been estimated in numerical simulations of non-Gaussian cosmologies. However, in order to get the normalization right, the box sizes of the simulations need to be very large, implying a generically poor mass resolution. This does not allow to estimate the matter power spectrum below quasi-linear scales. A similar limitation characterizes renormalized perturbation theory, where the computation of the matter power spectrum at non-linear scales would require many loop corrections, that have not been computed yet. Semi-analytic fit to numerical simulations have been obviously calibrated in Gaussian scenarios, and it is not clear how they could be meaningfully extended to non-Gaussian models without new calibrations. Therefore, the halo model, being physically motivated, seems the only reasonable way to describe the matter power spectrum down to fully non-linear scales, despite the various limitations that have been discussed in detail in Section \[sct:modeling\].
In relation to this, a line of investigation that is certainly worth exploring in the future is the definition of a unique prescription for computing the matter power spectrum on all scales, that would be in agreement with simulated power spectra and the mean structure of $n$-body dark matter halos. Such prescription should also be physically motivated, in order to be straightforwardly able to comprehend the effect of baryons, dark energy and primordial non-Gaussianity. This task is going to become increasingly important, as the precision of galaxy redshift and weak lensing surveys increases.
Summarizing our results, we found that primordial non-Gaussianity has little effect on the matter power spectrum, and hence also on the cosmic shear power spectrum. This conclusion is unaffected by the choice of the background source redshift distribution, as long as the latter is observationally reasonable. Also, the shape of the primordial bispectrum seem not to have significant incidence on this qualitative conclusion. Summing the signal over a large number of multipoles can help to beat down the noise, providing a $1-$sigma detection for a level of non-Gaussianity $|f_\mathrm{NL}|\simeq 17$, if local shape for the primordial bispectrum is assumed and all other cosmological parameters are held fixed. Including weak lensing tomography can increase the constraining power of cosmic shear of $\sim 20\%$.
These constraints are probably still looser than those that will be put with the study of e.g., number counts and correlation function of galaxy clusters in future X-ray surveys (Sartoris et al., in preparation). However, it is likely that combining these probes with cosmic shear can help breaking the degeneracy between, for instance, $f_\mathrm{NL}$ and $\sigma_8$. A more complete statistical analysis than that performed here is necessary in order to understand if and at what level this is confirmed. Finally, it is our future plan to compare the constraints given in this paper with those that can be obtained from the abundance of the S/N peaks in cosmic shear maps, a cosmological probe that has attracted some attention in the literature recently (@BE09.1; @DI09.1 [@MA09.1]). The occurrence of shear peaks depends not only on the power spectrum of large scale matter distribution, but also on the abundance of massive dark matter halos, hence it is expected to have a more constraining power with respect to cosmic shear alone.
Acknowledgments {#acknowledgments .unnumbered}
===============
We acknowledge financial contributions from contracts ASI-INAF I/023/05/0, ASI-INAF I/088/06/0, and ASI “EUCLID-DUNE” I/064/08/0. We are grateful to M. Bartelmann, C. Carbone, and G. Zamorani for very useful discussions. We also wish to thank A. Amara and A. Refregier for reading the manuscript and for helpful comments. We acknowledge the anonymous referee for very useful comments that helped improving the presentation of our work.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We exploit the intrinsic difference between disordered and crystalline solids to create systems with unusual and exquisitely tuned mechanical properties. To demonstrate the power of this approach, we design materials that are either virtually incompressible or completely auxetic. Disordered networks can be efficiently driven to these extreme limits by removing a very small fraction of bonds via a selected-bond removal procedure that is both simple and experimentally relevant. The procedure relies on the nearly complete absence of any correlation between the contributions of an individual bond to different elastic moduli. A new principle unique to disordered solids underlies this lack of correlation: independence of bond-level response.'
author:
- 'Carl P. Goodrich'
- 'Andrea J. Liu'
- 'Sidney R. Nagel'
title: 'Tuning by pruning: exploiting disorder for global response and the principle of bond-level independence'
---
The properties of amorphous solids are essentially and qualitatively different from those of simple crystals [@Goodrich:2014fl]. In a crystal, identical unit cells are interminably and symmetrically repeated, ensuring that all cells make identical contributions to the solid’s global response to an external perturbation [@Ashcroft:1976ud; @kittel2004introduction]. Unless a crystal’s unit cell is very complicated, all particles or inter-particle bonds contribute nearly equally to any global quantity, so that each bond plays a similar role in determining the physical properties of the solid. For example, removing a bond in an ordered array or network decreases the overall elastic strength of the system, but in such a way that the resistance to shear and the resistance to compression drop in tandem [@Feng:1985vr] so that their ratio is nearly unaffected. Disordered materials are not similarly constrained. We will show that as a consequence, one can exploit disorder to achieve a unique, varied, textured and tunable global response.
A tunable global response is a corollary to a new principle that emerges for disordered matter: independence of bond-level response. This independence refers not only to the dearth of strong correlations between the response of different bonds, but also, and more importantly, to the response of any specific bond to different external perturbations. We will demonstrate this by constructing selected-bond-removal networks, where individual bonds, or springs, are successively removed to drive the overall system into different regimes of behavior, characterized by ratios of different mechanical responses. Starting from the same initial network, we can remove as few as 2% of the bonds to produce a network with a ratio of the shear to bulk modulus, $G/B$, that is either nearly zero (incompressible limit) or nearly infinite (maximally auxetic [@Greaves:2011ku]) merely by removing different sets of bonds. Moreover, by using different algorithms or starting with different configurations, we find that the region within which the bonds are removed can be confined to strips of controllable size, ranging from a few bond lengths to the size of the entire sample. This has the practical consequence that one can achieve precise spatial control in tuning properties of the material from region to region within the network–as is needed for creating origami [@Witten:2007cq; @Mahadevan:2005hr] or kirigami [@Castle:2014jg] materials.
We construct networks numerically by starting with a configuration of particles produced by a standard jamming algorithm [@OHern:2003vq; @Liu:2010jx]. We place $N$ soft repulsive particles at random in a box of linear size $L$ and minimize the total energy until there is force balance on each particle. We work in either two or three dimensions and start with a packing fraction, $\phi$, that is above the jamming density. After minimizing the energy of a configuration, we create a network by replacing each pair of interacting particles with an unstretched spring of unit stiffness between nodes at the particle centers [@Wyart:2005jna]. We characterize the network by the excess coordination number ${\Delta Z}\equiv Z - Z_\text{iso}$, where $Z$ is the average number of bonds at each node and $Z_\text{iso} \equiv 2d - 2d/N$ is the minimum for a system to maintain rigidity in $d$ dimensions [@Goodrich:2012ck].
For each network, we use linear response to calculate the contribution $B_i$ of each bond $i$ to the bulk modulus, $B=\sum_i B_i$ (see Appendix for details). The distribution of $B_i$ in three dimensions is shown in blue in Fig. \[fig:Ri\_distributions\_3d\], where data are averaged over 500 configurations, each with approximately 4000 nodes and an initial excess coordination number ${\Delta Z_\text{initial}}\approx 0.127$ (corresponding to a total number of bonds that is about 2% above the minimum needed for rigidity).
Similarly, we can start with the same initial network and calculate $G_i$, the contribution of each bond to the angle-averaged shear modulus, $G=\sum_i G_i$. (A finite system is not completely isotropic, so the shear modulus varies with direction [@DagoisBohy:2012dh]; we calculate the angle-averaged shear modulus, which approaches the isotropic shear modulus in the infinite system size limit [@Goodrich:2014iu].) The resulting distribution for $G_i$ is shown in purple in Fig. \[fig:Ri\_distributions\_3d\]. Note that the distributions of the bond contributions to $B$ and $G$ are continuous, very broad, and non-zero in the limit $B_i,G_i \rightarrow 0$. That is, some bonds have nearly zero contribution to the bulk or shear modulus while others contribute disproportionately. For both $B$ and $G$, the distribution decays as a power law at low values of $B_i$ or $G_i$. These power laws are terminated above ${\left< B_i \right>}$ and ${\left< G_i \right>}$ by approximately exponential cut-offs. In comparison, the distributions for a perfect crystal would be composed of discrete delta functions.
![\[fig:Ri\_distributions\_3d\]Bond-level response. Distribution on a log-log scale (inset: log-linear scale) of the contribution of each bond to the macroscopic bulk and shear moduli, $B_i$ and $G_i$, for $3d$ networks with ${\Delta Z_\text{initial}}\approx 0.127$. Here $i$ indexes bonds. At low $B_i$ or $G_i$, the distributions follow power-laws with exponents $-0.51$ and $-0.38$, respectively. At high values, the distributions decay over a range that is broad compared to their means, ${\left< B_i \right>}$ and ${\left< G_i \right>}$.](fig1.pdf){width="\linewidth"}
![\[fig:bond\_level\_independence\]Independence of bond-level response. ([**A**]{}) Joint probability distribution of $B_i$ and $G_i$ for $3d$ networks with ${\Delta Z_\text{initial}}\approx 0.127$. There is little apparent correlation between the response to compression ($B_i$) and to shear ($G_i$) for a given bond $i$. ([**B**]{}) The value of $G$ when bonds with the largest (purple squares) and smallest (purple circles) $B_i$ are removed is nearly indistinguishable from $G$ when bonds are removed at random (purple crosses). Similarly, $B$ is very similar whether bonds with the largest $G_i$ (blue triangles) are removed or bonds are removed at random (blue pluses).](fig2_combined2.pdf){width="0.9\linewidth"}
![\[fig:Global\_response\]Tuning global response in three dimensions. The ratio of shear to bulk modulus, $G/B$, for four pruning algorithms. Error bars (included) are smaller than the symbols. Lines are fits to the data over the indicated range and have slopes, from top to bottom, of -7.96, -0.01, 1.01, and 1.82. Starting with the same initial conditions, we can tune global response by 16 orders of magnitude by pruning of order 2% of the bonds.](fig3.pdf){width="0.9\linewidth"}
![\[fig:Global\_response\_2d\]Tuning global response in two dimensions. The ratio of shear to bulk modulus, $G/B$, for four pruning algorithms. Error bars (included) are smaller than the symbols. Lines are fits to the data over the indicated range and have slopes, from top to bottom, of -5.36, -0.26, 1.27, and 3.05. Starting with the same initial conditions, we can tune global response by 17 orders of magnitude by pruning of order 1% of the bonds.](fig3_2d.pdf){width="0.9\linewidth"}
We next ask if there is a correlation between how an individual bond responds to shear and how it responds to compression. Do bonds with a large contribution to the bulk modulus also have a proportionately large contribution to the shear modulus? Fig. \[fig:bond\_level\_independence\]a shows the joint probability distribution $P(B_i,G_i)$. There is a nearly vanishing (but not identically zero) correlation between how individual bonds respond to shear and how they respond to compression. This is qualitatively different from what one would find for a simple crystal. Thus, Fig. \[fig:bond\_level\_independence\]a illustrates a previously-unrecognized property that is very well obeyed by disordered networks: independence of bond-level response.
This new property suggests that one can tailor the behavior of the network by selectively removing (pruning) those bonds that contribute more or less than the average to one of the moduli. By so doing, one can decrease one modulus with respect to the other.
First, we consider the known case of *rigidity percolation* [@Feng:1985vr; @Ellenbroek:2009to; @Ellenbroek:2014uh], where a bond is picked at random and removed. This pruning is repeated until the system becomes unstable at ${\Delta Z}= 0$. We have implemented a slight variation to this procedure: at each step, a bond is removed only if each node connected to this bond has at least $d+1$ remaining bonds in $d$ dimensions. This is the condition for local stability of a particle in the original jammed packing [@Levine2001]. As the excess coordination number decreases, the bulk and shear moduli vanish together, so that $G \sim B \sim {\Delta Z}$ [@Feng:1985vr; @Ellenbroek:2009to; @Ellenbroek:2014uh] (see Fig. \[fig:bond\_level\_independence\]b). Therefore, as shown in Fig. \[fig:Global\_response\], the ratio $G/B$ is independent of ${\Delta Z}$.
We now implement the idea of *selected*-bond removal in a variety of ways. First we remove the bond with the smallest $B_i$, namely the weakest contribution to the bulk modulus (provided, as above, that each node connected to this bond has at least $d+1$ remaining bonds). Since the distribution $P(B_i)$ is continuous and nonzero as $B_i \rightarrow 0$, the bond removal has almost no effect on the bulk modulus. However, since there is little correlation between the contribution of each bond to the bulk and shear moduli, there is a much larger effect on the shear modulus. The contributions $B_i$ and $G_i$ of the remaining bonds to the moduli are then recalculated and the procedure is repeated to remove the bond with the smallest $B_i$. Figure \[fig:bond\_level\_independence\]b shows that when bonds with the smallest $B_i$ are successively removed, the *shear* modulus linearly proportional to $\Delta Z$. Furthermore, it is quantitatively identical, within numerical precision, to when bonds are removed at random. The ability to alter the scaling of the bulk modulus without affecting the scaling of the shear modulus is a clear demonstration that the principle of independence of bond-level response allows for very precise tuning of global properties. Since removing bonds with the smallest $B_i$ has little effect on the bulk modulus, we would expect $G/B \rightarrow 0$ as ${\Delta Z}\rightarrow 0$. As shown in Fig. \[fig:Global\_response\], we find that $G/B \sim {\Delta Z}^{{\mu_{B_-}}}$, with ${{\mu_{B_-}}}= 1.01 \pm 0.01$. This behavior is identical to the scaling found in the original jammed sphere packings, where ${\Delta Z}$ is lowered by decompressing the system. In decompressing a jammed packing, this suggests that the contacts most likely to disappear are those which contribute minimally to the bulk modulus, providing theoretical insight into why jamming has anomalous $G/B$ behavior.
We can drive the same initial network to the opposite limit, $G/B \rightarrow \infty$, by successively removing bonds with the *largest* contribution to $B$. As before, independence of bond-level response predicts that the shear modulus will again decrease linearly with ${\Delta Z}$, as we indeed find (see Fig. \[fig:bond\_level\_independence\]b). However, the bulk modulus will decrease more quickly, as prescribed by the high $B_i$ tail of the distribution, suggesting that the ratio $G/B$ should *increase*. The result of this successive bond-removal algorithm is shown by the blue squares in Fig. \[fig:Global\_response\]. We find that $G/B \sim {\Delta Z}^{{{\mu_{B_+}}}}$, where ${{\mu_{B_+}}}= -7.96 \pm 0.01$. Thus, the increase in $G/B$ occurs with a *much* steeper power law than the decrease of $G/B$ when the bond with the smallest contribution to $B$ is removed. This power law implies that the distribution $P(B_i/\left<B_i\right>)$ evolves as bond pruning proceeds.
The algorithms mentioned above can be extended in a number of ways. For example, one can remove the bond with the largest contribution to the shear modulus to drive $G/B$ towards zero. In this case, independence of bond-level response implies that the bulk modulus would respond as if bonds were removed randomly, so that $B \sim {\Delta Z}$ (see Fig. \[fig:bond\_level\_independence\]b). However, the shear modulus decreases more rapidly; we find $G/B \sim {\Delta Z}^{{{\mu_{G_+}}}}$, where ${{\mu_{G_+}}}= 1.82 \pm 0.01$ (purple diamonds in Fig. \[fig:Global\_response\]).
We can also tune two-dimensional networks with equal ease. We construct spring networks in two dimensions with approximately 8000 nodes and an initial coordination number of ${\Delta Z_\text{initial}}\approx 0.047$, which is about 1% above the minimum needed for rigidity. Figure \[fig:Global\_response\_2d\] shows $G/B$ as bonds are pruned towards ${\Delta Z}\rightarrow 0$ for the same four selected-bond removal algorithms as in Fig. \[fig:Global\_response\]. When bonds with the smallest $B_i$ are removed, we find that $G/B \sim {\Delta Z}^{{\mu_{B_-}}}$ with ${{\mu_{B_-}}}= 1.27 \pm 0.01$. This is close to the behavior known for jammed packings ($G/B \sim {\Delta Z}^1$), though it is certainly not as clean as in three dimensions. When we prune bonds that resist compression the most (largest $B_i$), we find that $G/B \sim {\Delta Z}^{{\mu_{B_+}}}$, where ${{\mu_{B_+}}}= -5.36 \pm 0.01$. At the smallest ${\Delta Z}$, $G/B \sim 10^{10}$. Finally, when bonds with the largest $G_i$ are removed we find that $G/B \sim {\Delta Z}^{{\mu_{G_+}}}$, with ${{\mu_{G_+}}}= 3.05 \pm 0.01$.Although $G/B$ diverges/vanishes with slightly different power laws in two dimensions, the overall effect is no less drastic.
Note that our procedures are remarkably efficient in tuning $G/B$. Figures \[fig:Global\_response\] and \[fig:Global\_response\_2d\] show that by removing less than 2% of the bonds in three-dimensional networks we can obtain a difference of more than 16 orders of magnitude in the tuned value of $G/B$, depending on which bonds we prune. In two dimensions, pruning is similarly efficient; starting with the same initial configuration we are able to obtain differences in $G/B$ that span over 17 orders of magnitude by pruning only $\sim 1\%$ of the bonds.We also note that our bond-cutting procedures do not create any zero-frequency vibrational modes in the system, which would herald an instability in the structure. The limit $G/B \rightarrow 0$ corresponds to the incompressible limit of a solid where the Poisson ratio, $\nu= (d - 2 G/B)/ [d (d-1)+ 2 G/B ]$ in $d$ dimensions, reaches its maximum value of $\nu=+1$ (in $2d$) or $+1/2$ (in $3d$). The limit $G/B \rightarrow \infty$ corresponds to the auxetic limit where the Poisson ratio reaches its minimum value of $\nu=-1$. By using these different pruning algorithms, we can tailor networks to have any Poisson ratio between these two limits. This ability provides great flexibility in the design of network materials.
We turn now to spatial correlations between cut bonds. Driscoll [*et al.*]{} [@Driscoll:2015vf] have conducted numerical simulations in which they removed bonds with the [*largest*]{} strain under uniaxial or isotropic compression or shear. They showed that the cut bonds form a damage zone whose width increases and diverges as the initial excess coordination number, ${\Delta Z_\text{initial}}\rightarrow 0$; for sufficiently small ${\Delta Z_\text{initial}}$, the pruned bonds are homogeneously distributed throughout the entire system. Outside this zone, they found that the network is essentially unaffected.
When pruning bonds with the [*largest*]{} contribution to $B$ or $G$, all the data presented thus far are for systems with a sufficiently small ${\Delta Z_\text{initial}}$ so that the distribution of the cut bonds appears homogeneous. In our simulations with large ${\Delta Z_\text{initial}}$, where the damage zone is smaller than the size of our system, we find that $G/B$ still diverges/vanishes, but does so when $\Delta Z >0$. When we remove the bond with the [*smallest*]{} contribution to $B$ or $G$, the bonds are initially removed homogeneously throughout the system, independent of ${\Delta Z_\text{initial}}$. The existence of tunable strong spatial correlations in the cut bonds, as found by Driscoll [*et al.*]{} [@Driscoll:2015vf], allows one to create textured materials spatially varying mechanical properties. One region may be highly incompressible while a nearby region may be highly auxetic. This offers a great variety in the mechanical response of these networks.
For many materials [@Greaves:2011ku] the Poisson ratio decreases with increased connectivity of the constituent particles and increases with packing density. We note that neither of these correlations hold for the algorithms we have introduced for tuning the Poisson ratio (or ratio of shear and bulk moduli). We can reach $G/B \rightarrow \infty$ (minimum Poisson ratio) or $G/B \rightarrow 0$ (maximum Poisson ratio) by removing the same number of bonds from the same starting configuration. Neither the overall connectivity nor the overall density is different in the two final states. Thus, our procedures for producing tunable Poisson ratio materials are fundamentally different from correlations considered in the literature.
We have presented a number of ways of tuning $G/B$. Our results suggest that these ideas may be extended to other global properties ([*e.g.*]{}, thermal expansion or electrical response [@DEARCANGELIS:1985uh; @DEARCANGELIS:1986vk]) where the response can be written in terms of sums over bond contributions. As long as there is independence of bond-level response, one should be able to tune the ratio of global properties by using the same protocol of removing bonds that are especially susceptible (or especially unsusceptible) to a given global perturbation.
Our results demonstrate that disordered networks provide particularly elegant opportunities for constructing mechanical metamaterials with tunable, flexible and spatially textured response. However, the algorithms we have presented may not be restricted to artificially constructed materials. For example, compressing a network composed of springs that fail when stressed past a given threshold would result in the same network as removing springs with the largest $B_i$, provided that the threshold is sufficiently small. It is also not beyond imagination that one could selectively break bonds at the nano-scale level in response to global perturbations in complex solids. Indeed, biology appears to be able to target structures in networks that are under particularly high stress and to enhance their strength (such as in trabecular bone [@Keyak:2013ga]). Alternatively, there may be mechanisms to buckle or sever strongly stressed fibers (such as in actin networks [@Lenz:2012df]). It is interesting to ask if such selective repair or destruction of biological structures changes ratios of different mechanical responses such as the Poisson ratio.
Calculation of bond-level elastic response
==========================================
We consider networks of nodes connected by unstretched central-force springs with stiffness $k=1$. Let $\vec{\delta r}_i$ be the total strain on bond $i$ when the system is deformed according to some strain tensor $\epsilon_{\alpha\beta}$. The change in energy of the network is then given to lowest order by [$$\begin{aligned}
\Delta E = \sum_{i} k_i \delta r_{i,\parallel}^2, \label{eq:energy_change} \end{aligned}$$]{} where $\delta r_{i,\parallel}$ is the component of $\vec {\delta r}_i$ that is parallel to the bond direction. Thus, the bond that contributes the most (least) to the response to a given boundary deformation is the one with the largest (smallest) $\delta r_{i,\parallel}^2$. To remove the bond that contributes the most to the bulk modulus, for example, one would remove the bond with the largest $\delta r_{i,\parallel}^2$ under compression. This procedure can be implemented in either a simulation or an experiment.
In practice, for our computations, we use linear algebra to calculate the response of each bond more efficiently, as follows. The bulk elasticity of a system is described to linear order by the elastic modulus tensor $c_{\alpha\beta\gamma\delta}$, so that if the system is distorted by the symmetric strain tensor $\epsilon_{\alpha\beta}$, the change in energy is given to leading order by [$$\begin{aligned}
\Delta E/V = \frac 12 \epsilon_{\alpha\beta}c_{\alpha\beta\gamma\delta}\epsilon_{\gamma\delta}, \end{aligned}$$]{} where $V$ is the volume of the system. In general, there are 6 (21) independent components of the elastic modulus tensor in two (three) dimensions, but in the isotropic limit this reduces to just the bulk modulus $B$ and the shear modulus $G$.
The components of $c_{\alpha\beta\gamma\delta}$ are calculated from the change in energy of the system under various boundary deformations using Eq. \[eq:energy\_change\]. The strain $\vec{\delta r}_i$ can be decomposed into two distinct parts. First there is an affine strain set directly by the strain tensor. However, this results in a nonzero net force, $\vec f_m$, on each node $m$, leading to a secondary non-affine response. This non-affine response is calculated by solving the following system of equations [$$\begin{aligned}
\mathcal{M}_{mn} \vec u^\text{NA}_{m} = \vec f_n, \end{aligned}$$]{} where $\mathcal{M}_{mn}$ is the Hessian matrix and $\vec u^\text{NA}_m$ is the non-affine displacement of each node. The total strain $\vec {\delta r}_i$ of bond $i$ is calculated from the sum of the affine and non-affine displacements of the two nodes that the bond connects. Since $\Delta E$ can be written as a sum over bonds, so too can the elastic modulus tensor: [$$\begin{aligned}
c_{\alpha\beta\gamma\delta} = \sum_i c_{i,\alpha\beta\gamma\delta}. \end{aligned}$$]{} Under the deformation $\epsilon_{\alpha\beta}$, the change in energy of bond $i$ is [$$\begin{aligned}
\Delta E_i = \frac 12 \epsilon_{\alpha\beta}c_{i,\alpha\beta\gamma\delta}\epsilon_{\gamma\delta}. \end{aligned}$$]{} $c_{i,\alpha\beta\gamma\delta}$ thus completely describes the bond-level elastic response for bond $i$, and can be used to calculate the quantities $B_i$ and $G_i$ considered in the main text.
The global bulk and shear moduli are linear combinations of the components of the elastic modulus tensor. In two dimensions, they are [$$\begin{aligned}
B &= \tfrac 14 \left( c_{xxxx} + c_{yyyy} + 2c_{xxyy}\right) \\
G &= \tfrac 18 \left( 4c_{xyxy} + c_{xxxx} + c_{yyyy} - 2c_{xxyy}\right),
\end{aligned}$$]{} while in three dimensions they are [$$\begin{aligned}
B ={}& \tfrac 19 \left( c_{xxxx} + c_{yyyy} + c_{zzzz} + 2c_{yyzz} + 2c_{xxzz} + 2c_{xxyy}\right) \\
G ={}& \tfrac 1{15} \left( 3c_{yzyz} + 3c_{xzxz} + 3c_{xyxy} \right. \nonumber \\
& \left. +\, c_{xxxx} + c_{yyyy} + c_{zzzz} - c_{yyzz} - c_{xxzz} - c_{xxyy}\right).
\end{aligned}$$]{} Finite disordered systems are never perfectly isotropic, so the shear modulus always has some dependence on the angle of shear. The above expressions for $G$ represent the angle-averaged shear modulus, which reduces to the shear modulus in the isotropic limit of infinite system size. We calculate the contribution of bond $i$ to the bulk and shear moduli in exactly the same way: [$$\begin{aligned}
B_i &= \tfrac 14 \left( c_{i,xxxx} + c_{i,yyyy} + 2c_{i,xxyy}\right) \\
G_i &= \tfrac 18 \left( 4c_{i,xyxy} + c_{i,xxxx} + c_{i,yyyy} - 2c_{i,xxyy}\right),
\end{aligned}$$]{} in two dimensions, and [$$\begin{aligned}
B_i ={}& \tfrac 19 \left( c_{i,xxxx} + c_{i,yyyy} + c_{i,zzzz} + 2c_{i,yyzz} + 2c_{i,xxzz} + 2c_{i,xxyy}\right) \\
G_i ={}& \tfrac 1{15} \left( 3c_{i,yzyz} + 3c_{i,xzxz} + 3c_{i,xyxy} \right. \nonumber \\
& \left. +\, c_{i,xxxx} + c_{i,yyyy} + c_{i,zzzz} - c_{i,yyzz} - c_{i,xxzz} - c_{i,xxyy}\right)
\end{aligned}$$]{} in three dimensions.
We thank Bryan Chen, Michelle Driscoll, Heinrich Jaeger and Vincenzo Vitelli for important discussions. This research was supported by the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Awards DE-FG02-05ER46199 (A.J.L., C.P.G.) and DE-FG02-03ER46088 (S.R.N.). This work was partially supported by a grant from the Simons Foundation (\#305547 to A.J.L.).
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We show that a star orbiting close enough to an adiabatically grown supermassive black hole (SMBH) can capture weakly interacting massive particles (WIMPs) at an extremely high rate. The stellar luminosity due to annihilation of captured WIMPs in the stellar core may be comparable to or even exceed the luminosity of the star due to thermonuclear burning. The model thus predicts the existence of unusual stars, essentially WIMP burners, in the vicinity of a SMBH. We find that the most efficient WIMP burners are stars with degenerate electron cores, e.g. white dwarfs (WDs); such WDs may have a very high surface temperature. If found, such stars would provide evidence for the existence of particle dark matter and can possibly be used to establish its density profile. On the other hand, the lack of such unusual stars may provide constraints on the WIMP density near the SMBH, as well as the WIMP-nucleus scattering and pair annihilation cross-sections.'
author:
- 'Igor V. Moskalenko'
- 'Lawrence L. Wai'
title: Dark matter burners
---
Introduction
============
The nature of the non-baryonic dark matter, which dominates the visible matter by about 4:1, is perhaps the most interesting experimental challenge for contemporary particle astrophysics. A hint for a solution has been found in particle physics where the WIMPs arise naturally in supersymmetric extensions of the Standard Model [e.g., @haber-kane], among other possibilities. The WIMP is typically defined as a stable, electrically neutral, massive particle. Assuming that non-baryonic dark matter is dominated by WIMPs, the pair annihilation cross-section is related to the observed relic density [@jkg96; @bergstrom00]. A pair of WIMPs can annihilate producing ordinary particles and [$\gamma$-rays]{}.
WIMPs are expected to form high density clumps according to N-body simulations of test particles with only gravitational interactions [@Navarro97; @Moore99]. The highest density “free space” dark matter regions occur for dark matter particles captured within the gravitational potential of adiabatically grown SMBHs [@gs99; @gp04; @bm05]. Higher dark matter densities are possible for dark matter particles captured inside of stars or planets. Any star close enough to a SMBH can capture a large number of WIMPs during a short period of time. Annihilation of captured WIMPs may lead to considerable energy release in stellar cores thus affecting the evolution and appearance of such stars.
Such an idea has been first proposed by @salati89 and further developed by @bouquet89 who applied it to main-sequence stars. The model led to the conclusion of suppression of stellar core convection, thus predicting a concentration of stars in the Galactic Center masquerading as cold red giants.
An order-of-magnitude estimate of the WIMP capture rates for stars of various masses and evolution stages [@MW06] lead us to the conclusion that WDs, fully burned stars without their own energy supply, are the most promising candidates to look for. In this paper we calculate the WIMP capture by WDs located in a high density dark matter region, and discuss their observational features. We use current limits on WIMP-nucleus interaction and WIMP annihilation cross sections, as well as recent estimates of WIMP energy density near an adiabatically grown SMBH.
{width="98.00000%"}
WIMP accumulation in stars
==========================
In a steady state the WIMP capture rate $C$ is balanced by the annihilation rate [@gs87] $$C=A N_\chi^2,
\label{balance}$$where $$A=\frac{{\langle\sigma_a v\rangle}}{\pi^{3/2} r_\chi^3},
\label{A}$$${\langle\sigma_a v\rangle}$ is the velocity averaged WIMP pair annihilation cross-section, the effective radius $$r_\chi=c \left(\frac{3T_c}{2\pi G\rho_c m_\chi}\right)^{1/2}$$is determined by matching the star core temperature $T_c$ with the gravitational potential energy (assuming thermal equilibrium), $c$ is the speed of light, $G$ is the gravitational constant, $\rho_c$ is the star core density, and $m_\chi$ is the WIMP mass. The total number of WIMPs captured by a star is $$N_\chi=C \tau_{eq} \tanh(\tau_*/\tau_{eq}),
\label{Nchi}$$where $\tau_*$ is the star’s age, and the equilibrium time scale is given by $$\tau_{eq}=(CA)^{-1/2}.
\label{taueq}$$ The number density distribution of WIMPs can be estimated as [@ps85; @gs87; @bottino02]: $$n_\chi(r)=n_\chi^c \exp(-r^2/r_\chi^2),
\label{chidensity}$$where $n_\chi^c=N_\chi/V_{\rm eff}$ is the central WIMP number density. In thermal equilibrium, the effective radius $r_\chi=r_T$ is determined by the core temperature $T_c$ and density $\rho_c$ $$r_T=c \left(\frac{3T_c}{2\pi G\rho_c m_\chi}\right)^{1/2},\nonumber
\label{rT}$$where $c$ is the speed of light, $G$ is the gravitational constant, and $m_\chi$ is the WIMP mass. Limits from direct detection of dark matter on the WIMP-nucleon cross-section imply that only a fraction of the WIMPs crossing the star will scatter and be captured. The capture rate for a Maxwellian WIMP velocity distribution (in the observer’s frame) by a star moving with an arbitrary velocity $v_*$ relative to the observer is given by [@gould87]: $$C=4\pi \int_0^{R_*} dr\, r^2\, \frac{dC(r)}{dV},
\label{gould2.27}$$where $$\begin{aligned}
\frac{dC(r)}{dV}&=&
\left(
\frac{6}{\pi}
\right)^{1/2}
\sigma_0 A_n^4 \frac{\rho_*}{M_n}\frac{\rho_\chi}{m_\chi}
\frac{v^2(r)}{\bar{v}^2} \frac{\bar{v}}{2\eta A^2}
\label{gould2.24}\\
&\times&
\left\{
\left(
A_+A_- -\frac12
\right)
\left[
\chi(-\eta,\eta) -\chi(A_-,A_+)
\right]
\right.\nonumber\\
&+&\left.
\frac12A_+e^{-A_-^2} -\frac12A_- e^{-A_+^2}-\eta e^{-\eta^2}
\right\},\nonumber\\
A^2&=& \frac{3v^2(r)\mu}{2\bar{v}^2\mu_-^2},\nonumber\\
A_\pm&=&A\pm\eta,\nonumber\\
\eta&=&\frac{3v_*^2}{2\bar{v}^2}, \nonumber\\
\chi(a,b)&=&\int_a^b dy\, e^{-y^2}=
\frac{\sqrt\pi}{2}[{\rm erf}(b)-{\rm erf}(a)],
\nonumber\end{aligned}$$$\rho_\chi$ is the ambient WIMP energy density, $A_n$ is the atomic number of the star’s nuclei, $M_n$ is the nucleus mass, $\bar{v}$ is the WIMP velocity dispersion, and $\mu=m_\chi/M_n$, $\mu_-=(\mu-1)/2$. The escape velocity at a given radius $r$ *inside* of a star is given by $$v(r)=
\left[
2G\int_{V_*} dV\, \frac{\rho_*(r)}{r}
\right]^{1/2}
=
\left[
\frac{GM_*}{R_*}
\left(
3 -\frac{r^2}{R_*^2}\right)
\right]^{1/2},
\label{vesc}$$where we assumed the same mass density $\rho_*=M_*/V_*$ and the same chemical composition over the entire scattering volume $V_*$. This is a reasonable assumption for a degenerate electron core. Near a SMBH, where orbital motion around a single mass dominates, the test particle (WIMP or star) velocities are Keplerian $v_*=\bar{v}$; in this case $\eta=3/2$, although the exact value does not significantly change the result. The value of the spin-independent WIMP-nucleon scattering cross-section $\sigma_0$ is limited by direct detection experiments, i.e. less than $10^{-43}$ cm$^2$ [@cdms06]. If the star is composed of nuclei with atomic number $A_n$, the cross section increases by a coherent factor of $A_n^4$.
If a WD is heavy ($M\ga M_\odot$) and/or $A_n\gg1$, almost all WIMPs crossing the star will be captured. In this case, the WIMP capture rate is determined by the geometrical limit $\pi R_*^2$ rather than the total interaction cross section $\sigma_0 A_n^4 M_*/M_n$. We thus use a modified interaction cross section $\sigma'_0$ defined as $$\sigma'_0 A_n^4\frac{M_*}{M_n}
=\min\left(\sigma_0 A_n^4\frac{M_*}{M_n},\pi R_*^2\right).
\label{geom_limit}$$
{width="98.00000%"}
Figure \[C\_rate\] shows the capture rate by Oxygen WDs ($A_n=16$) vs. WIMP velocity dispersion for several masses of WDs, assuming Keplerian orbits around the SMBH, $m_\chi=100$ GeV, $\sigma_0=10^{-43}$ cm$^2$, $\rho_\chi \sim \rho_\chi^{\max}\sim
m_\chi/({\langle\sigma_a v\rangle}\tau_{\rm bh}) \sim 10^{10}$ GeV cm$^{-3}$ which corresponds to the maximal central particle dark matter density allowed by the age of the SMBH $\tau_{\rm bh}\sim10$ Gyr and our selected value ${\langle\sigma_a v\rangle}= 3\times10^{-26}$ cm$^3$ s$^{-1}$ [@gs99; @bm05]. The left panel corresponds to WD effective temperature of $T=100,000$ K (without an envelope), where the masses and radii used in the calculation are [Fig. 5 in @panei00]: $M_*/M_\odot$ = 0.6, 0.8, 1.0, 1.2, 1.4, and $R_*/R_\odot$ = 0.02, 0.012, 0.0085, 0.006, 0.0045, correspondingly. The right panel corresponds to the zero-temperature approximation [@hs61], where the mass-radius relation has been obtained by fitting the numerical results for Carbon WDs with a function $R_*/R_\odot=0.94-0.67\tan(1.49[M_*/M_\odot-0.85])$ in the interval $M_*/M_\odot=0.15-1.4$. The solid lines show the capture rate calculated using the modified interaction cross section (eq. \[\[geom\_limit\]\]), and the dashed lines are calculated for $\sigma'_0=\sigma_0$. In the case of Oxygen WDs the geometrical limit is reached for $M_*\sim1.2M_\odot$; the larger mass WDs have smaller radii and therefore smaller capture rates. For $\bar{v}\la 10^3$ km s$^{-1}$ our geometrical limit calculations agree well with the results of @bottino02, their equation (26). For $\bar{v}\ga 10^3$ km s$^{-1}$ the @bottino02 formula, derived under the assumption that each WIMP crossing the star is captured, gives a systematically larger capture rate, up to a factor of 10 for $10^4$ km s$^{-1}$. This can be treated as an upper limit, whereas our approximation to the geometrically limited case can be considered as a lower limit.
It can be seen (Figure \[C\_rate\]) that cooler WDs have a capture rate (eqs. \[\[gould2.27\]\],\[\[gould2.24\]\]) larger than hot ones of the same mass because of the larger escape velocity (eq. \[\[vesc\]\]). The latter is the result of a smaller radius $R_*$ and consequently stronger gravity. This effect may be explained in terms of the “focusing factor” [@gould87] or simply because WIMPs can be captured from the larger volume of the Maxwellian velocity phase space. A larger capture rate by a cooler WD will lead to accelerated heating until the WD radius increases due to increased temperature. A hotter WD will be less efficient for WIMP capture and cool down. This mechanism will thus lead to fast self-regulation of the WD temperature and capture rate.
The capture rate for a different ($A_n$) composition WD can be estimated from scaling the Oxygen WD curves by a factor of $(A_{n}/16)^3$, e.g., the curves for a Carbon WD can be obtained from scaling the Oxygen WD curves down by a factor of $(3/4)^3$. WDs with heavier nuclei (up to iron) may exist [@panei00]; in this case, the capture rate is restricted mostly by the geometrical limit.
Figure \[C\_profile\] shows the capture rate for Oxygen (left panel) and Iron (right panel) WDs vs. distance from the central black hole with $M_{\rm bh}=3.7\times10^6 M_\odot$ [@ghez05]; this includes effects of the radial dependence of the WIMP velocity dispersion and the WD orbital (Keplerian) velocity. Following @gp04 and @bm05, the WIMP mass density is normalized as $\rho_\chi(2\
{\rm pc})=100M_\odot$ pc$^{-3}$. For the central spike we assume a power-law profile with indices (top to bottom): 7/3, 3/2, 4/3; these profiles are predicted for different scenarios of the black hole growth, adiabatic [@ullio01], quasi-equilibrium [@gp04], and instanteneous [@ullio01], correspondingly. Here we use the same estimate for $\rho_\chi^{\max}$ as for Figure \[C\_rate\]. The mass-radius relation for Oxygen and Iron WDs of $T=100,000$ K is taken from @panei00.
The capture rate scales linearly with the WIMP density, so that the largest capture rate is reached with the adiabatic profile. The capture rate increases toward the SMBH until the maximal WIMP mass density $\rho_\chi^{\max}$ is reached; then the capture rate decreases due to increases in the WIMP velocity dispersion and the orbital velocity of the star. For the quasi-equilibrium profile, $\rho_\chi^{\max}$ is reached only at $\sim$$10^{-4}$ pc, while the instanteneous profile is even flatter.
As can be seen from Figure \[C\_profile\] and a simple inspection of the capture rate formulae, in the geometrically limited case (eq. \[\[geom\_limit\]\]) the capture rate becomes essentially independent of the WIMP-nucleon scattering cross-section and degenerate core parameters. Observationally, the brightest WIMP burners may be the geometrically limited ones. The main uncertainty in the geometrically limited capture rate is the dark matter density; thus it may be possible to perform largely “model independent" measurements of the dark matter density profile by measuring the luminosity of different WIMP burners orbiting within a particular dark matter spike.
A smaller annihilation cross section ${\langle\sigma_a v\rangle}<3\times10^{-26}$ cm$^3$ s$^{-1}$ would allow for larger ambient WIMP densities near the SMBH. This would lead to a larger capture rate and consequently larger burning rate at the innermost radii. The energy release due to WIMP annihilation in the stellar core is $L_\chi\sim 0.16\, C (m_\chi/100\ \rm GeV)$ erg s$^{-1}$ which is actually independent of the WIMP mass $m_\chi$.
Discussion
==========
Where does the energy released during the WIMP annihilation go? Table 1 in @MW06 shows that the effective radius of the thermal distribution of WIMPs in the stellar core is much smaller than the radius of the star $r_\chi\ll R_*$, therefore, the products of WIMP annihilation cannot propagate to the stellar surface and are converted into thermal energy and neutrino emission. A WD, a star without its own energy supply consisting of Carbon and Oxygen, may emit up to $L_\chi\sim3\times10^{34}$ erg s$^{-1}$, i.e. $\sim$10 times the luminosity of the sun, burning WIMPs only and this energy source will last forever! (Note that this estimate is based on our *approximation* to the geometrically limited case and larger luminosities are possible even for the given set of parameters.) At such a luminosity, the surface temperature of the WD would be close to $\sim$140,000 K, assuming $M_*=1.2M_\odot$, $R_*=0.006R_\odot$. The maximum of the black body emission falls into the UV band making such stars strong thermal UV emitters concentrated in the inner $\sim$0.01 pc. A smaller annihilation cross section ${\langle\sigma_a v\rangle}<3\times10^{-26}$ cm$^3$ s$^{-1}$ and/or larger WIMP density normalization $\rho_\chi(2\
{\rm pc})>100M_\odot$ pc$^{-3}$ would allow for a larger ambient WIMP density near the SMBH, thus increasing the capture and burning rates further.
The energy transport in the interiors of WDs is dominated by degenerate electrons and is very efficient [see @hansen04 for a recent review]; therefore, the large number of captured WIMPs and their annihilation in the core would not change the internal structure of WDs. A recently published catalog of spectroscopically confirmed WDs from the Sloan Digital Sky Survey (SDSS) [@e06] contains several hot WDs with surface temperature in the range of 100,000 K, thus providing observational evidence that high temperature does not change the appearance of WDs. A bare WD with an effective temperature as high as 170,000–200,000 K has also been observed [H1504+65, @ww99].
The number of very hot WDs in the SDSS catalog is small, just a handful out of 9316. This means that observation of a concentration of very hot WDs at the Galactic Center would be extremely unlikely unless they are “dark matter burners.” The spectra of confirmed hot WDs can serve as templates for spectroscopic analysis of WDs at the Galactic Center where only a limited part of the near-IR band can be used. An independent determination of the $M_*/R_*$ ratio is possible using the gravitational redshift that has to be equivalent to a radial velocity of about 50 km s$^{-1}$ [@gt67].
A bare WD with a highly eccentric orbit around the central black hole may exhibit variations in brightness correlated with the orbital phase (Figure \[C\_profile\]). To have this working, the orbital period should exceed the equilibrium time scale $\tau_{eq}$ (eq. \[\[taueq\]\]). Carbon burning stars have $\tau_{eq}\sim10$ yr, and it is even shorter $\sim$0.5 yr in case of a WD [@MW06]. If a WD appears in a high-density WIMP region, the WIMP density in its material would quickly reach equilibrium; thus, the surrounding WIMP density variation as the WD orbits would result in variations of brightness. This makes bare WDs ideal objects to test the WIMP density in the environment in which they are orbiting. Since $L_\chi\propto\rho_\chi$, a population of WDs, bare or with envelopes, located at different distances from the SMBH would exhibit a luminosity correlated with the radial WIMP density profile. Geometrically limited WIMP burners have the highest luminosities and therefore will be the easiest to observe. Their luminosity is largely independent of WIMP-nucleon scattering cross-section, WIMP pair annihilation cross-section, and degenerate core parameters.
Advances in near-IR instrumentation have made possible observations of stars in the inner parsec of the Galaxy [@genzel00; @ghez03; @ghez05]. The apparent K-band brightness of these stars is 14–17 mag. The observed absorption line widths imply high temperatures and lead to a “paradox of youth:” apparently young stars in the region whose current conditions seem to be inhospitable to star formation. One of the possibilities is that they are old stars masquerading as youths. Assuming a central spike with index 7/3, the K-band brightness for Oxygen WDs with $T\sim100,000$ K and $R_*/R_\odot\sim0.01$ is about 22–23 mag not including extinction, which may be as large as 3.3 mag [@rrp89]. It is, therefore, unlikely that the currently observed stars in the K-band are WDs burning WIMPs; however, stars with degenerate electron cores plus envelopes cannot be ruled out.
We thank R. Blandford, J. Edsjö, J. Faulkner, S. Kahn, J. Primack, and R. Romani for interesting discussions and the anonymous referee for useful comments. L. L. W. would like to thank S. Nagataki for interesting discussions on massive stars. I. V. M. acknowledges partial support from NASA Astronomy and Physics Research and Analysis Program (APRA) grant. A part of this work was done at Stanford Linear Accelerator Center, Stanford University, and supported by Department of Energy contract DE-AC03-768SF00515.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We present a search for Trojan companions to 25 transiting exoplanets. We use the technique of Ford & Gaudi, in which a difference is sought between the observed transit time and the transit time that is calculated by fitting a two-body Keplerian orbit to the radial-velocity data. This technique is sensitive to the imbalance of mass at the L4/L5 points of the planet-star orbit. No companions were detected above 2$\sigma$ confidence. The median 2$\sigma$ upper limit is 56 $M_\earth$, and the most constraining limit is 2.8 $M_\earth$ for the case of GJ 436. A similar survey using forthcoming data from the [ *Kepler*]{} satellite mission, along with the radial-velocity data that will be needed to confirm transit candidates, will be sensitive to 10–50 $M_\earth$ Trojan companions in the habitable zones of their parent stars. As a by-product of this study, we present empirical constraints on the eccentricities of the planetary orbits, including those which have previously been assumed to be circular. The limits on eccentricity are of interest for investigations of tidal circularization and for bounding possible systematic errors in the measured planetary radii and the predicted times of secondary eclipses.'
author:
- 'N. Madhusudhan & Joshua N. Winn'
title: Empirical Constraints on Trojan Companions and Orbital Eccentricities in 25 Transiting Exoplanetary Systems
---
Introduction
============
Trojan companions are bodies in a 1:1 mean-motion resonance with a planet, librating around one of the two triangular Lagrange points (L4 and L5) of the planet’s orbit around the star. The archetypal example is the population of Trojan asteroids in resonance with Jupiter. Trojan companions to Neptune and Mars have also been detected (Sheppard and Trujillo 2006, Rivkin et al. 2007). Another interesting example is the pair of Saturnian satellites Calypso and Telesto, which are in 1:1 resonance with their fellow satellite Tethys (Reitsema 1981). The presence of Trojan companions and their orbital and physical characteristics have been considered as clues to processes in planet formation and migration. Several recent studies have examined the capture and survival of Trojans in the context of suspected changes in the orbital architecture of the Solar system (Morbidelli et al. 2005, Chiang and Lithwick 2005, Kortenkamp et al. 2004).
Although the Trojan-to-planet mass ratios in the Solar system are very small ($m_T/m_P \sim 10^{-7}$ for Jupiter), it is conceivable that Trojans with much higher mass ratios exist in exoplanetary systems. For circular orbits, even very massive Trojans can be dynamically stable. Laughlin & Chambers (2002) explored the viability of Trojans with mass ratios of unity (i.e., co-orbital planets of equal mass), finding that such configurations can be dynamically stable over time scales comparable to or longer than stellar lifetimes. More generally, the stability of the L4/L5 points depends on the orbital eccentricity and the relative masses of the Trojan, planet, and star (see, e.g., Nauenberg 2002, Dvorak et al. 2004). For many of the known exoplanets, considerations of dynamical stability allow for massive Trojan companions. For example, at least 7 of the known gas giant planets that are within the habitable zones of their parent stars could have dynamically-stable, terrestrial-mass Trojan companions (Schwarz et al. 2007).
Several methods have been proposed to detect Trojan companions to exoplanets. A Trojan may be massive enough to perturb the stellar motion by an amount that is detectable in the radial-velocity (RV) orbit of the star (Laughlin & Chambers 2002). A Trojan in a nearly edge-on orbit may be large enough for its transit to be detected photometrically (see, e.g., Croll et al. 2007). For a transiting planet, the gravitational perturbations from a Trojan companion may cause a detectable pattern in the recorded transit times (Ford & Holman 2007). Alternatively, Ford & Gaudi (2006) proposed comparing the measured transit times with the times that would be expected based only on the RV data and the assumption of a two-body orbit.
An important virtue of the latter technique is that a sensitive search for Trojans can be performed using only the RV and photometric data that are routinely obtained while confirming transit candidates and characterizing the planets. This is in contrast to the first three methods, for which it is generally necessary to gather new and highly specialized data (very precise RVs, continuous space-borne photometry, and a long sequence of precisely-measured transit times, respectively). For example, Ford & Gaudi (2006) and Narita et al. (2007) placed upper limits on Trojan companions of approximately Neptune mass to the transiting planets HD 209458b, HD 149026b and TrES-1b, using data gathered for other purposes.
In this paper, we present a search for Trojan companions to 25 known transiting exoplanetary systems for which suitable data are available, using the method of Ford & Gaudi (2006, hereafter, “FG”). This paper is organized as follows. The method is described in § 2. The compilation and analysis of the data is described in § 3. The results are given in § 4. These results are summarized and discussed in § 5, which also looks ahead to the prospects for a similar search using data from the [*Kepler*]{} mission (Borucki et al. 2008).
As will be explained in § 2, the orbital eccentricity of the planet-star orbit affects the interpretation of the data. Hence, a necessary part of our analysis was the determination of the orbital eccentricity for each system, or the justification of the common assumption that the orbit is circular due to tidal effects. These issues are investigated systematically in § 3. Our findings may be of interest independently of our results on Trojan companions, not only because of the connection to the theory of tidal circularization, but also because the orbital eccentricity affects estimates of the planetary radius via transit photometry, as well as the predicted times of planetary occultations (secondary eclipses). We discuss these points in § 5.
Method {#sec:method}
======
The basic idea of the FG method is to compare the measured transit time with the expected transit time that is calculated by fitting a two-body Keplerian orbit to the RV data. We will denote by $t_O$ the observed transit time, and by $t_C$ the calculated transit time, in which the calculation is based on fitting a two-body Keplerian orbit to the RV data. The presence of a Trojan companion as a third body would cause a timing offset $\Delta t = t_O - t_C$.
This is most easily understood for the case of a planet on a circular orbit. In such a case, if there is no Trojan companion, the force vector on the star points directly at the planet, and the observed transit time $t_O$ coincides with the time $t_V$ when the orbital velocity of the star is in the plane of the sky (i.e., the time corresponding to the null in the RV variation). If instead there is a single Trojan located at L4 or L5 (or librating with a small amplitude), then the force vector on the star does not point directly at the planet; it is displaced in angle toward the Trojan companion, given by $\tan(\phi) \simeq \sqrt{3}
\epsilon/(2-\epsilon)$ where, $\epsilon = m_T/(m_P + m_T)$ for a Trojan mass $m_T$ and a planet mass $m_P$ (Ford & Gaudi 2006). As a result, $t_O$ occurs earlier or later than $t_V$, and the time difference is given by $\Delta t = \pm \phi P/2 \pi$. For small values of the Trojan-to-planet mass ratio, the magnitude of $t_O - t_V$ is proportional to $m_T$, (Ford & Gaudi 2006): $$\label{eq:deltat-mt-circular}
\Delta t \simeq \pm 37.5~\text{min}~\bigg( \frac{P}{3 \, \textrm{days}} \bigg)
\bigg( \frac{m_T}{10 \, M_\earth} \bigg)
\bigg( \frac{0.5\,M_{\rm Jup}}{m_P + m_T} \bigg) .$$ The positive sign corresponds to a mass excess at the L4 point (leading the planet) while the negative sign corresponds to a mass excess at the L5 point (lagging the planet). Thus, given a $\Delta t$, the mass excess can be estimated using Eq. (\[eq:deltat-mt-circular\]), assuming small Trojan-to-planet mass ratio.
More generally, the mass excess is given by: $$\label{eq:deltat-mt-x}
m_T = m_P \bigg(\frac{2\, \tan(2\pi \Delta t/P)}{\sqrt{3} - |\tan(2\pi \Delta t/P)|}\bigg).$$
For an eccentric two-body orbit, the transit time does not generally coincide with the time of null RV variation, and hence in general $t_C
\neq t_V$. To first order, $t_C-t_V \approx (e\,\cos\omega)P/2\pi$, where $e$ is the eccentricity and $\omega$ is the argument of pericenter, and hence one may use the statistic $\Delta t = t_O - t_V
- (e\,\cos\omega)P/2\pi$ to search for Trojan companions. This is how the problem was described by FG, although we find it useful to cast the problem more generally as a comparison between $t_O$ and $t_C$. We emphasize here that $t_O$ depends solely on photometric observations of transits, while $t_C$ depends almost entirely on RV observations.[^1]
Specifically, one calculates $t_C$ by fitting a two-body Keplerian orbit to the RV data and calculating the expected transit time based on the the fitted orbital parameters (see, e.g., Kane et al. 2008). The true anomaly ($f$) corresponding to the transit time is $$f = \frac{\pi}{2} - \omega,$$ from which the eccentric anomaly $E$ can be calculated using $$\tan \frac{E}{2} = \sqrt{\frac{1-e}{1+e}} \, \tan \frac{f}{2},$$ which in turn leads to the mean anomaly $M$ of the transit using Kepler’s equation, $$M = E - e\sin E.$$ Finally, the calculated transit time $t_C$ is obtained from the definition of the mean anomaly, $M = 2\pi(t - t_P)/P$, where $t_P$ is the time of pericenter passage.
Our basic procedure is therefore to determine $t_O$ from published transit ephemerides, calculate $t_C$ by fitting a two-body Keplerian orbit to the available RV data, and calculate $\Delta t = t_O - t_C$. For circular orbits, we use Eq. (\[eq:deltat-mt-x\]) to determine the Trojan companion mass $m_T$ corresponding to a given value of $\Delta t$. For eccentric orbits, the relationship between $t_C$ and $m_T$ is determined using direct numerical integrations of 3-body systems using a Bulirsch-Stoer algorithm (Varadi et al. 1996). In these integratons, we hold fixed $P$, $e$, $\omega$, and the stellar mass $m_S$ at the values given in the literature, and select a Trojan mass $m_T$ and planetary mass $m_P$ such that the RV semi-amplitude (Nauenberg 2002) $$K = \bigg(\frac{2\,\pi\,G}{P}\bigg)^{1/3} \frac{\sqrt{m_P^2+m_T^2+m_Pm_T}}{(m_S + m_P + m_T)^{2/3}\sqrt{1-e^2}},
\label{eqn:3body}$$ is equal to the observed value. Hence we simulate the case in which the observed RV variation is due to the combined force of a planet and a Trojan, rather than a planet alone, but the RV data alone are insufficiently precise to discern the difference.[^2] We compute the transit time $t_C$, repeat the analysis for an increasing sequence of $m_T$, and fit a polynomial function to the resulting relationship $t_C(m_T)-t_C(0)$. We found a quadratic function, $m_T = a_1 \Delta t + a_2 (\Delta t)^2$, to give a good fit to the results. Taking $m_T$ to be in Earth masses and $\Delta
t$ in minutes, the coefficients ($a_1$,$a_2$) are (0.044,-1.17$\times$ 10$^{-5}$) for GJ 436b and (6.787,-0.001) for XO-3b. For the cases of HAT-P-2b and HD 17156b, we find that even very low-mass Trojan companions are dynamically unstable, owing to the large orbital eccentricities (see § \[subsec:dynamical\]). Thus, for those systems, the requirement of dynamical stability is more constraining than the empirical upper limit on $m_T$ based on the FG method. (As will be described in § 4, this also proved to be true for XO-3b based on the current data.)
Data Analysis {#sec:data}
=============
The RV data were taken from the available literature on each system. The references are given in Table 1. These data were generally obtained for the purpose of discovering or confirming the planet, although in a few cases the data were obtained for other reasons, such as precisely measuring the orbital eccentricity (Laughlin et al. 2005) or for measuring the Rossiter-McLaughlin effect (Winn et al. 2006). Regarding the latter, the data that were obtained while a transit was in progress were not used, to avoid the needless complication of incorporating the Rossiter-McLaughlin effect into the RV model. However, in those cases the investigators usually gathered additional data outside of the transit which are useful for refining the spectroscopic orbit.
Our RV model for an eccentric Keplerian orbit has $4
+ N$ free parameters, where $N$ is the number of independent data sets. Those parameters are the projected planet mass ($m_P\sin i$), orbital eccentricity ($e$), argument of pericenter ($\omega$), calculated time of midtransit ($t_C$), and a constant additive velocity ($\gamma$) for each data set. In practice we use parameters $e\cos\omega$ and $e\sin\omega$ instead of $e$ and $\omega$ because for small $e$, the errors in $e\cos\omega$ and $e\sin\omega$ are uncorrelated (see, e.g., Winn et al. 2005, Shen & Turner 2008). The orbital period $P$ is held fixed at the photometrically determined value, but of course the transit time $t_C$ is not constrained by the photometric data, since it is the difference between $t_C$ and the actual transit time $t_O$ that we are trying to measure. In two cases for which a long-term acceleration has been identified in the RV data (GJ 436b and CoRoT-Exo-1b), we include an additional free parameter, $\dot{\gamma}$[^3] We assign a different $\gamma$ to each RV data set, to allow for telescope-specific velocity offsets. The stellar masses for the systems were taken from the homogeneous analysis of Torres et al. (2008) when possible, and otherwise from the discovery paper.
Estimation of Jitter {#subsec:jitter}
--------------------
For each system, we first analyzed the data with the goal of reproducing the quoted results in the literature. We fitted a Keplerian model to the RV data by minimizing the $\chi^2$ statistic using the AMOEBA algorithm (Press et al. 1992). The initial conditions for the free parameters were taken to be the literature values, and for consistency, for this step we used the exact same choices of $P$ and $m_S$ as in the literature. We define $\chi^2$ as $$\chi^2 = \sum_{i = 1}^{N_{v}} \bigg( \frac{v_{i,O} - v_{i,C}}{\sigma_i} \bigg)^2,
\label{eq:chisqr}$$ where $v_{i,O}$ and $v_{i,C}$ are the observed and calculated radial velocities, respectively, and $\sigma_i$ is the corresponding uncertainty. The uncertainty should include the statistical uncertainty $\sigma_{\rm stat}$, as well as the systematic error $\sigma_{\rm sys}$ due to unmodeled instrumental systematic errors and intrinsic variations of the stellar photosphere, often referred to as “stellar jitter” in this context (Wright 2006). To estimate the appropriate values of $\sigma_i$ for this project, we determined the value of $\sigma_{\rm sys}$ such that $\chi^2/N_{\rm dof} = 1$ when using $$\sigma_i = \sqrt{\sigma_{\rm stat}^2+\sigma_{\rm sys}^2}$$ in Eq. (\[eq:chisqr\]). Our estimates of the stellar jitter using this procedure are given in Table 1.
Data Selection
--------------
Our desire was to perform as wide a search as possible, using all publicly available data, but for some systems the RV data is so sparse that meaningful constraints cannot yet be obtained. To guide our selection of systems, we used a figure-of-merit based on the results of the Fisher information analysis presented by FG. Those authors showed that in the limit of continuous RV sampling with uniform errors, the uncertainty in $\Delta t$ will approach $\sigma_{\Delta t}
= (1/2 \pi^2 N_v)^{1/2}P\,\sigma_v/K$, where $N_v$ is the number of radial-velocity data points and $\sigma_v$ is the error per point. For a circular orbit the corresponding uncertainty in $m_T$ will approach $\sigma_{m_T} = (8/3 N_v)^{1/2} m_P\,\sigma_{v}/K$. Thus, it is possible to estimate the expected uncertainty in $\Delta t$ and $m_T$, given the published system parameters, without fitting the actual data. We calculated the figure of merit $$\xi \equiv \frac{\sqrt{(3N_v/8)}\,K}{\sigma_v} \approx 1/\sigma_{(m_T/m_P)}
\label{eq:goodness}$$ for each system in the literature at the outset of this project, and ranked the systems accordingly. Table 1 shows $\xi$ for all the systems, along with $N_v$, $\sigma_v$, and $K$. For systems where multiple data sets are available, the effective $\sigma_{v}/N_{v}^{1/2}$ is obtained by adding in quadrature the corresponding terms from the different data sets. A higher value of $\xi$ reflects better quality of data. From MCMC analyses of all the systems, we find that for systems with $\xi < 6$, the fitting algorithm is susceptible to poor convergence and allows for unphysical parameter ranges. Since the scientific return on such low-$\xi$ systems is comparatively poor we decided to remove them from consideration rather than eliminate these fitting problems. In what follows we focus exclusively on the 25 systems with $\xi > 6$.
Assumptions for orbital eccentricity {#subsec:ecc}
------------------------------------
As explained in § 2, the orbital eccentricity affects the calculation of $t_C$ and also affects the relation between $\Delta t$ and the possible Trojan companion mass $m_T$. Hence it is imperative to consider the possiblity of eccentric orbits. Most of the currently known transiting planets are in very close orbits, where the effects of tidal interactions between the star and planet—and orbital circularization in particular—are expected to be significant (Rasio et al. 1996, Trilling 2000, Dobbs-Dixon et al. 2004). A common practice is to assume that, in the absence of positive evidence for an eccentric orbit, the orbital eccentricity has been reduced to insignificance by the action of tides.
If the assumption of a circular orbit could be justified, it would be advantageous for the present study because it would remove 2 free parameters from the Keplerian model ($e$ and $\omega$) and thereby strengthen the determination of the other parameters, including the key parameter $t_C$. Our approach was to assume the orbit to be circular only when (1) a circular orbit is consistent with the RV data, (2) the estimated stellar age is more than 20 times larger than the estimated timescale for tidal circularization, and (3) no constraint on $e\cos\omega$ is available because no planetary occultations (secondary eclipses) have been observed. These points are explained in detail in the paragraphs to follow.
To test whether the RV data are consistent with a circular orbit, we fitted a Keplerian model to the RV data using a Markov Chain Monte Carlo (MCMC) technique, employing a Metropolis-Hastings algorithm within the Gibbs sampler (see, e.g., Tegmark et al. 2004; Ford 2005; Holman et al. 2006; Winn et al. 2007a). For this step, the free parameters were $m_P\sin i$, $e\cos\omega$, $e\sin\omega$ and a $\gamma$ for each data set. Uniform priors were used for all parameters. The fitting statistic, $\chi^2$, was defined in Eq. (\[eq:chisqr\]). A single chain of $\sim 10^6$ links was used for each system. The jump sizes for the various parameters were set such that the acceptance rate for each parameter was $\sim$20%. For each parameter, we found the mode of the [*a posteriori*]{} distribution (marginalized over all other parameters), and the 68.3% confidence interval, defined as the range that excludes 15.9% of the probability at each extreme of the [*a posteriori*]{} distribution. For cases when $e\cos\omega$ and $e\sin\omega$ were both consistent with zero, we also found the 95.4%-confidence upper limit on $e$. Table 2 gives the results. All of the systems were found to be consistent with a circular orbit, except for the 4 well-known eccentric systems GJ 436, HAT-P-2, HD 17156, and XO-3.
For the estimated timescale for tidal circularization, we used (Goldreich & Soter 1966): $$\tau_{\rm circ} = \frac{4}{63} Q\bigg(\frac{a^3}{G\,m_S}\bigg)^{1/2} \frac{m_P}{m_S}\bigg(\frac{a}{R_p}\bigg)^5,
\label{eq:tcirc}$$ which is based on the highly simplified, widely-used model of tidal dissipation in which the tidal bulge experiences a constant phase lag due to tidal friction. Here, $a$ is the orbital separation, and $R_p$ is the radius of the planet. The dimensionless number $Q$ is inversely proportional to the dissipation rate. In the solar system, Jupiter is thought to have $Q\sim 10^5$ (Ioannou & Lindzen 1993) to the extent that this simplified model is applicable. For our purpose, a necessary condition for assuming the orbit to be circular was that $\tau_\star/\tau_{\rm circ} > 20$, i.e., there have been at least 20 $e$-foldings of tidal circularization, according to this model. In calculating $\tau_{\rm circ}$ we assumed $Q=10^6$, which is conservative in the sense that a larger $Q$ corresponds to a longer calculated timescale for circularization, and a smaller risk that we are assuming a circular orbit when this assumption is not justified.
![Constraints on $e\cos\omega$ and $e\sin\omega$ based on fitting a Keplerian orbit to the RV data, and using constraints from the observed time of the secondary eclipse when available. The four clearly eccentric systems are labeled. The dotted circles show the eccentricity contours corresponding to $e=0.2$, 0.4 and 0.6.[]{data-label="fig:eccen"}](f1.eps){width="50.00000%"}
![The inner region of Fig \[fig:eccen\]. The dotted circles show the eccentricity contours corresponding to $e=0.05$ and 0.1.[]{data-label="fig:eccen_zoom"}](f2.eps){width="50.00000%"}
For a few systems, additional constraints on the orbital eccentricity are available because a planetary occultation (secondary eclipse) has been observed. For small eccentricities, the variables $e\cos\omega$ and $e\sin\omega$ are directly related to the interval between the transit and occultation, and the relative durations of those two events (Kallrath & Milone 1999, Charbonneau et al. 2005): $$e\,cos\omega = \frac{\pi}{2P} \bigg( t_{\rm occ} - t_{\rm tra} - \frac{P}{2} \bigg),
\label{eq:ecosw}$$ $$e\,sin\omega = \frac{\Theta_{\rm tra} - \Theta_{\rm occ}}{\Theta_{\rm tra} + \Theta_{\rm occ}}$$ where $t_{\rm tra}$ and $t_{\rm occ}$ are the times of transit and occultation, and $\Theta_{\rm tra}$ and $\Theta_{\rm occ}$ are the corresponding durations. In all cases to date, the bounds on $e\sin\omega$ that follow from this relation are weaker than bounds from the RV data (see, e.g., Winn et al. 2005), and the bounds on $e\cos\omega$ are more constraining. Thus, for those cases in which an occultation has been observed, we add a term to our $\chi^2$ statistic to enforce the corresponding constraint on $e\cos\omega$: $$\label{eq:chi2-ecosw}
\chi^2 = \sum_{n = 1}^{N_v} \bigg( \frac{v_O - v_C}{\sigma_v} \bigg)^2 + \bigg( \frac{(e\,cos\omega)_O -
(e\,cos\omega)_C}{\sigma_{e\,cos\omega}} \bigg)^2,$$ where $(e\cos\omega)_o$ and $(e\cos\omega)_c$ are the “observed” and calculated values of $e\cos\omega$. By “observed” we mean the value that follows from the measured interval between transits and occultations when inserted into Eq. (\[eq:ecosw\]). The 5 systems in this category, and the constraints on $e\cos\omega$, are listed in Table 3. In those cases, because such a powerful empirical constraint is available, we do not assume the orbit to be circular even if the RV data are consistent with a circular orbit and $\tau_\star/\tau_{\rm circ} >
20$.
Constraints on the masses of Trojan companions {#subsec:mcmc}
----------------------------------------------
For each system, after determining the appropriate level of stellar jitter and deciding whether or not the assumption of a circular orbit is justified, we determined the key parameter $t_C$ and its uncertainty using the same MCMC code that was described in the previous section. In all cases the free parameters included $m_P\sin i$, $\gamma$, and $t_C$. In cases for which the a circular orbit was not assumed, we also fitted for $e\cos\omega$ and $e\sin\omega$. For the special cases of GJ 436 and CoRoT-Exo-1, a velocity gradient $\dot{\gamma}$ was also included as a free parameter. The basic fitting statistic, $\chi^2$, was defined in Eq. (\[eq:chisqr\]), and for the systems in Table 3 an [*a priori*]{} constraint on $e\cos\omega$ was applied as in Eq. (\[eq:chi2-ecosw\]).
To determine the photometric transit time $t_O$, we used the most precise published photometric ephemeris to compute a predicted transit time close to the midpoint of the RV time series. We then computed the key parameter $\Delta t = t_O - t_C$, the difference between the photometrically observed transit time and the transit time calculated from the RV data assuming zero Trojan mass. In all cases, the uncertainty in $t_O$ is negligible in comparison to the uncertainty in $t_C$. The results for $\Delta t$ are translated into constraints on the Trojan mass $m_T$ using Eq. (\[eq:deltat-mt-x\]) for circular orbits, and using the numerical integrations described in § \[sec:method\] for eccentric orbits.
Results {#sec:results}
=======
Constraints on Trojan Masses
----------------------------
Table 4 gives the 68.3% (1$\sigma$) confidence intervals for $\Delta
t$ and $m_T$ for all 25 systems under consideration, as well as the 95.4% (2$\sigma$) upper limits on $m_T$ and $m_T/m_P$. In all the cases, the result for $\Delta t$ was consistent with zero within 2$\sigma$. The system that was closest to a 2$\sigma$ detection was WASP-2, for which $\Delta t = -123^{+64}_{-53}$ minutes. The result for WASP-2 is therefore worth following up with additional RV data. However, in a sample of 25 systems, even if $\Delta t$ is always consistent with zero, one expects approximately one 2$\sigma$ outlier. Hence our survey has not produced compelling evidence for a Trojan companion in this ensemble.
The 2$\sigma$ upper limits on $m_T$ and on $m_T/m_P$ are shown in Figures \[fig:trojans\] and \[fig:ratios\], respectively. The systems are ordered from least-constrained to best-constrained, going from left to right. The median upper limit on $m_T$ is 56 $M_\earth$, with the most constraining limit of 2.8 $M_\earth$ holding for the Neptune-sized planet GJ 436. Such a powerful upper limit is possible in this case because of the small stellar and planetary masses, and the copious RV data that is available for this system. The median upper limit on the mass ratio $m_T/m_P$ is 0.1.
It is possible to compare our results to those obtained previously for 3 particular systems. For HD 209458, FG found $\Delta t = 13 \pm 9$ minutes and we find $2^{+11}_{-9}$ min. For HD 149026, FG found $\Delta t =
-13\pm 27$ min and we find $26^{+29}_{-38}$ min. For TrES-1, Narita et al. (2007) found $\Delta t = - 3.2 \pm 11.8$ min, assuming a circular orbit, and we find $-4^{+13}_{-11}$ min, allowing the orbit to be eccentric but using the constraint on $e\cos\omega$ from secondary eclipse. These results are all consistent with zero with approximately the same range of uncertainty. Minor differences in the quoted central values are probably attributable to minor differences in the fitting procedures and in reporting median values of the [*a posteriori*]{} distributions rather than modes. We also find our uncertainties to be in general agreement with the forecasted uncertainties based on the Fisher information analysis of FG.
Considerations of dynamical stability {#subsec:dynamical}
-------------------------------------
For a planet on a circular orbit, non-librating Trojan companions are stable as long as the masses satisfy the condition (Laughlin & Chambers 2002): $$\frac{m_P + m_T}{(m_S + m_P + m_T)} \leq 0.03812,
\label{eqn:mt_circ}$$ where $m_S$, $m_P$ and $m_T$ are the masses of the star, planet and Trojan companion, respectively. This criterion allows for Trojan “companions” that are just as massive as the planet itself, even for planets as massive as 10 $M_{\rm Jup}$ around a Sun-like star.
However, the condition for stability of Trojan companions depends strongly on the eccentricity of the orbit. Nauenberg (2002) reported just such a study, showing the stability domain of bodies in 1:1 resonance as a function the eccentricity of the orbit and the Routh parameter ($\gamma_R$) given by: $$\gamma_R = \frac{m_S m_P + m_P m_T + m_S m_T }{(m_S + m_P + m_T)^2}.
\label{eqn:gammaR}$$ Given the masses of the three bodies, and the eccentricity of the system, one can calculate $\gamma_R$ and determine from Fig. 5 of Nauenberg (2002) whether or not the system is stable in 1:1 resonance. Conversely, given the eccentricity of the system, Fig. 5 of Nauenberg (2002) gives the the maximum $\gamma_R$ allowed for stability which, along with $m_S$ and $m_P$, gives the maximum Trojan mass allowed in the system.
![95.4%-confidence upper limits on masses of Trojan companions. The systems are ordered from the weakest to the strongest upper bound, from left to right.[]{data-label="fig:trojans"}](f3.eps){width="50.00000%"}
![95.4%-confidence upper limits on Trojan-to-planet mass ratios. The systems are ordered from the weakest to the strongest upper bound, from left to right.[]{data-label="fig:ratios"}](f4.eps){width="50.00000%"}
We calculated such limits on the Trojan masses in the four eccentric systems that we analyzed in this work. We find the mass limits to be zero for HAT-P-2b and HD 17156b (in agreement with the results of our 3-body integrations that were described in § \[sec:method\]), 105 M$_\oplus$ for XO-3b, and 3030 M$_\oplus$ for GJ 436b. Thus, for HAT-P-2b, HD 17156b, and XO-3b, the upper limit on $m_T$ based on considerations of dynamical stability is more constraining than the empirical upper limit using the FG method. For GJ 436b, the upper limit on $m_T$ using the FG method ( 2.8 M$_\oplus$) is much stronger than the upper limit imposed by the stability requirement.
Discussion
==========
Exoplanetary science has provided enough surprises that an appropriate maxim for observers is: If you can look for a novel effect or phenomenon that is at least physically plausible, then you should do so, especially when this can be done with existing data. We have obeyed this maxim by conducting a search for Trojan companions to 25 transiting planets. Specifically we have put the technique of FG into practice with a much larger ensemble than has been previously analyzed. We have conducted a search for planets in particular locations (L4/L5) with a median sensitivity of $\sim$56 $M_\earth$, without gathering any new data. Instead, we asked the RV data: when should the transit occur if there is no Trojan companion? Then we consulted the photometric ephemeris to determine when a transit actually did occur, and interpreted the time difference as a measurement or constraint on Trojan companions. Our results must be understood as constraints on the imbalance of mass residing at the L4 and L5 positions. Equally massive Trojan companions at those positions would produce opposite effects, and no net FG signal, when averaged over the libration periods.
For some systems such as HAT-P-3 and WASP-2, the existing RV data are sparse and noisy enough to allow only the barest constraints on Trojan companions, with masses comparable to the planetary mass. These constraints are nevertheless physically meaningful, in the sense that Laughlin & Chambers (2002) have shown that equal-mass planets in a 1:1 mean-motion resonance and circular orbits can be dynamically stable. In one case, WASP-2, we found a near - 2$\sigma$ evidence for a timing offset that could be interpreted as a Trojan companion. This is not compelling evidence, especially given that we examined a total of 25 systems, but this system is worthy of follow-up. In one case, GJ 436, we have found a 2$\sigma$ upper limit of $ 2.8$ $M_\earth$ on $m_T$. A positive detection at this level would have represented the least-massive planet detection to date, which is remarkable considering that we did not gather any new data. In no case was there evidence for a timing offset at the 2$\sigma$ level.
As explained in § \[subsec:ecc\], as part of this study we assessed the justification for assuming that a given planetary orbit is circular, given the existing RV data and reasonable estimates of the stellar age and the timescale for tidal circularization. One part of this assessment was the determination of empirical constraints on $e\cos\omega$ and $e\sin\omega$ based on the RV data. These results may be interesting to other investigators, independently of our results on Trojan companions. The compilation in Table 2 of the results for $\tau_\star$ and $\tau_{\rm circ}$ may be useful to those who are interested in making inferences about the tidal circularization process from the ensemble of transiting planets (see, e.g., Rasio et al. 1996, Trilling 2000, Dobbs-Dixon et al. 2004, Jackson et al. 2008, Mazeh 2008). The limits on $e\cos\omega$ are also useful for bounding the possible error in the predicted times of occultations (secondary eclipses), using the relationship given in Eqn. (\[eq:ecosw\]). In addition, although the planetary radius that is determined from transit photometry depends mainly on the observed transit depth, there is a secondary dependence on the transit timescales (the total duration, and the duration of ingress or egress) and the sky-projected orbital speed of the planet during the transit. The latter quantity is not directly observable; it depends on the orbital period, the stellar mass, and the orbital eccentricity and argument of pericenter. Thus there is a secondary dependence of the inferred planetary radius on $e$ and $\omega$ (see, e.g., Barnes 2007, or McCullough et al. 2008 for a particular example). The results of Table 2 can be used to bound the systematic error that could arise from this effect.
Having completed this survey using the existing transit data, one may wonder about the achievable limits on Trojan companions using data from ambitious future transit surveys. We consider here the particular case of the [*Kepler*]{} satellite mission (Borucki et al. 2008), whose primary goal is the detection of Earth-like planets in the habitable zones of Sun-like stars. [*Kepler*]{} is also likely to find larger planets such as gas giants in the habitable zones of their parent stars, and such planets may have lower-mass Trojan companions that are perhaps “more habitable” than the gas giants. Such companions might be detectable photometrically if they are very nearly coplanar with the transiting planet. However, even if they do not transit, they can be detected with the FG method. It is therefore natural to ask what constraints on Trojan companions will be possible for a given planet that [ *Kepler*]{} detects in the habitable zone of a Sun-like star, using only the photometric data and the RV data that are routinely gathered for the purposes of confirming and characterizing transiting planets.
![Simulated radial velocities and the corresponding 2$\sigma$ upper limits on Trojan companions to potential *Kepler* detections in the habitable zone around a Sun-like star with V = 12.[]{data-label="fig:kep"}](f5.eps){width="50.00000%"}
To answer this question, we consider four different cases, in which [*Kepler*]{} finds a planet of mass 300 $M_\earth$ (case A), 100 $M_\oplus$ (B), 30 $M_\oplus$ (C) or 10 $M_\oplus$ (D), orbiting a star of solar mass and apparent magnitude $V=12$, with a period of 1 yr and an orbital eccentricity of 0.1. We simulate RV data for each system with $\sigma_v = 1$ m s$^{-1}$, and a number of data points $N_v$ that seems realistic for the [*Kepler*]{} follow-up program. For cases A and B we assume $N_v=15$. For case C we assume $N_v=30$, which is sufficient to measure the planetary mass to at least 10% according to the expression for the signal-to-noise ratio from Gaudi & Winn (2007), $$\begin{aligned}
S/N &\simeq &0.6 \, \bigg(\frac{m_P}{M_\oplus}\bigg) \bigg(\frac{N_{v}}{100}\bigg)^{1/2} \bigg(\frac{a}{\textrm{AU}}\bigg)^{-1/2} \nonumber \\
& & \times \bigg(\frac{m_S}{M_\odot}\bigg)^{-1/2} 10^{- 0.2\,(V - 12)}\frac{D}{3.6\, \textrm{m}},
\label{equn:snr}\end{aligned}$$ which was intended to approximate the case of the HARPS instrument on the ESO 3.6 m telescope. For the challenging case D we assume $N_v=70$, corresponding to a 20% uncertainty in the planetary mass. We perform a MCMC analysis on the simulated data for each system, just as was done for the 25 transiting systems in this study, and obtain the corresponding constraints on Trojan companion masses.
Fig. \[fig:kep\] shows the simulated data and the results for the four cases. We find the 2$\sigma$ upper limits on the mass of Trojan companions to be 25 $M_\oplus$, 26 $M_\oplus$, 21 $M_\oplus$, and 24 $M_\oplus$, for cases A, B, C and D respectively. The results are limited by the RV follow-up program; the superb photometric precision of [*Kepler*]{} does not lead to correspondingly superb constraints on Trojan masses.
The results for cases A–D are all of the same order of magnitude. This is because Trojan detectability depends primarily on $\sigma_v/\sqrt{N_v}$, which only varies by a factor of 2.2 between case A and D. For small $m_T/m_P$ and large $\xi$ (see Eq. 9), Trojan detectability is indeed independent of planet mass for fixed $\sigma_v/\sqrt{N_v}$. A larger planet produces a larger RV semi-amplitude, and hence offers greater sensitivity in measuring $\Delta t$, but the conversion from $\Delta t$ to $m_T$ varies inversely with planetary mass.
However, for cases C and D, this scaling is not precisely obeyed. One reason is that for the larger values of $m_T/m_P$ that are relevant in those cases, the relation between $m_T$ and $\Delta t$ is nonlinear (see Eq. \[eq:deltat-mt-x\]), and therefore the error in $m_T$ is not Gaussian. For case D, there is an additional source of non-Gaussianity: the signal-to-noise ratio is low enough that the correlations between $\Delta t$, $K$, $e$, and $\omega$ become important. This means that the upper limit on $m_T$ is less constraining than one would predict based only on $\sigma_v/\sqrt{N_v}$.
We conclude that a “serendipitous” search for Trojan companions to the habitable-zone planets that will be detected and confirmed as part of the *Kepler* program will be sensitive to planets of approximately Neptunian mass or larger. Of course the sensitivity could be improved by obtaining additional RV data as part of a more focused search effort.
Considering the HARPS spectrograph (Pepe et al. 2002), mounted on the 3.6 m ESO telescope, as a fiducial instrument for precise RV measurements, and assuming the noise to be limited by photon-counting statistics, the measurement uncertainty can be obtained by scaling current results (Lovis et al. 2005, Gaudi & Winn, 2007): $$\sigma_{v} = \frac{10^{0.2\,(V - 12)}}{D/3.6\,\textrm{m}} \textrm{m/s},$$ where $V$ is the apparent visual magnitude of the star, and $D$ is the aperture of the telescope. Here we have assumed a 60 min exposure and a G-type star.
For small Trojan-to-planet mass ratios, and assuming a circular orbit, one can determine a nominal estimate of the sensitivity of current observational facilities to detect Trojan companions to *Kepler* planets. Under these asumptions, the signal-to-noise ratio $\Delta t/\sigma_{\Delta t}$ for detecting a Trojan companion is given by: $$\begin{aligned}
S/N &\simeq &0.56 \, \bigg(\frac{m_T}{M_\oplus}\bigg) \bigg(\frac{N_{v}}{100}\bigg)^{1/2} \bigg(\frac{a}{\textrm{AU}}\bigg)^{-1/2} \nonumber \\
& & \times \bigg(\frac{m_S}{M_\odot}\bigg)^{-1/2} 10^{- 0.2\,(V - 12)}\frac{D}{3.6\, \textrm{m}}
\label{equn:snr}\end{aligned}$$ Let us assume a *Kepler* detection of a Jupiter-mass transiting planet at 1 AU around a sun-like star with $V = 12$, observed with the HARPS instrument. Then, considering 100 observations evenly spaced in orbital phase, a 3$\sigma$ Trojan detection (i.e. $\Delta t > 3 \,
\sigma_{\Delta t}$) can be made, if a Trojan mass imbalance of $m_T
\gtrsim 5.36 M_\earth$ existed in the orbit. If the planet were orbiting instead at 0.03 AU (a “hot Jupiter” orbit), then a detection would be possible for $m_T \gtrsim 0.93 M_\earth$. It would seem that searching for Trojan companions is a promising alternate channel for finding small and potentially habitable bodies in the habitable zones of their parent stars, even if the transiting planets themselves are too massive to be habitable.
We thank Jack Wisdom, Scott Gaudi and Eric Ford for helpful conversations. We further thank Scott Gaudi for providing a detailed and helpful review of the manuscript. We are grateful to the William S. Edgerly Innovation Fund for partial support of this work.
[^1]: As explained in § \[sec:data\], the only sense in which $t_C$ depends on photometric data is that we used the photometrically-determined orbital period $P$ when fitting the RV data, to reduce the number of free parameters.
[^2]: We verified that this discernment is indeed impossible, for $m_T/m_P < 0.5$, for the systems with eccentric orbits considered in this paper.
[^3]: For the remaining systems, we have assumed that the acceleration term is zero. Any real acceleration (and any other failures of the single-Keplerian model) will appear as “noise” in our analysis and will be reflected in a larger estimate of stellar “jitter” (see § \[subsec:jitter\]). We find that our results for $\Delta t$ do not depend much on whether or not a long-term acceleration is allowed as an additional parameter, because the error in this parameter is not strongly correlated with the error in $\Delta t$.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
Understanding the difference between group orbits and their closures is a key difficulty in geometric complexity theory (GCT): While the GCT program is set up to separate certain orbit closures, many beautiful mathematical properties are only known for the group orbits, in particular close relations with symmetry groups and invariant spaces, while the orbit closures seem much more difficult to understand. However, in order to prove lower bounds in algebraic complexity theory, considering group orbits is not enough.
In this paper we tighten the relationship between the orbit of the power sum polynomial and its closure, so that we can separate this orbit closure from the orbit closure of the product of variables by just considering the symmetry groups of both polynomials and their representation theoretic decomposition coefficients. In a natural way our construction yields a multiplicity obstruction that is neither an occurrence obstruction, nor a so-called vanishing ideal occurrence obstruction. All multiplicity obstructions so far have been of one of these two types. Our paper is the first implementation of the ambitious approach that was originally suggested in the first papers on geometric complexity theory by Mulmuley and Sohoni (SIAM J Comput 2001, 2008): Before our paper, all existence proofs of obstructions only took into account the symmetry group of one of the two polynomials (or tensors) that were to be separated. In our paper the multiplicity obstruction is obtained by comparing the representation theoretic decomposition coefficients of both symmetry groups. Our proof uses a semi-explicit description of the coordinate ring of the orbit closure of the power sum polynomial in terms of Young tableaux, which enables its comparison to the coordinate ring of the orbit.
author:
- 'Christian Ikenmeyer[^1] and Umangathan Kandasamy[^2]'
date: 'November 10, 2019'
title: 'Implementing geometric complexity theory: On the separation of orbit closures via symmetries'
---
[0.9]{} **Acknowledgements:** This paper was written partially when CI was at the Max Planck Institute for Informatics, at the Max Planck Institute for Software Systems, and at the University of Liverpool. This paper contains results that are present in UK’s master’s thesis at the Universität des Saarlandes.
[0.9]{} **2012 ACM CCS:** Theory of computation $\rightarrow$ Algebraic complexity theory
[0.9]{} **Keywords:** Geometric complexity theory, group orbit, orbit closure, multiplicity obstruction
Motivation: Geometric complexity theory
=======================================
#### Symmetries
The idea of using the symmetries of the determinant ${\textup{det}}_n := \sum_{\pi \in {\mathfrak{S}}_n}{\textup{sgn}}(\pi)\prod_{i=1}^n x_{i,\pi(i)}$ and the permanent ${\textup{per}}_m := \sum_{\pi \in {\mathfrak{S}}_m} \prod_{i=1}^m x_{i,\pi(i)}$ to separate algebraic complexity classes was pioneered by Mulmuley and Sohoni in 2001 [@MS:01]. This approach is based on the observation that ${\textup{det}}_n$ and ${\textup{per}}_m$ are both characterized by their respective symmetry groups. For example, consider homogeneous degree $n$ polynomials in $n^2$ variables $x_{1,1},\ldots,x_{n,n}$. Let $X$ denote the $n \times n$ matrix whose entry in row $i$ and column $j$ is $x_{i,j}$. Then ${\textup{det}}(X)={\textup{det}}_n$. Now, for matrices $A,B \in {\textup{SL}}_n({\mathbb{C}})$ the entries of the matrix $AXB$ are homogeneous linear polynomials in the $n^2$ variables. The crucial fact is that every homogeneous degree $n$ polynomial $q$ in $n^2$ variables that satisfies $q(AXB)=q(X)$ equals $\alpha \cdot {\textup{det}}_n$ for some scalar $\alpha \in {\mathbb{C}}$. This means that ${\textup{det}}_n$ is *characterized by its symmetries*. For the permanent polynomial, an analogous statement holds, and also for many other structurally simpler polynomials, for example for the power sum polynomial $x_1^D+\cdots+x_m^D$ and for the product of variables $x_1 x_2 \cdots x_D$, see [@Ike:19].
#### Algebraic complexity theory
An *affine projection* of a polynomial is its evaluation at a point whose coordinates are given by affine linear polynomials, e.g., $(x_1+x_2+1)^2 = x_1^2 + 2 x_1 x_2 + x_2^2 + 2 x_1 + 2 x_2 + 1$ is an affine projection of $x_1^2$. Kayal proved that it is ${\textup{\textsf{NP}}}$-hard to decide whether a polynomial is an affine projection of another polynomial [@Kay:12]. Valiant proved [@Val:79b] that every polynomial $p$ is an affine projection of some ${\textup{det}}_n$ for $n$ large enough. The smallest $n$ for which this is possible is called the *determinantal complexity* ${\textup{\textsf{dc}}}(p)$. The class of sequences of polynomials $(p_m)$ whose sequence of natural numbers ${\textup{\textsf{dc}}}(p_m)$ is polynomially bounded is called ${\ensuremath{\textup{\textsf{VP}}_{\textsf{s}}}}$. For the permanent we can define the *permanental complexity* ${\textup{\textsf{pc}}}(p)$ in a completely analogous manner: ${\textup{\textsf{pc}}}(p)$ is the smallest $n$ such that $p$ is an affine projection of ${\textup{per}}_n$. The class of sequences of polynomials $(p_m)$ whose ${\textup{\textsf{pc}}}(p_m)$ is polynomially bounded is called ${\textup{\textsf{VNP}}}$. Since ${\textup{\textsf{pc}}}({\textup{det}}_n)$ is polynomially bounded, ${\ensuremath{\textup{\textsf{VP}}_{\textsf{s}}}}\subseteq {\textup{\textsf{VNP}}}$. Valiant’s flagship conjecture in algebraic complexity theory, which is also known as the *determinant versus permament* conjecture can be succinctly phrased as ${\ensuremath{\textup{\textsf{VP}}_{\textsf{s}}}}\neq {\textup{\textsf{VNP}}}$. This is equivalent to conjecturing that ${\textup{\textsf{dc}}}({\textup{per}}_m)$ grows superpolynomially fast.
#### Homogeneous projections and endomorphism orbits
It will be beneficial to phrase Valiant’s conjecture in a homogeneous setting: A *homogeneous projection* of a homogeneous polynomial (i.e., all monomials have the same total degree) is its evaluation at a point whose coordinates are given by homogeneous linear polynomials. The set of all homogeneous projections of ${\textup{det}}_n$ to polynomials in the variables $x_1,\ldots,x_N$ can then be written as $\{{\textup{det}}_n(\ell_1,\ldots,\ell_{n^2}) \mid \text{$\ell_i$ is a homogeneous linear polynomial in $x_1,\ldots,x_N$}\}$. Note that we put the $n \times n = n^2$ inputs of the determinant in a linear order. The polynomial function $(x_{1,1},\ldots,x_{n,n}) \mapsto {\textup{det}}_n(\ell_1,\ldots,\ell_{n^2})$ equals the composition ${\textup{det}}_n \circ A$, where $A$ is the linear map $(x_{1,1},\ldots,x_{n,n}) \mapsto (\ell_1,\ldots,\ell_{n^2})$. As it is common in representation theory, we write $A \cdot {\textup{det}}_n$ or just $A {\textup{det}}_n$ for ${\textup{det}}_n \circ A$. The *endomorphism orbit* ${\textup{End}}_{n^2}{\textup{det}}_n$ is defined as $\{A {\textup{det}}_n \mid A \in {\mathbb{C}}^{n^2 \times n^2}\}$, which is the set of all homogeneous projections of ${\textup{det}}_n$ to polynomials in at most $n^2$ variables. Since all polynomials in $A {\textup{det}}_n$ are homogeneous of degree $n$, we have ${\textup{per}}_m \notin {\textup{End}}_{n^2} {\textup{det}}_n$ for any $m \neq n$. This slight technicality is treated by a procedure called *padding*: For fixed $m$, $n$ with $m < n$, define the *padded permanent* ${\textup{per}}_{m,n} := (x_{n,n})^{n-m}\cdot {\textup{per}}_m$. Let ${\textup{\textsf{dc}}}'({\textup{per}}_{m,n})$ denote the smallest $n$ such that ${\textup{per}}_{m,n} \in {\textup{End}}_{n^2} {\textup{det}}_n$. A short calculation shows that Valiant’s conjecture is equivalent to the conjecture that ${\textup{\textsf{dc}}}'({\textup{per}}_m)$ grows superpolynomially.
#### Group orbits
It turns out that if we restrict ${\textup{End}}_{n^2} {\textup{det}}_n$ to only the points $A{\textup{det}}_n$ for which $A$ is invertible, we get the much simpler *group orbit* ${\textup{GL}}_{n^2}{\textup{det}}_n := \{g {\textup{det}}_n \mid g \in {\textup{GL}}_{n^2}\} \subseteq {\textup{End}}_{n^2} {\textup{det}}_n$. The group orbit of the determinant consists of “determinants in disguise”, i.e., determinants after a base change. The question whether a polynomial $p$ lies in ${\textup{GL}}_{n^2}{\textup{det}}_n$ can be answered in randomized polynomial time [@Kay:12]. Finding $g \in {\textup{GL}}_{n^2}$ such that $p = g {\textup{det}}_n$ is called the *reconstruction* problem, which is the focus of several recent papers, where the determinant is replaced with other algebraic computational models, see e.g.[@GKQ:14], [@KS:19], [@KNS:19].
#### Representation theory
Let $H_{{\textup{det}}_n}\subseteq {\textup{GL}}_{n^2}$ denote the symmetry group of ${\textup{det}}_n$. From the viewpoint of algebraic geometry, the set ${\textup{GL}}_{n^2}{\textup{det}}_n$ is an affine variety ([@BLMW:11 Sec 4.2], [@Mat:60 Cor., p. 206]) and a homogeneous space that is isomorphic to the quotient ${\textup{GL}}_{n^2}/H_{{\textup{det}}_n}$. It is crucial to note here that the group $H_{{\textup{det}}_n}$ does not carry any information about the fact that we study a space of polynomials! In this way, we study the orbit ${\textup{GL}}_{n^2}{\textup{det}}_n$ independently of its embedding into the space of polynomials. This gives a particularly beautiful description of its coordinate ring via invariant theory: $${\mathbb{C}}[{\textup{GL}}_{n^2}{\textup{det}}_n] = {\mathbb{C}}[{\textup{GL}}_{n^2}]^{H_{{\textup{det}}_n}},$$ where ${\mathbb{C}}[{\textup{GL}}_{n^2}]^{H_{det_n}}$ is the set of regular $H_{{\textup{det}}_n}$-invariant functions on the variety ${\textup{GL}}_{n^2}$ [@BLMW:11 (5.2.6)]. The coordinate ring of ${\textup{GL}}_{n^2}$ has classically been studied: It is a ${\textup{GL}}_{n^2} \times {\textup{GL}}_{n^2}$-representation whose representation theoretic decomposition is multiplicity free: $$\label{eq:algpeterweyl}
{\mathbb{C}}[{\textup{GL}}_{n^2}] = \bigoplus_{{\lambda}}\{{\lambda}\} \otimes \{{\lambda}^*\},$$ where ${\lambda}$ runs over all nondecreasing lists of $n^2$ integers and $\{{\lambda}\}$ denotes the irreducible rational representation of ${\textup{GL}}_{n^2}$ corresponding to ${\lambda}$ (see e.g. Section \[sec:prelrepr\] or the textbooks [@FH:91; @MaD:95; @fult:97] for more information). Eq. is known as the algebraic Peter-Weyl theorem. It implies that the multiplicity of ${\lambda}^*$ in ${\mathbb{C}}[{\textup{GL}}_{n^2}{\textup{det}}_n]$ equals the dimension of the $H_{{\textup{det}}_n}$-invariant space $\{{\lambda}\}^{H_{{\textup{det}}_n}}$. This coefficient $\dim \{{\lambda}\}^{H_{{\textup{det}}_n}}$ is known as the symmetric rectangular Kronecker coefficient and has been the object of several papers [@BOR:09; @Man:11; @IP:17]. Even though Kronecker coefficients are mysterious, many properties are known (see e.g. [@CDW:12; @Bla:12; @SS:16; @IMW:17; @Liu:17; @BCMW:17] for some recent advances), and character theory is available to compute their values in many cases, see e.g. [@ike:12b Appendix]. A better understanding of Kronecker coefficients could lead to a better understanding of ${\textup{GL}}_{n^2}{\textup{det}}_n$, which could eventually help to separate points from ${\textup{End}}_{n^2}{\textup{det}}_n$.
If we replace ${\textup{det}}_n$ by other polynomials, we get an analogous theory that is often equally beautiful. The power sum polynomial and the product $x_1 \cdots x_D$ will be of particular interest in this paper. The corresponding coefficients are not Kronecker coefficients, but plethysm coefficients and related coefficients that appear in algebraic combinatorics, see Proposition \[pro:gctandsymmetries\] for the power sum and for the product of variables.
#### Closures
As we just have seen, group orbits have several desirable properties and can be understood directly via symmetry groups and algebraic combinatorics. But endomorphism orbits do not behave that nicely. In general, ${\textup{End}}_{n^2}{\textup{det}}_n$ is not a variety. In order to enable the study of ${\textup{End}}_{n^2}{\textup{det}}_n$ with methods from algebraic geometry, we go to the closure (Euclidean closure and Zariski closure coincide here by general principles, see [@Mum:95 §2.C]): $\overline{{\textup{End}}_{n^2}{\textup{det}}_n}$, which coincides with the group orbit closure $\overline{{\textup{GL}}_{n^2}{\textup{det}}_n}$, see e.g. [@BI:18 Sec. 3.5]. Hence we have a chain of inclusions ${\textup{GL}}_{n^2}{\textup{det}}_n \subseteq {\textup{End}}_{n^2}{\textup{det}}_n \subseteq \overline{{\textup{GL}}_{n^2}{\textup{det}}_n}$. The border determinantal complexity ${\underline{\textup{\textsf{dc}}}}({\textup{per}}_m)$ is defined as the smallest $n$ such that ${\textup{per}}_{m,n} \in \overline{{\textup{GL}}_{n^2}{\textup{det}}_n}$. Mulmuley and Sohoni’s conjecture (closely related to Bürgisser’s conjecture [@Bue:01 hypothesis (7)]) is that ${\underline{\textup{\textsf{dc}}}}({\textup{per}}_m)$ grows superpolynomially. Since ${\underline{\textup{\textsf{dc}}}}({\textup{per}}_m) \leq {\textup{\textsf{dc}}}'({\textup{per}}_m)$, this would imply Valiant’s conjecture. Mulmuley and Sohoni’s conjecture can be attacked by representation theoretic multiplicities (see Section \[sec:preliminaries\]) as follows. If we assume for the sake of contradiction that $\overline{{\textup{GL}}_{n^2}{\textup{per}}_{m,n}} \subseteq \overline{{\textup{GL}}_{n^2}{\textup{det}}_n}$, then by Schur’s lemma (see [@FH:91 Lemma 1.7]) the multiplicities satisfy ${\textup{mult}}_{\lambda}{\mathbb{C}}[\overline{{\textup{GL}}_{n^2}{\textup{per}}_{m,n}}] \leq {\textup{mult}}_{\lambda}{\mathbb{C}}[\overline{{\textup{GL}}_{n^2}{\textup{det}}_n}]$. Thus, if there exists $({\lambda},d)$ that satisfies $$\label{eq:ineq}
{\textup{mult}}_{\lambda}{\mathbb{C}}[\overline{{\textup{GL}}_{n^2}{\textup{per}}_{m,n}}]_d > {\textup{mult}}_{\lambda}{\mathbb{C}}[\overline{{\textup{GL}}_{n^2}{\textup{det}}_n}]_d,$$ then ${\underline{\textup{\textsf{dc}}}}(m)>n$. Such a pair $({\lambda},d)$ is called a *multiplicity obstruction*.
#### Orbits vs orbit closures
The algebraic geometry of $\overline{{\textup{GL}}_{n^2}{\textup{det}}_n}$ and the representation theory of its coordinate ring are rather difficult to understand, see e.g. [@Kum:15; @BHI:17]. But the close relationship between orbit and orbit closure gives hope that results can be transferred from the orbit to the closure. Indeed, ${\mathbb{C}}[\overline{{\textup{GL}}_{n^2}{\textup{det}}_n}] \subseteq {\mathbb{C}}[{\textup{GL}}_{n^2}{\textup{det}}_n]$ is a subalgebra, and hence we have ${\textup{mult}}_{\lambda}{\mathbb{C}}[{\textup{GL}}_{n^2} {\textup{det}}_n] \geq {\textup{mult}}_{\lambda}{\mathbb{C}}[\overline{{\textup{GL}}_{n^2} {\textup{det}}_n}]$. Getting lower bounds on multiplicities in ${\mathbb{C}}[\overline{{\textup{GL}}_{n^2} {\textup{det}}_n}]$ seems much harder. For example, the result in [@Kum:15] holds for those $n$ for which the Alon-Tarsi property holds, in particular if $n$ is an odd prime number $\pm 1$, see Section \[sec:notoccobsorvanidoccobs\]. The occurrence results in [@BIP:19] use explicit constructions using Young symmetrizers, which is a laborious process. But as a first step towards lower bounds on ${\textup{mult}}_{\lambda}{\mathbb{C}}[\overline{{\textup{GL}}_{n^2} {\textup{det}}_n}]$, [@BI:17] proved that ${\textup{GL}}_{n^2} {\textup{det}}_n$ is open in its closure and that the ring ${\mathbb{C}}[{\textup{GL}}_{n^2} {\textup{det}}_n]$ is a localization of ${\mathbb{C}}[\overline{{\textup{GL}}_{n^2} {\textup{det}}_n}]$.
Our contribution
================
In this paper we tighten the results from [@BI:17] in the case of the power sum polynomial. For $m \geq D$ let $p := x_1^D + x_2^D + \cdots + x_m^D$ and let $q := x_1 x_2 \cdots x_D$. Let $G := {\textup{GL}}_{m}$. For $m=D$ we separate the two families of orbit closures $\overline {Gp} \not\subseteq \overline {Gq}$ of polynomials $p$ and $q$ using multiplicity obstructions ${\lambda}$, i.e., ${\textup{mult}}_{\lambda}{\mathbb{C}}[\overline{Gp}]>{\textup{mult}}_{\lambda}{\mathbb{C}}[\overline{Gq}]$. Our key contribution is a proof method that for the first time implements closely the strategy in [@MS:01; @MS:08]: Both the lower bound on ${\textup{mult}}_{\lambda}{\mathbb{C}}[\overline{Gp}]$ and the upper bound on ${\textup{mult}}_{\lambda}{\mathbb{C}}[\overline{Gq}]$ are obtained directly from the symmetry groups of $p$ and $q$ and the dimension of the spaces of $H_p$- and $H_q$-invariants in irreducible ${\textup{GL}}_m$-representations. This is the result of our tightening of the relationship between ${\textup{mult}}_{\lambda}{\mathbb{C}}[Gp]$ and ${\textup{mult}}_{\lambda}{\mathbb{C}}[\overline{Gp}]$, see Theorem \[thm:proofofconcept\].
Before our paper, all existence proofs of multiplicity obstructions $\overline{Gp} \not\subseteq \overline{Gq}$ for any $p$ and $q$ required to explicitly construct (with multilinear algebra) copies of irreducible representations in ${\textup{mult}}_{\lambda}{\mathbb{C}}[\overline{Gp}]$. These papers only took into account the symmetry group of $q$ instead of both symmetry groups, see [@BI:11; @BI:13; @GIP:17; @DIP:19].
In particular, we prove $\overline{Gp}\not\subseteq\overline{Gq}$ by explicitly constructing a multiplicity obstruction ${\lambda}$ in Theorem \[thm:proofofconcept\] using the symmetry groups of $p$ and $q$ and their representation theoretic decomposition coefficients, but we do *not* construct an explicit function that separates the two orbit closures! Since our obstruction is neither an occurrence obstruction, nor a vanishing ideal occurrence obstruction (see next paragraph), the separating function is quite nontrivial to recover. This is a step in the right direction, since the explicit construction of separating functions for Valiant’s conjecture could turn out to be problematic because of the algebraic natural proofs barrier [@FSV:17; @GKSS:17; @BIJL:18].
Occurrence obstructions and vanishing ideal occurrence obstructions {#occurrence-obstructions-and-vanishing-ideal-occurrence-obstructions .unnumbered}
-------------------------------------------------------------------
The classical approach of Mulmuley and Sohoni [@MS:01; @MS:08] conjectures the existence of so-called *occurrence obstructions*, which are types ${\lambda}$ for which the stronger property ${\textup{mult}}_{\lambda}{\mathbb{C}}[\overline{G{\textup{per}}_{m,n}}] > 0 = {\textup{mult}}_{\lambda}{\mathbb{C}}[\overline{G{\textup{det}}_n}]$ holds. These obstructions are not enough to prove strong complexity lower bounds (at least not in the classical setting of ${\textup{det}}_n$ vs ${\textup{per}}_{m,n}$), see [@IP:17; @BIP:19]. Recently it was shown that there are settings in which multiplicity obstructions are provably stronger than just occurrence obstructions [@DIP:19]. The types ${\lambda}$ that are used in [@DIP:19] occur in the vanishing ideal of one orbit closure, but not in the vanishing ideal of the other, hence we call them *vanishing ideal occurrence obstructions*. How useful vanishing ideal occurrence obstructions are for separating orbit closures is an open question, but it seems unlikely that strong complexity lower bounds can be proved by using only vanishing ideal occurrence obstructions.
The techniques that we develop in this paper study multiplicities and go beyond just occurrence obstructions and vanishing ideal occurrence obstructions. Theorem \[thm:proofofconcept\] gives the first family of multiplicity obstructions that are neither occurrence obstructions nor vanishing ideal occurrence obstructions, see Proposition \[pro:alontarsi\]. To prove this fact, we make use of Drisko’s and Glynn’s progress on the Alon-Tarsi conjecture about the difference between the number of even and odd Latin squares of a given size.
The toy setting as a starting point {#the-toy-setting-as-a-starting-point .unnumbered}
-----------------------------------
Our separation $\overline{Gp}\not\subseteq\overline{Gq}$ with $p=x_1^D+\cdots+x_m^D$ and $q=x_1\cdots x_D$ is clearly a toy problem, but even though its complexity theoretic relevance is quite limited (multivariate factorization of a power sum) it shares all (as far as we know) crucial geometric and representation theoretic features with the determinant versus permanent problem: Both problems are problems about orbit closures of polynomials, and the group action is the same canonical action. The only difference between the two setups are the specific polynomials $p$ and $q$. Even though $p$ and $q$ do not share the complexity theoretic properties of the determinant and the permanent, $p$ and $q$ are characterized by their symmetry groups and are stable points (see [@BI:17]). Therefore this setup can be seen a starting point from which $p$ and $q$ could now be gradually adjusted until some orbit closure separations can be obtained that give lower bounds in algebraic complexity theory.
Structure of the paper {#structure-of-the-paper .unnumbered}
----------------------
In Section \[sec:preliminaries\] we start with preliminaries that are necessary to state our results precisely in Section \[sec:resultdetails\]. The main connection between representation theory and tableau combinatorics is discussed in Sections \[sec:prelrepr\]–\[sec:hworbitclosure\]. Section \[sec:threetableaux\] proves a technical result about plethysm coefficients that was postponed from Section \[sec:resultdetails\]. Section \[sec:proofmain\] proves the main technical theorem under the assumption that the so-called *Tableau Lifting Theorem* \[thm:prolongation\] is true. The rest of the paper (Sections \[EVENsec:hypergraphsevenD\] to \[sec:psipropertiesDodd\]) is then used to prove the Tableau Lifting Theorem using elementary but subtle Young tableau combinatorics. Even and odd degrees $D$ are treated mostly independently, where the odd degree case is much more involved.
Preliminaries {#sec:preliminaries}
=============
A *partition* ${\lambda}$ is a nonincreasing finite sequence of natural numbers $({\lambda}_1,{\lambda}_2,\ldots)$. We identify a partition with its *Young diagram*, which is a top-left aligned array of boxes with ${\lambda}_i$ boxes in row $i$. For example, the Young diagram for ${\lambda}=(4,3,1)$ is $\tiny\yng(4,3,1)$. The *length* $\ell({\lambda})$ is the number of rows in the Young diagram of ${\lambda}$, formally $\ell({\lambda})=\max\{i \mid {\lambda}_i>0\}$. The *number of boxes* of ${\lambda}$ is defined as $|{\lambda}|:=\sum_{i\in{\mathbb{N}}} {\lambda}_i$. We also define $|\varrho|:=\sum_{i\in{\mathbb{N}}} \varrho_i$ in the case where $\varrho$ is not a partition. The *transpose* ${\lambda}^t$ of a partition ${\lambda}$ is the partition corresponding to the Young diagram that is the reflection of ${\lambda}$ at the main diagonal, e.g., $(4,3,1)^t = (3,2,2,1)$. The entries of ${\lambda}^t$ are the column lengths of ${\lambda}$. For a partition of length $\leq m$ and $\delta$ many boxes, we write ${\lambda}\vdash_m \delta$. For two partitions ${\lambda}$ and $\mu$ we define their sum ${\lambda}+\mu$ in a row-wise fashion: $({\lambda}+\mu)_i:={\lambda}_i+\mu_i$. For natural numbers $a,b$ let $a \times b$ denote the partition that corresponds to the rectangular Young diagram with $a$ rows and $b$ columns, i.e., $a \times b := (b,b,\ldots,b)$.
Fix $m \in {\mathbb{N}}$. For a partition ${\lambda}\vdash_m \delta$ we write $\{{\lambda}\}$ to denote the irreducible ${\textup{GL}}_m$-representation of type ${\lambda}$. These $\{{\lambda}\}$ form a pairwise non-isomorphic list of irreducible polynomial ${\textup{GL}}_m$-representations, see [@fult:97]. The dual ($=$contragredient) representation of $\{{\lambda}\}$ is denoted by $\{{\lambda}^*\}$. Since $G$ is a reductive group, every finite dimensional $G$-representation ${\mathscr{V}}$ can be decomposed into a direct sum of irreducible $G$-representations, and we write ${\textup{mult}}_{\lambda}{\mathscr{V}}$ for the multiplicity of $\{{\lambda}\}$ in such a decomposition (although the decomposition might not be unique, the multiplicity is the same in any decomposition).
A *tableau of shape* $\lambda$ over some finite *alphabet* $\mathcal{A}$ is a mapping $\lambda \rightarrow \mathcal{A}$ of boxes to elements in $\mathcal{A}$. For example a tableau of shape $(4,3,1)$ over the alphabet $\mathbb{N}$ is given by $\Yvcentermath1\tiny\young(1322,232,1)$. For a tableau $T$ let ${\textup{\textsf{sh}}}(T):= \lambda$ denote its *shape*, i.e., its vector of row lengths. For a tableau $T$ of shape $\lambda \vdash_m \delta$ over the alphabet $\{1,\dots,m\}$ we define its *content* to be the vector $\varrho \in (\mathbb{N}_{\geq 0})^m$ where $\varrho_i$ counts the number of occurrences of $i$ in $T$. For example, the tableau $\Yvcentermath1\tiny\young(1322,232,1)$ has content $(2,4,2)$. The *sorted content* of a tableau is the partition obtained by sorting the entries in the content in a decreasing order, e.g. $(4,2,2)$ in the preceding example. For two tableaux $T$ and $S$ we define their *concatenation* $T+S$ by concatenating rows. The resulting tableau $T+S$ has shape ${\textup{\textsf{sh}}}(T)+{\textup{\textsf{sh}}}(S)$. For example, $\Yvcentermath1\tiny\young(1111,2222,3333) + \Yvcentermath1\tiny\young(444,55,6) = \Yvcentermath1\tiny\young(1111444,222255,33336)$. A tableau with entries from $\mathbb{N}$ is called *semistandard* if the entries of each row are nondecreasing from left to right and the entries of each column are strictly increasing from top to bottom. A semistandard tableau of shape ${\lambda}$ is called *standard* if every number $1,\ldots,|{\lambda}|$ appears exactly once. A column of a tableau $T$ is called *regular* if it does not have a repeated entry. A tableau $T$ is called *regular* if each of its columns is regular. A tableau $L$ is called *duplex* if each column in $L$ appears an even number of times. For example, $L={\Yvcentermath1\tiny\young(11113344,2222,3344)}$ is duplex, while $L={\Yvcentermath1\tiny\young(11112344,2223,3344)}$ is not duplex. Duplex tableaux are the main idea behind the construction in [@bci:10], see also [@ike:12b Sec. 6.2].
Let ${\textup{Poly}}^D {\mathbb{C}}^m$ denote the vector space of homogeneous degree $D$ polynomials in $m$ variables. Let $G := {\textup{GL}}_m$. The group $G$ acts on ${\textup{Poly}}^D {\mathbb{C}}^m$ via $(gp)(x) := p(g^t x)$ for all $g \in G$, $p \in {\textup{Poly}}^D {\mathbb{C}}^m$, $x \in {\mathbb{C}}^m$, where the transpose makes this a left action. We consider the power sum polynomial $p := x_1^D + \cdots + x_m^D \in {\textup{Poly}}^D {\mathbb{C}}^m$. Clearly $p$ is fixed by permuting variables and by rescaling any variable by $D$th roots of unity. For $D\geq 3$ these group elements generate the whole stabilizer of $p$, see e.g. [@CKW:10]. Let $H := {\text{stab\,}}p = {\mathbb{Z}}_D^m \rtimes {\mathfrak{S}}_m \subseteq G$ denote the stabilizer of $p$. Let $q := x_1 \cdots x_D$. Clearly $q$ is fixed under permuting the variables and under rescaling each variable by a scalar $\alpha_i$ such that their product $\prod_{i=1}^D \alpha_i$ equals 1. These elements generate the stabilizer of $q$, see e.g. [@Ike:19 Prop. 3.1].
Considering ${\textup{Poly}}^D {\mathbb{C}}^m$ as a vector space with a $G$-action, let ${\mathbb{C}}[{\textup{Poly}}^D {\mathbb{C}}^m]_d$ be the vector space of homogeneous degree $d$ polynomials on ${\textup{Poly}}^D {\mathbb{C}}^m$. In a natural way, ${\mathbb{C}}[{\textup{Poly}}^D {\mathbb{C}}^m]_d$ is a $G$-representation: $(gf)(q) := f(g^{-1} q)$ for all $g \in G$, $f \in {\mathbb{C}}[{\textup{Poly}}^D {\mathbb{C}}^m]_d$, $q \in {\textup{Poly}}^D {\mathbb{C}}^m$. The multiplicity ${\textup{mult}}_{\nu^*} {\mathbb{C}}[{\textup{Poly}}^D {\mathbb{C}}^m]_d$ is called the *plethysm coefficient*, denoted by $a_\nu(d,D)$. Note that it does not depend on $m$ for $m \geq \ell({\lambda})$, see e.g. [@ike:12b Sec. 4.3]. For the empty partition $(0)$ we define $a_{(0)}(0,i)=1$. Whether or not the plethysm coefficient has a nice combinatorial description is an open research question in algebraic combinatorics, see Problem 9 in [@sta:00]. Among computer scientists, this question is commonly phrased as whether or not the map $(\nu,d,D)\mapsto a_\nu(d,D)$ is in the complexity class $\#{\textup{\textsf{P}}}$.
Given two irreducible $G$-representations $\{\mu\}$ and $\{\nu\}$, their tensor product $\{\mu\}\otimes\{\nu\}$ is a $G \times G$-representation. Embedding $G \hookrightarrow G \times G$ diagonally via $g \mapsto (g,g)$ the tensor product $\{\mu\}\otimes\{\nu\}$ becomes a $G$-representation that decomposes into irreducibles. The multiplicity ${\textup{mult}}_{\lambda}\{\mu\}\otimes\{\nu\}$ is called the *Littlewood-Richardson coefficient* $c^{\lambda}_{\mu,\nu}$. This quantity has numerous beautiful combinatorial interpretations, see e.g. [@bz:92], [@fult:97 Sec. 5], [@knta:99], [@buc:00], [@ike:12b Sec. 10] and many more. In particular, the map $({\lambda},\mu,\nu)\mapsto c^{\lambda}_{\mu,\nu}$ is in the complexity class $\#{\textup{\textsf{P}}}$. Even though the exact computation of $c^{\lambda}_{\mu,\nu}$ is ${\textup{\textsf{NP}}}$-hard [@nara:06], deciding its positivity is possible in polynomial time [@DLM:06], [@MNS:12], [@BI:13LRC]. Completely analogous properties hold when we take tensor products of polynomially many irreducible $G$-representations $\{\mu^1\}\otimes \cdots \otimes \{\mu^d\}$ and embed $G \hookrightarrow G \times G \times \cdots \times G$. The corresponding coefficient is called the *multi-Littlewood-Richardson coefficient* $c_{\mu^1,\mu^2,\ldots,\mu^d}^{\lambda}$.
For a partition $\varrho \vdash_m d$ the *frequency notation* $\hat\varrho \in {\mathbb{N}}^m$ is defined via $\hat\varrho_i := |\{j \mid \varrho_j = i\}|$. For example, the frequency notation of $\varrho=(3,3,2,0)$ is $\hat\varrho=(0,1,2,0)$. We observe that $|\varrho|=\sum_i i \hat\varrho_i$.
Result details {#sec:resultdetails}
==============
The following Proposition \[pro:gctandsymmetries\] writes the multiplicity ${\textup{mult}}_{{\lambda}^*} {\mathbb{C}}[G p]$ as a nonnegative sum of products of multi-Littlewood-Richardson coefficients and plethysm coefficients.
\[pro:gctandsymmetries\] Let ${\lambda}\vdash_m dD$. Define $$b({\lambda},\varrho,D,d) := \sum_{\mu^1,\mu^2,\ldots,\mu^d \atop \mu^i \vdash D i \hat\varrho_i} c_{\mu^1,\mu^2,\ldots,\mu^d}^{\lambda}\prod_{i=1}^d a_{\mu^i}(\hat\varrho_i,iD).$$ Then $${\textup{mult}}_{{\lambda}^*} {\mathbb{C}}[G p] = \sum_{\varrho\vdash_m d} b({\lambda},\varrho,D,d).$$
The proof technique is based on the technique in [@BI:11]. The proof is postponed to Section \[sec:gctandsymmetries\].
We remark that if Problem 9 in [@sta:00] is resolved positively, then Proposition \[pro:gctandsymmetries\] implies that the multiplicity ${\textup{mult}}_{{\lambda}^*}{\mathbb{C}}[Gp]$ has a combinatorial description, i.e., the map $({\lambda},m,d,D)\mapsto {\textup{mult}}_{{\lambda}^*}{\mathbb{C}}[Gp]$ is in $\#{\textup{\textsf{P}}}$. The same holds also for its summands $b(\lambda,\varrho,D,d)$. It is known that ${\textup{mult}}_{{\lambda}^*}{\mathbb{C}}[Gq] = a_{\lambda}(D,d)$ (see e.g. [@Lan:17 Sec. 9.2.3]), so the same holds for ${\textup{mult}}_{{\lambda}^*}{\mathbb{C}}[Gq]$.
Our main technical theorem that enables us to find obstructions is the following.
\[thm:main\] Let $m, d, D \in {\mathbb{N}}$. If $D$ is odd, we assume that ${{2(D-1)}\choose{D-1}} \geq 2(m-1)$. Let ${\lambda}\vdash_m dD$. For each $\varrho\vdash_m d$ define the number $e_\varrho$ as follows:
- if $D$ is even, let $e_\varrho := \sum_{i=1}^{m} \lceil \frac{\varrho_i}{D-2} \rceil$.
- if $D$ is odd, let $e_\varrho := \sum_{i=1}^{m} 2\lceil \frac{\varrho_i}{2(D-2)} \rceil$.
Let $\Xi$ be a subset of the set of all partitions $\varrho\vdash_m d$. Let $e_\Xi := \max\{e_\varrho \mid \varrho \in \Xi\}$. Then $${\textup{mult}}_{({\lambda}+(m \times e_\Xi D))^*} {\mathbb{C}}[\overline{Gp}] \geq \sum_{\varrho \in \Xi} b({\lambda},\varrho,D,d).$$
The proof of Theorem \[thm:main\] is postponed to Section \[sec:proofmain\].
Explicit obstructions via symmetries {#explicit-obstructions-via-symmetries .unnumbered}
------------------------------------
We use Theorem \[thm:main\] as follows to construct obstructions.
\[thm:proofofconcept\] Let $m=D\geq 4$ be even. Let $d=2$. Let ${\lambda}= (2m)$ and let $\nu = (2m)+(m\times 2m)$. Let $p:=x_1^m+\cdots+x_m^m$ and $q := x_1 x_2 \cdots x_m$. Let $\Xi = \{(2),(1,1)\}$. In the notation of Theorem \[thm:main\] we have $e_{(2)}=1$ and $e_{(1,1)}=2$, thus $e_\Xi = 2$. Note that $\nu = {\lambda}+ (m \times e_\Xi D)$. We have
- $b((2m),(2),m,2)=1$ and $b((2m),(1,1),m,2)=1$ and hence with Theorem \[thm:main\] we have ${\textup{mult}}_{\nu^*} {\mathbb{C}}[\overline{Gp}] \geq 2$.
- $1 \geq {\textup{mult}}_{\nu^*} {\mathbb{C}}[G q] \geq {\textup{mult}}_{\nu^*} {\mathbb{C}}[\overline{G q}]$.
In particular ${\textup{mult}}_{\nu^*} {\mathbb{C}}[\overline{Gp}] \geq 2 > 1 \geq {\textup{mult}}_{\nu^*} {\mathbb{C}}[\overline{Gq}]$ and hence $\nu$ is a multiplicity obstruction that proves the separation $\overline{Gp}\not\subseteq\overline{Gq}$.
We use the following well-known properties about Littlewood-Richardson coefficients (see e.g. [@fult:97 Ch. 5]).\
$\bullet$ The Littlewood-Richardson coefficient is symmetric in the subscript parameters: $c_{\mu,\nu}^{\lambda}= c_{\nu,\mu}^{\lambda}$.\
$\bullet$ $c_{\mu,(0)}^{\lambda}$ equals 1 iff $\mu={\lambda}$, otherwise $c_{\mu,(0)}^{\lambda}=0$.
Recall that $\hat\varrho$ is the frequency notation of $\varrho$. We calculate $$\begin{aligned}
b({\lambda},(2),m,2) &=& \sum_{\mu^1,\mu^2 \atop \mu^i \vdash m i \widehat{(2)}_i} c_{\mu^1,\mu^2}^{\lambda}\prod_{i=1}^2 a_{\mu^i}(\widehat{(2)}_i,im) \\
&=&
\sum_{\mu^2 \vdash 2D} c_{(0),\mu^2}^{\lambda}\cdot
a_{(0)}(\widehat{(2)}_1,m) \cdot a_{\mu^2}(\widehat{(2)}_2,2m)
\underbrace{=}_{c_{(0),\mu}^{\lambda}= \delta_{\mu,{\lambda}}}
a_{(0)}(\widehat{(2)}_1,m) \cdot a_{{\lambda}}(\widehat{(2)}_2,2m) \\
&=&
a_{(0)}(0,m) \cdot a_{(2m)}(1,2m) = 1 \cdot 1 = 1.\end{aligned}$$ Here we used the trivial fact that $a_{(\delta)}(1,\delta)=1$ for all $\delta$. $$\begin{aligned}
b({\lambda},(1,1),m,2) &=& \sum_{\mu^1,\mu^2 \atop \mu^i \vdash m i \widehat{(1,1)}_i} c_{\mu^1,\mu^2}^{\lambda}\prod_{i=1}^2 a_{\mu^i}(\widehat{(1,1)}_i,im) \\
&=&
\sum_{\mu^1 \vdash m} c_{\mu^1,(0)}^{\lambda}\cdot
a_{\mu^1}(\widehat{(1,1)}_1,m) \cdot a_{(0)}(\widehat{(1,1)}_2,2m)
\underbrace{=}_{c_{\mu,(0)}^{\lambda}= \delta_{\mu,{\lambda}}}
a_{{\lambda}}(\widehat{(1,1)}_1,m) \cdot a_{(0)}(\widehat{(1,1)}_2,2m) \\
&=&
a_{(2m)}(2,m) \cdot a_{(0)}(0,2m) = 1 \cdot 1 = 1.\end{aligned}$$ Here we used the classical fact that $a_{{\lambda}}(2,m)=1$ if ${\lambda}$ has $2m$ boxes and at most 2 rows and both rows have even length (see the formula for $h_2 \circ h_n$ in [@MaD:95 I.8, Exa. 9(a), p. 140]). This proves the first part of the claim.
${\mathbb{C}}[\overline{Gq}] \subseteq {\mathbb{C}}[Gq]$ is a subalgebra, so ${\textup{mult}}_{\nu^*} {\mathbb{C}}[\overline{Gq}] \leq {\textup{mult}}_{\nu^*} {\mathbb{C}}[Gq]$. To show ${\textup{mult}}_{\nu^*} {\mathbb{C}}[Gq] \leq 1$ we use that $$\label{eq:reverseplethysm}
{\textup{mult}}_{\nu^*} {\mathbb{C}}[Gq] = a_\nu(m,2m+2),$$ see e.g. [@Lan:17 Sec. 9.2.3]. We apply the upper bound given by the Kostka numbers $a_\nu(m,2m+2) \leq K(\nu,m \times (2m+2))$, which is the number of semistandard tableaux of shape $\nu$ and content $m \times (2m+2)$. This classical upper bound can be deduced for example from [@Gay:76], see also [@ike:12b Sec. 4.3(A)]. It is clear from the special shape of $\nu$ that this Kostka number is 1. As an illustration, we give an example of this tableau in the case where $m=4$:
[(1111111111223344,22222222,33333333,44444444) ]{}
In Section \[sec:notoccobsorvanidoccobs\] we prove that the obstructions in Theorem \[thm:proofofconcept\] are neither occurrence obstructions nor vanishing ideal occurrence obstructions (in infinitely many cases. This holds in *all* cases if the Alon-Tarsi conjecture is true).
#### A remark on plethysm coefficients
As far as we know, Theorem \[thm:main\] is the first result of its type in the literature so far. Even the following direct corollary about plethysm coefficient positivity is new.
\[cor:plethpos\] Let $D\geq 3$ be odd and let $m$ be arbitrary with $\binom{2(D-1)}{D-1}\geq 2(m-1)$. Let $d$ be arbitrary. Let $e=2\lceil\frac d {2(D-2)}\rceil$. Then $a_{(dD)+m\times eD}(d+me,D)\geq 1$.
Let $\lambda=(dD)$ and $\varrho=(d)$, $\Xi = \{(d)\}$. Theorem \[thm:main\] implies ${\textup{mult}}_{((dD)+(m \times e D))^*} {\mathbb{C}}[\overline{Gp}] \geq b((dD),(d),D,d) = 1$. Since $\overline{Gp}$ is a subvariety of ${\textup{Poly}}^D{\mathbb{C}}^m$, it follows $a_{(dD)+m\times eD}(d+me,D)\geq 1$.
Many additional direct corollaries of this type can be drawn from Theorem \[thm:main\].
Neither occurrence obstructions nor vanishing ideal occurrence obstructions {#sec:notoccobsorvanidoccobs}
===========================================================================
The main new property of our obstructions in Theorem \[thm:proofofconcept\] is that they use both the symmetry group of $p$ and the symmetry group of $q$. This is a fundamentally new way of constructing obstructions. To highlight the novelty of the approach, in this section we prove that the obstructions in Theorem \[thm:proofofconcept\] are not vanishing ideal occurrence obstructions. We also prove that they are not occurrence obstructions, provided a property of Latin squares is true (which we know is true for an infinite number of cases). The novelty of our method gives hope that more and stronger results can be proved in a similar way.
Recall the following proposition from [@BI:17]:
\[lem:shift\] If $D$ is even, then ${\textup{mult}}_{{\lambda}^*} {\mathbb{C}}[G p] = {\textup{mult}}_{({\lambda}+(m \times D))^*} {\mathbb{C}}[G p]$. If $D$ is odd, then ${\textup{mult}}_{{\lambda}^*} {\mathbb{C}}[G p] = {\textup{mult}}_{({\lambda}+(m \times 2D))^*} {\mathbb{C}}[G p]$.
\[lem:cormain\] Let ${\lambda}\vdash_m dD$. If $D$ is even, let $$e := \begin{cases}
d & \text{ if } d\leq m \\
m + \lfloor \frac{d-m}{D-2} \rfloor & \text{ if } d\geq m
\end{cases}.$$ If $D$ is odd and ${{2(D-1)}\choose{D-1}} \geq 2(m-1)$, let $$e := \begin{cases}
2d & \text{ if } d\leq m \\
2m + 2\lfloor \frac{d-m}{2(D-2)} \rfloor & \text{ if } d\geq m
\end{cases}.$$ In both cases we have ${\textup{mult}}_{({\lambda}+(m \times e D))^*} {\mathbb{C}}[\overline{Gp}] = {\textup{mult}}_{{\lambda}^*} {\mathbb{C}}[G p] = {\textup{mult}}_{({\lambda}+(m \times e D))^*} {\mathbb{C}}[Gp]$.
The second equality follows from Lemma \[lem:shift\].
Let $\Xi$ denote the set of *all* partitions $\varrho\vdash_m d$. Using Theorem \[thm:main\], then according to Proposition \[pro:gctandsymmetries\] we have $${\textup{mult}}_{({\lambda}+(m \times e_\Xi D))^*} {\mathbb{C}}[\overline{Gp}] \geq {\textup{mult}}_{{\lambda}^*} {\mathbb{C}}[G p].$$ Using Lemma \[lem:shift\] (and the fact that $e_\Xi$ is even if $D$ is odd) we see that $${\textup{mult}}_{({\lambda}+(m \times e_\Xi D))^*} {\mathbb{C}}[\overline{Gp}] = {\textup{mult}}_{{\lambda}^*} {\mathbb{C}}[G p].$$ It remains to show that $e_\Xi = e$. Recall that for natural numbers $a,b$ we have $\lceil\frac{1+a}{b}\rceil-1=\lfloor\frac a b\rfloor$.
Let $D$ be even and $d \leq m$. Then the number of nonzero $\varrho_i$ is at most $d$. Hence $e_\varrho \leq d$. On the other hand, $\varrho=(1,1,\ldots,1,0,0,\ldots,0) \vdash_m d$ provides $e_\varrho = d$, so the bound is tight. The argument for $D$ odd and $d \leq m$ is completely analogous and yields $e = 2d$.
Let $D$ be even and $d \geq m$. Then $\varrho = (1+d-m,1,1,\ldots,1)\vdash_m d$ provides $e_\varrho = m+(\lceil\frac{1+d-m}{D-2}\rceil-1) = m + \lfloor \frac{d-m}{D-2}\rfloor$. The upper bound is provided via $$\textstyle e_\varrho = {\displaystyle\sum_{i=1}^m}\lceil\frac{\varrho_i}{D-2}\rceil = {\displaystyle\sum_{i=1}^m}\left(\lceil\frac{\varrho_i+1-1}{D-2}\rceil+1-1\right) = {\displaystyle\sum_{i=1}^m} \left( \lfloor\frac{\varrho_i-1}{D-2}\rfloor +1 \right) = m+\underbrace{\textstyle{\displaystyle{\displaystyle\sum_{i=1}^m}}\lfloor\frac{\varrho_i-1}{D-2}\rfloor}_{\leq \lfloor\frac{d-m}{D-2}\rfloor} \leq m+\lfloor\frac{d-m}{D-2}\rfloor.$$
Let $D$ be odd and $d \geq m$. Then $\varrho = (1+d-m,1,1,\ldots,1)\vdash_m d$ provides $e_\varrho = 2m+2(\lceil\frac{1+d-m}{2(D-2)}\rceil-1) = 2m + 2\lfloor \frac{d-m}{2(D-2)}\rfloor$. The upper bound is provided via $$\begin{aligned}
\textstyle e_\varrho &=& {\displaystyle\sum_{i=1}^m}2\lceil\tfrac{\varrho_i}{2(D-2)}\rceil = {\displaystyle\sum_{i=1}^m}2\left(\lceil\tfrac{\varrho_i+1-1}{2(D-2)}\rceil+1-1\right) = {\displaystyle\sum_{i=1}^m} 2\left( \lfloor\tfrac{\varrho_i-1}{2(D-2)}\rfloor +1 \right) \\
&=& 2m+2\underbrace{\textstyle{\displaystyle{\displaystyle\sum_{i=1}^m}}\lfloor\tfrac{\varrho_i-1}{2(D-2)}\rfloor}_{\leq \lfloor\tfrac{d-m}{2(D-2)}\rfloor}
\leq 2m+2\lfloor\tfrac{d-m}{2(D-2)}\rfloor.\end{aligned}$$
A Latin square of dimension $m$ is an $m \times m$ matrix for which in each row and in each column each number $1,\ldots,m$ appears exactly once. The *sign* of a column is 1 if the permutation in the column is even, and $-1$ otherwise. The *sign* of a Latin square is defined as the product of all column signs. A Latin square is called *even* if its sign is 1, and *odd* otherwise. If $m$ is odd, then it is easy to construct an involution on the set of all $m \times m$ Latin squares that pairs each even Latin square with an odd one. Hence, if $m$ is odd, then the number of even $m \times m$ Latin squares equals the number of odd $m \times m$ Latin squares. If $m$ is even, then Alon and Tarsi [@AT:92] conjecture that the number of even $m \times m$ Latin squares *differs from* the number of odd $m \times m$ Latin squares (see also [@HR:94] and [@KL:15] for equivalent formulations). This is proved for all $m=\tau+1$ [@Dri:97] for an odd prime number $\tau$ and for all $m=\tau-1$ [@Gly:10] for an odd prime number $\tau$, making $m=26$ the smallest open case. If the number of even $m \times m$ Latin squares differs from the number of odd $m \times m$ Latin squares, then we say that $m$ satisfies the *Alon-Tarsi condition*. The Alon-Tarsi condition first appeared in connection with geometric complexity theory in [@Kum:15].
\[pro:alontarsi\] Let $\nu$, $m$, $D$, $d$ be as in Theorem \[thm:proofofconcept\].
- ${\textup{mult}}_{\nu^*} {\mathbb{C}}[\overline{Gp}]< a_\nu(2(m+2),m)$ and hence $\nu$ is not a vanishing ideal occurrence obstruction.
- If $m$ satisfies the Alon-Tarsi condition, then ${\textup{mult}}_{\nu^*} {\mathbb{C}}[\overline{Gq}] > 0$ and hence $\nu$ is not an occurrence obstruction. In particular this is true for $m=\tau\pm1$ for all odd primes $\tau$.
The first bullet point is treated as follows. ${\textup{mult}}_{\nu^*} {\mathbb{C}}[\overline{Gp}] = {\textup{mult}}_{\nu^*} {\mathbb{C}}[{Gp}]$ by Lemma \[lem:cormain\]. Moreover, ${\textup{mult}}_{\nu^*} {\mathbb{C}}[{Gp}]=2$ (Proposition \[pro:gctandsymmetries\] and Theorem \[thm:proofofconcept\]). It remains to prove that $a_\nu(2(m+1),m)\geq 3$, which is postponed to Proposition \[pro:threetableaux\].
The second bullet point is treated as follows. Kumar proved [@Kum:15] that ${\textup{mult}}_{(m)^*} {\mathbb{C}}[\overline{G(x_1\cdots x_m)}] \geq 1$ and that ${\textup{mult}}_{(m \times m)^*} {\mathbb{C}}[\overline{G(x_1\cdots x_m)}] \geq 1$, provided that $m$ satisfies the Alon-Tarsi condition. Since $\nu = (m \times m) + (m \times m) + (m) + (m)$, the semigroup property for occurrences in ${\mathbb{C}}[\overline{G(x_1\cdots x_m)}]$ (see e.g. [@DIP:19]) yields that ${\textup{mult}}_{\nu^*} {\mathbb{C}}[\overline{G(x_1\cdots x_m)}] \geq 1$.
The rest of this paper (besides Section \[sec:threetableaux\]) is dedicated to the proof of Theorem \[thm:main\].
Preliminaries - Representation Theory {#sec:prelrepr}
=====================================
In the remainder of this paper we write $G:=\textup{GL}_m(\mathbb{C})$ to denote the general linear group for some fixed natural number $m$. A *representation* of $G$ is a finite dimensional complex vector space $\mathscr{V}$ together with a group homomorphism $\xi: G \rightarrow \textup{GL}(\mathscr{V})$. If $\xi$ is given by a polynomial map, then we call $(\mathscr{V},\xi)$ a *polynomial representation*. We write $gf := \xi(g)(f)$ for all $g \in G$ and $f \in \mathscr{V}$. A linear subspace $\mathscr{W} \subseteq \mathscr{V}$ that is closed under the action of $G$ is called a *subrepresentation*. A representation $\mathscr{V}$ is called *irreducible* iff it has only two subrepresentations, namely the zero-dimensional subspace and $\mathscr{V}$ itself.
For a $G$-representation $\mathscr{V}$, a *highest weight vector* $f \in \mathscr{V}$ of weight $\lambda \in {\mathbb{Z}}^m$ is defined to be a vector that satisfies the following two properties:
- for all upper triangular matrices $g$ with 1s on the main diagonal we have $gf=f$, and
- for all diagonal matrices $g := \textup{diag}(a_1,\dots, a_m)$ we have $gf = a_1^{\lambda_1}\dots a_m^{\lambda_m} f$.
We denote by ${\textup{HWV}}_\lambda(\mathscr{V})$ the vector space of highest weight vectors of weight $\lambda$ in $\mathscr{V}$. It turns out that for an irreducible polynomial $G$-representation $\mathscr{V}$ there is exactly one partition $\lambda \in \mathbb{N}^m$ such that ${\textup{HWV}}_\lambda(\mathscr{V})$ has dimension $=1$, while for all $\mu\neq{\lambda}$ we have $\dim{\textup{HWV}}_\mu(\mathscr{V})=0$. In this case we write $\mathscr{V} = \{\lambda\}$. Moreover, for each partition $\lambda \vdash_m$ there is an irreducible polynomial $G$-representation $\{\lambda\}$ and we call it the irreducible $G$-representation of *isomorphism type* $\lambda$. Furthermore, these $\{\lambda\}$ are pairwise non-isomorphic. We can now define the *multiplicity* $\textup{mult}_\lambda(\mathscr{V})$ of $\{\lambda\}$ in $\mathscr{V}$ as $$\begin{aligned}
\textup{mult}_\lambda(\mathscr{V}) = \dim {\textup{HWV}}_\lambda(\mathscr{V}),\end{aligned}$$ which is the same as the multiplicity with which $\{{\lambda}\}$ appears as a summand in a decomposition of $\mathscr{V}$ into a direct sum of irreducible $G$-representations.
The irreducible $G$-representations {#the-irreducible-g-representations .unnumbered}
-----------------------------------
In the following exposition we closely follow [@fult:97].
For a tablean $T$ let $T(r,c)$ denote the entry of $T$ in row $r$ and column $c$. Let $(e_i)_i$ be the standard basis of ${\mathbb{C}}^m$. Let $\mu = \lambda^t$.
To each tableau $T : {\lambda}\to \{1,\ldots,m\}$ we assign a tensor $$e_{T(1,1)} \otimes e_{T(2,1)} \otimes \cdots \otimes e_{T(\mu_1,1)} \otimes e_{T(1,2)} \otimes e_{T(2,2)} \otimes \cdots \otimes e_{T(\mu_2,2)} \otimes \cdots \cdots \otimes e_{T(\mu_{{\lambda}_1},{\lambda}_1)}.$$
These form a basis of ${\textstyle\bigotimes}^{|{\lambda}|}{\mathbb{C}}^m$. In this way, the space ${\textstyle\bigotimes}^{|{\lambda}|}{\mathbb{C}}^m$ is isomorphic to the space of formal linear combinations of tableaux $T : {\lambda}\to \{1,\ldots,m\}$. Using this isomorphism, the space of tableaux inherits the natural $G$-action on ${\textstyle\bigotimes}^{|{\lambda}|}{\mathbb{C}}^m$, which is given by $$\begin{aligned}
g(v_1 \otimes \dots \otimes v_{|\lambda|}) = g(v_1) \otimes \dots \otimes g(v_{|\lambda|}).\end{aligned}$$ The following vectors span the linear subspace $K({\lambda})$ of *Grassmann-Plücker relations* (sometimes called *shuffle relations*), which is invariant under the $G$-action:
- $T+T'$, where $T'$ is a tableau that arises from $T$ by switching two numbers within one column.
- $T-\Sigma S$, where for two fixed columns $j,j'$ and a fixed number of entries $k$ the sum is over all tableaux $S$ that arise from $T$ by exchanging the top $k$ entries in column $j$ with any $k$ entries in column $j'$, preserving the internal vertical order.
It turns out that $\{{\lambda}\} \simeq ({\textstyle\bigotimes}^{|{\lambda}|}{\mathbb{C}}^m) / K({\lambda})$, see [@fult:97 Ch. 8]. A basis of $\{\lambda\}$ is given by the semistandard tableaux of shape $\lambda$ with entries from $\{1,\dots,m\}$. The unique highest weight vector (up to scale) in $\{\lambda\}$ is the *superstandard tableau* of shape $\lambda$, which is the semistandard tableau that has only entries $i$ in row $i$. It has weight $\lambda$.
Two similar projections of tableaux {#sec:twoprojections}
===================================
In this section we present two symmetrizations that look quite similar on the basis of tableaux. We crucially use this similarity in Section \[sec:proofmain\] in the proof of the Main Theorem \[thm:main\]. Indeed, this peculiarity is the driving force behind our result.
We embed $\mathfrak{S}_m \subseteq G$ via permutation matrices. Given a tableau $S:\lambda \rightarrow \{1,\dots,m\}$ we define: $$\begin{aligned}
P_m S := \sum_{\pi \in \mathfrak{S}_m} \pi S\end{aligned}$$ Interpreting a tableau $S:\lambda \rightarrow \{1,\dots,m\}$ as an element in $\{\lambda\}$ this can be seen as a map $P_m: \{\lambda\} \rightarrow \{\lambda\}^{\mathfrak{S}_m}$, where $\{\lambda\}^{\mathfrak{S}_m}$ consists of the elements in $\{\lambda\}$ that are invariant under the action of $\mathfrak{S}_m$. For example, using the Grassmann-Plücker relations we get that: $$\begin{aligned}
P_{3}\,{\Yvcentermath1\small\young(1122,33)} &=& {\Yvcentermath1\small\young(1122,33)}+\underbrace{{\Yvcentermath1\small\young(2211,33)}}_{=\,{\Yvcentermath1\tiny\young(1133,22)}\,-2\,{\Yvcentermath1\tiny\young(1123,23)}\,+\,{\Yvcentermath1\tiny\young(1122,33)}}+\underbrace{{\Yvcentermath1\small\young(3322,11)}}_{=\,{\Yvcentermath1\tiny\young(1122,33)}}\\
&+&{\Yvcentermath1\small\young(1133,22)} + \underbrace{{\Yvcentermath1\small\young(2233,11)}}_{=\,{\Yvcentermath1\tiny\young(1133,22)}}+\underbrace{{\Yvcentermath1\small\young(3311,22)}}_{=\,{\Yvcentermath1\tiny\young(1133,22)}\,-2\,{\Yvcentermath1\tiny\young(1123,23)}\,+\,{\Yvcentermath1\tiny\young(1122,33)}} \\
&=& 4\,{\Yvcentermath1\small\young(1122,33)}-4\,{\Yvcentermath1\small\young(1123,23)}+4\,{\Yvcentermath1\small\young(1133,22)}\end{aligned}$$
Now we consider tableaux that have $\delta$ many symbols. Let $\varphi: \{1,\dots,\delta\} \rightarrow \{1,\dots,m\}$ be a map. For a tableau $T: \lambda \rightarrow \{1,\dots,\delta\}$, we define $\varphi T$ as the tableau that is the result of replacing the entries from $\{1,\dots,\delta\}$ in $T$ with entries from $\{1,\dots,m\}$. Let $$\mathcal{M}_{\delta,m} := \{\varphi \mid \varphi: \{1,\dots,\delta\} \rightarrow \{1,\dots,m\} \; \text{is a map}\}.$$ We use this to define $$\begin{aligned}
M_{\delta,m}T:=\sum_{\varphi \in \mathcal M_{\delta,m}} \varphi T \in \{{\lambda}\}^{{\mathfrak{S}}_m}.\end{aligned}$$
Tableau contraction {#sec:tableaucontraction}
===================
Let ${\mathbb{S}}\in \{{\lambda}\}$ be the superstandard tableau. Let $\gamma \in \{{\lambda}\}^*$ denote the vector dual to ${\mathbb{S}}$, i.e., the linear map that satisfies $\gamma({\mathbb{S}})=1$ and $\gamma(T)=0$ for every semistandard tableau $T \neq {\mathbb{S}}$.
In the following our goal is to understand $\gamma$ explicitly in terms of determinants, see eq. below. For a matrix $g$ let $g_{1..j,i} \in {\mathbb{C}}^j$ denote the vector that consists of the top $j$ elements in the $i$th column of $g$. We interpret a list of $j$ vectors in ${\mathbb{C}}^i$ as an $i \times j$ matrix of column vectors. A determinant of a matrix with more rows than columns is defined as the determinant of the square matrix of its top rows. For a tableau $T : {\lambda}\to \{1,\ldots,m\}$ let $T(r,c) \in \{1,\ldots,m\}$ denote the number in row $r$ and column $c$.
\[lem:coldets\] For $g\in{\mathbb{C}}^{m \times m}$ we have $$\label{eq:gammadet}
\gamma(gT) = \prod_{c=1}^{{\lambda}_1}{\textup{det}}(g_{1..\mu_c, T(1,c)},\ldots,g_{1..\mu_c, T(\mu_c,c)}),$$ where $\mu={\lambda}^t$.
$$gT = \sum_{S:{\lambda}\to\{1,\ldots,m\}} \left(\prod_{(r,c)\in {\lambda}} g_{S(r,c),T(r,c)} \right) S = \sum_{S:{\lambda}\to\{1,\ldots,m\}\atop S \text{ regular}} \left(\prod_{(r,c)\in {\lambda}} g_{S(r,c),T(r,c)} \right) S$$ The fact that ${\mathbb{S}}$ is superstandard implies that $\gamma(S)=0$ for all $S$ that do not have a permutation of $\{1,\ldots,\mu_i\}$ in column $i$ for all $1 \leq i \leq {\lambda}_1$. Therefore $$\gamma(gT) = \sum_{S:{\lambda}\to\{1,\ldots,m\}\atop S \text{ is a col-perm.\ of ${\mathbb{S}}$}} \left(\prod_{(r,c)\in {\lambda}} g_{S(r,c),T(r,c)} \right) \gamma(S) =
\sum_{S:{\lambda}\to\{1,\ldots,m\}\atop S \text{ is a col-perm.\ of ${\mathbb{S}}$}} \left(\prod_{(r,c)\in {\lambda}} g_{S(r,c),T(r,c)} \right) \prod_{\text{column }c} {\textup{sgn}}(S,c),$$ where ${\textup{sgn}}(S,c)$ is the sign of the permutation of column $c$ of $S$. We conclude $$\gamma(gT) = \prod_{c=1}^{{\lambda}_1} \sum_{\pi \in {\mathfrak{S}}_{\mu_c}} {\textup{sgn}}(\pi) \left(\prod_{r=1}^{\mu_c} g_{\pi(r),T(r,c)} \right),$$ which finishes the proof.
We draw three quick corollaries:
\[cor:gammafactorization\] $\gamma(gT) = \prod_{c=1}^{{\lambda}_1}\gamma(g {\textup{\textsf{col}}}_c),$ where ${\textup{\textsf{col}}}_c$ is the $c$-th column of $T$.
Obvious from Lemma \[lem:coldets\].
\[cor:pm1\] For a tableau $T$ that consists of a single regular column of $m$ boxes, we have $\gamma(gT) \in \{-{\textup{det}}(g),{\textup{det}}(g)\}$. In particular, if $g \in {\textup{SL}}_m$, we have $\gamma(gT) \in \{-1,1\}$.
Since $T$ is regular, the entries of $T$ are precisely all numbers $1,\ldots,m$, not necessarily sorted. Thus according to Lemma \[lem:coldets\], $\gamma(gT)\in\{-{\textup{det}}(g),{\textup{det}}(g)\}$.
\[cor:gammazero\] If $T$ is not regular, then $\gamma(gT)=0$, independent of $g$.
$\gamma(gT)$ factorizes according to Cor. \[cor:gammafactorization\]. Since $T$ is not regular, there is a column with a repeated entry. Thus, according to Lemma \[lem:coldets\], one of the factors of $\gamma(gT)$ equals the determinant of a matrix with a repeated column, and hence is zero.
Highest weight functions on the orbit {#sec:hworbit}
=====================================
In this section we prove the following theorem.
\[thm:functionsonorbit\] The vector space ${\textup{HWV}}_{{\lambda}^*}({\mathbb{C}}[Gp]_d)$ decomposes into a direct sum of vector spaces ${\textup{HWV}}_{{\lambda}^*}({\mathbb{C}}[Gp]_d) = \bigoplus_{\varrho\vdash_m d} {\mathscr{W}}_\varrho$, and each ${\mathscr{W}}_\varrho$ is generated by the functions $$g \mapsto \gamma(g P_m S),$$ where $S$ runs over all semistandard tableaux $S$ of shape ${\lambda}$ and content $\varrho D$.
Let ${\mathscr{W}}_\varrho'$ denote the linear space spanned by all semistandard tableaux $S$ of shape ${\lambda}$ whose sorted content is $\varrho D$. Then ${\mathscr{W}}_\varrho$ is isomorphic to the ${\mathfrak{S}}_m$-invariant subspace of ${\mathscr{W}}_\varrho'$.
Explicit algebraic Peter-Weyl theorem {#explicit-algebraic-peter-weyl-theorem .unnumbered}
-------------------------------------
$G \subseteq {\mathbb{C}}^{m \times m}$ is the nonvanishing set of a polynomial and thus $G$ is a variety. It carries a canonical action of $G\times G$ via $(h',h)g := h' g h^{-1}$ for all $g,h,h'\in G$. This action lifts to to the coordinate ring ${\mathbb{C}}[G]$ via the canonical pullback: $((h',h)f)(g) := f({h'}^{-1} g h)$ for all $g,h,h'\in G$, $f \in {\mathbb{C}}[G]$.
\[thm:explalgpeterweyl\] As a $G \times G$-representation we have $${\mathbb{C}}[G] = \bigoplus_{{\lambda}} \{{\lambda}\}^* \otimes \{{\lambda}\}$$ If we embed $G \hookrightarrow G \times G$ via $g \mapsto (g,\textup{id})$, then the ${\lambda}^*$-isotypic component of ${\mathbb{C}}[G]$ is $\{{\lambda}\}^* \otimes \{{\lambda}\}$, which is spanned by functions $$g \mapsto l(gv),$$ where $l\in\{{\lambda}\}^*$ and $v \in \{{\lambda}\}$.
$H$-invariants {#h-invariants .unnumbered}
--------------
Recall the definition of the symmetry subgroup $H \leq G$ from Section \[sec:preliminaries\]. Again, consider the action of $G\times G$ on $G$ via $(h',h)g := h' g h^{-1}$ for all $g,h,h'\in G$. The action of $G\times G$ lifts to the coordinate ring ${\mathbb{C}}[G]$ and we denote by ${\mathbb{C}}[G]^{\vv H}$ the linear subspace of right $H$-invariants. ${\mathbb{C}}[G]^{\vv H}$ carries a left $G$-action. There is a bijection $gp \sim gH$ between points in the orbit $Gp$ and left cosets of $H$. This leads to the following explicit $G$-equivariant algebra isomorphism [@TY:05 25.4.6, Prop.]: $$\begin{aligned}
{\mathbb{C}}[G p] &\stackrel{\sim}{\longrightarrow}& {\mathbb{C}}[G]^{\vv H} \label{eq:algiso}\\
F & \mapsto & [g \mapsto F(gp)] \nonumber\\
\mbox{~} [gp \mapsto f(g)] & \reflectbox{\ensuremath{\mapsto}} & f \nonumber\end{aligned}$$
Taking right $H$-invariants in Theorem \[thm:explalgpeterweyl\] yields: $${\mathbb{C}}[G]^{\vv H} = \bigoplus_{{\lambda}} \{{\lambda}\}^* \otimes \{{\lambda}\}^H,$$ and explicitly: the left ${\lambda}^*$-isotypic component of ${\mathbb{C}}[G]^{\vv H}$ is spanned by functions $$g \mapsto l(gv),$$ where $l\in\{{\lambda}\}^*$ and $v \in \{{\lambda}\}^H$.
Taking left highest weight functions, we see that ${\textup{HWV}}_{{\lambda}^*}({\mathbb{C}}[G]^{\vv H}) \simeq \{{\lambda}\}^H$ and that ${\textup{HWV}}_{{\lambda}^*}({\mathbb{C}}[G]^{\vv H})$ is spanned by functions $$\label{eq:ggammagv}
g \mapsto \gamma(gv),$$ where $v \in \{{\lambda}\}^H$. In the next subsection we will use the special structure of $H$ to finish the proof of Theorem \[thm:functionsonorbit\].
The coordinate ring of the orbit of the power sum {#the-coordinate-ring-of-the-orbit-of-the-power-sum .unnumbered}
-------------------------------------------------
Considering eq. , in order to understand ${\textup{HWV}}_{{\lambda}^*}({\mathbb{C}}[G]^{\vv H})$ we need to understand the $H$-invariant space $\{{\lambda}\}^H$. The group $H$ is the semidirect product $\{{\lambda}\}^H = (\{{\lambda}\}^{{\mathbb{Z}}_D^m})^{{\mathfrak{S}}_m}$, which enables us to analyze $\{{\lambda}\}^H$ via a two-step process by first analyzing $\{{\lambda}\}^{{\mathbb{Z}}_D^m}$ and then taking ${\mathfrak{S}}_m$-invariants. We consider the basis of $\{{\lambda}\}$ given by the semistandard tableaux $S:{\lambda}\to\{1,\ldots,m\}$. Note that for $g \in {\mathbb{Z}}_D^m$ we have that $S$ and $gS$ coincide up to rescaling with a $D$th root of unity. Indeed, if any symbol in the tableau $S$ does not occur a multiple of $D$ times, then $S$ vanishes under the symmetrization $$S \mapsto \frac{1}{D^m} \sum_{g \in {\mathbb{Z}}_D^m} \pi S.$$ Moreover, each $S$ in which every number appears a multiple of $D$ many times is fixed under this symmetrization map. Hence $\{{\lambda}\}^{{\mathbb{Z}}_D^m}$ has a basis given by semistandard tableaux of shape ${\lambda}$ in which each number appears a multiple of $D$ many times. For a partition $\varrho$ let ${\mathscr{W}}_\varrho'$ denote the linear space spanned by all semistandard tableaux of shape ${\lambda}$ whose sorted content is $\varrho D$. This gives a decomposition $\{{\lambda}\}^{{\mathbb{Z}}_D^m} = \bigoplus_{\varrho}{\mathscr{W}}_\varrho'$ as a direct sum. The action of ${\mathfrak{S}}_m$ leaves ${\mathscr{W}}_\varrho'$ fixed. Hence $(\{{\lambda}\}^{{\mathbb{Z}}_D^m})^{{\mathfrak{S}}_m} = \bigoplus_\varrho ({\mathscr{W}}_\varrho')^{{\mathfrak{S}}_m}$. A generating set for $\{{\lambda}\}^H$ is thus obtained by taking the ${\mathscr{W}}_\varrho'$ and their tableau bases and independently symmetrizing over $\mathfrak S_m$, which is done (up to scale) by applying $P_m$. Using eq. we thus obtain the following proposition.
The vector space ${\textup{HWV}}_{{\lambda}^*}({\mathbb{C}}[G]^{\vv H}_d)$ decomposes into a direct sum of vector spaces ${\textup{HWV}}_{{\lambda}^*}({\mathbb{C}}[G]^{\vv H}_d) = \bigoplus_{\varrho\vdash_m d} {\mathscr{W}}_\varrho$, and each ${\mathscr{W}}_\varrho$ is generated by the functions $$g \mapsto \gamma(g P_m S),$$ where $S$ runs over all semistandard tableaux $S$ of shape ${\lambda}$ and content $\varrho D$.
Let ${\mathscr{W}}_\varrho'$ denote the linear space spanned by all semistandard tableaux $S$ of shape ${\lambda}$ whose sorted content is $\varrho D$. Then ${\mathscr{W}}_\varrho$ is isomorphic to the ${\mathfrak{S}}_m$-invariant subspace of ${\mathscr{W}}_\varrho'$.
Theorem \[thm:functionsonorbit\] now follows immediately by applying the algebra isomorphism .
A formula for the multiplicities in the coordinate ring of the orbit of the power sum (Proof of Prop. ) {#sec:gctandsymmetries}
=======================================================================================================
Crucial parts of this section are the result of a collaboration with Greta Panova and appeared in the first author’s unpublished lecture notes for a winter 2017/2018 course on geometric complexity theory [@Ike:19].
In this section we prove Proposition \[pro:gctandsymmetries\]. In fact, we prove a slightly stronger result as follows.
\[pro:precisedecomposition\] Let $\{{\lambda}\}_{\varrho} \subseteq \{{\lambda}\}$ denote the linear subspace spanned by the tableaux whose sorted content is $\varrho D$. Then $\dim (\{{\lambda}\}_{\varrho})^{{\mathfrak{S}}_m} = b({\lambda},\varrho,D,d)$.
Let $\{{\lambda}\}^\varrho$ denote the $\varrho$-weight space, i.e., the linear space spanned by tableaux of shape ${\lambda}$ and content $\varrho$. Then $\{{\lambda}\}_\varrho = \bigoplus_{\gamma \in {\mathfrak{S}}_m\varrho} \{\lambda\}^\gamma$.
For a partition $\varrho \vdash_m d$ let ${\mathfrak{S}}_m\varrho \subseteq {\mathbb{N}}^m$ denote the orbit of $\varrho$. Note that $\varrho$ is the only partition in its orbit, while the other lists are not in the correct order. Let ${\text{stab\,}}\varrho \leq {\mathfrak{S}}_m$ denote the stabilizer of $\varrho$.
$\dim (\{{\lambda}\}_\varrho)^{{\mathfrak{S}}_m} = \dim\left(\{\lambda\}^\varrho\right)^{{\text{stab\,}}\varrho}.$
We construct an isomorphism of vector spaces.
Let $W := \{{\lambda}\}$. Let $\pi_1,\ldots,\pi_r$ be a system of representatives of left cosets for ${\text{stab\,}}\varrho \leq {\mathfrak{S}}_m$ with $\pi_1 = \text{id}$, i.e., ${\mathfrak{S}}_m = \pi_1 {\text{stab\,}}\varrho \ \dot\cup \ \cdots \ \dot\cup \ \pi_r{\text{stab\,}}\varrho$ and we have ${\mathfrak{S}}_m \varrho = \{\pi_1 \varrho , \ldots, \pi_r \varrho\}$. Therefore we have the decomposition $$W_\varrho = \bigoplus_{j=1}^r \pi_j W^\varrho.$$ Let $\overline p : W_\varrho \twoheadrightarrow W^\varrho$ be the projection according to this decomposition. We claim that the restriction $$p : \left(W_\varrho\right)^{{\mathfrak{S}}_m} \to \left(W^\varrho\right)^{{\text{stab\,}}\varrho}$$ is an isomorphism of vector spaces. This then finishes the proof. We verify well-definedness, injectivity, and surjectivity of $p$.
Well-definedness: The spaces $\pi_1 W^\varrho,\ldots,\pi_r W^\varrho$ are permuted by ${\mathfrak{S}}_m$. Every $\sigma \in {\text{stab\,}}\varrho$ fixes $W^\varrho$, thus $\sigma v_1 = v_1$ if $v_1 \in W^\varrho$. Thus the map $v = \sum_{j=1}^r v_j \stackrel{\overline p}{\mapsto} v_1$ maps $W_\varrho$ to $(W^\varrho)^{{\text{stab\,}}\varrho}$.
Injectivity: If $v \in (W_\varrho)^{{\mathfrak{S}}_m}$, then $v = \pi v = \sum_j \pi v_j$. Therefore $v_j = \pi_j v_1$. If $p(v)=0$, then $v_1=0$, thus all $v_j=0$, which proves injectivity.
Surjectivity: Let $v_1 \in (W^\varrho)^{{\text{stab\,}}\varrho}$. Set $v_j := \pi_j v_1$ and put $v:=\sum_j v_j$. Clearly $p(v)=v_1$. It remains to verify that $v$ is ${\mathfrak{S}}_m$-invariant. $$v = \sum_{j=1}^r \pi_j v_1 = \sum_{j=1}^r \tfrac{1}{|{\text{stab\,}}\varrho|} \sum_{\tau \in {\text{stab\,}}\varrho} \pi_j \tau v_1 = \tfrac{1}{|{\text{stab\,}}\varrho|} \sum_{\pi \in {\mathfrak{S}}_m}\pi v_1,$$ which is ${\mathfrak{S}}_m$-invariant.
We are left with determining $\dim\left(\{{\lambda}\}^{\varrho}\right)^{{\text{stab\,}}\varrho}$. We use a detour via Specht modules: the Specht module $[{\lambda}]$ is an irreducible ${\mathfrak{S}}_{|{\lambda}|}$-representation that can be constructed as the subrepresentation of $\{{\lambda}\}$ spanned by all standard tableaux. Define the Young subgroup $G_\varrho \subseteq {\mathfrak{S}}_{dD} := {\mathfrak{S}}_{\varrho_1 D} \times \cdots \times {\mathfrak{S}}_{\varrho_m D}$. We use that $\{{\lambda}\}^{\varrho} \simeq [{\lambda}]^{G_\varrho}$, see e.g. [@ike:12b Sec. 4.3(A)].
Schur-Weyl duality implies that $$\begin{aligned}
\dim\left([{\lambda}]^{G_\varrho}\right)^{{\text{stab\,}}\varrho} = \dim{\textup{HWV}}_{\lambda}(\{{\lambda}\}\otimes ([{\lambda}]^{G_\varrho})^{{\text{stab\,}}\varrho} ) = \dim{\textup{HWV}}_{\lambda}((\otimes^{dD}V)^{G_\varrho\rtimes{{\text{stab\,}}\varrho}})\end{aligned}$$ for $V$ having large enough dimension.
$$\begin{aligned}
(\otimes^{dD}V)^{G_\varrho\rtimes{{\text{stab\,}}\varrho}} &=& ({\textup{Poly}}^{D\varrho_1}V \otimes \cdots \otimes {\textup{Poly}}^{D\varrho_m}V)^{{\text{stab\,}}\varrho}\\
&=& \left(\bigotimes^{\hat\varrho_1}{\textup{Poly}}^{D}V \otimes \bigotimes^{\hat\varrho_2}{\textup{Poly}}^{2D}V \otimes \cdots \otimes \bigotimes^{\hat\varrho_d}{\textup{Poly}}^{dD}V\right)^{{\text{stab\,}}\varrho} \\
&=& \underbrace{{\mathbb{C}}[{\textup{Poly}}^{D}V]^*_{\hat\varrho_1}}_{=\bigoplus_{\mu^1} \{\mu^1\}^{\oplus a_{\mu^1}(\hat\varrho_1,D)}} \otimes {\mathbb{C}}[{\textup{Poly}}^{2D}V]^*_{\hat\varrho_2} \otimes \cdots \otimes \underbrace{{\mathbb{C}}[{\textup{Poly}}^{dD}V]^*_{\hat\varrho_d}}_{{=\bigoplus_{\mu^d} \{\mu^d\}^{\oplus a_{\mu^d}(\hat\varrho_d,dD)}}} \quad\quad\quad\hfill(\dagger).\end{aligned}$$
The multiplicity of $\{\mu^i\}^*$ in ${\mathbb{C}}[{\textup{Poly}}^{iD}V]_{\hat\varrho_i}$ is $a_{\mu^i}(\hat\varrho_i,iD)$. Using distributivity we obtain that the multiplicity of $\{{\lambda}\}$ in the representation $(\dagger)$ equals $$\sum_{\mu^1,\mu^2,\ldots,\mu^d \atop \mu^i \vdash \hat\varrho_i D i} c_{\mu^1,\mu^2,\ldots,\mu^d}^{\lambda}\prod_{i=1}^d a_{\mu^i}(\hat\varrho_i,iD)$$
${\textup{mult}}_{\lambda}{\mathbb{C}}[Gp]_d \stackrel{\text{Thm. }\ref{thm:functionsonorbit}}{=} \sum_{\varrho\vdash_m d}({\mathscr{W}}_\varrho)^{{\mathfrak{S}}_m} \stackrel{\text{Prop.~\ref{pro:precisedecomposition}}}{=} \sum_{\varrho\vdash_m d} b({\lambda},\varrho,D,d)$.
Highest weight functions on the orbit closure {#sec:hworbitclosure}
=============================================
In this section we study the coordinate ring ${\mathbb{C}}[\overline{Gp}]$. Much less is known about this ring compared to ${\mathbb{C}}[Gp]$, in particular we do not have formulas for ${\mathbb{C}}[\overline{Gp}]$ that are comparable to Prop. \[pro:gctandsymmetries\]. Nevertheless, the following theorem provides a way to analyze ${\mathbb{C}}[\overline{Gp}]$ and connect it to ${\mathbb{C}}[Gp]$.
For an $m \times n$ matrix $A$ let $A p := \ell_1^D+\ell_2^D+\cdots+\ell_n^D$, where $\ell_i$ is the linear form given by the $i$-th column of $A$. Note that this is a generalization of $gp$ for $g \in {\textup{GL}}_m$.
\[thm:functionsonorbitclosure\] If $|{\lambda}|$ is not divisible by $D$, then ${\textup{HWV}}_{{\lambda}^*}({\mathbb{C}}[\overline{Gp}])=0$.
Let ${\lambda}\vdash \delta D$. The vector space ${\textup{HWV}}_{{\lambda}^*}({\mathbb{C}}[\overline{Gp}])$ is generated by the functions $$g \mapsto \gamma(g M_{\delta,m} T),$$ where $T$ runs over all semistandard tableaux of shape ${\lambda}$ in which each entry $1,\ldots,\delta$ appears exactly $D$ many times.
Additionally, for a semistandard tableau $T$ of shape ${\lambda}$ in which each entry $1,\ldots,\delta$ appears exactly $D$ many times, the function $$A \mapsto \gamma(A M_{\delta,m} T)$$ is either zero or a HWV of weight ${\lambda}^*$ in ${\mathbb{C}}[{\textup{Poly}}^D {\mathbb{C}}^m]$.
This is a rephrasing of [@AIR:16], which is a special case of [@BIP:19 Prop. 4.5 and Thm. 4.7]. Indeed, [@BIP:19] covers more general cases. For the sake of completeness and to highlight that the proof technique is very different from the technique in Section \[sec:hworbit\], we prove Theorem \[thm:functionsonorbitclosure\].
The first observation follows from the fact that $\overline{Gp} \subseteq {\textup{Poly}}^D{\mathbb{C}}^m$ is a closed subvariety that is closed under rescaling, and hence ${\mathbb{C}}[\overline{Gp}]$ is a graded subalgebra of ${\mathbb{C}}[{\textup{Poly}}^D{\mathbb{C}}^m]$. We know that in each degree $\delta$ component ${\mathbb{C}}[{\textup{Poly}}^D{\mathbb{C}}^m]_\delta$ of ${\mathbb{C}}[{\textup{Poly}}^D{\mathbb{C}}^m]$ the only types ${\lambda}^*$ that occur satisfy ${\lambda}\vdash \delta D$, see e.g. [@ike:12b Lemma 4.3.3].
Schur-Weyl duality yields that $$\label{eq:schurweyl}
{\textstyle\bigotimes}^{\delta D} {\mathbb{C}}^{m*} = \bigoplus_{{\lambda}\vdash_m \delta D} \{{\lambda}^*\} \otimes [{\lambda}].$$ A highest weight vector of weight ${\lambda}^*$ in ${\textstyle\bigotimes}^{\delta D} {\mathbb{C}}^{m*}$ is given for example by $$v_{\lambda}:= x_1 \wedge x_2 \wedge \cdots \wedge x_{\mu_1} \otimes x_1 \wedge x_2 \wedge \cdots \wedge x_{\mu_2} \otimes \cdots \cdots \otimes x_1 \wedge x_2 \wedge \cdots \wedge x_{\mu_{{\lambda}_1}},$$ where $\mu = {\lambda}^t$ and $\{x_i\}_i$ is the basis of ${\mathbb{C}}^{m*}$. Let $T_{\lambda}$ denote the column-standard tableau of shape ${\lambda}$, i.e., the tableau that is filled with the numbers $1,\ldots,|{\lambda}|$ in a columnwise fashion from left to right, top to bottom. Since $[{\lambda}]$ is irreducible, from we see that ${\textup{HWV}}_{{\lambda}^*}({\textstyle\bigotimes}^{\delta D}{\mathbb{C}}^{m*})$ is generated by the set $\{v_{\lambda}\pi \mid \pi \in {\mathfrak{S}}_{\delta D} \text{ s.t. $\pi T_{\lambda}$ is standard}\}$.
By the polarization principle (see e.g. [@ike:12b Claim 4.2.13]), all functions $f$ in ${\mathbb{C}}[{\textup{Poly}}^D{{\mathbb{C}}^{m*}}]_\delta$ can be obtained via some tensor $v_f \in {\textstyle\bigotimes}^{\delta D}{\mathbb{C}}^m$ and defining $$f(y) := \langle v_f , y^{\otimes \delta}\rangle.$$ The resulting function $f$ is a highest weight function iff $v_f$ is a HWV. Thus we see that ${\textup{HWV}}_{{\lambda}^*}({\mathbb{C}}[{\textup{Poly}}^D{\mathbb{C}}^m]_\delta)$ is generated by the functions $$f(y) := \langle \pi v_{\lambda}, y^{\otimes \delta}\rangle.$$ In the following we analyze how to restrict these functions to $\overline{Gp}$. When $y = gp = \ell_1^D+\cdots+\ell_m^D$ is in the orbit of the power sum, then clearly $$y^{\otimes \delta} = \sum_{\varphi:\{1,\ldots,\delta\}\to\{1,\ldots,m\}} \ell_{\varphi(1)}^D \otimes \cdots \otimes \ell_{\varphi(\delta)}^D.$$ The linear forms $\ell_i$ correspond to the vectors $g_{1..m,i}$. We evaluate $$f(y) := \langle \pi v_{\lambda}, y^{\otimes \delta}\rangle = \langle v_{\lambda}, \pi(y^{\otimes \delta})\rangle$$ $$= \sum_{\beta:\{1,\ldots,\delta D\}\to\{1,\ldots,m\}\atop\text{respecting $\pi T_{\lambda}$}} \prod_{c=1}^{{\lambda}_1}{\textup{det}}(g_{1..\mu_c, \beta(\pi T_{\lambda}(1,c))},\ldots,g_{1..\mu_c, \beta(\pi T_{\lambda}(\mu_c,c))}),$$ where $\beta$ *respects* a tableau $S$ if all numbers $1,\ldots,D$ are mapped to the same value, and all numbers $D+1,\ldots,2D$ are mapped to the same value, and so on.
Therefore the vector space ${\textup{HWV}}_{{\lambda}^*}({\mathbb{C}}[\overline{Gp}])$ is generated by the functions $$\label{eq:respectS}
g \mapsto \sum_{\beta:\{1,\ldots,Dd\}\to\{1,\ldots,m\}\atop\text{respecting $S$}} \prod_{c=1}^{{\lambda}_1}{\textup{det}}(g_{1..\mu_c, \beta(S(1,c))},\ldots,g_{1..\mu_c, \beta(S(\mu_c,c))}),$$ where $S$ runs over all standard tableaux of shape ${\lambda}$.
Given a standard tableau $S$ we define a tableau $T$ by replacing the first $D$ entries $1,\ldots,D$ by the number $1$, the next $D$ entries $D+1,\ldots,2D$ by the number $2$, and so on. It is easy to check that if $T$ is not regular, then the function corresponding to $S$ describes the zero function, because each summand in has a zero factor that is the determinant of a matrix with a repeating column. We assume from now on that $T$ is regular. Since $T$ is regular and $S$ is standard, $T$ is semistandard. We rewrite as follows:
$$\label{eq:sumsofprodofdets}
g \mapsto \sum_{\varphi:\{1,\ldots,\delta\}\to\{1,\ldots,m\}} \prod_{c=1}^{{\lambda}_1}{\textup{det}}(g_{1..\mu_c, \varphi(T(1,c))},\ldots,g_{1..\mu_c, \varphi(T(\mu_c,c))})$$
Using , we can write this in terms of $\gamma$ as follows:
$$g \mapsto \sum_{\varphi:\{1,\ldots,\delta\}\to\{1,\ldots,m\}} \gamma(g \varphi T).$$
By definition of $M_{\delta,m}$, this can be rewritten as:
$$g \mapsto \gamma(g M_{\delta,m} T),$$ which finishes the proof of the second part of Theorem \[thm:functionsonorbitclosure\].
For the last claim, we note that in this construction of highest weight functions we did not use that $g$ is a square matrix. A rectangular matrix $A$ works in the same way.
An equation for Waring rank (Proof of the missing part in Prop. ) {#sec:threetableaux}
=================================================================
In this section we prove the following proposition that was used in the proof of Proposition \[pro:alontarsi\].
\[pro:threetableaux\] Let $\nu = (2m)+m\times 2m$. Then $a_\nu(2(m+1),m) > 2$.
By Proposition \[pro:gctandsymmetries\] and Lemma \[lem:cormain\] we have ${\textup{mult}}_{\nu^*} {\mathbb{C}}[\overline{Gp}] = {\textup{mult}}_{\nu^*} {\mathbb{C}}[{Gp}] = 2.
$ This means that there are two linearly independent HWVs of weight $\nu^*$ that do not vanish on $Gp$. To finish the proof it suffices to construct a nonzero HWV of weight $\nu^*$ that vanishes on $Gp$, because then these three HWVs are linearly independent. Note that in particular we construct an equation that vanishes on all polynomials of Waring rank at most $m$.
We use the last part of Theorem \[thm:functionsonorbitclosure\] to construct this third function. Let $n := 2m+2$. Let $T_\textsf{left}$ be the $m \times (m+2)$ tableau that is filled in a rowwise fashion from top to bottom and from left to right with $m$ many 1s, then $m$ many 2s, and so on, until $m$ many $(m+2)$s. For example, if $m=6$, then $$T_\textsf{left} =
{\ytableausetup{boxsize=1.1em}
\ytableaushort{
11111122,
22223333,
33444444,
55555566,
66667777,
77888888
}}$$ We remark that the function corresponding to $T_\textsf{left}$ via Theorem \[thm:functionsonorbitclosure\] is a generalization of Aronhold’s degree 4 invariant on ternary cubics, see also [@BI:17] for other (related) generalizations.
Let $T_\textsf{right}$ be the $(m \times 2)+(m^2-2m)$ tableau whose first two columns are equal and consist of the entries $m+3,m+4,\ldots,2m+2$ from top to bottom. The remaining singleton columns get filled with $m-2$ many entries $m+3$, $m-2$ many entries $m+4$, $m-2$ many entries $m+5$, and so on, until $m-2$ many entries $2m+2$. For example, if $m=6$, then $$T_\textsf{right} =
{\ytableausetup{boxsize=1.1em}
\ytableaushort{
999999{10}{10}{10}{10}{11}{11}{11}{11}{12}{12}{12}{12}{13}{13}{13}{13}{14}{14}{14}{14},
{10}{10},
{11}{11},
{12}{12},
{13}{13},
{14}{14}
}}$$ The tableau $T$ is defined as the concatenation $T := T_\textsf{left}+T_\textsf{right}$. We observe that $T$ is duplex.
By Theorem \[thm:functionsonorbitclosure\] the function $f : A \mapsto \gamma(A M_{n,n} T)$ is either zero or a HWV of weight $\nu^*$ in ${\mathbb{C}}[{\textup{Poly}}^m {\mathbb{C}}^m]$.
$f$ does not vanish on ${\textup{Poly}}^m{\mathbb{C}}^m$.
This is due to the fact that $T$ is duplex, in complete analogy to [@bci:10]. Choose $A$ to be an $m \times n$ matrix whose entries are real numbers chosen generically (one can alternatively think of the entries being chosen uniformly at random for example from a Gaussian distribution). Since $T$ is duplex, each summand in $\gamma(A M_{n,n}T)$ is a product of determinants (see ), but each factor in the product appears an even number of times and hence the product is nonnegative. Since $A$ was chosen generically, for the identity map $\textsf{id} \in \mathcal M_{n,n}$ we have $\gamma(A \,\textsf{id}\, T) > 0$. Any finite sum of nonnegative numbers that contains at least one positive number is nonzero, so $\gamma(A M_{n,n}T)$ is nonzero. This finishes the proof.
The preceding claim implies that $f$ is a nonzero HWV of weight $\nu^*$ in ${\mathbb{C}}[{\textup{Poly}}^m {\mathbb{C}}^m]$. To finish the proof of Proposition \[pro:threetableaux\] it suffices to prove that $f$ vanishes on $Gp$. The crucial property is that no tableau in $\mathcal M_{m+2,m}T_\textsf{left}$ is regular: Since $T_\textsf{left}$ is rectangular with the maximum number of rows, a regular tableau in $\mathcal M_{m+2,m}T_\textsf{left}$ has $m+2$ many 1s, $m+2$ many 2s, and so on, but every symbol in $\mathcal M_{m+2,m}T_\textsf{left}$ appears a multiple of $m$ many times. Since no tableau in $\mathcal M_{m+2,m}T_\textsf{left}$ is regular, no tableau in $\mathcal M_{n,m}T$ is regular. Hence all summands in $\gamma(g M_{n,m}T)$ are zero, see Corollary \[cor:gammazero\]. This finishes the proof of Proposition \[pro:threetableaux\].
Proof of the Main Technical Theorem \[thm:main\] using the Tableau Lifting Theorem {#sec:proofmain}
==================================================================================
In this section we prove Theorem \[thm:main\], based on the following combinatorial Tableau Lifting Theorem \[thm:prolongation\] whose long and combinatorial proof we will develop during the remaining sections of this paper. Much simpler forms of other tableau lifting theorems appeared in [@KL:12; @BIP:19].
Fix a shape ${\lambda}$ and natural numbers $m$ and $e$. For a tableau $T$ of shape $(m \times e) + {\lambda}$ we define ${\textup{\textsf{leftpart}}}(T)$ to be the $m\times e$ rectangular subtableau consisting of the leftmost $e$ columns, and we define ${\textup{\textsf{rightpart}}}(T)$ to be the shape ${\lambda}$ subtableau consisting of the rightmost ${\lambda}_1$ columns. In particular, we have $T = {\textup{\textsf{leftpart}}}(T)+{\textup{\textsf{rightpart}}}(T)$. For $\varrho \in ({\mathbb{N}}_{\geq 0})^m$ a tableau $S$ has content $D\varrho$ if each number $i$ appears exactly $D\varrho_i$ many times in $S$.
\[thm:prolongation\] Let $D \geq 3$ and $m \geq 2$. Given a regular tableau $S$ of shape $\lambda \vdash_m dD$ and content $D \varrho$ for $\varrho\vdash_m d$. Let
- $e_\varrho:=\sum_{i=1}^{m} \lceil \frac{\varrho_i}{D-2} \rceil$ in the case where $D$ is even
- and $e_\varrho:=\sum_{i=1}^{m} 2\lceil \frac{\varrho_i}{2(D-2)} \rceil$ in the case where $D$ is odd and ${{2(D-1)}\choose{D-1}} \geq 2(m-1)$.
Let $\delta := m e_\varrho+d$. In both cases there exists a tableau $T:\lambda +(m \times e_\varrho D) \rightarrow \{1,\dots,\delta\}$ in which every entry appears exactly $D$ many times such that
1. \[enum:rightpartinSmS\] For each $\varphi \in \mathcal{M}_{\delta,m}$ for which $\varphi(T)$ is regular we have that ${\textup{\textsf{rightpart}}}(\varphi(T))\in{\mathfrak{S}}_m S$,
2. \[enum:duplex\] For each $\varphi \in \mathcal{M}_{\delta,m}$ for which $\varphi(T)$ is regular we have that ${\textup{\textsf{leftpart}}}(\varphi(T))$ is duplex,
3. \[enum:existspreimage\] there exists $\varphi\in\mathcal{M}_{\delta,m}$ such that $\varphi(T)$ is regular and ${\textup{\textsf{rightpart}}}(\varphi(T))=S$.
By Theorem \[thm:functionsonorbit\] and Proposition \[pro:precisedecomposition\] we know that there exists a set of regular tableaux $\{S_{\varrho,i}\mid \varrho\in\Xi, \ 1 \leq i \leq b({\lambda},\varrho,D,d)\}$ of shape ${\lambda}$ such that each $S_{\varrho,i}$ has content $D\varrho$ and the set of corresponding functions $$f^{S_{\varrho,i}}\in {\mathbb{C}}[Gp], \quad gp \mapsto \gamma(g P_m S_{\varrho,i}).$$ is linearly independent. Since all these functions $f^{S_{\varrho,i}}$ are homogeneous of the same degree $d$, they are not only linearly independent as functions on ${\textup{GL}}_m p$, but also their restrictions to ${\textup{SL}}_m p$ are linearly independent. Using the Tableau Lifting Theorem \[thm:prolongation\], for each $S_{\varrho,i}$ we construct a tableau $T_{\varrho,i}$ of shape ${\lambda}+(m\times e_\varrho D)$ satisfying the properties listed in Theorem \[thm:prolongation\]. We claim that for all $\varrho,i$ there exists $\alpha\neq 0$ such that $$\label{eq:alpha}\tag{$\ast$}
\begin{minipage}{\textwidth-2cm}
under the map $\psi : \varphi\mapsto{\textup{\textsf{rightpart}}}(\varphi T_{\varrho,i})$ that maps from $\mathcal M_{\delta,m}$ to the set of tableaux of shape $\lambda$,
each tableau in ${\mathfrak{S}}_m S_{\varrho,i}$
has exactly $\alpha$ many preimages in $\mathcal{M}_{\delta,m}$ for which $\varphi T_{\varrho,i}$ is regular.
\end{minipage}$$ Proof of : Clearly $\varphi T_{\varrho,i}$ is regular iff $\pi \varphi T_{\varrho,i}$ is regular. Note that $\psi$ is ${\mathfrak{S}}_m$-equivariant in the following sense: $\psi(\pi \circ \varphi) = \pi\psi(\varphi)$. Hence taking the preimage $\psi^{-1}$ is also ${\mathfrak{S}}_m$-equivariant. Thus for all $\hat S \in {\mathfrak{S}}_m S_{\varrho,i}$ we have $\varphi \in \psi^{-1}(\hat S)$ iff $\pi \circ \varphi \in \psi^{-1}(\pi \hat S)$. Thus the application of $\pi$ gives a bijection between the preimages of $\hat S$ and $\pi \hat S$. To prove claim it remains to show that $\alpha \neq 0$. This follows from Thm. \[thm:prolongation\]. $\square$
According to [@BI:17 Prop. 3.25] there exists an ${\textup{SL}}_m$-invariant function $\Phi$ in ${\mathbb{C}}[\overline{Gp}]$ with $\Phi(p)=1$, called the fundamental invariant. Moreover, $$\begin{cases}
\text{$\Phi$ has degree $m$ and $\Phi(gp) = {\textup{det}}(g)^D p$ for $g \in {\textup{GL}}_m$} & \text{ if $D$ is even} \\
\text{$\Phi$ has degree $2m$ and $\Phi(gp) = {\textup{det}}(g)^{2D} p$ for $g \in {\textup{GL}}_m$} & \text{ if $D$ is odd and $2m \leq \binom{2D}{D}$}.
\end{cases}$$ Since orbit closures are irreducible varieties, given two highest weight vectors $f$ of weight ${\lambda}$ and $\tilde f$ of weight $\tilde {\lambda}$ in ${\mathbb{C}}[\overline{Gp}]$, their product $f \cdot \tilde f$ is nonzero and a highest weight vector of weight ${\lambda}+\tilde {\lambda}$. Let $\overline{f}^{S_{\varrho,i}}$ be the product of $\Phi^{e_\Xi-e_\varrho}$ ($\Phi^{(e_\Xi-e_\varrho)/2}$ if $D$ is odd) and the function corresponding to $T_{\varrho,i}$: $$\overline{f}^{S_{\varrho,i}}(gp) = {\textup{det}}(g)^{D(e_\Xi-e_\varrho)}\cdot \gamma(g M_{m,d} T_{\varrho,i}) \quad\quad\text{(holds for $D$ even and odd.)}$$ We claim that $\overline{f}^{S_{\varrho,i}}$ coincides with $f^S_{\varrho,i}$ when restricted to ${\textup{SL}}_m p$ (up to nonzero a factor), which can be seen as follows. For $g \in {\textup{SL}}_m$ we have $$\begin{aligned}
\overline{f}^{S_{\varrho,i}}(gp) &=& 1 \cdot \gamma(g M_{m,d} T_{\varrho,i})
= \sum_{\varphi \in \mathcal M_{m,d}}\gamma(g \varphi T_{\varrho,i}) \\
&\stackrel{\text{Cor.}~\ref{cor:gammazero}}{=}& \sum_{\substack{\varphi \in \mathcal M_{m,d}\\\varphi T_{\varrho,i}\text{ regular}}}\gamma(g \varphi T_{\varrho,i}) \\
&\stackrel{\text{Cor.}~\ref{cor:gammafactorization}\text{ and Cor.}~\ref{cor:pm1}\text{ and Thm.}~\ref{thm:prolongation}\eqref{enum:duplex}}{=}&(\pm 1)^2\sum_{\substack{\varphi \in \mathcal M_{m,d}\\\varphi T_{\varrho,i}\text{ regular}}}\gamma(g {\textup{\textsf{rightpart}}}(\varphi T_{\varrho,i})) \\
&\stackrel{\text{Thm.~}\ref{thm:prolongation}\eqref{enum:rightpartinSmS}\text{ and }\eqref{eq:alpha}}{=}& \alpha \sum_{\hat S \in {\mathfrak{S}}_m S_{\varrho,i}}\gamma(g \hat S)
= \tfrac{|{\mathfrak{S}}_m S_{\varrho,i}|\alpha}{m!} \gamma(g P_m S_{\varrho,i}) \\
&=& \tfrac{|{\mathfrak{S}}_m S_{\varrho,i}|\alpha}{m!} f^{S_{\varrho,i}}(gp).\end{aligned}$$ Since each $\overline{f}^{S_{\varrho,i}}$ coincides with $f^{S_{\varrho,i}}$ when restricted to ${\textup{SL}}_m p$ (up to a nonzero factor), the $\overline f^{S_{\varrho,i}}$ are linearly independent. It follows from Theorem \[thm:functionsonorbitclosure\] that the $\overline{f}^{S_{\varrho,i}}$ are restrictions of functions in ${\textup{HWV}}_{({\lambda}+(m\times e_\Xi D))^*}{\mathbb{C}}[\overline{Gp}]$. This proves the main Theorem \[thm:main\].
It remains to prove the Tableau Lifting Theorem \[thm:prolongation\], whose purely combinatorial proof will be the focus of the rest of the paper. The construction for odd $D$ is more complicated than for even $D$, which is why we focus on the case where $D$ is even first.
We will construct ${\textup{\textsf{leftpart}}}(T)$ and ${\textup{\textsf{rightpart}}}(T)$ mainly independently.
For even $D$, the alphabet that we are using for $T$ is not $\{1,\ldots,\delta\}$, but a more descriptive alphabet using the symbols $i_\ell$ and $j_k^i$. For each box $\square$ in $S$, if $\square$ has the entry $i$, then the box corresponding to $\square$ in ${\textup{\textsf{rightpart}}}(T)$ has the symbol $i_\ell$ for some $\ell$. The only other constraints for ${\textup{\textsf{rightpart}}}(T)$ are concerned with how often the different symbols $i_\ell$ appear. The symbols $j_k^i$ do not appear in ${\textup{\textsf{rightpart}}}(T)$, but only in ${\textup{\textsf{leftpart}}}(T)$. The tableau ${\textup{\textsf{leftpart}}}(T)$ is constructed in several steps, starting with a tableau obtained from a set of hypergraphs $H^{(i)}$, and then reordering entries within the rows.
For odd $D$ the situation is similar. The alphabet that we are using for $T$ is not $\{1,\ldots,\delta\}$, but a more descriptive alphabet using the symbols $i_\ell$, $j_k^i$, $j_{\overline k}^i$. For each box $\square$ in $S$, if $\square$ has the entry $i$, then the box corresponding to $\square$ in ${\textup{\textsf{rightpart}}}(T)$ has the symbol $i_\ell$ for some $\ell$. The only other constraints for ${\textup{\textsf{rightpart}}}(T)$ are concerned with how often the different symbols $i_\ell$ appear. Symbols $j_k^i$ and $j_{\overline k}^i$ do not appear in ${\textup{\textsf{rightpart}}}(T)$, but only in ${\textup{\textsf{leftpart}}}(T)$. The tableau ${\textup{\textsf{leftpart}}}(T)$ is constructed in several steps, starting with a tableau obtained from a set of hypergraphs $H^{(i)}$ (similar to those hypergraphs in the case where $D$ is even), and then reordering entries within the rows.
The construction for odd $D$ has many more subtleties than the construction for even $D$ and there are numerous slight differences between the two cases. We think that the readability would suffer greatly if we did not explain the whole construction again for odd $D$ in a self-contained manner, including the parts that are very similar to the even case. Therefore in the following sections we first treat the case for even $D$ and then treat the case for odd $D$ in a fairly self-contained manner. The reader will see that much more care and attention to the details is necessary in the case where $D$ is odd.
The hypergraphs $H^{(i)}$ for even $D$ {#EVENsec:hypergraphsevenD}
======================================
Let $I := \{i \mid \varrho_i \neq 0\}$. In order to construct ${\textup{\textsf{leftpart}}}(T)$ we will first constuct a so-called $(D,\varrho_i)$-hypergraph for each $i \in I$. We refer to that hypergraph as $H^{(i)}$. The number of columns in ${\textup{\textsf{leftpart}}}(T)$ will precisely be the number of vertices in all these hypergraphs together. So we want the hypergraphs to be as small as possible.
We first recall some basic terms. Let $H = (V,E)$ by a hypergraph and $e \in E$ be an edge. Then we define the *size* of an edge ${\textup{\textsf{size}}}(e)$ as the number of vertices in $e$. Let $v,w \in V$. Then a *path* between $v$ and $w$ is a sequence of edges $(e_1,e_2,\ldots,e_l)$ such that $v \in e_1$, $w \in e_l$, and $e_i \cap e_{i+1} \neq \emptyset$. We say that two vertices are connected in $H$ iff there exists a path between them. We say that a hypergraph is *connected* iff every pair of vertices is connected. For a nonempty set $X$ a *set partition* $P$ of $X$ is a set of pairwise disjoint subsets whose union is $X$.
\[def:dkhypergraph\] Let $D,K$ be integers. A *$(D,K)$-hypergraph* is defined to be a hypergraph $H=(V,E)$ that satisfies the following properties:
1. $H$ is connected. \[eq:def:connected\]
2. $H$ has two different types of hyperedges: the *block edges* and the *name edges*. \[eq:def:blockandnameedges\]
3. Each block edge has size $D$, and the set of block edges $E_{\text{Block}} \subseteq E$ is a set partition of $V$. \[eq:def:blockedgessizeD\]
4. Each name edge has size strictly less than $D$, but at least size $1$, and the set of name edges $E_{\text{Name}} \subseteq E$ is a set partition of $V$. \[eq:def:nameedgessizelessthanD\]
5. $|E_{\text{Name}}| - |E_{\text{Block}}| = K$.\[eq:def:nameblockdifference\]
6. There exists a name edge $e_{\text{Name}}$ and a block edge $e_{\text{Block}}$ whose intersection contains at least 2 vertices. We choose one of these two vertices and call it the *link vertex*. \[prop:hypergraphshareblockname\]
Several examples for $(6,K)$-hypergraphs are given in Figure \[EVENfig:6Khypergraphs\].
\[EVENprop:hypergraph\] For even $D \geq 4$, $K\neq 0$, there exists a $(D,K)$-hypergraph that has exactly $\lceil \frac{K}{D-2} \rceil$ many block edges.
Let $n := \lceil \frac{K}{(D-2)} \rceil$. We start the construction by considering $n$ many disjoint block edges with $D$ many vertices each. We arrange the vertices in a linear fashion as in Figure \[EVENfig:6Khypergraphs\]. The leftmost vertex is the link vertex. We now place the vertices in $K+n$ many name edges as follows. As in Figure \[EVENfig:6Khypergraphs\], the rightmost vertex of every block edge but the last shall be placed in a size 2 name edge with the leftmost vertex of the next block edge. The resulting hypergraph is connected. At this point we have $nD-2(n-1)=n(D-2)+2$ vertices that are not in name edges yet; and we have $K+n-(n-1) = K+1$ name edges left to put vertices in. Since $$(n(D-2)+2)-(K+1) =
(\lceil \tfrac{K}{(D-2)} \rceil(D-2)+2)-(K+1)
\geq
(K+2)-(K+1) = 1 > 0,$$ we can position the name edges so that the link vertex is in a name edge of size at least $2$. Moreover, we position that name edge of size at least 2 in such a way that the link vertex has a vertex that not only lies in the same name edge, but also in the same block edge.
Construction of ${\textup{\textsf{leftpart}}}(T)$ for even $D$ {#EVENsec:rightpartevenD}
==============================================================
For each $i \in I$ let $H^{(i)}$ be a $(D,\varrho_i)$-hypergraph from Proposition \[EVENprop:hypergraph\]. We write $E_{\text{Block}}^{(i)}$ to denote its set of block edges and $E_{\text{Name}}^{(i)}$ to denote its set of name edges.
In this section, for every $i \in I$ and every $e \in E_{\text{Block}}^{(i)}$ we construct an $m \times D$ block tableau $\check B_e$ such that ${\textup{\textsf{leftpart}}}(T)$ is constructed as the concatenation $$\label{EVENeq:leftpartmadefromblocks}
{\textup{\textsf{leftpart}}}(T) := \sum_{i\in I} \sum_{e \in E_{\text{Block}}^{(i)}} \check B_e.$$ Notice that since every block edge has size $D$ (see Def. \[def:dkhypergraph\]), this implies that the number of columns in ${\textup{\textsf{leftpart}}}(T)$ is equal to the total number of vertices in the hypergraphs $H^{(i)}$, $i \in I$.
Each $m\times D$ block tableau $\check B_e$ is constructed in three steps: First we construct an $m\times D$ block tableau $B_e$ in which each column corresponds to a vertex in $H^{(i)}$, then we exchange entries between columns that correspond to link vertices.
Let $\zeta^{(i)}$ denote the link vertex in $H^{(i)}$. We attach some additional data to each $H^{(i)}$ as follows. We put a linear order on the set of name edges $E_{\text{name}}^{(i)}$ and for each vertex $v$ in $H^{(i)}$ we define $\ell(v)$ to be the index of its corresponding name edge. Here $\ell(v)=1$ if $v$ lies in the first name edge, $\ell(v)=2$ for the next name edge, and so on. We ensure that $$\label{EVENeq:ellzetaione}
\ell(\zeta^{(i)})=1.$$ In the same way, we put a linear order on the set of block edges; for each block edge $e$ we write $k(e)$ for its index and for each vertex $v$ in $H^{(i)}$ we define $k(v)$ to be the index of its corresponding block edge. We ensure that $$\label{EVENeq:kzetaione}
k(\zeta^{(i)})=1.$$ Moreover, for every vertex $v$ in any $H^{(i)}$ we define $i(v):=i$.
In the following, for each vertex $v$ we define an $m \times 1$ rectangular tableau (i.e., a column of length $m$) called $B_v$. Concatenating them results in $B_e := \sum_{v \in e} B_v$. The order of columns does not matter, but it is convenient to have the vertices of $H^{(i)}$ ordered from left to right in the same way as the columns of ${\textup{\textsf{leftpart}}}(T)$. Later we define $\check B_e$, from which we can extract $\check B_v$ for $v \in e$ as follows: If $B_v$ is the $n$-th column of $B_e$, then $\check B_v$ is the $n$-th column of $\check B_e$.
Starting with $B$ {#starting-with-b .unnumbered}
-----------------
Let $e$ be in the $k$-th block edge in $H^{(i)}$ and let $v \in e$. The column $B_v$ is defined by the following properties. $$\begin{aligned}
\text{ the $i$-th entry of } B_v \text{ is } i_{\ell(v)} \label{EVENeq:niBv}\\
\text{ the $j$-th entry ($j \neq i$) of } B_v \text{ is } j_{k}^{i}
\label{EVENeq:njBv}\end{aligned}$$ An example is given in Figure \[EVENfig:exampleB\].
From $B$ to $\check B$ {#from-b-to-check-b .unnumbered}
----------------------
Most columns $B_v$ and $\check B_v$ coincide, as we define $\check B_v := B_v$ if $v \notin \{\zeta^{(i)} \mid i \in I\}$. This means that only the columns corresponding to link vertices are adjusted.
Let $h$ denote the smallest number in $I$. For $i \in I$, $i \neq h$, the column $\check B_{\zeta^{(i)}}$ arises from $B_{\zeta^{(i)}}$ by switching the $i$-th entry with the $i$-th entry in $B_{\zeta^{(h)}}$. This means that the column $\check B_{\zeta^{(h)}}$ arises from $B_{\zeta^{(i)}}$ by switching the $i$-th entry with the $i$-th entry in $B_{\zeta^{(i)}}$ for all $i \in I$.
The columnwise description of $\check B_v$ thus as follows.
- If $v$ is not a link vertex, then $$\begin{aligned}
\text{ the $i$-th entry of } \check B_v \text{ is } i_{\ell(v)} \label{EVENeq:iBv}\\
\text{ the $j$-th entry ($j \neq i$) of } \check B_v \text{ is } j_{k(v)}^{(i)} \label{EVENeq:jBv}\end{aligned}$$
- If $i \neq h$, then $$\begin{aligned}
\text{ the $i$-th entry of } \check B_{\zeta^{(i)}} \text{ is } i_{k(v)}^h \label{EVENeq:iBzetaioneh}\\
\text{ the $j$-th entry ($j \neq i$) of } \check B_{\zeta^{(i)}} \text{ is } j_{k(v)}^i \label{EVENeq:iBzetajoneh}\end{aligned}$$
- Moreover, $$\begin{aligned}
\text{ the $h$-th entry of } \check B_{\zeta^{(h)}} \text{ is } h_1 \label{EVENeq:hBzetah}\\
\text{ for $j \neq h$, $j \in I$, the $j$-th entry of } \check B_{\zeta^{(h)}} \text{ is } j_1 \label{EVENeq:jBzetahneq}\\
\text{ for $j \neq h$, $j \notin I$, the $j$-th entry of } \check B_{\zeta^{(h)}} \text{ is } j_{1}^h\label{EVENeq:jBzetaheq}\end{aligned}$$
An example is provided in Figure \[EVENfig:examplecheckB\].
at (0.05,0) ; at (-0.05,-1.6) [ ]{}; at (-0.05,-4.5) [ ]{};
We quickly observe the following.
\[EVENcla:acutecheck\] For $i \in I$ and a block edge $e$ in $H^{(i)}$ we have that
- if $\zeta^{(i)} \notin e$, then $B_e = \check B_e$,
- if $\zeta^{(i)} \in e$, $i \neq h$, then $B_e$ and $\check B_e$ differ only in a single entry: The $i$-th entry of the column $\check B_{\zeta^{(i)}}$ is $i^{h}_{1}$ instead of $i_{1}$.
This follows from and using and .
\[EVENcla:leftpart\] In each row $j$ of ${\textup{\textsf{leftpart}}}(T)$ there are only entries $j_\ell$ for some $\ell$, or $j_k^i$ for some $k$, $i$.
Recall . The claim now follows from combining , , , , , , and .
\[EVENcla:symbolsleft\] If $i \notin I$, then no symbol $i_\ell$ appears in ${\textup{\textsf{leftpart}}}(T)$ for any $\ell$. For a fixed $i \in I$, the symbol $i_\ell$ appears in ${\textup{\textsf{leftpart}}}(T)$ iff there is a vertex $v$ in $H^{(i)}$ with $\ell(v)=\ell$. Moreover, $i_\ell$ appears exactly as many times as there are vertices $v$ in $H^{(i)}$ with $\ell(v)=\ell$.
Consider and observe that ${\textup{\textsf{leftpart}}}(T)$ is obtained by a permutation of the entries of the tableau $$\sum_{i\in I} \sum_{e \in E_{\text{Block}}^{(i)}} B_e.$$ Now use and .
Construction of ${\textup{\textsf{rightpart}}}(T)$ for even $D$ {#EVENsec:constructC}
===============================================================
The tableau ${\textup{\textsf{rightpart}}}(T)$ is constructed in any way (for example in a greedy fashion) such that the following constraints are satisfied:
\[EVENeq:sameshape\] &&
$$\label{EVENeq:directreplacement}
\begin{minipage}{15.1cm}
a box in $S$ has entry $i$ iff there is some $\ell$ for which the corresponding box in ${\textup{\textsf{rightpart}}}(T)$ has entry $i_\ell$,
\end{minipage}$$
\[EVENeq:symbolsappearDtimes\] &&
\[EVENeq:allsymbolsappearleftiffright\] . &&
Such a tableau might not be unique, but we only care about its existence. The existence can be shown as follows.
Let $n(i_\ell)$ denote the number of times the symbol $i_\ell$ appears in ${\textup{\textsf{leftpart}}}(T)$. If $n(i_\ell)>0$, then Claim \[EVENcla:symbolsleft\] implies that there are $n(i_\ell)>0$ many vertices $v$ in $H^{(i)}$ with $\ell(v)=i$. By Def. \[def:dkhypergraph\] we know that $n(i_\ell)<D$. We construct ${\textup{\textsf{rightpart}}}(T)$ by arbitrarily replacing $D-n(i_\ell)$ many entries $i$ in $S$ by the symbol $i_\ell$ for each $i$, $\ell$ for which $n(i_\ell)>0$. Claim \[EVENcla:divisibility\] below shows that this procedure replaces exactly all entries of $S$ (recall that $i$ appears in $S$ exactly $D\varrho_i$ many times). It is clear that this construction satisfies , and . Since $0 < n(i_\ell) < D$ iff $0 < D-n(i_\ell) < D$, we conclude .
\[EVENcla:divisibility\] $$\forall i\in I: \quad \sum_{\ell \textup{ with } n(i_\ell)>0} (D-n(i_\ell)) = D\varrho_i.$$
Since $H^{(i)}$ satisfies Def. \[def:dkhypergraph\]\[eq:def:nameblockdifference\] we have that $$|E^{(i)}_{\text{Name}}|-|E^{(i)}_{\text{Block}}| = \varrho_i$$ and hence $$\label{EVENproof_C_*}
D|E^{(i)}_{\text{Name}}|-D|E^{(i)}_{\text{Block}}| = D\varrho_i \tag{*}.$$ Moreover Def. \[def:dkhypergraph\]\[eq:def:blockedgessizeD\] states that block edges form a set partition of $V$ and each block edge has size $D$. Together with the fact that the name edges form a set partition of $V$ (Def. \[def:dkhypergraph\]\[eq:def:nameedgessizelessthanD\]) we see that $
D|E^{(i)}_{\text{Block}}| = \sum_{e \in E_{\text{Name}}^{(i)}} {\textup{\textsf{size}}}(e).
$ Together with we obtain $
D|E^{(i)}_{\text{Name}}| - \sum_{e \in E_{\text{Name}}^{(i)}} {\textup{\textsf{size}}}(e) = D\varrho_i
$ and hence $$\sum_{e \in E_{\text{Name}}^{(i)}} (D-{\textup{\textsf{size}}}(e)) = D\varrho_i.$$
Since for each vertex $v$ in a name edge $e$ the value $\ell(v)$ is the same, we write $\ell(e):=\ell(v)$. From Claim \[EVENcla:symbolsleft\] we know that for all $e\in E_{\text{Name}}^{(i)}$ we have $n(i_{\ell(e)})={\textup{\textsf{size}}}(e)$. Therefore $$\sum_{e \in E_{\text{Name}}^{(i)}} (D-n(i_{\ell(e)})) = D\varrho_i$$
All numbers $\ell(e)$, $e \in E_{\text{Name}}^{(i)}$, are distinct by definition. Hence all symbols $i_{\ell(e)}$ are distinct. All $i_{\ell(e)}$ satisfy $n(i_{\ell(e)})>0$ by Claim \[EVENcla:symbolsleft\]. Moreover, for each $\ell$ with $n(i_\ell)>0$ there exists some $e$ with $\ell(e)=\ell$ also by Claim \[EVENcla:symbolsleft\]. Therefore we can rewrite the sum as $$\sum_{\ell \textup{ with } n(i_\ell)>0} (D-n(i_\ell)) = D\varrho_i,$$ which concludes the proof.
An example of the whole construction can be seen in Figure \[EVENfig:fullexample\].
at (-3,0) [ ]{}; at (0,0) [ ]{};
We draw some quick corollaries.
\[EVENcla:symbols\] If $i \notin I$, then the symbol $i_\ell$ does not appear in $T$ for any $\ell$. For a fixed $i \in I$, the symbol $i_\ell$ appears in $T$ iff $i_\ell$ appears in ${\textup{\textsf{leftpart}}}(T)$ iff $i_\ell$ appears in ${\textup{\textsf{rightpart}}}(T)$ iff there is a vertex $v$ in $H^{(i)}$ with $\ell(v)=\ell$.
We combine Claim \[EVENcla:symbolsleft\] and .
\[EVENcla:jki\] If a symbol $j_k^i$ appears in $T$, then it appears exactly $D$ many times in $T$.
By the symbols $j_k^i$ only appear in ${\textup{\textsf{leftpart}}}(T)$. Consider and observe that ${\textup{\textsf{leftpart}}}(T)$ is obtained by a permutation of the entries of the tableau $$\sum_{i\in I} \sum_{e \in E_{\text{Block}}^{(i)}} B_e.$$ Now use Def. \[def:dkhypergraph\], , and .
Proof of the Tableau Lifting Theorem \[thm:prolongation\] for even $D$ {#EVENsec:psipropertiesDodd}
======================================================================
In this section we prove the Tableau Lifting Theorem \[thm:prolongation\] for even $D$.
First we observe that the shape of $T$ is indeed the required shape: This follows from Proposition \[EVENprop:hypergraph\], , and the fact that $B_e$ and $\check B_e$ have the same rectangular shape $m \times D$.
We remark that every symbol in $T$ appears exactly $D$ many times: For the symbols $j_k^i$ this follows from Claim \[EVENcla:jki\]. For the symbols $i_\ell$ this follows from .
It remains to prove the parts , , and of Theorem \[thm:prolongation\]. We start with part , then build up insights that then eventually lead to the proof of parts and .
Proof of part of Theorem \[thm:prolongation\] {#proof-of-part-of-theoremthmprolongation .unnumbered}
----------------------------------------------
Part of Theorem \[thm:prolongation\] is proved as follows. We choose $\varphi(i_\ell) := i$ and $\varphi(j_k^i) := j$. We observe that ${\textup{\textsf{rightpart}}}(\varphi(T))=S$, see . Since $S$ is regular, ${\textup{\textsf{rightpart}}}(\varphi(T))$ is regular. It remains to show that ${\textup{\textsf{leftpart}}}(\varphi(T))$ is also regular. From Claim \[EVENcla:leftpart\] we see that every column of ${\textup{\textsf{leftpart}}}(\varphi(T))$ contains all entries $1,\ldots,m$, sorted from top to bottom. Thus ${\textup{\textsf{leftpart}}}(\varphi(T))$ is regular. Since ${\textup{\textsf{leftpart}}}(\varphi(T))$ and ${\textup{\textsf{rightpart}}}(\varphi(T))$ are both regular, we conclude that $\varphi(T)$ is regular, which finishes the proof of part of Theorem \[thm:prolongation\].
Parts and of Theorem \[thm:prolongation\]: Preliminaries {#parts-and-of-theoremthmprolongation-preliminaries .unnumbered}
----------------------------------------------------------
In order to prove parts and of Theorem \[thm:prolongation\], we start with some preliminary observations.
\[EVENcla:varphii\] If $\varphi(T)$ is regular, then for each $i \in I$ we have: For every $i_\ell$ that appears in $T$, $\varphi(i_\ell)$ only depends on $i$ and does not depend on $\ell$.
By definition, for every name edge $e$ in $H^{(i)}$ the values $\ell(v)$ coincide for all $v \in e$. This trivially implies that $$\label{EVENeq:nameedgevarphicoincideNEW}
\text{for every name edge $e$ in $H^{(i)}$:
the values $\varphi(i_{\ell(v)})$ coincide for all $v \in e$.}$$
We claim that $$\label{EVENeq:blockedgevarphicoincideNEW}
\text{for every block edge $e$ in $H^{(i)}$:
the values $\varphi(i_{\ell(v)})$ coincide for all $v \in e$.}$$ Proof of : Let $k:=k(e)$. According to Def. \[def:dkhypergraph\] there exists a vertex $\xi^{(i)}\neq\zeta^{(i)}$ that has the same name edge and block edge as $\zeta^{(i)}$, i.e., $\ell(\zeta^{(i)}) = \ell(\xi^{(i)})$ and $k=k(\zeta^{(i)}) = k(\xi^{(i)})$. For each $v \in e$, $v \neq \zeta^{(i)}$ we have that the $j$-th entry ($j \neq i$) of $\check B^{(i)}_v$ is $j_k^i$, see . Moreover, the symbol that appears as the $i$-th entry of $\check B^{(i)}_v$ is $i_{\ell(v)}$, see . By construction of $T$, we have that $\check B^{(i)}_v$ is a column in $T$. Since by assumption $\varphi(T)$ is regular, it follows that $\varphi(\check B^{(i)}_v)$ is regular. Hence if $v \neq \zeta^{(i)}$, the $\varphi(j_k^i)$ are pairwise distinct. Thus $\varphi(i_{\ell(v)})$ equals the one element in $\{1,\ldots,m\} \setminus \{\varphi(j_k^i) \mid 1 \leq j \leq m, \ j\neq i\}$. This is independent of $\ell$. Hence the values $\varphi(i_{\ell(v)})$ coincide for all $v \in e$, $v \neq \zeta^{(i)}$. This proves for all $v\in e$, $v \neq \zeta^{(i)}$. Now, if $\zeta^{(i)} \in e$, then $\xi^{(i)} \in e$, for which we have $\ell(\zeta^{(i)}) = \ell(\xi^{(i)})$, and thus clearly $\varphi(i_{\ell(\zeta^{(i)})})=\varphi(i_{\ell(\xi^{(i)})})$. This proves the claim .
Since $H^{(i)}$ is connected (Def. \[def:dkhypergraph\]\[eq:def:connected\]), we conclude with and : The values $\varphi(i_{\ell(v)})$ coincide for all $v$ in $H^{(i)}$. Since the symbol $i_\ell$ appears in $T$ iff there is some vertex $v$ in $H^{(i)}$ with $\ell(v)=\ell$ (see Claim \[EVENcla:symbols\]), Claim \[EVENcla:varphii\] follows.
For $i\in I$ we define $$\label{EVENeq:defvarphii}
\varphi^{\circ}(i):=\varphi(i_1).$$ This definition is natural, because we saw in Claim \[EVENcla:varphii\] that if $\varphi(T)$ is regular, then $$\varphi^{\circ}(i)=\varphi(i_1)=\varphi(i_2)=\ldots$$
\[EVENcla:differentphiNEW\] Let $\varphi(T)$ be regular. Let $i, j \in I$, $i\neq j$. Then $\varphi^{\circ}(i)\neq \varphi^{\circ}(j)$.
The column $\check B_{\zeta^{(h)}}$ contains the symbol $i_1$ in row $i$ and the symbol $j_1$ in row $j$, see and . The fact that $\varphi(T)$ is regular implies that $\varphi(i_1)\neq\varphi(j_1)$. By this concludes the proof.
\[EVENcla:droph\] Let $\varphi(T)$ be regular. Let $i \in I$, $i \neq h$. Then $\varphi(i_1^h) = \varphi(i_1) = \varphi^{\circ}(i)$.
The last equality is . We now prove the first equality. Let $e$ be the block edge in $H^{(i)}$ that contains the link vertex $\zeta^{(i)}$. Then $\check B_e^{(i)}$ is an $m \times D$ subtableau of ${\textup{\textsf{leftpart}}}(T)$, which differs from $B_e^{(i)}$ only in a single entry in the length $m$ column corresponding to $\zeta^{(i)}$: The $i$-th entry of the column $\check B_{\zeta^{(i)}}$ is $i^{h}_{1}$ instead of $i_{1}$, see Claim \[EVENcla:acutecheck\]. Hence $\varphi(B_{\zeta^{(i)}})$ and $\varphi(\check B_{\zeta^{(i)}})$ are columns that coincide in all but at most this single box. Since $\varphi(T)$ is regular and the $\varphi(T)$ only contains entries from $\{1,\ldots,m\}$ and the columns $\varphi(B_{\zeta^{(i)}})$ and $\varphi(\check B_{\zeta^{(i)}})$ are of length $m$, we conclude that $\varphi(i^{h}_{1}) = \varphi(i_{1})$.
Proof of part of Theorem \[thm:prolongation\] {#proof-of-part-of-theoremthmprolongation-1 .unnumbered}
----------------------------------------------
We now prove part of Theorem \[thm:prolongation\]. The tableau ${\textup{\textsf{rightpart}}}(T)$ only contains entries $i_\ell$ and no entries $j_k^i$, see . As also seen in , if ${\textup{\textsf{rightpart}}}(T)$ contains an entry $i_\ell$, then the corresponding entry of $S$ is $i$. Therefore $\varphi^{\circ}(S) = \varphi({\textup{\textsf{rightpart}}}(T))$, where we lifted the map $\varphi^{\circ} : I \to \{1,\ldots,m\}$ to a map with the same name that is defined on tableaux with entries from $I$. Claim \[EVENcla:differentphiNEW\] proves property of Theorem \[thm:prolongation\].
Proof of part of Theorem \[thm:prolongation\] {#proof-of-part-of-theoremthmprolongation-2 .unnumbered}
----------------------------------------------
The rest of this section is devoted to proving part \[enum:duplex\] of Theorem \[thm:prolongation\]. A rectangular tableau whose columns all coincide is called *uniform*. In the following proof we will crucially use that a uniform tableau with an even number of columns is duplex. Indeed, we prove part \[enum:duplex\] of Theorem \[thm:prolongation\] by showing that if $\varphi(T)$ is regular, then for every block edge $e$:
- $\varphi(\check B_{e})$ is uniform if $e$ does not contain any link vertex $\zeta^{(i)}$,
- $\varphi(\check B_{e})$ is uniform if $\zeta^{(i)} \in e$ for $i \neq h$, and
- $\varphi(\check B_{e})$ is uniform if $\zeta^{(h)} \in e$.
It is clear that these three properties cover all cases and hence $\varphi(T)$ is uniform by construction . This implies part \[enum:duplex\] of Theorem \[thm:prolongation\].
We start with proving (I).
\[EVENcla:ithentry\] Let $\varphi(T)$ be regular. Given a block edge $e$ in $H^{(i)}$. For all $v \in e$, $v \neq \zeta^{(i)}$, we have that the $i$-th entry of $\varphi(\check B_v)$ is $\varphi^{\circ}(i)$.
Combine and Claim \[EVENcla:varphii\].
\[EVENcla:jBvneqzeta\] Let $\varphi(T)$ be regular. Given a block edge $e$ in $H^{(i)}$. For all $j \neq i$ we have that the set $$\{
\text{$j$-th entry of $\varphi(\check B_v)$} \mid v \in e, v \neq \zeta^{(i)}
\}$$ consists of the single element $\varphi(j_{k(e)}^{i})$.
This follows from .
Combining Claim \[EVENcla:ithentry\] and Claim \[EVENcla:jBvneqzeta\] we see that (I) is true.
We now prove (II). Let $i \neq h$ and let $e$ be the block edge in $H^{(i)}$ that contains $\zeta^{(i)}$. Note that $k(e)=1$.
\[EVENcla:almostexceptionalcolumn\] Let $\varphi(T)$ be regular. Then $\varphi(\check B_{\zeta^{(i)}})$ coincides with $\varphi(\check B_v)$, $v \in e$, $i \neq h$.
We compare the columns entrywise. Note that $k(v)=k(\zeta^{(i)})=1$. We make a case distinction.
Case 1: Let $j \neq i$. The $j$-th entry of $\check B_v$ is $j_{1}^i$, see . The $j$-th entry of $\check B_{\zeta^{(i)}}$ is $j_{1}^i$, see . Hence the $j$-th entry of $\varphi(\check B_v)$ equals the $j$-th entry of $\varphi(\check B_{\zeta^{(i)}})$.
Case 2: The $i$-th entry of $\check B_v$ is $i_1$, see . The $i$-th entry of $\check B_{\zeta^{(i)}}$ is $i_1^h$, see . Hence Claim \[EVENcla:droph\] implies that the $i$-th entry of $\varphi(\check B_v)$ equals the $i$-th entry of $\varphi(\check B_{\zeta^{(i)}})$.
It follows from Claim \[EVENcla:almostexceptionalcolumn\] that all columns in $\varphi(\check B_e)$ coincide, i.e., $\varphi(\check B_e)$ is uniform. Thus (II) is proved.
It remains to show (III), i.e., that $\varphi(\check B_{e})$ is uniform if $\zeta^{(h)} \in e$.
\[EVENcla:hthentry\] Let $\varphi(T)$ be regular and $\zeta^{(h)}$ the link vertex in the block edge $e$. For all $v \in e$, $v \neq \zeta^{(h)}$, we have that the $h$-th entry of $\varphi(\check B_v)$ is $\varphi^{\circ}(h)$.
This is a direct implication of Claim \[EVENcla:ithentry\].
\[EVENcla:exceptionalcolumn\] Let $\varphi(T)$ be regular and $\zeta^{(h)} \in e$. Then $\varphi(\check B_{\zeta^{(h)}})$ coincides with $\varphi(\check B_{v})$, $v \in e$.
We compare the columns entrywise, considering three cases.
Case 1: We compare the $h$-th entry: According to Claim \[EVENcla:hthentry\], the $h$-th entry of $\varphi(\check B_{v})$ is $\varphi^{\circ}(h)$. According to the $h$-th entry of $\check B_{\zeta^{(h)}}$ is $h_1$, so the $h$-th entry of $\varphi(\check B_{\zeta^{(h)}})$ is $\varphi(h_1)=\varphi^{\circ}(h)$, see .
Case 2: We compare the $j$-th entry, $j \neq h$, in the case $j \notin I$: According to , the $j$-th entry of $\check B_{v}$ is $j_1^{h}$. The $j$-th entry of $\check B_{\zeta^{(h)}}$ is also $j_1^{h}$, see . Therefore the $j$-th entry of $\varphi(\check B_{v})$ equals the $j$-th entry of $\varphi(\check B_{\zeta^{(h)}})$.
Case 3: We compare the $j$-th entry, $j \neq h$, in the case $j\in I$: According to , the $j$-th entry of $\check B_{v}$ is $j_1^{h}$. The $j$-th entry of $\check B_{\zeta^{(h)}}$ is $j_1$, see . Claim \[cla:droph\] shows that the $j$-th entry of $\varphi(\check B_{v})$ equals the $j$-th entry of $\varphi(\check B_{\zeta^{(h)}})$.
It follows from Claim \[EVENcla:exceptionalcolumn\] that all columns in $\varphi(\check B_e)$ coincide, i.e., $\varphi(\check B_e)$ is uniform. Thus (III) is proved. This finishes the proof of part \[enum:duplex\] of Theorem \[thm:prolongation\].
Theorem \[thm:prolongation\] is now completely proved for even $D$.
The hypergraphs $H^{(i)}$ for odd $D$ {#sec:hypergraphsoddD}
=====================================
Let $I := \{i \mid \varrho_i \neq 0\}$. In order to construct ${\textup{\textsf{leftpart}}}(T)$ we will first construct a so-called $(D,\varrho_i)$-paired-hypergraph for each $i \in I$. The number of columns in ${\textup{\textsf{leftpart}}}(T)$ will be precisely the number of vertices in all these hypergraphs together. So we want the hypergraphs to be as small as possible.
We will need the basic terms from section \[EVENsec:hypergraphsevenD\]. Moreover, we will need the definition of a $(D,K)$-hypergraph (Def. \[def:dkhypergraph\]).
\[def:dkpairedhypergraph\] Let $D,K$ be integers. A *$(D,K)$-paired-hypergraph* is defined to be a $(D,K)$-hypergraph $H=(V,E)$ that satisfies the following additional property: $$\begin{minipage}{\textwidth-2cm}
Each block edge $e \in E_{\text{Block}}$ is paired with another block edge $\overline{e} \in E_{\text{Block}}$ such that they are connected by a name edge, i.e., there are vertices $v \in e$ and $\overline{v} \in \overline{e}$ called \emph{bridge vertices} and a name edge $e_{\text{Name}} \in E_{\text{Name}}$ such that $v,\overline{v} \in e_{\text{Name}}$. In other words, the set of block edges can be written as a disjoint union of sets of cardinality two such that the elements of each of the sets are connected by a name edge.
\end{minipage}
\label{eq:hypergraphoddpairs}$$
Several examples for $(5,K)$-paired-hypergraphs are given in Figure \[fig:5Kpairedhypergraphs\]. For a block edge $e$ we write $\overline e$ to denote the other block edge in its pair, and $\overline{\overline e} = e$.
\[prop:hypergraphodd\] For odd $D \geq 3$, $K \neq 0$, there exists a $(D,K)$-paired-hypergraph that has exactly $2\lceil \frac{K}{2(D-2)} \rceil$ many block edges.
Let $n := 2\lceil \frac{K}{2(D-2)} \rceil$. We start the construction by considering $n$ many disjoint block edges with $D$ many vertices each. We arrange the vertices in a linear fashion as in Figure \[fig:5Kpairedhypergraphs\]. The leftmost vertex is the link vertex. We now place these vertices in $K+n$ many name edges as follows. As in Figure \[fig:5Kpairedhypergraphs\], the rightmost vertex of every block edge but the last shall be placed in a size 2 name edge with the leftmost vertex of the next block edge. The resulting hypergraph is connected. The rightmost vertex of every odd block edge and the leftmost vertex of every even block edge are bridge vertices. At this point we have $nD-2(n-1)=n(D-2)+2$ vertices that are not in name edges yet; and we have $K+n-(n-1) = K+1$ name edges left to put vertices in. Since $$(n(D-2)+2)-(K+1) =
(2\lceil \tfrac{K}{2(D-2)} \rceil(D-2)+2)-(K+1)
\geq
(K+2)-(K+1) = 1 > 0,$$ we can position the name edges so that the link vertex is in a name edge of size at least $2$. Moreover, we position that name edge of size at least 2 in such a way that the link vertex has a vertex that not only lies in the same name edge, but also in the same block edge.
Construction of ${\textup{\textsf{leftpart}}}(T)$ for odd $D$ {#sec:rightpartoddD}
=============================================================
For each $i \in I$ let $H^{(i)}$ be the $(D,\varrho_i)$-paired-hypergraph from Proposition \[prop:hypergraphodd\]. We write $E_{\text{Block}}^{(i)}$ to denote its set of block edges and $E_{\text{Name}}^{(i)}$ to denote its set of name edges.
In this section, for every $i \in I$ and every $e \in E_{\text{Block}}^{(i)}$ we construct an $m \times D$ block tableau $\check B_e$ such that ${\textup{\textsf{leftpart}}}(T)$ is constructed as the concatenation $$\label{eq:leftpartmadefromblocks}
{\textup{\textsf{leftpart}}}(T) := \sum_{i\in I} \sum_{e \in E_{\text{Block}}^{(i)}} \check B_e.$$ Notice that since every block edge has size $D$ (see Def. \[def:dkhypergraph\]), this implies that the number of columns in ${\textup{\textsf{leftpart}}}(T)$ is equal to the sum of numbers of vertices in the hypergraphs $H^{(i)}$, $i \in I$.
Each $m\times D$ block tableau $\check B_e$ is constructed in three steps: First we construct an $m\times D$ block tableau $B_e$, then we modify its entries to $\acute B_e$, and the we make final adjustments to the entries to obtain $\check B_e$.
Let $\zeta^{(i)}$ denote the link vertex in $H^{(i)}$. We attach some additional data to each $H^{(i)}$ as follows. We put a linear order on the set of name edges $E_{\text{name}}^{(i)}$ and for each vertex $v$ in $H^{(i)}$ we define $\ell(v)$ to be the index of its corresponding name edge. Here $\ell(v)=1$ if $v$ lies in the first name edge, $\ell(v)=2$ for the next name edge, and so on. We ensure that $$\label{eq:ellzetaione}
\ell(\zeta^{(i)})=1.$$ In the same way, we put a linear order on the set of block edge pairs; for each block edge $e$ we write $k(e)$ for the index of its corresponding block edge pair and for each vertex $v$ in $H^{(i)}$ we define $k(v)$ to be the index of its corresponding block edge pair. We ensure that $$\label{eq:kzetaione}
k(\zeta^{(i)})=1.$$ Moreover, for every vertex $v$ in any $H^{(i)}$ we define $i(v):=i$.
In the following, for each vertex $v$ we define an $m \times 1$ rectangular tableau (i.e., a column of length $m$) called $B_v$. Concatenating them results in $B_e$: $B_e := \sum_{v \in e} B_v$. The order of columns does not matter. Analogously, later we define $\acute B_e := \sum_{v \in e} \acute B_v$ and $\check B_e := \sum_{v \in e} \check B_v$.
Starting with $B$ {#starting-with-b-1 .unnumbered}
-----------------
For each block edge pair we choose one block edge to be the *barred* block edge, and the other one to be the *unbarred* block edge.
Let $e$ be in the $k$-th pair of block edges in $H^{(i)}$ and let $v \in e$. The column $B_v$ is defined by the following properties. $$\begin{aligned}
\text{ the $i$-th entry of } B_v \text{ is } i_{\ell(v)} \label{eq:niBv}\\
\text{ the $j$-th entry ($j \neq i$) of } B_v \text{ is } \begin{cases}
j_{k}^{i} & \text{ if } e \text{ is unbarred} \\
j_{\overline k}^{i} & \text{ if } e \text{ is barred} \\
\end{cases}
\label{eq:njBv}\end{aligned}$$ An example is given in Figure \[fig:exampleB\].
From $B$ to $\acute B$ {#from-b-to-acute-b .unnumbered}
----------------------
Fix $i \in I$. To go from $B_e$ to $\acute B_e$ we switch some entries $j_k^{i}$ to $j_{\overline k}^{i}$ and vice versa. We do this by considering the concatenation $B_e+B_{\overline{e}}$ and permuting some entries within the rows of this $m \times (2D)$ block tableau to obtain $\acute B_e+\acute B_{\overline{e}}$. For each $1 \leq k \leq |E_{\text{Block}}^{(i)}|$, let $\{e,\overline e\}$ denote the $k$-th pair of block edges in $H^{(i)}$ and choose a set of $m-1$ many distinct cardinality $D$ subsets ${\textup{\textsf{barred}}}(i,j,k)$ of the vertex set $e \cup \overline e$ such that $$\label{eq:bridgebar}
\begin{minipage}{16cm}
one of the two bridge vertices is contained in \emph{all} the $m-1$ many sets ${\textup{\textsf{barred}}}(i,j,k)$, $1 \leq j \leq m$, $j \neq i$, and the other bridge vertex is contained in \emph{none} of those sets.
\end{minipage}$$
We define $${\textup{\textsf{k-bar}}}(i,j,v) := \begin{cases}
\overline{k(v)} & \text{ if } v \in {\textup{\textsf{barred}}}(i,j,k(v)) \\
k(v) & \text{otherwise}
\end{cases}.$$ Now we define $$\begin{aligned}
\text{ the $i$-th entry of } \acute B_v \text{ is } i_{\ell(v)} \label{eq:acuteiBv}\\
\text{ the $j$-th entry ($j \neq i$) of } \acute B_v \text{ is } j_{{\textup{\textsf{k-bar}}}(i,j,v)}^{i}\label{eq:acutejBv}\end{aligned}$$
An example is provided in Figure \[fig:exampleacuteB\].
Since the sets ${\textup{\textsf{barred}}}(i,j,k)$ have cardinality $D$, it follows: For $i \in I$ and a block edge $e\in H^{(i)}$ we have that $$\label{eq:shuffleacute}
B_e+B_{\overline e} \text{ and } \acute B_e+\acute B_{\overline e} \text{ differ only by permutations of boxes within rows, while row $i$ stays the same.}$$ For $1 \leq j \leq m$, $j \neq i$, it is now straightforward to see that $$\label{eq:Dmanysymbols}
\text{${B}_{e} + {B}_{\overline e}$ contains exactly the symbols $j_k^i$ and $j_{\overline k}^i$, both exactly $D$ many times.}$$
We state another insight at this point: In $H^{(i)}$, for the pair of bridge vertices $v$, $w$ of the $k$-th block edge pair pair we have that $$\label{eq:bridge}
\text{$\acute B_v + \acute B_w$ contains all symbols from $S$},$$ where $S := \{j_k^i \mid 1 \leq j \leq m, j \neq i\} \cup \{j_{\overline k}^i \mid 1 \leq j \leq m, j \neq i\}$.
This follows from combining and : For all $j \neq i$ we have that $j_k^i$ is the $j$-th entry of $\acute B_v$ and $j_{\overline k}^i$ is the $j$-th entry of $\acute B_w$ (or vice versa).
From $\acute B$ to $\check B$ {#from-acute-b-to-check-b .unnumbered}
-----------------------------
Most columns $\acute B_v$ and $\check B_v$ coincide, as we define $\check B_v := \acute B_v$ if $v \notin \{\zeta^{(i)} \mid i \in I\}$. This means that only the columns corresponding to link vertices are adjusted.
Let $h$ denote the smallest number in $I$. For $i \in I$, $i \neq h$, the column $\check B_{\zeta^{(i)}}$ arises from $\acute B_{\zeta^{(i)}}$ by switching the $i$-th entry with the $i$-th entry in $\acute B_{\zeta^{(h)}}$. This means that the column $\check B_{\zeta^{(h)}}$ arises from $\acute B_{\zeta^{(i)}}$ by switching the $i$-th entry with the $i$-th entry in $\acute B_{\zeta^{(i)}}$ for all $i \in I$.
The columnwise description of $\check B_v$ thus as follows.
- If $v$ is not a link vertex, then $$\begin{aligned}
\text{ the $i$-th entry of } \check B_v \text{ is } i_{\ell(v)} \label{eq:iBv}\\
\text{ the $j$-th entry ($j \neq i$) of } \check B_v \text{ is } j_{{\textup{\textsf{k-bar}}}(i,j,v)}^{(i)} \label{eq:jBv}\end{aligned}$$
- If $i \neq h$, then $$\begin{aligned}
\text{ the $i$-th entry of } \check B_{\zeta^{(i)}} \text{ is } i_{{\textup{\textsf{k-bar}}}(h,i,\zeta^{(i)})}^h \label{eq:iBzetaioneh}\\
\text{ the $j$-th entry ($j \neq i$) of } \check B_{\zeta^{(i)}} \text{ is } j_{{\textup{\textsf{k-bar}}}(i,j,\zeta^{(i)})}^i \label{eq:iBzetajoneh}\end{aligned}$$
- Moreover, $$\begin{aligned}
\text{ the $h$-th entry of } \check B_{\zeta^{(h)}} \text{ is } h_1 \label{eq:hBzetah}\\
\text{ for $j \neq h$, $j \in I$, the $j$-th entry of } \check B_{\zeta^{(h)}} \text{ is } j_1 \label{eq:jBzetahneq}\\
\text{ for $j \neq h$, $j \notin I$, the $j$-th entry of } \check B_{\zeta^{(h)}} \text{ is } j_{{\textup{\textsf{k-bar}}}(h,j,\zeta^{(h)})}^h\label{eq:jBzetaheq}\end{aligned}$$
An example is provided in Figure \[fig:examplecheckB\].
We quickly observe the following.
\[cla:acutecheck\] For $i \in I$ and a block edge $e\in H^{(i)}$ we have that
- if $\zeta^{(i)} \notin e$, then $\acute B_e = \check B_e$,
- if $\zeta^{(i)} \in e$, $i \neq h$, then $\acute B_e$ and $\check B_e$ differ only in a single entry: The $i$-th entry of the column $\check B_{\zeta^{(i)}}$ is $i^{h}_{1}$ or $i^{h}_{\overline 1}$ instead of $i_{1}$.
This follows from and using and .
\[cla:leftpart\] In each row $j$ of ${\textup{\textsf{leftpart}}}(T)$ there are only entries $j_\ell$ for some $\ell$, or $j_k^i$ or $j_{\overline k}^i$ for some $k$, $i$.
Recall . The claim now follows from combining , , , , , , and .
\[cla:symbolsleft\] If $i \notin I$, then no symbol $i_\ell$ appears in ${\textup{\textsf{leftpart}}}(T)$ for any $\ell$. For a fixed $i \in I$, the symbol $i_\ell$ appears in ${\textup{\textsf{leftpart}}}(T)$ iff there is a vertex $v$ in $H^{(i)}$ with $\ell(v)=\ell$. Moreover, $i_\ell$ appears exactly as many times as there are vertices $v$ in $H^{(i)}$ with $\ell(v)=\ell$.
Consider and observe that ${\textup{\textsf{leftpart}}}(T)$ is obtained by a permutation of the box entries of the tableau $$\sum_{i\in I} \sum_{e \in E_{\text{Block}}^{(i)}} B_e.$$ Now use and .
Construction of ${\textup{\textsf{rightpart}}}(T)$ for odd $D$ {#sec:constructC}
==============================================================
The tableau ${\textup{\textsf{rightpart}}}(T)$ is constructed in any way (for example in a greedy fashion) such that the following constraints are satisfied:
\[eq:sameshape\] &&
$$\label{eq:directreplacement}
\begin{minipage}{15.1cm}
a box in $S$ has entry $i$ iff there is some $\ell$ for which the corresponding box in ${\textup{\textsf{rightpart}}}(T)$ has entry $i_\ell$,
\end{minipage}$$
\[eq:symbolsappearDtimes\] &&
\[eq:allsymbolsappearleftiffright\] . &&
Such a tableau might not be unique, but we only care about its existence. The existence can be shown as follows.
Let $n(i_\ell)$ denote the number of times the symbol $i_\ell$ appears in ${\textup{\textsf{leftpart}}}(T)$. If $n(i_\ell)>0$, then Claim \[cla:symbolsleft\] implies that there are $n(i_\ell)>0$ many vertices $v$ in $H^{(i)}$ with $\ell(v)=i$. Using Def. \[def:dkhypergraph\] we see that $n(i_\ell)<D$. We construct ${\textup{\textsf{rightpart}}}(T)$ by arbitrarily replacing $D-n(i_\ell)$ many entries $i$ in $S$ by the symbol $i_\ell$ for each $i$, $\ell$ for which $n(i_\ell)>0$. This replaces exactly all entries of $S$, as Claim \[cla:divisibility\] below shows (recall that $i$ appears in $S$ exactly $D\varrho_i$ many times). It is clear that this construction satisfies , and . Since $0 < n(i_\ell) < D$ iff $0 < D-n(i_\ell) < D$, we conclude .
\[cla:divisibility\] $$\forall i\in I: \quad \sum_{\ell \textup{ with } n(i_\ell)>0} (D-n(i_\ell)) = D\varrho_i.$$
Since $H^{(i)}$ satisfies Def. \[def:dkhypergraph\]\[eq:def:nameblockdifference\] we have that $$|E^{(i)}_{\text{Name}}|-|E^{(i)}_{\text{Block}}| = \varrho_i$$ and hence $$\label{proof_C_*}
D|E^{(i)}_{\text{Name}}|-D|E^{(i)}_{\text{Block}}| = D\varrho_i \tag{*}.$$ Moreover Def. \[def:dkhypergraph\]\[eq:def:blockedgessizeD\] states that block edges form a set partition of $V$ and each block edge has size $D$. Together with the fact that the name edges form a set partition of $V$ (Def. \[def:dkhypergraph\]\[eq:def:nameedgessizelessthanD\]) we see that $
D|E^{(i)}_{\text{Block}}| = \sum_{e \in E_{\text{Name}}^{(i)}} {\textup{\textsf{size}}}(e).
$ Together with we obtain $
D|E^{(i)}_{\text{Name}}| - \sum_{e \in E_{\text{Name}}^{(i)}} {\textup{\textsf{size}}}(e) = D\varrho_i
$ and hence $$\sum_{e \in E_{\text{Name}}^{(i)}} (D-{\textup{\textsf{size}}}(e)) = D\varrho_i.$$
Since for each vertex $v$ in a name edge $e$ the value $\ell(v)$ is the same, we write $\ell(e):=\ell(v)$. From Claim \[cla:symbolsleft\] we know that for all $e\in E_{\text{Name}}^{(i)}$ we have $n(i_{\ell(e)})={\textup{\textsf{size}}}(e)$. Therefore $$\sum_{e \in E_{\text{Name}}^{(i)}} (D-n(i_{\ell(e)})) = D\varrho_i$$
All numbers $\ell(e)$, $e \in E_{\text{Name}}^{(i)}$, are distinct by definition. Hence all symbols $i_{\ell(e)}$ are distinct. All $i_{\ell(e)}$ satisfy $n(i_{\ell(e)})>0$ by Claim \[cla:symbolsleft\]. Moreover, for each $\ell$ with $n(i_\ell)>0$ there exists some $e$ with $\ell(e)=\ell$ also by Claim \[cla:symbolsleft\]. Therefore we can rewrite the sum as $$\sum_{\ell \textup{ with } n(i_\ell)>0} (D-n(i_\ell)) = D\varrho_i,$$ which concludes the proof.
An example of the whole construction can be seen in Figure \[fig:fullexample\].
We draw some quick corollaries.
\[cla:symbols\] If $i \notin I$, then the symbol $i_\ell$ does not appear in $T$ for any $\ell$. For a fixed $i \in I$, the symbol $i_\ell$ appears in $T$ iff $i_\ell$ appears in ${\textup{\textsf{leftpart}}}(T)$ iff $i_\ell$ appears in ${\textup{\textsf{rightpart}}}(T)$ iff there is a vertex $v$ in $H^{(i)}$ with $\ell(v)=\ell$.
We combine Claim \[cla:symbolsleft\] and .
\[cla:jki\] If a symbol $j_k^i$ or $j_{\overline{k}}^i$ appears in $T$, then it appears exactly $D$ many times in $T$.
By the symbols $j_k^i$ and $j_{\overline{k}}^i$ only appear in ${\textup{\textsf{leftpart}}}(T)$. Consider and observe that ${\textup{\textsf{leftpart}}}(T)$ is obtained by a permutation of the box entries of the tableau $$\sum_{i\in I} \sum_{e \in E_{\text{Block}}^{(i)}} B_e.$$ Now use Def. \[def:dkpairedhypergraph\], , and .
Proof of the Tableau Lifting Theorem \[thm:prolongation\] for odd $D$ {#sec:psipropertiesDodd}
=====================================================================
In this section we prove the Tableau Lifting Theorem \[thm:prolongation\] for odd $D$.
First we observe that the shape of $T$ is indeed the required shape: This follows from Proposition \[prop:hypergraphodd\], , and the fact that $B_e$ and $\check B_e$ have the same rectangular shape $m \times D$.
We remark that every symbol in $T$ appears exactly $D$ many times: For the symbols $j_k^i$ and $j_{\overline{k}}^i$ this follows from Claim \[cla:jki\]. For the symbols $i_\ell$ this follows from .
It remains to prove the parts , , and of Theorem \[thm:prolongation\]. We start with part , then build up insights that then eventually lead to the proof of part and .
Proof of part of Theorem \[thm:prolongation\] {#proof-of-part-of-theoremthmprolongation-3 .unnumbered}
----------------------------------------------
Part of Theorem \[thm:prolongation\] is proved as follows. We choose $\varphi(i_\ell) := i$ and $\varphi(j_k^i) := j$ and $\varphi(j_{\overline k}^i) := j$. We observe that ${\textup{\textsf{rightpart}}}(\varphi(T))=S$, see . Since $S$ is regular, ${\textup{\textsf{rightpart}}}(\varphi(T))$ is regular. It remains to show that ${\textup{\textsf{leftpart}}}(\varphi(T))$ is also regular. From Claim \[cla:leftpart\] we see that every column of ${\textup{\textsf{leftpart}}}(\varphi(T))$ contains all entries $1,\ldots,m$, sorted from top to bottom. Thus ${\textup{\textsf{leftpart}}}(\varphi(T))$ is regular. Since ${\textup{\textsf{leftpart}}}(\varphi(T))$ and ${\textup{\textsf{rightpart}}}(\varphi(T))$ are both regular, we conclude that $\varphi(T)$ is regular, which finishes the proof of part of Theorem \[thm:prolongation\].
Parts and of Theorem \[thm:prolongation\]: Preliminaries {#parts-and-of-theoremthmprolongation-preliminaries-1 .unnumbered}
----------------------------------------------------------
In order to prove parts and of Theorem \[thm:prolongation\], we start with some preliminary observations.
\[cla:barcoincide\] If $\varphi(T)$ is regular, then for each $i \in I$, $j \neq i$, $k$ we have: $\varphi(j_k^i) = \varphi(j_{\overline k}^i)$.
Fix $i,k$, but do not fix $j$. For notational convenience we define $\acute B := \acute B_{e} + \acute B_{\overline e}$. Let $$S := \{j_k^i \mid j \neq i\} \cup \{j_{\overline k}^i \mid j \neq i\}$$ denote the set of symbols in $\acute B$ in all rows $j \neq i$ (which is the same as the set of symbols in $B_e + B_{\overline e}$ in all rows $j \neq i$, see ). Note that $|S| = 2(m-1) = 2m-2$. Since ${B}_{e} + {B}_{\overline e}$ contains exactly the symbols $j_k^i$ and $j_{\overline k}^i$, both exactly $D$ many times (see ), the same is true for row $j$ of $\acute{B}$ (again, by ).
The tableau $\check B_{e} + \check B_{\overline e}$ differs from $\acute B$ iff $\zeta^{(i)} \in e \cup \overline e$. In this case all differences are in the column that corresponds to the link vertex $\zeta^{(i)}$ (see Claim \[cla:acutecheck\]).
If $\zeta^{(i)} \in e \cup \overline e$, then define $\acute B'$ as the $m \times (2D-1)$ tableau that is obtained from $\acute B$ by removing the column corresponding to $\zeta^{(i)}$. Note that $\zeta^{(i)}$ is not a bridge vertex. If $\zeta^{(i)} \notin e \cup \overline e$, then define $\acute B'$ as the $m \times (2D-1)$ tableau that is obtained from $\acute B$ by removing a single arbitrary column that does not correspond to a bridge vertex.
A symbol $s \in S$ is called *abound* if it appears $D$ many times in $\acute B'$. If $s \in S$ appears $D-1$ many times in $\acute B'$, then $s$ is called *scarce*. Note that each $s\in S$ is either abound or scarce.
Note that $\acute B'$ is a subtableau of $T$ and hence $\varphi(\acute B')$ is regular. Let $v,w$ denote the two bridge vertices. Since by definition they lie in the same name edge, we have $\ell(v)=\ell(w)$. Let $\ell := \ell(v)$. Note that $\acute B_v$ and $\acute B_w$ both have the symbol $i_\ell$ in row $i$, see . Since $\varphi(\acute B_v + \acute B_w)$ is regular and since the rows $j\neq i$ of $\acute B_v + \acute B_w$ contains all symbols from $S$ (see ), it follows that $$\label{eq:P}
\forall s \in S: \ \varphi(s) \in \underbrace{\{1,\ldots,m\}\setminus\{\varphi(i_\ell)\}}_{=: P}.$$ Note that $|P| = m-1$.
Two symbols $s,t \in S$ are called *partners* if $s$ appears exactly in those columns of $\acute B$ in which $t$ does not appear. In this case (by construction) we have $s=j_k^i$ and $t=j_{\overline k}^i$, or $t=j_k^i$ and $s=j_{\overline k}^i$. If two symbols $s,t$ are not partners, then there exists a column in $\acute B$ that contains both $s$ and $t$, because both $s$ and $t$ appear in exactly $D$ many columns in the shape $m\times (2D)$ tableau $\acute B$.
Clearly, for each partnership one partner is abound and the other is scarce.
Since $\varphi(B')$ has shape $m \times (2D-1)$, every symbol from $\{1,\ldots,m\}$ appears exactly $2D-1$ times in $\varphi(B')$. Therefore for every symbol $p \in P$ there is at most one abound symbol $s \in S$ with $\varphi(s)=p$. Since $S$ contains $m-1$ many abound symbols, from it follows that for each $p \in P$ there is *exactly* one abound symbol $s \in S$ with $\varphi(s)=p$. Having seen this, it follows that for each $p\in P$ there is at most one scarce symbol $t \in S$ with $\varphi(s)=p$. Again, since $S$ contains $m-1$ many scarce symbols and because of , there is *exactly* one scarce symbol $t \in S$ with the property that $\varphi(t)=p$.
Given an abound symbol $s \in S$ and a symbol $t\in S$ that is not the partner of $s$, then in $\acute B'$ there is a column that contains both $s$ and $t$. Therefore $\varphi(s)\neq \varphi(t)$ if $s$ is abound and $t$ is not the partner of $s$. We conclude that for every $p \in P$ there is exactly one abound $s \in S$ and its scarce partner $t\in S$ that have $\varphi(s)=\varphi(t) = p$. This implies the claim.
\[cla:varphii\] If $\varphi(T)$ is regular, then for each $i \in I$ we have: For every $i_\ell$ that appears in $T$, $\varphi(i_\ell)$ only depends on $i$ and does not depend on $\ell$.
By definition, for every name edge $e$ in $H^{(i)}$ the values $\ell(v)$ coincide for all $v \in e$. This trivially implies that $$\label{eq:nameedgevarphicoincideNEW}
\text{for every name edge $e$ in $H^{(i)}$:
the values $\varphi(i_{\ell(v)})$ coincide for all $v \in e$.}$$
We claim that $$\label{eq:blockedgevarphicoincideNEW}
\text{for every block edge $e$ in $H^{(i)}$:
the values $\varphi(i_{\ell(v)})$ coincide for all $v \in e$.}$$ Proof: Let $k:=k(e)$. According to Def. \[def:dkhypergraph\] there exists a vertex $\xi^{(i)}\neq\zeta^{(i)}$ that has the same name edge and block edge as $\zeta^{(i)}$, i.e., $\ell(\zeta^{(i)}) = \ell(\xi^{(i)})$ and $k=k(\zeta^{(i)}) = k(\xi^{(i)})$. For each $v \in e$, $v \neq \zeta^{(i)}$ we have that from each of the $m-1$ sets $\{j_k^i,j_{\overline k}^i\}$, $1\leq j \leq m$, $j \neq i$, there is one symbol in the column $\check B^{(i)}_v$, see . Moreover, the symbol that appears as the $i$-th entry of $\check B^{(i)}_v$ is $i_{\ell(v)}$, see . Since $\varphi(T)$ is regular, Claim \[cla:barcoincide\] implies that $\varphi(j_k^i)=\varphi(j_{\overline k}^i)$, which we will use implicitly in the upcoming argument. For each $v \neq \zeta^{(i)}$ we have that $\check B^{(i)}_v$ is a column in $T$. In this case, since by assumption $\varphi(T)$ is regular, it follows that $\varphi(\check B^{(i)}_v)$ is regular and hence the $\varphi(j_k^i)$ are pairwise distinct. Thus $\varphi(i_{\ell(v)})$ equals the one element in $\{1,\ldots,m\} \setminus \{\varphi(j_k^i) \mid 1 \leq j \leq m, \ j\neq i\}$, which is independent of $\ell$. This implies that the values $\varphi(i_{\ell(v)})$ coincide for all $v \in e$, $v \neq \zeta^{(i)}$. This proves for all $v\in e$, $v \neq \zeta^{(i)}$. Now, if $\zeta^{(i)} \in e$, then $\xi^{(i)} \in e$, for which we have $\ell(\zeta^{(i)}) = \ell(\xi^{(i)})$, and thus clearly $\varphi(i_{\ell(\zeta^{(i)})})=\varphi(i_{\ell(\xi^{(i)})})$. This proves the claim .
Since $H^{(i)}$ is connected (Def. \[def:dkhypergraph\]\[eq:def:connected\]), we conclude with and : The values $\varphi(i_{\ell(v)})$ coincide for all $v$ in $H^{(i)}$. Since the symbol $i_\ell$ appears in $T$ iff there is some vertex $v$ in $H^{(i)}$ with $\ell(v)=\ell$ (see Claim \[cla:symbols\]), Claim \[cla:varphii\] follows.
For $i\in I$ we define $$\label{eq:defvarphii}
\varphi^{\circ}(i):=\varphi(i_1).$$ This definition is natural, because we saw in Claim \[cla:varphii\] that if $\varphi(T)$ is regular, then $$\varphi^{\circ}(i)=\varphi(i_1)=\varphi(i_2)=\ldots$$
\[cla:differentphiNEW\] Let $\varphi(T)$ be regular. Let $i, j \in I$, $i\neq j$. Then $\varphi^{\circ}(i)\neq \varphi^{\circ}(j)$.
The column $\check B_{\zeta^{(h)}}$ contains the symbol $i_1$ in row $i$ and the symbol $j_1$ in row $j$, see and . The fact that $\varphi(T)$ is regular implies that $\varphi(i_1)\neq\varphi(j_1)$. By definition , this concludes the proof.
\[cla:droph\] Let $\varphi(T)$ be regular. Let $i \in I$, $i \neq h$. Then $\varphi(i_{\overline 1}^h) = \varphi(i_1^h) = \varphi(i_1) = \varphi^{\circ}(i)$.
The first equality follows from Claim \[cla:barcoincide\]. The last equality is . We now prove the second equality. Let $e$ be the block edge in $H^{(i)}$ that contains the link vertex $\zeta^{(i)}$. Then $\check B_e^{(i)}$ is an $m \times D$ subtableau of ${\textup{\textsf{leftpart}}}(T)$, which differs from $\acute B_e^{(i)}$ only in a single entry in the length $m$ column corresponding to $\zeta^{(i)}$: The $i$-th entry of the column $\check B_{\zeta^{(i)}}$ is $i^{h}_{1}$ or $i^{h}_{\overline 1}$ instead of $i_{1}$, see Claim \[cla:acutecheck\]. Hence $\varphi(\acute B_{\zeta^{(i)}})$ and $\varphi(\check B_{\zeta^{(i)}})$ are columns that coincide in all but at most this single box. Since $\varphi(T)$ is regular and the $\varphi(T)$ only contains entries from $\{1,\ldots,m\}$ and the columns $\varphi(\acute B_{\zeta^{(i)}})$ and $\varphi(\check B_{\zeta^{(i)}})$ are of length $m$, we conclude with Claim \[cla:barcoincide\] (i.e., $\varphi(i^{h}_{1})=\varphi(i^{h}_{\overline 1})$) that $\varphi(i^{h}_{1}) = \varphi(i_{1})$.
Proof of part of Theorem \[thm:prolongation\] {#proof-of-part-of-theoremthmprolongation-4 .unnumbered}
----------------------------------------------
We now prove part of Theorem \[thm:prolongation\]. The tableau ${\textup{\textsf{rightpart}}}(T)$ only contains entries $i_\ell$ and no entries $j_k^i$ or $j_{\overline k}^i$, see . As also seen in , if ${\textup{\textsf{rightpart}}}(T)$ contains an entry $i_\ell$, then the corresponding entry of $S$ is $i$. Therefore $\varphi^{\circ}(S) = \varphi({\textup{\textsf{rightpart}}}(T))$, where we lifted the map $\varphi^{\circ} : I \to \{1,\ldots,m\}$ to a map with the same name that is defined on tableaux with entries from $I$. Claim \[cla:differentphiNEW\] proves property of Theorem \[thm:prolongation\].
Proof of part of Theorem \[thm:prolongation\] {#proof-of-part-of-theoremthmprolongation-5 .unnumbered}
----------------------------------------------
The rest of this section is devoted to proving part \[enum:duplex\] of Theorem \[thm:prolongation\]. A rectangular tableau whose columns all coincide is called *uniform*. In the following proof we will crucially use that a uniform tableau with an even number of columns is duplex. Indeed, we prove part \[enum:duplex\] of Theorem \[thm:prolongation\] by showing that if $\varphi(T)$ is regular, then:
- $\varphi(\check B_{e})$ is uniform if $e$ does not contain any link vertex $\zeta^{(i)}$,
- $\varphi(\check B_{e})$ is uniform if $\zeta^{(i)} \in e$ for $i \neq h$, and
- $\varphi(\check B_{e})$ is uniform if $\zeta^{(h)} \in e$.
It is clear that these three properties cover all cases and hence $\varphi(T)$ is uniform by construction . This implies part \[enum:duplex\] of Theorem \[thm:prolongation\].
We start with proving (I).
\[cla:ithentry\] Let $\varphi(T)$ be regular. Given a block edge $e$ in $H^{(i)}$. For all $v \in e$, $v \neq \zeta^{(i)}$, we have that the $i$-th entry of $\varphi(\check B_v)$ is $\varphi^{\circ}(i)$.
Combine and Claim \[cla:varphii\].
\[cla:jBvneqzeta\] Let $\varphi(T)$ be regular. Given a block edge $e$ in $H^{(i)}$. For all $j \neq i$ we have that the set $$\{
\text{$j$-th entry of $\varphi(\check B_v)$} \mid v \in e, v \neq \zeta^{(i)}
\}$$ consists of the single element $\varphi(j_{k(e)}^{i})$.
Combine and Claim \[cla:barcoincide\].
Combining Claim \[cla:ithentry\] and Claim \[cla:jBvneqzeta\] we see that (I) is true.
We now prove (II). Let $i \neq h$ and let $e$ be the block edge in $H^{(i)}$ that contains $\zeta^{(i)}$. Note that $k(e)=1$.
\[cla:almostexceptionalcolumn\] Let $\varphi(T)$ be regular. Then $\varphi(\check B_{\zeta^{(i)}})$ coincides with $\varphi(\check B_v)$, $v \in e$, $i \neq h$.
We compare the columns entrywise. Note that $k(v)=k(\zeta^{(i)})=1$. We make a case distinction.
Case 1: Let $j \neq i$. The $j$-th entry of $\check B_v$ is either $j_{1}^i$ or $j_{\overline 1}^i$, see . The $j$-th entry of $\check B_{\zeta^{(i)}}$ is either $j_{1}^i$ or $j_{\overline 1}^i$, see . Using Claim \[cla:barcoincide\] we see that the $j$-th entry of $\varphi(\check B_v)$ equals the $j$-th entry of $\varphi(\check B_{\zeta^{(i)}})$.
Case 2: The $i$-th entry of $\check B_v$ is $i_1$, see . The $i$-th entry of $\check B_{\zeta^{(i)}}$ is $i_1^h$, see . Hence Claim \[cla:droph\] implies that the $i$-th entry of $\varphi(\check B_v)$ equals the $i$-th entry of $\varphi(\check B_{\zeta^{(i)}})$.
It follows from Claim \[cla:almostexceptionalcolumn\] that all columns in $\varphi(\check B_e)$ coincide, i.e., $\varphi(\check B_e)$ is uniform. Thus (II) is proved.
It remains to show (III), i.e., that $\varphi(\check B_{e})$ is uniform if $\zeta^{(h)} \in e$.
\[cla:hthentry\] Let $\varphi(T)$ be regular and $\zeta^{(h)}$ the link vertex in the block edge $e$. For all $v \in e$, $v \neq \zeta^{(h)}$, we have that the $h$-th entry of $\varphi(\check B_v)$ is $\varphi^{\circ}(h)$.
This is a direct implication of Claim \[cla:ithentry\].
\[cla:exceptionalcolumn\] Let $\varphi(T)$ be regular and $\zeta^{(h)} \in e$. Then $\varphi(\check B_{\zeta^{(h)}})$ coincides with $\varphi(\check B_{v})$, $v \in e$.
We compare the columns entrywise, considering three cases.
Case 1: We compare the $h$-th entry: According to Claim \[cla:hthentry\], the $h$-th entry of $\varphi(\check B_{v})$ is $\varphi^{\circ}(h)$. According to the $h$-th entry of $\check B_{\zeta^{(h)}}$ is $h_1$, so the $h$-th entry of $\varphi(\check B_{\zeta^{(h)}})$ is $\varphi(h_1)=\varphi^{\circ}(h)$, see .
Case 2: We compare the $j$-th entry, $j \neq h$, in the case $j \notin I$: According to , the $j$-th entry of $\check B_{v}$ is either $j_1^{h}$ or $j_{\overline 1}^{h}$. The $j$-th entry of $\check B_{\zeta^{(h)}}$ is $j_1^{h}$, see . Therefore Claim \[cla:barcoincide\] shows that the $j$-th entry of $\varphi(\check B_{v})$ equals the $j$-th entry of $\varphi(\check B_{\zeta^{(h)}})$.
Case 3: We compare the $j$-th entry, $j \neq h$, in the case $j\in I$: According to , the $j$-th entry of $\check B_{v}$ is either $j_1^{h}$ or $j_{\overline 1}^{h}$. The $j$-th entry of $\check B_{\zeta^{(h)}}$ is $j_1$, see . Combining Claim \[cla:barcoincide\] and Claim \[cla:droph\] shows that the $j$-th entry of $\varphi(\check B_{v})$ equals the $j$-th entry of $\varphi(\check B_{\zeta^{(h)}})$.
It follows from Claim \[cla:exceptionalcolumn\] that all columns in $\varphi(\check B_e)$ coincide, i.e., $\varphi(\check B_e)$ is uniform. Thus (III) is proved. This finishes the proof of part \[enum:duplex\] of Theorem \[thm:prolongation\].
Theorem \[thm:prolongation\] is now completely proved for odd $D$.
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[^2]: Universität des Saarlandes, umangathan.kandasamy@outlook.de
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'Adam Back\'
- 'Iddo Bentov\'
title: Note on fair coin toss via Bitcoin
---
Introduction
============
In this short note we show that the Bitcoin network can allow remote parties to gamble with their bitcoins by tossing a fair or biased coin, with no need for a trusted party, and without the possibility of extortion by dishonest parties who try to abort. The superfluousness of having a trusted party implies that there is no house edge, as is the case with centralized services that are supposed to generate a profit.
One simple way to accomplish a coin toss protocol with Bitcoin is via a protocol fork that adds to the Bitcoin scripting language an opcode that puts on the stack the hash of the block in which the transaction resides. However, this implies that the parties have to wait for 10 minutes on average until the result of the bet becomes known. Worse still, such an opcode should have a maturity time of e.g. 100 blocks due to possible reorgs, thus the winning party will have to wait for more than 16 hours before being able to spend the coins that she won.
We propose an alternative coin toss protocol that utilizes the current Bitcoin implementation, i.e. with no need for a protocol fork. Further, with our protocol it is not necessary to wait for the next solved block, and instead the amount of coins of the bet can dictate the appropriate confidence level that the parties require. This means that 0-confirmations security for low value bets does not use the PoW irreversibility property, and instead the mining race degrades into a network race. Hence this is similar to Point of Sale for low value transactions with Bitcoin, as merchants can take a small risk by accepting unconfirmed transactions, while listening on the network to detect double-spending attempts.
Protocol
========
The reason why we can resist malicious adversaries who abort upon discovering that they lost the bet is that the Bitcoin scripting language allows us to have a primitive with which Alice locks a certain amount of her coins until a specified time in the future, and Bob can spend these coins to an arbitrary address at any time upon meeting certain arbitrary conditions that were specified in advance via a Bitcoin script, otherwise the locked coins are returned to Alice.
This primitive can be implemented in Bitcoin as follows: Alice creates a “principle” transaction that takes inputs that she controls, and can be spent according to “(Alice’s signature AND Bob’s signature) OR (arbitrary conditions)”. Alice keeps the “principle” transaction private, and creates another “refund” transaction that spends the ”principle” transaction to an output address that she controls, but has locktime set in the future. Alice then signs the “refund” transaction, and sends Bob a private message with the the “refund” transaction, asking Bob to sign it. Notice that since Bob only sees the hash of the “principle” transaction, he can protect himself from being tricked into signing a malevolent transaction that steals his other coins by generating a fresh secret key and asking Alice to create the “principle” transaction with the corresponding public address of this key. Hence, Bob sends Alice a private message with his signature for the “refund” transaction. Now Alice broadcasts the “principle” transaction to the Bitcoin network. If Bob (or anyone) cannot meet the conditions that the “principle” transaction specified, Alice will recover her coins after the locktime expires. [@g01]
Suppose that Alice and Bob wish to do a fair coin toss where each of them inputs X coins and the winner gets the 2X coins. This can be done by selecting the winner according to the least significant bit of two committed secrets, with the following protocol:
1. Alice picks some random secret $A1$ and sends a private message to Bob with the value $A2=\texttt{SHA256}(A1)$.
2. Bob picks some random secret $B1$ and sends a private message to Alice with the value $B2=\texttt{SHA256}(B1)$.
3. Bob creates a “bet”transaction that takes as input $2X$ of his own coins, and can be spent by: \[Alice’s signature AND Bob’s signature\] OR \[$\texttt{SHA256}(A)==A2$ AND $\texttt{SHA256}(B)==B2$ AND (($(A\ \texttt{xor}\ B)\ \texttt{mod}\ 2 == 0$ AND Alice’s signature) OR ($(A\ \texttt{xor}\ B)\ \texttt{mod}\ 2 == 1$ AND Bob’s signature))\]
4. Bob asks Alice to sign a “refund\_bet” transaction which spends his $2X$ coins to an address that he controls, and has locktime of (say) 20 blocks into the future.
5. Bob broadcasts the “bet” transaction to the Bitcoin network.
6. Alice creates a “reveal” transaction that takes as input $X$ of her own coins, and can be spent by: \[Alice’s signature AND Bob’s signature\] OR \[$\texttt{SHA256}(B)==B2$ AND Bob’s signature\]
7. Alice asks Bob to sign a “refund\_reveal” transaction which spends her $X$ coins to an address that she controls, and has locktime of (say) 10 blocks into the future.
8. Alice broadcasts the “reveal” transaction to the Bitcoin network (when she is confident enough that the “bet” transaction will not be reversed).
9. Bob redeems the “reveal” transaction by revealing B1 (when he is confident enough that the “reveal” transaction will not be reversed).
10. Alice redeems the “bet” transaction if she won, otherwise she sends $A1$ to Bob so that he could redeem the “bet” transaction without waiting for the locktime to expire.
This protocol is sound because the locktime in step (7) is shorter than the locktime of step (4), therefore Bob cannot cheat by broadcasting the transaction that reveals $B1$ in step (9) right before the locktime of step (4) expires.
By using more bits from the two committed secrets, Alice and Bob can bet with a biased coin so that the party with the worse odds wins the larger amount.
Currently `OP_MOD` is considered nonstandard in the Bitcoin protocol, but we can for example combine `OP_SHA1` and `OP_GREATERTHAN` to gain a similar effect.
[1]{} A. Back, I. Bentov, et al. , [*Bitcoin forum thread*]{}, August 2013. <https://bitcointalk.org/index.php?topic=277048.0>. `Last accessed on December 5, 2013`. G. Maxwell. , [*Bitcoin wiki*]{}. <https://en.bitcoin.it/wiki/Zero_Knowledge_Contingent_Payment>. `Last accessed on December 5, 2013`.
[revision 7]{}
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Starting from rotational invariance we derive sum rules for the single–spin asymmetries in inclusive production and binary processes. We also get sum rules for spin correlation parameters in elastic $pp$–scattering.'
author:
- |
S.M. Troshin, N.E. Tyurin\
*Institute for High Energy Physics,\
*Protvino, Moscow Region, 142280, Russia**
title: Sum rules for spin asymmetries
---
An important role of spin effects for analysis of hadron interaction dynamics is widely recognized nowadays. The space–time structure of the strong interactions provides a number of constraints for the spin observables (cf. [@bks]) and, as it will be shown further, allows us to get a useful sum rule for the single–spin asymmetries and spin correlation parameters.
Let us consider first single–spin asymmetry in hadron production $$h_1+h_2\to h_3+X,$$ where the beam or target hadron $h_{1,2}$ is transversely polarized. Let $\xi$ stands for the set of variables related to the hadron $h_3$. The definition of asymmetry $A_N$ is well known $$A_N(s,\xi)=
\left[\frac{d\sigma^\uparrow}{d\xi}(s,\xi)-\frac{d\sigma^\downarrow}{d\xi}(s,\xi)\right]/
\left[\frac{d\sigma^\uparrow}{d\xi}(s,\xi)+\frac{d\sigma^\downarrow}{d\xi}(s,\xi)\right].$$ The equality of the integrated inclusive cross-sections follows from rotational invariance in a straightforward way, i.e. $$\int
\frac{d\sigma^\uparrow}{d\xi}(s,\xi)d\xi=\int \frac{d\sigma^\downarrow}{d\xi}(s,\xi)d\xi.$$ Then from the definition of $A_N$ we have the following sum rule $$\label{srincl}
\int A_N(s,\xi)\frac{d\sigma}{d\xi}(s,\xi)d\xi=0,$$ where ${d\sigma}/{d\xi} $ is the unpolarized cross-section. Eq. (\[srincl\]) should be taken into account at the construction of models intended to explain the significant single–spin asymmetries observed in the inclusive processes. Similar sum rule (with replacement $A_N$ $\to$ $P$) takes place when the polarization of the final hadron $h_3$ can be measured ($\Lambda$–hyperon for example).
The above arguments can be applied to analyzing power in elastic and binary processes, e.g. from the equality $$\sigma_{el}^\uparrow(s)=\sigma_{el}^\downarrow(s)$$ we should have $$\label{srelas}
\int_{-s+4m^2}^0 A(s,t)\frac{d\sigma}{dt}(s,t)dt=0,$$ where $A(s,t)$ is the analyzing power in elastic scattering and ${d\sigma}/{dt}$ is the unpolarized cross–section for the elastic scattering of the particles with equal masses. Sum rule for the inelastic binary processes has the similar form with minor kinematical changes of the integration limits. [*From Eq. (\[srelas\]) we arrive to conclusion that $A(s,t)$ should have sign–changing $t$-dependence since ${d\sigma}/{dt}$ is positive*]{}. This conclusion on the $t$–dependence is useful for the planning of the future experiments on the analyzing power measurements in elastic scattering at higher values of $t$ [@adk]. It should be noted that change of sign of $A$ in elastic $pp$–scattering was revealed for the first time in the measurements at 40 $GeV/c$ [@kaz] and considered as a new experimental regularity in the analyzing power $t$–dependence that time.
Oscillating pattern of analyzing power $t$-dependence with amplitude of oscillations increasing with $t$ observed in various experiments in elastic and binary processes [@bin], is in conformity with the sum rule Eq. (\[srelas\]). Such oscillating dependence has obtained model explanation in [@osa]. It should be noted that the Eq. (\[srincl\]) does not imply similar $p_{\perp}$–dependence for the single–spin asymmetries in the inclusive processes and the corresponding experimental data have not revealed oscillations (cf. e.g. [@bks]).
Using rotational invariance combined with particle identity we can obtain similar sum rules for the spin correlation parameters in elastic and inclusive $pp$–scattering. Spin correlation parameters are the spin observables which describe dependence of the interaction on the relative orientations of the spins of the two particles (cf. [@bks]). We will consider scattering when both protons in [*the initial state*]{} are polarized. Definition of spin correlation parameter $A_{nn}$ is the following $$\label{ann}
A_{nn}=\frac{{\frac{d\sigma_{^{\uparrow\uparrow }} }{dt}} +
{\frac{d\sigma_{^{\downarrow\downarrow }} }{dt}} -
{\frac{d\sigma_{^{\uparrow\downarrow }}}{dt}} -
{\frac{d\sigma_{^{\downarrow\uparrow }}}{dt}}}
{{\frac{d\sigma_{^{\uparrow\uparrow }}}{dt}}
+{\frac{d\sigma_{^{\downarrow\downarrow }} }{dt}}
+ {\frac{d\sigma_{^{\uparrow\downarrow }}}{dt}}
+ {\frac{d\sigma_{^{\downarrow\uparrow }} }{dt}}},$$ where index $n$ means that spins polarized along a normal to the scattering plane. Other parameters $A_{ll}$, $A_{ss}$ and $A_{sl}$ have the similar to Eq. (\[ann\]) structure and differ by the orientation of spins in the initial state. Rotational invariance and particle identity leads to the following equality $$\label{srcor}
\Delta\sigma^{el}_T(s)=-4\int_{-s+4m^2}^0
A_{nn}(s,t)\frac{d\sigma}{dt}dt,$$ where $\Delta\sigma^{el}_T(s)$ is the cross section difference with protons polarized along normal to beam direction: $$\Delta\sigma^{el}_T(s)\equiv \sigma^{el}_{\uparrow\downarrow}(s)-
\sigma^{el}_{\uparrow\uparrow}(s)$$ Parity conservation combined with particle identity allows us to get another relation $$\label{srcoral}
\Delta\sigma^{el}_L(s)=-4\int_{-s+4m^2}^0
A_{ll}(s,t)\frac{d\sigma}{dt}dt,$$ where $\Delta\sigma^{el}_L(s)$ is the cross section difference for the protons polarized along beam direction: $$\Delta\sigma^{el}_L(s)\equiv
\sigma^{el}_{{^\rightarrow_\leftarrow}}(s)-
\sigma^{el}_{{^\rightarrow_\rightarrow}}(s)$$ We also have due to rotational invariance that $$\label{srcoreq}
\int_{-s+4m^2}^0 [A_{nn}(s,t)- A_{ss}(s,t)]\frac{d\sigma}{dt}dt=0,$$ And parity conservation and rotational invariance provide $$\label{srcoral1}
\int_{-s+4m^2}^0 A_{sl}(s,t)\frac{d\sigma}{dt}dt=0.$$
Similar relations can be written for the spin correlation parameters in the inclusive processes.
All above sum rules should be, of course, in agreement with the experimental data and therefore they can be used for the extrapolation to the region where data are absent at the moment. These sum rules are also interesting as a test ground for the models and must be obeyed under their construction.
[9]{} C. Bourrely, E. Leader, J. Soffer, Phys. Rep. 59, 95 1980;\
S.M. Troshin, N.E. Tyurin, [*Spin Phenomena in Particle Interactions*]{}, World Scientific Publishing Co., Singapore, 1994;\
E. Leader, [*Spin in Particle Physics*]{}, Cambridge University Press, UK, 2001. V.G. Luppov, et al., AIP Conf. Proc. 675, 538, 2003. Yu. M. Kazarinov, et al., Nucl. Phys. B 124, 391, 1977. I. Auer, et al., Phys. Lett. 70, 475, 1977;\
G. Fidecaro, et al., Phys. Lett. 76, 369, 1978;\
J. Antille, et al., Nucl. Phys. 185, 1, 1981;\
V.D. Apokin, et al. Sov. Journal of Nucl. Phys. 45, 1355, 1987;\
D.G. Crabb, et al., Phys. Rev. Lett. 41, 1350, 1978; Phys. Rev. Lett. 65, 3241, 1990. S.M. Troshin, N.E. Tyurin, Proceedings “Polarization Phenomena In Nuclear Physics”, Osaka 1985, 954.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We develop a new abstract derivation of the observability inequalities at two points in time for Schrödinger type equations. Our approach consists of two steps. In the first step we prove a Nazarov type uncertainty principle associated with a non-negative self-adjoint operator $H$ on $L^2(\mathbb{R}^n)$. In the second step we use results on asymptotic behavior of $e^{-itH}$, in particular, minimal velocity estimates introduced by Sigal and Soffer. Such observability inequalities are closely related to unique continuation problems as well as controllability for the Schrödinger equation.'
address:
- ' Shanlin Huang, School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, P.R.China '
- ' Avy Soffer, Department of Mathematics, Rutgers University, Piscataway, 08854-8019, USA '
author:
- 'Shanlin Huang, Avy Soffer'
---
Introduction
============
In a recent paper by Wang, Wang and Zhang [@WWZ], they established a new type of observability inequality at two points in time for the free Schrödinger equation. More precisely, let $u(x, t)$ satisfy $$\begin{aligned}
\label{equ1.1}
\begin{cases}
i\partial_{t}u +\Delta u=0,\,\,\,(x, t)\in \mathbb{R}^n\times (0, \infty),\\
u(0,x)=u_{0}\in L^2(\mathbb{R}^n).
\end{cases}\end{aligned}$$ Then given any $r_1,\,r_2>0$, and $t_1>t_2\ge 0$, there exists a positive constant $C$ depending only on $n$ such that $$\begin{aligned}
\label{equ1.2}
\int_{\mathbb{R}^n}|u_0|^2\,dx\leq Ce^{C\frac{r_1r_2}{t_2-t_1}}\left(\int_{|x|\ge r_1}|u(x, t_1)|^2\,dx+\int_{|x|\ge r_2}|u(x, t_2)|^2\,dx\right), \,\,u_0\in L^2(\mathbb{R}^n).\end{aligned}$$ The proof in [@WWZ] is based on the fact that in the free case, one has the identity $$\begin{aligned}
\label{equ1.3}
(2it)^{\frac{n}{2}}e^{-i|x|^2/4t}u(x, t)=\widehat{e^{-i|\cdot|^2/4t}u_0}(x/2t),\,\,\, \text{for all}\,\,t>0,\end{aligned}$$ where $\widehat{\cdot}$ denotes the Fourier transform. After applying with a scaling argument, it’s easy to see that the estimate is equivalent to the following Nazarov’s uncertainty principle built up in [@Jam]: $$\begin{aligned}
\label{equ1.4}
\int_{\mathbb{R}^n}|f(x)|^2\,dx\leq Ce^{Cr_1r_2}(\int_{|x|\ge r_1}|f(x)|^2\,dx+\int_{|\xi|\ge r_2}|\hat{f}(\xi)|^2\,d\xi), \,\,\,\,f\in L^2(\mathbb{R}^n).\end{aligned}$$
A natural question is whether such kind of observability inequalities still hold for more general Hamiltonian. We mention that the approach in [@WWZ] is restricted to the free Laplacian, since the argument there is essentially relying on the formula , which in turn follows from the fundamental solution of $e^{it\Delta}$. For general $H$, no such explicit solutions are available, thus one needs to proceed differently.
The motivation of this paper is to develop an abstract approach to obtain observability inequalities at two points in time for $e^{-itH}$ under some general assumptions on $H$. Then we apply it to special cases including Schrödinger equation with potentials and fractional Schrödinger equations.
We first point out that may fail if $H$ has eigenvalues. Indeed if $H\phi=\lambda\phi$, for some $\lambda\in \mathbb{R}$ and $\phi\in L^2$. Then $u(x, t)=e^{-i\lambda t}\phi(x)$ is a solution of the following Cauchy problem $$i\partial_{t}u =H u, \qquad u(0,x)=u_{0}(x)\in L^2(\mathbb{R}^n).$$ After choosing $r_1=r_2=\sqrt{t_2-t_1}$ in , we find that the RHS of is equal to $C\int_{|x|\ge \sqrt{t_2-t_1}}{|\phi|^2\,dx}$ with some fixed constant $C$, which goes to zero as $t_2-t_1\rightarrow \infty$. Hence estimate can’t hold for such $\phi$. Therefore, we only expect to hold for vectors lying in the continuous subspace of $H$.
We proceed to illustrate the key idea and main tools used in our approach. To simplify matters, we change the uncertainty principle into a form concerning two projection operators on $L^2$, i.e., for any $r>0$ $$\begin{aligned}
\label{equ1.5}
\|f\|^2\leq C\left(\|\chi(|x|\ge r)f\|^2+\|\chi(H\ge r^{-2})f\|^2\right), \,\,\,\,f\in L^2(\mathbb{R}^n).\end{aligned}$$ where $H=-\Delta$ and $C$ is a constant depending only on the dimension. We mention that inequality indicates that if the initial data is localized in a ball, then its “energy” must have a positive lower bound. Actually, it’s easy to see is equivalent to the following $$\|\chi(|x|\le r)f\|\leq C\|\chi(H\ge r^{-2})f\|,\,\,\,\text{for any}\,\, r>0.$$
Having established this type of uncertainty principle for $H$, we can use propagation estimates, in particular [**minimal velocity estimates**]{} to further study the asymptotic behavior of $e^{-itH}f$. To provide intuition in understanding of this method, let us consider the simple case $H=-\Delta$, and assume $f$ is a Schwartz function such that $f\in Ran\, \chi(H\ge \delta )$ with some $\delta>0$, hence $\hat{f}$ is smooth and $\text{supp}\,\hat{f}\subset \{\xi\in \mathbb{R}^n,\,|\xi|\ge \sqrt{\delta}\}$. Then a integration by parts argument yields $$\int_{\frac{|x|}{t}<\sqrt{\delta}}{|e^{-itH}f|^2\,dx}=O(t^{-m}),\,\,\,\text{as}\,\, t\rightarrow\infty$$ for any $m>0$. In this sense, the evolution $e^{-itH}f$ is said to have a minimal velocity $v_{min}=\sqrt{\delta}>0$. Roughly speaking, the goal of minimal velocity estimates is to obtain similar results for general Hamiltonian via an abstract way. And it’s based on choosing observable (self-adjoint operator) $A$ so that the commutator $i[H, A]$ is positive definite, see section \[sec2.2\] for further discussion. Such estimates are crucial in our proof, since it provides quantitative information about the rate with which the wave $e^{-itH}f$ moves out to spacial infinity. As a comparison, we recall that the RAGE theorem (see e.g. [@RS] ) indicates that for certain Schrödinger operators $H=-\Delta+V$, $e^{-itH}f$ is escaping any fixed ball in a mean ergodic sense: $$\lim_{T\rightarrow\infty}\frac{1}{T}\int_0^T{\,dt}\int_{|x|\leq R}{|e^{-itH}f|^2\, dx}=0.$$ We mention that minimal velocity estimates were first appeared in the work of Sigal and Soffer [@SS], which turned out to be very useful in scattering theory and theory of resonances, we refer to [@Ski; @SS90; @SW; @HSS] and references therein for further extensions and applications. One of the novelties in our paper is that we establish the close relationship between observability inequalities and minimal velocity estimates for Schrödinger type equations.
Now we turn to some applications. As is pointed out in [@WWZ], estimate can be used to derive controllability for Schrödinger equations. It is also closely related to quantitative unique continuation problems for Schrödinger equations. In Sect. \[sec4\], we shall use observability inequalities built up in this paper to obtain results concerning unique continuation properties of Schrödinger equations with potentials as well as fractional Schrödinger equations. Such kind of results for certain linear and nonlinear Schrödinger equations were considered by Ionescu and Kenig [@IK] based on the use of Carleman estimates. For the uncertainty principle and unique continuation inequalities for Schrödinger equations, we would like to refer a series of paper by Escauriaza, et al. [@EKPV1; @EKPV2; @EKPV3; @EKPV4] and references therein.
The rest of the paper is organized as follows. Section \[sec2\] is divided into two subsections, where we discuss the related uncertainty principle and minimal velocity estimates. In Sect. \[sec3.1\], the observability inequalities are proved based on tools established in Sect. \[sec2\]. Furthermore we show in Sect. \[sec3.2\] that the observability inequalities may not hold by observing the solution at two different points in time, one time in a ball, while the other outside a ball. Section \[sec4\] is devoted to applications to unique continuation property as well as controllability for the Schrödinger equation.
Main tools {#sec2}
==========
Uncertainty principle {#sec2.1}
---------------------
In this subsection, we first present an abstract version of the Nazarov type uncertainty principle for a non-negative self-adjoint operator $H$ on $L^2(\mathbb{R}^n)$, assuming some $L^2-L^{\infty}$ decay estimates of the corresponding heat semigroup $e^{-tH}$. Then in the case of Schrödinger operator $H=-\Delta+V$, as well as the fractional Laplacian ($(-\Delta)^{\frac{s}{2}}$, $s>0$), we shall obtain more quantitative results for $n\ge 3$, which will be used in Sect. \[sec3\].
\[lem2.1\] Let $H$ be a non-negative self-adjoint operator on $L^2(\mathbb{R}^n)$, $n\ge 1$. Assume that there is some $\gamma>0$ such that $$\begin{aligned}
\label{equ2.1}
\|e^{-tH}\|_{L^2\rightarrow L^{\infty}}\leq Ct^{-\gamma}.\end{aligned}$$ Then for any $R>0$, there is a constant $\delta_R>0$, such that for all $f\in L^2(\mathbb{R}^n)$ $$\begin{aligned}
\label{equ2.2}
\|f\|^2\leq C\left(\|\chi(|x|\ge R)f\|^2+\|\chi(H\ge \delta_R)f\|^2\right).\end{aligned}$$
We first point out that \[equ2.2\] is equivalent to proving that there is some $C>0$ such that for any $R>0$, $$\begin{aligned}
\label{equ2.3}
\|\chi(|x|\leq R)f\|^2\leq C\|\chi(H\ge \delta_R)\chi(|x|\leq R)f\|^2, \,\,\,\,f\in L^2,\end{aligned}$$ which in turn is equivalent to prove that for any $R>0$, there exists a constant $\delta_R>0$, $$\begin{aligned}
\label{equ2.4}
\|\chi(|x|\leq R)\chi(H\le \delta_R)\|_{L^2\rightarrow L^2}<1.\end{aligned}$$ In order to prove , we note that it follows from and Laplace transform $$\begin{aligned}
\|(H+\epsilon_0)^{-\alpha}\|_{L^2\rightarrow L^{\infty}}\leq& \frac{1}{\Gamma(\alpha)}\int_0^{\infty}\|e^{-tH}\|_{L^2\rightarrow L^{\infty}}e^{-t\epsilon_0}t^{\alpha-1}\,dt\\
\leq& \frac{1}{\Gamma(\alpha)}\int_0^{\infty}e^{-t\epsilon_0}t^{\alpha-1-\gamma}\,dt\\
\leq& C\epsilon_0^{\gamma-\alpha},\end{aligned}$$ provided $\alpha>\gamma$. Hence if we denote by $K(x, y)$ the kernel of the operator $\chi(|x|\leq R)(H+\epsilon_0)^{-\alpha}$, we deduce that $$\begin{aligned}
\int{|K(x, y)|^2\,dxdy}\leq& \int_{\mathbb{R}^n}{\chi(|x|\leq R)\,dx}\int_{\mathbb{R}^n}{|(H+\epsilon_0)^{-\alpha}(x, y)|^2\,dy}\\
\leq& CR^n\epsilon_0^{2(\gamma-\alpha)},\end{aligned}$$ which implies that $\chi(|x|\leq R)(H+\epsilon_0)^{-\alpha}$ is a Hilbert-Schmitd operator and furthermore $$\begin{aligned}
\label{equ2.5}
\|\chi(|x|\leq R)(H+\epsilon_0)^{-\alpha}\|_{L^2\rightarrow L^2}\leq CR^{\frac n2}\epsilon_0^{\gamma-\alpha}.\end{aligned}$$ Therefore for any $R>0$, we can choose $\epsilon_0>0$ small enough, and then let $\delta_R=\epsilon_0$, we obtain $$\begin{aligned}
\|\chi(|x|\leq R)\chi(H\le \delta_R)\|\leq& \|\chi(|x|\leq R)(H+\epsilon_0)^{-\alpha}\|\cdot\|(H+\epsilon_0)^{\alpha}\chi(H\le \delta_R)\|\\
\leq& CR^{\frac n2}\cdot\epsilon_0^{\gamma-\alpha}\cdot(\delta_R+\epsilon_0)^{\alpha}\\
\leq& CR^{\frac n2}\epsilon_0^{\gamma}<1,\end{aligned}$$ which proves .
We mention that the result above doesn’t imply sharp relationship between $R$ and $\delta_R$. However, in the case $H=-\Delta+V$, with suitable class of potentials or $H=(-\Delta)^{\alpha}$, $\alpha>0$. Instead of using the heat kernel estimate , we shall establish sharp results for $n\ge 3$ by studying the limiting behavior of $(H+\epsilon)^{-1}$ as $\epsilon\rightarrow 0$.
We first recall the definition of Kato class and the related global Kato norm.
Let $n\ge 3$, a real measurable function $V(x)$ is said to lied in the Kato class if $$\lim_{\delta \to 0}\sup_{x\in\mathbb{R}^n}\int_{|x-y|<\delta}{\frac{|V(y)|}{|x-y|^{n-2}}\,dy}=0.$$ Moreover, the global Kato norm of $V(x)$ is defined as $$\|V\|_K=\sup_{x\in\mathbb{R}^n}\int_{\mathbb{R}^n}{\frac{|V(y)|}{|x-y|^{n-2}}\,dy}$$
Our assumption on $V$ is the following $$\begin{aligned}
\label{equ2.6}
\begin{cases}
V=V_+-V_-,\,\,\, \text{where}\,\,V_+=\max\{V, 0\}\\
V_+\,\,\text{is of Kato class},\,\,\, \|V_-\|_K<\pi^{\frac{n}{2}}/\Gamma(\frac{n}{2}-1).
\end{cases}\end{aligned}$$
It’s known that (see [@DP Lemma 3.1]) under this assumption, $-\Delta+V$ defined on $C_0^{\infty}(\mathbb{R}^n)$ extends to a unique nonnegative self-adjoint operator. Furthermore, we shall prove
\[lem2.2\] Let $n\ge 3$ and assume that $V$ satisfies condition . Then for any $R>0$, there are uniform constants $C, \delta>0$ such that $$\begin{aligned}
\label{equ2.7}
\|f\|^2\leq C\left(\|\chi(|x|\ge R)f\|^2+\|\chi(H\ge \delta R^{-2})f\|^2\right),\,\,\, f\in L^2(\mathbb{R}^n).\end{aligned}$$
We first point out that it suffices to prove the case $R=1$ via a scaling argument. To this end, we consider the scaling operator $$U_Rf=R^{\frac n2}f(R\cdot),\,\,\,\,\, R>0.$$ Clearly, $U_R$ is an isometry on $L^2(\mathbb{R}^n)$. Now set $H_R=\Delta+V_R$, where $V_R=R^2V(R\cdot)$. A direct computation yields $$\begin{aligned}
\label{equ2.8}
U_R^{-1}H_RU_R=R^2H.\end{aligned}$$ Thus follows by proving that for all $R>0$, there exists a $\delta>0$ such that $$\begin{aligned}
\label{equ2.9}
\|\chi(|x|\leq 1)\chi(H_R\leq \delta)\|_{L^2\rightarrow L^2}<1.\end{aligned}$$ In order to show , we point out that the key is to verify the following $$\begin{aligned}
\label{equ2.10}
\||x|^{-1}H_R^{-\frac12}\|_{L^2\rightarrow L^2}\leq C_H<\infty.\end{aligned}$$ Indeed, applying , we have $$\begin{aligned}
\label{equ2.10'}
\|\chi(|x|\leq 1)\chi(H_R\leq \delta)\|\leq& \|\chi(|x|\leq 1)|x|\|\cdot\||x|^{-1}H_R^{-\frac12}\|\cdot\|H^{\frac12}\chi(H_R\leq \delta)\|\nonumber \\
\leq& C_H\cdot\delta^{\frac12}<1,\end{aligned}$$ provided $\delta<C_H^{-2}$, which implies . Therefore it remains to prove . Note that when $n\ge 3$, in view of the Hardy’s inequality $$\begin{aligned}
\label{hardy}
\||x|^{-1}(-\Delta)^{-\frac12}\|_{L^2\rightarrow L^2}\leq C.\end{aligned}$$ follows if one can prove that $$\begin{aligned}
\label{equ2.11}
\|(-\Delta)^{\frac12}H_R^{-\frac12}\|_{L^2\rightarrow L^2}\leq C\,\,\, \text{for all}\,\,R>0.\end{aligned}$$ In order to prove , we denote the operator $$T_V=|V_{R-}|^{\frac12}(-\Delta)^{-\frac12},$$ then by a $TT^*$ argument (see [@DP Lemma 3.1]) and observing that the global Kato norm is invariant under the scaling, i.e., $\|V_R\|_K=\|V\|_K$, for any $R>0$, we deduce that $$\begin{aligned}
\|T_VT_V^*f\|^2\leq& \frac{\|V_{R-}\|_K}{\alpha_n^2}\int\int{\frac{|V_{R-}||f(y)|^2}{|x-y|^{n-2}}\,dxdy} \\
\leq& \frac{\|V_{R-}\|^2_K}{\alpha_n^2}\|f\|^2\\
=& \frac{\|V_{-}\|^2_K}{\alpha_n^2}\|f\|^2,\end{aligned}$$ where in the first inequality, we used the fact that the kernel of $(-\Delta)^{-1}$ satisfies $$|(-\Delta)^{-1}(x,y)|\leq \frac{1}{\alpha_n|x|^{n-2}},\,\,\,\,\, \alpha_n=4\pi^{\frac n2}/\Gamma(\frac n2-1).$$
The method in the proof of Lemma \[lem2.2\] can be applied to other situations directly, such as $H=-\Delta-\frac{c_n}{|x|^2}$, $c_n<\frac{(n-2)^2}{4}$ and $H=(-\Delta)^{\alpha}$, $\alpha>0$. More precisely we have
Let $n\ge 3$, $H=-\Delta-\frac{c_n}{|x|^2}$, where $c_n<\frac{(n-2)^2}{4}$. Then Then for any $R>0$, there are uniform constants $C, \delta>0$ such that $$\begin{aligned}
\label{equ2.12}
\|f\|^2\leq C\left(\|\chi(|x|\ge R)f\|^2+\|\chi(H\ge \delta R^{-2})f\|^2\right),\,\,\, f\in L^2(\mathbb{R}^n).\end{aligned}$$
We follow the proof in Lemma \[lem2.2\], and note that in this case, we have $$H_R=H,\,\,\,\, \text{for any}\,\, R>0.$$ Meanwhile, it follows from the Hardy’s inequality that $$(\varphi, \frac{c_n}{|x|^2}\varphi)\leq \|\nabla\varphi\|^2,\,\,\,\,\, c_n<\frac{(n-2)^2}{4},$$ which implies that $$\|(-\Delta)^{\frac12}H^{-\frac12}\|_{L^2\rightarrow L^2}\leq C$$
\[cor2.5\] Let $n\ge 3$, $H=(-\Delta)^{\frac{s}{2}}$, $s>0$. Then Then for any $R>0$, there are uniform constants $C, \delta>0$ such that $$\begin{aligned}
\label{equ2.14}
\|f\|^2\leq C\left(\|\chi(|x|\ge R)f\|^2+\|\chi(H\ge \delta R^{-s})f\|^2\right),\,\,\, f\in L^2(\mathbb{R}^n).\end{aligned}$$
We note that the two types of estimates and are equivalent to each other. Thus the proof is followed from $$\begin{aligned}
\|\chi(|x|\leq R)\chi(H\leq \delta R^{-s})\|\leq& \|\chi(|x|\leq R)|x|\|\cdot\||x|^{-1}(-\Delta)^{-\frac12}\|\cdot\|H^{\frac{1}{s}}\chi(H\leq \delta R^{-s})\| \\
\leq& C_H\cdot\delta^{\frac{1}{s}}<1,\end{aligned}$$ provided $\delta<C_H^{-s}$.
Minimal escape velocities {#sec2.2}
-------------------------
In this subsection, we first collect some known minimal velocity estimates for the unitary evolutions $e^{-itH}$. Then we discuss examples of operators which these estimates apply to. As already pointed out in the introduction, results established in this subsection play an essential role in our proof of the observability estimates in Sect. \[sec3\].
The starting point is Mourre’s inequality [@Mo], whose fundamental idea is to find observable (self-adjoint operator) $A$ such that the commutator $i[H,\, A]$ is conditionally positive, in the sense that $$\begin{aligned}
\label{equ2.15}
E_{\Delta}i[H,\, A]E_{\Delta}\ge \theta E_{\Delta},\,\,\, \theta>0\end{aligned}$$ for some compact interval $\Delta\subset\mathbb{R}$, where $E_{\Delta}$ is the corresponding spectral projection of $H$. To provide further intuition in the understanding of condition , let us consider $H=-\frac12\Delta$, and $A$ is the generator of dilation: $$\begin{aligned}
\label{equ2.150}
A=\frac12(x\cdot p+p\cdot x), \,\,\,\, i[H,\, A]=2H, \,\, p=-\imath\nabla_x.\end{aligned}$$ Observe that $A=i[H,\, \frac{x^2}{2}]$, then can be written as $\partial_t^2\langle x^2\rangle_t\ge 2\theta$, where $\langle x^2\rangle_t=\langle \psi_t,\, x^2\psi_t\rangle$, and $\psi_t=e^{-itH}\psi\in E_{\Delta}$, which in turn implies that $$\begin{aligned}
\label{equ2.15'}
\langle x^2\rangle_t\ge \theta t^2+O(t),\,\,\, t\rightarrow \infty.\end{aligned}$$
The second key ingredient is that the multiple commutator of $H$ and $A$ are well behaved. More precisely, we assume that for any $g\in C_0^{\infty}(\mathbb{R})$ $$\begin{aligned}
\label{equ2.16}
\|ad_{A}^{(k)}g(H)\|\leq C_k,\,\,\, k=1,2.\end{aligned}$$ We regard as a regularity assumption and refer to the monograph [@AMG] for extensive discussion on this. We mention here that the commutator method, used in the proof of minimal velocity estimates, can be viewed as an abstract version of the integration by parts arguments. Hence for higher value of $k$ that is satisfied, the faster decay (for $t$) is expected. Here, we only assume that $k\le 2$, which is good enough for our applications.
Having discussed the main assumptions, we now present the following type of minimal velocity estimates.
[@SS Theorem 5.2]\[lem2.3\] Assume $H$ and $A$ satisfy and . Furthermore, if $$\begin{aligned}
\label{equ2.17}
\|(1+|A|^2)^{\alpha/2}(H+i)^{-1}(1+|x|^2)^{-\alpha/2}\|\leq C\,\,\,\, \text{for}\,\,\, 0\leq\alpha\leq 1.\end{aligned}$$ Then for any $v<v_{min}=\sqrt{\theta}$, and $0<m<1$ $$\begin{aligned}
\label{equ2.18}
\|F(\frac{|x|}{t}<v)e^{-itH}\psi\|\leq C(1+|t|)^{-m}\left(\|\psi\|+\||A|\psi\|\right),\end{aligned}$$ where $\psi=E_{\Delta}\psi\in D(|A|)$.
A few remarks are given in order. First, the condition is not hard to verify in applications, see, e.g. [@PSS] for the case of Schrödinger operators. Next, the estimate , can be thought as a quantitative version of the estimate , shows that if the initial data is localized in the sense that $\psi=E_{\Delta}\psi\in D(|A|$), then the support of the distribution $|e^{-itH}\psi|^2$ is asymptotically contained in the region $|x|\ge t\sqrt{\theta}$, as $t\rightarrow \infty$, up to a remainder of order $t^{-m}$, with any $m<1$.
We proceed with another type of minimal velocity estimates. Before stating the result, we briefly illustrate that the main idea is to decompose the state into outgoing and incoming waves by means of the spectral decomposition of $A$. Such idea was introduced by Enss [@En] in order to prove asymptotic completeness, More precisely, we say that a state $\psi$ is outgoing/incoming if $\psi\in \text{Ran} P^{\pm}(A)$, where $P^{\pm}$ denotes the projection on $\mathbb{R}^{\pm}$, see e.g. [@Mo82; @Jen; @RT]. Roughly speaking, the advantage of this decomposition is that outgoing components will evolve towards spatial infinity (and never come back) as $t\rightarrow\infty$, whereas incoming parts will evolve towards spatial infinity as $t\rightarrow-\infty$. In particular, we have the following
[@HSS Theorem 1.1, 1.2]\[lem2.4\] Assume $H$ and $A$ satisfy and . Let $\chi^{\pm}$ be the characteristic function of $\mathbb{R}^{\pm}$. Then for $0<m<1$, $$\begin{aligned}
\label{equ2.20}
\|\chi^{-}(A-a-vt)e^{-itH}g(H)\chi^{+}(A-a)\|\leq C(1+|t|)^{-m}\end{aligned}$$ holds for any $g\in C_0^{\infty}(\Delta)$, any $0<v<\sqrt{\theta}$, uniformly in $a\in \mathbb{R}$. In particular, if $H=-\Delta+V$, and $A=\frac12(x\cdot p+p\cdot x)$ satisfy the assumption above, then $$\begin{aligned}
\label{equ2.21}
\|\chi^{-}(|x|^2-2at-vt^2)e^{-itH}g(H)\chi^{+}(A-a)\|\leq C(1+|t|)^{-m}.\end{aligned}$$
\[rmk2.8\] (1) We note that the estimates above are uniform with respect to $a\in \mathbb{R}$. Note that when $a<0$, thus the state $\chi^{+}(A-a)\psi$ contains incoming component. However, the estimate indicates that after finite time ($\approx\frac{-a}{v}$), the incoming part turns out to be outgoing.\
(2) In our applications, we shall further investigate the behavior of the constants in and when $g$ varies in a suitable way. In particular, we shall prove that (see Corollary) it’s also uniform when $g$ is replaced by $g_k(\cdot)=g(\frac{\cdot}{2^k})$, $k=1,2,\ldots$.
Now we turn to concrete examples. First we consider Schrödinger operators $H=-\frac12\Delta+V$. We note that in the next section, we shall work with $H_R=-\frac12\Delta+R^2V(R\cdot)$, $R>0$ via a scaling argument. Hence we make the following assumption on $V_R=R^2V(R\cdot)$. Assume that there are constants $a_k, b_k>0$ ($k=0,1,2$) independent with $R>0$ and $0<a_0<1$ such that $$\begin{aligned}
\label{equ2.23}
\begin{cases}
\|(x\cdot\nabla)^k V_Rf\|\leq \frac{a_k}{2}\|\Delta f\|+b_k\|f\|,\,\,\, \text{for any}\,\,R>0,\,\,k=0, 1, 2.\\
-(x\cdot\nabla)V_R\ge 0,\,\,\, \text{for any}\,\,R>0.
\end{cases}\end{aligned}$$
Under this assumption, we have
\[cor2.9\] Let $V_R=R^2V(R\cdot)$ satisfy the condition above. Then there exists a constant $C$ uniformly in $R>0, k\ge 1$ and $a\in \mathbb{R}$ such that for $g_k(\cdot)=g(\frac{\cdot}{2^k})$, $k=1,2,\ldots$ $$\begin{aligned}
\label{equ2.24}
\|\chi^{-}(|x|^2-2at-vt^2)e^{-itH_R}g_k(H_R)\chi^{+}(A-a)\|\leq C(1+|t|)^{-m}.\end{aligned}$$
We first note that $$\begin{aligned}
\label{equ2.25}
i^kad_A^{(l)}(H_R)=-2^{l-1}\Delta+(-x\cdot\nabla)^l V_R,\end{aligned}$$ then it’s easy to check that the Mourre’s inequality is satisfied. In the following, we claim that $$\begin{aligned}
\label{equ2.26}
\|ad_A^{(l)}g_k(H_R)\|\leq C, \,\,\text{for}\,\, l=1,2,\end{aligned}$$ where the constant can be chosen independent with $R>0$ and $k=1,2,\ldots$. To show , we will make use of the Helffer-Sjöstrand formula (see e.g. [@Da p.24]) $$g_k(H_R)=-\frac{1}{\pi}\int_{\mathbb{C}}{\frac{\partial\tilde{g_k}(z)}{\partial\bar{z}}(H_R-z)^{-1}\,L(dz)},$$ which implies $$\begin{aligned}
\label{equ2.27}
[A,\, g_k(H_R)]=-\frac{1}{\pi}\int_{\mathbb{C}}{\frac{\partial\tilde{g_k}(z)}{\partial\bar{z}}[A,\, (H_R-z)^{-1}]\,L(dz)},\end{aligned}$$ where $L(dz)$ denotes the Lebesgue measure on $\mathbb{C}$ and $\tilde{g_k}\in C_0^{\infty}(\mathbb{C})$ is an almost analytic continuation of $g_k$ supported in a small neighborhood of $\text{supp}\,g_k\subset 2^k\Delta$. More explicitly, one can take $$\tilde{g_k}(z)=\sum_{r=0}^{N}\frac{g_k^{(r)}(x)(iy)^r}{r!}\cdot \tau(\frac{y}{\langle x\rangle}),\,\,\,\,\, z=x+iy,$$ where $\tau\in C_0^{\infty}(\mathbb{R})$, $\tau(s)=1$, if $|s|<1$, and $\tau(s)=0$, if $|s|>2$. Note that $$|g_k^{(r)}(x)|\leq C,$$ which is uniform with $k=1,2,\ldots$. It then follows that $$\begin{aligned}
\label{equ2.29}
|\partial\tilde{g_k}(z)|\leq C|{\operatorname{Im}}z|^N.\end{aligned}$$ Using and our assumption on $V_R$, we obtain that $$\begin{aligned}
\label{equ2.30}
\|[A,\, (H_R-z)^{-1}]\|&=\|(H_R-z)^{-1}[A,\, H_R](H_R-z)^{-1}\| \nonumber\\
&\leq C|{\operatorname{Im}}z|^{-1}.\end{aligned}$$ Thus the claim follows by combining and . Therefore the commutator condition $\eqref{equ2.16}$ is also verified with a uniform upper bound. Now the conclusion is followed by a step-by-step repetition of the proof in [@HSS].
Next, we consider applications to the fractional Laplacian. Let $H=(-\Delta)^{\frac{s}{2}}$ with domain the Sobolev space $H^{s}(\mathbb{R}^n)$. It’s well-known that $\sigma(H)=\sigma_c(H)=[0,\,\infty)$. We have the following
\[cor2.10\] For $s\ge 1$, then estimates and are valid with $H=H_R=(-\Delta)^{\frac{s}{2}}$.
We first observe that $$\begin{aligned}
\label{equ2.31}
[iH, A]=sH,\,\,\,\text{where}\,\,\, A=\frac12(x\cdot p+p\cdot x).\end{aligned}$$ For $b>a>0$, it then follows that $$\begin{aligned}
E_{[a, b]}i[H,\, A]E_{[a, b]}\ge as E_{[a, b]}\end{aligned}$$ Hence the Mourre’s inequality is satisfied.
We proceed to verify the commutator estimates with $g$ replaced by $g_k(\cdot)=g(\frac{\cdot}{2^k})$, $k=1,2,\ldots$. Similar to the proof in Corollary \[cor2.9\], it suffices to check the following $$\begin{aligned}
\label{equ2.32}
\|[A,\, (H_R-z)^{-1}]\|\leq C|{\operatorname{Im}}z|^{-1},\,\,\,\text{for any}\,\,\, R>0,\end{aligned}$$ which in turn follows by observing (note that $H=H_R=(-\Delta)^{\frac{s}{2}}$) $$[A,\, (H-z)^{-1}]=is(H-z)^{-1}H(H-z)^{-1}.$$
We further point out that the estimate is also satisfied for $(-\Delta)^{\frac{s}{2}}$, $s\ge 1$. Indeed, by interpolation, it suffices to prove $\alpha=1$. This follows by using the fact $$\begin{aligned}
\label{equ2.33}
\||p|(H+i)^{-1}\|\leq C,\,\,\,\,\,\,s\ge 1\end{aligned}$$ and writing $$p\cdot x(H+i)^{-1}(1+|x|^2)^{-\frac12}=S_1+S_2,$$ where $$S_1=\left(p(H+i)^{-1}\right)\cdot\left(x(1+|x|^2)^{-\frac12}\right),$$ and $$S_2=\left(-isp(H+i)^{-1}\right)\cdot\left(p(-\Delta)^{\frac s2-1}(H+i)^{-1}(1+|x|^2)^{-\frac12}\right).$$ Having checked all the needed conditions, the result then follows by applying Lemma \[lem2.3\], Lemma \[lem2.4\] and combining the proof in Corollary \[cor2.9\].
Sharp observability inequalities for Schrödinger type equations {#sec3}
===============================================================
Observability inequalities {#sec3.1}
--------------------------
In this subsection, we shall show how tools established in section \[sec2\] could be used to control the initial data at two different points in time. More precisely, For Schrödinger equations with potentials, we shall prove the following
\[thm3.1\] Let $n\ge 3$, $H=\Delta+V$ and assume that $V$ satisfies and . Let $u(x,t)$ be the solution of the following Cauchy problem $$\label{equ3.1}
i\partial_{t}u =H u, \qquad u(0,x)=u_{0}(x).$$ Then there exists some constant $\delta>0$ and $T_0$ large enough, such that for any $R>0$, $t_2>t_1\ge 0$ with $t_2-t_1>R^2T_0$, we have $$\begin{aligned}
\label{equ3.2}
\|u_0\|^2\leq C\left(\int_{|x|\ge R}{|u(x,t_1)|^2\,dx}+\int_{|x|\ge \frac{\sigma(t_2-t_1)}{R}}{|u(x,t_2)|^2\,dx}\right),\end{aligned}$$ where the constant $C$ depends only on the dimension.
Since $e^{-itH}$ is unitary, it follows that is equivalent to the case $t_1=0$ $$\begin{aligned}
\label{equ3.3}
\|u_0\|^2\leq C\left(\int_{|x|\ge R}{|u_0|^2\,dx}+\int_{|x|\ge \frac{\sigma t}{R}}{|u(x,t)|^2\,dx}\right),\,\,\, t>R^2T_0.\end{aligned}$$ In order to prove , we note that by scaling and using , it suffices to prove that for $H_R=-\frac12\Delta+V_R$, $R>0$, there exists a uniform constant $C, T_0>0$ such that for any $u_0\in L^2$ $$\begin{aligned}
\label{equ3.4}
\|u_0\|^2\leq C\left(\int_{|x|\ge 1}{|u_0|^2\,dx}+\int_{|x|\ge \sigma t}{|e^{-itH_R}f|^2\,dx}\right),\,\,\, t>T_0.\end{aligned}$$ We proceed to observe that can be easily deduced from the following $$\begin{aligned}
\label{equ3.5}
\|u_0\|^2\leq C_1\int_{|x|\ge \sigma t}{|e^{-itH_R}u_0|^2\,dx},\,\,\, t>T_0,\,\,\,\text{supp}\,u_0\subset B(0, 1).\end{aligned}$$ Indeed, assuming that is true, then for any $u_0\in L^2$, we write $f=u_{01}+u_{02}$, where $u_{01}=\chi(|x|\le 1)f$, then applying to $u_{01}$ and using Minkowski inequality we obtain with $C=C_1+1$. The advantage of this reduction is that it allows us to choose initial data localized in the unit ball.
Now we apply the uncertainty principle associated with $H_R$ established in Lemma \[lem2.2\]. More precisely, if $\text{supp}\,u_0\subset B(0, 1)$, then it follows from that there exists some fixed $\delta_1>0$ such that $$\begin{aligned}
\label{equ3.6}
\|u_0\|^2\leq C\|\chi(H_R\geq \delta_1)u_0\|^2,\,\,\,\, \text{for any}\,\, R>0.\end{aligned}$$ Thus would be followed if we can prove that there exists a uniform constant $\sigma$ doesn’t depend on $R$ and $t$, such that $$\begin{aligned}
\label{equ3.7}
\|\chi(H_R\geq \delta_1)u_0\|^2\leq C_2\int_{|x|\ge \sigma t}{|e^{-itH_R}u_0|^2\,dx}+\varepsilon\|u_0\|^2,\,\,\,\, t>T_0,\end{aligned}$$ with $C_1\varepsilon<1$.
The role of the uncertainty principle is to make sure that the energy of the initial data has a positive lower bound, which provides the possibility to use the method of minimal escape velocities. In order to use tools from section \[sec2.2\], we break the initial data into a sum of finite energy and write $$\chi(H_R\geq \delta_1)u_0=\chi(\delta_1\leq H_R\leq N )u_0+\sum_{k=n_0}^{\infty}\varphi(\frac{H}{2^k})u_0,$$ where $N=2^{n_0}$ is some large fixed number. Hence is valid if we can prove $$\begin{aligned}
\label{equ3.8}
\|\chi(\delta_1\leq H_R\leq N )u_0\|^2\leq C\|\chi(\delta_1\leq H_R\leq N )\chi(|x|\ge \sigma t)e^{-itH_R}u_0\|^2+\frac{\varepsilon}{2}\|u_0\|^2,\,\,\,\, t>T_0,\end{aligned}$$ and $$\begin{aligned}
\label{equ3.9}
\|\varphi(H/2^k)u_0\|^2\leq C\|\varphi(H/2^k)\chi(|x|\ge \sigma t)e^{-itH_R}u_0\|^2+\varepsilon2^{-k/4}\|u_0\|^2,\,\,\,\, t>T_0,\end{aligned}$$
We first investigate . Notice that it follows from Corollary \[cor2.9\] that for any fixed $\sigma<\sqrt{\delta_1}$ and $0<m<1$ $$\begin{aligned}
\label{equ3.10}
\|\chi(|x|\le \sigma t)e^{-itH_R}\chi(\delta_1\leq H_R\leq N )u_0\|^2\leq \frac{C}{\langle t\rangle^{2m}}\|\chi(\delta_1\leq H_R\leq N )u_0\|^2,\end{aligned}$$ then choose some fixed $T_0$ large enough, and for $t\ge T_0$, implies that $$\begin{aligned}
\label{equ3.11}
\|\chi(\delta_1\leq H_R\leq N )u_0\|^2\leq C\|\chi(|x|\ge \sigma t)e^{-itH_R}\chi(\delta_1\leq H_R\leq N)u_0\|^2,\,\,\,\, t>T_0.\end{aligned}$$ Compared to the desired form , we must commute the factor $\chi(\delta_1\leq H_R\leq N)$ to the left of the term $|\chi(|x|\ge \sigma t)$. To this end, we now apply Lemma \[lemA1\] with $A=H_R\chi(H_R)$, $B=\chi(|x|\ge \sigma t)$ and note that $$\|[H_R\chi(H_R), \chi(|x|\ge \sigma t)]\|\leq C,$$ where the constant is uniform with $R$ and $t$. Thus we have $$\begin{aligned}
\|\chi(|x|\ge \sigma t)e^{-itH}\chi(\delta_1\leq H_R\leq N)u_0\|^2&\leq \|\chi(\delta_1\leq H_R\leq N)\chi(|x|\ge \sigma te^{-itH}u_0\|^2\\
&+\|[\chi(\delta_1\leq H_R\leq N), \chi(|x|\ge \sigma t)]e^{-itH}u_0\|^2\\
&\leq \|\chi(\delta_1\leq H_R\leq N )\chi(|x|\ge \sigma t)e^{-itH_R}u_0\|^2+\frac{\varepsilon}{2}\|u_0\|^2,\end{aligned}$$ provided $t>T_0$, which proves .
We are left to prove . We set $g_k=\varphi(H_R/2^k)u_0$, notice that $\text{supp}\, f\subset B(0,1)$ and $H_R\sim 2^k$. Furthermore, under our assumption and on $V$, we have that $|p|\sim2^{k/2}$, thus classically, in phase space, we have $A\gtrsim -2^{k/2}$, without loss of generality, we write $$g_k=\chi^{+}(A+2^{k/2})g_k.$$ Then apply in Corollary \[cor2.9\] with $a=-2^{k/2}$, we find $$\begin{aligned}
\label{equ3.12}
\|\chi^{-}(|x|^2+2^{k/2+1}t-vt^2)e^{-itH_R}g_k(H_R)\chi^{+}(A+2^{k/2})\|\leq C(1+|t|)^{-m},\end{aligned}$$ which implies, after choosing sufficiently large $t$, that $$\begin{aligned}
\|\varphi(H/2^k)u_0\|^2&\leq \|\chi^{+}(|x|^2+2^{k/2+1}t-vt^2)e^{-itH_R}g_k(H_R)\chi^{+}(A+2^{k/2})u_0\|^2\\
&\leq \|\chi(|x|^2 \ge \sigma^2t^2 )\chi(|x|\ge \sigma t)e^{-itH_R}u_0\|^2+\frac{\varepsilon}{2}\|u_0\|^2,\end{aligned}$$ where in the last inequality, we have used the simple fact that $$vt^2-2^{k/2+1}t\ge \sigma^2t^2.$$ Then we apply Lemma \[lemA1\] with $A=H_R\varphi(H_R/2^k)$, $B=\chi(|x|\ge \sigma t)$ and note that $$\|[H_R\varphi(H_R/2^k), \chi(|x|\ge \sigma t)]\|\leq Ct^{-1}2^{-\frac k4},$$ which indicates and the proof is complete.
With no additional effort, the proof in Theorem \[thm3.1\] can be applied to the fractional Schrödinger equations by using Corollary \[cor2.5\] and Corollary \[cor2.10\].
\[thm3.2\] Assume $n\ge 3$, and let $u(x,t)$ be the solution of the following Cauchy problem $$\label{equ3.1}
i\partial_{t}u =(-\Delta)^{\frac{s}{2}} u, \qquad u(0,x)=u_{0}(x),\,\,\,\, s\ge 1.$$ Then there exists some constant $\delta>0$ and $T_0$ large enough, such that for any $R>0$, $t_2>t_1\ge 0$ with $t_2-t_1>R^sT_0$, we have $$\begin{aligned}
\label{equ3.14}
\|f\|^2\leq C\left(\int_{|x|\ge R}{|u(x,t_1)|^2\,dx}+\int_{|x|\ge \frac{\sigma(t_2-t_1)}{R^{s-1}}}{|u(x,t_2)|^2\,dx}\right),\end{aligned}$$ where the constant $C$ depends only on the dimension.
Sharpness of the observability inequalities {#sec3.2}
-------------------------------------------
The purpose of this subsection is to show the optimality of the inequalities established in section \[sec3.1\]. We recall that it was observed in [@WWZ] that for the free Schrödinger equation, the observability inequality can’t be replaced by $$\begin{aligned}
\label{equ3.15}
\int_{\mathbb{R}^n}|u_0|^2\,dx\leq C\left(\int_{|x|\ge r_1}|e^{it_1\Delta}u_0|^2\,dx+\int_{|x|\le r_2}|e^{it_2\Delta}u_0|^2\,dx\right), \,\,u_0\in L^2(\mathbb{R}^n)\end{aligned}$$ for any fixed $r_1, r_2>0$ and $t_2>t_1\ge 0$. In other words, we can’t expect to recover the solution by observing it at two different points in time, one point outside a ball while the other inside a ball with any fixed radius. However, since the argument in [@WWZ] again relies heavily on the representation formula for the solution $e^{it\Delta}u_0$, it doesn’t apply to other situations. We shall point out that by using [**minimal velocity estimates**]{}, one can treat more general $H$.
\[thm3.3\] Let $H=-\Delta+V$ satisfy the assumption in Theorem \[thm3.1\]. Then one can find a sequence of $L^2$ functions $\{f_k\}_{k\in\mathbb{Z}}$ with $$\begin{aligned}
\label{equ3.16}
\int_{\mathbb{R}^n}|f_k|^2\,dx=1\end{aligned}$$ and there exist some $\sigma>0$ and some large enough $T>0$, such that for any $t>T$ and any fixed $r_1>0$, $$\begin{aligned}
\label{equ3.17}
\lim_{k\rightarrow \infty}\int_{|x|\ge r_1}|f_k|^2\,dx=\lim_{k\rightarrow \infty}\int_{|x|\le \sigma t}|e^{itH}f_k|^2\,dx=0.\end{aligned}$$
Choose $f\in C_0^{\infty}(\mathbb{R}^n)$ such that $\|f\|_{L^2}=1$ and $\text{supp}\, f\subset B(0, 1)$. Then we set $f_k=U_kf=k^{\frac n2}f(k\cdot)$, $k=1,2,\cdots$, where $U_k$ is the scaling operator in . Since $U_k$ is an isometry on $L^2(\mathbb{R}^n)$, follows immediately. Moreover, a scaling argument shows that for any fixed $r_1>0$, $$\begin{aligned}
\label{equ3.18}
\lim_{k\rightarrow \infty}\int_{|x|\ge r_1}|f_k|^2\,dx=\lim_{k\rightarrow \infty}\int_{|x|\ge kr_1}|f|^2\,dx=0.\end{aligned}$$ Hence it suffices to prove the $L^2$ norm $\|\chi(|x|\le \sigma t)e^{itH}f_k\|$ goes to zero as $k\rightarrow \infty$. To this end, we write $$\begin{aligned}
\label{equ3.19}
\|\chi(|x|\le \sigma t)e^{itH}f_k\|\leq \|\chi(|x|\le \sigma t)e^{itH}\chi(H\le 1)f_k\|+\|\chi(|x|\le \sigma t)e^{itH}\chi(H\ge 1)f_k\|.\end{aligned}$$ On one hand, it follows from , and the fact $f_k=\chi(|x|\le \frac1k)f_k$ that $$\begin{aligned}
\label{equ3.20}
\|\chi(H\le 1)\chi(|x|\le \frac1k)f_k\|\leq C_H\cdot k^{-\frac12}\rightarrow 0,\,\,\,\text{as}\,\, k\rightarrow \infty.\end{aligned}$$ On the other hand, we observe that $$\begin{aligned}
\label{equ3.21}
\|\chi(|x|\le \sigma t)e^{itH}\chi(H\ge 1)f_k\|=\|\chi(|x|\le \frac{\sigma \tilde{t}}{k})e^{i \tilde{t}H_R}\chi(H\ge \frac {1}{k^2})f\|,\,\,\,\,\end{aligned}$$ where $\tilde{t}=k^2t,\,\, R=\frac1k$. Then by Lemma \[lem2.3\], we can choose $\sigma<1$ small enough and a uniform constant $C$ such that $$\begin{aligned}
\label{equ3.22}
\|\chi(|x|\le \frac{\sigma \tilde{t}}{k})e^{i \tilde{t}H_R}\chi(\frac {1}{k^2}\leq H\leq N)f\|\leq C(1+|\tilde{t}|)^{-m},\,\,\,\,\end{aligned}$$ where $N=2^{n_0}$ for some $n_0\in \mathbb{N}$. Also $$\begin{aligned}
\label{equ3.23}
\|\chi(|x|\le \frac{\sigma \tilde{t}}{k})e^{i \tilde{t}H_R}\chi(2^l\leq H\leq 2^{l+1})f\|\leq C(2^l|\tilde{t}|)^{-m},\,\,\,\,l=n_0, n_0+1,\cdots.\end{aligned}$$ Combining and $$\begin{aligned}
\label{equ3.24}
\|\chi(|x|\le \frac{\sigma \tilde{t}}{k})e^{i \tilde{t}H_R}\chi(H\ge \frac {1}{k^2})f\|\leq C|\tilde{t}|^{-m},\,\,\,\,t>T.\end{aligned}$$ Therefore it follows from , , and that $$\begin{aligned}
\label{equ3.25}
\lim_{k\rightarrow \infty}\|\chi(|x|\le \sigma t)e^{i tH}f_k\|=0,\end{aligned}$$ which completes the proof.
In the special case where $H=(-\Delta)^{\frac s2}$, $s\ge 1$. Though the solution can’t be written as the form like when $s\ne2$, we can still write the solution $e^{it(-\Delta)^{\frac s2}}f$ as a oscillatory integral and prove by a integration by parts argument. More precisely, we have
\[thm3.4\] Let $H=(-\Delta)^{\frac s2}$, $s\ge 1$. Then one can find a sequence of $L^2$ functions $\{f_k\}_{k\in\mathbb{Z}}$ with $$\begin{aligned}
\label{equ3.26}
\int_{\mathbb{R}^n}|f_k|^2\,dx=1\end{aligned}$$ and there exists some $\sigma>0$, such that for any $t>0$ and any fixed $r_1>0$, $$\begin{aligned}
\label{equ3.27}
\lim_{k\rightarrow \infty}\int_{|x|\ge r_1}|f_k|^2\,dx=\lim_{k\rightarrow \infty}\int_{|x|\le \sigma t}|e^{it(-\Delta)^{\frac s2}}f_k|^2\,dx=0.\end{aligned}$$
Let $f_k$ be as in Theorem \[thm3.3\]. By , it’s enough to prove that $$\begin{aligned}
\label{equ3.28}
\lim_{k\rightarrow \infty}\|\chi(|x|\le \sigma t)e^{itH}\chi(H\ge 1)f_k\|=0.\end{aligned}$$ We first observe that by scaling and the homogeneity of $(-\Delta)^{\frac s2}$ $$\begin{aligned}
\label{equ3.29}
\|\chi(|x|\le \sigma t)e^{itH}\chi(H\ge 1)f_k\|=\|\chi(|x|\le \frac{\sigma \tilde{t}}{k^{s-1}})e^{i \tilde{t}H}\chi(H\ge \frac {1}{k^s})f\|,\,\,\,\,\tilde{t}=t\cdot k^s.\end{aligned}$$ Then we write $$\begin{aligned}
\label{equ3.30}
e^{i \tilde{t}H}\chi(H\ge \frac{1}{k^s})f=\int_{|\xi|\ge\frac1k}{e^{-i( \tilde{t}|\xi|^s-x\cdot\xi)}\hat{f}(\xi)\,d\xi}\end{aligned}$$ To proceed, we notice that in the region $|x|\le \frac{\sigma \tilde{t}}{k^{s-1}}$ with $\sigma\le \frac12$, a direct computation shows that there exists a uniform constant $C$ such that $$\begin{aligned}
\label{equ3.31}
|\nabla_{\xi}\frac{(\tilde{t}|\xi|^s-x\cdot\xi)}{|x|+|\tilde{t}|}|\ge Ck^{1-s}\end{aligned}$$ Then we obtain after an integration by parts (see e.g. ) $$\begin{aligned}
\label{equ3.32}
\|\chi(|x|\le \frac{\sigma \tilde{t}}{k^{s-1}})e^{i \tilde{t}H}\chi(H\ge \frac {1}{k^s})f\|\leq C_N|tk|^{-N}\rightarrow 0,\,\,\,\text{as}\,\,k\rightarrow \infty,\end{aligned}$$ which implies by , hence completes the proof.
\[rmk3.1\]
The above proof extends to the case where for a general $H$, one can construct $\tilde A$ satisfying $\imath [H,\tilde A]=H$, as well as regularity assumptions as before. Furthermore, we require that the principal symbol of $\tilde A$ will be the same as that of $A$. Such $\tilde A$ were constructed for large class of potentials, without the repulsive assumption on $V$. See [@GLS; @LS].
Applications {#sec4}
============
Now we turn to the application. First we mention that the observability inequalities established in Sect. \[sec3.1\] may also be regarded as a kind of quantitative unique continuation property for the corresponding solutions. In particular, we consider $$\label{equ4.1}
i\partial_{t}u =H u, \qquad u(0,x)=u_{0}(x)\in L^2(\mathbb{R}^n),\,\,\,\,n\ge 3.$$ Based on Theorem \[thm3.1\], we can derive the following
\[thm4.1\] Let $u(x,t)$ be the solution of the Cauchy problem with $H$ satisfying the assumption in Theorem \[thm3.1\]. Moreover, for any $R>0$, if $$\begin{aligned}
\label{equ4.2}
\text{supp}\,u_0\subset B(0,R),\,\,\, \text{and}\,\,\,\text{supp}\,u(x, R^2t)\subset B(0, \sigma t/R),\end{aligned}$$ where $\sigma>0$ is some fixed constant and $t>T_0$ for some $T_0$ large enough (see ). Then $u(x,t)\equiv 0$.
The proof follows immediately by combining estimate and our assumption .
Similarly, concerning the fractional Schrödinger equations, Theorem \[thm3.2\] gives
\[thm4.2\] Let $u(x,t)$ be the solution of the Cauchy problem with $H=(-\Delta)^{\frac s2}$, $s\ge 1$. Moreover, for any $R>0$, if $$\begin{aligned}
\label{equ4.3}
\text{supp}\,u_0\subset B(0,R),\,\,\, \text{and}\,\,\,\text{supp}\,u(x, R^st)\subset B(0, \sigma t/R^{s-1}),\end{aligned}$$ where $\sigma>0$ is some fixed constant and $t>T_0$ for some $T_0$ large enough (see ). Then $u(x,t)\equiv 0$.
\[rmk4.3\] We remark that for Schrödinger operators $H=-\Delta+V$, stronger uniqueness results are valid by only assuming certain Gaussian type decay of the solution at two different points in time, see e.g. in [@EKPV2; @EKPV3; @EKPV4; @EKPV4]. However, it seems that our method can be applied to more general Hamiltonian.
Next, we consider applications to controllability for Schrödinger type equations. Based on an abstract lemma [@WWZ Lemma 5.1] concerning the equivalence between observability and controllability, we can obtain the following result from Theorem \[thm3.1\].
\[thm4.3\] Let $H$ satisfy the assumption in Theorem \[thm3.1\]. Consider the following impulse controlled Schrödinger equation $$\begin{aligned}
\label{equ4.4}
\begin{cases}
i\partial_{t}u -H u=\delta_{t=\tau_1}\chi(|x|\ge R)h_1+\delta_{t=\tau_2}\chi(|x|\ge\frac{\sigma(\tau_2-\tau_1)}{R})h_2,\,\,\,(x, t)\in \mathbb{R}^n\times (0, T),\\
u(0,x)=u_{0}\in L^2(\mathbb{R}^n),
\end{cases}\end{aligned}$$ where $\sigma>0$ is some fixed constant and $\tau_2-\tau_1>R^2T_0$ for some $T_0$ large enough (see ). Denote $u(\cdot,\cdot, u_0, h_1, h_2)$ the solution to the equation . Then for any $u_0, u_T\in L^2$, there exists a pair of controls $(h_1, h_2)\in L^2\times L^2$ such that $$\begin{aligned}
\label{equ4.5}
u(x,T, u_0, h_1, h_2)= u_T\end{aligned}$$ and for some $C>0$ $$\begin{aligned}
\label{equ4.6}
\|h_1\|^2+\|h_2\|^2\leq C\|u_T-e^{-itH}u_0\|^2.\end{aligned}$$
This is a direct consequence of [@WWZ Lemma 5.1]. We sketch the proof here for the sake of self-containment. Consider the following dual equation $$\begin{aligned}
\label{equ4.7}
\begin{cases}
i\partial_{t}\varphi -H\varphi=0,\,\,\,(x, t)\in \mathbb{R}^n\times (0, T),\\
u(x, T)=f\in L^2(\mathbb{R}^n),
\end{cases}\end{aligned}$$ and denote $\varphi(\cdot, \cdot, T, f)$ the solution to . Then Theorem \[thm3.1\] implies that $$\begin{aligned}
\label{equ4.8}
\|f\|^2\leq C\left(\int_{|x|\ge r_1}{|\varphi(\cdot, \tau_1, T, f)|^2\,dx}+\int_{|x|\ge \frac{\sigma(\tau_2-\tau_1)}{r_1}}{|\varphi(\cdot, \tau_2, T, f)|^2\,dx}\right),\end{aligned}$$ provided $\tau_2-\tau_1>r_1^2T_0$. Now we define the state transformation operator $R: L^2\rightarrow L^2$ and the observation operator $O: L^2\rightarrow L^2\times L^2$ as follows: $$\begin{aligned}
\label{equ4.9}
Rf=f;\,\,\, Of=\left(\chi(|x|\ge r_1)\varphi(\cdot, \tau_1, T, f),\,\,\, \chi(|x|\ge \sigma(\tau_2-\tau_1)/r_1)\varphi(\cdot, \tau_2, T, f)\right)\end{aligned}$$ Thus by and , we have for any $f\in L^2$ $$\begin{aligned}
\label{equ4.10}
\|Rf\|^2\leq C\|Of\|_{L^2\times L^2}^2+\frac1k\|f\|^2,\,\,\,k\in\mathbb{N}^+.\end{aligned}$$ According to Lemma 5.1 in [@WWZ], there exists a pair $(h_{1k}, h_{2k})\in L^2\times L^2$, $k\in\mathbb{N}^+$ such that the following dual inequality holds $$\begin{aligned}
\label{equ4.11}
C\|(h_{1k}, h_{2k})\|_{L^2\times L^2}^2+k\|R^*f-O^*(h_{1k}, h_{2k})\|^2\leq \|f\|^2,\,\,\,k\in\mathbb{N}^+,\end{aligned}$$ where $$\begin{aligned}
\label{equ4.12}
R^*f=f;\,\,\, O^*(h_{1k}, h_{2k})=u(\cdot, T, 0, h_{1k}, h_{2k}).\end{aligned}$$ Here the dual operator $O^*$ is viewed as the control operator. Then and are followed by choosing a weak convergence subsequence in and a limiting procedure.
Similarly, combining Theorem \[thm3.2\] with Lemma 5.1 in [@WWZ], we obtain the following controllability for fractional Schrödinger equations
\[thm4.3\] Let $H=(-\Delta)^{\frac s2}$, $s\ge 1$. Consider the the following impulse controlled Schrödinger equation $$\begin{aligned}
\label{equ4.13}
\begin{cases}
i\partial_{t}u -H u=\delta_{t=\tau_1}\chi(|x|\ge R)h_1+\delta_{t=\tau_2}\chi(|x|\ge\sigma(\tau_2-\tau_1)/R^{s-1})h_2,\,\,\,(x, t)\in \mathbb{R}^n\times (0, T),\\
u(0,x)=u_{0}\in L^2(\mathbb{R}^n),
\end{cases}\end{aligned}$$ where $\sigma>0$ is some fixed constant and $\tau_2-\tau_1>R^sT_0$ for some $T_0$ large enough (see ). Denote $u(\cdot,\cdot, u_0, h_1, h_2)$ the solution to the equation . Then for any $u_0, u_T\in L^2$, there exists a pair of controls $(h_1, h_2)\in L^2\times L^2$ such that $$\begin{aligned}
\label{equ4.14}
u(x,T, u_0, h_1, h_2)= u_T\end{aligned}$$ and $$\begin{aligned}
\label{equ4.15}
\|h_1\|^2+\|h_2\|^2\leq C\|u_T-e^{-itH}u_0\|^2\end{aligned}$$
Commutator estimates {#sec5}
====================
\[lemA1\] Let $A$ and $B$ be two operators on a Hilbert space $\mathfrak{X}$ with $A$ self-adjoint and $B$ bounded. Assume that $D(A)\cap D(B)$ is dense and $[A, B]$ extends to a bounded operator. Further there is a constant $M_{AB}$ such that $$\|[A, B]\|\leq M_{AB}.$$ Let $0\leq \varphi\in C_0^{\infty}(\mathbb{R})$ with $\text{supp} \varphi\subset [\frac12, 2]$, $\varphi=1$ on $[\frac34, \frac54]$ and denote by $\varphi_N=\varphi(\frac{\cdot}{N})$. Then we have $$\begin{aligned}
\label{equA.1}
\|[\varphi_N(A), B]\|\leq CM_{AB}N^{-\frac34}.\end{aligned}$$
Let $g(\lambda)$ denote the Fourier transform of $\varphi_N$ and set $\psi_N=i\frac{d}{dx}\varphi_N$. Thus we have $\lambda g(\lambda)=\hat{\psi_N}$. Note that in the sense of quadratic forms on $D(A)\cap D(B)$ $$[\varphi_N(A), B]=-i\int{g(\lambda)e^{-i\lambda A}(\int_0^{\lambda}e^{i\mu A}[A, B]e^{-i\mu A}\,d\mu)\,d\lambda}$$ Hence $$\begin{aligned}
\label{equA.2}
\|\left(f,\,[\varphi_N(A), B]g\right)\|&\leq M_{AB}\int_{\mathbb{R}}{|\lambda g(\lambda)|\,d\lambda}\|f\|\cdot\|g\|\nonumber\\
&\leq M_{AB}\|\psi_N\|_{\mathcal{F}L^1}\|f\|\cdot\|g\|\end{aligned}$$ In order to estimate the norm in , we apply Bernstein’s inequality, i.e., $H^{\alpha}(\mathbb{R}^n)\hookrightarrow \mathcal{F}L^1(\mathbb{R}^n)$, $\alpha>\frac n2$ (see e.g., [@Hie Lemma 3.2]) $$\begin{aligned}
\label{equA.3}
\|\psi_N\|_{\mathcal{F}L^1}&\leq C\|\psi_N\|_{L^2}^{\frac12}\cdot\|\frac{d}{dx}\psi_N\|_{L^2}^{\frac12}\nonumber\\
&\leq CN^{-\frac34},\end{aligned}$$ where the constant $C$ doesn’t depend on $N$. Therefore is followed by Combining and .
Acknowledgements {#acknowledgements .unnumbered}
================
S.H. would like to thank C. Kenig for useful discussions about topics on uncertainty principle and unique continuation for Schrödinger equation. Part of this work was done while A. Soffer was a visiting Professor at CCNU (Central China Normal University). A. Soffer is partially supported by NSFC grant No.11671163 and NSF grant DMS01600749.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We study an extended QCD model in $2D$ obtained from QCD in $4D$ by compactifying two spatial dimensions and projecting onto the zero-mode subspace. This system is found to induce a dynamical mass for transverse gluons – adjoint scalars in $QCD_2$, and to undergo a chiral symmetry breaking with the full quark propagators yielding non-tachyonic, dynamical quark masses, even in the chiral limit. We construct the hadronic color singlet bound-state scattering amplitudes and study quark-antiquark bound states which can be classified in this model by their properties under Lorentz transformations inherited from $4D$.'
title: 'Quark-Antiquark Bound States in an Extended $QCD_2$ Model[^1]'
---
[Pedro Labra[ñ]{}a, Jorge Alfaro]{}\
Facultad de Física, Pontificia Universidad Católica de Chile.\
plabrana@puc.cl, jalfaro@puc.cl\
\
St.Petersburg State University and INFN, Sezione di Bologna.\
andrianov@bo.infn.it
We study a QCD reduced model in $2D$ which can be formally obtained from QCD in $4D$ by means of a classical dimensional reduction from $4D$ to $2D$ and neglecting heavy K-K (Kaluza-Klein) states. Thus only zero-modes in the harmonic expansion in compactified dimensions are retained. As a consequence, we obtain a two dimensional model with some resemblances of the real theory in higher dimension, that is, in a natural way adding boson matter in the adjoint representation to $QCD_2$ [@light; @Alfaro:2003yy]. The latter fields being scalars in $2D$ reproduce transverse gluon effects [@adjoint]. Furthermore this model has a richer spinor structure than just $QCD_2$ giving a better resolution of scalar and vector states which can be classified by their properties inherited from $4D$ Lorentz transformations. The model is analyzed in the light cone gauge and using large $N_c$ limit. The contributions of the extra dimensions are controlled by the radiatively induced masses of the scalar gluons as they carry a piece of information of transverse degrees of freedom. We consider their masses as large parameters in our approximations yet being much less than the first massive K-K excitation. This model might give more insights into the chiral symmetry breaking regime of $QCD_4$. Namely, we are going to show that the inclusion of solely lightest K-K boson modes catalyze the generation of quark dynamical mass and allows us to overcome the problem of tachyonic quarks present in $QCD_2$.
We start with the $QCD$ action in $(3+1)$ dimensions for one flavor (extension to more flavors is straightforward): $$S_{QCD}= \int d^4x \left[-\frac{1}{2{\tilde g}^2}tr(G_{\mu\nu}^2)
+ {\bar \Psi}\,(i\gamma^\mu \, D_\mu - m)\,\Psi \right].$$
Follow the scheme of [@Alfaro:2003yy] we proceed to make a dimensional reduction of $QCD$, at the classical level, from $4D$ to $2D$. For this we consider the coordinates $x_{2,3}$ being compactified in a 2-Torus, respectively the fields being periodic on the intervals ($0\leq x_{2,3}\leq L=2\pi R$). Next we assume $L$ to be small enough in order to get an effective model in $2D$ dimensions. Then by keeping only the zero K-K modes, we get the following effective action in $2D$, after a suitable rescaling of the fields: $$\begin{aligned}
\label{Lmodelo2D}
S_2 &=& \int \!d^2x\,\,tr \!\left[-\frac{1}{2}F_{\mu\nu}^2+ (D_\mu
\phi_1)^2 + (D_\mu \phi_2)^2 \right] +
{\bar \psi}_1\,(i\gamma^\mu \, D_\mu - m)\,\psi_1 \nonumber\\
%
&+& {\bar \psi}_2\,(i\gamma^\mu \, D_\mu - m)\,\psi_2 -
i\frac{g}{\sqrt{N_c}}\left({\bar \psi}_1\,\gamma^5\,\phi_1\,\psi_2 +
{\bar \psi}_2\,\gamma^5\,\phi_1\,\psi_1\right) \\
%
&-& i\frac{g}{\sqrt{N_c}}\left({\bar \psi}_1\,\gamma^5\,\phi_2\,\psi_1 -
{\bar \psi}_2\,\gamma^5\,\phi_2\,\psi_2\right) +
\frac{g^2}{N_c}\,tr[\phi_1,\phi_2]^2\,,
\nonumber\end{aligned}$$ where we have defined the coupling constant of the model $g^2= N_c\,{\tilde g}^2/L^2$. We expect [@Coleman1] the infrared mass generation for the two-dimensional scalar gluons $\phi_i$. To estimate the masses of scalar gluons $\phi_i$ we use the Schwinger-Dyson equations as self-consistency conditions, we get: $$M^2=\frac{2 N_c {\tilde g}^2}{L^2} \int^\Lambda \!\!
\frac{d^2p}{(2\pi)^2} \,\frac{1}{p^2+M^2} \,
= \frac{N_c {\tilde g}^2\,\Lambda^2}{8\pi^3}\,\,
\log\frac{\Lambda^2+M^2}{M^2}\,, \label{glumass}$$
![Inhomogeneous Bethe-Salpeter equation for quark-antiquark scattering amplitude $T_5$.[]{data-label="F1"}](Diagramas3b.eps){width="90.00000%"}
thus $M^2$ brings an infrared cutoff as expected. We notice that the gluon mass remains finite in the large-$N_c$ limit if the QCD coupling constant decreases as $1/N_c$ in line with the perturbative law of $4D$ QCD. We adopt the approximation $M \ll \Lambda\simeq 1/R$ to protect the low-energy sector of the model and consider the momenta $|p_{0,1}| \sim M$. Thereby we retain only leading terms in the expansion in $p^2/\Lambda^2$ and $M^2/\Lambda^2$, and also neglect the effects of the heavy K-K modes in the low-energy Wilson action. We observe that the limit $M \ll \Lambda$ supports consistently both the fast decoupling of the heavy K-K modes and moderate decoupling of scalar gluons [@Alfaro:2003yy], the latter giving an effective four-fermion interaction different from [@Burkardt]. Also allow us to define the “heavy-scalar” expansion parameter $A=g^2/(2\pi\,M^2)=1/log \frac{\Lambda^2}{M^2} \ll 1$. Now we proceed to the study of bound states of quark-antiquark. In our reduction we have four possible combinations of quark bilinears to describe these states with valence quarks: $(\psi_1\,{\bar \psi}_1),(\psi_1\,{\bar \psi}_2),$ $
(\psi_2\,{\bar \psi}_1),(\psi_2\,{\bar \psi}_2 )$. We need to compute the full quark-antiquark scattering amplitude $T$ in the different channels. As an example we are going to show the computing of $T_5$ which correspond to the scattering ($q_1 + \bar{q}_2 \longrightarrow q_1 + \bar{q}_2$). It satisfies, the equation given graphically in , in the large $N_c$ limit and in ladder exchange approximation (non-ladder contribution are estimated to be of higher order in the $A$ expansion). Notice that in the equation for $T_5$ the amplitude $T_8$ appears, which correspond to the process $q_2 + \bar{q}_1 \longrightarrow q_1 + \bar{q}_2$. This means that the equations for $T_5$ and $T_8$ are coupled. In Figure \[F1\] all internal fermion lines correspond to dressed quark propagators which were determined in [@Alfaro:2003yy] by using the large $N_c$ limit and the one-boson exchange approximation. There are two kind of solutions for the dressed quark propagators. In one solution, the perturbative one, we have tachyonic quarks in the chiral limit as in $QCD_2$ and our model could be interpreted as a perturbation from the result of chiral QCD in $2D$. But we are not allowed to consider that possibility because the spectrum for the lowest $q {\bar q}$ bound states becomes imaginary if one takes into account the scalar field exchange. The second solution, the non-perturbative one, supports non-tachyonic quarks with masses going to zero, in the chiral limit and also yield real masses for the $q {\bar q}$ bound states.
![$q\bar{q}$ scattering $T_5$ in the color-singlet channel.[]{data-label="F2"}](Diagramas4.eps){width="90.00000%"}
Having found the full quark propagator we proceed to solve the inhomogeneous Bethe-Salpeter equation for $T_5$ and $T_8$. We obtain for $T_5$ : $${\displaystyle T_5}\!
{\textstyle \stackrel{\alpha \beta, \gamma \delta ~~~~~~} {(q,q';p)}}
=
-i\frac{g^2}{N}\,\frac{\gamma^{\alpha \delta}_-\,\gamma^{\beta
\gamma}_-}{(q-q')^2_-} - i\frac{g^2}{N}\,\frac{\sigma^{\alpha
\gamma}_3\,\sigma^{\beta \delta}_3}{[(q-q')^2 - M^2+i\epsilon]} \,+$$ $$+\, {\displaystyle \sum_j}\,\frac{i}{\big(p^2-\bar{m}^2_j + i\epsilon \big)}\,
\Theta^{\alpha\gamma}_j(q;p)\,\Theta^{\beta\delta}_j(q';p)
+\, {\displaystyle \sum_k}\,\frac{i}{\big(p^2-\tilde{m}^2_k +i\epsilon \big)}\,
\Lambda^{\alpha\gamma}_k(q;p)\, \Lambda^{\beta\delta}_k(q';p)\, .$$
There are no continuum states in the quark-antiquark amplitude– only bound states at $p^2=\bar{m}^2$ and $p^2=\tilde{m}^2$, whose residue yield the bound state wave functions $\Theta(p,r)$ and $\Lambda(p,r)$. These bound states have a direct interpretation in term of Dirac Bilinears of the theory in $4D$ [@Alfaro:2003yy]. In particular, the pseudoscalar $4D$ states are related with $\Lambda_k(r;p)$ which could be interpreted as the vertex: (quark)-(antiquark)-($4D$ pseudoscalar meson). The masses of the bound states and the vertex functions are fixed by the solution of the homogeneous Bethe-Salpeter equation which, in our model, yield to a eigenvalue problem, generalization of the integral equation found by ’t Hooft [@thooft1] in $QCD_2$. For example the following eigenvalue integral equation determine the mass spectrum of the pseudoscalar bound states (in the chiral limit): $$\label{n1} \tilde{m}^2 \, \phi(x) = \, -\frac{g^2}{\pi}\,
\int^{1}_{0} \! dy \, \frac{{\bf P}}{(x - y)^{2}}\,\, \phi(y) -\,
A\,\frac{\Sigma^{2}_{0}}{x^{1-\beta}} \int^{1}_{0} \! dy \,
\frac{\phi(y)}{(1 - y)^{1-\beta}}$$ $$- A\,\frac{\Sigma^{2}_{0}}{(1 - x)^{1-\beta}}
\int^{1}_{0} \! dy \, \frac{\phi(y)}{y^{1-\beta}} +
A\,\frac{\Sigma^{2}_{0}}{[(1 - x)\,x]^{1-\beta}} \int^{1}_{0} \!
dy \, \phi(y) + A\,\Sigma^{2}_{0}\, \int^{1}_{0} \! dy \,
\frac{\phi(y)}{[(1 - y)\,y]^{1-\beta}} ,$$ where $\beta = A/2$. To explore solutions to Eq.(\[n1\]) we examine small $A$. Evidently Eq.(\[n1\]) does not mix even and odd functions with respect to the symmetry $x \Longleftrightarrow
1 - x$. On the other hand the ground state should be an even function. When inspecting the wave function end-point asymptotics from the integral equation (\[n1\]) one derives the following even function as a ground state solution for $A \rightarrow 0$ limit: $\phi_0(x)=(4x[1-x])^{\frac{A}{2}}-\frac{1}{\pi}\,(4x[1-x])^{\frac{1}{2}}$. This is basically a non-perturbative result that differs from ’t Hooft solution in the $A \rightarrow 0$ limit, giving $p^2=0$, as we would expect from spontaneous chiral symmetry breaking in $4D$. For the other massive states we are unable to find analytic solutions, as happens with equation, but we could estimate them working with the Hamiltonian matrix elements $\tilde{m}^2(\phi,\phi)=(\phi,H\phi)$ and using the regular perturbation theory, starting from solutions ($A=0$). In Table \[1\] we show lowest values of mass spectra for scalar and pseudoscalar states.
Discussions {#discussions .unnumbered}
===========
Quantum Chromodynamics at low energies has been decomposed by means of dimensional reduction from $4D$ to $2D$ and a low energy effective model in $2D$ has been derived. In this model we did an explicit analysis of meson bound states by solving the inhomogeneous and the homogeneous Bethe-Salpeter equations, in the large $N_c$ limit and in the ladder exchange approximation. We found that the $2D$ model has four types of bound states which can be classified by their properties under Lorentz transformations inherited from $4D$. The $4D$ pseudoscalar and scalar sectors of the theory were analyzed and in the quiral limit a massless solution for the pseudoscalar ground state was found. We interpreted this solution as the “pion” of the model.\
Also, in solving the Bethe-Salpeter equations we found a solution to the full quark propagators yielding non-tachyonic dynamical quark masses, in contrast to what happen in $QCD_2$.
-------------- --------
Pseudoscalar Scalar
0 974
1286 1531
1741 1929
2166 2356
-------------- --------
Table \[1\]: Some values for the masses of bound states in $[MeV]$,\
where we have taken $\frac{g}{\sqrt{\pi}}=267 [MeV]$ and $A=0.22$.
**Acknowledgments:** P.L. thanks the organizers of the V-SILAFAE. The work of A.A. is partially supported by RFBR Grant and the Program “Universities of Russia: Basic Research”. The work of P.L. is supported by a Conicyt Ph. D. fellowship (Beca Apoyo Tesis Doctoral). The work of J.A. is partially supported by Fondecyt \# 1010967.
[0]{} G. ’t Hooft, Nucl. Phys. [**B75**]{} (1974) 461.
F. Antonuccio and S. Dalley Nucl. Phys. [**B461**]{} (1996) 275; H.-C. Pauli and S.J. Brodsky, Phys. Rev. [**D32**]{} (1985) 1993 and 2001.
F. Antonuccio and S. Dalley, Phys. Lett. [**B376**]{} (1996) 154.
J. Alfaro, A. A. Andrianov and P. Labra[ñ]{}a, JHEP [**0407**]{} (2004) 067.
S. Coleman, Comm. Math. Phys. [**31**]{} (1973) 259.
M. Burkardt, Phys. Rev. [**D56**]{} (1997) 7105.
[^1]: alk given at 5th Latin American Symposium on High Energy Physics , Lima Per[ú]{},
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We utilized the Very Large Array to make multifrequency polarization measurements of 20 radio sources viewed through the IC 1805 HII region and “Superbubble”. The measurements at frequencies between 4.33 and 7.76 GHz yield Faraday rotation measures (RMs) along 27 lines of sight to these sources. The RMs are used to probe the plasma structure of the IC 1805 HII region and to test the degree to which the Galactic magnetic field is heavily modified (amplified) by the dynamics of the HII region. We find that IC 1805 constitutes a “Faraday rotation anomaly”, or a region of increased RM relative to the general Galactic background value. The $|$RM$|$ due to the nebula is commonly 600 – 800 rad m$^{-2}$. However, the observed RMs are not as large as predicted by simplified analytic models that include substantial amplification of the Galactic magnetic field within the shell. The magnitudes of the observed RMs are consistent with shells in which the Galactic field is unmodified, or increased by a modest factor, such as due to magnetic flux conservation. We also find that with one exception, the sign of the RM is that expected for the polarity of the Galactic field in this direction. Finally, our results show intriguing indications that some of the largest values of $|$RM$|$ occur for lines of sight that pass outside the fully ionized shell of the IC 1805 HII region but pass through the Photodissociation Region associated with IC 1805.'
author:
- 'Allison H. Costa and Steven R. Spangler'
bibliography:
- 'FRbib.bib'
- 'LBVbib.bib'
- 'compbib.bib'
- 'FollowupBib.bib'
title: A Faraday Rotation Study of the Stellar Bubble and Region Associated with the W4 Complex
---
Introduction\[intro\]
=====================
Young massive stars in OB associations photoionize the surrounding gas, creating an region, and their powerful stellar winds can inflate a bubble around the star cluster. Magnetic fields are important to the dynamics of these structures [@Tomisaka:1990; @Ferriere:1991; @Vallee:1993; @Tomisaka:1998; @Haverkorn:2004; @Sun:2008; @Stil:2009], and they can elongate the cavity preferentially in the direction of the magnetic field and thicken the shell perpendicular to the field [@Ferriere:1991; @deAvillez:2005; @Stil:2009], causing deviations from the classical structure of the @Weaver:1977 wind-blown bubble. Knowledge of the magnitude and direction of the magnetic field within stellar bubbles and regions is important for simulations and for understanding how the magnetic field interacts with and modifies these structures.
In previous work (i.e., @Savage:2013 and @Costa:2016), we investigated whether the Galactic magnetic field is amplified in the shell of the Rosette Nebula, an region and stellar bubble associated with NGC 2244 ($\ell$ = 206.5, $b$ = –2.1). Other similar work investigating magnetic fields near massive star clusters has been done by @Harvey:2011 and @Purcell:2015. In this work, we continue our investigation of how regions and stellar bubbles modify the ambient Galactic magnetic field by considering another example of a young star cluster and an region that appears to be formed into a shell by the effect of stellar winds.
Faraday Rotation and Magnetic Fields in the Interstellar Medium
---------------------------------------------------------------
Faraday rotation measurements probe the line of sight (LOS) component of the magnetic field in ionized parts of the interstellar medium (ISM), provided there is an independent estimate of the electron density. Faraday rotation is the rotation in the plane of polarization of a wave as it passes through magnetized plasma and is described by the equation $$\chi=\chi_{0}+\left[\left(\frac{e^{3}}{2\pi m_{e}^{2}c^{4}}\right)\int{n_e\ \mathbf{B}\cdot \textrm{d}\mathbf{s}}\right]\lambda^{2},
\label{eq:rmorg}$$ where $\chi$ is the polarization position angle, $\chi_0$ is the intrinsic polarization position angle, the quantities in the parentheses are the usual standard physical constants in cgs units, *n$_{\textrm{e}}$* is the electron density, **B** is the vector magnetic field, d**s** is the incremental path length interval along the LOS, and $\lambda$ is the wavelength. We define the terms in the square bracket as the rotation measure, RM, and we can express the RM in mixed but convenient interstellar units as $$\textnormal{RM}=0.81\intn_{e} \ (\text{cm$^{-3}$}) \ \mathbf{B} \ (\mu\text{G})\cdot \textrm{d}\mathbf{s} \text{ (pc) rad m$^{-2}$.}
\label{eq:rmprat}$$
The Region and Stellar Bubble Associated with the W4 Complex\[sec:structure\]
-----------------------------------------------------------------------------
The region and stellar bubble of interest for the present study is IC 1805, which is located in the Perseus Arm. The star cluster responsible for the region and stellar bubble is OCl 352, which is a young cluster (1–3 Myr) [@Basu:1999]. OCl 352 has 60 OB stars [@Shi:1999]. Three of these are the O stars HD 15570, HD 15558, and HD 15629, and they have mass loss rates between 10$^{-6}$ and 10$^{-5}$ [@Massey:1995] and terminal wind velocities of 2200 – 3000 [@Garmany:1988; @Groenewegen:1989; @Bouret:2012]. We adopt the nominal center of the star cluster to be R.A.(J2000) = 02$^h$ 23$^m$ 42$^s$, decl.(J2000) = +6127$'$ 0$''$ ($\ell$ = 134.73, *b* = +0.92) [@Guetter:1989] and a distance of 2.2 kpc to IC 1805 to conform with previous studies of the region (e.g., @Normandeau:1996 [@Dennison:1997; @Reynolds:2001; @Terebey:2003; @Gao:2015]). In the literature, other distance values include: 2.35 kpc [@Massey:1995; @Basu:1999; @West:2007; @Lagrois:2012], 2 kpc [@Dickel:1980], 2.04 kpc [@Feigelson:2013; @Townsley:2014], and 2.4 $\pm$ 0.1 kpc [@Guetter:1989].
We refer to the region between –0.2 $<$ *b* $<$ 2 as IC 1805. This structure is also known as the Heart Nebula for its appearance at optical wavelengths. We differentiate this region from the northern latitudes that constitute the W4 Superbubble [@Normandeau:1996; @West:2007; @Gao:2015], and we use the nomenclature of W4 to describe the entire region, which includes IC 1805 and the W4 Superbubble. Below we summarize the structure of IC 1805 and Figure \[fig:cartoon\] is a cartoon diagram of the structure described here.
- *South.* On the southern portion of IC 1805, there is a loop structure of ionized material at 134$<$ $\ell$ $<$ 136, *b* $<$ 1, which we call the southern loop. @Terebey:2003 find that at far infrared and radio wavelengths, the shell structure is well defined and ionization bounded, since the ionized gas lies interior to the dust shell. However, they also find that there is warm dust that extends past the southern loop and a faint ionized halo (see their Figure 6). @Terebey:2003 argue that the shell is patchy and inhomogeneous in density, which allows ionizing photons to escape. @Gray:1999 discuss extended emission surrounding IC 1805 and suggest that it may be evidence of an extended region [@Anantharamaiah:1985]. Also surrounding IC 1805 are patchy regions of [@Braunsfurth:1983; @Hasegawa:1983; @Sato:1990] and CO [@Heyer:1998; @Lagrois:2009].
@Terebey:2003 model the structure of the southern loop using radio continuum data. They assume a spherical shell and place OCl 352 at the top edge of the bubble instead of at the center to accommodate spherical symmetry (see their Figures 4 and 5). The center of their shell model is at ($\ell$, $b$) = (135.02, 0.42). They find an inner radius of 30 arcmin (19 pc) and a shell thickness of 10 arcmin (6 pc) and 2.5 arcmin (2 pc) for a thick and thin shell model, respectively. @Terebey:2003 report electron densities of 10 and 20 for the thick and thin shell models, respectively (see Section 3.5 and Table 3 of @Terebey:2003). While we utilize and discuss these models in the following sections, the center position of the shell in @Terebey:2003 was selected to fit the ionized shell, and as such, the shell parameters should only be used to describe the bottom of IC 1805. For latitudes near the star cluster, the model fails, as the star cluster is at the top edge of the bubble instead of at the center.
- *East.* On the eastern edge of IC 1805 ($\ell$ $>$ 134.6, *b* $<$ 0.9), @Terebey:2003 find that warm dust extends outside the loop boundary and suggest that if the warm dust is associated with the ionized gas, then the bubble has blown out on the eastern side of IC 1805. At the Galactic latitude equal to the star cluster, the ionized gas appears to be pinched [@Basu:1999], which is usually caused by higher densities. There is a clump of CO emission in the vicinity of the eastern pinch at ($\ell$, $b$) = (135.2, 1.0) [@Lagrois:2009], and there is emission on the eastern edge at ($\ell$, $b$) $\geq$ (136, 0.5) (see Figure 1 of @Sato:1990).
- *West.* On the western edge of IC 1805 is the W3 molecular cloud and the W3 complex, which hosts a number of compact regions and young stellar objects (see @Bik:2012 and their Figure 1). @Dickel:1980 modeled the structure of W3, which is thought to be slightly in front of W4, and they argue that the advancement of the IC 1805 ionization front and shock front into the W3 molecular cloud may have triggered star formation. @Moore:2007 similarly conclude that the W3 molecular cloud has been compressed on one side by the expansion of IC 1805. While infrared sources nestled between the western edge of IC 1805 and eastern edge of the W3 molecular cloud are thought to be the product of this interaction , W3 Main, W3 (OH), and W3 North are thought to be sites of triggered star formation from IC 1795, which is part of W3 as well and not from the expansions of the ionization front [@Nakano:2017; @Jose:2016; @Kiminki:2015]. There is therefore uncertainty regarding a physical connection between W3 and IC 1805.
- *North.* North of OCl 352, the bubble opens up into what is called the W4 Superbubble [@Normandeau:1997; @Dennison:1997; @West:2007; @Gao:2015], which is a sealed “egg-shaped” structure that extends up to *b* $\sim$ 7 [@Dennison:1997; @West:2007]. At the latitude of the star cluster, @Lagrois:2009 estimate the distance between the eastern and western shell to be $\sim$ 1.2 (46 pc) that increases in size up to 1.6 (61 pc) at *b* = 1.8 (see Figure 11 of @Lagrois:2009). At higher latitudes, @Dennison:1997 model the thickness of the shell to be between 10–20 pc (16 – 31 arcminutes) from H$\alpha$ observations.
The “v”-shaped feature seen in Figure \[fig:w4\] at ($\ell$, $b$) $\sim$ (134.8, 1.35) is prominent in the ionized emission, and @Heyer:1996 report a cometary-shaped molecular cloud near ($\ell$, $b$) $\sim$ (134.8, 1.35). The alignment of the cometary cloud, as it is pointed towards IC 1805, suggests that the UV photons from the star cluster are responsible for the “v” shaped feature in the ionized emission on the side closest to the star cluster [@Dennison:1997; @Taylor:1999]. @Lagrois:2009 argue, from radial velocity measurements, that the cloud is located on the far side of the bubble wall, and while it may appear to be a cap to the bubble connecting to the southern loop, it is simply a projection effect. As such, the ridge of ionized material directly north of OCl 352 is not the outer radius of the shell but is part of the rear bubble wall.
- *PDR.* The and molecular emission near the southern ($\ell$ $<$ 0.9) portions of IC 1805 suggest that a [Photodissociation Region (PDR)]{} has formed exterior to the region. PDRs are the transition layer between the fully ionized region and molecular material, where far UV photons can propagate out and photodissociate molecules. We discuss the importance and observational evidence of a PDR in Section \[sec:pdr\].
There is an extensive literature on the W4 region and its relationship to W3, dealing with the morphology [@Dickel:1980; @Dickel:1980b; @Braunsfurth:1983; @Normandeau:1996; @Dennison:1997; @Heyer:1998; @Taylor:1999; @Basu:1999; @Terebey:2003; @Lagrois:2009; @Lagrois:2009b; @Stil:2009] and star formation history [@Carpenter:2000; @Oey:2005]. In the following paragraphs, we summarize those results from the literature that are most relevant to our polarimetric study and inferences on magnetic fields in this region.
Measurements of the total intensity and polarization of the Galactic nonthermal emission in the vicinity of regions are of interest because the regions and environs act as a Faraday-rotating screen inserted between the Galactic emission behind the region and that in front. Few radio polarimetric studies exist in the literature to date of the IC 1805 stellar bubble. @Gray:1999 present their polarimetric results of the W3/W4 region at 1420 MHz with the Dominion Radio Astrophysical Observatory (DRAO) Synthesis Telescope. They find zones of strong depolarization near the regions, particularly in the south, where there is a halo of extended emission around IC 1805. They conclude that RM values on order 10$^3$ and spatial RM gradients must exist to explain the depolarization near the region. More recently, @Hill:2017 present results of their polarimetric study of the Fan region ($\ell$ $\sim$ 130, –5$\leq$ *b* $\leq$ +10), which is a large structure in the Perseus arm that includes W3/W4. While the focus of their study was not on W4 specifically, they find similar results to @Gray:1999 in that there is sufficient Faraday rotation to cause beam depolarization in the regions of extended emission. In the W4 Superbubble, @West:2007 determined the LOS magnetic field strength by estimating depolarization effects along adjacent lines of sight. Using estimates of the shell thickness and the electron density from @Dennison:1997, @West:2007 estimate $\sim$ 3.4 – 9.1 $\mu$G for lines of sight at *b* $>$ 5. @Gao:2015 also report estimates in the W4 Superbubble by assuming a passive Faraday screen model [@Sun:2007] and measuring the polarization angle for lines of sight interior and exterior to the screen. For the western shell ($\ell$ $\sim$ 132.5, 4$<$ *b* $<$ 6) and the eastern shell ($\ell$ $\sim$ 136, 6$<$ *b* $<$ 7.5) in the superbubble, @Gao:2015 report negative RMs between –70 and –300 in the western shell and positive RMs on order +55 in the eastern shell. @Gao:2015 conclude that the sign reversal is expected in the case of the Galactic magnetic field being lifted out of the plane by the expanding bubble. With H$\alpha$ estimates from @Dennison:1997 for the electron density and geometric arguments for the shell radii of the W4 Superbubble, @Gao:2015 estimate $|$$|$ $\sim$ 5 $\mu$G.
@Stil:2009 compare their magnetohydrodynamic simulations of superbubbles to the W4 Superbubble. In general, they find that the largest Faraday rotation occurs in a thin region around the cavity, and inside the cavity, it would be smaller. They also present two limiting cases for the orientation of the Galactic magnetic field with respect to the line of sight, and the consequences for the RMs through the shell. If the Galactic magnetic field is perpendicular to the observer’s line of sight, then the contributions to the RM from the front and rear bubble wall would be of equal but opposite magnitude, except for small asymmetries which would lead to low RMs ( 20 ) through the cavity. This requires the magnetic field to be bent by the bubble to have a non-zero line of sight component. If the Galactic magnetic field is parallel to the line of sight, then the RMs through the front and rear bubble wall reinforce each other, and there are high RMs for lines of sight through the shell. In this case, there are higher RMs ( 3 $\times$ 10$^3$ ) everywhere.
There are also studies of the magnetic field for W3. From Zeeman observations, @vanderWerf:1990 conclude that the has small-scale structures that can vary on order of 50 $\mu$G over $\sim$ 9 arcsec scales. @Roberts:1993 report values of the LOS magnetic field from Zeeman observations towards three resolved components of W3. The three components are near ($\ell$, $b$) $\sim$ (133.7, 1.21), with a maximum separation of 1.5 arcmin, and the LOS magnetic field is between –50 $\mu$G and +100 $\mu$G. @Balser:2016 observed carbon radio recombination line (RRL) widths to estimate the total magnetic field strength in the photodissociation region (see @Roshi:2007 for details). They report B$_{\textrm{tot}}$ = 140 – 320 $\mu$G near W3A (133.72, 1.22) and argue that for a random magnetic field, B$_{\textrm{tot}}$ = 2 $|$$|$, which would then be consistent with the @Roberts:1993 estimates of the . It should be noted that these magnetic field strengths are substantially larger than those inferred for the W4 Superbubble on the basis of polarimetry of the Galactic background (see text above).
In this paper, we present new Faraday rotation results for IC 1805 to investigate the role of the magnetic field in the region and stellar bubble. As in @Savage:2013 and @Costa:2016, we utilize an arguably simpler and more direct method of inferring the LOS component of the magnetic field in regions. This is the measurement of the Faraday rotation of nonthermal background sources (usually extragalactic radio sources) whose lines of sight pass through the region and its vicinity. In Section \[sec:obs\], we describe the instrumental configuration and observations, including source selection. Section \[sec:dataredux\] details the data reduction process, including the methods used to determine RM values. In Section \[sec:obsres\], we report the results of the RM analysis and discuss Faraday rotation through the W4 complex in Section \[sec:fr\]. We present models for the RM within the region and stellar bubble in Section \[sec:models\]. We discuss our observational results and their significance for the nature of IC 1805 in Section \[sec:results\] and compare the results of this study with our previous study of the Rosette nebula in Section \[sec:rosette\]. We discuss future research in Section \[sec:fut\], and present our conclusions and summary in Section \[sec:sum\].
Observations {#sec:obs}
============
Source Selection\[sec:sourceselect\]
------------------------------------
![Mosaic of IC 1805 from the Canadian Galactic Plane Survey at 1.42 GHz, with Galactic longitude and latitude axes. The lines of sight listed in Table \[tab:sources\] are the red and blue symbols, where positive RMs are blue and negative RMs are red. The green and purple symbols are RM values from @Taylor:2009 or @Brown:2003, where positive RMs are green and negative RMs are purple. We utilize the naming scheme from Table \[tab:sources\] for the RM values from the literature for ease of reference, but we omit the “W4-” prefix in this image for clarity. The size of the plotted symbols is proportional to the $|$RM$|$ value.[]{data-label="fig:w4"}](f2.pdf){width="90.00000%"}
Our criteria for source selection were identical to @Savage:2013 and @Costa:2016 in that we searched the National Radio Astronomy Observatory Very Large Array Sky Survey (NVSS, @Condon:1998) database for point sources within 1 of OCl 352 (the “I” sources) with a minimum flux density of 20 mJy. We also searched in an annulus centered on the star cluster with inner and outer radii of 1 and 2 for outer sources (“O”) to measure the background RM due to the general ISM. We identified 31 inner sources and 26 outer sources in the region. We then inspected the NVSS postage stamps to ensure that they were point sources at the resolution of the NVSS ( 45 arcseconds). We discarded sources that showed extended structure similar to Galactic sources. We selected 24 inner sources and 8 outer sources from this final list.
-------- ----------------- ----------------- ------------ ------------ ---------- ------------------------ -------- --
Source $\alpha$(J2000) $\delta$(J2000) $\emph{l}$ $\emph{b}$ $\xi$ S$_{4.33\textrm{GHz}}$ m
Name h m s $^o$ $'$ $''$ ($^o$) ($^o$) (arcmin) (mJy) ($\%$)
W4-I1 02 30 16.2 +62 09 37.9 134.19 1.47 46.0 77 4
W4-I2 02 38 34.2 +61 08 46.6 135.49 0.91 46.2 5 12
W4-I3 02 36 45.5 +60 55 48.8 135.38 0.63 42.8 82 3
W4-I4 02 27 59.8 +62 15 44.0 133.91 1.47 58.9 46 10
W4-I5 02 28 01.6 +62 02 16.7 133.99 1.26 48.4 — —
W4-I6 02 38 19.9 +61 08 03.5 135.47 0.89 44.8 12 11
W4-I7 02 27 33.8 +61 55 58.1 133.98 1.14 46.6 – —
W4-I8 02 38 10.1 +62 08 57.0 135.05 1.81 57.1 47 9
W4-I9 02 28 21.6 +61 28 36.5 134.23 0.75 31.1 — —
W4-I10 02 29 13.0 +61 00 53.4 134.50 0.36 36.2 — —
W4-I11 02 25 15.2 +61 19 14.4 133.94 0.47 54.0 39 3
W4-I12 02 35 20.6 +62 16 02.3 134.70 1.79 52.5 77 2
W4-I13 02 28 25.1 +60 56 20.2 134.44 0.25 43.5 16 6
W4-I14 02 36 19.2 +61 44 05.5 135.01 1.35 31.4 2 16
W4-I15 02 33 36.1 +60 37 40.4 135.14 0.20 49.8 27 4
W4-I16 02 36 56.8 +61 57 58.6 134.99 1.59 43.3 35 0
W4-I17 02 34 08.8 +61 40 35.5 134.80 1.19 17.1 9 16
W4-I18 02 31 56.3 +61 25 50.9 134.65 0.87 5.6 24 2
W4-I19 02 40 31.7 +61 13 45.9 135.68 1.09 57.8 11 3
W4-I20 02 27 03.9 +61 52 24.9 133.94 1.07 47.6 457 0
W4-I21 02 30 44.5 +61 05 30.2 134.64 0.50 25.9 5 7
W4-I22 02 26 07.8 +61 56 43.7 133.82 1.09 55.4 — —
W4-I23 02 40 30.9 +61 47 10.1 135.45 1.59 59.3 3 0
W4-I24 02 37 45.1 +60 37 31.4 135.61 0.40 61.5 20 10
W4-O1 02 41 33.9 +61 26 29.5 135.70 1.33 63.5 377 0
W4-O2 02 35 37.8 +59 56 29.5 135.64 -0.33 93.0 107 0
W4-O4 02 44 57.7 +62 28 06.5 135.64 2.43 105.8 747 0
W4-O5 02 21 52.6 +60 10 03.2 133.96 -0.75 110.3 94 0
W4-O6 02 31 59.2 +62 50 34.1 134.12 2.18 83.7 120 4
W4-O7 02 43 35.6 +61 55 54.6 135.72 1.88 82.7 52 0
W4-O8 02 23 04.5 +60 58 19.6 133.82 0.05 75.2 40 0
W4-O10 02 20 26.2 +61 34 46.2 133.31 0.51 88.0 64 3
-------- ----------------- ----------------- ------------ ------------ ---------- ------------------------ -------- --
: List of Sources Observed
\[tab:sources\]
Angular distance between the line of sight and a line of sight through the center of the star cluster.
NVSS position. No source detected in the Stokes I map in any frequency bin.
High Mass X-Ray Binary .
The sources are listed in Table \[tab:sources\], where the first column lists the source name in our nomenclature. The second and third columns list the right ascension ($\alpha$) and declination ($\delta$) of the observed sources. The positions are determined with the [imfit]{} task in CASA, which fits a 2D Gaussian to the intensity distribution at 4.33 GHz. Columns four and five give the Galactic longitude ($\ell$), Galactic latitude (*b*), which is converted from $\alpha$ and $\delta$ using the Python *Astropy* package, and the angular separation from the center of the nebula ($\xi$) is given in column six. Column seven lists the flux density at 4.33 GHz calculated with [imfit]{}, and column eight gives the mean percent linear polarization (m = *P*/*I*) as measured across the eight 128 MHz maps and assuming a source. Figure \[fig:w4\] is a radio continuum mosaic from the Canadian Galactic Plane Survey (CGPS) [@Taylor:2003; @Landecker:2010] with the location of the sources, along with the names, indicated with filled circles.
VLA Observations\[sec:vlaobs\]
------------------------------
----------------------------------------------- ------------------------------
VLA Project Code 13A-035
Date of Observations 2013 July 10, 13, 16, and 17
Number of Scheduling Blocks 4
Duration of Scheduling Blocks (h) 4
Frequencies of Observation (GHz) 4.850; 7.250
Number of Frequency Channels per IF 512
Channel Width (MHz) 2
VLA array C
Restoring Beam (diameter) 481
Total Integration Time per Source 18–25 minutes
RMS Noise in Q and U Maps ($\mu$Jy/beam) 39
RMS Noise in RM Synthesis Maps ($\mu$Jy/beam) 23
----------------------------------------------- ------------------------------
: Log of Observations \[tab:logofobs\]
The observations had 1.024 GHz wide intermediate frequency bands (IFs) centered on the frequencies listed, each composed of eight 128 MHz wide subbands.
The “O” sources (see Table \[tab:sources\]) averaged 18 minutes, and the “I” sources, being weaker, were between 22–25 minutes.
This number represents the average rms noise level for all the Q and U maps.
Polarized sensitivity of the combined RM Synthesis maps.
We observed 32 radio sources with the NSF’s Karl G. Jansky Very Large Array (VLA)[^1] whose lines of sight pass through or near to the shell of the IC 1805 stellar bubble. Table \[tab:logofobs\] lists details of the observations. Traditionally, polarization observations require observing a polarization calibrator source frequently over the course of an observation to acquire at least 60 of parallactic angle coverage. This is done to determine the instrumental polarization (D-factors, leakage solutions). Since the completion of the upgraded VLA, shorter scheduling blocks, typically less than 4 hours in duration, have become a common mode of observation. It is difficult, if not impossible, with very short scheduling blocks to acquire enough parallactic angle coverage to measure the instrumental calibration with a polarized source. Another method of determining the instrumental polarization is to observe a single scan of an unpolarized source. This technique can be used with shorter scheduling blocks.
In this project we calibrated the instrumental polarization using both techniques. We used the source J0228+6721, observed over a wide range of parallactic angle, and also made a single scan of the unpolarized source 3C84. Use of the CASA task on the J0228+6721 data solved for the instrumental polarization, determined by the antenna-specific D factors [@Bignell:1982], which are complex, as well as the source polarization (*Q* and *U* fluxes). In the case of 3C84, solves only for the D factors. We find no significant deviations between these two calibration methods, indicating accurate values for the instrumental polarization parameters. 3C138 and 3C48 functioned as both flux density and polarization position angle calibrators. J0228+6721 was used to determine the complex gain of the antennas as a function of time as well to as serve as a check, as described above, for the D-factors. We observed the program sources for 5 minute intervals and interleaved the observations of J0228+6721. There was one observation of 3C138, 3C48, and 3C84 each. For our final data products, we utilized 3C84 as the primary leakage calibrator and 3C138 as the flux density and polarization position angle calibrator.
Data Reduction\[sec:dataredux\]
===============================
The data were reduced and imaged using the NRAO Common Astronomy Software Applications (CASA)[^2] version 4.5. The procedure for the data reduction as described in Section 3 of @Costa:2016 is identical to the procedure we employed in this study. The only difference for the current data set is that in the CASA task , we utilized *Briggs* weighting with the “robust” parameter set to 0.5, which adjusts the weighting to be slightly more *natural* than *uniform*. *Natural* weighting has the best signal/noise ratio at the expense of resolution, while *uniform* is the opposite. *Briggs* weighting allows for intermediate options. As in our previous work, we also implemented a cutoff in the (*u*, *v*) plane for distances $<$ 5000 wavelengths to remove foreground nebular emission.
Similar to @Costa:2016, we had two sets of data products after calibration and imaging. The first set of images consisted of radio maps (see Figures \[fig:I18\] and \[fig:I24\]) of each Stokes parameter, formed over a 128 MHz wide subband for each source. These images were inputs to the analysis (Section \[sec:chilam\]), and there were typically 14 individual maps for each source per Stokes parameter.
The second set of images consisted of maps of *I*, *Q*, and *U* in 4 MHz wide steps across the entire bandwidth using the mode “channel”, which averages two adjacent 2 MHz channels. Ideally, changes in and should only be due to Faraday rotation; however, the spectral index can affect and independently of the RM, which can be interpreted as depolarization. RM Synthesis does not, by default, account for the spectral index, so a correction must be applied prior to performing RM Synthesis (see Section 3 of @Brentjens:2005). We first determine the spectral index, $\alpha$, of each source from a least-squares fit to the log of the flux density, $S_\nu$, and the log of the frequency, $\nu$. We adopt the convention that $S_\nu$ $\nu^{-\alpha}$. We use the center frequency, $\nu_c$, of the band and the measured value of *Q* and *U* at each frequency, $\nu$, to find *Q$_o$* and *U$_o$* using the relationship $$Q = Q_o\left(\frac{\nu}{\nu_c}\right)^{-\alpha} \textrm{ and } \ U = U_o\left(\frac{\nu}{\nu_c}\right)^{-\alpha}.$$ The final images consisted of approximately 336 maps per source, per Stokes parameter, as inputs for the RM Synthesis analysis (Section \[sec:rmsyn\]).
Rotation Measure Analysis via a Least-Squares Fit to $\chi$ vs $\lambda^2$ \[sec:chilam\]
-----------------------------------------------------------------------------------------
The output of the CASA task produces images in Stokes *I, Q, U,* and *V*. From these images, we generated maps with the task of the linear polarized intensity *P*, $$P = \sqrt{Q^2+U^2}$$ and the polarization position angle $\chi$, $$\chi=\frac{1}{2}\tan^{-1}{\left(\frac{U}{Q}\right)}$$ for each source over a 128 MHz subband. Data that are below the threshold of 5$\sigma_{\textrm{Q}}$ are masked in the *P* and $\chi$ maps, where $\sigma_{\textrm{Q}}$ = $\sigma_{\textrm{U}}$ is the rms noise in the *Q* data. This threshold prevents noise in the *Q* and *U* data from generating false structure in the *P* and $\chi$ maps. Examples of images are shown in Figure \[fig:MAP\], which displays the total intensity, polarized intensity, and polarization position angle for sources W4-I18 and W4-I24. W4-I18 is an example of a point source, or slightly resolved source. Twelve of the sources in Table \[tab:sources\] were of this type and unresolved to the VLA in C array. Eight sources were like W4-I24, showing extended structure in the observations and potentially yielding RM values on more than one line of sight.
\
In the case of a single foreground magnetic-ionic medium responsible for the rotation of an incoming radio wave, the relation between $\chi$ and $\lambda^2$ is linear, and we calculate the RM through a least-squares fit of . To measure $\chi$, we select the pixel that corresponds to the highest value of *P* on the source in the 4338 MHz map, and we then measure $\chi$ at that location in each subsequent 128 MHz wide subband. Figure \[fig:newpol\] shows two examples of the least-squares fit to . The $\chi$ errors are $\sigma_{\chi} = \frac{\sigma_{Q}}{2P},$ (@Everett:2001, Equation 12).
Rotation Measure Synthesis \[sec:rmsyn\]
----------------------------------------
In additional to the least-squares fit to , we performed Rotation Measure Synthesis [@Brentjens:2005]. The inputs to RM Synthesis are images in Stokes *I*, *Q*, and *U* across the entire observed spectrum in 4 MHz spectral intervals. We refer the reader to Section 3.1.2 of @Costa:2016 for a detailed account of our procedure, which follows the implementation of RM Synthesis as developed by @Brentjens:2005. The goal of RM synthesis is to recover the Faraday dispersion function $F(\phi)$. Here $\phi$, the Faraday depth, is a variable which is Fourier-conjugate to $\lambda^2$ (see @Costa:2016, Equations 3 and 4), and has units of . We also refer to $F(\phi)$ as the “Faraday spectrum”.
[p[2cm]{} p[4cm]{} p[9.5cm]{}]{} $\Delta \lambda^2$ & 3.2 $\times$ 10$^{-3}$ (m$^2$) & Total bandwidth.\
$\lambda^2_{min}$ & 1.5 $\times$ 10$^{-3}$ (m$^2$) & Shortest observed wavelength squared.\
$\delta \lambda^2$ & 4.8 $\times$ 10$^{-6}$ (m$^2$) & Width of a channel; Eq (35) @Brentjens:2005.\
$\delta \phi$ & 1072 () & FWHM of RMSF; Eq (61) @Brentjens:2005.\
\
& & Sensitivity to extended Faraday structures; Eq (62) @Brentjens:2005.\
\
& & Maximum detectable Faraday depth before bandwidth depolarization; Eq (63) @Brentjens:2005.\
\
This bandwidth includes the frequencies not observed that lie between our two IFs. They are set to 0 via the weighting function, W($\lambda^2$).
Since flagging for RFI and bad antennas were done individually for each scheduling block, the FWHM of the RMSF can vary slightly from source to source. However, these slight variations are not significant in our interpretation of the RM values report in this paper.
is recovered via an algothrim [@Heald:2009; @Bell:2012], and we applied a 7$\sigma$ cutoff, which is above the amplitude at which peaks due to noise are likely to arise [@Brentjens:2005; @Macquart:2012; @Anderson:2015]. The <span style="font-variant:small-caps;">rmsynthesis</span> algorithm initially searched for peaks in the Faraday spectrum using a range of $\phi$ $\pm$ 10,000 at a resolution of 40 to determine if there were significant peaks at large values of $|\phi|$. Then, we performed a finer search at $\phi$ = $\pm$ 3000 at a resolution of 10 . The RM Synthesis parameters, such as the full-width-at-half-maximum (FWHM) of the rotation measure spread function (RMSF) and the maximum detectable Faraday depth, are given in Table \[tab:rmpar\].
As in @Costa:2016, we utilized an IDL code for the <span style="font-variant:small-caps;">rmsynthesis</span> and algorithms. The output of the IDL code is a data cube in Faraday depth space that is equal in range to the range of $\phi$ that was searched over in the <span style="font-variant:small-caps;">rmsynthesis</span> algorithm. The data cube contains, for example, 500 maps of the polarized intensity as a function of spatial coordinates and $\phi$, which ranges between $\pm$ 10,000 at intervals of 40 . Initially, we generated these maps for a 1024 $\times$ 1024 pixel image. We then used the Karma package [@Gooch:1995] tool <span style="font-variant:small-caps;">kvis</span> to review the maps to search for sources or source components away from the phase center that, while being too weak to detect in the 128 MHz maps, may be detectable in the RM Synthesis technique since it uses the entire bandwidth to determine the Faraday spectrum[^3]. However, no such sources were identified above the cutoff. From the 1024 $\times$ 1024 maps of the Faraday spectrum, we identified the *P$_{\textrm{max}}$* for the observed sources and extracted the Faraday spectrum at that location. Figure \[fig:psynmap\] shows an example of a *P$_{\textrm{max}}$* map that has been flattened along the $\phi$ axis, i.e., the gray scale in the image represents the full range of $\phi$. From this map, it is easy to identify the spatial location of *P$_{\textrm{max}}$* for the source, which agrees with the location of the peak linear polarized intensity in the analysis. We obtained this same result in @Costa:2016 for the Rosette Nebula.
To determine the RM, we fit a 2 degree polynomial to the Faraday spectrum at each pixel in the 1024 x 1024 image above the 7$\sigma$ cutoff. The gray scale in Figure \[fig:rmsynmap\] shows the RM value from the fit to each pixel. The image is zoomed and centered on the source. While we can mathematically determine the RM at each pixel, the sources are not resolved, so we only select the RM at the spatial location of *P$_{\textrm{max}}$*. Figure \[fig:RMSYN\] plots the Faraday spectrum and components for W4-I18, and Figure \[fig:RMSF\] shows the RMSF.
@Anderson:2015 describe two cases for the behavior of the Faraday spectrum. A source is considered when is non-zero at only one value of $\phi$, *Q* and *U* as a function of vary sinusoidally with equal amplitude, and is constant. The case has the physical meaning of a uniform Faraday screen in the foreground that is responsible for the Faraday rotation, and $\chi$ is linearly dependent on . If a source is , then is a delta function at a Faraday depth equal to the RM. The second behavior @Anderson:2015 describe for the Faraday spectrum is a source, which is any spectrum that deviates from the criteria set for the case. A spectrum can be the result of depolarization in form of beam depolarization, internal Faraday dispersion, multiple interfering Faraday rotating components, etc. [@Sokoloff:1998].
Observational Results\[sec:obsres\]
===================================
Measurements of Radio Sources Viewed Through the W4 Complex\[sec:obs2\]
-----------------------------------------------------------------------
![ Plot of RM values derived from the analysis vs the RM Synthesis analysis. The blue markers are the RM from primary component and the red are the secondary component. The straight line represents perfect agreement between the two sets of measurements.[]{data-label="fig:comprm"}](f7.pdf){width="60.00000%"}
We measured 27 RM values for 20 lines of sight, including secondary components, through or near to IC 1805. In Table \[tab:results\], the first column lists the source name using our naming scheme, and column two gives the component, if the source had multiple resolved components for which we could determine a RM value. Columns three and four list the RM value from the least-squares method and the reduced chi-squared value, respectively. Column five lists the RM value determined from the RM Synthesis technique and the associated error (Equation 7 of @Mao:2010).
Figure \[fig:comprm\] shows the agreement between the two techniques for determining the RM. As in @Costa:2016, we find good agreement between the two techniques, and the good agreement between the results using the two techniques gives us confidence in our RM measurements.
-------- --- --------------- ------------ --------------- ------- -------
RM Reduced RM $\xi$ $\xi$
() $\chi^{2}$ () (pc) (pc)
W4-I1 a -277 $\pm$ 1 29 -258 $\pm$ 3 29 53
a -1042 $\pm$ 7 1.5 -930 $\pm$ 30
b -935 $\pm$ 6 1.5 -954 $\pm$ 11
W4-I3 a -876 $\pm$ 2 1.9 -878 $\pm$ 8 27 16
a -139 $\pm$ 3 1.9 -153 $\pm$ 12
b -91 $\pm$ 6 2 -68 $\pm$ 15
W4-I6 a -990 $\pm$ 8 1.3 -961 $\pm$ 23 29 25
W4-I8 a -276 $\pm$ 2 4.4 -337 $\pm$ 8 37 54
W4-I11 a -377 $\pm$ 8 12 -141 $\pm$ 18 35 41
W4-I12 a -315 $\pm$ 4 2.8 -306 $\pm$ 10 34 54
a -777 $\pm$ 8 1.2 -801 $\pm$ 24
b -701 $\pm$ 28 0.5 -772 $\pm$ 66
W4-I14 a -678 $\pm$ 27 0.6 -666 $\pm$ 66 20 36
W4-I15 a -157 $\pm$ 9 0.8 -124 $\pm$ 14 32 10
a -492 $\pm$ 8 1.6 -440 $\pm$ 40
b -509 $\pm$ 15 0.6 -464 $\pm$ 40
W4-I18 a +514 $\pm$ 12 1.1 +501 $\pm$ 33 4 22
W4-I19 a -407 $\pm$ 14 0.3 -431 $\pm$ 36 37 34
a -53 $\pm$ 26 1.4 -167 $\pm$ 67
b -98 $\pm$ 28 0.5 -79 $\pm$ 62
c -173 $\pm$ 34 0.5 -232 $\pm$ 70
a -658 $\pm$ 5 1.4 -678 $\pm$ 14
b -675 $\pm$ 12 0.4 -716 $\pm$ 30
W4-O4 a -31 $\pm$ 12 24 -178 $\pm$ 18 68 81
W4-O6 a -95 $\pm$ 1 3.6 -96 $\pm$ 4 54 76
W4-O7 a -175 $\pm$ 24 0.8 -256 $\pm$ 56 53 62
W4-O10 a -379 $\pm$ 5 1.8 -343 $\pm$ 16 56 66
-------- --- --------------- ------------ --------------- ------- -------
: Faraday Rotation Measurement Values through the W4 Complex \[tab:results\]
RM value obtained from a least-squares linear fit to $\chi(\lambda^2)$. The errors are 1$\sigma$.
Reduced $\chi^{2}$ for the fit.
Effective RM derived from RM Synthesis.
Distance from center of OCl 352.
Distance from @Terebey:2003 center.
Report on Faraday Complexity and Unpolarized Lines of Sight\[sec:depol\]
------------------------------------------------------------------------
In the last paragraph of Section \[sec:rmsyn\], we discuss Faraday complexity. If a source is , then the RM is equal to a delta function in at the Faraday depth. If a source is , then the interpretation of the RM is not as straightforward. There is extensive literature (e.g. @Farnsworth:2011, @OSullivan:2012, @Anderson:2015, @Sun:2015, @Purcell:2015) to understand Faraday complexity. One indicator of a source is a decreasing fractional polarization, *p* = *P*/*I*, as a function of . The ways in which this can arise are discussed at the end of Section \[sec:rmsyn\]. Although depolarization does not necessarily lead to a net rotation of the source $\chi$, its presence indicates the potential for a $\chi$ rotation independent of the plasma medium through which the radio waves subsequently propagate. This could result in an error in our deduced RMs. Nine of the sources, W4-I1, -I3, -I11, -I15, I21b, -O4, -O6, -O7, and -O10 show a decreasing *p* with increasing .
A rough estimate of the potential position angle rotation associated with depolarization may be obtained using the analysis in @Cioffi:1980. These calculations assume that depolarization arises from Faraday rotation within the synchrotron radiation source, and we can estimate the effect of internal depolarization from the changes in fraction polarization. Given the fractional polarization at the shortest and longest wavelength, we obtain the corresponding polarization angle change from Figure 1 of @Cioffi:1980 and then calculate a RM due to internal depolarization. If the calculated RM due to depolarization (RM$_{depol}$) is larger than the observed RM, then the RM is potentially affected by depolarization. W4-I1, -O6, -O7, and -O10 show RM$_{depol}$ RM$_{obs}$, which indicates that internal depolarization could affect the observed RM. The observed RM of W4-I3 is 3 times larger than RM$_{depol}$, so it is not affected by internal depolarization.
Depolarization due to internal Faraday rotation makes predictions for the form of $\chi (\lambda^2)$ which would not have $\chi \propto \lambda^2$ [@Cioffi:1980]. For all of the sources mentioned above, we compared the observed behavior of to the predicted behavior (Equation 4b of @Cioffi:1980). Within the errors, only W4-O7 is consistent with the non-linear behavior of a RM affected by internal depolarization. We interpret this result as meaning that our deduced RM values for most of the sources are not significantly in error due to internal depolarization, and we consider the measurement of depolarization as providing a cautionary flag.
We also considered whether our measurements could have been affected by bandwidth depolarization or beam depolarization. Bandwidth depolarization occurs when the polarization angle varies over frequency averaged bins, $\Delta\nu$. For example in this study, we use values of $\chi$ in 128 MHz wide bins (Section \[sec:chilam\]) and 4 MHz (Section \[sec:rmsyn\]). For a center frequency of the lowest frequency bin, we use $\nu_c$ = 4466 MHz, and the relationship between the change in polarization position angle, $\Delta\chi$, is $$|\Delta\chi| = 2|\textrm{RM}| \; c^2 \; \frac{\Delta\nu}{\nu_c^3},
\label{eq:bandwidth}$$ where c is the speed of light. This formula shows that even for $|$RM$|$ = 10$^4$ , which is far larger than any RMs we measure, the Faraday rotation across the band is 0.41 radians. This is insufficient to cause substantial bandwidth depolarization. Beam depolarization occurs when there are small scale variations of the electron density or the magnetic field within a beam. It is unlikely that the RMs are affected by beam depolarization as the beam at 6 cm for the VLA in C array is 5 arcseconds. We interpret these RMs as a characteristic value due to the plasma medium (primarily the Galactic ISM) between the source and the observer. In the analysis that follows, we choose the RM values from the RM Synthesis method.
When the data were mapped and inspected, we found that a few sources that had passed our criteria for flux density and compactness to the VLA D array at L band (1.42 GHz) were completely unpolarized. W4-I16 and W4-O8 are not polarized at any frequency, and the RM Synthesis technique does not show significant ($>$ 7$\sigma$) peaks at any $\phi$. Three of the lines of sight, W4-I5, W4-I10, and W4-I22, have no source in the field. Despite appearing to be point sources in the NVSS postage stamps (see Section \[sec:obs\]), we do not observe a source at these locations, and they may have been clumpy foreground nebular emission that was filtered out during the imaging process.
Subsequent investigations determined that some of the selected sources were previously cataloged ultra compact regions associated with the W3 star formation region. These sources are W4-I7 (W3(OH)-C), W4-I9 (AFGL 333), and W4-I20 (W3(OH)-A) [@Feigelson:2008; @Navarete:2011; @Roman:2015]. The W4-I7 field has no source at the observed $\alpha$ and $\delta$, despite it being identified as W3(OH)-C. We do not observe a source at this location in any frequency bin. W3(OH)-A, however, is observed and is a point source in our maps at all frequencies. Similarly, W4-I9 is detected in each frequency bin and is an extended source. These sources are unpolarized and do not feature in our subsequent analysis.
A Unique Line of Sight Through the W4 Region: LSI +61303\[sec:HMXB\]
--------------------------------------------------------------------
W4-I19 has a spectrum which is inconsistent with an optically-thin extragalactic radio source. It is linearly polarized, and we measure RM = –431 $\pm$ 36 . Investigation of this source during the data analysis phase revealed that it is not an extragalactic source, although it passed our selection criteria for flux and compactness. W4-I19 is the high mass X-ray binary (HMXB) LSI +61303 [@Gregory:1979; @Bignami:1981], which is notable for being one of five known gamma ray binary systems [@Frail:1991]. This system has been extensively studied, and as a result, much is known about the nature of the compact object [@Massi:2004; @Massi:2004b; @Dubus:2006; @Paredes:2007; @Massi:2017], the stellar companion [@Casares:2005; @Dubus:2006; @Paredes:2007], orbital period [@Gregory:2002], radio structure [@Albert:2008], and radial velocity [@Gregory:1979; @Lestrade:1999].
The spatial location of LSI +61303 is important for understanding the RM we determined for this source. @Frail:1991 argue that since signatures of the Perseus arm shock are present in the absorption spectrum to LSI +61303 but not the post-shock gas from the Perseus arm, LSI +61303 must lie between the two features at a distance of 2.0 $\pm$ 0.2 kpc. They also report that they do not see absorption features due to the IC 1805 ionization front and shock front. The estimated distance to is consistent with distance estimates to OCl 352. The position relative to the nebula has consequences for the interpretation of the RM that we measure. The possibilities are:
1. LSI +61303 is in front of the stellar bubble and region, so it is exterior to a region modified by OCl 352. The RM is then an estimate of the foreground ISM between us and the nebula.
2. If LSI +61303 is at the same distance as IC 1805 or slightly behind (greater distance), then the RM is unique among our sources in that it is not affected by Faraday rotation from material in the outer Galaxy. The RM is then probing at least a part of the Faraday rotating material due to the nebula.
To further determine the position of with respect to IC 1805, we review the current state of knowledge on the subject from the literature. @Dhawan:2006 observed with the Very Long Baseline Array (VLBA) and report a proper motion of ($\mu_{\alpha}$, $\mu_{\delta}$) = (-0.30 $\pm$ 0.07, -0.26 $\pm$ 0.05) mas yr$^{-1}$. @Aragona:2009 report a radial velocity for of $V_{rad}$ = –41.4 $\pm$ 0.6 , which agrees with previous estimates by @Casares:2005. For OCl 352, @Dambis:2001 estimate the radial velocity to be –41 $\pm$ 3 , and more recent estimates by @Kharchenko:2005 ($V_{rad}$ = –47 $\pm$ 18 ) agree within the errors. Both and OCl 352 have similar radial velocities, and the proper motion estimates by @Dhawan:2006 indicate that is moving similarly on the plane of the sky to OCl 352, which has a proper motion of ($\mu_{\alpha}$, $\mu_{\delta}$) = (–1.0 $\pm$ 0.4, –0.9 $\pm$ 0.4) mas yr$^{-1}$ [@Dambis:2001].
From proper motion and radial velocity estimates, appears to be moving in relatively the same direction and speed as OCl 352. Using a distance of 2 kpc to and 2.2 kpc to OCl 352, the transverse velocities are 3 and 14 , respectively. If originally belonged to OCl 352, then it is unlikely that it is in front of IC 1805, given that both are moving at the same radial velocity. While appears to be outside the obvious shell structure of IC 1805, it is more likely that it is probing material modified by OCl 352. We discuss this possibility further in Section \[sec:pdr\]. If did not originate in OCl 352, then it is possible to still be in front of the nebula, despite the similar velocities. In such a case, the RM we obtained for this line of sight is due to the ISM between us and IC 1805. The RM value we find for is nearly 3 times larger than the background RM, which we discuss in Section \[sec:bkgrm\]. This would require a magneto-ionic medium between the observer and the nebula capable of producing 400 along this line of sight. As may be seen from Table \[tab:results\] and Figure \[fig:w4\], other lines of sight near IC 1805, but exterior to the shell, do not have as large of RM values (e.g. W4-O26, -O19, -O7, -I11). It therefore seems most probable that the RM for W-I19 () is dominated by plasma in W4. In summary, there is evidence in the literature that suggests may lie within a region modified by OCl 352, particularly if did indeed once belong to OCl 352. If this is the case, then the RM we find is unaffected by the ISM in the outer galaxy and is due to the material near IC 1805.
Results on Faraday Rotation Through the W4 Complex\[sec:fr\]
============================================================
The Rotation Measure Sky in the Direction of W4
-----------------------------------------------
@Whiting:2009, @Savage:2013, and @Costa:2016 compared observations to a model of the ionized shell in which the RM depended only on $\xi$, the impact parameter, or closest approach of a line of sight to the center of the shell. In anticipation of a similar analysis in this study, we show Figures \[fig:northb\] and \[fig:southb\], which plot the RM versus distance from the center of star cluster for the lines of sight through the W4 Superbubble (W4-I1, -I4, -I8, -I12, -I14, -I17, -O4, -O6, and -O7) and the ones through or close to the southern loop (W4-I2, -I3, -I6, -I11, -I13, -I15, -I18, -I19, -I21, -I24, -O10).
In Section \[sec:structure\], we discussed the morphology of the region around IC 1805 and made the distinction between the southern latitudes and the northern latitudes, so in the following sections, we address each region near IC 1805 separately.
The Galactic Background RM in the Direction of W4\[sec:bkgrm\]
--------------------------------------------------------------
In @Savage:2013, we determined the background RM in the vicinity of the Rosette Nebula ($\ell$ $\sim$ 206) by finding the median value of the RM for sources outside the obvious shell structure of the Rosette. Determining the background RM near IC 1805 is difficult, however, due to proximity of W3, the W3 molecular cloud, and the W4 Superbubble. Given the morphological difference between the northern and southern parts of IC 1805, we assume that sources south of OCl 352 (*b* $<$ 0.9) should be modeled independently of the northern sources, since the W4 Superbubble extends up to *b* $\sim$ 7 [@West:2007]. The lines of sight north of the star cluster are intersecting the W4 Superbubble and are not probing the RM due to the general ISM independent of IC 1805. Therefore, the only lines of sight that are potentially probing the RM in the vicinity of IC 1805 are those exterior to the shell structure of the southern loop.
-------- ----------------- ----------------- ---------------
Source $\alpha$(J2000) $\delta$(J2000) RM
Name h m s $^o$ $'$ $''$ (rad m$^2$)
W4-O3 02 35 43.0 +63 22 33.0 –138 $\pm$ 18
W4-O19 02 46 23.9 +61 33 19.9 –157 $\pm$ 15
W4-O26 02 42 32.3 +60 02 31.0 +61 $\pm$ 41
W4-O27 02 25 48.7 +59 53 52.0 –145 $\pm$ 22
-------- ----------------- ----------------- ---------------
: List of Sources with RM values from Catalogs\[tab:taylor\]
RM values from @Brown:2003 unless otherwise noted.
@Taylor:2009 give –75 $\pm$ 9 for this line of sight.
RM value from @Taylor:2009.
If we apply the thick shell model from @Terebey:2003 (see Section \[sec:structure\] for details), then the lines of sight with RM values exterior to the shell are W4-I2, -I11, and -O10. For the thin shell case, W4-I6, -I13 and -I24 are also exterior sources. The mean RM value for the background using these sources is –554 and –670 for the thick or thin shell, respectively. In Table \[tab:taylor\], we list RM values from the literature for lines of sight near IC 1805 that we include in our estimate of the background RM. The mean RM value for these sources (excluding W4-O3 for being in the superbubble) is –80 . The sources W4-I2, -I6, -I13, and -I24 are seemingly outside the obvious ionized shell structure; however, they are also the lines of sight for which we measure some of the highest RM values. This is a surprising result, and one we did not observe in the case of the Rosette Nebula. It strongly suggests that the lines of sight to W4-I2, -I6, -I13, and -I24 have RMs that are dominated by the W4 complex, despite the fact that they are outside the obvious ionized shell of IC 1805. We discuss this further in the next section. For the present discussion, we exclude these sources from the estimate of the background. Using W4-I11, -O10, -O19, -O26, and -O27, we find a mean value for the background RM due to the ISM of –145 . While this value is similar in magnitude to the value of the background RM we found in our studies on the Rosette Nebula, we have significantly fewer lines of sight, and only two of the lines of sight were observed in this study. Due to a low number of lines of sight exterior to IC 1805, we utilize the model of a Galactic magnetic field by @vanEck:2011 to estimate the background RM due to the ISM. From their Figure 6, they find the Galactic magnetic field is best modeled by an almost purely azimuthal, clockwise field. @vanEck:2011 use their model to predict the RM values in the Galaxy, and in the vicinity of IC 1805, their model predicts RMs of order –100 . Using this as an estimate of the background RM, we find an excess RM due to IC 1805 of +600 to –860 .
High Faraday Rotation Through Photodissociation Regions\[sec:pdr\]
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The lines of sight with the highest RM values, W4-I2, -I6, and -I24, appear to be outside the obvious shell of the southern loop. These sources are very near to the bright ionized shell. @Terebey:2003 and @Gray:1999 discuss a halo of ionized gas that surrounds IC 1805, which may be causing the high RM values. @Gray:1999 speculate that the diffuse extended structure is an extended envelope as suggested by @Anantharamaiah:1985. Another possibility is that these high RMs arise in the PDR surrounding the IC 1805 region. PDRs are the regions between ionized gas, which is fully ionized by photons with *h$\nu$* $>$ 13.6 eV, and neutral or molecular material. PDRs can be partially ionized and heated by far-ultraviolet photons (6 eV $<$ *h$\nu$* $<$ 13.6 eV) [@Tielens:1985; @Hollenbach:1999]. Typically, the PDR consists of neutral hydrogen, ionized carbon, and neutral oxygen nearest to the ionization front, and with increasing distance, molecular species (e.g., CO, H$_2$, and O$_2$) dominate the chemical composition of a PDR [@Hollenbach:1999]. One tracer of PDRs is polycyclic aromatic hydrocarbon (PAH) emission at infrared (IR) wavelengths. @Churchwell:2006 identify more than 300 bubbles at IR wavelengths in the Galactic Legacy Infrared Mid-Plane Survey Extraordinaire (GLIMPSE), and 25$\%$ of these bubbles coincide with known regions. @Watson:2008 examine three bubbles from the @Churchwell:2006 catalog with the *Spitzer* Infrared Array Camera (IRAC) bands 4.5, 5.8, and 8.0 and the 24 $\mu$m band from the *Spitzer* Multiband Imaging Photometer (MIPS) to determine the extent of the PDR around three young regions. One of their main results is that the 8 emission, which is due to PAHs, encloses the 24 emission, which traces hot dust. @Kerton:2013 discuss similar observations near the W 39 region. @Watson:2008 use ratios between the 4.5, 5.8, and 8.0 bands to determine the extent of the PDRs, as the 4.5 emission does not include PAHs but the 5.8 and 8.0 bands do (see their Section 1 for details). To determine the presence and extend of a potential PDR around IC 1805, we analyze Wide-field Infrared Survey Explorer (WISE) data from the IPAC All-Sky Data Release[^4] at 3.6, 4.6, 12, and 22 $\mu$m. The 4.6 WISE bands is similar in bandwidth and center frequency to the IRAC 4.5 band, and the WISE 22 band is also similar to the MIPS 24 band [@Anderson:2014]. The 12 WISE band does not overlap with the 8.0 band of IRAC, but the WISE band traces PAH emission at 11.2 and 12.7 . @Anderson:2012 note, however, that the 12 flux is on average lower than the 8.0 IRAC band, which is most likely due to the WISE band sampling different wavelengths of PAH emission instead of the 7.7 and 8.6 PAH emission in the IRAC band.
Figure \[fig:wise\] is a RGB image of the southern loop of IC 1805 at 4.6 (blue), 12 (green), and 22 $\mu$m (red). The 1.42 GHz radio continuum emission is shown in the white contours at 8.5, 9.5, and 10 K, and the lines of sight that intersect this region are labeled as well. Similar to the results of @Watson:2008, the majority of the 22 emission is located inside the bubble. The radio contours trace the ionized shell of the region, which show a patchy ionized shell. Outside of the radio contours, there is a shell of 12 (green) PAH emission that encloses the 22 emission as well. In the northeastern portion of the image, there is extended 22 (hot dust) emission, which is spatially coincident with a CO clump [@Lagrois:2009].
The PDR model predicts the presence of neutral hydrogen and molecular CO (see Figure 3 of @Hollenbach:1999) at increasing distance from the exciting star cluster. Figure 1 of @Sato:1990 and Figure 2 of @Hasegawa:1983 show contours in the vicinity of IC 1805, and the emission appears to completely enclose the southern loop except near 135.5 $\leq$ $\ell$ $\leq$ 136, 0.2$\leq$ $b$ $\leq$ 0.9. @Braunsfurth:1983 report emission near IC 1805, and he notes that the hole could be due to cold gas or the lack of gas if the winds have sufficiently swept the material away or ionized it. Figure 6 of @Digel:1996 shows the CO emission, with the W3 molecular cloud on the western side of IC 1805, CO emission along the southern loop of IC 1805, and the molecular material associated with the W5 ($\ell$ = 137.1, $b$ = +0.89) region on the eastern side of IC 1805. We interpret the WISE data, the radio contours, and the CO and maps as a patchy ionized shell surrounded by a PDR.
If there is a PDR surrounding IC 1805, then the highest RM values from our data set, RM = –954 and –961 for W4-I2 and -I6, respectively, lie outside the ionized shell of the region and in the PDR. Similarly, the sources W4-I19 and -I24 are also outside the radio continuum contours but appear to be within the 12 (green) emission. This is a surprising result compared with our results from the Rosette Nebula, where we found the highest RM values for lines of sight that pass through the ionized shell. @Gray:1999 note zones of depolarization near the southern portion of IC 1805, which require RMs on order 10$^3$ , and spatial RM gradients. The RMs for W4-I2 and -I6 are on this order, but those for W4-I24 and -I19 are not, and we do not find that these lines of sight are affect by depolarization. W4-I24 has two components for which we measure RMs, and the components are separated by 18 arcseconds. The $\Delta$RM, which is the difference in RM between the two components is 38 , which is not a large change in the RM and is consistent within the errors.
The presence of the PDR is complicated, however, by the extended diffusion ionized emission reported by @Terebey:2003 and @Gray:1999. At lower contours, the high RM sources do lie within the radio continuum emission. To fully understand the presence and extent of a PDR or an extended envelope, observations of radio recombination lines on the eastern side of IC 1805 would clarify the structure as well as observations of other tracers of PDRs (e.g., fine structure lines of C and C$^+$, H$_2$, and CO). It may be the case that the ionized shell is patchy along the shell wall, which allows photons $>$ 13.6 eV to escape the shell at places, but the shell is sufficiently ionization-bounded at other places such that a PDR can form.
![Inset from Figure \[fig:w4\]. A RGB image of archive WISE data at 4.6 (blue), 12 (green), and 22 (red) with CGPS contours at 8.5, 9.5, and 10 K in white. The lines of sight from the present study are shown with circles and are labeled according to Table \[tab:sources\].[]{data-label="fig:wise"}](f9.pdf){width="95.00000%"}
Faraday Rotation Through the Cavity and Shell of the Stellar Bubble
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There are four lines of sight through the cavity of the stellar bubble, assuming an inner radius from the @Terebey:2003 model. The sources W4-I3, -I15, -I18, and -I21 are through the cavity, and including multiple components, we find 6 RM values. W4-I3 has a high RM (–878 $\pm$ 18 ), and W4-I15 and -I21 have comparatively low RM values ( –79 to –232 ). Examination of Figures \[fig:w4\] and \[fig:wise\] does not reveal enhanced emission near W4-I3 in comparison to W4-I21. W4-I15, however, is in a region of relatively low emission, which may explain why W4-I15 has a RM value at least 4 times smaller than W4-I3.
W4-I13 is outside the shell, assuming a shell radius from either @Terebey:2003 model. From Figure \[fig:w4\], it does appear to be outside the ionized shell. However W4-I13 is within a 8.5 K contour on the 1.42 GHz radio continuum map, which may indicate that it is probing the ionized shell. We find a high RM for both components of this source, which is similar to the RM values for W4-I2 and -I6.
Across IC 1805, we observe negative RM values for all lines of sight except one: W4-I18, which is 5.6 arcmin (4 pc) from the center of the star cluster. The absolute value of the RM for W4-I18 is also large (+501 $\pm$ 33 ), indicating a large change in RM along this line of sight relative to other lines of sight in this part of the sky. This line of sight is probing the space close to the massive O and B stars responsible for IC 1805. In the @Weaver:1977 model for a stellar bubble, the hypersonic stellar wind dominates the region between the star responsible for the bubble and the inner termination shock. Equation (12) of @Weaver:1977 states that the distance of the inner shock, $R_t$ is
$$R_{\textrm{t}} = 0.90\; \alpha^{3/2} \left(\frac{1}{\rho_0}\frac{\textrm{d}M_w}{\textrm{d}t}\right)^{3/10} \ V_w^{1/10} \ t^{2/5},
\label{eq:weaver}$$
where $\alpha$ is a constant equal to 0.88, $\rho_0$ is the mass density in the external ISM, d$M_w$/d$t$ is the mass loss rate, $V_w$ is the terminal wind speed, and $t$ is time. For a rough estimate of the inner shock distance, we utilize general stellar parameters for OCl 352 of d$M_w/dt$ = 10$^{-5}$ , $t$ = 10$^6$ yr, and V$_w$ = 2200 (see Section \[sec:structure\] or Table \[tab:stellarpar\]). From the discussion in Section \[sec:ferriere\], we adopt $n_0$ = 4.5 for $\rho_0$ = $n_0 m_p$, where $m_p$ is the mass of a proton. With these values in the appropriate SI units, $R_t$ $\sim$ 6 pc. It is possible that the line of sight to W4-I18 passes inside the inner shock, and the large, positive RM is due to material modified by the hypersonic stellar wind and not the shocked interstellar material. Because the inner shock is interior to the contact discontinuity between the stellar wind and the ambient ISM, the magnetic field close to the star cluster may be oriented in any direction relative to the exterior (upstream) field. With a positive value of the RM for W4-I18, the line of sight component of the field points toward us while the remaining lines of sight in the cavity are negative, meaning points away.
Low Rotation Measure Values Through the W4 Superbubble
------------------------------------------------------
North of IC 1805 is the W4 Superbubble, which is an extended “egg-shaped” structure closed at $b$ $\sim$ 7 [@West:2007]. @Basu:1999 utilize an H$\alpha$ map to define the shape, which would include the southern loop (134$<$ $\ell$ $<$ 136, *b* $<$ 0.5); @Normandeau:1996 examine the distribution, however, and place the base of the structure at OCl 352. Similarly, @West:2007 place an offset bottom of the “egg” at OCl 352. The southern loop of IC 1805 is seemingly sufficiently different from the northern latitudes, as it is often not included in the discussion of the W4 Superbubble in spite of the fact that OCl 352 is thought to be responsible for the formation of both structures [@Terebey:2003; @West:2007].
Nine lines of sight in the present study are north of OCl 352 in the W4 Superbubble. These sources are W4-I1, -I4, -I8, -I12, -I14, -I17, -O4, -O6, and -O7, and they have a mean RM of –293 and a standard deviation of 178 . Of these sources, W4-I14 and -I17 have the largest RM values, –666 and –460 , respectively, and they are close to OCl 352, with distances of 31 arcminutes (20 pc) and 17 arcminutes (11 pc), respectively. As discussed in Section \[sec:structure\], @Lagrois:2009 argue that the ionized “v” structure north of OCl 352 is part of the bubble wall and not a cap to southern loop structure, but examination of Figure \[fig:w4\] suggests that the bubble walls are denser, or thicker, at latitudes $<$ 1.5 than higher latitudes, which may explain the larger RM associated with W4-I14 and -I17. The remaining lines of sight, however, in the W4 Superbubble have some of the lowest RM values in the data set and are consistently lower RM values than the lines of sight through the PDR.
At higher latitudes, @Gao:2015 modeled the polarized emission and applied a Faraday screen model to the W4 Superbubble. They report RMs on the western side of W4 ($\ell$ $\sim$ 132, $b$ $\sim$ 4.8) between –70 and –300 and $\sim$ +55 for the eastern shell ($\ell$ $\sim$ 136, $b$ $\sim$ 7). @Gao:2015 argue that since W4 is tilted at an angle towards the observer [@Normandeau:1997], a change in the sign of the RM is consistent with a scenario in which the superbubble lifts up a clockwise running Galactic magnetic field [@Han:2006] out of the Galactic plane. The magnetic field would go up the eastern side of the superbubble and then down the western side, resulting in the field being pointed toward the observer in the east and away from the observer in the west. While the lines of sight reported in this paper are at $b$ $<$ 2, we find a similar range of RM values as reported by @Gao:2015 for the western side. However, we measure RM values 3 – 4.5 times higher on the eastern side, and we do not observe a sign reversal on the eastern side as suggested by @Gao:2015.
@West:2007 report positive values of the magnetic field for the western side from a change in polarization position angle of $\sim$ 60 at 21 cm, which gives a RM value on order of 20 . We do not observe RM values this low for any of our lines of sight through the northern latitudes. Our lines of sight, however, do not probe the same regions as the @West:2007 and @Gao:2015 studies. The line of sight W4-I4 is arguably within the W4 Superbubble; however, it is also 8 arcmin (5 pc) on the sky from W3-North (G133.8 +1.4), which is a star forming region within W3. W4-I4 has two components, separated by 15 arcsec (0.2 pc), and a difference in RM values between the two components of $\Delta$RM = 85 . The RM values for both components are low (–153 and –68 ) despite being in the superbubble and near to W3, which may have variable but potentially large magnetic fields [@vanderWerf:1990; @Roberts:1993] (see Section \[sec:structure\]).
Models for the Structure of the region and Stellar Bubble\[sec:models\]
=======================================================================
@Whiting:2009 Model of the Rotation Measure in the Shell of a Magnetized Bubble\[sec:whitingmod\]
-------------------------------------------------------------------------------------------------
@Whiting:2009 developed a simple analytical shell model intended to represent the Faraday rotation due to a @Weaver:1977 solution for a wind-blown bubble. We employed this model in @Savage:2013 and @Costa:2016 to model the magnitude of the RM in the shell of the Rosette Nebula as a function of distance from the exciting star cluster. Figure 6 of @Whiting:2009 and their Section 5.1 give the details of the model, and Sections 4.1 of @Savage:2013 and 5 of @Costa:2016 describe the application of the model to the Rosette Nebula. This model takes as inputs the general interstellar magnetic field ($\textbf{B}$) in $\mu$G, the inner ($R_1$) and outer ($R_0$) radii of the shell in parsecs, and the electron density in the shell, $n_e$ (). $R_0$ represents the shock between the ambient ISM and the shocked, compressed ISM, and $R_1$ separates the shocked ISM from the hot, diffuse stellar wind in the cavity. Only the component of the ambient interstellar magnetic field that is perpendicular to the shock normal is amplified by the density compression ratio, X. The resulting expression for the RM through the shell is $$\textrm{RM}=C\, n_{e}\, L(\xi)\, B_{0z} \left(1+(X-1)\left(\frac{\xi}{R_{0}}\right)^{2}\right),
\label{eq:rmmodelW}$$ where L($\xi$) is the cord length through the shell in parsecs (see Equation 10 in @Whiting:2009 or Equation 6 in @Costa:2016), and $B_{0z}$ is the z-component of **B$_0$**, the magnetic field in the ISM. If $n_e$ has units of cm$^{-3}$, $B_{0z}$ is in $\mu$G, and $L$ is in parsecs, $C=0.81$ (see Equation \[eq:rmprat\]). $B_{0z}$ is at an angle $\Theta$ with respect to the LOS and is written as
$$B_{0z}=B_{0}\cos{\Theta}.$$
In our previous work, we presented two cases for the behavior of the magnetic field in the shell. The first is that the magnetic field is amplified by a factor of 4 in the shell. The second case, in which there is not an amplification of the magnetic field in the shell, sets X = 1. Equation \[eq:rmmodelW\] then simplifies to $$\textrm{RM}(\xi)=0.81\, n_{e} \, L(\xi) \, B_{0z}.
\label{eq:rmmodelH}$$ In @Costa:2016, we employed a Bayesian analysis to determine which of the two models better reproduces the observed dependence of the RM as a function of distance. We found that neither model was strongly favored in the case of the Rosette. The model given in Equation (\[eq:rmmodelW\]) is subject to the criticism that it applies shock jump conditions for **B** over a large volume of a shell, and that the outer radius of an observed region need not be the outer shock of a Weaver bubble (see remarks in Section 5.1.1 of @Costa:2016). It is worth including this model, however, in our analysis of IC 1805 for completeness and in order to compare our results to those of the Rosette Nebula.
In Section \[sec:structure\], we discussed the the structure of IC 1805, and we present evidence from the literature that north of OCl 352 is part of the W4 Superbubble. Thus, lines of sight north of OCl 352 may have different model parameters for the shell radii and electron density than the southern loop. For the southern latitudes, we utilize the @Terebey:2003 thin and thick shell values for the shell radii and electron density. The remaining parameters in Equation (\[eq:rmmodelH\]) are $B_0$ and $\Theta$. As in @Savage:2013 and @Costa:2016, we adopt $B_0$ = 4 $\mu$G for the general Galactic field in front of the region. The angle $\Theta$ is calculated as follows. Assuming a distance of 8.5 kpc to the Galactic center, a distance to OCl 352 of 2.2 kpc, and given a Galactic longitude of 135, the angle between the line of sight and an azimuthal magnetic field is $\Theta$ = 55. We discuss our comparison of this model with the data in Section \[sec:results\].
[ccccccc]{}\
Center & R$_i$ & $R_o$ & $n_e$ & X & $\Theta$ & Figure\
() & (pc) & (pc) & () & & () &\
(135, +0.42) & 19 & 25 & 10 & 1, 4 & 55 & \[fig:southshell-thick\]\
(135, +0.42) & 19 & 21 & 20 & 1, 4 & 55 & \[fig:southshell-thin\]\
\
Center & $\Delta R$ & $R_s$ & $n_s$ & $\epsilon$ & $\Theta$ &Figure\
() & (pc) & (pc) & () & & () &\
(135, +0.42) & 6 & 25 & 10 & 0.25 & 55 & \[fig:steve2\]\
Position of model center in Galactic coordinates in the format of ($\ell$, $b$).
The model uses either X = 1 or X = 4.
Analytical Approximation to Magnetized Bubbles of @Ferriere:1991\[sec:ferriere\]
--------------------------------------------------------------------------------
![Illustration of a simplified version of the shell and cavity produced by a stellar wind, as discussed by @Ferriere:1991. The z direction is that of the interstellar magnetic field, and $\Theta$ is the angle between the magnetic field and the line of sight. A line of sight passes at a closest distance $\xi$ from the center of the cavity (the “impact parameter”). Other parameters in the figure are defined in the text. A Faraday rotation measurement is along a line of sight offset a linear distance $\xi$ from the center of the bubble. The quantity d$s$ represents an incremental spatial interval along the LOS. []{data-label="fig:steve1"}](f10.pdf){width="60.00000%"}
@Ferriere:1991 presented a semi-analytic discussion of the evolution of a stellar bubble in a magnetized interstellar medium. The theoretical object discussed by @Ferriere:1991 could describe a shock wave produced by a supernova explosion or energy input due to a stellar wind. The main features of the model were an outer boundary (e.g. outer shock) which was the first interface between the undisturbed ISM and the bubble, and an inner contact discontinuity between ISM material, albeit modified by the bubble, and matter that originated from the central star or star cluster.
The main feature of the model is that plasma passing through the outer boundary is concentrated in a region between the outer boundary and the contact discontinuity. In what follows, we will refer to this region as the shell of the bubble. The equation of continuity then indicates that there will be higher plasma density in the shell, and the law of magnetic flux conservation indicates that there will be an increase in the strength of the magnetic field in the shell relative to the general ISM field. @Ferriere:1991 were interested in the structure of the bubble, and their results have been corroborated by the fully numerical studies of @Stil:2009. However, @Ferriere:1991 did not calculate the Faraday rotation measure through their model for diagnostic purposes. @Stil:2009 explicitly considered the model RMs from their calculations, but only for a couple of cases and for two values of bubble orientation. It is our goal in this section to use a simplified, fully analytic approximation of the results of @Ferriere:1991, that permits RM profiles RM($\xi$) for a wide range of bubble parameters and orientation with respect to the LOS.
The geometry of the bubble is shown in Figure \[fig:steve1\], which is an adaptation of, and approximation to Figure 1 and Figure 4 from @Ferriere:1991. An important feature of Figure \[fig:steve1\], not present in @Ferriere:1991, is the orientation of the line of sight at an angle $\Theta$ with respect to the ISM magnetic field at the position of the bubble, and the impact parameter $\xi$ indicating the separation of the LOS from the center of the bubble. The region interior to the contact discontinuity is referred to as the cavity, and for the purposes of our discussion will be considered a vacuum. Another important shell parameter is the thickness $\Delta R$ $\equiv$ $R_s$ - $R_i$, where $R_s$ and $R_i$ are the outer radius of the bubble and the radius of the contact discontinuity, respectively (see Figure \[fig:steve1\]). We also define and use the dimensionless shell thickness $$\epsilon \equiv \frac{\Delta R}{R_s}
\label{eq:steve8}$$ A major simplification that we adopt, based on an approximation of the results of @Ferriere:1991, is that the magnetic field in the shell ($\textbf{B}_s$) is entirely in the azimuthal direction, and that we ignore radial variations within the shell, i.e. $$\textbf{B}_s(r, \theta) \equiv \pm B_s(\theta) \hat{e}_{\theta}
\label{eq:steve9}$$ where $\hat{e}_{\theta}$ is a unit vector in the azimuthal direction, and the $\pm$ is selected by the polarity of the interstellar field at the bubble.
We need expressions for the electron density and vector magnetic field within the shell, as well as the geometry of the line of sight. The most important aspect of the @Ferriere:1991 theory is the conservation of magnetic flux as the magnetic field in the external medium is swept up and accumulated in the shell. This results in the azimuthal component of the magnetic field increasing as $\theta$ increases from $0$ to $\frac{\pi}{2}$, as given by Equation (40) of @Ferriere:1991. In @Ferriere:1991 the shell thickness also depends on $\theta$ (Equation 46 of @Ferriere:1991), and as a consequence, so does the plasma density in the shell $n_s$ (Equation 38 of that paper).
In the initial version of this paper, we calculated the RM through model bubbles in which $B_s$, $n_s$, and $\Delta$ R all varied with $\theta$ as prescribed by @Ferriere:1991. These calculations utilized an approximate form for lines of sight that intersected the bubble in two segments (passing through the central cavity between), and a form that contained a numerically-evaluated expression for lines of sight that remained within the shell from ingress to egress. The algebraic distinction between these 2 cases is discussed below (Sections \[sec:walls\] and \[sec:allshell\]). These expressions for RM($\xi$), including a comparison with our RM measurements, are given in @Costa:2018phd. After examining the results of these calculations, it was decided to simplify our bubble model to that of a spherical shell with constant $\epsilon$. The motivation for this suggestion was the very limited success of the more general model in representing our data, which did not justify the extensive algebraic presentation and non-compact expressions that resulted. The calculations with the approximation of constant $\epsilon$ are presented below. Due to magnetic flux conservation, the expanding shell (now approximated as spherical) will have a magnetic field that is larger than in the external medium, and increases with $\theta$, as in the original discussion of @Ferriere:1991. For our spherical case, it may be shown that the magnetic field in the shell is $$B_s(\theta) = \frac{B_0}{2 \epsilon} \sin \theta
\label{eq:steve10}$$ where $B_0$ is the magnitude of the magnetic field in the external medium. The dimensionless shell thickness $\epsilon$ remains a free parameter of the model, or one that can be determined by observations. Finally, the plasma density in the shell, determined by mass conservation, is $$n_s = \frac{n_0}{3 \epsilon}
\label{eq:steve11}$$ where $n_0$ is the plasma density in the external medium. is a valid approximation for $\epsilon \ll 1$.
### RM Calculation for Lines of Sight Through the Walls of the Shell\[sec:walls\]
In evaluating the integral Equation (\[eq:rmorg\]) or (\[eq:rmprat\]) through the model shell shown in Figure \[fig:steve1\], we consider two cases. The first calculation is for lines of sight that pass through a portion of the shell, emerge into the cavity, and then reenter the shell on the opposite side before exiting the shell entirely. This is the case illustrated in Figure \[fig:steve1\]. The incremental RM for a spatial interval d$s$ along the line of sight is $$\textrm{d(RM)} = \pm \, C \, n_s \, B_s(\theta) \, (\hat{e}_s \cdot \hat{e}_{\theta})\, \textrm{d}s
\label{eq:steve12}$$
The $\pm$ in front of the RHS indicates that the polarity of the field in the external medium determines the sign of the measured RM. We introduce the variable $s$ as a coordinate along the line of sight; d$s$ is an incremental vector along the line of sight from the source to the observer, and $\hat{e}_s$ is the corresponding unit vector. The constant $C$ is the same as introduced in Equation (\[eq:rmmodelW\]).
It is convenient to change the variable of integration over the LOS from $s$ to $\phi$, an angle defined in Figure \[fig:steve1\]. With the introduction of this variable, the term $(\hat{e}_s \cdot \hat{e}_{\theta}) = -\sin \phi$. Integration through the shell segments along the line of sight then corresponds to an appropriate integration over $\phi$. The shell segment closest to the observer corresponds to an integration from $\phi_1$ to $\phi_2$, and the segment furthest from the observer is given by an integration from $\phi_3$ to $\phi_4$.
Substitution of Equations (\[eq:steve10\]) and (\[eq:steve11\]) into (\[eq:steve12\]), followed by integration over $\phi$ and straightforward algebraic manipulation yields the following expression for the RM $$\textrm{RM}(x) = \pm \left( \frac{C \, n_0\, B_0\, R_s}{3 \epsilon^2} \right) x \left[ \arcsin \left(\frac{x}{1 - \epsilon}\right) - \arcsin (x) \right] \cos \Theta
\label{eq:steve13}$$
where the new dependent variable is the normalized impact parameter $x \equiv \frac{\xi}{R_s}$. The identity (\[eq:steve11\]) may be used to convert Equation (\[eq:steve13\]) into a form in which the observed plasma density in the shell ($n_s$) is the density parameter rather than that in the external medium ($n_0$). This substitution makes Equation (\[eq:steve13\]) more directly comparable to Equation (\[eq:rmmodelW\]).
### RM for Lines of Sight Entirely Within the Shell \[sec:allshell\]
If the “impact parameter” $\xi$ is sufficiently large, the entire line of sight is within the shell from the point of ingress to that of egress. From Figure \[fig:steve1\], it can be seen that this occurs if $$x \equiv \frac{\xi}{R_s}\, \geq\, x_{min} = 1 -\epsilon
\label{eq:steve14}$$ The RM in this case is a simple generalization of the algebra involved in obtaining Equation (\[eq:steve13\]) via an integration over the angular variable $\phi$; the upper limit of integration in the segment closest to the observer $\phi_2 \rightarrow \frac{\pi}{2}$, and the lower limit of integration for the shell segment further from the observer $\phi_3 \rightarrow \frac{\pi}{2}$. $$\textrm{RM}(x) = \pm \left( \frac{C\, n_0\, B_0\, R_s}{3 \epsilon^2} \right)\, x\, \left[ \frac{\pi}{2} - \arcsin (x) \right] \cos \Theta \mbox{ , if: } x_{min} \leq x \leq 1
\label{eq:steve15}$$ A plot of the expression RM($x$) given by (\[eq:steve13\]) and (\[eq:steve15\]) is shown in Figure \[fig:steve2\] for a set of parameters that are representative for the IC 1805 region (see Table \[tab:model\]). The curve is very similar in form to the Whiting model, for the case of no magnetic compression, Equation (\[eq:rmmodelW\]) with $X = 1$ or Equation (\[eq:rmmodelH\]). The model expression for RM($x$) is dependent on $n_0$ (or the shell density $n_s$), $B_0$, $R_s$, $\Theta$, and $\epsilon$, the shell thickness parameter. For comparison with observations, we also need to specify the background Galactic rotation measure, RM$_{off}$.
Our simple model contained in Equations (\[eq:steve13\]) and (\[eq:steve15\]) immediately accounts for one of the main results emergent from the numerical simulations of @Stil:2009. The RM through a bubble is maximized when the LOS is parallel to **B**$_0$ ($\cos \Theta = 1$) and small or zero when the LOS is $\perp \mbox{ to } \textbf{B}_0$ ($\cos \Theta = 0$).
![Model for the analytic approximation to the bubble model of @Ferriere:1991, Equations (\[eq:steve13\]) and (\[eq:steve14\]). The model RM is function of the normalized impact parameter $x = \frac{\xi}{R_s}$. The plotted points represent measured RMs presented in this paper. []{data-label="fig:steve2"}](f12.pdf){width="60.00000%"}
Discussion of Observational Results\[sec:results\]
==================================================
Comparison of Models with Observations in the Region
----------------------------------------------------
In this section we discuss the results of the two models presented in Sections \[sec:whitingmod\] and \[sec:ferriere\]. In both cases, we adopt the @Terebey:2003 center for geometric ease and spherical symmetry as well as the parameters given in their Table 3 for a thick shell.
Figures \[fig:southshell-thick\] and \[fig:southshell-thin\] show model RM values for lines of sight south of IC 1805 (*b* $<$ 0.9) with the @Whiting:2009 model for the RM as a function of distance and the shell parameters from @Terebey:2003. Table \[tab:model\] gives the values of the center of the bubble, the shell radii, the electron density, X, and $\Theta$ for Figure \[fig:shell\]. Neither model reliably reproduces the observed RM as a function of distance, and as in @Costa:2016, the model can not account for the dispersion of RM values at similar distances. Generally, the lines of sight in the cavity are low and are more consistent with the background RM. In the thin shell approximation, the largest RM values are associated with lines of sight outside the shell. While the model without amplification of the magnetic field in the shell can marginally account for the magnitude of the RM, the model with amplification (Equation \[eq:rmmodelW\]) predicts far too high values for the RM for $\Theta$ = 55. The analysis contained here mildly supports a result from @Costa:2016 for the Rosette Nebula; Faraday rotation values through these regions do not permit a substantial increase in $|$B$|$ over the general Galactic field.
To reproduce the observed RM in the shell at $\xi$ 20 pc, the angle between the magnetic field and the observer would need to be tilted more into the plane of the sky for the X = 4 case or into the line of sight for the X = 1 case. For the former case, an angle of 75 would reproduce the magnitude of the RM in the shell; such an angle is greater than that expected from a geometric argument, even accounting for a magnetic field pitch angle of 8. Also, no one angle can account for the range of the RM values in the cavity.
With our analytic solution for the RM due to a magnetized bubble as described by @Ferriere:1991, we can examine the dependence of RM on $\Theta$ as well as $\xi$. The most obvious choice for the latter parameter is $\Theta = 55^{\circ}$, based on the geometry as described in Section \[sec:whitingmod\]. Figure \[fig:steve2\] shows our model RM(x) for $\Theta = 55^{\circ}$, with other parameters given in Table 6. Data for sources south of IC 1805 are superposed on the model. Although the model obviously does not reproduce the measurements in detail, it can describe the overall scale of the “rotation measure anomaly” associated with W4, as well as the approximate magnitude of the largest measured RMs ($|$RM$|$ $\sim$ 1000 ). The peak model RM values shown in Figure 12 do not significantly exceed the measured values, unlike the case for the Whiting model with X = 4 (see Figure \[fig:shell\]). It should be kept in mind that the shell modeled in Figure \[fig:steve2\] is the “thick shell model” of @Terebey:2003; the center of that shell is not the star cluster OCl 352, as might be expected.
@Stil:2009 carried out numerical MHD simulations of the Ferrière bubbles, which are obviously more accurate than our analytic approximations. Furthermore, they specifically consider and calculate the Faraday rotation through their models. However, @Stil:2009 only consider $\Theta$ = 0 and $\Theta$ = 90, so the calculations reported in that paper can not explore the changes in RM structure with $\Theta$. Furthermore, the Faraday rotation calculation of @Stil:2009 is done when the outer radius $R_s$ 200 pc (see Figure 14 of @Stil:2009), which is much larger than the structure we are modeling in Section \[sec:ferriere\] of this paper. In what follows, we compare our observations with the results presented in Section 6 of @Stil:2009.
If LOS $||$ , then the highest values of RM will be through the shell closest to the Galactic plane, but the mean RM across the region will be similar to the mean RM exterior to the bubble (see Figure 14 of @Stil:2009). Out of the Galactic plane, the RM is 20 – 30$\%$ of the mean RM exterior to the bubble. Effectively, the largest RMs will always be found in the Galactic plane, and different lines of sight through the bubble will have varying RM values.
In comparing the simulations of @Stil:2009 to our observational results, we find low RM measures for lines of sight through the cavity, though not always low (e.g., W4-I14 and -I17 vs -I15 and -I21). Lines of sight through the shell have generally large RMs, which is inconsistent with a perpendicular to the LOS. The case of LOS $||$ **B$_{\textrm{ext}}$** is inconsistent as well because far from the bubble, the RM is low (e.g., W4-O26 vs -I24) even at similar latitudes, and the lines of sight at $b$ $>$ 1 are consistent with the background RM instead of being reduced by 70 – 80$\%$. Unsurprisingly, our results indicate a case somewhere between these two predictions. As a reminder, we note that the largest values of the RM are for lines of sight exterior to the shell, which is not a prediction from @Stil:2009, most likely due to their simulations modeling the ionized bubble and not a PDR structure.
Magnetic Fields in the PDR
--------------------------
In Section \[sec:pdr\], we examine evidence for a PDR outside the southern loop of IC 1805. @Brogan:1999, @Troland:2016, and @Pellegrini:2007 report large ( 150 $\mu$G) magnetic fields in PDRs associated with the Orion Veil and M17. In the analysis that follows, we attempt to understand the large RM values for lines of sight through the IC 1805 PDR.
If we consider the PDR and the region to be in pressure equilibrium and include magnetic pressure in the PDR, then $$\mPhii = \mPpdr \; + P^{\textrm{PDR}}_{\textrm{mag}},
\label{eq:pbal}$$ where and are the thermal pressures in the region and PDR, respectively, and $P^{\textrm{PDR}}_{\textrm{mag}}=\frac{B^2}{8\pi} $ is the magnetic pressure in the PDR. In the region, $\mPhii = 2n_e^{\mhii}\, k \,T_{\mhii} $, where $n_e^{\mhii}$ and T$_{\mhii}$ are the electron density and temperature, $k$ is the Boltzmann constant, and the factor of 2 accounts for the contribution from both ions and electrons. For P$^{\textrm{PDR}}_{\textrm{th}}$ = $N_{\textrm{PDR}}$ $k$ $T_{\textrm{PDR}}$, $N_{\textrm{PDR}}$ and $T_{\textrm{PDR}}$ are the neutral hydrogen density and the temperature in the PDR.
Near the interface of the PDR and the region, the electron density in the PDR is governed by photoioniziation of carbon [@Tielens:1985], so we estimate $n_e^{\textrm{PDR}}$ by $$n_e^{\textrm{PDR}} = N_{\textrm{PDR}}X_C,$$ where $X_C$ is the cosmic abundance of carbon given in Table 1.4 of @Draine:2011 ($X_C$ 2.95 $\times$ 10$^{-4}$). Solving for $B$ in gives $$B = \sqrt{8\pi \; k(2\; n_e^{\mhii} \;T_{\mhii} - N_{\textrm{PDR}}\;T_{\textrm{PDR}})},
\label{eq:bpbal}$$ and inserting it into , we express the RM in the PDR as $$RM = 0.81\; L \; X_C N_{\textrm{PDR}} \sqrt{8\pi \; k \; (2\;n_e^{\mhii} T_{\mhii} - N_{\textrm{PDR}}\; T_{\textrm{PDR}})}.
\label{eq:rmpdr}$$ It should be emphasized that this RM estimate is in the nature of an upper limit to the rotation measure through the PDR. The reason is that it is obtained from a value of $B$, given by Equation (\[eq:rmpdr\]), which is based on the magnetic pressure $\frac{B^2}{8 \pi}$. The magnetic pressure includes contributions from turbulent fluctuations on all scales, as well as that from a mean or large scale field that produces the net Faraday rotation. In general then, a magnetic field value obtained from an estimate of the magnetic pressure will exceed that obtained from a Faraday rotation measurement.
We differentiate with respect to $N_{\textrm{PDR}}$ to find the value of $N_{\textrm{PDR}}$ that maximizes the RM, which is $$N_{\textrm{PDR}} = \frac{4}{3}\frac{n_e^{\mhii}\;T_{\mhii}}{T_{\textrm{PDR}}}.
\label{eq:maxN}$$ Inserting values of $T_{\mhii}$ = 8000 K, $n_e^{\mhii}$ = 10 [@Terebey:2003], and $T_{\textrm{PDR}}$ = 100 K [@Tielens:1985], gives $N_{\textrm{PDR}}$ 1000 , $B$ 14 $\mu$G (Eq \[eq:bpbal\]), and RM 100 . The electron density in the region is governing the maximum $B$ expected in the PDR given pressure balance. For the IC 1805 region, $n_e$ is low compared to M17 ($n_e$ 560 ) [@Pellegrini:2007], which suggests that a high density (pressure) region is needed to explain large magnetic fields in the PDR.
Our analysis suggests that a simple pressure balance analysis predicts low RM values from the PDR that are inconsistent with our observations. It appears that a different mechanism is required to achieve the magnetic fields strengths observed in @Brogan:1999.
@Terebey:2003 discuss an extended halo of ionized emission around the southern loop, which may indicate that there are more free electrons present outside the obvious ionized shell as seen in Figure \[fig:w4\]. This may account for the larger values of the RM we observe. It is clear that knowing the electron density in this region and determining the presence of a PDR through observations, such as carbon radio recombination lines, is necessary to understand how the magnetic field is modified in this complex region.
A Comparison of IC 1805 and the Rosette Nebula as “Rotation Measure Anomalies” \[sec:rosette\]
==============================================================================================
-- ---------- ---------- ---------------------- ------ -----------------------
HD 46223 O4V(f) 1.6$\times$10$^{-6}$ 3100 4.8$\times$10$^{36}$
HD 46150 O5.5V 2.0$\times$10$^{-6}$ 3100 6.0$\times$10$^{36}$
HD 46202 O9V(f) 6.3$\times$10$^{-8}$ 1150 2.6$\times$10$^{34}$
HD 46149 O8.5V(f) 2.0$\times$10$^{-7}$ 1700 1.8$\times$10$^{35}$
HD15570 O4I 1.0$\times$10$^{-5}$ 2200 1.5$\times$10$^{37} $
HD15558 O4III 6.3$\times$10$^{-6}$ 3000 1.8$\times$10$^{37} $
HD 15629 O5V 2.0$\times$10$^{-6}$ 2900 5.3$\times$10$^{36}$
-- ---------- ---------- ---------------------- ------ -----------------------
: Stellar Parameters \[tab:stellarpar\]
Calculated mechanical wind luminosity based on cited mass loss rates and terminal velocities.
@Massey:1995
@Howarth:1989
@Chlebowski:1991
@Roman:2008
@Garmany:1988
@Bouret:2012
@Groenewegen:1989
We are interested in how the Galactic magnetic field is modified by OB associations via their stellar winds and ionizing photons, and we started our study with the Rosette Nebula, where we found large ( 10$^{3}$ ) RM measurements through the ionized shell of the region [@Costa:2016]. In the case of the Rosette, we find positive RM across the region, and for IC 1805, we find negative values. If the Galactic magnetic field follows the spiral arms in a clockwise direction, then we would expect the LOS magnetic field component to be pointed towards us (positive B) for $\ell$ $>$ 180, and pointed away from us (negative B) for $\ell$ $<$ 180. Except for one line of sight in each nebula, we find that the polarity of the Galactic magnetic field is preserved across each nebula and is consistent with the large scale field through the arm.
In our study of the Rosette, we investigated whether the magnetic field is amplified in the shell of the nebula. We found that the model without amplification was weakly favored over the case when the magnetic field is amplified in the shell. When we applied the same model to IC 1805, however, it is difficult to conclude in favor of either model, but in both cases, the model with an enhanced magnetic field overpredicts the RM. From inspection of Figures \[fig:southshell-thick\] and \[fig:southshell-thin\], it seems that the model without amplification better accounts for the magnitude of the observed RMs, but the observations do not conform to the model prediction of RM($\xi$), and the model can not account for the wide range in observed values of RM at a given $\xi$.
In the present study, we find the highest RMs for lines of sight outside the obvious shell structure, though one line of sight (W4-I13) does appear to intersect the ionized shell and it has a large RM. These lines of sight may be probing the magnetic field within the PDR. In the case of the Rosette, we found that the highest RM values were for lines of sight through the bright ionized shell. However with our work on IC 1805 and the PDR associated with it, we have briefly revisited our results in the Rosette, particularly Figure 1 from @Costa:2016. There are a few lines of sight with RM of order a few 10$^2$ that appear to be outside the ionized shell. These lines of sight were included in the background estimate for the Rosette, but if the Rosette also has a PDR, then these lines of sight may actually be probing that material.
Table \[tab:stellarpar\] lists spectral type, mass loss rate, terminal wind velocity, and calculated wind luminosity from the literature for O stars with the largest wind luminosities in both NGC 2244, which is associated with the Rosette Nebula, and OCl 352. The sum of the wind luminosities of the three main stars in OCl 352 is 3.8 $\times$ 10$^{37}$ ergs s$^{-1}$, while the corresponding number for NGC 2244 (4 stars) is 1.1 $\times$ 10$^{37}$ ergs s$^{-1}$. In addition, OCl 352 appears to have more luminous stars. As such, OCl 352 might be expected to produce a more energetic stellar bubble than NGC 2244. Our Faraday rotation measurements show no indication of this, in that the largest RMs observed are similar for the two objects. In fact, higher RMs were measured for the Rosette than for any line of sight through IC 1805. A number of factors can control the impact a star cluster has on the ISM. If some relationship exists between the total wind luminosity of a star cluster and properties of an interstellar bubble that can be measure with Faraday rotation, it will apparently require a large sample of clusters/ regions to reveal it.
Future Research\[sec:fut\]
==========================
In the future, we will continue our investigation of regions and how they modify their surroundings and the Galactic magnetic field. An immediate investigation will be centered on observations of the region IC 1396. This will provide a third region with different age, stellar content, and Galactic location. The observations are similar to those we have made of the Rosette Nebula and IC 1805. The observations of IC 1396 have been made with the VLA and are awaiting analysis. By adding more regions to our study, we can begin to address questions such as
1. Since the electron density distributions in regions are known from radio continuum observations, we can inquire what conditions would result in an RM $>$$>$ 10$^3$ through the shell of an region.
2. Is it a general property of regions and stellar bubbles that the polarity of the Galactic magnetic field is preserved within the region? The answer to this question has implications for the amplitude of MHD turbulence in the ISM on scales of the order of the regions, $\sim 10 - 30$ pc.
3. Do PDRs around other nebulae produce high RMs? What is the magnitude of the RM due to the PDR relative to that of the shell of an region?
In addition to increasing the number of regions, understanding Faraday complexity and how to interpret the associated RM measurements is important to studies of Galactic magnetic fields, particularly with large polarization surveys like the and with the in the near future.
Summary and Conclusions\[sec:sum\]
==================================
1. We performed polarimetric observations using the VLA for 27 lines of sight through or near the shell of the region and stellar bubble associated with the OB association OCl 352.
2. We obtain RM measurements for 20 sources using two methods. The first is through the traditional least-squares fit to , and the second is using RM Synthesis. Including components that are resolved, we report 27 RM values, and we find good agreement between the two methods. We find the same sign of the RM across the entire region with the exception of one source, W4-I18. We estimate a background RM due to the general ISM of –145 in this part of the Galactic plane. We measure an excess of RM of +600 to –800 due to W4.
3. Only one line of sight has a positive RM value, W4-I18. It has a RM of +501 $\pm$ 33 , and it is located 5.6 arcminutes from the center of OCl 352. This line of sight may be probing the material close to the massive stars. The orientation of the line of sight component of the magnetic field is directed towards the observer, whereas in the rest of the region, the magnetic field is directed away.
4. We find that some of the lines of sight with the largest RM values occur just outside the obvious ionized shell of IC 1805 and are potentially probing the magnetic field in the PDR. The lines of sight through the cavity of the bubble have lower RM values than those through the shell. In the W4 Superbubble, which is north of OCl 352, we find RM values consistent with the background RM.
5. We discuss two shell models to reproduce the magnitude of the RM and its dependency on distance from the center of the star cluster. We employed the first of these models in @Savage:2013 and @Costa:2016, and it is based on the @Weaver:1977 solution for a stellar bubble, which includes a shock expanding into an ambient medium. The second model uses magnetic flux conservation to describe how the magnetic field is modified in the shell and consists of a simplified analytic approximation to the results presented by @Ferriere:1991. Neither of these simplified models satisfactorily accounts for the dependence of RM on spatial location within the shell, although the Whiting model without field amplification (X = 1) and the simplified Ferrière model approximately reproduce the magnitude of the largest RMs. However, both models predict a single-valued dependence of RM on $\xi$, the separation of the line of sight from the center of the nebula, whereas the observations show a large range of RM for sources with similar values of $\xi$.
6. Because we have independent information on the electron density from radio continuum observations of both IC 1805 and the Rosette Nebula, our observations can limit the magnitude of the magnetic field in the regions. Our RM measurements indicate that the field does not greatly exceed the value in the general ISM.
7. We compare our results from the current study of IC 1805 and our previous study of the Rosette Nebula. Notably, we find the same order of magnitude for the RM for the two nebulae, but the sign of the RM in each region is opposite. Since IC 1805 and the Rosette are at different Galactic longitudes and on either side of $b$ = 180, the sign difference between the two nebula is consistent with a Galactic magnetic field that follows the spiral arm structure in a clock-wise direction, as suggested in models [@vanEck:2011].
This research was partially supported at the University of Iowa by grants AST09-07911 and ATM09-56901 from the National Science Foundation. This publication makes use of data products from the Wide-field Infrared Survey Explorer [@2010AJ....140.1868W], 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. Additionally, the research presented in this paper uses data from the Canadian Galactic Plane Survey, a Canadian project with international partners, supported by the Natural Sciences and Engineering Research Council. This research also uses the Python packages Astropy, a community-developed core Python package for Astronomy and NumPy [@van2011numpy]. Finally, we thank the referee of this paper for a helpful and collegial review.
[^1]:
[^2]:
[^3]: private communication, L. Rudnick
[^4]: http://wise2.ipac.caltech.edu/docs/release/allsky/
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The work on large-scale graph analytics to date has largely focused on the study of static properties of graph snapshots. However, a static view of interactions between entities is often an oversimplification of several complex phenomena like the *spread of epidemics*, *information diffusion*, *formation of online communities*, and so on. Being able to find temporal interaction patterns, visualize the evolution of graph properties, or even simply compare them across time, adds significant value in reasoning over graphs. However, because of lack of underlying data management support, an analyst today has to manually navigate the added temporal complexity of dealing with large evolving graphs. In this paper, we present a system, called *Historical Graph Store*, that enables users to store large volumes of historical graph data and to express and run complex temporal graph analytical tasks against that data. It consists of two key components: a [*Temporal Graph Index*]{} (TGI), that compactly stores large volumes of historical graph evolution data in a partitioned and distributed fashion; it provides support for retrieving snapshots of the graph as of any timepoint in the past or evolution histories of individual nodes or neighborhoods; and a Spark-based [*Temporal Graph Analysis Framework*]{} (TAF), for expressing complex temporal analytical tasks and for executing them in an efficient and scalable manner. Our experiments demonstrate our system’s efficient storage, retrieval and analytics across a wide variety of queries on large volumes of historical graph data.'
author:
- |
Udayan Khurana\
\
Amol Deshpande\
\
bibliography:
- 'TGIdraft-extended.bib'
title:
- Historical Graph Data Management
- Storing and Analyzing Historical Graph Data at Scale
---
Introduction
============
Graphs are useful in capturing behavior involving interactions between entities. Several processes are naturally represented as graphs – social interactions between people, financial transactions, biological interactions among proteins, geospatial proximity of infected livestock, and so on. Many problems based on such graph models can be solved using well-studied algorithms from graph theory or network science. Examples include finding driving routes by computing shortest paths on a network of roads, finding user communities through dense subgraph identification in a social network, and many others. Numerous graph data management systems have been developed over the last decade, including specialized graph database systems like Neo4j, Titan, etc., and large-scale graph processing frameworks such as Pregel [@pregel], Giraph, GraphLab [@distgraphlab], GraphX [@graphx], GraphChi [@kyrola2012graphchi], etc. However much of the work to date, especially on cloud-scale graph data management systems, focuses on managing and analyzing a single (typically, current) static snapshot of the data. In the real world, however, interactions are a dynamic affair and any graph that abstracts a real-world process changes over time. For instance, in online social media, the friendship network on Facebook or the “follows” network on Twitter change steadily over time, whereas the “mentions” or the “retweet” networks change much more rapidly. Dynamic cellular networks in biology, evolving citation networks in publications, dynamic financial transactional networks, are few other examples of such data. Lately, we have seen an increasing merit in dynamic modeling and analysis of network data to obtain crucial insights in several domains such as cancer prediction [@taylor2009dynamic], epidemiology [@gross2006epidemic], organizational sociology [@gulati1999interorganizational], molecular biology [@eisenberg2000protein], information spread on social networks [@lerman2010information] amongst others.
In this work, our focus is on providing the ability to analyze and to reason over the entire history of the changes to a graph. There are many different types of analyses of interest. For example, an analyst may wish to study the evolution of well-studied static graph properties such as centrality measures, density, conductance, etc., over time. Another approach is through the search and discovery of temporal patterns, where the events that constitute the pattern are spread out over time. Comparative analysis, such as juxtaposition of a statistic over time, or perhaps, computing aggregates such as *max* or *mean* over time, possibly gives another style of knowledge discovery into temporal graphs. Most of all, a primitive notion of just being able to access past states of the graphs and performing simple static graph analysis, empowers a data scientist with the capacity to perform analysis in arbitrary and unconventional patterns.
Supporting such a diverse set of temporal analytics and querying over large volumes of historical graph data requires addressing several data management challenges. Specifically, there is a want of techniques for storing the historical information in a compact manner, while allowing a user to retrieve graph snapshots as of any time point in the past or the evolution history of a specific node or a specific neighborhood. Further the data must be stored and queried in a distributed fashion to handle the increasing scale of the data. We must also develop an expressive, high-level, easy-to-use programming framework that will allow users to specify complex temporal graph analysis tasks, while ensuring that the specified tasks can be executed efficiently in a data-parallel fashion across a cluster.
In this paper, we present a graph data management system, called [*Historical Graph Store (HGS)*]{}, that provides an ecosystem for managing and analyzing large historical traces of graphs. HGS consists of two key distinct components. First, the [*Temporal Graph Index (TGI)*]{}, is an index that compactly stores the entire history of a graph by appropriately partitioning and encoding the differences over time (called [*deltas*]{}). These deltas are organized to optimize the retrieval of several temporal graph primitives such as neighborhood versions, node histories, and graph snapshots. TGI is designed to use a distributed key-value store to store the partitioned deltas, and can thus leverage the scalability afforded by those systems (our implementation uses Apache Cassandra[^1] key-value store). TGI is a tunable index structure, and we investigate the impact of tuning the different parameters through an extensive empirical evaluation. TGI builds upon our prior work on DeltaGraph [@icdepaper], where the focus was on retrieving individual snapshots efficiently; we discuss the differences between the two in more detail in Section \[sec:tgi\].
The second component of HGS is a *Temporal Graph Analysis Framework (TAF)*, which provides an expressive library to specify a wide range of temporal graph analysis tasks and to execute them at scale in a cluster environment. The library is based on a novel set of *temporal graph operators* that enable a user to analyze the history of a graph in a variety of manners. The execution engine itself is based on Apache Spark [@zaharia2010spark], a large-scale in-memory cluster computing framework.
[ ]{} The rest of the paper is organized as follows. In Section \[sec:related\], we survey the related work on graph data stores, temporal indexing, and other topics relevant to the scope of the paper. In Section \[sec:overview\], we provide a sketch of the overall system, including key aspects of the underlying components. We then present the Temporal Graph Index and the Temporal Graph Analytics Framework in detail in Section \[sec:tgi\] and Section \[sec:taf\], respectively. In Section \[sec:experiments\], we provide an empirical evaluation of the various system components such as the graph retrieval, scalability of temporal analytics, etc. We conclude with a summary and a list of future directions in Section \[sec:conclusion\].
Related Work {#sec:related}
============
In the recent years, there has been much work on graph storage and graph processing systems and numerous systems have been designed to address various aspects of graph data management. Some examples include Neo4J, AllegroGraph [@aasman2006allegro], Titan[^2], GBase [@kang2011gbase], Pregel [@pregel], Giraph, GraphChi [@kyrola2012graphchi], GraphX [@graphx], GraphLab [@distgraphlab], and Trinity [@shao2013trinity]. These systems use a variety of different models for representation, storage, and querying, and there is a lack of standardized or widely accepted models for the same. Most graph querying happens through programmatic access to graphs in languages such as Java, Python or C++. Graph libraries such as Blueprints[^3] provide a rich set of implementations for graph theoretic algorithms. SPARQL [@perez2006semantics] is a language used to search patterns in linked data. It works on an underlying RDF representation of graphs. T-SPARQL [@grandi2010t] is a temporal extension of SPARQL. He et al. [@he:sigmod08], provide a language for finding sub-graph patterns using a graph as a query primitive. Gremlin[^4] is a graph traversal language over the property graph data model, and has been adopted by several open-source systems. For large-scale graph analysis, perhaps the most popular framework is the vertex-centric programming framework, adopted by Giraph, GraphLab, GraphX, and several other systems; there have also been several proposals for richer and more expressive programming frameworks in recent years. However, most of these prior systems largely focus on analyzing a single snapshot of the graph data, with very little support for handling dynamic graphs, if any.
A few recent papers address the issues of storage and retrieval in dynamic graphs. In our prior work, we proposed DeltaGraph [@icdepaper], an index data structure that compactly stores the history of all changes in a dynamic graph and provides efficient snapshot reconstruction. G\* [@gstar] stores multiple snapshots compactly by utilizing commonalities. Chronos [@hant2014chronos; @immortalgraph] is an in-memory system for processing dynamic graphs, with objective of shared storage and computation for overlapping snapshots. Ghrab et al. [@ghrab2013analytics] provide a system of network analytics through labeling graph components. Gedik et al. [@6702469], describe a block-oriented and cache-enabled system to exploit spatio-temporal locality for solving temporal neighborhood queries. Koloniari et al. also utilize caching to fetch selective portions of temporal graphs they refer to as partial views [@koloniari2013partial]. LLAMA [@llama] uses multiversioned arrays to represent a mutating graph, but their focus is primarily on in-memory representation. There is also recent work on streaming analytics over dynamic graph data [@kineograph; @graphinc], but it typically focuses on analyzing only the recent activity in the network (typically over a sliding window). Our work in this paper focuses on techniques for a wide variety of temporal graph retrieval and analysis on entire graph histories.
Temporal graph analytics is an area of growing interest. Evolution of shortest paths in dynamic graphs has been studies by Huo et al. [@huo2014efficient], Ren et al. [@RenEvolvGraph11], and Xuan et al. [@xuan2003computing]. Evolution of community structures in graphs has been of interest as well [@asur2009event; @berger2006framework; @greene2010tracking; @tang2008community]. Change in page rank with evolving graphs [@desikan2005incremental; @bahmani2010fast], and the study of change in centrality of vertices, path lengths of vertex pairs, etc. [@pan2011path], also lie under the larger umbrella of temporal graph analysis. Ahn et al. [@ahn2014task] provide a taxonomy of analytical tasks over evolving graphs. Barrat et al. [@barrat2008dynamical], provide a good reference for studying several dynamic processes modeled over graphs. Our system significantly reduces the effort involved in building and deploying such analytics over large volumes of graph data.
Temporal data management for relational databases was a topic of active research in the 80s and early 90s. Snapshot index [@Tsotras1995] is an I/O optimal solution to the problem of snapshot retrieval for transaction-time databases. Salzberg and Tsotras [@Salzberg1999] present a comprehensive survey of temporal data indexing techinques, and discuss two extreme approaches to supporting snapshot retrieval queries, referred to as the *Copy* and *Log* approaches. While the copy approach relies on storing new copies of a snapshot upon every point of change in the database, the log approach relies on storing everything through changes. Their hybrid is often referred to as the *Copy+Log* approach. We omit a detailed discussion of the work on temporal databases, and refer the interested reader to a representative set of references [@Bolour92; @DBLP:conf/sigmod/SnodgrassA85; @Ozsoyoglu1995; @Tansel1993; @date2002temporal; @tsql2; @Salzberg1999]. Other data structures, such as Interval Trees [@Arge1996] and Segment trees [@Blankenagel1994] can also be used for storing temporal information. Temporal aggregation in scientific array databases [@soroush2013time] is another related topic of interest, but the challenges there are significantly different.
Overview {#sec:overview}
========
In this section, we introduce key aspects related to HGS. We begin with the data model, followed by the key challenges and concluding with an overview of the system.
Data Model
----------
Under a discreet notion of time, a time-evolving graph $G^T=(V^T,E^T)$ may be expressed as a collection of graph *snapshots* over different time points, $\{G^0 = (V^0, E^0), G^1, \dots, G^t \}$. The vertex set $V^i$ for a snapshot consists of a set of vertices (nodes), each of which has a unique identifier, and an arbitrary number of key-value attribute pairs. The edge sets $E^i$ consist of edges that each contain references to two valid nodes in the corresponding vertex set $V^i$, information about the direction of the edge, and an arbitrary list of key-value attribute pairs. A temporal graph can also be equivalently described by a set of changes to the graph over time. We call an atomic change at a specific timepoint in the graph an *event*. The changes could be structural, such as the addition or the deletion of nodes or edges, or be related to attributes such as an addition or a deletion or a change in the value of a node or an edge attribute. These approaches as well as certain hybrids have been used in the past for the physical and logical modeling of temporal data. Our approach to temporal processing in this paper is best described using a *node-centric* logical model, i.e., the historical graph is seen as a collection of evolving vertices over time; the edges are considered as attributes of the nodes.
![The scope of temporal graph analytics can be represented across two different dimensions - time and entity. The chart lists retrieval tasks (black), graph operations (red), example queries (magenta) at different granularities of time and entity size.[]{data-label="fig:entity-time"}](entity-time1.pdf){width="\linewidth"}
Challenges
----------
The nature of data management tasks in historical graph analytics can be categorized based on the scope of analysis using the dual dimensions of *time* and *entity* as illustrated with examples in Figure \[fig:entity-time\]. The temporal scope of an analysis task can range from a single point in time to a long interval; the entity scope can range from a single node to the entire graph. While the diversity of analytical tasks provides a potential for a rich set of insights from historical graphs, it also poses several challenges in constructing a system that can perform those tasks. To the best of our knowledge, none of the existing systems address a majority of those challenges that are described below:
[ ]{} An natural tradeoff between index size and access latencies can be seen in the Log and Copy approaches for snapshot retrieval. Log requires minimal information for encoding the graph’s history, but incurs large reconstruction costs. Copy, on the other hand, provides direct access, but at the cost of excessive storage. The desirable index should consume space of the order of Log index but provide near direct access like Copy.
[ ]{} For [*point*]{} access such as past snapshot retrieval, a time-centric indexing such as DeltaGraph or Copy+Log is suitable. However, for version retrieval tasks such as retrieving a [*node’s history*]{}, entity-centric indexing is the correct choice. Neither of the indexing approaches, however, are feasible in the opposite scenarios. Given the diversity of access needs, we require an index that works well with both styles of lookup at the same time.
[ ]{} Query latencies for a graph also depends on the size of chunks in which the data is indexed. While larger granularities of storage incur wasteful data read for “node retrieval”, a finely chunked graph storage would mean higher number of lookups and aggregation for a 2-hop neighborhood lookup. The physical and logical arrangement of data should take care of access needs of queries of all granularities.
[ ]{} It is evident that graph partitioning is inevitable in the storage and processing of large graphs. However, finding the appropriate strategy to maintain workable partitioning on a constantly [*changing*]{} graph is another challenge while designing a historical graph index.
[ ]{} A platform for expressing a wide variety of historical graph analytics requires an appropriate amalgam of temporal logic and graph theory. Additionally, utilizing a vast body of existing tools in network science is an engineering challenge and opportunity.
[ ]{} Parallelization is the key to scale up analytics for large network datasets. It is essential that the underlying data-representations and operators in the analytical platform be designed for parallel computing.
System Overview
---------------
Figure \[fig:overview\] shows the architecture of our proposed Historical Graph Store. It consists of two main components:
[ ]{} records the entire history of a graph compactly while enabling efficient retrieval of several temporal graph primitives. It encodes various forms of differences (called *deltas*) in the graph, such as atomic events, changes in subgraphs over intervals of time, etc. It uses specific choices of graph partitioning, data replication, temporal compression and data placement to optimize the graph retrieval performance. TGI uses the Apache Cassandra distributed key-value store as the backend to store the deltas. In Section \[sec:tgi\], we describe the design details of TGI and the access algorithms. [ ]{} provides a *temporal node*-centric abstraction for specifying and executing complex temporal network analysis tasks. We provide a Java and Python based library to specify the retrieval, computation and analysis on a *set of (temporal) nodes (SoN)*. Computational scalability is achieved by distributing tasks by node and time. TAF is built on top of Apache Spark for supporting scalable, in-memory, cluster computation and provides an option to utilize GraphX for static graph computation. In Section \[sec:taf\], we describe the details of the library, query processing, parallel data fetch aspects of the system, along with a few examples of analytics.
![System Overview[]{data-label="fig:overview"}](TGIFoverview.pdf){width="\linewidth"}
Temporal Graph Index {#sec:tgi}
====================
In this section, we investigate the issue of indexing temporal graphs. First, we introduce a *delta framework*[^5] to define any temporal index as a set of different changes or *deltas*. Using this framework, we are able to qualitatively compare the access costs and sizes of different alternatives for temporal graph indexing, including our proposed approach. We then present the Temporal Graph Index (TGI), that stores the entire history of a large evolving network in the cloud, and facilitates efficient parallel reconstruction for different graph primitives. TGI is a generalization of both entity and time-based indexing approaches and can be tuned to suit specific workload needs. We claim that TGI is the minimal index that provides efficient access to a variety of primitives on a historical graph, ranging from past snapshots to versions of a node or neighborhood. We also describe the key partitioning strategies instrumental in scaling to large datasets across a cloud storage.
Preliminaries
-------------
We start with a few preliminary definitions that help us formalize the notion of the delta framework.
A *static node* refers to the state of a vertex in a network at a specific time, and is defined as a *set* containing: (a) *node-id*, denoted *I* (an integer), (b) an edge-list, denoted *E* (captured as a set of node-ids), (c) attributes, denoted *A*, a map of key-value pairs.
A *static edge* is defined analogously, and contains the node-ids for the two endpoints and the edge direction in addition to a map of key-value pairs. Finally, a *static graph component* refers to either a static edge or a static node.
\[Delta\] A Delta ($\Delta$) refers to either: (a) a static graph component (including the empty set), or (b) a difference, sum, union or intersection of two deltas.
\[Cardinality and Size\] The cardinality\
and the size of a $\Delta$ are the unique and total number of static node or edge descriptions within it, respectively.
\[$\Delta$ Sum\] A sum (+) over two deltas, $\Delta_1$ and $\Delta_2$, i.e., $\Delta_s = \Delta_1 + \Delta_2$ is defined over graph components in the two deltas as follows: (1) $\forall gc_1 \in \Delta_1$, if $\exists gc_2 \in \Delta_2\ s.t.\ gc_1.I = gc_2.I$, then we add $gc_2$ to $\Delta_s$, (2) $\forall gc_1 \in \Delta_1\ s.t.\ \nexists gc_2 \in \Delta_2\ s.t.\ gc_1.I = gc_2.I$, we add $gc_1$ to $\Delta_s$, and (3) analogously the components present only in $\Delta_2$ are added to $\Delta_s$.
Note that: $\Delta_1 + \Delta_2 = \Delta_2 +\Delta_1$ is not necessarily true due the order of changes. We also note that: $\Delta_1 + \emptyset = \Delta_1$, and $(\Delta_1 + \Delta_2) + \Delta_3 = \Delta_1 + (\Delta_2 + \Delta_3)$. Analogously, difference(-) is defined as a set difference over different components of the two deltas. $\Delta_1 - \phi = \Delta_1$ and $\Delta_1 - \Delta_1 = \phi$, are true, while, $\Delta_1 - \Delta_2 = \Delta_2 - \Delta_1$, does not necessarily hold.
\[$\Delta$ Intersection\] An intersection of two $\Delta$s is defined as a set intersection over the the components of two deltas. $\Delta_1 \cap \phi = \phi$, is true for any delta. Similarly, union of two deltas $\Delta_{\cup} = \Delta_1 \cup \Delta_2$, consists of all elements from $\Delta_1$ and $\Delta_2$. The following is true for any delta: $\Delta_1 \cup \phi = \Delta_1$.
Next we discuss and define some specific types of $\Delta$s:
An *event* is the smallest change that happens to a graph, i.e., addition or deletion of a node or an edge, or a change in an attribute value. An event is described around one time point. As a $\Delta$, an event concerning a graph component $c$, at time point $t_e$, is defined as the difference of state of $c$ at and before $t_e$, i.e., $\Delta_{event}(c,t_e) = c(t_e) - c(t_e-1)$.
An *eventlist* delta is a chronologically sorted set of event deltas. An eventlist’s scope may be defined by the time duration, $(t_s,t_e]$, during which it defines all the changes that happened to the graph.
An *partitioned eventlist* delta is an eventlist constrained by the scope of a set of nodes (say a set of nodes, $\mathcal N = \{N_1, N_2, \dots\})$ apart from the time range constraint $(t_s,t_e]$.
\[Snapshot\] A snapshot, $\mathcal G^{t_a}$ is the state of a graph $\mathcal G$ at a time point $t_a$. As a $\Delta$, it is defined as the difference of the state of the graph at $t_a$ from an empty set, $\Delta_{snapshot}({\mathcal G}, t_a)= G(t_a)-G(-\infty)$.
\[Partitioned Snapshot\] A partitioned snapshot is a subset of a snapshot. It is identified by a subset of all nodes, $\mathcal P$ in graph, $\mathcal G$ at time, $t_a$. It consists of the state of all nodes at time $t_a$ and all the edges whose at least one of the end points lies in $\mathcal P$ at time, $t_a$.
Prior Techniques
----------------
The prior techniques for temporal graph indexing use changes or differences in various forms to encode time-evolving datasets. We can express them in the $\Delta$ framework as follows. The [***Log***]{} index is equivalent to a set of all [*event*]{} deltas (equivalently, a single [*eventlist*]{} delta encompassing the entire history). The [***Copy+Log***]{} index can be represented as combination of: (a) a finite number of distinct [*snapshot*]{} deltas, and (b) [*eventlist*]{} deltas to capture the change between successive snapshots. Although we are not aware of a specific proposal for a [***vertex-centric***]{} index, however, a natural approach would be to maintain a set of [*partitioned eventlist*]{} deltas, one for each node (with edge information replicated with the endpoints). The [***DeltaGraph***]{} index, proposed in our prior work, is a tunable index with several parameters. For a typical setting of parameters, it can be seen as equivalent to taking a Copy+Log index, and replacing the [*snapshot*]{} deltas in it with another set of deltas constructed hierarchically as follows: for every $k$ successive [*snapshot*]{} deltas, replace them with a single delta that is the intersection of those deltas and a set of difference deltas from the intersection to the original snapshots, and recursively apply this till you are left with a single delta.
Table \[tab:deltacompare\] estimates the cost of fetching different graph primitives as the number and the cumulative size of deltas that need to be fetched for the different indexes. The first column shows an estimate of the total storage space, which varies considerably across the techniques.
-------------- --------------------- --------------------------- --------------------- --------------------------------- --------------------- --------------------------- --------------------- --------------------------- --------------------- --------------------------- ---------------------
Index
Size $\sum_{\Delta}{|\Delta|}$ $\sum_{\Delta}{1} $ $\sum_{\Delta}{|\Delta|}$ $\sum_{\Delta}{1} $ $\sum_{\Delta}{|\Delta|}$ $\sum_{\Delta}{1} $ $\sum_{\Delta}{|\Delta|}$ $\sum_{\Delta}{1} $ $\sum_{\Delta}{|\Delta|}$ $\sum_{\Delta}{1} $
Log $|G|$ $|G|$ $|G| \over |E|$ $|G|$ $|G| \over |E|$ $|G|$ $|G| \over |E|$ $|G|$ $|G| \over |E|$ $|G|$ $|G| \over |E|$
Copy $|G|^2$ $|S|$ $1$ $|S|$ $1$ $|S||G|$ $|G|$ $|S|$ $1$ $|S||G|$ $|G|$
Copy+Log ${|G|^2 \over |E|}$ $|S|+|E|$ $2$ $|S|+|E|$ 2 $|G|$ $|G| \over |E|$ $|S|+|E|$ 2 $|G|$ $|G| \over |E|$
Node Centric $2|G|$ $2.|G|$ $|N|$ $|C|$ $1$ $|C|$ 1 $|R|.|V|$ $|R|$ $|R|.|V|$ $|R|$
DeltaGraph $X_1^*$ h.$|S|+|E|$ $2h$ $h.|S|+|E|$ $2h$ $|G|$ $|G| \over |E|$ $h.(|S|+|E|)$ 2h $|G|$ $|G| \over |E|$
TGI $X_2^{**}$ h.$|S|+|E|$ $2h$ ${h.|S| \over p}+{|E| \over p}$ $2h$ $|V|(1+{|S| \over p})$ $|V|+1$ ${h.(|S|+|E|) \over p}$ 2h $|V|(1+{|S| \over p})$ $|V|+1$
-------------- --------------------- --------------------------- --------------------- --------------------------------- --------------------- --------------------------- --------------------- --------------------------- --------------------- --------------------------- ---------------------
Temporal Graph Index: Definition
--------------------------------
Given the above formalism, a Temporal Graph Index for a graph $\mathcal G$ over a time period $T=[0, t_c]$ is described by a collection of different $\Delta$s as follows:
- Partitioned Eventlists: A set of partitioned eventlist $\Delta$s, $\{E_{tp}\}$, where $E_{tp}$ captures the changes during the time interval $t$ belonging to partition $p$.
- Derived Partitioned Snapshots: Consider $r$ distinct time points, $t_i$, where $1 \le i \le r$, $t_i \in T$, For each $t_i$, we consider $l$ partition $\Delta$s, $P_{j}^i$, $1< j <l$, such that $\cup_{j} P_{j}^i = {\mathcal G^{t_i}}$. There exists a function that maps any node-id(I) in $\mathcal G^{t_i}$ to a unique partition-id($P_j^i$), $f_i: I \rightarrow P_j^{i}$. With a collection of $P_j^{i}$ over $T$ as leaf nodes, we construct a hierarchical tree structure where a parent is the intersection of children deltas. The difference of each parent from its child delta is called as a [*derived partitioned snapshot*]{} and is explicitly stored. Note that $P_j^{i}$’s are not explicitly stored. This is the same as DeltaGraph, with the exception of partitioning.
- Version Chain: For all nodes $\mathcal N$ in the graph $\mathcal G$, we maintain a chronologically sorted list of pointers to all the references for that node in the delta sets described above (a and b). For a node $I$, this is called a *version chain*($VC_I$).
In short, the TGI stores *deltas* or *changes* in three different forms, as follows. The first one is the atomic changes in a chronological order through partitioned eventlists. This facilitates direct access to the changes that happened to a part or whole of the graph at specified points in time. Secondly, the state of nodes at different points in time is stored indirectly in form of the derived partitioned snapshot deltas. This facilitates direct access to the state of a neighborhood or the entire graph at a given time. Thirdly, a meta index stores node-wise pointers to the list of chronological changes for each node. This gives us a direct access to the changes occurring to individual nodes. Figure \[fig:tgi-stuff\](a) shows the arrangement of eventlist, snapshot and derived snapshot partitioned deltas. Figure \[fig:tgi-stuff\](b) shows a sample version chain.
TGI utilizes the concept of temporal consistency which was optimally utilized by DeltaGraph. However, it differs from DeltaGraph in two major ways. First, it uses a partitioning for eventlists, snapshots or deltas instead of a large monolithic chunks. Additionally, it maintains a list of version chain pointers for each node. The combination of these two novelties along with DeltaGraph’s temporal compression generalizes the notion of entity-centric and time-centric indexing approaches in an efficient way. This can be seen by the qualitative comparison shown in Table \[tab:deltacompare\] as well as empirical results in Section \[sec:experiments\].
\
TGI: Design and Architecture
----------------------------
In the previous subsection, we presented the logical description of TGI. We now describe the strategies for physical storage on a cloud which enables high scalability. In a distributed index, we desire that all graph retrieval calls achieve maximum parallelization through equitable distribution. A distribution strategy based on pure node-based key is good idea for snapshot style access, however, it is bad for a subgraph history style of access. A pure time-based key strategy on the other hand, has complementary qualities and drawbacks. An important related challenge for scalability is dealing with two different skews in a temporal graph dataset – temporal and topological. They refer to the uneven density of graph activity over time and the uneven edge density across regions of the graph, respectively. Another important aspect to note is that for a retrieval task, it is desirable that all the required micro-deltas on a particular machine be proximally located to minimize latency of lookups[^6].
Based on the above constraints and desired properties, we describe the physical layout of TGI as follows:
1. The entire history of the graph is divided into *time spans*, keeping the number of changes to the graph consistent across different time spans, $f_t: e.time \rightarrow tsid$, where $e$ is the event and $tsid$ is the unique identifier for the time span. This is illustrated in Figure \[fig:timespans\].
2. A graph at any point is horizontally partitioned into a fixed number of *horizontal partitions* based upon a random function of the node-id, $f_h: nid \rightarrow sid$, where $nid$ is the node-id and $sid$ is unique identifier of for the horizontal partition.
3. The micro-deltas (including eventlists) are stored as a key-value pairs, where the delta-key is composed of\
$\{tsid, sid, did, pid\}$, where $did$ is a delta-id, and $pid$ is the partition-id of the micro-delta.
4. The placement-key is defined as a subset of the composite deltas key described above, as $\{tsid, sid\}$, which defines the chunks in which data is placed across a set of machines on a cluster. A combination of the $tsid$ and $sid$ ensure that a large fetch task, whether snapshot or version oriented, seeks data distributed across the cluster and not just one machine.
5. The micro-deltas are clustered by the delta key. The given order of the delta-key besides the placement-key elements, means that all the micro-partitions of a delta are stored contiguously, which makes it efficient to scan and read all micro-partitions belonging to a delta in a snapshot query. On the other hand, if the order of $did$ and $pid$ is reversed, it makes fetching a micro-partition across different deltas more efficient.
![The graph history is divided into non-overlapping periods of time. Such division is based on time intervals after which the locality-based graph partitioning is updated. It is also used as a partial key for data chunking and placement.[]{data-label="fig:timespans"}](TimeSpans.pdf){width="\linewidth"}
Irrespective of a temporal or a topological skew in the graph, the index is spread out across a cluster in a balanced manner. This also makes it possible to fetch the graph primitives of large sizes in a naturally parallel manner. For instance, a snapshot query would demand all micro-partitions for a specific set of deltas in a particular timespan across all horizontal partitions. Given an equitable distribution of the deltas across all machines of a cluster, we retrieve the data in parallel on each storage machine, without a considerable skew.
[ ]{} TGI uses Cassandra for its delta storage. There are 5 tables that contain TGI data and metadata:\
(1) `Deltas(tsid, sid, did, pid, dval)` table stores the deltas as described above, where `dval` contains serialized value of the micro-delta as a binary string.\
(2) `Versions(nid, vchain)` consists of each node’s version chain as a hash-table with keys for each timespan.\
(3) `Timespans(tsid, start, end, checkpts, k, df)` stores, for each timespan, start and end times, a list of snapshot checkpoints, and arity.\
(4) `Graph(start, end, events, tscount, gtype)` contains global information about the graph and TGI.\
(5) `Micropartitions(nid, tsid, pid)` contains micro-delta partitioning information about nodes. It is not utilized in case of random partitioning.\
The graph construction and fetching modules are written in Python, using Pickle and Twisted libraries for serialization and communication.
[ ]{} TGI architecture can be seen in Figure \[fig:tgi-arch\], where *Query Mananger (QM)* is responsible for planning, dividing and delegating the query to one or more *Query Processors (QP)*. The QPs query the datastore in parallel and process the raw deltas into the required result. Depending on the query specification, the distributed result is either aggregated at a particular QP (the QM) or returned to the client which made the request without aggregation. The *Index Manager* is responsible for the construction and maintenance activities of the index. The cloud represents the distributed datastore.
[ ]{} The construction process involves three different stages. First, we analyze the input data using the index construction parameters including the timespan length (ts), number of horizontal partitions (ns), number of likely datastore nodes (m), eventlist size (l), and micro-delta partition size (psize). In the second stage, the input data is split into horizontal partitions. In the third stage, parallel construction workers of the IM work on separate horizontal partitions, and build the index, a time span at a time. The process of construction of each timespan is similar to that of DeltaGraph, albeit more fine-grained due to delta partitioning and version chain construction as well. The TGI accepts updates of events in batches of timespan length. The update process involves creating an independent TGI with the new events, and merging it with the original TGI. The merger of TGIs involves updates of corresponding deltas, VC index and the metadata.
Dynamic Graph Partitioning
--------------------------
Partitioning deltas into micro-deltas is an essential aspect of TGI and provides cheaper access to subgraph elements when compared to DeltaGraph or similar indexes. In a time-evolving graph, however, the size and topology of the graph change with time. The key is to keep the size of each micro-delta (and each micro-eventlist) about the same and bounded by a number that dictates the latency for fetching a node or neighborhood. The two traditional approaches to partitioning a static graph are random (node-id hash-based) or locality-based (min-cut max-flow) partitioning. Random partitioning is simpler and involves minimal bookkeeping. However, since it loses locality, it is unsuitable for neighborhood-level granularity access. Locality-aware partitioning, on the other hand, preserves locality but incurs extra bookkeeping in form of a {node-id:partition-id} map. TGI is designed to work with either configuration as desired, as well as different partition size specifications. TGI also supports replication of edge-cuts for further speed up of 1-hop neighborhoods. It uses a separate [*auxiliary micro-delta*]{} besides each micro-delta to store the replication, thereby preventing extra read cost for snapshot or node centric queries. This is illustrated in Figure \[fig:micrdeltasaux\].
Locality-aware partitioning, however, faces an additional challenge with time-evolving graphs. With the change in size and topology of a graph, a partitioning deemed good (with respect to locality) at an instant may cease to be good at a later time. A probable approach of frequent repartitioning over time would maintain partitioning quality, but leads to excessive amounts of bookkeeping, which in turn leads to degradation of performance while accessing different node or neighborhood versions. Maintaining and looking up that map as frequently as the changes in the graph is highly inefficient. Hence, we divide the history of the graph into [*time spans*]{}, where we keep the partitioning consistent within each time span, but perform it afresh it at the beginning of each new time span. This gives rise to two problems, described briefly as follows. Firstly, given a graph over time span, $T \in [t_s, t_e)$, find the graph partitioning that minimizes the edge cuts across all time points combined. Secondly, to determine the appropriate points for the end of a time span and the beginning of a new one, with respect to over all query performance. We discuss these problems below.
Static graph partitioning for an undirected and unweighted graph $G=(V, E)$ into $k$ partitions is defined as follows. Each node $v_i \in V$ is assigned a partition set $P_r$ such that $0 \le r < k$. The constraint is that $ {\left\lfloor |V| \over k \right\rfloor} \le |P_r| \le {\left\lceil |V| \over k \right\rceil}$, i.e., the partitions are more or less equal in size. The number of edge cuts across partitions are intended to be minimized, i.e., a count of all edges whose end points lie in different partitions. For a weighted graph, the edge cut cost is counted as the sum of the edge weights, which pushes stronger relationships (with higher edge weights) to be preferred for being in the same partition over the lesser weighted ones. Also, in case of a node weighted graph, the partition set count can be determined using the node weight. Different graph partitioning algorithms work under these constraints using one or the other heuristic, as described before.
For a dynamic graph partitioning, we consider an edge and node weighted, undirected time evolving graph, without the loss of generality. Consider the following: graph $G^T=(V^\tau, E^\tau, W_E^\tau, W_N^\tau)$ where, $\tau \in [t_s, t_e)$, is the time range for which we find a single partitioning; $V^\tau, E^\tau, W_E^\tau, W_N^\tau$ are the set of vertices, edges, edge weights, node weights over time $\tau$, respectively. Our partitioning strategy involves projecting the graph over time range $T$ to a single point in time using a *time collapsing* function $\Omega$, there by reducing the graph $G^\tau$ to a static graph, $G_\tau = \Omega(G^\tau)$. The constraint on function $\Omega$ is that $G_\tau$ must contain all the vertices that existed in $G^\tau$ at least once in $G^\tau$. Using $G_\tau$, we can employ static graph partitioning to find a suitable partitioning technique in the following manner.
The choice of $\Omega$ function determines how well the $G_T$ is a representation of $G^\tau$. Let us consider a few different options. (1) Median: consider the time point $t$ which is the median of the end points of $\tau$. The edges and weights in $G_\tau$ are the edge weights in $G^t$. (2) Union-Max: for an edge that existed at any time in $G^\tau$, we include it in $G_\tau$ such that its weight is the maximum value from all time points in $G^T$. (3) Union-Mean: for an edge that existed at any time in $G^\tau$, we include it in $G_\tau$, where its weight is the weighted average (time fraction) of the edge weights in $G^\tau$. Non existence of an edge during a time period counts as a $0$ contribution for the respective time period. (4). For any of the cases above, the node weight, $w_n$, can be defined independently of the edge set and edge weights. We consider three options as follows. (1) $w_n = 1$ for each nodes $n$ in $G_\tau$; (2) $w_n=degree(n)$ for each node $n$ in $G_T$; (3) $w_n={\bar{degree(n)}}$, i.e., average degree over $\tau$.
Given these different heuristic combinations, we plan to study their empirical behavior and use the apparently most suitable one for TGI partitioning. The default TGI partitioning uses Union-Max for edge weights and uniform node weights.
We argue that this style of partitioning that involves first projecting a temporal graph to a static one, followed by conventional forms of static graph partitioning, is better than other conceivable alternatives. One such alternative way of doing it is to determine the partitioning at different time points in $\tau$, say $P^\tau$ and then reducing $P^\tau$ to $P_\tau$, a single partitioning scheme. This approach has the following major disadvantages. Firstly, the output partitions from a a static graph partitioning algorithm for two versions of graph $G^\tau$, say $G^1$ and $G^2$ are not aligned, even when the two snapshots are similar to a large extent. This is attributed to some degree of randomness associated with graph partitioning algorithms. This makes it infeasible to combine $P^1$ and $P^2$ in to a single result. Secondly, this approach is much more expensive compared to our approach, because it involves computing $\tau$ orders of partitions. Another alternative approach is to use one of the *online graph partitioning* algorithms, which updates a partition set for a graph upon a small change in the graph. However, the output of such an approach only gives us partitioning schemes at different time points. The partitions across time are better aligned to each other than the previous approach, but we would still need to compute a combined partitioning from all available partitions, and the notion of time collapsing is inevitable. Secondly, the partitioning results from incremental graph partitioning are often inferior compared to the batch mode of partitioning for obvious reasons.
Determining the appropriate number and the exact boundaries of time-spans is another important issue. The need for creating higher number of time-spans and hence reducing the duration of a time-span is to maintain healthier partitioning. Let the hit taken on query latencies (assuming a certain query load Q) due to a subpar snapshot partitioning be, $f(T)$. This hit is generally incurred on k-hop queries, without replication, due to higher number of micro-delta seeks. In case of replication across partitions, the degree of replication increases with inferior partitioning, and leads to indirect impact on query latencies. On the other hand, there is need to create longer time-spans because the version queries require multiple micro-deltas, at different time points. Higher the changing number of partitions over query’s time interval, say $t$, higher the query latency. Let us say that for an average query time interval (again, as per a specific query load), the gain due to longer time spans is, $g(T)$. The appropriate length of a average time-span hence is the solution of the maxima of $g(T) - f(T)$. In practice, uniform time-span length in numbers of the number of events is perhaps the most convenient. While the models of $f$ and $g$ are complex, a good number for size of $T$ can be observed empirically.
\
Fetching Graph Primitives
-------------------------
We briefly describe access methods for different graph primitives. The algorithms provided here use primitive TGI fetch methods whose description should be self-explanatory from their nomenclature. [ ]{} In snapshot retrieval, the state of a graph at a time point is retrieved. Given a time $t_s$, the query manager locates the appropriate time span $T$ such that $t_s \in T$, within which, it figures out the path from the root of the TGI to the leaf closest to the given time point. All the snapshot deltas, $\Delta_{s1}, \Delta_{s2}, \dots, \Delta_{sm}$, (i.e., all their micro-partitions) along that path and the eventlists from the leaf node to the time point, $\Delta_{e1}, \Delta_{e2}, \dots, \Delta_{en}$ are fetched and merged appropriately as: $\sum_{i=1}^m{\Delta_{si}} + \sum_{i=1}^n{\Delta_{ei}}$ (notice the order). This is performed across different query processors covering the entire set of horizontal partitions. The procedure for snapshot retrieval is specified in Algorithm \[algo:sr\].
$t'\gets {\sf GetNearestPartTime}(t)$ $K\gets {\sf GetNearestPartKeys}(t)$ $D \gets {\sf GetDeltas}(K)$ $g \gets \emptyset$ $g \gets g + d$ $ B \gets {\sf GetEventLists}(t',t)$ $b \gets {\sf FilterByTime}(b, t',t)$ $g \gets g +b$ **return** $g$
[ ]{} Retrieving a node’s history during time interval, $[t_s, te)$ involves finding the state of the graph at point $t_s$, and all changes during the time range $(t_s, t_e)$. The first one is done in a similar manner to snapshot retrieval except the fact that we look up only a specific micro-partition in a specific horizontal partition, that the node belongs to. The second part happens through fetching the node’s version chain to determine its points of changes during the given range. The respective eventlists are fetched and filtered for the given node. The procedure for node-history retrieval is specified in Algorithm \[algo:nh\].
$C\gets {\sf GetVC}(I)$ $ C \gets {\sf FilterByTime}(C, t_s, t_e)$ $D \gets {\sf GetDeltas}(C)$
$I_N \gets \emptyset$ $D \gets {\sf FilterByTime}(D,t_s,t_e)$ $D \gets {\sf FilterById}(D,I)$ $I_N \gets I_N \cup d$ **return** $I_N$
[ ]{} In order to retrieve the k-hop neighborhood of a node, we can proceed in two possible ways. One of them is to fetch the whole graph snapshot and filter the required subgraph. The other is to fetch the given node, and then determine its neighbors, fetch them, and recurse. It is easy to see that the performance of the second method will deteriorate fast with growing $k$. However for lower values, typically $k \le 2$, the latter is faster or at least as good, especially if we are using neighborhood replication as discussed in a previous subsection. In case of a neighborhood fetch, the query manager automatically fetches the auxiliary portions of deltas (if they exist), and if the required nodes are found, further lookup is terminated. Two different procedures for fetching a k-hop neighborhood are specified in Algorithm \[algo:khop1\] and Algorithm \[algo:khop2\], respectively.
$g \gets {\sf GetSnapshot}(t)$ $ C \gets \{I\}$ $ R \gets \{I\}$
$C \gets C \cup N$ $S \gets S \cup N$ $g' \gets {\sf FilterByID}(g, C)$ **return** $g'$
$N \gets {\sf GetNode}(I,t)$ $M \gets { \sf GetNeighbors}(N)$ $G \gets \emptyset$ $L \gets \emptyset$ $N \gets {\sf GetNode(m)}$ $G \gets G + R$ $L \gets L \cup {\sf GetNeighbors}(m)$
**return** $G$
[ ]{} Neighborhood evolution queries can be posed in two different ways. First, requesting all changes for a described neighborhood, in which case the query manager fetches the initial state of the neighborhood followed by the events indicating the change. Second, requesting the state of the neighborhood at multiple specific time points. This translates to the retrieval of multiple single neighborhoods fetch tasks. Algorithm \[algo:1hophist\] specifies the procedure to fetch one hop neighborhood history. The general k-hop evolution process can be seen as a combination of the 1-hop evolution procedure along with the k-hop (static) neighborhood retrieval.
**return** $G$
Analytics Framework {#sec:taf}
===================
In this section, we describe the *Temporal Graph Analysis Framework (TAF)*, that enables programmers to express and execute complex analytical tasks on time-evolving graphs. We present details of the novel model of computation, including a library of temporal graph operators and operands (exposed through Python and Java APIs); we also present the details of implementation on top of Apache Spark, which enables scalable, parallel, in-memory execution. Finally, we describe TAF’s coordination with TGI to provide a complete ecosystem for historical graph management and analysis.
Temporal Graph Analysis Library {#subsec:tafcore}
-------------------------------
In this section, we describe a set of operators for analyzing large historical graphs. At the heart of this library is a data model where we view the historical graph as a *set of nodes or subgraphs evolving over time*. The choice of temporal nodes as a primitive helps us describe a wide range of fetch and compute operations in an intuitive manner. More importantly, it provides us an abstraction to parallelize computation. The *temporal nodes* and *set of temporal nodes* bear a correspondence to *tuples* and *tables* of the relational algebra, as the basic unit of data and the prime operand, respectively. [ ]{} The two central data types are defined below:
A *temporal node* (NodeT), $ N^T$, is defined as a sequence of all and only the states of a node $ N$ over a time range, $T=[t_s, t_e)$. All the $k$ states of the node must have a valid time duration $T_i$, such that $\cup_i^k T_i = T$ and $\cap_i^k T_i = \phi$.
A *SoN*, is defined as a set of $r$ temporal nodes $\{ N_1^T, N_2^T \dots N_r^T \}$ over a time range, $T=[t_s, t_e)$, as depicted in Figure \[fig:son\].
The [*NodeT*]{} class provides a range of methods to access the state of the node at various time points, including: [getVersions()]{} which returns the different versions of the node as a list of static nodes (NodeS), [getVersionAt()]{} which finds a specific version of the node given a timepoint, [getNeighborIDsAt()]{} which returns IDs of the neighbors at the specified time point, and so on.
A [*Temporal Subgraph (SubgraphT)*]{} generalizes NodeT and captures a sequence of the states of a subgraph (i.e., a set of nodes and edges among them) over a period of time. Typically the subgraphs correspond to $k$-hop neighborhoods around a set of nodes in the graph. An analogous [getVersionAt()]{} function can be used to retrieve the state of the subgraph as of a specific time point as an in-memory [Graph]{} object (the user program must ensure that any graph object so created can fit in the memory of a single machine). A Set of Temporal Subgraphs (SoTS) is defined analogously to SoN as a set of temporal subgraphs.
[ ]{} Below we discuss the important temporal graph algebra operators supported by our system.\
1. [***Selection***]{} accepts an SoN or an SoTS along with a boolean function on the nodes or the subgraphs, and returns an SoN or SoTS. Selection performs *entity-centric filtering* on the operand, and does not alter temporal or attribute dimensions of the data.\
2. [***Timeslicing***]{} accepts an SoN or an SoTS along with a timepoint (or time interval) $t$, finds the state of each of individual nodes or subgraphs in the operand as of $t$, and returns it as another SoN or SoTS, respectively (SoN/SoTS can represent sets of static nodes or subgraphs as a well). The operator can accept a list of timepoints as input and return a list.\
3. [***Graph***]{} accepts an SoN and returns an in-memory Graph object containing the nodes in the SoN (with only the edges whose both endpoints are in the SoN). An optional parameter, $t_p$, may be specified to get a GraphS valid at time $t_p$.\
4. [***NodeCompute***]{} is analogous to a [*map*]{} operation; it takes as input an SoN (or an SoTS) and a function, and applies the function to all the individual nodes (subgraphs) and returns the results as a set.\
5. [***NodeComputeTemporal***]{}. Unlike [NodeCompute]{}, this operator takes as input a function that operates on a static node (or subgraph) in addition to an SoN (or an SoTS); for each node (subgraph), it returns a sequence of outputs, one for each different state (version) of that node (or subgraph). Optionally, the user may specify another function (NodeComputeDelta) that operates on the delta between two versions of a node (subgraph), which the system can use to compute the output more efficiently. An optional parameter is a method describing points of time at which computation needs to be performed; in the absence of it, the method will be evaluated at all the points of change.\
6. [***NodeComputeDelta***]{} operator takes as input: (a) a function that operates on a static node (or subgraph) and produces an output quantity, (b) an SoN (or an SoTS) like\
[NodeComputeTemporal]{}, (c) a function that operates on the following: a static node (or subgraph), some auxiliary information pertaining to that state of the node (or subgraph), the value of the quantity at that state, and an *update* (event) to it. This operator returns a sequence of outputs, one for each state of the node (or subgraph), similar to\
[NodeComputeTemporal]{}. However, the method of computation in this method is different because it updates the computed quantity for each version incrementally instead of computing it afresh. An optional parameter is the method describing points of time at which to base the comparison. An optional parameter is a method describing points of time at which computation needs to be performed; in the absence of it, the method will be evaluated at all the points of change.\
7. [***Compare***]{} operator takes as input two SoNs (or two SoTSs) and a scalar function (returning a single value), computes the function value over all the individual components, and returns the differences between the two as a set of [*(node-id, difference)*]{} pairs. This operator tries to abstract the common operation of comparing two different snapshots of a graph at different time points. A simple variation of this operator takes a single SoN (or SoTS) and two timepoints as input, and does the compare on the timeslices of the SoN as of those two timepoints. An optional parameter is the method describing points of time at which to base the comparison.\
8. [***Evolution***]{} operator samples a specified quantity (provided as a function) over time to return evolution of the quantity over a period of time. An optional parameter is the method describing points of time at which to base the evolution.\
9. [***TempAggregation***]{} abstractly represents a collection of temporal aggregation operators such as [Peak]{}, [Saturate]{}, [Max]{}, [Min]{}, and [Mean]{} over a scalar timeseries. The aggregation operations are used over the results of temporal evaluation of a given quantity over an SoN or SoTs. For instance, finding “times at which there was a [*peak*]{} in the network density” is used to find eventful timepoints of high interconnectivity such as conversations in a cellular network, or high transactional activity in a financial network.
System Implementation
---------------------
The library is implemented in Python and Java and is built on top of the Spark API. The choice of Spark provides us with an efficient in-memory cluster compute execution platform, circumventing dealing with the issues of data partitioning, communication, synchronization, and fault tolerance. We provide a GraphX integration for utilizing the capabilities of the Spark based graph processing system for static graphs.
The key abstraction in Spark is that of an RDD, which represents a collection of objects of the same type, stored across a cluster. SoN and SoTS are implemented as RDDs of NodeT and SubgraphT respectively (i.e., as [RDD<NoteT>]{} and [RDD<SubgraphT>]{}). The in-memory graph objects may be implemented using any popular graph representation, specially the ones that support useful libraries on top. We now describe in brief the implementation details for NodeT and SubgraphT, followed by details of the incremental computational operator, and the parallel data fetch operation.
Figure \[fig:taf-code-ex\] shows sample code snippets for three different analytical tasks – (a) finding the node with the *highest clustering coefficient* in a historical snapshot; (b) *comparing different communities* in a network; (c) finding the *evolution of network density* over a sample of ten points.
\
\
[ ]{} A set of temporal nodes is represented with an `RDD` of `NodeT` (temporal node). A temporal node contains the information for a node during a specified time interval. The question of the appropriate physical storage of the `NodeT` (or `SubgraphT`) structure is quite similar to storing a temporal graph on disk such as the one using a DeltaGraph or a TGI, however, in-memory instead of disk. Since NodeT is fetched at query time, it is preferable to avoid creating a complicated index, since the cost to create the index at query time is likely to offset any access latency benefits due to the index. An intuitive guess based upon examination of certain temporal analysis tasks is that its access pattern is most likely going to be in a chronological order, i.e., the query requesting the subsequent versions or changes, in order of time. Hence, we store `NodeT` (and `SubgraphT`) as an initial snapshot of the node (or subgraph), followed by a list of chronologically sorted events. It provides methods such as `GetStartTime()`, `GetEndTime()`, `GetStateAt()`, `GetIterator()`, `Iter- ator.GetNextVersion()`, `Iterator.GetNextEvent()`, and so on. We omit the details of these methods as their functionality is apparent from the nomenclature.
[ ]{} `NodeComputeDelta` evaluates a quantity over each NodeT (or SubgraphT) using two supplied methods, $f()$ which computes the quantity on a state of the node or subgraph, and, $f_\Delta()$, which updates the quantity on a state of the node or subgraph for a given set of event updates. Consider a simple example of finding the fraction of nodes with a specific attribute value in a given `SubgraphT`. If this were to be performed using\
`NodeComputeTemporal`, the quantity will be computed afresh on each new version of the subgraph, which would cost $\mathcal{O}(N.T)$ operations where $N$ is the size of the operand (number of nodes) and $T$ is the number of versions. However, using the incremental computation, each new version can be processed in constant time after the first snapshot, which adds up to, $\mathcal{O}(N + T)$. While performing the incremental computation, the corresponding $f_\Delta ()$ method is expected to be defined so as to evaluate the nature of the event – whether it brings about any changing the output quantity or not, i.e., a scalar change value based upon the actual event and the concerned portions of the state of the graph, and also update the auxiliary structure, if used. Code in Figure \[fig:computedelta\_taf\] illustrates the usage of [NodeComputeTemporal]{} and [NodeComputeDelta]{} in a similar example.
\
Consider a somewhat more intricate example, where one needs to find counts of a small pattern [*over time*]{} on an [SoTS]{}, such as finding the occurrence of a subgraph pattern in the data graph’s history. In order to perform such pattern matching over long sequences of subgraph versions, it is essential to maintain certain inverted indexes which can be looked up to answer in constant time whether an event has caused a change in the answer from a previous state or caused a change in the index itself, or both. Such inverted indexes, quite common to subgraph pattern matching, are required to be updated with every event; otherwise, with every new event update, we would need to look up the new state of the subgraph afresh which would simply reduce it to performing non-indexed subgraph pattern matching over new snapshots of a subgraph at each time point, which is a fairly expensive task. In order to utilize a constantly updated set of indices, the auxiliary information, which is a parameter and a return type for $f_\Delta()$, can be utilized. Note that such an incremental computational operator opens up possibilities for using a considerable amount of algorithmic work available in literature on online and streaming graph query evaluation, respectively, to be applied to historical graph analysis. For instance, there is work on pattern matching in streaming [@wang2009continuous; @gao2014continuous] and incremental computing [@fan2013incremental; @varro2006incremental] contexts, respectively.
[ ]{} In the -oriented operators on an [SoN]{} or an [SoTS]{}, the time points of evaluation, by default, are all the points of change in the given operand. However, a user may choose to provide a definition of which points to select. This can be as simple as returning a constant set of timepoints, or based on a more complex function of the operand(s). Except the [Compare]{} operator, which accepts two operands, other operators allow an optional function, which works on a singe temporal operand; the compare accepts a similar function that operates on two such operands. Two such examples can be seen in Figure \[fig:timepts\_taf\].
\
[ ]{} In a temporal graph analysis task, we first need to instantiate a *TGI connection handler* instance. It contains configurations such as address and port of the TGI *query manager host*, *graph-id*, and a *SparkContext* object. Then, a SON (or SOTS) object is instantiated by passing the reference to the TGI handler, and any query specific parameters (such as k-value for fetching 1-hop neighborhoods with SOTS). The next few instructions specify the semantics of the graph to be fetched from the TGI. This is done through the commands explained in Section \[subsec:tafcore\], such as the `Select`, `Filter`, `Timeslice`, etc. However, the actual retrieval from the index doesn’t happen until the first statement following the specification instructions. A `fetch()` command can be used explicitly to tell the system to perform the fetch operation. Upon the `fetch()` call, the analytics framework sends the combined instructions to the query planner of the TGI, which translates those instructions into an optimal retrieval plan. This prevents the system from retrieving large amounts of data from the index that is a superset of the required information and prune it later.
The analytics engine runs in parallel on a set of machines, so does the graph index. The parallelism at both places speeds up and scales both the tasks. However, if the retrieval graph at the TGI cluster was aggregated at the Query Manager and sent serially to the master of the analytical framework engine after which it was distributed to the different machines on the cluster, it would create a space and time bottleneck at the Query Manager and the master, respectively, for large graphs. In order to bypass this situation, we have designed a parallel fetch operation, in which there is a direct communication between the nodes of the analytics framework cluster and the nodes of the TGI cluster. This happens through a protocol that can be seen in Figure \[fig:parallel-fetch\]. The protocol is briefly described in the following ordered steps:
1. Analytics query containing fetch instructions is received by the TAF master.
2. A handshake between the TAF master and TGI query manager is established. The latter receives fetch instructions and the former is made aware of the active TGI query processor nodes.
3. Parallel fetch starts at the TGI cluster.
4. The TAF master instantiates a TGIDriver instance at each of its cluster machines wrapped in a RDD.
5. Each node at the TAF performs a handshake with one or more of the TGI nodes.
6. Upon completion of fetch at TGI, the individual TGI nodes transfer the SoN to an RDDs on the corresponding TAF nodes.
Experimental Evaluation {#sec:experiments}
=======================
In this section, we empirically evaluate the efficiency of TGI and TAF. To recap, TGI is a persistent store for entire histories of large graphs, that enables fast retrieval for a diverse set of graph primitives – snapshots, subgraphs, and nodes at past time points or across intervals of time. We primarily highlight the performance of TGI across the entire spectrum of retrieval primitives. We are not aware of a baseline that may compete with TGI across all or a substantial subset of these retrieval primitives. Specialized alternatives such as DeltaGraph for snapshot retrieval is highly unsuitable for node or neighbor version retrieval; a version centric index may be specialized for node-version retrieval but is highly unsuitable for snapshot or neighborhood-version style retrieval. Also note that TGI generalizes all the known approaches including those two; using appropriate parameter configurations, it can even converge to any specific alternative. Secondly, we demonstrate the scalability of TGI design through experiments on parallel fetching for large and varying data sizes. Finally, we also report experiments demonstrating computational scalability of the TAF for a graph analysis task, as well as the benefits of our incremental computational operator.
[ ]{} We use four datasets: (1) Wikipedia citation network consisting of 266,769,613 edge addition events from Jan 2001 to Sept 2010. At its largest point, the graph consists of 21,443,529 nodes and 122,075,026 edges; (2) We augment Dataset 1 by adding around 333 million synthetic events which randomly add new edges or delete existing edges over a period of time, making a total of 700 million events; (3) Similarly, we add 733 million events, making the total around 1 billion events; (4) Using a Friendster gaming network snapshot, we add synthetic dates at uniform intervals to 500 million events with a total of approximately 37.5 million nodes and 500 million edges.
Following key parameters that are varied in the experiments: data store machine count ($m$), replication across dataset ($r$), number of parallel fetching clients ($c$), eventlist size ($l$), snapshot or eventlist partition size ($ps$), and Spark cluster size ($m_a$).
We conducted all experiments on an Amazon EC2 cluster. Cassandra ran on machines containing 4 cores and 15GB of available memory. We did not use row caching and the actual memory consumption was much lower that the available limit on those machines. Each fetch client ran on a single core with up to 7.5GB available memory. The machines with TAF nodes running Spark workers ran on a single core and 7.5GB of available memory each.
![Snapshot retrieval times for varying parallel fetch factor ($c$), on Dataset 1; $m=4$; $r=1, ps=500$.[]{data-label="fig:ss1"}](SS267m-MultiCore_4MachDB.pdf){width="\linewidth"}
[ ]{} Figure \[fig:ss1\] shows the snapshot retrieval times for Dataset 1 for different values of the parallel fetch factor, $c$. As we can see, the retrieval cost is directly proportional to the size of the output. Further, using multiple clients to retrieve the snapshots in parallel gives near-linear speedup, especially with low parallelism. This demonstrates that TGI can exploit available parallelism well. We expect that with higher values of $m$ (i.e., if the index were distributed across a larger number of machines), the linear speedup would be seen for larger values of $c$ (this is also corroborated by the next set of experiments). The snapshot retrieval times for dataset 4 can be seen in Figure \[fig:friendsterss\].
Figure \[fig:multss\] shows snapshot retrieval performance for three different sets of values for $m$ and $r$. We can see that while there is no considerable difference in performance across the different configurations, using two storage machines slightly decreases the query latency over using one machine, in the case of a single query client, $c=1$. For higher $c$ values, we see that $m=2$ has a slight edge over $m=1$. Also, the behavior for the two $m=1$ and $m=2;r=2$ cases are quite similar for same $c$ values. However, we observed that the latter case allows a higher possibility of $c$ value whereas the former peaks out at a lower $c$ value. Further, the net effect of Cassandra compression for deltas is negligible for TGI. We omit the detailed points of our investigation, but Figure \[fig:compression\] is representative of the general behavior.
Size of the delta partitions (or the number) affects the performance the snapshot retrieval performance only to a small degree as seen in Figure \[fig:ss-varp\]. This occurs due to a the TGI design which makes sure that all the partitions of a delta (micro-deltas) are stored contiguously in a cluster. This demonstrates that TGI is a superset of DeltaGraph where we are able to handle other queries along with efficient snapshot retrieval. Note that we do not provide experimental results on the internals of snapshot retrieval which have been thoroughly explored in our prior work [@icdepaper].
[ ]{} Smaller eventlists or partition sizes provide a lower latency time for retrieving different versions of a node, which can be seen in Figure \[fig:nver-varl\] and Figure \[fig:nver-varp\], respectively. This is primarily due to the reduction in work required for fetching and deserialization. A higher parallel fetch factor is effective in reducing the latency for version retrieval (Figure \[fig:nver-varc\]). Note that the performance of version retrieval and snapshot retrieval with respect to varying partition sizes is contrary and represents a trade-off. However, smaller eventlist sizes benefit both version retrieval and snapshots. Node version retrieval for Dataset 4 shows a similar behavior, which can be seen in Figure \[fig:friendsterNV\].
[ ]{} We compared the performance of retrieving 1-hop neighborhoods, both static and specific versions, using different graph partitioning and replication choices. A topological, flow-based partitioning accesses fewer graph partitions compared to a random partitioning scheme, and a 1-hop neighborhood replication restricts the access to a single partition.This can be seen in Figure \[fig:part\_replication\] for 1-hop neighborhood retrieval latencies. As discussed in Section \[sec:tgi\], the 1-hop replication does not affect other queries involving snapshots or individual nodes, as the replicated portion is stored separately from the original partition. In case of a 2-hop neighborhood retrieval, there are similar performance benefits over random partitioning, which can be reasoned based upon similar speed-ups for 1-hop neighborhoods.
[ ]{} We observed the fetch performance of TGI with an increasing size of the index. We measured the latencies for retrieving certain snapshots upon varying the time duration of the graph dataset, as shown in Figure \[fig:datascale\]. Datasets 2 and 3 contain additional 333 million and 733 million events over dataset 1, respectively. Only a marginal difference in snapshot retrieval performance demonstrates TGI’s scalability for large datasets.
[ ]{} We examined TAF’s performance through an analytical task for determining the highest local clustering coefficient in historical graph snapshot. Figure \[fig:cc\_taf\] shows compute times for the given task on different graph sizes, as well as varying size of the Spark cluster. Speedups due to parallel execution can be observed, especially for larger datasets.
[ ]{} Earlier in the chapter, we presented two separate ways of computing a quantity over changing versions of a graph (or node). Those include, evaluating the quantity on different versions of the graph separately, and alternatively, performing it in an incremental fashion, utilizing the result for the previous version and updating it with respect to the graph updates. This can be seen for a simple node label counting task in Figure \[fig:computedelta\_taf\]. the benefits due to the incremental ([NodeComputeDelta]{} operator) computation over a version-based computation ([NodeComputeTemporal]{} operator) can be seen in Figure \[fig:incr\_exp\_taf\].
Conclusion {#sec:conclusion}
==========
Graph analytics are increasingly considered crucial in obtaining insights about how interconnected entities behave, how information spreads, what are the most influential entities in the data, and many other characteristics. Analyzing the history of how a graph evolved can provide significant additional insights, especially about the future. Most real-world networks however, are large and highly dynamic. This leads to creation of very large histories, making it challenging to store, query, or analyze them. In this paper, we presented a novel Temporal Graph Index that enables compact storage of very large historical graph traces in a distributed fashion, supporting a wide range of retrieval queries to access and analyze only the required portions of the history. We also present a distributed analytics framework, built on top of Apache Spark, that allows analysts to quickly write complex temporal analysis tasks. Our experiments show that our temporal index exhibits very efficient retrieval performance across a wide range of queries, and can effectively exploit the available parallelism in a distributed setting.
[^1]: https://cassandra.apache.org
[^2]: http://thinkaurelius.github.io/titan/
[^3]: https://github.com/tinkerpop/blueprints/wiki
[^4]: https://github.com/tinkerpop/gremlin
[^5]: A delta formalism provided by Ghandeharizadeh et al. [@Ghandeharizadeh:1996:HED:232753.232801] is an interesting related read on this topic.
[^6]: In general, this depends on the underlying storage mechanism. While the physical placement of micro-deltas is irrelevant for a memory-based storage, it is significant for any disk-based storage due to seek times.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We study the nonequilibrium dynamics in the pump-probe spectroscopy of excitonic insulators using the spinless two-orbital model with phonon degrees of freedom in the time-dependent mean-field approximation. We introduce the pulse light as a time-dependent vector potential via the Peierls phase in the Hamiltonian. We find that, in the Bose-Einstein condensation regime where the normal state is semiconducting, the excitonic order is suppressed when the frequency of the pulse light is slightly larger than the band gap, while the order is enhanced when the frequency of the pulse is much larger than the band gap. We moreover find that the excitonic order is completely destroyed in the former situation if the intensity of the pulse is sufficiently strong. In the BCS regime where the normal state is semimetallic, we find that the excitonic order is always suppressed, irrespective of the frequency of the pulse light. The quasiparticle band structure and optical conductivity spectrum after the pumping are also calculated for the instantaneous states.'
author:
- Tetsuhiro Tanabe$^1$
- Koudai Sugimoto$^2$
- Yukinori Ohta$^1$
title: 'Nonequilibrium dynamics in the pump-probe spectroscopy of excitonic insulators'
---
Introduction
============
Nonequilibrium dynamics induced by applying the intense laser pulse have recently been a new way of investigating a variety of quantum condensed phases. Recent achievement of the time resolution of a femto-second order enables one to perform experiments studying the ultrafast dynamics of materials. Examples include a success of observing light-induced superconductivity [@Fausti2011Science; @Mitrano2016Nature] and a pump-probe measurement of melting of charge-density-wave orders [@Perfetti2006PRL; @Schmitt2008Science].
The pump-probe measurement is also applicable to the study of excitonic condensation. In the excitonic phase, holes in the valence band and electrons in the conduction band form pairs called excitons, just like Cooper pairs of electrons in superconductivity, and they undergo quantum condensation at low temperatures [@Jerome1967PR; @Halperin1968RMP]. The realization of such condensation has been suggested in transition-metal chalcogenides $1T$-TiSe$_2$ [@Cercellier2007PRL; @Kogar2017Science] and Ta$_2$NiSe$_5$ [@Wakisaka2009PRL; @Seki2014PRB]. Here, we should note that, since the spin-singlet excitonic state necessarily couples to the phonon degrees of freedom [@Phan2013PRB; @Zenker2014PRB; @Kaneko2013PRB; @Sugimoto2016PRB; @Kaneko2018PRB], it is difficult to single out the excitonic contributions at least in equilibrium state experiments. There are, however, some attempts to distinguish between the excitonic and phononic contributions using the nonequilibrium dynamics induced by laser pulse in 1$T$-TiSe$_2$ [@Rohwer2011Nature; @Mohr-Vorobeva2011PRL; @Hellmann2012NC; @Monney2016PRB]. In Ta$_2$NiSe$_5$, Mor *et al.* found that the band gap can be controlled by the excitation density [@Mor2017PRL] and argued that its nonequilibrium phenomena come from the exciton dynamics [@Mor2018PRB]. Coherent order parameter oscillations caused by the induced phonons were also observed [@Werdehausen2018JPC; @Werdehausen2018SA].
The pump-probe spectroscopy experiments in the excitonic phases have been interpreted from the theoretical point of view. While the GW calculations showed that the excitonic order vanishes after applying the laser pulse in the BCS regime where the normal state is semimetallic [@Golez2016PRB], Murakami *et al.* [@Murakami2017PRL] recently showed that the excitonic order can be enhanced by the laser pulse in the Bose-Einstein condensation (BEC) regime where the normal state is semiconducting. Tanaka *et al.* [@Tanaka2018PRB] also showed that the switching between the melting and enhancement of excitonic orders can occur when the order varies from the BCS regime to BEC regime.
Note that all of these calculations [@Golez2016PRB; @Murakami2017PRL; @Tanaka2018PRB] assumed that the laser pulse excites the electrons in the valence-band orbital directly to the conduction-band orbital via the dipole transition. However, as was discussed in Ref. \[\], the matrix elements of the dipole transition can be small in the case where the valence-band and conduction-band orbitals are well-localized and are spatially separated in distant positions, just as in Ta$_2$NiSe$_5$ [@Kaneko2013PRB]. The Peierls term, on the other hand, can survive even in such situations [@Wissgott2012PRB]. Thus, there is another way of treating the laser pulse, which is to introduce a time-dependent vector potential via the Peierls phase in a tight-binding Hamiltonian.
In this paper, we study the nonequilibrium dynamics of excitonic insulator states applying the time-dependent mean-field approximation to the spinless two-orbital model in one-dimension (1D) with phonon degrees of freedom, whereby we simulate the situation where an optical laser pulse is applied to the system as a pump light. Unlike preceding studies, we here introduce the pulse light as a time-dependent vector potential via the Peierls phase in the Hamiltonian of the external field, assuming the situations where the dipole matrix elements are small. We note that the spontaneous hybridization between the valence-band and conduction-band orbitals occurs in the symmetry-broken excitonic insulator state, so that the interband excitations by the Peierls mechanism can work in the present model. We thus investigate the time evolution of the excitonic order parameter in both the BEC and BCS regimes, paying particular attention to its dependence on the frequency and intensity of the laser light.
We will show that, in the BEC regime where the normal state is semiconducting, the excitonic order is suppressed when the frequency of the pulse light is slightly larger than the band gap, while the order is enhanced when the frequency of the pulse is much larger than the band gap. In the BCS regime where the normal state is semimetallic, we will show that the excitonic order is always suppressed, irrespective of the frequency of the pulse light. We will demonstrate that the excitonic order parameter oscillation occurs in agreement with experiment. We will also calculate the optical conductivity spectrum assuming a single-time, instantaneous response for a quasi-steady state after pumping and demonstrate the measurement is a useful way for probing the nonequilibrium dynamics of excitonic insulator states.
The rest of this paper is organized as follows. In Sec. II, we introduce the spinless two-orbital model, define the laser pulse light, and derive the equations of motions for the excitonic order parameters in the time-dependent mean-field approximation. In Sec. III, we present results for nonequilibrium dynamics induced by laser pulse in both BEC and BCS regimes. We also present results for the optical conductivity spectra in the nonequilibrium state. We summarize our results and discuss their experimental significance in Sec. IV.
Model and Method
================
Spinless two-orbital model
--------------------------
As a minimum model for describing the spin-singlet excitonic insulator state coupled with phonon degrees of freedom, we consider the spinless two-orbital model (or extended Falicov-Kimball model [@Ihle2008PRB; @Zenker2011PRB; @Seki2011PRB; @Ejima2014PRL]) defined on the 1D lattice \[see Fig. \[fig1\](a)\], interacting with Einstein phonons of frequency $\omega_{0}$ [@Murakami2017PRL]. This model may be relevnt to the electronic state of an excitonic insulator candidate Ta$_2$NiSe$_5$ with a quasi-1D crystal structure [@Seki2014PRB], although the method discussed below is applicable to higher dimensional systems as well. Our model is defined by the Hamiltonian $$H = H_\mathrm{e} + H_{\mathrm{e,int}} + H_{\mathrm{ph}} + H_{\mathrm{e-ph}}
\label{eq:Hamiltonian}$$ with $$\begin{aligned}
&H_\mathrm{e}
= -\sum_{i, \alpha} \big( J_{\alpha} c^\dagger_{i + 1, \alpha} c_{i, \alpha} + \mathrm{H.c.} \big)
+ \sum_{i, \alpha} \Delta_{\alpha} c^\dagger_{i, \alpha} c_{i, \alpha}
\label{eq:H_e}
\\
&H_{\mathrm{e,int}}
= U \sum_i c^\dagger_{i, 0} c_{i, 0} c^\dagger_{i, 1} c_{i, 1}
\\
&H_{\mathrm{ph}}
= \omega_{0} \sum_{i} b^\dagger_{i} b_{i}
\\
&H_{\mathrm{e-ph}}
= g \sum_{i} \big( b^\dagger_{i} + b_{i} \big) \big( c^\dagger_{i, 1} c_{i, 0} + \mathrm{H.c.} \big),\end{aligned}$$ where $c_{i, \alpha}$ ($c^\dagger_{i, \alpha}$) is the annihilation (creation) operator of an electron at site $i$ with orbital $\alpha$ $(=0,1)$ and $b_{i}$ ($b_i^\dagger$) is the annihilation (creation) operator of a phonon at site $i$. $J_\alpha$, $\Delta_\alpha$, and $U$ are the hopping integral, on-site energy, and interorbital repulsive interaction, respectively, and $\omega_0$ and $g$ are the phonon frequency and electron-phonon coupling constant, respectively. Throughout the paper, we set $J_\alpha = (-1)^\alpha J$, assuming a direct-gap semiconductor or semimetal \[see Figs. \[fig1\](b) and \[fig1\](c)\], so that we create a momentum $q=0$ condensate appropriate to Ta$_2$NiSe$_5$, rather than $q=\pi$ assumed in, e.g., Ref. \[\]. We keep the relation $U=-(\Delta_0+\Delta_1)$, so that the condition that the number of electrons per site (containing two orbitals) is one is satisfied at the chemical potential $\mu=0$. We define the energy level difference $D=\Delta_0-\Delta_1$. We introduce the effective electron-phonon interaction $\lambda=2g^2/\omega_0$ and assume the values $\lambda=0.1$ and $\omega_0=0.1$ throughout the paper. We use the values $J=1$ (unit of energy), $\Delta_0=-0.36$, $\Delta_1=-2.44$, $D=2.08$, and $U=2.8$ unless otherwise indicated. The noninteracting band dispersions used are illustrated in Figs. \[fig1\](b) and \[fig1\](c).
![ (a) Schematic representation of the spinless two-band model used. In (b) and (c), we show the quasiparticle band dispersions calculated for the model (a) in the equilibrium state (solid lines), together with the noninteracting band dispersions (dotted lines). We assume $D=2.08$ in (b) and $D=1.1$ in (c). []{data-label="fig1"}](fig1.eps){width="0.85\columnwidth"}
Optical laser pulse
-------------------
To treat the optical laser pulse as a pump light, we introduce the time-dependent vector potential as a Peierls phase to the Hamiltonian. We assume that the vector potential is approximately independent of the spatial position since the wavelength of the light is much longer than the lattice spacing $a$. The hopping-integral term in Eq. (\[eq:H\_e\]) is then replaced by [@Freericks2017PS] $$J_{\alpha} c^\dagger_{i + 1, \alpha} c_{i, \alpha}
\to
J_{\alpha} e^{i ea A(t)} c^\dagger_{i + 1, \alpha} c_{i, \alpha},$$ where $e$ $(<0)$ is the elementary charge, for which we set $e=-1$. We use the units $\hbar = c = 1$ and the lattice constant $a=1$ throughout the paper. We consider the Gaussian-type laser pulse defined by the time-dependent vector potential as [@Matsueda2012JPSJ] $$A (t)
= \theta (t) A_0 e^{- \frac{\left( t - t_{\mathrm{p}} \right)^2}{2 \sigma_{\mathrm{p}}^2}} \sin \Omega t ,$$ where $\Omega$ and $A_0$ are the frequency and intensity of the pulse light, respectively, and $$\theta (t) =
\begin{cases}
0 & (t \leq 0) \\
1 & (t > 0)
\end{cases}$$ is the step function. The time and width of the pulse are described by $t_{\mathrm{p}}$ and $\sigma_{\mathrm{p}}$, respectively. In this paper, we set $\sigma_{\mathrm{p}} = 30$ and $t_{\mathrm{p}} = 100$. By the Fourier transformation $c_{i, \alpha} = \frac{1}{\sqrt{N}} \sum_{k} e^{i k r_i} c_{k, \alpha}$, where $N$ is the number of lattice sites in the system, Eq. (\[eq:H\_e\]) may be rewritten as $$H_\mathrm{e}
= \sum_{k, \alpha} \varepsilon_{\alpha} ( ka - ea A(t) ) c^\dagger_{k, \alpha} c_{k, \alpha}$$ where $\varepsilon_{\alpha} (ka) = -2 \left( -1 \right)^\alpha \cos ka + \Delta_{\alpha}$.
Equations of motion
-------------------
The time evolution of the system induced by the laser pulse is calculated in the time-dependent mean-field approximation [@Murakami2017PRL; @Barankov2004PRL; @Yezbashyan2005PRB]. We define the uniform excitonic order parameter as $\phi (t) = \langle c^\dagger_{i, 0} (t) c_{i, 1} (t) \rangle$, the uniform phonon displacement as $X (t) = \langle b^\dagger_i (t) + b_i (t) \rangle$, and the uniform electron density as $n_\alpha (t) = \langle c^\dagger_{i, \alpha} (t) c_{i, \alpha} (t) \rangle$. The time dependence of the operators is given by the Heisenberg representation.
For convenience, we use the pseudospin representation for the spinless electron in the two orbitals [@Murakami2017PRL]. We define the pseudospin as $
S_k^\gamma = \frac{1}{2} \bm{\Psi}_k^\dagger \sigma_\gamma \bm{\Psi}_k
$, where $
\bm{\Psi}_k^\dagger
=\begin{pmatrix}
c^\dagger_{k, 0} & c^\dagger_{k, 1}
\end{pmatrix}
$ is the spinor, $\sigma_\gamma$ ($\gamma = x,y,z$) is the $\gamma$ component of Pauli matrix $\bm{\sigma}$, and $\sigma_0$ is the identity matrix. Introducing the pseudomagnetic field as
$$\begin{aligned}
&B^x_k (t)
= 2g X (t) - 2 U \mathrm{Re} \, \phi (t)
\\
&B^y_k (t)
= - 2 U \mathrm{Im} \, \phi (t)
\\
&B^z_k (t)
= \left[ \varepsilon_{0} ( ka - ea A(t) ) - \varepsilon_{1} ( ka - ea A(t) ) \right] \notag \\
&~~~~~~~~~~- U \left( n_0 (t) - n_1 (t) \right)
\\
&B^0_k (t)
% = \sum_{\alpha} \left[ \Delta_{\alpha} + U n_{\alpha} (t) \right],
= \sum_\alpha \left[ \varepsilon_\alpha (ka-eaA(t)) + U n_{\alpha} (t) \right],\end{aligned}$$
\[eq:pseudoB\]
we may write the mean-field Hamiltonian as $$H^{\mathrm{MF}} (t) = H^{\mathrm{MF}}_\mathrm{e} (t) + H^{\mathrm{MF}}_{\mathrm{ph}} (t)$$ with $$\begin{aligned}
&H^{\mathrm{MF}}_\mathrm{e} (t)
= \sum_{k} \sum_{\gamma=0,x,y,z} B^{\gamma}_k (t) S^{\gamma}_k
\label{eq:H^MF_e}
\\
&H^{\mathrm{MF}}_{\mathrm{ph}} (t) =
\omega_0 \sum_{i} b_i^\dagger b_i + 2g \, \mathrm{Re} \, \phi (t) \sum_{i} \big( b_i + b_i^\dagger \big).\end{aligned}$$
The time-dependent variables $\phi (t)$, $X (t)$, and $n_{\alpha} (t)$ may be calculated from the Heisenberg equations of motion. We thus obtain the equations
$$\begin{aligned}
&\frac{\partial \left\langle \bm{S}_k (t) \right\rangle }{\partial t}
= \bm{B}_k (t) \times \left\langle \bm{S}_k (t) \right\rangle
\\
&\frac{\partial \left\langle S^0_k (t) \right\rangle }{\partial t}
= 0
\\
&\frac{\partial X (t) }{\partial t}
= \omega_0 P (t)
\label{eq:EOM_X}
\\
&\frac{\partial P (t) }{\partial t}
= - \omega_0 X (t) - 4g \, \mathrm{Re} \, \phi (t) ,
\label{eq:EOM_P}\end{aligned}$$
\[eq:EOM\]
where $P (t) = i \langle b^\dagger_i - b_i \rangle$ is the momentum of the phonon. We solve Eqs. (\[eq:EOM\]) numerically using the Runge-Kutta fourth-order method, and substitute the solutions into $$\begin{pmatrix}
n_0 (t) & \phi^* (t) \\
\phi (t) & n_1 (t)
\end{pmatrix}
= \frac{1}{N} \sum_k \left[ \left\langle \bm{S}_k (t) \right\rangle \cdot \bm{\sigma} + \left\langle S^0_k (t) \right\rangle \sigma_0 \right]
\label{eq:order_parameters_and_pseudospins}$$ to obtain the excitonic order parameter and the number of electrons.
Equilibrium state
-----------------
To solve the above differential equations, we need to set the initial conditions. At $t = 0$, where the external field is absent, the system is in equilibrium. Since we consider the system at zero temperature, the phonons are at rest, or $P(0) = 0$. From Eq. (\[eq:EOM\_P\]), the phonons satisfy $X(0) = - \frac{4g}{\omega_0} \phi (0)$. We note that the excitonic order parameter is real when the electron-phonon coupling is present [@Zenker2014PRB; @Kaneko2015PRB].
Diagonalizing Eq. (\[eq:H\^MF\_e\]), we obtain $$H^{\mathrm{MF}}_\mathrm{e} (t)
= \sum_k \big(
E_{k, +} (t) \gamma^\dagger_{k, +} \gamma_{k, +}
+ E_{k, -} (t) \gamma^\dagger_{k, -} \gamma_{k, -}
\big),$$ where $\gamma_{k, \pm}$ ($\gamma^\dagger_{k, \pm}$) is the annihilation (creation) operator of the quasiparticle with dispersions $E_{k, \pm} (t) = \left[ B^0_k (t) \pm \left| \bm{B}_k (t) \right| \right]/2$. At $t=0$, the expectation value of the pseudospin is given by $$\begin{aligned}
& \left\langle S^\gamma_k (0) \right\rangle = \nonumber \\
& \begin{cases}
\displaystyle{\frac{B^\gamma_k(0)}{2 \left| \bm{B}_k (0) \right|}} \left[ f (E_{k, +}(0)) - f (E_{k, -}(0)) \right]
& (\gamma = x,y,z)
\\
\displaystyle{\frac{1}{2}} \left[f (E_{k, +} (0)) + f (E_{k, -} (0)) \right]
& (\gamma = 0)
\end{cases}
\label{eq:self-consistent_equation}\end{aligned}$$ where $f (E)$ is the Fermi distribution function. Solving the self-consistent equations for the pseudospins, we obtain Eq. (\[eq:self-consistent\_equation\]) with Eqs. (\[eq:pseudoB\]) and (\[eq:order\_parameters\_and\_pseudospins\]) at $t=0$, which we use as the initial conditions for Eqs. (\[eq:EOM\]). The calculated quasiparticle band dispersions at $t=0$ are shown in Figs. \[fig1\](b) and \[fig1\](c).
Results of calculations
=======================
Here, we discuss the results of calculations mainly when the system is semiconducting in the normal phase. Our parameter values give the quasiparticle band gap of size 1.15 in the presence of the excitonic order, which is considerably enhanced in comparison to the band gap of a size 0.88 in the absence of the excitonic order \[see Fig. \[fig1\](b)\]. We also discuss the results when the system is semimetallic in the last part of this section. All the calculations are made at absolute zero temperature.
![ Calculated time evolution of the absolute value of the excitonic order parameter $|\phi(t)|$ for a variety of frequency of the laser pulse $\Omega$ at $A_0=0.05$. The BEC (or semiconducting) regime is assumed. (a) The case where $\Omega$ is smaller than the quasiparticle band gap $1.15$ and (b) the case where $\Omega$ is larger than the quasiparticle band gap. []{data-label="fig2"}](fig2.eps){width="1.0\columnwidth"}
![ Time averages of the real part of the excitonic order parameter $\mathrm{Re}\,\bar{\phi}$ calculated as a function of (a) the frequency of the laser light $\Omega$ and (b) intensity of the laser light $A_0$. []{data-label="fig3"}](fig3.eps){width="1.0\columnwidth"}
![ Calculated trajectory of the excitonic order parameter in the complex plane for $0\le t<800$. The laser pulse is applied at $t=0$, where $\mathrm{Re}\,\phi(t)=0.163$ and $\mathrm{Im}\,\phi(t)=0$. We assume $\Omega=1.15$. []{data-label="fig4"}](fig4.eps){width="1.0\columnwidth"}
Evolution of the order parameter
--------------------------------
First, let us discuss the evolution of the excitonic order parameter. In Fig. \[fig2\], we show the results for the time evolution of the excitonic order parameter. We find in Fig. \[fig2\](a) that, when the frequency of the laser light is smaller than the band gap, the order parameter remains unchanged against the laser pulse. This is because the pulse cannot excite the electrons in the valence band to the conduction band. Once the frequency of the laser light becomes comparable to the size of the band gap, the excitations of electrons occur and the order parameter drastically changes as seen in Figs. \[fig2\](a) and \[fig2\](b). After the laser pulse passes through the system, the absolute value of the order parameter decreases rapidly in comparison to the equilibrium value, and it oscillates with the frequency of the phonon. This result indicates that the excitonic order partially melts by the laser pulse. Note that, since the electron-electron correlations and electron-phonon couplings are treated in the mean-field approximation, the thermalization process of the nonequilibrium state after the pulse light passes through the system [@Tsuji2013PRL; @Kemper2015PRB; @Sentef2016PRB; @Schuler2018PRL] is excluded in our calculations. As a result, the order parameter oscillates persistently without damping.
On the other hand, when the frequency of the laser light is much larger than the size of the band gap, we find that the opposite result occurs. As seen in Fig. \[fig2\](b), we find that, after the laser pulse passes through the system, the absolute value of the order parameter is enhanced. This result can be explained as the effect of the Hartree shift [@Murakami2017PRL]: The strong excitations induced by the pulse lead to the reduction of the number of valence electrons and increase in the number of conduction electrons, which reduces the size of the band gap by the Hartree shift, and therefore the excitonic order parameter is enhanced.
To see the dependence of the excitonic order parameter on the frequency and intensity of the laser pulse more clearly, we introduce the time average of the order parameter $\bar{\phi} = \frac{1}{t_2 - t_1} \int^{t_2}_{t_1} \phi(t) dt$, which is calculated in a sufficiently long time interval between $t_1$ and $t_2$ after the pulse is applied. The oscillations due to the phonons are thus obliterated. In Fig. \[fig3\](a), we show the real part of the averaged order parameter $\textrm{Re}\,\bar{\phi}$ as a function of the frequency of the laser light. We find that, while the order parameter is suppressed for the frequencies slightly larger than the band gap, it is enhanced for the frequencies much larger than the band gap. This result can be interpreted as the competition between the melting of the excitonic order and the reduction of the band gap caused by the Hartree shift.
We also calculate the real part of the averaged order parameter as a function of the intensity of the laser light. As shown in Fig. \[fig3\](b), we find that, when the frequency of the light is around the size of the band gap, there appears a critical value of the intensity at which the averaged order parameter completely vanishes, retaining only the oscillation of the phonons. Note that, when the frequency of the laser light is much larger than the band gap, such suppression of the excitonic order does not occur.
Although not shown here, we also calculate the time average of the phonon displacement $\bar{X}$ and find that the behavior is very similar to that of the order parameter $\bar{\phi}$ shown in Fig. \[fig3\].
We can understand the present result clearly from the trajectory of the excitonic order parameter in the complex plane. The results are shown in Fig. \[fig4\], where we find the following: At $t=0$, the excitonic order parameter is a real number since the electron-phonon coupling term in the Hamiltonian fixes the phase of the order parameter to zero [@Zenker2014PRB; @Kaneko2015PRB]. After the laser pulse is applied, the excitonic order parameter goes around a certain point where the imaginary part is zero and the real part is smaller than that in the equilibrium state. As the intensity of the laser pulse increases, the central point around which the order parameter oscillates approaches the origin, and finally it reaches the origin when the intensity become larger than the critical value. Thus, the strong laser pulse destroys the excitonic order.
![ Quasiparticle band dispersion below the Fermi level $E_{k,-}(t)$ calculated at time $t$. We assume (a) $A_0=0.05$ and $\Omega=1.15$ and (b) $A_0=0.05$ and $\Omega=3$. The dashed line indicates the quasiparticle band dispersion at $t=0$ (or in the equilibrium state). Insets enlarge the dispersions. At $t>0$, the dispersion oscillates between the red solid line and blue solid line. []{data-label="fig5"}](fig5a.eps "fig:"){width="0.95\columnwidth"} ![ Quasiparticle band dispersion below the Fermi level $E_{k,-}(t)$ calculated at time $t$. We assume (a) $A_0=0.05$ and $\Omega=1.15$ and (b) $A_0=0.05$ and $\Omega=3$. The dashed line indicates the quasiparticle band dispersion at $t=0$ (or in the equilibrium state). Insets enlarge the dispersions. At $t>0$, the dispersion oscillates between the red solid line and blue solid line. []{data-label="fig5"}](fig5b.eps "fig:"){width="0.95\columnwidth"}
Quasiparticle band dispersion
-----------------------------
We calculate the quasiparticle band dispersion in the time-dependent mean-field approximation of the model after the laser pulse is applied. The time- and angle-resolved photoemission spectroscopy experiment can in principle observe this dispersion. The results are illustrated in Figs. \[fig5\](a) and \[fig5\](b). We find in Fig. \[fig5\](a) that the size of the energy gap decreases when the frequency of the laser light is slightly larger than the energy gap (or at $\Omega=1.15$). This is because the laser pulse leads to the partial melting of the excitonic order as shown in Fig. \[fig3\](b), of which the behavior seems consistent with recent experiment [@Mor2017PRL].
We then find in Fig. \[fig5\] (b) that the size of the energy gap also decreases even when the frequency of the laser light is much larger than the energy gap (or at $\Omega=3$). Thus, the top of the valence band goes up in the entire frequency region of the pulse light. However, the bottom of the valence band behaves differently depending on the frequency of the laser light: it goes down at $\Omega=1.15$ but it goes up at $\Omega=3$.
![ Real part of the optical conductivity spectrum $\mathrm{Re}\,\sigma(\omega,t)$ calculated at time $t$. The dashed line indicates the optical conductivity spectrum at $t=0$ (or in the equilibrium state). We assume (a) $A_0=0.05$ and $\Omega=1.15$ and (b) $A_0=0.05$ and $\Omega=3$. At $t>0$, the spectrum oscillates between the red solid line and blue solid line. []{data-label="fig:optcond"}](fig6a.eps "fig:"){width="0.95\columnwidth"} ![ Real part of the optical conductivity spectrum $\mathrm{Re}\,\sigma(\omega,t)$ calculated at time $t$. The dashed line indicates the optical conductivity spectrum at $t=0$ (or in the equilibrium state). We assume (a) $A_0=0.05$ and $\Omega=1.15$ and (b) $A_0=0.05$ and $\Omega=3$. At $t>0$, the spectrum oscillates between the red solid line and blue solid line. []{data-label="fig:optcond"}](fig6b.eps "fig:"){width="0.95\columnwidth"}
Optical conductivity spectrum
-----------------------------
To make a connection between our theory and experiment, we calculate the real part of the optical conductivity spectrum $\mathrm{Re} \, \sigma (\omega,t)$ measured at the frequency $\omega$ of the probe light after the pulse light passes through the system. The assumption used here is that the double-time retarded response that is actually required in a true pump-probe setting [@Eckstein2008PRB; @Lenarcic2014PRB; @Shao2016PRB] can be approximated by a single-time, instantaneous response for a quasi-steady state after pumping [@Fukaya2015NC]. In this approximation, the real part of the optical conductivity of the present nonequilibrium state at a certain moment $t$ is given by $$\begin{gathered}
\mathrm{Re} \, \sigma (\omega,t)
= -\frac{\pi}{N \omega}
\sum_{k}
\left[ f (E_{k, +} (t)) - f (E_{k, -} (t)) \right]
\\ \times
\left| J_{+-} (k,t) \right|^2
\delta (E_{k, +}(t) - E_{k, -}(t) - \omega) , \end{gathered}$$ where $$J_{+-}(k,t) = 2ea \frac{B^x_k(t)+iB^y_k(t)}{\left| \bm{B}_k (t) \right|} \sin ka$$ is the off-diagonal matrix element of the electric current. When the excitonic order is absent, the optical conductivity completely vanishes in the present model. This is because the orbital off-diagonal elements of the current operator are all zero due to the orthogonality of the orbitals, prohibiting the excitations of electrons from the valence to the conduction bands. Only when the excitonic order appears, or the hybridization between the valence and conduction bands occurs, the optical conductivity acquires the finite spectral weight.
The calculated results are shown in Fig. \[fig:optcond\] at $\Omega=1.15$ and $\Omega=3$. We find that the optical conductivity spectrum acquires the finite spectral weight for the probe frequency $\omega$ larger than the quasiparticle band gap. At $\Omega=1.15$, where the excitonic order is suppressed and the size of the band gap is reduced, we find that the peak position of the optical conductivity spectrum shifts to lower energy side and the peak height decreases in comparison to that of the equilibrium state \[see Fig. \[fig:optcond\](a)\]. At $\Omega=3$, where the excitonic order is enhanced but the size of the band gap remains nearly unchanged, we find that the peak position of the optical conductivity spectrum remains unchanged but the peak height slightly increases in comparison to that of the equilibrium state \[see Fig. \[fig:optcond\](b)\].
We note again that, in the actual measurement of the optical conductivity in the nonequilibrium state, the spectra may be affected by the phonon oscillations, so that the double-time retarded response should be taken into account in future improved calculations.
![ Contour plot of the time average of the real part of the excitonic order parameter ${\rm Re}\,\bar{\phi}$ calculated in the parameter space of $(D,\Omega)$ at $U=2.8$. We assume $A_0=0.05$. The excitonic order is suppressed (enhanced) in the blue (red) region of the parameter space. []{data-label="fig7"}](fig7.eps){width="1.0\columnwidth"}
Semimetallic case
-----------------
Finally, let us discuss the case where the quasiparticle band structure is semimetallic in the normal state (or in the absence of the excitonic order). To see this, we calculate the time average of the real part of the excitonic order parameter ${\rm Re}\,\bar{\phi}$ as a function of the level difference $D=\Delta_0-\Delta_1$. The semiconducting and semimetallic regions in the normal state are separated at $D=1.61$. The calculated results are summarized as a phase diagram, which is shown in Fig. \[fig7\]. We find that the excitonic order is not stable at $D>2.20$, that either the suppression or enhancement of the excitonic order occurs depending on the frequency of the laser light in the semiconducting region (or at $1.61<D<2.20$) as shown in Fig. \[fig3\](a), and that only the suppression of the order parameter occurs in the semimetallic region (or at $D<1.61$). This result is consistent with the preceding study assuming the semimetallic band structure [@Golez2016PRB].
Summary and Discussion
======================
In summary, we have studied the nonequilibrium dynamics of excitonic insulator states using the spinless two-orbital model with phonon degrees of freedom in the time-dependent mean-field approximation. Unlike preceding studies where the dipole transition of electrons between the valence and conduction bands was assumed, we have introduced the pulse light as a time-dependent vector potential via the Peierls phase in the Hamiltonian.
We have then found that, in the BEC regime where the normal state is semiconducting, the excitonic order is suppressed when the frequency of the pulse light is slightly larger than the band gap, while the order is enhanced when the frequency of the pulse is much larger than the band gap. We have moreover found that the excitonic order is completely destroyed in the former situation if the intensity of the pulse is sufficiently strong. In the BCS regime where the normal state is semimetallic, we have found that the excitonic order is always suppressed, irrespective of the frequency of the pulse light. We have also calculated the time-dependent quasiparticle band dispersion and optical conductivity spectrum of the model.
Finally, let us discuss possible experimental significance of our results, taking a quasi-1D direct-gap semiconductor Ta$_2$NiSe$_5$ as an example. The energy of the laser light so far used in experiment is 1.55 eV [@Mor2017PRL; @Mor2018PRB; @Werdehausen2018JPC; @Werdehausen2018SA], which is very large in comparison with the observed band gap 160 meV [@Seki2014PRB; @Lu2017NC; @Larkin2017PRB]. Thus, our two-band model seems to be too simple to take into account the excitations in such a high-energy scale where other orbitals such as Se $4p$ become relevant. In other words, the use of lower-energy laser lights may be more informative to clarify the nonequilibrium dynamics of Ta$_2$NiSe$_5$ in the lowest energy scales. Also, it was pointed out that Ta$_2$NiSe$_5$ is in the strong coupling BEC regime, despite the fact that the noninteracting band structure is semimetallic [@Sugimoto2018PRL]. Such a situation cannot be treated in our simple mean-field approximation. Nevertheless, we may point out that our calculations could yield some experimental aspects such as the enhancement of the excitonic order [@Mor2017PRL; @Mor2018PRB], coherent order parameter oscillations [@Werdehausen2018JPC; @Werdehausen2018SA], as well as the insulator-to-metal transitions reported recently [@Okazaki2018NC]. The improved method of calculations beyond the mean-field approximation based on more realistic models will be required for future quantitative studies of the nonequilibrium dynamics of the excitonic insulator states. We may note here that, in the case of superconductivity under periodically oscillating phonons driven by an external field, the order parameter is reduced in the dynamical mean-field-theory calculation [@Murakami2017PRB], while it is enhanced in the effective electron-electron interaction scheme [@Knap2016PRB; @Komnik2016EPJB]; the opposite results are obtained depending on the approximations used. Thus, a careful treatment of strongly interacting systems will be required in future calculations of nonequilibrium phenomena observed in the excitonic phases as well.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank T. Kaneko, T. Mizokawa, Y. Murakami, K. Okazaki, Y. Yamada, and K. Yonemitsu for enlightening discussions. This work was supported in part by a Grant-in-Aid for Scientific Research (No. JP17K05530) from JSPS of Japan.
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| {
"pile_set_name": "ArXiv"
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---
abstract: 'Gas accretion is necessary to maintain star formation, spiral and bar structure, and secular evolution in galaxies. This can occur through tidal interaction, or mass accretion from cosmic filaments. Different processes will be reviewed to drive gas towards galaxy centers and trigger starbursts and AGN. The efficiency of these dynamical processes can be estimated through simulations and checked by observations at different redshift, across the Hubble time. Large progress has been made on galaxies at moderate and high redshifts, allowing to interpret the star formation history and star formation efficiency as a function of gas content, dynamical state and galaxy evolution.'
author:
- Francoise Combes
title: Gas accretion in disk galaxies
---
Introduction
============
In the last decade, cosmological simulations have emphasized the importance of cold gas accretion onto galaxies in the mass assembly, in particular at high redshift (e.g. Keres et al. 2005, Dekel et al. 2009, Devriendt et al. 2010). This mode of accretion is thought to be about one order of magnitude more important than galaxy mergers in mass assembly, unlike what was assumed in the hierarchical scenario.
In parallel, simulations of isolated galaxies show how important is the gas accretion to maintain star formation at a constant level, as is observed in spiral galaxies, to explain abundance gradients, and also to maintain the spiral structure. In the following, the impact of gas accretion is first described, and then we will review the evidence of circumgalactic gas inflow.
Role of gas accretion: secular evolution, bars
==============================================
Secular evolution involves mainly the disk galaxies of the Hubble sequence: all spirals and irregulars. As for the ellipticals, their formation scenario is still heavily relying on mergers, either a few major mergers, or more likely a series of minor mergers, to heat the stellar component, destroy disks progressively, and cancel out any angular momentum, explaining its low values in this class.
In disk galaxies, the main motor of evolution is non-axisymmetries and bars. When the disk is abundant in gas, as are most high redshift disky objects, then the bars are not long-lived, but weakened and destroyed by the accumulation of mass in the centers, and by the exchange of angular momentum between the gas and the stars of the bar (Friedli & Benz 1993, Berentzen et al. 1998, Bournaud & Combes 2002). A weakened bar would transiently look like a lens inside its inner ring (e.g. Laurikainen et al. 2009).
The frequency of bars in disk galaxies as a function of mass shows an interesting bimodality (Nair & Abraham 2010), with two maxima in the blue cloud and in the red sequence. This was also found by Masters et al (2011) with the Galaxy Zoo, although not by the S4G consortium (see K. Sheth, this meeting), but this could be due to selection effects.
Bars can act in conjunction with spirals, to redistribute the angular-momentum across galaxies, and to modify significantly stellar radial profiles. The non-linear interactions at resonances overlap multiply the effects (Minchev et al 2011). In less then 3 Gyrs, the effective sizes of galaxy disks may be multiplied by 3, and the corresponding radial migration brings high-velocity dispersion stars in the outer parts. Disk thickening is also substantial (Minchev et al 2012). When gas accretion is considered, the strength of bars can be re-boosted, and new stars formed out of the accreted gas continuously re-shape the radial stellar profiles. It is then possible to obtain the three observed types: Type I as a single exponential disk, Type II as a truncated one, when star formation is newly forming at the break, and beyond the break the star formation threshold is not yet reached, and Type III, the anti-truncated profile can be obtained in case of strong gas accretion in the outer parts (cf Fig \[fig:minchev-12\]). The Type III morphology appears relatively transient, and able to evolve into Type II or Type I. Its presence could be a tracer of accretion events. There is some correlation of these Type II with weakened bars, as expected from strong gas accretion (Bournaud & Combes 2002, Combes 2011).
When the gas accretion occurs essentially from non-aligned cosmic filaments, characteristic signatures may occur, such as inclined and warped rings (Roskar et al 2010), or even polar rings, when the accretion is near polar. Brook et al (2008) have shown how a lenticular system is first formed from matter accretion, which suddenly stops when the birth filament is consumed out. The next filament in the perpendicular direction then fuels a relatively stable polar ring system. Gas accretion may mimick galaxy interactions, since it can produce asymmetries, lopsidedness, clumpiness, and sfatbursts. Even if the accretion is globally symmetric and isotropic, it may be temporarily on one side only.
Gas accretion replenishes the extended gas reservoirs present around most spiral galaxies. This gas slowly spirals in, when in quiescent state. However, the first tidal interaction may drive the gas violently towards the center, and strongly affects abundance gradients, that can even be reversed (Montuori et al 2010): low-metallicity gas flows into the center and dilutes the abundance of the central gas, in a time-scale shorter than the time required for this gas to re-enrich through the triggered nuclear starburst. Such gradient reversals have been observed at high redshift in the MASSIV survey (Queyrel et al 2012).
Inside-out disk formation, inflow/outflow, metallicity
======================================================
Gas accretion occurs in the outer parts of disks, and is a way to explain inside-out disk formation. There are now multiple evidence of this progressive mode of disk formation. Through observations of galaxies as a function of redshift, it is possible to track the evolution directly. However, finding the progenitors of today galaxies is not easy. Statistically, this is solved by matching the galaxies at a given cumulative number density, for instance 1.4 10$^{-4}$ Mpc$^{-3}$. Plotting the mass of these matched galaxies at a given redshift, Patel et al (2013) follow the mass increase of galaxies over z=3 to 0. They notice that this mass increase involves only the mass of the outer parts, while the mass inside 2kpc is stable in all the same galaxies. This fact is related to the observations that the normalised galaxy radius, at a given mass, increases by a factor 3 from z=3 to 0 (Newman et al 2012). It is possible that dry minor mergers explain this increase of size, without much mass accretion, in the case of quenched galaxies. For star forming disk galaxies, external gas accretion, accompanied by secular evolution, is necessary. In the GASS sample of local galaxies detected in HI-21cm, Wang et al (2011) find that galaxies richer in HI are bluer, and the more so in the outer parts: the radial gradient of color is a function of HI-mass fraction. They conclude that the gas must be accreted slowly, and relaxed, since there is no correlation between HI-fraction and lopsidedness in this sample.
More directly, looking at H$\alpha$ in the 3D-HST project in 57 galaxies at z$\sim$ 1, Nelson et al (2012) find that the effective radii of galaxies in ionised gas and new stars is about 1.3 larger than the effective radius of the rest-frame R-band stellar continuum, representing older stars. The effet is larger for massive galaxies, that certainly are the first to form inside-out.
In this inside-out scenario, with gas accretion, the radial migration will produce a typical reversal of stellar ages in the outer parts: the gas is accreted at the radius of the break, which is where the new stars are formed, accentuating the negative age gradient from the center. After the break, only old stars migrated from the center are expected, and there is now a positive gradient (Roskar et al 2008). This age gradient reversal has been observed in M33 by Williams et al (2009). Some other galaxies do not show any break, but a flat age gradient in the outer parts (Vlajic et al 2011).
Evidence of gas accretion: HVC, warps, QSO absorption
=====================================================
Most of the gas from cosmic filaments is accreted at large scales, settles down to the disk, and spirals in progressively. Since the alignment process occurs through precession and dissipation, with a time-scale of the order of a few dynamical times at these large radii, this can take some Gyrs, during which galaxies appear warped or perturbed in the outer parts. Warps and polar rings are therefore the best tracers of external accretion, and indeed most spiral galaxies are observed to be warped (e.g. Briggs 1990, Binney 1992, Reshetnikov & Combes 1998). Also the frequency of asymmetries and lopsidedness in spiral galaxies cannot be explained but with external accretion (Jog & Combes 2009).
Searches have been done of extra-planar gas in the halo of spiral galaxies (Fraternali et al. 2002, Heald et al 2011, Gentile et al. 2013) and the quantities found are relatively small, NGC 891 being the most remarkable for its gas entension. The origin of this gas is multiple. Some gas can be ejected into the halo by stellar feedback (fountain effect), or through tidal disruption of satellites. In the Milky Way, the High Velocity Clouds (HVC) and the Magellanic stream are good examples. This gas is accreted progressively, with an interface of multiphase gas, but at a rate lower than the star formation rate (0.4 M$\odot$/yr in the Galaxy, Putman et al. 2012).
The way external hot gas is accreted might be complex (Fraternali & Binney 2008). The fountain effect ejects ionised gas into the halo, where it encounters the hot coronal gas. Merging with this assumed non-rotating gas, it looses angular momentum, and cold gas condensates in the shock. Finally more gas is infalling down, that was uplifted by star formation feedback. The metallicity of the infalling gas is relatively high, since it is a mixture of gas from very different origins. The same kind of processes is invoked in cool core clusters, as schematically shown in Figure \[fig:CF-halo\]. The hot X-ray gas in the cluster halo is dense enough at the center to cool down and fuel a central AGN. But the radio jets of the AGN re-heat the medium, in creating two cavities of plasma, pushing the X-ray gas out at the cavity boundaries. There the gas is cooling down to low temperatures, and molecular clouds are observed through their CO emission (Salome et al 2006, 2008). In parallel, the AGN jets drag some molecular gas previously settled in the central galaxy, and this uplifts some high-metallicity gas at 20-30kpc, which explains the more efficient cooling in filaments far from the center, and the gas abundance sufficient to produce CO emission.
High velocity clouds have been found around M31 and M33 in the local group, and also along a gas bridge between M31 and M33 (Lockman et al. 2012). About 121 HVC have been detected around the isolated galaxy NGC 6946 (Boomsma et al. 2008). In nearly face-on galaxies, these HVC re traced by the numerous HI holes they created in the disk plane, as demonstrated in M101 (Kamphuis et al. 1991). Some of these highly perturbed galaxies are aslo lopsided, like UGC7989 (Noordermeer et al. 2005). More generally, 30% of galaxies show an asymmetry larger than 10%, as quantified by the Fourier decomposition of the density. In poor environments, low surface brightness (LSB) galaxies are dwarfs particularly rich in HI gas, and revealing spectacular warps, like NGC 2915 (Meurer et al. 1006), NGC 5055 (Battaglia et al. 2006), or NGC 3741 (Begum et al. 2005, Gentile et al. 2007).
Another important way to discover the circum-galactic gas around galaxies is through absorption measurements in front of remote quasars. The various systems can be sorted by their column density, from the Ly$\alpha$ forest at very low columns, to the damped Ly$\alpha$ (DLA) at NHI $>$ 10$^{20}$ cm$^{-2}$, passing through the Lyman limit systems (LLS). The abundance of the systems varies as a power law (Prochaska et al. 2010). Fumagalli et al. (2011) have derived from cosmological simulations the probability to observe absorptions in front of backgound sources, at z=2.3, and according to the distance from the center of the galaxy up to the virial radius. The result is that there are only 10% of line of sights optically thick in HI, and 1% probability to find a DLA. About 30% of absorbers come from massive galaxies and their streams. There has been some debate about the filling factor of gaseous filaments around galaxies. While Dekel et al. (2009) claim a filling factor for cold filaments of 25% between 20 and 100kpc around a halo of total mass 10$^{12}$ M$_\odot$ at z=2.5, in the same conditions, Faucher-Giguère & Keres (2011) or Kimm et al. (2011) find 2% or 5 % respectively. So the observation of cold gas accretion is difficult, for the small filling fators, but also due to the confusion with the galaxy host, when the line-of-sights are too close, or the low metallicity of the circum-galactic gas, if carbon lines are used.
Another possibility would be to detect Ly$\alpha$ photons emitted when the gas is infalling in dark matter haloes, in some way when it re-radiates its gravitational energy. It has been proposed that the frequently observed Ly$\alpha$ blobs at high redshift are precisely due to this radiation. But the simulations from Faucher-Giguère et al. (2010), taking into account the proper self-shileding, etc., have shown that the gravitational emission in these blobs is negligible. Ly$\alpha$ blobs are powered by star formation or AGN. Now, several hundreds of Ly$\alpha$ blobs have been discoverd (Matsuda et al. 2006, 2011). The Ly$\alpha$ line is resonant and requires radiative transfer to better understand the shapes of the profiles observed. Verhamme et al. (2006) have showned that in case of outflowing material, a P-cygni profile is expected, with emission on the red side, and absorption on the blue side. The reverse is expected for inflowing gas. However, in all Ly$\alpha$ blobs observed until now, there are always P-cygni profiles with emission in the red, therefore only outflows have been detected.
Another way to detect the filaments could then be through fluorescence of Ly$\alpha$ photons emitted by a starburst or a luminous quasar. This has been done in a few cases (Rauch et al. 2011, Cantalupo et al. 2012), but this clever technique should be developed more. A powerful quasar can illuminate dark gas, or dark galaxies, up to 100kpc distance.
Finally, when absorption lines are detected in front of quasars, what are the methods to distringuish inflows from outflows in the circumgalactic medium (CGM)? Two methods have been used, and are illustrated in Figure \[fig:Flows-stat\].
Using an H I-selected sample of 28 Lyman limit systems (LLS) at z$<$1, observed in absorption with the HST-COS spectrograph, Lehner et al. (2013) are able to determine their metallicity from weakly ionized metal species (e.g., O II, Si II, Mg II) and find that the metallicity distribution of the LLS is bimodal with metal-poor and metal-rich branches peaking at about 2.5% and 50% solar metallicities. Both branches have comparable number of absorbers. The metal-rich branch likely traces winds, recycled outflows, and tidally stripped gas, while the metal-poor branch has properties consistent with cold accretion streams
When the galaxy host is detected in the proximity of the absorbant lines of sight, it is possible to determine the azimuthal orientation of the gas: Is it towards the minor axis? more likely to be an outflow, or the major axis? then it is probably an inflow. Bouché et al. (2012, 2013) have shown that the number of absorbants as the function of angle with respect to the major-axis, reveals a bimodal distribution also. These are MgII absorbants at z$\sim$ 0.1 (Fig \[fig:Flows-stat\]). The outflow speeds are found to be 150-300 km/s, i.e. of the order of the circular velocity, and smaller than the escape velocity by a factor of $\sim$2. The outflow rates are typically two to three times the instantaneous SFRs.
Evolution with redshift of gas content
======================================
It is now well established that the global SFR in the Universe evolves towards a maximum at about z=1-2, and then decreases by a factor 10 to the present (e.g. Bouwens et al. 2012). When the SFR is normalized to the stellar mass, the specific SFR (called sSFR) is derived, which is the inverse of the growth time-scale for the galaxy. This typical time-scale has been establied to be 2 Gyr at z=0, and then regularly decreases with increasing z. There was a recent debate about a possible plateau of the sSFR after z=2, but this has been corrected (Smit et al 2013). The first galaxies are so active in forming stars, that their nebular emission dominates their total emission. Without precise spectroscopy, it was not possible to disentangle the continuum from the lines, and the continuum was over-estimated. With the recent re-normalisation, the sSFR increases at all redshift, which corresponds better to the simulations predictions.
Why galaxies in the main sequence of star formation are more efficient to form stars at high redshift? According to our PHIBSS survey, detecting about 52 galaxies at z=1.2 and 2.3 with the IRAM interferometer, we determine that the gas fraction over all baryons in these galaxies increase with redshift up to 50% at least. The gas fraction is 34% at z=1.2 and 44% at z=2.3, while it is only of the order of 5% at z=0 (Tacconi et al. 2010, 2013).
The evolution of the specific SFR is shown in Figure \[fig:phibss\], where we have assumed that the depletion time-scale for star formation in the main sequence can be modelled as t$_{dep}$ = 1.5 (1+z)-1.05 Gyr. We have shown that the gas fraction was strongly correlated to the sSFR, and explored the whole range from 10 to 90%. In the Kennicutt-Schmidt diagram, the surface density of gas and the SFR surface density follows the main sequence branch. With respect to the gas surface density divided by the dynamical time, all objects (main sequence and starbursts) align on the same almost linear curve. This is due to the smaller dynamical time-scale of nuclear starbursts.
When only ULIRGS and starbursts are considered, the strong increase of star formation efficiency rate is explained at high redshift by two factors with comparable weight: first the gas fraction is higher by a factor 3 at z=1 with respect to z=0, and the star formation efficiency (SFR per unit gas mass), is also higher by a factor 3 (Combes et al 2013).
Conclusion
==========
External gas supply is fundamental for secular evolution: bars drive gas towards the very center, and accretion is necessary to replenish the disk, to reform a bar, or for the galaxy to stay on the blue cloud, instead of directly move on the red sequence.
The gas accretion is slow, progressive, from gas reservoirs in the outer parts of galaxies. Matter and angular momentum is redistributed by non-axisymmetries and bars, including radial migration. These processes explain the inside-out disk formation, that is now currently observed at all redshifts. The evolution of the disk sizes at a given mass can also be explained by slow matter accretion and dry minor mergers. In particular secular evolution can multiply the effective radii by up to a factor 3.
Gas accretion was more intense in the past. It could be the explanation, together with galaxy interactions and fly-bys, to abundance gradients reversal in galaxy disks. Tidal forces drive the gas reservoir into the center, and this nearly primordial gas dilutes the metallicity of the central gas, on a time-scale shorter, than the enrichment time-scale due to the triggered starburst.
Warps and polar rings might be the best evidence of gas accretion, since their setlling takes a few Gyr, and they reflect non-aligned accretion of matter. Other evidence of accretion can be seen in high velocity clouds, high-column density absorbants in front of quasars. There is a bimodality between the gas inflowing and outflowing, with almost equal weights, through the metallicity and the angle of accretion.
There is a strong redshift evolution of gas fraction in galaxies, which can explain the SFR history, peaking at z=1-2. The disk dynamics at high z is different from what is observed locally, since the high gas fraction makes disks highly unstable and turbulent.
Thanks to Mark Seigar and the organisers for such an interesting and nicely located meeting.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We construct an invariant of certain open four-manifolds using the Heegaard Floer theory of Ozsvath and Szabo. We show that there is a manifold $X$ homeomorphic to ${\mathbb{R}}^4$ for which the invariant is non-trivial, showing that $X$ is an exotic ${\mathbb{R}}^4$. This is the first invariant that detects exotic ${\mathbb{R}}^4$’s.'
address: |
Stat Math Unit,\
Indian Statistical Institute,\
Bangalore 560059, India
author:
- Siddhartha Gadgil
title: 'Open manifolds, Ozsvath-Szabo invariants and Exotic ${\mathbb{R}}^4$’s'
---
Introduction
============
In this paper, we construct invariants of certain open $4$-manifolds using the Heegaard Floer theory of Ozsvath and Szabo, and show that our invariants can detect exotic ${\mathbb{R}}^4$s. Previous constructions of exotic ${\mathbb{R}}^4$’s used indirect arguments to establish exoticity.
Given an $(n+1)$-dimensional field theory, a direct limit construction can be used to construct an invariant of open $(n+1)$-dimensional manifolds (which we see in detail later). The subtlety in the case of Ozsvath-Szabo invariants is that they do not give a field theory, but satisfy a more complicated composition law. However if we restrict to a class of cobordisms, which we call *admissible cobordisms*, we do get a field theory. Using this, we construct our invariants.
Recall that the Ozsvath-Szabo invariants of a smooth, oriented $3$-manifold $M$ associate homology groups to $M$ equipped with a $Spin^c$ structure $t$. Further, given a smooth cobordism $W$ between $3$-manifolds $M_1$ and $M_2$ and a $Spin^c$ structure ${\mathfrak{s}}$ on $W$, we get an induced map on the groups associated to the restrictions of ${\mathfrak{s}}$ to $M_1$ and $M_2$. To make this into a field theory, one needs a composition rule for a cobordism $W_1$ from $M_1$ to $M_2$ equipped with a $Spin^c$ structure ${\mathfrak{s}}_1$ and a cobordism $W_2$ from $M_2$ to $M_3$ equipped with a $Spin^c$ structure ${\mathfrak{s}}_2$ with ${\mathfrak{s}}_1|_{M_2}={\mathfrak{s}}_2|_{M_2}$. However, such $Spin^c$ structures ${\mathfrak{s}}_1$ and ${\mathfrak{s}}_2$ do not in general uniquely determine a $Spin^c$ structure on the composition $W=W_1\coprod_{M_2} W_2$ of $W_1$ and $W_2$. We do have a weaker composition law, where we sum over $Spin^c$ structures on $W$ restricting to ${\mathfrak{s}}_1$ and ${\mathfrak{s}}_2$.
We now find sufficient conditions under which ${\mathfrak{s}}_1$ and ${\mathfrak{s}}_2$ uniquely determine a $Spin^c$ structure ${\mathfrak{s}}$ on $W$. The $Spin^c$ structures on a manifold $X$ are a torseur of $H^2(X,{\mathbb{Z}})$. Consider the Mayer-Vietoris sequence for $W=W_1\cup W_2$ $$\to H^1(W_1)\oplus H^1(W_2)\to H^1(M_2)\overset{\delta}\to H^2(W)\to
H^2(W_1)\oplus H^2(W_2)\to H^2(M_2)$$
From this sequence, it follows that, given ${\mathfrak{s}}_1$ and ${\mathfrak{s}}_2$ as above, there is a unique $Spin^c$ structure ${\mathfrak{s}}$ on $W$ which restricts to ${\mathfrak{s}}_1$ and ${\mathfrak{s}}_2$ if and only if the coboundary map $\delta:H^1(M_2)\to
H^2(W)$ is trivial. This is equivalent to the map induced by inclusions $H^1(W_1)\oplus H^1(W_2)\to H^1(M_2)$ being surjective. Motivated by this, we make the following definition.
A smooth $4$-dimensional cobordism $W$ from $M_1$ to $M_2$ is admissible if the map induced by inclusion $H^1(W)\to H^1(M_2)$ is surjective.
We shall see basic properties of such cobordisms in Section \[cnvx\]. We now turn to the corresponding notions for open manifolds. Let $X$ be an open $4$-manifold which we assume for simplicity has one end. Let $K_1\subset K_2\subset \dots$ be an exhaustion of $X$ by compact manifolds and let $M_i={\partial}K_i$. We assume here and henceforth (for all exhaustions) that $K_i\subset
int(K_{i+1})$. For $i<j$, let $W_{ij}=K_j-int(K_i)$ be cobordisms from $M_i$ to $M_j$.
The exhaustion $\{K_i\}$ of $X$ is said to be admissible if each cobordism $W_{ij}$, $i,j\in{\mathbb{N}}$, $i<j$, is admissible. The manifold $X$ is said to be admissible if it has an admissible exhaustion.
We shall need to consider the appropriate notion of $Spin^c$ structures for the ends of $4$-manifolds.
An asymptotic $Spin^c$ structure ${\mathfrak{s}}$ on $X$ is a $Spin^c$ structure on $X-K$ for a compact subset $K\subset X$. Two asymptotic $Spin^c$ structures ${\mathfrak{s}}_1$ and ${\mathfrak{s}}_2$, defined on $X-K_1$ and $X-K_2$, are said to be equal if there is a compact set $K_0\supset K_1,K_2$ with ${\mathfrak{s}}_1|_{M-K_0}={\mathfrak{s}}_2|_{M-K_0}$.
Given an admissible open $4$-manifold $X$ and an asymptotic $Spin^c$ structure ${\mathfrak{s}}$, we can define invariants of $X$, which we call the *End Floer Homology*, using direct limits. We shall see in Section \[inv\] that an admissible exhaustion gives a directed system.
There is an invariant $HE(X,{\mathfrak{s}})$ which is the direct limit of the reduced Heegaard Floer homology groups $HF^+_{red}(M_i,{\mathfrak{s}}|_{M_i})$ under morphisms induced by the cobordisms $W_{ij}$. Furthermore this is independent of the admissible exhaustion of $X$.
We shall also need a *twisted* version of these invariants. Let $K\subset X$ be a compact set, ${\mathfrak{s}}$ a $Spin^c$-structure on $X-K$ and $\omega$ a $2$-form on $X-K$. Then we consider the reduced Floer theory with $\omega$-twisted coefficients (as in [@OZ4]). Once more we get a directed system whose limit gives an invariant $\underline{HE}(X,{\mathfrak{s}})$.
By taking an exhaustion of ${\mathbb{R}}^4$ by balls, we have the following proposition.
For the unique asymptotic $Spin^c$ structure ${\mathfrak{s}}$ on ${\mathbb{R}}^4$ (and any $2$-form $\omega$ on ${\mathbb{R}}^4-K$ with $K$ compact), we have $\underline{HE}({\mathbb{R}}^4,{\mathfrak{s}})=0$.
Our main result is that there are manifolds homeomorphic to ${\mathbb{R}}^4$ but with non-vanishing end Floer homology.
\[exot\] There is a $4$-manifold $X$ homeomorphic to ${\mathbb{R}}^4$ such that there is a compact set $K\subset X$, a $Spin^{{\mathbb{C}}}$ structure ${\mathfrak{s}}$ on $X-K$ and a closed $2$-form $\omega$ on $X-K$ with $\underline{HE}(X,{\mathfrak{s}})\neq 0$ with $\omega$-twisted coefficients.
Thus, $X$ is an exotic ${\mathbb{R}}^4$. Previous constructions of exotic ${\mathbb{R}}^4$’s used indirect arguments to show that they are exotic. The *End Floer homology* is the first invariant that detects exotic ${\mathbb{R}}^4$’s.
Admissible cobordisms and admissible ends {#cnvx}
=========================================
We henceforth assume that all our manifolds are smooth and oriented and all cobordisms are compact and $4$-dimensional. By $W:M_1\to M_2$ we mean a smooth cobordism from the closed $3$-manifold $M_1$ to the closed $3$-manifold $M_2$. Given $W_1:M_1\to M_2$ and $W_2:M_2\to
M_3$, $W_2\circ W_1$ denotes the composition of the cobordisms $W_1$ and $W_2$.
In this section we prove some simple results concerning admissible cobordisms and admissible ends.
\[comp\] Suppose $W_1:M_1\to M_2$ and $W_2:M_2\to M_3$ are admissible cobordisms, then $W=W_2\circ W_1$ is admissible.
We need to show that the map $H^1(W)\to H^1(M_3)$ induced by inclusion is surjective. This is the composition of maps $H^1(W)\to H^1(W_2)$ and $H^1(W_2)\to H^1(M_3)$ induced by inclusion, with the latter surjective by hypothesis. We shall show that the map $H^1(W)\to
H^1(W_2)$ is surjective.
Let $\alpha\in H^1(W_2)$ be a class. Let $i_j:M_2\to W_j$, $j=1,2$, be inclusion maps. Consider the Mayer-Vietoris sequence $$\dots\to H^1(W)\to H^1(W_1)\oplus H^1(W_2)\overset{i_1^*+i_2^*}\to
H^1(M_2)\to\dots$$
By admissibility of $W_1$, there is a class $\beta\in H^1(W_1)$ with $i_1^*(\beta)=i_2^*(\alpha)$. Hence the image of the class $(-\beta,\alpha)\in H^1(W_1)\oplus H^1(W_2)$ in $H^1(M_2)$ is zero, and so $(-\beta,\alpha)$ is the image of a class $\varphi\in
H^1(W)$. In particular $\alpha$ is the image of $\varphi$ under the map induced by inclusion.
\[comp2\] Suppose $W_1:M_1\to M_2$ and $W_2:M_2\to M_3$ are cobordisms with $W=W_2\circ W_1$ admissible. Then $W_2$ is admissible.
By hypothesis the map $H^1(W)\to H^1(M_3)$ is surjective. This factors through the map $H^1(W_2)\to H^1(M_3)$, which must also be surjective.
We need criteria for when cobordisms corresponding to attaching handles are admissible.
\[handl\] Let $M=M_1$ be a $3$-manifold, $W$ the cobordism corresponding to a handle addition and $M_2$ the other boundary components of $W$. The following hold.
1. A product cobordism is admissible.
2. The cobordism corresponding to attaching a $1$-handle to a closed $3$-manifold $M$ is admissible.
3. If $K$ is a knot in a closed $3$-manifold which represents a primitive, non-torsion element in $H_1(M)$, then the cobordism corresponding to attaching a $2$-handle to $M$ along $K$ is admissible.
We shall show that the map induced by the inclusion from $H_1(M_2)$ to $H_1(W)$ is an isomorphism in each case. As the map on cohomology is the adjoint of this map, it follows that it is a surjection.
The case of a product cobordism is immediate. In the second case we see that $H_1(M_2)=H_1(W)=H_1(M)\oplus {\mathbb{Z}}$ with the isomorphism induced by inclusion. In the third case we have $H_1(M)=H\oplus{\mathbb{Z}}$, with $[K]$ generating the ${\mathbb{Z}}$ component and $H$ isomorphic to the homology of the $3$-manifold obtained by surgery about $K\subset M$. It is easy to see that $H_1(W)=H_1(M_2)=H$.
Now let $X$ be an open manifold and let $K_1\subset K_2\subset\dots $ be an exhaustion of $X$ and $M_i$ and $W_{ij}$ be as before.
The exhaustion $\{K_i\}$ is admissible if and only if each of the manifolds $K_{j+1}-int(K_j)$ is admissible.
Each $W_{ij}$ is the composition of cobordisms $K_{j+1}-int(K_j)$. The result follows by Lemmas \[comp\] and \[comp2\].
Thus, if $X$ is obtained from a compact manifold $K$ by attaching handles as in Lemma \[handl\] then $X$ is admissible. Our examples of exotic ${\mathbb{R}}^4$s will be of this form.
It is immediate from the definition that for any admissible exhaustion $K_i$, the exhaustion obtained by passing to a subsequence $K_{i_j}$ is admissible. To show independence of our invariants under exhaustions, we need the following lemma.
\[refin\] Let $K_1\subset L_1\subset K_2\subset L_2\dots$ be an exhaustion of $X$ with $K_1\subset K_2\subset \dots$ and $L_1\subset L_2\subset
\dots$ admissible exhaustions. Then the exhaustion $L_1\subset
K_2\subset L_2\subset K_3\dots$ is admissible.
It suffices to show that the cobordisms $K_{j+1}-int(L_j)$, $j\geq 1$ and $L_j-int(K_j)$, $j\geq 2$ are admissible. This follows from Lemma \[comp2\] as the cobordisms $K_{j+1}-int(K_j)$ and $L_{j+1}-int(L_j)$ are admissible and we have $K_{j+1}-int(K_j)=(K_{j+1}-int(L_j))\circ (L_{j}-int(K_j))$ and $L_{j+1}-int(L_j)=(L_{j+1}-int(K_j))\circ (K_{j}-int(L_j))$.
Invariants for admissible ends {#inv}
==============================
We are now ready to define our invariants for an admissible open $4$-manifold $X$. We shall construct invariants based on reduced Heegaard Floer theory $HF_{red}^+$. First we recall some facts about Ozsvath-Szabo theory.
Associated to each closed, oriented $3$-manifold $M$ and $Spin^c$ structure $t$ on $M$ we have abelian groups $HF^+(M,t)$, $HF^-(M,t)$ and $HF^{\infty}(M,t)$ that fit in an exact sequence $$\dots\to HF^-(M,t)\to HF^{\infty}(M,t)\to HF^+(M,t)\to\dots$$
Further, a cobordism $W:M_1\to M_2$ with a $Spin^c$ structure ${\mathfrak{s}}$ on $W$ such that $t_i=s|_{M_i}$ induces homomorphisms $F_{W,{\mathfrak{s}}}$ on these abelian groups which commute with the maps in the above exact sequence.
The group $HF_{red}^+(M,t)$ is defined as the quotient of $HF^+(M,t)$ by the image of $HF^{\infty}(M,t)$. This is isomorphic to the kernel $HF_{red}^-(M,t)$ of the map from $HF^-(M,t)$ to $HF^{\infty}(M,t)$. Further, given a cobordism $W:M_1\to M_2$ with a $Spin^c$ structure ${\mathfrak{s}}$ on $W$ such that $t_i=s|_{M_i}$, we get an induced homomorphism on the abelian groups $F_{W,{\mathfrak{s}}}:HF_{red}^+(M_1,t_1)\to HF_{red}^+(M_2,t_2)$ induced by the corresponding homomorphism on $HF^+$ as the image of $HF^{\infty}(M_1,t)$ is contained in $HF^{\infty}(M_2,t)$. This homomorphism is well defined up to choice of sign. We shall denote the above cobordism with its $Spin^c$ structure by $(W,{\mathfrak{s}}):(M_1,t_1)\to
(M_2,t_2)$.
Further, if $(W_1,{\mathfrak{s}}_1):(M_1,t_1)\to (M_2,t_2)$ and $(W_2,{\mathfrak{s}}_2):(M_2,t_2)\to (M_3,t_3)$, with $W=W_2\circ W_1$, we have the composition formula $$F_{W_2,{\mathfrak{s}}_2}\circ F_{W_1, s_1}=\sum _{s|_{W_i}=s_i} \pm F_{W,{\mathfrak{s}}}$$
We shall consider the special case when $W_1$ is admissible.
\[fact\] If $W_1$ is admissible then there is a unique $Spin^c$ structure ${\mathfrak{s}}$ on $W$ with ${\mathfrak{s}}|_{W_i}=s_i$. For this $Spin^c$ structure $F_{W_2,{\mathfrak{s}}_2}\circ
F_{W_1, s_1}= \pm F_{W,{\mathfrak{s}}}$
Recall that $Spin^c$ structures are a torseur of $H^2(\cdot,{\mathbb{Z}})$. Consider the Mayer-Vietoris sequence for $W=W_1\cup
W_2$ $$\to H^1(W_1)\oplus H^1(W_2)\to H^1(M_2)\overset{\delta}\to H^2(W)\to
H^2(W_1)\oplus H^2(W_2)\to H^2(M_2)$$
By admissibility the map $H^1(W_1)\oplus H^1(W_2)\to H^1(M_2)$ is a surjection, hence $H^2(W)\to H^2(W_1)\oplus H^2(W_2)$ is an injection. This shows uniqueness of the $Spin^c$ structure. As ${\mathfrak{s}}_1|_{M_2}=t_2=s_2|_{M_2}$, existence follows from the same exact sequence.
The second statement follows from the first using the composition formula.
For an admissible exhaustion, it follows that we get a directed system of abelian groups up to sign. We next see that we can choose signs to get a directed system, and the direct limit of the system does not depend on the choice of signs.
\[sign\] Assume $A_i$ is a sequence of Abelian groups and maps $f_{ij}:A_i\to
A_j$, such that for $i<j<k$, $f_{ik}=\pm f_{jk}\circ f_{ij}$. Then we can choose $g_{ij}=\pm f_{ij}$ such that we get a directed system. Furthermore the limit is independent, up to isomorphism, of the choices.
Let $g_{1j}=f_{1j}$. For $i<j$, the composition law $g_{1j}=g_{ij}\circ g_{1i}$ uniquely determines sign of $g_{ij}=\pm
f_{ij}$, and such a $g_{ij}$ exists as $f_{1j}=\pm f_{ij}\circ
f_{1i}$. It is easy to see that this gives a directed system.
For a different choice the maps $g_{1j}$ are replaced by $g'_{ij}=\epsilon_j g_{1j}$, $\epsilon_j=\pm 1$. We get in general a different directed system, with the groups $A_i$ . However, using the isomorphisms $\epsilon_i:A_i\to A_i$ (i.e., $x\mapsto \epsilon_i\times
x$ for $x\in A_i$), we get an isomorphism of directed systems. Hence the limits are isomorphic.
The End Floer homology $HE(X,{\mathfrak{s}})$ is the direct limit of the directed system constructed above.
\[welldef\] The End Floer homology is independent of the admissible exhaustion chosen.
By elementary properties of direct limits, the limit does not change on passing to a subsequence of an exhaustion. Given two admissible exhaustions $K_1\subset K_2\subset \dots$ and $L_1\subset L_2\subset
\dots$, by passing to subsequences we can assume that $K_1\subset
L_1\subset K_2\subset L_2\subset \dots$ for the two exhaustions. By Lemma \[refin\] the exhaustion $L_1\subset K_2\subset L_2\subset
K_3\dots$ is admissible. As $L_1\subset L_2\subset \dots$ and $K_2\subset K_3\subset \dots$ are subsequences of this exhaustion, the direct limits for the exhaustions $K_1\subset K_2\subset \dots$ and $L_1\subset L_2\subset \dots$ are the same (as they are both isomorphic to the direct limit corresponding to the exhaustion $L_1\subset K_2\subset L_2\subset K_3\dots$).
We see that this depends only on the diffeomorphism class of the end of $X$. More precisely, we have the following.
\[enddiff\] Suppose $X$ and $Y$ are admissible smooth $4$-manifolds and $K\subset X$ and $L\subset Y$ are compact sets so that there is a diffeomorphism $f:X-K\to Y-L$. Then the End Floer homology groups of $X$ and $Y$ are isomorphic.
Consider an admissible exhaustion $K_1\subset K_2\subset \dots$ with $K\subset K_1$. We define an exhaustion $L_1\subset L_2\subset \dots$ of $Y$ by $L_i=L\cup f(K-K_i)$. The map $f$ induces isomorphisms between the terms of the directed systems corresponding to the two exhaustions. Thus, the End Floer homology groups, which are the limits of these directed systems, are isomorphic.
We consider the $\omega$-twisted version of this as in [@OZ4]. Let $K\subset X$ be a compact manifold and $\omega$ a $2$-form on $X-K$. We call such a $2$-form $\omega$ on $X-K$, for $K$ compact, an asymptotic $2$-form. Given two closed $2$-forms $\omega_i$, $i=1,2$, on the complements $X-K_i$ of smooth compact sets $K_i$, $1=1,2$, we say that $\omega_1$ and $\omega_2$ are asymptotically cohomologous if, for some compact set $K$, $K_i\subset K$ for $i=1,2$, the restrictions of the forms are cohomologous on $X-K$. We can thus speak of asymptotic cohomology classes of asymptotic $2$-forms.
We consider an admissible exhaustion with the first term $K_1$ satisfying $K\subset K_1$. For this, we can define the twisted groups $\underline{HF}_{red}^+(M_i,t_i)$ and homomorphisms associated to $W_{ij}$ which are well defined up to sign and multiplication by powers of $T$. For any composition $W=W_2\circ W_1$ associated with the exhaustion as above, the coboundary map $\delta:H^1(M_2)\to
H^2(W)$ is zero. It follows by the composition rule for $\omega$-twisted coefficients that we have a directed system up to multiplication by powers of $T$ and sign. As in Lemma \[sign\], we can make choices for the homomorphisms to get a directed system and the direct limit is independent of the choices.
The direct limit is the End Floer homology $\underline{HE}(X,{\mathfrak{s}})$ with $\omega$-twisted coefficients. The following propositions are ananlogous to Propositions \[welldef\] and \[enddiff\].
For an aymptotic $2$-form $\omega$, the $\omega$-twisted End Floer homology is independent of the choice of admissible exhaustion.
Let $X$ and $Y$ are smooth $4$-manifolds with admissible ends and $\omega_X$ and $\omega_Y$ are asymptotic $2$-forms on $X$ and $Y$. If there are compact sets $K\subset X$ and $L\subset Y$, with $\omega_X$ and $\omega_Y$ defined on $X-K$ and $Y-L$, and a diffeomorphism $f:X-K\to Y-L$ so that $\omega_X$ is asymptotically cohomologous to $f^*(\omega_Y)$, then the End Floer homology with $\omega_X$-twisted coefficients of $X$ is homologous to the End Floer homology with $\omega_Y$ twisted coefficients of $Y$.
Exotic ${\mathbb{R}}^4$’s
=========================
We now construct a manifold $X$ homeomorphic to ${\mathbb{R}}^4$ with $\underline{HE}(X)\neq 0$. This is done by first constructing a convex symplectic manifold $W$ with one convex boundary component $N_0$ and one convex end and then gluing a compact manifold $Y$ to $W$ along $N_0$.
Construction of $X$
-------------------
Let $K$ be a non-trivial slice knot in $S^3$ and let $N$ be obtained by $0$-frame surgery about $K$. Then $N$ admits a taut foliation by [@Ga], and hence $N\times [0,1]$ admits a symplectic structure with both ends convex by [@ET]. The symplectic structure induces a contact structure $\xi$ on $N$. We shall construct a symplectic manifold $Q$ with one concave boundary component contactomorphic to $(N,\xi)$ and one convex end. The manifold $W$ is obtained by gluing $Q$ to $N\times [0,1]$.
Let $P$ be the manifold obtained by attaching a $2$-handle $H$ to $N\times
\{1\}$ corresponding to the surgery cancelling the $0$-frame surgery about $K$. The manifold $P$ has boundary $S\cup N_0$ with $N_0=N\times \{0\}$ and $S$ a $3$-sphere. Let $P_0$ be $P-S$. Then $P_0$ has one boundary component, which is diffeomorphic to $N$, and one end.
\[casson\] There is a symplectic manifold $Q$ properly homotopy equivalent to $P_0$ so that the end of $Q$ is convex and the boundary component identified with $N_0$ is concave with induced contact structure $\xi$.
We construct $Q$ as a Stein cobordism as in [@EH]. Firstly, by a theorem of Eliashberg [@El1] (Lemma 2.2 in [@EH]), there is a Stein cobordism from $(N,\xi)$ to itself, which is thus a Stein structure on $N\times [0,1]$ with $N\times\{0\}$ a concave boundary component and $N\times \{1\}$ a convex boundary component. We construct the manifold $Q$ by attaching $1$-handles and $2$-handles starting with the convex boundary component, with the $2$-handles attached with framing $1$ less than the Thurston-Bennequin framing (we call this Legendrian handle addition). By Eliashberg’s characterisation of Stein domains [@El2] (see also [@El3] and [@Go]), $Q$ is Stein.
The $1$-handles and $2$-handles are attached as in Theorem 3.1 of [@Go], so that the handle $H$ is replaced by a Stein Casson handle. Specifically, by taking a Legendrian representative of $\kappa={\partial}H$, we can perform Legendrian handle addition about $\kappa$ but with incorrect framing, differing from that of $H$ by an integer $k$. If we attach a handle to $\kappa$ with this farming but with $k$ self-plumbings (a so called *kinky handle*), then the self-intersection pairing coincides with that obtained by attaching $H$. As in [@Go] (where there is an explicit construction in Figure 22), one can attach $1$-handles and Legendrian $2$-handles to obtain a Stein manifold diffeomorphic to that obtained by attaching a $2$-handle with $k$ self-plumbings to $\kappa$ so that we have the same intersection pairing as adding $H$.
Thus, we obtain a Stein cobordism with the same intersection pairing as attaching the handle $H$, but with non-trivial fundamental group. By a lemma of Casson, we can find a family of curves on the boundary of the attached kinky handle, hence the convex boundary of the Stein cobordism, so that attaching $2$-handles to these curves (with appropriate framing) gives the manifold obtained on attaching $H$. As before, we can instead attach kinky handles to obtain a Stein cobordism.
Iterating this procedure gives a non-compact Stein cobordism $Q$ with one concave boundary component and one convex end, which is diffeomorphic to the manifold obtained by attaching a Casson handle in place of $H$. As Casson handles are properly homotopy equivalent to the interiors of handles, $Q$ is properly homotopy equivalent to $P_0$.
Let $W$ be the symplectic manifold obtained by gluing $N\times [0,1]$ with its sympectic structure obtained by the Gabai-Eliashberg-Thurston theorem, to the symplectic manifold $Q$, with $N\times \{1\}$ identified with the (concave) boundary of $Q$. Observe that $W$ is simply-connected as the Casson handle corresponding to the $2$-handle $H$ is attached along the meridian of $K$, which normally generates $\pi_1(N)$. Also observe that in the proof of Lemma \[casson\], following Theorem 3.1 of [@Go], the handles attached are as in Lemma \[handl\], and hence the corresponding exhaustion is admissible.
Next, let $Y'$ be obtained from $B^4$ by attaching a $2$-handle along $K$ with framing $0$. Then ${\partial}Y'=N$. As $K$ is slice, the generator of $H_2(Y)={\mathbb{Z}}$ can be represented by an embedded sphere $\Sigma$. Let $Y$ be obtained from $Y'$ by performing surgery along $\Sigma$. Glue $W$ to $Y$ along ${\partial}Y=N=N\times\{0\}$ to obtain $X$.
By a Mayer-Vietoris argument, $X$ has the homology of ${\mathbb{R}}^4$. Further, as $\pi_1(Y)$ is normally generated by a meridian of $K$, to which a Casson handle is attached, $\pi_1(X)=1$. Finally, the end of $X$ is properly homotopic to the end of $P_0=P-S$, and hence $Y$ is simply-connected at infinity. Thus $Y$ is homeomorphic to ${\mathbb{R}}^4$ by Freedman’s theorem [@Fr].
Non-Vanishing of End Floer homology
-----------------------------------
Finally, we show that the End Floer homology for $X$ does not vanish. Consider the exhaustion of $X$ with $K_1=Y$, hence $M_1=N$ and $K_2$, $K_3$, …being the level sets after attaching successive handles as above. Note that $X-K_1$ is symplectic with symplectic form $\omega$, and each of the cobordisms $W_{1j}$ is a convex symplectic manifold with two convex boundary components $M_1$ and $M_j$. Hence $W_{1j}$ embeds in a symplectic $4$-manifold $Z=X_1\cup W_{1j}\cup X_j$ with both components of $Z-W_{1j}$ having $b_2^+>0$ by results of Eliashberg [@El] and Kronheimer-Mrowka [@KM]. Here $X_1$ and $X_j$ are manifolds with boundaries $M_1$ and $M_j$, respectively.
We shall consider $\omega$-twisted coefficients and the $Spin^c$ structure ${\mathfrak{s}}$ associated to $\omega$. Recall that $\omega$-twisted coefficients are coefficients determined by $\omega$ as follows: for a $3$-manifold $P\subset M$, we consider ${\mathbb{Z}}[{\mathbb{R}}]$ as a module over ${\mathbb{Z}}[H^1(N,{\mathbb{Z}})]$ via the ring homomorphism $[\gamma]\mapsto T^{\int_N
[\gamma]\wedge\omega}$. Ozsvath and Szabo show that we have induced maps with $\omega$-twisted coefficients satisfying an appropriate composition formula. By an application of Stokes theorem, we deduce the relation $$\int_N [\gamma]\wedge\omega=\int_Z \delta[\gamma]\wedge\omega$$
Let $t_i$ be the $Spin^c$ structure on $M_i$ induced by ${\mathfrak{s}}$. We first construct an element $x_1\in \underline{HF}^+(M_1,t_1)$ whose image $z_1\in \underline{HF}_{red}^+(M_1,t_1)$ will be shown to have non-zero image in the direct limit giving the End Floer homology.
Let $P\subset X_1$ be an admissible cut in the terminology of Ozsvath and Szabo. Then as $\delta H^1(P)=0$, $\omega$-twisted coefficients coincide with untwisted coefficients(as $\int_P
[\gamma]\wedge\omega=\int_Z \delta[\gamma]\wedge\omega=0$). Let the closures of the components of $X_1-P$ be $U$ and $V$, with $M_1\subset{\partial}V$. Let $B_1\subset U$ be a ball. As in the construction of the closed $4$-manifold invariants, we obtain an element $\xi\in \underline{HF}^+(P,{\mathfrak{s}})={HF}^+(P,{\mathfrak{s}})$ as the image of the generator of $HF^-(S^3)$ using the isomorphism between $HF^-_{red}$ and $HF^+_{red}$. We define $x_1$ to be the image $\underline{F}_V(\xi)$ of $\xi$ in $\underline{HF}^+(M_1,t_1)$ under the map induced by the cobordism $V$ and let $z_1$ be its image in reduced Floer homology.
Let $x_j\in\underline{HF}^+(M_j,t_j)$ be the image of $x_1$ under the cobordism induced by $W_{1j}$ and let $z_j\subset
\underline{HF}_{red}^+(M_j,t_j)$ be corresponding image of $z_1$.
For every $j\geq 0$, $z_j\neq 0$.
Let $j>1$ be fixed. Let $W=W_{1j}\cup X_j$ and let $B_2$ be a ball in $X_j$. We shall show that the image of $x_1$ in $HF^+(S^3,{\mathfrak{s}}_0)$ under the map induced by $W-B_2$ is non-zero.
The image $\underline{F}_{W-B_2}(x_1)$ of $x_1$ in $HF^+(S^3,{\mathfrak{s}}_0)$ under the map induced by $W-B_2$ is non-zero.
Our proof is based on the proof of Theorem 4.2 in [@OZ4]. We use the product formula with $\omega$-twisted coefficients $$\sum_{\eta\in H^1(M_1,{\mathbb{Z}})} \Phi_{M,{\mathfrak{s}}+\delta\eta}T^{<\omega\cup
c_1(s+\delta\eta),[M]>}=\underline{F}_{W-B_2}\circ
\underline{F}_V(\xi)=\underline{F}_{W-B_2}(x_1)$$
Thus it suffices to show that the left hand side does not vanish. By results of Ozsvath and Szabo on the closed four-manifold invariants for symplectic manifolds (as in [@OZ4], Theorem 4.2), the lowest order term of the left hand side, which is a polynomial in $T$, is $1$. It follows that $\underline{F}_{W-B_2}(x_1)\neq 0$, completing the proof.
Now, by Lemma \[fact\], as $W_{1j}$ is admissible, this factors through the map induced by $W_{1j}$, and hence the image of $x_j$ in $HF^+(S^3,{\mathfrak{s}}_0)$ is non-zero. But as the cobordism $X_j-int(B_2)$ has $b_2^+>0$, the induced map on $\underline{HF}^{\infty}$ is zero. It follows that $x_j$ is not in the image of $\underline{HF}^{\infty}(M_i,t_i)$, i.e. $z_j\neq 0$, as claimed.
Thus, the End Floer homology of $X$ does not vanish. We have seen that $X$ is homeomorphic to ${\mathbb{R}}^4$. This completes the proof of Theorem \[exot\].
[10]{}
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Ozsvath, Peter; Szabo, Zoltan *Holomorphic discs and topological invariants for closed three-manifolds*, to appear in the Annals of Mathematics.
Ozsvath, Peter; Szabo, Zoltan *Holomorphic triangles and invariants of smooth four-manifolds*, preprint
Ozsvath, Peter; Szabo, Zoltan *Holomorphic triangle invariants and the topology of symplectic four-manifolds*. Duke Math. J. 121(2004), no. 1, 1–34
Ozsvath, Peter; Szabo, Zoltan *Holomorphic disks and genus bounds*, Geom. Topol. **8** (2004), 311–334
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We show that the usual sufficient criterion for a very general hypersurface in a smooth projective manifold to have the same Picard number as the ambient variety can be generalized to quasi-smooth hypersurfaces in complete simplicial toric varieties. This sufficient condition always holds for very general K3 surfaces embedded in Fano toric 3-folds.'
address:
- |
$^\P$ Department of Mathematics, University of Pennsylvania,\
David Rittenhouse Laboratory, 209 S 33rd Street,\
Philadelphia, PA 19104, USA[^1][^2]
- '$^\S$ Istituto Nazionale di Fisica Nucleare, Sezione di Trieste'
author:
- 'Ugo Bruzzo$^{\P\S\dag}$ and Antonella Grassi$^\P$'
title: |
Picard group of hypersurfaces\
in toric 3-folds
---
SISSA Preprint 78/2010/fm\
[arXiv:1011.1003]{}
[^3]
Introduction
============
In this paper we study the Noether-Lefschetz problem for hypersurfaces in complete simplicial toric threefolds, namely, we prove that under a certain condition, a very general hypersurface in an ample linear system in such a toric threefold ${{{{\mathbb P}}_\Sigma}}$ has the same Picard number as ${{{{\mathbb P}}_\Sigma}}$. In particular, this holds for a very general K3 hypersurface in the anticanonical system of a simplicial toric Fano threefold. (A property is very general if it holds in the complement of countably many proper closed subvarieties [@Lazarsfeld].)
This result can be regarded on one hand as a first step towards the study of Noether-Lefschetz loci of the moduli space of $K3$ hypersurfaces in a toric Fano threefolds; see also the recent works of [@KMPS; @MPP; @Kloos07]. On the other hand, this completes the picture for computing the Picard number for certain hypersurfaces in the anticanonical system of a toric Fano variety, by handling the unknown case in dimension $3$.
Recall that the Picard number, $\rho(Y)$ of a variety $Y$ is the rank of the Néron-Severi group, that is of the image of the Picard group in the second cohomology group with integer coefficients. The Picard group and the Picard number of a toric variety ${{{{\mathbb P}}_\Sigma}}$ can be easily computed from the combinatorial data of $\Sigma$. Let $X$ be a nondegenerate hypersurface in the anticanonical system of a simplicial toric Fano variety ${{{{\mathbb P}}_\Sigma}}$, with $\dim {{{{\mathbb P}}_\Sigma}}\geq 4$ (note that a general hypersurface is also nondegenerate). In the 80s and 90s it was shown [@CoKa99; @Bat94; @DK; @AGM], that the Picard number of any such $X$ can be explicitly computed from combinatorial data. This result was a pivotal ingredient in describing the toric version of mirror symmetry (see for example [@CoKa99]). The argument in the above papers is essentially topological and computes the dimension of the second cohomology group of $X$, which happens to be equal to $\rho(X)$ if $\dim (X) \geq 3$, but not necessarily if $\dim (X)=2$. In addition, even the statement in the above papers does not hold when $\dim {{{{\mathbb P}}_\Sigma}}=3$, as we see from the case of Fermat’s quartic in ${{\mathbb P}}^3$, which is nondegenerate.
This type of result was generalized by Roan to the case of toric varieties (not necessarily Fano) also for the case when the ambient variety has dimension $d\ge 4$ [@Roan96], and by Ravindra and Srinivas to general normal varieties, still with the restriction $d\ge 4$ [@RaviSri06].
This paper then fills the gap for $\dim {{{{\mathbb P}}_\Sigma}}= 3$. It was already known that $\rho (X)= \rho({{{{\mathbb P}}_\Sigma}})$ for particular cases of toric Fano threefolds, namely certain weighted projective spaces [@CoxWeighted; @SJ; @deJongSteen], as in the higher dimensional case. The techniques used in the case of weighted projective spaces are very much tailored to that specific case [@Dolgy; @SJ; @deJongSteen]. On the other hand, the classical infinitesimal techniques introduced in the 70s by Griffiths, Steenbrink and collaborators to solve the Noether-Lefschetz problem in the smooth case (see for example [@CMP03]) cannot be used due to the presence of singularities. Our argument is partly inspired by Cox’s paper [@CoxWeighted]: it generalizes the classical infinitesimal techniques and combines them with more recent results about toric varieties, their Cox ring and their cohomology [@BaCox94]. In fact, $X$ and ${{{{\mathbb P}}_\Sigma}}$ are projective orbifolds, and a pure Hodge structure can be defined for them; this will be a key tool in the proof.
In Section \[background\] we mostly recall some relevant results from [@BaCox94], and adapt them to the set up of [@CMP03]. We start with basic properties of simplicial toric varieties and general hypersurfaces defined by ample divisors. Moreover we note that the exact sequence defining the primitive cohomology in middle dimension of such a hypersurface splits orthogonally with respect to the intersection pairing. The middle cohomology is the sum of the primitive cohomology and the “fixed” cohomology, i.e., the cohomology inherited from the ambient toric variety; the splitting is consistent with the Hodge decomposition. We then state some results of [@BaCox94] which express the primitive cohomology in middle degree in terms of the Jacobian ring of the hypersurface; here we assume that ambient space has odd dimension.
Section \[Pic\] contains the bulk of the argument: we proceed along the lines of the infinitesimal arguments of Griffiths for smooth varieties and adapt it to the toric case. We start from the moduli space of quasi-smooth hypersurfaces constructed in [@BaCox94], consider a natural Gauss-Manin connection, proceed to prove an infinitesimal Noether-Lefschetz theorem and then the needed global Noether-Lefschetz theorem. Finally, we focus on the case of $K3$ hypersurfaces in the anticanonical system of a simplicial toric Fano threefold.
The suggestion that a very general hypersurface in a toric Fano threefold ${{{{\mathbb P}}_\Sigma}}$ has the same Picard number as the ambient variety can be found, in a different language, in an unpublished paper by Rohsiepe [@Rohsiepe] (see the formula and Remark in the middle of page 3), based on some dimension counting arguments and trying to generalize to the case $\dim {{{{\mathbb P}}_\Sigma}}=3 $ a formula that Batyrev proved for $\dim{{{{\mathbb P}}_\Sigma}}= 4$ [@Bat94].
[**Acknowledgements.**]{} We thank Eduardo Cattani, Alberto Collino, David Cox, Igor Dolgachev, Luca Migliorini, Vittorio Perduca, Domingo Toledo and the referee for useful discussions and suggestions. We are grateful for the hospitality and support offered by the University of Pennsylvania and SISSA. The first author would also like to thank the staff and the scientists at Penn’s Department of Mathematics for providing an enjoyable and productive atmosphere.
Hypersurfaces in simplicial complete toric varieties {#background}
====================================================
In this section we recall some basic facts about hypersurfaces in toric varieties and their cohomology. We mainly follow the notation in [@BaCox94]. All schemes are schemes over the complex numbers.
Preliminaries and notation
--------------------------
Let $M$ be a free abelian group of rank $d$, let $N={\operatorname{Hom}}(M,{{\mathbb Z}})$, and $N_{{\mathbb R}}=N\otimes_{{\mathbb Z}}{{\mathbb R}}$.
[@BaCox94 Def. 1.1 and 1.3]
1. A convex subset $\sigma\subset N_{{\mathbb R}}$ is a rational $k$-dimensional simplicial cone if there exist $k$ linearly independent primitive elements $e_1,\dots,e_k\in N$ such that $\sigma = \{\mu_1e_1+\dots+\mu_ke_k\}$, with $\mu_i$ nonnegative real numbers. The generators $e_i$ are said to be [*integral*]{} if for every $i$ and any nonnegative rational number $\mu$, the product $\mu\,e_i$ is in $N$ only if $\mu$ is an integer.
2. Given two rational simplicial cones $\sigma$, $\sigma'$, one says that $\sigma'$ is a face of $\sigma$ (we then write $\sigma' < \sigma$) if the set of integral generators of $\sigma'$ is a subset of the set of integral generators of $\sigma$.
3. A finite set $\Sigma=\{\sigma_1,\dots,\sigma_s\}$ of rational simplicial cones is called a rational simplicial complete $d$-dimensional fan if
1. all faces of cones in $\Sigma$ are in $\Sigma$;
2. if $\sigma,\sigma'\in\Sigma$, then $\sigma\cap\sigma'<\sigma$ and $\sigma\cap\sigma'<\sigma'$;
3. $N_{{\mathbb R}}= \sigma_1\cup\dots\cup\sigma_s$.
A rational simplicial complete $d$-dimensional fan $\Sigma$ defines a toric variety ${{{{\mathbb P}}_\Sigma}}$ of dimension $d$ having only Abelian quotient singularities. Moreover, $
{{{{\mathbb P}}_\Sigma}}$ is simply connected, and is an orbifold. We shall use the term “orbifold” in the following sense (see, e.g., [@CoKa99], Def. A.2.1): an $n$-dimensional variety $Y$ is an orbifold if every point $y\in Y$ has a neighborhood which is isomorphic to $U/G$ as an analytic space, where $G$ is a subgroup of $Gl_n({{\mathbb C}})$ with no nontrivial complex reflections, and $U$ is a $G$-invariant neighborhood of the origin of ${{\mathbb C}}^n$. (A complex reflection is an element in $Gl_n({{\mathbb C}})$ with $n-1$ eigenvalues equal to 1.) A sub-orbifold of an orbifold $Y$ is a subvariety $Y'\subset Y$ with the property that for every $y\in Y'$ there is a local chart $(U/G,0)$ of $Y$ at $y$ such that the inverse image of $Y'$ in $U$ is smooth at 0. Intuitively, a sub-orbifold is a subvariety whose only singularities come from the ambient variety. These notions of orbifold and sub-orbifold are synonymous to those of $V$-manifold and sub-$V$-manifold, which is indeed the terminology used in [@BaCox94]. The notion of $V$-manifold is originally due to Satake [@Satake].
\[picards\]Let $Cl(\Sigma)$ be the group of Weil divisors in ${{{{\mathbb P}}_\Sigma}}$ modulo rational equivalence, and let $\operatorname{Pic}(\Sigma)$ be the group of line bundles on ${{{{\mathbb P}}_\Sigma}}$ modulo isomorphism. As the notation suggests, both are intrinsic to the fan $\Sigma$. Both are finitely generated Abelian groups, and $\operatorname{Pic}(\Sigma)$ is actually free. Moveover, under our assumptions the toric variety ${{{{\mathbb P}}_\Sigma}}$ is ${{\mathbb Q}}$-factorial, i.e., the natural inclusion $\operatorname{Pic}(\Sigma) \hookrightarrow Cl(\Sigma)$ becomes an isomorphism if one tensors by ${{\mathbb Q}}$. The rank of the two groups, denoted by $\rho(\Sigma)$, is also the [Picard number]{}, the rank of the Néron-Severi group of ${{{{\mathbb P}}_\Sigma}}$.
Recall that the Néron-Severi group of a variety $Y$ is the image of the Picard group in the second cohomology group with integer coefficients. One can define its rank as $\rho(Y) \stackrel{def}{=}
\dim_{{{\mathbb Q}}} NS(Y) \otimes _{{{\mathbb Z}}} {{\mathbb Q}}= \dim_{{{\mathbb Q}}} H^{2}(Y,{{\mathbb Q}}) \cap H^{1,1}(Y,{{\mathbb C}})$.
The group $\mathbf D(\Sigma)=\operatorname{Spec}{{\mathbb C}}[Cl(\Sigma)]$ is an affine algebraic group whose character group is isomorphic to $Cl(\Sigma)$. Since there is a surjection $\mathbb Z^n \twoheadrightarrow Cl(\Sigma)$, we have an embedding $\mathbf D(\Sigma)\hookrightarrow ({{\mathbb C}}^\ast)^n$, and a natural action of $\mathbf D(\Sigma)$ on the affine space ${{\mathbb A}}^n$. The quotient $\mathbf T(\Sigma)=({{\mathbb C}}^\ast)^n/\mathbf D(\Sigma)$ is an algebraic torus. Below we shall show that this group acts naturally on ${{{{\mathbb P}}_\Sigma}}$.
([@Cox95]) Given a fan $\Sigma$, consider a variable $z_i$ for each 1-dimensional cone $\varsigma_i$ in $\Sigma$, and let $S(\Sigma)$ be the polynomial ring ${{\mathbb C}}[z_1,\dots,z_n]$. For every $\sigma\in\Sigma$, let $z_\sigma = \prod_{\varsigma_i\not\subset \sigma} z_i$, and let $B(\Sigma)$ the ideal in $S(\Sigma)$ generated by the $z_\sigma$’s.
$S(\Sigma)$ is called the *Cox ring*.
$S(\Sigma)$ is a graded ring, with grading provided by the class group, $S(\Sigma) = \oplus_{\beta\in Cl(\Sigma) }S_\beta$. We identify the affine space ${{\mathbb A}}^n$ with $\operatorname{Spec}S(\Sigma)$, and denote by $Z(\Sigma)$ the affine variety in ${{\mathbb A}}^n$ given by the ideal $B(\Sigma)$. If we set $U(\Sigma)={{\mathbb A}}^n-Z(\Sigma)$, the group $\mathbf D(\Sigma)$ acts on $U(\Sigma)$, and the toric variety ${{{{\mathbb P}}_\Sigma}}$ may be represented as $U(\Sigma)/\mathbf D(\Sigma)$. This yields an action of $\mathbf T(\Sigma)$ on ${{{{\mathbb P}}_\Sigma}}$. For every face $\tau$ in $\Sigma$ we shall denote by $\mathbf T_\tau\subset {{{{\mathbb P}}_\Sigma}}$ the orbit of $\tau$ in ${{{{\mathbb P}}_\Sigma}}$ under this action.
Quasi-smooth hypersurfaces
--------------------------
From now on we assume that ${{{{\mathbb P}}_\Sigma}}$ is projective. Let $L$ be an ample line bundle on ${{{{\mathbb P}}_\Sigma}}$, and denote by $\beta\in Cl(\Sigma)$ its degree; a section of $L$ is a polynomial in $S_\beta$.
[@BaCox94 Def. 3.1] Let $f$ be a section of $L$, and let $\mathbf V(f)$ be the zero locus of $f$ in $\operatorname{Spec}S(\Sigma)$. We say that the hypersurface $X$ cut in ${{{{\mathbb P}}_\Sigma}}$ by the equation $f=0$ is [*quasi-smooth*]{} if $\mathbf V(f)$ is smooth outside $Z(\Sigma)$.
\[nondegenerate\][@BaCox94 Def. 4.13] If $L$ is an ample line bundle on ${{{{\mathbb P}}_\Sigma}}$, a hypersurface $X$ is said to be [*nondegenerate*]{} if $X\cap\mathbf T_\tau$ is a smooth 1-codimensional subvariety of $\mathbf T_\tau$ for all $\tau$ in $\Sigma$.
[@BaCox94 Prop. 3.5, 4.15] If $f$ is the general section of an ample invertible sheaf, then $X$ is nondegenerate. Moreover, every nondegenerate hypersurface $X\subset{{{{\mathbb P}}_\Sigma}}$ is quasi-smooth. Thus, if $f$ is a general section of $L$, its zero locus is a quasi-smooth hypersurface $X$ in ${{{{\mathbb P}}_\Sigma}}$, hence it is an orbifold.
An important fact is that the complex cohomology of an orbifold has a pure Hodge structure in each dimension [@Stee77; @Tu86].
We also note that in view of the homotopy hyperplane Lefschetz theorem, which holds for the embedding $ X \hookrightarrow {{{{\mathbb P}}_\Sigma}}$ [@GoMac88 Thm. 1.2 Part II], $X$ is simply connected if $\dim ({{{{\mathbb P}}_\Sigma}}) \geq 3$.
Primitive cohomology of a hypersurface
--------------------------------------
Let $L$ be an ample line bundle on ${{{{\mathbb P}}_\Sigma}}$, and let $X$ be a hypersurface in ${{{{\mathbb P}}_\Sigma}}$ cut by a section $f$ of $L$ (note that by [@BaCox94], Proposition 10.8, $f$ lies in $B(\Sigma)$). Denote by $i\colon X \to {{{{\mathbb P}}_\Sigma}}$ the inclusion, and by $i^\ast\colon H^\bullet( {{{{\mathbb P}}_\Sigma}},{{\mathbb C}})
\to H^\bullet(X,{{\mathbb C}})$ the associated morphism in cohomology; $i^\ast\colon H^{d-1}( {{{{\mathbb P}}_\Sigma}},{{\mathbb C}})
\to H^{d-1}(X,{{\mathbb C}})$ is injective by Proposition 10.8 in [@BaCox94].
[@BaCox94 Def. 10.9] The primitive cohomology group $PH^{d-1}(X)$ is the quotient $H^{d-1}(X,{{\mathbb C}})/$ $i^\ast(H^{d-1} ({{{{\mathbb P}}_\Sigma}},{{\mathbb C}}))$.
\[ortho\] The exact sequence $$0 \to i^\ast(H^{d-1} ({{{{\mathbb P}}_\Sigma}},{{\mathbb C}})) \to H^{d-1}(X,{{\mathbb C}}) \to PH^{d-1}(X) \to 0$$ splits orthogonally with respect to the intersection pairing in $H^\bullet(X,{{\mathbb C}})$. The same is true with coefficients in ${{\mathbb Q}}$.
The hard Lefschetz theorem holds also for projective orbifolds (this follows from the results in [@SaitoKyoto]; a simple proof is given in [@Zaf09]).
Then cupping by $c_1(L)$ we get an isomorphism $\ell\colon H^{d-1}({{{{\mathbb P}}_\Sigma}},{{\mathbb C}}) \to H^{d+1}({{{{\mathbb P}}_\Sigma}},{{\mathbb C}})$. Let $i_\ast \colon H^{d-1}(X,{{\mathbb C}})\to
H^{d+1}({{{{\mathbb P}}_\Sigma}},{{\mathbb C}})$ be the Gysin map. We claim that the following commutative diagram $$\xymatrix{
&& 0\ar[d] & 0 \ar[d]\\
&0 \ar[d] \ar[r] & H^{d-1}({{{{\mathbb P}}_\Sigma}},{{\mathbb C}}) \ar[d]_{i^\ast} \ar[r] & H^{d-1}({{{{\mathbb P}}_\Sigma}},{{\mathbb C}}) \ar[d]_\ell \ar[r] & 0 \\
0 \ar[r] & \ker i_\ast \ar[d] \ar[r] & H^{d-1}(X,{{\mathbb C}}) \ar[d] \ar[r]_{i_\ast} & H^{d+1}({{{{\mathbb P}}_\Sigma}},{{\mathbb C}}) \ar[d] \ar[r] & 0 \\
0 \ar[r] & \ker i_\ast \ar[d] \ar[r] & PH^{d-1}(X) \ar@<-2mm>@/_10pt/[u]_s \ar[r]\ar[d] & 0 \\
& 0 & 0}$$ provides a straightforward splitting $s$ of the above exact sequence. Let $\langle\,,\rangle$ be the intersection pairing in cohomology both in $H^\bullet(X,{{\mathbb C}})$ and $H^\bullet({{{{\mathbb P}}_\Sigma}},{{\mathbb C}})$, and recall that $i^\ast$ and $i_\ast$ are adjoint with respect to the intersection pairing. The upper-right square commutes since by Poincaré duality $$\langle i_\ast i^\ast\alpha,\beta\rangle = \langle i^\ast\alpha,i^\ast\beta\rangle
=\langle c_1(L) \cup\alpha,\beta\rangle=\langle \ell(\alpha),\beta\rangle\,.$$ If $\alpha\in H^{d-1}({{{{\mathbb P}}_\Sigma}},{{\mathbb C}})$ and $\beta \in PH^{d-1}(X)$, we have $$\langle i^\ast(\alpha),s(\beta)\rangle=\langle \alpha, i_\ast(s (\beta ))\rangle =0\,.$$ If the statement is true with coefficients in ${{\mathbb C}}$ it also true with coefficients in ${{\mathbb Q}}$ since $H^\bullet(X,{{\mathbb C}})\simeq H^\bullet(X,{{\mathbb Q}})\otimes_{{\mathbb Q}}{{\mathbb C}}$.
The kernel of $i_\ast$ in $H^{d-1}(X,{{\mathbb C}})$ is sometimes called the “variable cohomology” $H^{d-1}_{\rm var}(X,{{\mathbb C}})$; in degree $d-1$ the variable and primitive cohomologies of $X$ are then isomorphic.
Both $H^{d-1}( {{{{\mathbb P}}_\Sigma}},{{\mathbb C}})$ and $ H^{d-1}(X,{{\mathbb C}})$ have pure Hodge structures, and the morphism $i^\ast$ is compatible with them, so that $PH^{d-1}(X)$ inherits a pure Hodge structure. We shall write $$PH^{d-1}(X) = \bigoplus_{p=0}^{d-1} PH^{p,d-1-p}(X).$$
The following Proposition \[iso\] implicitly uses a generalization of Bott’s vanishing theorem, called the Bott-Steenbrink-Danilov theorem, which indeed holds under our assumptions. The exact statement is that $H^i({{{{\mathbb P}}_\Sigma}},\Omega^p_{{{{\mathbb P}}_\Sigma}}(L))$ $=0$ for all $i>0$ and $p\ge 0$ if $L$ is an ample line bundle on ${{{{\mathbb P}}_\Sigma}}$. This was stated without proof by Danilov [@Dan78] and proved in [@BaCox94] (Theorem 7.1).
\[iso\] There is a natural isomorphism $$PH^{p,d-p-1}(X) \simeq \frac{H^0({{{{\mathbb P}}_\Sigma}},\Omega^d_{{{{\mathbb P}}_\Sigma}}((d-p+1)X)}{
H^0({{{{\mathbb P}}_\Sigma}},\Omega_{{{{\mathbb P}}_\Sigma}}^d((d-p)X)+dH^0({{{{\mathbb P}}_\Sigma}},\Omega_{{{{\mathbb P}}_\Sigma}}^{d-1}((d-p)X)}$$
This follows from comparing Corollaries 10.2 and 10.12 in [@BaCox94].
The resulting projection map, multiplied by the factor $(-1)^{p-1}/(d-p+1)!$, will be denoted by $$\label{residue} r_p \colon H^0({{{{\mathbb P}}_\Sigma}},\Omega^d_{{{{\mathbb P}}_\Sigma}}((d-p+1)X) \to PH^{p,d-p-1}(X)$$ and is called the [*$p$-th residue map*]{} in analogy with the classical case.
Let $X$ be any hypersurface in ${{{{\mathbb P}}_\Sigma}}$ cut by a section $f$ of $L$ and let $J(f)$ be the ideal of the Cox ring generated by the derivatives of $f$. The ring $R(f)=S(\Sigma)/J(f)$ is the [*Jacobian ring*]{} of $S(\Sigma)$.
The Jacobian ring encodes all the information about the primitive cohomology of $X$:
If $p\ne d/2-1$, $ PH^{p,d-p-1}(X) \simeq R(f)_{(d-p)\beta-\beta_0 }$, where $\beta_0 = - \deg K_{{{{{\mathbb P}}_\Sigma}}}$, $\beta=\deg L$. \[isoring\]
[@BaCox94] Theorem 10.13.
The Picard group of the general toric threefold {#Pic}
===============================================
The Gauss-Manin connection
--------------------------
Let ${\mathcal Z}$ be the open subscheme of $\vert L \vert$ parametrizing the quasi-smooth hypersurfaces in $\vert L \vert$, and let $\pi\colon \mathscr F \to {\mathcal Z}$ be the tautological family on ${\mathcal Z}$; we denote by $X_z$ the fiber of $\mathscr F$ at $z\in \mathcal Z$. Let $\mathscr H^{d-1}$ be the local system on ${\mathcal Z}$ whose fiber at $z$ is the cohomology $H^{d-1}(X_z)$, i.e., $\mathscr H^{d-1}=R^{d-1}\pi_\ast{{\mathbb C}}$. It defines a flat connection $\nabla$ in the vector bundle $\mathscr E^{d-1} = \mathscr H^{d-1} \otimes_{{\mathbb C}}{{\mathcal O}}_{\mathcal Z}$, the [*Gauss-Manin connection*]{} of $\mathscr E^{d-1}$. Since the hypersurfaces $X_z$ are quasi-smooth, the Hodge structure of the fibres $H^{d-1}(X_z)$ of $\mathscr E^{d-1}$ varies analytically with $z$ [@Stee77]. The corresponding filtration defines holomorphic subbundles $F^p\mathscr E^{d-1}$, and the graded object of the filtration defines holomophic bundles $Gr_F^p(\mathscr E^{d-1})$. The bundles $\mathscr E^{p,d-p-1}$ given by the Hodge decomposition are not holomorphic subbundles of $\mathscr E^{d-1}$, but are diffeomorphic to $Gr_F^p(\mathscr E^{d-1})$, and as such they have a holomorphic structure. The quotient bundles $\mathscr {PE}^{p,d-p-1}$ of $\mathscr E^{p,d-p-1}$ correspond to the primitive cohomologies of the hypersurfaces $X_z$. Let $\pi_p: \mathscr E^{d-1}\to \mathscr{PE}^{p,d-p+1}$ be the natural projection.
We denote by $\tilde\gamma_p$ the cup product $$\tilde\gamma_p\colon H^0({{{{\mathbb P}}_\Sigma}}, {{\mathcal O}}_{{{{\mathbb P}}_\Sigma}}(X)) \otimes H^0({{{{\mathbb P}}_\Sigma}},\Omega_{{{{\mathbb P}}_\Sigma}}^d((d-p)X))
\to H^0({{{{\mathbb P}}_\Sigma}},\Omega_{{{{\mathbb P}}_\Sigma}}^d((d-p+1)X))\,.$$ If $z_0$ is the point in ${\mathcal Z}$ corresponding to $X$, the space $ H^0({{{{\mathbb P}}_\Sigma}}, {{\mathcal O}}_{{{{\mathbb P}}_\Sigma}}(X))/{{\mathbb C}}(f)$, where ${{\mathbb C}}(f)$ is the 1-dimensional subspace of $H^0({{{{\mathbb P}}_\Sigma}}, {{\mathcal O}}_{{{{\mathbb P}}_\Sigma}}(X))$ generated by $f$, can be identified with $T_{z_0}{\mathcal Z}$. The morphism $\tilde\gamma_p$ induces in cohomology the Gauss-Manin connection:
Let $\sigma_0$ be a primitive class in $PH^{p,d-p-1}(X)$, let $v\in T_{z_0}{\mathcal Z}$, and let $\sigma$ be a section of $\mathscr E^{p,d-p-1}$ along a curve in $\mathcal Z$ whose tangent vector at $z_0$ is $v$, such that $\sigma(z_0)=\sigma_0$.
Then $$\label{GM} \pi_{p-1}( \nabla_v(\sigma)) =r_{p-1}( \tilde\gamma_p(\tilde v\otimes \tilde\sigma))$$ where $r_p$, $r_{p-1}$ are the residue morphisms defined in equation , $\tilde\sigma$ is an element in $H^0({{{{\mathbb P}}_\Sigma}},\Omega_{{{{\mathbb P}}_\Sigma}}^d((d-p+1)X))$ such that $r_p(\tilde\sigma)=\sigma_0$, and $\tilde v$ is a pre-image of $v$ in $H^0({{{{\mathbb P}}_\Sigma}}, {{\mathcal O}}_{{{{\mathbb P}}_\Sigma}}(X))$.
In particular the following diagram commutes: $$\label{commuta}
\xymatrix{
\displaystyle H^0({{{{\mathbb P}}_\Sigma}}, {{\mathcal O}}_{{{{\mathbb P}}_\Sigma}}(X))
\otimes H^0({{{{\mathbb P}}_\Sigma}},\Omega_{{{{\mathbb P}}_\Sigma}}^d((d-p)X))
\ar[r]^{\ \ \ \ \ \ \ \ \ \tilde\gamma_p}\ar[d]_{\phi\otimes r_p} & H^0({{{{\mathbb P}}_\Sigma}},\Omega_{{{{\mathbb P}}_\Sigma}}^d((d-p+1)X))
\ar[d]_{r_{p-1}} \\
T_{z_0}{\mathcal Z} \otimes PH^{p,d-1-p}(X) \ar[r]^{\ \ \ \ \ \ \ \gamma_p }& PH^{p-1,d-p}(X) }$$ where $\gamma_p$ is the morphism that maps $v\otimes \alpha$ to $\nabla_v\alpha$ and $\phi$ is the projection $\phi\colon$ $ H^0({{{{\mathbb P}}_\Sigma}}, {{\mathcal O}}_{{{{\mathbb P}}_\Sigma}}(X)) \to T_{z_0}{\mathcal Z}$.
This is a standard computation, see [@CMP03], Proposition 5.4.3. Let $f_i$ be local representatives, with respect to a suitable cover $\{U_i\}$ of ${{{{\mathbb P}}_\Sigma}}$, of the section $f$. Via the isomorphism of Proposition \[iso\], we locally represent $\sigma_0$ by the meromorphic differential forms $\omega_i/f_i^{d-p+1}$. A tangent vector $v\in T_{z_0}{\mathcal Z}$ represents a deformation $f_i \mapsto f_i+tg_i$ where $t$ is a complex parameter, and $g_i$ are holomorphic functions. Then $\nabla_v(\sigma)$ is represented by $$\left[\frac{d}{dt} \frac{\omega_i}{(f_i+tg_i)^{d-p+1}}\right]_{t=0} =
-(d-p+1) \frac{g_i\,\omega_i}{f_i^{d-p+2}}\,.$$ But the right-hand side of this equation is, up to a suitable factor, the argument of the map $r_{p-1}$ in the right-hand side of equation .
If $\alpha$ and $\eta$ are sections of $\mathscr E^{p,d-p-1}$ and $\mathscr E^{d-p,p-1}$ respectively, then for every tangent vector $v\in T_{z_0}{\mathcal Z}$, $$\label{compa}\nabla_v\alpha\cup \eta= - \alpha \cup \nabla_v\eta\,.$$
The Gauss-Manin connection is compatible with the cup product by definition, i.e., $$\nabla_v (\alpha\cup\eta) = \nabla_v\alpha\cup \eta + \alpha \cup \nabla_v\eta\,.$$ But $\alpha\cup\eta=0$ because it is an element in $\mathscr E^{d,d-2}$.
The moduli space of hypersurfaces in ${{{{\mathbb P}}_\Sigma}}$
---------------------------------------------------------------
Let ${\operatorname{Aut}}_\beta({{{{\mathbb P}}_\Sigma}})$ be the subgroup of $ \operatorname{Aut}({{{{\mathbb P}}_\Sigma}}) $ which preserves the grading $\beta$. The coarse moduli space $\mathcal M_\beta$ for the general quasi-smooth hypersurfaces in ${{{{\mathbb P}}_\Sigma}}$ with divisor class ${\beta}$ may be constructed as a quotient $$\label{moduli} U/\widetilde{\operatorname{Aut}}_\beta({{{{\mathbb P}}_\Sigma}})\,,$$ [@BaCox94; @AGM], where $U$ is an open subset of $H^0({{{{\mathbb P}}_\Sigma}},{{\mathcal O}}_{{{{{\mathbb P}}_\Sigma}}}(X))$, and $\widetilde{\operatorname{Aut}}_\beta({{{{\mathbb P}}_\Sigma}})$ is the unique nontrivial extension $$1 \to D(\Sigma) \to \widetilde{\operatorname{Aut}}_\beta({{{{\mathbb P}}_\Sigma}}) \to {\operatorname{Aut}}_\beta({{{{\mathbb P}}_\Sigma}}) \to 1\,.$$ By differentiating, we have a surjective map $$\kappa_\beta\colon H^0({{{{\mathbb P}}_\Sigma}}, {{\mathcal O}}_{{{{\mathbb P}}_\Sigma}}(X)) \to T_X \mathcal M_{\beta}\,,$$ which is the analogue of the Kodaira-Spencer map.
The local system $\mathscr H^{d-1}$ and its various sub-systems do not descend to the moduli space $\mathcal M_{\beta}$, because the group ${\operatorname{Aut}}_\beta({{{{\mathbb P}}_\Sigma}})$ is not connected. Nevertheless, this group has a connected subgroup $\operatorname{Aut}^0_\beta({{{{\mathbb P}}_\Sigma}})$ of finite order, and, perhaps after suitably shrinking $U$, the quotient $\mathcal M^0_\beta \stackrel{def}{=} U/\operatorname{Aut}^0_\beta({{{{\mathbb P}}_\Sigma}})$ is a finite étale covering of $\mathcal M_\beta$ [@CoxDon; @AGM].
There is a morphism $$\label{gamma2} \gamma_p\colon T_X \mathcal M_{\beta} \otimes PH^{p,d-1-p}(X)\to PH^{p-1,d-p}(X)$$ such that the diagram $$\label{commuta2}\xymatrix{
H^0({{{{\mathbb P}}_\Sigma}}, {{\mathcal O}}_{{{{\mathbb P}}_\Sigma}}(X))
\otimes H^0({{{{\mathbb P}}_\Sigma}},\Omega_{{{{\mathbb P}}_\Sigma}}^d((d-p)X))
\ar[r]^{\ \ \ \ \ \ \ \ \ \cup }\ar[d]_{\kappa_\beta\otimes r_p} & H^0({{{{\mathbb P}}_\Sigma}},\Omega_{{{{\mathbb P}}_\Sigma}}^d((d-p+1)X))
\ar[d]_{r_{p-1}} \\
T_X \mathcal M_{\beta} \otimes PH^{p,d-1-p}(X) \ar[r]^{\ \ \ \ \ \ \ \gamma_p }& PH^{p-1,d-p}(X) }$$ commutes.
It suffices to prove the Proposition with $\mathcal M_\beta$ replaced by $\mathcal M^0_\beta$; in fact the tangent spaces at points $\mathcal M^0_\beta$ are canonically isomorphic to the tangent spaces at the image points in $\mathcal M_\beta$.
If $\rho\colon {\mathcal Z} \to \mathcal M^0_\beta$ is the induced map (where ${\mathcal Z}$ has been suitably restricted), the local system $\mathscr H^{d-1}$ descend to a local system $\rho_\ast \mathscr H^{d-1}$ on $\mathcal M^0_\beta$, and $\rho^\ast \rho_\ast \mathscr H^{d-1}\simeq \mathscr H^{d-1}$ (the natural morphism $\mathscr H^{d-1} \to \rho^\ast \rho_\ast \mathscr H^{d-1}$ is an isomorphism on the stalks due to topological base change; note that $\rho$ is proper). Thus we obtain on $\mathcal M^0_\beta$ holomorphic bundles that are equipped with a Gauss-Manin connection, which is trivial in the direction of the fibers of $\rho$. If we define again $\gamma_p$ by $\gamma_p(v\otimes \alpha)=\nabla_v(\alpha)$ (where $\nabla$ is now the Gauss-Manin connection on $\mathcal M^0_\beta$), the commutavity of the diagram in the statement follows from the commutativity of the diagram .
The tangent space $T_X\mathcal M_{\beta}$ at a point representing a hypersurface $X$ is naturally isomorphic to the degree ${\beta}$ summand of the Jacobian ring of $f$, that is, $T_X\mathcal M_{\beta}\simeq R(f)_{\beta}$ [@BaCox94]. Moreover, by Proposition \[isoring\], $ PH^{p,d-p-1}(X) \simeq R(f)_{(d-p)\beta-\beta_0 }$.
\[gammaisring\] Under these isomorphisms, the morphism $\gamma_p$ in equation coincides with the multiplication in the ring $R(f)$, $$R(f)_\beta \otimes R(f)_{(d-p)\beta-\beta_0} \to R(f)_{(d-p+1)\beta-\beta_0}\,.$$
Theorem 9.7 in [@BaCox94] implies $$H^0({{{{\mathbb P}}_\Sigma}},\Omega_{{{{\mathbb P}}_\Sigma}}^d((d-p)X) / H^0({{{{\mathbb P}}_\Sigma}},\Omega_{{{{\mathbb P}}_\Sigma}}^d((d-p-1)X) \simeq S_{(d-p)\beta-\beta_0}\,,$$ and, moreover, $H^0({{{{\mathbb P}}_\Sigma}},\co_{{{{\mathbb P}}_\Sigma}}(X))$ $\simeq S_\beta$; the cup product corresponds to the product in the ring $S$. This implies that the “top square” of the 3-dimensional diagram $$\xymatrix@C=-25pt{
\displaystyle {H^0({{{{\mathbb P}}_\Sigma}}, {{\mathcal O}}_{{{{\mathbb P}}_\Sigma}}(X)) \otimes \atop H^0({{{{\mathbb P}}_\Sigma}},\Omega_{{{{\mathbb P}}_\Sigma}}^d((d-p)X)) }
\ar[rr]^\cup\ar[dr]\ar[dd]_{\kappa_\beta\otimes r_p} &&
H^0({{{{\mathbb P}}_\Sigma}},\Omega_{{{{\mathbb P}}_\Sigma}}^d((d-p+1)X)) \ar'[d]_{r_{p-1}}[dd]\ar[dr] \\ &
S_\beta\otimes S_{(d-p)\beta-\beta_0}\ar[dd] \ar[rr] &&
S_{(d-p+1)\beta-\beta_0}\ar[dd] \\
T_X \mathcal M_{\beta} \otimes PH^{p,d-1-p}(X) \ar'[r][rr]\ar[dr] &&
PH^{p-1,d-p}(X) \ar[dr] \\ &
R(f)_\beta\otimes R(f)_{(d-p)\beta-\beta_0} \ar[rr] &&
R(f)_{(d-p+1)\beta=\beta_0}
}$$ commutes. We need to show that the “bottom square” commutes as well, which will follow from the commutativity of the “side squares”, and the surjectivity of the morphism $\kappa_\beta\otimes r_p$. The commutativity of the diagram on the right is contained in the proof of Theorem 10.6 in [@BaCox94]. The commutativity of the diagram on the left follows from the commutativity of the previous diagram, with $d-p+1$ replaced by $d-p$, and the commutativity of $$\xymatrix{
H^0({{{{\mathbb P}}_\Sigma}}, {{\mathcal O}}_{{{{\mathbb P}}_\Sigma}}(X)) \ar[d]\ar[r]^(.7){\sim} & S_\beta\ar[d] \\ T_X \mathcal M_{\beta} \ar[r]^{\sim} & R(f)_\beta
}$$ which is shown in the proof of Proposition 13.7 in [@BaCox94].
Picard group
------------
Our aim is now to prove the following result. Let us recall that a property is said to be [*very general*]{} if it holds in the complement of a countable union of subschemes of positive codimension [@Lazarsfeld]. Also recall that the Picard number $\rho(X)$ is the rank of the Néron-Severi group, i.e., $\rho(X) = \dim_{{\mathbb Q}}(H^{1,1}(X,{{\mathbb C}}) \cap H^2(X,{{\mathbb Q}}))$.
Let ${{{{\mathbb P}}_\Sigma}}$ be a 3-dimensional complete simplicial toric variety, $L$ an ample line bundle on ${{{{\mathbb P}}_\Sigma}}$, and $X$ a very general quasi-smooth hypersurface in the linear system $\vert L\vert$. If the morphism $\gamma_2\colon T_X\mathcal M_\beta \otimes PH^{2,0} (X) \to PH^{1,1} (X)$ is surjective, then $X$ and ${{{{\mathbb P}}_\Sigma}}$ have the same Picard number. \[picard\]
Theorem \[picard\] will follow from two Lemmas. In the first Lemma no restriction on the dimension $d$ of ${{{{\mathbb P}}_\Sigma}}$ needs to be made, in the second we shall assume that $d$ is odd.
The first Lemma is an “infinitesimal Noether-Lefschetz theorem", such as Theorem 7.5.1 in [@CMP03].
Denote by $H^{d-1}_T(X) \subset H^{d-1}(X)$ the subspace of the cohomology classes that are annihilated by the action of the Gauss-Manin connection. Coefficients may be taken in ${{\mathbb C}}$ or ${{\mathbb Q}}$. Note that $H^{d-1}_T(X)$ has a Hodge structure.
For a given $p$ with $ 1 \le p \le d-1$, assume that the morphism $$\gamma_p\colon T_X \mathcal M_{\beta} \otimes PH^{d-p,p-1}(X)\to PH^{d-p-1,p}(X)$$ is surjective. Then $H^{p,d-1-p}_T(X)=i^\ast(H^{p,d-1-p}({{{{\mathbb P}}_\Sigma}}))$.
Replace $ \mathcal M_\beta$ by $\mathcal M^0_\beta$, and consider the local systems $\mathscr E^{d-1}$ and $\mathscr{PE}^{p,d-p-1}$ on $\mathcal M^0_\beta$. Take $$\alpha\in H^{p,d-1-p}_T(X)\cap PH^{p,d-1-p}(X).$$ We regard classes in $PH^{p,d-1-p}(X)$ as elements in the fiber of $\mathscr{PE}^{p,d-p-1}$ at the point $[X]\in M^0_\beta$. By hypothesis $\beta\in PH^{d-p-1,p}(X)$ can be written as $\beta = \sum_i \gamma_p(t_i\otimes\eta_i)$ with $\eta_i\in PH^{d-p,p-1}(X)$. Then by equations and $$\langle \alpha,\beta\rangle = \sum_i \langle \alpha, \gamma_p(t_i\otimes\eta_i) \rangle
= \sum_i \langle \alpha,\nabla_{t_i}\eta_i\rangle
= - \sum_i \langle \nabla_{t_i} \alpha,\eta_i\rangle=0.$$ So $\alpha$ is orthogonal to $PH^{d-1-p,p}(X)$. By Lemma \[ortho\], this means that $\alpha$ is orthogonal to the whole group $H^{d-1-p,p}(X)$, hence it is zero. Therefore $H^{p,d-1-p}_T(X)=i^\ast(H^{p,d-1-p}({{{{\mathbb P}}_\Sigma}}))$.
For any variety $Y$ we define $H^{m,m}(Y,{{\mathbb Q}}) = H^{m,m}(Y,{{\mathbb C}})\cap H^{2m}(Y,{{\mathbb Q}})$.
\[verygeneral\] Let $d=2m+1\ge 3$, and assume that the hypotheses of the previous Lemma hold for $p=m$. Then for $z$ away from a countable union of subschemes of ${\mathcal Z}$ of positive codimension one has $$H^{m,m}(X_z,{{\mathbb Q}}) = \operatorname{im}[ i^\ast\colon H^{m,m}({{{{\mathbb P}}_\Sigma}},{{\mathbb Q}}) \to H^{2m}(X_z.{{\mathbb Q}})].$$
Let $\bar {\mathcal Z}$ be the universal cover of ${\mathcal Z}$. On it the (pullback of the) local system $\mathscr H^{d-1}$ is trivial. Given a class $\alpha\in H^{m,m}(X)$ we can extend it to a global section of $\mathscr H^{d-1}$ by parallel transport using the Gauss-Manin connection. Define the subset $\bar {\mathcal Z}_\alpha$ of $\bar {\mathcal Z}$ as the common zero locus of the sections $\pi_p(\alpha)$ of $\mathscr E^{p,d-1-p}$ for $p\ne m$ (i.e., the locus where $\alpha$ is of type $(m,m)$).
If $\bar {\mathcal Z}_\alpha=\bar {\mathcal Z}$ we are done because $\alpha$ is in $H^{d-1}_T(X)$ hence is in the image of $i^\ast$ by the previous Lemma. If $\bar {\mathcal Z}_\alpha \ne \bar {\mathcal Z}$, we note that $\bar {\mathcal Z}_\alpha$ is a subscheme of $\bar {\mathcal Z}$.
We subtract from ${\mathcal Z}$ the union of the projections of the subschemes $\bar {\mathcal Z}_\alpha$ where $\bar {\mathcal Z}_\alpha \ne \bar {\mathcal Z}$. The set of these varieties is countable because we are considering rational classes.
[*Proof of Theorem \[picard\]*]{}. Lemma \[verygeneral\], for $d=3$, implies that $H^{1,1}(X_z,{{\mathbb Q}})$ and $H^{1,1}({{{{\mathbb P}}_\Sigma}},{{\mathbb Q}})$ have the same dimension for a very general $z$. These two numbers are the Picard numbers of $X_z$ and ${{{{\mathbb P}}_\Sigma}}$, respectively (see Definition/Proposition \[picards\]).
We assume now that ${{{{\mathbb P}}_\Sigma}}$ is Fano, and that $L = - K_{{{{{\mathbb P}}_\Sigma}}}$, so that the hypersurfaces in the linear system $\vert L \vert $ are K3 surfaces. We have $PH^{2,0}\simeq R(f)_{0} \simeq {{\mathbb C}}$, $PH^{1,1} (X) \simeq R(f)_{\beta}$, and $T_X\mathcal M_{\beta} \simeq R(f)_\beta$, where $\beta = - \deg K_{{{{{\mathbb P}}_\Sigma}}}$. By Propositions \[isoring\] and \[gammaisring\], the morphism $\gamma_2$ corresponds to the multiplication $R(f)_\beta\otimes R(f)_0\to R(f)_\beta$, and since $R(f)_0\simeq{{\mathbb C}}$, this is an isomorphism. From Theorem \[picard\] we have:
Let ${{{{\mathbb P}}_\Sigma}}$ be a 3-dimensional Fano complete simplicial toric variety, and $X$ a very general hypersurface in the linear system $\vert -K_{{{{{\mathbb P}}_\Sigma}}}\vert$. Then $X$ has the same Picard number as ${{{{\mathbb P}}_\Sigma}}$.
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[^1]: $^\dag$ On leave of absence from Scuola Internazionale Superiore di Studi Avanzati, Via Bonomea 265, 34136 Trieste, Italy
[^2]: Support for this work was provided by the NSF Research Training Group Grant DMS-0636606, by [prin]{} “Geometria delle varietà algebriche e dei loro spazi dei moduli ” and the [infn]{} project [pi14]{} “Nonperturbative dynamics of gauge theories”. U.B. is a member of the [vbac]{} group.
[^3]: E-mail: [bruzzo@math.upenn.edu, bruzzo@sissa.it, grassi@sas.upenn.edu]{}
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'From Crofton’s formula for Minkowski tensors we derive stereological estimators of translation invariant surface tensors of convex bodies in the $n$-dimensional Euclidean space. The estimators are based on one-dimensional linear sections. In a design based setting we suggest three types of estimators. These are based on isotropic uniform random lines, vertical sections, and non-isotropic random lines, respectively. Further, we derive estimators of the specific surface tensors associated with a stationary process of convex particles in the model based setting.'
author:
- 'Astrid Kousholt[^1]'
- Markus Kiderlen
- Daniel Hug
bibliography:
- 'litteratur.bib'
title: Surface tensor estimation from linear sections
---
**Keywords** Crofton formula, Minkowski tensor, stereology, isotropic random line, anisotropic random line, vertical section estimator, minimal variance estimator, stationary particle process, stereological estimator\
\
**MSC2010** 60D05, 52A22, 53C65, 62G05, 60G55
Introduction
============
In recent years, there has been an increasing interest in Minkowski tensors as descriptors of morphology and shape of spatial structures of physical systems. For instance, they have been established as robust and versatile measures of anisotropy in [@Beisbart2002; @SM10; @Schroder-Turk2013]. In addition to the applications in materials science, [@Beisbart] indicates that the Minkowski tensors lead to a putative taxonomy of neuronal cells. From a pure theoretical point of view, Minkowski tensors are, likewise, interesting. This is illustrated by Alesker’s characterization theorem [@Alesker1999], stating that the basic tensor valuations (products of the Minkowski tensors and powers of the metric tensor) span the space of tensor-valued valuations satisfying some natural conditions.
This paper presents estimators of certain Minkowski tensors from measurements in one-dimensional flat sections of the underlying geometric structure. We restrict attention to translation invariant Minkowski tensors of convex bodies, more precisely, to those that are derived from the top order surface area measure; see Section \[prelim\] for a definition. As usual, the estimators are derived from an integral formula, namely the Crofton formula for Minkowski tensors. We adopt the classical setting where the sectioning space is affine and integrated with respect to the motion invariant measure. Rotational Crofton formulae where the sectioning space is a linear subspace and the rotation invariant measure on the corresponding Grassmannian is used, are established in [@A-C2013]. The latter formulae were the basis for local stereological estimators of certain Minkowski tensors in [@Jensen2013] (for $j \in \{1, \dots, n-1\}, s,r \in \{0,1\}$ and $j=n, s=0, r \in {{\mathbb N}}$ in the notation of and , below).
Kanatani [@Kanatani1984; @Kanatani1984a] was apparently the first to use tensorial quantities to detect and analyse structural anisotropy via basic stereological principles. He expresses the expected number $N(m)$ of intersections per unit length of a probe with a test line of given direction $m$ as the cosine transform of the spherical distribution density $f$ of the surface of the given probe in ${{\mathbb R}}^n$ for $n=2,3$. The relation between $N$ and $f$ is studied by expanding $f$ into spherical harmonics and by using the fact that these are eigenfunctions of the cosine transform. In order to express his results independently of a particular coordinate system, Kanatani uses tensors. For a fixed $s$, he considers the vector space $V_s$ of all symmetric tensors spanned by the elementary tensor products $u^{\otimes s}$ of vectors $u$ from the unit sphere $S^{n-1}$. Let $\hat T$ denote the deviator part (or trace-free part) of some symmetric tensor $T$. The tensors $\widehat{(u^{\otimes k})}$, for $k\le s$ and $u\in S^{n-1}$, then span $V_s$ and the components of $\widehat{(u^{\otimes k})}$ with respect to an orthonormal basis of ${{\mathbb R}}^n$ are spherical harmonics of degree $k$, when considered as functions of $u$. Hence, $u\mapsto \widehat{(u^{\otimes k})}$ is an eigenfunction of the cosine transform (Kanatani calls it ‘Buffon transform’), which in fact is the underlying integral transform when considering Crofton integrals with lines, as we shall see below in . In [@Kanatani1984b; @Kanatani1985], he suggests to use these ‘fabric tensors’ to detect surface motions and the anisotropy of the crack distribution in rock.
General Crofton formulas in ${{\mathbb R}}^n$ with arbitrary dimensional flats and for general Minkowski tensors (defined in ) of arbitrary rank are given in [@Hug]. Theorem \[thm\] is a special case of one of these results, for translation invariant surface tensors and one-dimensional sections, that is, sections with lines. In comparison to [@Hug], we get simplified constants in the case considered and obtain this result by an elementary independent proof. In contrast to Kanatani’s approach, our proof does not rely on spherical harmonics. Here we focus on relative Crofton formulas in which the Minkowski tensors of the sections with lines are calculated relative to the section lines and not in the ambient space (Crofton formulas of the second type may be called extrinsic Crofton formulas). A quite general investigation of integral geometric formulas for translation invariant Minkowski tensors, including extrinsic Crofton formulas, is provided in [@BernigHug].
In Theorem \[thm\] we prove that the relative Crofton integral for tensors of arbitrary even rank $s$ of sections with lines is equal to a linear combination of surface tensors of rank at most $s$. From this we deduce by the inversion of a linear system that any translation invariant surface tensor of even rank $s$ can be expressed as a Crofton integral. The involved measurement functions then are linear combinations of relative tensors of rank at most $s$. This implies that the measurement functions only depend on the convex body through the Euler characteristic of the intersection of the convex body and the test line.
Our results do not allow to write surface tensors of odd rank as Crofton integrals based on sections with lines. This drawback is not a result of our method of proof. Indeed, apart from the trivial case of tensors of rank one, there does not exist a translation invariant or a bounded measurement function that expresses a surface tensor of odd rank as a Crofton integral; see Theorem \[s ulige\] for a precise statement of this fact.
In Section \[sec estimation\] the integral formula for surface tensors of even rank is transferred to stereological formulae in a design based setting. Three types of unbiased estimators are discussed. Section \[IUR\] describes an estimator based on isotropic uniform random lines. Due to the structure of the measurement function, it suffices to observe whether the test line hits or misses the convex body in order to estimate the surface tensors. However, the resulting estimators possess some unfortunate statistical properties. In contrast to the surface tensors of full dimensional convex bodies, the estimators are not positive definite. For convex bodies, which are not too eccentric (see ), this problem is solved by using $n$ orthogonal test lines in combination with a measurement of the projection function of order $n-1$ of the convex body.
In applications it might be inconvenient or even impossible to construct the isotropic uniform random lines, which are necessary for the use of the estimator described above. Instead, it might be a possibility to use vertical sections; see Definition \[VUR\]. A combination of Crofton’s formula and a result of Blaschke-Petkantschin type allows us to formulate a vertical section estimator. The estimator, which is discussed in Section \[sec VUR\], is based on two-dimensional vertical flats.
The third type of estimator presented in the design based setting is based on non-isotropic linear sections; see Section \[Sec noniso\]. For a fixed convex body in ${{\mathbb R}}^2$ there exists a density for the distribution of test line directions in an importance-sampling approach that leads to minimal variance of the non-isotropic estimator, when we consider one component of a rank 2 tensor, interpreted as a matrix. In practical applications, this density is not accessible, as it depends on the convex body, which is typically unknown. However, there does exist a density independent of the underlying convex body yielding an estimator with smaller variance than the estimator based on isotropic uniform random lines. If *all* components of the tensor are sought for, the non-isotropic approach requires three test lines, as two of the four components of a rank 2 Minkowski tensor coincide due to symmetry. It should be avoided to use a density suited for estimating one particular component of the tensor to estimate any other component, as this would increase variance of the estimator. In this situation, however, a smaller variance can be obtained by applying an estimator based on *three* isotropic random lines (each of which can be used for the estimation of *all* components of the tensor).
In Section \[SecModel\] we turn to a model-based setting. We discuss estimation of the *specific (translation invariant) surface tensors* associated with a stationary process of convex particles; see for a definition. In [@RSRS06] the problem of estimating the area moment tensor (rank $2$) associated with a stationary process of convex particles via planar sections is discussed. We consider estimators of the specific surface tensors of arbitrary even rank based on one-dimensional linear sections. Using the Crofton formula for surface tensors, we derive a rotational Crofton formula for the specific surface tensors. Further, the specific surface tensor of rank $s$ of a stationary process of convex particles is expressed as a rotational average of a linear combination of specific tensors of rank at most $s$ of the sectioned process.
Preliminaries {#prelim}
=============
We work in the $n$-dimensional Euclidean vector space ${{\mathbb R}^n}$ with inner product $ {\langle {\cdot} , {\cdot} \rangle} $ and induced norm $ {\|}\cdot {\|}$. Let $B^n:=\{x \in {{\mathbb R}^n}\mid {\|}x {\|}\leq 1\}$ be the unit ball and $ S^{n-1}:=\{x \in {{\mathbb R}^n}\mid {\|}x {\|}=1\} $ the unit sphere in ${{\mathbb R}^n}$. By $\kappa_n$ and $\omega_{n}$ we denote the volume and the surface area of $ B^n $, respectively. The Borel $\sigma$-algebra of a topological space $X$ is denoted by ${\mathcal{B}}(X)$. Further, let $\lambda$ denote the $n$-dimensional Lebesgue measure on ${{\mathbb R}}^n$, and for an affine subspace $E$ of ${{\mathbb R}}^n$, let $\lambda_E$ denote the Lebesgue measure defined on $E$. The $k$-dimensional Hausdorff measure is denoted by ${\mathcal{H}}^k$. For $A \subseteq {{\mathbb R}}^n$, let $\dim A$ be the dimension of the affine hull of $A$.
Let ${\mathbb{T}}^p$ be the vector space of symmetric tensors of rank $p$ over ${{\mathbb R}}^n.$ For symmetric tensors $a\in {\mathbb{T}}^{p_1}$ and $b\in {\mathbb{T}}^{p_2}$, let $ab\in {\mathbb{T}}^{p_1+p_2}$ denote the symmetric tensor product of $a$ and $b$. Identifying $x \in {{\mathbb R}}^n$ with the rank 1 tensor $z \mapsto {\langle {z} , {x} \rangle}$, we write $x^p \in {\mathbb{T}}^p$ for the $p$-fold symmetric tensor product of $x$. The metric tensor $Q \in {\mathbb{T}}^2$ is defined by $Q(x,y)={\langle {x} , {y} \rangle}$ for $x,y \in {{\mathbb R}}^n$, and for a linear subspace $L$ of ${{\mathbb R}}^n$, we define $Q(L) \in {\mathbb{T}}^2$ by $Q(L)(x,y)={\langle {p_L(x)} , {p_L(y)} \rangle}$, where $p_L\colon {{\mathbb R}}^n \rightarrow L$ is the orthogonal projection on $L$.
As general references on convex geometry and Minkowski tensors, we use [@Schneider93] and [@Hug]. Let ${{\mathcal K}}^n$ denote the set of convex bodies (that is, compact, convex sets) in ${{\mathbb R}}^n$. In order to define the Minkowski tensors, we introduce the support measures $\Lambda_0(K, \cdot), \dots, \Lambda_{n-1}(K, \cdot)$ of a non-empty, convex body $ K \in{{\mathcal K}}^n$. Let $ p(K,x) $ be the metric projection of $x \in {{\mathbb R}^n}$ on a non-empty convex body $ K $, and define $ u(K,x):=\frac{x-p(K,x)}{{\|}x-p(K,x){\|}} $ for $x \notin K$. For $ \epsilon >0 $ and a Borel set $\nobreak{A \in {\mathcal{B}}({{\mathbb R}}^n \times S^{n-1}})$, the Lebesgue measure of the local parallel set $$M_\epsilon(K,A):=\{x \in (K+\epsilon B^n) \setminus K \mid (p(K,x),u(K,x)) \in A \}$$ of $K$ is a polynomial in $\epsilon$, hence $$\lambda(M_\epsilon(K,A))=\sum_{k=0}^{n-1}\epsilon^{n-k}\kappa_{n-k}\Lambda_k(K,A).$$ This local version of the Steiner formula defines the support measures $ \Lambda_0(K, \cdot), \allowbreak \dots, \Lambda_{n-1}(K,\cdot) $ of a non-empty convex body $ K\in{{\mathcal K}}^n$. If $K=\emptyset$, we define the support measures to be the zero measures. The intrinsic volumes $ V_0(K), \dots, V_{n-1}(K)$ of $ K $ appear as total masses of the support measures, $V_j(K)=\Lambda_j(K,{{\mathbb R}^n}\times S^{n-1})$ for $j=0, \dots, n-1$. Furthermore, the area measures $ S_0(K, \cdot), \dots, S_{n-1}(K,\cdot) $ of $ K $ are rescaled projections of the corresponding support measures on the second component. More explicitly, they are given by $$\binom{n}{j}S_j(K,\omega)=n \kappa_{n-j}\Lambda_j(K,{{\mathbb R}^n}\times \omega)$$ for $ \omega \in {\mathcal{B}}(S^{n-1}) $ and $j=0, \dots, n-1$.
For a non-empty convex body $K\in{{\mathcal K}}^n$, $r,s \in {{\mathbb N}}_0$, and $ j \in \{0,1,\dots, n-1\} $, we define the *Minkowski tensors* as $$\label{Mtensor}
\Phi_{j,r,s}(K) :=\frac{\omega_{n-j}}{r!s! \omega_{n-j+s}} \int_{{{\mathbb R}}^n \times S^{n-1}} x^r u^s\, \Lambda_j (K,d(x,u))$$ and $$\label{volumeTensor}
\Phi_{n,r,0}(K):=\frac{1}{r!}\int_{K}x^r \,\lambda(dx).$$ The definition of the Minkowski tensors is extended by letting $\Phi_{j,r,s}(K)=0$, if $j \notin\{0,1,\dots, n\}$, or if $r$ or $s$ is not in ${{\mathbb N}}_0$, or if $j=n$ and $s \neq 0$. For $j=n-1$, the tensors are called surface tensors. In the present work, we only consider translation invariant surface tensors which are obtained for $r=0$. In [@Hug] the functions $Q^m\Phi_{j,r,s}$ with $m,r,s \in {{\mathbb N}}_0$ and either $ j \in \{0,\dots, n-1\} $ or $ (j,s)=(n,0) $ are called the basic tensor valuations.
For $k \in \{1, \dots,n \}$, let ${\mathcal{L}}^n_k$ be the set of $k$-dimensional linear subspaces of ${{\mathbb R}}^n$, and let ${\mathcal{E}^n}_k$ be the set of $ k $-dimensional affine subspaces of ${{\mathbb R}}^n$. For $L \in {\mathcal{L}}^n_k$, we write $L^\perp$ for the orthogonal complement of $L$. For $E \in {\mathcal{E}^n}_k$, let ${\pi(E)}$ denote the linear subspace in ${\mathcal{L}}^n_k$ which is parallel to $E$, and we define $ E^\perp :={\pi(E)}^\perp$. The sets ${\mathcal{L}}^n_k$ and ${\mathcal{E}}^n_k$ are endowed with their usual topologies and Borel $\sigma$-algebras. Let $\nu^n_k$ denote the unique rotation invariant probability measure on ${\mathcal{L}}^n_k$, and let $\mu_k^n$ denote the unique motion invariant measure on ${\mathcal{E}^n}_k$ normalized so that $\nobreak{\mu^n_k(\{E \in {\mathcal{E}^n}_k \vert E \cap B^n \neq \emptyset\})}=\kappa_{n-k}$ (see, e.g., [@Weil]).
If $K \in {{\mathcal K}}^n$ is non-empty and contained in an affine subspace $E \in {\mathcal{E}^n}_k$, for some $k \in \{1, \dots, n\}$, then the Minkowski tensors can be evaluated in this subspace. For a linear subspace $ L \in {\mathcal{L}}_k^n $, let $ \pi_L \colon S^{n-1} \setminus L^\perp \rightarrow L \cap S^{n-1}$ be given by $$\pi_L(u):=\frac{p_L(u)}{{\|}p_L(u){\|}}.$$ Then we define the $ j $th support measure $ \Lambda_j^{(E)}(K, \cdot) $ of $ K $ relative to $ E $ as the image measure of the restriction of $\Lambda_j(K, \cdot)$ to $ {{\mathbb R}^n}\times (S^{n-1} \setminus E^\perp) $ under the mapping $ {{\mathbb R}^n}\times (S^{n-1} \setminus E^\perp) \rightarrow {{\mathbb R}^n}\times ({\pi(E)}\cap S^{n-1}) $ given by $(x,u) \mapsto (x,\pi_{{\pi(E)}}(u))$.
For a non-empty convex body $K \in {{\mathcal K}}^n$, contained in an affine subspace $ E \in {\mathcal{E}^n}_k $, for some $k \in \{1, \dots, n\}$, we define $$\Phi_{j,r,s}^{(E)}(K) :=\frac{\omega_{k-j}}{r!s! \omega_{k-j+s}} \int_{E \times (S^{n-1} \cap {\pi(E)}) } x^r u^s \, \Lambda^{(E)}_j (K,d(x,u))$$ for $ r,s \in {{\mathbb N}}_0 $ and $ j \in \{0,\dots, k-1\} $, and $$\Phi_{k,r,0}^{(E)}(K):=\frac{1}{r!}\int_{K} x^r \,
\lambda_E(dx).$$ As before, the definition is extended by letting $\Phi^{(E)}_{j,r,s}(K)=0$ for all other choices of $j,r$ and $s$, and for $K=\emptyset$.
In [@Hug], Crofton integrals of the form $$\int_{{\mathcal{E}^n}_k} \Phi_{j,r,s}^{(E)}(K \cap E) \, \mu_k^n (dE),$$ where $K \in {{\mathcal K}}^n$, $r,s \in {{\mathbb N}}_0$ and $0 \leq j \leq k \leq n-1$, are expressed as linear combinations of the basic tensor valuations. When $j=k$ the integral formula becomes $$\label{j=k}
\int_{{\mathcal{E}^n}_k} \Phi_{k,r,s}^{(E)}(K \cap E) \, \mu_k^n (dE)=
\begin{cases}
\Phi_{n,r,0}(K) & \text{if } s=0, \\
0 & \text{otherwise},
\end{cases}$$ see [@Hug Theorem 2.4]. In the case where $j < k$, the formulas become lengthy with coefficients in the linear combinations that are difficult to evaluate, see [@Hug Theorem 2.5 and 2.6]. In the following, we are interested in using the integral formulas for the estimation of the surface tensors, and therefore we need more explicit integral formulas. We only treat the special case where $k=1$, that is, we consider integrals of the form $$\int_{{\mathcal{E}^n}_1} \Phi_{j,r,s}^{(E)}(K \cap E) \, \mu_1^n (dE).$$ Since $\dim(E)=1$, the tensor $\Phi_{j,r,s}^{(E)}(K)$ is by definition the zero function when $j>1$, so the only non-trivial cases are $j=0$ and $j=1$. When $j=1$ formula gives a simple expression for the integral. In the case where $j=0$ and $r=0$, we provide an independent and elementary proof of the integral formula, which also leads to explicit and fairly simple constants.
Linear Crofton formulae for tensors {#CroftonSec}
===================================
We start with the main result of this section, which provides a linear Crofton formula relating an average of tensor valuations defined relative to varying section lines to a linear combination of surface tensors.
\[crofton-like\] Let $K \in {{\mathcal K}}^n$. If $s \in {{\mathbb N}}_0$ is even, then $$\label{thm}
\int_{\mathcal{E}^n_1}\Phi^{(E)}_{0,0,s}(K\cap E)\, \mu_1^n(dE) = \frac{2 \omega_{n+s+1}}{ \pi s!\omega_{s+1}^2\omega_n} \sum_{k=0}^{\frac{s}{2}} c_{k}^{(\frac{s}{2})} Q^{\frac{s}{2}-k} \Phi_{n-1,0,2k}(K),$$ with constants $$c_{k}^{(m)}=(-1)^{k} \binom{m}{k} \frac{(2k)! \, \omega_{2k+1}}{1-2k}$$ for $m \in {{\mathbb N}}_0$ and $k=0, \dots, m$.
For odd $s \in {{\mathbb N}}_0$ the Crofton integral on the left-hand side is zero.
Before we give a proof of Theorem $\ref{crofton-like}$, let us consider the measurement function $\MT[s]$ on the left-hand side of . Let $k \in \{1, \dots, n\}$. Slightly more general than in , we choose $s \in {{\mathbb N}}_0$ and $E \in {\mathcal{E}}_k^n$. Then $$\Phi^{(E)}_{0,0,s}(K\cap E)
=\frac{1}{s!\omega_{k+s}}\int_{S^{n-1}\cap {\pi(E)}}u^s\, {\mathcal{H}}^{k-1}(du)\, V_0(K\cap E),$$ since the surface area measure of order 0 of a non-empty set is up to a constant the invariant measure on the sphere. From calculations equivalent to [@TVCB (24)-(26)] (or from a special case of Lemma 4.3 in [@Hug]) we get that $$\label{metric}
\int_{S^{n-1}\cap {\pi(E)}}u^s\, \mathcal{H}^{k-1}(du)
=\begin{cases}
2\frac{\omega_{s+k}}{\omega_{s+1}}Q({\pi(E)})^{\frac{s}{2}} & \text{if $s$ is even}, \\
0 & \text{if $s$ is odd}.
\end{cases}$$ Hence $$\label{Simpel form for T}
\Phi^{(E)}_{0,0,s}(K\cap E)
=\frac{2}{s! \omega_{s+1}}\cdot Q({\pi(E)})^{\frac{s}{2}}V_0(K\cap E),$$ when $s$ is even, and $\Phi^{(E)}_{0,0,s}(K\cap E)=0$ when $s$ is odd. This implies that the Crofton integral in is zero for odd $s$, and the tensors $\T[s]$ are hereby not accessible in this situation. This is even true for more general measurement functions; see Theorem \[s ulige\]. To show Theorem \[crofton-like\] we can restrict to even $s$ from now on.
In the proof of Theorem \[crofton-like\] we use the following identity for binomial sums.
\[binomial\] Let $m,n \in {{\mathbb N}}_0$. Then $$\sum_{j=0}^m(-1)^j \frac{\binom{2n}{2j}\binom{n-j}{m-j}}{\binom{n-\frac{1}{2}}{j}}
=
\frac{\binom{n}{m}}{1-2m}.$$
Lemma \[binomial\] can be proven by using the identity $$\label{hjaelpebinomial}
\sum_{j=0}^k (-1)^j \frac{\binom{2n}{2j} \binom{n-j}{m-j}}{\binom{n-\frac{1}{2}}{j}}
=
\frac{(-1)^k(2k+1)\binom{2n}{2(k+1)} \binom{n-k-1}{m-k-1}}{(2m-1)\binom{n-\frac{1}{2}}{k+1}}-\frac{\binom{n}{m}}{(2m-1)},$$ where $n, k \in {{\mathbb N}}_0$, and $m \in {{\mathbb N}}$ such that $k<m$. Identity follows by induction on $k$.
Let $K \in {{\mathcal K}}^n$ and let $s \in {{\mathbb N}}_0$ be even. If $n=1$, formula follows from the identity $$\label{binom2}
\sum_{j=0}^m (-1)^j\frac{\binom{m}{j}}{1-2j}=\frac{\sqrt{\pi} \,\Gamma(m+1)}{\Gamma(m+\frac{1}{2})}$$ with $m=\frac{s}{2}$. The left-hand side of is a sum of alternating terms of the same form as the right-hand side of the binomial sum in Lemma \[binomial\]. Using Lemma \[binomial\] and then changing the order of summation yields .
Now assume that $n\geq2$. Using we can rewrite the integral as $$\begin{aligned}
&\int_{\mathcal{E}^n_1}\Phi^{(E)}_{0,0,s}(K\cap E)\, \mu_1^n(dE)\\
&\qquad =\frac{2}{s!\omega_{s+1}}
\int_{{\mathcal{L}}_1^n} Q(L)^{\frac{s}{2}} \int_{{L^{\perp}}} V_0(K \cap (L+x)) \,\lambda_{{L^{\perp}}}(dx)\,\nu^n_1(dL)
\\
&\qquad =\frac{2}{s!\omega_{s+1}\omega_n} \int_{S^{n-1}}u^sV_{n-1}(K\mid u^\perp)\, {\mathcal{H}}^{n-1}(du),\end{aligned}$$ by the convexity of $K$ and an invariance argument for the second equality. Cauchy’s projection formula (see, e.g., [@Gardner (A.43)]) and Fubini’s theorem then imply that $$\begin{aligned}
\label{eq1}
&\int_{\mathcal{E}^n_1}\Phi^{(E)}_{0,0,s}(K\cap E)\, \mu_1^n(dE)\nonumber \\
&\qquad =\frac{1}{s!\omega_{s+1}\omega_n}
\int_{S^{n-1}}\int_{S^{n-1}} u^s |\langle u,v\rangle|\, \mathcal{H}^{n-1}(du)\, S_{n-1}(K,dv).\end{aligned}$$
We now fix $v \in S^{n-1}$ and simplify the inner integral by introducing spherical coordinates (see, e.g, [@SH66]). Then $$\begin{aligned}
&\int_{S^{n-1}}u^s |\langle u,v\rangle|\, \mathcal{H}^{n-1}(du)\\
&\qquad =\int_{-1}^1\int_{S^{n-1}\cap v^\perp}(1-t^2)^{\frac{n-3}{2}}(tv+\sqrt{1-t^2}w)^s|t|\,{\mathcal{H}}^{n-2}(dw) \, dt
\\
&\qquad=\sum_{j=0}^s \binom{s}{j}v^j \int_{-1}^1 (1-t^2)^{\frac{n-3}{2}}t^j\sqrt{1-t^2}^{s-j} |t| \,dt \int_{S^{n-1}\cap v^\perp} w^{s-j}\,{\mathcal{H}}^{n-2}(dw).\end{aligned}$$ The integral with respect to $t$ is zero if $j$ is odd. If $j$ is even, then it is equal to the beta integral $$B\bigg(\frac{j+2}{2},\frac{n+s-j-1}{2}\bigg)
=\frac{2\omega_{n+s+1}}{\omega_{j+2}\,\omega_{n+s-j-1}}.$$ Hence, since $s$ is even, we conclude from that $$\begin{aligned}
&\int_{S^{n-1}}u^s |\langle u,v\rangle|\, {\mathcal{H}}^{n-1}(du)=4 \omega_{n+s+1} \sum_{j=0}^{\frac{s}{2}} \binom{s}{2j}v^{2j} \frac{1}{\omega_{2j+2} \, \omega_{s-2j+1}}Q(v^{\perp})^{\frac{s-2j}{2}}
\\
&\qquad =4 \omega_{n+s+1} \sum_{j=0}^{\frac{s}{2}} \sum_{i=0}^{\frac{s}{2}-j} (-1)^{i} \binom{s}{2j} \binom{\frac{s}{2}-j}{i} \frac{1}{\omega_{2j+2} \, \omega_{s-2j+1}} Q^{\frac{s}{2}-j-i}v^{2(i+j)},\end{aligned}$$ where we have used that $Q(v^{\perp})=Q-v^2$. Substituting this into and by the definition of $\T[2(i+j)]$, we obtain that $$\label{S}
\int_{\mathcal{E}^n_1}\Phi^{(E)}_{0,0,s}(K\cap E)\, \mu_1^n(dE)
=\frac{4 \omega_{n+s+1}}{s!\omega_{s+1}\omega_n} \, S,$$ where $$S= \sum_{j=0}^{\frac{s}{2}} \sum_{i=0}^{\frac{s}{2}-j} (-1)^{i} \binom{s}{2j} \binom{\frac{s}{2}-j}{i} \frac{(2(i+j))!\omega_{2(i+j)+1}}{\omega_{2j+2} \, \omega_{s-2j+1}} Q^{\frac{s}{2}-j-i} \Phi_{n-1,0,2(i+j)}(K).$$ Re-indexing and changing the order of summation, we arrive at $$\begin{aligned}
S&=
\frac{\Gamma(\frac{s}{2}+\frac{1}{2})}{4 \pi^{\frac{s+3}{2}}} \sum_{k=0}^{\frac{s}{2}} (-1)^k (2k)! \omega_{2k+1} Q^{\frac{s}{2}-k}\Phi_{n-1,0,2k}(K)
\\
&\qquad \times \sum_{j=0}^{k}(-1)^{j}\binom{s}{2j}\binom{\frac{s}{2}-j}{k-j} \binom{\frac{s-1}{2}}{j}^{-1}
\\
&=
\frac{1}{2 \pi \omega_{s+1}}\sum_{k=0}^{\frac{s}{2}} (-1)^k \binom{\frac{s}{2}}{k}\frac{(2k)! \, \omega_{2k+1}}{1-2k} Q^{\frac{s}{2}-k}\Phi_{n-1,0,2k}(K),\end{aligned}$$ where we have used Lemma \[binomial\] with $n=\frac{s}{2}$ and $m=k$.
Setting $s=2$ we immediately get the following corollary.
Let $K \in {{\mathcal K}}^n$. Then $$\int_{\mathcal{E}^n_1}\Phi^{(E)}_{0,0,2}(K\cap E)\, \mu_1^n(dE)
=a_n\bigg(\Phi_{n-1,0,2}(K) + \frac{1}{4\pi} Q V_{n-1}(K)\bigg),$$ where $$a_n=\frac{\Gamma(\frac{n}{2})}{2\Gamma(\frac{n+3}{2})\sqrt{\pi}}.$$
The Crofton formula in Theorem \[crofton-like\] expresses the integral of the measurement function $\MT[s]$ as a linear combination of certain surface tensors of $K \in {{\mathcal K}}^n$. This could, in principle, be used to obtain unbiased stereological estimators of the linear combinations. However, it is more natural to ask what measurement one should use in order to obtain $\T[s]$ as a Crofton-type integral. For even $s$ the tensor $\T[s]$ appears in the last term of the sum on the right-hand side of $\eqref{thm}$. But surface tensors of lower rank appear in the remaining terms of the sum. Therefore, we need to express the lower rank tensors $\T$ for $k=0, \dots, \frac{s}{2}-1$ as integrals. This can be done by using Theorem $\ref{crofton-like}$ with $s=2k$ for $k=0, \dots, \frac{s}{2}-1$. This way, we get $\frac{s}{2} + 1$ linear equations, which give rise to the linear system $$\begin{aligned}
&\begin{pmatrix}
C_0\integral[0] \\
C_2\integral[2]\\
\vdots \\
\\
C_s\integral
\end{pmatrix}
= C
\begin{pmatrix}
\T[0] \\
\T[2] \\
\\
\vdots \\
\\
\T[s]
\end{pmatrix}\end{aligned}$$ where $$C=\begin{pmatrix}
c_0^{(0)} & 0 & 0 & \dots & 0 \\
c_0^{(1)}Q & c_1^{(1)} & 0 & & \vdots \\
\vdots &&\ddots & & 0\\
&&&&\\
c_0^{(\frac{s}{2})}Q^{\frac{s}{2}} & c_1^{(\frac{s}{2})}Q^{\frac{s}{2}-1} & \dots &c_{\frac{s}{2}-1}^{(\frac{s}{2})}Q & c_{\frac{s}{2}}^{(\frac{s}{2})}
\end{pmatrix}$$ and $C_j=\frac{\pi j! \omega_{j+1}^2 \omega_n}{2\omega_{n+j+1}}$ for $j=0,2,4, \dots, s$. Our aim is to express $\T[s]$ as an integral, hence we have to invert the system. Notice that the constants $c_i^{(i)}$ are non-zero, which ensures that the system actually is invertible. The system can be inverted by the matrix $$\label{invers matrix}
D=
\begin{pmatrix}
d_{00} & 0 & 0 & \dots && 0 \\
d_{10}Q & d_{11} & 0 & &&\vdots \\
d_{20}Q^2 & d_{21}Q & d_{22} & 0 \\
\vdots & & & \ddots & & 0 \\
d_{\frac{s}{2}0}Q^{\frac{s}{2}} & d_{\frac{s}{2}1}Q^{\frac{s}{2}-1} & \dots & & & d_{\frac{s}{2} \frac{s}{2}}
\end{pmatrix},$$ where $d_{ii}=\frac{1}{c_i^{(i)}}$ for $i=0, \dots, \frac{s}{2}$, and $d_{ij}=-\frac{1} {c_i^{(i)}}\sum_{k=j}^{i-1}c_k^{(i)}d_{kj}$ for $i=1, \dots, \frac{s}{2}$ and $j=0, \dots, i-1$. In particular, we have $$\label{reffinal}
\T[s] = \sum_{j=0}^\frac{s}{2} d_{\frac{s}{2}j} Q^{\frac{s}{2}-j}C_{2j} \integral[2j].$$ Notice that only the dimension of the matrix depends on $s$, hence we get the same integral formulas for the lower rank tensors for different choices of $s$. Formula and the above considerations give the following ‘inverse’ version of the Crofton’s formula.
\[crofton2\] Let $K \in {{\mathcal K}}^n$ and let $s \in {{\mathbb N}}_0$ be even. Then $$\label{thm2}
\int_{{\mathcal{E}^n}_1} G_s({\pi(E)})V_0(K \cap E)\, \mu_1^n (dE) = \T[s],$$ where $$G_{2m}(L):=\sum_{j=0}^m \frac{2 d_{mj} C_{2j} }{(2j)! \, \omega_{2j+1}} Q^{m-j}Q(L)^j$$ for $L \in {\mathcal{L}_1^n}$ and $m \in {{\mathbb N}}_0$.
It should be remarked that the measurement function in is just a linear combination of the relative tensors of even rank at most $s$, but we prefer the present form to indicate the dependence on $K$ more explicitly.
\[s er 4\] For $s=4$ the matrices are $$C=
\begin{pmatrix}
2 & 0 & 0 \\
2Q & 8 \pi & 0 \\
2Q^2 & 16 \pi Q & -\frac{64 \pi^2}{3}
\end{pmatrix}$$ and $$\label{invers matrix ex}
D=
\begin{pmatrix}
\frac{1}{2} & 0 & 0 \\[1ex]
-\frac{1}{8 \pi}Q & \frac{1}{8 \pi} & 0 \\[1ex]
-\frac{3}{64 \pi^2}Q^2 & \frac{3}{32 \pi^2}Q & -\frac{3\pi^2}{64}
\end{pmatrix}.$$ Since $C_0=\frac{2 \pi \omega_n}{\omega_{n+1}}$, $C_2=\frac{16 \pi^3 \omega_{n}}{\omega_{n+3}}$ and $C_4=\frac{256 \pi^5 \omega_n}{3 \omega_{n+5}}$, we have $$G_4(L)= -\frac{\omega_n}{32 \pi \omega_{n+1}}\big(3Q^2 - 6(n+1)QQ(L) + \pi^4(n+1)(n+3)Q(L)^2\big) ,$$ and $$G_2(L)=\frac{\omega_n}{4\omega_{n+1}} \bigg((n+1)Q(L)-Q \bigg)$$ for $L \in {\mathcal{L}_1^n}.$
In Theorem \[crofton2\] we only considered the situation, where $s$ is even. It is natural to ask whether $\T[s]$ can also be written as a linear Crofton integral when $s$ is odd. The case $s=1$ is trivial, as the tensor $\Phi_{n-1,0,1}(K)=0$ for all $K\in{{\mathcal K}}^n$. If $n=1$, then $\T[s]=0$ for all odd $s$, since the area measure of order 0 is the Hausdorff measure on the sphere. Apart from these trivial examples, $\Phi_{n-1,0,s}$ cannot be written as a linear Crofton-type integral, when $s$ is odd and the measurement function satisfies some rather weak assumptions. This is shown in Theorem \[s ulige\].
\[s ulige\] Let $n \geq 2$ and let $s>1$ be odd. Then there exists neither a translation invariant nor a bounded measurable measurement function $\nobreak{\alpha \colon {{\mathcal K}}^n \rightarrow {\mathbb{T}}^s}$ such that $$\label{eq}
\int_{{\mathcal{E}^n}_1} \alpha(K \cap E) \, \mu_1^n(dE)=\T[s]$$ for all $K \in {{\mathcal K}}^n.$
Let $\alpha \colon {{\mathcal K}}^n \rightarrow {\mathbb{T}}^s$ be a measurable and bounded function that satisfies equation . Since $\mu_1^n(\{E \in {\mathcal{E}^n}_1 \mid E \cap K = \emptyset\})=\infty$ for $K \in {{\mathcal K}}^n$, we have $\alpha(\emptyset)=0$. Now define the averaged function $$\alpha_r(M)=\frac{1}{V_n(rB^n)}\int_{rB^n}\alpha(M+x) \, \lambda (dx), \qquad M \in {{\mathcal K}}^n,$$ for $r > 0$. Since $\alpha$ is measurable and bounded, the average function $\alpha_r$ is well-defined. Clearly $\alpha_r(\emptyset)=0$. Using Fubini’s theorem, the invariance of $\mu_1^n$ and the fact that $\Phi_{n-1,0,s}$ is translation invariant, we get that $$\int_{{\mathcal{E}^n}_1} \alpha_r(K \cap E) \, \mu_1^n (dE)
=\frac{1}{V_n(rB^n)} \int_{rB^n} \Phi_{n-1,0,s}(K + x) \, \lambda(dx)
=\T[s].$$ Let $K \in {{\mathcal K}}^n$ be such that $K \subseteq B^n$. Since $K \cap E$ is either the empty set or a a line segment in $B^n$ when $E \in {\mathcal{E}^n}_1$, there exists a vector $z_E \in {{\mathbb R}}^n$ with ${\|}z_E {\|}\leq 2$ such that $-(K \cap E)=(K \cap E) + z_E$. Let ${\mathcal{A}}=\{E \in {\mathcal{E}^n}_1 \mid B^n \cap E \neq \emptyset\}$, let $B_1\Delta B_2$ denote the symmetric difference of two sets $B_1,B_2$, and assume that $|\alpha|\le M$ for some constant $M$. Then $$\begin{aligned}
& \big|\T[s] - \Phi_{n-1,0,s}(-K)\big|
=\bigg| \int_{{\mathcal{A}}} \alpha_r(K \cap E) - \alpha_r(-(K \cap E)) \, \mu_1^n (dE) \bigg|
\\
&\qquad \leq \frac{1}{V_n(rB^n)} \int_{\mathcal{A}}\bigg| \int_{rB^n} \alpha((K \cap E) + x)\, \lambda(dx) \\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad - \int_{rB^n+z_E} \alpha((K \cap E) + x)\, \lambda (dx) \bigg| \,\mu_1^n (dE)\\
&\qquad \leq \frac{1}{V_n(rB^n)} \int_{{\mathcal{A}}}\int_{(rB^n+z_E)\Delta (rB^n)} \left|\alpha((K\cap E)+x)\right|\, \lambda(dx)\, \mu_1^n (dE)\\
&\qquad\leq \frac{2M}{V_n(rB^n)} \int_{{\mathcal{A}}} V_n((rB^n+z_E)\setminus (rB^n))\, \mu_1^n (dE)\\
&\qquad\leq 2M\frac{ (r+2)^n-r^n}{r^n}\kappa_{n-1}\longrightarrow 0\qquad\text{as $r \rightarrow \infty$. }\end{aligned}$$ Here we used that $(rB^n+z_E)\setminus (rB^n)\subseteq (r+2)B^n\setminus (rB^n)$ and $\mu^n_1({\mathcal{A}})=\kappa_{n-1}$.
Hence, we get $\T[s]=\Phi_{n-1,0,s}(-K)$. Since $s$ is odd, we also have $\T[s]=-\Phi_{n-1,0,s}(-K)$. Therefore $\T[s]=0$, which is not the case for all $K \subseteq B^n$, since $s>1$. Then, by contradiction, cannot be satisfied by a bounded measurement function, when $s>1$ is odd.
Now assume that $\alpha$ is translation invariant and satisfies equation . As $-(K \cap E)$ is a translation of $K \cap E$, we have $$\int_{{\mathcal{E}^n}_1} \alpha(-K \cap E) \, \mu_1^n (dE)
= \int_{{\mathcal{E}^n}_1} \alpha(-(K \cap E)) \, \mu_1^n (dE)
=\int_{{\mathcal{E}^n}_1} \alpha(K \cap E) \, \mu_1^n (dE),$$ implying $\Phi_{n-1,0,s}(-K)=\T[s]=-\Phi_{n-1,0,s}(-K)$, and hereby we obtain that $\T[s]=0$ for all $K \in {{\mathcal K}}^n$. This is a contradiction as before.
Design based estimation {#sec estimation}
=======================
In this section we use the integral formula in Theorem \[crofton2\] to derive unbiased estimators of the surface tensors $\T[s]$ of $K \in {{\mathcal K}}^n$, when $s$ is even. We assume throughout this chapter that $n \geq 2$. Three different types of estimators based on 1-dimensional linear sections are presented. First, we establish estimators based on isotropic uniform random lines, then estimators based on random lines in vertical sections and finally estimators based on non-isotropic uniform random lines.
Estimation based on isotropic uniform random lines {#IUR}
--------------------------------------------------
In this section we construct estimators of $\T[s]$ based on isotropic uniform random lines. Let $K \in {{\mathcal K}}^n.$ We assume that (the unknown set) $K$ is contained in a compact reference set $A \subseteq {{\mathbb R}}^n$, the latter being known. Now let $E$ be an *isotropic uniform random (IUR) line in ${{\mathbb R}}^n$ hitting $A$*, i.e., the distribution of $E$ is given by $$\label{IURdistribution}
{\mathbb{P}}(E \in {\mathcal{A}})= c_1(A) \int_{{\mathcal{A}}} {\textbf{1}}(E' \cap A \neq \emptyset)\,\mu_1^n(dE')$$ for ${\mathcal{A}}\in {\mathcal{B}}({\mathcal{E}^n}_1),$ where $c_1(A)$ is the normalizing constant $$c_1(A)=\bigg(\int_{{\mathcal{E}^n}_1} {\textbf{1}}(E' \cap A \neq \emptyset)\,\mu_1^n(dE')\bigg)^{-1}.$$ By with $ s=0 $ the normalizing constant becomes $c_1(A)=\frac{\omega_n }{2 \kappa_{n-1}}V_{n-1}(A)^{-1}$, when $A$ is a convex body. Then Theorem \[crofton2\] implies that $$\label{Estimator}
c_1(A)^{-1} G_{s}({\pi(E)})V_0(K \cap E)$$ is an unbiased estimator of $\T[s]$, when $s$ is even.
\[example\] Using the expressions of $G_2$ and $G_4$ in Example $\ref{s er 4}$ we get that $$\begin{aligned}
-\frac{V_{n-1}(A)}{32\pi^2}\big(3Q^2 - 6(n+1)QQ(L) + \pi^4(n+1)(n+3)Q(L)^2\big) V_0(K \cap E)\end{aligned}$$ is an unbiased estimator of $\T[4],$ and $$\begin{aligned}
\label{s er 2}
\frac{V_{n-1}(A)}{4\pi}\bigg((n+1)Q({\pi(E)})-Q \bigg) V_0(K \cap E)\end{aligned}$$ is an unbiased estimator of $\T[2]$, when $A$ is a convex body. For $n=3$, these estimators read $$-\frac{V_2(A)}{32 \pi^2}\bigg(3 Q^2-24QQ({\pi(E)})+24\pi^4Q({\pi(E)})^2\bigg)V_0(K \cap E)$$ and $$\label{IURn3}
\frac{V_2(A)}{\pi} \bigg(Q({\pi(E)})-\frac{1}{4}Q\bigg)V_0(K \cap E).$$
An investigation of the estimators in Example \[example\] shows that they possess some unfavourable statistical properties. If $K \cap E = \emptyset$ the estimators are simply zero. Furthermore, if $K \cap E \neq \emptyset,$ the matrix representation of the estimator of $\T[2]$ is, in contrast to $ \T[2] $, not positive semi-definite. In fact, the eigenvalues of the matrix representation of $(n+1)Q({\pi(E)})-Q$ are $n$ (with multiplicity $ 1 $) and $-1$ (with multiplicity $n-1$). It is not surprising that estimators based on the measurement of one single line, are not sufficient, when we are estimating tensors with many unknown parameters. To improve the estimators, they can be extended in a natural way to use information from $N$ *IUR* lines for some $N \in {{\mathbb N}}$. In addition, the integral formula can be rewritten in the form $$\begin{aligned}
\label{Alternativ}
\T[s]&=\int_{{\mathcal{L}_1^n}} \int_{L^\perp} G_s(L)V_0(K \cap (L+x))\, \lambda_{{L^{\perp}}}(dx) \, \nu_1^n (dL) \nonumber \\
&=\int_{{\mathcal{L}_1^n}} G_s(L) V_{n-1}(K \vert L^\perp) \, \nu_1^n (dL),\end{aligned}$$ which implies that $$\label{projestimator}
\frac{1}{N}\sum_{i=1}^NG_s(L_i)V_{n-1}(K \vert L_i^\perp)$$ is an unbiased estimator of $\T[s]$, when $L_1, \dots L_N \in {\mathcal{L}_1^n}$ are $ N $ isotropic lines (through the origin) for an $ {N \in {{\mathbb N}}} $. When $K$ is full-dimensional this estimator never vanishes. In the case where $s=2$ the estimator becomes $$\label{EstimatorN}
\frac{1}{N}\frac{\omega_n}{4\omega_{n+1}}\sum_{i=1}^N\big((n+1)Q(L_i)-Q \big) V_{n-1}(K \vert L_i^\perp).$$ In stereology it is common practice to use orthogonal test lines. If we set $ N=n $ and let $ L_1, \dots, L_n $ be isotropic, pairwise orthogonal lines, then the estimator becomes positive definite exactly when $$\label{posdefcondition}
(n+1)V_{n-1}(K \vert L_i^\perp) > \sum_{j=1}^n V_{n-1}(K \mid L_j^\perp)$$ for all $i=1, \dots, n.$ This is a condition on $ K $ requiring that $ K $ is not too eccentric. A sufficient condition for to hold makes use of the radius $ R(K) $ of the smallest ball containing $ K $ and the radius $ r(K) $ of the largest ball contained in $ K $. If $$\frac{r(K)}{R(K)}> \bigg(1-\frac{1}{n}\bigg)^{\frac{1}{n-1}},$$ then is satisfied, and hence the estimator with $ n $ orthogonal, isotropic lines is positive definite. In $ {{\mathbb R}}^2 $ this means that $ 2r(K) > R(K) $ is sufficient for a positive definite estimator , and in particular for all ellipses for which the length of the longer main axis does not exceed twice the length of the smaller main axis, yields positive definite estimators. For ellipses, this criterion is also necessary as the following example shows.
Consider the situation where $ n=2 $ and $ K $ is an ellipse, $ K=\{x \in {{\mathbb R}}^2 \mid x^\top B x \leq 1\} $, given by the matrix $$B=\begin{pmatrix}
\alpha^{-2} & 0 \\ 0 & (k\alpha)^{-2}
\end{pmatrix},$$ where $ \alpha > 0 $ and $ k \in (0,1] $. The parameter $ k $ determines the eccentricity of $ K $. If $ k \in (\frac{1}{2},1] $, and $ L_1 $ and $ L_2 $ are orthogonal, isotropic random lines in $ {{\mathbb R}}^2 $, the estimator becomes positive definite by the above considerations. Now let $ k \in [0,1/2] $. Since $ n=2 $, each pair of orthogonal lines is determined by a constant $ \phi \in [0, \frac{\pi}{2}) $ by letting $ L_1 = u_{\phi}^\perp $ and $ L_2=u_{\phi+ \frac{\pi}{2}}^\perp $, where $ u_{\phi}=(\cos(\phi),\sin(\phi))^\top $. Then $$V_{n-1}(K \mid L_1^\perp)=2h(K,u_{\phi})=2 \alpha \sqrt{\cos^2(\phi)+k^2 \sin^2(\phi)}$$ and $$V_{n-1}(K \mid L_2^\perp ) = 2 \alpha \sqrt{\sin^2(\phi)+k^2 \cos^2(\phi)}.$$ Condition is satisfied if and only if $$\phi \in [\sin^{-1}\bigg(\sqrt{\frac{1-4k^2}{5(1-k^2)}}\bigg), \cos^{-1}\bigg(\sqrt{\frac{1-4k^2}{5(1-k^2)}}\bigg) ],$$ and the probability that the estimator is positive definite, when $ L_1 $ and $ L_2 $ are orthogonal, isotropic lines (corresponding to $ \phi $ being uniformly distributed on $ [0,\frac{\pi}{2}] $) is $$\frac{2}{\pi}\bigg(\cos^{-1}\bigg(\sqrt{\frac{1-4k^2}{5(1-k^2)}}\bigg)-\sin^{-1}\bigg(\sqrt{\frac{1-4k^2}{5(1-k^2)}}\bigg)\bigg),$$ which converges to $ \frac{2}{\pi} \big(\cos^{-1}(\sqrt{\frac{1}{5}})- \sin^{-1}(\sqrt{\frac{1}{5}}) \big) \approx 0.41 $ as $ k $ converges to 0.
In $ {{\mathbb R}}^2 $ the estimator can alternatively be combined with a systematic sampling approach with $ N $ isotropic random lines. Let $ N \in {{\mathbb N}}$, and let $ \phi_0 $ be uniformly distributed on $ [0,\frac{\pi}{N}] $. Moreover, let $ \phi_i=\phi_0 + i \frac{\pi}{N} $ for $ i=1, \dots, N-1 $. Then $ u_{\phi_0}, \dots, u_{\phi_{N-1}} $ are $ N $ systematic isotropic uniform random directions in the upper half of $ S^1 $, where $ u_{\phi}=(\cos(\phi),\sin(\phi))^\top $. As the estimator is a tensor of rank 2, it can be identified with the symmetric $2 \times 2$ matrix, where the $(i,j)$’th entry is the estimator evaluated at $(e_i,e_j)$, where $(e_1,e_2)$ is the standard basis of ${{\mathbb R}}^2$. The estimator becomes $$\label{syst}
S_N(K, \phi_0)=\frac{1}{N}\sum_{i=0}^{N-1} \begin{pmatrix}
3\cos^2(\phi_i)-1 & 3\cos(\phi_i) \sin(\phi_i) \\
3\cos(\phi_i) \sin(\phi_i) & 3\sin^2(\phi_i)-1
\end{pmatrix} V_1(K \mid u_{{\phi}_i}^\perp).$$
\[exsyst\] To investigate how the estimator $ S_N(K, \phi_0) $ performs we estimate the probability that the estimator is positive definite for three different origin-symmetric convex bodies in $ {{\mathbb R}}^2 $; a parallelogram, a rectangle, and an ellipse. Thus let $$\begin{aligned}
K_1&=\mathrm{conv}\{ (1,\epsilon),(-1,\epsilon),(-1,-\epsilon),(1,-\epsilon)\}, \\
K_2&=\mathrm{conv}\{ (1,0),(0,\epsilon),(-1,0),(0,-\epsilon)\} \\
\intertext{and}
K_3&= \{x \in {{\mathbb R}}^2 \mid x^\top \begin{pmatrix}
1 & 0 \\ 0 & \frac{1}{\sqrt{\epsilon}}
\end{pmatrix} x \leq 1\} \end{aligned}$$ with $\epsilon=0.1$. The support functions, and hence the intrinsic volumes $V_1(K_i \vert u_\phi^\perp)$, of $ K_1, K_2$ and $ K_3 $ have simple analytic expressions, and the estimator $ S_N(K_i, \phi_0) $ can be calculated for $ \phi_0 \in [0,\frac{\pi}{N}] $ and $ i=1,2,3 $. The eigenvalues of the estimators can be calculated numerically, and the probability that the estimators $ S_N(K_i,\phi_0) $ are positive definite, when $ \phi_0 $ is uniformly distributed on $ [0, \frac{\pi}{N}] $, can hereby be estimated. For each choice of $ N $, the estimate of the probability is based on $ 500 $ equally spread values of $ \phi_0 $ in $ [0,\frac{\pi}{N}] $. The estimate of the probability that $ S_N(K_i,\phi_0) $ is positive definite is plotted against the number of equidistant lines $ N $ for $i=1,2,3$ in Figure \[Plot of prob for posdef\]. The plots in Figure \[Plot of prob for posdef\] show that even though we consider rather eccentric shapes, the number $N$ of lines needed to get a positive definite estimator with probability $1$ is in all cases less than $7$.
![The probability that $ S_N(K_i,\phi_0) $ is positive definite for $ i=1,2,3 $, when $ \phi_0 $ is uniformly distributed on $ [0,\frac{\pi}{N}] $ plotted against the number of equidistant lines $ N $.[]{data-label="Plot of prob for posdef"}](R)
![The probability that $ S_N(K_i,\phi_0) $ is positive definite for $ i=1,2,3 $, when $ \phi_0 $ is uniformly distributed on $ [0,\frac{\pi}{N}] $ plotted against the number of equidistant lines $ N $.[]{data-label="Plot of prob for posdef"}](Rectangle)
\
![The probability that $ S_N(K_i,\phi_0) $ is positive definite for $ i=1,2,3 $, when $ \phi_0 $ is uniformly distributed on $ [0,\frac{\pi}{N}] $ plotted against the number of equidistant lines $ N $.[]{data-label="Plot of prob for posdef"}](Par)
![The probability that $ S_N(K_i,\phi_0) $ is positive definite for $ i=1,2,3 $, when $ \phi_0 $ is uniformly distributed on $ [0,\frac{\pi}{N}] $ plotted against the number of equidistant lines $ N $.[]{data-label="Plot of prob for posdef"}](Parallelogram)
\
![The probability that $ S_N(K_i,\phi_0) $ is positive definite for $ i=1,2,3 $, when $ \phi_0 $ is uniformly distributed on $ [0,\frac{\pi}{N}] $ plotted against the number of equidistant lines $ N $.[]{data-label="Plot of prob for posdef"}](E)
![The probability that $ S_N(K_i,\phi_0) $ is positive definite for $ i=1,2,3 $, when $ \phi_0 $ is uniformly distributed on $ [0,\frac{\pi}{N}] $ plotted against the number of equidistant lines $ N $.[]{data-label="Plot of prob for posdef"}](Ellipse)
To apply the estimator it is only required to observe whether the test line hits or misses the convex body $K$. The estimator requires more sophisticated information in terms of the projection function. In the following example the coefficient of variation of versions of the estimators and are estimated and compared in a three-dimensional set-up.
\[3dimex\] Let $K_{l}'$ be the prolate spheroid in ${{\mathbb R}}^3$ with main axis parallel to the standard basis vectors $e_1,e_2$ and $e_3$, and corresponding lengths of semi-axes $\lambda_1=\lambda_2=1$ and $\lambda_3=l$. For $l=1, \dots, 5$, let $K_l$ denote the ellipsoid obtained by rotating $K_l'$ first around $e_1$ with an angle $\frac{3\pi}{16}$, and then around $e_2$ with an angle $\frac{5\pi}{16}$. Note, that the eccentricity of $K_l$ increases with $l$. In this example, based on simulations, we estimate and compare the coefficient of variation *(CV)* of the developed estimators of $\Phi_{2,0,2}(K_l)$ for $l=1, \dots, 5$.
Formula provides an unbiased estimator of the tensor $\Phi_{2,0,2}(K_l)$ for $l=1, \dots, 5$. The estimator is based on one *IUR* line hitting a reference set $A$, and can in a natural way be extended to an estimator based on three orthogonal *IUR* lines hitting $A$. We estimate the variance of both estimators. Let, for $l=1, \dots,5$, the reference set $A_l$ be a ball of radius $R_l>0$. The choice of the reference set influences the variance of the estimator. In order to minimize this effect in the comparison of the CV’s, the radii of the reference sets are chosen such that the probability that a test line hits $K_l$ is constant for $l=1, \dots, 5$. By formula the probability that an *IUR* line hitting $A_l$ hits $K_l$ is $\frac{V_{2}(K_l)}{V_2(A_l)}$. The radius is chosen, such that this probability is $\frac{1}{7}$. We further estimate the variance of the projection estimator based on one isotropic line and on three orthogonal isotropic lines.
As $\Phi_{2,0,2}(K_l)$ is a tensor of rank 2, it can be identified with the symmetric $3 \times 3$ matrix $\{\Phi_{2,0,2}(K_l)(e_i,e_j)\}_{i,j=1}^3$. Thus, in order to estimate $\Phi_{2,0,2}(K_l)$, the matrix $\{\hat{\Phi}_{2,0,2}(K_l)(e_i,e_j)\}_{i,j=1}^3$ is calculated. Here, $\hat{\Phi}_{2,0,2}(K_l)$ refers to any of the four estimators described above. Due to symmetry, there are six different components of the matrices.
The estimates of the variances are based on 1500-10000 estimates of the tensor, depending on the choice of the estimator and the eccentricity of $K_l$. Using the estimates of the variances, we estimate the absolute value of the CV’s by $$\widehat{CV}_{ij}=\frac{\sqrt{\widehat{\text{Var}}(\hat{\Phi}_{2,0,2}(K_l)(e_i,e_j))}}{|\Phi_{2,0,2}(K_l)(e_i,e_j)|},$$ for $i,j=1,2,3$ and $l=1, \dots, 5$. As $K_l$ is an ellipsoid, the tensor $\Phi_{2,0,2}(K_l)$ can be calculated numerically. The CV’s of the four estimators are plotted in Figure \[4variances\] for each of the six different components of the associated matrix. As $K_1$ is a ball, the off-diagonal elements of the matrix associated with $\Phi_{2,0,2}(K_1)$ are zero. Thus, the CV is in this case calculated only for the estimators of the diagonal-elements.
The projection estimators give, as expected, smaller CV’s, than the estimators based on the Euler characteristic of the intersection between the test lines and the ellipsoid. For the estimators based on one test line the CV of the projection estimator is typically around $38\%$ of the corresponding estimator . For the estimators based on three orthogonal test lines, the CV of the projection estimator is typically $9\%$ of the estimator , when $l=2,\dots,5$. Due to the fact that $K_1$ is a ball, the variance of the projection estimator based on three orthogonal lines is 0, when $l=1$.
It is interesting to compare the increase of efficiency when using the estimator based on three orthogonal test lines instead of three i.i.d. test lines. The CV of an estimator based on three i.i.d. test lines is $\frac{1}{\sqrt{3}}$ of the CV of the estimator , (the “$+$” signs in Figure \[4variances\]). The CV, when using three orthogonal test lines, is typically around $92\%$ of that CV. For $l=2,\dots, 5$, the CV’s of the projection estimator based on three orthogonal lines, are typically $20\%$ of the CV, when using three i.i.d lines, indicating that spatial random systematic sampling increases precision without extra workload.
The CV’s of the estimators of the diagonal-elements $\Phi_{2,0,2}(K_l)(e_i,e_i)$ are almost constant in $l$. Hence the eccentricity of $K_l$ does not affect the CV’s for these choices of $l$. There is a decreasing tendency of the CV’s of the estimators of the off-diagonal elements. This might be explained by the fact that the true value of $\Phi_{2,0,2}(K_l)(e_i,e_j)$ is close to zero, when $i \neq j$ and $l$ is small.
![The estimated coefficients of variation $\widehat{CV}_{ij}$ of the estimators of $\Phi_{2,0,2}(K_l)(e_i,e_j)$ plotted against $l$ for $\nobreak{i,j \in \{1,2,3\}}$. The CV of the estimator based on *one* line is designated by “$+$”, while the CV of the corresponding estimator based on *three* lines is designated by “$\bullet$”. The CV of the projection estimator is designated by “$\circ$” and “$\square$” for one and three lines, respectively.[]{data-label="4variances"}](CV11ny)
![The estimated coefficients of variation $\widehat{CV}_{ij}$ of the estimators of $\Phi_{2,0,2}(K_l)(e_i,e_j)$ plotted against $l$ for $\nobreak{i,j \in \{1,2,3\}}$. The CV of the estimator based on *one* line is designated by “$+$”, while the CV of the corresponding estimator based on *three* lines is designated by “$\bullet$”. The CV of the projection estimator is designated by “$\circ$” and “$\square$” for one and three lines, respectively.[]{data-label="4variances"}](CV22ny)
\
![The estimated coefficients of variation $\widehat{CV}_{ij}$ of the estimators of $\Phi_{2,0,2}(K_l)(e_i,e_j)$ plotted against $l$ for $\nobreak{i,j \in \{1,2,3\}}$. The CV of the estimator based on *one* line is designated by “$+$”, while the CV of the corresponding estimator based on *three* lines is designated by “$\bullet$”. The CV of the projection estimator is designated by “$\circ$” and “$\square$” for one and three lines, respectively.[]{data-label="4variances"}](CV33ny)
![The estimated coefficients of variation $\widehat{CV}_{ij}$ of the estimators of $\Phi_{2,0,2}(K_l)(e_i,e_j)$ plotted against $l$ for $\nobreak{i,j \in \{1,2,3\}}$. The CV of the estimator based on *one* line is designated by “$+$”, while the CV of the corresponding estimator based on *three* lines is designated by “$\bullet$”. The CV of the projection estimator is designated by “$\circ$” and “$\square$” for one and three lines, respectively.[]{data-label="4variances"}](CV12ny)
\
![The estimated coefficients of variation $\widehat{CV}_{ij}$ of the estimators of $\Phi_{2,0,2}(K_l)(e_i,e_j)$ plotted against $l$ for $\nobreak{i,j \in \{1,2,3\}}$. The CV of the estimator based on *one* line is designated by “$+$”, while the CV of the corresponding estimator based on *three* lines is designated by “$\bullet$”. The CV of the projection estimator is designated by “$\circ$” and “$\square$” for one and three lines, respectively.[]{data-label="4variances"}](CV13ny)
![The estimated coefficients of variation $\widehat{CV}_{ij}$ of the estimators of $\Phi_{2,0,2}(K_l)(e_i,e_j)$ plotted against $l$ for $\nobreak{i,j \in \{1,2,3\}}$. The CV of the estimator based on *one* line is designated by “$+$”, while the CV of the corresponding estimator based on *three* lines is designated by “$\bullet$”. The CV of the projection estimator is designated by “$\circ$” and “$\square$” for one and three lines, respectively.[]{data-label="4variances"}](CV23ny)
The above example shows that only the projection estimator based on three orthogonal test lines has a satisfactory precision. For $l=2$ the CV’s are approximately $\frac{1}{3}$ for the diagonal-elements and $1$ for the off-diagonal elements. Further variance reduction of the projection estimator can be obtained by using a larger number of systematic random test directions. For $n=2$ this can be effectuated by choosing equidistant points on the upper half circle; see . For $n=3$ the directions must be chosen evenly spread; see [@Leopardi2006] for details.
If the projections are not available or too costly to obtain, systematic sampling in the position of the test lines with given orientations can be applied. In ${{\mathbb R}}^2$ this corresponds to a Steinhaus-type estimation procedure (see e.g. [@Jensen2005]). In ${{\mathbb R}}^3$ the fakir method described in [@L.1998] can be applied.
Estimation based on vertical sections {#sec VUR}
-------------------------------------
In the previous section we constructed an estimator of $\T[s]$ based on isotropic uniform random lines. As described in [@Markus], it is sometimes inconvenient or impossible to use the *IUR* design in applications. For instance, in biology when analysing skin tissue, it might be necessary to use sample sections, which are normal to the surface of the skin, so that the different layers become clearly distinguishable in the sample. Instead of using *IUR* lines it is then a possibility to use vertical sections introduced by Baddeley in [@Baddeley1983]. The idea is to fix a direction (the normal of the skin surface), and only consider flats parallel to this direction. After randomly selecting a flat among these flats, we want to pick a line in the flat in such a way that this line is an isotropic uniform random line in ${{\mathbb R}}^n$. Like in the classical formulae for vertical sections, we select this line in a non-uniform way according to a Blaschke-Petkantschin formula (see ). This idea is used to deduce estimators of $\T[s]$ from the Crofton formula .
When introducing the concept of vertical sections we use the following notation. For $ 0 \leq k \leq n $ and $ L \in {\mathcal{L}}^n_k $, let $$\cLL=
\begin{cases}
\{M \in {\mathcal{L}}_r^n \mid M \subseteq L \} & \text{if } 0 \leq r \leq k \\
\{M \in {\mathcal{L}}_r^n \mid L \subseteq M \} & \text{if } k < r \leq n,
\end{cases}$$ and, similarly, let ${\mathcal{E}}^E_r= \{F \in {\mathcal{E}^n}_r \mid F \subseteq E \} $ for $ E \in {\mathcal{E}}^n_k $ and $ 0 \leq r \leq k$. Let $ \nu^L_r$ denote the unique rotation invariant probability measure on $ {\mathcal{L}}^L_r $, and let $ \mu^E_r $ denote the motion invariant measure on $ {\mathcal{E}}^E_r $ normalized as in [@Weil].
Let $L_{0} \in {\mathcal{L}_1^n}$ be fixed. This is the *vertical axis* (the normal of the skin surface in the example above). Let the reference set $A \subseteq {{\mathbb R}}^n$ be a compact set.
\[VUR\] Let $1 < k < n$. A random $k$-flat $H$ in ${{\mathbb R}}^n$ is called **a vertical uniform random (VUR) $k$-flat hitting $A$** if the distribution of $H$ is given by $$P(H \in {\mathcal{A}}) = c_2(A) \int_{\cLLn[k]} \int_{A \mid L^\perp} {\textbf{1}}(L+x \in {\mathcal{A}})\, \lambda_{{L^{\perp}}}(dx) \, \nu_k^{L_0}(dL)$$ for ${\mathcal{A}}\in {\mathcal{B}}({\mathcal{E}^n}_k)$, where $c_2(A)>0$ is a normalizing constant.
The distribution of $H$ is concentrated on the set $$\{E \in {\mathcal{E}^n}_{k} \mid E \cap A \neq \emptyset, L_0 \subseteq {\pi(E)}\}.$$ When the reference set $A$ is a convex body, the normalizing constant becomes $$c_2(A)=\binom{n-1}{k-1}\frac{\kappa_{n-1}}{\kappa_{k-1}\kappa_{n-k}}\frac{1}{V_{n-k}(A\vert L_0^\perp)}.$$ (Note that we do not indicate the dependence of $c_2(A)$ on $k$ by our notation.) This can be shown, e.g., by using the definition of $\nu_k^{L_0}$ together with [@Weil (13.13)], Crofton’s formula in the space $L_0^{\perp}$, and the equality $$\label{v0 lighed}
{\textbf{1}}_{A \vert L^\perp}(x)=V_0((A\vert L_0^\perp) \cap (x+L))$$ for $A \in {{\mathcal K}}^n$, $L \in \cLLn[k]$ and $x \in L^\perp$. For later use note that when $k=2$ the normalizing constant becomes $$\label{normalizing constant}
c_2(A)=\frac{\omega_{n-1}}{2 \kappa_{n-2}V_{n-2}(A \vert L_0^\perp)}.$$
To construct an estimator, which is based on a vertical uniform random flat, we cannot use Theorem \[crofton2\] immediately as in the *IUR*-case. It is necessary to use a Blaschke-Petkantschin formula first; see [@Markus (2.8)]. It states that for a fixed $L_0 \in {\mathcal{L}_1^n}$ and an integrable function $f \colon {\mathcal{E}}^n_1 \rightarrow {{\mathbb R}}$, we have $$\begin{aligned}
\label{B-P}
\int_{{\mathcal{E}^n}_1} f(E) \, \mu_1^n(dE)& = \frac{\pi \omega_{n-1}}{\omega_n} \int _{{\mathcal{L}}^{L_0}_2} \int_{M^\perp} \int _{{\mathcal{E}}_1^{M+x}}
f(E) \sin (\angle(E,L_0))^{n-2} \nonumber
\\
&\qquad\qquad\times \mu_1^{M+x}(dE) \, \lambda_{M^{\perp}}(dx) \, \nu_2^{L_0} (dM),\end{aligned}$$ where $\angle(E_1,E_2)$ is the (smaller) angle between $\pi(E_1)$ and $ \pi(E_2)$ for two lines $E_1,E_2 \in {\mathcal{E}}^n_1$. For $K \in {{\mathcal K}}^n$ and even $s \in {{\mathbb N}}_0$, equation can be applied coordinate-wise to the mapping $E \mapsto \MT[s]$ and combined with the Crofton formula in Theorem \[crofton-like\]. The result is an integral formula for two-dimensional vertical sections.
\[VUR crofton\] Let $L_0 \in {\mathcal{L}_1^n}$ be fixed. If $K \in {{\mathcal K}}^n$ and $s \in {{\mathbb N}}_0$ is even, then $$\begin{aligned}
\label{VUR crofton formel}
&\int _{{\mathcal{L}}^{L_0}_2} \int_{M^\perp} \int _{{\mathcal{E}}_1^{M+x}} \Phi^{(E)}_{0,0,s}(K \cap E) \sin(\angle(E,L_0))^{n-2} \, \mu_{1}^{M+x} (dE) \, \lambda_{M^{\perp}} (dx) \, \nu_{2}^{L_0} (dM)
\nonumber \\
&\qquad\qquad = \frac{2 \omega_{n+s+1}}{ s! \pi^2 \omega_{n-1} \omega_{s+1}^2 } \sum_{k=0}^{\frac{s}{2}} c_k^{(\frac{s}{2})} Q^{\frac{s}{2}-k} \Phi_{n-1,0,2k}(K),\end{aligned}$$ where the constants $c_k^{(m)}$ are given in Theorem $\ref{crofton-like}$. For odd $s \in {{\mathbb N}}_0$ the integral on the left-hand side is zero.
If Theorem \[crofton-like\] is replaced by Theorem \[crofton2\] in the above line of arguments, we obtain an explicit measurement function for vertical sections leading to one single tensor.
\[VURcrofton2\] Let $L_0 \in {\mathcal{L}_1^n}$ be fixed. If $K \in {{\mathcal K}}^n$ and $s\in {{\mathbb N}}_0$ is even, then $$\begin{aligned}
\frac{\omega_n}{\pi \omega_{n-1}}\T[s]&=\int _{{\mathcal{L}}^{L_0}_2} \int_{M^\perp} \int _{{\mathcal{E}}_1^{M+x}} G_s({\pi(E)}) V_0(K \cap E)
\\
& \qquad \times \sin(\angle(E,L_0))^{n-2} \, \mu_1^{M+x}(dE) \, \lambda_{M^{\perp}}(dx) \, \nu_2^{L_0}(dM),\end{aligned}$$ where $G_s$ is given in Theorem $\ref{crofton2}$.
Let $s \in {{\mathbb N}}_0$ be even and assume that $K \in {{\mathcal K}}^n$ is contained in a reference set $A \in {{\mathcal K}}^n$. Using Theorem \[VURcrofton2\] we are able to construct unbiased estimators of the tensors $\T[s]$ of $K$ based on a vertical uniform random 2-flat. If $H$ is an *VUR* 2-flat hitting $A$ with vertical direction $L_0 \in {\mathcal{L}_1^n}$, then it follows from Theorem \[VURcrofton2\] and that $$V_{n-2}(A \vert L_0^\perp)
\int_{{\mathcal{E}}_1^{H}} G_s({\pi(E)})V_0(K \cap E)\sin(\angle(E,L_0))^{n-2}\, \mu_1^H(dE)$$ is an unbiased estimator of $\T[s]$. Hence the surface tensors can be estimated by a two-step procedure. First, let $H$ be a *VUR* 2-flat hitting the convex body $A$ with vertical direction $L_0$. Given $H$, the integral $$\label{integral}
\int_{{\mathcal{E}}_1^{H}} G_s({\pi(E)})V_0(K \cap E)\sin(\angle(E,L_0))^{n-2} \,\mu_1^H(dE)$$ is estimated in the following way. Let $E \in {\mathcal{E}}^H_1$ be an *IUR* line in $H$ hitting $A$, i.e. the distribution of $E$ is given by $$P(E \in {\mathcal{A}})= c_3(A) \int_{{\mathcal{A}}} {\textbf{1}}(A \cap E \neq \emptyset) \, \mu_1^H (dE), \qquad {\mathcal{A}}\in {\mathcal{B}}({\mathcal{E}}_1^H),$$ where $$c_3(A)=\frac{\pi}{2}V_{1}(A \cap H)^{-1}$$ is the normalizing constant. The integral is then estimated unbiasedly by $$\label{Integral estimator}
c_3(A)^{-1}G_s({\pi(E)})V_0(K \cap E) \sin(\angle(E,L_0))^{n-2}.$$
Consider the case $s=2$. Let $H$ be a *VUR* 2-flat hitting $A \in {{\mathcal K}}^n$ with vertical direction $L_0$. Given $H$, let $E$ be an *IUR* line in $H$ hitting $A$. Then $$\frac{ \kappa_{n-2}V_{n-2}(A \vert L_0^\perp)V_1(A \cap H)}{ \omega_{n+1} }\bigg((n+1)Q({\pi(E)})-Q \bigg)V_0(K \cap E) \sin(\angle(E,L_0))^{n-2}$$ is an unbiased estimator of $\T[2]$.
Using [@Weil (13.13)] and an invariance argument, the integral can alternatively be expressed by means of the support function of $K$ in the following way $$\begin{aligned}
&\quad ~ \int_{{\mathcal{E}}_1^{H}} G_s({\pi(E)})V_0(K \cap E)\sin(\angle(E,L_0))^{n-2} \,\mu_1^H (dE)
\\
&=\frac{1}{\omega_2}\int_{S^{n-1} \cap {\pi (H)}} G_s(u^\perp \cap {\pi (H)})\sin(\angle(u^\perp \cap {\pi (H)},L_0))^{n-2}
\\ & \qquad \qquad \quad \times \int_{[u]} V_0(K \cap H \cap (u^\perp + x)) \, \lambda_{[u]}(dx) \, {\mathcal{H}}^{1}(du) \nonumber
\\
&=\frac{1}{\omega_2}\int_{S^{n-1} \cap {\pi (H)}} G_s(u^\perp \cap {\pi (H)}) \cos(\angle(u,L_0))^{n-2} w(K \cap H, u) \,{\mathcal{H}}^{1}(du),\end{aligned}$$ where $ [u] $ denotes the linear hull of a unit vector $ u $, and $$w(M,u)=h(M,u)+h(M,-u)$$ is the width of $M \in {{\mathcal K}}^n$ in direction $u$. Hence, given $H$, $$\label{Alternative integral estimator}
G_s(U^\perp \cap {\pi (H)})\cos(\angle(U,L_0))^{n-2} w(K \cap H, U)$$ is an unbiased estimator of the integral if $U$ is uniform on $S^{n-1} \cap {\pi (H)}$. As in the *IUR* set-up in Section \[IUR\] we have two estimators: an estimator , where it is only necessary to observe whether the random line $E$ hits or misses $K$, and the alternative estimator $\eqref{Alternative integral estimator}$, which requires more information. The latter estimator has a better precision at least when the reference set $A$ is large. Variance reduction can be obtained by combining the estimators with a systematic sampling approach.
Estimation based on non-isotropic random lines {#Sec noniso}
----------------------------------------------
In this section we consider estimators based on non-isotropic random lines. It is well-known from the theory of importance sampling, that variance reduction of estimators can be obtained by modifying the sampling distribution in a suitable way (see, e.g., [@Asmussen]). The estimators in this section are developed with inspiration from this theory. Let again $K \in {{\mathcal K}}^n$, and let $f \colon {\mathcal{L}_1^n}\rightarrow [0,\infty)$ be a density with respect to the invariant measure $\nu_1^n$ on ${\mathcal{L}_1^n}$ such that $f$ is positive $\nu^n_1$-almost surely. Then by Theorem \[crofton2\] we have trivially $$\label{density integral}
\int_{{\mathcal{E}^n}_1} \frac{G_s({\pi(E)})V_0(K \cap E)}{f({\pi(E)})} \, f({\pi(E)})\,\mu_1^n(dE) = \T[s].$$ Let $A \subseteq {{\mathbb R}}^n$ be a compact reference set containing $K$, and let $E$ be an $f$-weighted random line in ${{\mathbb R}}^n$ hitting A, that is, the distribution of $E$ is given by $$\begin{aligned}
P(E \in {\mathcal{A}}) = c_4(A) \int_{{\mathcal{A}}} {\textbf{1}}(E \cap A \neq \emptyset) f({\pi(E)}) \, \mu_1^n (dE)\end{aligned}$$ for ${\mathcal{A}}\in {\mathcal{B}}({\mathcal{E}^n}_1)$, where $$c_4(A)=\bigg(\int_{{\mathcal{E}^n}_1} {\textbf{1}}(E \cap A \neq \emptyset) f({\pi(E)})\, \mu_1^n (dE)\bigg)^{-1}$$ is a normalizing constant. Then $$\frac{c_4(A)^{-1} G_{s}({\pi(E)})V_0(K \cap E)}{f({\pi(E)})}$$ is an unbiased estimator of $\T[s]$. Notice that if we let the density $ f $ be constant, then this procedure coincides with the *IUR* design in Section \[IUR\].
Our aim is to decide, which density $f$ should be used in order to decrease the variance of the estimator of $\T[s]$. Furthermore, we want to compare this variance with the variance of the estimator based on an *IUR* line. From now on, we restrict the investigation to the situation where $ n=2 $ and $ s=2 $. Furthermore, we assume that the reference set $ A $ is a ball in $ {{\mathbb R}}^2 $ of radius $ R $ for some $ R > 0 $. Then $ c_4(A)=(2R)^{-1} $ independently of $f$.
Since $ \Phi_{1,0,2}(K) $ can be identified with a symmetric $ 2 \times 2 $ matrix, we have to estimate three unknown components. We consider the variances of the three estimators separately. The components of the associated matrix of $G_2(L)$ for $L \in {\mathcal{L}}^n_1$ is defined by $$g_{ij}(L)=G_2(L)(e_i,e_j),$$ for $i,j=1,2$, where $(e_1,e_2)$ is the standard basis of ${{\mathbb R}}^2$. More explicitly, by Example \[s er 4\], the associated matrix of $G_2(L)$ of the line $L=[u]$, for $u \in S^1$, is $$\{g_{ij}([u])\}_{ij}=
\frac{3}{8}
\begin{pmatrix}
u_1^2 - \frac{1}{3} & u_1 u_2 \\
u_1u_2 & u_2^2-\frac{1}{3}
\end{pmatrix}.$$ Now let $${\hat{\varphi}_{ij}(K \cap E)}:= 2R \, g_{ij}({\pi(E)})V_0(K \cap E).$$ Then $$\label{estnon}
\frac{{\hat{\varphi}_{ij}(K \cap E)}}{f({\pi(E)})}$$ is an unbiased estimator of $ \Phi_{1,0,2}(K)(e_i,e_j) $, when $ E $ is an $ f $-weighted random line in $ {{\mathbb R}}^2 $ hitting $ A $.
For a given $K \in {{\mathcal K}}^2$ the weight function $f$ minimizing the variance of the estimators of the form can be determined.
\[lemmaopt\] For a fixed $K \in {{\mathcal K}}^2$ with $\dim K \geq 1$ and $i,j \in \{1,2\}$, the estimator has minimal variance if and only if $f=f_K^*$ holds $\nu^2_1-a.s.$, where $$f_K^*(L)\propto \sqrt{2RV_1(K\vert L^{\perp})}\, \vert g_{ij}(L) \vert$$ is a density with respect to $\nu^2_1$ that depends on $i,j$ and $K$.
As $K$ is compact, $f^*_K$ is a well-defined probability density, and since $\dim K \geq 1$, the density $f^*_K$ is non-vanishing $\nu^2_1$-almost surely. The second moment of the estimator is $$\label{2momentberegning}
{\mathbb{E}}_f \bigg(\frac{{\hat{\varphi}_{ij}(K \cap E)}}{f({\pi(E)})}\bigg)^2
= 2R \int_{{\mathcal{L}}^2_1}V_1(K \vert L^\perp) \frac{g_{ij}(L)^2}{f(L)}\, \nu_1^2(dL),$$ where ${\mathbb{E}}_f$ denotes expectation with respect to the distribution of an $f$-weighted random line in ${{\mathbb R}}^2$ hitting $A$. The right-hand side of is the second moment of the random variable $$\frac{\sqrt{2RV_1(K \vert {L^{\perp}})}\,g_{ij}(L)}{f(L)},$$ where the distribution of the random line $L$ has density $f$ with respect to $\nu^2_1$. By [@Asmussen Chapter 5, Theorem 1.2] the second moment of this variable is minimized, when $f$ is proportional to $\sqrt{2RV_1(K \vert {L^{\perp}})}\, |g_{ij}(L)|$. Since the proof of [@Asmussen Chapter 5, Theorem 1.2] follows simply by an application of Jensen’s inequality to the function $t \mapsto t^2$, equality can be characterized due to the strict convexity of this function, (see, e.g., [@Gardner (B.4)]). Equality holds if and only if $\sqrt{2RV_1(K \vert {L^{\perp}})}\,|g_{ij}(L)|$ is a constant multiple of $f(L)$ (or equivalently $f = f_K^*$) almost surely.
The proof of Lemma \[lemmaopt\] generalizes directly to arbitrary dimension $n$. As a consequence of Lemma \[lemmaopt\], we obtain that for any convex body $K \in {{\mathcal K}}^2$, optimal non-isotropic sampling provides a strictly smaller variance of the estimator than isotropic sampling. Indeed, noting that with a constant function $f$ reduces to the usual estimator (with $n=2$, $A=RB^2$) based on *IUR* lines, this follows from the fact that $f^*_K$ cannot be constant. If $f^*_K$ was constant almost surely, then $V_1(K \vert u^\perp) \propto |g_{ij}([u])|^{-2} $ for almost all $u \in S^1$. The left-hand side is essentially bounded, whereas the right-hand side is not. This is a contradiction.
A further consequence of Lemma \[lemmaopt\] is that there does not exist an estimator of the form independent of $K$ that has uniformly minimal variance for all $K \in {{\mathcal K}}^2$ with $\dim K \geq 1$. Unfortunately, $ f_K^* $ is not accessible, as it depends on $ K $, which is typically unknown. Even though estimators of the form cannot have uniformly minimal variance for all $K \in {{\mathcal K}}^2$ with $\dim K \geq 1$, we now show that there is a non-isotropic sampling design which always yields smaller variance than the isotropic sampling design. Let $$f^*(L) \propto |g_{ij}(L)|$$ be a density with respect to $\nu^2_1$. As $ |g_{ij}(L)| $ is bounded and non-vanishing for $ \nu^2_1$-almost all $ L $, $ f^* $ is well-defined and non-zero $ \nu^2_1$-almost everywhere. For convex bodies of constant width, the density $f^*$ coincides with the optimal density $f^*_K$.
\[onecomponent\] Let $ K \in {{\mathcal K}}^2 $, and let $ A = RB^2 $ for some $ R>0 $ be such that $ K \subseteq A $. Then $$\label{varineq}
\text{Var}_{f^*}\bigg(\frac{{\hat{\varphi}_{ij}(K \cap E)}}{f^*({\pi(E)})}\bigg)
<
\text{Var}_{IUR}\big({\hat{\varphi}_{ij}(K \cap E)}\big).$$
Using the fact that both estimators are unbiased, it is sufficient to show that there is a $0 < \lambda < 1$ with $$\label{Eineq}
{\mathbb{E}}_{f^*}\bigg(\frac{{\hat{\varphi}_{ij}(K \cap E)}}{f^*({\pi(E)})}\bigg)^2 \leq
\lambda \, {\mathbb{E}}_{IUR}\big({\hat{\varphi}_{ij}(K \cap E)}\big)^2,$$ for all $K \in {{\mathcal K}}^2$. Using , the left-hand side of this inequality is $$2R \int_{{\mathcal{L}}^2_1}|g_{ij}(L)| \, \nu_1^2 (dL) \int_{{\mathcal{L}}^2_1} |g_{ij}(L)|V_{1}(K \vert L^\perp) \, \nu_1^2 (dL)$$ and the right-hand side is $$2R \int_{{\mathcal{L}}^2_1} g_{ij}(L)^2 \, V_{1}(K \vert L^\perp) \, \nu_1^2 (dL).$$
Since $u \mapsto V_1(K \vert u^{\perp})$ is the support function of an origin-symmetric zonoid, the inequality holds if $$\begin{aligned}
\label{support ineq}
& \int_0^{2 \pi}|g_{ij}([u_{\phi}])| \, \frac{d\phi}{2\pi} \, \int_0^{2\pi} |g_{ij}([u_{\phi}])|\, h(Z,u_{\phi}) \, \frac{d\phi}{2\pi} \nonumber
\\
& \quad \leq \lambda \, \int_0^{2 \pi} g_{ij}([u_{\phi}])^2 h(Z,u_{\phi}) \, \frac{d\phi}{2\pi}\end{aligned}$$ for any origin-symmetric zonoid $ Z $. Here $ u_{\phi} = (\cos(\phi),\sin(\phi))^\top $ for $ \phi \in [0,2\pi] $. As support functions of zonoids can be uniformly approximated by support functions of zonotopes (see, e.g., [@Schneider93 Theorem 1.8.14]) and the integrals in depend linearly on these support functions, it is sufficient to show for all origin-symmetric line segments $Z$ of length two. Hence, we may assume that $Z$ is an origin-symmetric line segment with endpoints $\pm(\cos(\gamma), \sin(\gamma))^\top$, where $\gamma \in [0,\pi)$. We now substitute the support function $$h(Z,u_{\phi})=|\cos(\phi-\gamma)|$$ for $\phi \in [0,2\pi),$ into .
First, we consider the estimation of the first diagonal element of $\Phi_{1,0,2}(K)$, that is, $i,j=1$ and $g_{ij}([u_\phi])= \frac{3}{8}(\cos^2(\phi)-\frac{1}{3}) $ for $ \phi \in [0,2\pi] $. The integrals in then become $$P_{f^*}(\gamma):= \frac{3}{8}\int_0^{2\pi}|\cos^2(\phi)-\frac{1}{3}| \, \frac{d\phi}{2 \pi} \, \frac{3}{8} \int_{0}^{2\pi}|\cos^2(\phi)-\frac{1}{3}| |\cos(\phi-\gamma)| \, \frac{d\phi}{2\pi}$$ and $$P_{IUR}(\gamma):=\frac{9}{64}\int_0^{2\pi}\bigg(\cos^2(\phi)-\frac{1}{3}\bigg)^2 |\cos(\phi- \gamma)| \, \frac{d\phi}{2 \pi}.$$ Let $ \kappa = \arccos(\frac{1}{\sqrt{3}}) $. Then $$M:= \frac{3}{8}\int_0^{2\pi}|\cos^2(\phi)-\frac{1}{3}| \, \frac{d\phi}{2 \pi}
=\frac{\sqrt{2}+\kappa}{4\pi}-\frac{1}{16},$$ and elementary, but tedious calculations show that $$\begin{aligned}
P_{f^*}(\gamma)&=\frac{M}{ \pi} \bigg(\frac{2\sqrt{2}}{3\sqrt{3}}\cos(\gamma) - \frac{1}{4}\cos^2(\gamma) \bigg){\textbf{1}}_{[0,\frac{\pi}{2}-\kappa]}(\gamma)
\\
&\qquad +
\frac{M}{\pi}\bigg(\frac{1}{4}\cos^2(\gamma) + \frac{1}{3\sqrt{3}}\sin(\gamma) \bigg){\textbf{1}}_{(\frac{\pi}{2}-\kappa,\frac{\pi}{2}]}(\gamma)\end{aligned}$$ for $ \gamma \in [0,\frac{\pi}{2}] $. Further, $P_{f^*}(\gamma)=P_{f^*}(\pi-\gamma) $ for $\gamma \in [\frac{\pi}{2},\pi] $. For the *IUR* estimator we get that $$P_{IUR}(\gamma)=\frac{1}{20\pi}\bigg(-\frac{3}{8}\cos^4(\gamma) + \cos^2(\gamma) + \frac{1}{2} \bigg)$$ for $ \gamma \in [0,\frac{\pi}{2}] $, and $ P_{IUR}(\gamma)=P_{IUR}(\pi -\gamma) $ for $ \gamma \in [\frac{\pi}{2},\pi] $. The functions $ P_{f^*} $ and $ P_{IUR} $ are plotted in Figure \[Secondmoments\]. Basic calculus for the comparison of these two functions shows that $P_{f^*} < P_{IUR}$. This implies that $P_{f^*} \leq \lambda P_{IUR}$, where $\nobreak{\lambda=\max_{\gamma \in [0,\pi]} \frac{P_{f^*(\gamma)}}{P_{IUR}(\gamma)}}$ is smaller than one as $P_{f^*}$ and $P_{IUR}$ are continuous on the compact interval $[0,\pi]$. Hereby is satisfied for $i=j=1$. Interchanging the roles of the coordinate axes in yields the same result for $i=j=2$.
We now consider estimation of the off-diagonal element, that is, $i=1$, $j=2$. Then the left-hand and the right-hand side of become $$\label{Qfs}
Q_{f^*}(\gamma)=\frac{3}{8} \int_0^{2\pi}|\cos(\phi)\sin(\phi)| \, \frac{d\phi}{2 \pi}
\,\frac{3}{8} \int_{0}^{2\pi} |\cos(\phi)\sin(\phi)| |\cos(\phi-\gamma)| \, \frac{d\phi}{2 \pi}$$ and $$\label{M}
Q_{IUR}(\gamma)=\frac{9}{64}\int_0^{2\pi}\cos^2(\phi)\sin^2(\phi) |\cos(\phi- \gamma)| \,\frac{d\phi}{2 \pi}$$ for $ \gamma \in [0,\pi] $. We have $$\frac{3}{8} \int_0^{2\pi}|\cos(\phi)\sin(\phi)| \, \frac{d\phi}{2 \pi} = \frac{3}{8\pi},$$ and then $$Q_{f^*}(\gamma) = \frac{3}{32\pi^2}\bigg(\sin(\gamma)+\cos(\gamma)-\sin(\gamma)\cos(\gamma) \bigg)$$ for $ \gamma \in [0,\frac{\pi}{2}] $, and $ Q_{f^*}(\gamma)=Q_{f^*}(\gamma-\frac{\pi}{2}) $ for $ \gamma \in [\frac{\pi}{2},\pi] $. For $ \gamma \in [0,\pi]$ we further find that $$Q_{IUR}(\gamma) = \frac{3}{320 \pi}\bigg( 4-\frac{1}{2}\sin^2(2\gamma) \bigg).
$$ The functions $ Q_{IUR} $ and $ Q_{f^*} $ are plotted in Figure \[Secondmoments2\]. Basic calculus shows that $$\label{minmaxf}
\min_{0 \leq \gamma \leq \pi} Q_{f^*} = \frac{3}{32 \pi^2}\left(\sqrt{2}-\frac{1}{2}\right),
\qquad
\max_{0 \leq \gamma \leq \pi}Q_{f^*}=\frac{3}{32 \pi^2},$$ and $$\label{maxminIUR}
\min_{0 \leq \gamma \leq \pi} Q_{IUR} = \frac{21}{640 \pi},
\qquad
\max_{0 \leq \gamma \leq \pi} Q_{IUR} = \frac{3}{80 \pi}.$$ Hence $$Q_{f^*}(\gamma) \leq \frac{3}{32 \pi^2} \le \lambda \frac{21}{640 \pi} \leq \lambda \, Q_{IUR}(\gamma)$$ for $\gamma \in [0,\pi]$ with $\lambda=\frac{3}{\pi}< 1$. Hereby holds for all zonotopes $Z$ and $i=1,j=2$, and the claim is shown.
![The straight line is $ P_{IUR} $, the dashed line is $ P_{f^*} $, and the dash-dotted line is $ P_{opt} $.[]{data-label="Secondmoments"}](P.pdf)
![The straight line is $ Q_{IUR} $, the dashed line is $ Q_{f^*} $, and the dash-dotted line is $ Q_{opt} $.[]{data-label="Secondmoments2"}](Q.pdf)
If $ E $ is an $ f^* $-weighted random line suited for estimating one particular component of $ \Phi_{1,0,2}(K) $, then $ E $ should not be used to estimate any of the other components, as this would increase the variance of these estimators considerably. Hence, if we estimate all of the components of the tensor using the estimator based on $ f^* $-weighted lines, we need three lines; one for each component. If we want to compare this approach with an estimation procedure based on *IUR* lines, requiring the same workload, we will use *three* *IUR* lines. Note however, that all three *IUR* lines can be used to estimate *all* three components of the tensor. This implies that we should actually compare the variance of the estimator based on *one* $ f^* $-weighted random line with the variance of an estimator based on *three* *IUR* lines. It turns out that the estimator based on *three* independent *IUR* lines has always smaller variance, than the estimator based on *one* $ f$-weighted line, no matter how the density $f$ is chosen.
Let $ K \in {{\mathcal K}}^2 $, and let $ A =RB^2 $ with some $ R > 0 $ be such that $ K \subseteq A $. Let $ f $ be a density with respect to $ \nu^2_1 $, which is non-zero $ \nu^2_1 $-almost everywhere. Let $ E_1,E_2$ and $ E_3 $ be independent *IUR* lines in $ {{\mathbb R}}^2 $ hitting $ A $. Then $$Var\bigg(\frac{1}{3}\sum_{k=1}^3 \hat{\varphi}_{ij}(K \cap E_k) \bigg)
<
Var_{f}\bigg(\frac{{\hat{\varphi}_{ij}(K \cap E)}}{f(\pi(E))}\bigg)$$ for $i,j \in \{1,2\}$.
By Theorem \[onecomponent\], the variance of the estimator is bounded from below by the variance of the same estimator with $f=f_K^*$. Hence, it is sufficient to compare the second moments of $$\frac{1}{3}\sum_{k=1}^3 \hat{\varphi}_{ij}(K \cap E_k)$$ and with $f=f_K^*$. The latter is $$2R \bigg(\int_{{\mathcal{L}}^2_1} |g_{ij}(L)| \sqrt{V_1(K \vert L^\perp)} \, \nu^2_1 (dL) \bigg)^2,$$ so let $$P_{opt}(\gamma):= \bigg(\frac{3}{8} \int_0^{2\pi} |\cos^2(\phi)-\frac{1}{3}| \sqrt{|\cos(\phi - \gamma)|} \, \frac{d\phi}{2 \pi} \bigg)^2$$ and $$Q_{opt}(\gamma):= \bigg( \frac{3}{8} \int_0^{2\pi}|\cos(\phi)\sin(\phi)| \sqrt{|\cos(\phi - \gamma)|} \, \frac{d\phi}{2 \pi} \bigg)^2$$ for $ \gamma \in [0,\pi] $. Using the notation of the previous proofs, by , , and we have $$Q_{opt}(\gamma) \geq \bigg(\frac{8 \pi Q_{f^*}(\gamma)}{3}\bigg)^2 \geq \frac{9-4\sqrt{2}}{64 \pi^2} > \frac{1}{80\pi} \geq \frac{1}{3}Q_{IUR}(\gamma)$$ for $\gamma \in [0,\pi]$. Likewise, $ P_{opt}(\gamma) \geq \bigg(\frac{P_{f^*}(\gamma)}{M}\bigg)^2$. Elementary analysis shows that $$\min_{0\leq \gamma \leq \frac{\pi}{2}-\kappa}\bigg(\frac{P_{f^*}(\gamma)}{M}\bigg)^2=\frac{25}{324 \pi^2}
> \frac{3}{160\pi} = \max_{0\leq \gamma \leq \frac{\pi}{2} - \kappa} \frac{1}{3} P_{IUR}(\gamma),$$ and that $$\bigg(\frac{P_{f^*}(\gamma)}{M}\bigg)^2-\frac{1}{3}P_{IUR}(\gamma)
\geq
\bigg(\frac{P_{f^*}(\frac{\pi}{2})}{M}\bigg)^2-\frac{1}{3}P_{IUR}(\frac{\pi}{2}) > 0$$ on $[\frac{\pi}{2}-\kappa,\frac{\pi}{2}]$. Hence $P_{opt}>\frac{1}{3}P_{IUR}$ on $[0,\pi]$, and the assertion is proved.
This leads to the following conclusion: If one single component of the tensor $\T[2]$ is to be estimated for unknown $K$, the estimator with $f=f^*$ is recommended, as its variance is strictly smaller than the one from isotropic sampling (where $f$ is a constant). If all components are sought for, the estimator based on three *IUR* lines should be preferred.
Model based estimation {#SecModel}
======================
In this section we derive estimators of the specific surface tensors associated with a stationary process of convex particles based on linear sections. In [@RSRS06], Schneider and Schuster treat the similar problem of estimating the area moment tensor ($ s=2 $) associated with a stationary process of convex particles using planar sections.
Let $X$ be a stationary process of convex particles in ${{\mathbb R}}^n$ with locally finite (and non-zero) intensity measure, intensity $\gamma >0$ and grain distribution ${\mathbb{Q}}$ on ${{\mathcal K}}_0:=\{K \in {{\mathcal K}}^n \mid c(K)=0\}$; see, e.g., [@Weil] for further information on this basic model of stochastic geometry. Here $c \colon {{\mathcal K}}^n \setminus \{\emptyset\} \rightarrow {{\mathbb R}}^n$ is the center of the circumball of $K$. Since $ X $ is a stationary process of convex particles, the intrinsic volumes $V_0, \dots, V_n$ are $ {\mathbb{Q}}$-integrable by [@Weil Theorem 4.1.2]. For $j \in \{0, \dots, n-1\}$ and $s \in {{\mathbb N}}_0$ the tensor valuation $\Phi_{j,0,s}$ is measurable and translation invariant on ${{\mathcal K}}^n$, and since, by , $$|\jT[s](e_{i_1}, \dots, e_{i_s})| \leq \frac{\omega_{n-j}}{s!\omega_{n-j+s}}V_{j}(K),$$ it is coordinate-wise ${\mathbb{Q}}$-integrable. The *$j$th specific (translation invariant) tensor of rank s* can then be defined as $$\label{defmeansurface}
\joT[s]:=\gamma \int_{{{\mathcal K}}_0}\jT[s] \,{\mathbb{Q}}(dK)$$ for $j \in \{0, \dots, n-1\}$ and $s \in {{\mathbb N}}_0$. For $j=n-1$, the specific tensors are called the specific surface tensors. Notice that $\oT[2]=\frac{1}{8 \pi} \overline{T}(X)$, where $\overline{T}(X)$ is the mean area moment tensor described in [@RSRS06]. By [@Weil Theorem 4.1.3] the specific tensors of $X$ can be represented as $$\label{intuitive1}
\joT=\frac{1}{\lambda(B)} \; {\mathbb{E}}\, \sum_{\mathclap{\substack{K \in X \\ c(K)\in B}}} \, \jT[s],$$ where $B \in {\mathcal{B}}({{\mathbb R}}^n)$ with $0 < \lambda(B) < \infty$.
In the following we restrict to $j=n-1$ and discuss the estimation of $\oT[s]$ from linear sections of $X$. We assume from now on that $n\geq 2$. For $L \in {\mathcal{L}_1^n}$ we let $X \cap L :=\{K \cap L \mid K \in X, K \cap L \neq \emptyset\}$ be the stationary process of convex particles in $L$ induced by $X$. Let $\gamma_L$ and ${\mathbb{Q}}_L$ denote the intensity and the grain distribution of $ X \cap L $, respectively. The tensor valuation $\Phi_{0,0,s}^{(L)}$ is measurable and ${\mathbb{Q}}_L$-integrable on $ K_0^{(L)} :=\{K \in {{\mathcal K}}_0 \mid K \subseteq L\}$. We can thus define $$\oMT:=\gamma_L \int_{{{\mathcal K}}_0^{(L)}} \Phi_{0,0,s}^{(L)}(K) \, {\mathbb{Q}}_L (dK).$$ This deviates in the special case $\overline{T}^{(L)}(X \cap L) = 8 \pi \, \overline{\Phi}_{0,0,2}^{(L)}(X \cap L)$ from the definition in [@RSRS06] due to a misprint there. An application of yields, $$\label{simpel form, model}
\oMT= \frac{2}{s! \omega_{s+1}}Q(L)^{\frac{s}{2}} \gamma_L$$ for even $s$, and $\oMT=0$ for odd $s$.
\[modelthm\] Let $X$ be a stationary process of convex particles in ${{\mathbb R}}^n$ with positive intensity. If $s \in {{\mathbb N}}_0$ is even, then $$\label{model integral}
\int_{{\mathcal{L}_1^n}} \oMT \, \nu_1^n (dL)= \frac{2 \omega_{n+s+1}}{\pi s! \omega_{s+1}^2 \omega_n} \sum_{k=0}^{\frac{s}{2}}c_k^{(\frac{s}{2})}Q^{\frac{s}{2}-k}\,\oT[2k],$$ where the constants $c_k^{(\frac{s}{2})}$ for $k=0, \dots, \frac{s}{2}$ are given in Theorem \[crofton-like\].
Let $L \in {\mathcal{L}_1^n}$, and let $\gamma_L$ be the intensity of the stationary process $X \cap L$. If $B \subseteq L$ is a Borel set with $\lambda_L(B)=1$, then an application of Campbell’s theorem and Fubini’s theorem yields $$\begin{aligned}
\gamma_L&=
{\mathbb{E}}\; \sum_{\mathclap{\substack{K \in X \\ K \cap L \neq \emptyset}}} \; {\textbf{1}}(c(K \cap L) \in B)
\\
&=\gamma \int_{{{\mathcal K}}_0} \int_{{L^{\perp}}} V_0(K \cap (L + x)) \, \lambda_{{L^{\perp}}}(dx) \, {\mathbb{Q}}(dK),\end{aligned}$$ where $\gamma$ and ${\mathbb{Q}}$ are the intensity and the grain distribution of $X$. Then, implies that $$\oMT=\gamma \int_{{{\mathcal K}}_0} \int_{{L^{\perp}}} \Phi_{0,0,s}^{(L+z)}(K \cap (L+z)) \, \lambda_{{L^{\perp}}}(dz) \,{\mathbb{Q}}(dK),$$ and by Fubini’s theorem we get $$\begin{aligned}
\int_{{\mathcal{L}_1^n}} \oMT \, \nu^n_1(dL)
=\gamma \int_{{{\mathcal K}}_0} \int_{{\mathcal{E}^n}_1} \Phi^{(E)}_{0,0,s}(K \cap E) \,\mu_1^n(dE)\, {\mathbb{Q}}(dK). \label{Eq}\end{aligned}$$ Now Theorem \[crofton-like\] yields the stated integral formula .
A combination of equation and equation immediately gives the following Theorem \[model2\], which suggests an estimation procedure of the specific surface tensor $\oT$ of the stationary particle process $X$.
\[model2\] Let $X$ be a stationary process of convex particles in ${{\mathbb R}}^n$ with positive intensity. If $s \in {{\mathbb N}}_0$ is even, then $$\label{inversmodel}
\int_{{\mathcal{L}_1^n}} \sum_{j=0}^{\frac{s}{2}} d_{\frac{s}{2}\,j}C_{2j} Q^{\frac{s}{2}-j} \oMT[2j] \, \nu^n_1(dL)= \oT,$$ where $d_{\frac{s}{2}\,j}$ and $ C_{2j} $ for $ j=0, \dots, \frac{s}{2} $ are given before Theorem \[crofton2\].
Using , we can reformulate the integral formula in the form $$\int_{{\mathcal{L}_1^n}} G_s(L) \gamma_L \, \nu^n_1(dL)= \oT,$$ where $G_s$ is given in Theorem \[crofton2\].
In the case where $ s=2 $ formula becomes $$\int_{{\mathcal{L}}_1^n} \frac{2 \pi^2\omega_n}{\omega_{n+3}}\oMT[2]-\frac{\omega_n}{4 \omega_{n+1}}Q \oMT[0] \, \nu_1^n(dL)=\oT[2].$$ Up to a normalizing factor $ 2 \pi $ in the constant in front of $ \overline{\Phi}_{0,0,2}^{(L)} $, this formula coincides with formula (7) in [@RSRS06], when $ n=2 $. Apparently the normalizing factor got lost, when Schneider and Schuster used [@TVCB (36)], which is based on the spherical Lebesgue measure. In [@RSRS06], Schneider and Schuster use the *normalized* spherical Lebesgue measure.
**Acknowledgements** The authors acknowledge support by the German research foundation (DFG) through the research group “Geometry and Physics of Spatial Random Systems” under grants HU1874/2-1, HU1874/2-2 and by the Centre for Stochastic Geometry and Advanced Bioimaging, funded by a grant from The Villum Foundation.
[^1]: E-mail: kousholt@imf.au.dk
| {
"pile_set_name": "ArXiv"
} |
---
author:
- Christopher Broadbent
- Arnaud Carayol
- 'Matthew Hague[^1]'
- Olivier Serre
bibliography:
- 'references.bib'
title: ' Emptiness of Stack Automata is NEXPTIME-complete: A Correction [^2] '
---
[^1]: Supported by EPSRC \[EP/K009907/1\].
[^2]: We thank an anonymous reviewer for pointing out the error in the original PSPACE algorithm.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Suppose that $E$ and $E''$ denote real Banach spaces with dimension at least $2$ and that $D\varsubsetneq E$ and $D''\varsubsetneq E''$ are uniform domains with homogeneously dense boundaries. We consider the class of all $\varphi$-FQC (freely $\varphi$-quasiconformal) maps of $D$ onto $D''$ with bilipschitz boundary values. We show that the maps of this class are $\eta$-quasisymmetric. As an application, we show that if $D$ is bounded, then maps of this class satisfy a two sided Hölder condition. Moreover, replacing the class $\varphi$-FQC by the smaller class of $M$-QH maps, we show that $M$-QH maps with bilipschitz boundary values are bilipschitz. Finally, we show that if $f$ is a $\varphi$-FQC map which maps $D$ onto itself with identity boundary values, then there is a constant $C\,,$ depending only on the function $\varphi\,,$ such that for all $x\in D$, the quasihyperbolic distance satisfies $k_D(x,f(x))\leq C$.'
address:
- 'Yaxiang. Li, Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, People’s Republic of China'
- 'Matti. Vuorinen, Department of Mathematics and Statistics, University of Turku, FIN-20014 Turku, Finland'
- 'Xiantao. Wang, Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, People’s Republic of China'
author:
- 'Y. Li'
- 'M. Vuorinen'
- 'X. Wang ${}^{~\mathbf{*}}$'
title: Quasiconformal maps with bilipschitz or identity boundary values in Banach spaces
---
Introduction and main results {#intro}
=============================
Many results of classical function theory have their counterparts in the context of quasiconformal maps in the Euclidean $n$-dimensional space $\mathbb{R}^n$. J. Väisälä [@Vai6-0; @Vai6; @Vai5] has developed a theory of quasiconformality in the Banach space case which differs from the finite dimensional theory in many respects because tools such as conformal invariants and measures of sets are no longer available. These classical tools are replaced by fundamental objects from metric space geometry such as curves, their lengths, and approximately length minimizing curves. Väisälä used these notions in the setup of several metric space structures on the same underlying Banach space and developed effective methods based on these basic notions. In addition to the norm metric he considered two hyperbolic type metric structures, the quasihyperbolic metric and the distance ratio metric. The quasihyperbolic metric $k_D$ of a domain $D$ has a key role as quasiconformality is defined in terms of it in the Banach space case. Only recently some basic properties of quasihyperbolic metric have been studied: the convexity of quasihyperbolic balls was studied by R. Klén [@k; @k2], A. Rasila and J. Talponen [@rt; @krt], Väisälä [@Vai6']. Rasila and Talponen also proved the smoothness of quasihyperbolic geodesics in [@rt2] applying now stochastic methods.
Given domains $D, D'$ in Banach spaces $E$ and $E'$, respectively, our basic problem is to study the class of homeomorphisms $f\in QC^L_{\varphi}(D,D')$, where $$\label{intro-eq-1}
QC^L_{\varphi}(D,D') = \{f: \overline{D}\to \overline{D}'\; { \rm homeo}\;\Big| f|_{D} \;{\rm is}\; {\rm a}\;\varphi{\rm -FQC}\;{\rm map}\; {\rm and}\; f|_{\partial D}\; {\rm is}\; L{\rm -bilipschitz}\}\, .$$ For the definition of $\varphi$-FQC and $L$-bilipschitz maps see Section \[sec-2\]. The class $QC^L_{\varphi}(D,D')$ is very wide and many particular cases of interest are obtained by choosing $D, D', \varphi, L$ in a suitable way as we will see below.
Our first result deals with the case when both $D$ and $D'$ are uniform domains. In this case we prove that the class consists of quasisymmetric maps. More precisely, we prove the following theorem.
\[thm1.2\] Let $D\subsetneq E$, $D'\subsetneq E'$ be $c$-uniform domains. If $f\in QC^L_{\varphi}(D,D')$, then $f$ is $\eta$-QS in $\overline{D}$ with $\eta$ depending on $c$, $L$ and $\varphi$ only.
Applying this result to the case of a bounded domain $D$ we obtain the second result. Recall that in the case of ${\mathbb R}^n$ results of this type have been proved by R. Näkki and B. Palka [@np]. For the definitions, see Section \[sec-2\].
\[thm1.3\] Let $D\subsetneq E$, $D'\subsetneq E'$ be $c$-uniform domains. If $f\in QC^L_{\varphi}(D,D')$ and $D$ is bounded, then for all $x,y\in D$, $$\frac{|x-y|^{1/{\alpha}}}{C}\leq|f(x)-f(y)|\leq C|x-y|^{\alpha},$$ where $C\geq 1$ and $\alpha\in(0,1)$ depend on $c$, $L$, $\varphi$ and $\operatorname{diam}(D)$.
Our third result concerns the case when both $D$ and $D'$ are uniform domains and $\varphi(t)=Mt$ for some fixed $M \ge 1\,.$ We also require a density condition of the boundary of a domain. This $(r_1,r_2)$-HD condition will be defined in Section 2.
\[thm1.4\] Let $D\subsetneq E$, $D'\subsetneq E'$ be $c$-uniform domains and the boundary of $D$ be $(r_1,r_2)$-HD. If $f\in QC^L_{\varphi}(D,D')$ with $\varphi(t)=Mt$, then $f$ is $M'$-bilipschitz in $\overline{D}$, where $M'$ depends only on $c$, $r_1$, $r_2$, $L$ and $M$.
Our fourth result deals with the case when $D=D'$, $L=1$ and, moreover, the boundary mapping $f|_{\partial D}:\partial D\to\partial D$ is the identity. This problem has been studied very recently in [@MV; @M2; @VZ]. Originally, the problem was motivated by Teichmüller’s work on plane quasiconformal maps [@K; @T] and then extended to the higher dimensional case by several authors: [@AV], [@M2], [@MV; @VZ]. Our result is as follows.
\[thm1.1\] Let $D\subsetneq E$ be a $c$-uniform domain with $(r_1,r_2)$-HD boundary. If $f$ is a $\varphi$-FQC map which maps $D$ onto itself with identity boundary values, then for all $x\in D$, $$k_D(x, f(x))\leq C,$$ where $C$ is a constant depending on $r_1$, $r_2$, $c$ and $\varphi$ only.
For the case $n=2$, when $D$ is the unit disk, the sharp bound is due to Teichmüller [@K; @T]. For the case of unit ball in $\mathbb{R}^n, n\ge 2,$ nearly sharp results appear in [@MV; @VZ]. In both of these cases one uses the hyperbolic metric in place of the quasihyperbolic metric.
We do not know whether there are sharp results for the Banach spaces, too. For instance, it is an open problem whether Theorem \[thm1.1\] could be refined for the case $D=\mathbb{B}$, the unit ball, to the effect that $C\rightarrow 0$ when $\varphi$ approaches the identity map.
The organization of this paper is as follows. In Section \[sec-4\], we will prove Theorems \[thm1.2\], \[thm1.3\], \[thm1.4\] and \[thm1.1\]. In Section \[sec-2\], some preliminaries are stated.
Preliminaries {#sec-2}
=============
We adopt mostly the standard notation and terminology from Väisälä [@Vai6-0; @Vai5]. We always use $E$ and $E'$ to denote real Banach spaces with dimension at least $2$. The norm of a vector $z$ in $E$ is written as $|z|$, and for every pair of points $z_1$, $z_2$ in $E$, the distance between them is denoted by $|z_1-z_2|$, the closed line segment with endpoints $z_1$ and $z_2$ by $[z_1,
z_2]$. Moreover, we use $\mathbb{B}(x, r)$ to denote the ball with center $x\in E$ and radius $r$ $(> 0)$, and its boundary and closure are denoted by $\mathbb{S}(x,\; r)$ and $\overline{\mathbb{B}}(x,\; r)$, respectively. In particular, we use $\mathbb{B}$ to denote the unit ball $\mathbb{B}(0,\; 1)$. The one-point extension of $E$ is the Hausdorff space $\dot{E}=E\cup\{\infty\}$, where the neighborhoods of $\infty$ are the complements of closed bounded sets of $E$. The boundary $\partial A$ and the closure $\overline{A}$ of a set $A\subset E$ are taken in $\dot{E}$.
The [*quasihyperbolic length*]{} of a rectifiable arc or a path $\alpha$ in the norm metric in $D$ is the number (cf. [@Geo; @GP; @Vai6-0]):
$$\ell_k(\alpha)=\int_{\alpha}\frac{|dz|}{d_{D}(z)},$$ where $d_D(z)$ denotes the distance from $z$ to the boundary $\partial D$ of $D$.
For each pair of points $z_1$, $z_2$ in $D$, the [*quasihyperbolic distance*]{} $k_D(z_1,z_2)$ between $z_1$ and $z_2$ is defined in the usual way: $$k_D(z_1,z_2)=\inf\ell_k(\alpha),$$ where the infimum is taken over all rectifiable arcs $\alpha$ joining $z_1$ to $z_2$ in $D$.
For each pair of points $z_1$, $z_2$ in $D$, the [*distance ratio metric*]{} $j_D(z_1,z_2)$ between $z_1$ and $z_2$ is defined by $$j_D(z_1,z_2)=\log\Big(1+\frac{|z_1-z_2|}{\min\{d_D(z_1),d_D(z_2)\}}\Big).$$
For all $z_1$, $z_2$ in $D$, we have (cf. [@Vai6-0])
$$\label{eq(0000)} k_{D}(z_1, z_2)\geq
\inf\left\{\log\Big(1+\frac{\ell(\alpha)}{\min\{d_{D}(z_1), d_{D}(z_2)\}}\Big)\right\}\geq j_D(z_1, z_2)$$
$$\geq
\Big|\log \frac{d_{D}(z_2)}{d_{D}(z_1)}\Big|,$$ where the infimum is taken over all rectifiable curves $\alpha$ in $D$ connecting $z_1$ and $z_2$. Moreover, if $|z_1-z_2|\le d_D(z_1)$, we have [@Vai6-0; @Mvo1] $$\label{vu1}
k_D(z_1,z_2)\le \log\Big( 1+ \frac{
|z_1-z_2|}{d_D(z_1)-|z_1-z_2|}\Big).$$
Gehring and Palka [@GP] introduced the quasihyperbolic metric of a domain in $\mathbb{R}^n$ and it has been recently used by many authors in the study of quasiconformal mappings and related questions [@HIMPS; @krt; @rt] etc.
A domain $D$ in $E$ is called $c$-[*uniform*]{} in the norm metric provided there exists a constant $c$ with the property that each pair of points $z_{1},z_{2}$ in $D$ can be joined by a rectifiable arc $\alpha$ in $ D$ satisfying (see [@Martio-80; @Vai; @Vai4])
1. \[wx-4\] ${\displaystyle}\min_{j=1,2}\ell (\alpha [z_j, z])\leq c\,d_{D}(z)
$ for all $z\in \alpha$, and
2. \[wx-5\] $\ell(\alpha)\leq c\,|z_{1}-z_{2}|$,
where $\ell(\alpha)$ denotes the length of $\alpha$ and $\alpha[z_{j},z]$ the part of $\alpha$ between $z_{j}$ and $z$. Moreover, $\alpha$ is said to be a [*uniform arc*]{}.
In [@Vai6], Väisälä characterized uniform domains as follows.
\[pre-lem-1\] For a domain $D$, the following are quantitatively equivalent:
1. $D$ is a $c$-uniform domain;
2. $k_D(z_1,z_2)\leq c'\; j_D(z_1,z_2)$ for all $z_1,z_2\in D$;
3. $k_D(z_1,z_2)\leq c'_1\; j_D(z_1,z_2) +d$ for all $z_1,z_2\in D$.
In the case of domains in $ {\mathbb R}^n \,,$ the equivalence of items (1) and (3) in Theorem D is due to Gehring and Osgood [@Geo] and the equivalence of items (2) and (3) due to Vuorinen [@Mvo1]. Many of the basic properties of this metric may be found in [@Geo; @krt; @rt; @Vai6-0; @Vai6].
In [@Vai5], Väisälä proved the following examples for some special uniform domain.
\(1) Each ball $B\subset E$ is $2$-uniform;
\(2) Every bounded convex domain $G\subset E$ is uniform;
\(3) Half space $H\subset E$ is $c$-uniform for all $c>2$.
Suppose $G\varsubsetneq E\,,$ $G'\varsubsetneq
E'\,,$ and $M \ge 1\,.$ We say that a homeomorphism $f: G\to G'$ is [*$M$-bilipschitz*]{} if $$|x-y|/M \leq |f(x)-f(y)|\leq M\,|x-y|$$ for all $x$, $y\in G$, and [*$M$-QH*]{} if $$k_{G}(x,y)/M\leq k_{G'}(f(x),f(y))\leq M\,k_{G}(x,y)$$ for all $x$, $y\in G$.
Clearly, if $f$ is $M$-bilipschitz or $M$-QH, then also $f^{-1}$ has the same property.
Let $G\not=E$ and $G'\not=E'$ be metric spaces, and let $\varphi:[0,\infty)\to [0,\infty)$ be a growth function, that is, a homeomorphism with $\varphi(t)\geq t$. We say that a homeomorphism $f: G\to G'$ is [*$\varphi$-semisolid*]{} if $$k_{G'}(f(x),f(y))\leq \varphi(k_{G}(x,y))$$ for all $x$, $y\in G$, and [*$\varphi$-solid*]{} if both $f$ and $f^{-1}$ satisfy this condition.
We say that $f$ is [*fully $\varphi$-semisolid*]{} (resp. [*fully $\varphi$-solid*]{}) if $f$ is $\varphi$-semisolid (resp. $\varphi$-solid) on every subdomain of $G$. In particular, when $G=E$, the corresponding subdomains are taken to be proper ones. Fully $\varphi$-solid maps are also called [*freely $\varphi$-quasiconformal maps*]{}, or briefly [*$\varphi$-FQC maps*]{}.
Clearly, if $f$ is freely $\varphi$-quasiconformal, then so is $f^{-1}\,.$
If $E=\mathbb{R}^n=E'$, then $f$ is $FQC$ if and only if $f$ is quasiconformal (cf. [@Vai6-0]). See [@Vai1; @Mvo1] for definitions and properties of $K$-quasiconformal maps, or briefly $K$-QC maps.
Let $X$ be a metric space and $\dot{X}=X\cup \{\infty\}$. By a triple in $X$ we mean an ordered sequence $T=(x,a,b)$ of three distinct points in $X$. The ratio of $T$ is the number $$\rho(T)=\frac{|a-x|}{|b-x|}.$$ If $f: X\to Y$ is an injective map, the image of a triple $T=(x,a,b)$ is the triple $fT=(fx,fa,fb)$.
Let $X$ and $Y$ be two metric spaces, and let $\eta: [0, \infty)\to [0, \infty)$ be a homeomorphism. An embedding $f: X\to Y$ is said to be [*$\eta$-quasisymmetric*]{}, or briefly $\eta$-$QS$, if $\rho(f(T))\leq \eta(\rho(T))$ for each triple $T$ in $X$.
It is known that an embedding $f: X\to Y$ is $\eta$-$QS$ if and only if $\rho(T)\leq t$ implies that $\rho(f(T))\leq \eta(t)$ for each triple $T$ in $X$ and $t\geq 0$ (cf. [@TV]).
A quadruple in $X$ is an ordered sequence $Q=(a,b,c,d)$ of four distinct points in $X$. The cross ratio of $Q$ is defined to be the number $$\tau(Q)=|a,b,c,d|=\frac{|a-b|}{|a-c|}\cdot\frac{|c-d|}{|b-d|}.$$ Observe that the definition is extended in the well known manner to the case where one of the points is $\infty$. For example, $$|a,b,c,\infty|= \frac{|a-b|}{|a-c|}.$$ If $X_0 \subset \dot{X}$ and if $f: X_0\to \dot{Y}$ is an injective map, the image of a quadruple $Q$ in $X_0$ is the quadruple $fQ=(fa,fb,fc,fd)$.
Let $X$ and $Y$ be two metric spaces and let $\eta: [0, \infty)\to [0, \infty)$ be a homeomorphism. An embedding $f: X\to Y$ is said to be [*$\eta$-quasimöbius*]{} (cf. [@Vai2]), or briefly $\eta$-$QM$, if the inequality $\tau(f(Q))\leq \eta(\tau(Q))$ holds for each quadruple $Q$ in $X$.
Observe that if $\infty\in X$ and if $f:X\to Y$ is $\eta$-quasimöbius with $f(\infty)=\infty$, then $f$ is $\eta$-quasisymmetric (see [@Vai5 6.18]). Conversely, the following result holds.
\[pre-lem-2\]$($[@Vai2 Theorem 3.12]$)$ Suppose that $X$ and $Y$ are bounded spaces, that $\lambda>0$, that $z_1,z_2,z_3\in X$, and that $f:X \to Y$ is $\theta$-quasimöbius such that $$|z_i-z_j|\geq \operatorname{diam}(X)/{\lambda}\, \mbox{,}\, |f(z_i)-f(z_j)|\geq\operatorname{diam}(Y)/{\lambda}$$ for $i\neq j$. Then $f$ is $\eta$-quasisymmetric with $\eta=\eta_{\theta,\lambda}$.
Concerning the relation between the class of uniform domains and quasimöbius maps, Väisälä proved the following result.
\[pre-lem-3\] Suppose that $D\varsubsetneq E$ and $D'\varsubsetneq E'$, that $D$ and $D'$ are $c$-uniform domain, and that $f:D\to D'$ is a $\varphi$-FQC map. Then $f$ extends to a homeomorphism $\overline{f}: \overline{D}\to
\overline{D}'$ and $\overline{f}$ is $\theta_1$-QM in $\overline{ D}$.
Finally we introduce the concept of homogeneous density from [@TV].
[([@TV Definition 3.8])]{} A space $X$ is said to be [*homogeneously dense*]{}, abbreviated HD, if there are numbers $r_1$, $r_2$ such that $0<r_1\leq r_2<1$ and such that for each pair of points $a,b\in X$ there is $x\in X$ satisfying the condition $$r_1|b-a|\leq|x-a|\leq r_2|b-a|.$$ We also say that $X$ is $(r_1,r_2)$-HD or simply $r$-HD, where $r=(r_1,r_2)$.
By the definition, obviously, a HD space has no isolated point. And for all $0<r_1\leq r_2<1$, every connected domain is $(r_1,r_2)$-HD, $[0,1]\cup [2,3]$ is $(\frac{1}{6},\frac{1}{4})$-HD (see [@TV]). Particularly, a finite union of connected nondegenerate sets (i.e. the set is not a point) is $(r_1,r_2)$-HD with some constants $0<r_1\leq r_2<1$. For a HD space, Tukia and Väisälä proved the following properties in [@TV].
\[pre-lem-4\][([@TV Lemma 3.9])]{} $(\textit{1})$ Let $X$ be $(r_1,r_2)$-HD and let $m$ be a positive integer. Then $X$ is $(r_1^m,r_2^m)$-HD.
$(\textit{2})$ Let $X$ be $r$-HD and let $f:X\to Y$ be $\eta$-QS. Then $fX$ is $\mu$-HD, where $\mu$ depends only on $\eta$ and $r$.
Moreover, we prove the following property.
\[lem-2-2\] Let $D\subsetneq E $ be a domain with $(r_1,r_2)$-HD boundary and let $x\in D$. Then for all $x_0\in \partial D$ with $|x-x_0|\leq 2d_D(x)$ there exists some point $x_1\in \partial D$ such that $$\label{eq-th-ll}\frac{1}{2}d_D(x)
\leq |x_0-x_1|\leq \big(2+\frac{17}{2r_1}\big) d_D(x).$$
By Lemma \[pre-lem-4\] we may assume that $0<r_1\leq r_2<\frac{1}{3}$. For example, if $r_2\geq \frac{1}{3}$, then there exists a positive integer $m$ such that $r_2^m<\frac{1}{3}$. In fact we can choose $m-1$ to be the integer part of $\log_{r_2}\frac{1}{3}$, and by Lemma \[pre-lem-4\] the $(r_1,r_2)$-HD property of $\partial D$ implies that $\partial D$ is $(r_1^m,r_2^m)$-HD with $r_2^m<\frac{1}{3}$.
For a given $x\in D$, let $x_0\in \partial D$ be such that $|x-x_0|\leq 2d_D(x)\,.$ We divide the proof into three cases.
[*Case I*]{}: $\partial D\subset \overline{\mathbb{\mathbb{B}}}\big(x,\frac{5}{2}d_D(x)\big)$.
Obviously, $D$ is bounded. Let $x_1\in \partial D$ be such that $|x_0-x_1|\geq \frac{1}{3}\operatorname{diam}(D)$. Then $$\frac{2}{3}d_D(x)\leq |x_0-x_1|\leq 5d_D(x),$$ which shows that $x_1$ is the desired point and satisfies .
[*Case II*]{}: $\partial D\cap \Big(\mathbb{B}\big(x, \frac{1}{r_1}d_D(x)\big)\setminus \overline{\mathbb{B}}\big(x,\frac{5}{2}d_D(x)\big)\Big)\neq \emptyset$.
Let $x_2\in \partial D\cap \mathbb{\mathbb{B}}\big(x, \frac{1}{r_1}d_D(x)\big)\setminus \overline{\mathbb{B}}\big(x,\frac{5}{2}d_D(x)\big)$. Then $$|x_0-x_2|\geq |x_2-x|-|x-x_0|\geq \frac{1}{2}d_D(x)$$ and $$|x_0-x_2|\leq |x_0-x|+|x-x_2|\leq \big(\frac{1}{r_1}+2\big)d_D(x).$$
Obviously, $x_2$ is the needed point.
[*Case III*]{}: $\partial D\cap \Big( \mathbb{B}\big(x, \frac{1}{r_1}d_D(x)\big)\setminus \overline{\mathbb{B}}\big(x,\frac{5}{2}d_D(x)\big)\Big)= \emptyset$.
Let $\omega=\partial D \cap (E\setminus \mathbb{B}\big(x, \frac{1}{r_1}d_D(x))\big)$ and $d_1$ denote the distance from $\omega$ to $\mathbb{B}\big(x, \frac{1}{r_1}d_D(x)\big)$, i.e., $d_1=d\Big(\omega, \mathbb{B}\big(x, \frac{1}{r_1}d_D(x)\big)\Big)$. If $d_1=0$, let $x_3\in \omega$ be such that $d(x_3, \mathbb{B}\big(x, \frac{1}{r_1}d_D(x))\big)\leq \frac{1}{2}d_D(x)$. Hence $$(\frac{1}{r_1}-2)d_D(x)\leq |x_0-x_3|\leq |x_0-x|+|x-x_3|\leq (\frac{1}{r_1}+\frac{5}{2})d_D(x).$$ So $x_3$ is the desired point.
On the other hand, if $d_1>0$, let $x_4\in \omega$ be such that $$\label{lem-2-eq1}d(x_4, \mathbb{B}\big(x, \frac{1}{r_1}d_D(x))\big)\leq \frac{3}{2}d_1.$$ We claim that the point $x_4$ satisfies . To see this, we first prove $$\label{eq-th-ss} d_1<\frac{5}{r_1}d_D(x).$$ Suppose on the contrary that $d_1\geq \frac{5}{r_1}d_D(x).$ Then by there exists some point $u\in \partial D$ such that $$\begin{aligned}
|u-x_0|&\geq& r_1|x_0-x_4|\geq r_1(|x_4-x|-|x-x_0|)\\&\geq& r_1\big(\frac{6}{r_1}-2\big)d_D(x)= (6-2r_1)d_D(x)\end{aligned}$$ and $$\begin{aligned}
|u-x_0|&\leq& r_2|x_0-x_4|\leq r_2(|x_0-x|+|x-x_4|)\\&\leq& r_2\big(2+\frac{1}{r_1}\big)d_D(x)+\frac{3r_2}{2}d_1\leq d_1,\end{aligned}$$ which shows that $u\in \partial D\cap \Big(\mathbb{B}\big(x, \frac{1}{r_1}d_D(x)\big)\setminus \overline{\mathbb{B}}\big(x,\frac{5}{2}d_D(x)\big)\Big)$. This is a contradiction. Hence holds.
By , we have $$\big(\frac{1}{r_1}-2\big)d_D(x)\leq|x_1-x_0|\leq \big(2+\frac{1}{r_1}\big)d_D(x)+\frac{3}{2}d_1\leq \big(2+\frac{17}{2r_1}\big)d_D(x).$$ Hence the point $x_4$ has the required properties, and so the proof of the lemma is complete.
The discussions in the case III also follows from [@H Lemma 11.7].
Proofs of Theorems \[thm1.2\], \[thm1.3\], \[thm1.4\] and \[thm1.1\] {#sec-4}
====================================================================
For convenience, in the following, we always assume that $x$, $y$, $z$, $\ldots$ denote points in $D$ and $x'$, $y'$, $z'$, $\ldots$ the images in $D'$ of $x$, $y$, $z$, $\ldots$ under $f$, respectively. We start with some known results that are necessary for the following proofs.
\[proof-lem-1\] Suppose that $x,y\in D\neq E$ and that either $|x-y|\leq \frac{1}{2}d_D(x)$ or $k_D(x,y)\leq 1$. Then $$\frac{1}{2}\frac{|x-y|}{d_D(x)}\leq k_D(x,y)\leq 2\frac{|x-y|}{d_D(x)}.$$
\[proof-lem-2\] Suppose that $X$ is connected, that $f:X\to Y$ is $\eta$-quasisymmetric, and that $A\subset X$ is bounded. Then $f|_A$ satisfies a two-sided Hölder condition $$|x-y|^{{1}/{\alpha}}/M\leq |fx-fy|\leq M|x-y|^{\alpha}\;\;\;\;\; for \; x,y \in A,$$ where $\alpha=\alpha(\eta)\leq1$ and $M=M(\eta, d(A),d(fA))\geq 1.$
Since $f:\partial D\to \partial D'$ is $L$-bilipschitz, we know that the boundedness of $D$ (resp. $D'$) implies the boundedness of $D'$ (resp. $D$). In fact, suppose on the contrary that $D$ is bounded and $D'$ is unbounded. Then let $w_1', w_2'\in \partial D'$ such that $|w_1'-w_2'|\geq 4L \operatorname{diam}(D)$. Then we have $$\operatorname{diam}(D)\geq \frac{1}{2}|w_1-w_2|\geq \frac{1}{2L}|w_1'-w_2'|\geq 2\operatorname{diam}(D),$$ which is a contradiction.
If $D$ is unbounded, then $\infty\in \partial D$, by auxiliary inversions we normalize the situation such that $f(\infty)=\infty.$ Hence by Lemma \[pre-lem-3\], $f$ is $\eta$-QS in $\overline{D}$ with $\eta$ depending on $c$, $L$ and $\varphi$.
In the following, we assume that $D$ is bounded. Then $$\label{thm-1-4}\frac{1}{4L}\operatorname{diam}(D)\leq\operatorname{diam}(D')\leq 4L\operatorname{diam}(D).$$ Let $z_1, z_2\in \partial D$ be such that $|z_1-z_2|\geq \frac{1}{2}\operatorname{diam}(D)$ and let $z_3'\in \partial D' $ be such that $$\min\{|z_1'-z_3'|,|z_2'-z_3'|\}\geq \frac{1}{6}\operatorname{diam}(D').$$ Then by , we have $$|z_1'-z_2'|\geq \frac{1}{L}|z_1-z_2|\geq \frac{1}{2}\operatorname{diam}(D)\geq \frac{1}{8L^2}\operatorname{diam}(D')$$ and $$\min\{|z_1-z_3|,|z_2-z_3|\}\geq\frac{1}{L}\min\{|z_1'-z_3'|,|z_2'-z_3'|\}\geq\frac{1}{24L^2}\operatorname{diam}(D),$$ which, in combination with Lemma \[pre-lem-2\] and Lemma \[pre-lem-3\], shows that $f$ is $\eta$-QS in $\overline{D}$ with $\eta$ depending on $c$, $L$ and $\varphi$.
The proof of Theorem \[thm1.3\] easily follows from Theorem \[thm1.2\] and Lemma \[proof-lem-2\].
In the remaining part of this paper, we always assume that $D$ and $D'$ are $c$-uniform subdomains in $E$ and $E'$, respectively, that the boundary of $D$ is $(r_1,r_2)$-homogeneously dense, that $f: D\to D'$ is a $\varphi$-FQC map, and that $f$ extends to a homeomorphism $\overline{f}:
\overline{D}\to \overline{D'}$ such that $\overline{f}:\partial
D\to \partial D'$ is $L$-bilipschitz.
We first show that the following lemma holds.
\[lem-1\] There is a constant $M_1=M_1(c,L,\varphi,r_1,r_2)$ such that for given $x\in D$ the following hold:\
$(1)$ For $x_0\in \partial D$ with $|x-x_0|\leq 2d_D(x)$, we have $$|x_0'-x'|\leq
M_1d_{D}(x).$$ $(2)$ For all $x_1\in \partial D$, we have $$\label{eq-lem-ls}\frac{1}{2(2L+M_1)}|x_1-x|\leq|x'_1-x'|\leq 2(2L+M_1)|x_1-x|.$$
We first prove $(1)$.
For a fixed $x\in D$, let $x_0\in \partial D$ be such that $|x-x_0|\leq 2d_D(x)$. Let $x_2$ be the intersection point of $\mathbb{S}(x, \frac{1}{2}d_{D}(x))$ with $[x_0, x]$. Then by we have $$k_{D}(x_2,x)\leq
\log\Big(1+\frac{|x-x_2|}{d_{D}(x)-|x-x_2|}\Big)=\log 2,$$ which implies that
$$\log\frac{|x'_2-x'|}{|x'_2-x'_0|}\leq k_{D'}(x'_2,x')\leq \varphi(k_{D}(x_2,x))=\varphi(\log 2).$$ Hence $$\label{lem-1-0}|x'_2-x'|\leq e^{\varphi(\log 2)} |x'_2-x'_0|,$$ and so $$\label{lem-1-1}|x'-x'_0|\leq|x'-x'_2|+|x'_2-x'_0|
\leq(e^{\varphi(\log 2)}+1)|x'_2-x'_0|.$$
Since $\partial D$ is $(r_1,r_2)$-HD, we see from Lemma \[lem-2-2\] that there must exist some point $x_3\in \partial D$ such that $$\label{lem-1-16}\frac{1}{2}d_D(x)\leq |x_3-x_0|\leq \big(2+\frac{17}{2r_1}\big)d_D(x).$$ Hence
$$\label{lem-1-6'}|x-x_3|\leq |x-x_0|+|x_0-x_3|\leq
\big(4+\frac{17}{2r_1}\big)d_D(x)$$
and $$\label{lem-1-6}\frac{1}{2L}d_{D}(x)\leq
\frac{1}{L}|x_3-x_0|\leq |x'_3-x'_0|\leq L|x_3-x_0|\leq L\big(2+\frac{17}{2r_1}\big)d_{D}(x).$$ By Lemma \[pre-lem-3\] we see that $f^{-1}$ is $\theta$-quasimöbius in $\overline{D}$, where $\theta=\theta(c,\varphi)$. It follows from (\[lem-1-0\]), (\[lem-1-16\]), (\[lem-1-6’\]) and (\[lem-1-6\]) that $$\begin{aligned}
\frac{1}{6\big(4+\frac{17}{2r_1}\big)}&\leq&\frac{|x_3-x_0|}{|x_2-x_0|}\cdot\frac{|x_2-x|}{|x-x_3|}\leq
\theta
\Big(\frac{|x'_3-x'_0|}{|x'_2-x'_0|}\cdot\frac{|x'_2-x'|}{|x'-x'_3|}\Big)\\&\leq&
\theta\Big(\frac{L\big(2+\frac{17}{2r_1}\big)e^{\varphi(\log 2)}d_D(x)}{|x'-x'_3|}\Big) ,\end{aligned}$$ which, together with (\[lem-1-1\]), shows that $$\begin{aligned}
|x'-x'_0|&\leq& |x'-x'_3|+|x'_3-x'_0|\leq (\lambda L\big(2+\frac{17}{2r_1}\big)e^{\varphi(\log 2)}+1)d_D(x)\\&\leq & 2\lambda L\big(2+\frac{17}{2r_1}\big)e^{\varphi(\log 2)}d_D(x),\end{aligned}$$ where $\lambda={1}/{\theta^{-1}(\frac{1}{6\big(4+\frac{17}{2r_1}\big)})}$. Thus the proof of $(1)$ is complete by taking $M_1=2\lambda L\big(2+\frac{17}{2r_1}\big)e^{\varphi(\log 2)}$.
Now we are going to prove $(2)$.
We first observe that $f:\partial D\to \partial D'$ is $\eta$-QS with $\eta(t)=L^2t$. Hence Lemma \[pre-lem-4\] shows that $\partial D'$ is $(\lambda_1,\lambda_2)$-HD with $\lambda_1, \lambda_2$ depending only on $L$, $r_1$ and $r_2.$ Since $f^{-1}$ is also a $\varphi$-FQC map, it is easily seen that we only need to prove the right hand side of . For $x\in D$, we let $y_1\in
\partial D$ be such that $$\label{lem-2-13}|x-y_1|\leq 2d_D(x).$$ Then it follows from Lemma \[lem-1\] $(1)$ that $$\label{lem-2-7}|x'-y'_1|\leq M_1d_D(x)\leq M_1|x-y_1|.$$
For $x_1\in \partial D$, on one hand, if $|y_1-x_1|\leq 2 |x-y_1|$, then by , $$\begin{aligned}
|x'-x'_1|&\leq&
|x'-y'_1|+|y'_1-x'_1|\leq M_1|x-y_1|+L|y_1-x_1|\\ \nonumber &\leq&(2L+M_1)|x-y_1|\leq 2(2L+M_1)d_D(x)\\
\nonumber &\leq& 2(2L+M_1)|x-x_1|.\end{aligned}$$
On the other hand, if $|y_1-x_1|>2|x- y_1|$, then we have $$|x-x_1|>|y_1-x_1|-|x-y_1|>\frac{1}{2}|y_1-x_1|,$$ which, together with (\[lem-2-7\]), shows that $$\begin{aligned}
|x'-x'_1|&\leq&
|x'-y'_1|+|y'_1-x'_1|\leq M_1|x-y_1|+L|y_1-x_1|\\ \nonumber &\leq& 2M_1d_D(x)+2L|x-x_1|\leq 2(L+M_1)|x-x_1|.\end{aligned}$$ Hence the proof of is complete.
Supposing that $f \in QC^L_{\varphi}(D,D')$ is $M$-QH, we show that $f$ is $M'$-bilipschitz from $\overline{D}$ to $\overline{D}'$. Lemma \[pre-lem-4\] yields that $\partial D'$ is $(\lambda_1,\lambda_2)$-HD with $\lambda_1, \lambda_2$ depending only on $L$, $r_1$ and $r_2$. Then by Lemma \[lem-1\] and the fact that $``f^{-1}$ is also $M$-QH and a $M$-QH map is a $\varphi$-FQC map with $\varphi(t)=Mt$" we know that it suffices to show that for all $z_1,z_2\in D$, the following holds: $$\label{thm-1-2}|z_1'-z_2'|\leq M'|z_1-z_2|.$$
Fix $z_1,z_2\in D\,.$ Without loss of generality, we may assume that $$\max\{d_D(z_1),d_D(z_2)\}=d_D(z_1).$$
Consider first the case $|z_1-z_2|\leq \frac{1}{2M}d_D(z_1)\,.$ Then by Lemma \[proof-lem-1\], $$k_{D'}(z_1',z_2')\leq M k_D(z_1,z_2)\leq 2M \frac{|z_1-z_2|}{d_D(z_1)}\leq 1,$$ which shows that $$\frac{1}{2}\frac{|z_1'-z_2'|}{d_{D'}(z_1')}\leq k_D(z_1',z_2')\leq M k_D(z_1,z_2)\leq 2M\frac{|z_1-z_2|}{d_D(z_1)}.$$ Hence Lemma \[lem-1\] shows that $$\label{thm-1-2-proof1}|z_1'-z_2'|\leq 4M\frac{|z_1-z_2|}{d_D(z_1)}d_{D'}(z_1')\leq 4MM_1|z_1-z_2|.$$
Next we consider the case $|z_1-z_2|> \frac{1}{2M}d_D(z_1)$. We let $z\in \partial D$ be such that $|z_1-z|\leq 2 d_D(z_1)$. If $|z_1-z|\leq \frac{1}{2}|z_2-z|$, then $$|z_1-z_2|\geq |z_2-z|-|z_1-z|\geq \frac{1}{2}|z_2-z|,$$ and so Lemma \[lem-1\] yields $$\label{thm-1-2-proof2}|z_1'-z_2'|\leq |z_1'-z'|+|z_2'-z'|\leq M_1 d_D(z_1)+ 2(2L+M_1)|z_2-z|$$$$\begin{aligned}
\leq 2(MM_1+4L+2M_1)|z_1-z_2|.\end{aligned}$$ On the other hand, if $|z_1-z|\geq \frac{1}{2}|z_2-z|$, then by Lemma \[lem-1\] we have $$\label{thm-1-2-proof3}|z_1'-z_2'|\leq |z_1'-z'|+|z_2'-z'|\leq M_1 d_D(z_1)+ 2(2L+M_1)|z_2-z|$$$$\begin{aligned}
\leq M_1 d_D(z_1)+ 4(2L+M_1)|z_1-z|\leq 2M(9M_1+16L)|z_1-z_2|.\end{aligned}$$ By taking $M'= 2M(9M_1+16L)$ we see from , and that holds. Hence the proof of Theorem \[thm1.4\] is complete.
\[remark\]
1. In Theorem \[thm1.4\], the hypothesis “$f$ is FQC" alone does not imply the conclusion “$f$ is bilipschitz". As an example, we consider the radial power map $f_{\alpha}: \mathbb{B}\to \mathbb{B}$ with $f_{\alpha}(x)=|x|^{\alpha-1}x$ and $\alpha\geq 1$. By [@Vai6-0 6.5] we see that $f_{\alpha}$ is a FQC map and $f_{\alpha}|_{\partial \mathbb{B}}$ is the identity on the boundary, but $f_{\alpha}$ is not bilipschitz (see [@Vai5 6.8]).
2. If the boundary of $D$ is not HD, then “$f$ being QH" does not always imply that “$f$ is bilipschitz". We still consider the radial power map $f_{\alpha}: E\setminus\{0\}\to E\setminus\{0\}$ with $f_{\alpha}(x)=|x|^{\alpha-1}x$ and $\alpha\geq 1$. On one hand, the domain $E\setminus\{0\}$ has only two boundary components: $\{0\}$ and $\{\infty\}$, and so the boundary is not HD. On the other hand, $f$ is $\alpha$-QH (see [@Vai5 5.21]) and it is the identity on the boundary. But it is not bilipschitz.
Given $x\in D=D'$, let $z'\in \partial D'$ be such that $d_{D'}(x')\geq \frac{1}{2}|x'-z'|$. Then Lemma \[lem-1\] yields $$d_{D'}(x')\geq \frac{1}{4(2L+M_1)}|x-z|\geq \frac{1}{4(2L+M_1)}d_D(x).$$
Let $z_1\in \partial D$ be such that $|x-z_1|\leq 2 d_D(x)$. Then it follows from Lemma \[lem-1\] that $$|x-x'|\leq |x-z_1|+|x'-z_1|\leq (2+M_1)d_D(x).$$
Hence by Lemma \[pre-lem-1\] we see that $$k_D(x,x')\leq c'\log\Big(1+\frac{|x-x'|}{\min\{d_D(x),d_D(x')\}}\Big)\leq c'\log\big(1+4(2+M_1)(2L+M_1)\big).$$
[**Acknowledgement.**]{} This research was finished when the first author was an academic visitor in Turku University and the first author was supported by the Academy of Finland grant of Matti Vuorinen with the Project number 2600066611. She thanks Department of Mathematics in Turku University for hospitality.
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} |
---
abstract: 'This paper considers the problem of acquiring an unknown target location (among a finite number of locations) via a sequence of measurements, where each measurement consists of simultaneously probing a group of locations. The resulting observation consists of a sum of an indicator of the target’s presence in the probed region, and a zero mean Gaussian noise term whose variance is a function of the measurement vector. An equivalence between the target acquisition problem and channel coding over a binary input additive white Gaussian noise (BAWGN) channel with state and feedback is established. Utilizing this information theoretic perspective, a two-stage adaptive target search strategy based on the sorted Posterior Matching channel coding strategy is proposed. Furthermore, using information theoretic converses, the fundamental limits on the target acquisition rate for adaptive and non-adaptive strategies are characterized. As a corollary to the non-asymptotic upper bound of the expected number of measurements under the proposed two-stage strategy, and to non-asymptotic lower bound of the expected number of measurements for optimal non-adaptive search strategy, a lower bound on the adaptivity gain is obtained. The adaptivity gain is further investigated in different asymptotic regimes of interest.'
author:
- 'Anusha Lalitha, Nancy Ronquillo, and Tara Javidi, [^1] [^2]'
bibliography:
- 'HypTest.bib'
title: Improved Target Acquisition Rates with Feedback Codes
---
Introduction {#sec:intro}
============
Consider a single target acquisition over a search region of width $B$ and resolution up to width $\delta$. Mathematically, this is the problem of estimating a unit vector $\mbf{W} \in \{0,1\}^{\frac{B}{\delta}}$ via a sequence of noisy linear measurements $$\label{eq:linear1}
Y_n = \langle {\mbf{S}_n},\mbf{W} + \mbs{\Xi}_n\rangle, \quad n = 1,2, \ldots, \tau,$$ where a binary measurement vector $\mbf{S}_n \in \{0,1\}^{\frac{B}{\delta}}$ denotes the locations inspected and the vector $\mbs{\Xi}_n \in \mathbb{R}^{\frac{B}{\delta}}$ denotes the additive measurement noise per location. More generally, the observation $Y_n$ at time $n$ can be written as $$\label{eq:linear2}
Y_n = \langle\mbf{S}_n, \mbf{W}\rangle + Z_n(\mbf{S}_n),$$ where $Z_n(\mbf{S}_n)$ is a noise term whose statistics are a function of the measurement vector $\mbf{S}_n$. The goal is to design the sequence of measurement vectors $\{\mbf{S}_n\}_{n = 1}^{\tau}$, such that the target location $\textbf{W}$ is estimated with high reliability, while keeping the (expected) number of measurements $\tau$ as low as possible.
In this paper, we first consider the linear model when the elements of $\mbf{\Xi}_n$ are i.i.d Gaussian with zero mean and variance $\delta \sigma^2$. This means that $Z_n(\mbf{S}_n)$ in are distributed as $\mathcal{N}(0, |\mbf{S}_n| \delta \sigma^2)$, and show that the problem of searching for a target under measurement dependent Gaussian noise $Z_n(\mbf{S}_n)$ is equivalent to channel coding over a binary additive white Gaussian noise (BAWGN) channel with state and feedback (in Section 4.6 [@Gallager]). This allows us not only to retrofit the known channel coding schemes based on sorted Posterior Matching (sort PM) [@SungEnChiu] as adaptive search strategies, but also to obtain information theoretic converses to characterize fundamental limits on the target acquisition rate under both adaptive and non-adaptive strategies. As a corollary to the non-asymptotic analysis of our sorted Posterior-Matching-based adaptive strategy and our converse for non-adaptive strategy, we obtain a lower bound on the adaptivity gain.
Our Contributions
-----------------
Our main results are inspired by the analogy between target acquisition under measurement dependent noise and channel coding with state and feedback. This connection was utilized in [@DBLP:journals/corr/KaspiSJ16] under a Bernoulli noise model. In this paper, in Proposition \[prop:connection\], we formalize the connection between our target acquisition problem with Gaussian measurement dependent noise and channel coding over a BAWGN channel with state. Here, the channel state denotes the variance of the measurement dependent noise $ |\mbf{S}_n| \delta \sigma^2$. Since feedback codes i.e., adapting the codeword to the past channel outputs, are known to increase the capacity of a channel with state and feedback. This motivates us to use adaptivity when searching, i.e., to utilize past observations $\{Y_1, Y_2, \ldots, Y_{n-1}\}$ when selecting the next measurement vector $\mbf{S}_n$. Furthermore, this information theoretic perspective allows us to quantify the increase in the adaptive target acquisition rate. Our analysis of improvement in the target acquisition rate as well as the adaptivity gain, measured as the reduction in expected number of measurements, while using an adaptive strategy over a non-adaptive strategy has two components. Firstly, we utilize information theoretic converse for an optimal non-adaptive search strategy to obtain a non-asymptotic lower bound on the minimum expected number of measurements required while maintaining a desired reliability. As a consequence, this provides the best non-adaptive target acquisition rate. Secondly, we utilize a feedback code based on Posterior Matching as a two-stage adaptive search strategy and obtain a non-asymptotic upper bound on the expected number of measurements while maintaining a desired reliability. These two components of our analysis allow us to characterize a lower bound on the increased target acquisition rate due to adaptivity.
Our non-asymptotic analysis of adaptivity gain reveals two qualitatively different asymptotic regimes. In particular, we show that adaptivity gain depends on the manner in which the number of locations grow. We show that the adaptivity grows logarithmically in the number of locations, i.e., $O\left(\log \frac{B}{\delta} \right)$ when refining the search resolution $\delta$ ($\delta$ going to zero) and while keeping total search width $B$ fixed. On the other hand, we show that as the search width $B$ expands while keeping search resolution $\delta$ fixed, the adaptivity gain grows in the number of locations as $O\left(\frac{B}{\delta} \log \frac{B}{\delta} \right)$.
The problem of searching for a target under a binary measurement dependent noise, whose crossover probability increases with the weight of the measurement vector was studied by [@DBLP:journals/corr/KaspiSJ16] and analyzed under sort PM strategy in [@SungEnChiu]. In particular, [@DBLP:journals/corr/KaspiSJ16] and [@SungEnChiu] provide asymptotic analysis of the adaptivity gain for the case where $B = 1$ and $\delta $ approaches zero. Our prior work [@8007098] by utilizing a (suboptimal) hard decoding of Gaussian observation $Y_n$, strengthens [@DBLP:journals/corr/KaspiSJ16] and [@SungEnChiu] by also accounting for the regime in which $B$ grows. While the analysis in [@8007098] strengthens the non-asymptotic bounds in [@SungEnChiu] with Bernoulli noise it failed to provide tight analysis for our problem with Gaussian observations. In this paper, by strengthening our analysis in [@8007098] we extend the prior work in three ways: (i) we consider the soft Gaussian observation $Y_n$, (ii) we obtain non-asymptotic achievability and converse analysis, and (iii) we characterize tight non-asymptotic adaptivity gain in the two asymptotically distinct regimes of $B \to \infty$ and $\delta \to 0$.
Applications
------------
Our problem formulation addresses two challenging engineering problems which arise in the context of modern communication systems. We will discuss the two problems in the following examples and then provide the details of the state of art.
Consider the problem of detecting the direction of arrival for initial access in millimeter wave (mmWave) Communications. In mmWave communication, prior to data transmission the base station is tasked with aligning the transmitter and receiver antennas in the angular space. In other words, the base station’s antenna pattern can be viewed as a measurement vector $\mbf{S}_n$ searching the angular space $B \subset (0, 360^{\circ})$. At each time $n$, the noise intensity depends on the base station’s antenna pattern $\mbf{S}_n$ and the noisy observation $Y_n$ is a function of measurement dependent noise $Z_n(\mbf{S}_n)$. Here it is natural to characterize the fundamental limit on the measurement time as a function of asymptotically small $\delta$.
Consider the problem of opportunistically searching for a vacant subband of bandwidth $\delta$ over a total bandwidth of $B$. In this problem secondary user desires to locate the single stationary vacant subband quickly and reliably, by making measurements $\mbf{S}_n$ at every time $n$. At each time instant $n$, the noise intensity depends on the number of subbands probed as dictated by $\mbf{S}_n$ and noisy observation $Y_n$ is is a function of measurement dependent noise $Z_n(\mbf{S}_n)$. Here it is natural to characterize fundamental limit of the measurement time required for a secondary user to acquire the vacant subband as a function of the asymptotically large bandwidth $B$.
Giordani et al. [@7460513] compare the exhaustive search like the Sequential Beamspace Scanning considered by Barati et al. [@7421136], where the base station sequentially searches through all angular sectors, against a two stage iterative hierarchical search strategy. In the first stage an exhaustive search identifies a coarse sector by repeatedly probing each coarse region for a predetermined SNR to be achieved. In the second stage an exhaustive search over all locations identifies the target. Giordani et al. show that in general the adaptive iterative strategy reduces the number of measurements over exhaustive search except when desired SNR is too high, forcing the number of measurements required at each stage to get too large. We observe this in through our simulations in Section \[sec:num\_results\]-A. In fact, as confirmed by our simulations random-coding-based non-adaptive strategies including the Agile-Link protocol [@Abari_AgileLink], outperform the repetition based adaptive strategies.
Past literature on spectrum sensing for cognitive radio [@42_35Multibandjoint; @50AdaptiveMultiband; @55AdaptiveAgileCR] and support vector recovery [@Nowak_CompSensing; @Y_Kim_MACSensing] have focused on the problem where $\textbf{S}_n$ can be real or complex, with measurement independent noise applying both exhaustive search and multiple adaptive search strategies. In contrast, our work considers a simple binary model, $\textbf{S}_n \in \{0,1\}^{\frac{B}{\delta}}$, but captures the implications of measurement dependence of the noise, which is known in the spectrum sensing literature as noise folding. The problem of measurement dependent noise (known as noise folding) has been investigated in [@Treichler_NoiseFolding] where non-adaptive design of complex measurements matrix satisfying RIP condition has been investigated. Our work compliments this study by characterizing the gain associated with adaptively addressing the measurement dependent noise (noise folding), albeit for the simpler case of binary measurements. We note that the case of adptively finding a subset of a sufficiently large vacant bandwidth with noise folding is considered in [@Sharma_Murthy], where ideas from group testing and noisy binary search have been utilized. The solutions however depend strongly on the availability of sufficiently large consective vacant band and does not apply to our setting.
Vectors are denoted by boldface letters $\mbf{A}$ and $\mbf{A}{(j)}$ is the $j^{th}$ element of a vector. Matrices are denoted by overlined boldface letters. Let $\mathcal{U}_M$ denote the set $\{\textbf{u}\in \mathbb{R}^M: u(j)\in\{0,1\} \}$. Bern$(p)$ denotes the Bernoulli distribution with parameter $p$, $h(p) = - p\log p -(1-p)\log(1-p)$ denotes the entropy of a Bernoulli random variable with parameter $p$. Let $G(x; \mu, \sigma^2)$ denote the pdf of Gaussian random variable with mean $\mu$ and variance $\sigma^2$ at $x$. Logarithms are to the base 2. Let $[g]_a = g$ if $g \geq a$ otherwise $[g]_a = 0$.
Problem Setup {#sec:prob_setup}
=============
In this section, we describe the mathematical formulation of the target acquisition problem followed by the performance criteria.
Problem Formulation
-------------------
We consider a search agent interested in quickly and reliably finding the true location of a single stationary target by making measurements over time about the target’s presence. In particular, we consider a total search region of width $B$ that contains the target in a location of width $\delta$. In other words, the search agent is searching for the target’s location among $ \frac{B}{\delta}$ total locations. Let $\mbf{W} \in \mathcal{U}_{\frac{B}{\delta}}$ denote the true location of the target, where $\mbf{W}(j)=1$ if and only if target is located at location $j$. The target location $\mbf{W}$ can take $\frac{B}{\delta}$ possible values uniformly at random whose value remains fixed during the search. A measurement at time $n$ is given by a vector $\mbf{S}_n \in \mathcal{U}_{\frac{B}{\delta}}$, where $\mbf{S}_n(j)=1$ if and only if location $j$ is probed. Each measurement can be imagined to result in a clean observation $X_{n} = \textbf{W}^{\intercal} \mbf{S}_{n} \in \{0,1 \}$ indicating of the presence of the target in the measurement vector $\textbf{S}_n$. However, only a noisy version of the clean observation $X_n$ is available to the agent. The resulting noisy observation $Y_n \in \mathbb{R}$ is given by the following linear model with additive measurement dependent noise $$Y_n = X_{n}+ {Z}_{n}(\mbf{S}_n).
\label{eq:noisysearch}$$ Here, we assume ${Z}_{n} \sim \mathcal{N}(0, |\textbf{S}_n|\delta \sigma^2)$ which corresponds to the case of i.i.d white Gaussian noise with $\sigma^2$ denotes the noise variance per unit width. Conditioned on the measurement vector $\mbf{S}_n$, the noise $Z_{n}$ is independent over time.
A search consisting of $\tau$ measurements can be represented by a measurement matrix $\overline{\mbf{S}}^{\tau} = [\mbf{S}_1, \mbf{S}_2, \ldots , \mbf{S}_{\tau}]$ which yields the observation vector $\mbf{Y}^{\tau} = [Y_1, Y_2, \ldots, Y_{\tau}]$. At any time instant $n = 1,2 \ldots, \tau$, the agent selects the measurement vector in general as a function of the past observations and measurements. Mathematically, $$\begin{aligned}
\mbf{S}_{n} = g_{n}\left(\mbf{Y}^{n-1}, \overline{\mbf{S}}^{n-1}\right),\end{aligned}$$ for some causal (possibly random) function $g_{n}: \mathbb{R}^{n-1} \times \mathcal{U}^{n-1}_{\frac{B}{\delta}} \to \mathcal{U}_{\frac{B}{\delta}}$. After observing the noisy observations $\mbf{Y}^{\tau}$ and measurement matrix $\overline{\mbf{S}}^{\tau}$, the agent estimates the target location $\mbf{W}$ as follows $$\begin{aligned}
\hat{\mbf{W}} = d\left( \mbf{Y}^{\tau}, \overline{\mbf{S}}^{\tau}\right),\end{aligned}$$ for some decision function $d: \mathbb{R}^{\tau} \times \mathcal{U}^{\tau}_{\frac{B}{\delta}} \to \mathcal{U}_{\frac{B}{\delta}}$. The probability of error for a search is given by $\Pe = \P(\hat{\mbf{W}} \neq \mbf{W} | \mbf{Y}, \overline{\mbf{S}})$ and the average probability of error is given by $\overline{\Pe} = \P(\hat{\mbf{W}} \neq \mbf{W})$.
Now we define the measurement strategy:
For some $\epsilon \in(0, 1)$, an *$\epsilon$-reliable search strategy*, denoted by $\mathfrak{c}_{\epsilon}$, is defined as a sequence of $\tau$ (possibly random) number of causal functions $\{g_1, g_2, \ldots, g_{\tau}\}$, according to which the measurement matrix $\overline{\mbf{S}}^{\tau}$ is selected, and a decision function $d$ which provides an estimate $\mbf{\hat{W}}$ of $\mbf{W}$, such that the average probability of error $\overline{\Pe}$ is at most $\epsilon$.
A target acquisition rate $R$ is said to be an *$\epsilon$-achievable*, if for any small $\xi > 0$ and $n$ large enough, there exists an $\epsilon$-reliable search strategy $\mathfrak{c}_{\epsilon}$ such the following holds $$\begin{aligned}
\expe_{\mathfrak{c}_{\epsilon}}[\tau] &\leq n, \\
\frac{B}{\delta} &\geq 2^{n(R-\xi)}.\end{aligned}$$ A targeting rate $R$ is said to be *achievable target acquisition rate* if it is $\epsilon$-achievable for all $\epsilon \in (0,1)$.
The above definition is motivated by information theoretic notion of transmission rate over a communication channel, which captures the exponential rate at which the number of messages grow with the number of channel uses while the receiver can decode with a small average error probability. Similarly, the target acquisition rate captures the exponential rate at which the number of target locations grow with the number of measurement vectors while a search strategy can still locate the target with a diminishing average error probability.
The supremum of achievable target acquisition rates is called the target acquisition capacity.
Types of Search Strategies and Adaptivity Gain
----------------------------------------------
Each measurement vector $\mbf{S}_n$ and the number of total measurements $\tau$ can be selected either based on the past observations $\mbf{Y}^{n-1}$, or independent of them. Based on these two choices, strategies can be divided into four types i) having fixed length versus variable length number of the measurement matrix $\overline{\mbf{S}}$, and ii) being adaptive versus non-adaptive. A *fixed length $\epsilon$-reliable strategy* $\mathfrak{c}_{\epsilon}$ uses a fixed number of measurements $\tau$ predetermined offline independent of the observations, to obtain estimate $\hat{\mbf{W}}$. On the other hand, a *variable length $\epsilon$-reliable strategy* $\mathfrak{c}_{\epsilon}$ uses a random number of measurements $\tau$ (possibly determined as a function of the observations $\mbf{Y}^{\tau}$) to obtain estimate, $\hat{\mbf{W}}$. For example, $\tau$ can be selected such that agent achieves $\Pe \leq \epsilon$ in every search and hence $\tau$ is a random variable which is a function of the past noisy observations. Under an *adaptive strategy* $\mathfrak{c}_{\epsilon} \in \mathcal{C}^A_{\epsilon}$ the agent designs the measurement vector $\textbf{S}_n$ as a function of the past observations $\mbf{Y}^{n-1}$, i.e., $g_n$ is a function of both $\textbf{S}^{n-1}$ and $\mbf{Y}^{n-1}$.
Let $\mathcal{C}^A_{\epsilon}$ be a class of all $\epsilon$-reliable adaptive strategies.
Under a *non-adaptive strategy*, the agent designs the measurement vector $\textbf{S}_n$ offline independent of past observations, i.e., $g_n$ does not depend on $\textbf{S}^{n-1}$ or $\mbf{Y}^{n-1}$.
Let $\mathcal{C}^{NA}_{\epsilon}$ be a class of all $\epsilon$-reliable non-adaptive strategies.
For any $\epsilon$-reliable strategy $\mathfrak{c}_{\epsilon}$, the performance is measured by the expected number of measurements $\expe_{\mathfrak{c}_{\epsilon}}[\tau]$. To achieve better reliability, i.e., smaller $\epsilon$, in general the agent requires larger $\expe_{\mathfrak{c}_{\epsilon}}[\tau]$.
The adaptivity gain is defined as the best reduction in the expected number of measurements when searching with an $\epsilon$-reliable adaptive strategy $\mathfrak{c}^{\prime}_{\epsilon} \in \mathcal{C}^{A}_{\epsilon}$, over an $\epsilon$-reliable non-adaptive strategy $\mathfrak{c}_{\epsilon} \in \mathcal{C}^{NA}_{\epsilon}$. Mathematically, it is given as $$\begin{aligned}
\min_{\mathfrak{c}_{\epsilon} \in \mathcal{C}^{NA}_{\epsilon}}\expe[\tau] - \min_{\mathfrak{c}^{\prime}_{\epsilon} \in \mathcal{C}^{A}_{\epsilon}}\expe[\tau^{\prime}].\end{aligned}$$
Hence, characterizing adaptivity gain allows us to characterize the improvement in target acquisition rate when using adaptive strategies over non-adaptive strategies.
Preliminaries: Channel Coding with State and Feedback {#sec:prelim}
=====================================================
In this section, we review fundamentals of channel coding with state and feedback and relevant literature to connect these information theoretic concepts to the problem of searching under measurement dependent noise discussed in the previous section. The aim is to formulate an equivalent model of channel coding with state and feedback for comparison to (\[eq:noisysearch\]).
\[ReviewChannelCoding\]
A communication channel is specified by a set of inputs $\tilde{X} \in \tilde{\mathcal{X}}$, a set of outputs $\tilde{Y} \in \tilde{\mathcal{Y}}$, and a channel transition probability measure $\P(\tilde{y}|\tilde{x})$ for every $\tilde{x} \in \tilde{\mathcal{X}}$ and $\tilde{y} \in \tilde{\mathcal{Y}}$ that expresses the probability of observing a certain output $\tilde{y}$ given that an input $\tilde{x}$ was transmitted [@CoverBook2nd]. Throughout this work, we will concentrate on coding over a channel with state and feedback (section 4.6 in [@Gallager]). Formally, at time $n$ the channel state, $\tilde{\mbf{S}}_n$ belongs to a discrete and finite set $\tilde{\mathcal{A}}$. We assume that the channel state is known at both the encoder and the decoder. For a channel with state, the transition probability at time $n$ is specified by the conditional probability assignment $\P_n\left(\tilde{Y}_n |\tilde{X}_n, \tilde{\mbf{S}}_n \right)$. Transmission over such a channel is shown in Figure \[fig:Basic\]. In general, the channel state $\tilde{\mbf{S}}_n$ at time $n$ evolves as a function of all past outputs and all past states, $$\label{eq:state}
\tilde{\mbf{S}}_n = \tilde{g}_n(\tilde{Y}_1, \tilde{Y}_2,\ldots, \tilde{Y}_{n-1}, \tilde{\mbf{S}}_1, \tilde{\mbf{S}}_2,\ldots, \tilde{\mbf{S}}_{n-1}).$$ The goal is to encode and transmit a uniformly distributed message $\tilde{\mbf{W}} \in [M] $ over the channel. The encoding function $\phi_n$ at any time $n$ depends on the message to be transmitted $\tilde{\mbf{W}}$, all past states, and all the past outputs. Thus the next symbol to be transmitted is given by $$\tilde{X}_n = \phi_n(\tilde{Y}_1, \tilde{Y}_2, \ldots, \tilde{Y}_{n-1}, \tilde{\mbf{S}}_1, \tilde{\mbf{S}}_2, ..., \tilde{\mbf{S}}_{n}, \tilde{\mbf{W}}).$$ The encoder obtains the past outputs from the decoder due to the availability of a noiseless feedback channel from decoder to encoder. In this paper, we assume that both encoder and decoder know the evolution of the channel state, i.e., the sequence $\{\mbf{\mbf{S}}_n\}_{n \geq 1}$. After $\tau$ channel uses, the decoder uses the noisy observations $\tilde{\mbf{Y}}^{\tau}$ and state information $\{\tilde{\mbf{S}}_1, \tilde{\mbf{S}}_2, \ldots, \tilde{\mbf{S}}_{\tau}\}$ to find the best estimate $\tilde{\mbf{W}}^{\prime}$, of the message $\tilde{\mbf{W}}$. The probability of error at the end of message transmission is given by $\Pe = \P(\tilde{\mbf{W}}^{\prime} \neq \tilde{\mbf{W}} | \tilde{\mbf{Y}}, \{\tilde{\mbf{S}}_1, \tilde{\mbf{S}}_2, \ldots, \tilde{\mbf{S}}_{\tau}\})$ and the average probability of error is given by $\overline{\Pe} = \P(\tilde{\mbf{W}}^{\prime} \neq \tilde{\mbf{W}})$.
\[ex:bawgn\] Consider a Binary Additive White Gaussian Noise (BAWGN) channel with noisy output $\tilde{Y}_n$ given as the sum of input $\tilde{X}_n \in \{0,1\}$ and Gaussian random variable $\tilde{Z}_n \in \mathbb{R}$ whose distribution is a function of the channel state $\tilde{\mbf{S}}_n$. Specifically, $\tilde{Z}_n$ is a Gaussian random variable with state dependent noise variance $|\tilde{\mbf{S}}_n|\delta\sigma^2$ for some $\delta> 0$. In other words, we have $$\tilde{Y}_n = \tilde{X}_{n} + \tilde{Z}_{n}(\tilde{\mbf{S}}_n),
\label{eq:Gaussianoutput}$$ where $\tilde{Z}_n \sim \mathcal{N}(0, |\tilde{\mbf{S}}_n|\delta\sigma^2)$, and the state evolves as $\tilde{\mbf{S}}_n = \tilde{g}_n(\tilde{Y}_1, \tilde{Y}_2, \ldots, \tilde{Y}_{n-1}, \tilde{\mbf{S}}_1, \tilde{\mbf{S}}_2, \ldots, \tilde{\mbf{S}}_{n-1})$. Transmission over a BAWGN channel is illustrated in Figure \[fig:Gaussian\].
\[prop:connection\] The problem of searching under measurement dependent Gaussian noise is equivalent to the problem of channel coding over a BAWGN channel with state and feedback. Specifically,
- The true location vector $\mbf{W}$ can be cast as a message $\tilde{\mbf{W}}$ to be transmitted over the BAWGN. Therefore, there are $\frac{B}{\delta}$ possible messages.
- An $\epsilon$-reliable search strategy $\mathfrak{c}_{\epsilon}$ provides a sequence of $\{g_1, g_2, \ldots, g_{\tau}\}$ such that $\P(\tilde{\mbf{W}}^{\prime} \neq \tilde{\mbf{W}}) \leq \epsilon$. Hence, setting $\tilde{g}_i = g_i$ for all $i \in \{1, 2, \ldots, \tau\}$, the search strategy dictates the evolution of channel states $\tilde{\mbf{S}}_n$.
- The measurement matrix $\overline{\mbf{S}}^{\tau}$ can be used as the codebook, i.e., by setting $\{\tilde{\mbf{S}}_1, \tilde{\mbf{S}}_2, \ldots, \tilde{\mbf{S}}_{\tau}\} = \overline{\mbf{S}}$. Specifically, codewords are obtained by setting $\tilde{X}_n = \phi_n(\tilde{\mbf{Y}}^{n-1},\tilde{ \mbf{S}}_1, \tilde{\mbf{S}}_2, \ldots, \tilde{\mbf{S}}_n, \tilde{\mbf{W}}) = \tilde{\mbf{W}}^{\intercal}\tilde{\mbf{S}}_n$.
- The measurement vector fixes the channel transition probability measure as $\P(\tilde{Y}_n| \tilde{x}_n,\tilde{\textbf{S}}_n) = \mathcal{N}(\tilde{x}_n, |\tilde{\textbf{S}}_n|\delta\sigma^2)$ since noise distribution is $\tilde{Z}_n \sim \mathcal{N}(0, |\tilde{\mbf{S}}_n|\delta \sigma^2)$ for $\tilde{x}_n \in \{ 0, 1\}$. Hence, the channel state depends on measurement vector.
A coding scheme for a channel with state and feedback can double as a search strategy. This general approach of search using channel codes provides an efficient way to design and compare non-adaptive and adaptive search strategies. This also implies that feedback can improve the capacity of a channel with state which is what we characterize as our adaptivity gain for the problem of searching under measurement dependent noise.
The BAWGN capacity with input distribution $\ber(q)$ and noise variance $\sigma^2$ is defined as $$\begin{aligned}
C_{\text{BAWGN}}\left(q, \sigma^2\right)
&:= -\int_{-\infty}^{\infty} \left( (1-q) G(y; 0, \sigma^2) + q G(y; 1, \sigma^2)\right) \times
\nonumber
\\
&\hspace{0.5cm}\times \log \left( (1-q) G(y; 0, \sigma^2) + q G(y; 1, \sigma^2)\right)
\nonumber
\\
& \hspace{0.5cm}- \frac{1}{2}\log(2\pi e \sigma^2).\end{aligned}$$
\[cor:channelstate\] From channel coding over a BAWGN channel with state and feedback, we obtain that for any small $\xi > 0$ and $n$ large enough, there exists an $\epsilon$-reliable search strategy $\mathfrak{c}_{\epsilon}$ such the following holds $$\begin{aligned}
\expe_{\mathfrak{c}_{\epsilon}}[\tau] &\leq n, \\
2^{n(C_{\text{BAWGN}}(\frac{1}{2}, \frac{B\sigma^2}{2})-\xi)} &\overset{(a)}\leq \frac{B}{\delta} \overset{(b)}< 2^{nC_{\text{BAWGN}}(\frac{1}{2}, \delta \sigma^2)},\end{aligned}$$ where $(a)$ follows from Theorem 4.6.1 in [@Gallager] and $(b)$ follows by combining the fact that the best channel is obtained when noise variance is the least, i.e., $\delta \sigma^2$, with the converse of the noisy channel coding theorem [@CoverBook2nd].
Main Results {#sec:main_results}
============
In this section, we characterize a lower bound on the adaptivity gain $\min_{\mathfrak{c}_{\epsilon} \in \mathcal{C}^{NA}_{\epsilon}}\expe[\tau] - \min_{\mathfrak{c}^{\prime}_{\epsilon} \in \mathcal{C}^{A}_{\epsilon}}\expe[\tau^{\prime}]$; the performance improvement measured in terms of reduction in the expected number of measurements for searching over a width $B$ among $\frac{B}{\delta}$ locations under measurement dependent Gaussian noise.
\[thm:gain\_lower\_bound\] Let $\epsilon \in (0,1)$. For any $\epsilon$-reliable non-adaptive strategy $\mathfrak{c}_{\epsilon} \in \mathcal{C}^{NA}_{\epsilon}$ searching over a search region of width $B$ among $\frac{B}{\delta}$ locations with $\tau$ number of measurements, there exists an $\epsilon$-reliable adaptive strategy $\mathfrak{c}^{\prime}_{\epsilon} \in \mathcal{C}^{A}_{\epsilon}$ with $\tau^{\prime}$ number of measurements, such that for some small constant $\eta > 0$ the following holds $$\begin{aligned}
\expe_{\mathfrak{c}_{\epsilon}}[\tau] - \expe_{\mathfrak{c}^{\prime}_{\epsilon}}[\tau^{\prime}]
&
\geq \max_{\alpha \in \mathcal{I}_{\frac{B}{\delta}}} \left\{
\log \frac{1}{\alpha} \left(\frac{(1-\epsilon) }{C_{\text{BAWGN}}(q^{\ast}, q^{\ast}B \sigma^2)} \right. -\frac{1}{C_{\text{BAWGN}} \left( q^{\ast}, q^{\ast} B \sigma^2\right) - \eta}
\right)
\nonumber
\\
&
\quad + \log \frac{\alpha B}{\delta} \left(\frac{(1-\epsilon) }{
C_{\text{BAWGN}}(q^{\ast}, q^{\ast}B\sigma^2)}
- \frac{1}{
C_{\text{BAWGN}} \left(\frac{1}{2}, \frac{\alpha B \sigma^2}{2}\right)-\eta} \right)
\\
\nonumber
&\left. \quad
- h(B, \delta, \sigma^2, \alpha, \epsilon, \eta) \right\},\end{aligned}$$ where $$\begin{aligned}
h(B, \delta, \sigma^2, \alpha, \epsilon, \eta)
&=
\frac{\log \left( \frac{2}{\epsilon}\right) + \log \log \left( \frac{1}{\alpha}\right) + a_{\eta}}{C_{\text{BAWGN}} \left( q^{\ast}, q^{\ast}B \sigma^2\right) -\eta}
+ \frac{\log \left( \frac{2}{\epsilon}\right) + \log \log \left( \frac{\alpha B}{\delta }\right) + a_{\eta}}{C_{\text{BAWGN}}\left(\frac{1}{2}, \frac{\alpha B \sigma^2}{2} \right) - \eta}
\nonumber
\\
& \quad + \frac{h(\epsilon)}{C_{\text{BAWGN}}^{(B, \delta, \sigma^2)}(q^{\ast}, q^{\ast}B \sigma^2)} ,\end{aligned}$$ $
q^{*}
=
\operatorname*{arg\,max}_{q \in \mathcal{I}_{\frac{B}{\delta}}} C_{\text{BAWGN}}(q, qB\sigma^2),
$ and $a_{\eta}$ is the solution of the following equation $$\begin{aligned}
\eta =\frac{a}{a-3}\max_{q \in \mathcal{I}_{\frac{B}{\delta}}}\int_{-\infty}^{\infty} \frac{e^{-\frac{y^2}{2Bq\sigma^2}}}{\sqrt{2 \pi qB \sigma^2}} \left[ \frac{2y-1}{2qB\sigma^2}\right]_{(a-3)} dy .\end{aligned}$$
Proof of Theorem \[thm:gain\_lower\_bound\] is obtained by combining Lemma \[lemma:converse\_k\_1\] and Lemma \[lemma:achv\]. Theorem \[thm:gain\_lower\_bound\] provides a non-asymptotic lower bound on adaptivity gain. The bound can be viewed as two parts corresponding to two stages. Intuitively, the first part corresponds to the initial stage of the search, where the agent narrows down the target’s location to some coarse $\alpha$ fractions of the total search region, i.e., narrows to a section of width $\alpha B$ with high confidence. The second stage corresponds to refined the search within one of the coarse sections $\alpha B$ obtained from initial stage. This implies that an adaptive strategy can zoom in and confine the search to a smaller section to reduce the noise intensity. Whereas, a non adaptive strategy does not adapt to zoom in, and thus performs equally in both stages. We formalize this intuition in Lemma \[lemma:achv\]. Optimizing over $\alpha$ fraction of the first search we obtain a bound on expected number of measurements. We obtain the following corollary as a consequence of Theorem \[thm:gain\_lower\_bound\].
\[cor:two\_regime\_gains\] Let $\epsilon \in (0,1)$. For any $\epsilon$-reliable non-adaptive strategy $\mathfrak{c}_{\epsilon} \in \mathcal{C}^{NA}_{\epsilon}$ searching over a search region of width $B$ among $\frac{B}{\delta}$ with $\tau$ number of measurements, there exists an $\epsilon$-reliable adaptive strategy $\mathfrak{c}^{\prime}_{\epsilon} \in \mathcal{C}^{A}_{\epsilon}$ with $\tau^{\prime}$ number of measurements, such that for a fixed $B$ the asymptotic adaptivity gain grows logarithmically with the total number of locations, $$\begin{aligned}
\label{eq:delta_gain}
\lim_{\delta \to 0} \frac{\expe_{\mathfrak{c}_{\epsilon}}[\tau] - \expe_{\mathfrak{c}^{\prime}_{\epsilon}}[\tau^{\prime}]}{\log \frac{B}{\delta}}
\geq
\frac{1-\epsilon}{ C_{\text{BAWGN}}(q^{\ast}, q^{\ast}B \sigma^2)} - 1.\end{aligned}$$ For a fixed $\delta$, the asymptotic adaptivity gain grows at least linearly with total number of locations, $$\begin{aligned}
\label{eq:B_gain}
\lim_{B \to \infty} \frac{\expe_{\mathfrak{c}_{\epsilon}}[\tau] - \expe_{\mathfrak{c}^{\prime}_{\epsilon}}[\tau^{\prime}]}{\frac{B}{\delta} \log \frac{B}{\delta}}
\geq
\frac{(1-\epsilon) \sigma^2 \delta }{\log e}.\end{aligned}$$ Furthermore, we have $$\begin{aligned}
\label{eq:B_NA}
\lim_{B \to \infty} \frac{\min_{\mathfrak{c}_{\epsilon} \in \mathcal{C}_{\epsilon}^{NA}}\expe_{\mathfrak{c}_{\epsilon}}[\tau]}{\frac{B}{\delta} \log \frac{B}{\delta}}
\geq
\frac{(1-\epsilon) \sigma^2 \delta }{\log e},\end{aligned}$$ and $$\begin{aligned}
\label{eq:B_A}
\lim_{B \to \infty} \frac{\min_{\mathfrak{c}_{\epsilon} \in \mathcal{C}_{\epsilon}^{A}} \expe_{\mathfrak{c}^{\prime}_{\epsilon}}[\tau^{\prime}]}{\frac{B}{\delta} }
= 0.\end{aligned}$$
The proof of the above corollary is provided in Appendix-C.
The above corollary characterizes the two qualitatively different regimes previously discussed. For fixed $B$, as $\delta$ goes to zero the asymptotic adaptivity gain scales as only $\log \frac{B}{\delta}$, whereas for fixed $\delta$, as $B$ increases the asymptotic adaptivity gain scales as $\frac{B}{\delta} \log \frac{B}{\delta}$. In other words, target acquisition rate improves by a constant for fixed $B$ as $\delta$ decreases while it grows linearly with $B$ for a fixed $\delta$. In other words, adaptivity provides a larger gain in target acquisition rate for the regime where the total search width is growing than in the case where we fix the total width and shrink the location widths. In Section \[sec:num\_results\] we related this phenomenon to the diminishing capacity of BAWGN channel when the total noise $\frac{B\sigma^2}{2}$ grows.
Next we provide the main technical components of the proof of Theorem \[thm:gain\_lower\_bound\].
Converse: Non-Adaptive Search Strategies
----------------------------------------
\[lemma:converse\_k\_1\]
The minimum expected number of measurements required for any $\epsilon$-reliable non-adaptive search strategy can be lower bounded as $$\begin{aligned}
\min_{\mathfrak{c}_{\epsilon} \in \mathcal{C}^{NA}_{\epsilon}}\expe_{\mathfrak{c}_{\epsilon}}[\tau] \geq \frac{(1-\epsilon) \log \left(\frac{B}{\delta}\right) -h(\epsilon)}{C_{\text{BAWGN}}\left(q^{\ast}, q^{\ast} B \sigma^2 \right)}.\end{aligned}$$
Proof of the Lemma \[lemma:converse\_k\_1\] is provided in Appendix-A. The proof follows from the fact that clean signal $X_i$ and noise $Z_i$ are independent over time and independent of past observations for $i = 1,2, \ldots, n$, due to the non-adaptive nature of the search strategy. In the absence of information from past observation outcomes, the agent tries to maximize the mutual information $I(X_i, Y_i)$ at every measurement. Since $X_i \sim \ber(q_i)$ and $Z_i \sim \mathcal{N}(0, q_i B\sigma^2)$, the mutual information $I(X_i, Y_i) = C_{\text{BAWGN}}\left(q_i, q_i B \sigma^2 \right)$ is maximized at $q_i = q^{\ast}$.
Achievability: Adaptive Search Strategy
---------------------------------------
Consider the following two stage search strategy.
### First Stage (Fixed Composition Strategy $\mathfrak{c}^{1}_{\frac{\epsilon}{2}}$)
We group the $\frac{B}{\delta}$ locations of width $\delta$ into $\frac{1}{\alpha}$ sections of width $\alpha B$. Let $\mbf{W}^{\prime}$ denote the true location of the target among the sections of width $ \alpha B $. Now, we use a non-adaptive strategy to search for the target location among $\frac{1}{\alpha}$ sections of width $\alpha B$. In particular, we use a fixed composition strategy where at every time instant $n$, the fraction of total locations probed is fixed to be $q^{\ast}$. In other words, the measurement vector $\mbf{S}^{\prime}_n$ at every instant $n$ is picked uniformly randomly from the set of measurement vectors $\{\mbf{S}^{\prime} \in \mathcal{U}_{\frac{1}{\alpha}}: |\mbf{S}^{\prime}| = \lfloor \frac{q^{\ast}}{\alpha} \rfloor \}$. For the ease of exposition, we assume that $\frac{q^{\ast}}{\alpha}$ is an integer. Hence, for this strategy, at every $n$, $X_n \sim \ber(q^{\ast})$ and $Z_n \sim \mathcal{N}(0, q^{\ast}B\sigma^2)$. For all $i \in \{1,2, \ldots, \frac{1}{\alpha} \}$, let $\mbs{\rho}^{\prime}_n(i)$ be the posterior probability of the estimate $\hat{\mbf{W}}^{\prime}(i) = 1$ after reception of $\mbf{Y}^{n-1}$, i.e., $\mbs{\rho}^{\prime}_n(i): = \P \left( \hat{\mbf{W}}^{\prime}(i) = 1| \mbs{Y}^{n-1} \right)$ and let $\mbs{\rho}^{\prime}_n: = \left\{\mbs{\rho}^{\prime}_n(1), \mbs{\rho}^{\prime}_n(2), \ldots, \mbs{\rho}^{\prime}_{n}\left(\frac{1}{\alpha}\right) \right\}$. Assume that agent begins with a uniform probability over the $\frac{1}{\alpha}$ sections, i.e., $\mbs{\rho}^{\prime}_0 = \{\alpha, \alpha, \ldots, \alpha \}$. The posterior probability $\mbs{\rho}^{\prime}_{n+1}(i)$ at time $n+1$ when $Y_n = y$ is obtained by the following Bayesian update: $$\begin{aligned}
\label{eq:rho_update1}
\mbs{\rho}^{\prime}_{n+1}(i)
=
\left\{
\begin{array}{ll}
\frac{\mbs{\rho}^{\prime}_{n}(i) G(y; 1, q^{\ast}B\sigma^2)}{\mathcal{D}^{\prime}_n} & \text{if } \mbs{S}^{\prime}_n(i) = 1,\\
\frac{\mbs{\rho}^{\prime}_{n}(i) G(y; 0, q^{\ast}B\sigma^2)}{\mathcal{D}^{\prime}_n} & \text{if } \mbs{S}^{\prime}_n(i) = 0,
\end{array}
\right.\end{aligned}$$ where $$\begin{aligned}
\label{eq:rho_norm1}
\mathcal{D}^{\prime}_n
= \sum_{j: \mbf{1}_{\{\mbs{S}_n(j) = 1\}}}\mbs{\rho}^{\prime}_{n}(j) G(y; 1, q^{\ast}B\sigma^2)
+ \sum_{j: \mbf{1}_{\{\mbs{S}_n(j) = 0\}}}\mbs{\rho}^{\prime}_{n}(j) G(y; 0, q^{\ast}B\sigma^2).\end{aligned}$$
Let $\tau^{1} : = \inf\left\{n: \max_{i} \mbs{\rho}^{\prime}_n(i) \geq 1- \frac{\epsilon}{2} \right\}$ be the number of measurements used under stage 1. Note that $\tau^{1}$ is a random variable. Hence, first stage is a non-adaptive variable length strategy. Now, the expected stopping time $\expe_{\mathfrak{c}^{1}_{\frac{\epsilon}{2}}}[\tau^{1}]$ can be upper bounded using Lemma \[lemm:stage\_1\_time\] from Appendix-B.
### Second Stage (Sorted Posterior Matching Strategy $\mathfrak{c}_{\frac{\epsilon}{2}}^2$)
In the second stage, the agent zooms into the $\alpha B$ width section obtained from the first stage and uses an adaptive strategy to search only within this $\alpha B$ section. The agent searches for the target location of width $\delta$ among the remaining $\frac{\alpha B}{\delta}$ locations. In particular, we use the sorted posterior matching strategy proposed in [@SungEnChiu] which we describe next. Let $\mbf{W}^{\prime \prime}$ denote the true target location of width $\delta$. For all $i \in \{1,2, \ldots, \frac{\alpha B}{\delta} \}$, let $\mbs{\rho}^{\prime \prime}_n(i)$ be the posterior probability of the estimate $\hat{\mbf{W}}^{\prime \prime}(i) = 1$ after reception of $\mbf{Y}^{n-1}$, i.e., $\mbs{\rho}^{\prime}_n(i): = \P \left( \hat{\mbf{W}}^{\prime \prime}(i) = 1| \mbf{Y}^{n-1} \right)$ and let $\mbs{\rho}^{\prime \prime}(n): = \{\mbs{\rho}^{\prime \prime}_n(1), \mbs{\rho}^{\prime \prime}_n(2), \ldots, \mbs{\rho}^{\prime \prime}_{n}\left(\frac{\alpha B}{\delta}\right) \}$. Assume the agent begins with a uniform probability over the $\frac{\alpha B}{\delta}$ sections, i.e., $\mbs{\rho}^{\prime \prime}_0 = \left\{\frac{\delta}{\alpha B}, \frac{\delta}{\alpha B}, \ldots, \frac{\delta}{\alpha B} \right\}$. At every time instant $n$, we sort the posterior values in descending order to obtain the sorted posterior vector $\mbs{\rho}^{\downarrow}_n$. Let vector $I_n$ denote the corresponding ordering of the location indices in the new sorted posterior. Define $$\begin{aligned}
k^{\ast}_n:= \operatorname*{arg\,min}_{i} \left| \sum_{j = 1}^{i} \mbs{\rho}^{\downarrow}_n(j)- \frac{1}{2}\right|.
\end{aligned}$$ We choose the measurement vector $\mbf{S}_n^{\prime \prime}$ such that $\mbf{S}_n^{\prime \prime}(j) = 1$ if and only if $j \in \{I_n(1),I_n(2), \ldots, I_n(k^{\ast}_n)\}$. Note that for this strategy, at every $n$, the noise is $Z_n \sim \mathcal{N}(0, |\mbf{S}^{\prime \prime}_n|\delta \sigma^2)$ and the worst noise intensity is $\mathcal{N}(0, \frac{\alpha B \sigma^2}{2})$. The posterior probability $\mbs{\rho}^{\prime \prime}_{n+1}(i)$ at time $n+1$ when $Y_n = y$ is obtained by the following Bayesian update: $$\begin{aligned}
\label{eq:rho_update2}
\mbs{\rho}^{\prime \prime}_{n+1}(i)
=
\left\{
\begin{array}{ll}
\frac{\mbs{\rho}^{\prime \prime}_{n}(i) G(y; 1, |\mbf{S}^{\prime \prime}_n|\delta \sigma^2)}{\mathcal{D}^{\prime \prime}_n} & \text{if } \mbs{S}^{\prime \prime}_n(i) = 1,\\
\frac{\mbs{\rho}^{\prime \prime}_{n}(i) G(y; 0, |\mbf{S}^{\prime \prime}_n|\delta \sigma^2)}{\mathcal{D}^{\prime \prime}_n} & \text{if } \mbs{S}^{\prime \prime}_n(i) = 0,
\end{array}
\right.\end{aligned}$$ where $$\begin{aligned}
\label{eq:rho_norm2}
\mathcal{D}^{\prime \prime}_n
= \sum_{j: \mbf{1}_{\{\mbs{S}_n(j) = 1\}}}\mbs{\rho}^{\prime \prime}_{n}(j) G\left(y; 1,|\mbf{S}^{\prime \prime}_n|\delta \sigma^2\right)
+ \sum_{j: \mbf{1}_{\{\mbs{S}_n(j) = 0\}}}\mbs{\rho}^{\prime \prime}_{n}(j) G\left(y; 0, |\mbf{S}^{\prime \prime}_n|\delta \sigma^2\right).\end{aligned}$$ Let $\tau^{2} : = \inf\left\{n: \max_{i} \mbs{\rho}^2_n(i) \geq 1- \frac{\epsilon}{2} \right \}$ be the number of measurements used under stage 2. Note that $\tau^{2}$ is a random variable. Hence, the second stage is an adaptive variable length strategy. The expected number of measurements $\expe_{\mathfrak{c}^{2}_{\frac{\epsilon}{2}}}[\tau^{\prime \prime}]$ can be upper bounded using Lemma \[lemm:sortPM\_tau\] from Appendix-B.
Noting that the total probability of error of the two stage search strategy is less than $\epsilon$ and that the expected stopping time is $\expe_{\mathfrak{c}^{\prime}_{\epsilon}}[\tau^{\prime}] = \expe_{\mathfrak{c}^1_{\frac{\epsilon}{2}}}[\tau^{1}]+ \expe_{\mathfrak{c}^2_{\frac{\epsilon}{2}}}[\tau^{2}]$, we have the assertion of the following lemma.
\[lemma:achv\] The minimum expected number of measurements required for the above $\epsilon$-reliable adaptive search strategy $\mathfrak{c}^{\prime}_{\epsilon}$ can be upper bounded as $$\begin{aligned}
\expe_{\mathfrak{c}^{\prime}_{\epsilon}}[\tau^{\prime}]
\leq
\min_{\alpha \in \mathcal{I}_{\frac{B}{\delta}}}\left\{\frac{\log \frac{1}{\alpha} + \log \frac{2}{\epsilon} + \log \log \frac{1}{\alpha} + a_{\eta}}{C_{\text{BAWGN}}\left(q^{\ast}, q^{\ast} B \sigma^2 \right) -
\eta}
+
\frac{\log \frac{\alpha B}{\delta} + \log \frac{2}{\epsilon} + \log \log \frac{\alpha B}{\delta} + a_{\eta}}{C_{\text{BAWGN}}\left(\frac{1}{2}, \frac{\alpha B \sigma^2}{2} \right) -
\eta} \right\}.\end{aligned}$$
For an $\epsilon$-reliable adaptive search strategy $\mathfrak{c}^{\prime}_{\epsilon} \in \mathcal{C}^{A}_{\epsilon}$ using the two stage strategy, the non-asymptotic upper bound provided by Lemma \[lemma:achv\] for $\min_{\mathfrak{c}^{\prime}_{\epsilon} \in \mathcal{C}_{\epsilon}^{A}} \expe^{\prime}_{\mathfrak{c}_{\epsilon}}[\tau^{\prime}]$ is tighter than the upper bound provided in [@SungEnChiu] using the sorted posterior matching strategy. In fact, for any given $\alpha$, our bound is significantly smaller than the upper bound in [@SungEnChiu]. In the asymptotically dominating terms of the order $\log \frac{B}{\delta}$, our upper bound closely follows the simulations as illustrated in Section \[sec:num\_results\].
In the regime of fixed $B$ and diminishing $\delta$, Lemma \[lemma:achv\] together with Corollary \[cor:channelstate\] establishes the optimality of our proposed algorithm. Further, it characterizes a lower bound on the increase in targeting capacity when utilizing an adaptive strategy over the non-adaptive strategies.
Extensions and Generalizations
==============================
Generalization to other noise models
------------------------------------
The main results presented in this paper consider the setup where the noise $Z_n$ is distributed as $\mathcal{N}(0, |\mbf{S}_n|\delta \sigma^2)$. In other words, the variance of the noise given by $(|\textbf{S}_n|\delta \sigma^2)$ is a linear function of the size of a measurement vector $|\mbf{S}_n|$. This model assumption holds when each target location adds noise equally and independently of other locations when probed together. In general, due to correlation across locations the additive noise variance can be assumed to scale as a non-decreasing function $f(\cdot)$ of the measurement vector $|\mbf{S}_n|$. In this section, we extend our model to a general formulation for the noise $Z_n \sim \mathcal{N}(0, f(|\mbf{S}_n|)\delta \sigma^2)$, where $f(\cdot)$ is a non-decreasing function of $|\mbf{S}_n|$. For example, $f(\textbf{S}_n) = |\mbf{S}_n|^{\gamma}$ for some $\gamma > 0$. Figure \[fig:capacity\], shows that the effect of the noise function $f(|\mbf{S}_n|)$ on the capacity.
![Behavior of capacity of BAWGN channel with $\sigma^2=0.25$ over a total search region of width $B=10$, location width $\delta =0.1$, as a function of the size of a measurement $|S_n|$.[]{data-label="fig:capacity"}](Capacity_func_gamma){width="70.00000%"}
\[thm:fgain\_lower\_bound\] Let $\epsilon \in (0,1)$ and let $f(\cdot)$ be a non-decreasing function. For any $\epsilon$-reliable non-adaptive strategy $\mathfrak{c}_{\epsilon} \in \mathcal{C}^{NA}_{\epsilon}$ searching over a search region of width $B$ among $\frac{B}{\delta}$ locations with $\tau$ number of measurements, there exists an $\epsilon$-reliable adaptive strategy $\mathfrak{c}^{\prime}_{\epsilon} \in \mathcal{C}^{A}_{\epsilon}$ with $\tau^{\prime}$ number of measurements, such that for some small constant $\eta > 0$ the following holds $$\begin{aligned}
\expe_{\mathfrak{c}_{\epsilon}}[\tau] - \expe_{\mathfrak{c}^{\prime}_{\epsilon}}[\tau^{\prime}]
&
\geq \left(\max_{\alpha \in \mathcal{I}_{\frac{B}{\delta}}} \left\{
\log \frac{1}{\alpha} \left(\frac{(1-\epsilon) }{C_{\text{BAWGN}}(q^{\ast}, f(\frac{q^{\ast} B}{\delta})\delta \sigma^2)} \right. \right. -\frac{1}{C_{\text{BAWGN}} \left( q^{\ast}, f(\frac{q^{\ast} B}{\delta})\delta \sigma^2\right) - \eta}
\right)
\nonumber
\\
&
\quad + \log \frac{\alpha B}{\delta} \left(\frac{(1-\epsilon) }{
C_{\text{BAWGN}}(q^{\ast}, f(\frac{q^{\ast} B}{\delta})\delta\sigma^2)} \right.
\nonumber
\\
& \left. \left. \left. \hspace{1 cm} - \frac{1}{
C_{\text{BAWGN}} \left(\frac{1}{2}, f(\frac{\alpha B}{2\delta})\delta \sigma^2\right)-\eta} \right) \right\} \right) (1+o(1)),\end{aligned}$$ where $
q^{*}
=
\operatorname*{arg\,max}_{q \in \mathcal{I}_{\frac{B}{\delta}}} C_{\text{BAWGN}}(q, f(\frac{qB}{\delta})\delta\sigma^2),
$ and $o(1)$ goes to 0 as $\frac{B}{\delta} \to \infty$.
Multiple Targets {#generalsetup}
----------------
The problem formulation and the main results of this paper consider the special case when there exists a single stationary target. Suppose instead the agent aims to find the true location of $r$ unique targets quickly and reliably. Our problem formulation is easily extended to the general case where there may exist multiple targets. In our generalization to multiple targets under the linear noise model (\[eq:noisysearch\]), the clean signal indicates the the number of targets present in the measurement vector $\textbf{S}_n$. In particular, let $\mbf{W}^{(i)} \in \mathcal{U}_{\frac{B}{\delta}}$ be such that $\mbf{W}^{(i)}(j) = 1$ if and only if $j$-th location contains the $i$-th target. Then, the noisy observation is given as $$\begin{aligned}
Y_n = \sum_{i = 1}^{r} (\mbf{W}^{(i)})^{\intercal}\mbf{S}_n + Z_n,\end{aligned}$$ where $Z_n \sim \mathcal{N}(0, |\mbf{S}_n|\delta \sigma^2)$. Setting $X_n^{(i)} = (\mbf{W}^{(i)})^{\intercal}\mbf{S}_n $ for $i \in [r]$, we have $$\begin{aligned}
Y_n = \sum_{i = 1}^{r} X^{(i)}_n + Z_n.\end{aligned}$$ The problem of searching for multiple targets is equivalent to the problem of channel coding over a Multiple Access Channel (MAC) with state and feedback [@nancy_asilomar]. In other words, we can extend the Proposition 1, to channel coding over a MAC with state and feedback with the following constraints: (i) $\mbf{W}^{(i)}$ can be viewed as the message to be transmitted by the $i$-th transmitter, (ii) the measurement matrix $\overline{\mbf{S}}_n$ can be viewed as the common codebook shared by all the transmitters, and (iii) a search strategy dictates the evolution of the MAC state. The channel transition is then fixed by the channel state which is measurement dependent.
**Example $\mathbf{1^{\prime}}$** (Establishing initial access in mm-Wave communications). In the deployment of mm-Wave links into a cellular or 802.11 network, the base station needs to to quickly switch between users and accommodate multiple mobile clients. In this setup at time $n$ the noisy observation, $Y_n$, is a function of multiple users in the network, in addition to a measurement dependent noise.
**Example $\mbs{2^{\prime}}$** (Spectrum Sensing for Cognitive Radio). Consider the problem of opportunistically searching for $r$ vacant subbands of bandwidth $\delta$ over a total bandwidth of $B$. In this problem we desire to locate $r$ stationary vacant subbands quickly and reliably, by making measurements over time. Here again the noise intensity depends on the number of subbands probed, $\mbf{S}_n$, at each time instant $n$.
Searching for multiple targets with measurement dependent noise is a significantly harder problem compared to a single target case and achievability strategies for this problem even in the absence of noise are far more complex [@Bshouty_kTargetsweighing; @Chang_kTargetsCode] .
Numerical Results {#sec:num_results}
=================
In this section we provide numerical analysis.
Comparing Search Strategies
---------------------------
In this section, we numerically compare four strategies proposed in the literature. Besides the sort PM strategy $\mathfrak{c}_{\epsilon}^2$ and the optimal variable length non-adaptive strategy i.e., the fixed composition strategy $\mathfrak{c}_{\epsilon}^1$, we also consider two noisy variants of the binary search strategy. The noisy binary search applied to our search proceeds by selecting $\mbf{S}_n$ as half the width of the previous search region $\mbf{S}_{n-1}$ with higher posterior probability. The first variant we consider is fixed length noisy binary search, resembles the adaptive iterative hierarchical search strategy [@7460513], where each measurement vector $\mbf{S}_n$ is used $\alpha_{\epsilon}(\mbf{S}_n)|\mbf{S}_n|$ times where $\alpha_{\epsilon}(\mbf{S}_n)$ is chosen such that entire search result in an $\epsilon$-reliable search strategy. The second variant is variable length noisy binary search where each measurement vector $\mbf{S}_n$ is used until in each search we obtain error probability less than $\epsilon_p:=\frac{\epsilon}{\log{B/\delta}}$. Table I provides a quick summary of the search strategies.
[|l|l|]{} &\
Strategies $\mathfrak{c}_{\epsilon} \in \mathcal{C}_{\epsilon}$& Description of $\mbf{S}_n$ selection\
Variable Length Random & $\bullet$ Select $\mbf{S}_n$ s.t. $|\mbf{S}_n| = \frac{q^{\ast}B}{\delta}$
------------------------------------------------------------------------
\
& as dictated by strategy $\mathfrak{c}_{\epsilon}^1$\
Fixed Length Noisy Binary& $\bullet$ Select $\mbf{S}_n$ as dictated by
------------------------------------------------------------------------
\
& binary search strategy\
&$\bullet$ Repeat $\alpha_{\epsilon}(\mbf{S}_n)|\mbf{S}_n|$ times\
Variable Length Noisy Binary& $\bullet$ Select $\mbf{S}_n$ as dictated by
------------------------------------------------------------------------
\
& binary search strategy\
&$\bullet$ Repeat $\tau$ times s.t.\
& $\tau = \min \{n: \|\mbs{\rho}_n \|{_{\scalebox{0.5}{$\infty$}}} \geq 1- \epsilon_p\}$\
Sorted Posterior Matching&$\bullet$ Select $\mbf{S}_n$ as dictated by
------------------------------------------------------------------------
\
& Sort PM strategy $\mathfrak{c}_{\epsilon}^2$\
Figure \[fig:EN\_strategies\], shows the performance of each $\epsilon$-reliable search strategy, when considering fixed parameters $B$, $\delta$, and $\epsilon$. We note that the fixed length noisy binary strategy performs poorly in comparison to the optimal non-adaptive strategy. This shows that randomized non-adaptive search strategies such as the one considered in [@Abari_AgileLink] perform better than both exhaustive search and iterative hierarchical search strategy. In particular, it performs better than variable length noisy binary search since when SNR is high since each measurement is repeated far too many times in order to be $\epsilon$-reliable. The performance of the optimal fully adaptive variable length strategies sort PM [@SungEnChiu] is superior to all strategies even in the non-asymptotic regime.
![$\mathbb{E}_{\mathfrak{c}_{\epsilon}}[\tau]$ with $\epsilon = 10^{-4}$, $B=16$, and $\delta=1$, as a function of $\sigma^2$ for various strategies.[]{data-label="fig:EN_strategies"}](Strategies2){width="70.00000%"}
Two Distinct Regimes of Operation {#sect:mainsimulations}
---------------------------------
In this section, for a fixed $\sigma^2$ we are interested in the expected number of measurements required $\mathbb{E}_{\mathfrak{c}_{\epsilon}}[\tau]$ by an $\epsilon$-reliable strategy $\mathfrak{c}_{\epsilon}$, in the following two regimes: varying $\delta$ while keeping $B$ fixed, and varying $B$ while keeping $\delta$ fixed. Figures \[fig:EN\_varyB\] and \[fig:EN\_varyDelta\] show the simulation results of $\mathbb{E}_{\mathfrak{c}_{\epsilon}}[\tau]$ as a function of width $B$ and resolution $\delta$ respectively, for the fixed composition non adaptive strategy $\mathfrak{c}_{\epsilon}\in \mathcal{C}_{\epsilon}^{NA}$ and for the sort PM adaptive strategy $\mathfrak{c}_{\epsilon}\in \mathcal{C}_{\epsilon}^{A}$, along with dominant terms of the lower bound of Lemma \[lemma:converse\_k\_1\], and the upper bound of Lemma \[lemma:achv\] for a fixed noise per unit width $\sigma^2=0.25$. For both of these cases, we see that the adaptivity gain grows as the total number of locations increases; however in distinctly different manner as seen in Corollary \[cor:two\_regime\_gains\].
![$\mathbb{E}_{\mathfrak{c}_{\epsilon}}[\tau]$ with $\epsilon = 10^{-4}$, $\sigma^2=0.25$, and $\delta=1$, as a function of B.[]{data-label="fig:EN_varyB"}](Vary_B){width="70.00000%"}
![$\mathbb{E}_{\mathfrak{c}_{\epsilon}}[\tau]$ with $\epsilon = 10^{-4}$, $\sigma^2=0.25$ and $B=1$, as a function of $\delta$.[]{data-label="fig:EN_varyDelta"}](Vary_delta){width="70.00000%"}
Relating the Regimes of Operation to Capacity
---------------------------------------------
In this section, we attempt to relate these two regimes of operation to the manner in which the capacity of a BAWGN channel varies. Let noise parameter $Z_n \sim \mathcal{N}(0, 2q\sigma^2_{\text{Total}})$, where $q = \frac{|\textbf{S}_n|\delta}{B}$ is the fraction of the search region measured and $\sigma^2_{\text{Total}} = \frac{B \sigma^2}{2}$ is the half bandwidth variance. Figure \[fig:EN\_sigma\] show the effects of the half bandwidth variance on the capacity of a search as a function of $q$. Intuitively, the target acquisition rate of the adaptive strategy relates to the time spent searching sets of size $q$ as $q$ varies from $\frac{1}{2}$ to $\frac{\delta}{B}$. This means for sufficiently small $\sigma^2_{\text{Total}}$ ($\leq 0.005$ in this example), the adaptivity gain is negligible since $C_{\text{BAWGN}}(\frac{1}{2}, 2q\sigma^2_{\text{Total}})$ is about 1 for all $q$. For medium range $\sigma^2_{\text{Total}}$ (for e.g., $ 0.05$ in this example), the adaptivity effects the target acquisition rate from $C_{\text{BAWGN}}(\frac{1}{2}, 2q^{\ast}\sigma^2_{\text{Total}})$ to $C_{\text{BAWGN}}(\frac{1}{2}, 2\frac{\delta}{B}\sigma^2_{\text{Total}})$. When $\sigma^2_{\text{Total}}$ grows significantly, however, the capacity drops rather quickly to zero, forcing the non-adaptive strategies to operate close to exhaustive search, whose measurement time increases linearly in $\frac{B}{\delta}$. This is the regime with most significant adaptivity gain as predicted by Corollary \[cor:two\_regime\_gains\].
![For arbitrary $B$ and $\delta$, and with $\epsilon = 10^{-4}$, $\mathbb{E}_{\mathfrak{c}_{\epsilon}}[\tau]$ as a function of $q$ for different values of total noise variance ($\sigma^2_{Total}$)[]{data-label="fig:EN_sigma"}](Capacity_Func_SigmaTotal){width="70.00000%"}
Beyond i.i.d
------------
In this section, we analyze $\mathbb{E}_{\mathfrak{c}_{\epsilon}}[\tau]$ under a general noise model, as presented in section (VI-A). Recall, $Y_n \sim \mathcal{N}(X_n, f(|\mbf{S}_n|)\delta \sigma^2)$, where $f$ is a non-decreasing function of the measurement vector $|\textbf{S}_n|$. Figure \[fig:capacity\] shows that the behavior of the capacity range of a search with fixed parameters $B$, $\delta$, $\textbf{S}_n$ can be significantly affected by the function $f(\cdot)$. Let us consider the noise function $f(\cdot)$ to be of the form $ |\mbf{S}_n|^{\gamma}$. Figure \[fig:EN\_varygamma\] shows the plot of dominant terms of the lower bound of Lemma \[lemma:converse\_k\_1\], and the upper bound of Lemma \[lemma:achv\] as a function of $\sigma^2$ for the values of $\gamma \in \{0.5, 1, 2\}$. The adaptivity gain is clearly more significant for larger values of gamma and hence, validates the need for generalizing the noise function.
![$\mathbb{E}_{\mathfrak{c}_{\epsilon}}[\tau]$ with $\epsilon = 10^{-4}$, $\sigma^2=0.25$ and $B=25$, $\delta =1$, as a function of $\gamma$ when $Z_n \sim \mathcal{N}(0, |\mbf{S}_n|^{\gamma}\delta \sigma^2)$.[]{data-label="fig:EN_varygamma"}](varygamma_varysigma){width="70.00000%"}
Conclusion and Future Work
==========================
We considered the problem of searching for a target’s unknown location under measurement dependent Gaussian noise. We showed that this problem is equivalent to channel coding over a BAWGN channel with state and feedback. We used this connection to utilize feedback code based adaptive search strategies. We obtained information theoretic converses to characterize the fundamental limits on the target acquisition rate under both adaptive and non-adaptive strategies. As a corollary, we obtained a lower bound on the adaptivity gain. We identified two asymptotic regimes with practical applications where our analysis shows that adaptive strategies are far more critical when either noise intensity or the total search width is large. In contrast, in scenarios where neither the total width nor noise intensity is large, non-adaptive strategies might perform quite well. The immediate step is the extension of this work to a model with $r>1$ target locations, where the problem has been shown to be equivalent to MAC encoding with feedback [@nancy_asilomar].
\[appendix\]
Proof of Lemma 1
----------------
Applying Fano’s inequality [@CoverBook2nd] to any non-adaptive search strategy that locates the target among $\frac{B}{\delta}$ locations with $P_e \leq \epsilon$, we have $$\begin{aligned}
\log \left(\frac{B}{\delta}\right)
& \overset{(a)} \leq \frac{1}{1-\epsilon} \sup_{X^n} \sum_{i = 1}^{n} I(X_i, Y_i) +\frac{h(\epsilon)}{1-\epsilon}
\nonumber
\\
&\overset{(b)}\leq \frac{1}{1-\epsilon}\sum_{i = 1}^{n} C_{\text{BAWGN}} \left(q_i, q_i B \sigma^2 \right) +\frac{h(\epsilon)}{1-\epsilon}
\nonumber
\\
&\leq \frac{n}{1-\epsilon} \max_{q \in \mbf{I}_{\frac{B}{\delta}}} C_{\text{BAWGN}}(q, q B \sigma^2) +\frac{h(\epsilon)}{1-\epsilon},\end{aligned}$$ where $(a)$ follows from the fact that $X_i$ and $Z_i$ for $i = 1,2, \ldots, n$ are independent over time and independent of past observations due to the non-adaptive nature of the search strategy. Since $X_i \sim \ber(q_i)$ and $Z_i \sim \mathcal{N}(0, q_i B\sigma^2)$, $(b)$ follows from the fact that $I(X_i, Y_i) = C_{\text{BAWGN}}\left(q_i, q_i B \sigma^2 \right)$. Rearranging the above equation, we have the assertion of the lemma.
Proof of Lemma 2
----------------
For any $q \in \mathcal{I}_{\frac{B}{\delta}}$ and under any query vector $\mbf{S}_n \in \mathcal{U}_{\frac{B}{\delta}}$ such that $|\mbf{S}_n| = \frac{qB}{\delta}$ we have the following $$\begin{aligned}
\left| \log \frac{\P(y| \mbf{S}_n, \mbf{W}(i) = 1)}{\P(y| \mbf{S}_n, \mbf{W}(j) = 1)} \right|
=\left\{
\begin{array}{cl}
0 & \text{if $\mbf{S}_n(i) = 1$ and $\mbf{S}_n(j) =1$}, \\
0 & \text{if $\mbf{S}_n(i) \neq 1$ and $\mbf{S}_n(j) \neq 1$},\\
\left| \frac{2y-1}{2qB\sigma^2}\right| &
\text{Otherwise}.\\
\end{array}
\right.\end{aligned}$$ Hence, we have $$\begin{aligned}
\max_{i,j \in [\frac{B}{\delta}]} \max_{\mbf{S}_n \in \mathcal{U}_{\frac{B}{\delta}}}\int_{-\infty}^{\infty}\P(y| \mbf{S}_n, \mbf{W}(i) = 1)
\left|\log \frac{\P(y| \mbf{S}_n, \mbf{W}(i) = 1)}{\P(y| \mbf{S}_n, \mbf{W}(j) = 1)} \right|^{1+\gamma}
\nonumber
\\
=
\max_{q \in \mathcal{I}_{\frac{B}{\delta}}}\left\{\int_{-\infty}^{\infty} \frac{e^{-\frac{y^2}{2qB\sigma^2}}}{\sqrt{2 \pi qB \sigma^2}} \left| \frac{2y-1}{2qB\sigma^2}\right|^{1+\gamma} dy \right\}.\end{aligned}$$ Therefore, there exists $\xi_{\frac{B}{\delta}} < \infty$ and $\gamma > 0$ such that $$\begin{aligned}
&\max_{i,j \in [\frac{B}{\delta}]} \max_{\mbf{S}_n \in \mathcal{U}_{\frac{B}{\delta}}}\int_{-\infty}^{\infty}\P(y| \mbf{S}_n, \mbf{W}(i) = 1)
\left| \log \frac{\P(y| \mbf{S}_n, \mbf{W}(i) = 1)}{\P(y| \mbf{S}_n, \mbf{W}(j) = 1)} \right|^{1+\gamma}
\leq \xi_{\frac{B}{\delta}}.\end{aligned}$$ Define $$\begin{aligned}
\psi_{\frac{B}{\delta}}(a)
: =
\max_{q \in \mathcal{I}_{\frac{B}{\delta}}}\left\{\int_{-\infty}^{\infty} \frac{e^{-\frac{y^2}{2Bq\sigma^2}}}{\sqrt{2 \pi qB \sigma^2}} \left[ \frac{2y-1}{2qB\sigma^2}\right]_a dy \right\},\end{aligned}$$ and recall that $$\begin{aligned}
[g]_a =
\left\{
\begin{array}{ll}
g & \text{if $g \geq a$,}\\
0 & \text{if $g < a$.}
\end{array}
\right.\end{aligned}$$ Note that $\psi_{\frac{B}{\delta}}(a)$ is non-increasing in a, and we have $\psi_{\frac{B}{\delta}}(a) \leq a^{-\gamma}\xi_{\frac{B}{\delta}}$. Hence, $\psi_{\frac{B}{\delta}}(a) \to 0$ as $a \to \infty$.
### Stage I
\[lemm:stage\_1\_time\] Under the fixed composition search strategy while searching over a search region of width $B$ among $\frac{1}{\alpha}$ locations such that $|\mbf{S}^{\prime}_n|\alpha = q^{\ast}$ for $n \geq 1$, the following holds true for all $n \geq 1$ $$\begin{aligned}
\expe\left[U(\mbs{\rho}^{\prime}_{n+1}) - U(\mbs{\rho}^{\prime}_n)| \mathcal{F}_n , \mbf{S}^{\prime}_n \right]
\geq C_{\text{BAWGN}}\left(q^{\ast}, q^{\ast} B \sigma^2 \right),\end{aligned}$$ where define $
U(\mbs{\rho}^{\prime}_n)
:= \sum_{i = 1}^{\frac{1}{\alpha}} \mbs{\rho}^{\prime}_n(i)\log \frac{\mbs{\rho}^{\prime}_n(i)}{1-\mbs{\rho}^{\prime}_n(i)}.
$
The proof follows closely the proof of inequality (9) in [@6400990]. There are $\frac{1}{\alpha}$ locations of length $\alpha B$ and hence query vector $\mbf{S}^{\prime}_n \in \mathcal{U}_{\frac{1}{\alpha}}$. At every time instant under the fixed composition strategy $K^{\ast} = |\mbf{S}_n|= \frac{q^{\ast}}{\alpha}$ number of locations are searched. i.e., a region of length $q^{\ast}B$ is searched. Let $\mathcal{P}_{K^{\ast}}$ denote the collection of all partitions $p$ of search locations $1$ to $\frac{1}{\alpha}$ into sets $A_n^0$ and $A_n^1$ such that $|A_n^1| = K^{\ast}$. The probability of picking a partition $p \in \mathcal{P}_{K^{\ast}}$ is $\lambda_p = {{\frac{1}{\alpha}}\choose{K^{\ast}}}^{-1}$. For simplicity of exposition let $M = \frac{1}{\alpha}$. Also, we have $\sum_{p \in \mathcal{P}_{K^{\ast}}}\lambda_p\mbf{1}_{ \{i \in A^0_n\}} = \pi^{\ast}_{0} := \frac{M - K^{\ast}}{M}$, and $\sum_{p \in \mathcal{P}_{K^{\ast}}}\lambda_p\mbf{1}_{\{i \in A^1_n\}} = \pi^{\ast}_{1} :=\frac{K^{\ast}}{M}$.
Since a region of $q^{\ast}B$ is searched at every time instant, the noise variance is fixed at $q^{\ast}B \sigma^2$. Hence, let $\P_k = \P(Y|X=k, |A^1_n|= K^{\ast}) = \mathcal{N}(k, q^{\ast} B \sigma^2)$ for $k\in \{0,1 \}$. Consider $$\begin{aligned}
&\expe\left[U(\mbs{\rho}^{\prime}_{n+1}) - U(\mbs{\rho}^{\prime}_n)| \mathcal{F}_n , \mbf{S}_n \right]
\nonumber
\\
&=
\sum_{p \in \mathcal{P}_{ K^{\ast}}} \lambda_p \sum_{i = 1}^M \sum_{k = 0}^{1} \mbs{\rho}^{\prime}_n(i) \mbf{1}_{\{i \in A^k_n\}}{D\left( \P_k \left\| \sum_{j \neq i} \sum_{l = 1}^{1}\frac{\mbs{\rho}^{\prime}_n(j)}{1-\mbs{\rho}^{\prime}_n(i)}\mbf{1}_{\{i \in A^l_n\}} \P_l \right. \right)}
\nonumber
\\
& = \sum_{i = 1}^M \mbs{\rho}^{\prime}_n(i) \sum_{k = 0}^{1} \pi^{\ast}_{k} \sum_{p \in \mathcal{P}_{ K^{\ast}}} \frac{\lambda_p }{\pi^{\ast}_{k}} \mbf{1}_{\{i \in A^k_n\}} {D\left( \P_k \left\| \sum_{j \neq i} \sum_{l = 1}^{1}\frac{\mbs{\rho}^{\prime}_n(j)}{1-\mbs{\rho}^{\prime}_n(i)}\mbf{1}_{\{i \in A^l_n\}} \P_l \right. \right)}
\nonumber
\\
&\overset{(a)}\geq
\sum_{i = 1}^M \mbs{\rho}^{\prime}_n(i) \sum_{k = 0}^{1} \pi^{\ast}_{k} D\left( \P_k \left\| \sum_{j \neq i} \sum_{l = 1}^{1}\frac{\mbs{\rho}^{\prime}_n(j)}{1-\mbs{\rho}^{\prime}_n(i)} \right. \sum_{p \in \mathcal{P}_{ K^{\ast}}} \frac{\lambda_p}{\pi^{\ast}_{k}}\mbf{1}_{\{i \in A^k_t\}}\mbf{1}_{\{i \in A^l_t\}} \P_l\right)
\nonumber
\\
&\overset{(b)}=
\sum_{i = 1}^M \mbs{\rho}^{\prime}_n(i) \left(
\pi^{\ast}_1{D\left( \P_1 \left\| \frac{ K^{\ast}-1}{M-1}\P_1 + \frac{M- K^{\ast}}{M-1} P_0 \right. \right)} \right.
\nonumber
\\
& \hspace{1cm}+
\left.
\pi^{\ast}_0 {D\left( \P_0 \left\| \frac{M- K^{\ast}-1}{M-1}\P_0 + \frac{ K^{\ast}}{M-1} P_1 \right. \right)}
\right]
\nonumber
\\
&\geq
\sum_{i = 1}^M \mbs{\rho}^{\prime}_n(i) \left(
\pi^{\ast}_1 {D\left( \P_1 \left\| \frac{ K^{\ast}}{M}\P_1 + \frac{M- K^{\ast}}{M} P_0 \right. \right)}
\pi^{\ast}_0 {D\left( \P_0 \left\| \frac{M- K^{\ast}}{M}\P_0 + \frac{ K^{\ast}}{M} P_1 \right. \right)}
\right)
\nonumber
\\
&\overset{(c)} = C_{\text{BAWGN}} \left( q^{\ast}, q^{\ast} B \sigma^2 \right),\end{aligned}$$ where $(a)$ follows from Jensen’s inequality $(b)$ follows from the definition of $\pi_0^{\ast}$, $\pi_1^{\ast}$ and $\sum_{p \in \mathcal{P}_{K^{\ast}}}\lambda_p\mbf{1}_{\{i \in A^0_n\}}\mbf{1}_{\{j \in A^1_n\}}
= \sum_{p \in \mathcal{P}_{K^{\ast}}}\lambda_p\mbf{1}_{\{i \in A^1_n\}}\mbf{1}_{\{j \in A^0_n\}} = \frac{K^{\ast}(M-K^{\ast})}{M(M-1)}$, $\sum_{p \in \mathcal{P}_{K^{\ast}}}\lambda_p\mbf{1}_{\{i \in A^0_n\}}\mbf{1}_{\{j \in A^0_n\}} = \frac{\pi^{\ast}_0(M-K^{\ast}-1)}{M-1}$, $\sum_{p \in \mathcal{P}_{K^{\ast}}}\lambda_p\mbf{1}_{\{i \in A^1_n\}}\mbf{1}_{\{j \in A^1_n\}} = \frac{\pi_1^{\ast}(K^{\ast}-1)}{M-1}$, and $(c)$ is the definition of non-adaptive BAWGN channel capacity with input distribution $\text{Bern}(q^{\ast})$ and noise variance $q^{\ast}B \sigma^2$.
Under the fixed composition search strategy while searching over a search region of width $B$ among $\frac{1}{\alpha}$ locations such that $|\mbf{S}^{\prime}_n|\alpha = q^{\ast}$ for $n \geq 1$, the following holds true for the expected number of queries while searching with $\P_e \leq \frac{\epsilon}{2}$ $$\begin{aligned}
\expe_{\mathfrak{c}_{\epsilon}^1}[\tau^1]
\leq
\frac{\log \frac{1}{\alpha} + \log \frac{2}{\epsilon} + \log \log \frac{B}{\delta} + a_{\eta}}{C_{\text{BAWGN}}\left(q^{\ast}, q^{\ast} B \sigma^2 \right) -
\eta}.\end{aligned}$$
Proof is similar to the proof of Lemma \[lemm:sortPM\_tau\].
### Stage II
Note that BAWGN capacity for all $q \in \mathcal{I}_{\frac{B}{\delta}}$ $$\begin{aligned}
C_{\text{BAWGN}}\left( \frac{1}{2}, qB\sigma^2\right)
&=
D\left( \mathcal{N}(0, qB\sigma^2) \left\| \frac{1}{2}\mathcal{N}(0, qB \sigma^2) + \frac{1}{2}\mathcal{N}(1, qB\sigma^2)\right. \right)
\nonumber
\\
&=
D\left(\mathcal{N}(1, qB \sigma^2) \left\| \frac{1}{2}\mathcal{N}(0, qB\sigma^2) + \frac{1}{2}\mathcal{N}(1, qB\sigma^2)\right. \right).\end{aligned}$$ Hence, the following Lemma follows from Proposition 3 in [@7031961].
\[lemm:EJSGaussian\] Under the sortPM search strategy while searching over a search region of width $\alpha B$ among $\frac{\alpha B}{\delta}$ locations, the following holds true for all $n \geq 1$ $$\begin{aligned}
\expe\left[U(\mbs{\rho}^{\prime \prime}_{n+1}) - U(\mbs{\rho}^{\prime \prime}_n)| \mathcal{F}_n , \mbf{S}_n \right]
\geq C_{\text{BAWGN}}\left(\frac{1}{2}, \frac{\alpha B \sigma^2}{2} \right),\end{aligned}$$ where define $
U(\mbs{\rho}^{\prime \prime}_n)
:= \sum_{i = 1}^{\frac{\alpha B}{\delta}} \mbs{\rho}^{\prime \prime}_n(i)\log \frac{\mbs{\rho}^{\prime \prime}_n(i)}{1-\mbs{\rho}^{\prime \prime}_n(i)}.
$
\[lemm:sortPM\_tau\] Under the sortPM search strategy, the following holds true for the expected number of queries while searching over the search width $\alpha B$ among $\frac{\alpha B}{\delta}$ locations with $\P_e \leq \frac{\epsilon}{2}$ $$\begin{aligned}
\expe_{\mathfrak{c}_{\epsilon}^2}[\tau^{2}]
\leq
\frac{\log \frac{\alpha B}{\delta} + \log \frac{2}{\epsilon} + \log \log \frac{\alpha B}{\delta} + a_{\eta}}{C_{\text{BAWGN}}\left(\frac{1}{2}, \frac{\alpha B \sigma^2}{2} \right) -
\eta},\end{aligned}$$ where $a_{\eta}$ is the solution of the following equation $$\begin{aligned}
\eta =\frac{a}{a-3}\psi_{\frac{B}{\delta}}(a-3).\end{aligned}$$
Let $M = \frac{\alpha B}{\delta}$. Let $\tilde{\rho}^{\prime} = 1 - \frac{1}{1+\max\{\log M, \frac{2}{\epsilon}\}}$. Now, define $U^{\prime}(\mbs{\rho}^{\prime \prime}_0) = U(\mbs{\rho}^{\prime \prime}_0)- \log \frac{\tilde{\rho}^{\prime}}{1-\tilde{\rho}^{\prime}}$ and define $U^{\prime}(\mbs{\rho}^{\prime \prime}_n)$ as follows: if $U^{\prime}(\mbs{\rho}^{\prime \prime}_n) < 0$, then $$\begin{aligned}
\label{eq:UPrimeCase1}
U^{\prime}(\mbs{\rho}^{\prime \prime}_{n+1})
=
\left\{
\begin{array}{ll}
U(\mbs{\rho}^{\prime \prime}_{n+1}) - U(\mbs{\rho}^{\prime \prime}_{n}) + U^{\prime}(\mbs{\rho}^{\prime \prime}_n) & \text{if $U(\mbs{\rho}^{\prime \prime}_{n+1}) - U(\mbs{\rho}^{\prime \prime}_{n}) < a - U^{\prime}(\mbs{\rho}^{\prime \prime}_n)$,} \\
a & \text{if $U(\mbs{\rho}^{\prime \prime}_{n+1}) - U(\mbs{\rho}^{\prime \prime}_{n}) \geq a - U^{\prime}(\mbs{\rho}^{\prime \prime}_n)$,}
\end{array}
\right.\end{aligned}$$ and if $U^{\prime}(\mbs{\rho}^{\prime \prime}_n) \geq 0$, then $$\begin{aligned}
\label{eq:UPrimeCase2}
U^{\prime}(\mbs{\rho}^{\prime \prime}_{n+1})
=
\left\{
\begin{array}{ll}
U(\mbs{\rho}^{\prime \prime}_{n+1}) - U(\mbs{\rho}^{\prime \prime}_{n}) + U^{\prime}(\mbs{\rho}^{\prime \prime}_n) &
\text{if $U(\mbs{\rho}^{\prime \prime}_{n+1}) - U(\mbs{\rho}^{\prime \prime}_{n}) < a$,} \\
a + U^{\prime}(\mbs{\rho}^{\prime \prime}_n)
& \text{if $U(\mbs{\rho}^{\prime \prime}_{n+1}) - U(\mbs{\rho}^{\prime \prime}_{n}) \geq a $.}
\end{array}
\right.\end{aligned}$$ By induction we can show that $$\begin{aligned}
\label{eq:UPrimeAndU}
\log \frac{\tilde{\rho}^{\prime}}{1-\tilde{\rho}^{\prime}}
\leq U(\mbs{\rho}^{\prime \prime}_n) - U^{\prime}(\mbs{\rho}^{\prime \prime}_n).\end{aligned}$$ We have $$\begin{aligned}
&\expe\left[U^{\prime}(\mbs{\rho}^{\prime \prime}_{n+1}) - U^{\prime}(\mbs{\rho}^{\prime \prime}_n)| \mathcal{F}_n \right]
\nonumber
\\
&=
\expe\left[U(\mbs{\rho}^{\prime \prime}_{n+1}) - U(\mbs{\rho}^{\prime \prime}_n)| \mathcal{F}_n \right]
+
\expe\left[ \left[-b - U(\mbs{\rho}^{\prime \prime}_{n+1}) + U(\mbs{\rho}^{\prime \prime}_n) -U^{\prime}(\mbs{\rho}^{\prime \prime}_n) \mbf{1}_{\{U^{\prime}(\mbs{\rho}^{\prime \prime}_n) < 0 \}} \right]^{+}| \mathcal{F}_n\right]
\nonumber
\\
& \overset{(a)}\geq \expe\left[U(\mbs{\rho}^{\prime \prime}_{n+1}) - U(\mbs{\rho}^{\prime \prime}_n)| \mathcal{F}_n \right] -
\frac{a}{a-3}\psi_{\frac{B}{\delta}}(a-3)
\nonumber
\\
& \overset{(b)}\geq C_{\text{BAWGN}}\left(\frac{1}{2}, \frac{\alpha B \sigma^2}{2} \right) -
\frac{a}{a-3}\psi_{\frac{B}{\delta}}(a-3),\end{aligned}$$ where $(a)$ follows from MN thesis eq (4.140) and $(b)$ follows Lemma \[lemm:EJSGaussian\]. Let $\tau^{\prime} = \min\{n: U^{\prime}(\mbs{\rho}^{\prime \prime}_n) \geq 0\}$ and $\tau_{\frac{\epsilon}{\epsilon}} = \min\{n: U(\mbs{\rho}^{\prime \prime}_n) \geq \log \frac{\tilde{\rho}}{1-\tilde{\rho}}\}$ where $\tilde{\rho} = 1- \frac{2}{\epsilon}$. From equation and since $\tilde{\rho}^{\prime} > \tilde{\rho}$, we have $$\begin{aligned}
\label{eq:stopping_time_comp}
\expe_{\mathfrak{c}_{\epsilon}^2}[\tau_{\frac{\epsilon}{2}}] \leq \expe_{\mathfrak{c}_{\epsilon}^2}|\tilde{\tau}^{\prime}].\end{aligned}$$ The sequence $\frac{U^{\prime}(\mbs{\rho}^{\prime \prime}_n)}{C_{\text{BAWGN}}\left(\frac{1}{2}, \frac{\alpha B \sigma^2}{2} \right) -
\frac{a}{a-3}\psi_{\frac{B}{\delta}}(a-3)} - n$ forms a submartingale with respect to filtration $\mathcal{F}_n$. Now by Doob’s Stopping Theorem we have $$\begin{aligned}
\frac{U^{\prime}(\mbs{\rho}^{\prime \prime}_{0})}{C_{\text{BAWGN}}\left(\frac{1}{2}, \frac{\alpha B \sigma^2}{2} \right) -
\frac{a}{a-3}\psi_{\frac{B}{\delta}}(a-3)}
\leq
\expe \left[
\frac{U^{\prime}(\mbs{\rho}^{\prime \prime}_{\tilde{\tau}^{\prime}})}{C_{\text{BAWGN}}\left(\frac{1}{2}, \frac{\alpha B \sigma^2}{2} \right) -
\frac{a}{a-3}\psi_{\frac{B}{\delta}}(a-3)}
- \tilde{\tau}^{\prime}
\right].\end{aligned}$$ Hence, we have $$\begin{aligned}
\label{eq:doon_ineq}
\expe_{\mathfrak{c}_{\epsilon}^2}[\tilde{\tau}^{\prime}]
&\leq
\frac{-U^{\prime}(\mbs{\rho}^{\prime \prime}_0) + \expe[U^{\prime}(\mbs{\rho}^{\prime \prime}_{\tilde{\tau}^{\prime}})]}{C_{\text{BAWGN}}\left(\frac{1}{2}, \frac{\alpha B \sigma^2}{2} \right) -
\frac{a}{a-3}\psi_{\frac{B}{\delta}}(a-3)}
\nonumber
\\
&
= \frac{-U(\mbs{\rho}^{\prime \prime}_0) + \log \frac{\tilde{\rho}^{\prime}}{1-\tilde{\rho}^{\prime}} + \expe[U^{\prime}(\mbs{\rho}^{\prime \prime}_{\tilde{\tau}^{\prime}})]}{C_{\text{BAWGN}}\left(\frac{1}{2}, \frac{\alpha B \sigma^2}{2} \right) -
\frac{a}{a-3}\psi_{\frac{B}{\delta}}(a-3)}
\nonumber
\\
& \overset{(a)}\leq
\frac{\log \frac{\alpha B}{\delta} +\log \log \frac{\alpha B}{\delta} + \log \frac{2}{\epsilon} + \expe[U^{\prime}(\mbs{\rho}^{\prime \prime}_{\tilde{\tau}^{\prime}})]}{C_{\text{BAWGN}}\left(\frac{1}{2}, \frac{\alpha B \sigma^2}{2} \right) -
\frac{a}{a-3}\psi_{\frac{B}{\delta}}(a-3)}
\nonumber
\\
& \overset{(b)}\leq
\frac{\log \frac{\alpha B}{\delta} + \log \log \frac{\alpha B}{\delta} + \log \frac{2}{\epsilon} + a}{C_{\text{BAWGN}}\left(\frac{1}{2}, \frac{\alpha B \sigma^2}{2} \right) -
\frac{a}{a-3}\psi_{\frac{B}{\delta}}(a-3)},\end{aligned}$$ where $(a)$ follows from the fact that $U(\mbs{\rho}^{\prime \prime}_0) = -\log (\frac{B}{\delta} - 1)$ and $(b)$ follows from the fact that for all $n < \tau^{\prime}$, $U^{\prime}(\mbs{\rho}^{\prime \prime}_n) < 0$ and hence from equation we have $U^{\prime}(\mbs{\rho}^{\prime \prime}_{\tilde{\tau}^{\prime}}) < a$. Let $\eta> 0$ such that $\eta
\ll C_{\text{BAWGN}}\left(\frac{1}{2}, \frac{\alpha B \sigma^2}{2} \right)$. Choose $a_{\eta}$ such that $$\begin{aligned}
\eta =\frac{a}{a-3}\psi_{\frac{B}{\delta}}(a-3).\end{aligned}$$ We have the assertion of the lemma by combining above equation with equations and .
Proof of Corollary 2
--------------------
Choose $a_{\eta} = \log \log \frac{B}{\delta}$ so that $\eta$ goes to zero as $\frac{B}{\delta} \to \infty$, and choose $\alpha(\frac{B}{\delta}) = \frac{1}{\log \frac{B}{\delta}}$. Note that $\alpha (\frac{B}{\delta})$ goes to $0$ slower than $\delta$ goes to $0$. Combining this with Theorem \[thm:gain\_lower\_bound\] and using the fact $\lim_{\delta \to 0}C_{\text{BAWGN}}\left( \frac{1}{2}, \frac{1}{2} \alpha\left( \frac{B}{\delta}\right) B \sigma^2 \right) = 1$, we have equation (\[eq:delta\_gain\]). Similarly, note that $\alpha (\frac{B}{\delta})$ goes to $0$ slower than $B$ goes to $\infty$. Using loose approximations $C_{\text{BAWGN}}(q^{\ast}, q^{\ast}B\sigma^2)\leq \frac{\log e}{B \sigma^2}$ and $C_{\text{BAWGN}}\left(\frac{1}{2}, \alpha(\frac{B}{\delta}) B \sigma^2 \right) \geq \frac{\log(\frac{B}{\delta})}{16B \sigma^2} \left( 1 - \frac{\log(\frac{B}{\delta})}{16B \sigma^2} \right)$ with Theorem \[thm:gain\_lower\_bound\] we have equations (\[eq:B\_NA\]–\[eq:B\_gain\]).
[^1]: Preliminary versions of this work were presented at the 50th Asilomar Conference on Signals, Systems, and Computers, and at 2017 International Symposium on Information Theory.
[^2]: A. Lalitha N. Ronquillo, and T. Javidi are with the Department of Electrical and Computer Engineering, University of California San Diego, La Jolla, CA 92093, USA. (e-mail: alalitha@ucsd.edu; nronquil@ucsd,edu; tjavidi@ucsd.edu).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We present a new approach to equivariant version of the topological complexity, called a symmetric topological complexity. It seems that the presented approach is more adequate for the analysis of an impact of symmetry on the motion planning algorithm than the one introduced and studied by Colman and Grant. We show many bounds for the symmetric topological complexity comparing it with already known invariants and prove that in the case of a free action it is equal to the Farber’s topological complexity of the orbit space. We define the Whitehead version of it.'
address:
- |
Theoretical Computer Science Department\
Faculty of Mathematics and Computer Science\
Jagiellonian University\
30-348 Kraków, Poland
- 'Faculty of Mathematics and Computer Science, Adam Mickiewicz University of Poznań, ul. Umultowska 87, 61-614 Poznań, Poland'
author:
- Wojciech Lubawski
- 'Wacław Marzantowicz$^{1}$'
title: A new approach to the equivariant topological complexity
---
[^1]
Introduction
============
A topological invariant introduced by Farber in [@farber1; @farber2], and called the topological complexity, was the first to estimate a complexity of motion planning algorithm. With the configuration space $X$ of a mechanical robot he associated a natural number $TC(X)$ called topological complexity of $X$. To be more precise he considered the natural fibration $$\label{eq1}\pi\colon PX \rightarrow X\times X$$ from the free path space in $X$ which assigns to a path $\gamma$ defined on the unit interval its ends $(\gamma(0), \gamma(1))$. The topological complexity is the least $n$ such that $X\times X$ can be covered by $n$ open sets $U_1 , \ldots , U_n$ such that for each $i$ there is a homotopical section $s_i\colon U_i\rightarrow PX$ to $\pi$. This invariant is a special case of the well known Lusternik-Schnirelmann (or LS for short) category of $X\times X$ (cf [@colman-grant] for more detailed exposition of this notion and other references).
In this paper we discuss the following question: If the mechanical robot admits a symmetry with respect to a compact Lie group (and therefore the configuration space $X$ admits it too) what is an appropriate definition of the topological complexity that takes into account that symmetry? An answer is not that simple as it may look like and it is not unique. We define an invariant, different than the equivariant topological complexity introduced by H. Colman and A. Grant in [@colman-grant], called the symmetric topological complexity. By showing its properties we would like to demonstrate that in many situations it is better than that of [@colman-grant].
Let $G$ be a compact Lie group. Let us assume that $X$ is a $G$ space, i.e. $G$ acts continuously on $X$ (therefore we assume that $G$ is the “symmetry group” that appears in $X$). The formulae for topological complexity uses the natural fibration \[eq1\]. If the space $X$ is a $G$ space then $PX$ is a $G$ space in a natural way, and so does $X\times X$ by the diagonal action. It would be natural to define the equivariant complexity by assuming that all maps are $G$ maps. Actually this approach has been studied in [@colman-grant]. We will use the notation introduced there $TC_G(X)$ to denote this invariant. In spite of its mathematical naturalness this approach has some disadvantages that we present below.
![A symmetric robot arm with an action of $\tau$[]{data-label="fig:1"}](robotarm.png)
Let us consider a mechanical robot arm that admits a symmetry. For simplicity let us assume that $G = \mathbb Z / 2=\{ 1,\tau \}$ as showed in the picture \[fig:1\]. The element $\tau$ acts by interchanging the part $A$ of the arm with $B$. Assume we are given a path $\xi$ between points $x$ and $y$ in the configuration space $X$, as noted in picture \[fig:2\].
![A path in configuration space[]{data-label="fig:2"}](robotarmMove.png)
Note that although points $x$ and $\tau x\in X$ are distinct in the configuration space there is no physical difference between these two states of a mechanical arm as can be observed from picture \[fig:1\]. Therefore it is natural to require that the path $\xi$ determines a path between $\tau x$ and $\tau y$ – namely $\tau \xi$. This natural requirement leads us to a definition of equivariant topological complexity $TC_G(X)$. On the other hand if the task the mechanical arm is supposed to perform is symmetric we would like the path $\xi$ to determine the following four paths – between $x$ and $y$, $\tau x$ and $\tau y$ as well as between $x$ and $\tau y$, $\tau x$ and $y$. In other words we would like to exploit the $G\times G$ structure of the space $X\times X$. The main problem is that usually $PX$ is not a $G\times G$ space. We will show in section \[2.\] how to deal with this problem by defining so called symmetric topological complexity, $STC_G(X)$.
Unlike many other equivariant versions of numerical invariants the equivariant topological complexity does not have the required mathematical properties – for example when the group $G$ acts freely on $X$ then in general $TC_G(X)\neq TC(X/G)$ where $X / G$ is the orbit space and $TC(X/G)$ is the topological complexity of $X/G$. We will show that in our case $STC_G(X) = TC(X/G)$. A bridge to apply advanced homotopy theory in the theory of Lusternik-Schnirelmann category is the Whitehead version of it (cf. [@whitehead1]). We will show that for the symmetric topological complexity we can define a Whitehead version of it and for a finite group $G$ it gives the original symmetric topological complexity. We conjecture that the same holds for any compact Lie group. Finally we provide examples which distinguish the equivariant topological complexity and the symmetric topological complexity and calculate the latter in several cases.
Lusternik-Schnirelmann category
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Basic definitions
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In this section we define and give some basic properties of a version of an equivariant Lusternik-Schnirelmann category for topological spaces that we will use later on in our considerations. We shall the standard notations of the theory of compact Lie group transformations of [@bredon].
Let $G$ be a Lie group and let $A$ be a closed $G$ subset of a $G$ space $X$.
A metrizable $G$-space $X$ we call a $G$-ANR if for every equivariant imbedding $\iota: X \to Y $ as a closed $G$-subset of a metrizable $G$-space $ Y$ there exists a $G$ neighborhood $U$ of $\iota(X)$ in $Y$ and a continuous equivariant retraction $U\to
\iota(X)$.
Throughout this paper we assume that $X$ is a compact $G$-ANR (see [@murayama] for the properties of $G$-ANRs). The class of $G$-ANRs includes $G$-ENRs (cf. [@jaworowski] for the definition), countable $G$-CW complexes thus smooth $G$-manifolds with smooth action of $G$.
\[1:1\] We call on open $G$ set $U\subseteq X$ *$G$-compressable* into $A$ whenever the inclusion map $\iota_U\colon U\subseteq X$ is $G$ homotopic to $c\colon U\rightarrow X$ such that $c(U)\subseteq A$.
This allows us to define our main tool
\[1:2\]\[Adfi3\] An $A$-Lusternik-Schnirelmann $G$-category of a $G$ space $X$ is the least $n$ such that $X$ can be covered by $U_1,\ldots , U_n$ open $G$ subsets of $X$ each $G$-compressable into $A$. We denote in by $_Acat_G(X)$.
Remind, we say that a $G$-space $X$ is $G$-path-connected if for every closed subgroup $H\subset G$ the space $X^H$ is path-connected.
Note that we have a relation to the standard Lusternik-Schnirelmann category . By $\ast $ we denote a fixed one point subset of $X$ provided it is invariant, i.e $\ast
\in X^G$.
\[1:3\] If $X$ is path connected and $G$ is the trivial group then we have $$_\ast cat_G(X) = cat(X).$$ If $\ast\in X^G$ and $X^H$ is path connected for all closed subgroups $H\subseteq G$ then $$_{\ast} cat_G(X) = cat_G(X)\,$$ where $cat_G(X)$ denote the equivariant Lusternik-Schnirelmann category of $X$ (cf. [@marzan]).
This version of the LS category has many similarities to the standard category. We say that $(X,A)$ $G$-dominates $(Y,B)$ if there are $G$-maps $$f\colon (X,A)\rightarrow (Y,B)\text{ and }g\colon (Y,B)\rightarrow (X,A)$$ such that $fg\simeq id_{(Y,B)}$ in the equivariant topological category of pairs of spaces.
\[1:4\] If $(X,A)$ $G$-dominates $(Y,B)$, then $$_Acat_G(X) \geqslant _Bcat_G(Y).$$
The proof is similar to that of Lemma 1.29 in [@cornea] after a suitable change of categories.
The Whitehead definition of the category
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As the classical LS category, the defined above notion of a category (Definition \[Adfi3\]) has its Whitehead counterpart.
We recall that a pair of $G$-spaces $A\subset X$ is called a $G$-cofibration (or the Borsuk pair) if it has the equivariant homotopy extension property, i.e. for any $G$ -space $Y$ and $G$-homotopy $h: A \times I \to Y$ there exists an equivariant homotopy $H: X \times I \to Y$ extending $h$.
\[1:7\] Let $A\subseteq X$ be a closed $G$-cofibration with $A$ invariant. By a *[fat $A$-sum]{} we mean for every $n\in \mathbb N$ a $G$-space $F_A^n(X) \subset X^{n}:=X\times \ldots \times X$ defined as follows:*
- $F_A^1 := A$
- $F_A^n(X)$ is the colimit in the category of $G$-spaces of the following diagram: $$\xymatrix{A\times F^{n-1}_A(X)\ar[rr]\ar[d] && X\times F^{n-1}_A(X)\ar[d]\\
A\times X^{n-1}\ar[rr] && F_A^n(X)
}$$
\[1:8\] We say that the $G$-Whitehead $A$-category, denoted by $_Acat^{Wh}_G (X)$, is less or equal $n$ if and only if there is a $G$-mapping $\xi_n \colon X\rightarrow F^n_A(X)$ such that the following diagram is commutative:
$$\xymatrix{
X\ar[drr]^{\Delta_n} \ar[rr]^{\xi_n} && F^n_A(X)\ar[d]^{\subseteq}\\
&& X^n}$$ where $\Delta_n\colon X\rightarrow X^n$ is the diagonal mapping.
\[thm1\]\[1:9\] Let $X$ be a $G$-space and $\iota\colon A\subseteq X$ closed $G$-cofibration. Then $_Acat_G(X) = _Acat^{Wh}_G(X)$
Before giving the proof of the theorem (which follows the proof of theorem 1.55 in [@cornea]) we need a technical lemma:
\[1:10\] Under the assumptions of the theorem, if $\{U_i\}_{i=1}^n$ is an invariant open covering of $X$ such that for each $i$ there exists $G$-map $s_i\colon U_i\rightarrow A$ such that $G_i\colon \iota\circ s_i$ is $G$-homotopic to $(U_i\subseteq X)$ then there exist an open and invariant covering $\{V_i\}_{i=1}^n \leqslant \{U_i\}_{i=1}^n$ such that for each $i$ there exists a $G$-homotopy $H_i\colon X\times I\rightarrow X$ with $H_i(x,0) = x$ for each $x\in X$ and $H_i(x,1) = \iota\circ s_i$ for $x\in V_i$.
Here $\{V_i\}_{i=1}^n \leqslant \{U_i\}_{i=1}^n$ means that for every $1\leq i\leq n$, $V_i \subset U_i$. By a direct argument, we can find invariant coverings $\{V_i\}$ and $\{W_i\}$ of $X$ such that $$V_i\subseteq \bar{V_i}\subseteq W_i\subseteq \bar{W_i}\subseteq
U_i.$$ Since $X$ is a $G$-ANR, $X/G$ is normal. Moreover $(\bar{V}_i/G) \cap ((X\setminus W_i)/G) =\emptyset$ in $X/G$. Consequently, by normality of $X$ there exists a $G$-invariant continuous function $\lambda\colon X\rightarrow I$ be such that $\lambda(\bar{V_i}) = 1$ and $\lambda (X\backslash W_i) = 0$. For each $i$ we define the $G$-homotopy by: $$H_i\colon X\times I\ni (x,t)\mapsto \begin{cases} x & ,x\in X\backslash W_i \\ G_i(x , t\cdot \lambda (x)) & ,x\in \bar{W_i}\end{cases}$$
\[1:11\] Of course the converse implication in the lemma above also holds. Given a family of $G$-homotopies $H_i$ with an invariant covering $\{V_i\}$ of $X$ it is sufficient to set $s_i := H_i(- ,t)|_{V_i}$.
If $_Acat_G(X)\leqslant n$ then we have $n$ $G$-homotopies $H_i\colon X\times I\rightarrow X$ satisfying conditions of the lemma \[1:10\]. Now to show that $_Acat^{Wh}_G(X)\leqslant n$ it is sufficient to put $$\xi_n\colon X\ni x\mapsto (H_i(x,1))_{i=1}^n\in
F^n_A.$$
Conversely, if $_Acat^{Wh}_G(X)\leqslant n$ then we are given $\xi_n\colon X\rightarrow F^n_A(X)$ such that $\Delta$ and $(F^n_A(X)\subseteq X^n)\circ \xi_n$ are $G$-homotopic by a homotopy $\zeta$. We denote by $\zeta^i$ the $i$-th coordinate of $\zeta$. Since $A\subseteq X$ is a $G$-cofibration then there exists $N=N(A)$ an invariant and open neighborhood of $A$ in $X$ such that $A$ is a $G$-deformation retract of $N$. Let us denote this equivariant deformation retraction by $R$. Then $R(x,0) = x$ and $R(x,1)\in A$ for $x\in N$. Set $U_i := H ^{-1}(N,1)$. It is easy to see that $\{U_i\}$ is an invariant open covering of $X$. Moreover setting $$H_i\colon X\times I\ni \mapsto \begin{cases} \zeta^i(x, 2t) & ,0\leqslant t\leqslant 1/2\\ R(\zeta^i(x,1),2t-1) & ,1/2\leqslant t \leqslant 1.\end{cases}$$ we obtain the required family of $G$-homotopies with $s_i\colon U_i\rightarrow A$ equal to $H_i(-,1)|_{U_i}$.
For the case without symmetry, i.e. $G=e$, we have (comp. [@hatcher Chapter 4]) that a space $X$ is n-connected if $\pi_i(X) = 0$ for all $i \leq n$. Likewise, a pair $(X;A)$ is $n$-connected if $\pi_i(X;A) = 0$ for all $i \leq n$. A natural analog of the definition in the equivariant case is the following:
\[G-n-connected\] We call a $G$-space $X$ $G$-$n$-connected if $X^G\neq\emptyset$ and $\pi_i(X^H) = 0$ for all $i \leq n$ and all closed subgroups $H\subset G$. Likewise, a $G$-pair $(X,A)$ is $n$-connected if $A^G\neq\emptyset$ and $\pi_i(X^H,A^H) = 0$ for all $i
\leq n$ and all closed subgroups $H\subset G$.
The following fact is well-known in the non-equivariant case, e.g. [@hatcher Proposition 4.13 and Corollary 4.16]. Since we could not find any direct reference of the equivariant case we reprove the CW approximation theorem as stated in [@may theorem XI.3.6] making some minor changes.
\[Corollary 4.16\] If $(X,A)$ is an $n$-connected $G$-CW pair for a discrete $G$ then there exists a $G$-CW pair $(Z;A) \sim (X;A) \; rel\; A$ such that all cells of $Z \setminus A$ have dimension greater than $n$.
We construct a family of $G$-CW complexes $A\subseteq Y_0\subseteq Y_1\subseteq Y_2\subseteq \ldots $ together with maps $\gamma_i\colon (Y_i,A)\rightarrow (X,A)$ such that $\pi_q(\gamma_i)$ is a surjection for $q>i+n$ and an isomorphism for $q\leqslant i+n$. Let $b\in A^G$. We choose a representative map $$s^q_H\colon\mathbb (I^{q+1},\partial I^{q+1})\rightarrow (X^H,A^H)$$ for each element of $\pi_q (X^H, A^H, b)$ where $q>0$ and $H$ runs over conjugacy classes of closed subgroups of $G$. Let $\tilde{Y_0}$ be the disjoint union of spaces $G/H\times I^{q}$, one for each chosen $s^q_H$ and of $A$. Let $\tilde{\gamma_0}$ be a map induced by all $s^d_H$. For each $s^q_H\colon\mathbb (I^{q},\partial I^{q})\rightarrow (X^H,A^H)$ we identify each $x\in\partial I^{q}$ with $s^q_H(x)\in A$ in $\tilde{Y_0}$ hence obtaining $$\gamma_0\colon Y_0:= (\bigsqcup_{s^q_H} G/H\times I^{q}) \cup_{\sqcup {s^q_H | \partial I^{q}}} A.$$ Note that $\gamma_0^H$ is an isomorphisms on $\pi_i$ for $i\leqslant n$ and a surjection for $i>n$. Indeed, surjectivity is obvious for all $i$, for injectivity for $i\leqslant n$ let $H$ be a closed subgroup of $G$. Note that $(Y_0)^H$ is of the form $(\bigsqcup_{s^q_{H'}} I^q) \cup_{\sqcup s^q_{H'}| \partial I^q} A^H$ where the sum is taken over all $s^q_{H'}$ such that there is a $G$-map $G/H\rightarrow G/H'$. We have a cofibration $$A^H\rightarrow (Y_0)^H\rightarrow (\bigvee_{s^q_{H'}} \ S^q)$$ which shows that $H_i((Y_0)^H, A^H)$ for $i\leqslant n$. Therefore by [@whitehead corollary 7.10] we get that $\pi_i((Y_0)^H , A^H) = 0$.
Now assume that $\gamma_i\colon (Y_i,A)\rightarrow (X,A)$ has been constructed. We choose representative maps $(f,g)$ for each pair of elements in $\pi_{q}((Y_i)^H,A^H, b)$ that are equalized by $\pi_q(\gamma_i)$ (note that then $q>i+n$). Using the cellular approximation theorem [@may theorem 3.4] we assume that $f$ and $g$ have image in the $q$ skeleton of $Y_i$. Let $(Y_{i+1},A)$ be the homotopy coequalizer of the disjoint union of all such maps – i.e. $(Y_{i+1},A)$ is obtained by attaching $G/H_+\wedge (I^q,\partial I^q)\times I_+$ via each chosen pair. Note that such operation does not affect $\pi_\ast(Y_i,A)$ for $\ast\leqslant d+i$ and kills the kernel of $\pi_{i+d+1} (\gamma_i)$. We define $\gamma_{i+1}$ with the use of homotopies $\gamma_if\simeq \gamma_ig$ based at $b$. Now it is enough to triangulate $Y_{i+1}$ as a $G$-CW complex containing $Y_i$. We set $(Z,A) = \cup_i (Y_i,A)$ and $\gamma=\cup\gamma_i\colon (Z,A)\rightarrow (Y,A)$
For a compact, i.e. finite $G$-CW complex $X$ by $\dim_G X $ we mean the the maximal dimension of $G$-cells of the form $G/H
\times I^n$ that appear in the construction of $X$. Consequently, $\dim_G X = \dim X/G$ where the dimension on the right hand side is the cell dimension equals to the covering dimension of $X/G$. Furthermore, if $G$ is discrete then $\dim_G X = \dim X = \dim X/G$.
\[1:12\] Let $A\subseteq X$ be a pair of $G$-CW-complexes for a discrete $G$. pair $(X,A)$ is $G$ $n$-connected then $_Acat(X)\leqslant \dim_G X / (n+1) +1$.
Since $(X,A)$ is $G$ $n$-connected we may assume that $X\backslash
A$ has no $k$ dimensional $G$-cells for $k\leqslant n$. Then $F^s_A(X)$ and $X$ have similar $s(n+1)-1$ skeleton. Let $s$ satisfy $(s-1)(n+1) \leqslant \dim_G X \leqslant s(n+1)$ and using the equivariant cellular approximation (comp. [@may theorem 3.4]) theorem we get that the diagonal map $\Delta_s\colon X\rightarrow X^s$ is $G$ homotopic to a $G$ cellular map $\xi\colon X\rightarrow F^s_A(X)$.
At the end of this subsection we prove a technical lemma that will be used later on to prove the product formula for the category (theorem \[1:5\]).
\[1:15\] Let $X$ and $Y$ be $G$ spaces and $A\subseteq X$ and $B\subseteq Y$ its closed $G$ subsets. There is a commutative diagram $$\xymatrix{
F^n_A(X)\times F^m_B(Y)\ar[rr]\ar[d] && F^{n+m-1}_{A\times B}(X\times Y)\ar[d]\\
X^n\times Y^m \ar[rr]^{\omega^{n,m}} && (X\times Y)^{n+m-1}}$$ such that $\omega^{n,m}\circ(\Delta_n(X)\times \Delta_m(Y)) = \Delta_{n+m-1}(X\times Y)$.
We prove the theorem inductively. If $(n,m) = (n,1)$ then our diagram is of the form $$\xymatrix{
F^n_A(X)\times B\ar[rr]^\alpha\ar[d] && F^{n}_{A\times B}(X\times Y)\ar[d]\\
X^n\times Y \ar[rr]^\omega^{n,1} && (X\times Y)^{n}}$$ and it is easy to see that it is commutative whenever we set $$\alpha (x_1 , \ldots , x_n , b)= (x_1 , \ldots , x_n , b,\ldots , b).$$ The condition on the diagonal is also satisfied. Similar argument prove the statement for $(n,m) = (1,m)$. Now let us assume that $n,m\geqslant 2$. Since in the category of $G$ CW complexes the product of two pushouts is a pushout of products therefore we have a pushout $$\xymatrix{
A\times B\times F^n_A(X)\times F^m_B(Y)\ar[rr]\ar[d] && X\times Y\times F^n_A(X)\times F^m_B(Y)\ar[d]\\
A\times B\times X^n \times Y^m \ar[rr] && F^{n+1}_A(X) \times F^{m+1}_B(Y)}.$$ We get a commutative diagram
where $\eta$ is the universal map between two pushouts. The whole diagram is over the map $$\omega^{n+1,m+1}:= id_{X\times Y}\times \omega^{n,m}\colon X\times Y \times X^n\times Y^m\rightarrow X\times Y\times (X\times Y)^{n+m-1}$$ and the assertion follows.
Bounds for the category
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\[1:13\] Let $X$ be a $G$ set, $H\subseteq G$ closed subgroup and assume that $A\subseteq B$ are its closed invariant subsets. Then:
1) $_Bcat_G(X)\leqslant _Acat_G(X)$;
2) $_Acat_G(X)\leqslant _Bcat_G(X)\cdot _Acat_G(B)$;
3) $_{A/G}cat(X/G)\leqslant _Acat_G(X)$;
4) $_Acat_H(X)\leqslant _Acat_G(X)$;
For the proof of 2 let us assume that $_Bcat_G(X)\leqslant n$ and $_Acat_G(B)\leqslant m$. Let $U_1,\ldots ,U_n$ be open invariant subsets of $X$, each compressable by $s_i$ into $B$ and $V_1,\ldots ,V_m$ open invariant subsets of $B$, each compressable by $t_j$ into $A$. Let $$W^j_i := s_i^{-1}(V_j).$$ We know that $\{W^j_i\}$ is an invariant open covering of $X$. We define $r^j_i := t_j\circ s_i |_{W^j_i}$ then it can be readily seen that $W^j_i$ is compressable into $A$ by $r^j_i$. Since the cardinality of $\{W^j_i\}$ is $n\cdot m$ the inequality follows.
The rest of the points are obvious (or were mentioned before).
The category behaves well (i.e. similar to the standard LS category) under taking products.
\[1:5\] Let $X$ and $Y$ be two $G$ spaces, $A\subseteq X$, $B\subseteq Y$ their closed $G$ subsets. Then $$_{A\times B}cat_G(X\times Y)\leqslant _Acat_G(X) + _Bcat_G(Y) - 1$$ where $X\times Y$ is given the diagonal action.
We prove the theorem using lemma \[1:15\]. Note that whenever we have a commutative diagrams $$\xymatrix{
X\ar[drr]^{\Delta_n(X)} \ar[rr]^{\xi_n} && F^n_A(X)\ar[d]^{\subseteq} & Y\ar[drr]^{\Delta_m(Y)} \ar[rr]^{\xi'_m} && F^m_B(Y)\ar[d]^{\subseteq}\\
&& X^n &&& Y^m}$$ Then commutative is also the following diagram $$\xymatrix{
X\times Y \ar[drr]\ar[rr]^{\xi_n\times \xi'_m} && F^n_A(X)\times F^m_B(Y)\ar[d]^{\subseteq}\ar[rr] && F^{n+m-1}_{A\times B} (X\times Y)\ar[d]\\
&& X^n\times Y^m \ar[rr]^{\omega^{n,m}} && (X\times Y)^{n+m-1}}$$ Now $\omega^{n,m}\circ(\Delta_n(X)\times \Delta_m(Y)) = \Delta_{n+m-1}(X\times Y)$ which ends the proof.
Note that we do not need any additional assumption for the action of $G$ on $X$ and $Y$. In case $A$ and $B$ consists of a single point and $X$ and $Y$ are $G$ connected our result is equivalent to that obtained in [@colman-grant theorem 3.15] nevertheless our approach is much more general.
\[1:6\] Let $X$ be a $G$ space and $Y$ be a $H$ space for Lie groups $G$ and $H$. Then for a closed $G$ subspace $A\subseteq X$ and a closed $H$ subset $B\subseteq Y$ we have (where we consider $X\times Y$ as a standard $G\times H$ space) $$_{A\times B}cat_{G\times H}(X\times Y)\leqslant _Acat_G(X) + _Bcat_H(Y) - 1.$$
Follows directly from \[1:5\] since we can consider $X$ as a $G\times H$ space with trivial $H$ action and $Y$ as a $G\times H$ space with trivial $G$ action.
We end this section with a remark concerning the category of $H$ invariant elements for a closed subgroup $H\subseteq G$.
\[1:16\] Let $X$ be a $G$ space, $A\subseteq X$ its closed $G$ subset, $H$ closed subgroups of $G$ then $$_{A^H}cat_H(X^H) \leqslant _{A}cat_G(X).$$
If $U\subseteq X$ is $G$ compressable into $A$ then $V:= U\cap X^H$ is $H$ compressable into $A^H$ (which follows from the equivariant condition for the homotopy).
Topological robotics in presence of a symmetry {#2.}
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Basic classical concepts
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Let $X$ be a topological space with an action of a compact Lie group $G$. Consider the space of all continuous paths $s\colon I\rightarrow X$ with compact open topology denoted by $PX$. $PX$ posses a natural action of $G$.
Observe that the natural projection $$p\colon PX\ni s\mapsto (s(0),s(1))\in X\times X$$ is a continuous, $G$-equivariant $G$-fibration. Whenever we talk about$X\times X$ we consider it as a $G$-space (via the diagonal action) unless explicitly stated.
\[2:1\] By a motion planning algorithm on an open set $U\subseteq X\times X$ we mean a section $$s\colon U \rightarrow PX$$ of the fibration p, i.e. $p\circ s = (U\subseteq X\times X)$.
\[dfi1\]\[2:2\] An equivariant motion planning algorithm on an open set $U\subseteq X\times X$ is a $G$-equivariant section $$s\colon U \rightarrow PX$$ of the $G$-fibration $p$, i.e. $p\circ s =(U\subseteq X\times X)$.
*An invariant motion planning algorithm* is a motion planning algorithm of the orbit space.
Farber’s topological complexity
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\[dfi2\]\[2:3\] Topological complexity of $X$, denoted by $TC(X)$, is the smallest $n$ such that $X\times X$ can be covered by $U_1,\ldots ,U_n$ – open subsets such that for each $i$ there exists $s_i\colon
U_i\rightarrow PX$ a motion planing algorithm on $U_i$.
Similarly, equivariant topological complexity, denoted by $TC_G(X)$, of a $G$-space $X$ is the smallest $n$ such that $X\times X$ can be covered by $U_1,\ldots ,U_n$ – invariant open subsets such that for each $i$ there exists $s_i\colon U_i\rightarrow PX$ an equivariant motion planing algorithm on $U_i$.
Note that equivariant motion planning algorithm does not have to induce an invariant one – free path space is not $G\times G$-space unlike $X\times X$.
\[2:4\] Let $X = G = \mathbb S^1$ with $G$ acting by multiplication from the left. Note that $X/G$ is trivial so that $TC(X/G) = 1$ whereas $p\colon (\mathbb S^1)^I\rightarrow \mathbb S^1\times \mathbb S^1$ cannot have a section, in particular cannot have an equivariant one, so $TC_G(X)\geqslant 2$. This shows that topological complexity of an orbit space and equivariant topological complexity does not have to coincide, even in the simplest examples of a free action.
Our aim is to give a suitable definition of a motion planning algorithm in a equivariant setting which induces an invariant motion planning algorithm and as mentioned in the introduction have a reasonable geometric meaning. Moreover we want this motion planning algorithms to give a topological complexity witch coincides with the topological complexity of an orbit space for free $G$ spaces. In order to do so we will translate it into the language of the Lusternik-Schnirelmann category.
Let $\Delta_n\colon X\rightarrow X^n$ be the diagonal. We denote the image of $\Delta_2$ in $X\times X$ by $\Delta(X)$.
Let $X$ be a $G$ space. The map $$\pi\colon PX\rightarrow X\times X$$ is a $G$-fibration (satisfies the homotopy lifting property for $G$-maps).
Since $\{0,1\}\subseteq I$ is a closed $G$ cofibration (satisfies $G$ equivariant version of homotopy extension property) where we consider $I$ as a trivial $G$ space and functors $map_G(I, - )$ and $-\times I$ as well as $map_G(\{0,1\},-)$ and $-\times \{0,1\}$ are conjugate the proof follows from [@whitehead] theorem 7.8.
\[2:5\] \[lmm1\] For a $G$-space $X$ the following statements are equivalent:
1) $TC_G(X) \leqslant n$;
2) there exist $n$ invariant open sets $U_1,\ldots ,U_n$ which cover $X\times X$ and $\bar{s_i}\colon U_i\rightarrow PX$ such that $p\circ \bar{s_i}$ is $G$-homotopic to $id$ (as mappings $U_i\rightarrow X\times X$);
3) $_{\Delta (X)}cat_G (X\times X)\leqslant n$.
1)$\Rightarrow$2) is obvious.
2)$\Rightarrow$1). Let $s\colon U\rightarrow PX$ be such that $H\colon p\circ s\simeq (\colon U\subseteq X\times X)$ (as $G$-maps), where $p\colon PX\rightarrow X\times X$. Then from the equivariant homotopy lifting property there exists a $G$-homotopy $\hat{H}\colon U\times I\rightarrow PX$ such that the following diagram is commutative: $$\xymatrix{
U\times \{ 0\}\ar[rr]^s\ar[d]^\subseteq && PX\ar[d]^p\\
U\times I \ar[urr]^{\hat{H}} \ar[rr]^H && X\times X
}$$ now it is sufficient to set $\bar{s}(a,b):= \hat{H}(a,b;1)$.
2)$\Leftrightarrow $3). Let $H\colon PX\times I\rightarrow PX$ be given as: $$H(\omega; t) (s) = \omega (s(1-t))\text{ for }\omega\in PX,\, s,t\in I.$$ It is a $G$-deformation retraction between $PX$ and $\iota (X)\subseteq PX$, where $\iota(x)\equiv x$ assigns to every point $x\in X$ the constant map defined by it; $\iota$ in this case is a $G$-homeomorphism onto the image. Now given $\bar{s}\colon U\rightarrow PX$ we can compose it with $p\circ H_1$ to get $\hat{s}\colon U\rightarrow \Delta(X)$ $G$-homotopic in $X$ to the inclusion $U\subseteq X\times X$. On the other hand given $\hat{s}\colon U\rightarrow X$ we see that $\hat{s} = \Delta_2\circ \hat{s}'$, where $\hat{s}'\colon U\rightarrow X$ is a $G$-map. We can compose it with $\iota$ to get $\bar{s}\colon U\rightarrow PX$ such that $p\circ \bar{s}$ is homotopic in $X\times X$ to the inclusion $U\subseteq X\times X$. Note that $\Delta_2 = p\circ \iota$. These processes are mutually inverse up to $G$-homotopy so that we proved the equivalence.
Hence we obtained a characterization of topological complexity in terms of a suitable version of LS-category.
Symmetric topological complexity
--------------------------------
The main problem arising from geometric interpretation, as mentioned in the introduction, is that $PX$ is not a $G\times G$-space – which is equivalent to the problem that $\Delta(X)$ is not a $G\times G$-space. But the latter can be easily fixed.
For a given $G$-space $X$ by $\daleth (X)$ we denote the saturation $$\daleth (X):= (G\times G)\cdot\Delta(X)\subseteq X\times X$$ of the diagonal with respect to the group $G \times G$.
Now instead of $\Delta(X)$ in the definition of equivariant topological complexity we consider $\daleth (X) \subset X\times X$ and instead of considering open subsets $G$-compressable into $\Delta(X)$ we consider open subsets of $X\times X$ which are $G\times G$-compressable into $\daleth(X)$.
\[dfi4\]\[2:6\] For a $G$-space $X$ we define symmetric topological complexity as $$STC_G(X) = _{\daleth(X)}cat _{G\times G} (X\times X).$$
One should distinguish this notion with the symmetric motion planning algorithms studied in [@farber-grant] where a natural symmetry (action) of the group $\mathbb Z/2$ comes from the time reverse of the motion.
Let us state a lemma similar to \[lmm1\] but formulated for the symmetric topological complexity. For a $G$ space $X$ we consider $$PX\times_{\daleth(X)} PX := \{ (\gamma,\delta)\in PX\times PX\colon G\cdot\gamma(1) = G\cdot \delta(0) \}$$ as a $G\times G$ space with the obvious multiplication $(g_1,g_2)\cdot (\gamma ,\delta) = (g_1\gamma ,g_2\delta)$. Note that we have a natural $G\times G$ map $p\colon PX\times_{\daleth(X)} PX \rightarrow X\times X$ given by $p(\gamma ,\delta) = (\gamma(0), \delta (1))$.
Let $X$ be a $G$ space. The map $$p\colon PX\times_{\daleth(X)} PX\rightarrow X\times X$$ is a $G\times G$-fibration (satisfies the homotopy lifting property for $G$-maps).
Note that $\{(0,0), (0,1), (1,0), (1,1)\}\subseteq I\times I$ is a closed $G\times G$-cofibration where we consider $\{(0,0), (0,1), (1,0), (1,1)\}$ and $I\times I$ as trivial $G\times G$ spaces. Therefore from [@whitehead] theorem 7.8 the following map $$\overline{p}\colon (X\times X)^{I\times I}\rightarrow X\times X\times X\times X$$ where $\overline{p}(f) = (f(0,0),f(0,1),f(1,0),f(1,1))$ is a $G\times G$ fibration. We know also that any projection from the product of two $G\times G$ spaces is $G\times G$ fibration and that the composition of two $g\times G$ fibrations is a $G\times G$ fibration. Now it is enough to note that $$p = pr_{1,4}\circ \overline{p} | _{\overline{p}^{-1}(X\times \daleth(X)\times X)}\colon \overline{p}^{-1}(X\times \daleth(X)\times X)\rightarrow X\times X$$ which ends the proof.
\[2:7\] \[lmm2\] For a $G$-space $X$ the following statements are equivalent:
1) $STC_G(X) \leqslant n$;
2) there exist $n$ $G\times G$ invariant open sets $U_1,\ldots ,U_n$ which cover $X\times X$ and $G\times G$ maps $\bar{s_i}\colon U_i\rightarrow PX\times_{\daleth(X)} PX$ such that $p\circ \bar{s_i}$ is $G\times G$-homotopic to $id$ (as mappings $U_i\rightarrow X\times X$);
3) $_{\daleth (X)}cat_{G\times G} (X\times X)\leqslant n$.
1\) $\Leftrightarrow$ 3) by the definition.
3\) $\Rightarrow$ 2). Let $\iota_U\colon U\subseteq X\times X$ be an open $G\times G$ invariant subset that in $G\times G$ compressable into $\daleth(X)$. Let $H\colon \iota_U\simeq c$ be a $G\times G$ homotopy where $c(U)\subseteq \daleth(X)$. From the equivariant homotopy lifting property we get that $$\xymatrix{
U\times \{ 0\}\ar[rr]^s\ar[d]^\subseteq && PX\times_{\daleth(X)} PX\ar[d]^p\\
U\times I \ar[urr]^{\hat{H}} \ar[rr]^H && X\times X
}$$ where $s(u_1,u_2) = (c_{u_1},c_{u_2})$ and $c_u$ is the constant path equal to $u$. Now it is enough to set $s_i := \hat{H}(-,1)$.
2\) $\Rightarrow $ 3). Let $H\colon PX\times_{\daleth(X)} PX\times I\rightarrow PX\times_{\daleth(X)} PX$ be given as: $$H(\gamma, \delta,t)(\tau,\tau ') = (\gamma (\tau + t(1-\tau)), \delta((1-t)\tau ) ).$$ It is a $G\times G$ deformation retraction between $PX\times_{\daleth(X)} PX$ and $\iota(\daleth(X))\subseteq PX\times_{\daleth(X)} PX$ where $\iota$ assigns to $(u_1,u_2)$ the constant maps defined by it. Then if $s_i\colon U\rightarrow X$ is a $G\times G$ map such that $F\colon p\circ s_i\simeq id_ U$ for a $G\times G$ homotopy $F$ then $U$ is $G\times G$ compressable into $\daleth (X)$ as $$id_ U\simeq p\circ s_i \sim p\circ id_{PX\times_{\daleth(X)PX}} \circ s_1\simeq p\circ H(-,1)\circ s_i$$ and $H(-,1)\circ s_i\colon U\rightarrow \daleth(X)$.
\[2:8\] For a $G$-space $X$ we have inequality $TC(X/G)\leqslant STC_G(X)$.
One of our main requirements was that our version of equivariant topological complexity of $X$ should be equal to the topological complexity of the orbit space $X/G$. The symmetric topological complexity satisfies this condition.
\[2:9\] \[thm0\] Let $X$ be a free $G$-space. Then $TC(X/G) = STC_G(X)$.
Let us recall the Covering Homotopy Theorem of Palais:
\[dfi5\]\[2:10\] Let $G$ be a compact Lie group, $X$, $Y$ $G$-spaces, $f\colon X\rightarrow Y$ a $G$-equivariant map. Denote by $f'\colon X/G\rightarrow Y/G$ the map induced by $f$. Let $F'\colon X/G\times I\rightarrow Y/G$ be a homotopy which preserves the orbit structure and starts at $f'$. Then there exists an equivariant homotopy $X\times I\rightarrow Y$ covering $F'$ starting at $f$.
Assume that $TC(X/G)\leqslant n$ then there exists a $G\times G$ invariant covering $U_1,\ldots ,U_n$ of $X\times X$ and $s_i\colon U_i\rightarrow \Delta(X/G)$ such that $s_i$ is homotopic to $U_i\subseteq X$ via the homotopy $H\colon U_i\times I\rightarrow X\times X$ (we assume it starts at the identity). Since the action of $G\times G$ is free on $X\times X$, the homotopy $H$ preserves the orbit structure. Hence from the Covering Homotopy Theorem we get a $G\times G$-equivariant homotopy $\tilde{H}\colon U_i \times I\rightarrow X\times X$ starting at $U_i\subseteq X\times X$. For the orbit map $\pi\colon X\times X\rightarrow X/G\times X/G$ we have $\pi^{-1}(\Delta(X/G)) = \daleth(X)$ hence $G\times G$-map $\tilde{s_i}\colon U_i\rightarrow \daleth(X)$ can be defined by the formula $\tilde{s_i}(z) = \tilde{H}(z,1)$.
As a direct consequence of computation of the topological complexity of real projective space by M. Farber, S. Tabachnikov and S. Yuzvinsky ([@farber-tab-yuzv]) we get the following
\[symmetric complexity for projective space\]
If $n\neq 1,3,7,$ then $STC_{{\mathbb{Z}}_2}(S^n)$ is equal to the smallest $k$ for which ${\mathbb{R}}P^n$ admits an immersion in ${\mathbb{R}}^{k-1}$.
Whitehead symmetric topological complexity
------------------------------------------
From the classical theory (comp. [@dugundji] for the non-equivariant case, [@lewis] for short explanation how to pass to the equivariant one) we get that the map $$\Delta(X)\subseteq X\times X$$ for a $G$-CW complex $X$ is a closed $G$-cofibration; nevertheless the case of $$\daleth(X)\subseteq X\times X$$ and a question if it is a $G\times G$ cofibration is much more complicated and we do not know the answer for a general Lie group $G$. Here we will prove it for a finite $G$.
\[2:11\] From the theorem \[thm1\], lemma \[lmm1\] and the remark above we get that $$TC_G(X) = _{\Delta(X)}cat_G^{Wh}(X\times X)$$ In particular for the classical topological complexity we get that $$TC(X) = _{\Delta(X)}cat(X\times X) = _{\Delta(X)}cat^{Wh}(X\times X).$$ Moreover if $\daleth(X)\subseteq X\times X$ is a closed $G\times G$-cofibration then $$STC_G(X) = _{\daleth(X)}cat_G(X\times X) = _{\daleth(X)}cat_{G\times G}^{Wh}(X\times X).$$
Let us investigate closely the question if $\daleth(X)\subseteq X\times X$ is a $G\times G$ cofibration. Since we assume that $X$ is a compact $G$-CW-complex we have that $X$ is a compact $G$-ANR, i.e.
\[G-ANR\]\[2:12\] A metrizable $G$-space $X$ we call a $G$-ANR if for every equivariant imbedding $\iota: X \to Y $ as a closed $G$-subset of a metrizable $G$-space $ Y$ there exists a $G$ neighborhood $U$ of $\iota(X)$ in $Y$ and a continuous equivariant retraction $U\to
\iota(X)$.
It is known that a countable $G$-CW complex is a $G$-ANR.
\[2:13\]\[G-Borsuk pair\] A pair $(Y,X)$ of $G$-spaces $X{\overset{G}\hookrightarrow} Y$ is called $G$-cofibration if for any topological space $Z$, any $G$-map $f:Y \to Z$ every $G$-homotopy $f: X\times
I\to Z$, $f_0= f_{|X}$ extends to a $G$-homotopy $F_t:Y\times I \to
Z$.
We can use as well an equivalent formulation
\[2:14\] \[equivalent definitions\] Let $G$ be a compact Lie group. A compact metrizable space $G$ -space $X$ is a $G$-ANR if and only if every pair $(Y,X)$ is a closed $G$ cofibration for $Y$ and $X$ metrizable, $X$ closed and invariant in $Y$.
Jaworowski proved the following theorem
\[2:15\]\[Jaworowski\] Let $G$ be a compact Lie group. A compact $G$-space is a $G$-ANR if and only if for every closed subgroup $H\subset G$ the fixed point space $X^H$ is an ANR.
Denote by $\tilde{X}$ the Cartesian product $X\times X$. Recall that $\tilde{X}$ posses a natural action of the group $\tilde{G}:=G\times G$ induced by the action of $G$ on $X$.
We will show that $\daleth(X)$ is a $\tilde{G}$-ANR and consequently $\daleth(X)\subseteq \tilde{X}$ is a $\tilde{G}$-cofibration as follows from Proposition \[equivalent definitions\]
\[2:16\] Let $G$ be a compact Lie group and $X$ a compact $G$-ANR. Then $\daleth(X)$ is a $\tilde{G}$-ANR.
We are able to show the statement under an additional assumption that [**$G$ is finite**]{}.
We assume that $G$ is finite. First observe that $\daleth(X)$ can be represented as the saturation of $\Delta(X)\subseteq \tilde{X}$ with respect to the action of group $G_1:=G\times\{e\} \subseteq \tilde{G}$, i.e. $$\daleth(X)= \{(gx,x)\colon \;\ g\in G, \;x\in X\}$$ Indeed, since $G_1\subseteq G$, $\{(gx,x)\colon \;\ g\in G, \;x\in X\} \subset \daleth(X)$. On the other hand any $z=(x_1,x_2)=(g_1x,g_2x)$ can be represented as $(\tilde{g}x,x)\}$ where $ y=g_2x$ and $\tilde{g}=g_1\,g_2^{-1}$. This shows that $\daleth(X)\subseteq \{(gx,x)\}$.
Of course $\daleth(X)$ is $\tilde{G}$-invariant closed subset of $\tilde{X}$ as a continuous image of the compact space $\tilde{G}\times \Delta(X)$. In view of the Jaworowski theorem (\[Jaworowski\]) it is sufficient to show that for every closed subgroup $\tilde{H}\subseteq
\tilde{G}$ the space $\daleth(X)^{\tilde{H}}$ is an ANR.
Let $h=(h_1,h_2) \in \tilde{H}$. A point $(gx,x)$ belongs to $\daleth(X)^h$ (or equivalently to $X^{\{h\}}$, $\{h\}$ the cyclic group generated by $h$) if and only if $h(gx,x) = (gx,x)$. The latter is equivalent to $h_2 x=x$ and $h_1gx=gx$. The first equality gives $x\in X^{h_2}$, and the second $ gx \in X^{h_1}$. Since $G_{gx}= g G_x g^{-1}$ the latter means that $x\in X^{g^{-1}h_1g^{-1}}$. Consequently $(gx,x)\in \daleth(X)^h$ if and only if $x\in X^{h_2} \cap X^{g^{-1}h_1
g}$.
Next note that $X^{h} \cap X^{h'} = X^{\{h,h'\}}$, where $\{h,h'\}$ is a subgroup generated by $h$ and $h'$. Indeed since $h\subseteq \{h,h'\}$, $h'\subseteq \{h,h'\}$, $X^{\{h,h'\}} \subseteq X^{h}$ and $X^{\{h,h'\}} \subseteq X^{h'}$, thus $ X^{\{h,h'\}} \subseteq X^{h} \cap X^{h'}$.
Conversely, if $ x\in X^{h} \cap X^{h'}$ then $x\in
X^{h_1^{i_1^1} h_2^{i_2^1} \,\cdots \, h_1^{i^1_r}h_2^{i^2_r}}$ for any word $h_1^{i_1^1} h_2^{i_2^1} \,\cdots \,
h_1^{i^1_r}h_2^{i^2_r}$. This means that $x\in X^{\{h,h'\}}$, thus $X^{h} \cap X^{h'} \subset X^{\{h,h'\}}$.
From it follows that given $g\in G$ for $\daleth(X)_g:=
\{(gx,x)\}, \; x\in X$ and $h=(h_1'h_2)\in \tilde{H}$ the fixed point set $\daleth(X)_g^h$ is equal to $$\daleth(X)_g^h =X^{\{h',h_2\}}$$ where $h'=gh_1g^{-1}$. But such a set is an ANR, because $X$ is a $G$-ANR.
Observe next that $$\label{intersection}
\daleth(X)_{g_1}^h \cap \daleth(X)_{g_2}^h = X^{h_2}\cap X^{h_2}
\cap X^{g_1^{-1}h_1g_1} \cap X^{g_2^{-1}h_1g_2} =X^{\{h',h'',
h_2\}}\,,$$ with $ h'=g_1^{-1}h_1g_1$ and $h''=g_2^{-1}h_1g_2$. Consequently, it is an ANR.
Since $\daleth(X)= {\underset{g\in G}\cup} \daleth(X)_g$, $G$ is finite, we know that $\daleth(X)^h$ is a finite \[!\] union of ANRs such that the intersections of each two of them are ANRs. From the well known property of ANRs: *“$X$, $Y$, $X\cap Y$ are ANRs implies that $X\cup Y$ is an ANR ”* it follows that $\daleth(X)^h$ is an ANR.
Now lets take $h=(h_1,h_2)$, and $h'=(h_1',h_2')$. For a given $g\in G$ $$\label{second intersection}
\daleth(X)_{g}^h \cap \daleth(X)_{g}^{h'} = X^{h_2}\cap X^{h_2'}
\cap X^{g h_1g^{-1}} \cap X^{gh_1'g^{-1}} =X^{\{h_2,h_2', g
h_1g^{-1}\,,gh_1'g^{-1}\}}$$ Observe that for any $\tilde{G}$-subset $A$ of $\tilde{X}$ we have $$A^H = {\underset{h\in H}\cap} A^h$$
Now let $h^1=(h^1_1,h^1_2), \,\dots,\, h^s=(h^s_1,h^s_2)$ be all elements of $H\subset \tilde{G}$. For a given $g\in G$ $$(\daleth(X)_g)^H= {\underset{h\in H}\cap}(\daleth(X)_g)^h=
X^{\{h^1_2,h^2_2, ..., h^s_2,\, gh^1_1g^{-1}, ...,\,
gh^s_1g^{-1}\}}$$ and consequently this set is an ANR, since $X$ is a $G$-ANR.
Finally, by the same argument $\daleth(X)^H = {\underset{g\in
G}{\cup}} (\daleth(X)_g)^H$ is an ANR, because for two $g_1,\,g_2
\in G$ we have $$(\daleth(X)_{g_1})^H \cap (\daleth(X)_{g_2})^H=X^{\{h^1_2,h^2_2, ..., h^s_2, \,g_1h^1_1g_1^{-1}, ...\,,\, g_1h^s_1g_1^{-1}, g_2h^1_1g_2^{-1}, ...\,,
\,g_2h^s_1g_2^{-1}\}}$$ This shows that $(\daleth(X)_{g_1})^H \cap
(\daleth(X)_{g_2})^H$ is an ANR and consequently so is the union ${\underset{g\in G}{\cup}} (\daleth(X)_g)^H = \daleth(X)^H$.
\[2:17\] Let $H\subset \tilde{G}= G\times G$, $p_1:G\times G\to
G$, $p_2:G\times G \to G$ the projections onto the corresponding coordinates. Put $H_1=p_1(H)$, $H_2=p_2(H)$. Let $ \hat{H} =
H_1\times H_2\subset \tilde{G}$.
Since the subgroup $\hat{H}$ contains $H$ we have the corresponding inclusion $$\label{inclusion}
A^{\hat{H}} \subset A^H$$ for any $\tilde{G}$ subset $A$ of $\tilde{X}$. However, since the inclusion $H\subset \hat{H}$ is proper in general, also the inclusion (\[inclusion\]) is proper in general!
\[2:18\] Is it true that $$\daleth (X)\subseteq X\times X$$ is always a closed $G\times G$-cofibration for a compact $G$-CW complex $X$ and a compact Lie group $G$?
The above formulated problem seems be difficult in general.
Bounds for the symmetric topological complexity
-----------------------------------------------
We start with a product formula for the symmetric and equivariant topological complexity.
Let $X$ and $Y$ be any $G$ spaces. Then for $X\times Y$ treated as a $G$ space via the diagonal action we have $$TC_G(X\times Y) \leqslant TC_G(X)+TC_G(Y)$$ moreover if $\daleth(X)\subseteq X\times X$ and $\daleth (Y)\subseteq Y\times Y$ are $G\times G$ cofibrations then $$STC_G(X\times Y) \leqslant TC_G()+STC_G(Y)$$
Since $\Delta(X\times Y) = \Delta(X)\times \Delta(Y)$ and $\daleth(X\times Y)\subseteq \daleth(X)\times \daleth(Y)$, where we consider $X\times Y$ to be a $G$ space via the diagonal action, this is a direct consequence of theorem \[1:5\].
Let $X$ be a $G$ space and $Y$ a $H$ space. We consider $X\times Y$ as a $G\times H$ space. Then $$TC_{G\times H}(X\times Y) \leqslant TC_G(X)+TC_H(Y)$$ moreover if $\daleth(X)\subseteq X\times X$ is a $G\times G$ cofibration and $\daleth(Y)\subseteq Y\times Y$ is a $H\times H$ cofibration then $$STC_{G\times H}(X\times Y) \leqslant TC_G()+STC_H(Y)$$
Since $\Delta(X\times Y) = \Delta(X)\times \Delta(Y)$ and $\daleth(X\times Y)= \daleth(X)\times \daleth(Y)$ this is a direct consequence of corollary \[1:6\].
\[3:1\] Let $X$ be a $G$ set, $H\subseteq G$ closed subgroup and assume that $A\subseteq B$ are its closed invariant subsets. Then:
1) $STC_G(X)\leqslant _Ocat_{G\times G}(X\times X)$ for any $G\times G$ orbit $O\subseteq\daleth(X)$;
2) $_Ocat_{G\times G}(X\times X)\leqslant STC_G(X)\cdot _Ocat_{G\times G}(\daleth(X))$ for any $G\times G$ orbit $O\subseteq \daleth(X)$;
3) $TC(X/G)\leqslant STC_G(X)$;
This is a direct consequence of lemma \[1:13\].
From our point of view one of the most important properties of the symmetric topological complexity is that it is indeed finite for a large family of $G$ spaces $X$.
We have an obvious inequality $$TC(X)\leqslant cat(X\times X)$$ we will show that it passes to the equivariant case. For completeness let us first recall
\[3:2\] If $X$ is $G$ connected then $$TC_G(X) \leqslant cat_{G}X\times X$$
We give a similar result concerning the symmetric topological complexity.
\[3:3\] If $X$ is $G$-path-connected then $$STC_G(X) \leqslant _{O\times
O}cat_{G\times G}X\times X$$ where $O = G\cdot x_0$ for some $x_0\in
X$.
Let $U$ be a set $G\times G$ compressable into $O\times O$. We have a $G\times G$ homotopy $F\colon U\times I\rightarrow X\times X$ such that $F\colon id_U\simeq c$ where $c(U)\subseteq O\times O$. Let $H
= G_{x_0}\times G_{x_0}$. We know that $p((PX\times
_{\daleth(X)}PX)^H ) = (X\times X)^H$ which follows from the $G$ connectivity of $X$ hence $p(\gamma,\delta) = (x_0,x_0)$. Then we define $s\colon U\rightarrow PX\times_{\daleth (X)}PX$ by $s(y_0,y_1) = (g_0,g_1)\cdot (\gamma,\delta)$ whenever $c(y_0,y_1) =
(g_0,g_1)\cdot (x_0,x_0)$. Now note that $p\circ s \simeq c \simeq
id_U$.
The above theorem allows us to show that $STC_G(X)$ is in many cases finite – for example if $x_0\in X^G$ and $X$ is $G$ connected then we have $_{x_0\times x_0}cat _{G\times G} (X\times X) \leqslant 2_{x_0}cat_{G} (X\times X) - 1 = 2cat_G - 1$ by theorem \[1:6\].
Equivariant and symmetric topological complexity share some basic homotopical properties:
\[3:4\] Let $X$ $G$-dominates $Y$, that is there are $f\colon X\rightarrow Y$ and $g\colon Y\rightarrow X$ such that $fg\simeq id_Y$ are $G$-homotopic. Then $$TC_G(X)\geqslant TC_G(Y),\; STC_G(X) \geqslant STC_G(Y).$$
The part concerning $TC_G(X)$ can be found in [@colman-grant], theorem 5.2.
For the proof for $STC_G(X)$ let $H\colon fg\simeq id_Y$ be the $G$ homotopy. Note that then $$H\times H\colon (X\times X,\daleth(X))\times I \rightarrow (Y\times Y,\daleth(Y))$$ is the required homotopy between $(f\times f)\circ (g\times g)$ and $id_{(Y\times Y, \daleth (Y))}$. Now the assertion follows from \[1:4\].
\[3:5\] For a $G$ set $X$ and closed subgroup $H$ of $G$ we have $$TC_H(X^H)\leqslant TC_G (X),\; STC_H(X^H)\leqslant STC_G(X).$$
The part concerning $TC_G(X)$ follows from [@colman-grant], proposition 5.3.
For the proof for $STC_G(X)$ let $\tilde{H} = H\times H$. Note that $$\daleth(X)^{\tilde{H}} = \daleth (X^H)\supseteq (H\times H) \Delta(X^H)$$ therefore from theorem \[1:16\] we get that $$\begin{gathered}
TSC_H(X^H) = _{(H\times H) \Delta(X^H)}cat_{H\times H}(X^H\times X^H)\\ \leqslant \hbox{}_{\daleth (X)}cat_{G\times G}(X\times X) = TSC_G(X).\end{gathered}$$
For a $G$ space $X$ such that $X^G\neq \emptyset$ we have that $$STC_G(X)\geqslant TC_G(X^G) = TC(X^G).$$
Examples of calculations.
=========================
We end this article with calculations of $TSC_G(X)$ in some basic examples.
Let $G$ act on itself by left translations. The action of $G$ is free and therefore from theorem \[2:9\] we get that $$STC_G(G) = TC(G / G) = TC(\ast) = 1$$ which is in contrast to the case of equivariant topological complexity where we have that $TC_G(G) = cat(G)$ (comp. [@colman-grant], theorem 5.11).
Let $\mathbb Z / 2 = \{1 , \tau \}$ act on $\mathbb S^n$, $n\geqslant 1$ by reflecting the last coordinate. Note that for $n=1$ the set $\mathbb (S^1)^{\mathbb Z /2}$ is disconnected so that $TG_{\mathbb Z /2}(\mathbb S^1) = STC_{\mathbb Z /2} (\mathbb S^1) = \infty$. If $n>1$ then $\mathbb S^n$ is $\mathbb Z /2$ connected so that $$STC_{\mathbb Z /2}(\mathbb S^n)\leqslant cat_{\mathbb Z /2 \times \mathbb Z /2} (\mathbb S^n\times \mathbb S^n)\leqslant 2cat_{\mathbb Z /2}(\mathbb S^n) - 1 = 3$$ by theorem \[3:3\]. On the other hand, since $\mathbb (S^n)^{\mathbb Z /2}\cong \mathbb S^{n-1}$, we have that (comp. \[3:5\]) $STC_{\mathbb Z /2}(\mathbb S^n)\geqslant TC(\mathcal S^{n-1}) = 3$ for $n$ odd.
For an even $n$ let $U_1\subseteq \mathcal S^n\times S^n$ be defined as follows $$U_1 = \{(x,y)\in(\mathbb S^n)^2\colon x\neq -y\text{ if }x,y\in\mathbb S^{n-1}\}.$$ Then for each $(x_1,x_2)\in U_1$ there is a unique shortest path $s'(x_1,x_2)$ joining $x_1$ or $\tau x_1$ and $x_2$ or $\tau x_2$ in the upper hemisphere. Let $s_1(x_1,x_2)= (\alpha_1 s|_{[0,\tfrac{1}{2}]}, \alpha_2s|_{[\tfrac{1}{2} , 1]})$ in case we were joining $\alpha_1x_1$ with $\alpha_2x_2$ for $\alpha_i\in\mathbb Z / 2$.
Let $U_2\subseteq \mathcal S^n\times S^n$ be defined as follows $$V_2 = \{(x,y)\in(\mathbb S^n)^2\colon x, y\in\mathbb S^{n-1},\; x\neq y\}.$$ Now $V_2$ has a small $\mathbb Z/2$ invariant open neighborhood $U_2$ in $\mathbb S^n$ such that the projection $\pi\colon U_2\rightarrow V_2$ into the equator $\mathbb S^{n-1}$ is $\mathbb Z /2$ equivariant deformation retraction. We define for each $(x_1,x_2)$ a path from $x_1$ to $x_2$ as follows. First choose a non vanishing vector field $\nu$ on $\mathbb S^{n-1}$. The path $s'_2(x_1,x_2)$ consists of four parts. First by the shortest path we move $x_1$ to $\pi(x_1)$, then using the shortest path we move $\pi(x_1)$ to $-\pi(x_2)$ and using the vector field $\nu$ we move $-\pi(x_2)$ to $\pi(x_2)$ using a spherical arch defined by $\nu (\pi(x_2))$ and we end by moving through the shortest path $\pi(x_2)$ to $x_2$. We obtain $s$ from $s'$ by cutting it into two parts.
As it can be easily checked these two sets satisfy the definition of the symmetric topological complexity and prove that $STC_{\mathbb Z / 2} (\mathbb S^n) = 2$ for $n$ even.
Note that we have $TC_{\mathbb Z / 2} (\mathbb S^n) = 3$ for $n>1$ as shown in [@colman-grant], example 5.9.
[3]{}
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[^1]: $^{1}$Supported by the Polish Research Grant NCN 2011/03/B/ST1/04533
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Discovery of high-redshift ($z > 6$) supermassive black holes (BHs) may indicate that the rapid (or super-Eddington) gas accretion has aided their quick growth. Here, we study such rapid accretion of the primordial gas on to intermediate-mass ($10^2 - 10^5~M_\odot$) BHs under anisotropic radiation feedback. We perform two-dimensional radiation hydrodynamics simulations that solve the flow structure across the Bondi radius, from far outside of the Bondi radius down to a central part which is larger than a circum-BH accretion disc. The radiation from the unresolved circum-BH disc is analytically modeled considering self-shadowing effect. We show that the flow settles into a steady state, where the flow structure consists of two distinct parts: (1) bipolar ionized outflowing regions, where the gas is pushed outward by thermal gas pressure and super-Eddington radiation pressure, and (2) an equatorial neutral inflowing region, where the gas falls toward the central BH without affected by radiation feedback. The resulting accretion rate is much higher than that in the case of isotropic radiation, far exceeding the Eddington-limited rate to reach a value slightly lower than the Bondi one. The opening angle of the equatorial inflowing region is determined by the luminosity and directional dependence of the central radiation. We find that photoevaporation from its surfaces set the critical opening angle of about ten degrees below which the accretion to the BH is quenched. We suggest that the shadowing effect allows even stellar-remnant BHs to grow rapidly enough to become high-redshift supermassive BHs.'
author:
- |
Kazuyuki Sugimura,$^1$[^1] Takashi Hosokawa,$^{2,3,4}$ Hidenobu Yajima$^{1,5}$ and Kazuyuki Omukai$^{1,3}$\
$^1$Astronomical Institute, Tohoku University, Aoba, Sendai 980-8578, Japan\
$^2$Department of Physics, Kyoto University, Sakyo, Kyoto 606-8502, Japan\
$^3$Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA\
$^4$Department of Physics and Research Center for the Early Universe, the University of Tokyo, Bunkyo, Tokyo 113-0033, Japan\
$^5$Frontier Research Institute for Interdisciplinary Sciences, Tohoku University, Aoba, Sendai 980-8578, Japan
title: Rapid Black Hole Growth under Anisotropic Radiation Feedback
---
-1cm
quasars: supermassive black holes-cosmology: theory.
Introduction {#sec:intro}
============
Discovery of high-$z$ ($z\gtrsim6$) quasars suggests that supermassive black holes (SMBHs) already exist when the age the Universe is less than $1{\,\mathrm{Gyr}}$ [see, e.g., @Fan:2001aa; @Willott:2010aa; @Mortlock:2011aa; @Venemans:2013aa; @Wu:2015aa]. This poses a question about the formation mechanism of SMBHs in such a short interval. Among the scenarios for the SMBH seed formation [see, e.g., @Volonteri:2012ab; @Haiman:2013aa for a review], including the dense stellar cluster scenario [see, e.g., @Omukai:2008aa; @Devecchi:2009aa; @Katz:2015ab; @Yajima:2016aa and reference therein], following two are the most studied: the direct collapse BH (DCBH) and the population III (Pop III) remnant BH scenarios.
In the former scenario, supermassive stars of $\sim 10^5\,M_\odot$ collapse to form seed BHs with approximately the same mass. Specifically, supermassive stars are envisaged to form in exceptional environments in the high-$z$ Universe, for example, in atomic-cooling halos where the ${\mathrm{H_2}}$ cooling is totally suppressed by very strong far ultraviolet (FUV) irradiation [e.g., @Sugimura:2014aa]. While the seed BHs in this case are rather massive with $\sim
10^5\,M_\odot$, their number density might be too small to explain all the observed high-$z$ SMBHs due to the stringent necessary conditions [@Dijkstra:2008aa; @Dijkstra:2014aa; @Agarwal:2012aa; @Sugimura:2014aa; @Sugimura:2016aa; @Inayoshi:2015ab; @Chon:2016aa].
In the latter scenario, the remnant BHs of Pop III stars [@Yoshida:2008aa; @Hosokawa:2011aa; @Hosokawa:2016aa] are thought as SMBH seeds [@Alvarez:2009aa; @Jeon:2012aa]. Contrary to the DCBH scenario, they are abundant but the problem is whether they can actually grow to the SMBHs from smaller initial mass of $\lesssim 10^3\,M_\odot$ [@Susa:2014aa; @Hirano:2015aa] within the available time. Although BHs can acquire the mass by collisions with other BHs [@Tanikawa:2011aa], the BH collisions often result in ejection of the merged BHs from the host halo due to the recoil of gravitational wave emission [e.g., @Baker:2006aa; @Koppitz:2007aa]. Thus, the feasibility of this scenario relies on whether the rapid accretion on to seed BHs is possible or not [@Madau:2014aa; @Alexander:2014aa; @Volonteri:2015aa].
Recently, a number of authors have studied the BH accretion under radiation feedback [e.g., @Milosavljevic:2009aa; @Milosavljevic:2009ab; @Park:2011aa; @Park:2012aa; @Park:2013aa]. They solve the gas dynamics over the scale of the Bondi radius, where the accretion rate on to the circum-BH disc is physically determined. Although the central circum-BH disc is not spatially resolved, subgrid models that provide analytic prescriptions of its emissivity have been used. They have shown that the accretion rate is significantly reduced to $\lesssim1\%$ of that without radiation feedback (i.e., the Bondi rate) in case with modest BH mass and ambient density (e.g., $10^2\,M_\odot$ and $10^5{\,\mathrm{cm^{-3}}}$). Only in case with very high BH mass and/or ambient density (e.g., $10^4\,M_\odot$ and $10^5{\,\mathrm{cm^{-3}}}$), the accretion rate reaches to the Bondi value because of inefficient radiation feedback, as recently shown by [@Inayoshi:2016ac] [see also @Li:2011aa; @Pacucci:2015ab; @Park:2016ab for other mechanisms of efficient accretion]. However, all those calculations assume isotropic radiation (in either one- or two-dimensional simulations), whereas in reality the radiation from the BH accretion disc should be anisotropic. The flow structure will be significantly altered in such anisotropic radiation field. Although the BH accretion under anisotropic radiation has been studied in the context of active galactic nuclei (AGN) with the BH mass $\ga 10^6\,M_\odot$ [@Proga:2007aa; @Kurosawa:2009aa; @Novak:2011aa; @Barai:2012aa], the nature of accretion on to stellar-mass BHs would be quite different.
The anisotropic BH irradiation has been examined with different models of the BH accretion discs, including the “standard disc” for moderate accretion rates [@Shakura:1973aa], and “slim disc” for the higher rates [@Abramowicz:1988aa]. In particular, recent multi-dimensional simulations have investigated inner structure of the slim disc within roughly a hundred Schwarzschild radii, showing that the accretion rates can indeed exceed the Eddington-limited rate [e.g., @Ohsuga:2005aa; @Jiang:2014aa; @McKinney:2014aa; @Fragile:2014aa; @Takahashi:2015aa; @Sc-adowski:2016aa]. These studies show that the high-energy photons are predominantly emitted in polar directions from the inner part of the disc. However, the outer structure of the disc, which is not solved in the above simulations, should also modify the anisotropic radiation field. For instance, disc winds such as the line-driven AGN winds launched from the outer region will absorb a part of photons coming from the inner region [e.g., @Proga:2000aa; @Proga:2004aa; @Nomura:2016aa]. Since numerical simulations solving the whole structure of the disc are still infeasible, it is very uncertain how much anisotropy the BH accretion discs actually create.
In this paper, we will investigate accretion of the primordial gas on to BHs under the anisotropic radiation feedback from the central circum-BH accretion discs, considering the shadowing effect by the outer part of the discs. We perform a set of proof-of-concept two-dimensional (2D) radiation hydrodynamics (RHD) simulations, assuming that BHs are initially embedded in homogeneous and static media. We do not attempt to simulate the realistic directional dependence of BH irradiation in consideration of its high uncertainties; instead, we model it in a simple fashion to study how the anisotropy of radiation changes the nature of accretion flows. As confirmed later by our results, the shadowing effect dramatically enhances the accretion rate. This mechanism might give a new pathway from the remnant BHs of Pop III stars to SMBHs within a limited timescale of $\la 1{\,\mathrm{Gyr}}$ after the Big Bang.
The paper is organized as follows. In Sec. \[sec:sph\_acc\], we briefly review the basics of spherical gas accretion on to a BH. In Sec. \[sec:num-method\], we describe the numerical method and cases considered. In Sec. \[sec:result\], we present the main results of our simulations. The conclusions and discussions are given in Sec. \[sec:conclusion\].
Basics {#sec:sph_acc}
======
For later reference, we first briefly summarize the basics of spherical gas accretion on to a central BH under radiation feedback. We consider a system where a BH is embedded in a static and homogeneous medium. We take the BH mass $M_{\mathrm{BH}}=10^3\,M_\odot$, ambient density $n_\infty=10^5\,{\mathrm{cm^{-3}}}$ and ambient temperature $T_{\mathrm{HI}}=10^4\,{\mathrm{K}}$ as a fiducial parameter set.
If we ignore the effect of feedback, the mass accretion will proceed at the Bondi rate in this case, $$\begin{aligned}
\dot{M}_{\mathrm{B}}
&=\frac{4\pi\lambda_{\mathrm{B}}\rho_\infty G^2 M_{\mathrm{BH}}^2}{c_{\mathrm{s,HI}}^3}{\nonumber\\}&=1.7\times 10^{-3}
\,\left(\frac{n_\infty}{10^5\,{\mathrm{cm^{-3}}}}\right){\nonumber\\}&\qquad \times \left(\frac{M_{\mathrm{BH}}}{10^3\,M_\odot}\right)^{2}
\left(\frac{T_{\mathrm{HI}}}{10^4\,{\mathrm{K}}}\right)^{-3/2}
{\mathrm{M_\odot\, yr^{-1}}}
\,,\label{eq:1}\end{aligned}$$ where we take $\lambda_{\mathrm{B}}=(1/4)\left[2/(5-3\gamma)\right]^{(5-3\gamma)/2(\gamma-1)}
= 1.12$ assuming the gas is isothermal (the polytropic index $\gamma=1$). For a neutral primordial gas with helium-to-hydrogen ratio in the number of nuclei $y_{\mathrm{He}}=0.0972$, the mean molecular weight $\mu=(1+4y_{\mathrm{He}})/(1+y_{\mathrm{He}})=1.3$, the mass density of the medium $\rho_\infty=n_\infty(1+4y_{\mathrm{He}})m_{\mathrm{p}}=2.3\times10^{-19}{\,\mathrm{g\,cm^{-3}}}$ with $m_{\mathrm{p}}$ the proton mass and the (isothermal) sound speed $c_{\mathrm{s,HI}}=(k_{\mathrm{B}}T_{\mathrm{HI}}/\mu m_{\mathrm{p}})^{1/2}
=8.1\,\left(T_{\mathrm{HI}}/10^4\,{\mathrm{K}}\right)^{1/2} \,{\mathrm{km\, s^{-1}}}$. The Bondi radius, defined as $$\begin{aligned}
r_{\mathrm{B}}
&=\frac{GM_{\mathrm{BH}}}{c_{\mathrm{s,HI}}^2}{\nonumber\\}&=1.4\times 10^{4}
\left(\frac{M_{\mathrm{BH}}}{10^3\,M_\odot}\right)
\left(\frac{T_{\mathrm{HI}}}{10^4\,{\mathrm{K}}}\right)^{-1}
{\mathrm{AU}}\,,
\label{eq:2}\end{aligned}$$ demarcates the inner region where the gravitational energy dominates the thermal energy and the outer region where the thermal energy dominates. Correspondingly, the gas is approximately in free fall inside $r_{\mathrm{B}}$, whereas the pressure equilibrium is almost achieved outside.
The Eddington luminosity $L_{\mathrm{E}}$ is the critical luminosity above which the outward radiation force via the Thomson scattering exceeds the inward gravitational pull of the BH in fully ionized hydrogen gas, $$\begin{aligned}
L_{\mathrm{E}}=\frac{4\pi G M_{\mathrm{BH}} c m_{\mathrm{p}}}{\sigma_{\mathrm{T}}}
&=3.3\times 10^7
\left(\frac{M_{\mathrm{BH}}}{10^3M_\odot}\right) L_\odot\,,
\label{eq:12}\end{aligned}$$ where $\sigma_{{\mathrm{T}}}$ is the Thomson scattering cross section. Note that the Eddington luminosity does not always provide physical limit because gas pressure is not considered in the above argument. In addition, the radiation force becomes less effective in a partially ionized gas.
The (efficiency-independent) Eddington-limited accretion rate is defined as $$\begin{aligned}
\dot{M}_{\mathrm{E}}
&=\frac{L_{\mathrm{E}}}{c^2}
=2.2\times 10^{-6} \left(\frac{M_{\mathrm{BH}}}{10^3M_\odot}\right)\,M_\odot\, {\mathrm{yr^{-1}}}\,,
\label{eq:8}\end{aligned}$$ and the condition for the luminosity to be sub-critical can be rewritten as $\dot{M}<\dot{M}_{\mathrm{E}}/\eta$ with the radiative efficiency $\eta$. The radiative efficiency $\eta\approx 0.1$ for a standard accretion disc is widely used in the previous works. [see, e.g., @Milosavljevic:2009aa; @Park:2011aa; @Park:2012aa]. Note that in some literatures the efficiency-dependent Eddington-limited accretion rate, $\dot{M}_{\mathrm{E}}/\eta$ in our definition, is used instead. For large $M_{\mathrm{BH}}$ and/or $n_\infty$, the Bondi rate $\dot{M}_{\mathrm{B}}$ can be much larger than the Eddington rate $\dot{M}_{\mathrm{E}}$, e.g., $\dot{M}_{\mathrm{B}}/\dot{M}_{\mathrm{E}}=7.8\times 10^{2}
\,\left(n_\infty/10^5\,{\mathrm{cm^{-3}}}\right)
\left(M_{\mathrm{BH}}/10^3\,M_\odot\right)$, because $\dot{M}_{\mathrm{B}}$ is proportional to $M_{\mathrm{BH}}^2\,n_\infty$ while $\dot{M}_{\mathrm{E}}$ to $M_{\mathrm{BH}}$.
High energy photons emitted by the BH accretion disk create a surrounding [H[ii]{} ]{}bubble. With the power-law spectrum $L_\nu \propto
\nu^{-1.5}$, which is often postulated in the literature, the photoionized gas is heated up to $T_{\mathrm{HII}}\sim 7\times10^4{\,\mathrm{K}}$ owing to helium ionization heating. The high thermal pressure of the [H[ii]{} ]{}bubble, together with the outward radiation pressure, can significantly reduce the accretion rate [@Milosavljevic:2009aa; @Park:2011aa; @Park:2012aa]. The size of the [H[ii]{} ]{}bubble is estimated by the Strömgren radius, $$\begin{aligned}
r_{\mathrm{HII}} &=
6.8\times 10^{4}
\left(\frac{T_{\mathrm{HII}}}{7\times10^4{\,\mathrm{K}}}\right)^{\frac{1}{3}}{\nonumber\\}&\qquad\times\left(\frac{L}{3.3\times10^7L_\odot}\right)^{\frac{1}{3}}
\left(\frac{n_{\mathrm{HII}}}{10^5\,{\mathrm{cm^{-3}}}}\right)^{-\frac{2}{3}}
{\,\mathrm{AU}}\,,
\label{eq:13}\end{aligned}$$ which is obtained by equating the ionizing photon emissivity $\dot{N}_{\mathrm{ion}}=\int_{\nu_{\mathrm{T}}}^\infty d\nu L_\nu/h\nu$ ($=L/3h\nu_{\mathrm{T}}$ for the above spectrum with $L_\nu \propto
\nu^{-1.5}$) with the recombination rate within the [H[ii]{} ]{}bubble $\alpha_{\mathrm{B}}(4\pi/3)r_{\mathrm{HII}}^3 n_{\mathrm{HII}}^2$, where $n_{\mathrm{HII}}$ is the number density of hydrogen nuclei inside the bubble, $h\nu_{\mathrm{T}}=13.6{\,\mathrm{eV}}$ the hydrogen ionization energy, and $\alpha_{\mathrm{B}}=4.6\times10^{-14}{\,\mathrm{cm^3\,s^{-1}}}$ the case B hydrogen recombination coefficient at $7\times10^4{\,\mathrm{K}}$ [@Ferland:1992aa]. In the above, we take $n_{\mathrm{HII}}=n_\infty\,(=10^5{\,\mathrm{cm^{-3}}})$ and $L=L_{\mathrm{E}}$ as reference values.
Suppose that an ionizing source is suddenly turned on at the centre. The ionization front first propagates up to $r_{\mathrm{HII}}$ with $n_{\mathrm{HII}}$ kept almost constant. Then the bubble expands until pressure equilibrium with the surrounding medium is reached with $n_{\mathrm{HII}}=(c_{\mathrm{s,HI}}/c_{\mathrm{s,HII}})^2\,n_\infty \sim 0.07\,n_\infty$, where $c_{\mathrm{s,HII}} = (2T_{\mathrm{HII}}/T_{\mathrm{HI}})^{1/2}c_{\mathrm{s,HI}}$ is the sound speed of the ionized gas, with the factor of $2$ accounting for the increase of the particle number by ionization. Note that we have neglected the effect of helium in estimating $r_{\mathrm{HII}}$ and $n_{\mathrm{HII}}$, because it modifies them only slightly. For the flow from the [H[ii]{} ]{}bubble in pressure equilibrium with the surrounding neutral medium, the Bondi radius and rate are given by $r_{\mathrm{B,HII}}\sim
0.07\,r_{\mathrm{B}}$ and $\dot{M}_{\mathrm{B,HII}} \sim 1\times 10^{-3}
\dot{M}_{\mathrm{B}}$, respectively. This clearly shows that the photo-ionization feedback can considerably suppress the accretion.
In order for the [H[ii]{} ]{}bubble to be trapped around the BH, however, the condition $r_{\mathrm{HII}}>r_{\mathrm{B}}$ must be satisfied [@Inayoshi:2016ac]. Otherwise, the gas originally in between $r_{\mathrm{HII}}$ and $r_{\mathrm{B}}$ would accumulate around the periphery of the [H[ii]{} ]{}bubble. This leads to the enhancement of density $n_{\mathrm{HII}}$ and thus to shrinkage of the bubble. In the end, the [H[ii]{} ]{}bubble disappears and radiation feedback no longer affects the accretion. At that time, the accretion rate returns to the original Bondi value for the neutral gas $\dot{M}_{\mathrm{B}}$, instead of that for the [H[ii]{} ]{}bubble $\dot{M}_{\mathrm{B,HII}}$. For this to happen, a system with a massive BH and/or dense ambient medium, namely $(M_{\mathrm{BH}}/10^4
M_\odot)(n_\infty/10^5\,{\mathrm{cm^{-3}}}) \gtrsim 1$ [@Inayoshi:2016ac], is required when the BH luminosity is close to the Eddington value as $L\approx L_{\mathrm{E}}$.
Recall that the above argument is based on the assumption of the spherical symmetry, which should be modified in realistic situations with anisotropic BH irradiation. We expect that the flow through shadowed equatorial regions, if exist, enhance the accretion rate. In what follows, we will see what kind of the flow structure appears for such cases by using numerical simulations.
NUMERICAL METHOD {#sec:num-method}
================
We study accretion of primordial gas on to BHs under anisotropic radiation by performing a series of 2D RHD simulations (Sec. \[sec:rhd\_sim\]). Specifically, we solve the dynamics of the flow around the Bondi radius (see Fig. \[fig:acc\_whole\]b), where the accretion rate on to the BH and disc system is determined. We mask the inner circum-BH accretion disc (see Fig. \[fig:acc\_whole\]a) by the central sink region and inject ionizing photons at the inner boundary $R_{\rm in}$ according to a simple parametric sub-grid model that represents various directional dependences of BH irradiation (Sec. \[sec:radiation\]).
Two-dimensional radiation hydrodynamics simulations {#sec:rhd_sim}
---------------------------------------------------
We use a modified version of the public multi-dimensional magneto-hydrodynamics code [Pluto 3.0]{} [@Mignone:2007aa], which has been applied to studies on the present-day high-mass star formation [e.g., @Kuiper:2010aa; @Kuiper:2010ab; @Kuiper:2011aa; @Kuiper:2013aa] and Pop III star formation [@Hosokawa:2016aa].
Here, we have tuned the code used for the Pop III star formation [@Hosokawa:2016aa] to fit our study of the BH accretion. As in @Kuiper:2010aa, we adopt a 2D polar coordinate system assuming the axial symmetry. We calculate only the gravity of the central BH and neglect the gas self-gravity, as in the previous studies [e.g., @Park:2011aa; @Milosavljevic:2009ab; @Inayoshi:2016ac but also see Li 2011]. We assume that the outer edge of the accretion disc, i.e., the centrifugal radius $r_{\rm cen}$, is much smaller than the sink radius $R_{\mathrm{in}}$. We thus ignore the angular momentum of the flow in the computational domain. Other modifications we have added are summarized as follows.
### Chemical and thermal processes {#sec:chemistry}
To solve the chemical and thermal processes, we use the same methods developed in [@Hosokawa:2016aa] with several modifications. Unlike in [@Hosokawa:2016aa], we omit ${\mathrm{H}}_2$ chemistry assuming that ${\mathrm{H}}_2$ is completely photo-dissociated by the central FUV irradiation.[^2] We have added the ${\mathrm{He}}$ chemistry, since hard UV photons from BH accretion discs create a large helium photoionized region embedded in an [H[ii]{} ]{}region.
In summary, we solve the chemical network with six species: ${\mathrm{H}}$, ${\mathrm{H^+}}$, ${\mathrm{e}}$, ${\mathrm{He}}$, ${\mathrm{He^+}}$, and ${\mathrm{He^{2+}}}$, which consists of the following chemical processes: photoionization of ${\mathrm{H}}$, ${\mathrm{He}}$ and ${\mathrm{He^+}}$; collisional ionization of ${\mathrm{H}}$, ${\mathrm{He}}$ and ${\mathrm{He^+}}$; recombination of ${\mathrm{H^+}}$, ${\mathrm{He^+}}$ and ${\mathrm{He^{2+}}}$. Accordingly we consider the following thermal processes: photoionization heating of ${\mathrm{H}}$, ${\mathrm{He}}$, and ${\mathrm{He^+}}$; recombination cooling of ${\mathrm{H^+}}$, ${\mathrm{He^+}}$, and ${\mathrm{He^{2+}}}$; excitation cooling of ${\mathrm{H}}$, ${\mathrm{He}}$, and ${\mathrm{He^+}}$; collisional ionization cooling of ${\mathrm{H}}$, ${\mathrm{He}}$, and ${\mathrm{He^+}}$; free-free cooling of ${\mathrm{H}}$, ${\mathrm{He}}$, and ${\mathrm{He^{+}}}$; Compton cooling by cosmic microwave background (CMB) photons. Complete lists of our adopted chemical and thermal processes are available in Appendix \[sec:chem\_detail\].
We turn off the cooling when the temperature falls below $10^4{\,\mathrm{K}}$, as in the previous 2D simulations [e.g., @Park:2011aa]. We neglect secondary ionization and heating caused by X-ray photoionization [@Shull:1979ab; @Shull:1985aa; @Ricotti:2002aa]. We have confirmed with test calculations that these processes hardly affect the gas dynamics though the ionization degree is only slightly enhanced just outside the [H[ii]{} ]{}bubble.
### Transfer of ionizing photons {#sec:rad_transfer}
As in @Hosokawa:2016aa, we only solve the transfer of ionizing photons directly coming from the central accretion disc. Diffuse recombination photons are considered by way of the on-the-spot approximation. The radiation transfer is successively solved with the chemistry from the innermost cell, where photons are injected according to the sub-grid radiation model (see Sec. \[sec:radiation\] below). We do not consider the absorption between the radiation source and the inner boundary, which is currently masked by the sink cell.
Regarding the transport of ionizing photons, we have made the following major updates. First, we solve the frequency-dependent transfer with 128 logarithmically-spaced frequency bins between $13.6{\,\mathrm{eV}}$ and $1{\,\mathrm{keV}}$, to consider the photoionization of ${\mathrm{H}}$, ${\mathrm{He}}$ and ${\mathrm{He^+}}$ with different threshold energies. Second, we consider the radiation pressure via Thomson scattering and photoionization. As will be seen in Section \[sec:result\], the radiation pressure becomes important when the luminosity exceeds the Eddington limit.
We simply assume that photons with energy below $13.6{\,\mathrm{eV}}$ freely escape from the system [@Park:2011aa]. Although the radiation pressure of accumulated Ly$\alpha$ photons would affect the gas dynamics in spherically symmetric systems, it is probably not the case in realistic systems with channels for Ly$\alpha$ photons to escape [e.g., @McKee:2008aa; @Milosavljevic:2009aa].
Subgrid model for the irradiation by BH {#sec:radiation}
---------------------------------------
In our model of anisotropic BH irradiation, we assume that ionizing photons are emitted from the inner hot part of a circum-BH accretion disc but a portion of them are absorbed (or scattered) by outer structures (see Fig. \[fig:acc\_whole\]a). We inject ionizing photons at the inner boundary depending on the inflow rate into the sink cell, according to the model described here. We first describe the structure of the BH accretion disc in Section \[sec:acc\_disc\], which motivates our subgrid model. Then we give the expressions for the luminosity and directional dependence in Sections \[sec:efficiency\] and \[sec:direction\_dep\], respectively.
### BH accretion disc with shadowing effect {#sec:acc_disc}
Fig. \[fig:acc\_whole\](a) shows the expected inner structure including a BH accretion disc that motivates our subgrid model. Below we explain the inner and outer parts of the structure presented in Fig. \[fig:acc\_whole\](a) in this order. We also describe resulting directional dependences of the BH irradiation.
In the inner part, we see that the circum-BH disc consists of the two different types of accretion discs. One is the innermost geometrically thick (aspect ratio $\sim 1$) slim disc appearing inside the photon trapping radius $r_{\mathrm{tr}} \equiv (\dot{M}/\dot{M}_{\mathrm{E}})\,r_{\mathrm{Sch}}$, where the cooling via radial advection balances with the viscous heating [e.g., @Begelman:1978aa; @Abramowicz:1988aa]. The other is the geometrically thin (aspect ratio $\ll 1$) standard accretion disc appearing outside $r_{\mathrm{tr}}$, where the radiative loss from the disc surfaces is the main cooling process [e.g., @Shakura:1973aa]. When $\dot{M}/\dot{M}_{\mathrm{E}} < 1$, the slim disc disappears and the standard accretion disc extends all the way to the inner disk edge. We model the luminosity based on this consideration in Section \[sec:efficiency\]. Since the surface temperature of the disc increases with decreasing the radius $r$, ionizing photons mostly come from the hot innermost part.
In the outer part, a disc wind might be launched from the disc surface photo-heated by the high-energy photons from the inner region [see @Proga:2000aa; @Proga:2004aa; @Nomura:2013aa; @Nomura:2016aa for line-driven disc wind of AGNs]. In addition, around the outer edge of the disc, the vertically falling flow due to the centrifugal barrier might collide with the one coming from the opposite side of the equatorial plane and form a shocked region. Fig. \[fig:acc\_whole\](a) depicts these structures, both of which can absorb (or scatter) the ionizing photons coming from the inner part, forming a shadowed region behind them. We assume that the shadowing effect considered here is caused by the outer disc structures, and thus the appearance of a slim disc is not essential in forming the shadowed region. The outer structure of the disc is highly uncertain and probably varies depending on the BH mass, accretion rate, angular momentum, metallicity of inflowing gas, etc.. In Section \[sec:direction\_dep\], we model the shadowing effect with a simple parametric fashion.
### Luminosity {#sec:efficiency}
In our simulation, we determine the luminosity of the BH radiation $L$ depending on $\dot{M}$ evaluated at the inner boundary at each time step. To model the luminosity, we adopt the fitting formula [@Watarai:2000aa], $$\begin{aligned}
L =
\begin{cases}
{\displaystyle}2\, L_{\mathrm{E}}\,\left[1+\ln\left(\frac{\dot{m}}{20}\right)\right] & \dot{m} > 20\\
{\displaystyle}0.1\, L_{\mathrm{E}}\, \dot{m} & \dot{m} < 20
\end{cases}\,,
\label{eq:3}\end{aligned}$$ where $\dot{m}\equiv \dot{M}/\dot{M}_{\mathrm{E}}$. [@Watarai:2000aa] obtained this formula by fitting the dependence of the luminosity on the accretion rate in the 1D stationary disc model, taking into account the (dis-)appearance of the slim disc depending on $\dot{M}$ . When $\dot{M}$ is low ($\dot{m} < 20$), the radiative efficiency is fixed at 10%, which agrees with that of the standard disc. For rapid accretion with $\dot{m} > 20$, the second term $2 L_{\mathrm{E}}\, \ln(\dot{m}/20)$ represents the luminosity from the innermost slim disc, where the photon advection reduces the radiative efficiency. Note that the luminosity $L$ increases logarithmically with $\dot{M}$ and can even exceed $L_{\mathrm{E}}$ because a large fraction of the emitted photons escape from the disc surfaces in vertical directions [see e.g., @Abramowicz:1988aa; @Watarai:2000aa; @Ohsuga:2005aa; @Jiang:2014aa; @Sc-adowski:2016aa]. The first term $2 L_{\mathrm{E}}$ corresponds to the luminosity from the outer standard disc in $r > r_{\mathrm{tr}}$ (see Fig. \[fig:acc\_whole\]a), given approximately by the energy generation rate due to the gravitational energy released by $r_{\mathrm{tr}}$, $G\dot{M}/r_{\mathrm{tr}} \sim L_{\mathrm{E}}$ [e.g., @Begelman:1978aa; @Kato:1998aa].
The spectrum of the BH radiation is simply assumed to be the power-law with $L_\nu\propto \nu^{-1.5}$ for $h\nu>13.6{\,\mathrm{eV}}$, where $L=\int_{h\nu>13.6{\mathrm{eV}}}L_\nu d\nu$, as often assumed in the literature [e.g., @Park:2011aa; @Park:2012aa; @Milosavljevic:2009ab]. [@Park:2011aa] have shown that the qualitative properties of accretion do not depend on the spectral shape.
### Directional dependence {#sec:direction_dep}
We inject ionizing photons at the inner boundary with the directional dependence described below. Specifically, we multiply the anisotropy factor $\mathcal{F}(\theta)$ normalized as $\int
\mathcal{F}(\theta)d\Omega=4\pi$ with an isotropic radiation flux $L/4\pi R_{\mathrm{in}}^2$ at the inner boundary $R_{\mathrm{in}}$. With this definition, $\mathcal{F}(\theta) = 1$ represents the isotropic radiation (Fig. \[fig:dirdep\]). We use the latitudinal angle $\theta$ defined as the angle measured from the equatorial plane for our convenience.
Motivated by the expected disc structure described in Section \[sec:acc\_disc\] (also see Fig. \[fig:acc\_whole\]a), we model $\mathcal{F}(\theta)$ as $$\begin{aligned}
\mathcal{F}(\theta) &= C\, f_{\mathrm{disc}}(\theta)\, f_{\mathrm{shadow}}(\theta)\,,
\label{eq:5}\end{aligned}$$ where $C$ is the normalization factor. In this expression, the inner anisotropy factor $f_{\mathrm{disc}}$ that represents the directional dependences of the radiation emitted from the inner part of the disc is multiplied by the outer one $f_{\mathrm{shadow}}$ to take into account the outer shadowing effect.
For the inner anisotropy factor $f_{\mathrm{disc}}$, we simply assume $$\begin{aligned}
f_{\mathrm{disc}}(\theta)\propto\sin\theta\,,
\label{eq:11} \end{aligned}$$ which corresponds to radiation from an infinitely thin disc (recall that we define $\theta$ as the angle from the equatorial plane). Although numerical simulations suggest somewhat steeper $\theta$-dependence especially in the polar directions [e.g., @Ohsuga:2005aa; @Sc-adowski:2016aa], such deviations cause little effects on our results because the mass accretion predominantly occurs through the infalling region near the equatorial plane. For the disc radiation without the outer shadowing effect (i.e., $f_{\mathrm{shadow}}=1$), the normalized anisotropy factor is $\mathcal{F}(\theta) = 2\sin \theta$ (Fig. \[fig:dirdep\]).
We model the outer anisotropy factor $f_{\mathrm{shadow}}$ as $$\begin{aligned}
f_{\mathrm{shadow}}(\theta)
=
\begin{cases}
{\displaystyle}\exp\left[-\left(\frac{\theta-\tilde{\theta}_{\mathrm{shadow}}}{\delta\theta}\right)^2\right]
&0 < \theta < \tilde{\theta}_{\mathrm{shadow}}\\[0.4cm]
1 & \tilde{\theta}_{\mathrm{shadow}} < \theta < 90^\circ
\end{cases}
\label{eq:10}\end{aligned}$$ where $\tilde{\theta}_{\mathrm{shadow}}=\theta_{\mathrm{shadow}}+2\,\delta\theta$, $\theta_{\mathrm{shadow}}$ is the opening angle of the shadow, and $\delta\theta$ the thickness of the transition region. Here, we assume $f_{\mathrm{shadow}}$ is symmetric about the equatorial plane. We adopt the finite transition region setting $\delta\theta = 6^\circ$ to avoid artificial ionization structure that appears with $\delta\theta\rightarrow 0$. Our conclusions are independent of the arbitrary choice of a small value for $\delta\theta$. We show $\mathcal{F}(\theta)$ for the disc radiation with the outer shadowing effect with $\theta_{\mathrm{shadow}}=45^\circ$ and $22.5^\circ$ in Fig. \[fig:dirdep\]. With the expression given by equation , the outer anisotropy factor begins to decrease even for $\theta > \theta_{\mathrm{shadow}}$, and takes a value of $\sim 0.01$ at $\theta = \theta_{\mathrm{shadow}}$. Although we fix the shadowing profile $f_{\mathrm{shadow}}(\theta)$ during each simulation run for simplicity, it probably depends on accretion rates in reality.[^3] In view of large uncertainties in the shadowing effect, we perform a number of simulations varying $\theta_{\mathrm{shadow}}$ as a free parameter (see Sec. \[sec:parameters\]).
Cases considered {#sec:parameters}
----------------
[lccccccccccc]{} run& $M_{\mathrm{BH}}\,[M_\odot]$ & $n_\infty\,[{\mathrm{cm^{-3}}}]$ & $\theta_{\mathrm{shadow}}^{a}$& $N_r\times N_\theta$ & $R_{\mathrm{in}}\,[{\mathrm{AU}}]$ & $R_{\mathrm{out}}\,[{\mathrm{AU}}]$ & $t_{\mathrm{end}}\,[{\mathrm{yr}}]$\
Di & $10^3$ & $10^5$ & [**isotropic**]{}$^{b}$ & $512\times144$ & $3\times10^2$ & $6\times10^5$ & $5\times10^5$\
Ddn & $10^3$ & $10^5$ & [**disc**]{}$^{c}$ & $512\times144$ & $3\times10^2$ & $6\times10^5$ & $5\times10^5$\
Dds$^{d}$ & $10^3$ & $10^5$ & [$45^\circ$]{} & $512\times144$ & $2\times10^3$ & $3\times10^6$ & $2\times10^6$\
\
s075 & $10^3$ & $10^5$ & [$33.75^\circ$]{} & $256\times72$ & $2\times10^3$ & $3\times10^6$ & $2\times10^6$\
s050 & $10^3$ & $10^5$ & [$22.5^\circ$]{} & $256\times72$ & $2\times10^3$ & $3\times10^6$ & $2\times10^6$\
s025 & $10^3$ & $10^5$ & [$11.25^\circ$]{} & $256\times72$ & $2\times10^3$ & $3\times10^6$ & $2\times10^6$\
\
M1e2 & [$10^2$]{} & $10^5$ & $45^\circ$ & $256\times72$ & $2\times10^2$ & $1.5\times10^6$ & $2\times10^6$\
M1e4 & [$10^4$]{} & $10^5$ & $45^\circ$ & $256\times72$ & $2\times10^4$ & $2\times10^7$ & $2\times10^7$\
M1e5 & [$10^5$]{} & $10^5$ & $45^\circ$ & $256\times72$ & $2\times10^5$ & $1\times10^8$ & $5\times10^7$\
\
n1e3 & $10^3$ & [$10^3$]{} & $45^\circ$ & $256\times72$ & $2\times10^3$ & $1\times10^7$ & $5\times10^7$\
n1e4 & $10^3$ & [$10^4$]{} & $45^\circ$ & $256\times72$ & $2\times10^3$ & $6\times10^6$ & $2\times10^7$\
n1e6 & $10^3$ & [$10^6$]{} & $45^\circ$ & $256\times72$ & $2\times10^3$ & $2\times10^6$ & $2\times10^6$\
\
NOTES.$^{a}$Disc radiation with shadowing effect is assumed except for Di and Ddn runs; $^{b}$isotropic radiation; $^{c}$disc radiation without shadowing effect; $^{d}$Dds run is also called s100, M1e3 and n1e5 runs.
We perform a set of simulations to see how the directional dependence of BH irradiation affects the nature of accretion. Table \[tab:model\] summarizes model parameters and numerical settings adopted for the cases examined. In all the cases, we initially set a static and homogeneous neutral medium with the number density $n_\infty$ and the temperature $T_{\mathrm{HI}}=10^4{\,\mathrm{K}}$ around a central BH. The BH mass $M_{\mathrm{BH}}$ is fixed constant during the calculation for simplicity.
In Section \[sec:structure\], we perform three high-resolution simulations, called “D-series” (for “Directional”), with different types of the directional dependence of the BH irradiation. For “Di run” (“i” for “isotropic”), we assume the isotropic irradiation, i.e., $f_{\mathrm{disc}} = f_{\mathrm{shadow}}=1$ in equation . The anisotropic disc radiation without the outer shadowing effect, i.e., $f_{\mathrm{shadow}}=1$, is assumed for “Ddn run” (“dn” for “disc no-shadow”), and both $\theta$-dependences of $f_{\mathrm{disc}}$ and $f_{\mathrm{shadow}}$ are allowed for “Dds run” (“ds” for “disc shadow”). Below we take $\theta_{\mathrm{shadow}} = 45^\circ$ as the fiducial value for the shadow opening angle. For the other parameters, we take $M_{\mathrm{BH}}=10^3M_\odot$ and $n_\infty=10^{5}{\,\mathrm{cm^{-3}}}$. Note that, for this set of $M_{\mathrm{BH}}$ and $n_\infty$, previous studies with isotropic BH irradiation have shown that the accretion rate is significantly reduced by radiation feedback ([@Milosavljevic:2009ab; @Park:2012aa; @Inayoshi:2016ac]; see also Sec. \[sec:sph\_acc\]).
In Sec. \[sec:dependence\], we study how the BH accretion changes with different shadow size $\theta_{\mathrm{shadow}}$, BH mass $M_{\mathrm{BH}}$, and ambient density $n_\infty$. First, to see the $\theta_{\mathrm{shadow}}$-dependence, we perform three simulations of “s-series” (for “shadow”) with different values of $\theta_{\mathrm{shadow}}$ (Sec. \[sec:sdep\]). Specifically, we take $\theta_{\mathrm{shadow}}=11.25^\circ$, $22.5^\circ$, $33.75^\circ$ and $45^\circ$. Second, we study the $M_{\mathrm{BH}}$-dependence with the “M-series”, where we take $M_{\mathrm{BH}}=10^2$, $10^3$, $10^4$ and $10^5\,M_\odot$ (Sec. \[sec:Mdep\]). Finally, the $n_\infty$-dependence is examined with the “n-series”, where we take different values of $n_\infty=10^3$, $10^4$, $10^5$ and $10^6{\,\mathrm{cm^{-3}}}$ (Sec. \[sec:ndep\]). In the above simulations, we take the fiducial values of $M_{\mathrm{BH}}=10^3M_\odot$, $n_\infty=10^{5}{\,\mathrm{cm^{-3}}}$ and $\theta_{\mathrm{shadow}} =45^\circ$ unless otherwise stated. We discuss the parameters relevant to the growth of the remnant BHs of Pop III stars in Section \[sec:conclusion\].
For each case, the inner and outer boundaries $R_{\mathrm{in}}$ and $R_{\mathrm{out}}$ are determined in the following way. We choose small enough $R_{\mathrm{in}}$ to correctly evaluate $\dot{M}$. To be more specific, $R_{\mathrm{in}}$ is taken to be much smaller than the Bondi radius for a neutral (ionized) gas when the dominant component of the accreting gas is neutral (ionized). We choose large enough $R_{\mathrm{out}}$ to keep an [H[ii]{} ]{}bubble within a simulation region. We only allow the flow going out of the computational domain at the inner boundary at $R_{\mathrm{in}}$, where $\dot{M}$ is evaluated. Across the outer boundary at $R_{\mathrm{out}}$, however, both the inflow and outflow are allowed. In the angular direction, the computational domain is $0<\theta<90^\circ$ under the assumption of the equatorial symmetry.
The grid numbers are taken to be $N_r\times N_\theta=512\times144$ and $256\times 72$ for the high- and medium-resolution simulations, respectively (see Table \[tab:model\]). In order to simultaneously resolve the Bondi and Strömgren radii, which are different typically by $3-4$ orders of magnitude, we increase the radial cell size $\Delta
r$ with the fixed size ratio $\Delta r_i/\Delta r_{i-1}$ $(>1)$. We set $\Delta r_1 = 0.1 R_{\mathrm{in}}$ at the inner boundary. The grids in the angular direction are homogeneously distributed over $0<\theta<90^\circ$, and thus the grid size is $\Delta\theta =
90^\circ/N_\theta$.
We have tested the convergence of the numerical results by varying the grid numbers or inner boundary radius (see Appendix \[sec:res\_check\]). We follow the evolution over the duration $t_{\mathrm{end}}$, until the accretion reaches a steady state in Di and Ddn runs, or until $\dot{M}$ reaches almost constant in the other runs.
Results {#sec:result}
=======
Structures of flows {#sec:structure}
-------------------
. \[tab:d-model\]
run subgrid radiation type $\theta_{\mathrm{inflow}}(r_{\mathrm{B}})$$^{a}$ $\dot{M}/\dot{M}_{\mathrm{B}}$$^{d}$
----- ------------------------ -------------------------------------------------- --------------------------------------
Di isotropic ...$^{b}$ $0.17\%^{e}$
Ddn disc ...$^{c}$ $0.13\%^{e}$
Dds disc $+$ shadow $40^\circ$ $59\%^{f}$
: Summary of the results in Sec. \[sec:structure\]
\
NOTES. $^{a}$opening angle of equatorial neutral inflow region at $r_{\mathrm{B}}$ (see text); $^{b}$no equatorial neutral region; $^{c}$equatorial neutral region does not reach $R_{\mathrm{in}}$; $^{d}$accretion rate normalized by Bondi one; $^{e}$averaged between $t=4\times 10^5{\,\mathrm{yr}}$ and $5\times 10^5{\,\mathrm{yr}}$; $^{f}$evaluated at the end of simulation.
In this section, we perform the simulations of “D-series”, in order to see how the flow structure changes with different directional dependences of the radiation fields. The basic results for these cases are summarized in Table \[tab:d-model\].
### Case with isotropic radiation {#sec:spherical_sym}
We first describe “Di run”, for which we assume the isotropic BH irradiation with $M_{\mathrm{BH}}=10^3M_\odot$ and $n_\infty=10^5{\,\mathrm{cm^{-3}}}$. The behaviour of the accretion flow is qualitatively the same as those obtained in the earlier works [@Milosavljevic:2009ab; @Park:2011aa; @Park:2012aa], and so we refer the readers to the above literature for full details.
Fig. \[fig:mdot\](a) shows the time evolution of the accretion rate and luminosity in our 2D simulation, along with the result of 1D calculation with the same parameter set. As explained by [@Park:2011aa], the accretion rate oscillates by repeating the following three phases: (a) high thermal pressure of the hot [H[ii]{} ]{}bubble suppresses the gas inflow and forms a dense shell of the swept up neutral gas; (b) the accretion rate, and hence the luminosity, decreases because the density of the [H[ii]{} ]{}bubble decreases due to the bubble expansion and/or gas inflow into the sink, leading to the contraction of the [H[ii]{} ]{}bubble with the dense shell; (c) an accretion burst caused by the collapse of the shell dramatically increases the luminosity and revive the large [H[ii]{} ]{}bubble again. The interval time between bursts roughly corresponds to the sound crossing time across the [H[ii]{} ]{}bubble. The accretion history in the 2D simulation is identical to the 1D result in an early stage ($t\lesssim 5\times10^4{\,\mathrm{yr}}$), but deviates from it later on because the spherical symmetry breaks down due to the growth of numerical perturbations by the instability of expanding ionization front [e.g., @Garcia-Segura:1996aa; @Whalen:2008ac; @Whalen:2008aa; @Park:2014ab]. Although the qualitative features are similar in the 1D and 2D cases, the accretion variability is slightly weaker in the latter case. The peaks of accretion burst in different directions are smoothed out because they are not exactly synchronized in the 2D case.
The average accretion rate between $t=4\times 10^5{\,\mathrm{yr}}$ and $5\times
10^5{\,\mathrm{yr}}$ is only $\dot{M}=1.7\times10^{-3}\dot{M}_{\mathrm{B}}$. Such a low rate is consistent with the Bondi rate in the ionized medium, $\dot{M}_{\mathrm{B,HII}} \sim 1 \times 10^{-3} \dot{M}_{\mathrm{B}}$ (see Sec. \[sec:sph\_acc\]). Our result is in good agreement with the previous ones by @Milosavljevic:2009ab and @Park:2012aa, who provided $\dot{M}\sim 2\times10^{-3}\dot{M}_{\mathrm{B}}$ and $\dot{M}\sim
10^{-2}\dot{M}_{\mathrm{B}}$, respectively. The differences of a factor of a few might come from differences in the adopted chemistry, because the accretion rate is sensitive to the thermal structure within the [H[ii]{} ]{}bubble [@Park:2011aa; @Park:2012aa].
Fig. \[fig:snap\_Di\](a) and (b) show the structures of accretion flows before and after an accretion burst, respectively. As explained above, the [H[ii]{} ]{}bubble shrinks before the burst (Fig. \[fig:snap\_Di\]a) and expands again due to the enhanced luminosity after the burst (Fig. \[fig:snap\_Di\]b). Whereas the Bondi radius $r_{\mathrm{B}}$ for the ambient neutral gas is illustrated in the figure, $r_{\mathrm{B,HII}}$ for the ionized medium is, although resolved in our simulations, too small to be shown. The velocity field of the ionized gas does not exhibit a systematic inflow but subsonic turbulent structure since the gas pressure dominates the gravity outside the Bondi radius. These snapshots are also very similar to those shown in the previous works [@Milosavljevic:2009ab; @Park:2011aa; @Park:2012aa].
Note that there appears a thin finger-like structure of the neutral gas along the $z$ axis in Fig. \[fig:snap\_Di\](b). Since both initial condition and BH irradiation are spherically symmetric, the flow patterns should also be spherically symmetric at least in a statistical sense. Thus, this is an artifact of our 2D simulation, presenting its limitation. Any flows toward the $z$ axis inevitably collide each other on the axis due to the assumed axisymmetry, creating a high density neutral gas column that shadows the cells behind it. We expect this artifact vanishes in future 3D simulations.
### Case with disc radiation without shadowing effect {#sec:disc_rad}
Next, we consider “Ddn run”, for which we assume the disc radiation without shadowing effect. Specifically, we adopt the anisotropy factor $\mathcal{F}=2\sin\theta$ by taking $f_{\mathrm{shadow}}=1$ in equation .
Fig. \[fig:mdot\](b) presents that, as in the case with isotropic radiation, the accretion rate and luminosity initially show strong oscillatory behaviours. However, the oscillation settles down in $3\times10^5{\,\mathrm{Myr}}$, after which only weak variability remains. We consider that the initial strong oscillation occurs due to the artificial initial condition of a static homogeneous medium. The accretion variability gradually ceases, after which the flow structure reaches a quasi-steady state. In this case, burst accretions coming from different directions cannot be synchronized due to the aspherical shape of the [H[ii]{} ]{}bubble created by the anisotropic BH irradiation, resulting in the less variable accretion rate. The mean accretion rate between $t=4\times 10^5{\,\mathrm{yr}}$ and $5\times 10^5{\,\mathrm{yr}}$, when the oscillation has already abated, is $\dot{M}=1.3\times10^{-3}\dot{M}_{\mathrm{B}}$. This low rate is nearly the same as that in the isotropic irradiation case, and also well approximated by the Bondi rate from the ionized medium (see Sec. \[sec:sph\_acc\]). The inflows through the equatorial neutral region have little contribution to the accretion rate, as will be seen below.
![Same as Fig. \[fig:snap\_Di\] but for the case with disc radiation without shadowing effect. The structure of accretion flow at the end of the simulation is shown on the larger and smaller scales in panels (a) and (b), respectively. The white dashed square in panel (a) represents the region plotted in (b). In addition to the Bondi radius for the ambient neutral gas ($r_{\mathrm{B}}$, dashed black) shown in both panels (a) and (b), that for the ionized medium ($r_{\mathrm{B,HII}}$, dashed white) is shown in (b).[]{data-label="fig:snap_Dd"}](figure/snap_Ddn.eps){width="7cm"}
Fig. \[fig:snap\_Dd\] shows the structure of the accretion flow at the end of the simulation. The whole [H[ii]{} ]{}bubble is shown in the upper panel, while the central region on the scale of $r_{\mathrm{B}}$ is enlarged in the lower panel. In Fig. \[fig:snap\_Dd\](a), we see that the pressure equilibrium is approximately realized throughout the simulation region. The [H[ii]{} ]{}bubble is squeezed in the equatorial directions because of the anisotropic irradiation. The gas within the [H[ii]{} ]{}bubble moves upward until colliding with the ambient neutral medium. A high-density region near the $z$ axis just outside the [H[ii]{} ]{}bubble is again likely to be an artifact, as seen in Sec. \[sec:spherical\_sym\]. Hereafter, we will ignore this kind of features since it hardly affects our conclusion. In Fig. \[fig:snap\_Dd\](b), the gas is ionized in most of the region except the equatorial thin neutral layer, which extends inward across the Bondi radius but does not reach the sink. Most of the outflow within the [H[ii]{} ]{}region is launched from this equatorial neutral layer. In this figure, we do not clearly see the inflow, which is actually limited to a very central part, because the Bondi radius for the ionized medium $r_{\mathrm{B,HII}}$ is currently much smaller than the size of the plotted area.
With the current spatial resolution, the thickness of the equatorial neutral layer is limited by the angular cell size of $\Delta \theta =
0.6^\circ$. The ionizing photon flux injected into the cells closest to the equatorial plane ($0 \leq \theta \leq \Delta\theta$) is reduced to $\sim 0.01$ of the angle-averaged value for the assumed anisotropy. This flux is, however, still large enough to make the [H[ii]{} ]{}region extend beyond the sink radius. In fact, the Strömgren radius with $n=
n_\infty =10^5{\,\mathrm{cm^{-3}}}$ and $L = 10^{-3}L_{\mathrm{E}}$ gives $r_{\mathrm{HII}}
\simeq 7\times10^3{\,\mathrm{AU}}$ (equation \[eq:13\]), which is much larger than the sink radius $R_{\mathrm{in}} = 3 \times 10^2{\,\mathrm{AU}}$. The horizontal extension of the bubble is even larger than this because the actual bubble density is smaller than $n_\infty$. If we could perform a simulation with much higher resolution, a thinner equatorial neutral region would reach the sink as the less ionizing photon flux is injected for the smaller $\theta$. Such a very thin neutral region, however, is not expected to affect the overall accretion, because the mass that can be carried through such a very thin region is severely limited and is further reduced by the mass loss into the [H[ii]{} ]{}bubble, as will be shown later (Sec. \[sec:shadow\_rad\] - \[sec:mass\_loss\_inner\]). Moreover, diffuse recombination photons processed within the bubble, which are not considered in the current simulations, would eliminate such a very thin neutral region [see, e.g., @Hollenbach:1994aa; @Tanaka:2013aa].
### Case with disc radiation with shadowing effect {#sec:shadow_rad}
Finally, we describe “Dds run”, in which the inner disc radiation is modified by the outer shadowing effect, as $\mathcal{F}=C\,f_{\mathrm{disc}}\,f_{\mathrm{shadow}}a$ (equation \[eq:5\]). We adopt $\theta_{\mathrm{shadow}}=45^\circ$ for the outer anisotropy factor $f_{\mathrm{shadow}}$ given by equation .
Fig. \[fig:mdot\](c) shows that the accretion rate converges towards an constant value $\dot{M} \simeq 0.59~\dot{M}_{\mathrm{B}}$ in $1{\,\mathrm{Myr}}$, which is much higher than the value $\dot{M} \lesssim
2\times10^{-3}~\dot{M}_{\mathrm{B}}$ obtained in the former two cases with isotropic radiation and disc radiation without shadowing effect. This accretion rate is also “super-critical” and 400 times larger than the Eddington-limited rate $\dot{M}_{\mathrm{E}}$. By the end of the simulation, the luminosity also converges to $L \simeq 8L_{\mathrm{E}}$, i.e., a super-Eddington luminosity realized by the high accretion rate.
Fig. \[fig:snap\_Ds\] shows the structure of the accretion flow at the end of the simulation. The whole [H[ii]{} ]{}bubble is shown within the large plotted area of Fig. \[fig:snap\_Ds\](a), while the central regions over $\sim 10^6$ AU and $\sim 10^5$ AU scales are enlarged in Figs. \[fig:snap\_Ds\](b) and \[fig:snap\_Ds\](c). Owing to the shadowing effect, we see in Fig. \[fig:snap\_Ds\](a) that the large horizontal neutral region cuts into the central part with the [H[ii]{} ]{}bubbles bound to the bipolar regions. Similarly to the case with disc radiation without shadowing effect (Sec. \[sec:disc\_rad\]), the gas within the bipolar [H[ii]{} ]{}bubbles flows outward and collides with the ambient neutral medium. The pressure equilibrium is approximately achieved throughout the simulation region, although the thermal pressure slightly decreases before the collision because the total pressure including the ram pressure is balanced. The size of the bipolar [H[ii]{} ]{}bubbles is much larger than in the former cases owing to the much higher luminosity $L \simeq 8L_{\mathrm{E}}$. Note that the injected radiation is super-Eddington only in the polar region with $\theta \gtrsim 50^\circ$.
Fig. \[fig:snap\_Ds\](b) shows that the gas inside the equatorial neutral region is almost at rest in the pressure equilibrium, while that on the surfaces is photoevaporated to join the ionized outflow. In Fig. \[fig:snap\_Ds\](c), where the structure over the scale of the Bondi radius is presented, the gas flows into the central sink through the equatorial neutral region. The density and pressure increase with decreasing $r$ for $r \lesssim r_{\mathrm{B}}$, as expected for the Bondi flow. As seen in Fig. \[fig:snap\_Ds\](b), the neutral gas is photoevaporated into the [H[ii]{} ]{}regions, where the acceleration by the radiation pressure is stronger than the gravitational pull owing to the super-Eddington fluxes in the polar directions. The accretion proceeds only through the solid angle covered by the equatorial neutral region. Note that, in our simulations, we neglect possible photoevaporation outflow coming out from the sink. We discuss it later in Sec. \[sec:mass\_loss\_inner\]. In the next section, we investigate the structure of the flow in more detail.
### Analysis of flow structure in case with shadowing effect {#sec:anl_modeling}
In this section, we develop an analytical model and compare it with our result to examine the inflow-outflow structure presented in Fig. \[fig:snap\_Ds\]. The overall structure of our model is schematically depicted in Fig. \[fig:anl\_model\] and can be summarized as follows: in the equatorial neutral region where ionizing photons cannot penetrate, the gas inflows in a Bondi accretion fashion; in the bipolar [H[ii]{} ]{}regions where ionizing photons heat up the gas via photoionization, the outflows are launched due to the thermal and radiation pressure; through their boundaries, the photoevaporating gas is lost from the neutral region and supplied into the [H[ii]{} ]{}regions.
We begin with considering the density profiles of the inflow and outflow. The radial density profile in the equatorial neutral region $n_{\mathrm{inflow}}(r)$ is well approximated by that of the Bondi solution, which we further simplify as $$\begin{aligned}
n_{\mathrm{inflow}}(r) &=
\begin{cases}
{\displaystyle}n_\infty\left(\frac{r}{r_{\mathrm{B}}}\right)^{-3/2}&r < r_{\mathrm{B}}\\[0.4cm]
n_\infty&r > r_{\mathrm{B}}
\end{cases}\label{eq:9} \,.
\end{aligned}$$ This expression slightly over- and underestimates the obtained density profile at $r\ll r_{\mathrm{B}}$ and $r\sim r_{\mathrm{B}}$. As for the bipolar ionized outflows, the density profile $n_{\mathrm{outflow}}(r)$ can be estimated by assuming the pressure equilibrium at the conical boundaries between the neutral and ionized gas, $$\begin{aligned}
n_{\mathrm{outflow}}(r)&=n_{\mathrm{inflow}}(r)\left(\frac{T_{\mathrm{HI}}}{2T_{\mathrm{HII}}}\right)\,,
\label{eq:7}
\end{aligned}$$ where $T_{\mathrm{HI}} \simeq 10^4{\,\mathrm{K}}$ and $T_{\mathrm{HII}} \simeq
7\times10^4{\,\mathrm{K}}$ are the temperature in the neutral and ionized regions, respectively. Although not very precise, this simple expression captures the qualitative features of the bipolar ionized outflows.
![The opening angle of the equatorial neutral inflow region $\theta_{\mathrm{inflow}}$ as a function of $r$ (solid red), along with the angle-dependence of the radius of the [[H[ii]{} ]{}]{} region $r=r_{\mathrm{HII}}(\theta)$ (dashed orange; equation \[eq:6\]). The dotted part of the red line marks a few innermost cells that are artificially ionized as we neglect the absorption within the sink (also see the text). []{data-label="fig:th_in_Ds"}](figure/th_in_Ds.eps){width="8.5cm"}
Fig. \[fig:th\_in\_Ds\] shows the opening angle of the equatorial neutral region $\theta_{\mathrm{inflow}}$ as a function of $r$. In practice, we define the neutral region as the region where the ionization degree of hydrogen is less than $50\%$. Although $\theta_{\mathrm{inflow}}$ decreases as $r$ decreases, the neutral inflow region reaches the inner boundary at $r = R_{\mathrm{in}}$ with a finite angle, unlike in the case with disc radiation without shadowing effect. The sharp drop of $\theta_{\mathrm{inflow}}$ around $R_{\mathrm{in}} = 2000$ AU is caused by photoionization of a few innermost cells due to our ignorance of the consumption of ionizing photons within the sink, although this does not affect our conclusion. The opening angle at the Bondi radius, $\theta_{\mathrm{inflow}}(r_{\mathrm{B}})=40^\circ$, is similar to the assumed shadow opening angle $\theta_{\mathrm{shadow}} =45^\circ$ (equation \[eq:10\]).
In order to estimate $\theta_{\mathrm{inflow}}$, we calculate the radius of the [[H[ii]{} ]{}]{} region $r_{\mathrm{HII}}$ in each direction $\theta$ with the modeled density profiles. The supply rate of ionizing photons per unit solid angle is given by $\dot{N}_{\mathrm{ion}}\,\mathcal{F}(\theta)/4\pi$, where $\dot{N}_{\mathrm{ion}}$ is the total ionizing photon emissivity of the central accretion disc, given by $\dot{N}_{\mathrm{ion}}=L/3h\nu_{\mathrm{T}}$ for the assumed spectral shape of $L_\nu\propto \nu^{-1.5}$. Equating this supply rate with the recombination rate, we have $$\frac{L}{3h \nu_{\mathrm{T}}}\frac{\mathcal{F}(\theta)}{4 \pi}
= \int_{R_{\mathrm{in}}}^{r_{\mathrm{HII}}}(\theta)
\alpha_{\mathrm{B}}\,n_{\mathrm{outflow}}^2\,r^2\,{\mathrm{d}}r ,
\label{eq:uve}$$ where $n_{\mathrm{outflow}}$ is given by equation . Performing the integration in equation , we finally get $$\begin{aligned}
r_{\mathrm{HII}}(\theta)
=
\begin{cases}
{\displaystyle}R_{\mathrm{in}} \exp\left[
\frac{L\,\mathcal{F}(\theta)\,T_{\mathrm{HII}}^2}{3\pi h\nu_{\mathrm{T}}\, \alpha_{\mathrm{B}}\, n_\infty^2\, r_{\mathrm{B}}^3\, T_{\mathrm{HI}}^2}
\right]&r_{\mathrm{HII}} < r_{\mathrm{B}}\\[0.4cm]
{\displaystyle}\left[
\frac{L\,\mathcal{F}(\theta)\,T_{\mathrm{HII}}^2}
{\pi h\nu_{\mathrm{T}}\,\alpha_{\mathrm{B}}\,n_\infty^2\,T_{\mathrm{HI}}^2}
- 3r_{\mathrm{B}}^3 \ln\left(\frac{r_{\mathrm{B}}}{R_{\mathrm{in}}}\right)
+ r_{\mathrm{B}}^3
\right]^{1/3} \hspace{-2cm}\\[0.4cm]
&r_{\mathrm{HII}} > r_{\mathrm{B}}
\end{cases}\,.
\label{eq:6}\end{aligned}$$ Here, we show the relation $r=r_{\mathrm{HII}}(\theta)$, or equivalently $\theta=r_{\mathrm{HII}}^{-1}(r)$, in Fig. \[fig:th\_in\_Ds\] with $L =
8.2\,L_{\mathrm{E}}$ (Fig. \[fig:mdot\]). We see that equation qualitatively reproduces $\theta_{\mathrm{inflow}} (r)$ obtained in the simulation with a small deviation of a few degrees at each $r$. Such a deviation mainly comes from approximate modelling of $n_{\mathrm{outflow}}$ in equation . For example, in an outer part of the [H[ii]{} ]{}bubble where helium is not doubly ionized, the gas is no longer heated up to $7\times10^4{\,\mathrm{K}}$ by the ${\mathrm{He^+}}$ photoionization, resulting in the higher density than that estimated by equation with $T_{\mathrm{HII}} = 7 \times 10^4{\,\mathrm{K}}$. Nonetheless, our simple modelling with equation describes the numerical results well.
![The radial dependence of the equatorial inflow rate $\dot{M}_{\mathrm{inflow}}(r)$ (solid blue) and bipolar outflow rate $\dot{M}_{\mathrm{outflow}}(r)$ (solid red), defined in equation . The blue arrow marks the estimated value of the inflow rate at the inner boundary $\dot{M}_{\mathrm{inflow}}(R_{\mathrm{in}})$ (equation \[eq:17\]). The red dashed line represents the analytical estimate of the outflow rate $\dot{M}_{\mathrm{outflow}}(r)$ (equation \[eq:14\]). The values of the Bondi radius $r_{\mathrm{B}}$ and accretion rate $\dot{M}_{\mathrm{B}}$ are marked by the black arrows. []{data-label="fig:mdoio_Ds"}](figure/mdotio_Ds.eps){width="8.5cm"}
Finally, we investigate the flow rates in the neutral and ionized regions. Fig. \[fig:mdoio\_Ds\] shows the equatorial inflow rate $\dot{M}_{\mathrm{inflow}}$ and bipolar outflow rate $\dot{M}_{\mathrm{outflow}}$ through a spherical surface with radius $r$, $$\begin{aligned}
\dot{M}_{\mathrm{inflow}}(r) &= - 4\pi r^2\int_0^{\theta_{\mathrm{inflow}}} \rho\,v_r\,\cos \theta d\theta\,,{\nonumber\\}\dot{M}_{\mathrm{outflow}}(r) &= 4\pi r^2\int_{\theta_{\mathrm{inflow}}}^{\pi/2} \rho\,v_r\, \cos \theta d\theta\,,
\label{eq:16}\end{aligned}$$ where $v_r$ is the outward velocity. We have multiplied a factor of two to take into account the equatorial symmetry. The net accretion rate $\dot{M} \equiv \dot{M}_{\mathrm{inflow}}-\dot{M}_{\mathrm{outflow}}$ is almost constant with $r$, consistent with a quasi-steady flow structure. The value of $\dot{M}$ is equal to the inflow rate $\dot{M}_{\mathrm{inflow}}$ at the inner boundary $R_{\mathrm{in}}$, where $\dot{M}_{\mathrm{outflow}} = 0$ is imposed as the boundary condition.
We now estimate $\dot{M}_{\mathrm{inflow}}(R_{\mathrm{in}})$ ($=\dot{M}$) from the Bondi-like accretion through a solid angle corresponding to the opening angle $\theta_{\mathrm{inflow}}(r_{\mathrm{B}})$. With the solid angle $\Delta\Omega_{\mathrm{inflow}}(r_{\mathrm{B}})=4\pi
\,\sin\theta_{\mathrm{inflow}}(r_{\mathrm{B}})$, we obtain $$\begin{aligned}
\dot{M}_{\mathrm{inflow}}(R_{\mathrm{in}})=
\frac{\Delta\Omega_{\mathrm{inflow}}(r_{\mathrm{B}})}{4\pi}\dot{M}_{\mathrm{B}}\,.
\label{eq:17}\end{aligned}$$ Fig. \[fig:mdoio\_Ds\] demonstrates that equation well reproduces $\dot{M}_{\mathrm{inflow}}(R_{\mathrm{in}})$ measured in the simulation,[^4] although slightly overestimated owing to the photoevaporation mass loss. Note that the inflow rate $\dot{M}_{\mathrm{inflow}}$ increases with $r$ and exceeds the Bondi rate $\dot{M}_{\mathrm{B}}$ at $\sim 3000$ AU. This is because the circulation flows are generated to compensate the photoevaporation mass loss from the equatorial neutral region (see below) and contribute to the accretion rate together with the Bondi-like inflow.
The outflow rate can be modeled as the photoevaporation mass loss. The mass-loss flux from the surfaces of neutral region is estimated as $f_{\mathrm{outflow}}\,\rho_{\mathrm{b,HII}}\, c_{\mathrm{s,HII}}$, where $f_{\mathrm{outflow}}$ is an $O(1)$ correction factor, $\rho_{\mathrm{b,HII}}$ the density at the bottom of the ionized layer and $c_{\mathrm{s,HII}}$ the sound velocity for ionized gas [e.g., @Hollenbach:1994aa; @Tanaka:2013aa]. The outflow rate $\dot{M}_{\mathrm{outflow}}$ through a given radius $r$ is obtained by integrating the mass-loss fluxes between $R_{\mathrm{in}}$ and $r$ because the outflow is in a quasi-steady state. In reality, additional mass loss may happen even inside $R_{\mathrm{in}}$, as we will discuss in Sec. \[sec:mass\_loss\_inner\]. With $\rho_{\mathrm{b,HII}} \simeq m_{\mathrm{p}}
n_{\mathrm{outflow}}$ (equation \[eq:7\]) and $c_{\mathrm{s,HII}}=(2T_{\mathrm{HII}}/T_{\mathrm{HI}})^{1/2}c_{\mathrm{s,HI}}$, we obtain $$\begin{aligned}
\dot{M}_{\mathrm{outflow}}(r)
\simeq 4\pi\int_{R_{\mathrm{in}}}^r &f_{\mathrm{outflow}}\,m_{\mathrm{p}}\,c_{\mathrm{s,HII}}\,n_{\mathrm{outflow}} r' {\mathrm{d}}r'{\nonumber\\}=
2\pi\,f_{\mathrm{outflow}}\,& m_{\mathrm{p}}\, n_\infty\,c_{\mathrm{s,HI}}
\left(\frac{2T_{\mathrm{HII}}}{T_{\mathrm{HI}}}\right)^{1/2}{\nonumber\\}\times&
\begin{cases}
4 r_{\mathrm{B}}^{3/2} \left(r^{1/2}-R_{\mathrm{in}}^{1/2}\right)
& r < r_{\mathrm{B}} \\
r^2 + 3r_{\mathrm{B}}^2 -4r_{\mathrm{B}}^{3/2}R_{\mathrm{in}}^{1/2}
&
r > r_{\mathrm{B}}
\end{cases}\,,
\label{eq:14}\end{aligned}$$ where a factor of two is multiplied in the first equality to take into account both top and bottom surfaces. We see in Fig. \[fig:mdoio\_Ds\] that modeled $\dot{M}_{\mathrm{outflow}}$ with the best-fit value of $f_{\mathrm{outflow}}=0.7$ reproduces the simulation result with remarkable agreement.
### Possible mass loss from neutral inflow inside the sink {#sec:mass_loss_inner}
As mentioned above, we neglect the possible mass loss from the innermost part of the flow masked by the sink. Since the size of the accretion disc is supposed to be much smaller than the sink radius $R_{\mathrm{in}}$ (see Fig. \[fig:acc\_whole\]), we neglect the centrifugal effect and assume the similar flow structure extends inward. As an upper limit for the mass-loss rate, we evaluate the integral with the same integrand as equation but for the different range of $0< r <
R_{\mathrm{in}}$, and obtain $$\begin{aligned}
\dot{M}_{\mathrm{loss}}(<R_{\mathrm{in}})
\lesssim 0.1 \left[\frac{R_{\mathrm{in}}}{(r_{\mathrm{B}}/7)}\right]^{1/2} \dot{M}_{\mathrm{B}}\,.
\label{eq:4}\end{aligned}$$ Here, we take $R_{\mathrm{in}} \approx r_{\mathrm{B}}/7$ of our simulation setup (see Table \[tab:model\]) as a reference value.
The mass supply rate to the accretion disc can be conservatively estimated by $\dot{M} - \dot{M}_{\mathrm{loss}}(<R_{\mathrm{in}})$, meaning that $\dot{M}$ measured in the simulation slightly overestimates the true value. This would be alleviated by taking a smaller value for $R_{\mathrm{in}}$. We have performed a test run with the smaller sink radius (see Appendix \[sec:res\_check\]), but found no remarkable differences of the accretion rate. Note that the effect of the angular momentum becomes important on the smaller scale. It is not allowed to take an arbitrary small sink radius without considering the effect of the angular momentum.
If the accretion disc is spatially resolved, we expect further mass loss happens due to, e.g., the disc winds [e.g., @Blandford:1999aa; @Zahra-Zeraatgari:2016aa; @Begelman:2016aa] and/or jets from a close vicinity of the BH [e.g., @Ohsuga:2005aa; @Jiang:2014aa; @Yuan:2015aa; @Sc-adowski:2016aa]. Outflows from the sink caused by such phenomena may change the outer gas dynamics on the scale of the Bondi radius. This should be studied in future work.
Parameter dependence {#sec:dependence}
--------------------
Here, we study how the flow structure changes with variation of the simulation parameters: the shadow opening angle $\theta_{\mathrm{shadow}}$ (in Sec. \[sec:sdep\]), BH mass $M_{\mathrm{BH}}$ (in Sec. \[sec:Mdep\]), and ambient density $n_\infty$ (in Sec. \[sec:ndep\]). In Sec. \[sec:comp\_prev\], we compare our results with previous 1D calculations.
### Dependence on shadow size {#sec:sdep}
run $\theta_{\mathrm{shadow}}$$^{a}$ $\theta_{\mathrm{inflow}}(r_{\mathrm{B}})$$^{b}$ $\dot{M}/\dot{M}_{\mathrm{B}}$$^{c}$
------------ ---------------------------------- -------------------------------------------------- --------------------------------------
s100 (Dds) $45^\circ$ $40^\circ$ $59\%$
s075 $37.75^\circ$ $29^\circ$ $42\%$
s050 $25^\circ$ $19^\circ$ $25\%$
s025 $11.25^\circ$ $9^\circ$ $6.5\%$
: Summary of the $\theta_{\mathrm{shadow}}$ dependence.[]{data-label="tab:s-model"}
\
NOTES. $^{a}$shadow opening angle of our subgrid model (equation \[eq:10\]); $^{b}$opening angle of equatorial neutral inflow region at $r_{\mathrm{B}}$; $^{c}$accretion rate normalized by Bondi one.
Considering uncertainties in the anisotropic shadowing effect (Sec. \[sec:direction\_dep\]), we study the cases with different shadow opening angles $\theta_{\mathrm{shadow}}$ by reducing it from $45^\circ$ in “Dds run” (here we also call it “s100 run”) to $33.75^\circ$ (“s075 run”), $22.5^\circ$ (“s050 run”) and $11.25^\circ$ (“s025 run”). We call this series of runs as “s-series”. We take $M_{\mathrm{BH}}=10^3M_\odot$ and $n_\infty=10^5{\,\mathrm{cm^{-3}}}$ for the s-series.
Our main findings are as follows: in all the runs of the s-series, the overall flow structures are similar and the accretion rates $\dot{M}$ are much higher than in the cases without the shadow (i.e., Di and Ddn runs). The obtained accretion rates and opening angles of the equatorial neutral region at Bondi radius $\theta_{\mathrm{inflow}}(r_{\mathrm{B}})$ are summarized in Table \[tab:s-model\]. The values of $\theta_{\mathrm{inflow}}(r_{\mathrm{B}})$ agree well with the prediction by equation with only small offsets $\lesssim 3^\circ$. Note also that $\theta_{\mathrm{inflow}}(r_{\mathrm{B}}) \simeq \theta_{\mathrm{shadow}}$ despite the gradual transition between the shadowed and non-shadowed regions modeled as in equation .
The equatorial inflow rates $\dot{M}_{\mathrm{inflow}} (r)$ (see equation \[eq:16\]) are shown in Fig. \[fig:mdotio\_sdep\](a). The values of $\dot{M}_{\mathrm{inflow}}$ at $R_{\mathrm{in}}$ agree well with the rates estimated by the Bondi flow through the solid angle of $4\pi\sin\theta_{\mathrm{inflow}}(r_{\mathrm{B}})$ (equation \[eq:17\]; arrows in Fig. \[fig:mdotio\_sdep\] a), but with slight downward offset due to the photoevaporation mass loss. Fig. \[fig:mdotio\_sdep\](b) shows the outflow rates in polar directions $\dot{M}_{\mathrm{outflow}}(r)$ (again, see equation \[eq:16\]). As seen in Sec. \[sec:shadow\_rad\], the estimate by equation with $f_{\mathrm{outflow}}=0.7$ gives a good fit to the numerical results. Small differences of $\sim$ a few $\times$ 10 % among them are comparable to the intrinsic fluctuations of the outflow rates present in the quasi-steady states.
![The net accretion rate $\dot{M}$ normalized by the Bondi rate $\dot{M}_{\mathrm{B}}$ as a function of the opening angle of the horizontal neutral layer at the Bondi radius $\theta_{\mathrm{inflow}}(r_{\mathrm{B}})$. The crosses show the runs with shadow opening angles $\theta_{\mathrm{shadow}}=45^\circ$, $33.75^\circ$, $22.5^\circ$ and $11.25^\circ$. The solid line represents the relation given by equation assuming $\dot{M}_{\mathrm{loss}}=0.07\dot{M}_{\mathrm{B}}$. []{data-label="fig:mdot_th_indep"}](figure/mdot_th_indep.eps){width="8.5cm"}
The net accretion rates $\dot{M}$ are plotted as crosses in Fig. \[fig:mdot\_th\_indep\] against $\theta_{\mathrm{inflow}}(r_{\mathrm{B}})$. Since equation slightly overestimates $\dot{M}_{\mathrm{inflow}}(R_{\mathrm{in}})$ due to the photoevaporation mass loss, we modify equation assuming a constant mass loss rate $\dot{M}_{\mathrm{loss}}$ in all cases, as $$\begin{aligned}
\dot{M} = \dot{M}_{\mathrm{inflow}}(R_{\mathrm{in}}) =
\frac{\Delta\Omega_{\mathrm{inflow}}(r_{\mathrm{B}})}{4\pi}\dot{M}_{\mathrm{B}}
- \dot{M}_{\mathrm{loss}} \,.
\label{eq:18}\end{aligned}$$ We find that $\dot{M}_{\mathrm{loss}}=0.07\,\dot{M}_{\mathrm{B}}$ gives the best fit to the simulated results with errors less than 2% of $\dot{M}_{\mathrm{B}}$. This good agreement also supports the above assumption of constant $\dot{M}_{\mathrm{loss}}$. The value of $\dot{M}_{\mathrm{loss}}$ is similar to but smaller than $\dot{M}_{\mathrm{outflow}}(r_{\mathrm{B}})$ (equation \[eq:14\]) partly due to the contribution from the circulation flows, as mentioned in Sec. \[sec:anl\_modeling\]. Moreover, by setting $\dot{M} = 0$ in equation , we get the critical opening angle $\theta_{\mathrm{cr}} \simeq 4^\circ$, below which the equatorial neutral flow disappears by photoevaporation. This value will be raised up to $\theta_{\mathrm{cr}} \simeq 10^\circ$ if we include the mass loss inside the sink, which is currently ignored (see Sec. \[sec:mass\_loss\_inner\]).
### Dependence on BH mass {#sec:Mdep}
run $M_{\mathrm{BH}}\,[M_\odot]$ $\theta_{\mathrm{inflow}}(r_{\mathrm{B}})$ $\dot{M}/\dot{M}_{\mathrm{B}}$
------------ ------------------------------ -------------------------------------------- --------------------------------
M1e2 $10^2$ $38^\circ$ $55\%$
M1e3 (Dds) $10^3$ $40^\circ$ $59\%$
M1e4 $10^4$ $43^\circ$ $61\%$
M1e5 $10^5$ $48^\circ$ $67\%$
: Summary of the $M_{\mathrm{BH}}$ dependence.[]{data-label="tab:M-model"}
\
Next, we study the dependence on the BH mass $M_{\mathrm{BH}}$ by performing a set of simulations termed “M-series”, where $M_{\mathrm{BH}}=10^2\,M_\odot$ (“M1e2 run”), $10^3\,M_\odot$ (“M1e3 run” identical to “Dds run”), $10^4\,M_\odot$ (“M1e4 run” ) and $10^5\,M_\odot$ (“M1e5 run”). The other parameters are set to $n_\infty=10^5{\,\mathrm{cm^{-3}}}$ and $\theta_{\mathrm{shadow}}=45^\circ$.
We find that the accretion proceeds roughly at the Bondi rate in a quasi-steady fashion for all the runs. Flow properties at the end of calculation are summarized in Table \[tab:M-model\]. We see that, for all the runs, the neutral region spans the opening angle $\theta_{\mathrm{inflow}}(r_{\mathrm{B}})\simeq \theta_{\mathrm{shadow}} (=45^\circ)$ at the Bondi radius, and that the Bondi-like accretion proceeds through this solid angle with the rate $\dot{M} \simeq 0.5-0.7\,
\dot{M}_{\mathrm{B}}$.
Note, however, that the opening angle and thus the accretion rate increase gradually with the BH mass. This dependence can be understood as follows. Recall that the luminosity $L$ is approximately proportional to the Eddington value or the mass $M_{\mathrm{BH}}$ in the super-Eddington regime (equations \[eq:12\] and \[eq:3\]). The radius of the [H[ii]{} ]{}region thus varies as $r_{\mathrm{HII}}(\theta) \propto M_{\mathrm{BH}}^{2/3}$ (equation \[eq:13\]), while the Bondi radius follows $r_{\mathrm{B}} \propto
M_{\mathrm{BH}}$ (equation \[eq:2\]). Since $r_{\mathrm{HII}}(\theta)$ is an increasing function of $\theta$ (equation \[eq:6\]; see also Fig. \[fig:th\_in\_Ds\]), this means that $\theta_{\mathrm{inflow}}(r_{\mathrm{B}})$, obtained by solving $r_{\mathrm{B}}=r_{\mathrm{HII}}(\theta)$ with respect to $\theta$, increases with $M_{\mathrm{BH}}$. Equation indeed explains the variation of the opening angle $\theta_{\mathrm{inflow}}(r_{\mathrm{B}})$ in the numerical results within the error of $4^\circ$. Similarly, the accretion rates $\dot{M}/\dot{M}_{\mathrm{B}}$ estimated by equation reproduce the results with errors $\lesssim10\%$.
### Dependence on ambient density {#sec:ndep}
run $n_\infty\,[{\mathrm{cm^{-3}}}]$ $\theta_{\mathrm{inflow}}(r_{\mathrm{B}})$ $\dot{M}/\dot{M}_{\mathrm{B}}$
------------ ---------------------------------- -------------------------------------------- --------------------------------
n1e3 $10^3$ $36^\circ$ $59\%$
n1e4 $10^4$ $38^\circ$ $54\%$
n1e5 (Dds) $10^5$ $40^\circ$ $59\%$
n1e6 $10^6$ $44^\circ$ $71\%$
: Summary of the $n_\infty$ dependence.[]{data-label="tab:n-model"}
\
Motivated by a wide variety of the environment in the vicinity of BHs, we finally investigate the cases with different ambient densities, termed “n-series”, where $n_\infty$ is $10^3{\,\mathrm{cm^{-3}}}$ (“n1e3 run”), $10^4{\,\mathrm{cm^{-3}}}$ (“n1e4 run”), $10^5{\,\mathrm{cm^{-3}}}$ (“n1e5 run” identical to “Dds run”) and $10^6{\,\mathrm{cm^{-3}}}$ (“n1e6 run”). The other parameters are set to $M_{\mathrm{BH}}=10^3\,M_\odot$ and $\theta_{\mathrm{shadow}}=45^\circ$.
The flow characteristics are similar regardless of $n_\infty$ with the neutral-region opening angle $\theta_{\mathrm{inflow}}(r_{\mathrm{B}})\simeq
\theta_{\mathrm{shadow}} (=45^\circ)$ and the mass accretion rates comparable to the Bondi rates, $\dot{M}/\dot{M}_{\mathrm{B}} \simeq 0.6-0.7$, in all the cases (Table \[tab:n-model\]). The increasing trend of $\theta_{\mathrm{inflow}}(r_{\mathrm{B}})$ with $n_\infty$ can be understood again as in Sec. \[sec:Mdep\]. Now the luminosity $L$ is almost independent of $n_\infty$ in the super-Eddington regime (equation \[eq:3\]), which leads to $r_{\mathrm{HII}}(\theta) \propto n_\infty^{-2/3}$ (equation \[eq:6\]) while $r_{\mathrm{B}}$ is independent of $n_\infty$ (equation \[eq:2\]). Since $r_{\mathrm{HII}}(\theta)$ is an increasing function of $\theta$, it follows that $\theta_{\mathrm{inflow}}(r_{\mathrm{B}})$ increases with $n_\infty$. Again, the analytic estimates of $\theta_{\mathrm{inflow}}(r_{\mathrm{B}})$ with equation and $\dot{M}/\dot{M}_{\mathrm{B}}$ with equation agree well with the numerical results, with errors less than $4^\circ$ and $10\%$, respectively.
Comparison with previous works {#sec:comp_prev}
------------------------------
Recently, @Inayoshi:2016ac and @Sakurai:2016aa have investigated necessary conditions to overcome the radiative feedback to achieve the Bondi-like accretion using 1D calculations under the spherical symmetry. These authors simulated accretion on to the BH in the same setting but with different prescriptions on the BH irradiation, i.e., whether or not the BH luminosity is capped by $L_{\mathrm{E}}$. They concluded that the efficient Bondi-like accretion appears when the following condition is satisfied: $M_{\mathrm{BH}}\,n_\infty \gtrsim 10^9\,
M_\odot\,{\mathrm{cm^{-3}}}$ (see Sec. \[sec:sph\_acc\]).
However, our 2D simulations suggest that the above criterion needs to be modified. We find the efficient accretion at about the Bondi rate is always possible, as long as the shadow opening angle has a certain size ($\theta_{\mathrm{shadow}} \gtrsim O(10)^\circ$), because the inflows from equatorial shadowed regions are allowed in 2D simulations. We here emphasize that neither $M_{\mathrm{BH}}$ nor $n_\infty$ appears in this condition, and that the efficient accretion is possible even with $M_{\mathrm{BH}}\,n_\infty \ll 10^9\, M_\odot\,{\mathrm{cm^{-3}}}$. The accretion rate through the shadowed direction is set by the Bondi rate, which depends on $M_{\mathrm{BH}}$ and $n_\infty$. For the accretion rate to largely exceeds the Eddington rate, $\dot{M}_{\mathrm{B}}/\dot{M}_{\mathrm{E}}=
\left(M_{\mathrm{BH}}\,n_\infty/10^5\,M_\odot\,{\mathrm{cm^{-3}}}\right) \gg 1$ should be met in addition to the above condition for the shadow size. Note, however, this condition for $M_{\mathrm{BH}}\,n_\infty$ is much easier to be satisfied than the condition for the 1D calculations, i.e., $M_{\mathrm{BH}}\,n_\infty \gtrsim 10^9\, M_\odot\,{\mathrm{cm^{-3}}}$.
Our 2D simulations show that the flow structure qualitatively differs from that in 1D even in the cases with $M_{\mathrm{BH}}\,n_\infty \gtrsim
10^9\, M_\odot\,{\mathrm{cm^{-3}}}$. For example, with $M_{\mathrm{BH}}=10^5\,M_\odot$ and $n_\infty=10^5{\,\mathrm{cm^{-3}}}$ (M1e5 run), the large bipolar [H[ii]{} ]{}bubbles persist in a steady state, whereas the spherical Bondi-like accretion quenches the [H[ii]{} ]{}bubble in the 1D test run. This is partly because the enhanced ionizing radiation in the polar directions due to the assumed directional dependence increases the size of the [H[ii]{} ]{}bubbles.
As seen above, $M_{\mathrm{BH}}\,n_\infty$ is not the key parameter to demarcate the regimes for efficient/inefficient accretion in 2D simulations, unlike in the 1D cases. The efficient accretion is possible if only the shadow size is sufficiently large, irrespective of $M_{\mathrm{BH}}$ or $n_\infty$.
conclusions and discussion {#sec:conclusion}
==========================
We have studied the black hole (BH) accretion of the primordial gas under anisotropic irradiation by the circum-BH accretion disc. Using two-dimensional radiation hydrodynamics simulations, we have solved the dynamics of the accretion flow spatially resolving both Bondi radius and the size of the [H[ii]{} ]{}region, which can differ by 3-4 orders of magnitude. We do not resolve the central accretion disc which emits anisotropic radiation, but inject ionizing photons at the inner boundary of the computational domain according to the subgrid prescription. To see how the anisotropy of the BH irradiation affects the flow structure, we first perform simulations with the three different types of the directional dependence: isotropic radiation, anisotropic radiation from the disc with and without the shadowing effect. For the case with the anisotropic shadowing effect, we have also studied the dependence of the flow structure on the shadow opening angle $\theta_{\mathrm{shadow}}$, BH mass $M_{\mathrm{BH}}$, and ambient density $n_\infty$.
With the isotropic irradiation, the accretion rate varies periodically as a result of recurrent formation and collapse of a hot and low-density [H[ii]{} ]{}bubble around the BH. The time-averaged accretion rate is only 0.2% of the original Bondi rate $\dot{M}_{\mathrm{B}}$ for the neutral medium and roughly given by the Bondi rate for the ionized medium [e.g., @Park:2011aa; @Park:2012aa; @Milosavljevic:2009ab]. Even with the anisotropy of the BH irradiation, the accretion rate is still similar to that in the isotropic case unless the shadowing effect is included. The flow structure in this case, however, is qualitatively different from the isotropic case: the large periodic variation, which has been reported in previous studies, disappears.
Unlike in the former two cases, the accretion rate becomes much higher in the case with the shadowing effect. For example, in the case with $M_{\mathrm{BH}}=10^3\,M_\odot$, $n_\infty=10^5{\,\mathrm{cm^{-3}}}$ and $\theta_{\mathrm{shadow}}=45^\circ$, the accretion rate reaches as high as 60% of the Bondi rate $\dot{M}_{\mathrm{B}}$ and is “super-critical” with 400 times larger than the Eddington-limited rate $\dot{M}_{\mathrm{E}}$. The flow structure in the steady state consists of the equatorial Bondi-like neutral inflow and bipolar ionized outflow. Since the radiation is confined to the polar directions, the rapid accretion proceeds in spite of the BH luminosity eight times larger than the Eddington value.
We have investigated such steady flow structure with the analytical models. The opening angle of the equatorial neutral layer $\theta_{\mathrm{inflow}}$ is derived from the balance between the supply and consumption rates of ionizing photons in each direction (equation \[eq:6\]). In turn, the accretion rate $\dot{M}$ is modeled assuming a Bondi-like flow through this equatorial layer also considering the photoevaporation mass loss from its surfaces (equation \[eq:18\]). We have also found that, in order for the equatorial Bondi-like inflow to be maintained, $\theta_{\mathrm{inflow}}$ at the Bondi radius must be above a critical value, which is $\simeq
4^\circ$ for $M_{\mathrm{BH}}=10^3\,M_\odot$ and $n_\infty=10^5{\,\mathrm{cm^{-3}}}$. This value is raised up to $\sim 10^\circ$ if we account for the mass loss inside the sink. The parameter dependence of the flow structure found in our simulations is well reproduced by this analytical model.
Our results highlight the importance of the directional dependence of BH irradiation, especially in the equatorial directions, in determining $\dot{M}$. However, our current knowledge about the actual anisotropy is very limited. Although not exactly the system of our interest, line-driven disc winds around a supermassive BH (SMBH) with $M_{\mathrm{BH}}
\sim 10^8\,M_\odot$ is shown to create anisotropic radiation fields by blocking high-energy photons [e.g., @Proga:2000aa; @Proga:2004aa; @Nomura:2016aa]. @Proga:2000aa suggest that the opening angle of the resulting shadow is $\theta_{\mathrm{shadow}} \simeq 12^\circ$, which is larger than the critical angle. In reality, the accretion disc may undergo the precession with variable angular momentum of accreting gas and change the orientation of the shadowed region in time, resulting in the destruction of the pre-existing neutral inflowing region. In any case, it is clearly awaited to study the structure of the inner part and resulting anisotropy of the BH irradiation. Our current study is complementary to such future works, because our results provide outer boundary conditions for them.
We have found that the required condition for the rapid accretion is substantially relaxed from that obtained for the isotropic irradiation. For these cases, the accretion rate is reduced to $\lesssim
0.01\,\dot{M}_{\mathrm{B}}$ by the radiative feedback unless the condition $(M_{\mathrm{BH}}/10^4M_\odot)(n_\infty/10^5{\,\mathrm{cm^{-3}}})\gtrsim 1$ [e.g., @Inayoshi:2016ac] is satisfied. This condition requires the ambient density $n_\infty$ as high as $10^6{\,\mathrm{cm^{-3}}}$ even for the most massive Pop III remnants with $M_{\mathrm{BH}}\sim 10^3M_\odot$ in [@Hirano:2015aa]. Such high ambient density seems difficult to achieve because the typical central density of the first galaxies at $z\sim 15$ is estimated as $10^5{\,\mathrm{cm^{-3}}}$ [e.g., @Oh:2002aa; @Volonteri:2005aa], although it is theoretically possible if the BH resides at the very centre of a halo with the density profile $\rho \propto r^{-2}$ [e.g., @Wise:2007aa; @Inayoshi:2016ac]. With the shadowing effect, BHs in a central part of the first galaxies can grow much more quickly.
A long-term evolution of such fast mass growth would be as follows. Suppose that a seed BH with mass $M_{\mathrm{BH}}=10^3M_\odot$ is embedded in an ambient medium with $n_\infty= 10^5{\,\mathrm{cm^{-3}}}$. Using the shadow opening angle of $\theta_{\mathrm{shadow}}=12^\circ$ suggested by @Proga:2000aa for line-driven SMBH winds, we obtain the accretion rate of $\dot{M} \sim 0.1\,\dot{M}_{\mathrm{B}} \sim
2\times10^{-4}(M_{\mathrm{BH}}/10^3M_\odot)^2\, M_\odot{\,\mathrm{yr^{-1}}}$ with equations and . Integrating this expression, we obtain the growth history of the BH mass as $$M_{\mathrm{BH}}(t)\sim \frac{10^3\,M_\odot}{1-\left[(t-t_0)/5{\,\mathrm{Myr}}\right]}\,,$$ where $t_0$ is the initial time of the accretion. At a face value, the BH mass diverges within a short timescale of $5{\,\mathrm{Myr}}$. In reality, however, the BH mass growth via accretion should be limited by changes in the environmental conditions, such as exhaustion of the ambient gas by accretion. If the remnant BHs of Pop III stars with $\sim
10^{2-3}\,M_\odot$ grow immediately in a few Myr time-scale by accretion to $\sim 10^{5-6} M_\odot$, they subsequently evolve in the same way as direct collapse BHs and can eventually grow to $\sim 10^9M_\odot$ SMBHs via gas accretion and/or mergers by $z \sim 7$ [see, e.g., @Tanaka:2009aa]. With the shadowing effect, Pop III remnants can be seeds for high-$z$ SMBHs. It does not mean, however, that all the Pop III remnants experience such rapid growth. For example, if they stay in a low-density region with $n<10 {\,\mathrm{cm^{-3}}}$, as suggested by [@Alvarez:2009aa], they hardly grow in mass even at the Bondi accretion rate.
Although we have assumed the weak rotation of the ambient gas, with which a BH accretion disk should be much smaller than the Bondi radius, accreting gas may have higher amount of the angular momentum in general. Regarding the SMBH accretion, [@Li:2013aa] has shown that the accretion rate is considerably reduced by the rotation [see also @Proga:2003ab; @Proga:2003aa]. If the flow predominantly comes from the equatorial plane, the amount of the angular momentum carried, and hence the impact of the rotational support, would be increased. To investigate this effect, mechanisms for the angular momentum transfer should also be considered (see below).
Our simulations also neglect the gas self-gravity. The torque caused by the self-gravity can play an important role in the angular momentum transport [see, e.g., @Shlosman:1989aa; @Shlosman:2016aa]. In addition, the inward force of the self-gravity can enhance the gas accretion on to BHs [e.g., @Li:2011aa], on the scales larger than both of the following two: (1) the radius where the enclosed gas mass equals to the BH mass, $r_{\mathrm{eq}}=(3\,M_{\mathrm{BH}}/4\pi\rho)^{1/3}\sim
10^5\, (M_{\mathrm{BH}}/10^3\,
M_\odot)^{1/3}(n/10^5\,{\,\mathrm{cm^{-3}}})^{-1/3}\,{\mathrm{AU}}$, and (2) the Jeans length $\lambda_{\mathrm{J}}=\sqrt{\pi}c_{\mathrm{s}}/\sqrt{\rho G}\sim
10^6\,(n/10^5\,{\,\mathrm{cm^{-3}}})^{-1/2}\,{\mathrm{AU}}$. In our cases, however, the Bondi radii are smaller than those scales and thus the flow structures on the scale of Bondi radius are hardly affected by this effect, although the size of [H[ii]{} ]{}bubbles can exceed them in some cases.
To be realistic, the following improvements are needed. First of all, the gas dynamics should be followed in 3D, in particular, to see the effect of the gravitational torque. Next, diffuse recombination photons can modify the structure of neutral layer. Furthermore, solving the gas dynamics beyond our computational domain, i.e, in the outer molecular and photodissociation regions [e.g., @Ricotti:2001aa], will be needed to see the large-scale flow structure. Finally, it is crucial to perform numerical simulations for the inner part dedicated to resolving the generation of anisotropic radiation fields, which are inevitably coupled to our simulations through the outer boundary conditions for them. We have clearly shown that the interplay of multi-scale processes is essential in understanding the BH accretion.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to thank Kazumi Kashiyama, Rohta Takahashi, Sanemichi Takahashi and Kenji Toma for fruitful discussions. The numerical simulations were performed on the Cray XC30 at CfCA of the National Astronomical Observatory of Japan, as well as on the computer cluster, [Draco]{}, at Frontier Research Institute for Interdisciplinary Sciences of Tohoku University. This work is supported in part by MEXT/JSPS KAKENHI Grant Number 15J03873 (KS), 25800102, 15H00776 and 16H05996 (TH), 15H06022 (HY) and 25287040 (KO).
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details of chemical and thermal modelling {#sec:chem_detail}
=========================================
Reaction rates {#sec:reaction_rates}
--------------
No. Reaction Rate coeff. ${\mathrm{[cm^3\,s^{-1}]}}$ Ref.
--------- ------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------- ------
1 ${\mathrm{H}}+{\mathrm{e}} \rightarrow {\mathrm{H^+}} + 2{\mathrm{e}}$ $k_1=$ 1
2 ${\mathrm{He}}+{\mathrm{e}} \rightarrow {\mathrm{He^+}} + 2{\mathrm{e}}$ $k_2=$ 1
3 ${\mathrm{He^+}}+{\mathrm{e}} \rightarrow {\mathrm{He^{2+}}} + 2{\mathrm{e}}$ $k_3=$ 2
$4^{a}$ ${\mathrm{H^+}}+{\mathrm{e}} \rightarrow {\mathrm{H}} + h\nu$ $k_4=2.753\times10^{-14} (T_{\mathrm{K}}/315614)^{-3/2} (1+(T_{\mathrm{K}}/115188)^{-0.407})^{-2.242}$ 3
$5^{b}$ ${\mathrm{He^+}}+{\mathrm{e}} \rightarrow {\mathrm{He}} + h\nu$ $k_5=k_{\mathrm{5rr}}+k_{\mathrm{5di}}$
$k_{\mathrm{5rr}}=$ 4
$k_{\mathrm{5di}}=1.9\times10^{-3} T_{\mathrm{K}}^{-3/2} \exp[-473421/T_{\mathrm{K}}] (1 + 0.3\exp[-94684/T_{\mathrm{K}}])$ 5
$6^{c}$ ${\mathrm{He^{2+}}}+{\mathrm{e}} \rightarrow {\mathrm{He^+}} + h\nu$ $k_6=5.08\times10^{-13} (T_{\mathrm{K}}/40000)^{-0.8163-0.0208\log_{10}(T_{\mathrm{K}}/40000)}$ 6
7 ${\mathrm{He^+}}+{\mathrm{H}} \rightarrow {\mathrm{He}} + {\mathrm{H^+}}$ $k_7=1.25\times10^{-15}(T_{\mathrm{K}}/300)^{0.25}$ 7
8 ${\mathrm{H^+}}+{\mathrm{He}} \rightarrow {\mathrm{H}} + {\mathrm{He^+}}$ $k_8=$ 8
9 $2{\mathrm{H}} \rightarrow {\mathrm{H^+}} + {\mathrm{H}} + {\mathrm{e}}$ $k_9=1.7\times10^{-4}k_1$ 9
\
NOTES. The $T_{\mathrm{K}}$ and $T_{\mathrm{eV}}$ are the gas temperature in units of K and eV, respectively; $^{a}$Case B; $^{b}$radiative [Case B; singlet; our fit to @Hummer:1998aa] and dielectric [@Aldrovandi:1973aa] recombination; $^{c}$Case B [@Draine:2011aa with typo about the charge dependence corrected].\
REFERENCES. (1) [@Janev:1987aa]; (2) [@Abel:1997aa from Aladdin database 1989]; (3) [@Ferland:1992aa]; (4) [@Hummer:1998aa]; (5) [@Aldrovandi:1973aa]; (6) [@Draine:2011aa]; (7) [@Zygelman:1989aa]; (8) [@Kimura:1993aa]; (9) [@Palla:1983aa].
In Table \[tab:reaction\_rates\], we summarize the chemical reactions considered in this work, which are adopted following [@Glover:2007aa], [@Abel:1997aa] and [@Anninos:1997aa]. We adopt the Case B recombination rates for the recombination of ${\mathrm{H^+}}$, ${\mathrm{He^+}}$ and ${\mathrm{He^{2+}}}$. We neglect the ${\mathrm{He^+}}$ recombination through the quasi-stable triplet state $2^3S$ of ${\mathrm{He}}$, assuming that ${\mathrm{He}}$ in that state is easily photoionized by the BH irradiation [see, e.g., @Clegg:1989aa].
Cross sections {#sec:cross_sections}
--------------
No. Reaction Cross section ${\mathrm{[cm^2]}}$ Ref.
----- ---------------------------------------------------------------------- ----------------------------------- --------------------------------------------- ------
1 ${\mathrm{H}}+h\nu \rightarrow {\mathrm{H^+}} + {\mathrm{e}}$ $\sigma_{\nu,1}=$ $h\nu_{\mathrm{T,1}}= 13.60{\,\mathrm{eV}}$ 1
2 ${\mathrm{He}}+h\nu \rightarrow {\mathrm{He^+}} + {\mathrm{e}}$ $\sigma_{\nu,2}=$ $h\nu_{\mathrm{T,2}}= 24.58{\,\mathrm{eV}}$ 2
3 ${\mathrm{He^+}}+h\nu \rightarrow {\mathrm{He^{2+}}} + {\mathrm{e}}$ $\sigma_{\nu,3}=$ $h\nu_{\mathrm{T,3}}= 54.40{\,\mathrm{eV}}$ 1
\
REFERENCES. (1) [@Osterbrock:1989aa]; (2) [@Yan:1998aa].
In Table \[tab:cross\_sections\], we summarize the cross sections considered in this work.
Heating and cooling rates {#sec:heat_cool}
-------------------------
[lllc]{} No. & Process & Rate \[${\mathrm{erg\, cm^{-3}\, s^{-1}}}$\] & Ref.\
\
1 & ${\mathrm{H}}$ photoionization & $\Gamma_1$ (see text)&\
2 & ${\mathrm{He}}$ photoionization & $\Gamma_2$ (see text)&\
3 & ${\mathrm{He^+}}$ photoionization & $\Gamma_3$ (see text)&\
\
$1^{a}$ & ${\mathrm{H^+}}$ recombination & $\Lambda_1=$ & 1\
$2^{b}$ & ${\mathrm{He^+}}$ recombination & & 2,3\
$3^{c}$ & ${\mathrm{He^{2+}}}$ recombination & $\Lambda_3= 1.38\times10^{-16}T_{\mathrm{K}} \left(0.684-0.0416\,\ln(T_{\mathrm{K}}/40000)\right)
k_6\,n({\mathrm{e}})\,n({\mathrm{He^{2+}}})$ & 4\
4 & ${\mathrm{H}}$ excitation & $\Lambda_4=7.50\times10^{-19} \left(1 + (T_{\mathrm{K}}/100000)^{1/2}\right)^{-1} \exp[-118348/T_{\mathrm{K}}]
\,n({\mathrm{e}})\,n({\mathrm{H}})$ & 5\
$5^{d}$ & ${\mathrm{He}}$ excitation & $\Lambda_5=1.1\times10^{-19} T_{\mathrm{K}}^{0.082} \exp[-230000/T_{\mathrm{K}}]
\,n({\mathrm{e}})\,n({\mathrm{He}})$ & 6\
6 & ${\mathrm{He^+}}$ excitation & $\Lambda_6=5.54\times10^{-17} T_{\mathrm{K}}^{-0.397} \left(1 + (T_{\mathrm{K}}/100000)^{1/2}\right)^{-1} \exp[-473638/T_{\mathrm{K}}]
\,n({\mathrm{e}})\,n({\mathrm{He^{+}}})$ & 5\
7 & ${\mathrm{H}}$ ionization & $\Lambda_7=2.18\times10^{-11}k_1\,n({\mathrm{e}})\,n({\mathrm{H}})$ & 7\
$8^{c}$ & ${\mathrm{He}}$ ionization & $\Lambda_8=3.94\times10^{-11}k_2\,n({\mathrm{e}})\,n({\mathrm{He}})$ & 7\
9 & ${\mathrm{He^+}}$ ionization & $\Lambda_9=8.72\times10^{-11}k_3\,n({\mathrm{e}})\,n({\mathrm{He^{+}}})$ & 7\
10 & Free-free & & 8\
$11^{e}$ & Compton & $\Lambda_{11}=1.017\times10^{-37} T_{\mathrm{CMB}}^4 (T_{\mathrm{K}} - T_{\mathrm{CMB}}) \,n({\mathrm{e}})$ & 5\
\
NOTES. $^{a}$Case B, our fit to [@Ferland:1992aa]; $^{b}$radiative [Case B; singlet; our fit to @Hummer:1998aa] and dielectric [@Black:1981aa] recombination cooling; $^{c}$Case B [@Draine:2011aa with typo about the charge dependence corrected]; $^{d}$singlet; $^{e}$$T_{\mathrm{CMB}}=2.73(1 + z)$ with $z=15$.\
REFERENCES. (1) [@Ferland:1992aa]; (2) [@Hummer:1998aa]; (3) [@Black:1981aa] (4) [@Draine:2011aa]; (5) [@Cen:1992aa]; (6) [@Bray:2000aa]; (7) [@Anninos:1997aa]; (8) [@Shapiro:1987aa].
In Table \[tab:cool\_rates\], we summarize the heating and cooling processes considered in this work. Here, $n(X)$ is the number density of species $X$ in units of ${\mathrm{cm^{-3}}}$. We calculate the photoionization heating rates as $\Gamma_i=\int (4\pi j_\nu/h\nu)
n(X_i)\sigma_{\nu,i} (h\nu - h\nu_{{\mathrm{T}},i}){\mathrm{d}}\nu$ (see Table \[tab:cross\_sections\]), with $X_1={\mathrm{H}}$, $X_2= {\mathrm{He}}$ and $X_3={\mathrm{He^{+}}}$.
Resolution check {#sec:res_check}
================
![Same as Fig. \[fig:mdot\] but for the runs in App. \[sec:res\_check\]. The physical parameters are the same as Dds run but the resolution is different in each run. See the text for details.[]{data-label="fig:mdot_rdep"}](figure/mdot_rdep.eps){width="8.5cm"}
To check the resolution dependence of our results, we here see how the evolution of $\dot{M}$ is affected by numerical settings, namely the number of grids and sink size $R_{\mathrm{in}}$. Taking the same physical parameters as Dds run, we perform additional simulations with different resolutions, as shown in Fig. \[fig:mdot\_rdep\]. Here, we take $N_r
\times N_\theta = 512\times 144$; $N_r \times N_\theta = 256\times 72$; $N_r \times N_\theta = 128\times 36$; $N_r \times N_\theta = 256\times
72$ with $R_{\mathrm{in}}$ halved and doubled from the fiducial value. Note that our main results are obtained with the high- and medium-resolution simulations with $N_r \times N_\theta = 512\times 144$ and $256\times
72$, respectively.
The dependence on the number of grids is checked by comparing the results with $N_r \times N_\theta = 512\times 144$, $256\times 72$, and $128\times 36$ (Fig. \[fig:mdot\_rdep\]). The strong variability of $\dot{M}$ for $t \lesssim 10^6{\,\mathrm{yr}}$ seen with the highest-resolution is smoothed out with the lower resolutions. However, the values of $\dot{M}$ at the end of the simulations are almost the same in all three cases. This confirms that the conclusion of this paper does not depend on the number of grids.
Fig. \[fig:mdot\_rdep\] also shows the evolution of $\dot{M}$ for the cases with the different sink sizes. The differences of accretion rates are less than 10 % for $t > 1.5 \times 10^6$ years in all three cases. We also find that the values of $\dot{M}$ at the end of the simulations decrease only by 4% by halving the sink size from the fiducial value. Such a trend is consistent with the estimated mass-loss rate from the region between the halved and fiducial inner boundaries (see equation \[eq:14\]), although it is also within the numerical error. This ensures that the dependence of our results on the sink size is weak.
[^1]: E-mail: sugimura@astr.tohoku.ac.jp
[^2]: For a case with $M_{\mathrm{BH}}=10^3\,M_\odot$ and $L=L_{\mathrm{E}}$ with the spectrum $L_\nu \propto \nu^{-1.5}$, the specific FUV intensity at $r_{\mathrm{B}}$ is $J_{21}\sim 10^9$ (in units of $10^{-21}{\,\mathrm{erg\,s^{-1}\,Hz^{-1}\,sr^{-1}\,cm^{-2}}}$), while the critical intensity for totally suppressing ${\mathrm{H_2}}$ formation in atomic cooling halos is $J_{21,cr}\sim 10^3$ [see, e.g., @Sugimura:2014aa].
[^3]: Observations of Galactic stellar BHs support that disc winds and associated shadowed regions only exist in the high/soft state and disappear in the low/hard state [e.g., see @Ponti:2012aa].
[^4]: With equation , we are now able to explain why the accretion rate temporarily exceeds the Bondi rate in the early stage of the simulation in Fig. \[fig:mdot\](c). In the beginning, the hot bipolar [H[ii]{} ]{}bubbles compresses the equatorial neutral layer. Consequently, the corresponding Bondi-like accretion rate increases and becomes larger than the original Bondi rate $\dot{M}_{\mathrm{B}}$ even after multiplying the fraction of the neutral solid angle $\Delta\Omega_{\mathrm{inflow}}(r_{\mathrm{B}})/(4\pi)$.
| {
"pile_set_name": "ArXiv"
} |
---
address: |
Jefferson Physical Laboratory, Harvard University,\
Cambridge, MA 02138 USA
author:
- Minjae Cho
bibliography:
- 'open\_closed.bib'
title: 'Open-closed Hyperbolic String Vertices'
---
= .8mm
\
\
Introduction
============
In recent years, it has become clear that a well defined and consistent perturbative formulation of string theory requires the framework of string field theory (we refer readers to [@Zwiebach:1992ie; @Sen:2015uaa; @deLacroix:2017lif] and references therein for an overview of the subject). For example, traditional analytic continuation involved in the computation of string amplitudes to cure the divergences in moduli integration naturally arises in string field theory as explained in [@Sen:2019jpm]. Furthermore, prescriptions given by string field theory provide not only unambiguous recipe to compute physical quantities in a given background, but also descriptions of more general backgrounds arising as solutions to string field equations of motions. For example, this idea was used to study strings in Ramond-Ramond flux backgrounds [@Cho:2018nfn].
In practice, computations that appear in string field theory are those of worldsheet conformal field theories. Therefore, we in principle can compute relevant quantities in a rather strightforward manner. Being a field theory, string field theory carries vertices which are roughly speaking integration of worldsheet correlators of off-shell string fields over specific parts of the moduli spaces. However, such off-shell objects in general depend on how one coordinatizes Riemann surfaces and this ambiguity exactly amounts to string field redefinitions [@Hata:1993gf; @Sen:1993ic; @Sen:2014dqa; @Sen:2015hha]. Thus, the choice of vertices amounts to which coordinatization of Riemann surfaces to use and which parts of the moduli spaces to cover.
Of course, not all such arbitrary choices of vertices are consistent. There is a very natural requirement on string vertices when the homomorphism between Batalin-Vilkoviski (BV)-algebras of surfaces and string fields are considered [@Sen:1994kx; @Sen:1993kb]. The requirement is called geometric master equation and the job of finding the solutions is of fundamental interest in the framework of string field theory. In the past, such solutions were found using various metrics, an example being minimal area metrics [@Zwiebach:1990nh; @Zwiebach:1990qj; @Zwiebach:1992bw; @Headrick:2018ncs; @Headrick:2018dlw]. There were also approximate constructions using the hyperbolic metrics [@Moosavian:2017fta; @Moosavian:2017qsp; @Moosavian:2017sev; @Pius:2018pqr].
Recently, a nice explicit construction of closed string vertices using hyperbolic metric was achieved in [@Costello:2019fuh]. One starts with a bordered hyperbolic Riemann surface with specified border lengths and systolic constraints, and grafts flat semi-infinite cylinders to the borders to make them into punctures. Upon connecting such vertices using closed string propagator which is represented as a flat finite cylinder, the resulting metric is the Thurston metric (for an overview, see [@tanigawa1995grafting]). As string theory requires us to integrate over the moduli space, one possible advantage of hyperbolic vertices from string theory perspective is that there is a better understanding of moduli integration in such metrics [@1998InMat.132..607M; @Mirzakhani:2006fta; @Mirzakhani:2006eta].
In this work, we generalize the construction of hyperbolic string vertices to oriented open-closed string field theory [@Zwiebach:1990qj; @Zwiebach:1997fe] (we will omit the term “oriented” from now on and it is always assumed). Hyperbolic surfaces to be considered are bordered hyperbolic surfaces (we refer readers to Chapter 1 of [@Buser1992GeometryAS] for a gentle introduction to these surfaces). Their boundaries are piecewise geodesic and some of geodesic sides will correspond to open string punctures, while the other sides belong to boundaries. Some of the borders which are closed geodesics will correspond to closed string punctures as in [@Costello:2019fuh], while the others will correspond to boundaries. We will define a family of subsets of such bordered hyperbolic surfaces, and show that it solves the geometric master equation. The essential ingredients of the proof are collar theorems. Such theorems are well-known for hyperbolic bordered Riemann surfaces where boundaries are all smooth closed geodesics. We will extend them to the case of bordered hyperbolic surfaces under consideration. We will also give explicit description of all zero and one-dimensional open-closed hyperbolic string vertices. This description will show that the family of hyperbolic vertices we constructed do not include a point where the theory becomes Witten’s cubic theory [@Witten:1985cc; @Zwiebach:1992bw], as already discussed in [@Costello:2019fuh].
The paper is organized as follows. In section \[geometricmastereq\], we review the geometric master equation for open-closed string vertices [@Zwiebach:1997fe] and provide a proof that Feynman diagrams built out of the solutions to the geometric master equation represent fundamental classes in relative homologies of interest. Then, we proceed to discuss relevant geometric objects and theorems in section \[hyperbolicsurfaces\]. Using these objects, in section \[hyperbolicvertices\] we will define hyperbolic open-closed string vertices and prove that they solve the geometric master equation. We describe zero and one-dimensional hyperbolic vertices in section \[lowdimvertices\]. We conclude with remarks and discussions in section \[discussions\].
Open-closed geometric master equation and Feynman diagrams {#geometricmastereq}
==========================================================
In this section, we review the general framework of open-closed string field theory and the corresponding geometric master equation, which we will solve in later sections. All discussions are standard and we will closely follow [@Zwiebach:1997fe]. Then, we will show that the Feynman diagrams built out of the solutions cover the moduli space exactly once, following the ideas presented in [@Costello:2019fuh].
Moduli spaces, total spaces, and singular chains
------------------------------------------------
In open-closed string field theory, the geometric objects under consideration are bordered Riemann surfaces with marked bulk and boundary punctures. In order to specify the moduli space, one specifies genus $g$, number of bulk punctures $n$, number of boundary components $b$, and number $m_i$ of boundary punctures on the $i$-th boundary, with $i=1,2,...,b$. We will denote the corresponding moduli space as ${\cal M}^{g,n}_{b,\{m_i\}}$. Open-closed string vertices take all possible values of these parameters satisfying \[moduliconditions\] &i) n3 g=b=0,\
&ii) n1 g=1, b=0,\
&iii) m\_13 g=0, b=1.
Off-shell amplitudes of string fields are integration of worldsheet correlators over a given moduli space. As already mentioned, the result depends on the coordinatization of Riemann surfaces. We first introduce local coordinates around bulk and boundary punctures. For bulk punctures, we will take flat unit disk $\{z\in\mathbb{C}|~|z|\leq1\}$ whose origin is the location of the puncture. For boundary punctures, we will take semi-disk $\{z\in\mathbb{C}|~|z|\leq1~\text{and}~\text{Im}(z)\geq0\}$ on flat upper-half plane, where the origin is the location of the puncture and the real axis is the boundary. Then, the choice of embedding of disks and semi-disks corresponds to coordinatization of Riemann surfaces. Over the moduli space ${\cal M}^{g,n}_{b,\{m_i\}}$, we will take the fiber to be such embeddings modulo phase rotations for disk coordinates around bulk punctures. The resulting total space is denoted as $\hat{\cal P}^{g,n}_{b,\{m_i\}}$. By forgetting about coordinates, one can naturally project down to the moduli space, $\pi:\hat{\cal P}^{g,n}_{b,\{m_i\}}\rightarrow{\cal M}^{g,n}_{b,\{m_i\}}$.
Typically, one would choose a section in $\hat{\cal P}^{g,n}_{b,\{m_i\}}$ over ${\cal M}^{g,n}_{b,\{m_i\}}$ and compute off-shell amplitudes by integrating along the section. However, as pointed out in [@Costello:2019fuh], one in general can allow for singular chains with real coefficients, as chains are natural objects to integrate over. We also assume that chains are symmetrized over the punctures.
Vertices and Feynman diagrams
-----------------------------
Say we made a choice of chains for all zero dimensional moduli spaces, where chains are in the fundamental homology class of the corresponding moduli spaces when pushed forward to it. Roughly speaking, it means that chains cover the moduli space (which is a point here) exactly once taking into account the multiplicities. Then, we can construct Feynman diagrams by combining these vertices using either open string or closed string propagators. Explicitly, closed string propagator plumbs two bulk punctures with local disk coordinates $z$ and $w$ via $zw=e^{-s+i\theta}$ for all $s\geq0$ and $0<\theta\leq2\pi$, and open string propagator glues two boundary punctures with local semi-disk coordinates $z$ and $w$ via $zw=-e^{-s}$ for all $s\geq0$. As a result, such Feynman diagrams are equipped with specific coordinate systems and thus represent chains over higher dimensional moduli spaces. These chains, when pushed forward to the moduli space, in general do not belong to the relative homology $H_{\text{dim}({\cal M})}\left(\overline{\cal M}; \partial\overline{\cal M}\right)$, where $\overline{\cal M}$ is the Deligne-Mumford compactification [@Mumford1983TowardsAE] of the moduli space ${\cal M}$ under consideration. Roughly speaking, the moduli space is not covered with multiplicity one by these Feynman diagrams. The only known exception is Witten’s cubic theory [@Witten:1985cc] where Feynman diagrams built out of propagators and cubic open string vertex cover the entire moduli spaces of all disk diagrams with boundary punctures exactly once. Therefore, one needs to further add chains, which we call vertices, so that the sum of vertices and Feynman diagrams with propagators represents fundamental classes of the homology relative to the boundary, when pushed forward to the moduli space (there is an ambiguity in the notion of the fundamental class of the relative homology which we will properly discuss in section \[fundclass\]). The generalization to general ${\cal M}^{g,n}_{b,\{m_i\}}$ is straightforward.
In the discussion so far, there is no reason why vertex and Feynman diagrams with propagators should be disjoint and continuous across where they meet. Indeed, they may even share some of the regions over the moduli space. However, BV algebra defined over such chains is naturally homomorphic to that over string fields [@Sen:1994kx; @Sen:1993kb]. Thus, one requires that the homomorphic preimage of BV master equation for string fields to hold true for chains under consideration here. This is exactly the geometric master equation which we discuss now.
Geometric master equation
-------------------------
In open-closed string field theory, the operations appearing in the BV algebra of chains are $\{{\cal V}_1,{\cal V}_2\}_o,\{{\cal V}_1,{\cal V}_2\}_c,\Delta_o{\cal V},$ and $\Delta_c{\cal V}$ for chains ${\cal V},{\cal V}_1,{\cal V}_2$. For detailed definition and discussions of these operations, see [@Sen:1994kx; @Zwiebach:1997fe]. We will give a brief description of these operations.
$\{{\cal V}_1,{\cal V}_2\}_o$ glues an open string puncture in every element of the chain ${\cal V}_1$ to another open string puncture in every element of the chain ${\cal V}_2$ using the semi-disk local coordinate identification $zw=-1$. The result will be another chain of dimension same as sum of dimensions of ${\cal V}_1$ and ${\cal V}_2$.
$\{{\cal V}_1,{\cal V}_2\}_c$ twist-plumbs a closed string puncture in every element of the chain ${\cal V}_1$ to another closed string puncture in every element of the chain ${\cal V}_2$ using the disk local coordinate identification $zw=e^{i\theta}$ for all $0<\theta\leq2\pi$. The result will be another chain of dimension higher than the sum of dimensions of ${\cal V}_1$ and ${\cal V}_2$ by one, except for the case where one of the chains, say ${\cal V}_1$, is over ${\cal M}^{0,1}_{1,\{0\}}$, i.e. disk with a bulk puncture, whose moduli space is of dimension zero and has a conformal Killing vector. In the latter case, dimension of the resulting chain is the same as that of ${\cal V}_2$.
$\Delta_o{\cal V}$ glues an open string puncture in every element of the chain ${\cal V}$ to another open string puncture in the same element of the same chain ${\cal V}$ using the semi-disk local coordinate identification $zw=-1$. The result will be another chain of dimension same as that of ${\cal V}$.
$\Delta_c{\cal V}$ twist-plumbs a closed string puncture in every element of the chain ${\cal V}$ to another closed string puncture in the same element of the same chain ${\cal V}$ using the disk local coordinate identification $zw=e^{i\theta}$ for all $0<\theta\leq2\pi$. The result will be another chain of dimension higher than that of ${\cal V}$ by one.
For the convenience of notations, we introduce $\{~,~\}\equiv\{~,~\}_o+\{~,~\}_c$ and $\Delta\equiv\Delta_o+\Delta_c$. Also, there is a natural boundary operator acting on the chain in the homological sense, which we denote as $\partial\cal V$. Now, we introduce the following formal sum of chains over all moduli spaces obeying (\[moduliconditions\]) =\_[g,n,b,{m\_i}]{}\^p\^q[V]{}\^[g,n]{}\_[b,{m\_i}]{}, where $\kappa$ is the string coupling and ${\cal V}^{g,n}_{b,\{m_i\}}$ is a chain in $\hat{\cal P}^{g,n}_{b,\{m_i\}}$ whose dimension is the same as that of ${\cal M}^{g,n}_{b,\{m_i\}}$. The powers $p$ and $q$ are given by $p=2g+{n\over2}+b-1$ and $q=4g+2n+2b+\sum_im_i-4$. Then, the geometric master equation reads \[geometricmaster\] +[12]{}{[V]{},[V]{}}+=0. The solution $\cal V$ to (\[geometricmaster\]) is called string vertices. Geometric master equation (\[geometricmaster\]) is crucial in deriving Ward identities for off-shell amplitudes, from which null-state decoupling from on-shell amplitudes can be deduced. Therefore, it is tied to the gauge invariance of the string field action.
In case where $\cal V$ are sections solving (\[geometricmaster\]) and Feynman diagrams built by $\cal V$ are also sections, this condition is a matching condition at the boundary of vertices and Feynman diagrams with propagators. For general chains, the Feynman diagrams constructed by the solutions will represent fundamental classes in the homology relative to the boundary as we discuss now.
Fundamental class in homology relative to the boundary {#fundclass}
------------------------------------------------------
In this subsection, we discuss how Feynman diagrams built using the solution to the geometric master equation (\[geometricmaster\]) represent fundamental classes in the homology relative to the boundary. It roughly means that they cover the moduli space exactly once. The idea here closely follows similar ideas in [@Costello:2019fuh], but there will be differences arising from infinite length open string propagators and also infinite length closed string propagator connecting to a disk with a bulk puncture. Feynman diagrams $F_{b,\{m_i\}}^{g,n}$ built using vertices are chains in the total space whose dimension is the same as the base moduli space: $F_{b,\{m_i\}}^{g,n}\in C_{\text{dim}({\cal M}_{b,\{m_i\}}^{g,n})}(\hat{\cal P}_{b,\{m_i\}}^{g,n})$. Now, we further include to the chain infinite cylinder and infinite strip propagators. The resulting chain $\overline{F}_{b,\{m_i\}}^{g,n}$ is a chain in the compactified total space $\hat{\overline{{\cal P}}}_{b,\{m_i\}}^{g,n}$, which is a fibered space over the Deligne-Mumford compactification $\overline{\cal M}_{b,\{m_i\}}^{g,n}$ of the moduli space with fibers being the coordinates around bulk and boundary punctures modulo rotation around bulk punctures: $\overline{F}_{b,\{m_i\}}^{g,n}\in C_{\text{dim}({\cal M}_{b,\{m_i\}}^{g,n})}(\hat{\overline{\cal P}}_{b,\{m_i\}}^{g,n})$.
Projection map from the total space to the moduli space naturally extends to the compactified ones, $\pi:\hat{\overline{\cal P}}_{b,\{m_i\}}^{g,n}\rightarrow \overline{\cal M}_{b,\{m_i\}}^{g,n}$. Using this, we push forward $\overline{F}_{b,\{m_i\}}^{g,n}$ to the Delign-Mumford compactified moduli space: $\pi_*\overline{F}_{b,\{m_i\}}^{g,n}\in C_{\text{dim}({\cal M}_{b,\{m_i\}}^{g,n})}(\overline{\cal M}_{b,\{m_i\}}^{g,n})$. We now show that this chain belongs to the homology relative to the boundary $H_{\text{dim}({\cal M}_{b,\{m_i\}}^{g,n})}\left(\overline{\cal M}_{b,\{m_i\}}^{g,n}; \partial\overline{\cal M}_{b,\{m_i\}}^{g,n}\right)$, or in other words, $\partial\left(\pi_*\overline{F}_{b,\{m_i\}}^{g,n}\right)\in C_{\text{dim}({\cal M}_{b,\{m_i\}}^{g,n})-1}\left(\partial\overline{\cal M}_{b,\{m_i\}}^{g,n}\right)$.
Boundaries of the chains can come from three sources: boundaries of vertices, a propagator of length zero, and a propagator of length infinity. The first two cancel each other due to the geometric master equation (\[geometricmaster\]), and the last belongs to $C_{\text{dim}({\cal M}_{b,\{m_i\}}^{g,n})-1}\left(\partial\overline{\cal M}_{b,\{m_i\}}^{g,n}\right)$. Therefore, the chain under consideration indeed is a cycle in the relative homology. The question now is if it coincides with the fundamental class of the relative homology.
However, the relative homology $H_{\text{dim}({\cal M}_{b,\{m_i\}}^{g,n})}\left(\overline{\cal M}_{b,\{m_i\}}^{g,n}; \partial\overline{\cal M}_{b,\{m_i\}}^{g,n}\right)$ is a product of $\mathbb{R}$ rather than a single $\mathbb{R}$, because the moduli space ${\cal M}_{b,\{m_i\}}^{g,n}$ is disconnected generically. For example, the familiar disk four point function computation of the Veneziano amplitude involves summing over six disks, which correspond to marking inequivalent connected pieces of the disconnected moduli space ${\cal M}_{1,\{4\}}^{0,0}$. Nonetheless, there is a very natural definition of the fundamental class we can take. We denote each connected component of the moduli space as ${}_I{\cal M}_{b,\{m_i\}}^{g,n}$, where $I$ is an element of an appropriate index set, and consider its Deligne-Mumford compactification ${}_I\overline{\cal M}_{b,\{m_i\}}^{g,n}$. Then $H_{\text{dim}({\cal M}_{b,\{m_i\}}^{g,n})}\left({}_I\overline{\cal M}_{b,\{m_i\}}^{g,n}; \partial{}_I\overline{\cal M}_{b,\{m_i\}}^{g,n}\right)=\mathbb{R}$, since ${}_I\overline{\cal M}_{b,\{m_i\}}^{g,n}$ is a compact connected orbifold. Therefore, in each connected component, there is a unique fundamental class in the homology relative to the boundary. We simply define the fundamental class of $H_{\text{dim}({\cal M}_{b,\{m_i\}}^{g,n})}\left(\overline{\cal M}_{b,\{m_i\}}^{g,n}; \partial\overline{\cal M}_{b,\{m_i\}}^{g,n}\right)$ to be the unique cycle which is the sum of fundamental classes of each connected component of the moduli space with coefficents being 1. This notion of fundamental class is what we want, as this is the precise meaning of covering the moduli space exactly once. Thus, our goal is to show that the cycle $[\pi_*\overline{F}_{b,\{m_i\}}^{g,n}]\in H_{\text{dim}({\cal M}_{b,\{m_i\}}^{g,n})}\left(\overline{\cal M}_{b,\{m_i\}}^{g,n}; \partial\overline{\cal M}_{b,\{m_i\}}^{g,n}\right)$ is the same as the fundamental class we defined.
In order to show that, we should first disconnect the cycle. Over each ${}_I\overline{\cal M}_{b,\{m_i\}}^{g,n}$, the chain $\overline{F}_{b,\{m_i\}}^{g,n}$ will have corresponding part, which we denote ${}_I\overline{F}_{b,\{m_i\}}^{g,n}$, obtained by the restriction map. By similar arguments as before, we have $[\pi_*{}_I\overline{F}_{b,\{m_i\}}^{g,n}]\in H_{\text{dim}({\cal M}_{b,\{m_i\}}^{g,n})}\left({}_I\overline{\cal M}_{b,\{m_i\}}^{g,n}; \partial{}_I\overline{\cal M}_{b,\{m_i\}}^{g,n}\right)=\mathbb{R}$. To show that the multiplicative constant here is 1, it is enough to check one specific point on the chain $[\pi_*{}_I\overline{F}_{b,\{m_i\}}^{g,n}]$. We go back to the entire chain $\overline{F}_{b,\{m_i\}}^{g,n}$. There is a point on the chain over $\overline{\cal M}_{b,\{m_i\}}^{g,n}-{\cal M}_{b,\{m_i\}}^{g,n}$ where the surface is totally degenerate in that it is built by gluing zero-dimensional vertices using infinite length open and closed string propagators. We can consider its neighborhood whose coordinates are the lengths and twists (the latter applies only to closed string propagators) of the propagators. Points in this neighborhood project isomorphically to the uncompactified moduli space ${\cal M}_{b,\{m_i\}}^{g,n}$ and in particular to each connected components ${}_I{\cal M}_{b,\{m_i\}}^{g,n}$. This proves that $[\pi_*{}_I\overline{F}_{b,\{m_i\}}^{g,n}]$ indeed is the fundamental class in $H_{\text{dim}({\cal M}_{b,\{m_i\}}^{g,n})}\left({}_I\overline{\cal M}_{b,\{m_i\}}^{g,n}; \partial{}_I\overline{\cal M}_{b,\{m_i\}}^{g,n}\right)$, and thus the cycle $[\pi_*\overline{F}_{b,\{m_i\}}^{g,n}]$ is the fundamental class in $H_{\text{dim}({\cal M}_{b,\{m_i\}}^{g,n})}\left(\overline{\cal M}_{b,\{m_i\}}^{g,n}; \partial\overline{\cal M}_{b,\{m_i\}}^{g,n}\right)$ we defined above.
Note that the reason why we considered the homology relative to the boundary rather than the homology over the compactified moduli space is due to the infinite length open string propagator and also infinite length closed string propagator connecting to a disk with a bulk puncture. Unlike the other infinite length closed string propagators which do not contribute to the boundary of the chain (because it becomes codimension two as both length and twist parameters disappear), the infinite length open string propagator and also infinite length closed string propagator connected to a disk with a bulk puncture give contributions to the boundary of the chain. This is because the moduli for the open string propagator is one-dimensional and that for closed string propagator connected to a disk with a bulk puncture is also one-dimensional due to the conformal Killing vector. Therefore, the chain under consideration is not a cycle in $H_{\text{dim}({\cal M}_{b,\{m_i\}}^{g,n})}\left(\overline{\cal M}_{b,\{m_i\}}^{g,n}\right)$, even though it is a cycle in the relative homology $H_{\text{dim}({\cal M}_{b,\{m_i\}}^{g,n})}\left(\overline{\cal M}_{b,\{m_i\}}^{g,n}; \partial\overline{\cal M}_{b,\{m_i\}}^{g,n}\right)$. It may seem to suggest that in the actual computation of string amplitudes, total derivatives in the moduli integration give nonzero contributions and thus BRST exact states do not decouple from on-shell amplitudes. However, as long as the background solves the string field equations of motions, Ward identities derived in [@Moosavian:2019ydz] imply that null states decouple from on-shell amplitudes, regardless of the presence of boundary of moduli spaces[^1].
We have the following result implying that finding the solution to the geometric master equation (\[geometricmaster\]) is all we need
\[fundamentalhomology\] Feynman diagrams built by solutions to (\[geometricmaster\]) represent a fundamental class in the homology relative to the boundary when pushed forward to the moduli space.
In [@Costello:2019fuh], the existence and uniqueness of the solutions to (\[geometricmaster\]) were also discussed for the closed string vertices. We will not attempt to prove the analogous statements for open-closed vertices, but we believe that it should be a straightforward generalization, with a caveat that the boundary of the Deligne-Mumford compactified moduli space may give rise to subtleties.
Bordered hyperbolic hexagon surfaces {#hyperbolicsurfaces}
====================================
In this section, we collect mathematical ingredients relevant to the construction of open-closed hyperbolic string vertices. In the case of closed hyperbolic string vertices, the relevant objects were hyperbolic bordered Riemann surfaces [@Costello:2019fuh], where the borders correspond to closed string punctures. Now, we also allow for boundary borders and boundary geodesic sides which will correspond to open string punctures. Therefore our objects of interest are bordered hyperbolic surfaces. Note that these are more general than what is usually called bordered hyperbolic Riemann surfaces i.e. the latter is a proper subset of the former. For instance, the boundary components for bordered hyperbolic surfaces are piecewise geodesic, while those for bordered hyperbolic Riemann surfaces are smooth closed geodesics.
It will turn out that bordered hyperbolic surfaces are too general for our purposes. Thus, we will restrict to bordered hyperbolic hexagon surfaces (BHHS), which is a subset of what is sometimes called hyperbolic surfaces with boundaries and right angles in relevant literatures. Basic building blocks for BHHS are right-angled hexagons, which we describe in detail now.
![Right-angled hexagon drawn on a Poincare disk with three $p$-sides $a, b, c$ (in purple) and three $b$-sides $\alpha, \beta, \gamma$ (in blue), and its half-collars (gray region on the right). Collar theorems are based on the fact that hexagon half-collars do not overlap.[]{data-label="fig:hexagon"}](figures/hexagons.pdf){width="100.00000%"}
Right-angled hexagons and Y-pieces
----------------------------------
Much of the discussions in this subsection closely follow [@Buser1992GeometryAS]. A right-angled hexagon consists of six geodesic sides in the hyperbolic upper-half plane/Poincare disk, with angles between neighboring sides all being right angles, as drawn in Figure \[fig:hexagon\]. Lengths of three non-consecutive sides determine a right-angled hexagon completely and the following trigonometric identity holds (among several other identities) \[trigid\] (c)=(a)(b)()-(a)(b).
We introduce half-collars of hexagons, whose properties are basic building blocks for all collar theorems to be discussed later.
\[hexagoncollar\][(Proposition 3.1.8 in [@Buser1992GeometryAS])]{} Given a right-angled hexagon $H$ with sides $c_1,d_3,c_2,d_1,c_3,d_2$ labeled in order (say clockwise), define three half-collars associated to $c_i$’s \[halfcollar\] D\_[c\_i]{}={xH|((x,c\_i))1 }, where [dist]{}$(x,c_i)$ stands for the shortest hyperbolic distance between a point $x$ and a side $c_i$, and $c_i$ appearing as a number (as in $\sinh{c_i}$) stands for its length. Then, three half-collars $D_{c_1,c_2,c_3}$ do not overlap with each other. In particular, the collar $D_{c_i}$ does not overlap with the opposing side $d_i$.
This is described in Figure \[fig:hexagon\]. Of course, there is nothing special about choosing half-collars around $c_i$’s, and we could have considered half-collars around $d_i$’s instead and they will not overlap with each other.
In order to build more general surfaces of interest, we introduce labelings of the sides of hexagons.
A labeled right-angled hexagon is a right-angled hyperbolic hexagon with three non-consecutive sides labeled as $p$-sides and the other three non-consecutive sides labeled as $b$-sides.
We will later attach semi-infinite strips to $p$-sides to turn them into punctures on the boundary component corresponding to open string insertions, while $b$-sides will remain as part of the boundary component. From here on, by a hexagon, we mean a right-angled hexagon with $p$ and $b$-side labelings, unless specified otherwise.
![A Y-piece obtained by gluing three $p$-sides of two identical hexagons. The resulting Y-piece has three borders and no $p$-sides.[]{data-label="fig:ypiece"}](figures/ypiece_glued.pdf){width="100.00000%"}
Gluing two identical hexagons along three $p$-sides generates a Y-piece as illustrated in Figure \[fig:ypiece\]. It corresponds to a three-bordered sphere, where all borders consist completely of $b$-sides. By construction, a Y-piece is completely determined by lengths of three borders. The conventional hyperbolic bordered Riemann surfaces all can be formed by plumbing Y-pieces together, where plumbing is done by identifying two borders of same lengths taking into account the twist angle.
$p$-side gluing of hexagons and BHHS
------------------------------------
Now, we construct more general surfaces by gluing $p$-sides of hexagons, with the requirement that orientations of $b$-sides are preserved. We already saw an example of $p$-side gluing of two hexagons, which is a Y-piece. As another example, for a hexagon which has two $p$-sides of the same lengths, gluing of these two sides will result in an annulus with one $p$-side.
\
In general, we consider surfaces formed by gluing pairs of $p$-sides of same lengths preserving the orientation of $b$-sides, given some number of hexagons. The resulting hyperbolic surface defines a BHHS.
Bordered hyperbolic hexagon surfaces (BHHS) are connected surfaces obtained by gluing pairs of $p$-sides of same lengths of hexagons, where the gluing preserves the orientation of the $b$-sides of hexagons.
We will declare that glued $p$-sides are no longer $p$-sides of the BHHS and only the unglued $p$-sides will remain as its $p$-sides. For example, a Y-piece does not have any $p$-sides as all of the $p$-sides of two hexagons are glued with each other. Several examples are depicted in Figure \[fig:BHHS\]. The definition for BHHS given here is a constructive one. One can of course start by specifying the notion of moduli space and work backwards to see how hexagons arise. We will indeed discuss such a notion in section \[graftingsection\] after we further restrict to a more special subset of BHHS.
When a simple smooth closed boundary geodesic is formed under $p$-side gluing (which is possible due to right-angledness of hexagons), it consists only of $b$-sides, and we will call it a border. Also, multiple $b$-sides of hexagons may be smoothly connected to form a larger geodesic side which is not a border. In such cases, we will call the maximally smoothly connected $b$-sides as a single $b$-side of the BHHS, which then should neighbor $p$-sides. Therefore, $b$-sides of a BHHS are geodesic sides of non-border boundary components (in topological sense) which are not $p$-sides. A general BHHS will then carry some number of $p$-sides, some number of $b$-sides, and some number of borders. By construction, any BHHS allows for a hexagon decomposition. Of course, there may be different hexagon decompositions of a given BHHS. This is analogous to different possible pants decompositions of a given bordered Riemann surface. Note that the notion of $p$ and $b$-sides of a BHHS defined above is independent of such different hexagon decompositions. In summary,
Given a BHHS $S$, consider a hexagon decomposition of $S$. A border of $S$ is a simple smooth closed boundary geodesic of $S$. $p$-sides of $S$ are unglued $p$-sides of hexagons and $b$-sides of $S$ are boundary geodesic sides which are not $p$-sides of $S$ and not borders of $S$.
We are not considering plumbing of borders of same lengths at this point. If one allows for such plumbing in addition, the result still will be a BHHS i.e. it allows for hexagon decompositions, except for the case where the resulting surface has no boundary components, which is nothing but hyperbolic Riemann surfaces without borders with genus greater than or equal to two. Later when we discuss open-closed string vertices, such plumbing of borders will indeed be considered. But for the purpose of discussing a BHHS, such plumbing is not necessary.
Collar theorems for BHHS
------------------------
Since a BHHS is decomposable into hexagons, the collar theorem for a hexagon (Theorem \[hexagoncollar\]) has a straightforward generalization to BHHS. We first introduce important open geodesics of interest.
Given a BHHS $S$, a $p$-geodesic of $S$ is a simple (i.e. no self-intersection) nontrivial (i.e. not homotopic to a point) open geodesic satisfying the following conditions:\
i) its endpoints are on either $b$-sides or borders of $S$,\
ii) it is the shortest among its homotopy class, where the homotopy allows endpoints to glide along a given $b$-side or border of $S$ (but cannot glide to the other sides).
Given $b$-sides or borders where the endpoints may glide along, these $p$-geodesics are unique among the homotopy class and called perpendiculars since they end on $b$-sides or borders at right angles (Theorem 1.5.3 in [@Buser1992GeometryAS] which is also briefly reviewed in Appendix \[hyp\]). Note that there may be multiple homotopically different $p$-geodesics ending on the same $b$-sides or borders. By definition, all $p$-sides are also $p$-geodesics.
To find a hexagon decomposition of a BHHS, one simply draws a maximal set of pairwise disjoint $p$-geodesics and cut along them, where they become $p$-sides of hexagons after cutting. This works because $p$-geodesics meet $b$-sides or borders at right angles, always preserving the right-angledness and neighborness between $p$ and $b$-sides. In the intermediate steps of cutting, $(2n)$-gons arise with $n\geq3$. No odd-gons appear because they do not respect the notion of $p$ and $b$-sides, and rectangles do not appear because there is no right-angled rectangle in hyperbolic geometry. All $(2n)$-gons with $n\geq3$ can be decomposed into hexagons by cutting along $p$-geodesics.
Given a BHHS $S$ and a $p$-geodesic $\gamma$ on $S$, the collar $P_{\gamma}$ corresponding to it is defined as P\_={xS|((x,))1 }, where again, $\gamma$ appearing as a number means its length. Note that when $\gamma$ is a $p$-side, one gets a half-collar in the sense that the width of the collar is a half of that of the collar of a $p$-geodesic of the same length which is not a $p$-side. We have our first collar theorem for BHHS.
\[collar1\] Consider a BHHS $S$ and pairwise disjoint $p$-geodesics $\gamma_i$ on $S$. Their collars $P_{\gamma_i}$ do not overlap. Also, the only borders or $b$-sides having nonzero overlap with $P_{\gamma_i}$ are the ones on which $\gamma_i$ ends. Each $P_{\gamma_i}$ is homeomorphic to a strip.
The basic idea of the theorem is that there is a hexagon decomposition of $S$ where the $p$-geodesics will become $p$-sides of hexagons. Then, the collar theorem of a hexagon, Theorem \[hexagoncollar\], implies the above collar theorem for a BHHS. This is along the line of analogous collar theorems for closed geodesics for bordered Riemann surfaces (Theorem 4.1.1 in [@Buser1992GeometryAS]).
The second collar theorem considers collars of $b$-sides and borders. Given $\gamma$ which is either a $b$-side or a border on a BHHS $S$, we define the half-collars (since $b$-sides and borders belong to boundary components) as \[bcollar\] B\_={xS|((x,))1 }. When $\gamma$ is a border, the above definition has extra factor of 2 compared to collars in the literature of bordered hyperbolic Riemann surfaces, where $\sinh(\text{dist}(x,\gamma))\sinh(\gamma/2)\leq1$ is used instead. Thus, the half-collars of borders on BHHS are thinner than those of hyperbolic bordered Riemann surfaces. This is because in the latter case, surfaces are built using Y-pieces, which is gluing of two identical hexagons, while for the case of BHHS, a border in a hexagon decomposition may consist of any number of hexagon $b$-sides, including one.
There is a further important point about this. Let $\gamma$ be a $b$-side or a border of a BHHS. In a hexagon decomposition, $\gamma$ generically consists of multiple $b$-sides of hexagons, say $b_1,b_2,...,b_n$ for some $n\geq1$. The length of $\gamma$ is the sum of lengths of $b_i$’s, which implies that the half-collar width for $\gamma$ and half-collar widths for $b_i$’s are different unless $n=1$; they are different even among $b_i$’s of different lengths. However, since the width of a half-collar increases as the length of the associated side decreases, the half-collar of $\gamma$ defined as above is included in the union of the hexagon half-collars of $b_i$’s. Therefore, we have our second collar theorem.
\[collar2\] Given a BHHS $S$, consider all $b$-sides $\gamma_i$ and borders $\mu_I$ on $S$. Their half-collars $B_{\gamma_i}$ and $B_{\mu_I}$ do not overlap. Also, each $B_{\gamma_i}$ is homeomorphic to a strip and each $B_{\mu_I}$ is homeomorphic to an annulus.
One significant point of collar theorems is that any curve passing through a collar has to have a length greater than the width of the collar. This will be used extensively to construct open-closed hyperbolic string vertices in later sections.
Grafting: from BHHS to bordered Riemann surfaces with punctures {#graftingsection}
---------------------------------------------------------------
In this subsection, we describe how to go from BHHS to surfaces appearing in open-closed string field theory. In order to do so, we introduce one more label to our BHHS. Given a BHHS, say there are $n$ number of borders. We pick a subset of the borders and label them as $c$-borders ($c$ standing for closed strings), and label all the other borders as $b$-borders ($b$ standing for boundaries). We call such labeled BHHS as $c$-labeled BHHS.
A $c$-labeled BHHS is a BHHS where all borders are labeled either $b$ or $c$. We call them $b$-borders and $c$-borders.
Thus, for a given BHHS with $n$ borders, we generate at maximum $2^n$ $c$-labeled BHHS. For example, for a Y-piece with two of borders carrying the same lengths which are different from that of the third border, giving a single $c$-label to either of two same-length borders will result in the same $c$-labeled BHHS.
For a $c$-labeled BHHS, we will redefine $p$-geodesics. Among the $p$-geodesics of unlabeled BHHS, after $c$-labeling, we will throw away the ones whose either of endpoints lie on the $c$-border.
Given a $c$-labeled BHHS $S$, a $p$-geodesic of $S$ is a simple nontrivial open geodesic satisfying the following conditions:\
i) its endpoints are on either $b$-sides or $b$-borders of $S$,\
ii) it is the shortest among its homotopy class.
Again, such a geodesic is unique in its homotopy class by the condition that it is the shortest, and is also the unique common perpendicular to $b$-sides or $b$-borders on which it ends in its homotopy class. We further restrict to the following $c$-labeled BHHS with markings.
Given a pair of positive real numbers $L_o$ and $L_c$, the associated open-closed hyperbolic surfaces is a set defined as:\
$H_{L_o,L_c}$ = {$c$-labeled BHHS whose $p$-sides all have lengths $L_o$, $c$-borders all have lengths $L_c$, $~~~~~~~~~~~~$ and $p$-sides and $c$-borders are marked}.
So far we have not defined the notion of equivalance of elements in $H_{L_o,L_c}$. This is the content of the moduli space. The notion of moduli space associated with $H_{L_o,L_c}$ is specified by its genus $g\geq0$, number $n\geq0$ of $c$-borders, number $b\geq0$ of non-$c$-border boundary components (in topological sense), number $m_i\geq0$ of $p$-sides on the $i$-th non-$c$-border boundary component, and markings for $p$-sides and $c$-borders. We call this moduli space ${\cal{M}}_{b,\{m_i\}}^{g,n}(L_o,L_c)$. Note that our construction of BHHS implies that the possible values of $g, n, b$ and $m_i$ are the same as those of open-closed string vertices (\[moduliconditions\]), with the exception of a disk with a bulk puncture, an annulus with no punctures, and the cases $n=b=0$ with $g\geq2$ which are closed Riemann surfaces.
Given the numbers $g,n,b,\{m_i\}$ in (\[moduliconditions\]), excluding two cases $\{g=0,n=1,b=1,m=0\}$ and $\{g=0,n=0,b=2,\{m_1=0,m_2=0\}\}$ which we will separately deal with later, the precise formulation of the moduli space goes as follows. We consider an orientable connected surface $S$ of genus $g$ and $n+b$ number of boundaries. Teichmüller space ${\cal T}_{b,\{m_i\}}^{g,n}(L_o,L_c)(S)$ is given by the set of all hyperbolic metrics on the surface satisfying: i) $n$ boundaries (which we call $c$-borders) are simple smooth closed geodesics of lengths $L_c$ and marked, ii) other $b$ number of boundaries, say $b_i$ for $i=1...,n$, each consists of $2m_i$ number of simple geodesic sides which meet neighboring sides at right angles for $m_i>0$, while $b_i$ with $m_i=0$ is a simple smooth closed geodesic which we call a $b$-border, and iii) for each of $b_i$ with $m_i\neq0$, one nonconsecutive half of the sides are called $p$-sides all of lengths $L_o$ which are marked, and the other nonconsecutive half of them are called $b$-sides. As in the case of $c$-labeled BHHS, we define a $p$-geodesic to be the unique shortest simple nontrivial open geodesic in its homotopy class with the endpoints on either $b$-borders or $b$-sides of the surface. The mapping class group $\Gamma(S)$ is the quotient of all orientation preserving diffeomorphisms by all diffeomorphisms connected to the identity, where diffeomorphisms must preserve $b,c$-borders, $b,p$-sides, and markings. Then, the moduli space is defined by ${\cal{M}}_{b,\{m_i\}}^{g,n}(L_o,L_c)\equiv{\cal T}_{b,\{m_i\}}^{g,n}(L_o,L_c)(S)/\Gamma(S)$. Surfaces in ${\cal{M}}_{b,\{m_i\}}^{g,n}(L_o,L_c)$ are elements of $H_{L_o,L_c}$ except for the cases $n=b=0$ with $g\geq2$ as already noted above.
Even though closed hyperbolic Riemann surfaces with $g\geq2$ do not belong to $H_{L_o,L_c}$, once we cut along any nontrivial simple closed geodesic and label it $b$-borders, they become elements of $H_{L_o,L_c}$. For these surfaces, the lengths and twist angles of nontrivial simple closed geodesics associated with a pants decomposition provide a natural parameterization of the Teichmüller space, which are Fenchel-Nielsen coordinates. Or, one can also cut along a chosen simple closed geodesic and consider the hexagon decomposition of the resulting surface, where the length and twist angle of the closed geodesic and lengths of hexagon $p$-sides provide another coordinates of the Teichmüller space. For all the other cases, a hexagon decomposition of an open-closed hyperbolic surface provides the lengths of hexagon $p$-sides as coordinates for the Teichmüller space, once the conditions on the lengths of $p$-sides and $c$-borders of the open-closed hyperbolic surface are imposed.
![Grafting a surface in ${\cal{M}}_{2,\{2,0\}}^{0,1}(L_o,L_c)$. Flat semi-infinite strips of width $L_o$ (dotted purple) are attached to $p$-sides of lengths $L_o$ (solid purple) and a flat semi-infinite cylinder with circumference length $L_c$ (dotted cherry) is attached to a $c$-border of circumference length $L_c$ (solid cherry). The result is an element in ${\hat{\cal P}}_{2,\{2,0\}}^{0,1}$, which upon projection gives an element in ${\cal{M}}_{2,\{2,0\}}^{0,1}$ shown on the right. Boundaries are colored in blue.[]{data-label="fig:grafting"}](figures/grafting.pdf){width="100.00000%"}
Now, we introduce a map from ${\cal{M}}_{b,\{m_i\}}^{g,n}(L_o,L_c)$ to bordered Riemann surfaces with marked bulk (closed string) and boundary (open string) punctures, which are basic objects of open-closed string field theory. The map is called “grafting,” and as the name suggests, we will graft other surfaces to those in ${\cal{M}}_{b,\{m_i\}}^{g,n}(L_o,L_c)$. Given an element of ${\cal{M}}_{b,\{m_i\}}^{g,n}(L_o,L_c)$, to every $p$-side, we glue a flat semi-infinite strip of width $L_o$, and to every $c$-border, we glue a flat semi-infinite cylinder with circumference length $L_c$, as described in Figure \[fig:grafting\]. A grafted semi-infinite strip introduces a puncture on the boundary (open string puncture), while a grafted semi-infinite cylinder introduces a puncture in the bulk (closed string puncture). Note that gluing is done isometrically, so the metric is continuous across the glued parts, even though it is not smooth; for example, the curvature jumps.
Since grafting results in bordered Riemann surfaces with marked bulk and boundary punctures together with specific metric on it, it is a map to the total space over the moduli space of such punctured Riemann surfaces. Thus, grafting is a map (following the notations in [@Costello:2019fuh]) ’\_ : \_[b,{m\_i}]{}\^[g,n]{}(L\_o,L\_c) \_[b,{m\_i}]{}\^[g,n]{}. Composing with the projection map $\pi:{\hat{\cal P}}_{b,\{m_i\}}^{g,n}\rightarrow{\cal{M}}_{b,\{m_i\}}^{g,n}$, we also get the following map. \[graft\] \_’\_ : \_[b,{m\_i}]{}\^[g,n]{}(L\_o,L\_c) \_[b,{m\_i}]{}\^[g,n]{}. Note that in cases $n=0$ and $m_i=0$ for all $i$, where there is nothing to graft, the hyperbolic surfaces already provide chains in the total space defined by the hyperbolic structures.
In the case of hyperbolic bordered Riemann surfaces where $b=0$ in the above, such grafting map gr${}_\infty$ was shown to be a homeomorphism [@mondello2008riemann; @scannell1998grafting]. We will not try to prove that (\[graft\]) is a homeomorphism for general $b$ and $\{m_i\}$, but we believe it should be true and is a straightforward generalization of the story for hyperbolic bordered Riemann surfaces. So we will assume that the grafting map indeed is a homeomorphism.
Open-closed hyperbolic string vertices {#hyperbolicvertices}
======================================
In this section, we construct hyperbolic open-closed string vertices using $H_{L_o,L_c}$ and grafting. We will show that with appropriate conditions on $L_o$ and $L_c$, we get the solution to the open-closed geometric master equation (\[geometricmaster\]).
Critical length and open-closed vertex region
---------------------------------------------
Essentially, the region of $L_o$ and $L_c$ for vertices will be such that collars are wide enough to make curves passing through them lengthy enough. Here, we will define such a region without any explanation or motivation. But once we discuss the proof of the open-closed string vertex solutions, it will become clear why we defined such a region.
First, let us define the critical length $L_*\in\mathbb R_+$. It is defined to be the solution to the following equation \[critical\] ([L\_\*2]{})L\_\*=1.
Numerical value for $L_*$ is approximately 1.21876.... Note that this critical length is smaller than the critical length defined in [@Costello:2019fuh], where it was $2\sinh^{-1}1=1.76275...$. This is due to different collar widths around nontrivial simple closed geodesics for the case of BHHS and bordered hyperbolic Riemann surfaces, as already discussed below (\[bcollar\]). We define the following open-closed vertex region of $L_o$ and $L_c$ \[regionR\] [R]{}={(L\_o,L\_c)\_+\^2 | 0<L\_cL\_\*L\_c L\_o 1 }. It is depicted in Figure \[fig:regionR\]. For $L_o\leq {L_*\over2}$, it is a rectangular region given by $0<L_o\leq {L_*\over2}$ and $0<L_c\leq L_*$. For $L_o\geq {L_*\over2}$, we have a more strict upper bound on $L_c$ given by $\sinh L_c \sinh L_o \leq1$.
Even though it is a trivial result, we record the following which will be used later in the proof for string vertices. \[regionR2\] L\_c1, L\_o1.
A family of open-closed string vertices
---------------------------------------
We first define the subset ${\tilde{\cal V}}_{b,\{m_i\}}^{g,n}(L_o,L_c)$ of the moduli space ${\cal{M}}_{b,\{m_i\}}^{g,n}(L_o,L_c)$, excluding two cases $\{g=0,n=1,b=1,m=0\}$ and $\{g=0,n=0,b=2,\{m_1=0,m_2=0\}\}$, as follows \_[b,{m\_i}]{}\^[g,n]{}(L\_o,L\_c){\_[b,{m\_i}]{}\^[g,n]{}(L\_o,L\_c) | ()L\_c()L\_o }. Here, sys$(\Sigma)$ is the systole, which is the length of the shortest nontrivial simple closed geodesic in $\Sigma$ which is not a $c$-border. In particular, $b$-borders are included in the consideration of the systole. Also, psys$(\Sigma)$ is the length of the shortest $p$-geodesic of $\Sigma$ which is not a $p$-side. Therefore, surfaces in ${\tilde{\cal V}}_{b,\{m_i\}}^{g,n}(L_o,L_c)$ have no nontrivial simple closed geodesic of length smaller than $L_c$ and no $p$-geodesic of length smaller than $L_o$.
In cases $b=0$ where there is no $b$-border or $b$-side, there is no $p$-geodesic to consider and thus the condition $\text{psys}(\Sigma)\geq L_o$ is empty. These correspond to pure closed string processes and the systolic condition is imposed only on nontrivial simple closed geodesics, as in [@Costello:2019fuh]. For $b\geq1$, the surfaces are elements of $H_{L_o,L_c}$ with nontrivial $p$-geodesics to consider generically.
We now define the grafting of these surfaces \_[b,{m\_i}]{}\^[g,n]{}(L\_o,L\_c)’\_(\_[b,{m\_i}]{}\^[g,n]{}(L\_o,L\_c)). We also define the two cases which have not been considered so far. The first one is a disk with a bulk puncture \[disk1pt\] \_[b=1,{m=0}]{}\^[g=0,n=1]{}(L\_o,L\_c){ L\_c }. We will declare that the circle is a $c$-border, even though it is not bounding any surface. Then, grafting acting on the circle will simply result in a flat semi-infinite cylinder of circumference length $L_c$. The second one is an annulus without punctures \_[b=2,{m\_1=0,m\_2=0}]{}\^[g=0,n=0]{}(L\_o,L\_c). Now, we have all the ingredients to describe hyperbolic open-closed string vertices.
\[vertextheo\] The sets ${\cal V}_{b,\{m_i\}}^{g,n}(L_o,L_c)$ with any given pair $(L_o,L_c)\in{\cal R}$ solve the open-closed geometric master equation (\[geometricmaster\]).
The special case $\partial{\cal V}_{2,\{0,0\}}^{0,0}=-{1\over2}\{{\cal V}_{1,\{0\}}^{0,1},{\cal V}_{1,\{0\}}^{0,1}\}$ is trivially satisfied. From here on, we will elaborate on the proof of the general cases. The proof is in two parts. First, we prove that boundary $\partial\cal V$ of the candidate vertex set is contained in $-{1\over2}\{{\cal V},{\cal V}\}-\hbar\Delta\cal V$. Then, we prove the opposite direction that both $-{1\over2}\{{\cal V},{\cal V}\}$ and $-\hbar\Delta\cal V$ are contained in $\partial\cal V$.
The first part is easier than the second part. Boundary of $\cal V$ corresponds to surfaces $\Sigma$ where either a nontrivial non-$c$-border simple closed geodesic in $\Sigma$ becomes of length $L_c$, or a non-$p$-side $p$-geodesic becomes of length $L_o$. In the former case, if the closed geodesic were a $b$-border, then it is obtained by $\{{\cal V}_{b=1,\{m=0\}}^{g=0,n=1},{\chi}\}_c$ where ${\chi}$ is the same surface as $\Sigma$ except that the $b$-border of length $L_c$ in $\Sigma$ is a $c$-border of $\chi$. Such $\chi$ belongs to the vertex set because there are no $p$-geodesics shorter than $L_o$ and no nontrivial simple closed geodesics shorter than $L_c$, since $\Sigma$ satisfies the same condition by assumption.
If the nontrivial non-$c$-border simple closed geodesic of length $L_c$ is not a $b$-border, we cut along the closed geodesic to produce two new borders of lengths $L_c$ which we declare to be $c$-borders. If the result were two disjoint surfaces, then it belongs to $\{{\cal V,V}\}_c$ since no nontrivial simple closed geodesics are shorter than $L_c$, no $p$-geodesics are shorter than $L_o$, and the two new $c$-borders are both of lengths $L_c$. If the result of cutting were still a connected surface, say $\xi$, again no nontrivial simple closed geodesics are shorter than $L_c$ and no $p$-geodesics are shorter than $L_o$ and the original surface is obtained by acting $\Delta_c$ on $\xi$.
In the case where a non-$p$-side $p$-geodesic becomes of length $L_o$, the argument is similar and it is contained in $\{{\cal V,V}\}_o$ or $\Delta_o\cal V$, where two new sides of lengths $L_o$ arising from cutting are declared to be $p$-sides. This completes the first part of the proof.
The second part of the proof separates into four cases. In each of the cases where we obtain new surfaces from $\{\cal V,V\}$ or $\Delta{\cal V}$, we have to show that there are no new nontrivial simple closed geodesic of length smaller than $L_c$ and no new $p$-geodesic of length smaller than $L_o$. Then the resulting surfaces will be included in $\partial\cal V$.
![Cutting a possible new $p$-geodesic (in green) along the glued $p$-side of length $L_o$ (in purple in the middle). If it were shorter than $L_o$, the union of the curve $\gamma'$ and $\delta'$ has length shorter than $L_o$ leading to a contradiction.[]{data-label="fig:openglue1"}](figures/open_glue_1.pdf){width="100.00000%"}
\[case1\] $\{{\cal V,V}\}_o$ belongs to $\partial\cal V$.
The new surface is obtained by gluing two $p$-sides of two disjoint surfaces. Any new $p$-geodesics or nontrivial simple closed geodesic for the new surface should pass through the glued $p$-side of length $L_o$. Towards contradiction, assume that there is a new $p$-geodesic $\gamma$ which passes through the glued $p$-side, and it has length smaller than $L_o$. This is depicted in Figure \[fig:openglue1\]. Note that it may pass through the $p$-side multiple times. We cut $\gamma$ along the glued $p$-side and among possibly many pieces (at least two pieces), there will be two arcs with one endpoint on either a $b$-side or a $b$-border and the other endpoint on the glued $p$-side. At least one of the two arcs should be of length smaller than ${L_o\over2}$. Call it $\gamma'$. The endpoint of $\gamma'$ on the glued $p$-side divides the $p$-side into two segments, so one of them should have length smaller than or equal to ${L_o\over2}$. Call it $\delta'$. We define $\gamma''$ to be the union of two arcs $\delta'$ and $\gamma'$, and then $\gamma''$ has length strictly smaller than $L_o$. But $\gamma''$ is completely contained in one of the two vertices we glued together, and is an arc with both endpoints on $b$-sides or $b$-borders. Thus, in its homotopy class, there will be a $p$-geodesic having length shorter than $L_o$, which is a contradiction. One may wonder about the case where the arc $\gamma'$ is homotopic to a point. In this case, one can always glide the endpoint on $b$-side or $b$-border to the end of the glued $p$-side. Trigonometric identity (\[triangleid\]) guarantees that such glided arc has shorter length, so such case is not a $p$-geodesic to begin with. Another way to see this is that since a $p$-geodesic is a perpendicular to the $b$-sides or $b$-borders on which it ends, we cannot form a triangle with two sides being $\gamma'$ and $\delta'$, thus again making the curve under consideration not a $p$-geodesic.
![Possible new simple closed geodesics (in green) as $p$-sides of lengths $L_o$ (in purple in the middle) are glued. These new closed geodesics must pass through the collar of the $p$-side (in gray).[]{data-label="fig:openglue2"}](figures/open_glue_2.pdf){width="100.00000%"}
![A new simple closed geodesic cannot cross only one end of the collar and thus must pass through both ends of the collar. In the above case, the closed curve formed by the union of $\alpha$ and $\gamma'$ is completely contained in the left surface and thus is freely homotopic to a unique closed geodesic which also should be completely contained in the left surface, meaning that the candidate closed geodesic (in green) is not a geodesic.[]{data-label="fig:openglue3"}](figures/open_glue_3.pdf){width="80.00000%"}
Now we discuss new nontrivial simple closed geodesics. Any such closed geodesic should pass through the glued $p$-side. We consider the collar of this $p$-side, which will have width $w$ given by $\sinh{w\over2}={1\over\sinh L_o}$ as in Figure \[fig:openglue2\]. By Theorem \[collar1\], the closed geodesic under consideration cannot be fully contained in the collar and it also has to pass through both ends of the collar. Note that the case like Figure \[fig:openglue3\] cannot happen. If we cut along the glued $p$-side and look at the closed curve formed by the union of $\alpha$ and $\gamma'$, it is completely contained in one of the two glued surfaces. Theorem \[closedgeotehroem\] in Appendix \[hyp\] says that any closed curve is freely homotopic to a unique closed geodesic, meaning that in this case, the unique closed geodesic is also completely contained in one surface. Therefore, the original closed curve under consideration was not a closed geodesic to begin with.
Then, the length $L$ of the closed geodesic has to be greater than $w$: $L>w$. Using the relation between the width of the collar and $L_o$, we get \[newclosedopenglue\] >=[1]{}, where the last inequality follows from (\[regionR2\]). Therefore, the closed geodesic has length greater than $L_c$, which is what we wanted to prove.
$\Delta_o{\cal V}$ belongs to $\partial\cal V$.
A new surface is acquired by gluing two $p$-sides of a single surface. We first consider possibly new $p$-geodesics. They have to pass through the glued $p$-side. If their two ends lie on $b$-borders or $b$-sides which are not neighboring either of the two glued $p$-sides, we can assume that such a $p$-geodesic has length shorter than $L_o$ and show that it leads to a contradiction, exactly in the same way as in Case \[case1\]. If either of the two endpoints lie on a $b$-side neighboring the glued $p$-side, we again cut the curve along the glued $p$-side and consider the piece ending on the neighboring $b$-side. If that piece can be homotopically glided to the glued $p$-side, trigonometric identity for a right-angled triangle (\[triangleid\]) implies that the arc after the gliding is shorter, thus making the original curve not a geodesic. If not, we can again use the contradiction argument.
The case of a new nontrivial simple closed geodesic can be treated in the exactly same manner as Case \[case1\], again illustrated in Figure \[fig:openglue2\]. Such a closed geodesic should cross both ends of the collar of the glued $p$-side, thus being lengthier than the collar width, which is lengthier than $L_c$ by (\[regionR2\]) and (\[newclosedopenglue\]).
\[case3\] $\{{\cal V,V}\}_c$ belongs to $\partial\cal V$.
We first discuss a seemingly trivial, but very important special case, where one of the vertex is ${\cal V}_{b=1,\{m=0\}}^{g=0,n=1}$. Then, twist-plumbing it to a $c$-border of another surface simply turns the label into a $b$-border, still having length $L_c$. Thus, there are no new nontrivial simple closed geodesics having lengths smaller than $L_c$. However, there are new $p$-geodesics whose at least one of two endpoints lies on the new $b$-border of length $L_c$. Such $p$-geodesics should pass through the half-collar of the $b$-border by Theorem \[collar2\]. The width of the half-collar $w$ is given by $\sinh w={1\over\sinh L_c}$. By the definition of the vertex region $\cal R$ in (\[regionR\]), we have $w\geq L_o$, which shows that the new $p$-geodesics are lengthier than $L_o$. This is the only case where a possible new geodesic passes through only the half-collar, rather than a full collar. If this case were absent, the definition of the vertex region ${\cal R}$ in (\[regionR\]) could have been less restrictive, requiring $L_c\leq L_*$ and $\sinh{L_o}\sinh{L_c\over2}\leq1$ instead.
Now, we discuss all the other general cases where two disjoint surfaces are twist-plumbed along two $c$-borders, say $\gamma$. We first consider new $p$-geodesics. Since they end on $b$-sides or $b$-borders, they should pass through the collar of $\gamma$ by Theorem \[collar2\]. Since $\gamma$ is of length $L_c$, its collar has width $w$ given by $\sinh{w\over2}={1\over\sinh L_c}$. Then, we have =[1]{}, where the last inequality follows from (\[regionR2\]). Therefore, we conclude that $w\geq L_o$ and thus new $p$-geodesics all have lengths greater than $L_o$.
For new nontrivial simple closed geodesics, Theorem \[collar2\] again implies that they should be lengthier than the width of the collar $w$, given by $\sinh {w\over2}={1\over\sinh{L_c}}$. Since (\[regionR\]) requires $L_c\leq L_*$, the definition of $L_*$ given in (\[critical\]) implies 1 =. Therefore, new closed geodesics all have lengths greater than $L_c$.
\[case4\] $\Delta_c{\cal V}$ belongs to $\partial\cal V$.
Here, two $c$-borders of a single surface are twist-plumbed together to give a new surface. The proof is exactly the same as Case \[case3\]. This completes the proof of Theorem \[vertextheo\].
![Vertices (in gray) connected by flat finite strips corresponding to open string propagators (in skyblue) and flat finite cylinders corresponding to closed string propagators (in cherry). Open string propagators connect pairs of $p$-sides (in purple) and closed string propagators connect pairs of $c$-borders (in cherry). Grafting should also be applied (in dotted purple and cherry). $b$-sides and $b$-borders (in blue) remain as part of the boundaries. The result is the Feynman diagram on the bottom.[]{data-label="fig:feynman"}](figures/feynman1.pdf){width="100.00000%"}
Feynman diagrams
----------------
Now that open-closed string vertices are explicitly constructed, we discuss how to acquire Feynman diagrams. The discussion here closely follows Section 5 of [@Costello:2019fuh]. There are two kinds of propagators to consider: closed string and open string. As suggested in [@Costello:2019fuh], closed string propagators will correspond to flat finite cylinders of circumference $L_c$, height $t>0$, and twist angle $0\leq\theta<2\pi$. Similarly, open string propagators will correspond to flat finite strips of width $L_o$ and height $h>0$. Using these propagators, we can form Feynman diagrams by grafting cylinders to pairs $c$-borders of vertices, and strips to pairs of $p$-sides of vertices. This is illustrated in Figure \[fig:feynman\].
Therefore, the metric is hyperbolic over vertex regions while flat over propagator regions. Even though such a metric is not smooth over $c-$borders/$p$-sides used for plumbing/gluing as the curvature jumps, it is nonetheless continuous. It is named Thurston metric and we refer readers to [@tanigawa1995grafting] for a review.
Recall Theorem \[fundamentalhomology\] that chains acquired by Feynman diagrams using solutions to open-closed geometric master equation (\[geometricmaster\]) represent the fundamental class of the homology relative to the boundary upon push-forward to the moduli space. If it further happens that the Feynman diagrams built out of hyperbolic vertices are sections, then they provide a decomposition of the moduli space of bordered Riemann surfaces with bulk and boundary punctures. In particular, it means that the parameters of propagators and vertices become injective coordinates of the moduli space. We do not have any strong evidence for such an argument at this point, but it will be interesting to see if this is the case explicitly for some low genus, low number of boundaries, and low number of bulk and boundary punctures.
Description of zero and one-dimensional vertices {#lowdimvertices}
================================================
In this section, we describe low dimensional open-closed hyperbolic string vertices explicitly. For the convenience of discussions and presentations, we will describe the construction of open-closed hyperbolic surfaces in $H_{L_o,L_c}$ corresponding to vertices, without mentioning graftings for the punctures, which is always obvious and assumed.
Dimension zero vertices
-----------------------
We already constructed a dimension zero vertex: disk with one closed string puncture (\[disk1pt\]). It is important to note that this has one conformal Killing vector. Therefore, when we plumb the closed string puncture on this disk with a closed string puncture of another surface, the dimension of the moduli will increase only by one, rather than two. Other dimension zero vertices below will not have any conformal Killing vector.
![Disk with one bulk puncture and one boundary puncture formed by a $p$-side gluing of a hexagon. Two $p$-sides (in purple) of lengths $L$ are glued together and the side between them (cherry) of length $L_c$ becomes a $c$-border. Lengths $L$ and $L_c$ satisfy the relation (\[diskbulkbdryeqn\]).[]{data-label="fig:diskbulkbdry"}](figures/disk_bulk_bdry.pdf){width="100.00000%"}
**i) Disk with three open string punctures**
This is our favorite right-angled hexagon, with all $p$-sides being of lengths $L_o$. Note that there are two marking inequivalent hexagons to consider.
**ii) Sphere with three closed string punctures**
This is a Y-piece, with three closed boundary geodesics all being $c$-borders of lengths $L_c$.
**iii) Disk with one closed string puncture and one open string puncture**
This can be built using a hexagon drawn in Figure \[fig:diskbulkbdry\]. One of three $p$-sides has length $L_o$, corresponding to the open string puncture. The other two $p$-sides have lengths $L$, which is given by \[diskbulkbdryeqn\] \^2L=[L\_c+L\_oL\_c -1]{}. This is such that the $b$-side neighboring the two $p$-sides has length $L_c$, as can be checked using trigonometric identity (\[trigid\]). We then glue two $p$-sides of lengths $L$ and the $b$-side between the two becomes a $c$-border, which corresponds to a closed string puncture.
Dimension one vertices
----------------------
There are four vertices at dimension one, all of which have no conformal Killing vector.
![Disk with four $p$-sides of lengths $L_o$ (in purple). There are two $p$-geodesics to consider (in green) and either of their lengths provides a coordinate for the moduli space. Lengths of the top and bottom $b$-sides are denoted as $l$.[]{data-label="fig:open4pt"}](figures/open4pt.pdf){width="60.00000%"}
**i) Disk with four open string punctures**
For the convenience of discussion and presentation, we are going to work with only one ordering of markings, say $\{1,2,3,4\}$ for the open string punctures in the clockwise direction. The other five marking inequivalent diagrams can be obtained simply by permuting the marking labels. We take two identical hexagons whose two of the $p$-sides are of lengths $L_o$ corresponding to open string punctures, and the other $p$-side of length $L$. We then glue along the $p$-side of length $L$ as shown in Figure \[fig:open4pt\]. This $p$-side becomes a $p$-geodesic of the glued surface, which we call $\gamma_1$. Thus, we have the first condition for the vertex: $L\geq L_o$. However, this is not the only $p$-geodesic which is not a $p$-side. There is another $p$-geodesic $\gamma_2$ of length $L'$, which crosses $\gamma_1$, and thus we have the second vertex condition $L'\geq L_o$. Lengths described in Figure \[fig:open4pt\] have relations with each other. First, trigonometric identity for the hexagon (\[trigid\]) implies =[(1+)]{}. Secondly, trigonometric identity for pentagon (\[pentid\]) gives =. Combining the two, we get the following expression for $L'$ in terms of $L$ and $L_o$ =[2(2L\_o)+L+1L-1]{}. Thus, $L$ grows monotonically as $L'$ decreases and vice versa, and they become equal at L=L’=1+2L\_o>L\_o, where we wrote the last inequality to stress that at this length, both $p$-geodesics have lengths greater than $L_o$.
Two $p$-geodesics $\gamma_1$ and $\gamma_2$ are distinct due to markings and the moduli space is parameterized either by $L$ or $L'$. We choose to work with $L$ and the full moduli space is given by $L\in\mathbb{R}_+$. With the first and second vertex conditions and the expression for $L'$ as a function of $L$, we get the following result for the vertex region \[condition4pt\] [V]{}\^[0,0]{}\_[1,{4}]{}L\_oLe(L\_o), (e(L\_o)). Note that for any $L_o\in\mathbb{R}_+$, the vertex region for $L$ is nonempty. Again, there are five other marking inequivalent diagrams which essentially carry the exactly same form of the vertex conditions.
![Disk with two $p$-sides of lengths $L_o$ (in purple) and a $c$-border of length $L_c$ (in cherry). There are two $p$-geodesics of lengths $l$ and $L$ to consider (in green). Lengths $l$ and $L'$ can be expressed as functions of $L_o,L_c,$ and $L$, and the moduli space is parameterized by either $l$ or $L$.[]{data-label="fig:diskioneclosedtwoopen"}](figures/disk_one_closed_two_open.pdf){width="100.00000%"}
**ii) Disk with one closed string puncture and two open string punctures**
As shown in Figure \[fig:diskioneclosedtwoopen\], there are two $p$-geodesics to consider. We describe this case using two hexagons. The first hexagon on the left has three $p$-sides of lengths $L, L', L'$, so that two of them are equal in lengths. The $b$-side between them has length $L_c$ so that upon self-gluing two $p$-sides of lengths $L'$, we create a $c$-border of length $L_c$. This will correspond to the closed string puncture. (\[trigid\]) gives the relation \[relation1\] L\_c=[L+\^2L’\^2L’]{} \^2L’=[L\_c+LL\_c-1]{}.
The second hexagon on the right has $p$-sides of lengths $L, L_o, L_o$. Two $p$-sides with lengths $L_o$ correspond to two open string punctures. We now glue two $p$-sides of lengths $L$ of two hexagons together. Then, it becomes a $p$-geodesic, whose endpoints are on two $b$-sides neighboring $p$-sides of lengths $L'$ and $L_o$, and we have the first vertex condition $L\geq L_o$. The other $p$-geodesic has length $l$ and has its endpoints on a $b$-side between two $p$-sides of $L_o$. Therefore, the second vertex condition is given by $l\geq L_o$. We now find the relation between the two lengths $L$ and $l$.
Again using (\[trigid\]), we get =[L\_o(1+L)L\_oL]{}, =[L’(1+L)L’L]{}=[L+1L]{}, where we used (\[relation1\]) in the last equality. Also, trigonometric identity for pentagon (\[pentid\]) gives the following relation =L\_o(+). Combining altogether, we get \[relation2\] l=-1+2\^2[L2]{}(L\_o+L\_o )\^2. It is straighforward to see that as $L$ increases from $0$ to $\infty$, $l$ monotonically decreases from $\infty$ to $0$. The moduli space is parameterized by either $L$ or $l$ since these are distinct due to markings and the orientation. We choose to work with $L$. The full moduli space is given by $L \in\mathbb{R}_+$, and the first and second vertex conditions together with the expression of $l$ as a function of $L$ give the following vertex region \[condition1c2o\] &[V]{}\^[0,1]{}\_[1,{2}]{}L\_oLf(L\_o,L\_c), \
&(f(L\_o,L\_c))-1\
& +2\^2[L\_o2]{}(L\_o+L\_o )\^2. One can check that this region is nonempty for all $(L_o,L_c)\in{\cal R}$.
![Disk with two $c$-borders of lengths $L_c$ (in cherry). There is a single $p$-geodesic of length $L$ to consider (in green) which parameterizes the moduli space. $b$-border (in blue) has length $4l_1$ which is a function of $L$ and $L_c$.[]{data-label="fig:disktwoclosed"}](figures/disk_two_closed.pdf){width="100.00000%"}
**iii) Disk with two closed string punctures**
We take two identical hexagons with $p$-sides of lengths $L, l, l$ as shown in Figure \[fig:disktwoclosed\]. The $b$-side between two $p$-sides of length $l$ will have length $L_c$, and (\[trigid\]) implies that the other two left-over $b$-sides have equal lengths which we call $l_1$. Using (\[trigid\]), we have the relations $\cosh^2l={\cosh{L_c}+\cosh{L}\over\cosh L_c-1}$ and $\cosh l_1={\cosh l(1+\cosh L)\over\sinh l\sinh L}$. Combining these, we get \[l1id\] \^2l\_1=[L+L\_cL-1]{}.
For each hexagon, we glue two $p$-sides of lengths $l$ together to form a $c$-border of length $L_c$. This will correspond to a closed string puncture, so each hexagon has one closed string puncture. Now, we also glue $p$-sides of length $L$ of two hexagons together, which is the only $p$-geodesic of the resulting surface which is not a $p$-side. Therefore, the moduli space is given by $L\in\mathbb R_+$. We have the first vertex condition $L\geq L_o$. Also, we have a closed geodesic which is the boundary of the surface i.e. $b$-border. Its length is $4l_1$ so we require $4l_1\geq L_c$. Using (\[l1id\]), we get the following vertex condition \[condition2c\] [V]{}\^[0,2]{}\_[1,{0}]{} L\_oLg(L\_c), (g(L\_c)). Note that this is nonempty for all $(L_o,L_c)\in\cal R$.
![Annulus with one $p$-side of length $L_o$ (in purple) constructed from a hexagon with two $p$-sides of lengths $L$ glued together, which then become a $p$-geodesic (in green). $b$-border of length $l$ (in blue) is also formed and the moduli space can be parameterized by either $l$ or $L$. We choose to work with $L$.[]{data-label="fig:annulus1pt"}](figures/annulus1pt.pdf){width="100.00000%"}
**iv) Annulus with one open string puncture**
We take a hexagon with $p$-sides of lengths $L_o,L,L$ as shown in Figure \[fig:annulus1pt\]. The $p$-side with length $L_o$ correspond to the open string puncture. The $b$-side between two $p$-sides of lengths $L$ has length $l$ satisfying $\cosh l={\cosh L_o+\cosh^2L\over\sinh^2L}$ due to (\[trigid\]).
We glue two $p$-sides of lengths $L$, which then becomes the only $p$-geodesic which is not a $p$-side. Thus, the moduli space is parameterized by $L\in\mathbb R_+$. We have the first vertex condition $L\geq L_o$. Also, the $b$-side of length $l$ becomes a $b$-border, so it should satisfy $l\geq L_c$. Using the previous length relation, we get \[conditionAnn\] [V]{}\^[0,0]{}\_[2,{1,0}]{}L\_oLh(L\_o,L\_c), ( h(L\_o,L\_c)). Again, this is nonempty for all $(L_o,L_c)\in\cal R$.
This completes the construction of open-closed hyperbolic string vertices of dimension up to one.
![One-dimensional moduli spaces assuming that Feynman diagrams provide sections. Different regions are covered by either vertices (colored in gray) or Feynman diagrams with propagators. Note that i) is a single piece of six marking inequivalent configurations, while the cases ii), iii), and iv) have unique markings.[]{data-label="fig:modulispace"}](figures/modulimarked.pdf){width="110.00000%"}
Decomposition and Deligne-Mumford compactification of moduli spaces of dimension one
------------------------------------------------------------------------------------
In this subsection, we will assume that Feynman diagrams built by hyperbolic vertices indeed provide sections, and study which diagrams cover which parts of the moduli spaces for the case of dimension one. All four cases below are summarized in Figure \[fig:modulispace\] and notations for each case follow those in the previous subsection. We also discuss Deligne-Mumford compactifications of the moduli spaces.
**i) Disk with four open string punctures**
The regions $L<L_o$ and $L>e(L_o)$ should be covered by Feynman diagrams with two cubic open string vertices ${\cal V}^{0,0}_{1,\{3\}}$ and an open string propagator connecting the two. Two regions carry different markings as shown in Figure \[fig:modulispace\]. In a sense, open string propagator shortens the $p$-geodesic.
**ii) Disk with one closed string puncture and two open string punctures**
The regions $L<L_o$ and $L>f(L_o,L_c)$ are covered by Feynman diagrams with two vertices ${\cal V}^{0,1}_{1,\{1\}}$ and ${\cal V}^{0,0}_{1,\{3\}}$ with an open string propagator between them, where the markings on ${\cal V}^{0,0}_{1,\{3\}}$ are different between the two regions. Again, open string propagator effectively shortens the $p$-geodesic.
**iii) Disk with two closed string punctures**
There are two Feynman diagrams other than the vertex. The first one is given by two identical vertices ${\cal V}^{0,1}_{1,\{1\}}$ with an open string propagator. This effectively shortens the length $L$ of the $p$-geodesic and covers $L<L_o$. The second one is given by two vertices ${\cal V}^{0,1}_{1,\{0\}}$ and ${\cal V}^{0,3}_{0,\{\}}$ with a closed string propagator. Here, Y-piece with three $c$-borders of lengths $L_c$ acquires a boundary as the closed string propagator is attached to one of $c$-borders, since the other end of the propagator sticks to a circle of circumference $L_c$ and becomes a $b$-border. This effectively reduces the length of the plumbed $c$-border which is a $b$-border now, thus covering $4l_1<L_c$.
**iv) Annulus with one open string puncture**
There are again two Feynman diagrams other than the vertex. One is given by a vertex ${\cal V}^{0,0}_{1,\{3\}}$ with an open string propagator. This effectively shortens the length $L$ of the $p$-geodesic and covers the moduli space $L<L_o$. Another diagram consists of two vertices ${\cal V}^{0,1}_{1,\{0\}}$ and ${\cal V}^{0,1}_{1,\{1\}}$ with a closed string propagator, which effectively shortens the length $l$ of the $b$-border. Thus, it covers $l<L_c$.
Finally, note that we have simple descriptions of Deligne-Mumford compactifications for all four cases. In cases i) and ii), the compactification adds a point $L=0$ corresponding to the infinite open string propagator and another point $L=\infty$ also corresponding to the infinite open string propagator. In cases iii) and iv), the compactification similarly adds $L=0$ for the infinite open string propagator, but this time adds $L=\infty$ for the infinite closed string propagator. The latter is typically of codimension two and thus one might naively think that it cannot add a point to one-dimensional moduli space. However, due to the presence of one conformal Killing vector on a disk with a bulk puncture, it indeed adds a point which becomes a boundary of the moduli space. Therefore, all the one-dimensional moduli spaces $L\in(0,\infty)$ become $L\in[0,\infty]$ under Deligne-Mumford compactifications, where again, ${\cal M}^{0,0}_{1,\{4\}}$ strictly speaking should also include other five marking inequivalent configurations. As expected, Deligne-Mumford compactifications indeed get nontrivial boundary contributions, which is not the case for closed Riemann surfaces with punctures.
No cubic point in $\cal R$ {#cubic}
--------------------------
It is well known that classical open string field theory has a representation where it has only a cubic vertex [@Witten:1985cc]. In the case of open-closed string vertices, analogous cubic theory was constructed using minimal area metric in [@Zwiebach:1992bw], with one caveat that closed strings were always taken to be on-shell. The idea was that with a specific choice of a minimal area metric, all one-dimensional moduli spaces were covered by Feynman diagrams of zero dimensional vertices, meaning that one-dimensional vertices are empty. With the condition that closed string fields are on-shell and do not propagate in particular, it is obvious from geometric master equation that all higher-dimensional vertices whose surfaces have at least one boundary can be set to be empty, as it relates boundary of $d$-dimensional vertices to $\{~,~\}$ acting on two vertices whose dimensions add up to $(d-1)$ or $\Delta$ acting on a single $(d-1)$-dimensional vertex.
In our construction of open-closed hyperbolic vertices, we explicitly saw that all one-dimensional vertices are nonempty. For example, consider ${\cal V}^{0,0}_{1,\{4\}}$ given by (\[condition4pt\]). For any choice of $L_o\in\mathbb R_+$, it is nonempty meaning that one cannot set the open string quartic vertex to be empty. Therefore, we conclude that our construction of hyperbolic vertices do not have a point in $\cal R$ where the theory becomes cubic.
Nonetheless, there exists a similar limit as the one described in [@Zwiebach:1992bw]. We can consider $L_c\rightarrow0$ where closed string propagators become infinite. In this limit, closed string fields must be on-shell and do not propagate. Even though all the vertices involving both open and closed string fields are still present, moduli spaces get only infinitesimal contributions from Feynman diagrams involving closed string propagators, meaning that it is covered mostly by vertices and open string propagators. In particular, the description of dynamics of open string fields requires diagrams involving only open string fields without any closed string fields, modulo boundary contributions corresponding to infinite closed string propagators attached to vertices involving on-shell closed strings.
The other limit $L_o\rightarrow0$ describes the opposite situation where the open string propagators become infinitely lengthy. In this case, moduli spaces get infinitesimal contributions from Feynman diagrams involving open string propagators and in particular, dynamics of closed string fields are described by diagrams involving only closed string fields, modulo boundary contributions from infinite open string propagators attached to vertices involving on-shell open strings. Of course, it is important to keep in mind that these limits are only formal and the full open-closed string field theory requires finite nonzero $L_c$ and $L_o$.
Discussions
===========
In this work, we found a family of hyperbolic string vertices for open-closed string field theory, naturally generalizing the work of [@Costello:2019fuh] which considered the pure closed string vertices. Key ingredients were collar theorems of BHHS, which are responsible for restricting the systolic conditions on nontrivial simple closed geodesics and $p$-geodesics to region $\cal R$ (\[regionR\]). We then gave explicit descriptions of all zero and one-dimensional vertices.
There are two mathematical steps which one could naturally try to take. First is the existence and uniqueness of the solutions to the geometric master equation (\[geometricmaster\]), following similar ideas as [@Costello:2019fuh]. There may be subtleties due to the fact that Deligne-Mumford compactifications get nontrivial boundary contributions, but the general idea seems to allow for a rather straighfoward generalization to the open-closed case. The second is to check if the grafting map provides a homeomorphism between ${\cal{M}}_{b,\{m_i\}}^{g,n}(L_o,L_c)$ and ${\cal{M}}_{b,\{m_i\}}^{g,n}$, and if hyperbolic Feynman diagrams provide sections, which will then lead to coordinates for the entire moduli space.
On the computational side, in order to actually compute vertices with string field insertions, one needs to learn how to compute conformal field theory correlation functions on grafted BHHS, which will involve nontrivial Weyl transformations due to the presence of geodesic sides and borders. In addition, one then has to integrate such correlation functions over the vertex regions in moduli spaces. Both of these technical aspects are less explored in literature and thus remain as main obstacles if we were to use hyperbolic string vertices to perform string field theory computations.
In the case of moduli integration for hyperbolic surfaces, the results are known only for simple cases such as volumes over bordered hyperbolic Riemann surfaces via topological recursion relations [@1998InMat.132..607M; @Mirzakhani:2006fta; @Mirzakhani:2006eta]. Recall that the Weil-Petersson volume form for bordered hyperbolic Riemann surfaces is built from wedge products of a symplectic form expressed in terms of Fenchel-Nielsen coordinates, which is independent of different possible pants decompositions. In the case of BHHS however, it does not seem so straightforward to find a simple expression for the volume form which should be independent of the hexagon decompositions. For example, such a form for ${\cal M}^{0,0}_{1,\{4\}}$ is given by ${d(\cosh L)\over\cosh L-1}$, but if the lengths of the $p$-sides are different from each other, this volume form is no longer invariant under different hexagon decompositions. Nonetheless, hexagon decompositions for BHHS seem to provide another setup to study the moduli spaces of bordered hyperbolic Riemann surfaces in a rather direct way. Perhaps, it is not a coincidence that ${\cal N}=4$ super Yang-Mills integrability story also benefited by considering hexagons [@Basso:2015zoa; @Basso:2015eqa; @Fleury:2016ykk; @Fleury:2017eph]. Interestingly, the four-bordered sphere of Figure 1 in [@Fleury:2016ykk] is exactly $p$-side gluing of four hexagons, rather than typical plumbing construction of two Y-pieces.
On the physics side, open-closed string field theory has several interesting applications. Even at the level of computing perturbative on-shell ampltiudes involving D-branes [@Hashimoto:1996bf] or D-instanton contributions to type IIB closed string scattering amplitudes [@Polchinski:1994fq; @Green:1997tv], unambiguous procedures are presumably defined only through open-closed string field theory. Quantum master action for open-closed superstring field theory for instance was constructed only recently in [@Moosavian:2019ydz] and consistent computations of on-shell amplitudes using such a framework are yet to be performed. It will be interesting to explicitly see how open-closed string field theory resolves issues related to boundary contributions of the moduli space discussed in section \[fundclass\]. This issue has been recently discussed in the context of ZZ instanton contributions to $c=1$ string theory [@Balthazar:2019rnh] using the open-closed string field theory framework [@Sen:2019qqg].
Descriptions of open string tachyon dynamics [@Sen:2002nu; @Sen:2002in; @Sen:2003xs; @Sen:2003iv; @Sen:2004nf], S-brane solutions [@Gutperle:2002ai], and open-closed type dualities [@Kontsevich:1992ti; @Witten:1992fb; @Gopakumar:1998ki; @Khoury:2000hz; @Ooguri:2002gx; @Gaiotto:2001ji; @Gaiotto:2003rm; @Gaiotto_2005] may be formulated in the framework open-closed string field theory. For example, unstable D-brane decay is classically described by an appropriate time-dependent boundary state and it was conjectured [@Sen:2003iv] that the full physics describing the unstable D-brane is given by a quantum open string field theory, not necessarily needing to introduce closed strings into the description. It does not seem unrelated to the limit $L_c\rightarrow0$ discussed in section \[cubic\], even though the limit was only a formal one and the precise open string field theory is yet to be understood. On the open-closed duality side, one of typical ideas is that on-shell closed string processes can be constructed from open string diagrams. For example, one can take a vacuum open string amplitude and let the boundaries shrink to become on-shell closed string punctures. This corresponds to plumbing disks with one closed string puncture to pure closed string diagrams via infinite closed string propagators. Along the similar line, one can also take a solution to open-closed string field equations of motions and study how it interpolates between different looking solutions under field redefinitions induced by shifts in vertices e.g. taking different values of $L_o$ and $L_c$ in region $\cal R$ of our hyperbolic vertices. In particular, limits $L_c\rightarrow0$ and $L_o\rightarrow0$ would respectively give mostly open and mostly closed solution in the sense that $L_c\rightarrow0$ turns off off-shell closed string field deformation and $L_o\rightarrow0$ turns off off-shell open string field deformation from the original boundary conformal field theory around which open-closed string field theory was formulated[^2]. In many respects, it definitely appears that string vertices and string field theory in general provide a rich list of relevant mathematical problems and interesting physics questions to be answered.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank Ashoke Sen, Arnav Tripathy, Xi Yin, Max Zimet, and Barton Zwiebach for enlightening discussions and correspondences, and Ashoke Sen, Xi Yin, and Barton Zwiebach for comments on an early draft. This work is supported in part by Xi Yin’s Simons Investigator Award from the Simons Foundation and the Simons Collaboration Grant on the Non-Perturbative Bootstrap.
Hyperbolic surfaces {#hyp}
===================
Here, we collect relevant theorems and identities appearing in hyperbolic surfaces. All of these statements are explained in detail in [@Buser1992GeometryAS]. For the following theorems, we assume that we are on a general compact hyperbolic surface $S$.
\[perpendiculartheorem\] Given $A$ and $B$ which are either smooth closed boundary geodesics or geodesic sides which meet neighboring sides under an angle $\leq\pi/2$, and an open curve $c$ ending on them, there exists a unique shortest geodesic $\gamma$ in the homotopy class of $c$, and $\gamma$ is the unique common perpendicular to $A$ and $B$ in the homotopy class unless $\gamma$ is a point. In particular, if $c$ were simple, $\gamma$ is also simple.
This theorem is the reason why we favored right-angled hexagons and hexagon decompositions as they naturally consist of such perpendicular geodesics. The following theorem further justifies some of considerations of closed geodesics in our construction of hyperbolic vertices.
\[closedgeotehroem\][(Theorem 1.6.6 in [@Buser1992GeometryAS])]{} Given a nontrivial closed curve $c$ on $S$, it is freely homotopic to a unique closed geodesic $\gamma$. $\gamma$ is either contained in $\partial S$ or $\gamma\cap\partial S=\varnothing$.
We now state some of relevant trigonometric identities in hyperbolic geometry. There are more identities than the ones we discuss here, which can all be found in Chapter 2 of [@Buser1992GeometryAS]. First, consider a right-angled triangle of geodesic sides of lengths $a, b, c$. Assume that $a$ and $b$ are meeting at the right angle. Then, \[triangleid\] c=a b. This identity states that the side away from the right angle is lengthier than the other two sides.
We also consider a right-angled pentagon consisting of five geodesic sides of lengths $a,b,c,d,e$ where we took into account the ordering e.g. $b$ meets $a$ to the left and $c$ to the right. All sides meet their neighboring sides at right angles. Then, \[pentid\] c=ae, and likewise for the other sides.
[^1]: We thank Ashoke Sen for discussions on null-state decoupling in open-closed string field theory.
[^2]: We thank Xi Yin for suggesting viewing dual descriptions of open string and closed string field solutions as string field redefinitions interpolating between mostly open and mostly closed pictures.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We use an effective one-dimensional Gross–Pitaevskii equation to study bright matter-wave solitons held in a tightly confining toroidal trapping potential, in a rotating frame of reference, as they are split and recombined on narrow barrier potentials. In particular, we present an analytical and numerical analysis of the phase evolution of the solitons and delimit a velocity regime in which soliton Sagnac interferometry is possible, taking account of the effect of quantum uncertainty.'
author:
- 'J. L. Helm'
- 'S. L. Cornish'
- 'S. A. Gardiner'
title: 'Sagnac Interferometry Using Bright Matter-Wave Solitons'
---
A Bose–Einstein condensate (BEC) with attractive inter-atomic interactions can support soliton-like structures referred to as bright solitary matter-waves [@khaykovich_etal_science_2002; @strecker_etal_nature_2002; @cornish_etal_prl_2006; @Marchant_etal_2013; @Nguyen2014]. These propagate without dispersion [@morgan_etal_pra_1997], are robust to collisions with other bright solitary matter-waves and with slowly varying external potentials [@parker_etal_physicad_2008; @billam_etal_pra_2011], and have center-of-mass trajectories well-described by effective particle models [@martin_etal_prl_2007; @martin_etal_pra_2008; @poletti_etal_prl_2008]. Such soliton-like properties are due to the mean-field description of an atomic BEC reducing to the nonlinear Schrödinger equation in a homogeneous, quasi-one-dimensional (quasi-1D) limit, which for the case of attractive interactions supports the bright soliton solutions well-known in the context of nonlinear optics [@zakharov_shabat_1972_russian; @satsuma_yajima_1974; @gordon_ol_1983; @haus_wong_rmp_1996; @Helczynski_ps_2000]. The quasi-1D limit is experimentally challenging for attractive condensates [@billam_etal_variational_2011], but solitary wave dynamics remain highly soliton-like outside this limit [@cornish_etal_prl_2006; @billam_etal_pra_2011].
A bright solitary wave colliding with a narrow potential barrier is a good candidate mechanism to create two mutually coherent localised condensates, much as a beam-splitter splits the light of an optical interferometer. This has been extensively investigated in the quasi-1D, mean-field description of an atomic BEC [@HELM_PRA_2012; @kivshar_malomed_rmp_1989; @ernst_brand_pra_2010; @lee_brand_2006; @cao_malomed_pla_1995; @holmer_etal_cmp_2007; @holmer_etal_jns_2007; @POLO_etal_PRA_2013; @Molmer_arxiv_2012; @Minmar_thesis_2012; @abdullaev_brazhnyi_2012], and sufficiently fast collisions do lead to the desired beam-splitting effect [@holmer_etal_cmp_2007; @holmer_etal_jns_2007]. Consequently, bright solitary matter-waves, with their dispersion-free propagation, present an intriguing candidate system for future interferometric devices [@HELM_PRA_2012; @strecker_etal_nature_2002; @cornish_etal_physicad_2009; @weiss_castin_prl_2009; @streltsov_etal_pra_2009; @billam_etal_pra_2011; @al_khawaja_stoof_njp_2011; @martin_ruostekoski_njp_2010; @mcdonald_prl_2014]. Previous work [@martin_ruostekoski_njp_2010; @POLO_etal_PRA_2013; @HELM_PRA_2014] considered a Mach–Zehnder interferometer using a narrow potential barrier to split harmonically trapped solitary waves, based on the configuration of a recent experiment [@Nguyen2014]. These demonstrated one can also recombine solitary waves if they collide at the barrier; the collision dynamics are explained more fully in [@HELM_PRA_2012]. In these collisions, the relative atomic populations within the two outgoing solitary waves are governed by the relative phase $\Delta$ between the incoming ones. The mean-field nonlinearity can lead to the relative populations of the outgoing waves exhibiting greater sensitivity to small variations in the phase $\Delta$; however, simulations including quantum noise in the initial condition [@HELM_PRA_2014] or via the truncated Wigner method [@blakie_etal_ap_2008], demonstrated that enhanced number fluctuations counteract this improvement [@martin_ruostekoski_njp_2010].
![Stages of Sagnac interferometry. An incoming soliton splits at time $T_{\mathrm{s}}$ on a barrier into two solitons of equal amplitude and opposite velocity. After circumnavigating the ring trap, at time $T_{\mathrm{c}}$ the solitons recombine either at the same barrier (a), or a second barrier (b) antipodal to the first, illustrated in both cases with angular rotation $\Omega=1.875 \times 10^{-3}$, and ring circumference $L=40\pi$. The resulting phase difference, incorporating the Sagnac phase due to the rotating reference frame, is read out via the population difference in the final output products within the positive (shaded) and negative domains. (c) Final population in the positive domain $I_{+}$ as a function of $\Omega$, with $L=40\pi$ and initial soliton velocity $v=4$. The sensitivity of the single barrier case (dashed line) is twice that of the double barrier case (solid line) because the interrogation time $T_{\mathrm{c}}-T_{\mathrm{s}}$ is doubled.[]{data-label="fig:overview"}](fig_diagram.png){width="\columnwidth"}
We extend the framework of soliton interferometry to measurement of the Sagnac effect, first observed in an atom interferometer by Riehle [*et al.* ]{}[@Riehle_etal_prl_1991]. In this experiment the observation manifested as a shift in the Ramsey fringes produced by passing an atomic beam of $^{40}$Ca through four travelling waves in a Ramsey geometry, producing an atomic beam interferometer. What we present differs from the Riehle setup in two ways. Firstly, in [@Riehle_etal_prl_1991] some phase information is transported optically. In our system atom-light interactions serve only to coherently split the condensate; any resulting phase dynamics are incidental. Secondly, our system results, not in an interference fringe shift, but a population shift between the positive and negative domains of the interferometer. The Sagnac effect is inferred from measurements of particle numbers [@halkyard_etal_pra_2010] in the spatially distinct condensates on either side of the barrier, and not the structure of those condensates. (which are expected to remain soliton-like). We consider an experimental configuration, contained entirely within a rotating frame, where there is a smooth ring-shaped trapping potential (implemented by, e.g., using a spatial light modulator [@moulder_etal_pra_2012], time-averaging with acousto-optic deflectors [@henderson_etal_njp_2009], or imaging an intensity mask [@corman_etal_prl_2014]) and narrow barriers realised with optical light sheets, focussed using high numerical aperture lenses. Solitons, initially produced in an optical trap, can be adiabatically transferred into the ring, with the initial velocity set by moving the optical potential during the transfer [@rakonjac_etal_ol_2012]. Key sources of error include: uncertainty in the value of the soliton velocity relative to the barrier strength and, in turn, the barrier transmission level [@HELM_PRA_2012]; initial particle number, which determines the ground-state energy and so sets the low-energy splitting threshold, close to which the system becomes sensitive to otherwise small fluctuations in the velocity [@martin_ruostekoski_njp_2010]; and measurement of final particle number.
Within the Gross–Pitaevskii equation (GPE) framework, we consider $N$ bosonic atoms of mass $m$ and scattering length $a_{s}$, in an effective 1D configuration due to a tightly confining (frequency $\omega_{r}$) harmonic trapping potential in the degrees of freedom perpendicular to the direction of free motion, implying an interaction strength of ${g_{\mathrm{1D}}}= 2\hbar\omega_{r}a_{s}$ per particle. We use “soliton units” [@martin_etal_pra_2008] (equivalent to $\hbar = m = |{g_{\mathrm{1D}}}|N =1$), where position, time and energy are in units of $\hbar^{2}/m|{g_{\mathrm{1D}}}|N$, $\hbar^{3}/m{g_{\mathrm{1D}}}^{2}N^{2}$, and $m{g_{\mathrm{1D}}}^{2}N^{2}/\hbar^{2}$ [@Note1]. To describe a tightly confining toroidal trap geometry (or ring trap), we introduce periodic boundary conditions over the domain $-L/2<x\le L/2$, where $L$ is the dimensionless form of the circumference [@HELM_PRA_2014]. It is common to discuss Sagnac interferometry and ring systems in terms of an angle coordinate $\theta = 2\pi x/L$ [@halkyard_etal_pra_2010; @kanamoto_etal_pra_2003]; we choose not to, making it easier to draw on earlier work on splitting solitons at narrow barriers [@HELM_PRA_2012; @holmer_etal_cmp_2007; @holmer_etal_jns_2007; @HELM_PRA_2014]. Considering the dynamics within a frame rotating with dimensionless angular frequency $\Omega$ results in the following GPE: $$\begin{split}
{i}\frac{{\partial}\psi(x)}{{\partial}t} =&\Bigg[-\frac{1}{2}\frac{{\partial}^2}{{\partial}x^2}+{i}\Gamma\frac{{\partial}}{{\partial}x} +\frac{q}{\sigma \sqrt{2\pi}}e^{-x^2/2\sigma^2} \\
&+({n_{\mathrm{b}}}-1)\frac{q}{\sigma \sqrt{2\pi}}e^{-(x\pm L/2)^2/2\sigma^2} -\left|\psi(x)\right|^2\Bigg]\psi(x),
\end{split}
\label{eqn:SAGGPE}$$ where $\Gamma = \Omega L/2\pi$ \[which we can also write in terms of the dimensional circumference $L_{\mathrm{D}}$ and angular frequency $\Omega_{\mathrm{D}}$ as $\Gamma = (\hbar/|{g_{\mathrm{1D}}}|N)\Omega_{\mathrm{D}}L_{\mathrm{D}}/2\pi$\], and $\psi$ is the (unit norm) condensate wave function. Note the two barrier terms; presence or absence of the second barrier implies two different forms of Sagnac interferometry: one where both solitons perform full circumnavigations of the ring, enclosing the area within the ring twice; and one where each soliton circumnavigates a different half of the ring, enclosing the area once. These cases are distinguished by ${n_{\mathrm{b}}}=1$ for the first (single barrier) case and ${n_{\mathrm{b}}}=2$ for the second (two antipodal barriers) case; the second barrier term is zero for ${n_{\mathrm{b}}}=1$ \[see [Fig. \[fig:overview\](a)]{}\] and identical to the other barrier term, up to a spatial offset, for ${n_{\mathrm{b}}}=2$ \[see [Fig. \[fig:overview\](b)]{}\]. All simulations were carried out with $\sigma=0.2$; this width is suitably narrow to approximate a delta function for collisional velocities up to $v=4.0$ [@Note2], corresponding to a (variable, depending on the ring circumference) dimensionless angular velocity of $\omega=2\pi v/L$.
![Numerically calculated transmission into the positive domain after the second collision, $I_+$, for the two Sagnac interferometry geometries shown in Fig. \[fig:overview\]. Colormaps for the (b) two barrier and (d) single barrier cases show the $0.16<v<4$, $0<\Omega\times10^3<2.5$ parameter space. Panels (a) and (c) show specific curves of constant $v$ for these scenarios \[for $v=0.52$ (dashed-dotted line), $v=1$ (dashed line), and $v=4$ (solid line)\], and highlight how the different interrogation times result in a different Sagnac phase accumulation. The phase difference is varied by varying $\Omega$ while keeping $L=40\pi$ \[hence the $v$ ranged over in panels (b) and (d) correspond to dimensionless angular velocities of between $\omega = 0.008$ and $\omega = 0.2$\]. []{data-label="fig:int_s"}](fig_int_s.png){width="\columnwidth"}
We obtain soliton solutions to [[Eq. (\[eqn:SAGGPE\])]{}]{} (in the absence of splitting potentials and periodic boundary conditions), i.e., the usual nonlinear Schrödinger equation in a frame moving with velocity $\Gamma$, by the Galilean invariance of the standard soliton profile [@gordon_ol_1983]. The (amplitude $A$) invariant soliton solution is $\tilde\psi(\tilde x,t)=A{\mathrm{sech}}(A[\tilde x-Vt])\exp({i}V\tilde x + {i}[A^2-V^2]t/2)$; the tilde notation denotes the *stationary* frame of reference. A soliton moving with velocity $v$ in a frame moving with velocity $\Gamma$ is moving at velocity $V=v+\Gamma$ in the stationary frame. In the *moving* frame, where $x=\tilde x-\Gamma t$, we obtain $$\begin{split}
\psi(x,t)=&A{\mathrm{sech}}(A[x-vt])\\
&\times\exp({i}[v+\Gamma][x+\Gamma t]+{i}\{A^2-[v+\Gamma]^2\}t/2).
\end{split}
\label{eqn:tw_sol}$$ Assuming $L\gg1$ (a parameter regime far from the critical point described in [@kanamoto_etal_pra_2003]), [[Eq. (\[eqn:tw\_sol\])]{}]{} is a valid solution to [[Eq. (\[eqn:SAGGPE\])]{}]{}.
![Results of Monte Carlo simulations used to model effects of quantum uncertainty for a range of $v_0$, $N$ and $\Omega$. (a–d)(i) Scatter plot of the solitons’ collisional velocity $v_{\mathrm b}$ for ensembles of individual simulations. In (d)(i), the higher gradient of the curves through the points implies the detected transmission $I_{+}$ is less sensitive to quantum fluctuations. (a–d)(ii) Sample distributions of the simulation outcomes. For each $N,v_0$ pair we explored $\Omega\times10^3={0, 6.25, 12.50, 18.75, 25}$, corresponding to Sagnac phases of ${\delta_{\mathrm{S}}}={0{(\textcolor[rgb]{0,0,0}{\bm{+}})},\pi/2{(\textcolor[rgb]{0,0,0}{\bm{\circ}})},\pi{(\textcolor[rgb]{0,0,0}{\bm{\triangle}})},3\pi/2{(\textcolor[rgb]{0,0,0}{\bm{\square}})},2\pi{(\textcolor[rgb]{0,0,0}{\bm{\times}})}}$ respectively. The simulations were carried out in a two barrier system, with $L=40\pi$ (hence the $v_{0}$ values correspond to dimensionless angular velocities $\omega = 0.05$, $0.1$). The $I_{+}$ peak locations are consistent with the GPE-predicted nonlinear skew for these velocities [@HELM_PRA_2012] (see also Fig. \[fig:int\_s\]).[]{data-label="fig:mc"}](fig_mc.png){width="\columnwidth"}
We now outline the three-step process of soliton Sagnac-interferometry, common to both (${n_{\mathrm{b}}}=1,2$) configurations; later we will analyse the system phase evolution fully. First, a ground state soliton is split into two secondary solitons, of equal size and a specific relative phase, at a narrow potential barrier \[time ${T_{\mathrm{s}}}$ in [Fig. \[fig:overview\](a)(ii) and (b)(ii)]{}\]. We obtain an equal split by selecting the barrier’s strength $q$ [@Note3] for a given incident velocity $v$ [@loiko_etal_epjd_2014] and barrier width $\sigma$. It is the velocity $v$ in the frame *comoving* with the barrier that must be known; the value of the frame velocity $\Gamma$ (itself due to the angular frequency $\Omega$) does not affect the outcome. In the second step the secondary solitons accumulate a further relative phase difference ${\delta_{\mathrm{S}}}$. This is the $\Omega$-dependent quantity we wish to measure, gained as a result of the differing path lengths travelled by counter propagating waves in a moving frame \[time ${T_{\mathrm{s}}}<t<{T_{\mathrm{c}}}$ in [Fig. \[fig:overview\](a)(ii) and (b)(ii)]{}\]. Finally, the two solitons collide at a narrow barrier \[time ${T_{\mathrm{c}}}$ in [Fig. \[fig:overview\](a)(ii) and (b)(ii)]{}\]. After this collision the wave-function integrals on either side of the barrier, $I_{\pm}=\pm\int_0^{\pm L/2}|\psi(x)|^2 \mathop{dx}$, allow us to determine the value of ${\delta_{\mathrm{S}}}$ [@HELM_PRA_2012; @HELM_PRA_2014], where $I_+$ and $I_-$ are the positive and negative domain populations. These are ideally determined with an atom number variance below one particle, i.e., exact particle counting at output. This is a challenge facing the whole field of atom interferometry, particularly for experiments pursuing Heisenberg-limited measurements. Single atom resolution has been achieved using a variety of techniques [@ockeloen_etal_pra_2010; @muessel_etal_apb_2013; @bucker_etal_njp_2009; @hu_etal_ol_1994; @alt_etal_pra_2003; @serwane_etal_science_2011; @grunzweig_etal_np_2010; @puppe_etal_prl_2007; @gehr_etal_prl_2010; @goldwin_etal_nc_2011] for small numbers ($N\sim 10$) and has recently [@hume_etal_prl_2013] been extended to mesoscopic ensembles ($N\sim 1000$) typical of the output states of the soliton interferometer.
To determine how the Sagnac effect manifests in GPE soliton interferometry, we must describe the phase dynamics more fully. After the initial split at time ${T_{\mathrm{s}}}$, the transmitted soliton (in the positive domain) has peak phase ${\phi_{\mathrm{T}}}(t)$ (value of the phase at the position of the soliton’s peak amplitude), while that reflected (in the negative domain) has peak phase ${\phi_{\mathrm{R}}}(t)$. We wish to determine the phase difference $\Delta$ between the solitons before they collide with one another at a barrier at time ${T_{\mathrm{c}}}$, i.e., $\Delta=(-1)^{{n_{\mathrm{b}}}}[{\phi_{\mathrm{T}}}({T_{\mathrm{c}}}) - {\phi_{\mathrm{R}}}({T_{\mathrm{c}}})]$ \[the prefactor $(-1)^{{n_{\mathrm{b}}}}$ changes the sign of the phase difference to account for the solitons approaching the collisional barrier from different directions depending on the value of ${n_{\mathrm{b}}}$\]. In both cases we choose ${T_{\mathrm{s}}}=L/4v$ (the initial soliton starts at $x=-L/4$). If ${n_{\mathrm{b}}}=1$ the solitons created by the splitting event must both fully circumnavigate the ring before colliding at the same barrier, while for ${n_{\mathrm{b}}}=2$ the solitons only travel half the distance; hence ${T_{\mathrm{c}}}={T_{\mathrm{s}}}+L/{n_{\mathrm{b}}}v$. In the limiting case of a $\delta$-function barrier, the first (splitting) step causes the transmitted soliton to be phase shifted by $\pi/2$ ahead of the (equal amplitude) reflected soliton, as shown analytically in [@HELM_PRA_2014]. We use this figure as an estimate of the phase difference accumulated by splitting on a Gaussian barrier [@HELM_PRA_2012]; see [@POLO_etal_PRA_2013] for a discussion of phase shifts accumulated with finite-width barriers. We select a Gaussian profile for the barrier, as is typical for experimental setups involving off-resonant sheets of light [@Marchant_etal_2013], and take ${\phi_{\mathrm{T}}}({T_{\mathrm{s}}})={\phi_{\mathrm{R}}}({T_{\mathrm{s}}})+\pi/2$. We obtain the phase evolution at the peak of an individual soliton by taking the imaginary part of the exponent of [[Eq. (\[eqn:tw\_sol\])]{}]{} and setting $x=vt$, giving (up to an initial offset) ${\phi_{\mathrm{s}}}(t;v)=[A^2+(\Gamma+v)^2]t/2$. Hence, ${\phi_{\mathrm{R}}}(t)={\phi_{\mathrm{s}}}(t-{T_{\mathrm{s}}};-v)$, ${\phi_{\mathrm{T}}}(t)={\phi_{\mathrm{s}}}(t-{T_{\mathrm{s}}};v)+\pi/2$, and the final phase difference between the solitons is $$\Delta=(-1)^{{n_{\mathrm{b}}}}(2\Gamma L/{n_{\mathrm{b}}}+\pi/2).
\label{eqn:delta_fin}$$
Without a second barrier (${n_{\mathrm{b}}}=1$), the solitons mutually collide at the point antipodal to the splitting barrier. As this occurs in the absence of any axial potentials or barriers, the solitons are unaffected beyond asymptotic shifts to position and phase [@gordon_ol_1983; @zakharov_shabat_1972], given by $$A_j\delta x_j+{i}\delta\phi_j=(-1)^k\ln\left(\frac{A_j+A_k+{i}[v_j-v_k]}{A_j-A_k+{i}[v_j-v_k]}\right)
\label{eqn:asymptotic_2}$$ where $j,k \in \{1,2\}$ and $j\ne k$. The quantities $\delta x_j$ and $ \delta\phi_j$ are asymptotic position and phase shifts associated with the $j$th soliton, while $v_j$ and $A_j$ describe that soliton’s velocity and amplitude. Associating the soliton transmitted through the barrier at time ${T_{\mathrm{s}}}$ with $j=1$, we obtain the correct sign for our asymptotic shifts. In our case, noting that $A_1=A_2=1/4$ we determine the relative phase shift, and the relationship between the position shifts which arise as a result of this collision to be: $$\begin{split}
&{\phi_{\mathrm{C}}}=\delta \phi_2 - \delta \phi_1
=
{\operatorname{Im}}\{\ln(16v^2/[16v^2+1])\}
=0,
\\
&A_j\delta x_j =
[(-1)^{k}/2]\ln(1+1/16v^2)
=-A_k\delta x_k.
\end{split}
\label{eqn:pass}$$ Both results use the standard complex logarithmic identity $\ln(z)=\ln(|z|^2)/2+{i}\arg(z)$. [[Equation (\[eqn:pass\])]{}]{} shows us that ${\phi_{\mathrm{C}}}$ can be omitted from the calculation of $\Delta$, that $\delta x_j\to 0$ rapidly as $v\to \infty$, and also that whatever the size of the asymptotic position shift, the solitons are always shifted by equal amounts in opposite directions, and so will always meet at the collisional barrier situated at $x=0$. Hence, the antipodal collision in the absence of a barrier does not affect the outcome of Sagnac interferometry if we can assume that the solitons’ accelerations during the collision do not affect the Sagnac phase accumulation. The analysis supporting this assumption is beyond the scope of the current work but can be verified numerically. A potential experimental advantage of the single-barrier configuration is that there is no need to locate a second barrier with great precision relative to the first; that both splitting products traverse exactly the same path before recombining is also likely to “smooth over” effects of small asymmetries in the trapping potential.
We can now determine $I_\pm$ by recalling previous results pertaining to soliton collisions at narrow barriers [@HELM_PRA_2012]. Following the same procedure outlined in [@HELM_PRA_2014] we obtain $$I_\pm =[1 \pm (-1)^{{n_{\mathrm{b}}}} {\mathrm{cos}}({\delta_{\mathrm{S}}}-\epsilon)]/2,
\label{eqn:pop_pm}$$ where $\epsilon\to0$ as $v\to\infty$, and $
{\delta_{\mathrm{S}}}=|\Delta| - \pi/2 = 2\Gamma L/{n_{\mathrm{b}}}=\Omega L^2/\pi{n_{\mathrm{b}}}=(m/\hbar)(\Omega_{\mathrm D} L_{\mathrm D}^2/\pi{n_{\mathrm{b}}})
$ is the Sagnac phase we wish to determine. We show results of numerical GPE simulations in [Fig. \[fig:int\_s\](a,b)]{}. For very high velocities, $v\approx4$, the interference follows our prediction [[\[Eq. (\[eqn:pop\_pm\])\]]{}]{} closely, with very small skews arising from nonlinear effects during the final barrier collision, i.e., we can consider $\epsilon\approx0$ in this regime. The ${n_{\mathrm{b}}}=1$ (c,d) and ${n_{\mathrm{b}}}=2$ (a,b) cases have similar structures, however for ${n_{\mathrm{b}}}=1$ the phase varies twice as quickly, as the interrogation time per shot is twice as long. Otherwise, the similarity of the structures supports the assumption that accelerations during barrier-free collisions do not affect the Sagnac phase accumulation. As we reduce the velocity, and the necessary (to avoid complicating nonlinear effects arising from a slow interaction with the barrier) assumption of high initial kinetic energy [@holmer_etal_cmp_2007; @holmer_etal_jns_2007] breaks down, our numerics show that the preceding analysis no longer holds, and so we conclude that Sagnac interferometry is not practicable in the $v\lesssim1$ regime. This is consistent with previous work delimiting $1\geq v\geq 0.25$ as the high-to-low-energy transitional regime [@HELM_PRA_2014], and the results shown here are comparable to those obtained for the Mach–Zehnder configuration [@HELM_PRA_2014].
Figure \[fig:mc\] shows results of Monte Carlo simulations following the methodology described in [@HELM_PRA_2014], which accounts for quantum uncertainty in the initial soliton’s center of mass (CoM) position and velocity by adding Gaussian random offsets to the classical soliton’s initial velocity and peak position. Here we consider a two-barrier system where the soliton is initially accelerated by a harmonic trap, with frequency $\omega_x$ and its minimum at $x=-L/4$. The soliton is prepared and released from a position $x=-L/4-x_0$ (before quantum fluctuations in the CoM are considered). This harmonic trap is then switched off once the soliton reaches $x=-L/4$, and its velocity is $v_0=\omega_x x_0$. The CoM position and velocity uncertainties contribute to velocity uncertainty at the point of collision, giving collisional velocities $v_{\mathrm{b}}$ that follow a Rician distribution [@HELM_PRA_2014]. Increasing $N$ reduces the widths of the outcome distributions by reducing the relative significance of quantum fluctuations, hence making the transmission curves [\[Fig. \[fig:mc\](a-d)(i)\]]{} steeper. As the gradients of these curves asymptote upward, the distributions of the simulation outcomes [\[Fig. \[fig:mc\](a-d)(ii)\]]{} become narrower. The distributions for the ${\delta_{\mathrm{S}}}=\pi/2$ and $3\pi/2$ sets of simulations should, ideally, be centered on $I_{+}=0.5$; these distributions do not have the same location, but approach the ideal ($I_{+}=0.5$) with increasing $v_0$. This is due to the nonlinear skew interfering with the phase evolution during the final collision at time $T_{\mathrm{c}}$, as described in [@HELM_PRA_2012], and predicted by the GPE.
In conclusion, we have employed a GPE treatment to show how, using a moving bright matter-wave soliton as the initial condition, a matter-wave Sagnac interferometer can be realized within a quasi-1D toroidal trapping configuration (ring trap), in combination with one or two narrow Gaussian barriers due to off-resonant sheets of light. Although both configurations are in principle equally effective, we note that the single-barrier case is likely less susceptible to systematics due to small asymmetries in an experimental configuration. We have also explored the effects of quantum fluctuations in the atomic matter-wave’s center-of-mass position and velocity; we find that, so long as the initial soliton velocity is sufficiently fast, particle numbers of $N\gtrsim 1000$ suffice to give sharp transmission responses, which can then be interpreted to deduce a Sagnac phase.
We thank D. I. H. Holdaway, A. L. Marchant, and C. Weiss for useful discussions, and the UK EPSRC (grant no. EP/K030558/1) for support.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
The lack of detailed balance in active colloidal suspensions allows dissipation to determine stationary states. Here we show that slow viscous flow produced by polar or apolar active colloids near plane walls mediates attractive hydrodynamic forces that drive crystallization. Hydrodynamically mediated torques tend to destabilize the crystal but stability can be regained through critical amounts of bottom-heaviness or chiral activity. Numerical simulations show that crystallization is not nucleational, as in equilibrium, but is preceded by a spinodal-like instability. Harmonic excitations of the active crystal relax diffusively but the normal modes are distinct from an equilibrium colloidal crystal. The hydrodynamic mechanisms presented here are universal and rationalize recent experiments on the crystallization of active colloids.\
\
DOI: [10.1103/PhysRevLett.117.228002](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.117.228002)\
author:
- Rajesh Singh
- 'R. Adhikari'
title: Universal hydrodynamic mechanisms for crystallization in active colloidal suspensions
---
In active colloidal suspensions [@palacci2013living; @petroff2015fast], energy is continuously dissipated into the ambient viscous fluid. The balance between dissipation and fluctuation that prevails in equilibrium colloidal suspensions [@einstein1905theory; @kubo1966fluctuation] is, therefore, absent. Nonequilibrium stationary states in active suspensions, then, are determined by both dissipative and conservative forces, quite unlike passive suspensions where detailed balance prevents dissipative forces from determining phases of thermodynamic equilibrium. In this context, it is of great interest to enquire how thermodynamic phase transitions driven by changes in free energy are modified in the presence of sustained dissipation.
In two recent experiments disordered suspensions of active colloids have been observed to spontaneously order into two-dimensional hexagonal crystals when confined at a plane wall. Bottom-heavy synthetic active colloids which catalyze hydrogen peroxide when optically illuminated are used in the first experiment [@palacci2013living] while chiral fast-swimming bacteria of the species *Thiovulum majus* are used in the second experiment [@petroff2015fast]. Given this remarkably similar crystallization in two disparate active suspensions it is natural to ask if the phenomenon is universal and to search for mechanisms, necessarily involving dissipation, that drive it.
Our current understanding of phase separation in particulate active systems is derived from the coarse-grained theory of motility-induced phase separation (MIPS) where active particles are advected by a density-dependent velocity [@tailleur2008statistical; @cates2010arrested; @cates2013active; @cates2015]. Microscopic models with kinematics consistent with MIPS also show phase separation and crystallization of hard active disks have been reported in two dimensions [@henkes2011active; @fily2012athermal; @bialke2012crystallization; @redner2013structure]. However, these models ignore exchange of the locally conserved momentum of the ambient fluid with that of the active particles and are, thus, best applied to systems where such exchanges can be ignored. Fluid flow is an integral part of the physics in [@palacci2013living; @petroff2015fast] and a momentum-conserving theory, currently lacking, is essential to identify the dissipative forces and torques that drive crystallization.
In this Letter we present a microscopic theory of active crystallization that connects directly to the experiments described above. Specifically, we account for the *three-dimensional* active flow in the fluid and the effect of a plane wall on this flow. Representing activity by slip in a thin boundary layer at the colloid surface [@ghose2014irreducible; @singh2014many; @singh2016traction] we rigorously compute the long-ranged many-body hydrodynamic forces and torques on the colloids. Thus we estimate Brownian forces and torques to be smaller than their active counterparts by factors of order $10^{2}$ (for synthetic colloids in [@palacci2013living]) to $10^{4}$ (for bacteria in [@petroff2015fast]) making them largely irrelevant for active crystallization. We integrate the resulting deterministic balance equations numerically to obtain dynamical trajectories.
Our main numerical results are summarized in Fig. (\[fig:Dynamics-of-crystallization\]). Panels (a)-(c) show the spontaneous destabilization of the uniform state by attractive active hydrodynamic forces, the formation of multiple crystallites, and their coalescence into a single hexagonal crystal at late times. Panels (d)-(f) show the structure factor at corresponding times. The route to crystallization is not through activated processes that produce critical nuclei, but through a spinodal-like instability produced by the unbalanced long-ranged active attraction. The uniform state is, therefore, always unstable and crystallization occurs for all values of density, in contrast to the finite density necessary for crystallization in MIPS models [@cates2015]. Active hydrodynamic torques tend to destabilize the ordered state but stability is regained when these are balanced by external torques (from bottom-heaviness in [@palacci2013living]) or by chiral activity (from bacterial spin in [@petroff2015fast]). Crystallites of chiral colloids rotate at an angular velocity that is inversely proportional to the number of colloids contained in them, as shown in panel (g). This is in excellent agreement with the experiment [@petroff2015fast]. The critical values of bottom-heaviness and chirality above which orientational stability, and, hence, positional order, is ensured is shown in panel (h). We now present our model and detail the derivation of our results.
![Panels (a)-(c) are instantaneous configurations during the crystallization of $1024$ active colloids of radius $b$ at a plane wall. The colloids are colored by their initial positions. Panels (d)-(f) show the structure factor $S(\mathbf{k})$ at corresponding instants. Wavenumbers are scaled by the modulus of the reciprocal lattice vector $k_{0}$ and the contribution from $\mathbf{k}=0$ is discarded. Panel (g) shows the variation of the angular velocity $\mathbf{\Omega}_{c}$ of a crystallite with the number $N$ of colloids in it. A typical configuration is shown in the inset. Panel (h) is the state diagram for orientational stability in terms of the measure of chirality $V_{0}^{(3a)}$ and bottom-heaviness $T_{0}$ (see text). Each dot represents one simulation. Here $v_{s}$ is the self-propulsion speed of an isolated colloid, $\tau=b/v_{s}$, and $\epsilon$ is the scale of the repulsive steric potential.\[fig:Dynamics-of-crystallization\]](Figure1){width="45.00000%"}
*Model:* We consider $N$ spherical active colloids of radius $b$ near a plane wall with center-of-mass coordinates $\mathbf{R}_{i},$ orientation $\mathbf{p}_{i}$, linear velocity $\mathbf{V}_{i}$, and angular velocity $\boldsymbol{\Omega}_{i}$, where $i=1\ldots N$. Activity is imposed through a slip velocity $\mathbf{v}_{i}^{\mathcal{A}}$ which is a general vector field on the surface $S_{i}$ of the $i$-th colloid satisfying $\int{\bm{\hat{\rho}}_{i}\cdot}\mathbf{v}_{i}^{\mathcal{A}}\,d\text{S}_{i}=0$ [@slipConstraint], where $\boldsymbol{\rho}_{i}$ is the vector from the center of the colloid to a point on its surface. The fluid velocity $\mathbf{v}$ is subject to slip boundary conditions $$\mathbf{v}(\mathbf{R}_{i}+\boldsymbol{\rho}_{i})=\mathbf{V}_{i}+\bm{\Omega}_{i}\times\bm{\rho}_{i}+\mathbf{v}_{i}^{\mathcal{A}}(\bm{\rho}_{i}).\label{eq:slip-RBM-BC}$$ on the colloid surfaces, to a no-slip boundary condition $\mathbf{v}=0$ at the plane wall located at $z=0$, and to a quiescent boundary condition at large distances from the wall. The slip is conveniently parametrized by an expansion $\mathbf{v}^{\mathcal{A}}(\mathbf{R}_{i}+\boldsymbol{\rho}_{i})=\sum_{l=1}^{\infty}\tfrac{1}{(l-1)!(2l-3)!!}\,\mathbf{V}_{i}^{(l)}\cdot\mathbf{Y}^{(l-1)}(\bm{\hat{\rho}}_{i})$ in irreducible tensorial spherical harmonics $\mathbf{Y}^{(l)}(\bm{\hat{\rho}})=(-1)^{l}\rho^{l+1}\bm{\nabla}{}^{(l)}\rho^{-1}$, where $\bm{\nabla}^{(l)}=\bm{\nabla}_{\alpha_{1}}\dots\bm{\nabla}_{\alpha_{l}}$. The expansion coefficients $\mathbf{V}_{i}^{(l)}$ are $l$-th rank reducible Cartesian tensors with three irreducible parts of ranks $l,$ $l-1,$ and $l-2$, corresponding to symmetric traceless, antisymmetric and pure trace combinations of the reducible indices. We denote these by $\mathbf{V}_{i}^{(ls)}$, $\mathbf{V}_{i}^{(la)}$ and $\mathbf{V}_{i}^{(lt)}$ respectively. The leading contributions from the slip, $$\begin{aligned}
\mathbf{v}_{i}^{\mathcal{A}}(\bm{\rho}_{i}) & = & \underbrace{-\mathbf{V}_{i}^{\mathcal{A}}+\tfrac{1}{15}\mathbf{V}_{i}^{(3t)}\cdot\mathbf{Y}^{(2)}}_{\mathrm{achiral,\,polar}}-\underbrace{\tfrac{1}{9}\boldsymbol{\varepsilon}\cdot\mathbf{V}_{i}^{(3a)}\cdot\mathbf{Y}^{(2)}}_{\mathrm{chiral,\thinspace apolar}}\nonumber \\
& + & \underbrace{\mathbf{V}_{i}^{(2s)}\hspace{-0.1cm}\cdot\mathbf{Y}^{(1)}}_{\mathrm{achiral,\,apolar}}\hspace{-0.12cm}-\underbrace{\mathbf{\Omega}_{i}^{\mathcal{A}}\hspace{-0.05cm}\times\hspace{-0.05cm}\bm{\rho}_{i}-\tfrac{1}{60}\boldsymbol{\varepsilon}\cdot\mathbf{V}_{i}^{(4a)}\hspace{-0.12cm}\cdot\mathbf{Y}^{(3)}}_{\mathrm{chiral,\,polar}}\label{eq:slip-truncation}\end{aligned}$$ have coefficients of polar, apolar and chiral symmetry. Here $\boldsymbol{\varepsilon}$ is the Levi-Civita tensor. The retained modes have physical interpretations: for a single colloid in an unbounded fluid, $\mathbf{V}^{\mathcal{A}}$ ($l\sigma=1s)$ and $\bm{\Omega}^{\mathcal{A}}$ $(l\sigma=2a)$ are the linear and angular velocities in the absence of external forces and torques, $\mathbf{V}^{(2s)}$ is the active contribution to the stresslet, while $\mathbf{V}^{(3a)},\mathbf{V}^{(3t)}$, and $\mathbf{V}^{(4a)}$ are strengths of the chiral torque dipole, polar vector quadrupole, and chiral octupole respectively. The tensors are parametrized uniaxially, $\mathbf{V}_{i}^{\mathcal{A}}=v_{s}\mathbf{p}_{i}$, $\mathbf{\boldsymbol{\Omega}}_{i}^{\mathcal{A}}=\omega_{s}\mathbf{p}_{i}$, $\mbox{\ensuremath{{\bf V}_{i}^{(2s)}=V_{0}^{(2s)}(\mathbf{p}_{i}\mathbf{p}_{i}-\tfrac{\mathbf{I}}{3})}}$ and so on, where $v_{s}$ and $\omega_{s}$ are the speeds of active translation and rotation and $V_{0}^{(2s)}$ positive (negative) corresponds to a pusher (puller). [@supplementalM].
The synthetic active colloids in [@palacci2013living] are polar and achiral (they self-propel but do not spin) while the bacteria in [@petroff2015fast] are polar and chiral (they self-propel and spin). Both these cases are included in the leading contributions. In [@ghose2014irreducible] a procedure is outlined for estimating the leading coefficients from experimentally measured flows and it is shown that the active flow produced by flagellates and green algae can be modeled by slip. Our model is of sufficient generality, then, to include both synthetic and biological active colloids, and situations where swirling and time-dependent slip may be necessary [@drescher2010; @drescher2011; @guasto2010; @goldstein2015green].
{width="90.00000%"}
*Active forces and torques:* Newton’s equations of motion for the colloids reduce, when inertia is negligible, to instantaneous balance of forces and torques $$\mathbf{F}_{i}^{H}+\mathbf{F}_{i}^{P}+\boldsymbol{\xi}_{i}^{T}=0,\quad\mathbf{T}_{i}^{H}+\mathbf{T}_{i}^{P}+\boldsymbol{\xi}_{i}^{R}=0.\label{eq:Newton}$$ Here $\mathbf{F}_{i}^{H}=\int\mathbf{f}\,d\text{S}_{i}$, $\mathbf{F}^{P}$ and $\mathbf{\boldsymbol{\xi}^{T}}$ are respectively the hydrodynamic, body and Brownian forces while, $\mathbf{T}_{i}^{H}=\int\boldsymbol{\rho}_{i}\mathbf{\times}\mathbf{f}\,d\text{S}_{i}$, $\mathbf{T}_{i}^{P}$ and $\boldsymbol{\xi}_{i}^{R}$ are, corresponding torques, $\boldsymbol{\sigma}$ is the Cauchy stress in the fluid and $\mathbf{f}=\bm{\hat{\rho}_{i}}\cdot\boldsymbol{\sigma}$ is the traction. The linearity of the Stokes equation implies that these must be of the form
\[force-formulation\] $$\begin{aligned}
{1}
\mathbf{F}_{i}^{H}= & -\boldsymbol{\gamma}_{ij}^{TT}\mathbf{\cdot V}_{j}-\boldsymbol{\gamma}_{ij}^{TR}\mathbf{\cdot\boldsymbol{\Omega}}_{j}-\sum_{l\sigma=1s}^{\infty}\boldsymbol{\gamma}_{ij}^{(T,\,l\sigma)}\negmedspace\cdot\mathbf{V}_{j}^{(l\sigma)},\label{eq:linear-force-torque}\\
\mathbf{T}_{i}^{H}= & -\boldsymbol{\gamma}_{ij}^{RT}\mathbf{\cdot V}_{j}-\boldsymbol{\gamma}_{ij}^{RR}\mathbf{\cdot\boldsymbol{\Omega}}_{j}-\sum_{l\sigma=1s}^{\infty}\boldsymbol{\gamma}_{ij}^{(R,\,l\sigma)}\negmedspace\cdot\mathbf{V}_{j}^{(l\sigma)},\label{eq:torque-expression}\end{aligned}$$
where repeated particle indices are summed over. The $\boldsymbol{\gamma}_{ij}^{\alpha\beta}$, with $\alpha,\beta=T,R$, are the usual friction matrices associated with rigid body motion and $\mathcal{\boldsymbol{\gamma}}_{ij}^{(\alpha,\,l\sigma)}$ are friction tensors associated with the irreducible modes of the active slip. They are of rank $l+1$, $l$, and $l-1$, respectively, for $\sigma=s,a,t$. The forces and torques depend on relative position (through the $\mathcal{\boldsymbol{\gamma}}_{ij}^{(\alpha,\,l\sigma)}$) and on relative orientation (through the $\mathbf{V}_{j}^{(l\sigma)}$). Their signature under time-reversal shows that the active contributions are dissipative.
We calculate the friction tensors using a Galerkin discretization of the boundary integral equation [@singh2014many; @singh2016traction] with the Lorentz-Blake Green’s function [@blake1971c] which, by construction, vanishes at the plane wall. The $\boldsymbol{\gamma}_{ij}^{(T,\,l\sigma)}$ decay as $r_{ij}^{-(l+1)}$ and $r_{ij}^{-(l+2)}$ in the directions parallel and perpendicular to the wall. The $\boldsymbol{\gamma}_{ij}^{(R,\,l\sigma)}$ decay one power of $r_{ij}$ more rapidly. While the force and torque so obtained are sufficient to study colloidal motion, additional insight is obtained from studying the flow, which we compute from its boundary integral representation. Further details are given in [@supplementalM].
The modes $l\sigma=1s$ and $l\sigma=2a$ contribute most dominantly to forces and torques and they attain their lower bounds far away from the wall, where their magnitudes are $\textcolor{black}{\ensuremath{F=6\pi\eta bv_{s}}}$ and $T=8\pi\eta b^{3}\omega_{s}$. The bacteria in [@petroff2015fast] have radius $b\sim4\,\mu\text{m}$, swimming speed $v_{s}\sim500\,\mu\text{m/s}$ and angular speed $\omega_{s}\sim50\,s^{-1}$ in a fluid of viscosity $\eta=10^{-3}\,\text{kg/ms}$. This gives an estimate of $F\sim40\times10^{-12}\,\text{N}$ and$T\sim10^{-16}$ Nm. For the synthetic colloids in [@palacci2013living], $b\sim2\,\mu\text{m}$, $v_{s}\sim10\,\mu\text{m/s}$, which corresponds to $F_{\mathcal{}}\sim10^{-13}\,\text{N}$. Typical Brownian forces and torques are of order $\mathcal{O}\left(k_{B}\text{T/b}\right)\sim10^{-15}\,\text{N}$, and $\mathcal{O}\left(k_{B}\text{T}\right)\sim10^{-21}\,\text{Nm}$ respectively. Thus active forces and torques overwhelm Brownian contributions by factors of 100 or more in these experiments and, henceforth, we neglect their effects. Trajectories are obtained by integrating the kinematic equations $\mathbf{\dot{R}}_{i}=\mathbf{V}_{i}$ and $\dot{\mathbf{p}}_{i}=\mathbf{\Omega}_{i}\times\mathbf{p}_{i}$, where $\mathbf{V}_{i}$ and $\mathbf{\boldsymbol{\Omega}}_{i}$ satisfy Eq. (\[eq:Newton\]) with Brownian contributions removed. Integration methods and parameter choices are detailed in [@supplementalM].
*Crystallization kinetics:* The kinetics of crystallization obtained from numerical solutions is shown in Movie 1 [@supplementalM], together with the evolution of the structure factor $S(\mathbf{k})$. The uniform state is destabilized, most notably for any initial density, by attractive active hydrodynamic forces. Steric repulsion between particles balances these to produce crystallites with hexagonal positional order. Rings in the structure factor first appear at wavenumbers that correspond to Bragg vectors of the lattice, reminiscent of a spinodal instability, representing the averaged scattering from randomly oriented crystallites. These sharpen into Bragg peaks as the crystallites coalesce and orientational order grows. Finally particles assemble into a single crystallite which continues to rotate, while the structure factor shows a clear sixfold symmetry. In Movie 2 [@supplementalM] we show the formation of a hexagonal unit cell from the simulation of seven polar and chiral active colloids. The crystallite rotates with an angular velocity parallel to the chiral axis of the colloids.
*Universal mechanisms:* To better understand the mechanisms behind active crystallization we show, in Fig. (\[fig:distortion-of-active-flow\]) , the active flow near a wall and the dominant contributions to the flow-mediated forces and torques. The top three panels show the increasing distortion of the flow produced by the leading polar $(l\sigma=1s)$ and apolar $(l\sigma=2s)$ modes for $\mathbf{p}_{i}$ normal to the wall and $V_{0}^{(2s)}<0.$ The flow develops a monopolar character as the colloid is brought to rest at a height $h$ by the balance of hydrodynamic attraction, Fig. (\[fig:distortion-of-active-flow\]d), and steric repulsion from the wall. The induced monopole on the colloids leads to attractive forces between them below a critical height $h$ from the wall as shown in Fig. (\[fig:distortion-of-active-flow\]f). Nearby colloids entrained in this flow are attracted towards the central colloid as shown in the rightmost panel and in Movie 3 [@supplementalM]. The balance of the hydrodynamic attraction and steric repulsion determines the lattice spacing $d$. We note that even an apolar colloid is attracted to the wall, Fig. (\[fig:distortion-of-active-flow\]d), and induces hydrodynamic attractive forces. Thus, unlike MIPS [@cates2015], polarity is not necessary for crystallization. The induced monopole also tends to reorient the colloids, by generating a torque in the plane of wall, as shown by the curved red arrows in Fig. (\[fig:distortion-of-active-flow\]c) and quantified in Fig. (\[fig:distortion-of-active-flow\]g). **[@supplementalM].** Activity *and* body forces pointing away from the wall are necessary for positional order while bottom-heaviness *or* chirality is necessary for orientational stability.
*Harmonic excitations:* We now study harmonic excitations $\mathbf{u}_{i}$ of a perfect hexagonal crystal by expanding the positions as $\mathbf{R}_{i}=\mathbf{R}_{i}^{0}+\mathbf{u}_{i}$ around the stationary state $\mathbf{R}_{i}^{0}=(X_{i}^{0},\,Y_{i}^{0},\,h)$ and ignoring orientational fluctuations. Force balance to leading order gives $$\begin{aligned}
-\boldsymbol{\gamma}_{ij}^{TT}\cdot\dot{\mathbf{u}}_{j} & & +\left(\mathbf{\boldsymbol{\nabla}}_{j}\boldsymbol{\gamma}_{ij}^{TT}\cdot\mathbf{V}^{\mathcal{A}}-\mathbf{D}_{ij}\right)\cdot\mathbf{u}_{j}=0,\label{eq:stability-eq-ac}\end{aligned}$$ where $\mathbf{D}_{ij}=-\boldsymbol{\nabla}_{j}\boldsymbol{\nabla}{}_{i}U\big|_{0}$ and $U$ is the sum of all steric potentials. **This shows that relaxation is determined by both activity and elasticity, unlike in an equilibrium colloidal crystal where elasticity alone relaxes strains. The normal modes of relaxation can be obtained by Fourier transforming in the plane and in time. The dispersion is found from the solutions of** $$\det\Big|-i\omega\boldsymbol{\gamma}_{\mathbf{k}}^{TT}+i\mathbf{k}\,\boldsymbol{\gamma}_{\mathbf{k}}^{TT}\cdot\mathbf{V}^{\mathcal{A}}-\mathbf{D}_{\mathbf{k}}\Big|=0.\label{eq:det-stability}$$ **Here $\mathbf{k}=(k_{1},\,k_{2})$ is the wavevector restricted to the first Brillouin zone [@supplementalM], $\omega$ is the frequency and $\mathbf{D}_{\mathbf{k}}$** is the dynamical matrix**. The pair of dispersion relations for motion parallel to the wall are shown in Fig. (\[fig:Normal-modes\]). The dispersion for $k\ll k_{0}$, where $k_{0}$ is the magnitude of the reciprocal lattice vector, is quadratic in wavenumber $$\omega_{\pm}=-i\tfrac{\gamma_{\perp}^{T}hv_{s}}{2\gamma_{{{\mkern3mu\vphantom{\perp}\vrule depth 0pt\mkern2.5mu\vrule depth 0pt\mkern3mu}}}^{T}}f_{\pm}(\theta)\,k^{2},\label{eq:small-k}$$ where $f_{\pm}(\theta)$ are angular factors, $\gamma_{{{\mkern3mu\vphantom{\perp}\vrule depth 0pt\mkern2.5mu\vrule depth 0pt\mkern3mu}}}^{T}$ and $\gamma_{\perp}^{T}$ are one-body frictions parallel and perpendicular to the wall, and $\tan\theta=\tfrac{k_{2}}{k_{1}}$. The small-$k$ approximation is compared with the numerical solution in Fig. (\[fig:Normal-modes\]) and it is found to hold for $k\lesssim0.1k_{0}$. These can be interpreted as overdamped phonon modes of the active crystal [@joannyDiscuss]. The presence of the active term** $i\mathbf{k}\,\boldsymbol{\gamma}_{\mathbf{k}}^{TT}\cdot\mathbf{V}^{\mathcal{A}}$ in Eq. (\[eq:det-stability\]) makes them differ from phonon modes of a colloidal crystal.
![Branches of the dispersion relation for the two planar normal modes of relaxation of a hexagonal active crystal. The curves in upper panel show the dispersion along high symmetry directions of the Brillouin zone (first inset). The surfaces in the second and third insets show the dispersion over the entire Brillouin zone. Polar plots in the lower panel, have comparisons of full numerical solution of Eq. (\[eq:det-stability\]) with the approximate solution at small $k$ of Eq. (\[eq:small-k\]) for $k=0.01k_{0}$ (left panel) and $k=0.3k_{0}$ (right panel).\[fig:Normal-modes\]](Figure3){width="47.00000%"}
*Discussion*: In this work, we have considered only hydrodynamic forces and torques, unlike the case of MIPS [@tailleur2008statistical; @cates2010arrested; @cates2013active; @cates2015] where Brownian torques drives reorientations [@henkes2011active; @fily2012athermal; @bialke2012crystallization; @redner2013structure]. We have shown that the latter are at least two orders of magnitude weaker than the former for experiments in the class of [@palacci2013living; @petroff2015fast]. However, it is conceivable that thermal fluctuations will play a more significant role when the activity is comparatively weak, modifying both the nature of crystallization transition and the stability of the crystalline phase. The spinodal-like instability appears due to the uncompensated long-ranged attractive active forces. These can be compensated by entropic forces to stabilize the disordered phase at finite temperatures. A nucleational route to crystallization, with activity-enhanced rates, is then possible in the regime where the active forces reduce the nucleation barrier without driving it to zero. In the crystalline phase, thermal fluctuations will excite both phonon and topological modes. Phonon fluctuations will destroy long-range translational order [@peierls1935quelques; @landau1937theorie], but due to the activity-enhanced stiffness of these modes, large system sizes (compared to equilibrium) will be needed to observe the power-law decay of correlations. Topological defects will be excited at higher temperatures and a defect unbinding transition [@kosterlitz1973ordering; @halperin1978theory; @nelson1979theory; @young1979; @chaikin2000principles], modified by activity, may destroy translational order entirely, producing instead an active hexatic phase. These present exciting avenues for future research. We remark that wall-bounded clustering phenomena in algae [@drescher2009] and charged colloids [@squires2001effective] are mediated by specific forms of the universal hydrodynamic mechanisms presented here.
Finally, we suggest that the flow-induced phase separation found here may provide a paradigm, complementary to MIPS, in which theoretical and experimental studies of momentum-conserving driven [@yeo2015collective] and active matter [@trau1996field; @solomentsev1997particle; @matas2014hydrodynamic; @pandey2014flow; @wang2015one; @wykes2016dynamic] may be situated.
We thank M. E. Cates, P. Chaikin, D. Frenkel, D. J. Pine, A. Laskar and T. V. Ramakrishnan for helpful discussions and IMSc for computing resources on the Nandadevi clusters.
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Active force, torque and flow\[sec:propulsion-matrices-in-wall-bounded\]
========================================================================
We derive, in this section, the expressions for the active forces, torques, and exterior flow in a suspension of $N$ active colloids bounded by a plane wall. The system of coordinates is shown in Fig. (\[fig:coordinate-system\]). The spheres are centered at $\mathbf{R}_{i}$ and their velocities and angular velocities are $\mathbf{V}_{i}$ and $\mathbf{\Omega}_{i}$ respectively. ${\bf p}_{i}$ denotes the orientation of the particle while points on the boundaries of the spheres is given by $\mathbf{s}_{i}=\mathbf{R}_{i}+\boldsymbol{\rho}_{i}$, where $\boldsymbol{\rho}_{i}$ is the radius vector. To ensure no-slip on the wall, we associate am image centered at ${\bf R}_{i}^{*}$ with the $i$-th colloid [@blake1971c], and a similar correspondence for all other quantities of the colloid and its image.
We closely follows our previous work [@singh2014many; @singh2016traction] where a boundary integral formulation has been used to solve the Stokes equation with arbitrary boundary conditions. The principal difference here is in the choice of the Green’s function which satisfies the no-slip condition at the plane wall [@blake1971c]. In the interest of being self-contained, we repeat certain key steps en route to the solution. A clear expression of the linearity of Stokes flow is found in its integral representation, where the flow in the bulk is given in terms of integrals of the tractions and velocities at the boundaries [@fkg1930bandwertaufgaben; @ladyzhenskaya1969; @pozrikidis1992; @kim2005], $$\begin{aligned}
{1}
v_{\alpha}(\mathbf{r})= & -\int G_{\alpha\beta}^{W}(\mathbf{r},\,\mathbf{s}_{j})f_{\beta}(\mathbf{s}_{j})\,d\mathrm{S}_{i}\nonumber \\
+ & \int K_{\beta\alpha\gamma}^{W}(\mathbf{r},\,\mathbf{s}_{j})\hat{\rho}_{\gamma}v_{\beta}(\mathbf{s}_{j})\,d\mathrm{S}_{i},\label{eq:BIE}\end{aligned}$$ where repeated particle indices are summed over, $\mathbf{s}_{j}=\mathbf{R}_{j}+\boldsymbol{\rho}_{j}$ is a point on the surface of $j$-th particle and $G_{\alpha\beta}^{W}(\mathbf{r},\,\mathbf{s}_{j})$ is the Green’s function of the Stokes system satisfying no-slip condition, $\mathbf{v}=0$ on the wall at $z=0$. The stress tensor $K_{\alpha\beta\gamma}^{W}(\mathbf{r},\,\mathbf{s}_{j})$ and the pressure vector $P_{\alpha}^{W}$($\mathbf{r},\,\mathbf{s}_{j})$ satisfy $K_{\alpha\beta\gamma}^{W}(\mathbf{r},\,\mathbf{s}_{j})=-\delta_{\alpha\gamma}P_{\beta}^{W}+\eta\left(\nabla_{\gamma}G_{\alpha\beta}^{W}+\nabla_{\alpha}G_{\beta\gamma}^{W}\right)$ and $-\nabla_{\alpha}P_{\beta}^{W}(\mathbf{r},\mathbf{r'})+\eta\nabla^{2}G_{\alpha\beta}^{W}=-\delta\left(\mathbf{r}-\mathbf{r'}\right)\delta_{ij}$ respectively.
We solve the Fredholm integral equation of Eq. (\[eq:BIE\]) by expanding the boundary fields in irreducible tensorial spherical harmonics, $\mathbf{Y}^{(l)}$, which are orthogonal basis function on the surface of the sphere $\frac{1}{4\pi b^{2}}\int\mathbf{Y}^{(l)}(\widehat{\bm{\rho}})\,\mathbf{Y}^{(l')}(\widehat{\bm{\rho}})\,d\mathrm{S}=\delta_{ll'}\,\frac{l!\,(2l-1)!!}{(2l+1)}\mathbf{\Delta}^{(l)},$ where $\mathbf{\Delta}^{(l)}$ is tensor of rank $2l$, projecting any $l$-th order tensor to its symmetric irreducible form [@mazur1982; @hess1980formeln]. The boundary velocity including active slip and its expansion in this basis has been provided above. The orthogonality of the basis functions gives the expansion coefficients in terms of surface integrals of traction and velocity as [@ladd1988; @ghose2014irreducible], $$\begin{aligned}
{1}
\mathbf{F}_{i}^{(l)} & =\frac{1}{(l-1)!(2l-3)!!}\int\mathbf{f}(\mathbf{R}_{i}+\bm{\rho}_{i})\mathbf{Y}^{(l-1)}(\bm{\hat{\rho}}_{i})\,d\mathrm{S}_{i},\nonumber \\
\mathbf{V}_{i}^{(l)} & =\frac{2l-1}{4\pi b^{2}}\int\mathbf{v}^{\mathcal{A}}(\mathbf{R}_{i}+\bm{\rho}_{i})\mathbf{Y}^{(l-1)}(\bm{\hat{\rho}}_{i})\,d\mathrm{S}_{i}.\end{aligned}$$ The coefficients of the traction and velocity are tensors of rank $l$ and can be written as irreducible tensor of rank $l,$ $l-1$ and $l-2$ [@singh2014many]. The first term in the traction expansion is the force $\mathbf{F}_{i}^{(1)}=\mathbf{F}_{i}^{H}$, while the antisymmetric part of the second term is the torque $b\boldsymbol{\varepsilon}\cdot\mathbf{F}_{i}^{(2)}=\mathbf{T}_{i}^{H}$. The first term in the velocity expansion is $\mathbf{V}_{i}^{(1)}=-\mathbf{V}_{i}^{\mathcal{A}}$ and the antisymmetric part of the second term is $\tfrac{1}{2b}\boldsymbol{\boldsymbol{\varepsilon}}\cdot\mathbf{V}_{i}^{(2)}=-\mathbf{\Omega}_{i}^{\mathcal{A}}$. Here $\mathbf{V}_{i}^{\mathcal{A}}=-\tfrac{1}{4\pi b^{2}}\int\mathbf{v}^{\mathcal{A}}(\bm{\rho}_{i})\,d\mathrm{S}_{i}$ denotes the self-propulsion, while $\bm{\Omega}_{i}^{\mathcal{A}}=-\frac{3}{8\pi b^{4}}\int\bm{\rho}_{i}\times\mathbf{v}^{\mathcal{A}}(\bm{\rho}_{i})\,d\mathrm{S}_{i}$ denotes the self-rotation of an isolated active colloid in unbounded flow. The expression for fluid flow can be obtained in terms of coefficients of traction and velocity, $$\begin{aligned}
{1}
\mathbf{v}(\mathbf{r})=\sum_{l=1}^{\infty}\Big(-\boldsymbol{G}_{j}^{(l)}\cdot\mathbf{F}_{j}^{(l)}+ & \boldsymbol{K}_{j}^{(l)}\cdot\mathbf{V}_{j}^{(l)}\Big),\label{eq:fluid-flow}\end{aligned}$$ where the boundary integrals $\boldsymbol{G}_{j}^{(l)}$ and $\boldsymbol{K}_{j}^{(l)}$ can be written in terms of Green’s function and its derivatives (Appendix \[sec:Expression-for-boundary\]). We multiply the fluid velocity by the $l$-th tensorial harmonic and integrate over the $i$-th boundary. Using the orthogonality of these basis functions, we obtain an infinite-dimensional linear system of equations for the unknown traction coefficients [@singh2014many], $$\begin{aligned}
{1}
\tfrac{1}{2}\,\mathbf{V}_{i}^{(l)}=\sum_{l'=1}^{\infty}\Big(-\boldsymbol{G}_{ij}^{(l,\,l')}\cdot\mathbf{F}_{j}^{(l')} & +\boldsymbol{K}_{ij}^{(l,\,l')}\cdot\mathbf{V}_{j}^{(l')}\Big),\label{eq:linear-system}\end{aligned}$$ where the matrix elements $\boldsymbol{G}_{ij}^{(l,\,l')}$ and $\boldsymbol{K}_{ij}^{(l,\,l')}$ can be evaluated in terms of the Green’s function and its derivatives, as given in Appendix \[sec:Expression-for-boundary\].
The traction and velocity coefficients are reducible and their irreducible decomposition is given as [@brunn1976effect; @schmitz1980force], $$\begin{aligned}
\mathbf{F}_{i}^{(ls)}=\overbracket[0.7pt][2.0pt]{\mathbf{F}_{i}^{(l)}},\quad & \mathbf{F}_{i}^{(la)}=\overbracket[0.7pt][2.0pt]{\bm{\varepsilon}\cdot\mathbf{F}_{i}^{(l)}},\quad & \mathbf{F}_{i}^{(lt)}={\bm{\delta}\cdot\mathbf{F}_{i}^{(l)}},\\
\mathbf{V}_{i}^{(ls)}=\overbracket[0.7pt][2.0pt]{\mathbf{V}_{i}^{(l)}},\quad & \mathbf{V}_{i}^{(la)}=\overbracket[0.7pt][2.0pt]{\boldsymbol{\mathbf{\varepsilon}}\cdot\mathbf{V}_{i}^{(l)}},\quad & \mathbf{V}_{i}^{(lt)}={\bm{\delta}\cdot\mathbf{V}_{i}^{(l)}}.\end{aligned}$$ Here the operator $\overbracket[0.7pt][2.0pt]{(\dots)}=\boldsymbol{\Delta}^{(l)}(\dots)$ extracts the symmetric irreducible part of the tensor it acts on. We use these irreducible coefficients and the linear system of equations to solve for the unknown traction in terms of the known boundary velocity [@singh2016traction]. The relations between the irreducible coefficients of the traction and velocity, then, becomes [@singh2016traction], $$\begin{aligned}
{1}
\mathbf{F}_{i}^{(l\sigma)}= & -\boldsymbol{\gamma}_{ij}^{(l\sigma,\,1s)}\cdot\mathbf{V}_{j}-\boldsymbol{\gamma}_{ij}^{(l\sigma,\,R)}\cdot\mathbf{\Omega}_{j}\nonumber \\
- & \sum_{l'\sigma'=1s}^{\infty}\boldsymbol{\gamma}_{ij}^{(l\sigma,\,l'\sigma')}\cdot\mathbf{V}_{j}^{(l'\sigma')}.\label{eq:main-traction-l}\end{aligned}$$ This infinite set of equations, called the traction laws [@singh2016traction], manifestly shows the linear relation between the traction and velocity coefficients and defines the friction tensors. The expressions for the friction tensors can be obtained by an iterative scheme [@singh2016traction]. We use the one-body solution as the initial guess for the iteration, $$\mathbf{F}_{i}^{H}=-\gamma^{T}(\mathbf{V}_{i}-\mathbf{V}_{i}^{\mathcal{A}}),\qquad\mathbf{T}_{i}^{H}=-\gamma^{R}(\mathbf{\Omega}_{i}-\mathbf{\Omega}_{i}^{\mathcal{A}}),$$ where $\gamma^{T}$ and $\gamma^{R}$ are one particle friction corresponding to translation and rotation. Near a wall no-slip wall, they are, ${\gamma}^{T}=\gamma_{\perp}^{T}\hat{\boldsymbol{z}}+\gamma_{{{\mkern3mu\vphantom{\perp}\vrule depth 0pt\mkern2.5mu\vrule depth 0pt\mkern3mu}}}^{T}(\hat{\boldsymbol{x}}+\hat{\boldsymbol{y}})$ and ${\gamma}^{R}=\gamma_{\perp}^{R}\hat{\boldsymbol{z}}+\gamma_{{{\mkern3mu\vphantom{\perp}\vrule depth 0pt\mkern2.5mu\vrule depth 0pt\mkern3mu}}}^{R}(\hat{\boldsymbol{x}}+\hat{\boldsymbol{y}})$ [@kim2005]. Here ${{\mkern3mu\vphantom{\perp}\vrule depth 0pt\mkern2.5mu\vrule depth 0pt\mkern3mu}}$ and $\perp$ subscripts indicating directions parallel and perpendicular to the wall. The expressions after one iteration, corresponding to the “first reflection” in Smoluchowski’s classical method, are shown in Appendix \[appendix:Evaluation-of-gamma\].
![Coordinate system used to describe active spherical particles and its images near a no-slip wall. The $i$-th particle and its image is shown. See text for description. \[fig:coordinate-system\]](SI-Figure1){width="47.00000%"}
Crystalline steady states \[sec:Crystalline-steady-states\]
===========================================================
In this section we work out the steady states of the active crystals using the leading terms of the force and torque equations. Using the leading order force balance for $i$-th particle, the steady state condition for position is given as $$\begin{aligned}
\bm{\gamma}_{ij}^{TT}\cdot\mathbf{V}_{j}^{\mathcal{A}}+\mathbf{F}_{i}^{P}=0.\label{eq:minimal-steady-state}\end{aligned}$$ Here $\mathbf{V}_{i}^{\mathcal{A}}=-v_{s}\hat{\mathbf{z}}$ is self-propulsion of the colloid at a speed $v_{s}$, assumed to be moving $\perp$ to the wall. The body force $\mathbf{F}_{i}^{P}=-\boldsymbol{\nabla}_{{\scriptscriptstyle \mathbf{R}_{i}}}U$ is due to a short-ranged repulsive potential $U,$ which depends on displacement $\mathbf{r}_{ij}=\mathbf{R}_{i}-\mathbf{R}_{j}$ and is given as, $U(r_{ij})=\epsilon\left(\frac{r_{min}}{r_{ij}}\right)^{12}-2\epsilon\left(\frac{r_{min}}{r_{ij}}\right)^{6}+\epsilon,$ for $r_{ij}<r_{min}$ and zero otherwise [@weeks1971role], where $\epsilon$ is the potential strength. The same potential has been used to model colloid-colloid repulsion $\mathbf{F}_{i}^{PP}$ and the colloid-wall repulsive force $\mathbf{F}_{i}^{PW}$.
*One- and two-body dynamics:* To estimate the height at which the particle is brought to rest close to the wall, we use the $z-$component of the force balance, $-\gamma_{\perp}^{T}v_{s}=F_{3}^{PW}$. Here $F_{3}^{PW}$ is the repulsive force from the wall in $\mathbf{\hat{z}}$ direction, while $\gamma_{\perp}^{T}v_{s}$ is the attractive force of the colloid to the wall in same direction. The balance between the attraction and repulsion sets the height $h$ at which the colloid is brought to rest. We now consider force balance for a pair of particles in planar direction, $$\begin{aligned}
{1}
-v_{s}\gamma_{{{\mkern3mu\vphantom{\perp}\vrule depth 0pt\mkern2.5mu\vrule depth 0pt\mkern3mu}}}^{T}\gamma_{\perp}^{T}\mbox{\ensuremath{\mathcal{F}}}_{i}^{0}\mbox{\ensuremath{\mathcal{F}}}_{j}^{0}G_{\alpha3}^{W}(\mathbf{R}_{i},\mathbf{R}_{j})+F_{\alpha}^{PP} & =0,\end{aligned}$$ where $$\mathcal{F}_{i}^{l}=\left(1+\tfrac{b^{2}}{4l+6}\boldsymbol{\nabla}_{{\scriptscriptstyle \mathbf{R}_{i}}}^{2}\right),$$ is an operator encoding the finite size of the sphere and $\alpha$ may takes either of the values 1 or 2 corresponding to two equivalent directions parallel to wall. We have used results provided in Appendix \[appendix:Evaluation-of-gamma\], to write the expression for friction. The solution of this equation gives the lattice spacing $d$. For fixed particle-wall potential, increasing $v_{s}$ decreases the resting height $h$ and separation between *pairs, $d$,* as show in Fig. (\[fig:steady-state\]).
*Rotational dynamics:* In Fig. (\[fig:Dynamics-of-crystallization\]), we show the state diagram, obtained from simulation, which shows that the crystal is stable over a critical strengths of either bottom-heaviness or chirality. For an initially symmetric distribution, a crystal stabilized by external torque alone *does not rotate*, while the crystal *stabilized by chirality does rotate*. When the crystal is rotating at an angular velocity $\mathbf{\Omega}_{c}$ about its center of mass $\mathbf{R}_{c}$, the velocity the $i$-th colloid at position $\mathbf{R}_{i}$ can be then written as $\mathbf{\dot{R}}_{i}=\mathbf{\Omega}_{c}\times\mathbf{R}_{i}$. Force balance parallel to the wall is then $$\begin{aligned}
\bm{\gamma}_{ij}^{TT}\cdot\left[\mathbf{\Omega}_{c}\times(\mathbf{R}_{j}-\mathbf{R}_{0}^{c})\right]+\bm{\gamma}_{ij}^{TR}\cdot\mathbf{\Omega}_{j}=0.\label{eq:minimal-steady-state-1}\end{aligned}$$ The angular speed perpendicular to wall is $\Omega=\Omega_{i}^{\mathcal{A}}$. This implies that in absence of chiral self-rotation there is *no* rotation of the crystal. The angular velocity of the crystal can be obtained by power counting - $\bm{\gamma}_{ij}^{TT}$ scales as $r_{ij}^{-3}$ in direction parallel to wall while $\bm{\gamma}_{ij}^{TR}$ scales as $r_{ij}^{-4}$. The angular velocity of the crystal, then, scales as $\Omega_{c}\propto1/R_{c}^{2}$. In Fig. (\[fig:Dynamics-of-crystallization\]) we show that rotation period of a crystal scales inversely as number of particles $N$ in the crystal for an assembly of chiral particles, which is an excellent agreement with a recent experiment [@petroff2015fast].
![Steady states of active crystallization. Left panel has the plot of leading terms for the analytical solution of height $h$, shown in solid line, along with the full numerical result, shown as dotted curve. Right panel has similar set of plots for lattice spacing $d$. The leading order estimates are found to be in agreement with the numerical solution.\[fig:steady-state\]](SI-Figure2){width="47.00000%"}
Harmonic excitations\[sec:dispersion-relation\]
================================================
In this section we study harmonic excitations $\mathbf{u}_{i}$ of the crystal about a stationary state $\mathbf{R}_{i}^{0}=(X_{i}^{0},\,Y_{i}^{0},\,h)$, such that $\dot{\mathbf{R}}_{i}^{0}=0$ and $\mathbf{\Omega}_{i}^{0}=0$ at this location. The force balance condition is then $\boldsymbol{\gamma}_{ij}^{TT}\cdot\mathbf{V}_{j}^{\mathcal{A}}\big|_{0}+\mathbf{F}_{i}^{P}\big|_{0}=0$. We, now, consider a small displacement about this state $\mathbf{R}_{i}=\mathbf{R}_{i}^{0}+\mathbf{u}_{i}$. Expanding the friction tensors about the stationarity point, we have, $$\boldsymbol{\gamma}_{ij}^{TT}=\boldsymbol{\gamma}_{ij}^{TT}\big|_{0}+(\mathbf{u}_{i}\cdot\mathbf{\boldsymbol{\nabla}}_{{\scriptscriptstyle \mathbf{R}_{i}}}-\mathbf{u}_{j}\cdot\mathbf{\boldsymbol{\nabla}}_{{\scriptscriptstyle \mathbf{R}_{j}}})\mathbf{}\boldsymbol{\gamma}_{ij}^{TT}\big|_{0}+\mathcal{O}(\mathbf{u}^{2}).$$ The force can be expanded in a similar way $\mathbf{F}_{i}^{P}=\mathbf{F}_{i}^{P}\big|_{0}-\mathbf{D}_{ij}\cdot\mathbf{u}_{j},$ where $\mathbf{D}_{ij}=-\boldsymbol{\nabla}_{{\scriptscriptstyle \mathbf{R}_{j}}}\mathbf{\boldsymbol{\nabla}}_{{\scriptscriptstyle \mathbf{R}_{i}}}U\big|_{0}$. Using the equations of motion and considering terms linear in the displacement, the equation becomes $$\begin{aligned}
-\boldsymbol{\gamma}_{ij}^{TT}\cdot\dot{\mathbf{u}}_{j} & & +\left(\mathbf{\boldsymbol{\nabla}}_{{\scriptscriptstyle \mathbf{R}_{j}}}\boldsymbol{\gamma}_{ij}^{TT}\cdot\mathbf{V}^{\mathcal{A}}-\mathbf{D}_{ij}\right)\cdot\mathbf{u}_{j}=0.\end{aligned}$$ We seek a solution of the form $\mathbf{u}_{i}(t)=\mathbf{u}_{\mathbf{k}}(t)\,e^{i\mathbf{k}\cdot\mathbf{R}_{i}}.$ Using this, the force balance condition becomes, $$-\boldsymbol{\gamma}_{\mathbf{k}}^{TT}\cdot\dot{\mathbf{u}}_{k}+\left(i\mathbf{k}\,\boldsymbol{\gamma}_{\mathbf{k}}^{TT}\cdot\mathbf{V}^{\mathcal{A}}-\mathbf{D}_{\mathbf{k}}\right)\cdot\mathbf{u}_{\mathbf{k}}=0.\label{eq:stability-fourier}$$ Here $\mathbf{D}_{\mathbf{k}}$ is the Fourier transform of $\mathbf{D}_{ij}$ and $\boldsymbol{\gamma}_{\mathbf{k}}^{TT}$ is the Fourier transform of the friction tensor
\_ & =\_[i=1]{}\^[N]{}\_[i1]{}e\^[i(\_[i]{}-\_[1]{})]{},\
\_\^[TT]{} & =\_[i=1]{}\^[N]{}\_[i1]{}\^[TT]{}e\^[i(\_[i]{}-\_[1]{})]{}.
Here, $\mathbf{D}_{\mathbf{k}}$ is called the dynamical matrix [@born1954dynamical]. We now write $\boldsymbol{\gamma}_{i1}^{TT}$ in terms of its planar Fourier transform $$\boldsymbol{\gamma}_{i1}^{TT}=\int\hat{\boldsymbol{\gamma}}_{\mathbf{k}}^{TT}(\mathbf{k};\,h)e^{-i\mathbf{k}'\cdot(\mathbf{R}_{i}-\mathbf{R}_{1})}\,\frac{d^{2}k'}{(2\pi)^{2}},$$ to obtain an expression for $\boldsymbol{\gamma}_{\mathbf{k}}^{TT}$, $$\begin{aligned}
\boldsymbol{\gamma}_{\mathbf{k}}^{TT}(\mathbf{k};\,h) & = & \sum_{i}\int\hat{\boldsymbol{\gamma}}_{\mathbf{k}}^{TT}(\mathbf{k};\,h)e^{-i(\mathbf{k}'-k)\cdot(\mathbf{R}_{i}-\mathbf{R}_{1})}\,\frac{d^{2}k'}{(2\pi)^{2}},\nonumber \\
& = & \frac{1}{A_{c}}\sum_{\lambda}\hat{\boldsymbol{\gamma}}_{\mathbf{k}}^{TT}(\mathbf{k}+\mathbf{q}_{\lambda};\,h).\end{aligned}$$ Here we have used the identity $$\sum_{i}e^{-i\mathbf{k}\cdot\mathbf{R}_{i}}=\frac{(2\pi)^{2}}{A_{c}}\sum\mathbf{\boldsymbol{\delta}}(\mathbf{k}-\mathbf{q}_{\lambda}),$$ where $A_{c}$ is area of the unit cell and $\mathbf{q}_{\lambda}$ are reciprocal lattice vectors. We now identify two parts of $\boldsymbol{\gamma}_{\mathbf{k}}^{TT},$ $$\begin{aligned}
{1}
& \boldsymbol{\gamma}_{\mathbf{k}}^{TT}(\mathbf{k};\,h)=\hat{\boldsymbol{\gamma}}_{{\scriptscriptstyle \mathbf{k}_{0}}}^{TT}(\mathbf{k};\,h)+\sum_{\lambda'}\hat{\boldsymbol{\gamma}}_{{\scriptscriptstyle \mathbf{k}_{q}}}^{TT}(\mathbf{k}+\mathbf{q}_{\lambda};\,h)\end{aligned}$$ Here $\hat{\boldsymbol{\gamma}}_{{\scriptscriptstyle \mathbf{k}_{0}}}^{TT}(\mathbf{k};\,h)$ corresponds to the $\mathbf{q}_{\lambda}=0$ and terms at arbitrary non-zero $q$ are denoted by $\hat{\boldsymbol{\gamma}}_{{\scriptscriptstyle \mathbf{k}_{q}}}^{TT}(\mathbf{k};\,h)$. Their leading order forms can be written as $$\begin{aligned}
{1}
\hat{\boldsymbol{\gamma}}_{{\scriptscriptstyle \mathbf{k}_{0}}}^{TT}(\mathbf{k};\,h) & =\gamma^{T}\mathbf{I}+\frac{\gamma^{T}\gamma^{T}}{A_{c}}\mathcal{F}^{k}\,\hat{\mathbf{G}}^{W}(\mathbf{k};\,h),\\
\hat{\boldsymbol{\gamma}}_{{\scriptscriptstyle \mathbf{k}_{q}}}^{TT}(\mathbf{k};\,h) & =\frac{\gamma^{T}\gamma^{T}}{A_{c}}\mathcal{F}^{k}\,\sum_{\lambda'}\hat{\mathbf{G}}^{W}(\mathbf{k}+\mathbf{q}_{\lambda};\,h).\end{aligned}$$ Here $\mathcal{F}^{k}=1-b^{2}k^{2}/3$ and $\hat{\mathbf{G}}^{W}(\mathbf{k};\,h)$ is the two-dimensional Fourier transform of $\mathbf{G}^{W}$ (see Appendix \[appendix:FT-GW\]). The prime on the summation on the right indicates that $\lambda=0$ is excluded from the sum. We now turn to the calculation of the dynamical matrix,
\_ & =\_[i=1]{}\^[N]{}(U’+U”)\_[0]{}(1-e\^[i\_[i]{}]{}).
Here $U'=-12\epsilon\,\big[\left(r_{min}/d\right)^{12}-\left(r_{min}/d\right)^{6}\big]$ and $U''=12\epsilon\,\big[14\left(r_{min}/d\right)^{12}-8\left(r_{min}/d\right)^{6}\big]$. We evaluate the above in the nearest neighbor approximation in the direction parallel to the wall. The expression for $\mathbf{D}_{\mathbf{k}}$ and $\boldsymbol{\gamma}_{\mathbf{k}}^{TT}$ can be evaluated numerically by summing over the reciprocal lattice vectors. The sum is unconditionally and rapidly convergent as the Green’s function decays as $r_{ij}^{-3}$ in the direction parallel to the wall. The dispersion is obtained numerically from Eq. (\[eq:stability-fourier\]) and is shown in Fig. (\[fig:Normal-modes\]).
*Long-wavelength approximation:* Analytical expression for the normal modes can be obtained in the $k\rightarrow0$ limit. Keeping terms of the $\mathcal{O}(k^{2})$, Eq. (\[eq:stability-fourier\]) becomes $$\begin{aligned}
\,\left(\begin{array}{c}
\dot{u}_{k_{1}}\\
\dot{u}_{k_{2}}
\end{array}\right) & = & -\frac{h\gamma_{\perp}^{T}v_{s}\,k^{2}}{\gamma_{{{\mkern3mu\vphantom{\perp}\vrule depth 0pt\mkern2.5mu\vrule depth 0pt\mkern3mu}}}^{T}}\boldsymbol{f}(\theta)\left(\begin{array}{c}
u_{k_{1}}\\
u_{k_{2}}
\end{array}\right),\end{aligned}$$ $$\boldsymbol{f}(\theta)=\left(\begin{array}{cc}
c_{1}\cos^{2}\theta+\mathcal{C}_{2}\sin^{2}\theta & c_{3}\sin\theta\cos\theta\\
c_{3}\sin\theta\cos\theta & c_{4}\sin^{2}\theta+c_{5}\cos^{2}\theta
\end{array}\right).$$ Here $k_{1}=k\cos\theta$, $k_{2}=k\sin\theta$ and $c_{i}$ are positive constants that can be determined in terms of the parameters of the steric potential and the friction tensors: $c_{1}=\gamma_{\perp}^{T}h/2\eta A_{c}+\left(\tfrac{3}{2}U'+\tfrac{9}{8}U''+\right)/h\gamma_{{{\mkern3mu\vphantom{\perp}\vrule depth 0pt\mkern2.5mu\vrule depth 0pt\mkern3mu}}}^{T}v_{s},$ $c_{2}=\left(\tfrac{3}{2}U'+\tfrac{3}{8}U''+\right)/h\gamma_{{{\mkern3mu\vphantom{\perp}\vrule depth 0pt\mkern2.5mu\vrule depth 0pt\mkern3mu}}}^{T}v_{s},\,$ $c_{3}=\gamma_{\perp}^{T}h/2\eta A_{c}+\tfrac{3}{4}U''/h\gamma_{{{\mkern3mu\vphantom{\perp}\vrule depth 0pt\mkern2.5mu\vrule depth 0pt\mkern3mu}}}^{T}v_{s}$, $c_{4}=\gamma_{\perp}^{T}h/2\eta A_{c}+3c_{5}$ and $c_{5}=\left(\tfrac{1}{2}U'+\tfrac{3}{8}U''+\right)/h\gamma_{{{\mkern3mu\vphantom{\perp}\vrule depth 0pt\mkern2.5mu\vrule depth 0pt\mkern3mu}}}^{T}v_{s}$. We can now diagonalize this matrix equation to obtain the relaxation of the overdamped modes after Fourier transforming in time. The eigenvalues of the resulting equations give the dispersion relation **$$\omega_{\pm}=-i\tfrac{\gamma_{\perp}^{T}hv_{s}}{2\gamma_{{{\mkern3mu\vphantom{\perp}\vrule depth 0pt\mkern2.5mu\vrule depth 0pt\mkern3mu}}}^{T}}f_{\pm}(\theta)\,k^{2},\label{eq:small-k-1}$$ in terms of an angular factor,** $f_{\pm}(\theta)=c_{15}\cos^{2}\theta+c_{24}\sin^{2}\theta\pm[(c_{15}\cos^{2}\theta+c_{24}\sin^{2}\theta)^{2}-4(c_{1}c_{5}\cos^{4}\theta+c_{2}c_{4}\sin^{4}\theta-c_{3}^{2}\cos^{2}\theta\sin^{2}\theta+(c_{1}c_{4}+c_{2}c_{5})\cos^{2}\theta\sin^{2}\theta)]{}^{1/2}$, with $c_{15}=c_{1}+c_{5}$ and $c_{24}=c_{2}+c_{4}$. The comparison of the long wavelength solution with the full numerical solution has been plotted in Fig. (\[fig:Normal-modes\]). For **$k\lesssim0.1k_{0}$,** the approximate solution shows excellent agreement.
Numerical method
================
In this section, we outline the method used to simulate the dynamics of active colloidal particles near a no-slip wall. We invert Eq. (\[eq:main-traction-l\]) to obtain rigid body motions in terms of the known slip modes, body forces and torques [@singh2016traction]. This gives the “mobility” formulation, $$\begin{aligned}
{1}
\mathsf{\mathbf{V}}_{i} & =\boldsymbol{\mu}_{ij}^{TT}\cdot\mathbf{F}_{j}^{P}+\boldsymbol{\mu}_{ij}^{TR}\cdot\mathbf{T}_{j}^{P}+\sum_{l\sigma=2s}^{\infty}\boldsymbol{\pi}_{ij}^{(T,l\sigma)}\cdot\mathbf{\mathsf{\mathbf{V}}}_{j}^{(l\sigma)}+\mathsf{\mathbf{V}}_{i}^{\mathcal{A}},\\
\mathsf{\mathbf{\Omega}}_{i} & =\boldsymbol{\mu}_{ij}^{RT}\cdot\mathbf{F}_{j}^{P}+\boldsymbol{\mu}_{ij}^{RR}\cdot\mathbf{T}_{j}^{P}+\sum_{l\sigma=2s}^{\infty}\boldsymbol{\pi}_{ij}^{(R,\,l\sigma)}\cdot\mathbf{\mathsf{\mathbf{V}}}_{j}^{(l\sigma)}+\mathsf{\mathbf{\Omega}}_{i}^{\mathcal{A}}.\end{aligned}$$ The mobility matrices $\boldsymbol{\mu}_{ij}^{\alpha\beta}$, with ($\alpha,\beta=T,R$), are inverses of the friction matrices $\boldsymbol{\gamma}_{ij}^{\alpha\beta}$ [@kim2005]. The propulsion tensors $\boldsymbol{\pi}_{ij}^{(\alpha,\,l\sigma)}$, first introduced in [@singh2014many], relate the rigid body motion to modes of the active velocity. They are related to the slip friction tensors by [@singh2016traction], $$\begin{aligned}
{1}
-\bm{\pi}_{ij}^{(\text{T},\,l\sigma)} & =\boldsymbol{\mu}_{ik}^{TT}\cdot\boldsymbol{\gamma}_{kj}^{(T,\,l\sigma)}+\boldsymbol{\mu}_{ik}^{TR}\cdot\boldsymbol{\gamma}_{kj}^{(R,\,l\sigma)},\\
-\bm{\pi}_{ij}^{(R,\,l\sigma)} & =\boldsymbol{\mu}_{ik}^{RT}\cdot\boldsymbol{\gamma}_{kj}^{(T,\,l\sigma)}+\boldsymbol{\mu}_{ik}^{RR}\cdot\boldsymbol{\gamma}_{kj}^{(R,\,l\sigma)}.\end{aligned}$$ We retain modes corresponding to $l\sigma=1s,\,2s,2a,\,3a,\,3t$ and $4a$ in the active slip. The role of these individual modes is summarized in Table (\[tab:slip-mode-verbose\]). The mobilities are calculated using the PyStokes [@pystokes] library. The initial distribution of particles is chosen to be the random packing of hard-spheres [@skoge2006packing]. We use an adaptive time step integrator using the backward differentiation formula (BDF) to integrate these equations of motion [@langtangen2012tutorial]. In Table (\[tab:Table-of-simulation-parameters\]) of Appendix, we present the parameters used to generate the figures.
**
[|>b[1.4cm]{}|>b[2cm]{}|>b[2cm]{}|>b[2cm]{}|]{} Slip mode & Positional clustering & Orientational stability & Cluster rotation [\
]{} $\mathbf{V}^{a}$ & Yes & No & No[\
]{} $\mathbf{\Omega}^{a}$ & No & Yes & No[\
]{} $\mathbf{V}^{(2s)}$ & Yes & No & No[\
]{} $\mathbf{V}^{(3t)}$ & Yes & No & No[\
]{} $\mathbf{V}^{(3a)}$ & No & Yes & Yes[\
]{} $\mathbf{V}^{(4a)}$ & No & Yes & Yes[\
]{}
[|>b[2cm]{}|c|>p[2cm]{}|>p[2cm]{}|>p[2cm]{}|c|c|]{} Figure & \# of colloids & $v_{s}$ & $V_{0}^{(3a)}/v_{s}$ & $T_{0}/\epsilon$ & Wall WCA & Inter-particle WCA[\
]{} 1 (1-f) & 1024 & 0.1 & 100 & 0 & $r_{min}=3.4b$,$\,$$\epsilon=0.083$ & $r_{min}=5b$,$\,$$\epsilon_{p}=0.004$[\
]{} 1 (g) & - & 0.01 & 100 & 0 & $r_{min}=3.4b$,$\,$$\epsilon=0.083$ & $r_{min}=5b$,$\,$$\epsilon_{p}=0.004$[\
]{} 1 (h) & 256 & 0.01 & - & - & $r_{min}=3.4b$,$\,$$\epsilon=0.083$ & $r_{min}=5b$,$\,$$\epsilon_{p}=0.004$[\
]{} 2 & 1 and 2 & 0.1 & 100 & - & - & -[\
]{} 3 & - & 0.1 & - & - & - & $r_{min}=5b$,$\,$$\epsilon_{p}=0.004$[\
]{}
Expression for boundary integrals and matrix elements\[sec:Expression-for-boundary\]
====================================================================================
The boundary integrals in fluid flow, Eq. (\[eq:fluid-flow\]), can be solved exactly. The resulting solution is given in terms of the Green’s function, Eq. (\[eq:wall-G\]), and its derivatives [@singh2014many], $$\begin{aligned}
{1}
\boldsymbol{G}_{j}^{(l)}(\mathbf{r},\mathbf{R}_{j}) & =\frac{2l-1}{4\pi b^{2}}\int\mathbf{G}^{W}(\mathbf{r},\mathbf{R}_{j}+\bm{\rho}_{j})\mathbf{Y}^{(l-1)}(\hat{\bm{\rho}}_{j})\,d\mathrm{S}_{i}=b^{l}\mathcal{F}^{l-1}\mathbf{\bm{\nabla}}_{{\scriptscriptstyle \mathbf{R}_{j}}}^{(l-1)}\mathbf{G}^{W}(\mathbf{r},\mathbf{R}_{j}),\\
\boldsymbol{K}_{j}^{(l)}(\mathbf{r},\mathbf{R}_{j}) & =\frac{1}{(l-1)!(2l-3)!!}\int\mathbf{K}^{W}(\mathbf{r},\mathbf{R}_{j}+\bm{\rho}_{j})\cdot\mathbf{n}\mathbf{Y}^{(l-1)}(\hat{\bm{\rho}}_{j})\,d\mathrm{S}_{i}=\frac{4\pi b^{l+1}}{(l-2)!(2l-1)!!}\mathcal{F}^{l-1}\mathbf{\bm{\nabla}}_{{\scriptscriptstyle \mathbf{R}_{j}}}^{(l-2)}\mathbf{K}^{W}(\mathbf{r},\mathbf{R}_{j}).\end{aligned}$$ The integrals appearing in the linear system of the equations, Eq. (\[eq:linear-system\]), are, $$\begin{aligned}
{1}
\boldsymbol{G}_{ij}^{(l,\,l')}(\mathbf{R}_{i},\mathbf{R}_{j}) & =\frac{(2l-1)(2l'-1)}{(4\pi b^{2})^{2}}\int\mathbf{Y}^{(l-1)}(\hat{\bm{\rho}}_{i})\mathbf{G}^{W}(\mathbf{R}_{i}+\bm{\rho}_{i},\,\mathbf{R}_{j}+\bm{\rho}_{j})\mathbf{Y}^{(l'-1)}(\hat{\bm{\rho}}_{j})\,d\mathrm{S}_{i}\,d\mathrm{S}_{j}\\
\boldsymbol{K}_{ij}^{(l,\,l')}(\mathbf{R}_{i},\mathbf{R}_{j}) & =\frac{2l-1}{4\pi b^{2}\,(l-1)!(2l-3)!!}\int\mathbf{Y}^{(l-1)}(\hat{\bm{\rho}}_{i})\mathbf{K}^{W}(\mathbf{R}_{i}+\bm{\rho}_{i},\,\mathbf{R}_{j}+\bm{\rho}_{j})\cdot\mathbf{n}\mathbf{Y}^{(l'-1)}(\hat{\bm{\rho}}_{j})\,d\mathrm{S}_{i}\,d\mathrm{S}_{j}.\end{aligned}$$ These integrals are solved exactly to give matrix elements in terms of the Green’s function and its derivatives [@singh2014many], $$\begin{aligned}
{1}
\boldsymbol{G}_{ij}^{(l,\,l')}(\mathbf{R}_{i},\mathbf{R}_{j}) & =\begin{cases}
{\displaystyle \mathcal{G}_{ii}^{(l,\,l')}+b^{l+l'-2}\mathcal{F}_{i}^{l-1}\mathcal{F}_{j}^{l'-1}\bm{\nabla}_{{\scriptscriptstyle \mathbf{R}_{i}}}^{(l-1)}\bm{\nabla}_{{\scriptscriptstyle \mathbf{R}_{j}}}^{(l'-1)}\mathbf{G}^{*}(\mathbf{R}_{i},\mathbf{R}_{j});} & \qquad\qquad\qquad\qquad{\displaystyle \qquad\quad}{\displaystyle j=i,}\\
{\displaystyle b^{l+l'-2}\mathcal{F}_{i}^{l-1}\mathcal{F}_{j}^{l'-1}\bm{\nabla}_{{\scriptscriptstyle \mathbf{R}_{i}}}^{(l-1)}\bm{\nabla}_{{\scriptscriptstyle \mathbf{R}_{j}}}^{(l'-1)}\mathbf{G}^{W}(\mathbf{R}_{i},\mathbf{R}_{j});} & \qquad\qquad\qquad\qquad{\displaystyle \qquad\quad{\displaystyle j\neq i},}
\end{cases}\\
\boldsymbol{K}_{ij}^{(l,\,l')}(\mathbf{R}_{i},\mathbf{R}_{j}) & =\begin{cases}
-{\displaystyle \tfrac{1}{2}\delta_{ll'}\bm{\Delta}^{(l-1)}+\frac{4\pi b^{(l+l'-1)}}{(l'-2)!(2l'-1)!!}\mathcal{F}_{i}^{l-1}\mathcal{F}_{j}^{l'-1}\bm{\nabla}_{{\scriptscriptstyle \mathbf{R}_{i}}}^{(l-1)}\bm{\nabla}_{{\scriptscriptstyle \mathbf{R}_{j}}}^{(l'-2)}\mathbf{K}^{*}(\mathbf{R}_{i},\mathbf{R}_{j});}\quad\qquad\qquad & {\displaystyle j=i,}\\
{\displaystyle \frac{4\pi b^{(l+l'-1)}}{(l'-2)!(2l'-1)!!}\mathcal{F}_{i}^{l-1}\mathcal{F}_{j}^{l'-1}\bm{\nabla}_{{\scriptscriptstyle \mathbf{R}_{i}}}^{(l-1)}\bm{\nabla}_{{\scriptscriptstyle \mathbf{R}_{j}}}^{(l'-2)}\mathbf{K}^{W}(\mathbf{R}_{i},\mathbf{R}_{j});}\qquad\qquad\qquad\qquad\qquad & {\displaystyle j\neq i,}
\end{cases}\end{aligned}$$ $$\mathcal{G}_{ii}^{(l,\,l')}=\delta_{ll'}\frac{2l-1}{2\pi b}\int\mathbf{Y}^{(l-1)}(\hat{\bm{\rho}})\left(\mathbf{\bm{I}}-\hat{\bm{\rho}}\hat{\bm{\rho}}\right)\mathbf{Y}^{(l-1)}(\hat{\bm{\rho}})\,d\Omega.$$
First order off-diagonal approximation for friction tensors\[appendix:Evaluation-of-gamma\]
===========================================================================================
The expressions for the friction tensor can be calculated from the solution of the linear system, provided above, using the Jacobi method [@singh2016traction]. The first order approximation to friction tensors used in this work are provide below, $$\begin{aligned}
\Big(\boldsymbol{\gamma}_{ij}^{(TT)} & \Big)^{[1]}= & \gamma^{T}\gamma^{T}\,\mathcal{F}_{i}^{0}\mathcal{F}_{j}^{0}\,\mathbf{G}^{W}(\mathbf{R}_{i},\mathbf{R}_{j}),\quad\Big(\boldsymbol{\gamma}_{ij}^{(RT)}\Big)^{[1]}=\tfrac{1}{2}\gamma^{T}\gamma^{R}\,\boldsymbol{\nabla}_{{\scriptscriptstyle \mathbf{R}_{i}}}\times\mathbf{G}^{W}(\mathbf{R}_{i},\mathbf{R}_{j}),\end{aligned}$$ $$\Big(\boldsymbol{\gamma}_{ij}^{(TR)}\Big)^{[1]}=\tfrac{1}{2}\gamma^{T}\gamma^{R}\,\boldsymbol{\nabla}_{{\scriptscriptstyle \mathbf{R}_{j}}}\times\left(\mathbf{G}^{W}(\mathbf{R}_{i},\mathbf{R}_{j})\right),\quad\Big(\boldsymbol{\gamma}_{ij}^{(RR)}\Big)^{[1]}=\tfrac{1}{4}\gamma^{R}\gamma^{R}\,\boldsymbol{\nabla}_{{\scriptscriptstyle \mathbf{R}_{i}}}\times\left(\boldsymbol{\nabla}_{{\scriptscriptstyle \mathbf{R}_{j}}}\times\mathbf{G}^{W}(\mathbf{R}_{i},\mathbf{R}_{j})\right),$$ $$\Big(\boldsymbol{\gamma}_{ij}^{(T,\,2s)}\Big)^{[1]}=\frac{28\pi\eta b^{2}}{3}\gamma^{T}\,\mathcal{F}_{i}^{0}\mathcal{F}_{j}^{1}\,\boldsymbol{\nabla}_{{\scriptscriptstyle \mathbf{R}_{j}}}\mathbf{G}^{W}(\mathbf{R}_{i},\mathbf{R}_{j}),\quad\Big(\boldsymbol{\gamma}_{ij}^{(R,\,2s)}\Big)^{[1]}=\frac{28\pi\eta b^{2}}{6}\gamma^{R}\,\boldsymbol{\nabla}_{{\scriptscriptstyle \mathbf{R}_{i}}}\times\left(\boldsymbol{\nabla}_{{\scriptscriptstyle \mathbf{R}_{j}}}\mathbf{G}^{W}(\mathbf{R}_{i},\mathbf{R}_{j})\right),$$ $$\Big(\boldsymbol{\gamma}_{ij}^{(T,\,3a)}\Big)^{[1]}=\frac{13\pi\eta b^{3}}{9}\gamma^{T}\,\boldsymbol{\nabla}_{{\scriptscriptstyle \mathbf{R}_{j}}}(\boldsymbol{\nabla}_{{\scriptscriptstyle \mathbf{R}_{j}}}\times\mathbf{G}^{W}(\mathbf{R}_{i},\mathbf{R}_{j})),\quad\Big(\boldsymbol{\gamma}_{ij}^{(R,\,3a)}\Big)^{[1]}=\frac{13\pi\eta b^{3}}{18}\gamma^{R}\boldsymbol{\nabla}_{{\scriptscriptstyle \mathbf{R}_{i}}}\times\left(\boldsymbol{\nabla}_{{\scriptscriptstyle \mathbf{R}_{j}}}(\boldsymbol{\nabla}_{{\scriptscriptstyle \mathbf{R}_{j}}}\times\mathbf{G}^{W}(\mathbf{R}_{i},\mathbf{R}_{j}))\right),$$ $$\Big(\boldsymbol{\gamma}_{ij}^{(T,\,3t)}\Big)^{[1]}=-\frac{4\pi\eta b^{3}}{5}\gamma^{T}\,\boldsymbol{\nabla}_{{\scriptscriptstyle \mathbf{R}_{j}}}^{2}\mathbf{G}^{W}(\mathbf{R}_{i},\mathbf{R}_{j}),\quad\Big(\boldsymbol{\gamma}_{ij}^{(T,\,4a)}\Big)^{[1]}=-\frac{121\pi\eta b^{4}}{10}\gamma^{T}\boldsymbol{\nabla}_{{\scriptscriptstyle \mathbf{R}_{j}}}\boldsymbol{\nabla}_{{\scriptscriptstyle \mathbf{R}_{j}}}(\boldsymbol{\nabla}_{{\scriptscriptstyle \mathbf{R}_{j}}}\times\mathbf{G}^{W}(\mathbf{R}_{i},\mathbf{R}_{j})),$$ $$\Big(\boldsymbol{\gamma}_{ij}^{(R,\,4a)}\Big)^{[1]}=-\frac{121\pi\eta b^{4}}{20}\gamma^{R}\boldsymbol{\nabla}_{{\scriptscriptstyle \mathbf{R}_{i}}}\times\left(\boldsymbol{\nabla}_{{\scriptscriptstyle \mathbf{R}_{j}}}\boldsymbol{\nabla}_{{\scriptscriptstyle \mathbf{R}_{j}}}(\boldsymbol{\nabla}_{{\scriptscriptstyle \mathbf{R}_{j}}}\times\mathbf{G}^{W}(\mathbf{R}_{i},\mathbf{R}_{j}))\right),\quad\Big(\boldsymbol{\gamma}_{ij}^{(R,\,3t)}\Big)^{[1]}=0.$$
Fourier transform of the Lorentz-Blake Green’s function\[appendix:FT-GW\]
=========================================================================
In this section, we derive the Fourier transform of the Green’s function for a fluid flow bounded by a plane infinite wall. Blake [@blake1971c] has derived the Green function of the Stokes equation which satisfies no-slip condition on the wall, $$\begin{aligned}
{1}
G_{\alpha\beta}^{\text{w}}(\mathbf{R}_{i},\,\mathbf{R}_{j}) & =G_{\alpha\beta}(\mathbf{R}_{i},\,\mathbf{R}_{j})+G_{\alpha\beta}^{*}(\mathbf{R}_{i},\,\mathbf{R}_{j}^{*})=G_{\alpha\beta}(\mathbf{R}_{i},\,\mathbf{R}_{j})-G_{\alpha\beta}(\mathbf{R}_{i},\,\mathbf{R}_{j})+G'_{\alpha\beta}(\mathbf{R}_{i},\,\mathbf{R}_{j})\label{eq:wall-G}\\
& =G_{\alpha\beta}(\mathbf{r}_{ij}^{*})-G_{\alpha\beta}(\mathbf{r}_{ij}^{*})-2h\nabla_{{\scriptscriptstyle \mathbf{r}_{ij}^{*}}}G_{\alpha3}(\mathbf{r}_{ij}^{*})\mathcal{M}_{\beta\gamma}+h^{2}\nabla_{{\scriptscriptstyle \mathbf{r}_{ij}^{*}}}^{2}G_{\alpha\gamma}(\mathbf{r}_{ij}^{*})\mathcal{M}_{\beta\gamma}.\nonumber \end{aligned}$$ Here $\mathbf{r}_{ij}=\mathbf{\mathbf{R}}_{i}-\mathbf{\mathbf{R}}_{j}$, $\mathbf{r}_{ij}^{*}=\mathbf{\mathbf{R}}_{i}-\mathbf{\mathbf{R}}_{j}^{*}$ and $\boldsymbol{\mathcal{M}}=\mathbf{I}-2\mathbf{\hat{z}}\mathbf{\hat{z}}$. $\mathbf{G}$ is the Green’s function in the unbounded fluid flow, $$\mathbf{G}(\mathbf{R}_{i},\,\mathbf{R}_{j})=\frac{1}{8\pi\eta}\left(\frac{\mathbf{I}}{r_{ij}}+\frac{\mathbf{r}_{ij}\mathbf{r}_{ij}}{r_{ij}^{3}}\right).$$ We define the Fourier transform in the plane of the wall as, $$\hat{\varphi}(k_{1},\,k_{2},\,r_{3})=\mathbb{\mathbb{F}}\left[\varphi\right]=\frac{1}{(2\pi)^{2}}\int\varphi(r_{1},\,r_{2},\,r_{3})e^{i(k_{1}r_{1}+k_{2}r_{2})}\,dr_{1}dr_{2}.$$ The Fourier transform of $G'_{ij}$, last term of Eq. (\[eq:wall-G\]), is then [@blake1971c], $$\hat{G}'_{\alpha\beta}(\mathbf{k};\,h)=\frac{h}{2\eta k}\left[ik_{\alpha_{1}}(\delta_{\alpha3}\delta_{j\alpha_{1}}+\delta_{\beta3}\delta_{\alpha\alpha_{1}})+h\left(ikk_{\alpha_{1}}\{\delta_{\alpha3}\delta_{j\alpha_{1}}-\delta_{\beta3}\delta_{\alpha\alpha_{1}}\}-k_{\alpha_{1}}k_{\alpha_{2}}\delta_{\alpha\alpha_{1}}\delta_{\beta\alpha_{2}}-\delta_{\alpha3}\delta_{\beta3}k^{2}\right)\right]e^{-2kh},$$ where $\alpha_{1}$ and $\alpha_{2}$ only take values 1 or 2 corresponding to directions parallel to wall. The rest of terms in Eq. (\[eq:wall-G\]), can be transformed using the relation $\mathbb{\mathbb{F}}\left[\frac{1}{r}\right]=\frac{2\pi\,e^{-kz}}{k}$. The two-dimensional Fourier transform of the wall Green’s function for a source at height $h$ from the wall is then, with $\mathcal{E}=1-e^{-2kh}$, $$\begin{aligned}
\hat{\mathbf{G}}^{W}(\mathbf{k};\,h) & =\frac{1}{4\eta k^{3}}\left(\begin{array}{ccc}
\mathcal{E}k_{2}^{2}+2hkk_{1}^{2}e^{-2kh}\quad & -\mathcal{E}k_{1}k_{2}-2hkk_{1}k_{2}e^{-2kh}\quad & -i2hk^{2}k_{1}e^{-2kh}\\
\\
-\mathcal{E}k_{1}k_{2}-2hkk_{1}k_{2}e^{-2kh}\quad & \mathcal{E}k_{1}^{2}+2hkk_{2}^{2}e^{-2kh}\quad & -i2hk^{2}k_{2}e^{-2kh}\\
\\
-i2hk^{2}k_{1}e^{-2kh}\quad & -i2hk^{2}k_{1}e^{-2kh}\quad & \mathcal{E}k^{2}-2hk^{3}e^{-2kh}
\end{array}\right)+\hat{\mathbf{G}}'(\mathbf{k};\,h).\label{eq:Gw-FT}\end{aligned}$$
{width="82.00000%"}
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The IceCube Neutrino Observatory detects high energy astrophysical neutrinos in two event topologies: tracks and cascades. Since the flavor composition of each event topology differs, tracks and cascades can be used to test the neutrino properties and the mechanisms behind the neutrino production in astrophysical sources. Assuming a conventional model for the neutrino production, the IceCube data sets related to the two channels are in $>3\sigma$ tension with each other. Invisible neutrino decay with lifetime $\tau/m=10^2$ s/eV solves this tension. Noticeably, it leads to an improvement over the standard non-decay scenario of more than $3\sigma$ while remaining consistent with all other multi-messenger observations. In addition, our invisible neutrino decay model predicts a reduction of $59\%$ in the number of observed $\nu_\tau$ events which is consistent with the current observational deficit.'
author:
- 'Peter B. Denton'
- Irene Tamborra
bibliography:
- 'Tracks\_Cascades.bib'
title: 'Invisible Neutrino Decay Resolves IceCube’s Track and Cascade Tension'
---
[*Introduction.*—]{}The IceCube Neutrino Observatory measures high energy astrophysical neutrinos with energies reaching up to few PeVs [@Aartsen:2013bka; @Aartsen:2013jdh; @Aartsen:2016xlq]. While numerous source candidates have been proposed to interpret the observed data, no clear picture has yet emerged [@Anchordoqui:2013dnh; @Meszaros:2015krr; @Waxman:2015ues; @Murase:2015ndr].
According to the conventional framework, adopted in this work, high energy astrophysical neutrinos are produced primarily by charged pion decay. Charged pions decay to a muon and a muon neutrino, and the muon in turn decays on to a positron, electron neutrino, and a muon antineutrino resulting in a neutrino flavor ratio at the source of $\nu_e:\nu_\mu:\nu_\tau = 1:2:0$, each with approximately the same energy. After neutrino oscillations, the flavor ratio at the Earth is roughly $1:1:1$ leading to the expectation that the spectral distributions of neutrinos will be the same for any flavor, see e.g. [@Anchordoqui:2013dnh; @Farzan:2008eg]. This is independent of the source class since any mechanism that produces high energy neutrinos will do so dominantly as a result of charged pion decays. Hence, within this picture, the only possible result is equal fluxes for each flavor.
Single power law (SPL) and broken power law (BPL) fits have been considered to interpret the neutrino data [@Aartsen:2015knd; @Chen:2014gxa; @Palladino:2016zoe; @Anchordoqui:2016ewn; @Palladino:2018evm; @Chianese:2017jfa; @Vincent:2016nut; @Palomares-Ruiz:2015mka; @Sui:2018bbh]. They favor a SPL, with a possible break to explain the excess of events below 100 TeV [@Chianese:2017jfa; @Denton:2018tdj].
IceCube is partially sensitive to the flavor state of the neutrino through two distinct event topologies: track events resulting dominantly from $\nu_\mu$ interactions [@Aartsen:2016xlq], and nearly spherical cascade events resulting dominantly from $\nu_e$ and $\nu_\tau$ interactions [@Niederhausen:2015svt]. The IceCube Collaboration has interpreted each of these data sets in terms of the true per-flavor neutrino flux at the Earth under the assumption that the flavor ratio remains constant at $1:1:1$ for all energies and that the flux follows a SPL [@Aartsen:2015knd]. It is found that those two different channels produce results in tension with each other [@Aartsen:2016xlq], as shown in Fig. \[fig:ICdata\].
![IceCube track [@Aartsen:2016xlq] and cascade [@Niederhausen:2015svt] data samples. The tension between the two data samples is driven on the high energy end by the observation of six tracks with energies $E_\nu>1$ PeV. On the low energy side there is an apparent excess of events in the cascade channel [@Denton:2018tdj].[]{data-label="fig:ICdata"}](Inu_Both){width="\columnwidth"}
IceCube finds that the best fit per-flavor astrophysical spectral index and normalization from the track analysis over $E_\nu \in[194$ TeV, $7.8$ PeV$]$ is $\gamma_{t,{\rm IC}}=2.13\pm0.13$, $\Phi_{t,{\rm IC}}=0.90^{+0.30}_{-0.27}$ [@Aartsen:2016xlq] and the best fit from the cascade analysis over $E_\nu \in[13$ TeV, $7.9$ PeV$]$ is $\gamma_{c,{\rm IC}}=2.67^{+0.12}_{-0.13}$, $\Phi_{c,{\rm IC}}=2.3^{+0.7}_{-0.6}$ [@Niederhausen:2015svt] where $\gamma_i$ is the spectral index and $\Phi_i$ is the flux normalization at $E_\nu = 100$ TeV in units of $10^{-18}$ GeV$^{-1}$ cm$^{-2}$ sr$^{-1}$ s$^{-1}$.
In this Letter, we combine spectral and flavor information simultaneously to investigate the tension between the data sets associated to the two event topologies. We explore several modifications to the standard picture of the high energy astrophysical neutrino flux beyond what is foreseen within the Standard Model [^1].
We determine the diffuse intensity at the Earth after oscillations, convert this into the per-flavor intensity from each of the track and cascade channels, and fit a power law to each assuming a $1:1:1$ flavor ratio to compare a model to IceCube’s observations. We then compare the normalizations and spectral indices to the measured ones by combining both tracks and cascades under the assumption that the correlation between the normalizations and spectral indexes are small. Invisible neutrino decay provides a good fit to the data and is preferred over the Standard Model at more than $3\sigma$ removing the tension. Our proposed solution is not in contradiction with existing multi-messenger constraints and also explains the current deficit in the observation of $\nu_\tau$ events.
[*Standard Neutrino Source Model.*—]{} For the sake of generality, we model the neutrino spectral distribution in such a way to be agnostic about the mechanism of the neutrino production, i.e. $p\gamma$ or $pp$ interactions. We consider a general BPL model at the source parameterized by the break energy in the source frame $\tilde E_{\nu,b}$ and the change in the spectral index $\Delta$, such that the spectral index below the break energy is $\gamma$ and it is $\gamma+\Delta$ above it [@Anchordoqui:2013dnh; @Meszaros:2015krr; @Waxman:2015ues; @Murase:2015ndr; @Meszaros:2001vi]. The SPL case is then recovered for $\Delta = 0$.
This model is further generalized to the case where the break energy for neutrinos coming from muon decay ($\nu_e$ and $\nu_\mu$) is different than that from pion decay ($\nu_\mu$). Pions and muons lose energy in $p\gamma$ sources, e.g. in the presence of magnetic fields due to synchrotron losses and they may have separate break energies, $\tilde E_{\nu,b,\mu}$ and $\tilde E_{\nu,b,\pi}$. For example, for synchrotron losses, the neutrino break energy scales like $m_i^{5/2}\tau_i^{-1/2}$ for $i\in\{\pi,\mu\}$ where $m$ ($\tau$) is the mass (lifetime) of the particle, so the ratio of the neutrino break energies is $R_{\pi,\mu} \equiv \tilde E_{\nu,b,\pi}/\tilde E_{\nu,b,\mu} \simeq 18.4$ when synchrotron cooling dominates. The simpler BPL model introduced above is recovered when $R_{\pi,\mu}=1$. Thus there are at most five free parameters in the BPL model: $\gamma$, $\Delta$, $\tilde E_{\nu,b}$, $R_{\pi,\mu}$, and the neutrino flux normalization $\Phi_\nu$.
The IceCube neutrino flux is considered to be dominantly extragalactic and compatible with a diffuse origin [@Denton:2017csz; @Aartsen:2017ujz; @Ando:2015bva; @Aartsen:2015knd]. Hence, the expected diffuse neutrino intensity at the Earth for the flavor $\nu_\beta$ ($\beta=e,\mu,\tau$) is $$\mathcal{I}_{\nu_\beta}=\sum_{\nu_\alpha} d_H\int_0^{z_{\max}}dz\frac{F_{\nu_\alpha}((1+z)E_\nu)\rho(z)}{h(z)}\bar P(\nu_\alpha \rightarrow \nu_\beta)\,,$$ where $d_H=c/H_0$, $h(z)=\sqrt{(1+z)^3\Omega_m+\Omega_\Lambda}$, with $\Omega_m=0.308$, $\Omega_\Lambda=1-\Omega_m$, and $H_0=67.8$ km s$^{-1}$ Mpc$^{-1}$ [@Ade:2015xua]. For the redshift evolution $\rho(z)$, we assume as benchmark case that the source luminosity density evolves as $(1+z)^\theta$ for $\theta=3$ up to a certain $z_c \simeq 1.5$ and it is constant for $z>z_c$ [@Gruppioni:2013jna]. Different redshift scalings for $\theta \in [0,4]$ and $z_c \in [0.5, 2]$ do not significantly affect our conclusions. The averaged oscillation probability is $\bar P(\nu_\alpha \rightarrow \nu_\beta) = \sum_i|U_{\alpha i}|^2|U_{\beta i}|^2$ where $U$ is the standard mixing matrix [@Maki:1962mu; @Pontecorvo:1967fh]. For the mixing angles we take the latest global fit results [@Esteban:2016qun; @nu-fit:v3.2]. The per-flavor flux from the source, $F_{\nu_\alpha}$, is either a SPL or a BPL.
We then compute the corresponding per-flavor intensity expected in the two event topologies; the track intensity roughly corresponds to the $\nu_\mu$ one, while the cascade one corresponds to the $\nu_e+\nu_\tau$ one (see the Appendix for technical details). A scan over all possible values of each model parameter is done to compare with the IceCube neutrino data through a $\chi^2$ test: $$\chi^2=\sum_{i\in\{t,c\}}\left(\frac{\Phi_i-\Phi_{i,{\rm IC}}}{\sigma_{\Phi_{\nu,i}}}\right)^2+\left(\frac{\gamma_i-\gamma_{i,{\rm IC}}}{\sigma_{\gamma_i}}\right)^2\,;
\label{eq:chisq}$$ where the sum runs on both neutrino event topologies, ($\Phi_i$, $\gamma_i$) are the normalization and spectral indices at the Earth which come from our calculations, and ($\Phi_{i,{\rm IC}}$, $\gamma_{i,{\rm IC}}$) fit the IceCube data. For the SPL case with two free parameters ($\Phi_\nu$, $\gamma$) we find $\chi^2=13.4$ which corresponds to $3.23\sigma$ of tension. When we expand the source model to the BPL case with four free parameters ($\gamma$, $\Delta$, $\tilde E_{\nu,b}$, $\Phi_\nu$) and $R_{\pi,\mu} = 1$, we find that the $\chi^2$ does not improve which results in $>3.66\sigma$ tension. That is the BPL case is not preferred by the data with respect to the SPL. In addition, letting $R_{\pi,\mu}$ float freely only improves the fit to $\chi^2=10.7$ which is disfavored at $>3.27\sigma$ and provides only marginal improvement ($1.64\sigma$) over the BPL case. In this case, the best fit point has $R_{\pi,\mu}>100$ and $\Delta$ large, similar to a damped muon source.
Our findings confirm that adding a break to the source spectra provides marginal improvement to the data fit and that a SPL fit is justified. Most importantly, the standard neutrino source scenario is disfavored at $>3.2\sigma$ by the IceCube track and cascade data (see the left columns of Table \[tab:chisq\] for a summary). While muon cooling does provide both an energy and flavor dependent effect, it is not enough to resolve the tension due to the large mixing angles. We expect that any mechanism which increases the relative number of $\nu_\mu$’s at the source (such as muon damping from synchrotron cooling) at high energy will equally increase the relative number of $\nu_\tau$’s after oscillations since $\theta_{23}\sim45^\circ$ is minimizing the effect.
[*Invisible Neutrino Decay.*—]{}To solve the tension between the fits provided by the two event topologies, an interesting model modifying the flavor ratio in an energy dependent fashion during propagation is neutrino decay [@Beacom:2002vi; @Shoemaker:2015qul; @Moss:2017pur; @Choubey:2017dyu]. The latter is described by a new interaction term: $\mathcal L\supset g_{ij}\nu_i\nu_j\phi$ where $\phi$ is a new light ($m_\phi\lesssim m_\nu$) or massless scalar known as the Majoron, which could provide neutrinos with their masses [@Acker:1991ej; @Gelmini:1980re; @Chikashige:1980ui]. Specifically, we here focus on the invisible decay scenario where the decay products are a Majoron and a right handed neutrino (left handed antineutrino) [@Gelmini:1980re; @Chikashige:1980ui]; another model of invisible neutrino decay is to unparticles [@Georgi:2007ek; @Zhou:2007zq]. Depending on the mass ordering and absolute mass scale, the decay products of visible neutrino decay may have significantly less energy. For a steeply falling spectrum ($\gamma\gtrsim2$), visible decay becomes effectively invisible.
We assume that $\nu_1$ is stable since it has the least $\nu_\mu$ fraction since this can suppress the $\nu_\mu$ fraction at low energies. This may be the case if the mass ordering is normal, as is currently favored at $2-3.4\sigma$ [@Esteban:2016qun; @nu-fit:v3.2; @deSalas:2017kay; @globalfit; @Capozzi:2018ubv], and the Majoron has a mass between $\nu_1$ and $\nu_2$, or if $\nu_1$ is massless (or very light) and has no (significant) coupling to the Majoron.
The oscillation averaged probability is $$\bar P(\nu_\alpha\to\nu_\beta)=\sum_{i=1}^3|U_{\alpha i}|^2|U_{\beta i}|^2e^{-\Lambda_i}\,,$$ where $\Lambda_i\equiv d_H f(z)m_i/E_\nu\tau_i$ and $f(z)=\int_0^zdz'(1+z')^{-2}h^{-1}(z')$ is the corrected cosmological distance scaling for neutrino decay [@Baerwald:2012kc]. We take $\Lambda_1=0$ and $\Lambda_2=\Lambda_3$; $\tau/m$, identical for $\nu_2$ and $\nu_3$, is our free parameter.
![The track to cascade ratio as a function of the neutrino energy. The invisible neutrino decay of $\nu_2$ and $\nu_3$ reduces the track and cascade ratio below 1 PeV up to $75\%$ with respect to the case where all neutrinos are stable. The deviation from the expected value of 0.5 for the standard case is mostly due to track misidentification wherein track events are sometimes misidentified as cascades (see the Appendix).[]{data-label="fig:nudecay"}](Rtc){width="\columnwidth"}
Figure \[fig:nudecay\] shows the modification of the track vs. cascade ratio due to invisible neutrino decay within the model introduced above. One can check that in order to have an effect ($\Lambda_2,\Lambda_3\sim1$) within the region of interest of IceCube, we should have $\tau/m \sim10^2$ s/eV.
Minimizing the $\chi^2$ in the SPL only case with neutrino decay, we find $\chi^2=1.57$ with $\log_{10}[({\tau/m})/({\rm s/eV})]=1.93^{+0.26}_{-0.40}$. At 1 d.o.f. this represents a good fit, consistent with the data at $1.25\sigma$. It is an improvement over the stable neutrino case of $\Delta\chi^2=11.8$ showing that the neutrino decay scenario is preferred by the data over the standard stable neutrino case by $3.4 \sigma$. The 2D $\chi^2$ projection of the source spectral index $\gamma$ and the neutrino lifetime $\tau/m$ is shown in Fig. \[fig:Triangle\]. We note that $\tau/m$ is fairly well determined since it must give observable consequences within IceCube’s region of interest. Varying the redshift evolution power $\theta$ produces a fairly small effect with the best fit value of $\tau/m$ and the $\chi^2$ slightly changes with $\tau/m$ increasing with $\theta$. If we extend our fit to the BPL source model, the best fit point does not change at all and $\Delta=0$ is preferred, see Table \[tab:chisq\] [^2].
![The 2D $\chi^2$ projection for neutrino decay with a single power law astrophysical flux. The shaded regions represent $1,2,3$ $\sigma$ for 2 d.o.f. The best fit point of $\gamma=2.73$ and $\log_{10}[(\tau/m)/($s/eV$)]=1.93$, indicated with the dot, has $\chi^2=1.57$. This includes a marginalization over the source normalization. The slight preference for the full decay case over the $\nu$SM is because it modifies the relative normalization of the track and cascade diffuse intensities.[]{data-label="fig:Triangle"}](Projection2D){width="\columnwidth"}
Our findings should be compared with existing bounds on invisible neutrino decay. The best terrestrial constraints on invisible $\nu_3$ decay come from atmospheric and long-baseline data: $\log_{10}[({\tau_3/m_3})/({\rm s/eV})]>-9.52$ [@GonzalezGarcia:2008ru; @Pagliaroli:2016zab]; the best terrestrial constraints on invisible $\nu_2$ decay are from solar neutrinos and are $\log_{10}[(\tau_2/m_2)/({\rm s/eV})]>-3.15$ [@Berryman:2014qha; @Picoreti:2015ika]. Hints for $\nu_3$ invisible decay exist at $\log_{10}[(\tau_3/m_3)/({\rm s/eV})]\sim-11$ [@Gomes:2014yua; @Choubey:2018cfz].
-------------------------------------- -------------- --------- ------------------------ ------------------------
SPL BPL SPL BPL
$\chi^2$ 13.4 13.4 1.57 1.57
$\sigma$ $3.23$ $>3.65$ $1.25$ $>1.25$
$\gamma$ $2.4\pm0.10$ - $2.73\pm0.10$ -
$\log_{10}(\frac{\tau/m}{\rm s/eV})$ - - $1.93^{+0.26}_{-0.40}$ $1.93^{+0.26}_{-0.40}$
-------------------------------------- -------------- --------- ------------------------ ------------------------
: The $\chi^2$ and significance for the single power law (SPL) and broken power law (BPL) models, along with the best fit source spectral index and neutrino lifetime. Here we fix $R_{\pi,\mu}=1$ for the BPL model, see text. The BPL models have as many or more parameters than data points; only a lower limit on the significance can be placed by taking 1 d.o.f.[]{data-label="tab:chisq"}
Strong constraints, in apparent contradiction with our findings, have been derived from SN 1987A: $\log_{10}[({\tau/m})/({\rm s/eV})]\gtrsim 5$ [@Hirata:1987hu]; however these constraints only apply to $\bar{\nu}_e$ measurements under the assumption that all neutrino mass eigenstates are decaying and should be considered with caution. Even in the case of full $\nu_2$ and $\nu_3$ decay, the $\bar{\nu}_e\to\bar{\nu}_e$ oscillation averaged probability would be suppressed by 16% which is still smaller than the SN 1987A statistical uncertainties ($\sim20\%$) and current theoretical uncertainties. IceCube data has been used to place a constraint on the neutrino lifetime at $\log_{10}[({\tau/m})/({\rm s/eV})] \gtrsim1$ by assuming that neutrinos do not fully decay within the IceCube energy range [@Pagliaroli:2015rca; @Bustamante:2016ciw] which is not the case considered here. The most stringent constraints on the lifetime of neutrinos have been derived from cosmic microwave background data at the level of $\log_{10}[({\tau/m})/({\rm s/eV})] \gtrsim 11$ [@Hannestad:2005ex]. Noticeably, these bounds can be alleviated in the event that only one or two neutrinos decay and the remaining ones are free streaming [@Bell:2005dr; @Archidiacono:2014nda; @Gariazzo:2014pja] and are therefore not in contradiction with our findings. Interestingly, neutrino decay with parameters similar to our model was proposed as an alternate solution to the solar neutrino problem [@Bahcall:1986gq].
[*Other Possible Interpretations.*—]{}Another possible explanation of the tension between the track and cascade data sets is the decay of dark matter (DM) [@Aartsen:2018mxl] to electron neutrinos ($\chi \rightarrow \nu_e \bar{\nu}_e$). We focus on DM decay instead of annihilation as the galactic anisotropy constraints [@Ahlers:2015moa; @Denton:2017csz; @Aartsen:2017ujz] are weaker for DM decay since the DM annihilation more peaked towards the galactic center. In order to estimate the expected track and cascade distribution, the galactic and extragalactic diffuse intensity of neutrinos is computed, including electroweak corrections, by using [Pythia]{} 8.2 [@Sjostrand:2014zea] and a Navarro-Frenk-White galactic DM profile [@Navarro:1995iw].
While a good quality of fit ($\chi^2<1$) is found in a SPL+DM model with 4 parameters ($\tau_\chi$ s, $m_\chi$ TeV, $\Phi_\nu$, $\gamma$), this model has a number of undesirable properties. The galactic contribution to the flux peaks at energies below the cascade flux sensitivity and contributes due to the typical energy uncertainty of cascades are $\gtrsim15\%$ [@Aartsen:2015zva]; this results in a contribution to the cascade flux at low energies due to the energy uncertainty, but a minimal contribution to the track flux (after oscillations). The resultant peak flux is larger than the measured flux at energies just below the region of interest for IceCube’s cascade analysis. From SU(2) symmetry there will be an $e^+e^-$ channel, leading to $\gamma$-rays from electroweak corrections constrained by [*Fermi*]{}-LAT [@Murase:2015gea]. Finally, this fit requires a short DM lifetime which is strongly constrained by the cosmic microwave background and bounds from the reionization epoch, the best fit values being $\tau_\chi\sim10^{23}$ s, $m_\chi\sim10$ TeV [@Liu:2016cnk; @Slatyer:2016qyl]. All considered, DM decay does not seem to resolve this tension.
Several additional effects could provide an energy and flavor dependent modification of the standard neutrino flux from an astrophysical source. For example, the Glashow resonance occurs when a $\bar\nu_e$ with $E_\nu=6.3$ PeV scatters off an electron in the ice creating an on-shell $W^-$ [@Glashow:1960zz] increasing IceCube’s sensitivity in that energy range considerably. IceCube performs their fits assuming that $\mathcal{I}_\nu=\mathcal{I}_{\bar\nu}$. While this is generally the case if neutrinos are mainly produced through $pp$ interactions, it won’t be the case if the main neutrino production channel is $p\gamma$ interactions [@Nunokawa:2016pop]. For the SPL case $\mathcal{I}_{\nu_e}/\mathcal{I}_{\bar\nu_e}\simeq3.5$ which would somewhat harden the cascade spectrum, but would not be enough to reduce the tension of the fit.
Another option that could alleviate the track vs. cascade fit tension is neutron decay sources. Neutrons decay to $\bar\nu_e$’s and are produced alongside charged pions in $p\gamma$ interactions (as well as in $pp$ interactions) and are thus expected to provide an additional contribution of $\nu_e$’s to the high energy astrophysical neutrino flux. The energy of neutrinos from neutrons is suppressed by about two orders of magnitude compared with those from pion decay. However, for a spectral index $\gtrsim2$ as in our case, this contribution is subleading. Neutrons also result from photodisintegration of heavy ions in dense sources, although this flux is also suppressed compared to the standard contribution by at least an order of magnitude [@Biehl:2017zlw; @Rodrigues:2017fmu].
In addition, non-standard neutrino interactions with ultralight mediators ($m_{Z'}\ll1$ eV) as well as pseudo-Dirac neutrino models [@Wolfenstein:1981kw; @Pakvasa:2012db] may also affect the track vs. cascade ratio. However, in both cases, we expect an impact on the neutrino data set that is smaller than the one induced by the invisible neutrino decay scenario.
[*A Solution to the $\nu_\tau$ Observational Deficit.*—]{}The IceCube detector is expected to observe 2 or 3 $\nu_\tau$ events in the energy range of interest [@Aartsen:2017mau; @Palladino:2018qgi]. However, currently no $\nu_\tau$ events are observed. The assumption of invisible neutrino decay for the $\nu_2$ and $\nu_3$ eigenstates would induce a reduction of $\mathcal{I}_{\nu_\tau}$ of $80\%$ below 1 PeV which convolved with the detection efficiency leads to a $\sim59\%$ reduction in the number of $\nu_\tau$ events for our best fit value $\tau/m = 10^2$ s/eV. The invisible neutrino decay could then also explain the current deficit of $\nu_\tau$ events.
[*Conclusions.*—]{}The IceCube Observatory detects high energy astrophysical neutrinos through two event topologies: tracks and cascades. By simultaneously taking advantage of the energy and flavor information present in the two data sets, for the first time we have placed strong constraints on the consistency of the data with the standard source picture. A conventional model for the neutrino production in astrophysical sources is unable to simultaneously explain the track and cascade data at $>3\sigma$.
We tested several New Physics models and found that the invisible neutrino decay of $\nu_2$ and $\nu_3$ with $\tau/m=10^2$ s/eV is preferred by the IceCube data by $3.4\sigma$ and is consistent with all other existing constraints. While this model is more natural in the normal mass ordering, it is consistent with either ordering. In addition, a model of visible decay in $\nu_1$ may provide additional improvements to the fit by producing additional $\nu_1$’s (mostly $\nu_e$’s) at lower energies. Interestingly, our model also predicts a $59\%$ reduction in the number of expected $\nu_\tau$ events reconciling the current observational deficit.
As more neutrino data arrives with the advent of IceCube-Gen2 [@Aartsen:2014njl] and KM3NeT [@Adrian-Martinez:2016fdl] and the spectral distributions will be defined more precisely for both event topologies, it will be possible to further test our result.
[*Acknowledgments.*—]{}We are grateful to Mauricio Bustamante, Steen Hannestad, Rebecca Leane, Orlando Peres, and Mohamed Rameez for useful discussions. PBD and IT acknowledge support from the Villum Foundation (Project No. 13164) and the Danish National Research Foundation (DNRF91). PBD thanks the Danish National Research Foundation (Grant No. 1041811001) for support. The work of IT has also been supported by the Knud Højgaard Foundation and the Deutsche Forschungsgemeinschaft through Sonderforschungbereich SFB 1258 “Neutrinos and Dark Matter in Astro- and Particle Physics” (NDM).
Appendix[\[sec:appendix\]]{}
============================
[*Per-flavor Flux Reconstruction.*—]{} Converting the flux observed at the Earth after oscillations and decay in a given channel to the per-flavor true flux includes corrections due to neutral current interactions, track misidentification in a given IceCube detection channel [@Aartsen:2015ivb], and $\tau\to\mu+2\nu$ decays, each of which is accounted for in our analysis and contributes only a sub-leading effect on our results. The details of these corrections are presented here.
The IceCube Collaboration reports the per-flavor flux from the track and cascade analyses. The track analysis is dominantly the result of $\nu_\mu$ charged current (CC) interactions and the cascade analysis is dominantly the result of $\nu_e$ and $\nu_\tau$ CC interactions.
For clarity, the relevant flux and intensity terms are now defined again. The flux of a single source of $\nu_\alpha$ is $F_{\nu_\alpha}$ and is normalized by $\Phi_\nu$ which is a free parameter in the fits. The diffuse intensity at the Earth after oscillations and decay of $\nu_\alpha$ is $\mathcal I_{\nu_\alpha}$ (note that $F_{\nu_\alpha}$ and $\mathcal I_{\nu_\alpha}$ both refer to the sum of neutrinos and antineutrinos unless otherwise mentioned). We take $f_{\rm mis}=0.3$ as the fraction of CC $\nu_\mu$ interactions that are misidentified as cascades [@Aartsen:2015ivb] and $f_{\rm CC}=0.7$ as the fraction of neutrino events that undergoes a CC interaction, while the rest undergo a NC interaction depositing $\sim1/3$ of the energy in the detector [@Gandhi:1995tf]. The branching ratio of $\tau\to\mu+2\nu$ is $f_{\tau\mu}=0.174$ [@Patrignani:2016xqp]. Thus the track ($t$) intensity is related to the neutrino flux by $$\mathcal{I}_t(E_\nu)=f_{\rm CC}(1-f_{\rm mis})\mathcal{I}_{\nu_\mu}(E_\nu)+f_{\rm CC}f_{\tau\mu}\mathcal{I}_{\nu_\tau}(3E_\nu)\,.
\label{eq:It}$$ In order to convert this into the per-flavor flux, we follow IceCube’s approach of assuming that $\mathcal{I}_{\nu_e}=\mathcal{I}_{\nu_\mu}=\mathcal{I}_{\nu_\tau}$ and that they are described by a SPL. Then the per-flavor intensity at the Earth from the track data set is $$\mathcal{I}_{t,{\rm pf}}(E_\nu)=\frac{\mathcal{I}_t(E_\nu)}{f_{\rm CC}(1-f_{\rm mis})+f_{\rm CC}f_{\tau\mu}3^{-\gamma_t}}\ ,
\label{eq:Itpf}$$ where $\gamma_t$ is the result of a power law fit to $\mathcal{I}_t(E_\nu)$.
Similarly, the cascade ($c$) intensity is related to the neutrino flux at the Earth by $$\begin{gathered}
\mathcal{I}_c(E_\nu)=f_{\rm CC}[\mathcal{I}_{\nu_e}(E_\nu)+f_{\rm mis}\mathcal{I}_{\nu_\mu}(E_\nu)+(1-f_{\tau\mu})\mathcal{I}_{\nu_\tau}(E_\nu)]\\
+(1-f_{\rm CC})\sum_{\alpha\in\{e,\mu,\tau\}}\mathcal{I}_{\nu_\alpha}(3E_\nu)\,.
\label{eq:Ic}\end{gathered}$$ Then the per-flavor flux at the Earth from the cascade data set is $$\mathcal{I}_{c,{\rm pf}}(E_\nu)=\frac{\mathcal{I}_c(E_\nu)}{f_{\rm CC}[1+f_{\rm mis}+(1-f_{\tau\mu})]+3(1-f_{\rm CC})3^{-\gamma_c}}\,.
\label{eq:Icpf}$$
If $\mathcal{I}_{\nu_\alpha}(E_\nu)$ is a power law and the diffuse intensity of each flavor is the same then these definitions recover the correct true neutrino intensity, while also allowing for different spectra at the Earth in terms of both deviations from a SPL and different intensities for different flavors. Compared with setting $\mathcal I_t \simeq \mathcal I_{\nu_\mu}$ and $\mathcal I_c \simeq \mathcal I_{\nu_e}+\mathcal I_{\nu_\tau}$, including these corrections is a $\lesssim1\%$ correction on the diffuse intensities. Hence the diffuse intensities of each flavor are related to the track and cascade intensities $\mathcal I_i$ for $i\in\{$t$,$c$\}$ by Eqs. \[eq:It\] and \[eq:Ic\]. Finally, these are converted into the per-flavor intensities $\mathcal I_{i,{\rm pf}}$ under the assumption of a SPL and $\mathcal{I}_{\nu_e}=\mathcal{I}_{\nu_\mu}=\mathcal{I}_{\nu_\tau}$ flavor ratio by Eqs. \[eq:Itpf\] and \[eq:Icpf\]; these have normalizations at 100 TeV of $\Phi_i$ which can then be compared with the data.
[^1]: Neutrino oscillations already provide evidence of physics beyond the Standard Model in that they have mass. In this work, New Physics refers to physics beyond both the Standard Model and the fact that neutrinos have mass.
[^2]: A newer unpublished analysis from the IceCube Collaboration [@Aartsen:2017mau] slightly changes the various qualities of fit related to the track and cascade datasets. Given the different energy ranges, the tension between the track and cascade data decreases to $2.5 \sigma$ for the SPL. However, we find that this does not significantly change our conclusions and neutrino decay is still preferred at $2.8 \sigma$. This trend has also been confirmed from the preliminary results presented at Neutrino 2018 [@taboada_ignacio_2018_1286919].
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'E. Ding'
- 'H. N. Chan'
- 'K. W. Chow'
- 'K. Nakkeeran'
- 'B. A. Malomed'
bibliography:
- 'ref.bib'
title: Exact states in waveguides with periodically modulated nonlinearity
---
Introduction
============
It is commonly known that optical spatial solitons arise in planar and bulk waveguides through the balance of the Kerr nonlinearity and transverse diffraction [@agrawal]. Modern fabrication technologies make it possible to create waveguides featuring spatially inhomogeneous nonlinearities that support novel classes of propagation patterns [@boris1]. In particular, spatially inhomogeneous waveguides with a defocusing nonlinearity, whose local strength grows toward the periphery, can support diverse species of fundamental and higher-order solitons, including vortices, necklace rings, vortex gyroscopes, *hopfions*, and complex hybrid modes [boris2,Lei\_Wu,wu, zhong,Radik,Yasha,hybrids]{}, as well as *localized dark solitons* [@Zeng]. Similar nonlinearity landscapes, featuring different growth rates of the local nonlinearity in opposite transverse directions, support strongly asymmetric bright solitons [@boris4]. Asymmetric solitons also appear spontaneously if the nonlinearity profile features a dual-well structure [@dual; @Nir]. Furthermore, a combination of the fast growing local strength of the defocusing nonlinearity with the usual $\mathcal{PT}$-symmetric gain-loss profile makes it possible to produce solitons that exhibit *unbreakable* $\mathcal{PT}$ symmetry [@unbreakable; @raju; @2D], which is essential for constructing robust solitons in such systems [@Demetri; @PTreview1; @PTreview2]. It is also relevant to mention that a combination of $\mathcal{PT}$-symmetric with competing nonlinearities supports spatiotemporal solitons [@ref1].
Considerable interest has also been drawn to models with uniform nonlinearity, either self-defocusing or focusing, and specially designed periodic potentials that support exact periodic wave solutions [Carr1,Carr2]{}. Although both the particular potentials and the corresponding exact periodic solutions are not generic, and the analysis of their stability can only be performed numerically, these models provide direct insight into the possibility to support periodic wave patterns by utilizing the interplay of periodic potentials and the ubiquitous cubic nonlinearity. Furthermore, nontrivial exact solutions serve as benchmarks which suggest the shape of generic solutions. The inverse problem, aimed at engineering waveguiding potentials adjusted to maintaining periodic waves with prescribed properties, is a physically relevant issue too [@Spain].
In this work, we introduce a model with a class of periodic modulations that represent spatially periodic *pseudopotentials* [@pseudo] induced by the local nonlinearity. This model admits exact solutions in the form of periodic wave patterns which, in the limiting case of an infinite modulation period, become bright solitons. Stability of these patterns is studied numerically. The same model can also be used to solve the inverse problem of engineering a nonlinearity-modulation profile needed to support a wave pattern with prescribed period and amplitude.
The Mathematical Model
======================
The light propagation in a planar waveguide with spatially modulated nonlinearity is described by the model that is based on the scaled nonlinear Schrödinger equation for the electromagnetic wave amplitude $%
\Psi (x,z)$, $$i\Psi _{z}+\Psi _{xx}+g(x)|\Psi |^{2}\Psi =0, \label{eq:nls}$$where $x$ and $z$ are the transverse and longitudinal coordinates, respectively. The periodically-modulated nonlinearity profile is defined by $%
g(x)$, chosen as $$g(x)=\frac{\alpha }{\mathrm{dn}^{2}(x)}+\beta +\gamma ~\mathrm{dn}^{2}(x),
\label{eq:g}$$where $\alpha $, $\beta $, and $\gamma $ are real constants, and $\mathrm{dn}%
(x)$ is the standard Jacobi elliptic function with modulus $\sqrt{m}$ and period $2K$ ($K$ being the complete elliptic integral of the first kind). It is relevant to mention that, in the general case, the periodic inhomogeneity affects not only the local nonlinearity, but also the local refractive index, which would generate an additional term $U(x)\Psi $ in Eq. (\[eq:nls\]), with an effective spatially-periodic potential, $U(x)$. Nevertheless, specific experimental methods, such as resonant doping, make it possible to create waveguides in which the nonlinearity is affected by the periodic modulation, while the refractive index remains nearly constant [@boris1].
The same model, with propagation distance $z$ replaced by time $t$, represents the scaled Gross-Pitaevskii equation for the mean-field wave function of an atomic Bose-Einstein condensate (BEC), for which the periodic nonlinearity modulation can be induced by means of the Feshbach resonance in a spatially non-uniform magnetic or optical field. In particular, the necessary periodic profile of the magnetic field can be accurately implemented by means of the known technique based on the use of appropriately designed magnetic lattices [@magnetic]. Furthermore, the spatially periodic distribution of the local nonlinearity coefficient in BEC has been experimentally realized by means of an optical lattice [Feshbach]{}. A particular anharmonic profile corresponding to Eq. (\[eq:g\]) can be effectively approximated by a superposition of several harmonics of its Fourier decomposition, represented by the respective optical lattices. In a real experiment, the setting also includes an overall parabolic trapping potential. However, in many situations the characteristic scale of this potential is much larger than the period of the spatial modulation, which makes it possible to neglect the trapping potential while analyzing the effects of periodic lattices [@Heidelberg].
We look for a stationary solution to Eq. (\[eq:nls\]) in the form of a $\mathrm{dn}$-wave": $$\Psi (x,z)\equiv \psi (x)e^{-i\Omega z}=\frac{A_{0}\;\mathrm{dn}(x)}{1+b\;%
\mathrm{dn}^{2}(x)}e^{-i\Omega z}, \label{eq:ansatz}$$where $A_{0}$ and $b$ are real constants, and $-\Omega $ is the propagation constant (or the chemical potential, in the BEC model). Note that this ansatz is nonsingular under the conditions $b>-1$ or $b<-1/(1-m)$. Substituting it into Eq. (\[eq:nls\]) gives rise to the following system of equations for the three ansatz parameters $A_{0}^{2}$, $b$, and $\Omega $: $$\left\{ \begin{aligned} 2+A_{0}^{2} \; \alpha -6b(m-1)-m+\Omega =0 \; , \\
A_{0}^{2} \; \beta +2b^{2}(m-1)+2b(\Omega +3m-6)-2=0 \; , \\ 6b+A_{0}^{2} \;
\gamma +b^{2}(2-m+\Omega )=0 \; . \end{aligned}\right. \label{eq:para}$$One can calculate any three parameters from this system for given values of the others. In particular, this allows one to address the above-mentioned inverse problem, aimed at determining the nonlinearity modulation profile (see Eq. (\[eq:g\])) needed for maintaining a particular wave pattern.
Stability Analysis
==================
The stability of the $\mathrm{dn}$-wave denoted by Eq. (\[eq:ansatz\]) is investigated by means of the standard linearization procedure [linstab1,linstab2]{}. Substituting $\Psi (x,z)=\left[ \psi (x)+u(x,z)\right]
e^{-i\Omega z}$ into Eq. (\[eq:nls\]), with the small complex perturbation defined as $u\left( x,z\right) \equiv R(x,z)+iI(x,z)$, we arrive at the linearized system, $$\left\{ \begin{aligned} \partial _{z}R &=&\left( -\Omega -g(x)\psi
^{2}(x)-\partial _{x}^{2}\right) I \; ,\\ \partial _{z}I &=&\left( \Omega
+3g(x)\psi ^{2}(x)+\partial _{x}^{2}\right) R \; . \end{aligned}\right.
\label{eq:linear}$$The stability of the $\mathrm{dn}$-wave is determined by substituting $%
\left\{ R(x,z),I(x,z)\right\} =\left\{ P(x),Q(x)\right\} \exp (\lambda z)$ into the above equations. The resulting problem for stability eigenvalue $%
\lambda $ is solved numerically using the finite-difference method. In particular, *modulational instability* of periodic states [agrawal]{} is accounted for by eigenvalues with $\mathrm{Re}(\lambda )>0$. Generic examples of stable and unstable $\mathrm{dn}$-waves are presented in Fig. \[fig:stab\]. The stability, as predicted by the calculation of the eigenvalue spectra, is corroborated by direct simulations of Eq. ([eq:nls]{}), using the Fourier transform in $x$ and a fourth-order Runge-Kutta algorithm in $z$.
The dependence of the stability of the $\mathrm{dn}$-waves on the system’s parameters can be explored with the help of numerical-continuation techniques [@Alan; @cont; @Doedel]. In particular, it is important to know how the stability is affected by varying the nonlinearity-modulation period $%
2K$, which is determined by the squared modulus, $m$. We fix $A_{0}$, $b$, and $\beta $ in Eq. (\[eq:para\]) as $A_{0}^{2}=1$, $b=-0.6$, and $\beta
=-1$, and determine the other parameters, *viz*., $\alpha $, $\gamma $, and $\Omega $, for each value of $m$. The results are summarized in Fig. \[fig:branch\]. In this case, the $\mathrm{dn}$-wave is found to be stable in the region of $0\leq m\leq 0.725$, where none of the eigenvalues in the spectrum has a positive real part (see the left panel). For $m>0.725$, the long-period $\mathrm{dn}$-waves are destabilized by at least one eigenvalue with $\mathrm{Re}(\lambda )>0$. The strongest instability is found at around $m=0.852$, where the $\mathrm{dn}$-wave is quickly destroyed by the instability. The evolution of the wave profiles at the onset of the instability, as well as the strongest-instability point, are also shown in the left panel. The *duty cycle* ($\mathrm{DC}$) of the modulation profile, i.e., the share of the region carrying a self-focusing nonlinearity per one period of $g(x)$, is shown in the right panel of Fig. [fig:branch]{}. The nonlinearity is entirely self-defocusing at $m<0.453$ where $\mathrm{DC}\equiv 0$. Once $m$ exceeds this threshold, the $\mathrm{DC%
}$ first increases to a maximum of $14.62\%$ at $m=0.755$, and then approaches zero in the long-wave limit of $m\rightarrow 1$ where the modulation period $2K$ becomes infinite. We have found that the mean value of $g(x)$ is always negative in the entire range of $m$ values, i.e., the nonlinearity is self-defocusing on average. It is worthy to note that the maximum of the $\mathrm{DC}$ roughly coincides with the onset of instability.
In the long-wave limit of $m\rightarrow 1$, the modulation profile (Eq. ([eq:g]{})) assumes the localized shape: $$g(x)=\alpha \cosh ^{2}x+\beta +\gamma \;\mathrm{sech}^{2}x\;.
\label{eq:sech}$$The left panel of Fig. \[fig:branch\] suggests that the instability of the -wave vanishes in this limit, with the corresponding stable exact solution (see Eq. (\[eq:ansatz\])) being a bright soliton: $$\Psi (x,z)=\frac{A_{0}\;\mathrm{sech}(x)}{1+b~\mathrm{sech}^{2}(x)}%
e^{-i\Omega z}\;. \label{eq:soliton}$$The stability of the soliton solution is analyzed here only for the case where $\alpha =0$, to ensure that the localized modulation profile (Eq. ([eq:sech]{})) is not singular at $|x|\rightarrow \infty $ (nevertheless, the self-defocusing singularity with $\alpha <0$ may readily support robust self-trapped modes [boris2,Lei\_Wu,wu,zhong,Radik,Yasha,hybrids,boris4,dual,Nir]{}). In this situation, i.e., with $\alpha =0$ and $m\rightarrow 1$, system (\[eq:para\]) yields $$\Omega =-1\;,\;A_{0}^{2}=\frac{2(1+4b)}{\beta }\;,\;\gamma =-\frac{3b\beta }{%
1+4b}\;. \label{eq:reduced}$$
The stability condition for the soliton pinned to the spatially modulated nonlinearity profile given by Eq. (\[eq:sech\]) with $\alpha =0$, is $%
\gamma >0$, as in that case the soliton is pulled to the local maximum of self-attraction. Equations (\[eq:soliton\]) and (\[eq:reduced\]) admit $%
\gamma >0$ in two cases:$$\beta >0,~0<-b<1/4 \; ; \label{beta>0}$$$$\beta <0,~1/4<-b<1 \; . \label{beta<0}$$In the former case, the nonlinearity is globally self-focusing, while in the latter one a finite self-focusing region (“defect") is embedded in a defocusing background.
An example of the latter situation is shown in Fig. \[fig:sech\], where the stability of the pinned bright soliton is confirmed by both the eigenvalue spectrum and direct simulations. In this case, $\beta =-2.8$ and $%
\gamma =3.6$,$\ g(x)$ being positive at $|x|<0.74$. In fact, stable solitons can be produced in a wide range of parameter values, as shown in Fig. [fig:soliton\_stab]{}. Unstable solitons are only found in the case of $\beta
>0>\gamma $ (represented by red dashed curves in the top panels). In this case, the self-focusing is stronger farther from the center, hence the soliton is repelled by the effective nonlinear potential. An example of such a nonlinearity profile is shown in the subplot in the top right panel of Fig. \[fig:soliton\_stab\].
Collisions between solitons play an important role in the study of their dynamics. In the case corresponding to condition (\[beta>0\]), one may consider the collision of a free bright soliton, with inverse width $\eta $ and velocity (slope) $c$,$$\Psi _{\mathrm{free}}\left( x,z\right) = \sqrt{\frac{2}{\beta}} \eta\;%
\mathrm{sech}\left( \eta \left( x-cz\right) \right) e^{\left( \frac{i}{2}%
cx+i\left( \eta ^{2}-\frac{c^{2}}{4}\right) z\right)} \; ,
\label{free-bright}$$with a pinned soliton. An example of such a collision is displayed in Fig. \[fig:collision\]. The incident soliton captures the pinned one, merging with it into a single soliton which continues to move with original velocity. This outcome may find applications to the design of soliton=based data-processing schemes.
In the case of $\gamma \ll \beta $, i.e., $-b\ll 1/4$ (see Eqs. ([eq:reduced]{}) and (\[beta>0\])), the collision can be considered by means of the perturbation theory [@RMP], which uses the exact result for the collision-induced soliton’s shift generated by the solution of the nonlinear Schrödinger equation. In this case, one may expects that the incident soliton will pass through, while the pinned one will start oscillating around the attractive nonlinear defect.
A completely novel situation arises in the case of Eq. (\[beta<0\]), when the model admits a freely moving dark soliton far from the defect. Its collision with the pinned bright soliton will be governed by the repulsive interaction, which may lead to various outcomes, such as rebound of the incident dark soliton and destruction of the bright one through its dislodgment from the pinned position. These possibilities call for systematic numerical simulations of the collisions, which is a subject for a separate work.
The Manakov System
==================
Lastly, Eq. (\[eq:nls\]) can be generalized to a coupled system $$\left\{ \begin{aligned} i\Psi _{z}+\Psi _{xx}+g(x)\left( |\Psi |^{2}+|\Phi
|^{2}\right) \Psi &=0 \;,\\ i\Phi _{z}+\Phi _{xx}+g(x)\left( |\Psi
|^{2}+|\Phi |^{2}\right) \Phi &=0 \end{aligned}\right. \label{eq:coupled}$$that describes the copropagation of light modes with orthogonal polarizations in a bimodal waveguide, under the Manakov’s condition that the self-phase- and cross-phase-modulation coefficients are equal [Manakov,Menyuk]{}, as well as a binary Bose-Einstein condensate composed of two hyperfine atomic states [@binary-BEC] (in the latter case, the relative nonlinearity is very close to the Manakov’s point). In the long-wave limit similar to Eq. (\[eq:sech\]) where $g(x)=\beta +\gamma ~%
\mathrm{sech}^{2}(rx)$, with $\gamma >0$ and parameter $r$ which determines the width of the attracting region, an exact solution of the coupled equations can be found in the form of a stable *symbiotic* dark-bright soliton complex [@symbio1; @symbio2]: $$\begin{aligned} \Psi (x,z) &=A_{0}\tanh (rx)e^{-i\Omega _{1}z} \;, \\ \Phi
(x,z) &=A_{0}\; \mathrm{sech}(rx)e^{-i\Omega _{2}z}\;, \end{aligned}
\label{eq:ms}$$with $$A_{0}=\sqrt{\frac{2}{\gamma }}\;r\;,\;\Omega _{1}=-\frac{2r^{2}\beta }{%
\gamma }\;,\;\Omega _{2}=-r^{2}-\frac{2r^{2}\beta }{\gamma }\;.$$In this case, the dark component $\Psi $ cannot exist without the interaction with the bright counterpart $\Phi $, and the background supporting the dark component is modulationally stable when $\beta <0$. An example of a stable dark-bright complex is displayed in Fig. [fig:darkbright]{}.
Conclusion
==========
In this work, we have studied the one-dimensional model for the wave transmission in a medium with a periodically-modulated local nonlinearity that is based on the Jacobi elliptic function. The model, which can be realized in optics and BEC [@ref2], admits both exact periodic solutions and bright solitons (in the long-wave limit). Stable solutions of these types provide a benchmark suggesting the shape of generic solutions that can be found numerically in the same model. The model also allows for the prediction of the modulation profile needed to support a particular periodic wave form with prescribed period and amplitude. The numerical analysis of the modulational stability has demonstrated that the periodic patterns can be unstable for sufficiently large periods. However, stability is retrieved in the limit of an infinite period which corresponds to bright solitons. In addition, we have found an exact solution for dark-bright soliton bound states in a similar two-component model that applies to periodically inhomogeneous bimodal planar optical waveguides and binary BEC. As an extension of the analysis, it will be relevant to study the periodic solutions, soliton solutions, localized structures [@ref3], and in particular bright-antidark soliton complex supported by this coupled model in detail. Also, as mentioned above, it may be interesting to systematically simulate collisions of a moving free bright or dark soliton with the pinned one.
Partial financial support has been provided by the Research Grants Council (Hong Kong) contract HKU 17200815.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
A famous result due to Grothendieck asserts that every continuous linear operator from $\ell_{1}$ to $\ell_{2}$ is absolutely $(1,1)$-summing. If $n\geq2,$ however, it is very simple to prove that every continuous $n$-linear operator from $\ell_{1}\times\cdots\times\ell_{1}$ to $\ell_{2}$ is absolutely $\left( 1;1,...,1\right) $-summing, and even absolutely $\left( \frac{2}{n};1,...,1\right) $-summing$.$ In this note we deal with the following problem:
Given a positive integer $n\geq2$, what is the best constant $g_{n}>0$ so that every $n$-linear operator from $\ell_{1}\times\cdots\times\ell_{1}$ to $\ell_{2}$ is absolutely $\left( g_{n};1,...,1\right) $-summing?
We prove that $g_{n}\leq\frac{2}{n+1}$ and also obtain an optimal improvement of previous recent results (due to Heinz Juenk $\mathit{et}$ $\mathit{al}$, Geraldo Botelho $\mathit{et}$ $\mathit{al}$ and Dumitru Popa) on inclusion theorems for absolutely summing multilinear operators.
address: 'UFRN/CERES - Centro de Ensino Superior do Seridó, Rua Joaquim Gregório, S/N, 59300-000, Caicó- RN, Brazil'
author:
- 'A. T. Bernardino'
title: 'On cotype and a Grothendieck-type theorem for absolutely summing multilinear operators'
---
Introduction
============
Grothendieck’s theorem for absolutely summing operators asserts that every continuous linear operator from $\ell_{1}$ to $\ell_{2}$ is absolutely $(1;1)$-summing (and hence absolutely $(p;p)$-summing for every $p\geq1$). For the linear theory of absolutely summing operators we refer to [@df; @djt] (see also [@bpr; @ku2; @seo] for recent developments).
In the multilinear setting, D. Pérez-García, in his PhD thesis [@dav] (see also [@bom] and [@botp] for a different proof), proved that every continuous $n$-linear operator from $\ell_{1}\times\cdots\times
\ell_{1}$ to $\ell_{2}$ is multiple $(1;1,...,1)$-summing (in fact, multiple $(p;p,...,p)$-summing for every $1\leq p\leq2$)$.$ This result can be regarded as the multilinear version of Grothendieck’s theorem.
Let us recall the notions.
The letters $X_{1},...,X_{n},X,Y$ will always denote Banach spaces over $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ and $X^{\ast}$ represents the topological dual of $X$.
For any $s>0,$ we denote the conjugate of $s$ by $s^{\ast}$. Given a positive integer $n$, the space of all continuous $n$-linear operators from $X_{1}\times\cdots\times X_{n}$ to $Y$ endowed with the $\sup$ norm is denoted by $\mathcal{L}(X_{1},...,X_{n};Y).$ For $p>0$, the vector space of all sequences $\left( x_{j}\right) _{j=1}^{\infty}$ in $X$ such that$$\left\Vert \left( x_{j}\right) _{j=1}^{\infty}\right\Vert _{p}=\left(
\sum_{j=1}^{\infty}\left\Vert x_{j}\right\Vert ^{p}\right) ^{\frac{1}{p}}<\infty$$ is denoted by $\ell_{p}\left( X\right) .$ We represent by $\ell_{p}^{w}\left( X\right) $ the linear space of the sequences $\left(
x_{j}\right) _{j=1}^{\infty}$ in $X$ such that $\left( \varphi\left(
x_{j}\right) \right) _{j=1}^{\infty}\in\ell_{p}\left( \mathbb{K}\right) $ for every $\varphi\in X^{\ast}$.
If $0<p,q_{1},...,q_{n}<\infty$ and $\frac{1}{p}\leq\frac{1}{q_{1}}+\cdots+\frac{1}{q_{n}},$ a multilinear operator $T\in\mathcal{L}(X_{1},...,X_{n};Y)$ is absolutely ** $(p;q_{1},...,q_{n})$-summing if $(T(x_{j}^{(1)},...,x_{j}^{(n)}))_{j=1}^{\infty}\in\ell_{p}(Y)$ for every $(x_{j}^{(k)})_{j=1}^{\infty}\in\ell_{q_{k}}^{w}(X_{k}),k=1,...,n.$ In this case we write $T\in\Pi_{\left( p;q_{1},...,q_{n}\right) }^{n}\left(
X_{1},...,X_{n};Y\right) $. For details we refer to [@am].
When $1\leq q_{1},...,q_{n}\leq p<\infty$ a multilinear operator $T\in\mathcal{L}(X_{1},...,X_{n};Y)$ is multiple ** $(p;q_{1},...,q_{n})$-summing if $(T(x_{j_{1}}^{(1)},...,x_{j_{n}}^{(n)}))_{j_{1},..,j_{n}=1}^{\infty}\in\ell_{p}(Y)$ for every $(x_{j}^{(k)})_{j=1}^{\infty}\in
\ell_{q_{k}}^{w}(X_{k}),k=1,...,n.$ In this case we write $T\in\Pi_{m\left(
p;q_{1},...,q_{n}\right) }^{n}\left( X_{1},...,X_{n};Y\right) $. For details we mention [@bom; @collec] and for recent developments and applications related to the multilinear and polynomial theory we refer to [@ag; @bbb; @bh; @an; @dgm; @se; @lit; @mat; @ppp] and references therein. For $n=1$ we write $\Pi$ instead of $\Pi^{1}$ and we recover the classical theory of absolutely summing linear operators.
For $1\leq q_{1},...,q_{n}\leq p<\infty,$ the inclusion $$\Pi_{m\left( p;q_{1},...,q_{n}\right) }^{n}\left( X_{1},...,X_{n};Y\right)
\subseteqq\Pi_{\left( p;q_{1},...,q_{n}\right) }^{n}\left( X_{1},...,X_{n};Y\right)$$ is obvious. So, the following coincidence result is an immediate consequence of Pérez-García multilinear version of Grothendieck’s theorem:
For every positive integer $n$, $$\Pi_{\left( 1;1,...,1\right) }^{n}\left( \ell_{1},...,\ell_{1};\ell
_{2}\right) =\mathcal{L}\left( \ell_{1},...,\ell_{1};\ell_{2}\right) .$$
However, using that $\ell_{1}$ has cotype $2$ it is easy to prove that the above result is far from being optimal. In fact, we have the following improvement (see [@irish; @irishd]):
For every positive integer $n\geq2$, $$\Pi_{\left( \frac{2}{n};1,...,1\right) }^{n}\left( \ell_{1},...,\ell
_{1};\ell_{2}\right) =\mathcal{L}\left( \ell_{1},...,\ell_{1};\ell
_{2}\right) . \label{dda}$$
So, the following problem is quite natural:
\[xc\]Given a positive integer $n\geq2$, what is the best constant $g_{n}>0$ so that$$\Pi_{\left( g_{n};1,...,1\right) }^{n}\left( \ell_{1},...,\ell_{1};\ell
_{2}\right) =\mathcal{L}\left( \ell_{1},...,\ell_{1};\ell_{2}\right) ?$$
If we test $n=1$ in (\[dda\]) we obtain $$\Pi_{(2;1)}\left( \ell_{1};\ell_{2}\right) =\mathcal{L}\left( \ell_{1};\ell_{2}\right)$$ which is not surprising at all, in view of Grothendieck’s Theorem. So, in some sense, we feel that the estimate $g_{n}\leq\frac{2}{n}$ for $n\geq2$ is probably not optimal. The optimistic reader will probably hope for an estimate for $g_{n}$ so that in the case $n=1$ we recover Grothendieck’s Theorem. Fortunately, in the last section we will precisely obtain such an estimate.
The problem of estimating $g_{n}$ is related to the generalization of certain results involving cotype and absolutely summing multilinear operators. The following result is a combination of [@junek Theorem $3$ and Remark $2$], [@pop Corollary $4.6$] and [@michels Theorem $3.8$ (ii)]:
\[Inclusion Theorem\]\[tt\] Let $X_{1},...,X_{n}$ be Banach spaces with cotype $s$ and $n\geq2$ be a positive integer:
\(i) If $s=$ $2,$ then $$\Pi_{\left( q;q,...,q\right) }^{n}(X_{1},...,X_{n};Y)\subseteqq\Pi_{\left(
p;p,...,p\right) }^{n}(X_{1},...,X_{n};Y) \label{uu}$$ holds true for $1\leq p\leq q\leq2$ and every $Y$.
\(ii) If $s>2,$ then $$\Pi_{\left( q;q,...,q\right) }^{n}(X_{1},...,X_{n};Y)\subseteqq\Pi_{\left(
p;p,...,p\right) }^{n}(X_{1},...,X_{n};Y) \label{vv}$$ holds true for $1\leq p\leq q<s^{\ast}$ and every $Y$.
The results above are clearly not always optimal since, for example,$$\Pi_{\left( 2;2,2,2\right) }^{3}(\ell_{2},\ell_{2},\ell_{2};\mathbb{K})\neq\mathcal{L}(\ell_{2},\ell_{2},\ell_{2};\mathbb{K})=\Pi_{\left( \frac
{2}{3};1,1,1\right) }^{3}(\ell_{2},\ell_{2},\ell_{2};\mathbb{K}).$$
So, another natural problem is:
\[xz\] Given $1\leq p\leq q<\infty$ and a positive integer $n\geq2$, what are the optimal $\alpha:=\alpha_{p,q,n}>0$ so that, under the same circumstances of (\[uu\]) and (\[vv\]), we have$$\Pi_{\left( q;q,...,q\right) }^{n}(X_{1},...,X_{n};Y)\subseteqq\Pi_{\left(
\alpha;p,...,p\right) }^{n}(X_{1},...,X_{n};Y) \label{ii}$$ for all Banach spaces $X_{1},...,X_{n},Y$ $?$
In this direction we extend Theorem \[tt\] and also recent results from [@4; @3] by showing that $$\alpha\leq\frac{qp}{n\left( q-p\right) +p}$$ and, in some sense, this constant is optimal, since for this value of $\alpha$ we have an equality in (\[ii\]).
An estimate for $\alpha$
========================
Let $1\leq k\leq n,$ where $n\geq2$ is a positive integer. If $X_{i}$ has cotype $s_{i}\geq2,i=1,...,k$ and $$1\leq p\leq q<\min_{1\leq i\leq k}s_{i}^{\ast}\text{ if }s_{i}>2\text{ for
some }i=1,...,k$$ or $$1\leq p\leq q\leq2\text{ if }s_{i}=2\text{ for all }i=1,...,k,$$ then $$\Pi_{(z;q,...,q,t,...,t)}^{n}(X_{1},...,X_{n};Y)=\Pi_{(\frac{zqp}{zk\left(
q-p\right) +qp};p,...,p,t,...,t)}^{n}(X_{1},...,X_{n};Y),$$ for all $X_{k+1},...,X_{n},Y$ and all $z,t\geq1$ (here $q$ and $p$ are repeated $k$ times). In particular, if $k=n$,$$\Pi_{(z;q,...,q)}^{n}(X_{1},...,X_{n};Y)=\Pi_{(\frac{zqp}{zk\left(
q-p\right) +qp};p,...,p)}^{n}(X_{1},...,X_{n};Y)$$
Since $X_{i}$ has finite cotype $s_{i}\geq2,i=1,...,k$, then we have$$\ell_{p}^{w}(X_{i})=\ell_{qp/\left( q-p\right) }\ell_{q}^{w}(X_{i})$$
for all $i=1,...,k$ with $$1\leq p\leq q<\min_{1\leq i\leq k}s_{i}^{\ast}\text{ if }s_{i}>2\text{ for
some }i=1,...,k$$ or $$1\leq p\leq q\leq2\text{ if }s_{i}=2\text{ for all }i=1,...,k.$$
Let $(x_{j}^{(i)})_{j=1}^{\infty}\in\ell_{p}^{w}(X_{i}),i=1,...,k$ and $(x_{j}^{(i)})_{j=1}^{\infty}\in\ell_{t}^{w}(X_{i})$ for $i=k+1,...,n$. So $x_{j}^{(i)}=\alpha_{j}^{\left( i\right) }y_{j}^{\left( i\right) },$ with $\left( \alpha_{j}^{\left( i\right) }\right) _{j=1}^{\infty}\in
\ell_{qp/\left( q-p\right) }$ and $\left( y_{j}^{\left( i\right)
}\right) _{j=1}^{\infty}\in\ell_{q}^{w}\left( X_{i}\right) ,$ for all $j$ and $i=1,...,k$. If $A\in\Pi_{(z;q...,q,t,...,t)}^{n}(X_{1},...,X_{n};Y)$, then$$\begin{aligned}
& \left( \sum_{j=1}^{\infty}\left\Vert A\left( x_{j}^{(1)},...,x_{j}^{\left( n\right) }\right) \right\Vert ^{\frac{zqp}{zk\left( q-p\right)
+qp}}\right) ^{\frac{zk\left( q-p\right) +qp}{zqp}}\\
& =\left( \sum_{j=1}^{\infty}\left( \left\vert \alpha_{j}^{\left(
1\right) }\cdots\alpha_{j}^{\left( k\right) }\right\vert \left\Vert
A\left( y_{j}^{(1)},...,y_{j}^{\left( k\right) },x_{j}^{(k+1)},...,x_{j}^{\left( n\right) }\right) \right\Vert \right) ^{\frac
{zqp}{zk\left( q-p\right) +qp}}\right) ^{\frac{zk\left( q-p\right)
+qp}{zqp}}\\
& \leq\left( \sum_{j=1}^{\infty}\left\Vert A\left( y_{j}^{(1)},...,y_{j}^{\left( k\right) },x_{j}^{(k+1)},...,x_{j}^{\left( n\right)
}\right) \right\Vert ^{z}\right) ^{\frac{1}{z}}\left( \sum_{j=1}^{\infty
}\left\vert \alpha_{j}^{\left( 1\right) }\cdots\alpha_{j}^{\left( k\right)
}\right\vert ^{\frac{qp}{k\left( q-p\right) }}\right) ^{k\left( \frac
{q-p}{qp}\right) }\\
& \leq\left( \sum_{j=1}^{\infty}\left\Vert A\left( y_{j}^{(1)},...,y_{j}^{\left( k\right) },x_{j}^{(k+1)},...,x_{j}^{\left( n\right)
}\right) \right\Vert ^{z}\right) ^{\frac{1}{z}}{\displaystyle\prod\limits_{i=1}^{k}}
\left( \sum_{j=1}^{\infty}\left\vert \alpha_{j}^{\left( i\right)
}\right\vert ^{\frac{qp}{\left( q-p\right) }}\right) ^{\frac{q-p}{qp}}<\infty\end{aligned}$$ and we conclude that$$\Pi_{(z;q,...,q,t,...,t)}^{n}(X_{1},...,X_{n};Y)\subseteqq\Pi_{(\frac
{zqp}{zk\left( q-p\right) +qp};p,...,p,t,...,t)}^{n}(X_{1},...,X_{n};Y).$$ The other inclusion is a consequence of the inclusion theorem for absolutely summing multilinear operators.
A similar result holds if $X_{j_{1}},...,X_{j_{k}},\left\{ j_{1},...,j_{k}\right\} \subseteqq\left\{ 1,...,n\right\} $ (instead of $X_{1},...,X_{k}$) have cotype $s_{j_{i}}\geq2,i=1,...,k.$
The following immediate corollary is an optimal (in the sense that we have an equality instead of an inclusion) generalization of Theorem \[tt\]:
\[ut\]If $n\geq2$ and $X_{1},...,X_{n}$ have finite cotype $s$ and $$1\leq p\leq q<s^{\ast}\text{ if }s>2$$ or$$1\leq p\leq q\leq2\text{ if }s=2,$$ then$$\Pi_{(q;q,...,q)}^{n}(X_{1},...,X_{n};Y)=\Pi_{(\frac{qp}{n\left( q-p\right)
+p};p,...,p)}^{n}(X_{1},...,X_{n};Y)$$ for every Banach space $Y$ and$$\alpha\leq\frac{qp}{n\left( q-p\right) +p}.$$
The above results were independently proved in [@Bla].
An estimate for $g_{n}$
=======================
From Corollary \[ut\] we know that $$\Pi_{(2;2,...,2)}^{n}\left( \ell_{1},...,\ell_{1};\ell_{2}\right)
=\Pi_{(\frac{2}{n+1};1,...,1)}^{n}\left( \ell_{1},...,\ell_{1};\ell
_{2}\right)$$ for all $n\geq2$. But, since$$\mathcal{L}\left( \ell_{1},...,\ell_{1};\ell_{2}\right) =\Pi_{m(2;2,...,2)}^{n}\left( \ell_{1},...,\ell_{1};\ell_{2}\right) \subseteqq\Pi
_{(2;2,...,2)}^{n}\left( \ell_{1},...,\ell_{1};\ell_{2}\right)$$ it readily follows that $$\Pi_{(\frac{2}{n+1};1,...,1)}^{n}\left( \ell_{1},...,\ell_{1};\ell
_{2}\right) =\mathcal{L}\left( \ell_{1},...,\ell_{1};\ell_{2}\right)$$ for all $n\geq2$. So we have:
If $n\geq2$, then$$g_{n}\leq\frac{2}{n+1}.$$
Note that Grothendieck’s Theorem asserts that $g_{1}=1$ and $1=\frac{2}{1+1};$ hence we conjecture that $\frac{2}{n+1}$ is in fact the optimal estimate for $g_{n}$.
This paper is a part of the doctoral thesis of the author which is being written under supervision of Prof. Daniel Pellegrino. The author thanks Prof. Pellegrino for introducing him to the subject and the main problem from this note and also for several suggestions and important insights.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We study two properties of two modules over a local hypersurface $R$: decency, which is close to proper intersection of the supports, and $\operatorname{Tor}$-rigidity. We show that the vanishing of Hochster’s function $\theta^R(M,N)$, known to imply decent intersection, also implies rigidity. We investigate the vanishing of $\theta^R(M,N)$ to obtain new results about decency and rigidity over hypersurfaces. Many applications are given.'
address: |
Department of Mathematics\
University of Kansas\
Lawrence, KS 66045-7523 USA
author:
- Hailong Dao
title: 'Decency and Tor-rigidity for modules over hypersurfaces'
---
[^1]
Introduction
============
Throughout this paper we will deal exclusively with a local, Noetherian, commutative ring $R$ and finitely generated modules over $R$.
Two $R$-modules $M$ and $N$ such that $l(M{\otimes}_RN)<\infty $ are said to *intersect decently* if $\dim M + \dim N \leq \dim
R$. We say that $M$ is *decent* if for all $N$ such that $l(M{\otimes}_RN)<\infty $, $M$ and $N$ intersect decently. This property arises naturally from Serre’s work on intersection multiplicity ([@Se]), which shows that over a regular local ring, any two modules intersect decently. In fact, to have a satisfying local intersection theory, one needs modules to intersect decently as a minimum requirement. However, sufficient conditions for decent intersection become much more elusive in general, even when $R$ is a hypersurface. For example, one can look at a famous open question in Commutative Algebra:
Let $A$ be a regular local ring, and suppose $R$ is a local ring such that $A\subset R$ and $R$ is module-finite over $A$. Then $A$ is a direct summand of $R$ as an $A$-module.
The Direct Summand Conjecture could be proved if one could show a certain module over a local hypersurface is decent (see [@Ho1]). As another example, it was conjectured by Peskine and Szpiro (cf. [@PS]) that in general, a module of finite projective dimension is decent. This question is open even for hypersurfaces of ramified regular local rings.
A pair of modules $(M,N)$ is called *rigid* if for any integer $i\geq0$, $\operatorname{Tor}_i^R(M,N)=0$ implies $\operatorname{Tor}_j^R(M,N)=0$ for all $j\geq i$. Moreover, $M$ is *rigid* if for all $N$, the pair $(M,N)$ is rigid. Auslander studied rigidity in order to understand torsion on tensor products ([@Au]). He also observed that rigidity of $M$ implies other nice properties, such as any $M$-sequence must be an $R$-sequence. To further demonstrate the usefulness of rigidity, let us recall the:
([@HW1], 2.5) Let $R$ be a complete intersection. Let $M,N$ be non-zero finitely generated modules over $R$ such that $\operatorname{Tor}_i^R(M,N)=0$ for all $i\geq 1$. Then: $$\operatorname{depth}(M) + \operatorname{depth}(N) = \operatorname{depth}(R)+\operatorname{depth}(M{\otimes}_RN)$$
Thus, rigidity allows us to force a very strong condition on the depths of the modules by proving the vanishing of one single $\operatorname{Tor}$ module. Auslander’s work, combined with that of Lichtenbaum ([@Li]), showed that modules over regular local rings are rigid. Huneke and Wiegand ([@HW1],[@HW2]) continued this line to study rigidity over hypersurfaces.
The classical condition conjectured to be sufficient for both rigidity and decency was that one of the modules must have finite projective dimension. In general, this is open for decency and false for rigidity (see [@He]). In any case, having finite projective dimension is too restrictive for the most interesting applications, so a key question arises: *Are there more flexible sufficient conditions for rigidity and decency?* A good answer here would not only make applications easier, it would also shed some light on the behaviour of modules of finite projective dimension with respect to the two conditions.In this paper, we try to answer this question for modules over hypersurfaces using a mixture of results and techniques from Commutative Algebra and Intersection Theory. Perhaps not surprisingly, our answers often involve some conditions about the classes of the modules in the Grothendieck group of finitely generated modules over $R$.
Throughout this note we will assume that our hypersurface $R$ comes from an unramified or equicharacteristic regular local ring (we call such hypersurfaces “admissible", following Hochster). Since we need to apply results such as Serre’s Positivity and Non-negativity of $\chi_i$, which are open in general for the ramified case, this is necessary. In some particular instances, such as in low dimensions, this restriction can be relaxed, however we feel it may disrupt the flow of the paper to comment on every such case. The reader will lose very little by thinking of the equicharacteristic (containing a field) case.
Section \[2\] is a review of basic notation and some preliminary results. Of particular interest is Hochster’s theta function. For a local hypersurface $R$ and a pair of finitely generated $R$ modules $M,N$ such that $l(\operatorname{Tor}_i^R(M,N))<\infty$ for all $i\gg0$, we can define: $$\theta^R(M,N) = l(\operatorname{Tor}_{2e+2}^R(M,N)) - l(\operatorname{Tor}_{2e+1}^R(M,N))$$ Here $e$ is any sufficiently large integer. The theta function was first introduced by Hochster \[Ho1\] in his study of the direct summand conjecture. We recall the basic properties of $\theta(M,N)$ and prove a key technical result: [the vanishing of $\theta^R(M,N)$ implies rigidity of (M,N)]{}(proposition \[rg1\]).
In section \[3\] we study the vanishing of $\theta^R(M,N)$ when $R$ is a hypersurface with isolated singularity. The key advantage with such rings is that $\theta^R(M,N)$ is always defined for a pair $M,N$, so we can “move" the modules easily into favorable positions where vanishing of $\theta$ is more evident. We give a fairly complete picture when the dimension of the ring is at most $4$ and obtain many results in higher dimension (see \[dim1\] to \[dim2,3\]). One of them (\[moving\]) states that when $R$ contains a field and $\dim M + \dim N \leq \dim R$, then $\theta^R(M,N)=0$. This result partly answers a question raised to the author by Roger Wiegand. Our results also point to a conjecture that $\theta^R(M,N)$ should always vanish if $\dim R$ is [*even*]{}.
In section \[4\] we study rigidity over hypersurfaces in general. We prove new criteria for rigidity (theorem \[rig1.1\]), as well as a connection to decency when one of the modules is Cohen-Macaulay(see theorem \[rigidandproper\]).
In section \[5\] we apply our results from previous sections to give new proofs of seemingly different results in the literature, as well as new results. First, we show that if $char(R)=p>0$, then $^eR$ is rigid, where $^eR$ is $R$ with the module structure defined by the $e$-th iteration of the Frobenius homeomorphism (theorem \[AM\]). This is the hypersurface case of a theorem by Avramov and Miller ([@AM]). We also apply our rigidity results to gain understanding of depth of tensor products, following the same line of investigation done by Auslander, Huneke and Wiegand(see \[vanishingiso\], \[HWmain\]). Finally, we switch our attention to projective hypersurfaces and prove a result on intersection of subvarieties(theorem \[projhyper\]) using what we know about decency.
Finally, section \[lastsection\] discusses our attempts to generalize the theta function and some additional results we gain on the way which improve on previous works by Murthy and Jorgensen. We also give numerous examples to illustrate our results throughout the paper and show that they are optimal in certain senses.
After this preprint appeared online, a few works by the authors and other researchers have been focused on extensions and applications of the ideas and results in here. For example, [@Da2] uses the $\operatorname{Tor}$-rigidity results here to extend Auslander’s theorems on $\operatorname{Hom}(M,M)$ over regular local rings to hypersurfaces, and [@Da3] applies such results on understanding Van den Bergh’s definition of non-commutative crepant resolutions. Papers [@CeDa; @Ce; @Da] deal with various generalizations to complete intersections as well as analogous results for vanishing of $\operatorname{Ext}$ modules. Another new interesting development is [@MPSW] where the Conjecture \[DaoConj\] is studied and settled in the graded, characteristic $0$ situation using sophisticated geometric techniques.
Part of this paper was included in the author’s PhD thesis at the University of Michigan. The author would like to thank his advisor, Melvin Hochster, for numerous invaluable discussions and encouragements. It is virtually impossible for the author to get to many results here without learning various techniques and insights from the experts. It is a pleasure to thank Luchezar Avramov, Ragnar-Olaf Buchweitz, William Fulton, Craig Huneke, Mircea Mustaţǎ, Paul Roberts and Roger Wiegand for their incredible patience with the author’s questions. Special thanks must go to Greg Piepmeyer for his careful reading of an earlier version of this paper and many suggestions for improvements.
Notation and preliminary results {#2}
================================
Unless otherwise specified, all rings are Noetherian, commutative and local, and all modules are finitely generated. A ring $(R,m,k)$ is a *hypersurface* if its completion $\hat R$ has the form $T/(f)$, where $T$ is a regular local ring and $f$ is in the maximal ideal of $T$. We say that $T$ is *admissible* (as a regular local ring) if it is a power series ring over a field or a discrete valuation ring. If $T$ is admissible we also say that $R$ is admissible (as a hypersurface). Note that an admissible hypersurface may be a ramified regular local ring, and thus not admissible as a regular local ring.
For a ring $R$ and a non-negative integer $i$, we set $X^iR := \{p\in \operatorname{Spec}(R)| \dim(R_p)\leq i\}$. We denote by $Y(R)$ the set $X^{\dim(R)-1}$, the punctured spectrum of $R$. We denote by $G(R)$ the Grothendieck group of finitely generated modules over $R$ and by $\overline{G}(R):= G(R)/[R]$, the reduced Grothendieck group. Also, we let $\operatorname{Sing}(R) := \{p\in \operatorname{Spec}(R)| R_p \ \text{is not regular} \}$ be the singular locus of $R$. For an abelian group $G$, we let $G_{\mathbb{Q}} = G{\otimes}_{\mathbb{Z}}\mathbb{Q}$.
Let the torsion submodule of $M$, $t(M)$, be the kernel of the map $M\to K{\otimes}_RM$, where $K$ is the total quotient ring of $R$. The module $M$ is *torsion* provided $t(M)=M$ and *torsion-free* provided $t(M)=0$. Let $M^{*} := \operatorname{Hom}(M,R)$ be the dual of $M$. The module $M$ is called *reflexive* provided the natural map $M\to M^{**}$ is an isomorphism. The module $M$ is called *maximal Cohen-Macaulay* if $\operatorname{depth}_RM = \dim R$.
A pair of modules $(M,N)$ is called [*rigid*]{} if for any integer $i\geq0$, $\operatorname{Tor}_i^R(M,N)=0$ implies $\operatorname{Tor}_j^R(M,N)=0$ for all $j\geq i$.
One defines the finite length index of the pair $(M,N)$ as : $$f_R(M,N) := \min\{ i |\ l(\operatorname{Tor}_j^T(M,N))<\infty \ \text{for $j \geq i$} \}$$
If $M,N$ are modules over a regular local ring $T$, then for any integer $i\geq0$ such that $f_T(M,N) \leq i$, we can define : $$\chi_i^T(M,N) = \sum_{j\geq i} (-1)^{j-i} l(\operatorname{Tor}_j^T(M,N))$$ When $i=0$ we simply write $\chi^T(M,N)$ or $\chi(M,N)$. Serre ([@Se]) introduced $\chi^R(M,N)$ forty years ago as a homological definition of intersection multiplicity for modules over a regular local ring and showed that it satisfied many of the properties which should hold for intersection multiplicities:
(Serre)\[Serre\] Let $T$ be a regular local ring such that $\hat T$ is admissible. Then for any pair of $T$-modules $M,N$ such that $l(M{\otimes}_TN)<\infty$, we have:
1. $\dim(M)+\dim(N) \leq \dim(T)$ (in other words, all modules are decent).
2. (Vanishing) If $\dim(M)+\dim(N)<\dim(T)$, then $\chi^{T}(M,N)=0$.
3. (Nonnegativity) It is always true that $\chi^{T}(M,N)\ge 0$.
4. (Positivity) If $\dim(M)+\dim(N)=\dim(T)$, then $\chi^{T}(M,N)>0$.
The following “long exact sequence for change of rings" plays a vital role in our proof of rigidity criterion (\[rg1\]). It follows from the famous Cartan-Eilenberg spectral sequence ([@Av2], 3.3.2).
\[longexact\] Let $R=T/f$ such that $f$ is a nonzerodivisor on $T$, and let $M,N$ be $R$-modules. Then we have the long exact sequence of $\operatorname{Tor}$s :
$$\begin{array}{ll}
...\to \operatorname{Tor}_{n}^R(M,N) \to \operatorname{Tor}_{n+1}^T(M,N) \to \operatorname{Tor}_{n+1}^R(M,N)\\
\to \operatorname{Tor}_{n-1}^R(M,N) \to \operatorname{Tor}_{n}^T(M,N) \to \operatorname{Tor}_{n}^R(M,N)\\
\to ... \\
\to \operatorname{Tor}_{0}^R(M,N) \to \operatorname{Tor}_{1}^T(M,N) \to \operatorname{Tor}_{1}^R(M,N) \to 0
\end{array}$$\
**The infinite projective dimension locus.**
Let $M$ be an $R$-module. One can define the infinite projective dimension locus of $M$ as $IPD(M):= \{ p\in \operatorname{Spec}(R) |\
\operatorname{pd}_{R_p}M_p =\infty \}$.
We gather some properties of this locus:
Let $R$ be a local hypersurface of dimension $d$. Let $M,N$ be $R$-modules. Let $\operatorname{Supp}_e(M,N)=\operatorname{Supp}(\operatorname{Tor}_{2d+2}^R(M,N))$ and $\operatorname{Supp}_o(M,N)=\operatorname{Supp}(\operatorname{Tor}_{2d+1}^R(M,N))$. Then we have :\
(1) $IPD(M)$ is a Zariski closed subset of $\operatorname{Spec}(R)$.\
(2) $IPD(M)\subseteq \operatorname{Supp}(M)\cap \operatorname{Sing}(R)$.\
(3) $\operatorname{Supp}_e(M,N) \cup \operatorname{Supp}_o(M,N) = IPD(M)\cap IPD(N)$.
\(1) Let $L = \operatorname{syz}_{d+1}^R(M)$. Let $F(M) = \{x\in R \ | \ L_{x} \ \text{is a free}
\ R_{x}\text{-module}\}$. For any prime $p$, $\operatorname{pd}_{R_p}M_p = \infty$ if and only if $L_p$ is not free if and only if $p \supset F(M)$. So $I(M) = V(F(M))$.\
(2) This is obvious.\
(3) Let $p\in \operatorname{Spec}(R)$ and localize at $p$. Then $R_p$ is a hypersurface. The assertion follows from the fact that $\operatorname{Tor}_{2d+2}^{R_p}(M_p,N_p)
= \operatorname{Tor}_{2d+1}^{R_p}(M_p,N_p)=0$ if and only if one of $M_p,N_p$ has finite projective dimension (see \[HW2\],theorem 1.9).
**The function $\theta^R(M,N)$.** \[\]
Let $R$ be a local hypersurface and assume $\hat R = T/(f)$ where $T$ is a regular local ring. The function $\theta^R(M,N)$ was introduced by Hochster (\[Ho1\]) for any pair of finitely generated modules $M,N$ such that $f_R(M,N)<\infty$ as: $$\theta^R(M,N) = l(\operatorname{Tor}_{2e+2}^R(M,N)) - l(\operatorname{Tor}_{2e+1}^R(M,N))$$ where $e$ is any integer $\geq (d+2)/2$. It is well known (see [@Ei]) that $\operatorname{Tor}^R(M,N)$ is periodic of period 2 after $d+1$ spots, so this function is well-defined. The theta function satisfies the following properties. First, if $M{\otimes}_RN$ has finite length, then: $$\theta^R(M,N) = \chi^T(M,N)$$ As a consequence of this fact and \[Serre\], we have :
\[decent\](Hochster) Let $R$ be an admissible hypersurface and $M,N$ be $R$-modules. Then $(M,N)$ intersect decently if and only if $\theta^R(M,N)=0$
Secondly, $\theta^R(M,N)$ is biadditive on short exact sequences, assuming it is defined. Specifically, for any short exact sequence: $$0 \to N_1 \to N_2 \to N_3 \to 0$$ and any module $M$ such that $f_R(M,N_i)<\infty$ for all $i=1,2,3$, we have $\theta^R(M,N_2) = \theta^R(M,N_1) +
\theta^R(M,N_3)$. Similarly, $\theta(M,N)$ is additive on the first variable. Hochster exploited these properties to give a sufficient condition for an cyclic module in $R$ to intersect decently. We will give his result here, in a slightly different form (his result was stated in terms of ideals):
(\[Ho1\],theorem 0.1)\[hochster\] Let $R$ be an admissible local hypersurface and $M,N$ be $R$-modules. Assume that :\
(1) $\operatorname{Supp}(M) \supset \operatorname{Sing}(R)$.\
(2) $[M] = 0$ in $\overline G(R)_{\mathbb{Q}}$.\
(3) $l(M {\otimes}_R N) <\infty$.\
Then $\theta^R(M,N)=0$ and $\dim M + \dim N \leq \dim R$ (In other words, $M$ is decent).
(sketch) Note that (1) and (3) imply $IPD(N) \subseteq \operatorname{Sing}(R)\cap
\operatorname{Supp}(N) \subseteq \operatorname{Supp}(M)\cap \operatorname{Supp}(N) = \{m_R\} $. In other words, $N$ has finite projective dimension on the punctured spectrum of $R$. Hence $f(M',N)<\infty$ for any $R$-modules $M'$. This together with the fact that $[M]=0$ in $\overline
G(R)_{\mathbb{Q}}$ and biadditivity of $\theta$ shows that $\theta(M,N)=0$. But over $T$, this means $\chi^T(M,N)=0$, so by Serre’s theorem, $ \dim M + \dim N \leq \dim T-1 = \dim R$.
It is worth noting that the direct summand conjecture could be solved if we have a “sufficiently good" criterion for a certain module in a certain hypersurface to intersect decently (see \[Ho1\]). We also note that Hochster’s theta function is closely related to the notion of “Herbrand difference" defined using stable cohomology by Buchweitz in [@Bu]. In fact, they agree up to sign.
**The depth formula.**
A result by Huneke and Wiegand showed that when all the positive $\operatorname{Tor}$ modules vanish, the depths of the modules satisfy a remarkable equation:
([@HW1], 2.5) Let $R$ be a complete intersection. Let $M,N$ be non-zero finitely generated modules over $R$ such that $\operatorname{Tor}_i^R(M,N)=0$ for all $i\geq 1$. Then: $$\operatorname{depth}(M) + \operatorname{depth}(N) = \operatorname{depth}(R)+\operatorname{depth}(M{\otimes}_RN)$$
**Vanishing of $\theta^R(M,N)$ implies rigidity.**
The main technical result of this section says that, when $\theta^R(M,N)$ can be defined and vanishes, then $(M,N)$ is rigid:
\[rg1\] Let $R$ be an admissible hypersurface and $M,N$ be $R$-modules such that $f_R(M,N)<\infty$ (so that $\theta^R(M,N)$ can be defined). Assume $\theta^R(M,N)=0$. Then $(M,N)$ is rigid.
Our main tool is a celebrated theorem first proved by Lichtenbaum (\[Li\]) except in a few cases. Those cases were completed by Hochster (\[Ho2\]):
(Lichtenbaum, Hochster)\[parchi\] Consider finitely generated modules $M,N$ over an admissible regular local ring $T$ and an integer $i$ such that $f_T(M,N)\leq i$ (so that $l(\operatorname{Tor}_j^T(M,N))<\infty$ for $j \geq i$). Then $\chi_i^T(M,N)\geq 0$ and it is $0$ if and only if $\operatorname{Tor}_j^T(M,N) = 0$ for all $j\geq i$.
In order to prove our rigidity result we will first need to prove a pivotal case, when all the relevant $\operatorname{Tor}$s have finite length, so we can apply theorem \[parchi\].
\[pivotal\] Let $R,M,N$ be as in \[rg1\]. Let $i$ be an integer such that $i
\geq f_R(M,N)$. Assume that $\theta^R(M,N)=0$ and $\operatorname{Tor}_i^R(M,N) =
0$. Then $\operatorname{Tor}_j^R(M,N) = 0$ for all $j\geq i$.
By completion we may assume $R=T/(f)$ where $T$ is an admissible regular local ring. We truncate the change of rings long exact sequence for $\operatorname{Tor}$s (\[longexact\]) as follows (note that all $\operatorname{Tor}^T(M,N)$ vanish after $d+1$ spots):\
$$\begin{array}{ll}
0 \to \operatorname{Tor}^R_{2e+2}(M,N) \\
\to \operatorname{Tor}_{2e}^R(M,N) \to \operatorname{Tor}_{2e+1}^T(M,N) \to \operatorname{Tor}_{2e+1}^R(M,N)\\
\to ... \\
\to \operatorname{Tor}_{i}^R(M,N) \to \operatorname{Tor}_{i+1}^T(M,N) \to \operatorname{Tor}_{i+1}^R(M,N) \to C \to 0
\end{array}$$\
It is easy to see that all the modules in this sequence have finite lengths. Therefore we can take the alternating sum of the lengths and get : $$l(C) + \chi_{i+1}^T(M,N) = (-1)^{2e+2-i}\theta^R(M,N) + l(\operatorname{Tor}_i^R(M,N)) = 0$$ This equation and theorem \[parchi\] forces $C=0$ and $ \operatorname{Tor}_j^T(M,N) =0$ for all $j\geq i+1$. The conclusion of the lemma now follows easily.
Now we can prove our rigidity result :
(of \[rg1\]) We use induction on $d = \dim R$. If $d=0$, $M,N$ both have finite length, so the previous lemma applies. Now assume $d \geq 1$ and $\operatorname{Tor}_i^R(M,N)=0$. Localizing at any $p\in Y(R)$, the punctured spectrum of $R$, and using the induction hypothesis (note that $R_p$ is at worst a hypersurface with dimension less than d, and $\operatorname{Tor}_j^{R_p}(M_p,N_p)=0$ for $j\geq f_R(M,N)$), we may conclude that $f_R(M,N) \leq i$. Again lemma \[pivotal\] can be applied to finish the proof.
Hypersurfaces with isolated singularity {#3}
=======================================
In this section we will investigate the vanishing of $\theta^R$ when $R$ is a local hypersurface with isolated singularity. Then $\theta^R(M,N)$ is defined for all pairs $(M,N)$ (since all higher $\operatorname{Tor}$ modules have finite length). In this situation $\theta^R$ defines a bilinear map from $G(R)_{\mathbb{Q}}\times
G(R)_{\mathbb{Q}}$ to $\mathbb{Q}$, hence by theorem \[rg1\] it vanishes whenever one of the modules is equivalent to $0$ in $\overline
G(R)_{\mathbb{Q}} = G(R)_{\mathbb{Q}}/{\mathbb{Q}}[R]$ (since $\theta^R(R,-) = 0$). We record this here for convenience.
Let $R$ be an admissible hypersurface with isolated singularity. Then $\theta^R(M,N)$ is always defined and vanishes if $M=0$ in $\overline G(R)_{\mathbb{Q}}$.
However, our investigation shall show that there are many more cases when $\theta^R$ vanishes. Our methods and inspirations come mostly from intersection theory. One key point is that we can often “move" the modules in to favorable position to show vanishing of $\theta^R$. Since moving in the Grothendieck group is much harder than in the Chow group, we need to make use of the Riemann-Roch map between the two groups.
We first need to review some facts about Chow groups. Let $X$ be a Noetherian scheme. Let $Z_iX$ be the free abelian group on the $i$-dimensional subvarieties (integral, closed subschemes) of $X$. For any $i+1$-dimensional subscheme $W$ of $X$, and a rational function $f$ on $W$, we can define an element of $Z_iX$ as follows: $$[{\textup{div}}(f,W)] = \sum_{V} {\textup{ord}}_V(f)[V]$$ summing over all codimension one subvarieties $V$ of $W$. Then the $i$-Chow group $\operatorname{CH}_i(X)$ is defined as the quotient of $Z_iX$ by the subgroup generated by all elements of the form $[{\textup{div}}(f,W)]$. Let $\operatorname{CH}_*(X) = \oplus \operatorname{CH}_i(X)$ and $\operatorname{CH}_*(X)_{\mathbb{Q}} = \operatorname{CH}_*(X){\otimes}_{\mathbb{Z}}{\mathbb{Q}}$. An $i$-cycle (resp. cycle class) is an element in $Z_i(X)$ (resp. $\operatorname{CH}_i(X)$) (by a slight abuse of notation, we also talk about a cycle as an element of $Z_i(X)_{\mathbb{Q}}$). When $X= \operatorname{Spec}(R)$, where $R$ is a ring, we simply write $\operatorname{CH}_*(R)$. We write $\operatorname{CH}^i(X)$ for $\operatorname{CH}_{d-i}(X)$, here $d =\dim X$.
If $R$ is local and is a homomorphic image of a regular local ring $T$ we have the important notion of [*[Todd class]{}*]{}. For any $R$-module $M$, let $F_*$ be the minimal free resolution of $M$ over $T$. The [*[Todd class]{}*]{} of $M$ is defined as: $$\tau_{R/T}(L) := \operatorname{ch}(F_*)([M])$$ Here $\operatorname{ch}()$ denotes the local Chern character. For much more details about the definition and properties of the Todd class, we refer to \[Fu\] or \[Ro\]. The Todd class has been very useful to prove such results as Serre’s Vanishing Conjecture and the New Intersection Theorem (\[Ro\]). The Todd class gives an isomorphism of $\mathbb{Q}$-vector spaces: $$\tau_{R/T} : G(R)_{\mathbb{Q}}\to \operatorname{CH}_*(R)_{\mathbb{Q}}$$ When $R,T$ are clear we will simply write $\tau$ for $\tau_{R/T}$. Recall that the Todd class satisfies the top terms property: $$\tau(M) = \sum_{\dim R/p =\dim M} l(M_p)[R/p] + \text{terms of lower dimension}$$
We collect below a number of facts that will be used frequently:
\[factschow\] Let $R$ be a local ring. Let $M$ be an $R$-module and $d=\dim R$.
1. If $d>0$ and $l(M)<\infty$ then $[M]=0$ in $G(R)_\mathbb{Q}$.
2. If $R$ is regular, $\operatorname{CH}^i(R)_\mathbb{Q} = 0$ for $0 < i \leq d$ and $\operatorname{CH}^0(R)_\mathbb{Q} = \mathbb{Q}$.
3. Assume that $R$ is a homomorphic image of a regular local ring.For any $P\in \operatorname{Spec}R$ we have $\tau^{-1}([R/P]) = [R/P] + \text{terms of lower dimension} $.
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1. It is easy to see that $[M] = l(M)[R/m]$ in $G(R)$. But take any $P\in \operatorname{Spec}R$ such that $\dim R/P =1$ and take $x \in m-P$, from the short exact sequence $$0 \to R/P \to R/P \to R/(P+(x)) \to 0$$ we then have $[R/(P+(x))] =0$ in $G(R)$. It follows that $[R/m]=0$ in $G(R)_\mathbb{Q}$, which gives what we want.
2. Since $R$ is regular, $G(R) = \mathbb{Z}[R]$ since any module has a finite resolution by free modules. The claim follows because $\tau([R]) = [R]$.
3. This follows immediately from the top term property.
Throughout the rest of this section we will assume that $R$ is a local hypersurface with isolated singularity. It is not hard to see that in all of our results we can safely pass to the completion of $R$ and the modules involved, so this is not a significant assumption.
\[dim1\] Assume $\dim R = 1$. Then $\theta^R(M,N)$ = 0 for all $N$ if and only if $M$ has constant rank,or equivalently, $[M]=0$ in $\overline{G}(R)$.
Let $p_1,p_2,...,p_n$ be the minimal primes of $R$. Then $G(R)_{\mathbb{Q}}$ has a basis consists of $[R/p_1],...,[R/p_n]$ by \[factschow\]. In particular, since $R$ has dimension 1 and is reduced, $[R] =
\sum_{1}^{n}[R/p_i]$. Let $\alpha_{ij} = \theta^R(R/p_i,R/p_j)$. For $i \neq j$, $p_i + p_j$ is $m$-primary, and it is not hard (for example, using \[ex2\]) to see that $\alpha_{ij} = l(R/(p_i+p_j)) >0$. Since $\theta^R(R,R/p_i) = 0$, we must have $\alpha_{ii} =
-\sum_{j\neq i} \alpha_{ij}$. Now, for a module $M$, let $[M] =
\sum a_i[R/p_i]$, here $a_i$ is the rank of $M_{p_i}$. If $a_1 =
a_2 = ... =a_n$, then $[M]=a_1[R]$, so $\theta^R(M,N) =
a_1\theta^R(R,N)= 0$. For the other direction, without loss of generality we may assume that $a_1$ is the largest among the $a_i$’s. Then since $0 = \theta^R(M,R/p_1) = \sum_{i=2}^{n}
\alpha_{1i}(a_i-a_1)$, we must have $a_i=a_1$ for all $i$.
Assume $\dim R =d > 1$ and $M,N$ are $R$-modules. Then $\theta^R(M,N)=0$ if $ \dim M \leq 1$.
Without affecting relevant issues we may complete and assume $R$ is a homomorphic image of a regular local ring. Since any module has a filtration by prime cyclic modules, we may assume that $M = R/P$ and $N=R/Q$ for some $P,Q \in \operatorname{Spec}R$. If $\dim R/P = 0$, so $P=m$, then $[R/P] = 0$ in $G(R)_\mathbb{Q}$, and $\theta$ vanishes. Also, we may assume $Q \neq 0$. If $Q$ is not contained in $P$, then $l(R/(P+Q)) <\infty$ because $\dim R/P
=1$, and since $\dim R/P +\dim R/Q \leq \dim R$ we have $\theta(R/P,R/Q) = 0$ by \[decent\]. So now we only need to consider the case $0 \neq Q \subset P$. We claim that there is cycle $\alpha = \sum l_i[R/Q_i] \in \operatorname{CH}^*(R)_\mathbb{Q}$ such that $\alpha = [R/Q]$ and $Q_i \nsubseteq P$. Consider the element $[R_P/Q] \in \operatorname{CH}^*(R_P)_\mathbb{Q}$. Since $R_P$ is regular, $[R_P/Q] = 0$. Therefore, formally, we have a collection of primes $q_i$ and elements $f_i$ and integers $n, n_i$ such that $n[R_P/Q]
= \sum {\textup{div}}(R_P/q_i,f_i)$. Now in $\operatorname{CH}^*(R)_\mathbb{Q}$ we will have $\sum {\textup{div}}(R/q_i,f_i) = n[R/Q] + \sum n_i[R/Q_i]$, with $Q_i
\nsubseteq P$, which proves our claim. The fact that $[R/Q] =
\sum_i l_i[R/Q_i]$ in $\operatorname{CH}^*(R)_\mathbb{Q}$ means that in $G(R)_\mathbb{Q}$, $[R/Q]=\sum_i l_i[R/Q_i]+ \text{terms of lower
dimension}$ (by part (3) of the proposition \[factschow\]). By the argument at the beginning of the proof, $\theta^R(R/P,R/Q_i)=0$ and we may conclude our proof by induction on $\dim R/Q$.
The above argument is true whenever $R_P$ is regular. This argument could be thought of as an algebraic “moving lemma" for $\operatorname{Spec}(R)$.
When the $R$ contains a field, we can of course apply the real moving lemma.
\[moving\] Suppose that $(R,m,k)$ contains a field. Then $\theta^R(R/P,R/Q)=0$ if $\dim R/P + \dim R/Q \leq \dim R$.
We first need to make some reductions. Assume we have a counterexample on $R$. We can first make a faithfully flat extension to replace $k$ by an algebraically closed field and then complete to get to the case of $R = k[[x_0,...x_d]]/(f)$, and $k$ is algebraically closed. The condition of isolated singularity is preserved by faithfully flat extension (cf Lemma 2.7, [@Wi]). Then by a theorem of Artin (see [@CS], theorem 1.6), $R =
\hat{S}$, where $S$ is local hypersurface with isolated singularity, and is essentially of finite type over $k$. But we can descend our example to $S^h$, the Henselization of $S$, by a standard argument (see [@Ho3] or [@Du1]). The only issue is how to descend the resolutions of the modules, which may be infinite. However, note that since the resolutions of our modules are eventually periodic, we only need to keep track of a finite part. Then since $S^h = \underrightarrow{\lim} S_n$, where each $S_n$ is an étale neighborhood of $S$, we have a counterexample in some $S_n$. Thus we may assume $R$ is essentially of finite type over an algebraically closed field $k$. Let’s say $R = A_m$, where $A$ is a finite $k$-algebra and $m$ is a maximal ideal in $A$. Since $R$ has isolated singularity, we have $\operatorname{Sing}(A)$ is a disjoint union of $\{m\}$ and some closed subset $Y \subset \operatorname{Spec}A$.
Let $X = \operatorname{Spec}(A) -\{m\}-Y$. Then $X$ is a smooth quasi-projective variety, so by the Moving Lemma (see [@Fu],11.4) one can find a cycle $\alpha = \sum_i n_i V(Q_i)$ in $\operatorname{CH}^*(X)$ such that for each $i$, $\dim (V(Q_i)\cup V(P))\leq 0$, that is the intersection consists only of points in $X$. When we restrict all the cycles and divisors to $\operatorname{Spec}(R)$ we will have $[R/Q] = \sum_j [R/Q_j]$ in $\operatorname{CH}^*(R)$ and for all $j$, $V(P)\cup V(Q_i) \subset \{m\}$. In $G(R)_{\mathbb{Q}}$ this means $[R/Q] = \sum_j [R/Q_j] +
\text{terms of lower dimension}$. Since $l(R/P+Q_j) <\infty$ and $\dim R/P + \dim R/Q_j \leq \dim R$ we have $\theta^R(R/P,R/Q_j)=0$ by \[decent\] and an induction on $\dim
R/Q$ finishes the proof.
\[dim2,3\] If $\dim R=2$ or $\dim R=3 $ and $\operatorname{CH}^1(R)_{\mathbb{Q}} = 0$ or $\dim R =4$ and $R$ contains a field, then $\theta^R(M,N)=0$ for all pairs $(M,N)$.
It suffices to assume that $M,N$ are cyclic prime modules, let’s say $M=R/P,N=R/Q$. Then by the previous theorems we only need to worry if both of them have dimension at least $2$. In the first case, they must be $R$ (note that since $R$ is normal and local, it is a domain), thus $\theta^R$ certainly vanishes. In the second case, assume that $\dim R/P=2$ (otherwise $R/P$ would be $R$). We can then complete $R$ without loss of generality. Then part (3) of \[factschow\] shows that in $G(R)$, $[R/P]$ is equal to a formal sum of cyclic primes of dimension $\leq 1$, forcing $\theta^R(R/P,R/Q)$ to be $0$. Finally, if $\dim R=4$ and $R$ contains a field we can apply \[moving\] and assume $\dim
R/P + \dim R/Q \geq 5$. Then one of the primes, say $P$ is height $1$ (if the minimal height is $0$ the assertion is trivial). We will be done if we can show that $\operatorname{CH}^1(R)=0$. But by the Grothendieck-Lefschetz theorem, the Picard group of $X = \operatorname{Spec}(R)-{m}$ is $0$. Since $X$ is regular, the Picard group of $X$ is the same as the $\operatorname{CH}^1(X)=
\operatorname{CH}^1(R)$.
Let $A=\mathbb{R}[x,y,z,w]/(F)$ where $F$ is a non-degenerate quadratic form of signature $2$ (for example, if $F=x^2+y^2+z^2+2w^2$). Let $m=(x,y,z,w)$ and $R=A_m$. By 10.3 and 11.7 of [@Fo], $Cl(R)=0$. So $\theta$ vanishes on any pair of modules over $R$.
Our next result is an algebraic Bertini type theorem for the vanishing of $\theta^R$. It is most useful when $M=N$ (so in a sense when moving them apart is the hardest).
Assume that $\dim R \geq 2$. Suppose $M,N$ are $R$-modules such that there is an element $x \in {\textup{Ann}}{M} \cap {\textup{Ann}}(N)$ such that $R/(x)$ is still a hypersurface with isolated singularity. Then $\theta^R(M,N)=0$.
Take $L = \operatorname{syz}_R^{2d}(N)$. Then $\theta^R (M,N) = \theta^R(M,L)$. Also, $L$ is maximal Cohen-Macaulay: in particular, $x$ is a nonzerodivisor on $L$. Thus $\operatorname{Tor}_i^R(M,L) = \operatorname{Tor}_i^{R/x}(M,L/(x))$, so it is enough to prove that $\theta^{R/(x)}(M,L/(x)) = 0$ (the assumption that $R/(x)$ is still a hypersurface with isolated singularity ensures that $\theta^{R/(x)}$ is well-defined). We define a map $\alpha : G(R) \to G(R/(x))$ as follows: $$\alpha([M]) = [\operatorname{Tor}_0^R(M,R/(x))] - [\operatorname{Tor}_1^R(M,R/(x))]$$ We need to show that $\alpha$ is well-defined. The only thing that needs to be checked is that if $0 \to M_1 \to M_2 \to M_3 \to 0$ is a short exact sequence of $R$-modules, then $\alpha([M_2]) =
\alpha([M_1]) + \alpha([M_3])$. But this follows from tensoring the exact sequence with $R/(x)$, and note that $\operatorname{Tor}_i^R(M,R/(x))
=0$ for all $R$-modules $M$ and $i>1$.
It is easy to see that $\alpha([N]) = [0]$ and $\alpha([L]) = [L/(x)]$ (since $x$ kills $N$ and is $L$-regular, respectively). But since $L = \operatorname{syz}^{2d}(N)$ we have $[L] = [N] + n[R]$ for some integer $n$. Applying $\alpha$ we get: $ [L/(x)] = n[R/(x)]$ in $G(R/(x))$. But then $\theta^{R/(x)}(M,L/xL) = 0$, which is what we want.
Now we will consider a special case, when $R$ is a local ring of an affine cone of a projective variety. That is, $R=A_{\mathfrak
m}$, where $A$ is a graded hypersurface over a field $k$ whose homogeneous maximal ideal is $\mathfrak m$. The following result by Kurano is very helpful (cf. [@Ku1],Thm 1.3).
(Kurano)\[kurano\] Let $A,R$ be as above. Let $X = \operatorname{Proj}(A)$. Assume that $R$ has isolated singularity.Then $X$ is smooth and $\operatorname{CH}^*(X)$ becomes a (graded) commutative ring with the intersection product. Let $h
\in \operatorname{CH}^1(X)$ represent the hyperplane section (alternatively, the first Chern class of the invertible sheaf ${\mathcal O}_X(1)$). Then there is a graded isomorphism of ${\mathbb{Q}}$-vector spaces from $\operatorname{CH}^*(X)/h\operatorname{CH}^*(X)$ to $\operatorname{CH}^*(R)$.
Let $X,R$ as above and $i\geq 0$ an integer. If $\operatorname{CH}^{i+1}(X)_{\mathbb{Q}} = \mathbb{Q}$ then $\operatorname{CH}^{i+1}(R)_{\mathbb{Q}} = 0$.
We only need to observe that in $\operatorname{CH}^*(X)_{\mathbb{Q}}$, multiplication by $h$ is a nonzero map from $\operatorname{CH}^i(X)_{\mathbb{Q}}$ to $\operatorname{CH}^{i+1}(X)_{\mathbb{Q}}$ (just take $h^{i+1}$). Hence if $\operatorname{CH}^{i+1}(X)_\mathbb{Q} = \mathbb{Q}$ then $ \operatorname{CH}^{i+1}(X)_\mathbb{Q} =h\operatorname{CH}^{i}(X)_\mathbb{Q}$, so $\operatorname{CH}^{i+1}(R)_\mathbb{Q}=0$ by Kurano’s result.
This allow us to exploit many results in the literature about the Chow groups of projective varieties. For example, we have the following:
([@ELV],2.3) Let $X \subset \mathbb{P}^d_k$ be an irreducible hypersurface of degree $s$. If: $${{i+s} \choose {i+1}} \leq d-i$$ Then $\operatorname{CH}^{d-1-i}(X)_{\mathbb{Q}} = \mathbb{Q}$.
Let $X$ be a smooth hypersurface of degree $s$ in $\mathbb{P}^d_k$. Let $R$ be the local ring at the homogenous maximal ideal of the affine cone of $X$. Let $n$ be the biggest integer such that: $${{n+s} \choose {n+1}} \leq d-n$$ Then $\theta^R(M,N) = 0$ if $\dim M \leq n+1$.
By the two previous results we have $\operatorname{CH}^{d-1-i}(R)_{\mathbb{Q}} = 0$ for $i\leq n$, in other words, $\operatorname{CH}_{i}(R)_{\mathbb{Q}} = 0$ for $i\leq n+1$. Thus $\tau([M]) = 0$, hence $[M] = 0$ in $G(R)_{\mathbb{Q}}$.
When $s=2$, we have $n+1 = \lfloor d/2 \rfloor $. This gives a strong version of \[moving\].
Next we want to discuss a conjecture, attributed to Hartshorne (see [@Ha1], page 142), which could be relevant to our interest:
\[HarConj\](R.Hartshorne) Let $X$ be a smooth projective, complete intersection variety in $\mathbb{P}^n_k$. Then $\operatorname{CH}^i(X)_{\mathbb{Q}}= \mathbb{Q}$ for $i<{\dim X}/2$.
It is interesting to observe that Hartshorne’s conjecture, together with \[moving\], show that if $\dim R$ is even, when $R$ is the local ring at the origin of the affine cone of $X$, then $\theta^R$ always vanishes.
(of Hartshorne’s conjecture) Let $X$ be a smooth hypersurface in $\mathbb{P}^d_k$. Assume that $d$ is even. Let $R$ be the local ring at the origin of the affine cone of $X$. Then $\theta^R$ always vanishes.
Let $d=2n$. We have $\dim X = 2n-1$, so by Hartshorne’ conjecture with $Y=\mathbb{P}^d_k$ we have $\operatorname{CH}^i(X)_{\mathbb{Q}} ={\mathbb{Q}}$ for $i \leq n-1$. Thus $\operatorname{CH}^i(R)_{\mathbb{Q}} = 0$ for $i\leq n-1$, in other words, $\operatorname{CH}_i(R)_{\mathbb{Q}} = 0$ for $i \geq n+1$ . So in the Grothendieck group $G(R)_{\mathbb{Q}}$, any module can be represented as a sum of cyclic prime modules of dimension $\leq n$. But since $\dim R = 2n$, for any such pair of modules $(R/P,R/Q)$ we must have $\theta^R(R/P,R/Q)=0$ by \[moving\].
In view of this and our knowledge of dimensions $2$ and $4$, we feel it is reasonable to make:
\[DaoConj\] Let $R$ be a hypersurface with isolated singularity. Assume that $\dim R$ is even and $R$ contains a field. Then $\theta^R$ always vanishes.
Finally we present some classes of hypersurfaces on which the above conjecture can be readily verified. First we observe that the values of $\theta^R$ only depends on its values on pairs of maximal Cohen-Macaulay (MCM) modules (as one can replace the modules by their high syzygies). The group of isomorphism classes of indecomposable MCM modules has been a subject of intense study for a while, and in the case when it has been computed, we may exploit those results for our purpose. We note that the theta function is closely related to the notion of “Herbrand difference" defined using stable cohomology by Buchweitz in [@Bu]. In fact, it is not hard to see that they agree up to sign (We would like to thank Ragnar-Olaf Buchweitz for explaining to us the connection).
Finally, we want to discuss the case of simple singularities. We say that a complete local ring $R$ is a simple singularity if it is isomorphic to $T/(f)$, where $T=k[[x_0,x_1,...,x_d]]$, $k$ is algebraically closed of charateristic $0$, and $f$ has one of the following forms :\
$ (A_n)\ \ \ \ x_0^2+x_1^{n+1} + x_2^2+...+x_d^2 \ \ \ \ \ \ (n\geq 1)$\
$ (D_n)\ \ \ \ x_0^2x_1 + x_1^{n-1} +x_2^2+...+x_d^2 \ \ \ \ (n\geq 4)$\
$ (E_6)\ \ \ \ x_0^3 + x_1^4 +x_2^2+...+x_d^2 \ \ \ \ $\
$ (E_7)\ \ \ \ x_0^3 + x_0x_1^3 +x_2^2+...+x_d^2 \ \ \ \ $\
$ (E_8)\ \ \ \ x_0^3 + x_1^5 +x_2^2+...+x_d^2 \ \ \ \ $\
In the hypersurface case, simple singularity is the same as finite representation type, that is, the group of isomorphism classes of indecomposable MCM modules is finite. In this case, the Grothendieck group of MCM modules can be computed completely (see [@Yo], 13.10). One striking feature is that in even dimensions, all the Grothendieck groups are torsion after we kill the class of $[R]$. Thus we have the following result, which confirm Conjecture \[DaoConj\]:
\[MainConj\] Let $R$ be a hypersurface with isolated, simple singularity of even dimension. Then $\theta^R$ always vanishes.
Rigidity over hypersurfaces {#4}
===========================
By the virtue of theorem \[rg1\] and the results in the previous section, we have a lot of results about rigidity of modules when the hypersurface has an isolated singularity. We collect them in a theorem:
\[rigidiso\] Let $R$ be an admissible hypersurface with isolated singularity and $M,N$ be $R$-modules.
1. If $[M] = 0$ in $\overline G(R)_{\mathbb{Q}}$, then $M$ is rigid.
2. If $\dim R=1$ and $M$ has constant rank then $M$ is rigid.
3. If $\dim R=2$ or $\dim R=3 $ and $\operatorname{CH}^1(R)_{\mathbb{Q}} = 0$ or $\dim R =4$ and $R$ contains a field, then any module is rigid.
4. If $\dim M \leq 1$ then $M$ is rigid.
5. If $R$ contains a field and $\dim M +\dim N \leq \dim R$, then $(M,N)$ are rigid.
6. If there is an element $x \in {\textup{Ann}}{M} \cap {\textup{Ann}}(N)$ such that $R/(x)$ is still a hypersurface with isolated singularity. Then $(M,N)$ are rigid.
On general hypersurfaces, however, we need to be more careful about using the function $\theta^R$. Typically, we need some extra conditions to show that $\theta$ is defined for all the modules in the short exact sequences involved. We will give plenty of examples to show that these conditions are unavoidable. One very effective method is to use induction on dimension to show that the $\operatorname{Tor}$ modules become $0$ on the punctured spectrum of $R$, which means that they have finite lengths and we can use $\theta^R$.
We will now state two immediate corollaries of \[rg1\]. The first appeared implicitly in the work of Lichtenbaum, (\[Li\]) :
\[lichten\] Let $R$ be an admissible hypersurface and $M,N$ be finitely generated $R$-modules. If $M$ or $N$ has finite projective dimension, then $(M,N)$ are rigid.
\[rg2\] Let $R$ be an admissible hypersurface and $M,N$ be finitely generated $R$-modules. Assume that $\operatorname{pd}_{R_p}M_p <\infty$ for all $p\in Y(R)$ (the punctured spectrum of $R$) and $[N]=0$ in $\overline{G}(R)_{\mathbb{Q}}$. Then $(M,N)$ is rigid.
The first assumption ensures that $f_R(M,N)<\infty$ for all $N$, hence $\theta^R(M,N)$ can be defined for all $N$. Then the second assumption forces $\theta^R(M,N) = 0$.
Another immediate corollary of our result is the first “rigidity" theorem in a paper of Huneke and Wiegand (\[HW1\].
\[HWrg\] (\[HW1\],2.4). Let $R$ be an admissible hypersurface and $M,N$ be $R$-modules. Assume:\
(1) $M{\otimes}_RN$ has finite length.\
(2) $\dim(M) + \dim(N) \leq \dim(R)$.\
Then $(M,N)$ are rigid.
Suppose $R=T/(f)$ where $T$ is regular local. In this case $\theta^R(M,N) =\chi^T(M,N)$, so by Serre’s vanishing theorem, it must be $0$.
The next result introduces a class of rigid modules not necessarily having finite projective dimension. To state it, recall the definition: $IPD(M):= \{ p\in \operatorname{Spec}(R) |\ \operatorname{pd}_{R_p}M_p =\infty \}$ (section 1)\
\[rig1.1\] Let $R$ be an admissible hypersurface, and $M$ be an $R$-module such that $[M] =0$ in $\overline G(R)_{\mathbb{Q}}$. Assume that $IPD(M)$ is either $\emptyset$ or is equal to $\operatorname{Sing}(R)$. Then $M$ is rigid.
If $IPD(M)=\emptyset$ then $\operatorname{pd}_RM<\infty$, so there is nothing to prove. Assume that $IPD(M)= \operatorname{Sing}(R) \neq 0$. We again use induction on $d = \dim R$. If $d=0$ the condition that $[M]=[0]$ in $\overline G(R)_{\mathbb{Q}}$ implies that $\theta^R(M,N)$ is defined and equal to $0$ for any $R$-module $N$. Suppose $d>0$ and $\operatorname{Tor}_i^R(M,N)=0$ for some $i$. We localize at any prime $p\in Y(R)$. Both conditions on $M$ localize, so by the induction hypothesis, $\operatorname{Tor}_j^{R_p}(M_p,N_p)=0$ for $j \geq i$. This forces either $\operatorname{pd}_{R_p}M_p$ or $\operatorname{pd}_{R_p}N_p$ to be finite (see theorem 1.9,\[HW2\]). But since $I(M) = \operatorname{Sing}(R)$, $N$ must has finite projective dimension on $Y(R) \cap \operatorname{Sing}(R)$. So $N$ has finite projective dimension on $Y(R)$, hence $\theta(M,N)=0$, finishing the proof.
The following will be useful for our application to torsion of tensor products.
\[vanishinglem\] Let $R$ be an admissible hypersurface and $M,N$ be $R$-modules. Assume that:
1. $\operatorname{Tor}_1^R(M,N)=0$
2. $\operatorname{depth}(N)\geq 1$ and $\operatorname{depth}(M{\otimes}_RN)\geq 1$.
3. $f_R(M,N)<\infty$.
Then $\operatorname{Tor}_i^R(M,N)=0$ for $i\geq 1$.
The depth assumptions ensure that we can choose $t$ a nonzerodivisor for both $N$ and $M{\otimes}_RN$. Let $\overline{N} = N/tN$. Tensoring the short exact sequence : $$\xymatrix {0 \ar[r] &N \ar[r]^{t} &N \ar[r] &\overline{N} \ar[r] &0}$$ with $M$ and using (1) we get : $$\xymatrix{0 \ar[r] &\operatorname{Tor}_1^R(M,\overline{N}) \ar[r] &M{\otimes}_RN \ar[r]^{t} &M{\otimes}_RN \ar[r] &M{\otimes}_R \overline{N} \ar[r] &0}$$ which shows that $\operatorname{Tor}_1^R(M,\overline{N}) = 0$. But condition (3) is satisfied for both of the pairs $(M,N)$ and $(M,\overline{N})$ and so : $$\theta^R(M,\overline{N}) = \theta^R(M,N) - \theta^R(M,N)= 0$$ The conclusion then follows from theorem \[rg1\] and Nakayama’s lemma.
\[dim1rg\] Suppose $\dim R=1$ and $M$ is an $R$ module of constant rank. Then $\theta^R(M,N)$ is defined and equal $0$ for all $N$. Consequently, $M$ is rigid.
By ([@HW1],1.3), we can find a homeormorphism : $\beta: M \to
F$ such that $F$ is free and the kernel $K$ and cokernel $C$ are torsion module. Since $\dim R=1$, both $K,C$ have finite length. Let $D = \operatorname{Im}\beta$. From the short exact sequence: $$0 \to D \to F \to C \to 0$$ we know that for any module $N$, $f_R(M,N) <\infty$ and $\theta^R(D,N) = \theta^R(F,N)-\theta^R(C,N)=0$ ($\theta^R(C,N)=0$ by \[HWrg\]). Now we look at the short exact sequence: $$0 \to K \to M \to D \to 0$$ and it follows that $f_R(M,N) <\infty$ and $\theta^R(M,N) = \theta^R(K,N)+ \theta^R(D,N) = 0$.
The next result shows some connection between rigidity, decency and a property of modules first studied by Auslander ([@Au]).
\[rigidandproper\] Let $R$ be an admissible hypersurface. For a Cohen-Macaulay $R$-module $M$, the following are equivalent:
1. $(M,N)$ is rigid for all $N$ such that $l(M{\otimes}_RN)<\infty$.
2. Every $M$-sequence is an $R$-sequence.
3. $M$ is decent ($\dim M +\dim N \leq \dim R$ for all $N$ such that $l(M{\otimes}_RN)<\infty$).
4. $\theta^R(M,N)=0$ for any $N$ such that $l(M{\otimes}_RN)<\infty$.
Assume (1). Then we can prove (2) by adapting the argument in Auslander paper ([@Au],4.1) (which assumed that $(M,N)$ is rigid for all $N$ but did not need $M$ to be Cohen-Macaulay). We give a sketch here. Let $X$ be the free resolution of $M$. Let $\bold x$ be a full $M$-sequence (so its length is $\dim M$ and $l(M/(\bold {x})) <\infty$). Let $Y$ be the Koszul complex on $\bold x$. Then the total complex $X{\otimes}_RY$ is acyclic. Filtering that complex by $F_p(X{\otimes}Y) = \sum_{p\geq r}\sum
_qX_r{\otimes}_RY_q$ ,we obtain a spectral sequence with $E^2_{p,q}
= H_p(X{\otimes}_RH_q(Y))$. By assumption $H_n(X{\otimes}_RY)=0$, so $E^{\infty}_{p,q}=0$ for $p,q>0$. We also have $E^2_{p,q}=0$ for $p,q<0$. Hence $E^2_{1,0}=E^i_{1,0}$ for $i>1$. But $E^{\infty}_{1,0}=0$ which implies that $H_1(X{\otimes}_RH_0(Y)) =
E^2_{1,0}=0$. Since $M{\otimes}_RH_0(Y) = M/(\bold x)$ has finite length, we must have $0 = H_p(X{\otimes}_RH_0(Y)) = E^2_{p,0}$ for all $p \geq 1$. By induction we will have: $$0 = E^2_{p,q} = H_p(X{\otimes}_RH_q(Y))$$ for all $p\geq 1$ and $q\geq 0$ (note that since $\bold{x}R$ kills all the modules $H_q(Y)$, $M{\otimes}_RH_q(Y)$ has finite length so we can apply (1)). But since $E^{\infty}_{p,q}=0$ for $p,q>0$, we have: $$0 = E^2_{0,q} = H_0(X{\otimes}_RH_q(Y)) = M{\otimes}_RH_q(Y)$$ for each $q>0$. This forces $H_q(Y)=0$ for $q>0$, hence $\bold x$ is an $R$-sequence. If $\bold x$ is not a full $M$-sequence, we can always add more elements and reach the same conclusion.\
Assume (2). Let $N$ be an $R$ module such that $l(M{\otimes}_RN)<\infty$. Then we can find a full system of parameters $\bold x$ on $M$ such that $\bold x \subset {\textup{Ann}}(N)$. As $M$ is Cohen-Macaulay, $\bold x$ is also a full $M$-sequence. By assumption, $\bold x$ is an $R$-sequence. Thus: $$\dim N \leq \dim R/(\bold x)= \dim R -\dim M$$ Finally, $(3)\Rightarrow (4) \Rightarrow (1)$ is just the proof of \[HWrg\].
This result gives necessary conditions for rigidity that are easier to check then rigidity itself. The conditions (2) and (3) are quite familiar. They have played a vital role in a group of theorems and conjectures known as the “homological conjectures". Specifically we have (see [@Ho4], [@PS], [@Ro]):
\[FP1\] Let $R$ be a local ring and $M$ an $R$-module of finite projective dimension. Then every $M$-sequence is an $R$-sequence.
\[FP2\] Let $R$ be a local ring and $M$ an $R$-module of finite projective dimension. Then $\dim M +\dim N \leq \dim R$ for any $N$ such that $l(M{\otimes}_RN)<\infty$.
Theorem \[FP1\] is a consequence of the “intersection theorem", proved by Peskine and Szpiro for $R$ containing a field and by Roberts for the general case. Conjecture \[FP2\] is still wide open even for hypersurfaces in ramified regular local rings.
Applications {#5}
============
In this section we apply our results on a number of topics which involve decency or rigidity. Some of the results below are known, we just give a different proof. Some are new, to the best of our knowledge.\
**Rigidity of $^eR$.**\
Suppose $R$ is a hypersurface of characteristic $p>0$. Let $F:R\to
R$ be the Frobenius map. We denote by $F^e: R\to R$ the $e$-th iteration of $F$. Then $^eR$ is the $R$-module $R$ whose module structure is given by $F^e$. The following is a special case of Theorem 1 in [@AM] (is was proved for complete intersections):
\[AM\] $^eR$ is rigid. If $R$ is singular, and $M$ is an $R$-module such that $\operatorname{Tor}_i^R(M,^eR)=0$ for some $i>0$, then $pd_RM<\infty$.
To prove the first statement we will apply \[rig1.1\]. First, by making faithfully flat extensions we may assume that $R$ is complete and has a perfect residue field. Then we just need to make sure that the conditions in \[rig1.1\] are satisfied. Since $\operatorname{pd}_R{^eR}<\infty$ if and only if $R$ is regular, and $F$ commutes with localization, we may conclude that $IPD(^eR)=\operatorname{Sing}(R)$. And it is known that $[^eR] = p^{e\dim R}[R]$ in $G(R)$ (this is actually non-trivial, see [@Ku1], the fact that $R$ is complete with perfect residue field is needed here). The second statement follows because if $\operatorname{Tor}_i^R(M,^eR)=0$ for some $i>0$, then by rigidity all the higher $\operatorname{Tor}$s vanish. Thus one of the two modules $M,^eR$ has to have finite projective dimension, and since $R$ is singular, it must be $M$.
**Torsion on tensor products.**\
In this part we shall apply our rigidity results to show that tensor products rarely have good depths:
\[vanishingiso\] Let $R$ be an admissible hypersurface with dimension $d\geq2$. Let $M,N$ be $R$-modules. Assume that:\
(1) $R$ has isolated singularity.\
(2) $M{\otimes}_RN$ is torsion-free.\
(3) $\operatorname{depth}_R M{\otimes}_RN \geq 2$.\
Then $\operatorname{Tor}_i^R(M,N)=0$ for $i\geq 1$.
Condition (1) makes sure that $\theta^R(M,N)$ is defined for any pair of modules $(M,N)$. We may assume that $M,N$ are torsion-free (This argument was basically in the proof of (2.4,\[HW1\]), we repeat it here for the reader’s convenience). Let $t(M)$ be the torsion part of $M$ and $\overline{M}=M/t(M)$. Tensoring the exact sequence : $$0 \to t(M) \to M \to \overline{M} \to 0$$ with $N$, we get: $$\xymatrix{ \operatorname{Tor}_1^R(\overline{M},N) \ar[r] &t(M){\otimes}_RN \ar[r]^{\alpha} &M{\otimes}_RN \ar[r]^{\beta} &\overline{M}{\otimes}_RN \ar[r] &0}$$ Since $t(M){\otimes}_RN$ is torsion and $M{\otimes}_RN$ is torsion-free, $\alpha$ is the $0$ map and $\beta$ is an isomorphism. So $\overline{M}{\otimes}_RN$ satisfies all the condition (2) and (3). By the torsion-free case,$\operatorname{Tor}_1^R(\overline{M},N)=0$, so $t(M){\otimes}_RN=0$ and that can only mean $t(M)=0$. By symmetry $N$ is also torsion-free. Now there is an exact sequence: $$0 \to M \to R^{\lambda} \to M_1 \to 0$$ Here $\lambda = \lambda(M^*)$. This exact sequence is called the *pushforward* of $M$ (see [@HJW]). By tensoring the pushforward exact sequence of $M$ with $N$ , we get: $$0 \to \operatorname{Tor}_1^R(M_1,N) \to M{\otimes}_RN \to N^{\lambda} \to M_1{\otimes}_RN \to 0$$ By condition (1) $N$ is generically free, so $\operatorname{Tor}_1^R(M_1,N)$ is torsion, and it must be $0$ since $M{\otimes}_RN$ is torsion-free. Since $\operatorname{depth}_R M{\otimes}_RN \geq 2$, $\operatorname{depth}_RN\geq1$ (since $N$ is torsion-free and $d\geq2$), we must have $\operatorname{depth}_R M_1{\otimes}_RN \geq 1$. Now the theorem follows from lemma \[vanishinglem\].
In the dimension $2$ case, we can do a little bit better:
\[dim2normal\] Let $R$ be an admissible hypersurface of dimension $2$. Assume further that $R$ is normal. Let $M,N$ be $R$ modules such that $M{\otimes}_R N$ is torsion-free. Then $\operatorname{Tor}_i^R(M,N)=0$ for $i\geq 1$.
We may assume $M$ is torsion-free. Now, let $M_1$ be the pushforward of $M$. We have $\operatorname{Tor}_1^R(M_1,N)=0$. By the fact that $R$ is an isolated singularity of dimension 2, every module is rigid, so $\operatorname{Tor}_i^R(M_1,N)=0$ for $i>1$, which gives the desired conclusion.
It was asked in [@HW1] (4.1 and the discussion before 5.3), whether or not there are two non-free reflexive modules over a hypersurface of dimension $2$ such that their tensor product is torsion-free. In general, such pairs of modules exist. For example, let $R=k[[x,y,z]]/(xy)$ and $M=N=R/(x)$. But with the extra assumptions of the above, such modules can not exist. For by the conclusion, one of them must have finite projective dimension, and, being maximal Cohen-Macaulay, must be free.
To illustrate the efficiency of using $\theta^R$ for rigidity, we will give a short proof of one of the key results of [@HW1]:
\[HWmain\]\[[@HW1],2.7\] Let $R$ be an admissible hypersurface and $M,N$ be $R$-modules, at least one of which has constant rank. If $M{\otimes}_RN$ is reflexive, then $\operatorname{Tor}_i^R(M,N)=0$ for $i>0$.
We will use induction on $d = \dim R$. If $d=0$, the constant rank condition means one of the modules must be free, and the conclusion follows trivially. Now assume $d\geq 1$. By the induction hypotheses, $l(\operatorname{Tor}_i^R(M,N))<\infty$ for $i>0$. We can assume both $M,N$ are torsion free by standard arguments (see \[vanishingiso\]). In particular, they must have depth at least $1$. Let $M_1$ be the pushforward of $M$: $$0 \to M \to F \to M_1 \to 0$$ Then by the same reason as in proof of \[vanishingiso\], we have $\operatorname{Tor}_1^R(M_1,N)=0$. So we have: $$0 \to M_1 {\otimes}_RN \to F{\otimes}_RN \to M{\otimes}_RN \to 0$$ By the depth lemma, we get $\operatorname{depth}(M_1{\otimes}_RN)\geq 1$. Finally, since $l(\operatorname{Tor}_i^R(M,N))<\infty$ for $i>0$ we must have $f_R(M,N)<\infty$. Applying \[vanishinglem\] for $M_1$ and $N$, we get $\operatorname{Tor}_i^R(M_1,N)=0$ for $i>1$, which implies $\operatorname{Tor}_i^R(M,N)=0$ for $i>0$.
**Hypersurfaces in Projective spaces.**\
Our study so far has made use of some results from projective geometry. It is perhaps fair to try to give some thing back, so we will use our result to gain some information on projective hypersurfaces. We have:
\[projhyper\] Let $k$ be a field. Let $X \subset {\mathbf{P}}_k^n$ be a smooth hypersurface. Let $U,V$ be subvarieties of $X$ such that $\dim U +
\dim V \geq \dim X$. Assume that $[U] = h.[U']$ in $\operatorname{CH}^*(X)_{\mathbb{Q}}$, here $h$ is the hyperplane section. Then $U\cap V \ne \emptyset$.
Let $X = Proj(A)$ where $A = k[x_0,...,x_n]/(F)$. Let $R$ be the local ring at the origin of $A$. Suppose $P,Q \in \operatorname{Spec}(R)$ define $U,V$ respectively. Our assumption becomes : $R$ is a hypersurface with isolated singularity of dimension $n$, $\dim R/P +\dim
R/Q\geq n+1$, and $[R/P] = 0$ in $\operatorname{CH}^*(R)_{\mathbb{Q}}$ (by Kurano’s theorem \[kurano\]). We need to show that $R/P$, as a *module*, is decent. Now, by \[hochster\] we are done if we can show: there exist a module $M$ such that $M =0$ in $\overline
G(R)_{\mathbb{Q}}$, and $\operatorname{Supp}(M) = \operatorname{Supp}(R/P)$. We first pick $M
= R/P$. This may not guarantee that $M =0$ in $\overline
G(R)_{\mathbb{Q}}$, because by Riemann-Roch:
$$\tau(M) = [R/P] + \sum_{i}n_i[R/p_i]$$ Here the $p_i$s are in $\operatorname{Supp}(R/P)$, but have smaller dimensions. Our strategy will be to replace them one by one by elements of even smaller dimensions. Let’s look at $p_1$. Replacing $M$ by a multiple of $M$ if necessary, we may assume $n_1 \in {\mathbf{Z}}$. Next, we replace $M$ by $ M' = {p_1}^aM\oplus (R/p_1)^{\oplus{b}}$ for some $a,b \in {\mathbf{Z}}$. Then $$\tau(M') = \tau(M) - \tau(M/p_1^aM) + b \tau(R/p_1)$$ We now choose $a,b$ such that $b- l(M_{p_1}/{p_1}^aM_{p_1}) =
-n_1$. Then $p_1$ is replaced in the representation of $M$ by some elements of smaller dimension. Repeating this process, we will get to dimension $0$, which is $0$ in $\operatorname{CH}^*(R)_{\mathbb{Q}}$. So we get a module $M$ such that $\tau(M) = n[R/P] = 0$, which is what we need.
Miscellaneous results and examples {#lastsection}
==================================
In this section we will discuss our attempts to generalize the $\theta^R$ function and extend our class of rigid modules. We will end with a host of examples to illustrate our results.
An obvious drawback of our crucial rigidity theorem \[rg1\] is the requirement that all higher $\operatorname{Tor}$ modules have finite lengths. This prevents us from proving rigidity for a bigger class of modules (unless we impose extra conditions like isolated singularity on $R$). That raises a question: can we replace the length function in the definition of $\theta^R(M,N)$ by other functions with bigger domains? One possibility is taking the length at the minimal primes of the module. For any $R$-module $L$ , we define the class of $L$ in $Z_*(R)$ as : $$cl(L) := \sum_{p \in {\textup{Min}}(L)} l_{R_p}(L_p)[R/p]$$ and for a pair of modules $(M,N)$, let: $$\alpha^R(M,N):= cl(\operatorname{Tor}_{2e+2}^R(M,N)) - cl(\operatorname{Tor}_{2e+1}^R(M,N))$$ Here $e$ is any integer bigger than the dimension of $R$. Note that if the $\operatorname{Tor}$s have finite length then $\alpha^R(M,N)=\theta^R(M,N)[R/m_R]$. Then we have the following corollary of \[rg1\] :
Let $R$ be an admissible hypersurface, and $M,N$ be $R$-modules. Assume that $\alpha^R(M,N)=0$. Then $(M,N)$ is rigid.
We use induction on $d = \dim R$. If $d=0$ then $\alpha^R(M,N)$ coincides with $\theta^R(M,N)$. Suppose $d>0$ and $\operatorname{Tor}_i^R(M,N)=0$ for some $i>0$. Then by localizing at primes on the punctured spectrum and induction hypotheses $l(\operatorname{Tor}_j^R(M,N))<\infty$ for $j \geq i$. Then the vanishing of $\alpha$ again implies the vanishing of $\theta^R$, and we are done by \[rg1\].
Let $R$ be an admissible hypersurface, and $M$ be an $R$-modules. Assume that the minimal resolution of $M$ over $R$ is eventually periodic of period 1. Then $M$ is rigid.
Let $R$ be an admissible hypersurface, and $M,N$ be $R$-modules. Assume that there are integers $0<a<b$ such that $b-a$ is an odd integer and $\operatorname{Tor}_a^R(M,N)=\operatorname{Tor}_b^R(M,N)=0$. Then $\operatorname{Tor}_i^R(M,N)=0$ for $i\geq a$.
Let $M' = \operatorname{syz}_{a-1}M\oplus\operatorname{syz}_{b-1}M$. Since $b-a$ is odd and the resolution of $M$ is eventually periodic of period 2, it follows that the resolution of $M'$ is eventually periodic of period 1. Because $\operatorname{Tor}_i^R(M',N)=\operatorname{Tor}_{a+i-1}^R(M,N)\oplus\operatorname{Tor}_{b+i-1}^R(M,N)$, for $i\geq1$, the conclusion follows from the rigidity of $M'$.
The case $b=a+1$ is a well-known result by Murthy (\[Mu\]), and an asymptotic version (i.e when $a,b$ are big enough) was proved in (\[Jo2,3.1\]).
We could not get better results with $\alpha$ because it is not additive on short exact sequences. A much more refined version can be obtained replacing the function $cl$ in the definition of $\alpha$ by the Todd class $\tau$. We recall that $\tau$ is a map from $G(R)_{\mathbb{Q}}$ to $\operatorname{CH}_*(R)_{\mathbb{Q}}$. It is additive, so if we let : $$\beta^R(M,N) = \tau(\operatorname{Tor}_{2e+2}^R(M,N)) - \tau(\operatorname{Tor}_{2e+1}^R(M,N))$$ then we have a bilinear map from $G(R)_{\mathbb{Q}}\times G(R)_{\mathbb{Q}}$ to $\operatorname{CH}_*(R)_{\mathbb{Q}}$. Indeed, it is a bilinear map from $\overline G(R)_{\mathbb{Q}}\times \overline G(R)_{\mathbb{Q}}$ to $\operatorname{CH}_*(\operatorname{Sing}(R))_{\mathbb{Q}}$. The problem is that in $\operatorname{CH}_*(\operatorname{Sing}(R))_{\mathbb{Q}}$, the element $[R/m_R]$ is $0$ if $\dim \operatorname{Sing}(R)>0$. So when the $\operatorname{Tor}$s have finite length, the vanishing of $\beta^R(M,N)= \theta^R(M,N)[R/m_R]$ does not give you the vanishing of $\theta^R(M,N)$. To make this work, one possibility is to develop a version of “asymptotic chern character".
Another natural question is to extend the results in this paper to complete intersections. Typically, we will need more $\operatorname{Tor}$ modules to vanish. While for many results in this paper could be extended using some induction arguments with repeated use of the long exact sequence \[longexact\], the most efficient way is to define a generalized theta function for pair of modules over complete intersections. Since this required a bit of efforts, it will be dealt with in a separate paper (\[Da\]).
Finally, we will give some examples. The purpose is to demonstrate that many of our technical conditions can not be removed, and some statements can not be reversed.
\[ex2\] Let $R$ be a hypersurface and $M$ a Cohen-Macaulay $R$-module. Assume there exists a Cohen-Macaulay $R$-module $N$ such that:\
(1) $l(M{\otimes}_RN)<\infty$.\
(2) $\dim M + \dim N = \dim R + 1$.\
Then $\operatorname{Tor}_i^R(M,N)=0$ if and only if $i$ is an odd integer.
Suppose $R=T/(f)$, where $T$ is regular local. Then $\dim M + \dim N = \dim T$. Since both $M,N$ are Cohen-Macaulay, we have $\operatorname{depth}M + \operatorname{depth}N = \operatorname{depth}T$ as well. By \[HW2,2.2\] we have $\operatorname{Tor}_i^T(M,N)=0$ for all $i\geq 1$. A glance at the change of rings exact sequence gives $\operatorname{Tor}_1^R(M,N)=0$ and $\operatorname{Tor}_{i+2}^R(M,N)\cong \operatorname{Tor}_i^R(M,N)$ for $i\geq0$. But then $\operatorname{Tor}_2^R(M,N)\cong M{\otimes}_R N \neq 0$.
Now we are ready to give some examples:
Corollary \[dim2,3\] shows that one of the modules is $0$ in $\overline G(R)_{\mathbb{Q}}$ is not necessary for $\theta^R(-,-)$ to vanish because in an admissible hypersurface, normal domain of dimension $2$, $\theta^R$ always vanish. However, if we take our hypersurface $R$ to be the local ring at origin of $A=\mathbb{C}[x,y,z]/(x^3+y^3+z^3)$, then $X=\operatorname{Proj}(A)$ is an elliptic curve, therefore $\operatorname{CH}_*(R)_{\mathbb{Q}}$, hence $G(R)_{\mathbb{Q}}$ and $\overline G(R)_{\mathbb{Q}}$ , are infinite dimensional $\mathbb{Q}$-vector spaces.
In this example we will give a module $M$ such that $\theta^R(M,-)$ does not always vanish, but $M$ is still rigid and decent. Let $R=k[[x,y,u,v]]/(xu-yv)$ and $P=(x,y),Q=(x,v)$. Let $M
= R/P\oplus R/P\oplus R/Q$. It is easy to check that there is an exact sequence: $$0 \to Q \to R^2 \to P \to 0$$ which shows that $R/P+R/Q=0$ in $G(R)$. So $\theta^R(M,-) =
\theta^R(R/P,-)$. Clearly $\theta^R(R/P,-)$ is not always $0$, because $\theta^R(R/P,R/Q)=1$. It remains to show that $M$ is rigid and decent. Let $M'=R/P\oplus R/Q$. Then by the argument above $M'=0$ in $G(R)$, so it is rigid and decent since $R$ is an isolated singularity. So for any module $N$, $\operatorname{Tor}_i^R(M,N)=0$ $\Rightarrow$ $\operatorname{Tor}_i^R(R/P,N)=\operatorname{Tor}_i^R(R/Q,N)=0$ $\Rightarrow$ $\operatorname{Tor}_i^R(M',N)=0$ $\Rightarrow$ $\operatorname{Tor}_{i+1}^R(M',N)=0$ $\Rightarrow$ $\operatorname{Tor}_{i+1}^R(R/P,N)=\operatorname{Tor}_{i+1}^R(R/Q,N)=0$ $\Rightarrow$ $\operatorname{Tor}_{i+1}^R(M,N)=0$. As for decency, observe that $M$ and $M'$ have the same support, and as decency only depends on the support, $M$ must be decent as well.
In this example we will give concrete examples of isolated hypersurface singularities of dimension 3 such that $\theta$ always vanishes. $R=\mathbb{C}[[x,y,z,w]]/(x^2+y^2+z^p+w^q)$, where $p,q$ are coprime integers. Then $R$ is factorial and it follows from corollary \[dim2,3\] that $\theta^R$ must vanish. So over this hypersurface, rigidity and decency always hold.
Let $R=k[[x,y,u,v,t]]/(xu-yv)$ and let $M=R/(x,y,t)$. Then $M$ is not rigid (let $N=R/(u,v)$ and use \[ex2\]). However, the exact sequence: $$\xymatrix {0 \ar[r] &R/(x,y) \ar[r]^{t} &R/(x,y) \ar[r] &\overline{M} \ar[r] &0}$$ shows that $[M]=0$ in $\overline G(R)$. It is easy to check that $\operatorname{Sing}(R)=V((x,y,u,v))$ and $IPD(M) = \{(x,y,u,v,t)\} = \{m_R\}$. This example shows that the technical requirements for rigidity in theorem \[rig1.1\] can not be relaxed.
Let $R=k[[x,y,u,v]]/(xu-yv)$, $M=(x,y), N=(u,y)$. Then $M{\otimes}_RN \cong (x,y,u,v)$ is torsion free and has depth $1$. Also, $R$ is an isolated singularity. However, $\operatorname{Tor}_1^R(M,N)\neq 0$. This shows that the condition (3) of \[vanishingiso\] is critical.
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[^1]: The author is partially supported by NSF grant 0834050
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The aim of this paper is to study the Harnack type logarithmic submajorisation and Fuglede-Kadison determinant inequalities for operators in a finite von Neumann algebra. In particular, the Harnack type determinant inequalities due to Lin-Zhang[@LZ2017] and Yang-Zhang[@YZ2020] are extended to the case of operators in a finite von Neumann algebra.'
address:
- 'College of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, China'
- 'College of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, China'
author:
- Yazhou Han
- Cheng Yan
title: Harnack type inequalities for operators in logarithmic submajorisation
---
Introduction
============
The classical Harnack inequality, named after Carl Gustav Axel von Harnack, gives an estimate from above and an estimate from below for a positive harmonic function in a domain. Even though the classical Harnack inequality is almost trivially derived from the Poisson formula, the consequences that may be deduced from Harnack inequality are particularly of great importance. Later, these inequalities became an important tool in the general theory of harmonic functions and partial differential equations. There exist as yet extensive works on generalized Harnack inequalities in various forms, see [@M2007; @W2013; @YZ2020] for a nice introduction about the inequality. The purpose of this paper is to investigate the Harnack type determinant inequality for operators and matrices.
With the help of Language multiplier method, the following Harnack type determinant inequality was established by Tung[@T1964], as a tool to study Harnack inequality: If $Z\in \mathbb{M}_n$ is a complex matrix with singular values $r_k$ with $0\leq r_k<1, k=1, 2, ..., n$, then $$\label{inequ. T1}
\prod_{k=1}^n\frac{1-r_k}{1+r_k}\leq\frac{\det(\mathbb{I}-Z^*Z)}{|\det(\mathbb{I}-UZ)|^2}\leq
\prod_{k=1}^n\frac{1+r_k}{1-r_k}, U\in \mathbb{U}_n,$$ where $\mathbb{U}_n$ denote the set of all $n\times n$ unitary matrices $U$. From these bounds Tung obtained upper and lower bounds of a Poisson kernel on $\mathbb{U}$(see [@T1964]), hence that the so-called Harnack’s first and second theorems are established. Tung’s work drew immediate attention of Hua and Marcus. Using majorisation theory and singular value (eigenvalue) inequalities of Weyl, Marcus [@M1965] gave another proof of (\[inequ. T1\]) and gave an equivalent form of (\[inequ. T1\]). Almost at the same time, a proof of (\[inequ. T1\]) was also given by Hua[@H1965] based on the determinantal inequality he had previously obtained in[@H1955]. In the past several ten years, Tung’s work has attracted attentions of mathematicians and been extended to various cases (see [@M2007; @W2013; @Z2005; @JL2020; @L2015; @YZ2020] and the references therein for more details). Among these outstanding works we will be interested in Lin-Zhang’s and Yang-Zhang’s work. Specifically, with $A=UZ$, (\[inequ. T1\]) is equivalently rewritten in terms of eigenvalues ([@JL2020; @YZ2020]) as $$\label{inequ. YZ1}
\prod_{k=1}^n\frac{1-r_k}{1+r_k}\leq\prod_{k=1}^n
\lambda_k((\mathbb{I}-A^*)^{-1}(\mathbb{I}-A^*A)(\mathbb{I}-A)^{-1})
\leq\prod_{k=1}^n\frac{1+r_k}{1-r_k}.$$ (\[inequ. YZ1\]) leads to the study of inequalities of logarithmic submajorisation of eigenvalues and singular values. Following this line, an interesting generalization of (\[inequ. YZ1\]) is presented by Yang-Zhang[@YZ2020] and Jing-Lin[@JL2020] as follows: $$\label{FK det 1}
\prod_{k\in K}
\lambda_k((\mathbb{I}-A^*)^{-1}(\mathbb{I}-A^*A)(\mathbb{I}-A)^{-1})
\leq\prod_{k\in K}\frac{1+r_k}{1-r_k},$$ $$\label{FK det 2}
\prod_{i\in K}
\lambda_{n-k+1}((\mathbb{I}-A^*)^{-1}(\mathbb{I}-A^*A)(\mathbb{I}-A)^{-1})
\geq\prod_{k\in K}(1-r_k^2)\prod_{i=1}^{|K|}\frac{1}{(1+r_i)^2},$$ where $K$ is a subset of $\{r_1, r_2, \cdots, r_n\}$ and $|K|$ denote the number of terms in $K$. The main theme of the paper is to continue with Jiang-Lin and Yang-Zhang’s work and to show their results hold in the case of operators in finite von Neumann algebras.
We are concerned with the Harnack type logarithmic submajorisation inequality and Fuglede-Kadison determinant inequality for operators in a finite von Neumann algebra. The properties of the logarithmic submajorisation and Fuglede-Kadison determinant for operators in a finite von Neumann algebra was investigated by many authors, see for example [@B1983; @BL2008; @HSZ2020]. Those properties are important, for example, in investigation of noncommutative Hardy spaces and invariant subspaces for operators in von Neumann algebra. By adapting the techniques in [@YZ2020; @FK1986; @N1987], we obtain some inequalities which is related to the Harnack type logarithmic submajorisation inequality and Fuglede-Kadison determinant inequality. In particular, we show that the inequalities (\[FK det 1\]) and (\[FK det 2\]) hold for operators in a finite von Neumann algebra. We will conclude this paper with a series of logarithmic submajorisation submajorization inequalities which is related to Cayley transform.
Preliminaries
=============
von Neumann algebras
--------------------
Suppose that $\mathcal{H}$ is a separable Hilbert space over the field $\mathbb{C}$ and $\mathbb{I}$ is the identity operator in $\mathcal{H}$. We will denote by $\mathcal{B}(\mathcal{H})$ the $*$-algebra of all linear bounded operators in $\mathcal{H}$. Let $\mathcal{M}$ be a $*$-subalgebra of $\mathcal{B}(\mathcal{H})$ containing the identity operator $\mathbb{I}$. Then $\mathcal{M}$ is called a von Neumann algebra if $\mathcal{M}$ is weak operator closed. Let $\mathcal{M}^+$ denote the positive part of $\mathcal{M}$. We recall that a weight on $\mathcal{M}$ is a map $\tau: \mathcal{M}^+\rightarrow [0, \infty]$ satisfying
1. $\tau(x+y)=\tau(x)+\tau(y),$ for all $x, y\in \mathcal{M}^+$;
2. $\tau(\alpha x)=\alpha\tau(x)$ for all $x\in \mathcal{M}^+$ and $\alpha\in [0, \infty)$, with the convention $0\cdot\infty=0.$
The weight $\tau$ is called faithful if $\tau(x^*x)=0$ implies $x=0$, normal if $x_i\uparrow_i x$ in $\mathcal{M}^+$ implies that $0\leq\tau(x_i)\uparrow_i\tau(x)$, tracial if $\tau(x^*x)=\tau(xx^*)$ for all $x\in\mathcal{M}$. Note that since $(x_i)$ is bounded there is $x$ in $\mathcal{M}^+$ such that, for any $h$ in $\mathcal{H}$, $\langle x_ih, h\rangle\uparrow\langle xh, h\rangle$, which implies that $x_i$ tends to $x$ weak\* and hence $x\in \mathcal{M}^+$. The operator $x$ is obviously the least upper bound of $(x_i)$, it is natural to denote it by $\sup_i x_i$. The space $\mathcal{M}$ is a partially ordered vector space under the ordering $x\geq0$ defined by $\langle x\xi, \xi\rangle\geq0, \xi\in \mathcal{H}$. Recall that $x\in\mathcal{M}$ is contractive if $\|x\|\leq1$ and strict contractive if $\|x\|<1$. Moreover, if $x$ is strict contractive, then $\mathbb{I}-x^*x$ is invertible and $\mathbb{I}-x^*x\geq0$.
It is also customary to say trace instead of tracial weight. A trace $\tau$ is called finite if $\tau(\mathbb{I})<\infty.$ A finite trace $\tau$ is extended uniquely to a positive linear functional on $\mathcal{M}$ which will also be denoted by $\tau.$ A positive linear functional $\tau$ on a von Neumann algebra is said to be a state if $\tau(\mathbb{I})=1$.
A von Neumann algebra $\mathcal{M}$ is called finite if the family formed of the finite normal traces separates the points of $\mathcal{M}$. Clearly this happens if $\mathcal{M}$ admits a single faithful normal finite trace. But a finite $\mathcal{M}$ may fail to have any faithful finite trace, for instance $\mathcal{M} = \ell^\infty(\mathbb{R})$ where $\mathbb{R}$ is equipped with counting measure.
However, on a separable Hilbert space (i.e. if $\mathcal{M}$ is weak\*-separable) the converse is also true; then $\mathcal{M}$ is finite if and only if it admits a faithful normal finite trace.
In what follows, we will keep all previous notations throughout the paper, and $\mathcal{M}$ will always denote a finite von Neumann algebra acting on a separable Hilbert space $\mathcal{H}$, with a normal faithful finite tracial state $\tau$, i.e., a normal faithful finite trace $\tau$ satisfies that $\tau(\mathbb{I})=1$. We refer to [@T1979] for von Neumann algebras.
The eigenvalue function and generalized singular value function
---------------------------------------------------------------
Let $x\in \mathcal{M}$ and $t>0.$ The “$t$th singular number(or generalized singular number) of $x$" $\mu_t(x)$ is defined by $$\mu_t(x)=\inf\{\|xe\|: e~\mbox{is a projection in}~
\mathcal{M}~\mbox{with}~\tau(e^\bot)\leq t\}.$$
We will denote simply by $\mu(x)$ the function $t\rightarrow\mu_t(x)$. The generalized singular number function $t\rightarrow\mu_t(x)$ is decreasing right-continuous. For convenience to discuss the properties of $\mu_t(x)$ we define $\mu_t^\ell(x)$ by $$\mu_t^\ell(x)=\inf\{\|xe\|: e~\mbox{is a projection in}~
\mathcal{M}~\mbox{with}~\tau(e^\bot)< t\}.$$ Then $t\rightarrow\mu_t^\ell(x)$ is decreasing left-continuous and $\mu_t^\ell(x)=\mu_t(x)$ holds for almost every $t\in [0, 1]$. See [@DPS2019; @FK1986; @O19701; @O1970] for basic properties and detailed information on $\mu_t(x)$ and $\mu_t^\ell(x)$.
If $x$ is self-adjoint and $x=\int_{-\infty}^\infty tde_t(x)\in L_0(\mathcal{M})$ is the spectral resolution of $x$ then for any Borel subset $B\subseteq (-\infty, \infty)$ we denote by $e_B(x)$ the corresponding spectral projection. However, we write $e_s(x)=e_{(-\infty, s]}(x).$ Given $x\in L_0(\mathcal{M})^{sa}$, the spectral scale $\lambda_t(x)$ on $(0, \tau(\mathbb{\mathbb{I}}))$ is defined by $$\lambda_t(x)=\inf\{s\in \mathbb{R}: \tau(\mathbb{I}-e_s(x))\leq t\}.$$ Obviously, if $0\leq x\in L_0(\mathcal{M})$ then $\lambda_t(x) = \mu_t(x)$ for $0 <t<\tau(\mathbb{I}).$ The spectral scale $\lambda_t(x)$ is nonincreasing and right-continuous. For the properties of $\lambda_t(\cdot)$, it is important to note that $\lambda_t(x+a \mathbb{I})=\lambda_t(x)+a$ for every $x\in L_0(\mathcal{M})^{sa}$ and $a\in \mathbb{R}$. This property enables us to deduce estimations for $\lambda_t(x)$ from formulas on $\mu_t(x)$.
To achieve our main results, we state some properties of $\lambda_t(\cdot)$ and $\mu_t(\cdot)$ without proof( See [@H1987; @FK1986]).
\[proposition 2.2\](see [@H1987; @FK1986]) Let $x, y\in L_0(\mathcal{M})$ and $v\in\mathcal{M}$. Then
1. $\mu(|x|)=\mu(x)=\mu(x^*)$ and $\mu(\alpha x)=|\alpha|\mu_t(x),$ for $t>0$ and $\alpha\in \mathbb{C}$.
2. Let $f$ be a bounded continuous increasing function on $[0, \infty)$ with $f(0)=0$. Then $\mu(f(x))=f(\mu(x))$ and $\tau(f(x))=\int_0^{\tau(1)} f(\mu_t(x))dt.$
3. $\mu_{s+t}(x+y)\leq\mu_t(x)+\mu_s(y), s, t>0.$
4. If $0\leq x\leq y$, then $\mu_t(x)\leq\mu_t(y)$.
5. $\mu_{t+s}(xy)\leq\mu_t(x)\mu_s(y), s, t>0.$
6. If $x, y$ are self-adjoint, then $\lambda_{t+s}(x+y)\leq \lambda_t(x)+\lambda_s(y), t, s\geq0, t+s\leq1.$
7. If $0\leq t\leq1$ and $x, y$ are self-adjoint, then $\lambda_t(x)\geq0$ implies that $\lambda_t(v^*av)\leq\|v\|^2\lambda_t(x)$.
8. If $x, y$ are self-adjoint and $x\leq y$, then $\lambda_t(x)\leq \lambda_t(y)$.
9. If $x$ is self-adjoint, then $\lambda_t(f(x))=f(\lambda_t(x)), t\in (0, \tau(\mathbb{I}))$, for every increasing continuous function $f$ on $\mathbb{R}$.
Let $\mathcal{H}=\mathbb{C}^n$ and let $\mathcal{M}=\mathcal{B}(\mathcal{H})\cong \mathbb{M}_n(\mathbb{C})$ equipped with the normalized trace $\tau_n:\triangleq \frac{1}{n}tr_n$ where $tr_n$ is the standard trace on $\mathbb{M}_n(\mathbb{C})$. If $x\in \mathcal{B}(\mathcal{H})=\mathbb{M}_n(\mathbb{C})$ is self-adjoint, then $x$ can be written as $x=\sum_{i=1}^n\alpha_j p_j$, where $\alpha_1\geq\alpha_2\geq\cdots\geq\alpha_n$ is the sequence of eigenvalues of $x$ in which each is repeated according to its multiplicity and $\sum_{i=1}^n p_j=\mathbb{I}$. Therefore, $$\lambda_t(x)=\sum_{j=1}^n\alpha_j\chi_{[\frac{j-1}{n}, \frac{j}{n})}, t\in [0, 1).$$ If $x\geq0$, then $\alpha_1\geq\alpha_2\geq\cdots\geq\alpha_n\geq0$, $\lambda_t(x)=\mu_t(x)$ and $\mu_t^\ell(x)=\sum_{j=1}^n\alpha_j\chi_{(\frac{j-1}{n}, \frac{j}{n}]}$.
If $x\in \mathbb{M}_n(\mathbb{C})$ is arbitrary, then $\mu_t(x)=\mu_t(|x|)$ and the eigenvalues of $|x|$ are usual called the singular values of $x.$ It follows that $$\mu_t(x)=\sum_{j=1}^ns_j\chi_{[\frac{j-1}{n}, \frac{j}{n})}$$ and $$\mu_t^\ell(x)=\sum_{j=1}^ns_j\chi_{(\frac{j-1}{n}, \frac{j}{n}]},$$ where $s_1\geq s_2\geq\cdots\geq s_n\geq0$ is the sequence of singular values of $x$, repeated according to multiplicity. It is clear that $\mu_{\frac{j-1}{n}}(x)=\mu_{\frac{j}{n}}^\ell(x)$.
Note that if $x\in \mathbb{M}_n(\mathbb{C})$ is self-adjoint, then $x$ can also be written as $x=\sum_{i=1}^m\beta_j p_j$, where $\beta_1>\beta_2>\cdots>\beta_m(m\leq n)$. Then $$\lambda_t(x)=\sum_{j=1}^m\beta_j\chi_{[d_{j-1}, d_j)},$$ where $d_{j}=\sum_{i=1}^j\tau(p_i)$ for $j=1, 2, \cdots, m$ and $d_0=0.$ For each $j$, the length of the interval $[nd_{j-1}, nd_j)$ is $n\tau_n(p_j)$, which is the dimension of the eigenspace corresponding to $\beta_j.$ See [@H1997; @DPS2019] for more details of $\mu_t(\cdot)$ and $\lambda_t(\cdot)$ of operators and matrices (Note: the generalized singular values $\mu_{k}$, as defined in [@H1997], is denoted by $\mu_{\frac{k}{n}}^\ell$, in this paper; the generalized singular values $\mu_{\frac{k}{n}}$ and $\mu_{\frac{k}{n}}^\ell$ is noting but $\mu_{k+1}$ and $\mu_k$, respectively, in [@H1997])
Consider the algebra $\mathcal{M}=L^\infty([0, 1])$ of all Lebesgue measurable essentially bounded functions on $[0, 1]$. Algebra $\mathcal{M}$ can be seen as an abelian von Neumann algebra acting via multiplication on the Hilbert space $\mathcal{H}=L^2([0, 1])$, with the trace given by integration with respect to Lebesgue measure $m$. For a real measurable function $f\in L^\infty([0, 1])$, the decreasing rearrangement $f^*$ of the function $f$ is given by $$f^*(t)=\inf\{s\in \mathbb{R}: m(\{h\in[0,1]: f(h)>s\})\leq t\}, 0<t<1.$$ Then $\mu_t(f)=|f|^*(t)$ and $\lambda_t(f)=f^*(t)$. Suppose that $f=\sum_1^n\alpha_i \chi_{B_i}$, where $B_i\subseteq[0, 1]$ with $B_i\cap B_j=\emptyset$ whenever $i\neq j$, and $0<\alpha_j\in \mathbb{R} (j = 1,2,\cdots, n)$ are such that $\alpha_i\neq \alpha_j$ whenever $i\neq j$. For the computation of $\mu_t(f)$, it may be assumed that $\alpha_1>\alpha_2>\cdots>\alpha_n$. Then $$\lambda_t(f)=\sum_{j=1}^n\alpha_j\chi_{[d_{j-1}, d_j)},$$ where $d_{j}=\sum_{i=1}^jm(B_i)$ for $j=1, 2, \cdots, n$ and $d_0=0.$ If $f\geq0$, then $\alpha_1>\alpha_2>\cdots>\alpha_n\geq0$, $\lambda_t(f)=\mu_t(f)$ and $\mu_t^\ell(f)=\sum_{j=1}^n\alpha_j\chi_{(d_{j-1}, d_j]}$. See [@N1987; @DPS2019] for more details.
Fuglede-Kadison determinant
---------------------------
Let $\mathcal{M}$ be a finite von Neumann algebra acting on a separable Hilbert space $\mathcal{H}$, with a normal faithful finite tracial state $\tau$. Recall that the Fuglede-Kadison determinant $\Delta=\Delta_\tau:\mathcal{M}\rightarrow \mathbb{R}^+$ is defined by $\Delta_\tau(x)=\tau(\log|x|)$ if $|x|$ is invertible; and otherwise, we define $\Delta_\tau(x)=\inf\Delta_\tau(|x| + \varepsilon \mathbb{I})$, the infimum takes over all scalars $\varepsilon>0$. We define Fuglede-Kadison determinant-like function of $x$ by $$\Lambda_t(x)=\exp\{\int_0^t\log\mu_s(x)ds\}.$$ Since $\tau(\mathbb{I})=1$, if $|x|$ is invertible, then $$\Delta_\tau(x)=\Lambda_1(x)=\exp\{\int_0^1\log\mu_s(x)ds\}.$$ We understanding that $\Delta(x)=0$ if $$\int_0^{\tau(\mathbb{I})}\log\mu_s(x)ds=-\infty.$$ Recall that $x$ is said to be logarithmically submajorised by $y$(see [@HSZ2020]), denoted by $x\prec\prec_{\log}y$ (or $\mu_s(x)\prec\prec_{\log}\mu_s(y)$), if $\Lambda_t(x)\leq\Lambda_t(y)$ for all $t>0$.
We state for easy reference the following fact, obtained from [@A1967; @B1983] for Fuglede-Kadison determinant which will be applied below.
\[proposition 2.4\] Let $x, y\in\mathcal{M}$. Then
1. $\Delta_\tau(\mathbb{I})=1, \Delta_\tau(xy)=\Delta_\tau(x)\Delta_\tau(y),$
2. $\Delta_\tau(x)=\Delta_\tau(x^*)=\Delta_\tau(|x|),~\Delta_\tau(|x|^\alpha)
=(\Delta_\tau(|x|))^\alpha, \alpha\in \mathbb{R}^+$
3. $\Delta_\tau(x^{-1})=(\Delta_\tau(x))^{-1}, \mbox{if}~ x~\mbox{is invertible in}~\mathcal{M}$
4. $\Delta_\tau(x)\leq\Delta_\tau(y), \mbox{if}~ 0\leq x\leq y$
5. $\lim_{\varepsilon\rightarrow0^+}
\Delta_\tau(x+\varepsilon1)=\Delta_\tau(x), \mbox{if}~ 0\leq x.$
6. $\Delta_\tau(x)\leq\Delta_\tau(y), \mbox{if}~ x\prec\prec_{\log} y$.
See [@A1967; @B1983; @BL2008] for basic properties and detailed information on Fuglede-Kadison determinant of $x\in\mathcal{M}$.
Let $\mathcal{H}=\mathbb{C}^n$ and let $\mathcal{M}=\mathcal{B}(\mathcal{H})\cong \mathbb{M}_n(\mathbb{C})$ equipped with the normalized trace $\tau_n:\triangleq \frac{1}{n}tr_n$ where $tr_n$ is the standard trace on $\mathbb{M}_n(\mathbb{C})$. If $x\in \mathcal{B}(\mathcal{H})$, then $\Delta_{\tau_n}(x)=(\det(|x|))^{\frac{1}{n}}$. See [@H1997] for more information on determinant of matrices.
\[rk:determanint 2\] Let $x, y\in\mathcal{M}^+$ be invertible. Then the following conditions are equivalent:
1. $\mathbb{I}+rx\prec\prec_{\log} \mathbb{I}+ry,$ for all $ r\in \mathbb{R}^+$;
2. $x\prec\prec_{p} y,~~0<p<1$;
3. $x\prec\prec_{\log} y$;
4. $\int_0^t\varphi(\mu_s(x))ds\leq\int_0^t\varphi(\mu_s(y))ds$ for all $t>0$ and all nondecreasing functions $\varphi$ on $[0, \infty)$ such that $\varphi(0)=0$ and $t\rightarrow \varphi(e^t)$ is convex.
Indeed, let $\psi$ is a bounded positive measurable function on $[0, \infty)$ and $\pi_t(r)=\exp\{\int_0^t\log(1+r\psi(s))ds\}$. By [@F1983 Lemma 3.2], we have $$\int_0^t\psi(s)^pds=\frac{p\sin(\pi p)}{\pi}\int_0^\infty \frac{\log\pi_t(r)}{r^{p+1}}dr,$$ which implies that (1)$\Rightarrow$(2) holds.
Note that if $(\int_0^{\tau(\mathbb{I})}|\varphi(s)|^pds)^{\frac{1}{p}}<\infty$ for some $p>0$, then from [@R1974 p.71] we obtain $$\exp\{\int_0^{\tau(\mathbb{I})}\log|\varphi(s)|ds\}=\lim_{p\rightarrow0}
(\int_0^{\tau(\mathbb{I})}|\varphi(s)|^pds)^{\frac{1}{p}},$$ which yields (2)$\Rightarrow $(3). (3)$\Rightarrow$(4) follows from the fact that $t\rightarrow \varphi(e^t)$ is convex and $\varphi(e^{\log\mu_t(x)})=\varphi(\mu_t(x))$. It is easy to check that (4)$\Rightarrow$(1).
Unitary approximation and Logarithmic submajorisation
=====================================================
Our starting point is the following inequality for complex numbers: $$\label{ineq 3.1}
||z|-1|\leq||z|-v|\leq||z|+1|, z, v\in \mathbb{C}~\mbox{with}~|v|=1.$$ In this section, we will consider some Logarithmic submajorisation inequalities for operator case of (\[ineq 3.1\]). The Logarithmic submajorisation inequalities generalize that of some submajorisation inequalities in [@DD1992]. We start with a lemma which will be used in our proof.
\[lemma 3.1\] Let $x\in \mathcal{M}$. Then $$\mu_t(Rex)\prec\prec_{\log}\mu_t(x), \mu_t(Im x)\prec\prec_{\log}\mu_t(x),$$ with $-\infty$ allowed for values.
[**Proof.**]{} The proof is adapted from [@JL2020 Lemma 2.1]. For any $t>0,$ we have $(tx-\frac{1}{t}\mathbb{I})^*(tx-\frac{1}{t}\mathbb{I})\geq0$ and $(tx+\frac{1}{t}\mathbb{I})^*(tx+\frac{1}{t}\mathbb{I})\geq0$, which tell us that $$t^2x^*x+\frac{1}{t^2}\mathbb{I}\geq 2Rex\geq -(t^2x^*x+\frac{1}{t^2}\mathbb{I}).$$ [@HSZ2020 Lemma 4.2] now yields $\mu_s(Rex)\prec\prec_{\log}\mu_s(t^2x^*x+\frac{1}{t^2}\mathbb{I})$. If it were true that $\mu_s(x)=0$, there would be $\mu_s(Rex)=0$ by take $t\rightarrow\infty$. Otherwise we take $t=\frac{1}{\mu_s(x)^{\frac{1}{2}}}$, it follows that $\mu_s(Rex)\prec\prec_{\log}\mu_s(x).$
On the other hand, since $Re(ix)=-Im x$, from what has already been proved we see that $\mu_s(Imx)\prec\prec_{\log}\mu_s(ix)=\mu_s(x).$ $\Box$
\[rk:lemma 3.1\] Let $x\in \mathcal{M}$. A slight change in the proof of Lemma \[lemma 3.1\] actually shows that $$\lambda_t(Rex)\leq\mu_t(x).$$
\[corollary 3.2\] Let $x, y\in \mathcal{M}$ and let $\alpha\in \mathbb{R}$. If $x^{*}=x$, then $$\mu_t(y-Rey)\prec\prec_{\log}\mu_t(y-\alpha x), \mu_t(y-iIm y)\prec\prec_{\log}\mu_t(y-i\alpha x),$$ with $-\infty$ allowed for values.
[**Proof.**]{} The results follow from Lemma \[lemma 3.1\] along with the fact that $y-Rey=iImy=iIm(y-\alpha x)$ and $y-i Imy=Rey=Re(y-i\alpha x)$. $\Box$
If $x, y\in\mathcal{M}$ and $0<p<\infty$, then $x$ is said to be $p$-submajorised by $y$, denoted by $x\prec\prec_p y$, if $\int_0^t\mu_s(x)^pds\leq \int_0^t\mu_s(y)^pds$ for all $t>0.$
The following corollary is an easy consequence of Lemma \[lemma 3.1\] and Corollary \[corollary 3.2\] by using Remark \[rk:determanint 2\].
\[corollary 3.4\] Let $x, y\in \mathcal{M}$.
1. For any $0<p<\infty$, we have $$\mu_t(Rex)\prec\prec_{p}\mu_t(x), \mu_t(Im x)\prec\prec_{p}\mu_t(x).$$ Moreover, if $\alpha\in \mathbb{R}$ and $x^{*}=x$, then $$\mu_t(y-Rey)\prec\prec_{p}\mu_t(y-\alpha x), \mu_t(y-iIm y)\prec\prec_{p}\mu_t(y-i\alpha x).$$
2. Then $$\Delta_\tau(Rex)\leq\Delta_\tau(x), \Delta_\tau(Imx)\leq\Delta_\tau(x).$$ Moreover, if $\alpha\in \mathbb{R}$ and $x^{*}=x$, then $$\Delta_\tau(y-Rey)\leq\Delta_\tau(y-\alpha x), \Delta_\tau(y-iIm y)\leq\Delta_\tau(y-i\alpha x).$$
\[proposition 3.5\] Let $0\leq x\in \mathcal{M}$ such that $\|x\|>1$.
1. If $u\in \mathcal{M}$ is an unitary operator, then $$\mu_t(x-Reu)\prec\prec_{\log}\mu_t(x+\mathbb{I}),$$ which implies that $$\Delta_\tau(x-Reu)\leq\Delta_\tau(x+\mathbb{I}).$$
2. If $u\in \mathcal{M}$ is an unitary operator and $\tau(|x-\mathbb{I}|)=\tau(|x-u|)$, then $$\Delta_\tau(x-u)\leq\Delta_\tau(x-\mathbb{I}).$$
(1). From $-\mathbb{I}\leq -Reu\leq\mathbb{I}$, we deduce that $-(x+\mathbb{I})\leq x-\mathbb{I}\leq x-Reu\leq x+\mathbb{I}.$ Then we conclude from [@HSZ2020 Lemma 4.2] that $$\mu_t(x-Reu)\prec\prec_{\log}\mu_t(x+\mathbb{I}).$$ Hence we see that $$\Delta_\tau(x-Reu)\leq\Delta_\tau(x+\mathbb{I}).$$
(2). Note that [@DD1992 Corollary 2.6] leads to $$\mu_t(x-\mathbb{I})\prec\prec\mu_t(x-u).$$ Since $\tau(|x-\mathbb{I}|)=\tau(|x-u|)$, [@BR2014 Theorem 3.3] shows that $\tau(|x-u|^p)\leq\tau(|x-\mathbb{I}|^p), 0<p<1$, i.e. $$\int_0^{1}\mu_t(x-u)^p\leq
\int_0^{1}\mu_t(x-\mathbb{I})^pdt, 0<p<1.$$ Hence, from $\int_0^{1}|f(s)|^pds)^{\frac{1}{p}}<\infty$ and [@R1974 p.74] we obtain $$exp\{\int_0^{1}\log|f(s)|ds\}=\lim_{p\rightarrow0}
(\int_0^{1}|f(s)|^pds)^{\frac{1}{p}},$$ which force $$\Delta_\tau(x-u)\leq\Delta_\tau(x-\mathbb{I}).$$
\[lemma3.6\] Let $0\leq x\in \mathcal{M}$ be invertible. Then $$\mu_t^\ell(x^{-1})=\mu_{1-t}(x)^{-1}, 0<t<1.$$
Without loss of generality, we may assume that $\mathcal{M}$ has no minimal projections (Otherwise we consider the von Neumann algebra $\mathcal{M}\otimes L^{\infty}([0, 1])$). First we assume that $x=\sum_{i=1}^n\alpha_i p_i$ with $\alpha_1>\alpha_2>\cdots>\alpha_n>0$ and $\sum_{i=1}^n p_i=1, p_ip_j=0, i\neq j$. Thus $x^{-1}=\sum_{i=1}^n\frac{1}{\alpha_i} p_i$. Let $d_i=\sum_{j=1}^i\tau(p_j), 1\leq i\leq n$. Then $d_n=1$, $$\mu_t(x)=\alpha_1\chi_{(0, d_1)}+\sum_{i=2}^n\alpha_i\chi_{[d_{i-1}, d_i)}, 0<t<1,$$ and $$\mu_t^\ell(x^{-1})=
\sum_{i=2}^n\frac{1}{\alpha_i}\chi_{(1-d_{i}, 1-d_{i-1}]}+\frac{1}{\alpha_1}\chi_{(1-d_1, 1)}, 0<t<1.$$ Therefore, $$\mu_t^\ell(x^{-1})=\frac{1}{\mu_{\tau(\mathbb{I})-t}(x)}, 0<t<\tau(\mathbb{I}).$$ For the general case, let $0\leq x\in \mathcal{M}$. Since $x^{-1}\in \mathcal{M}$, there exists $\delta>0$ such that $x=\int_0^{\|x\|} \lambda de_\lambda(x)$ is the spectral decomposition of $x$. Put $$f_k(t)=\sum_{j=1}^{2^n}(\delta+\frac{(j-1)a}{2^n})
\chi_{[\delta+\frac{(j-1)a}{2^n}, \delta+\frac{ja}{2^n})},$$ where $a=\|x\|-\delta>0$. Then $0\leq f_k(t)\leq f_{k+1}(t)\leq t.$ We write $$x_n=f_n(|x|)=\sum_{j=1}^{2^n}(\delta+\frac{(j-1)a}{2^n})
e_{[\delta+\frac{(j-1)a}{2^n}, \delta+\frac{ja}{2^n})}(x).$$ Then $$x_n^{-1}=f_n(|x|)=\sum_{j=1}^{2^n}(\delta+\frac{(j-1)a}{2^n})^{-1}
e_{[\delta+\frac{(j-1)a}{2^n}, \delta+\frac{ja}{2^n})}(x).$$ It follows that $\|x-x_n\|\leq\frac{a}{2^n}$ and $$\|x^{-1}-x_n^{-1}\|\leq (\delta+\frac{(j-1)a}{2^n})^{-1}-
(\delta+\frac{ja}{2^n})^{-1}\leq\frac{1}{\delta^2}\frac{a}{2^{n}}.$$ Hence we infer from [@FK1986 Lemma 3.4] that $\mu_t(x)=\lim_{n\rightarrow\infty}\mu_t(x_n)$. On the other hand, picking up a small $\epsilon>0,$ we obtain $$\mu^\ell_t(x^{-1}_n)\leq\mu_{t-\epsilon}^\ell(x^{-1})+\mu_\epsilon^\ell(x^{-1}-x_n^{-1})
\leq\mu_{t-\epsilon}^\ell(x^{-1})+\|x^{-1}-x_n^{-1}\|.$$ Letting $\epsilon\downarrow0$ we get $$\mu^\ell_t(x^{-1}_n)
\leq\mu_{t}^\ell(x^{-1})+\|x^{-1}-x_n^{-1}\|.$$ In consequence, $\limsup_{n\rightarrow\infty}\mu^\ell_t(x^{-1}_n)\leq\mu_{t}^\ell(x^{-1})$. Therefore, $x^{-1}\leq x_n^{-1}$ tells us that $$\liminf_{n\rightarrow\infty}\mu^\ell_t(x^{-1}_n)\geq\mu_{t}^\ell(x^{-1}).$$ Hence $\mu_t^{\ell}(x)=\lim_{n\rightarrow\infty}\mu_t^\ell(x_n)$. This completes the proof.
Let $\mathcal{H}=\mathbb{C}^n$ and let $\mathcal{M}=\mathcal{B}(\mathcal{H})\cong \mathbb{M}_n(\mathbb{C})$ equipped with the normalized trace $\tau_n:\triangleq \frac{1}{n}tr_n$ where $tr_n$ is the standard trace on $\mathbb{M}_n(\mathbb{C})$. If $x\in \mathbb{M}_n(\mathbb{C})$ is positive and invertible, then $x$ can be written as $x=\sum_{i=1}^n\alpha_j p_j$, where $\alpha_1\geq\alpha_2\geq\cdots\geq\alpha_n>0$ is the sequence of eigenvalues of $x$ in which each is repeated according to its multiplicity and $\sum_{i=1}^n p_j=\mathbb{I}$. The proof of Lemma \[lemma3.6\] tells us that $$\mu_t^\ell(x^{-1})=\mu_{\tau(\mathbb{I})-t}(x)^{-1}, 0<t<\tau(\mathbb{I}),$$ i.e. $$\mu_{\frac{k}{n}}^\ell(x^{-1})=(\alpha_{n+1-k})^{-1}=\mu_{\tau(\mathbb{I})-\frac{k}{n}}(x)^{-1}.$$
We conclude this section with a series of inequalities of generalized singular value function.
\[lemma 3.7\] Let $x, y\in\mathcal{M}$.
1. If $x^*=x$, then $\lambda_t(x)\leq\mu_t(x).$
2. If $s, t>0$ such that $s+t<1$, then $1\leq\mu_t(x)+\mu_s(\mathbb{I}-x)$ and $1\leq\mu_t^\ell(x)+\mu_s^\ell(\mathbb{I}-x)$.
3. For any $t>0$ we have $1\leq\mu_t(x)+\mu_{1-t}^\ell(\mathbb{I}-x)$, $1\leq\mu_t^\ell(x)+\mu_{1-t}^\ell(\mathbb{I}-x)$ and $1\leq\mu_t^\ell(x)+\mu_{1-t}(\mathbb{I}-x)$.
4. For any $t>0$ we have $1\leq\mu_t(x)+\mu_{1-t}^\ell(x\pm i\mathbb{I})$, $1\leq\mu_t^\ell(x)+\mu_{1-t}^\ell(x\pm i\mathbb{I})$ and $1\leq\mu_t^\ell(x)+\mu_{1-t}(x\pm i\mathbb{I})$.
5. If $0\leq x\in \mathcal{M}$ and $\|x\|\leq1$, then $$\mu_t(1-x)=1-\mu_{1-t}^\ell(x),~~\mu_t^\ell(1-x)=1-\mu_{1-t}(x).$$
(1). Since $-|x|\leq x\leq |x|$, $\lambda_t(x)\leq\lambda_t(|x|)=\mu_x(x).$ (2)-(4) follow from the fact $\mu_{s+t}(x+y)\leq\mu_t(x)+\mu_s(y)$ and $\mu_{s+t}^\ell(x+y)\leq\mu_t^{\ell}(x)+\mu_s^\ell(y)$. (5). This follows by the same method as in Lemma \[lemma3.6\].
Harnack type inequality for operator
====================================
In this section Harnack type inequalities for operators in Logarithmic submajorisation are stated and proved. We will extend the results of Yang-Zhang[@YZ2020] and Lin-Zhang[@LZ2017] to the case of finite von Neumann algebra. We start with a lemma which follows by the same method as in [@YZ2020 Proposition 2].
\[lemma 3.8\] Let $x\in \mathcal{M}$. If $\mathbb{I}-x$ is invertible, then $$\begin{aligned}
(\mathbb{I}-x^*)^{-1}(\mathbb{I}-x^*x)(\mathbb{I}-x)^{-1}
=&2Re((\mathbb{I}-x)^{-1})-\mathbb{I}\\
=&2Re((\mathbb{I}-x)^{-1}-\frac{1}{2}\mathbb{I})\\
=&Re((\mathbb{I}+x)(\mathbb{I}-x)^{-1})=S^*S,\end{aligned}$$ where $S=(\mathbb{I}-x^*x)^{\frac{1}{2}}(\mathbb{I}-x)^{-1}$. Moreover, if $x\in\mathcal{M}$ with $\|x\|<1$, then $\mathbb{I}-x$ is invertible, which implies that the equalities above are hold.
\[theorem 3.9\] Let $x\in \mathcal{M}$ with $\|x\|<1$. Then $$\label{inequ. theorem 3.9}
\mu_t((\mathbb{I}-x^*)^{-1}(\mathbb{I}-x^*x)(\mathbb{I}-x)^{-1})\leq\frac{1+\mu_t(x)}{1-\mu_t(x)}.$$ Moreover, for any subset $K\subseteq[0, 1]$ we have $$\begin{aligned}
\int_K\log\mu_t((\mathbb{I}-x^*)^{-1}(\mathbb{I}-x^*x)(\mathbb{I}-x)^{-1})dt&\leq\int_K\log\frac{1+\mu_t(x)}{1-\mu_t(x)}dt\\
&\leq\int_0^1\log\frac{1+\mu_t(x)}{1-\mu_t(x)}dt.\end{aligned}$$ In particular, $$\begin{aligned}
\frac{\Delta_\tau(\mathbb{I}-x^*x)}{\Delta_\tau(\mathbb{I}-x)^2}\leq\exp\int_0^1\log\frac{1+\mu_t(x)}{1-\mu_t(x)}dt.\end{aligned}$$
We conclude from the definition of $\mu_t(\cdot)$ and $\lambda_t(\cdot)$ that $$\begin{aligned}
\mu_t((\mathbb{I}-x^*)^{-1}(\mathbb{I}-x^*x)(\mathbb{I}-x)^{-1})
=&\lambda_t((\mathbb{I}-x^*)^{-1}(\mathbb{I}-x^*x)(\mathbb{I}-x)^{-1})\\
=&\lambda_t(2Re((\mathbb{I}-x)^{-1})-\mathbb{I})~(Lemma~\ref{lemma 3.8})\\
=&\lambda_t(2Re((\mathbb{I}-x)^{-1}))-1~(Proposition~\ref{proposition 2.2}(9))\\
\leq&\mu_t(2 (\mathbb{I}-x)^{-1})-1~(Remark~\ref{rk:lemma 3.1})\\
=&\frac{2}{\mu_{1-s}^\ell(\mathbb{I}-x)}-1~(Lemma~\ref{lemma3.6})\\
\leq&\frac{2}{1-\mu_t(x)}-1~(Lemma~\ref{lemma 3.7})\\
=&\frac{1+\mu_t(x)}{1-\mu_t(x)}.\end{aligned}$$ Furthermore, since $\frac{1+\mu_t(x)}{1-\mu_t(x)}\geq1$, (\[inequ. theorem 3.9\]) means that $$\begin{aligned}
\int_K\log\mu_t((\mathbb{I}-x^*)^{-1}(\mathbb{I}-x^*x)(\mathbb{I}-x)^{-1})dt&\leq\int_K\log\frac{1+\mu_t(x)}{1-\mu_t(x)}dt\\
&\leq\int_0^1\log\frac{1+\mu_t(x)}{1-\mu_t(x)}dt.\end{aligned}$$ Finally, by (\[inequ. theorem 3.9\]) and Proposition \[proposition 2.4\](1)-(3), we have $$\begin{aligned}
\frac{\Delta_\tau(\mathbb{I}-x^*x)}{\Delta_\tau(\mathbb{I}-x)^2}&=\Delta_\tau((\mathbb{I}-x^*)^{-1}(\mathbb{I}-x^*x)(\mathbb{I}-x)^{-1})\\
&=\exp\int_0^1\log\mu_t((\mathbb{I}-x^*)^{-1}(\mathbb{I}-x^*x)(\mathbb{I}-x)^{-1})dt\\
&\leq\exp\int_0^1\log\frac{1+\mu_t(x)}{1-\mu_t(x)}dt.\end{aligned}$$
To achieve one of our main results, we state for easy reference the following fact, which will be applied below.
\[lemma 4.1\] Let $x, y\in\mathcal{M}$ be invertible. If $K$ is a Borel subset of $[0, 1]$ with $m(K)=t$ $(m(K)$ denotes the Lebesgue measure of $K)$, then $$\begin{aligned}
\int_K\log\mu_s(x)ds+\int_{0}^t\log\mu_{1-s}(y)ds
\leq
\int_K\log\mu_s(xy)ds.\end{aligned}$$
Let $K^c$ denote the set $\{t\in [0, 1]: t\notin E\}$. Then $m(K^c)=1-t$. We conclude from [@N1987 Theorem 2] that $$\label{equat. lemma 4.1(1)}
\int_{K^c}\log\mu_s(xy)ds\leq\int_{K^c}\log\mu_s(x)+\int_0^{1-t}\log\mu_s(y)ds.$$ Note that $x, y\in\mathcal{M}$ are invertible. By Proposition \[proposition 2.4\](1) and (3) we have $\Delta(x)\neq0, \Delta(y)\neq0$ and $$\label{equat. lemma 4.1(2)}
-\infty<\int_0^1\log(\mu_s(x))ds+\int_{0}^1\log\mu_{s}(y)ds=\int_{0}^1\log\mu_{s}(xy)ds<\infty$$ Subtracting (\[equat. lemma 4.1(1)\]) from (\[equat. lemma 4.1(2)\]) yields $$\int_E\log\mu_s(x)ds+\int_{1-t}^1\log\mu_{s}(y)ds\leq
\int_E\log\mu_s(xy)ds,$$ i.e., $$\int_E\log\mu_s(x)ds+\int_{0}^t\log\mu_{1-s}(y)ds\leq
\int_E\log\mu_s(xy)ds.$$
\[remark 4.2\]
1. Let $x, y\in\mathcal{M}$ and let $K$ be a Borel subset of $[0, 1]$ with $m(K)=t$ (here $m(K)$ denotes the Lebesgue measure of $K)$. Then $$\begin{aligned}
\int_K\log\mu_s(x)ds+\int_{0}^t\log\mu_{1-s}(y)ds
\leq
\int_K\log\mu_s(xy)ds.\end{aligned}$$ Indeed, if $x, y$ are invertible, then it follows from Lemma \[lemma 4.1\]. We write $x=u|x|$ and $y=v|y|$ for unitary operators $u, v\in \mathcal{M}$. Then $z=u|x||y^*|v^*$ and $\mu_t(x)=\mu_t(|x|)$, $\mu_t(y)=\mu_t(|y^*|)$, $\mu_t(z)=\mu_t(|x||y^*|)$. Thus, we may without loss of generality assume $x\geq0, y\geq0$ and let $$z(\epsilon_1, \epsilon_2)=(x+\epsilon_1\mathbb{I})(y+\epsilon_2\mathbb{I}).$$ Note that $\mu_s(x+\epsilon_1\mathbb{I})=\mu_s(x)+\epsilon_1$ and $\mu_s(y+\epsilon_2\mathbb{I})=\mu_s(y)+\epsilon_2$. From Lemma \[lemma 4.1\] we see that $$\label{inequ. remark 4.2}
\begin{split}
&\int_K\log(\mu_s(x)+\epsilon_1)ds+\int_{0}^t\log(\mu_{1-s}(y)+\epsilon_2)ds\\
\leq&
\int_K\log\mu_s(z(\epsilon_1, \epsilon_2))ds.
\end{split}$$ Moreover, for any projection operators $e\in \mathcal{M}$, we have $$\|z(\epsilon_1, \epsilon_2)e\|^2=\|e(y+\epsilon_2\mathbb{I})
(x^2+2\epsilon_1x+\epsilon_1^2\mathbb{I})
(y+\epsilon_2\mathbb{I})e\|,$$ which implies that $\mu_s(z(\epsilon_1, \epsilon_2))$ is decreasing in $\epsilon_1$. Similarly, $\mu_s(z(\epsilon_1, \epsilon_2))$ is decreasing in $\epsilon_2$. Letting $\epsilon_i\rightarrow0$ and using the monotone convergence theorem in (\[inequ. remark 4.2\]), we obtain the desired inequality.
2. Let $x, y\in\mathcal{M}$ and let $K$ be a Borel subset of $[0, 1]$ with $m(K)=t$ $(m(K)$ denotes the Lebesgue measure of $K)$. Combing [@N1987 Theorem 2] with Lemma \[lemma 4.1\] we can assert that $$\begin{aligned}
\int_K\log\mu_s(x)ds+\int_{0}^t\log\mu_{1-s}(y)ds
&\leq
\int_K\log\mu_s(xy)ds\\
&\leq\int_K\log\mu_s(x)+\int_0^t\log\mu_s(y)ds.\end{aligned}$$ In particular, if $K=[0, t]$, then $$\begin{aligned}
\int_0^t\log\mu_s(x)ds+\int_{0}^t\log\mu_{1-s}(y)ds
&\leq
\int_0^t\log\mu_s(xy)ds\\
&\leq\int_0^t\log\mu_s(x)+\int_0^t\log\mu_s(y)ds.\end{aligned}$$
\[theorem 3.10\] Let $x\in \mathcal{M}$ with $\|x\|<1$. If $K$ is a Borel subset of $[0, 1]$ with $m(K)=t$ $(m(K)$ denotes the Lebesgue measure of $K)$, then $$\begin{aligned}
&\int_K\log\mu_{s}((\mathbb{I}-x^*)^{-1}(\mathbb{I}-x^*x)(\mathbb{I}-x)^{-1})ds\\
&\geq
\int_{0}^{t}2\log\frac{1}{1+\mu_{s}(x)}ds+\int_K\log(1-\mu_{1-s}(x)^2)ds, t>0.\end{aligned}$$
For convenience, we write $A:=(\mathbb{I}-x^*)^{-1}(\mathbb{I}-x^*x)(\mathbb{I}-x)^{-1}$. Since $\|x\|<1$, $A$ is invertible, hence that $\Delta(A)>0.$ Therefore, $\int_0^{1}\log\mu_{s}(A)ds>-\infty$. Using Lemma \[lemma 4.1\] twice, we have $$\int_{K}\log\mu_{s}(A)ds\geq
\int_{0}^{t}2\log\mu_{1-s}((\mathbb{I}-x)^{-1})ds+\int_K\log\mu_{s}(\mathbb{I}-x^*x) ds.$$ It follows from Lemma \[lemma 3.7\](3)-(5) and Lemma \[lemma3.6\] that $$\begin{aligned}
\int_{K}\log\mu_{s}(A)ds&\geq
\int_{0}^{t}2\log\mu_{1-s}((\mathbb{I}-x)^{-1})ds+\int_K\log\mu_{s}(\mathbb{I}-x^*x) ds\\
&=\int_{0}^{t}2\log\frac{1}{\mu_{s}^\ell(\mathbb{I}-x)}ds+\int_K\log\mu_{s}(\mathbb{I}-x^*x) ds\\
&\geq\int_{0}^{t}2\log\frac{1}{1+\mu_{s}^\ell(x)}ds+\int_K\log(1-\mu_{1-s}^\ell(x)^2)ds\\
&=\int_{0}^{t}2\log\frac{1}{1+\mu_{s}(x)}ds+\int_K\log(1-\mu_{1-s}(x)^2)ds,\end{aligned}$$ because $\mu_{s}^\ell(x)=\mu_{s}(x)$ holds for almost every $t\in[0, 1].$
\[corollary 3.10\] Let $x\in \mathcal{M}$ with $\|x\|<1$. Then $$\int_0^t\log\mu_{1-s}((\mathbb{I}-x^*)^{-1}(\mathbb{I}-x^*x)(\mathbb{I}-x)^{-1})ds\geq
\int_0^t\log\frac{1-\mu_s(x)}{1+\mu_s(x)}ds, ~t>0.$$ In particular, $$\begin{aligned}
\frac{\Delta_{\tau}(\mathbb{I}-x^*x)}{\Delta_{\tau}(\mathbb{I}-x)^2}\geq\exp\int_0^1\log\frac{1-\mu_s(x)}{1+\mu_s(x)}ds.\end{aligned}$$
Replacing $K$ by $[1-t, 1]$, in Theorem \[theorem 3.10\] we have $$\begin{aligned}
\int_{0}^{t}\log\mu_{1-s}(A)ds&=\int_{1-t}^{1}\log\mu_{s}(A)ds\\
&\geq
\int_{0}^{t}2\log\frac{1}{1+\mu_{s}(x)}ds+\int_{1-t}^1\log(1-\mu_{1-s}(x)^2)ds\\
&=\int_{0}^{t}2\log\frac{1}{1+\mu_{s}(x)}ds+\int_0^t\log(1-\mu_{s}(x)^2) ds\\
&=\int_{0}^{t}\log\frac{1-\mu_{s}(x)}{1+\mu_{s}(x)}ds.\end{aligned}$$ Therefore, letting $t\rightarrow1$ yields $$\begin{aligned}
\int_{0}^{1}\log\mu_{s}(A)ds&=\int_{0}^{1}\log\mu_{1-s}(A)ds
\geq\int_{0}^{1} \log\frac{1+\mu_{s}(x)}{1-\mu_{s}(x)}ds.\end{aligned}$$ This completes the proof.
\[theorem 3.11\] Let $0\leq x_i\in \mathcal{M}$ with $\|x_i\|<1, i=1, 2,\cdots, n$. Then for any unitary operator $u\in\mathcal{M}$ and positive scalars $\omega_i, i=1, 2,\cdots, n,\sum^n_i\omega_i=1$, we have $$\begin{aligned}
\prod_{i=1}^n[\exp\int_0^1\log\frac{1-\mu_t(x_i)}{1+\mu_t(x_i)}dt]^{\omega_i}
\leq\frac{\Delta_\tau(\mathbb{I}-W^2)}{\Delta_\tau(\mathbb{I}-uW)^2}
\leq\prod_{i=1}^n[\exp\int_0^1\log\frac{1+\mu_t(x_i)}{1-\mu_t(x_i)}dt]^{\omega_i},\end{aligned}$$ where $W=\sum_{i=1}^n\omega_i x_i.$
An easy calculation shows that $1-W^2$ and $1-uW$ are invertible and $W\geq0$ with $\|W\|<1$. Theorem \[theorem 3.9\] and Corollary \[corollary 3.10\] tell us that $$\label{inequ. theorem 3.11-1}
\exp\int_0^1\log\frac{1-\mu_t(x)}{1+\mu_t(x)}dt\leq
\frac{\Delta(\mathbb{I}-x^*x)}{\Delta(\mathbb{I}-x)^2}
\leq\exp\int_0^1\log\frac{1+\mu_t(x)}{1-\mu_t(x)}dt.$$ Note that [@FK1986 Theorem 4.4] tells us that $$\int_0^t\mu_s(W)ds\leq \int_0^t\sum_{i=1}^n\omega_i\mu_s(x_i)ds.$$ The rest of the proof run as [@LZ2017 Theorem 5]. For the convenience of the reader, we add a proof. Indeed, the convexity and the monotonicity of the function $f(t)=\log\frac{1+t}{1-t}, 0\leq t<1$ mean that $$\int_0^tf(\mu_s(W))ds\leq \int_0^tf(\sum_{i=1}^n\omega_i\mu_s(x_i))ds.$$ On the other hand, by Lewent’s inequality( [@LZ2017; @L2013]), we obtain $$\frac{1+\sum_{i=1}^n\omega_i\mu_s(x_i)}{1-\sum_{i=1}^n\omega_i\mu_s(x_i)}
\leq\prod_{i=1}^n(\frac{1+\mu_s(x_i)}{1-\mu_s(x_i)})^{\omega_i}.$$ Thus $$\int_0^tf(\mu_s(W))ds\leq\int_0^t\log\prod_{i=1}^n(\frac{1+\mu_s(x_i)}{1-\mu_s(x_i)})^{\omega_i}ds
=\sum_{i=1}^n\omega_i\int_0^t\log(\frac{1+\mu_s(x_i)}{1-\mu_s(x_i)})ds.$$ It follows that $$\label{inequ. theorem 3.11-2}\exp\{\int_0^t\log(\frac{1+\mu_s(W)}{1-\mu_s(W)})ds\}\leq
\prod_{i=1}^n[\exp\int_0^1\log\frac{1+\mu_t(x_i)}{1-\mu_t(x_i)}dt]^{\omega_i}.$$ Moreover, the inequalities in (\[inequ. theorem 3.11-2\]) reverse by taking reciprocals, which implies $$\label{inequ. theorem 3.11-3}\exp\{\int_0^t\log(\frac{1-\mu_s(W)}{1+\mu_s(W)})ds\}\geq
\prod_{i=1}^n[\exp\int_0^1\log\frac{1-\mu_t(x_i)}{1+\mu_t(x_i)}dt]^{\omega_i}.$$ Combining (\[inequ. theorem 3.11-1\]) with (\[inequ. theorem 3.11-2\]) and (\[inequ. theorem 3.11-3\]) yields $$\begin{aligned}
\prod_{i=1}^n[\exp\int_0^1\log\frac{1-\mu_t(x_i)}{1+\mu_t(x_i)}dt]^{\omega_i}
\leq\frac{\Delta_\tau(\mathbb{I}-W^2)}{\Delta_\tau(\mathbb{I}-uW)^2}
\leq\prod_{i=1}^n[\exp\int_0^1\log\frac{1+\mu_t(x_i)}{1-\mu_t(x_i)}dt]^{\omega_i}.\end{aligned}$$
Cayley transform with logarithmic submajorisation
=================================================
In this section, we will consider some logarithmic submajorisation inequalities related to Cayley transform. We will extend some results of Yang-Zhang[@YZ2020] to the case of finite von Neumann algebra.
Let $x\in\mathcal{M}$. If $x+i\mathbb{I}$ is invertible, we call $\mathcal{C}(x)=(x-i\mathbb{I})(x+i\mathbb{I})^{-1}$ the Cayley transform of $x$.
\[theorem 5.3\] Let $x, y\in\mathcal{M}$ with $\|x\|<1, \|y\|<1$ and let $\mathcal{C}(x)$ and $\mathcal{C}(y)$ be the Cayley transforms of $x$ and $y$, respectively. If $K$ is a Borel subset of $[0, 1]$ with $m(K)=t$ $(m(K)$ denotes the Lebesgue measure of $K)$, then $$\begin{aligned}
&\int_K\log(1-\mu_{1-s}(x))ds-\int_0^t\log(1+\mu_{s}(x))ds\\
&\leq
\int_K\log\mu_s(\mathcal{C}(x))ds\\
&\leq\int_K\log(1+\mu_s(x))-\int_0^t\log(1-\mu_s(x))ds\end{aligned}$$ and $$\begin{aligned}
&\int_E\log\mu_s(\mathcal{C}(x)-\mathcal{C}(y))ds\\
\leq&
\int_E\log2\mu_s(x-y)ds-\int_0^t\log[(1-\mu_{s}(x))(1-\mu_{s}(y))]ds.\end{aligned}$$
Let us first compute the upper bounds. Remark \[remark 4.2\] shows that $$\begin{aligned}
\int_K\log\mu_s(\mathcal{C}(x))ds&=\int_K\log\mu_s((x-i\mathbb{I})(x+i\mathbb{I})^{-1})ds\\
&\leq\int_K\log\mu_s(x-i\mathbb{I})ds
+\int_0^t\log\mu_s((x+i\mathbb{I})^{-1})ds.\end{aligned}$$ Together with Lemma \[lemma 3.7\] this gives $$\begin{aligned}
\int_0^t\log\mu_s((x+i\mathbb{I})^{-1})ds&\leq \int_0^t\log[\mu_{1-s}^\ell(x+i\mathbb{I})]^{-1}ds\\
&\leq\int_0^t\log(1-\mu_{s}(x))^{-1}ds\\
&=-\int_0^t\log(1-\mu_{s}(x))ds.\end{aligned}$$ Thus $$\begin{aligned}
\int_K\log\mu_s(\mathcal{C}(x))ds
\leq\int_K\log(1+\mu_s(x))-\int_0^t\log(1-\mu_s(x))ds\end{aligned}$$
The lower bound follows easily by using Remark \[remark 4.2\]. Indeed, from Remark \[remark 4.2\] we obtain $$\begin{aligned}
\int_E\log\mu_s(\mathcal{C}(x))ds&\geq
\int_0^t\log\mu_{1-s}(x-i\mathbb{I})ds+\int_E\log\mu_{s}((x+i\mathbb{I})^{-1})ds\\
&\geq\int_0^t\log(1-\mu_{s}^\ell(x))ds-\int_E\log\mu_{1-s}^\ell(x+i\mathbb{I})ds\\
&\geq\int_0^t\log(1-\mu_{s}^\ell(x))ds-\int_E\log(1+\mu_{s}^\ell(x))ds\\
&=\int_0^t\log(1-\mu_{s}(x))ds-\int_E\log(1+\mu_{s}(x))ds.\end{aligned}$$
For the second part, an easy calculation shows that $\mathcal{C}(x)=1-2i(x+i\mathbb{I})^{-1}$ and $$\mathcal{C}(x)-\mathcal{C}(y)=2i(y+i\mathbb{I})^{-1}(x-y)(x+i\mathbb{I})^{-1}.$$ Hence, Remark \[remark 4.2\] implies that $$\begin{aligned}
\int_E\log\mu_s(\mathcal{C}(x)-\mathcal{C}(y))ds=&
\int_E\log2\mu_s((y+i\mathbb{I})^{-1}(x-y)(x+i\mathbb{I})^{-1})ds\\
\leq& \int_0^t\log\mu_s((y+i\mathbb{I})^{-1})ds+\int_E\log2\mu_s(x-y)ds\\
&+\int_0^t\log\mu_s((x+i\mathbb{I})^{-1})ds\\
\leq& \int_0^t\log[\mu_{1-s}^\ell(y+i\mathbb{I})]^{-1} ds+\int_E\log2\mu_s(x-y)ds\\
&+\int_0^t\log[\mu_{1-s}^\ell(x+i\mathbb{I})]^{-1}ds\\
\leq& \int_0^t\log[1-\mu_{s}(y)]^{-1} ds+\int_E\log2\mu_s(x-y)ds\\
&+\int_0^t\log[1-\mu_{s}(x)]^{-1}ds\\
=&\int_E\log2\mu_s(x-y)ds-\int_0^t\log[(1-\mu_{s}(x))(1-\mu_{s}(y))]ds\end{aligned}$$
If we replace $K$ by $[0, 1]$, in Theorem \[theorem 5.3\] we have the following corollary.
Let $x, y\in\mathcal{M}$ with $\|x\|<1, \|y\|<1$ and let $\mathcal{C}(x)$ and $\mathcal{C}(y)$ be the Cayley transforms of $x$ and $y$, respectively. Then $$\begin{aligned}
\int_0^1\log\frac{1-\mu_{1-s}(x)}{1+\mu_{s}(x)}ds
\leq
\int_0^1\log\mu_s(\mathcal{C}(x))ds \leq\int_0^1\log\frac{1+\mu_s(x)}{1-\mu_s(x)}ds\end{aligned}$$ and $$\begin{aligned}
\int_0^1\log\mu_s(\mathcal{C}(x)-\mathcal{C}(y))ds\leq
\int_0^1\log\frac{2\mu_s(x-y)}{(1-\mu_{s}(x))(1-\mu_{s}(y))}ds.\end{aligned}$$
Acknowledgments {#acknowledgments .unnumbered}
================
The first author wishes to express his thanks to Dr.Minghua Lin for suggesting the problem and for many stimulating conversations.
Funding {#funding .unnumbered}
========
This research was partially supported by the National Natural Science Foundation of China No. 11761067 and National Natural Science Foundation of China No. 11801486.
Availability of data and materials {#availability-of-data-and-materials .unnumbered}
==================================
Not applicable.
Competing interests {#competing-interests .unnumbered}
====================
The author declares that there is no conflict of interests regarding the publication of this paper.
Authors’s contributions {#authorss-contributions .unnumbered}
=========================
Both authors contributed equally and significantly in writing this article. Both authors read and approved the final manuscript.
[99]{}
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'A reduction of the transmission eigenvalue problem for multiplicative sign-definite perturbations of elliptic operators with constant coefficients to an eigenvalue problem for a non-selfadjoint compact operator is given. Sufficient conditions for the existence of transmission eigenvalues and completeness of generalized eigenstates for the transmission eigenvalue problem are derived. In the trace class case, the generic existence of transmission eigenvalues is established.'
address:
- |
M. Hitrik, Department of Mathematics\
UCLA\
Los Angeles\
CA 90095-1555\
USA
- |
K. Krupchyk, Department of Mathematics and Statistics\
University of Helsinki\
P.O. Box 68\
FI-00014 Helsinki\
Finland
- |
P. Ola, Department of Mathematics and Statistics\
University of Helsinki\
P.O. Box 68\
FI-00014 Helsinki\
Finland
- |
L. Päivärinta, Department of Mathematics and Statistics\
University of Helsinki\
P.O. Box 68\
FI-00014 Helsinki\
Finland
author:
- Michael Hitrik
- Katsiaryna Krupchyk
- Petri Ola
- Lassi Päivärinta
title: Transmission eigenvalues for elliptic operators
---
Introduction
============
Let $P_0(D)$ be an elliptic partial differential operator on ${\mathbb{R}}^n$, $n\ge 2$, of order $m\ge 2$ with constant real coefficients, $$P_0(D)=\sum_{|\alpha|\le m} a_{\alpha}D^\alpha, \quad a_\alpha\in{\mathbb{R}}, \quad D_j=-i\frac{\partial}{\partial x_j},\quad j=1,\dots,n.$$ Let $\Omega\subset {\mathbb{R}}^n$ be a bounded domain with a $C^\infty$-boundary and assume that $V\in C^\infty(\overline{\Omega}, {\mathbb{R}})$ with $V>0$ in $\overline{\Omega}$. The interior transmission problem associated to $P_0$ and $V$ is the following degenerate boundary value problem, $$\label{eq_TE_acoustic}
\begin{aligned}
(P_0-\lambda)v=0 \quad &\text{in} \quad \Omega,\\
(P_0-\lambda(1+ V))w=0 \quad &\text{in} \quad \Omega,\\
v-w \in H^{m}_0(\Omega).
\end{aligned}$$ Here $H^m_0(\Omega)$ is the standard Sobolev space, defined as the closure of $C^\infty_0(\Omega)$ in the Sobolev space $H^m(\Omega)$. We say that $\lambda\in {\mathbb{C}}$ is a transmission eigenvalue if the problem has non-trivial solutions $0\ne v\in L^2_{\textrm{loc}}(\Omega)$ and $0\ne w\in L^2_{\textrm{loc}}(\Omega)$.
In the recent paper [@HitKruOlaPai], we have studied the interior transmission problem and transmission eigenvalues for multiplicative sign-definite perturbations of linear partial differential operators with constant real coefficients. Sufficient conditions for the discreteness of the set of transmission eigenvalues and for the existence of real transmission eigenvalues were obtained. In particular, in the elliptic case, the set of transmission eigenvalues is discrete and in [@HitKruOlaPai], the existence of real transmission eigenvalues was obtained for certain elliptic operators such as the biharmonic operator and the Dirac system in ${\mathbb{R}}^3$.
The purpose of the present note is to point out an approach to the study of the transmission eigenvalues in the elliptic case, based on a reduction to the eigenvalue problem for a compact non-selfadjoint operator. By an application of Lidskii’s theorem, we obtain sufficient conditions for the existence of (possibly complex) transmission eigenvalues, and the completeness of the set of the generalized eigenvectors, as well as demonstrate the generic existence of transmission eigenvalues. Let us mention explicitly that in this approach, we were directly inspired by the recent works [@AboRob; @ChaHelLap04; @HelRobWang; @Rob2004], where similar ideas in dealing with quadratic eigenvalue problems have been used to study hypoelliptic partial differential operators which are not analytic hypoelliptic.
The significance of transmission eigenvalues and of the interior transmission eigenvalue problem comes from inverse scattering theory, and originally, this problem was introduced in [@ColMonk88] in this context. The real transmission eigenvalues can be characterized as those values for which the scattering amplitude is not injective, see [@ColPaiSyl; @HitKruOlaPai]. Furthermore, in reconstruction algorithms of inverse scattering theory [@CakColbook; @ColKir96; @KirGribook], transmission eigenvalues correspond to frequencies that one needs to avoid in the reconstruction procedure.
Recently there has been a large number of works devoted to the interior transmission eigenvalue problem [@CakColGint_complex; @CakColHous10; @CakDroHou; @ColKirPai; @kir07; @paisyl08], with the major part being concerned with the case $P_0=-\Delta$. The existing results establish the discreteness of the set of transmission eigenvalues, [@ColKirPai], and give sufficient conditions for the existence of an infinite set of real transmission eigenvalues, [@CakDroHou; @paisyl08]. We would particularly like to mention the recent paper [@CakColGint_complex], where the existence of complex transmission eigenvalues was shown, assuming that the perturbation $V$ in is constant and sufficiently small.
In this note, we have chosen to base our presentation on the generalized acoustic wave equation $(P_0-\lambda(1+V))u=0$. Under the assumption that the full symbol of $P_0$ is non-negative, all the results could equally well have been derived for the following interior transmission problem associated to the Schrödinger equation $(P_0+V-\lambda)u=0$, $$\begin{aligned}
(P_0-\lambda)v=0 \quad &\text{in} \quad \Omega,\\
(P_0+V-\lambda)w=0 \quad &\text{in} \quad \Omega,\\
v-w \in H^{m}_0(\Omega).\end{aligned}$$
The structure of this note is as follows. In Section 2 we reduce the interior transmission problem to an eigenvalue problem for a compact non-selfadjoint operator in a suitable Schatten class. As a consequence of this reduction, in Section 3, we derive sufficient conditions for the existence of transmission eigenvalues and completeness of the generalized eigenstates. Finally, in Section 4, we show the generic existence of transmission eigenvalues in the trace class case.
Reduction to an eigenvalue problem for a non-selfadjoint compact operator
=========================================================================
From [@HitKruOlaPai], let us recall the following characterization of transmission eigenvalues.
\[thm\_equivalence\] Assume that $V\in C^\infty(\overline{\Omega}, {\mathbb{R}})$ with $V>0$ in $\overline{\Omega}$. A complex number $\lambda\ne 0$ is a transmission eigenvalue if and only if there exists $0\ne u\in H^{m}_0(\Omega)$ satisfying $$T_\lambda u:=(P_0-\lambda(1+V))\frac{1}{V}(P_0-\lambda)u=0\quad \text{in}\quad \mathcal{D}'(\Omega).$$
The question of deciding whether $0\ne \lambda\in {\mathbb{C}}$ is a transmission eigenvalue is therefore equivalent to finding a non-trivial solution $u\in H^m_0(\Omega)$ of the following quadratic eigenvalue problem $$\label{eq_quadratic}
T_\lambda u=(A-\lambda B +\lambda^2 C)u=0,$$ where $$A=P_0\frac{1}{V}P_0,\quad B=\frac{1}{V}P_0+P_0\frac{1}{V}+P_0,\quad C=1+\frac{1}{V}.$$ Consider the following factorization $$\begin{aligned}
T_\lambda=C^{1/2}L_\lambda C^{1/2}, \quad L_\lambda&=\tilde A-\lambda \tilde B+\lambda^2,\\
\tilde A&=C^{-1/2}AC^{-1/2}, \quad \tilde B=C^{-1/2}BC^{-1/2}.\end{aligned}$$ In [@HitKruOlaPai] it was proved that the operator $\tilde A$, equipped with the domain $$\mathcal{D}(\tilde A)=H^{2m}(\Omega)\cap H^m_0(\Omega),$$ is a self-adjoint operator on $L^2(\Omega)$ with a discrete spectrum. Here the regularity assumption on $V$ can be relaxed to $V\in C^N(\overline{\Omega})$, with $N$ being large enough but finite.
\[prop\_properties\]
- The operator $\tilde A$ is positive, and $ \mathcal{D}(\tilde A^{1/2})=H_0^m(\Omega)$.
- The operators $\tilde B\tilde A^{-1/2}$ and $\tilde A^{-1/2}\tilde B$ are bounded in $L^2(\Omega)$.
- The operator $\tilde A^{-1/2}$ is in the Schatten class $\mathcal{C}^p$ for $p>n/m$.
We refer to [@Sim_book] for the definition and properties of the Schatten class operators.
(i). Let $u\in \mathcal{D}(\tilde A)\subset H^m_0(\Omega)$. Then $$(\tilde Au,u)=\int_{\Omega} \frac{1}{V}|P_0C^{-1/2}u|^2dx\ge C_{\Omega,V} \|u\|^2,\quad C_{\Omega,V}>0.$$ Here the last inequality follows from the estimate [@horbookII Theorem 10.3.7] $$\|P_0(D)u\|\ge C_{\Omega}\|u\|, \quad u\in H^m_0(\Omega).$$ We know from [@HitKruOlaPai] that the form domain of the positive self-adjoint operator $\tilde A$ is $H_0^m(\Omega)$ and thus, $$\mathcal{D}(\tilde A^{1/2})=H_0^m(\Omega).$$
(iii). The claim follows from the fact that the inclusion map $$i:H^m_0(\Omega)\to L^2(\Omega)$$ is in the Schatten class $\mathcal{C}^p$ for $p>n/m$. The latter can be concluded from the fact that the operator $(1-\Delta)^{-m/2}$ is in the Schatten class $\mathcal{C}^p$ for $p>n/m$, on $L^2(\mathbb{T}^n)$, where $\mathbb{T}^n$ is the $n$-dimensional torus. This concludes the proof of the proposition, as (ii) is clear.
Notice that $0\ne \lambda\in {\mathbb{C}}$ is an eigenvalue of the quadratic eigenvalue problem $T_\lambda u=0$ with an eigenstate $u\in H^m_0(\Omega)$ if and only if $\lambda$ is an eigenvalue of the quadratic eigenvalue problem $L_\lambda v=0$ with $v=C^{1/2} u\in H^m_0(\Omega)$.
The holomorphic family $L_\lambda:\mathcal{D}(\tilde A)\to L^2(\Omega)$ is Fredholm of index $0$, invertible at $\lambda=0$. Thus, by the analytic Fredholm theory, $$L^{-1}_\lambda:L^2(\Omega)\to \mathcal{D}(\tilde A), \quad \lambda\in {\mathbb{C}},$$ is a meromorphic family of operators, with residues of finite rank.
Following [@Rob2004], consider the closed operator $$\mathcal{A}=\begin{pmatrix} 0 & 1\\
-\tilde A & \tilde B
\end{pmatrix},$$ acting in the Hilbert space $$\mathcal{K}=\mathcal{D}(\tilde A^{1/2})\times L^2(\Omega)=H^m_0(\Omega)\times L^2(\Omega),$$ equipped with the domain $$\mathcal{D}(\mathcal{A})=\mathcal{D}(\tilde A)\times \mathcal{D}(\tilde A^{1/2})=(H^{2m}(\Omega)\cap H^m_0(\Omega))\times H^m_0(\Omega).$$ The spectrum of $\mathcal{A}$ is discrete, and as $$(\mathcal{A}-\lambda)^{-1}=\begin{pmatrix}
L_\lambda^{-1}(\tilde B-\lambda) & - L_\lambda^{-1}\\
L_\lambda^{-1}\tilde A & - L_\lambda^{-1}\lambda
\end{pmatrix},$$ it follows that $0\ne \lambda\in {\mathbb{C}}$ is a transmission eigenvalue if and only if $\lambda$ is an eigenvalue of the operator $\mathcal{A}$. The latter is equivalent to the fact that $1/\lambda$ is an eigenvalue of the operator $$\mathcal{A}^{-1}=\begin{pmatrix}
\tilde A^{-1}\tilde B & -\tilde A^{-1}\\
1 & 0
\end{pmatrix}:\mathcal{K}\to \mathcal{K}.$$ Given Proposition \[prop\_properties\], it follows from [@Rob2004] that $\mathcal{A}^{-1}$ is in the Schatten class $\mathcal{C}^p$ on $\mathcal{K}$, for $p>n/m$.
It will be more convenient to work in the Hilbert space $L^2(\Omega)\times L^2(\Omega)$ rather than $\mathcal{K}$. To this end, we introduce the operator $$T=\begin{pmatrix} \tilde A^{1/2} & 0\\
0 & 1
\end{pmatrix},$$ which defines an isomorphism $$T:\mathcal{K}\to L^2(\Omega)\times L^2(\Omega),$$ and set $$\label{eq_operator_P}
\mathcal{D}=T\mathcal{A}^{-1}T^{-1}=
\begin{pmatrix} \tilde A^{-1/2} \tilde B\tilde A^{-1/2} & -\tilde A^{-1/2}\\
\tilde A^{-1/2} &0
\end{pmatrix}: L^2(\Omega)\times L^2(\Omega)\to L^2(\Omega)\times L^2(\Omega).$$ The operator $\mathcal{D}$ is in the Schatten class $\mathcal{C}^p$ on $L^2(\Omega)\times L^2(\Omega)$. We summarize this section in the following result.
\[prop\_mathcal[D]{}\]
A complex number $\lambda\ne 0$ is a transmission eigenvalue for [(\[eq\_TE\_acoustic\])]{} if and only if $1/\lambda$ is an eigenvalue of the operator $\mathcal{D}$ in [(\[eq\_operator\_P\])]{}.
It was shown in [@ColKirPai; @ColPaiSyl; @HitKruOlaPai] that the set of transmission eigenvalues is discrete. The proof relied upon the analytic Fredholm theory. Our reduction of the transmission eigenvalue problem to the eigenvalue problem for the compact operator $\mathcal{D}$ gives another proof of the discreteness of the set of transmission eigenvalues in the elliptic case.
Existence of transmission eigenvalues and completeness of transmission eigenstates
==================================================================================
In this section, we continue to work under the assumptions made in the beginning of the paper, namely that $P_0=P_0(D)$ is elliptic, $V\in C^\infty(\overline\Omega)$, $V>0$ on $\overline\Omega$, and $\p \Omega\in C^\infty$.
In the previous section, we have reduced the transmission eigenvalue problem to a spectral problem for the operator $\mathcal{D}\in \mathcal{C}^p$, $p>n/m$. Recall from [@Sim_book] that this implies that $\mathcal{D}^p$ is of trace class, provided that $p\in {\mathbb{N}}$.
The following result is our main criterion for the existence of transmission eigenvalues. It is based on an application of Lidskii’s theorem, which we recall for the convenience of the reader, see e.g. [@GohGolKaa]: let $\mathcal{A}$ be a trace class operator. Then $$\sum_j\mu_j(\mathcal{A})=\textrm{tr}(\mathcal{A}),$$ where $\mu_j(\mathcal{A})$ are the non-vanishing eigenvalues of $\mathcal{A}$ counted with their algebraic multiplicities. In particular, if the spectrum $\textrm{spec}(\mathcal{A})=\{0\}$, then $\textrm{tr}(\mathcal{A})=0$.
\[thm\_trace\_L\] Assume that $p>n/m$, $p\in {\mathbb{N}}$, and $\emph{\textrm{tr}}(\mathcal{D}^{p})\ne 0$. Then the set of transmission eigenvalues is non-empty.
Assume that spectrum $\textrm{spec}(\mathcal{D})=\{0\}$. Then $\textrm{spec}(\mathcal{D}^p)=\{0\}$, since $$r(\mathcal{D}^p)=\lim_{n\to\infty}\|\mathcal{D}^{pn}\|^{1/n}=\lim_{n\to\infty}\|\mathcal{D}^{n}\|^{p/n}=r(\mathcal{D})^p=0,$$ where $r(\mathcal{D})$ is the spectral radius of $\mathcal{D}$. By an application of Lidskii’s theorem, we get $\textrm{tr}(\mathcal{D}^{p})=0$, which contradicts the assumption of the proposition.
In the case when $m>n$, the operator $\mathcal{D}$ is of trace class on $L^2(\Omega)\times L^2(\Omega)$, and $\textrm{tr}(\mathcal{D})=\textrm{tr}(\tilde A^{-1/2}\tilde B\tilde A^{-1/2})=\textrm{tr}(\tilde B\tilde A^{-1})$.
In the case when $m>n/2$, the operator $\mathcal{D}$ is of Hilbert-Schmidt class and $$\textrm{tr}(\mathcal{D}^{-2})=\textrm{tr}(\tilde A^{-1/2}(\tilde B\tilde A^{-1}\tilde B-2)\tilde A^{-1/2}).$$
The question of completeness of the eigenstates for the transmission eigenvalue problem for the Helmholtz equation has been posed in [@CakDroHou]. To the best of our knowledge, this issue remains unresolved in general. We shall now give a sufficient condition for completeness.
Following [@Rob2004] and [@Markus], we define the generalized eigenspace $\mathcal{E}_{\lambda_0}$ for the transmission eigenvalue $\lambda_0\in {\mathbb{C}}$ as the closed linear space spanned by the vectors $(u_j)_{j=0}^\infty$, $u_j\in H^m_0(\Omega)$, where $$\begin{aligned}
&L_{\lambda_0}u_0=0, \quad u_0\ne 0,\\
& L_{\lambda_0}u_j+L'_{\lambda_0}u_{j-1}+\frac{1}{2}L''_{\lambda_0}u_{j-2}=0, \quad j=1,2,\dots.\end{aligned}$$ Here we set $u_{-1}=0$.
\[thm\_complete\] Assume that the set $$\begin{aligned}
\{\langle \tilde A^{-1/2} \tilde B \tilde A^{-1/2} u_0,u_0 \rangle_{L^2}-2i \emph{{\hbox{Im}\,}} \langle \tilde A^{-1/2}v_0, u_0 \rangle_{L^2},\\
u_0,v_0\in L^2(\Omega),\|(u_0,v_0)\|_{L^2\times L^2}=1
\}
\end{aligned}$$ lies in a closed angle with vertex at zero and opening $\pi/p$, $p>n/m$. Then the space $\bigoplus_{\lambda\in {\mathbb{C}}}\mathcal{E}_\lambda$ is complete in $L^2(\Omega)$.
It follows from [@Rob2004] and Proposition \[prop\_mathcal[D]{}\] that to show that the space $\bigoplus_{\lambda\in {\mathbb{C}}}\mathcal{E}_\lambda$ is complete in $L^2(\Omega)$, it suffices to verify that the space of generalized eigenvectors $\bigoplus_{\lambda}\mathcal{E}_\lambda[\mathcal{D}]$ of the operator $\mathcal{D}$ is complete in $L^2(\Omega)\times L^2(\Omega)$. The latter can be obtained by an application of [@GohGolKaa Theorem 3.1, Chapter X.3], which states that if the set $$\{\langle \mathcal{D}\varphi,\varphi \rangle_{L^2\times L^2}:\varphi\in L^2(\Omega)\times L^2(\Omega),\|\varphi\|_{L^2\times L^2}=1\}$$ lies in a closed angle with vertex at zero and opening $\pi/p$, then the system of generalized eigenvectors of $\mathcal{D}$ is complete. The claim follows.
\[rem\_constant\_potential\]
In the case when $m>n$ and the operator $$B = \frac{1}{V} P_0 + P_0 \frac{1}{V} + P_0$$ is non-negative on $H^m_0(\Omega)$, it follows from Proposition \[thm\_complete\] that the space $\bigoplus_{\lambda\in {\mathbb{C}}}\mathcal{E}_\lambda$ is complete in $L^2(\Omega)$. In particular, if $V=\textrm{const}>0$ in $\overline{\Omega}$ and $P_0(\xi)\ge 0$, $\xi\in {\mathbb{R}}^n$, an application of Proposition \[thm\_complete\] shows that there exist infinitely many transmission eigenvalues and the corresponding generalized transmission eigenstates form a complete system in $L^2(\Omega)$. Notice that when $P_0=\Delta^2$ on ${\mathbb{R}}^3$, the existence of infinitely many real transmission eigenvalues has been established in [@HitKruOlaPai]. The completeness of the generalized transmission eigenstates in the case of a constant potential for this operator seems to be a new observation.
According to Proposition \[prop\_mathcal[D]{}\], $0\ne \lambda\in {\mathbb{C}}$ is a transmission eigenvalue if and only if $1/\lambda$ is an eigenvalue of the operator $\mathcal{D}$. Let us make explicit the connection between the generalized eigenvectors of $\mathcal{D}$ and the generalized transmission eigenstates. When doing so, since $\mathcal{D}=T\mathcal{A}^{-1}T^{-1}$, it will be convenient to consider the generalized eigenvectors of $\mathcal{A}$ directly. Let $$\begin{pmatrix}
u_0\\
v_0
\end{pmatrix}\in H^m_0(\Omega)\times L^2(\Omega)$$ be an eigenvector of $\mathcal{A}$ corresponding to $\lambda$, i.e. $$(\mathcal{A}-\lambda)\begin{pmatrix}
u_0\\
v_0
\end{pmatrix}=0\ \Longleftrightarrow \
v_0=\lambda u_0,\ L_{\lambda}u_0=0,$$ i.e. $u_0\in \mathcal{E}_{\lambda}$. Let $$(\mathcal{A}-\lambda)^2\begin{pmatrix}
u_1\\
v_1
\end{pmatrix}=0.$$ This is equivalent to the fact that $$(\mathcal{A}-\lambda)\begin{pmatrix}
u_1\\
v_1
\end{pmatrix}=\begin{pmatrix}
u_0\\
v_0
\end{pmatrix}$$ is an eigenvector of $\mathcal{A}$. The latter is equivalent to the fact that $$v_1=u_0+\lambda u_1,\quad L_{\lambda}u_1+L'_{\lambda}u_0=0,$$ i.e. $u_1\in \mathcal{E}_{\lambda}$. Continuing in the same fashion, for $j=2,3,\dots$, we have $$(\mathcal{A}-\lambda)^{j+1}\begin{pmatrix}
u_j\\
v_j
\end{pmatrix}=0$$ is equivalent to $$v_j=u_{j-1}+\lambda u_j,\quad L_{\lambda_0}u_j+L'_{\lambda }u_{j-1}+u_{j-2}=0,$$ i.e. $u_j\in \mathcal{E}_{\lambda}$. This shows that the first components of the generalized eigenvectors of $\mathcal{A}$, corresponding to the eigenvalue $\lambda$, are given by the generalized transmission eigenstates, corresponding to the transmission eigenvalue $\lambda$, and vice versa.
Generic existence of transmission eigenvalues in the trace class case
=====================================================================
In this section, we let $P_0=P_0(D)$ be a formally selfadjoint elliptic operator with constant coefficients of order $m$, with $m>n$, $V\in C^N(\overline{\Omega})$ where $N$ is large enough fixed, and $\p \Omega\in C^\infty$. Let us introduce the following open connected subset of the real Banach space $C^N(\overline{\Omega}, {\mathbb{R}})$, $$\mathcal{E}=\{V\in C^N(\overline{\Omega},{\mathbb{R}}):V>0\}.$$ When $V\in \mathcal{E}$, we shall be concerned with the quantity $\textrm{tr}(\mathcal{D})=\textrm{tr}(\tilde B\tilde A^{-1})$. In order to indicate the dependence of the operators $\tilde A$ and $\tilde B$ on the potential, we shall write $$\begin{aligned}
q=\frac{1}{V},&\quad V\in \mathcal{E},\quad A_q=P_0qP_0,\quad B_q=qP_0+P_0q+P_0,\\
\tilde A=\tilde A_q&=(1+q)^{-1/2}A_q(1+q)^{-1/2},\quad \tilde B= \tilde B_q=(1+q)^{-1/2}B_q(1+q)^{-1/2}.\end{aligned}$$ Using the cyclicity property of the trace, we have $$\textrm{tr}(\tilde B_q \tilde A_q^{-1})=\textrm{tr}((1+q)^{-1/2}B_q A_q^{-1}(1+q)^{1/2})=\textrm{tr}(B_q A_q^{-1}).$$
Assume that $m>n$ and that $P_0(\xi)\ge 0$, $\xi\in {\mathbb{R}}^n$. Then the set $$\mathcal{F}=\{V\in \mathcal{E}:\emph{tr}(B_qA^{-1}_q)\ne 0\}$$ is open and dense in $\mathcal{E}$.
The theorem above and Proposition \[thm\_trace\_L\] imply the existence of transmission eigenvalues in the trace class case, for an open and dense set of potentials.
Let us first show that the set $\mathcal{F}$ is open. To this end it suffices to prove that the function $
V\mapsto \textrm{tr}(B_qA^{-1}_q)$ is continuous on $\mathcal{E}$ in the topology of $C^N(\overline{\Omega}, {\mathbb{R}})$. We shall show that the map $V\mapsto B_qA^{-1}_q$ is continuous, with values in the space of trace class operators.
Let $V_j\to V$ in $\mathcal{E}$. Then $\p^\alpha q_j\to \p^\alpha q$ uniformly on $\overline{\Omega}$ for $|\alpha|\le N$. Let us write $$\label{eq_op_1}
\begin{aligned}
B_{q_j}A^{-1}_{q_j}-B_qA^{-1}_q=(B_{q_j}-B_q)A^{-1}_{q_j}+B_q(A_{q_j}^{-1}-A^{-1}_q)
\end{aligned}$$ When treating the first term in the right hand side of , we have $$(B_{q_j}-B_q)A^{-1}_{q_j}=((q_j-q)P_0+P_0(q_j-q))A^{-1}_{q_j}.$$ Thus, $$\begin{aligned}
&\|(q_j-q)P_0A^{-1}_{q_j}\|_{\textrm{tr}}\le \|q_j-q\|_{L^\infty}\|P_0A^{-1/2}_{q_j}\|\|A^{-1/2}_{q_j}\|_{\textrm{tr}},\\
&\|P_0(q_j-q)A^{-1}_{q_j}\|_{\textrm{tr}}\le \|P_0\|_{H^m_0\to L^2} \|q_j-q\|_{H^m_0\to H^m_0}\|A^{-1/2}_{q_j}\|_{L^2\to H^m_0}
\|A^{-1/2}_{q_j}\|_{\textrm{tr}},\end{aligned}$$ and hence, both expressions tend to zero as $j\to \infty$, provided that $N\ge m$. When considering the second term in the right hand side of , we write, using the resolvent identity, $$\begin{aligned}
B_q(A_{q_j}^{-1}-A^{-1}_q)&=B_qA^{-1}_{q_j}(A_q-A_{q_j})A^{-1}_q\\
&=(B_qA^{-1/2}_{q_j})(A^{-1/2}_{q_j}P_0)(q-q_j)(P_0A^{-1/2}_q)A^{-1/2}_q.\end{aligned}$$ The trace class norm of the above expression is easily seen to vanish as $j\to\infty$. If follows that the set $\mathcal{F}$ is open.
Let us now show that the set $\mathcal{F}$ is dense in $\mathcal{E}$. Let $V_0\in \mathcal{E}$ be fixed. Then there exists a complex neighborhood $U\subset C^N(\overline{\Omega}, {\mathbb{C}})$ of $V_0$ such that the map $$\label{eq_complex_pot}
U\to {\mathbb{C}},\quad
V\mapsto\textrm{tr}(B_qA^{-1}_q)$$ is well-defined on $U$. This follows from the fact that the operator $$A_q:H^{2m}(\Omega)\cap H^m_0(\Omega)\to L^2(\Omega), \quad q=\frac{1}{V},$$ is bijective for $V\in U$, since the operator norm of $$P_0{\hbox{Im}\,}q P_0 A_{\mathrm{Re}\, q}^{-1}:L^2(\Omega)\to L^2(\Omega)$$ is small.
We claim that the map is analytic. Since the arguments above show that the map is continuous, it therefore suffices to check the weak analyticity, [@postru_book]. To this end let $q_1=1/V_1$, $V_1\in U$, and $q_2$ be arbitrary, and consider the function $$\label{eq_holom}
z\mapsto \textrm{tr}(B_{q_1+zq_2}A_{q_1+zq_2}^{-1})$$ for $z$ near $0\in {\mathbb{C}}$. We have the convergent power series expansion $$A^{-1}_{q_1+zq_2}=A^{-1}_{q_1}\sum_{k=0}^\infty(-z)^k(P_0q_2P_0A_{q_1}^{-1})^k$$ for $z$ near $0\in {\mathbb{C}}$. Since the operator $(B_{q_1}+zB_{q_2})A^{-1}_{q_1}$ is of trace class, the operator $$B_{q_1+zq_2}A_{q_1+zq_2}^{-1}$$ is given by a power series in $z$ which converges in the trace class norm. Thus, it follows that the map is holomorphic near $0\in {\mathbb{C}}$.
We therefore conclude that the map $$V\mapsto\textrm{tr}(B_qA^{-1}_q)$$ is real-analytic on $\mathcal{E}$. We furthermore know from Remark \[rem\_constant\_potential\] that it does not vanish identically, for it is positive at $V=1$. Since $\mathcal{E}$ is connected, given $V_0\in \mathcal{E}$ it follows that for any neighborhood of $V_0$ there are points $V$ for which $\textrm{tr}(B_qA^{-1}_q)\ne 0$. This completes the proof.
Finally, concerning counting estimates for transmission eigenvalues, we have the following simple result.
Let $m>n$. Then the number of transmission eigenvalues in the disk of radius $R$ is $\mathcal{O}(R^2)$.
Recall that $0\ne\lambda\in {\mathbb{C}}$ is a transmission eigenvalue if and only if $1/\lambda$ is an eigenvalue of the operator $\mathcal{D}$ given by . The latter is equivalent to the fact that the operator $$I-\lambda(\tilde A^{-1/2}\tilde B\tilde A^{-1/2}) +\lambda^2 \tilde A^{-1}:L^2(\Omega)\to L^2(\Omega)$$ is not invertible. Here $\tilde A^{-1/2}\tilde B\tilde A^{-1/2}$ and $\tilde A^{-1}$ are of trace class. Thus, the latter is equivalent to the fact that $$\det(I-\lambda(\tilde A^{-1/2}\tilde B\tilde A^{-1/2}) +\lambda^2 \tilde A^{-1})=0.$$ The function $$f(\lambda)=\det(I-\lambda(\tilde A^{-1/2}\tilde B\tilde A^{-1/2}) +\lambda^2 \tilde A^{-1})$$ is entire holomorphic. Therefore, the number $N(R/2)$ of its zeros in the disk of radius $R/2$ can be estimated by Jensen’s formula, $$N(R/2)\le \frac{1}{\log 2}(\max_{|\lambda|=R }\log|f(\lambda)|-\log|f(0)|)=\mathcal{O}(R^2).$$ Here we have used that $|f(\lambda)|\le e^{C|\lambda|^2}$ with some constant $C$ and $|\lambda|\ge 1$.
Acknowledgements
================
The research of M.H. was partially supported by the NSF grant DMS-0653275 and he is grateful to the Department of Mathematics and Statistics at the University of Helsinki for the hospitality. The research of K.K. was financially supported by the Academy of Finland (project 125599). The research of P.O. and L.P. was financially supported by Academy of Finland Center of Excellence programme 213476.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We reanalyze the prompt muon neutrino flux from gamma-ray bursts (GRBs), at the example of the often used reference Waxman-Bahcall GRB flux, in terms of the particle physics involved. We first reproduce this reference flux treating synchrotron energy losses of the secondary pions explicitly. Then we include additional neutrino production modes, the neutrinos from muon decays, the magnetic field effects on all secondary species, and flavor mixing with the current parameter uncertainties. We demonstrate that the combination of these effects modifies the shape of the original Waxman-Bahcall GRB flux significantly, and changes the normalization by a factor of three to four. As a consequence, the gamma-ray burst search strategy of neutrino telescopes may be based on the wrong flux shape, and the constraints derived for the GRB neutrino flux, such as the baryonic loading, may in fact be already much stronger than anticipated.'
author:
- Philipp Baerwald
- 'Svenja H[ü]{}mmer'
- Walter Winter
title: 'Magnetic Field and Flavor Effects on the Gamma-Ray Burst Neutrino Flux'
---
Neutrino telescopes, such as IceCube [@Ahrens:2003ix] or ANTARES [@Aslanides:1999vq], are designed to detect neutrinos from astrophysical sources. There are numerous candidate sources, see [Ref.]{} [@Becker:2007sv] for a review and [Ref.]{} [@Rachen:1998fd] for the general theory. We focus on the prompt emission of gamma-ray bursts (GRBs) in this letter, where photohadronic interactions are expected to lead to a significant flux of neutrinos [@Waxman:1997ti]. So far, no extraterrestrial high energy neutrino flux has been detected yet. That is, for sources optically thin to neutrons, consistent with generic bounds [@Waxman:1998yy; @Mannheim:1998wp] which are just being touched by IceCube. The search for GRB neutrinos has been driven by analytical estimates for the shape and normalization, the simplest one being the Waxman-Bahcall (WB) flux [@Waxman:1998yy]. More recent analyses, such as the stacking analysis in [Ref.]{} [@Abbasi:2009ig], relating the neutrino flux to the observed gamma-ray flux, are based on the analytical generalization of this flux for arbitrary input parameters following [Ref.]{} [@Guetta:2003wi]. These calculations typically approximate the $\Delta(1232)$ resonance for the charged pion production $$p + \gamma \rightarrow \Delta^+ \rightarrow \left\{\begin{array}{lc} n + \pi^+ & \text{1/3 of all cases} \\ p + \pi^0 & \text{2/3 of all cases} \end{array} \right. \label{equ:Delta}$$ in some form. However, the GRB neutrino flux computation has been updated over the last ten years from the particle physics point of view by improving the description of the photo-meson production processes, and it has been obvious there is a substantial impact from magnetic field effects and flavor mixing on the neutrino flux as well; see, [[*e.g.*]{}]{}, [Refs.]{} [@Mucke:1999yb; @Murase:2005hy; @Kashti:2005qa; @Lipari:2007su; @Hummer:2010vx]. In this letter, we make the impact of these effects very explicit by revising the often used WB reference flux from [Ref.]{} [@Waxman:1998yy]. We include the relevant pion production modes and neutrinos from kaon and neutron decays. We treat the magnetic field effects on each charged particle species explicitly, and we include flavor effects/flavor mixing. Note that we keep our considerations as independent of the astrophysical source model as possible to factor out the particle physics effects, which are much better known than the details of the astrophysical model. The purpose of this letter is to demonstrate how the original WB flux changes in both shape and normalization effect by effect, and where the main impact comes from. We also discuss the impact on data analyses. The technology used in this letter is based on [Refs.]{} [@Hummer:2010vx; @Hummer:2010ai], where details can be found.
![\[fig:photo\] The WB flux from [[Eq.]{} (\[equ:WB\])]{} (thin dashed curve), the numerically reproduced flux using the $\Delta^+$ resonance only (lower solid curve), and the WB flux including higher resonances, direct production/$t$-channel processes, and multi pion production (high energy processes), which are successively switched on, leading to the final upper solid curve. Here the $\nu_\mu$ flux from $\pi^+$ and $\pi^-$ decays is considered. The normalization of our result to the numerically reproduced WB flux (gray dashed curve) is described in the main text.](Neutrinocontributionsplusminusz2){width="0.85\columnwidth"}
In the standard picture, protons collide with photons, possibly from synchrotron emission of co-accelerated electrons or positrons (see, [[*e.g.*]{}]{}, [Ref.]{} [@Dermer:2003zv]), leading to pion production by processes as, for instance, [[Eq.]{} (\[equ:Delta\])]{}. The charged pions then decay further into neutrinos, such as by $\pi^+ \rightarrow \mu^+ + \nu_\mu$, $\mu^+ \rightarrow e^+ + \nu_e + \bar{\nu}_\mu$. For the shape of the WB flux, consider only the $\nu_\mu$ from pion decays for the moment. It is often assumed that the target photon field corresponds to the observed prompt GRB flux, which is typically parameterized by $dN_{\gamma}(E)/dE \propto E^{\alpha_{\gamma}}$ for $E < \varepsilon_{\gamma,\text{break}}$ and $dN_{\gamma}(E)/dE \propto E^{\beta_{\gamma}}$ for $E > \varepsilon_{\gamma,\text{break}}$ in the observer’s frame, where $\alpha_\gamma \simeq -1$, $\beta_\gamma \simeq -2$, and the break $\varepsilon_{\gamma,\text{break}}$ at a few hundred keV. If the protons are injected with a power law with injection index two, one obtains for the prompt GRB neutrino flux, referred to as “WB flux”, $$E^2_{\nu} \frac{\mathrm{d}N_{\nu}}{\mathrm{d}E_{\nu}} \propto \left\{ \begin{array}{ll} (E_{\nu} / \varepsilon^b_{\nu})^{\alpha_{\nu}} & \text{for} \; E_{\nu} < \varepsilon^b_{\nu} \\ (E_{\nu} / \varepsilon^b_{\nu})^{\beta_{\nu}} & \text{for} \; \varepsilon^b_{\nu} \leq E_{\nu} < \varepsilon^s_{\nu} \\ (E_{\nu} / \varepsilon^b_{\nu})^{\beta_{\nu}} (E_{\nu} / \varepsilon^s_{\nu})^{-2} & \text{for} \; E_{\nu} \geq \varepsilon^s_{\nu} \end{array} \right. \label{equ:WB}$$ with $\alpha_\nu = -\beta_\gamma - 1 \simeq +1$, $\beta_\nu = - \alpha_\gamma - 1 \simeq 0$, $\varepsilon^b_{\nu} \simeq 10^5 \, \giga\electronvolt$ and $\varepsilon^s_{\nu} \simeq 10^7 \, \giga\electronvolt$. For the analytical estimates of the break energies, we follow the treatment in [Ref.]{} [@Guetta:2003wi], assuming that $\Gamma = 10^{2.5}$ and $\textit{z} = 2$; see, [[*e.g.*]{}]{}, [Refs.]{} [@Guetta:2003wi; @Wanderman:2009es]. The first break energy $\varepsilon^b_{\nu}$ can be related to $\varepsilon_{\gamma,\text{break}}$ from the threshold of the photohadronic interactions at the source. As a minor difference to [Ref.]{} [@Guetta:2003wi], where heads-on collisions between photons and protons are assumed for the threshold, we include the effect that the pion production efficiency peaks at higher center-of-mass energies (see Fig. 4 in [Ref.]{} [@Hummer:2010vx]) to match our numerical results. This leads to a factor of two higher photon energy break ($14.8 \, \mathrm{keV}$) in the source frame to match the $\varepsilon^b_{\nu} \simeq 10^5 \, \giga\electronvolt$ for the chosen parameter set. The second break comes from pion cooling in the magnetic field. It can be computed from the energy where the pion decay rate equals the synchrotron loss rate. In order to reproduce $\varepsilon^s_{\nu} \simeq 10^7 \, \giga\electronvolt$, one has $B \simeq 3 \cdot 10^5 \, \mathrm{G}$. Note that, in the light of recent Fermi data, it is not clear how “typical” this parameter set is, which, however, does not affect the logic of this letter. As another relevant parameter, we choose the maximum proton energy by balancing synchrotron loss and acceleration rates with an acceleration efficiency of 10% [@Hillas:1985is]. For the expected normalization of the flux in [[Eq.]{} (\[equ:WB\])]{}, we use [@Waxman:1998yy] (updated in [Ref.]{} [@Waxman:2002wp]) $$E_\nu^2 \phi_\nu = 0.45 \cdot 10^{-8} \, \frac{f_\pi}{0.2} \, \giga\electronvolt \, \centi\meter^{-2} \, \second^{-1} \, \steradian^{-1}
\label{equ:wb}$$ per neutrino species ($\nu_e$, $\nu_{\mu}$, or $\bar{\nu}_{\mu}$). After flavor mixing, the combined muon neutrino and antineutrino flux is, again, approximately given by [[Eq.]{} (\[equ:wb\])]{} [@Learned:1994wg]. This estimate is based on the assumption that GRBs are a dominant cosmic ray source in which the high energy protons dissipate a fraction $f_\pi < 1$ of energy into pion production before leaving the source. If no neutrino flux at the level of [[Eq.]{} (\[equ:wb\])]{} is observed, it means that effectively the product of $f_\pi$ and the fraction of energy in protons (baryonic loading) becomes stronger constrained [@Guetta:2003wi] – and therefore the hypothesis of GRBs being the dominant cosmic ray source. We choose $f_\pi=0.2$ for the following figures. For the numerical treatment of the photohadronic interactions, we follow [Ref.]{} [@Hummer:2010vx] (Sim-B), based on the physics of SOPHIA [@Mucke:1999yb] and the weak decays in [Ref.]{} [@Lipari:2007su], including the helicity dependence of the muon decays. The energy losses and other production modes are treated as described in [Ref.]{} [@Hummer:2010ai]. We assume that synchrotron losses are the leading energy loss mechanism, which means that only the product of the proton and photon densities is required, see [Eq.]{} (7) of [Ref.]{} [@Hummer:2010vx]. Therefore, our results are independent of the baryonic loading and GRB model details. We also assume that the source is optically thin to neutrons, which means that secondary interactions are neglected. There are limitations to these assumptions, such as if the protons cool significantly by photohadronic interactions (see, [[*e.g.*]{}]{}, discussion in [Ref.]{} [@Murase:2005hy]). However, such processes cannot be included in a model-independent way, because they require separate knowledge on the proton and photon densities.
{width="40.00000%"} {width="40.00000%"}
We show in [[Fig.]{} \[fig:photo\]]{} the WB reference flux from [[Eq.]{} (\[equ:WB\])]{} as thin dashed curve for $\nu_\mu$ from charged pion decays only. In order to normalize our flux to this curve, we need to take into account that the assumptions for the $\Delta$ resonance vary in the literature. Therefore, we first of all reproduce the WB flux numerically by including the synchrotron cooling of the pions explicitely, leading to the thick gray dashed curve “WB $\Delta^+$-approx.”. Here we use the same cross sections, pion multiplicities, and inelasticities as in [Ref.]{} [@Waxman:1997ti], and we choose the normalization such that the energy going into neutrinos is the same as for the analytical estimate. Note that for this curve, the second break is automatically reproduced by magnetic field effects and not put in by hand, which results in a small pile-up effect at the plateau. By this choice, the product of proton and photon density normalizations is fixed, and we can use the input spectra to compute the effects of the more refined interaction model. We show this by the solid curves, where higher resonances, direct ($t$-channel) production, and multi pion production are successively added to the actual $\Delta^+(1232)$ resonance process in [[Eq.]{} (\[equ:Delta\])]{}; see, [[*e.g.*]{}]{}, [Refs.]{} [@Mucke:1999yb; @Hummer:2010vx]. The final result exceeds the WB estimate by a factor of a few, especially at high energies. The additional tilt of the spectrum comes from the multi pion cross section staying approximately constant for high interaction energies. In addition, note that all processes other than the actual $\Delta$ resonance include $\pi^-$ production and two or multi pion production modes. From [[Fig.]{} \[fig:photo\]]{}, one can read off that the WB approximation basically includes the effect from direct production at low energies and higher resonances at high energies, but the high energy (or two and multi pion) contributions are clearly underestimated. In addition, one can read off that the effect of the additional production processes can lead to an up to one order of magnitude change of the flux, depending on the assumptions on the $\Delta$ resonance in the literature.
Apart from pion decays, neutrinos are produced from muon decays in the pion decay chain. In addition, kaons produced in the photohadronic interactions similar to pions may decay into neutrinos [@Asano:2006zzb]. The main qualitative difference among charged pions, muons, and kaons are their different masses and lifetimes, leading to different energies of the second (synchrotron) break in [[Eq.]{} (\[equ:WB\])]{}; see, [[*e.g.*]{}]{}, Fig. 3 in [Ref.]{} [@Hummer:2010ai]. This effect has interesting implications for the flavor ratio of the neutrinos, which changes as a function of energy [@Kashti:2005qa]; see also [Ref.]{} [@Kachelriess:2007tr]. Finally, any neutrino flux will be accompanied by a neutron flux, as it is obvious from [[Eq.]{} (\[equ:Delta\])]{}. These neutrons are, however, not stable. If the neutrons do not interact, they will decay either within or outside the source by $n \rightarrow p + e^- + \bar\nu_e$, leading to cosmic rays and (inevitably) to an additional (almost coincident) $\bar\nu_e$ neutrino flux. We show the total electron neutrino (left panel) and muon neutrino (right panel) flux before flavor mixing in [[Fig.]{} \[fig:source\]]{}, where the neutrino and antineutrino fluxes are added. The individual contributions to these neutrino fluxes from $\pi$, $\mu$, $n$, and $K^+$ decays (we only consider the leading kaon contribution mode) are shown as well. The WB flux from [[Eq.]{} (\[equ:wb\])]{} is given as reference for the corresponding number of neutrino species. In the right panel (muon neutrinos), one can clearly see the hierarchy in the second break energy among neutrinos from $\mu$, $\pi$, and $K$ decays. In the left panel (electron neutrinos), the main contribution comes from muon decays. However, neutron decays show up at low energies.
![\[fig:detector\] Total muon neutrino flux after flavor mixing (dark thick solid curve). The individual contributions to this flux from muon and electron neutrinos are shown as well (from thick curves in [[Fig.]{} \[fig:source\]]{}). The WB flux from [[Eq.]{} (\[equ:wb\])]{} is shown for reference, corrected by flavor mixing. In addition, a rescaled total flux is shown to illustrate the impact on the spectral shape. The shaded band shows the $3\sigma$ allowed range of the total flux from current mixing parameter uncertainties [@Schwetz:2008er]. Here also the 10 year (extrapolated) full-scale limits from IceCube [@Ahrens:2003ix] (for an $E^{-2}$ diffuse flux above the atmospheric neutrino background) and Auger [@Abraham:2009uy] (differential limit) are estimated at the 90% CL. ](Oscillationcontributionsmu_z2_rescaled.pdf){width="0.9\columnwidth"}
In order to obtain the final muon neutrino flux relevant for muon tracks in neutrino telescopes, the total electron and muon neutrino fluxes in [[Fig.]{} \[fig:source\]]{} are superimposed by flavor mixing (averaged neutrino oscillations) [@Learned:1994wg], see [[Fig.]{} \[fig:detector\]]{}. The shaded band indicates the $3\sigma$ allowed range of the total flux from current mixing parameter uncertainties [@Schwetz:2008er]. In fact, if the combined knowledge from the Double Chooz, Daya Bay, T2K, and NO$\nu$A is applied [@Huber:2009cw], as expected in about 2015, this band becomes hardly visible anymore. Therefore, mixing parameter uncertainties are less relevant for the GRB analysis, especially at the lower break (unless flavor ratios are considered). Comparing the final result (upper thick solid curve) with the WB flux (thin dashed curve), we notice that the expected neutrino flux is about a factor of three to four larger than the WB flux at 1 PeV. In addition, we show a rescaled version of the final result (thin solid curve) to illustrate the impact on the spectral shape compared to the WB flux. It is obvious from this comparison that the shape of [[Eq.]{} (\[equ:WB\])]{} cannot be used for realistic data analyses or to search for point source GRBs. For instance, the first break, to which AMANDA and IceCube are most sensitive to, has basically disappeared in its original form. On the other hand, magnetic field and flavor effects lead to a characteristic double peak structure, one could search for if a few bursts dominated. In addition, note the high energy excess coming from kaon decays, which increases the flux by at least one order of magnitude. At about $10^{8-9} \, \mathrm{GeV}$, horizontal air shower experiments, such as Auger [@Abraham:2009uy], in fact have the best sensitivity, where the flux shown in [[Fig.]{} \[fig:detector\]]{} is representative for $\nu_\mu$ or $\nu_\tau$ events. Although the full-scale diffuse flux sensitivity is considerably above the expected neutrino flux, Auger may detect a flux from GRB kaon decays especially if a bright burst out-shines the cosmogenic neutrino flux. Therefore, neutrino point source studies should be initiated by these experiments.
In summary, we have revised the WB neutrino flux, often used as a reference GRB flux, by including the most relevant neutrino production processes, and by treating magnetic field and flavor effects explicitly. We have used as few assumptions as possible on the astrophysical source model. We have demonstrated that the flux normalization increases by a factor of three to four with respect to the initial assumptions, and that the spectral shape exhibits a double peak structure qualitatively different from the WB flux. The main impact are additional neutrino production modes and magnetic field effects, which act differently on the charged secondary particle species. The revised spectral shape may allow for new search strategies for GRB neutrino fluxes. Since current state-of-the-art multi-messenger stacking analyses, such as [Ref.]{} [@Abbasi:2009ig], where the expected neutrino flux is computed from the observed gamma-ray flux on an event-by-event basis, rely on the assumptions on the photohadronic interactions, the non-observation of a flux may have stronger constraints on the energy equipartition between protons and electrons than anticipated. We would like to thank J. Becker, A. Kappes, and K. Murase for illuminating discussions and comments.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'This is a survey on the equivariant cohomology of Lie group actions on manifolds, from the point of view of de Rham theory. Emphasis is put on the notion of equivariant formality, as well as on applications to ordinary cohomology and to fixed points.'
address: |
Philipps Universität Marburg\
Fachbereich Mathematik und Informatik\
Hans-Meerwein-Straße\
35032 Marburg\
Germany
author:
- Oliver Goertsches
- Leopold Zoller
title: |
Equivariant de Rham cohomology:\
Theory and applications
---
Introduction
============
Equivariant cohomology is a topological invariant, not of spaces, but of group actions. It encodes in a subtle way information on the topology of the space, the isotropy groups of the action, and the orbit stratification, in particular on the fixed points of the action. In was introduced by Borel [@Borel] and H. Cartan [@Cartan1], [@Cartan2] in the 1950s and has found numerous applications wherever symmetries of geometric objects play a role. These purpose of these notes is twofold: they try to give a gentle introduction to this beautiful theory from the point of view of de Rham theory, and to survey both classical and more recent applications.
In the first few sections we introduce three different types of cohomology one can associate to a Lie group action on a manifold: cohomology of invariant forms, basic cohomology, and our main player, equivariant cohomology. After comparing them to each other and to ordinary (de Rham) cohomology we prove some basic results on equivariant cohomology like the homotopy axiom and the Mayer-Vietoris sequence.
We explain how equivariant cohomology can be used to gain information on both the ordinary cohomology of the manifold $M$ acted on, as well as on the fixed point set of the action. The main tool to relate equivariant cohomology to the fixed point set is the Borel localization theorem, which is the topic of Section \[sec:borellocalization\]. We explain how one uses it to show the equalities of the Euler characteristics of $M$ and the fixed point set $M^T$, as well as the inequality of total Betti numbers $\dim H^*(M^T)\leq \dim H^*(M)$, in Section \[sec:fixedpoints\].
Starting with Section \[sec:equivariantformality\] we make use of the spectral sequence of the Cartan model, as there we introduce another main topic of this survey, the notion of equivariant formality. All necessary knowledge on spectral sequences is contained in the appendix; in particular, there one can find details on the relation between the equivariant cohomology and the final page of the spectral sequence that are usually glossed over in the literature. Equivariant formality of an action is the condition that the spectral sequence of the Cartan model degenerates at $E_1$. In Theorem \[thm:bigthmequivformal\] we prove some equivalent formulations of this condition, one of which enables one to compute ordinary from equivariant cohomology. We apply this to obtain information on the cohomology of homogeneous spaces in Section \[sec:cohomhomspaces\], and of GKM manifolds in Section \[sec:HMHTM\].
In the last sections we give a short overview on some recent developments. The choice of material is rather biased and not meant to be exhaustive. We will explain some results surrounding the notions of Cohen-Macaulay actions and equivariant basic cohomology.
Throughout the paper we try to present the material in an easily accessible way, sometimes sacrificing greater generality for simplicity of the arguments. We do not give proofs for every result, but do so whenever we were not able to find a good reference in the literature; sometimes we provide a different proof. We will assume that the reader is familiar with the theory of actions of compact Lie groups on differentiable manifolds.
In preparation of this paper a wealth of literature was helpful, such as the monographs [@AlldayPuppe; @BeGeVe; @GuilleminSternberg; @Hsiang], as well as [@GGK Appendix C] and [@Bott].\
[*Acknowledgements.*]{} Parts of this paper stem from the first named author’s lectures at the University of Hamburg in 2012, and at the Philipps University of Marburg in 2018. We would like to thank the participants of these courses for their interest in the topic and their valuable comments. We are grateful to Michèle Vergne for several remarks on a previous version of this paper. We are especially indebted to Jeffrey Carlson for several enlightening discussions, as well as for a very thorough reading of a previous version and numerous suggestions that improved the presentation of this paper. The second named author is supported by the German Academic Scholarship foundation.
Invariant and basic differential forms
======================================
Let $G$ be a Lie group acting on a differentiable manifold $M$, with Lie algebra ${\mathfrak{g}}$. We denote, for $X\in {\mathfrak{g}}$, the induced fundamental vector field by $$\overline{X}_p:= \left.\frac{d}{dt}\right|_{t=0} \exp(tX)\cdot p.$$
A differential form $\omega\in \Omega(M)$ is called *$G$-invariant* if $g^*\omega=\omega$ for all $g\in G$. The space of $G$-invariant differential forms is denoted $\Omega(M)^G$.
The space $\Omega(M)^G$ is clearly invariant under the differential $d\colon\Omega(M)\to \Omega(M)$, i.e., $(\Omega(M)^G,d)$ is a subcomplex of $(\Omega(M),d)$ and we can consider its cohomology. However, if $G$ is connected and compact, this cohomology does not contain more information than the usual de Rham cohomology because of the following theorem due to É. Cartan [@ECartan]:
\[thm:cohomofinvariantforms\] If $G$ is a compact and connected Lie group acting on a differentiable manifold $M$, then the inclusion map $\Omega(M)^G\to \Omega(M)$ induces an isomorphism $H^*(\Omega(M)^G)\to H^*(M)$ in cohomology.
The proof of this result can be found e.g. in [@Onishchik §9]. One shows that the averaging operator $\mu\colon\Omega(M)\to \Omega(M)$ given by $$\mu(\omega)(v_1,\ldots,v_n):= \int_G (g^*\omega) (v_1,\ldots,v_n);$$ is chain homotopic to the identity. Of course, if $G$ is not connected, then this inclusion does not induce an isomorphism, see Examples \[ex:cohomfinitequotient\] and \[ex:cohomRPn\] below.
A different type of topological information is encoded in the complex of $G$-basic differential forms.
\[defn:basicforms\] Given an action of a Lie group $G$ on a smooth manifold $M$, a differential form $\omega\in \Omega(M)$ is called *($G$-)horizontal* if $i_{\overline{X}}\omega=0$ for all $X\in {\mathfrak{g}}$. It is called *$G$-basic* if it is both $G$-invariant and horizontal. The space of such differential forms is denoted $\Omega_{{\operatorname{bas}}G}(M)$.
Just like the $G$-invariant differential forms, also the basic differential forms comprise a subcomplex of the de Rham complex. In fact, for $\omega\in \Omega_{{\operatorname{bas}}G}(M)$, the form $d\omega$ is again (invariant and) horizontal because by the Cartan formula $i_{\overline{X}} d\omega = {{\mathcal L}}_{\overline{X}}\omega - di_{\overline{X}}\omega = 0$. Here, ${{\mathcal L}}$ denotes the Lie derivative.
We define the *basic cohomology* of the $G$-action to be $$H^*_{{\operatorname{bas}}G}(M):=H^*(\Omega_{{\operatorname{bas}}G}(M),d).$$
Recall that if the $G$-action on $M$ is free, then the orbit space $M/G$ is a smooth manifold, and the projection $\pi\colon M\to M/G$ is smooth. In general, for an arbitrary action of a compact Lie group, $M/G$ is just a topological Hausdorff space.
\[prop:basiccohomfreeaction\] Consider a free action of a (not necessarily connected) compact Lie group $G$ on a smooth manifold $M$, and consider the projection $\pi\colon M\to M/G$. Then $\pi^*$ defines an isomorphism of complexes $\pi^*\colon \Omega(M/G)\to \Omega_{{\operatorname{bas}}G}(M)$. In particular, $$H^*_{{\operatorname{bas}}G}(M)\cong H^*(M/G).$$
If $\omega\in \Omega(M/G)$, then $\pi^*\omega$ is $G$-invariant because for any $g\in G$ we have $$g^*\pi^*\omega = (\pi\circ g)^*\omega = \pi^*\omega.$$ At each $p\in M$, we have $\ker d\pi_p = T_pG\cdot p$. Thus, $\pi^*\omega$ is horizontal as well.
If conversely $\eta$ is a $G$-basic $k$-form on $M$, then we can define a $k$-form $\omega$ on $M/G$ as follows: if $v_1,\ldots,v_k$ are tangent vectors at $Gp\in M/G$, then let $w_1,\ldots,w_k$ be tangent vectors at $p\in M$ such that $d\pi_p(w_i)=v_i$, and define $$\omega(v_1,\ldots,v_k)=\eta(w_1,\ldots,w_n)$$ This is independent of both the choice of $p$ and the $w_i$ because $\eta$ is $G$-invariant and horizontal. Clearly, we have $\pi^*\omega = \eta$.
\[ex:cohomfinitequotient\] Consider a finite group $G$ acting freely on a smooth manifold $M$. Then being $G$-basic, for a differential form $\omega$ on $M$, is the same as being $G$-invariant. So in this case $\pi^*\colon \Omega(M/G) \to \Omega(M)^G$ is an isomorphism of complexes, so that $H^*(M/G) = H^*(\Omega(M)^G)$.
On the other hand, we have a well-defined action of $G$ on cohomology: for $g\in G$ and $[\omega]\in H^*(M)$, we put $g^*[\omega] := [g^*\omega]$. Then the inclusion $\Omega(M)^G\to \Omega(M)$ induces an injective homomorphism $H^*(\Omega(M)^G) \to H^*(M)$ whose image lies in the $G$-invariant cohomology: $$i_*\colon H^*(\Omega(M)^G)\longrightarrow H^*(M)^G.$$ We claim that this map is indeed surjective: let $[\omega]\in H^*(M)^G$, i.e., that for all $g\in G$ there exists $\eta_g\in \Omega(M)$ such that $g^*\omega = \omega + d\eta_g$. But then the average $\frac1{|G|} \sum_{g\in G} g^*\omega$ is a $G$-invariant form whose cohomology class is sent by $i_*$ to $[\omega]$. In total, we obtain isomorphisms $$H^*(M/G) \longrightarrow H^*(\Omega(M)^G) \longrightarrow H^*(M)^G.$$
\[ex:cohomRPn\] Let us give a concrete example: we consider the free ${{\mathbb Z}}_2$-action on the $n$-dimensional sphere $S^n$ given by sending a point to its antipodal map, with orbit space the real projective space ${{\mathbb R}}P^n$. To understand the cohomology of ${{\mathbb R}}P^n$ we therefore only have to understand the effect of the map $f(x)=-x$ on a volume form of $S^n$. A volume form on $S^n$ is given by $$\begin{aligned}
\omega_{(x_1,\ldots,x_{n+1})}&=i_{(x_1,\ldots,x_{n+1})} (dx_1\wedge \cdots \wedge dx_{n+1}),\end{aligned}$$ and as the radial vector field is invariant under $f$, it follows that $f^*\omega = \omega$ for odd $n$, and $f^*\omega = -\omega$ for even $n$. The action of ${{\mathbb Z}}_2$ on $H^n(S^n)={{\mathbb R}}\cdot [\omega]$ is thus trivial for odd $n$, and given by multiplication with $-1$ for even $n$, whence $$H^n({{\mathbb R}}P^n) = H^n(S^n)^{{{\mathbb Z}}_2} = \begin{cases} 0 & n \text{ even,} \\ {{\mathbb R}}& n \text{ odd}. \end{cases}$$
\[rem:basicsingularcohom\] We even have $H^*_{{\operatorname{bas}}G}(M)=H^*(M/G)$ for any action of a compact Lie group $G$, where the right hand side is understood as the singular cohomology of $M/G$. As singular cohomology is not the focus of these notes, we only refer to [@Michor Theorem 30.36] for the proof. This tells us that $H^*_{{\operatorname{bas}}G}(M)$ is, in many cases, not a very powerful invariant for group actions. For instance, there exist many nontrivial group actions for which the orbit space $M/G$ is contractible, so that $H^*_{{\operatorname{bas}}G}(M)={{\mathbb R}}$, e.g., the standard action of $S^1$ on $S^2$ by rotation. For free actions, however, the orbit space is again a manifold, so basic cohomology of free actions is an invariant as powerful as de Rham cohomology for manifolds.
The coadjoint representation {#sec:Coadjoint}
============================
Any Lie group $G$ acts on its Lie algebra by the adjoint representation. This is defined as follows: for any $g\in G$ conjugation with $g$ is denoted $$c_g\colon G\longrightarrow G;\, h\longmapsto ghg^{-1}.$$ Differentiating this at $e$, we obtain a map ${\operatorname{Ad}}_g\colon {\mathfrak{g}}\cong T_eG\to T_eG\cong {\mathfrak{g}}$ given by ${\operatorname{Ad}}_g:=(dc_g)_e$. In this way we obtain a homomorphism $${\operatorname{Ad}}\colon G\longrightarrow {\mathrm{GL}}({\mathfrak{g}})$$ which we call the *adjoint representation* of $G$.
Dualizing this representation, we obtain the *coadjoint representation* of $G$ on the dual vector space ${\mathfrak{g}}^*$ (which consists of linear forms $\xi\colon {\mathfrak{g}}\to {{\mathbb R}}$): $$({\operatorname{Ad}}_g^*\xi) (X):=\xi({\operatorname{Ad}}_{g^{-1}}(X))$$ We denote by $S({\mathfrak{g}}^*)$ the symmetric algebra on ${\mathfrak{g}}^*$, which we consider as the algebra of polynomials on ${\mathfrak{g}}$. The coadjoint representation naturally extends to $S({\mathfrak{g}}^*)$ via $({\operatorname{Ad}}_g^*f)(X):=f({\operatorname{Ad}}_{g^{-1}}X)$. Of particular importance will be the subspace of $G$-invariant polynomials $S({\mathfrak{g}}^*)^G$, i.e., those polynomials that are constant along adjoint orbits in ${\mathfrak{g}}$.
For compact and connected $G$, the ring of invariant polynomials is again a polynomial ring: Chevalley’s restriction theorem, see [@Bourbaki Chapitre VIII, §8.3, Théorème 1], [@Varadarajan Theorem 4.9.2] or [@DufloVergne] (it was mentioned by Chevalley without proof in [@ChevalleyProceedings Section IV]), states that the restriction map $$S({\mathfrak{g}}^*)^G\longrightarrow S({\mathfrak{t}}^*)^{W(G)},$$ where $T\subset G$ is a maximal torus and $W(G)$ the corresponding Weyl group, is an isomorphism. (See [@DufloVergne Proposition 30] for an explicit description of the inverse map.) Here, we define the Weyl group as the finite group $N_G(T)/T$, where $N_G(T) = \{g\in G\mid gTg^{-1}=G\}$ is the normalizer of $T$ in $G$. As the Weyl group acts on ${\mathfrak{t}}^*$ as a reflection group (it coincides with the algebraically defined Weyl group of the root system of ${\mathfrak{g}}^{{\mathbb C}}$, see [@Knapp Theorem IV.4.54]), the Chevalley-Shephard-Todd theorem [@Kane Section 18-1] states that the ring of invariants $S({\mathfrak{t}}^*)^{W(G)}$ is a polynomial ${{\mathbb R}}$-algebra.
\[ex:invpolyUn\] Consider $G={\mathrm{U}}(n)$, with maximal torus $T$ given by diagonal matrices, and corresponding Weyl group $S_n$, acting by permutations on the diagonal entries of ${\mathfrak{t}}$. Then $S({\mathfrak{g}}^*)^G \cong S({\mathfrak{t}}^*)^{W({\mathrm{U}}(n))}$ is the algebra of symmetric polynomials in $n$ variables, which is the polynomial algebra ${{\mathbb R}}[\sigma_1,\ldots,\sigma_n]$, generated by the elementary symmetric polynomials $\sigma_i$ of degree $i$. A direct proof of Chevalley’s restriction theorem for the case $G={\mathrm{U}}(n)$ can be found in [@GGK Example C.13].
For a disconnected compact Lie group $G$, the $G$-invariant polynomials do not necessarily form a polynomial ring. Consider, for example, the semidirect product $G= T^2\rtimes_\varphi {{\mathbb Z}}_2$, where $\varphi(1)$ acts as the inverse map on $T^2$. Then $S({\mathfrak{g}}^*)^G = {{\mathbb R}}[x,y]^{{{\mathbb Z}}_2}$, where ${{\mathbb Z}}_2$ acts on $x$ and $y$ by $\pm 1$, which is the algebra of polynomials in $x$ and $y$ of even degree. This is not a polynomial ring, because any generating set necessarily contains scalar multiples of $x^2,y^2$ and $xy$, and we have the relation $(xy)^2 = x^2y^2$.
The Cartan model {#sec:CartanModel}
================
In this section we introduce H. Cartan’s definition of equivariant cohomology [@Cartan1], [@Cartan2]. Let $G$ be a compact Lie group acting on a differentiable manifold $M$. We define the space of *equivariant differential forms on $M$* as $$C_G(M):=(S({\mathfrak{g}}^*)\otimes \Omega(M))^G.$$ Here, the superscript denotes taking the subspace of $G$-invariant objects, where $S({\mathfrak{g}}^*)\otimes \Omega(M)$ is endowed with the tensor product representation: $G$ acts on $S({\mathfrak{g}}^*)$ by the coadjoint representation described in the previous subsection and on $\Omega(M)$ by pull-back, i.e., the following representation: $$g\cdot \omega:=(g^{-1})^*\omega.$$ An equivariant differential form $\omega\in S({\mathfrak{g}}^*)\otimes \Omega(M)$ can be written as a finite sum $$\omega = \sum_i f_i\otimes \eta_i,$$ for $f_i\in S({\mathfrak{g}}^*)$ and $\eta_i\in \Omega(M)$. By abuse of notation, we will also denote the associated polynomial map ${\mathfrak{g}}\to \Omega(M);\, X\mapsto \sum_i f_i(X)\cdot \eta_i$ by $\omega$. Almost by definition, the $G$-invariance of the element $\omega\in S({\mathfrak{g}}^*)\otimes \Omega(M)$ translates to the equivariance of the polynomial map $\omega\colon {\mathfrak{g}}\to \Omega(M)$, i.e., to the condition $$\label{eq:Ad-invariantpolynomial}
\omega({\operatorname{Ad}}_g(X))=g\cdot (\omega(X)) = (g^{-1})^*(\omega(X))$$ for all $g\in G$ and $X\in {\mathfrak{g}}$. We think of $C_G(M)$ as the space of $G$-equivariant polynomial maps ${\mathfrak{g}}\to \Omega(M)$.
If $G=T$ is a torus, then the (co)adjoint action of $T$ is trivial, so $C_T(M) = S({\mathfrak{t}}^*)\otimes \Omega(M)^T$. A $T$-equivariant differential form is nothing but a polynomial $\omega\colon {\mathfrak{t}}\to \Omega(M)^T$.
Sometimes it is convenient to write equivariant differential forms in a basis: given a basis $\{X_i\}$ of the Lie algebra ${\mathfrak{g}}$, with dual basis $\{u_i\}$ of ${\mathfrak{g}}^*$, we can write an equivariant differential form $\omega\in C_G(M)$ as a finite sum $$\label{eq:localdescriptionofeqdiffform}
\omega = \omega_\emptyset + \sum_i \omega_i u_i + \sum_{i\leq j} \omega_{ij} u_iu_j + \cdots = \sum_I \omega_I u_I,$$ where $I$ runs over a finite set of multiindices.
There is a natural $S({\mathfrak{g}}^*)^G$-algebra structure on $C_G(M)$: first of all note that $C_G(M)$ is a ring with respect to the multiplication $$(\omega\wedge \eta)(X):=\omega(X)\wedge \eta(X),$$ where $\omega$ and $\eta$ are considered as polynomials ${\mathfrak{g}}\to \Omega(M)$. In other words, we give $C_G(M)$ the ring structure from the tensor product of the rings $S({\mathfrak{g}}^*)$ and $\Omega(M)$. The $S({\mathfrak{g}}^*)^G$-algebra structure is defined by the ring homomorphism $$\label{eq:algebrastroneqc}
i\colon S({\mathfrak{g}}^*)^G\to (S({\mathfrak{g}}^*)\otimes \Omega(M))^G = C_G(M);\, f\mapsto f\otimes 1.$$ As a polynomial ${\mathfrak{g}}\to \Omega(M)$, the equivariant differential form $f\otimes 1$ is $(f\otimes 1)(X)=f(X)$, where the real number $f(X)$ is regarded as a constant function on $M$.
\[def:equivariantdifferential\] We define the *equivariant differential* $d_G$ on $S({\mathfrak{g}}^*)\otimes\Omega(M)$ by $$d_G(\omega)(X)=d(\omega(X)) - i_{\overline{X}} \omega(X).$$
There are various sign conventions in the literature. Some authors use $+$ instead of $-$ in this definition; also, some authors use a sign in the definition of the fundamental vector field $\overline{X}$, to make the assignment $X\mapsto \overline{X}$ a Lie algebra homomorphism.
One directly verifies that $d_G$ maps $C_G(M)$ to itself. It is useful to write the equivariant differential $d_G\omega$ in case $\omega$ is given explicitly as in :
\[lem:d\_Gexplicit\] If $\omega= \sum_I \omega_Iu_I\in S({\mathfrak{g}}^*)\otimes \Omega(M)$, then $$\label{eq:d_Gexplicit}
d_G\omega = \sum_I (d\omega_I -\sum_i i_{\overline{X_i}}\omega_I u_i)u_I.$$
We only need to observe that for $X\in {\mathfrak{g}}$, we have $X=\sum_i u_i(X) X_i$, so that $i_{\overline{X}} = \sum_i u_i(X) i_{\overline{X_i}}$.
Let us introduce a grading on $C_G(M)$. For any integer $n\geq 0$ we define the space of equivariant differential forms of degree $n$ as $$C_G^n(M):=\bigoplus_{2k+l=n} (S^k({\mathfrak{g}}^*)\otimes \Omega^l(M))^G.$$ An element of $C_G^n(M)$ will be called an *equivariant differential form of degree $n$*. Note that the ring structure of $C_G(M)$ is graded in the sense that the product of elements in degree $n$ and $m$ is of degree $n+m$.
If $\omega=\sum_I \omega_I u_I$ is an equivariant differential form as in , then it is of degree $n$ if and only if for every $I=(i_1,\ldots,i_r)$ the differential form $\omega_I$ is of degree $n-2(i_1+\cdots+i_r)$.
In the following proposition we collect a few properties of the equivariant differential. We omit the straightforward proofs. The first item is the reason for our choice of grading on $C_G(M)$.
1. $d_G$ maps $C_G^n(M)$ to $C_G^{n+1}(M)$.
2. For $\omega\in C_G^n(M)$ and $\eta\in C_G^m(M)$ we have $$d_G(\omega\wedge\eta)=(d_G\omega)\wedge \eta + (-1)^n \omega\wedge (d_G\eta).$$
3. $d_G^2=0$.
If $d_G\omega=0$, then we say that $\omega$ is *equivariantly closed*, and a form of the type $d_G\eta$ is *equivariantly exact*.
The *equivariant cohomology* of the $G$-action on $M$ is defined as $H^*_G(M):=H^*(C_G^*(M),d_G)$.
The ring structure of $C_G(M)$ passes over to $H^*_G(M)$, and the ring homomorphism $i$ in induces a well-defined homomorphism of graded rings $i\colon S({\mathfrak{g}}^*)^G\to H^*_G(M)$. Thus, via $i$, the ring $H^*_G(M)$ becomes naturally a graded $S({\mathfrak{g}}^*)^G$-algebra, in the sense that the ring homomorphism $i$ respects the degree. In what follows, it will be extremely important to distinguish between this $S({\mathfrak{g}}^*)^G$-algebra structure on $H^*_G(M)$ and the induced structure as an $S({\mathfrak{g}}^*)^G$-module.
\[rem:Borelmodel\] There are other ways to introduce equivariant cohomology, most prominently the so-called Borel model, introduced first in [@Borel], which we now briefly explain. As was mentioned above in Remark \[rem:basicsingularcohom\], we consider the cohomology of the orbit space a reasonable invariant for free actions. In case of an arbitrary action on a topological space $X$, one now replaces the space $X$ acted on by a homotopy equivalent space with a free $G$-action, namely by $$EG\times X,$$ where $EG$ is a contractible space on which $G$ acts freely. Then, one defines the equivariant cohomology (with coefficients $R$) as the cohomology of the orbit space of the diagonal action: $$H^*_G(X;R) := H^*(EG\times_G G;R).$$ It admits the structure of a $H^*(BG;R)$-algebra, via the natural projection $EG\times_G X\to EG/G=:BG$. The equivariant de Rham theorem [@Cartan1], [@Cartan2], see also [@GuilleminSternberg Section 2.5], states that for manifolds and real coefficients, this Borel cohomology is isomorphic to the equivariant cohomology defined above. A further important model for equivariant cohomology is the Weil model. See [@Meinrenken] for a short overview of these models.
\[ex:trivialaction\] Let us consider an easy, yet very important example: that of a trivial $G$-action on a manifold $M$. In this case, any differential form on $M$ is automatically $G$-invariant, so we have $$C_G(M)=S({\mathfrak{g}}^*)^G\otimes \Omega(M).$$ All induced vector fields $\overline{X}$ are trivial, so the equivariant differential $d_G$ is nothing but the ordinary differential: $(d_G\omega)(X)=d(\omega(X))$. This means that the complex $(C_G(M),d_G)$ is obtained from the ordinary de Rham complex $(\Omega(M),d)$ by tensoring with $S({\mathfrak{g}}^*)^G$. Therefore, we have an $S({\mathfrak{g}}^*)^G$-algebra isomorphism $$\label{eq:H*trivial}
H^*_G(M) \cong S({\mathfrak{g}}^*)^G \otimes H^*(M),$$ where $S({\mathfrak{g}}^*)^G$ acts only on the first factor of the right hand side. In particular, $H^*_G(M)$ is a free module over $S({\mathfrak{g}}^*)^G$. Particularly important is the case where $M$ consists of a single point: we have $H^*_G({\mathrm{pt}}) = S({\mathfrak{g}}^*)^G$.
Later we will encounter classes of actions for which holds, but just as an isomorphism of $S({\mathfrak{g}}^*)^G$-modules.
One shows directly that any $G$-equivariant map $f\colon M\to N$ between $G$-manifolds $M$ and $N$ induces a pullback homomorphism between the Cartan complexes by $(f^*\omega)(X) = f^*(\omega(X))$ which descends to an $S({\mathfrak{g}}^*)^G$-algebra morphism $f^*\colon H^*_G(N)\to H^*_G(M)$. Then the following lemma follows directly from the definitions:
\[lem:algebrastructurefrommaptopoint\] The $S({\mathfrak{g}}^*)^G$-algebra structure $i\colon S({\mathfrak{g}}^*)^G\to H^*_G(M)$ is the same as the map in cohomology induced by the unique map $M\to \{pt\}$.
Let us have a look at the zeroth and first equivariant cohomology groups.
We have $C_G^0(M) = \Omega^0(M)^G$, the space of $G$-invariant smooth functions $f\colon M\to {{\mathbb R}}$. For such $f$, the equivariant differential computes as $d_Gf = df$, and therefore, closed equivariant $0$-forms are locally constant invariant functions. Hence, $H^0_G(M)=H^0(M/G)$ calculates the number of connected components of $M/G$. (In case $G$ is connected, this coincides with the number of connected components of $M$.)
\[ex:equivoneforms\] We have $C_G^1(M)=\Omega^1(M)^G$. For $\omega\in \Omega^1(M)^G$, the equivariant differential computes as $$(d_G\omega)(X)=d\omega - i_{\overline{X}}\omega$$ ($\omega$ is considered as a constant map ${\mathfrak{g}}\to \Omega(M);\, X\mapsto \omega$). Therefore, $d_G\omega=0$ if and only if $d\omega=0$ and $i_{\overline{X}}\omega=0$ for all $X\in {\mathfrak{g}}$, i.e., if $\omega$ is a closed basic form. We have computed $C_G^0(M)$ above, which implies that the exact equivariant one-forms are the same as the exact basic one-forms. This shows $H^1_G(M) = H^1_{{\operatorname{bas}}G}(M)$.
There is the following relation between basic and equivariant cohomology:
The ring homomorphism $\Omega_{{\operatorname{bas}}G}(M)\to C_G(M);\, \omega\mapsto 1\otimes \omega$ is an inclusion of complexes and therefore defines a homomorphism of ${{\mathbb R}}$-algebras $H^*_{{\operatorname{bas}}G}(M)\to H^*_G(M)$.
First of all note that $\omega= 1\otimes \omega\in S({\mathfrak{g}}^*)\otimes \Omega(M)$ really is an equivariant differential form because $\omega$ is $G$-invariant. Therefore, the map is well-defined. Clearly, it is an ${{\mathbb R}}$-algebra homomorphism. Moreover, we have $d_G(\omega) = d\omega$ because $\omega$ is horizontal, so it is a map between complexes.
\[ex:basicequivcohomrelation\] In general the natural map $H^*_{{\operatorname{bas}}G}(M)\to H^*_G(M)$ is neither injective nor surjective. Non-surjectivity is clear, as the basic cohomology always vanishes for degrees above the dimension of $M/G$, whereas $H^*_G(M)$ is in general nonzero in infinitely many degrees – see for instance Example \[ex:trivialaction\]. In degree $1$, the map is an isomorphism (see Example \[ex:equivoneforms\]), and in degree $2$ it is always injective: assuming that $\omega = d_G\alpha$, for a closed basic $2$-form $\omega$ and some $\alpha\in C^1_G(M) = \Omega^1(M)^G$, we have $$\omega = (d_G\alpha)(X) = d\alpha - i_{\overline{X}}\alpha.$$ This implies that $i_{\overline{X}}\alpha = 0$ for all $X\in {\mathfrak{g}}$, which, together with the $G$-invariance of $\alpha$ says that $\alpha$ is $G$-basic, and thus $d\alpha = \omega$ in $\Omega_{{\operatorname{bas}}G}(M)$.
The smallest degree in which non-injectivity can occur is $3$, see [@GGK Example C.18]: consider, on the $4$-sphere $$S^4=\{(a,z,w)\mid a^2 + |z|^2 + |w|^2=1\}\subset {{\mathbb R}}\times {{\mathbb C}}^2\cong {{\mathbb R}}^5$$ the circle action given by the product of the standard diagonal action on ${{\mathbb C}}^2$ and the trivial action on ${{\mathbb R}}$. Then one computes (using the equivariant Mayer-Vietoris sequence, Theorem \[thm:equivMV\] below) that $H^3_{S^1}(S^4)=0$. On the other hand, $H^3_{{\operatorname{bas}}S^1}(S^4)={{\mathbb R}}$: either using Remark \[rem:basicsingularcohom\], by observing that the action is the suspension of the Hopf action on $S^3$, so that the orbit space is homeomorphic to the suspension of $S^2$, which is $S^3$. Alternatively, if one would like to avoid using singular cohomology, one can use basic versions of the Mayer-Vietoris sequence and the homotopy axiom.
There is also a natural map from equivariant to ordinary de Rham cohomology:
The ring homomorphism $\Omega_G(M)\to \Omega(M);\, \omega\mapsto \omega(0)$ is a chain map and therefore defines a homomorphism of ${{\mathbb R}}$-algebras $H^*_G(M)\to H^*(M)$.
We just need to observe that $(d_G\omega)(0) = d(\omega(0)) - i_{\overline{0}} \omega(0) = d(\omega(0))$.
This map $H^*_G(M)\to H^*(M)$ is in general not injective (for example for trivial actions) and also not surjective (for example for nontrivial free actions). Note that the composition $$H^*_{{\operatorname{bas}}G}(M) \longrightarrow H^*_G(M) \longrightarrow H^*(M)$$ of the two natural maps just introduced is nothing but the map induced by the inclusion $\Omega_{{\operatorname{bas}}G}(M)\to \Omega(M)$.
Consider an Hamiltonian action of a compact, connected Lie group $G$ on a symplectic manifold $(M,\omega)$. In this situation we have a *momentum map*, i.e., a $G$-equivariant map $\mu\colon M\to {\mathfrak{g}}^*$ such that $i_{\overline{X}}\omega = d\mu^X$, where $\mu^X\colon M\to {{\mathbb R}}$ is defined by $\mu^X(p) = \mu(p)(X)$.
The momentum map defines an equivariant linear map (which we call $\mu$ again) $$\mu\colon {\mathfrak{g}}\longrightarrow C^\infty(M);\, X\longmapsto \mu^X.$$ In particular, $\mu$ can be regarded as an equivariant $2$-form on $M$: $\mu\in ({\mathfrak{g}}^*\otimes C^\infty(M))^G \subset C^2_G(M)$. For any element $f\in ({\mathfrak{g}}^*\otimes C^\infty(M))^G$ we can consider the equivariant $2$-form $\omega + f$ and compute $$\begin{aligned}
d_G(\omega+f)(X) &= (d_G\omega)(X) + (d_Gf)(X) \\
&= d\omega - i_{\overline{X}}\omega + df^X + i_{\overline{X}}f^X \\
&= df^X - i_{\overline{X}}\omega.\end{aligned}$$ This shows that $\omega + f$ is equivariantly closed if and only if $f\in C^2_G(M)$ is a momentum map for the $G$-action.
In particular, the cohomology class $[\omega]\in H^2(M)$ is in the image of the natural map $H^2_G(M)\to H^2(M)$. It is even true that for any Hamiltonian action on a compact manifold the map $H^*_G(M)\to H^*(M)$ is surjective, see Example \[ex:morsebotteqformal\] below.
Locally free actions
====================
The topic of this section is a theorem of H. Cartan [@Cartan2] that says that for (locally) free actions, equivariant cohomology is isomorphic to basic cohomology, hence (in the free case) isomorphic to the de Rham cohomology of the orbit space. Recall Remark \[rem:basicsingularcohom\] which heuristically explained that this is precisely this class of actions for which basic cohomology is a good invariant – later we will see that equivariant cohomology is a better invariant than basic cohomology for non-free actions.
We say that an action of a compact Lie group $G$ on a manifold $M$ is *locally free* if all isotropy groups $G_p$ of the action are finite.
\[thm:eqcohomlocallyfreeactions\] For a locally free action of a compact Lie group $G$ on a manifold $M$ the natural map $$H^*_{{\operatorname{bas}}G}(M)\longrightarrow H^*_G(M)$$ is an isomorphism.
There are many references for a proof of this statement. Besides the original source [@Cartan2] one can find it e.g. in [@GuilleminSternberg Section 5.1] or [@Nicolaescu]. A generalization to other coefficients can be found in [@DufloKumarVergne Section 1.7]. We will show the theorem only for the special case $G=S^1$.
The main tool in the proof is the following: because the $S^1$-action is free, $\overline X_p\neq 0$ for all $p\in M$. Thus, we find an $S^1$-invariant $1$-form $\alpha$ on $M$ such that $\alpha(\overline{X}) = 1$. (Choose an $S^1$-invariant Riemannian metric on $M$, and define $\alpha$, for any $p$, to be $1$ on $\overline{X}_p$, and zero on the orthogonal complement of $\overline{X}_p$.)
We first show surjectivity of the map $H^*_{{\operatorname{bas}}S^1}(M)\to H^*_{S^1}(M)$. Let $\omega\in C_{S^1}^n(M)={{\mathbb R}}[u]\otimes \Omega(M)^{S^1}$ be a closed $S^1$-equivariant differential form on $M$, and write $$\omega = \omega_0 + \omega_1 u + \cdots + \omega_k u^k,$$ where the $\omega_i$ are $S^1$-invariant differential forms, with $\deg \omega_i = n-2i$, and $\omega_k\neq 0$. We assume that $k>0$. Closedness of $\omega$ reads as $$0 = d_{S^1}\omega = d\omega_0 + (d\omega_1 - i_{\overline X} \omega_0) u + \cdots + (d\omega_k - i_{\overline X} \omega_{k-1}) u^k - i_{\overline{X}} \omega_k u^{k+1}.$$ In particular, $i_{\overline X} \omega_k = 0$. We now modify $\omega$ by an exact equivariant differential form: $$\begin{aligned}
\omega &+ d_{S^1} ((\alpha\wedge \omega_k) u^{k-1}) \\
&= \omega_0 + \omega_1 u + \cdots + (\omega_{k-1} + d(\alpha\wedge \omega_k))u^{k-1} +(\omega_k- i_{\overline X}(\alpha\wedge \omega_k)) u^k \\
&= \omega_0 + \omega_1 u + \cdots + (\omega_{k-1} + d(\alpha\wedge \omega_k))u^{k-1}\end{aligned}$$ because $i_{\overline X} \alpha=1$ and $i_{\overline X} \omega_k=0$. We have thus found, in the same equivariant cohomology class, a representative with polynomial degree one less. We can continue reducing the degree until we are left with a representative that is an ordinary differential form, which is at the same time equivariantly closed, i.e., closed and basic, and hence also defines an element in $H^n_{{\operatorname{bas}}S^1}(M)$.
Next, we show injectivity of the map $H^*_{{\operatorname{bas}}S^1}(M)\to H^*_{S^1}(M)$. So assume that $\eta\in \Omega^n_{{\operatorname{bas}}S^1}(M)$ is a closed basic form which is equivariantly exact, i.e., there exists $\omega = \omega_0 + \omega_1 u + \cdots + \omega_k u^k$ such that $$\eta = d_{S^1} \omega = d\omega_0 + (d\omega_1 - i_{\overline X} \omega_0) u + \cdots (d\omega_k - i_{\overline X} \omega_{k-1}) u^k - i_{\overline{X}} \omega_k u^{k+1}.$$ Im particular, $\omega_k$ is a basic differential form. If $k>0$, then we reduce the polynomial degree of $\omega$ successively as above, by replacing $\omega$ by $\omega + d_{S^1}((\alpha\wedge \omega_k)u^{k-1})$. Having reduced to the case $k=0$, we are done, because then $d\omega_0 = \eta$, i.e., $\eta$ is exact as a basic differential form.
Combining this theorem with Proposition \[prop:basiccohomfreeaction\] we obtain:
\[cor:orbitspacedings\] For a free action of a compact Lie group $G$ on a manifold $M$ the projection map $M\to M/G$ induces an isomorphism $H^*(M/G)\longrightarrow H^*_G(M)$.
One should note that in the Borel model, see Remark \[rem:Borelmodel\], the proof of this theorem is much easier, see e.g., [@GuilleminSternberg Section 1.1]: to see that $H^*_G(M;R) \cong H^*(M/G;R)$ one only needs to observe that in this case $EG\times_G M\to M/G$ is a fiber bundle with contractible fiber.
\[rem:commutingactionprinciple\] A more general version of this theorem states that for an action of a product $G\times H$ on a manifold $M$ such that the action of the subgroup $G$ is free, the natural map $$H^*_H(M/G) \longrightarrow H^*_{G\times H}(M)$$ is an isomorphism. In Proposition \[Appendix:Prop:QuotientShenanigans\] we will give a proof of this statement in case $G$ and $H$ are tori.
Equivariant homotopy and Mayer-Vietoris
=======================================
Many standard techniques and results from ordinary cohomology theory have an equivariant counterpart. In this section we prove two of them: the equivariant version of the homotopy axiom and of the Mayer-Vietoris sequence.
Assume that $G$ acts on $M$ and $N$, and let $f,g\colon M\to N$ be $G$-homotopic equivariant maps, i.e., there exists a smooth $G$-equivariant homotopy $F\colon M\times {{\mathbb R}}\to N$ such that $F(\cdot,0)=f$ and $F(\cdot,1)=g$, where we extend the $G$-action to $M\times {{\mathbb R}}$ trivially on the second factor. Then $f^*=g^*\colon H^*_G(N)\to H^*_G(M)$.
Recall the usual proof of the homotopy axiom for de Rham cohomology in the nonequivariant setting: one considers the operator $$Q\colon \Omega^k(M\times {{\mathbb R}})\to \Omega^{k-1}(M);\, \alpha\mapsto \int_0^1 i_{\partial_t} \alpha\, dt$$ and shows that it satisfies the equation $$\label{eq:oldhomotopy}
d\circ Q\circ F^*+Q\circ F^*\circ d = g^*-f^*\colon \Omega(N)\to \Omega(M),$$ i.e., that $Q\circ F^*$ is a chain homotopy between $f^*$ and $g^*$, see [@BottTu §I.4], [@Onishchik §7.5, Example 9]. We claim that this equation is still valid equivariantly, in the sense of Equation below. Define $A\colon C_G^k(M)\to C_G^{k-1}(M)$ by $$(A\omega)(X) = Q(F^*(\omega(X))).$$ First we need to show that $A$ is well-defined, i.e., that $A\omega$ is again a $G$-equivariant differential form. As $F$ is a $G$-homotopy, we have $F(gp,t)=gF(p,t)$ for all $g\in G$, $p\in M$ and $t\in {{\mathbb R}}$, i.e., $F\circ g = g \circ F$. Moreover, we have $Q\circ g^* = g^*\circ Q$. Putting this together, we obtain $$(A\omega)({\operatorname{Ad}}_g X) = Q(F^*(\omega({\operatorname{Ad}}_g X))) = Q(F^*((g^{-1})^*(\omega(X)))) = (g^{-1})^* ((A\omega)(X)).$$ We claim now that $$\label{eq:newhomotopy}
d_G\circ A + A\circ d_G = g^*-f^*\colon C_G(N)\to C_G(M).$$ For any $\omega \in C_G(N)$, we have $$\begin{aligned}
(d_G(A\omega))(X) &= d((A\omega)(X)) - i_{\overline{X}} ((A\omega)(X)) \\
&=d(Q(F^*(\omega(X)))) - i_{\overline{X}}(Q(F^*(\omega(X))))\\
&=d(Q(F^*(\omega(X)))) + Q(i_{\overline{X}} (F^*(\omega(X))))\\
&=d(Q(F^*(\omega(X)))) + Q(F^*(i_{\overline{X}}(\omega(X)))),\end{aligned}$$ where we used that $F$ is $G$-equivariant in the last line. Moreover, we have $$\begin{aligned}
(A(d_G\omega)(X)) &= Q(F^*(d(\omega(X)) - i_{\overline{X}}(\omega(X)))) \\
&= Q(F^*(d(\omega(X)))) - Q(F^*(i_{\overline{X}}(\omega(X)))).\end{aligned}$$ Adding up these two equations, implies . This proves the theorem.
It follows that if $M$ and $N$ are manifolds on which a compact Lie group $G$ acts, and which are $G$-homotopy equivalent, i.e., for which both $f\circ g$ and $g\circ f$ are equivariantly homotopic to the identity map, then $H^*_G(M)$ and $H^*_G(N)$ are isomorphic as graded $S({\mathfrak{g}}^*)^G$-algebras (via the maps $f^*$ and $g^*$).
\[thm:equivMV\] Let $U,V\subset M$ be open $G$-invariant subsets such that $U\cup V=M$. Denote the natural inclusions by $i_U\colon U\to M$, $i_V\colon V\to M$, $j_U\colon U\cap V\to U$, $j_V\colon U\cap V\to V$. Then there is a long exact sequence $$\cdots \longrightarrow H^*_G(M)\overset{i_U^*\oplus i_V^*}{\longrightarrow} H^*_G(U)\oplus H^*_G(V)\overset{j_U^*-j_V^*}{\longrightarrow} H^*_G(U\cap V)\overset{\delta}\longrightarrow H_G^{*+1}(M)\longrightarrow \cdots$$ The maps $i_U^*$ and $i_V^*$ are $S({\mathfrak{g}}^*)^G$-algebra homomorphisms, but $j_U^*-j_V^*$ and $\delta$ are only $S({\mathfrak{g}}^*)^G$-module homomorphisms.
Tensoring the short exact sequence $$\label{eq:equivmvproof}
0\longrightarrow \Omega(M) \overset{i_U^*\oplus i_V^*}\longrightarrow \Omega(U)\oplus \Omega(V)\overset{j_U^*-j_V^*}\longrightarrow \Omega(U\cap V)\longrightarrow 0$$ on the level of differential forms with $S({\mathfrak{g}}^*)$ preserves exactness. We take $G$-invariant forms in each term and and obtain a sequence $$0\longrightarrow C_G^*(M) \overset{i_U^*\oplus i_V^*}\longrightarrow C_G^*(U)\oplus C_G^*(V)\overset{j_U^*-j_V^*}\longrightarrow C_G^*(U\cap V)\longrightarrow 0$$ of which we need to show exactness. Injectivity at the first term is clear, as well as the inclusion of the image in the kernel at the second term. Let $(\omega,\eta)\in \ker(j_U^*-j_V^*)$. We find $\mu\in S({\mathfrak{g}}^*)\otimes \Omega(M)$ such that $(i_U^*\mu,i_V^*\mu) = (\omega,\eta)$, because the sequence , tensored with $S({\mathfrak{g}}^*)$, is exact. We define $\tilde\mu\in C_G(M)$ as $$\tilde\mu = \int_G g^*\mu\, dg,$$ where $g$ acts on $S({\mathfrak{g}}^*)\otimes \Omega(M)$ diagonally, i.e., $$\tilde\mu(X) = \int_G (g^{-1})^*\mu({\operatorname{Ad}}_{g^{-1}}X)\, dg$$ for $X\in {\mathfrak{g}}$, and claim that $(i_U^*\tilde\mu,i_V^*\tilde\mu) = (\omega,\eta)$ as well. For that, we compute $$\begin{aligned}
i_U^*\tilde\mu(X) &= \int_G i_U^* (g^{-1})^*\tilde \mu({\operatorname{Ad}}_{g^{-1}} X)\, dg = \int_G (g^{-1})^* i_U^*\tilde \mu({\operatorname{Ad}}_{g^{-1}} X)\, dg\\
&= \int_G (g^{-1})^* \omega({\operatorname{Ad}}_{g^{-1}} X)\, dg = \int_G \omega(X)\, dg = \omega(X)\end{aligned}$$ because $\omega$ is already $G$-invariant. Analogously, $i_V^*\tilde\mu = \eta$, so we have shown exactness at the second term.
For the surjectivity we argue similarly: we start with a possibly noninvariant preimage of an element in $C_G^*(U\cap V)$, and average (both components separately). Thus, we have an induced long exact sequence in equivariant cohomology.
\[ex:S1aufS2\] Consider the $S^1$-action on $S^2$ by rotation around the $z$-axis. Let $S^2=U\cup V$ be the covering of $S^2$ by upper and lower hemisphere. Then $U$ and $V$ are $S^1$-equivariantly homotopy equivalent to the north respectively to the south pole, and $U\cap V$ is $S^1$-equivariantly homotopy equivalent to the equator. Therefore, $$H^*_{S^1}(U)=H^*_{S^1}(V)=H^*_{S^1}({\mathrm{pt}})={{\mathbb R}}[u]$$ with $u$ in degree two, and using Theorem \[thm:eqcohomlocallyfreeactions\], $$H^*_{S^1}(U\cap V) = H^*_{S^1}(S^1) = {{\mathbb R}}$$ concentrated in degree zero. We obtain an exact sequence $$\cdots \longrightarrow H^*_{S^1}(S^2)\longrightarrow {{\mathbb R}}[u] \oplus {{\mathbb R}}[u] \overset{\varphi}\longrightarrow {{\mathbb R}}\longrightarrow \cdots$$ where the map $\varphi$ is given by $\varphi(f,g)=f(0)-g(0)$. It is surjective, so the sequence is in fact short exact and we obtain an isomorphism of ${{\mathbb R}}[u]$-algebras $$H^*_{S^1}(S^2)=\{(f,g)\in {{\mathbb R}}[u]\oplus {{\mathbb R}}[u]\mid f(0)=g(0)\}.$$ Note that $H^*_{S^1}(S^2)$ is a free ${{\mathbb R}}[u]$-module: a basis is given by $(1,1)$ and $(u,-u)$.
Note also the peculiar feature of this example that the map on equivariant cohomology induced by the inclusion of the fixed point set into the manifold is injective (the fixed point set is exactly the union of north and south pole). It will be a consequence of the Borel Localization Theorem that this is the case for a large class of actions.
Equivariant formality {#sec:equivariantformality}
=====================
Starting with this section, we will make use of the spectral sequence of the Cartan model, which is introduced in Section \[Appendix:Section:CartanSpeq\].
An action of a compact Lie group $G$ on a smooth manifold $M$ is *equivariantly formal* if the spectral sequence of the Cartan model collapses at the $E_1$-term.
The term *equivariant formality* was introduced twenty years ago in [@GKM]. In the context of the Borel model, see Remark \[rem:Borelmodel\], the Serre spectral sequence of the (Borel) fibration $EG\times_G M \to BG$ at and after $E_2$ is equivalent to the spectral sequence of the Cartan model at and after $E_2$. Since $E_1=E_2$ in the Cartan model (see Remark \[rem:doddvanishes\], the collapse of the Serre spectral sequence at the $E_2$-term is equivalent to equivariant formality of the action. This collapse is, in turn, equivalent to the surjectivity of the map induced in cohomology by the fiber inclusion (cf. Theorem \[thm:bigthmequivformal\] below), which is usually described by saying that the fiber is *totally nonhomologous to zero*, or that the fibration itself is *totally nonhomologous to zero*, abbreviated *TNHZ*, see e.g. [@BredonBook], [@AlldayPuppe], or [@FelixOpreaTanre]. Instead of the term *equivariant formality* many authors thus just speak about $M$ being (totally) nonhomologous to zero in the Borel fibration. This condition already appears in [@Borel Chapter XII].
It was shown in [@GKM Theorem 1.5.2] that equivariant formality implies formality properties of certain differential graded modules, which explains the choice of terminology. One might argue though that this nomenclature is not optimal as the formality aspect is just a consequence of the much stronger condition of equivariant formality and there are not many connections to the notion of formality from the point of view of rational homotopy theory. One such connection was given in [@CarlsonFok] where the authors prove that if the isotropy action of a homogeneous space is equivariantly formal, then the space is formal. Note that the other implication is not valid, see e.g. [@CarlsonFok Example 4.2].
The following theorem collects some equivalent formulations of equivariant formality, as well as some justification of its relevance: Condition $(5)$ says that for equivariantly formal actions the ordinary de Rham cohomology of $M$ is determined by the equivariant cohomology algebra. Note that the equivalence of $(1)$ and $(3)$ is not trivial: by Proposition \[prop:E1term\] the $E_1$-term of the spectral sequence is $S({\mathfrak{g}}^*)^G\otimes H^*(M)$, so equivariant formality tells us directly that $H^*_G(M)\cong S({\mathfrak{g}}^*)^G\otimes H^*(M)$, but this isomorphism is only one of graded vector spaces. In general, $H^*_G(M)$ and $E_\infty$ are not isomorphic as $S({\mathfrak{g}}^*)^G$-modules – see Section \[sec:counterexample\] for a counterexample.
\[thm:bigthmequivformal\] The following conditions are equivalent, for an action of a compact, connected Lie group $G$ on a compact manifold $M$:
1. The $G$-action is equivariantly formal.
2. The canonical map $H^*_G(M)\to H^*(M)$ is surjective.
3. There is an isomorphism of graded $S({\mathfrak{g}}^*)^G$-modules $$H^*_G(M) \cong S({\mathfrak{g}}^*)^G\otimes H^*(M).$$ (In particular $H^*_G(M)$ is a free module over $S({\mathfrak{g}}^*)^G$.)
If these conditions are satisfied, then also the following statements hold true:
1. The kernel of the canonical map $H^*_G(M)\to H^*(M)$ is the ideal generated by the image of $S^+({\mathfrak{g}}^*)^G\to H^*_G(M)$, i.e., $$S^+({\mathfrak{g}}^*)^G\cdot H^*_G(M) = \Big\{\sum_i f_i [\eta_i]\mid f_i\in S^+({\mathfrak{g}}^*)^G,\, [\eta_i]\in H^*_G(M)\Big\}.$$ Here, $S^+({\mathfrak{g}}^*)^G$ denotes the positive degree elements in $S({\mathfrak{g}}^*)^G$.
2. We have an isomorphism of ${{\mathbb R}}$-algebras $$\label{eq:ordinarycohomfromeqcohom}
H^*(M) \cong \frac{H^*_G(M)}{S^+({\mathfrak{g}}^*)^G\cdot H^*_G(M)}.$$
We first show that $(1)$ and $(2)$ are equivalent. Assuming $(1)$, we consider a cohomology class in $H^n(M)$, represented by a $G$-invariant differential form $\omega_0$. As $d_G\omega_0\in C_G^{2,n-1}(M)$ we have $\omega_0\in A_2^{0,n}$ and can consider the element $[\omega_0]\in E_2^{0,n}$, where we use the notation from Section \[Appendix:ConstrSec\]. The latter is annihilated by the differential $d_2$, because $d_2\colon E_2\to E_2$ is the zero map by assumption. Thus $d_G\omega_0$ lies in $d_G(A_1^{1,n-1})+A_1^{3,n-2}$. Consequently we find $\omega_1\in C^{1,n-1}_G(M)$ with $d_G\omega_1+d_G\omega_0\in C_G^{3,n-2}(M)$. Now the element $\omega_0+\omega_1$ lies in $A_3^{0,n}$ and induces an element of $E_3^{0,n}$. Using now that $d_3=0$ we inductively construct an element $\omega=\omega_0+\ldots+\omega_n$ with $d_G\omega=0$ and $\omega(0)=\omega_0$. We have shown that $H^*_G(M)\to H^*(M)$ is surjective.
Assume now that $(2)$ holds, i.e., that we can extend any closed $G$-invariant form $\omega_0$ to a closed equivariant differential form $\omega_0 + \omega_1 + \cdots$. But again by definition of the higher differentials in the spectral sequence this means that all $d_r$, $r=1,2,\ldots$, vanish. (Inductively; first they vanish on $E_r^{0,*}$, but because the $E_r$ are modules over $S({\mathfrak{g}}^*)^G$, and the $d_r$ are $S({\mathfrak{g}}^*)^G$-linear, they vanish completely.) Thus, $(1)$ holds.
We next show that $(2)$ implies $(4)$ and $(5)$. It is clear that $S^+({\mathfrak{g}}^*)^G\cdot H^*_G(M)$ is contained in the kernel of the canonical map $H^*_G(M)\to H^*(M)$. So let $\omega = \omega_0 + \omega_1+\cdots \in H^*_G(M)$ be an element in the kernel, where we use the same notation as above: the index $i$ refers to the polynomial degree of $\omega_i$. Being in the kernel means that $\omega_0=d\beta_0$ is exact as an ordinary invariant differential form. By replacing $\omega$ by $\omega - d_G \beta_0$ we can assume that $\omega_0=0$. Now consider $\omega_1$. Because $d\omega_1=0$, and the $E_1$-term is $S({\mathfrak{g}}^*)^G\otimes H^*(M)$, we can (by adding an appropriate exact form) assume that $\omega_1\in S^1({\mathfrak{g}}^*)^G\otimes \Omega(M)^G$, i.e., $\omega_1 = \sum_j f_j \gamma_j$, for $G$-invariant linear forms $f_j$, and closed $G$-invariant forms $\gamma_j$. Now, because $H^*_G(M)\to H^*(M)$ is surjective, we can extend the $\gamma_j$ to equivariantly closed differential forms $\tilde\gamma_j$, and subtract $\sum_j f_j \tilde\gamma_j$ from $\omega$ to obtain an element in the kernel of the form $\omega_2 + \omega_3 + \cdots$. By continuing in the same way, we have shown the desired expression for the kernel, i.e., $(4)$. Statement $(5)$ follows directly by combining $(2)$ with $(4)$.
Using this implication, we next show that $(1)$ and $(2)$ imply $(3)$: we construct a module isomorphism $H^*_G(M)\cong S({\mathfrak{g}}^*)^G\otimes H^*(M)$. More precisely, we fix a vector space basis $\{[\alpha_i]\}$ of $H^*(M)$, and preimages $[\beta_i]$ of the $[\alpha_i]$ under the canonical map $H^*_G(M)\to H^*(M)$, which exist by $(2)$. In other words, the $\beta_i$ are equivariant differential forms whose polynomial parts are cohomologous to $\alpha_i$. We wish to show that $H^*_G(M)$ is a free $S({\mathfrak{g}}^*)^G$-module with basis $\{[\beta_i]\}$.
Let us show that the $[\eta_i]$ span $H^*_G(M)$ as a module over $S({\mathfrak{g}}^*)^G$. We proceed by induction on the degree. For degree zero this is true, because $H^0_G(M) = H^0(M)$. So take an arbitrary class $[\omega]\in H^*_G(M)$. We write $[\omega(0)] = \sum_i a_i [\alpha_i]$, for $a_i\in {{\mathbb R}}$. By subtracting $\sum_i a_i [\beta_i]$ from $[\omega]$ we thus obtain an element in the kernel of $H^*_G(M)\to H^*(M)$. By $(4)$, this element is a linear combination $\sum_i f_i [\eta_i]$, for some $f_i$ of positive degree. By induction, the $[\eta_i]$ are contained in the span of the $[\beta_i]$, and hence also $[\omega]$.
Finally, we consider the $S({\mathfrak{g}}^*)^G$-module homomorphism $$S({\mathfrak{g}}^*)^G\otimes H^*(M) \longrightarrow H^*_G(M)$$ given by $f\otimes [\alpha_i]\longmapsto f[\beta_i]$. We have shown that it is surjective. But by the collapse of the spectral sequence (condition $(1)$), for every $n$ the degree $n$ part of the left and the right hand side are isomorphic (as abstract vector spaces). Because they are also finite-dimensional (we assumed that $M$ is a compact manifold, and we know also that the polynomial ring $S({\mathfrak{t}}^*)^G$ is finite-dimensional in each degree) this map has to be an isomorphism. We have shown $(3)$.
To conclude, we observe that $(3)$ implies $(1)$: if $H^*_G(M)\cong S({\mathfrak{g}}^*)^G\otimes H^*(M)$, then by Proposition \[prop:E1term\], $H^*_G(M)$ and the $E_1$-term of the spectral sequence are isomorphic as graded $S({\mathfrak{g}}^*)^G$-modules, and in particular as graded vector spaces. As both vector spaces are finite-dimensional in every degree, this forces all differentials of the spectral sequence to vanish, i.e., the action to be equivariantly formal.
Using more results from the appendix, one can shorten the argument. Without taking the detour through $(4)$ and $(5)$, the equivalent conditions $(1)$ and $(2)$ imply $(3)$ using Lemma \[Appendix:Lem:modulegenerators\]: a vector space basis of $H^*(M)$ is a module basis of $E_\infty \cong E_1 \cong S({\mathfrak{g}}^*)^G\otimes H^*(M)$, which induces by Lemma \[Appendix:Lem:modulegenerators\] a set of generators of the $S({\mathfrak{g}}^*)^G$-module $H^*_G(M)$ of the same cardinality. Then the same argument as in the proof above shows that this generating set is in fact a basis.
Having shown in this way that $(1)$, $(2)$ and $(3)$ are equivalent, the implication of $(4)$ and $(5)$ is immediate: $S^+({\mathfrak{g}}^*)^G\otimes H^*(M) \subset S({\mathfrak{g}}^*)^G\otimes H^*(M) \cong H^*_G(M)$ is a subspace of codimension $\dim H^*(M)$, contained in the kernel of the surjection $H^*_G(M)\to H^*(M)$. Thus, $S^+({\mathfrak{g}}^*)^G\cdot H^*_G(M)$ equals the kernel.
Any trivial action is equivariantly formal. For a trivial action, we have $H^*_G(M) = S({\mathfrak{g}}^*)^G\otimes H^*(M)$ even as an algebra over $S({\mathfrak{g}}^*)^G$.
\[ex:oddcohomzeroeqformal\] More generally, in Corollary \[thm:hoddcollapse\] we show that the spectral sequence of the action collapses at the $E_1$-term whenever $H^{{\operatorname{odd}}}(M)$ vanishes. Thus any Lie group action on such a manifold is equivariantly formal.
\[ex:S1aufS2equivformal\] The simplest nontrivial example of an action on a compact manifold with vanishing odd-dimensional cohomology is the standard circle action on the $2$-sphere. In Example \[ex:S1aufS2\] we identified its equivariant cohomology as $$H^*_{S^1}(S^2) \cong \{(f,g)\in {{\mathbb R}}[u]\oplus {{\mathbb R}}[u]\mid f(0)=g(0)\}.$$ Any element $(f,g)\in H^*_{S^1}(S^2)$ can be written in the form $$(f,g) = \frac12(f+g,f+g) + \frac12(f-g,g-f) = \frac12(f+g)(1,1) + \frac{f-g}{2u} (u,-u),$$ where we note that because $f(0)=g(0)$, the polynomial $f-g$ is divisible by $u$. Moreover, the elements $(1,1)$ and $(u,-u)$ are linearly independent over ${{\mathbb R}}[u]$. Thus, $H^*_{S^1}(S^2)$ is a free module over ${{\mathbb R}}[u]$, with basis $\{(1,1),(u,-u)\}$. Note that $H^*(S^2)$ is a graded vector space, with one-dimensional components in degree $0$ and $2$, which are precisely the degrees of the elements $(1,1)$ and $(u,-u)$.
By Theorem \[thm:bigthmequivformal\] we can recover the ordinary cohomology of $S^2$ from the equivariant one: $$H^*(S^2) \cong \frac{H^*_{S^1}(S^2)}{u\cdot {{\mathbb R}}[u]\cdot (1,1) \oplus u\cdot {{\mathbb R}}[u] \cdot (u,-u)}$$ As a vector space, $H^*(S^2)$ is spanned by the cosets of $(1,1)$ and $(u,-u)$. The ring structure is the obvious one, where $(1,1)$ is the unit.
The same argument works in full generality: if one is able to determine a basis $e_1,\ldots,e_k$ of $H^*_G(M)$ as an $S({\mathfrak{g}}^*)^G$-module, for any equivariantly formal $G$-action, then $H^*(M)$ is, as a vector space, isomorphic to the real vector space with the $e_i$ as basis. The multiplicative structure is encoded in the abstract quotient .
\[cor:subgroupeqformal\] Consider an equivariantly formal action of a compact, connected Lie group $G$ on a manifold $M$. Then, for any compact, connected Lie subgroup $H\subset G$, the induced $H$-action on $M$ is equivariantly formal as well.
Restiction of an equivariant differential form $\omega\colon {\mathfrak{g}}\to \Omega(M)$ to ${\mathfrak{h}}$ defines a natural morphism $C_G(M)\to C_H(M)$ which descends to a map $H^*_G(M)\to H^*_H(M)$. Then the statement follows directly from Theorem \[thm:bigthmequivformal\] because the canonical map $H^*_G(M)\to H^*(M)$ factors through $H^*_H(M)$.
Many important classes of actions are equivariantly formal.
\[ex:morsebotteqformal\] Consider an action of a torus $T$ on a compact manifold $M$. If there exists a $T$-invariant Morse-Bott function $f\colon M\to {{\mathbb R}}$ such that the critical set of $f$ is equal to the fixed point set $M^T$, then the action is equivariantly formal. Although not using precisely this formulation, the arguments to show this were given simultaneously by several authors, in [@Duflot], [@AtiyahBott], [@Ginzburg], and [@Kirwan]. Roughly, one shows, using an equivariant Thom isomorphism, that for every critical value $\kappa$ of $f$ one has a short exact sequence $$0 \longrightarrow H^*_T(M^{\kappa +\varepsilon},M^{\kappa-\varepsilon}) \longrightarrow H^*_T(M^{\kappa + \varepsilon})\longrightarrow H^*_T(M^{\kappa-\varepsilon})\longrightarrow 0$$ in (Borel) equivariant cohomology, where for any $a$ we denote the respective sublevel set by $M^a = \{p\in M\mid f(p)\leq a\}$. This implies, inductively, that all $H^*_T(M^a)$ are free $S({\mathfrak{t}}^*)$-modules. It was observed in [@GT] that the same argument goes through in the context of Cohen-Macaulay actions, see Section \[sec:CohenMacaulay\] below, for Morse-Bott functions whose critical set is the union of $b$-dimensional orbits, where $b$ is the lowest occurring orbit dimension.
For example, given any Hamiltonian torus action on a compact symplectic manifold, a generic component of the moment map $\mu\colon M\to {\mathfrak{t}}^*$ is a Morse-Bott function with this property, thus showing that any Hamiltonian torus action on a compact symplectic manifold is equivariantly formal.
A natural class of actions is given by isotropy actions of homogeneous spaces, i.e., the action of a connected Lie group $H$ on a homogeneous space of the form $G/H$. If $G$ and $H$ are of equal rank, then even the $G$-action on $G/H$ is equivariantly formal, see Theorem \[thm:homogeneousspacesequalrank\] below, so the $H$-action is, by Corollary \[cor:subgroupeqformal\], equivariantly formal as well. (In fact, in this case $H^{{\operatorname{odd}}}(G/H)$ vanishes, see again Theorem \[thm:homogeneousspacesequalrank\], so that any action on $G/H$ is automatically equivariantly formal.)
In general it is an open question for which homogeneous spaces $G/H$ the isotropy action is equivariantly formal. This question was considered by Shiga and Shiga–Takahashi in [@Shiga; @ShigaTakahashi], where they found several sufficient conditions for equivariant formality of isotropy actions (see also [@Carlson Section 2.1] for a summary of these results). It was shown in the affirmative for symmetric spaces [@G], more generally for spaces such that $H$ is the connected component of the fixed points of any automorphism of $G$ [@GH], and for ${{\mathbb Z}}_2\times {{\mathbb Z}}_2$-symmetric spaces in [@Hagh]. Some examples of homogeneous spaces whose isotropy action is not equivariantly formal were given in [@ShigaTakahashi] and [@Shiga], and the equivariantly formal homogeneous spaces with $H\cong S^1$ were classified in [@Carlson]. In [@CarlsonFok] it was shown that equivariant formality of the isotropy action of $G/H$ implies that $G/H$ is formal in the sense of rational homotopy theory.
Borel localization {#sec:borellocalization}
==================
Is this section, as well as the next, we consider only actions of tori on compact manifolds. Recall that for an equivariant smooth map $f\colon N\to M$ between $T$-manifolds, we can consider its induced map $f^*\colon H^*_T(M)\to H^*_T(N)$ in equivariant cohomology. Both its kernel and its cokernel, ${\operatorname{coker}}f^* = H^*_T(N)/{\operatorname{im}}f^*$, are naturally $S({\mathfrak{t}}^*)$-modules. Our goal in this section is to prove the following theorem (see [@GKM Section (1.7)] for information on the history of localization theorems):
\[thm:borellocalization\] Consider, for an action of a torus $T$ on a compact manifold $M$, the restriction map $$H^*_T(M)\longrightarrow H^*_T(M^T).$$ Its cokernel is a torsion module, and its kernel is the torsion submodule of $H^*_T(M)$.
The proof we give is a version of the proof in [@GuilleminSternberg Section 11], somewhat simplified by avoiding the usage of equivariant cohomology with compact support and the notion of support of a module. Note that there exist far more general versions of the Borel localization theorem, see e.g. [@AlldayPuppe Chapter 3] or [@Hsiang Chapter 3, §2].
Recall the notion of localization from commutative algebra [@AtiyahMac Chapter 3]. For a multiplicatively closed subset $S$ of a commutative ring with unit $R$ we denote the localized ring by $S^{-1}R$, and the localization of an $R$-module $A$ by $S^{-1}A$. We will need the fact that localization is an exact functor, see [@AtiyahMac Proposition 3.3]. In case $A$ is a finitely generated module over an integral domain, and $S=R\setminus \{0\}$, the localization $S^{-1}A$ is a finite-dimensional vector space over the field $S^{-1}R$, and we call its dimension the *rank* of $A$, denoted ${\operatorname{rank}}_R A$.
With this notion the statement in the Borel localization theorem that both kernel and cokernel of the restriction map are torsion can be reformulated as follows:
\[cor:localizedborel\] For any action of a torus $T$ on a compact manifold, the localized map $$S^{-1}H^*_T(M) \longrightarrow S^{-1}H^*_T(M^T),$$ where $S = S({\mathfrak{t}}^*)\setminus \{0\}$, is an isomorphism. The rank of the $S({\mathfrak{t}}^*)$-module $H^*_T(M)$ is $${\operatorname{rank}}_{S({\mathfrak{t}}^*)} H^*_T(M) = \dim H^*(M^T).$$
Before embarking on the proof, we need to calculate the equivariant cohomology of an orbit $Tp=T/T_p$. (Here we consider only tori – a more general statement about the equivariant cohomology of transitive actions is shown below in Proposition \[prop:eqcohomtransitive\].) Let ${\mathfrak{t}}'\subset {\mathfrak{t}}$ be a complement of ${\mathfrak{t}}_p$ in ${\mathfrak{t}}$ such that $\exp({\mathfrak{t}}')$ is a subtorus $T'$ of $T$. Then $S({\mathfrak{t}}^*) = S({\mathfrak{t}}_p^*)\otimes S({\mathfrak{t}}'^*)$. The Cartan complex $C_T(T/T_p)$ can be written as $$C_T(T/T_p) = S({\mathfrak{t}}_p^*)\otimes S({\mathfrak{t}}'^*)\otimes \Omega(T/T_p)^T,$$ and because $T_p$ acts trivially on all of $T/T_p$, the $T$-invariance of a differential form on $T/T_p$ is equivalent to the $T'$-invariance. Therefore, we have $$C_T(T/T_p) = S({\mathfrak{t}}_p^*)\otimes (S({\mathfrak{t}}'^*)\otimes \Omega(T/T_p)^{T'}).$$ The equivariant differential $d_T$ on $C_T(T/T_p)$ acts as $d_T = 1\otimes d_{T'}$, because the $T_p$-fundamental vector fields are zero on $T/T_p$. Thus, $$H^*_T(T/T_p) = S({\mathfrak{t}}_p^*)\otimes H^*_{T'}(T/T_p).$$ Because the $T'$-action on $T/T_p$ is locally free and transitive, we have $H^*_{T'}(T/T_p) = H^*(\{\mathrm{pt}\}) = {{\mathbb R}}$. Thus, $$H^*_T(T/T_p) = S({\mathfrak{t}}_p^*)$$ as $S({\mathfrak{t}}^*)$-algebras, where the $S({\mathfrak{t}}^*)$-algebra structure is induced by the natural restriction $S({\mathfrak{t}}^*)\to S({\mathfrak{t}}_p^*)$.
In particular, we see that if ${\mathfrak{t}}_p\neq {\mathfrak{t}}$ (i.e., if $p$ is not a $T$-fixed point), then $H^*_T(T/T_p)$ is a torsion module: Let $f\in S({\mathfrak{t}}^*)$ be a nonzero linear form on ${\mathfrak{t}}$ that vanishes on ${\mathfrak{t}}_{p}$; then multiplication with $f$ is the zero map on $H^*_T(T/T_p)$.
\[lem:maptoorbitsupport\] Let $M$ be a (not necessarily compact) manifold that admits a $T$-equivariant map $\varphi\colon M\to Tp$, where $p\in M$ is not a fixed point of the $T$-action. Then $H^*_T(M)$ is a torsion module.
We consider the maps $$M \overset{\varphi}\longrightarrow Tp \longrightarrow \{pt\}.$$ In equivariant cohomology they induce homomorphisms $$S({\mathfrak{t}}^*)\longrightarrow H^*_T(Tp)\overset{\varphi^*}\longrightarrow H^*_T(M).$$ Because of Lemma \[lem:algebrastructurefrommaptopoint\], the $S({\mathfrak{t}}^*)$-algebra structure of $H^*_T(M)$ is induced from the unique map to a point, which thus factors through $H^*_T(Tp)$. Above, we computed $H^*_T(Tp) \cong S({\mathfrak{t}}_p^*)$, where the $S({\mathfrak{t}}^*)$-algebra structure is given by the natural restriction map. Every $f\in S({\mathfrak{t}}^*)$ with $f|_{{\mathfrak{t}}_p}=0$ thus annihilates $H^*_T(M)$, because it already defines the zero element in $H^*_T(Tp)$.
Any tubular neighborhood $U$ of an orbit $T p$ admits a $T$-equivariant (retraction) map to $Tp$, so Lemma \[lem:maptoorbitsupport\] applies to any open $T$-invariant subset of $U$.
The idea of the proof is to use the equivariant Mayer-Vietoris sequence for a cover $M=U\cup V$, where $U$ is a tubular neighborhood of $M^T$, and $V$ an open $T$-invariant subset of $M\setminus M^T$, with the following property: both $V$ and $U\cap V$ can be covered by finitely many $T$-invariant open neighborhoods to which Lemma \[lem:maptoorbitsupport\] applies, in the sense that they admit an equivariant map to an orbit in $M\setminus M^T$. Let us first construct this covering: we choose two tubular neighborhoods $M^T\subset U'\subset U$ with $\overline{U'}\subset U$. We put $V:=M\setminus \overline{U'}$. As $M\setminus U'$ is compact, it can be covered by finitely many tubular neighborhoods of orbits in $M\setminus M^T$ (none of which intersects $M\setminus M^T$). This finite cover restricts to finite covers of $V$ and $U\cap V$. The open sets in this cover are open subsets of tubular neighborhoods of orbits of points in $M\setminus M^T$, so Lemma \[lem:maptoorbitsupport\] applies to them.
Now, consider any open subset $W\subset M$ which is a finite union $W = W_1\cup \cdots \cup W_r$ of open $T$-invariant open neighborhoods $W_i$ that admit an equivariant map $f_i\colon W_i\to Tp_i$, where $p_i\in M\setminus M^T$. By Lemma \[lem:maptoorbitsupport\] we have that $H^*_T(W_i)$ is a torsion module for all $i$. Put $Y_j:=W_1\cup \cdots \cup W_{j-1}$, so that $Y_{j+1} = Y_j\cup W_j$. It follows by induction that $H^*_T(Y_j)$ is a torsion module, using the portion $$H^*_T(Y_j\cap W_j) \longrightarrow H^*_T(Y_{j+1}) \longrightarrow H^*_T(Y_j) \oplus H^*_T(W_j)$$ of the equivariant Mayer-Vietoris sequence. Note that we used that with $W_j$ also the intersection $Y_j\cap W_j$ admits an equivariant map to an orbit in $M\setminus M^T$, hence Lemma \[lem:maptoorbitsupport\] also applies to this set. We have thus shown that $H^*_T(W)$ is a torsion module as well.
This observation in particular applies to the sets $V$ and $U\cap V$ from the open cover $M=U\cup V$ constructed above. Using that $H^*_T(U) \cong H^*_T(M^T)$, the equivariant Mayer-Vietoris sequence of this cover reads $$\cdots \longrightarrow H^*_T(U\cap V) \longrightarrow H^*_T(M) \overset{(i^*,j^*)}\longrightarrow H^*_T(M^T) \oplus H^*_T(V) \longrightarrow H^*_T(U\cap V)\longrightarrow \cdots,$$ where $j\colon V\to M$ is the natural inclusion map. Localizing this exact sequence at $S=S({\mathfrak{t}}^*)\setminus \{0\}$, the terms $S^{-1}H^*_T(U\cap V)$ and $S^{-1}H^*_T(V)$ vanish, so that we obtain an isomorphism $$S^{-1}H^*_T(M)\longrightarrow S^{-1}H^*_T(M^T)$$ as in the formulation in Corollary \[cor:localizedborel\]. That the kernel of the restriction map $H^*_T(M)\to H^*_T(M^T)$ contains the torsion submodule of $H^*_T(M)$ is clear because $H^*_T(M^T)$ is a free module.
\[rem:borelexplanationnofixedpoints\] In case the $T$-action has no fixed points, $M^T=\emptyset$. By convention, we understand $H^*_T(\emptyset)=0$.
$H^*_T(M)$ is a torsion module if and only if the $T$-action has no fixed points.
If the $T$-action has no fixed points, then we have just observed that $H^*_T(M)$ is torsion (see Remark \[rem:borelexplanationnofixedpoints\]). If there are fixed points, then $1\in H^*_T(M)$ is mapped to $1\neq 0\in H^*_T(M^T)$. Because $H^*_T(M^T)$ is a free and hence torsion-free $S({\mathfrak{t}}^*)$-module, $1$ is also not a torsion element in $H^*_T(M)$.
The Borel localization theorem is wrong without any assumptions on the space acted on. Consider the Borel model (see Remark \[rem:Borelmodel\]) of the free action of a torus $T$ on the contractible space $ET$. As the projection $$ET\times_{T} ET\longrightarrow BT$$ on the first factor is a homotopy equivalence (it is a fibration with contractible fiber $ET$), the map $$H^*(BT;{{\mathbb R}})\longrightarrow H^*_{T}(ET;{{\mathbb R}})$$ defining the $H^*(BT;{{\mathbb R}})$-algebra structure is an isomorphism. In particular, the equivariant cohomology $H^*_{T}(ET;{{\mathbb R}})$ is a free $H^*(BT;{{\mathbb R}})$-module although the $T$-action has no fixed points.
\[cor:eqformalinjection\] For an equivariantly formal action of a torus on a compact manifold $M$, the inclusion $M^T\to M$ induces an injective $S({\mathfrak{t}}^*)$-algebra homomorphism $$H^*_T(M) \longrightarrow H^*_T(M^T) = S({\mathfrak{t}}^*)\otimes H^*(M^T).$$
One can therefore try to understand the equivariant cohomology of an equivariantly formal action by understanding its image in $H^*_T(M^T)$.
We did this already for the standard circle action on $S^2$, with fixed point set the north and south pole $N,S$, see Example \[ex:S1aufS2\], in which we confirmed ad hoc that the inclusion $H^*_{S^1}(S^2)\to H^*(\{N,S\})= {{\mathbb R}}[u]\oplus {{\mathbb R}}[u]$ is injective, and has as image the ${{\mathbb R}}[u]$-subalgebra $\{(f,g)\mid f(0)=g(0)\}$.
We will give an example with nondiscrete fixed point set below (see Example \[ex:inclusionfixedsetconjugation\]).
In Example \[ex:oddcohomzeroeqformal\] we observed that any action on a manifold with vanishing odd-dimensional cohomology is equivariantly formal. If the fixed point set of the torus action is finite, then this is even an equivalence.
\[prop:equivformalfinitefixedpointset\] Consider an equivariantly formal action of a torus $T$ on a compact manifold $M$. If the fixed point set of the action is finite, then $H^{{\operatorname{odd}}}(M) = 0$.
By Corollary \[cor:eqformalinjection\] we have an injection $H^*_T(M) \to S({\mathfrak{t}}^*)\otimes H^*(M^T)$. As $M^T$ is a finite set, $H^*(M^T)$ is concentrated in degree zero. The polynomial ring $S({\mathfrak{t}}^*)$ is concentrated in even degrees, so that $H^{{\operatorname{odd}}}_T(M) = 0$. But by equivariant formality we have $$H^*_T(M) \cong S^*({\mathfrak{t}}^*)\otimes H^*(M),$$ so necessarily $H^{{\operatorname{odd}}}(M)=0$ as well.
Consequences for the fixed point set {#sec:fixedpoints}
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Recall that the *Euler characteristic* of a manifold $M$ with finite-dimensional cohomology $H^*(M)$ is defined as $$\chi(M):= \dim H^{{\operatorname{even}}}(M)-\dim H^{{\operatorname{odd}}}(M).$$ More generally, one can define the Euler characteristic for any finite-dimensional *${{\mathbb Z}}_2$-graded vector space* $V$, i.e., a vector space of the form $V=V^{{\operatorname{even}}}\oplus V^{{\operatorname{odd}}}$, where we call the elements of $V^{{\operatorname{even}}}$ and $V^{{\operatorname{odd}}}$ even and odd elements.
Let $V=V^{{\operatorname{even}}}\oplus V^{{\operatorname{odd}}}$ be a finite-dimensional ${{\mathbb Z}}_2$-graded vector space. Then the *Euler characteristic* of $V$ is $$\chi(V) = \dim V^{{\operatorname{even}}}-\dim V^{{\operatorname{odd}}}.$$
A fundamental property of the Euler characteristic is that it is preserved under taking cohomology. We omit the (standard) proof.
\[lem:eulercharofcohomology\] Let $V= V^{{\operatorname{even}}}\oplus V^{{\operatorname{odd}}}$ be a finite-dimensional vector space over a field $K$, and $d\colon V\to V$ a $K$-linear map that
1. is a differential, i.e., $d^2=0$, and
2. is an *odd* endomorphism, i.e., restricts to maps $d^{{\operatorname{even}}}:V^{{\operatorname{even}}}\to V^{{\operatorname{odd}}}$ and $d^{{\operatorname{odd}}}\colon V^{{\operatorname{odd}}}\to V^{{\operatorname{even}}}$.
Then $$\chi(V) = \chi(H(V,d)),$$ where $H(V,d) = \ker d/{\operatorname{im}}d$ (which naturally is a ${{\mathbb Z}}_2$-graded vector space).
The following theorem was originally shown by Kobayashi [@Kobayashi] without the usage of equivariant cohomology. We present it here as a corollary of the Borel localization theorem.
\[thm:eulercharfixedpoints\] Consider the action of a torus $T$ on a compact manifold $M$. Then $$\chi(M) =\chi(M^T).$$
By Corollary \[cor:localizedborel\] we have an isomorphism $$S^{-1}H^*_T(M) \longrightarrow S^{-1}H^*_T(M^T),$$ where $S=S({\mathfrak{t}}^*)\setminus \{0\}$. The localized equivariant cohomology is not ${{\mathbb Z}}$-graded anymore, but the dichotomy between even and odd degree elements survives after localization. This isomorphism thus restricts to isomorphisms of the respective even and odd parts. As $H^*_T(M^T) = S({\mathfrak{t}}^*)\otimes H^*(M^T)$, we therefore have (writing $R=S({\mathfrak{t}}^*)$) $$\begin{aligned}
\chi(M^T) &= \dim_{{\mathbb R}}H^{{\operatorname{even}}}(M^T) - \dim_{{\mathbb R}}H^{{\operatorname{odd}}}(M^T) \\
&= \dim_{S^{-1}R} S^{-1}R\otimes H^{{\operatorname{even}}}(M^T) - \dim_{S^{-1}R} S^{-1}R \otimes H^{{\operatorname{odd}}}(M^T)\\
&= \dim_{S^{-1}R} S^{-1}H^{{\operatorname{even}}}_T(M^T) - \dim_{S^{-1}R} S^{-1}H^{{\operatorname{odd}}}_T(M^T)\\
&= \dim_{S^{-1}R} S^{-1}H^{{\operatorname{even}}}_T(M) - \dim_{S^{-1}R} S^{-1}H^{{\operatorname{odd}}}_T(M).\end{aligned}$$ We now use the the spectral sequence of the Cartan model to relate this to $\chi(M)$. As observed in Section \[Appendix:Section:ModuleStructure\] each page $E_r$ of the spectral sequence naturally is an $R$-module, and the differentials are $R$-linear. We now forget the bigrading of the $E_r$, and keep only the total degree. The differential $d_r$, which was of bidegree $(r,-r+1)$, is then an ordinary differential which increases degree by one. Localizing each page of the spectral sequence, we then obtain ${{\mathbb Z}}_2$-graded vector spaces $E_r$, and the differentials $d_r\colon E_r\to E_r$ become odd endomorphisms. Then, each $E_{r+1}$ is the cohomology of $(E_r,d_r)$, in the category of ${{\mathbb Z}}_2$-graded vector spaces. Applying Lemma \[lem:eulercharofcohomology\] multiple times (noting that there can only be finitely many nontrivial differentials), we compute $$\begin{aligned}
\chi(M) &= \dim_{{\mathbb R}}H^{{\operatorname{even}}}(M) - \dim_{{\mathbb R}}H^{{\operatorname{odd}}}(M)\\
&= \dim_{S^{-1}R} S^{-1}R \otimes H^{{\operatorname{even}}}(M) - \dim_{S^{-1}R} S^{-1}R \otimes H^{{\operatorname{odd}}}(M)\\
&= \dim_{S^{-1}R} S^{-1}E_1^{{\operatorname{even}}} - \dim_{S^{-1}R} S^{-1}E_1^{{\operatorname{odd}}}\\
&= \dim_{S^{-1}R} S^{-1}E_\infty^{{\operatorname{even}}} - \dim_{S^{-1}R} S^{-1}E_\infty^{{\operatorname{odd}}},\end{aligned}$$ where we used Proposition \[prop:E1term\] for the third equality sign. This equals the result of the first chain of equations above, because the ranks of the even and odd parts of $H^*_T(M)$ and $E_\infty$ agree, as we show in Corollary \[Appendix:Cor:RankGleich\].
For any torus action with finitely many fixed points, their number is exactly $\chi(M)$. For example, consider orientable closed surfaces: any nontrivial circle action on the two-sphere has two fixed points, and any nontrivial circle action on the two-dimensional torus has no fixed points at all. Surfaces of higher genus do not admit any nontrivial circle actions.
\[ex:cpnaction\] By Example \[ex:oddcohomzeroeqformal\], any torus action on a manifold $M$ with $H^{{\operatorname{odd}}}(M)=0$ is equivariantly formal. For example, this is the case for ${{\mathbb C}}P^n$.
As a concrete example, consider the $T^2$-action on ${{\mathbb C}}P^2$ given by $$(t_0,t_1)\cdot [z_0:z_1:z_2]:=[t_0z_0:t_1z_1:z_2].$$ Because $\dim H^*({{\mathbb C}}P^2)=3$, we know that if this action has finitely many fixed points, then their number has to be equal to $3$. Indeed, we see that the fixed points are given by $[1:0:0]$, $[0:1:0]$ and $[0:0:1]$.
\[prop:aeqformalfixedpointset\] For any action of a torus $T$ on a compact manifold $M$, we have $\dim H^*(M^T)\leq \dim H^*(M)$. Moreover, the action is equivariantly formal if and only if $\dim H^*(M^T)=\dim H^*(M)$.
By the Borel Localization theorem we have $${\operatorname{rank}}H^*_T(M) = {\operatorname{rank}}H^*_T(M^T) = {\operatorname{rank}}H^*(M^T)\otimes S({\mathfrak{t}}^*) = \dim H^*(M^T).$$ On the other hand we know that ${\operatorname{rank}}H^*_T(M) \leq \dim H^*(M)$: the spectral sequence of the Cartan model has $E_1 = S({\mathfrak{t}}^*)\otimes H^*(M)$, which has rank $H^*(M)$. As submodules and quotients of a module cannot have greater rank than the original, we deduce that ${\operatorname{rank}}(E_\infty)\leq\dim H^*(M)$. Now the first claim follows by Corollary \[Appendix:Cor:RankGleich\].
If the action is equivariantly formal, then $H^*_T(M)$ is, as an $S({\mathfrak{t}}^*)$-module, isomorphic to $H^*(M)\otimes S({\mathfrak{t}}^*)$, hence its rank is equal to $\dim H^*(M)$. If the action is not equivariantly formal, then there exists a nontrivial differential; let $d_r$ be the first of these. As $E_r\cong E_1$ is a free $S({\mathfrak{t}}^*)$-module, it follows that $E_{r+1}$ has rank strictly smaller than $E_r$. As by Corollary \[Appendix:Cor:RankGleich\] the ranks of $H^*_T(M)$ and $E_\infty$ are equal, it follows that $\dim H^*(M^T) = {\operatorname{rank}}H^*_T(M) = {\operatorname{rank}}E_\infty < {\operatorname{rank}}E_1 = \dim H^*(M)$.
\[ex:conjugationeqformal\] Consider the action of a compact, connected Lie group $G$ on itself by conjugation. The action, restricted to a maximal torus $T\subset G$ (of dimension $r={\operatorname{rank}}G$), has $T$ as fixed point set. Therefore we have $2^r=\dim H^*(G^T)$ as the total dimension of the cohomology of the fixed point set. But on the other hand it is known that also $\dim H^*(G) = 2^r$: A classical theorem of Hopf, see e.g. [@FelixOpreaTanre Theorem 1.3.4], states that the de Rham cohomology of $G$ is an exterior algebra on generators of odd degree. The fact that the number of generators equals the rank of $G$ can be proven by various means; see [@FelixOpreaTanre Theorem 3.33] for an argument using rational homotopy theory, or [@Fok] for a more elementary argument using the degree of the squaring map $G\to G;\, g\mapsto g^2$. It follows that the $T$-action on $G$ by conjugation is equivariantly formal.
\[ex:inclusionfixedsetconjugation\] Consider, as a special case of Example \[ex:conjugationeqformal\], the case $G={\mathrm{SU}}(2)$, with maximal torus $S^1\subset {\mathrm{SU}}(2)$. As the action by conjugation is equivariantly formal, the inclusion $S^1\to {\mathrm{SU}}(2)$ induces an injection $$H^*_{S^1}({\mathrm{SU}}(2))\longrightarrow H^*_{S^1}(S^1) = {{\mathbb R}}[u]\otimes H^*(S^1).$$ By equivariant formality we know that, as an ${{\mathbb R}}[u]$-module, $H^*_{S^1}({\mathrm{SU}}(2))$ is generated by two elements in degree $0$ and $3$. As $H^n_{S^1}(S^1)$ is only one-dimensional for $n=0,3$ (in fact for all $n$), this implies that the restriction map induces an isomorphism of ${{\mathbb R}}[u]$-algebras $$H^*_{S^1}({\mathrm{SU}}(2))\cong {{\mathbb R}}[u]\oplus \alpha\cdot u{{\mathbb R}}[u],$$ where $\alpha$ is a generator of $H^1(S^1)$.
\[cor:actiononfixedseteqformal\] Consider an equivariantly formal action of a torus $T$ on a compact manifold $M$, and $H\subset T$ a subtorus. Then the $T$-action on (every component of) $M^H$ is again equivariantly formal.
By \[cor:subgroupeqformal\] the subtorus $H$ acts equivariantly formally on $M$. Thus, by Proposition \[prop:aeqformalfixedpointset\], $\dim H^*(M^H)=\dim H^*(M)$. Now, the fixed point set of the $T$-action on $M^H$ is again $M^T\subset M^H$, and by equivariant formality of the $T$-action on $M$, we have $$\dim H^*(M^T) = \dim H^*(M) = \dim H^*(M^H).$$ Applying Proposition \[prop:aeqformalfixedpointset\] again, we conclude that the $T$-action on $M^H$ is equivariantly formal.
Finally, a torus action on a disconnected manifold is equivariantly formal if and only if the action on every connected component is equivariantly formal.
Corollary \[cor:actiononfixedseteqformal\] in particular says that for an equivariantly formal torus action, every component of a fixed point submanifold $M^H$, where $H\subset T$ is a subtorus, contains a fixed point of the action. Let us give an example of a torus action with fixed points where this property is not satisfied, taken from [@Allday Example 2].
Consider $S^1$, embedded in $S^3 = {\mathrm{SU}}(2)$ as a maximal torus, as well as $S^2 = S^1 \times [0,1]/_\sim$, where we collapse the boundary circles to points. Elements of $S^2$ will thus be written as $[z,t]$, with $z\in S^1$, and $t\in [0,1]$; for $t=0,1$ the elements $[z,t]$ are identical for all $z$. As $S^3$ is simply-connected, we find a homotopy $h\colon S^1\times I\to S^3$ such that $h(z,0)=1$ (the identity element in $S^3$) and $h(z,1)=z$, for all $z\in S^1\subset S^3$.
Define an action of $T^2=S^1\times S^1$ on $M:=S^2\times S^3$ by $$(w_1,w_2)\cdot ([z,t],g) := ([zw_1^{-1},t],h(zw_1^{-1},t)w_2h(z,t)^{-1}gw_2^{-1}).$$ One directly verifies that this really defines an action. On the copy of $S^3$ where $t=0$ we have $$(w_1,w_2) \cdot ([z,0],g) = ([z,0],w_2gw_2^{-1}),$$ so the action is conjugation by $w_2$. On the copy of $S^3$ where $t=1$ we have $$(w_1,w_2) \cdot ([z,1],g) = ([z,1],zw_1^{-1}w_2z^{-1}gw_2^{-1}) = ([z,1],w_1^{-1}w_2gw_2^{-1}),$$ so the action is conjugation by $w_2$, followed by left multiplication with $w_1^{-1}$. We picture the whole action as an interpolation between these two actions.
The fixed point set of the full $T$-action is $M^T\cong S^1$, where $S^1$ is the maximal torus in $S^3$ embedded at $t=0$. The restricted action of the subcircle $H = \{(w^2,w)\} \subset T^2$ is given by $$(w^2,w)\cdot ([z,t],g) = ([zw^{-2},t],h(zw^{-2},t)wh(z,t)^{-1}gw^{-1}).$$ For $t\neq 0,1$ there cannot occur any $H$-fixed points, as $zw^{-2}$ cannot equal $z$ for all $w$. For $t=0$ again only the maximal torus is contained in $M^H$. For $t=1$ we have $$(w^2,w)\cdot ([z,1],g) = ([z,1],w^{-1}gw^{-1}),$$ and because $w^{-1}gw^{-1}=g$ is equivalent to $gwg^{-1}=w^{-1}$ we can only have $w^{-1}gw^{-1}=g$ for all $w\in S^1$ if $g$ is in the normalizer $N_{{\mathrm{SU}}(2)}(S^1)$. This normalizer is the union $S^1\cup A\cdot S^1$, where $A = \left(\begin{matrix}0 & 1 \\ -1 & 0\end{matrix}\right)$. For elements in the centralizer this equality is not satisfied, but it is satisfied for all elements in $A\cdot S^1$, so we have found another circle in the fixed point set. In total, $M^H$ has two connected components, each of which is diffeomorphic to a circle, and only one of them contains $T$-fixed points.
Concerning equivariant formality, this implies that the $H$-action on $M$ is equivariantly formal (as the total dimension of the cohomology $H^*(M^H)$ is $4$, which is the same as the dimension of $H^*(M)$), but the whole $T$-action is not.
Cohomology of homogeneous spaces {#sec:cohomhomspaces}
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In this section we will apply equivariant cohomology theory to obtain information on the cohomology of homogeneous spaces $G/H$, mostly for the case that the ranks of $G$ and $H$ are equal.
\[prop:eqcohomtransitive\] Given any two compact, connected Lie groups $H\subset G$, the equivariant cohomology of the $G$-action on $G/H$ by left multiplication is given by $$H^*_G(G/H) \cong S({\mathfrak{h}}^*)^H;$$ its algebra structure $S({\mathfrak{g}}^*)^G\to H^*_G(G/H) = S({\mathfrak{h}}^*)^H$ is given by restriction of polynomials.
Applying Theorem \[thm:eqcohomlocallyfreeactions\], or rather the generalization described in Remark \[rem:commutingactionprinciple\], twice gives isomorphisms $$H^*_G(G/H) \cong H^*_{G\times H}(G) \cong H^*_H({\mathrm{pt}}) = S({\mathfrak{h}}^*)^H$$ of graded ${{\mathbb R}}$-algebras. One needs to confirm that the $S({\mathfrak{g}}^*)^G$-algebra structure is as claimed. To this end, we consider these isomorphisms on the level of equivariant differential forms: $$(S({\mathfrak{g}}^*)\otimes \Omega(G/H))^G\longrightarrow (S({\mathfrak{g}}^*)\otimes S({\mathfrak{h}}^*)\otimes \Omega(G))^{G\times H} \longleftarrow S({\mathfrak{h}}^*)^H$$ where both maps are induced by the natural projection maps. On order to understand where a $G$-invariant polynomial on ${\mathfrak{g}}$ is mapped to on the level of cohomology, one needs a chain homotopy inverse of the map on the right, the so-called Cartan map, which is described explicitly in [@GuilleminSternberg Theorem 5.2.1] or [@Meinrenken Section 7]. One needs to fix the (in this case unique) connection one-form $\theta$ of the principal $G$-bundle $G\to {\mathrm{pt}}$, which is essentially given by the Maurer-Cartan form of $G$ (but note that $G$ acts by left multiplication on $G$ here). Then, for $Y\in {\mathfrak{h}}$ acting on $G$ from the right, we compute $$\theta_g(\overline{Y}_g) = \theta_g(dl_g(\overline{Y}_e)) = \theta_g(dr_g(\overline{{\operatorname{Ad}}_gY}_e)) = -{\operatorname{Ad}}_gY,$$ where $l_g$ and $r_g$ denote left and right multiplication with $g\in G$, respectively. Thus, the $H$-equivariant curvature $2$-form $F_H^\theta = d_H\theta + \frac12[\theta,\theta]\in C^2_H(G)\otimes {\mathfrak{g}}$ is given by $$F_H^\theta(Y)(g) = {\operatorname{Ad}}_gY,$$ for every $Y\in {\mathfrak{h}}$ and $g\in G$, because $\theta$ satisfies $d\theta + \frac12 [\theta,\theta] = 0$. Thus, for any $G$-invariant polynomial $f\in S({\mathfrak{g}}^*)^G$, replacing the ${\mathfrak{g}}$-variable by $F_H^\theta$ is the same as restricting the polynomial to ${\mathfrak{h}}$.
In [@DufloKumarVergne Théorème 24], the proposition is proved under relaxed conditions. Also, just as it is the case with Theorem \[thm:eqcohomlocallyfreeactions\], the proof is much easier in the Borel model. We have $$EG\times_G G/H = EG/H = BH,$$ inducing an isomorphism $H^*_G(G/H) = S({\mathfrak{h}}^*)^H$. When identifying $EG\times_G G/H = BH$, the projection map $EG\times_G G/H\to BG$ becomes the natural map $BH = EG/H \to EG/G = BG$, thus showing the claim about the algebra structure.
\[thm:homogeneousspacesequalrank\] For a homogeneous space $G/H$, where $G$ is a compact, connected Lie group and $H\subset G$ a connected closed subgroup, the $G$-action on $G/H$ is equivariantly formal if and only if ${\operatorname{rank}}G = {\operatorname{rank}}H$. In this case we have an ${{\mathbb R}}$-algebra isomorphism $$H^*(G/H) \cong \frac{S({\mathfrak{h}}^*)^H}{(S^+({\mathfrak{g}}^*)^G)}$$ and $H^*(G/H)$ vanishes in odd degrees.
If the $G$-action is equivariantly formal, then also a maximal torus in $G$ acts in an equivariantly formal fashion, by Corollary \[cor:subgroupeqformal\]. But the action of a maximal torus in $G$ on $G/H$ by left multiplication can only have fixed points if the ranks of $H$ and $G$ are equal.
Conversely, we consider first the case that $H=T$ is a maximal torus of $G$. In this case $G/T$ admits a CW structure with only even-dimensional cells, by the classical Bruhat decomposition – see e.g. [@Mare Section 7] (for a nice overview) and references therein, e.g. [@Kumar Theorems 5.1.3 and 5.1.5]. Thus, the odd cohomology of $G/T$ vanishes. By Example \[ex:oddcohomzeroeqformal\] the $G$-action on $G/T$ is equivariantly formal, and combining the description of the equivariant cohomology in Proposition \[prop:eqcohomtransitive\] with Theorem \[thm:bigthmequivformal\] we obtain $$H^*(G/T) \cong \frac{S({\mathfrak{t}}^*)}{(S^+({\mathfrak{g}}^*)^G)}.$$ For a general equal-rank homogeneous space $G/H$ we claim that the fibration $$H/T \longrightarrow G/T \longrightarrow G/H$$ satisfies that the map $H/T\to G/T$ induces a surjection in de Rham cohomology. Indeed, this map is the natural projection $$\frac{S({\mathfrak{t}}^*)}{(S^+({\mathfrak{g}}^*)^G)}\longrightarrow \frac{S({\mathfrak{t}}^*)}{(S^+({\mathfrak{h}}^*)^H)}$$ which is clearly surjective. Thus, the Leray-Hirsch theorem implies that the cohomology of $G/H$ also vanishes in odd degrees. Thus, in the same way as for $G/T$, the $G$-action on $G/H$ is equivariantly formal, and the desired description of the cohomology of $G/H$ follows.
There are various other ways to obtain this theorem, without using the Bruhat decomposition. Given a homogeneous space $G/H$ of equal rank, all isotropy groups of the $G$-action on $H$ have the same rank as that of $G$. For such actions equivariant formality is automatic, see [@GR Proposition 3.7]. Then, Proposition \[prop:eqcohomtransitive\] and Theorem \[thm:bigthmequivformal\] imply the description of the cohomology ring. The vanishing of the odd cohomology then follows directly from the fact $S({\mathfrak{h}}^*)^H$ is concentrated in even degrees, or equally directly from Proposition \[prop:equivformalfinitefixedpointset\], because by Lemma \[lem:homspacefixedpoints\] below, the equivariantly formal action of a maximal torus $T\subset G$ on $G/H$ has finite fixed point set.
Alternatively, one may also argue entirely algebraically and use that $S({\mathfrak{t}}^*)$ is a free module over $S({\mathfrak{g}}^*)^G$ (see e.g. [@Kane Section 18.3]) to prove equivariant formality of the $G$-action.
\[rem:modulestructureG/H\] By Corollary \[cor:orbitspacedings\] we have, for any connected closed subgroup $H\subset G$ of a compact, connected Lie group $G$ of equal rank, that $H^*_H(G) = H^*(G/H)$, where $H$ acts (freely) on $G$ by right multiplication. We claim that the $S({\mathfrak{h}}^*)^H$-algebra structure of this equivariant cohomology $$S({\mathfrak{h}}^*)^H \longrightarrow H^*_H(G) \cong H^*(G/H) \cong \frac{S({\mathfrak{h}}^*)^H}{(S^+({\mathfrak{g}}^*)^G)}$$ is given by the canonical projection map. To see this, we consider the following commutative diagram, whose upper horizontal isomorphisms are those from the proof of Proposition \[prop:eqcohomtransitive\], and whose vertical maps are given by restriction of the acting group: $$\xymatrix{
S({\mathfrak{h}}^*)^H \ar[rrd] \ar[r]^{\cong} & H^*_H({\mathrm{pt}}) \ar[r]^{\cong} & H^*_{G\times H}(G) \ar[r]^{\cong} \ar[d] & H^*_G(G/H) \ar[d]\\
& & H^*_H(G) \ar[r]^\cong & H^*(G/H) \ar[r]^\cong & \frac{S({\mathfrak{h}}^*)^H}{(S^+({\mathfrak{g}}^*)^G)}
}$$ Note that the square in the middle commutes because the inverses of the two horizontal maps are induced by the canonical projection $G\to G/H$. The claim follows because traversing the diagram from the top left to the bottom right via the upper path results in the canonical projection map.
\[cor:homspaceequivalences\] Consider a homogeneous space $G/H$, where $H\subset G$ are compact, connected Lie groups. Then $\chi(G/H)\geq 0$. Moreover, the following conditions are equivalent:
- ${\operatorname{rank}}G = {\operatorname{rank}}H$
- $\chi(G/H) >0$
- $H^{{\operatorname{odd}}}(G/H)=0$.
In Theorem \[thm:homogeneousspacesequalrank\] we showed that for homogeneous spaces with ${\operatorname{rank}}G = {\operatorname{rank}}H$ the odd degree cohomology vanishes, and hence also the Euler characteristic is positive.
Let us show that whenever ${\operatorname{rank}}G > {\operatorname{rank}}H$ the Euler characteristic is zero. Then, as we always have cohomology in degree zero, the odd cohomology cannot vanish either. To see this, we construct a circle action on $G/H$ without fixed points, and apply Theorem \[thm:eulercharfixedpoints\]: We choose a maximal torus $T_H\subset H$, as well as a maximal torus $T_G\subset G$ containing $T_H$. We can choose a circle $S^1\subset T_G$ which is not $G$-conjugate to a subgroup of $H$. (If this was not the case, then choose a sequence of subcircles $\{\exp(tX_n)\}$, with $X_n\to X\in {\mathfrak{g}}$, such that $\{\exp(tX)\}$ is dense in $G$. If there existed $g_n$ such that ${\operatorname{Ad}}_{g_n}X_n\in {\mathfrak{h}}$, then we could find a subsequence, converging to $g\in G$, and this element would satisfy ${\operatorname{Ad}}_gX\in {\mathfrak{h}}$. But then, by continuity, $gGg^{-1}\subset H$, a contradiction.) Then, this circle cannot fix any point $gH\in G/H$, as the $G$-isotropy of this point is $gHg^{-1}$ – if it fixed $gH$, then it would be conjugate to a subgroup of $H$.
We thus have found a circle action without fixed points, which shows that the Euler characteristic is zero.
We now neglect the ring structure of the cohomology of equal-rank homogeneous spaces obtained in Theorem \[thm:homogeneousspacesequalrank\], and concentrate on their Betti numbers. We first obtain a formula for the total Betti number in Proposition \[prop:dimGH\], and then describe explicitly the Poincaré polynomials in Proposition \[prop:homspaceequalrankbetti\].
\[lem:homspacefixedpoints\] Consider a homogeneous space $G/H$, where $H$ and $G$ are compact, connected Lie groups of equal rank, and $T\subset H$ a maximal torus. Then the inclusion $N_G(T)\to G$ induces an injection $$W(G)/W(H) \cong N_G(T)/N_H(T) \longrightarrow G/H$$ whose image is precisely the fixed point set of the $T$-action on $G/H$.
We observe that an element $gH\in G/H$ is fixed by $T$ if and only if $g^{-1}Tg\subset H$, i.e., by the conjugacy of maximal tori in $H$, if and only if there exists $h\in H$ such that $h^{-1}g^{-1}Tgh = T$. As $ghH = gH$, this means that the $T$-fixed point set is precisely the image of the composition $N_G(T) \to G \to G/H$ of the natural inclusion with the natural projection.
\[prop:dimGH\] For an equal-rank homogeneous space $G/H$, we have $$\label{eq:dimcohomGH}
\dim H^*(G/H) = \frac{|W(G)|}{|W(H)|}.$$
This follows from Proposition \[prop:aeqformalfixedpointset\] because the action of a maximal torus $T\subset H$ is equivariantly formal and has precisely $\frac{|W(G)|}{|W(H)|}$ fixed points.
The equality $\dim H^*(G/T)=|W(G)|$ follows also because the CW structure on $G/T$ given by the Bruhat decomposition has precisely $|W(G)|$ cells. Proposition \[prop:dimGH\] is then immediate from the observations on the fibration $H/T\to G/T\to G/H$ given in the proof of Theorem \[thm:homogeneousspacesequalrank\].
\[ex:totalBettiGrass\] For the complex Grassmannian of $k$-planes in ${{\mathbb C}}^n$
$${\mathrm{Gr}}_k({{\mathbb C}}^n) = {{\raisebox{.2em}{${\mathrm{U}}(n)$}}\ \!\!\big/\!\!\
{\raisebox{-.2em}{${\mathrm{U}}(k)\times {\mathrm{U}}(n-k)$}}}$$ we obtain $$\dim H^*({\mathrm{Gr}}_k({{\mathbb C}}^n)) = \frac{|W(U(n))|}{|W(U(k))| \cdot |W(U(n-k))|} = \frac{n!}{k!(n-k)!} = {n \choose k}.$$
\[prop:homspaceequalrankbetti\] Consider a homogeneous space $G/H$ of compact, connected Lie groups $H\subset G$ of equal rank $r$. If $$S({\mathfrak{g}}^*)^G \cong {{\mathbb R}}[\sigma_1,\ldots,\sigma_r]$$ and $$S({\mathfrak{h}}^*)^H \cong {{\mathbb R}}[\psi_1,\ldots,\psi_r]$$ with $\deg \sigma_i = p_i$ and $\deg \psi_i = q_i$ (usual degree of polynomials), then $$P_t(H^*(G/H)) = \prod_{i=1}^r \frac{1-t^{2p_i}}{1-t^{2q_i}},$$ where $P_t(H^*(G/H)) = \sum_{n=0}^{\dim G/H} b_n(G/H) t^n$ is the Poincaré polynomial of $G/H$.
In Theorem \[thm:homogeneousspacesequalrank\] we observed that the transitive $G$-action on $G/H$ is equivariantly formal. Using Proposition \[prop:eqcohomtransitive\] and Theorem \[thm:bigthmequivformal\] we conclude that $$\label{eq:freemoduleG/T}
S({\mathfrak{h}}^*)^H\cong H^*_G(G/H) \cong S({\mathfrak{g}}^*)^G \otimes H^*(G/H);$$ here we need these isomorphisms only as one of graded vector spaces (but note that as elements of equivariant cohomology, the $\sigma_i$ and $\psi_i$ have twice the degree they inherited from the polynomial rings). This equality helps to compute the Betti numbers of $G/H$: the Poincaré series of $S({\mathfrak{h}}^*)^H$ and $S({\mathfrak{g}}^*)^H$ (for a graded vector space $V = \bigoplus_{n\geq 0} V_n$ with $\dim V_n<\infty$ for all $n$, this is the formal power series $\sum_{n=0}^\infty t^n \dim V_n$) are $$P_t(S({\mathfrak{h}}^*)^H) = \prod_{i=1}^r \frac{1}{(1-t^{2q_i})},\qquad P_t(S({\mathfrak{g}}^*)^G) = \prod_{i=1}^r \frac{1}{(1-t^{2p_i})}.$$ Then implies that $$P_t(S({\mathfrak{h}}^*)^H) = P_t(S({\mathfrak{g}}^*)^G)\cdot P_t(H^*(G/H)),$$ so that $$P_t(H^*(G/H)) = \prod_{i=1}^r \frac{1-t^{2p_i}}{1-t^{2q_i}}.$$
\[ex:bettiG/T\] In the special case that $H=T$ is a maximal torus of $G$, the cohomology $H^*(G/T)$ is, as an ${{\mathbb R}}$-algebra, generated by the elements in $H^2(G/T)$. The Poincaré polynomial is $$P_t(H^*(G/T)) = \prod_{i=1}^r \frac{1-t^{2p_i}}{1-t^2} = \prod_{i=1}^r (1+t^2+t^4 \cdots + t^{2p_i-2}).$$ In particular, the total Betti number of $G/T$ is $$\dim H^*(G/T) = P_1(H^*(G/T)) = \prod_{i=1}^r p_i.$$ Comparing this with Equation , i.e., $\dim H^*(G/T) = |W(G)|$, we obtain the following general formula for the order of the Weyl group of $G$ in terms of the generators of the cohomology of $G$: $$|W(G)| = \prod_{i=1}^r p_i.$$
Consider the complex Grassmannian ${\mathrm{Gr}}_k({{\mathbb C}}^n)$ of $k$-planes in ${{\mathbb C}}^n$ as in Example \[ex:totalBettiGrass\]. In Example \[ex:invpolyUn\] we computed that for $G={\mathrm{U}}(n)$ we have $S({\mathfrak{g}}^*)^G = {{\mathbb R}}[\sigma_1,\ldots,\sigma_n]$, where $\deg \sigma_i = i$. Thus, Proposition \[prop:homspaceequalrankbetti\] gives $$\begin{aligned}
P_t({\mathrm{Gr}}_k({{\mathbb C}}^n)) &= \frac{(1-t^2)\cdots (1-t^{2n})}{(1-t^2)\cdots (1-t^{2k})(1-t^2)\cdots (1-t^{2(n-k)})}\\
&=\frac{(1-t^{2k+2})\cdots (1-t^{2n})}{(1-t^2)\cdots (1-t^{2(n-k)})}.\end{aligned}$$
For more information on the cohomology of homogeneous spaces $G/H$, where ${\operatorname{rank}}G > {\operatorname{rank}}H$, we only refer to the literature, e.g. [@GHV].
Computing $H^*(M)$ via $H^*_T(M)$ {#sec:HMHTM}
=================================
In Theorem \[thm:bigthmequivformal\] we have seen that for an equivariantly formal $G$-action on $M$ we have an isomorphism of ${{\mathbb R}}$-algebras $$H^*(M) \cong \frac{H^*_G(M)}{S^+({\mathfrak{g}}^*)^G\cdot H^*_G(M)}.$$ This means that whenever we know the equivariant cohomology $H^*_G(M)$ as an $S({\mathfrak{g}}^*)^G$-algebra, we can use this isomorphism to compute the ordinary cohomology $H^*(M)$.
For an equivariantly formal torus action, the Borel localization theorem \[thm:borellocalization\] states that the restriction map $$H^*_T(M) \longrightarrow H^*_T(M^T) = S({\mathfrak{t}}^*)\otimes H^*(M^T)$$ is injective, so one can try to compute $H^*_T(M)$ by understanding its image under this map. This is achieved by the Chang–Skjelbred Lemma, which describes the image only in terms of the $1$-skeleton $M_1 := \{p\in M\mid \dim T\cdot p\leq 1\}$ of the action, see [@ChangSkjelbred Lemma 2.3]. The original formulation used the Borel model; as $M_1$ is not a manifold, the formulation in terms of the Cartan model reads slightly differently – see [@GuilleminSternberg Section 11.5] for the proof.
\[thm:changskjelbred\] The image of the natural restriction map $i^*\colon H^*_T(M) \to H^*_T(M^T)$ is given by $$\label{eq:changintersection}
\bigcap_{H\subset T} i_H^*(H^*_T(M^H)),$$ where $H$ runs through all codimension-one subtori of $T$ and $i_H\colon M^T\to M^H$ is the inclusion.
Note that for almost all codimension-one subtori $H\subset T$ we have $M^H = M^T$; these $H$ are irrelevant for the intersection. The only relevant groups $H$ are the connected components of those isotropy groups of the $T$-action that are of codimension one – of these there are only finitely many. The one-skeleton $M_1$ of the action is the union of all the $M^H$, where $H$ runs through the codimension-one subtori as above.
\[ex:oneskeletoncp2\] Consider the $T^2$-action on ${{\mathbb C}}P^2$ from Example \[ex:cpnaction\]. The orbit space of this action is a triangle. The one-skeleton of the action is the preimage of the boundary of this triangle under the projection to the orbit space. It is the union of three $2$-spheres, any two of which meet in a single point.
One important special case in which this theorem yields explicitly computable results is that of so-called GKM actions, named after a paper by Goresky, Kottwitz, and MacPherson [@GKM]. There, one assumes that the structure of the one-skeleton is as simple as possible:
\[defn:GKM\] We call an action of a torus $T$ on a compact, connected manifold $M$ a *GKM action* if the following conditions are satisfied:
1. The action is equivariantly formal.
2. The fixed point set of the action is finite.
3. The one-skeleton $M_1$ is a finite union of $T$-invariant two-spheres.
Given the second condition, we know that the first one is equivalent to demanding that the odd cohomology groups of $M$ vanish, see Proposition \[prop:equivformalfinitefixedpointset\]. Easy examples of GKM actions are the standard circle action on $S^2$, or the $T^2$-action on ${{\mathbb C}}P^2$ (see Example \[ex:oneskeletoncp2\]). These can be generalized to the following class of examples:
All toric symplectic manifolds are GKM. Indeed, toric symplectic manifolds have vanishing odd cohomology groups [@Audin Theorem VII.3.5] and finite fixed point set, and at each fixed point the weights of the isotropy representation form a basis of ${\mathfrak{t}}^*$: if $M$ is $2n$-dimensional, then there are precisely $n$ weights of the isotropy representation at any given fixed point, which have to be linearly independent, as otherwise the common kernel of the weights would determine a positive-dimensional subtorus acting trivially on $M$.
Let $p\in M^T$ be a fixed point of a GKM action. Then the isotropy representation at $p$ decomposes into two-dimensional irreducible subrepresentations. If $\alpha$ is a weight of the isotropy representation – which is a linear form on ${\mathfrak{t}}$, well-defined up to sign – with weight space $V_\alpha$, and $T_\alpha\subset T$ the subtorus with Lie algebra $\ker \alpha$, then $V_\alpha$ is tangent to $M^{T_\alpha}\subset M_1$. The condition that $M_1$ is a finite union of two-dimensional submanifolds, is equivalent to the condition that the weights of the isotropy representation, at any fixed point, are pairwise linearly independent. Thus, for a GKM action on a manifold of dimension $2n$, in any given fixed point there meet precisely $n$ invariant two-spheres.
To any GKM action one associates, as follows, a labelled graph $\Gamma$, called the *GKM graph* of the action: the vertices $V(\Gamma)$ are given by the fixed points of the action, and we draw an edge (i.e., an element of the edge set $E(\Gamma)$) for any invariant $2$-sphere connecting two fixed points. The argument above shows that this graph, for $M$ of dimension $2n$, is $n$-valent. Additionally, we label the edge as follows: the tangent space of an invariant two-sphere in one of the two fixed points is a two-dimensional invariant submodule of the isotropy representation, and there is a codimension-one subtorus $H\subset T$ that acts trivially on it. We put any nonzero linear form $\alpha\in {\mathfrak{t}}^*$ that vanishes on ${\mathfrak{h}}$ as a label of the corresponding edge.
A classical result of Atiyah [@Atiyah2] and Guillemin–Sternberg [@GuiSte] states that the image of the momentum map $\mu\colon M\to {\mathfrak{t}}^*$ of an Hamiltonian torus action on a symplectic manifold $M$ is a convex polytope. For a toric symplectic manifold $M$, the dimension of an orbit $T\cdot p$ is precisely the smallest dimension of a face containing $\mu(p)$. It follows that the GKM graph of a toric symplectic manifold is precisely the one-skeleton of the polytope $\mu(M)$.
Consider a homogeneous space $G/H$, with ${\operatorname{rank}}G = {\operatorname{rank}}H$, equipped with the action of a maximal torus $T\subset H$ by left multiplication. We showed in Section \[sec:cohomhomspaces\] that this action is equivariantly formal, and that the fixed point set of this action is given by the finite set $W(G)/W(H)$. In [@GuilleminHolmZara] it was observed that the $T$-action is GKM, and the GKM graph was determined explicitly in terms of the root systems of $G$ and $H$ (see [@GuilleminHolmZara Theorem 2.4]).
The equivariant cohomology of a GKM action is encoded in the GKM graph:
Consider a GKM action of a torus $T$ on a compact, connected orientable manifold $M$. Then $$\begin{aligned}
&H^*_T(M) \cong \Big\{(f_p)\in \bigoplus_{p\in M^T} S({\mathfrak{t}}^*) \Bigm| f_p-f_q \in (\alpha) \text{ if there is an edge from } p \text{ to } q \text{ labelled } \alpha\Big\}.\end{aligned}$$ Here, $(\alpha)$ denotes the principal ideal generated by $\alpha$.
By Theorem \[thm:changskjelbred\] the image of the (injective) natural restriction map $H^*_T(M)\to H^*_T(M^T)$ is $$\bigcap_H i_H^*(H^*_T(M^H)),$$ where $H$ runs through the codimension one subgroups of $T$, and $i_H\colon M^T\to M^H$ is the inclusion. As observed before, under our assumptions each component $N$ of one of the $M^H$ is either a single fixed point or a two-sphere $S^2$ with an action of $T/H \cong S^1$. To compute the equivariant cohomology of $N$, we generalize Example \[ex:S1aufS2\] slightly: take $N = U\cup V$, where $U$ and $V$ are $T$-equivariantly homotopy equivalent to a fixed point. (Modulo the ineffective kernel, $N$ is equivariantly diffeomorphic to $S^2$ with the standard circle action. This follows from the theory of cohomogeneity-one actions, as such actions are determined by their group diagram.) So $H^*_T(U) = H^*_T(V) = S({\mathfrak{t}}^*)$. Moreover, $U\cap V$ is homotopy equivalent to an invariant circle, whose isotropy Lie algebra is ${\mathfrak{h}}$, so $H^*_T(U\cap V) = S({\mathfrak{h}}^*)$. We thus obtain an exact sequence $$\cdots \longrightarrow H^*_T(N) \longrightarrow S({\mathfrak{t}}^*)\oplus S({\mathfrak{t}}^*) \overset{\varphi}\longrightarrow S({\mathfrak{h}}^*)\longrightarrow \cdots,$$ where the map $\varphi$ is given by $\varphi(f,g) = f|_{\mathfrak{h}}- g|_{\mathfrak{h}}$. We thus obtain that $$H^*_T(N) \cong \{(f,g)\in S({\mathfrak{t}}^*)\oplus S({\mathfrak{t}}^*)\mid f|_{\mathfrak{h}}= g|_{\mathfrak{h}}\}.$$ Now, the condition that $f|_{\mathfrak{h}}=g|_{\mathfrak{h}}$ is equivalent to the condition that the polynomial $f-g$ is in the kernel of the restriction map $S({\mathfrak{t}}^*)\to S({\mathfrak{h}}^*)$. This kernel is a principal ideal, generated by any nonzero linear form that vanishes on ${\mathfrak{h}}$. This is precisely the relation prescribed by the edge corresponding to $N$.
\[ex:cp2gkm\] Consider the action of $T^2$ on ${{\mathbb C}}P^2$. We already understand the one-skeleton of the action, which consists of three invariant two-spheres. They are given by $\{[z:w:0]\}$, $\{[z:0:w]\}$ and $\{[0:z:w]\}$, whose isotropy groups are $\{(t,t)\mid t\in S^1\}$, $\{1\}\times S^1$, and $S^1\times \{1\}$, respectively. Choosing $\{u,v\}$ as the dual basis to the standard basis of ${\mathfrak{t}}\cong {{\mathbb R}}^2$, the labels of the graph (which is a triangle) are given by $u$, $v$, and $u-v$.
{width="117pt"}
The equivariant cohomology is thus given by $$H^*_{T^2}({{\mathbb C}}P^2) \cong \Big\{(f,g,h)\in {{\mathbb R}}[u,v]^3\Bigm| f-g\in (u),\, f-h\in (v),\, g-h\in (u-v)\Big\},$$ with the $S({\mathfrak{t}}^*)$-algebra structure induced from the equivariant cohomology of the fixed point set, i.e., componentwise multiplication.
From this, we can now determine the graded ring structure of the ordinary cohomology of ${{\mathbb C}}P^2$. One checks that $$(1,1,1),\qquad (v,v-u,0),\qquad (uv,0,0)$$ are ${{\mathbb R}}[u,v]$-module generators of the equivariant cohomology (which have degree $0,2,4$ as predicted by Theorem \[thm:bigthmequivformal\]). To understand the ring structure we have to multiply $$\begin{aligned}
(v,v-u,0)\cdot (v,v-u,0) &\equiv (v^2,(v-u)^2,0) \\
&\equiv (v^2,v^2-2uv+u^2,0) - v(v,v-u,0) + u(v,v-u,0)\\
&\equiv (uv,0,0)\end{aligned}$$ where we compute modulo $S^+({\mathfrak{t}}^*)\cdot H^*_{T^2}({{\mathbb C}}P^2)$, i.e., in the quotient $H^*_{T^2}({{\mathbb C}}P^2)/S^+({\mathfrak{t}}^*)\cdot H^*_{T^2}({{\mathbb C}}P^2)$. Also, $(v,v-u,0)^3 \equiv 0$. It follows that $$H^*({{\mathbb C}}P^2) \cong {{\mathbb R}}[\omega]/(\omega^3),$$ where $\omega$ is of degree $2$ (which we of course knew before).
A detailed introduction to GKM theory with many explicit computations can be found in [@Tymoczko].
One can not only apply GKM theory to concrete computations, but also to obtain structural results on certain classes of actions. For instance, in [@GW] it was shown that all known examples of even-dimensional positively curved Riemannian manifolds admit isometric GKM actions, and described their GKM graphs. The graphs that occur are simplices and the complete bipartite graph $K_{3,3}$, with possibly all edges doubled or quadrupled. As an example, see Figure \[figrp\] (which is taken from [@GW]) for the GKM graph of the action of the maximal torus of ${\mathrm{Spin}}(8)$ on $F_4/{\mathrm{Spin}}(8)$ by left multiplication.
![GKM graph of $F_4/{\mathrm{Spin}}(8)$[]{data-label="figrp"}](F4Spin8.eps){width="117pt"}
Restricting to GKM$_3$-actions (i.e., actions for which the two-skeleton of the action is the union of four-dimensional submanifolds) one obtains the following theorem.
Let $M$ be a compact, connected, positively curved, orientable Riemannian manifold. If $M$ admits an isometric GKM$_3$ torus action, then $M$ has the real cohomology ring of a compact rank one symmetric space.
To prove this theorem we determined all possible GKM graphs under the given curvature assumption, using the classification of four-dimensional positively curved $T^2$-manifolds by Grove and Searle [@GroveSearle].
Finally, we mention that GKM theory allows for various generalizations. One possibility to generalize is to allow a nonisolated fixed point set. This was considered in the context of Hamiltonian actions on symplectic manifolds [@GuilleminHolm], and for equivariantly formal torus actions with one-dimensional fixed point set [@Chen]. In He’s paper an important feature of the class of actions he considers is that the one-skeleton of the action is the union of (three-dimensional) submanifolds each containing an arbitrary number of fixed point components, contrary to the the classical case in which the invariant two-spheres always contain exactly two fixed points. GKM theory for actions without fixed points was considered in [@GNT], for a certain class of Cohen-Macaulay torus actions (see Section \[sec:CohenMacaulay\] below). Instead of the one-skeleton of the action one describes the equivariant cohomology of the action in terms of the $b+1$-skeleton $M_{b+1}$ of the action, where $b$ is the lowest occurring dimension of an orbit. The class of actions considered in [@GNT] has the property that $M_{b+1}$ is the union of submanifolds, each containing exactly two components of $M_b$. It is also possible to generalize GKM theory to actions of arbitrary compact Lie groups [@GM], as well as to possibly infinite-dimensional equivariant cell complexes [@HHH]. One can also abstract from torus actions on manifolds and consider GKM graphs as objects of independent interest (see e.g. [@GuilleminZara]).
Algebraic generalizations of equivariant formality {#sec:CohenMacaulay}
==================================================
An important property of equivariant formality of a torus action is that the restriction map $$\label{eq:restrictionmapCM}
H^*_T(M) \longrightarrow H^*_T(M^T)$$ is injective. Because the kernel of this map is the torsion submodule by the Borel Localization Theorem \[thm:borellocalization\], this property is in fact equivalent not to the freeness of $H^*_T(M)$ but to its torsion-freeness. One can therefore ask the question how different equivariantly formal actions are from actions whose equivariant cohomology is torsion-free.
It was shown in [@Allday] that for smooth actions of at most two-dimensional tori, torsion-freeness of the equivariant cohomology is equivalent to equivariant formality. The first example of a non-equivariantly formal torus action whose equivariant cohomology is torsion-free was given in [@FranzPuppe].
Recently, Allday–Franz–Puppe interpolated between torsion-freeness and freeness of the equivariant cohomology, by using the notion of syzygies [@AlldayFranzPuppe]: already Atiyah [@Atiyah Lecture 7] and Bredon [@Bredon Main Lemma] observed that equivariantly formal actions satisfy a stronger property than the Chang-Skjelbred Lemma, Theorem \[thm:changskjelbred\], namely the exactness of the so-called *Atiyah-Bredon sequence* $$0 \to H^*_T(M) \to H^*_T(M^T) \to H^{*+1}_T(M_1,M^T) \to \cdots \to H^{*+k}_T(M_k,M_{k-1})\to 0,$$ where $M_i$ is the union of the $T$-orbits of dimension at most $i$. Here, we use relative equivariant cohomology in the Borel model (cf. Remark \[rem:Borelmodel\]) to give meaning to the cohomologies occurring in the sequence. In [@FranzPuppe25] it was shown that exactness of this sequence is even equivalent to equivariant formality. More precise information was given in [@AlldayFranzPuppe], where the authors showed that exactness of this sequence at the first $i$ positions is equivalent to $H^*_T(M)$ being an $i$th syzygy. Examples of torus actions whose equivariant cohomologies vary among all possible syzygy orders are given by so-called big polygon spaces [@FranzBig].
A different way in which one can generalize the notion of equivariant formality is that of a Cohen-Macaulay action, introduced in [@GT]. The relevance of the Cohen-Macaulay property was already observed in [@Atiyah].
We say that an action of a compact Lie group $G$ on a compact manifold $M$ is *Cohen-Macaulay* if $H^*_G(M)$ is a Cohen-Macaulay module over $S({\mathfrak{g}}^*)^G$.
To motivate this notion, let us restrict to the action of a torus $T$. (Note as well that the Cohen-Macaulay property for the action of a compact, connected Lie group $G$ is equivalent to that of the restriction of the action to a maximal torus, see [@GR Proposition 2.9].) It turns out that the Cohen-Macaulay property is equivalent to the exactness of an Atiyah-Bredon-type sequence $$0 \to H^*_T(M) \to H^*_T(M_b) \to H^{*+1}_T(M_{b+1},M_b) \to \cdots \to H^{*+k}_T(M_k,M_{k-1})\to 0,$$ where $b$ is the lowest occurring orbit dimension, see [@GT] or [@FranzPuppe2 Section 5]. In particular, the equivariant cohomology algebra, for Cohen-Macaulay actions, is computable as for equivariantly formal actions, by determining the image of the restriction map $H^*_T(M)\to H^*_T(M_b)$. Note however that the natural map $H^*_T(M)\to H^*(M)$ is not surjective for Cohen-Macaulay actions, which is why this notion is less useful for computing the ordinary cohomology of a $T$-manifold (however, one may divide both the acting torus and the manifold by a locally freely acting $b$-dimensional subtorus to obtain an equivariantly formal action for which the considerations of Section \[sec:HMHTM\] hold true).
For torus actions with fixed points, or more generally for $G$-actions with points with maximal isotropy rank the notion of being Cohen-Macaulay coincides with equivariant formality [@GR Proposition 2.5].
Many geometrically important classes of actions are Cohen-Macaulay. Besides the already known classes of equivariantly formal actions, like Hamiltonian actions on symplectic manifolds, see Example \[ex:morsebotteqformal\], they include:
1. $G$-actions for which all points have the same isotropy rank [@GR Corollary 4.3], in particular, transitive $G$-actions.
2. Actions of cohomogeneity one [@GM1]. One can also determine the multiplicative structure of the equivariant cohomology of cohomogeneity one manifolds explicitly, see [@CGHM]. Note that cohomogeneity-two actions are not necessarily Cohen-Macaulay; an easy example is a $T^2$-action on $(S^1\times S^3)\#(S^2\times S^2)$ with exactly $2$ fixed point (see [@OrlikRaymond] and [@GM1 Example 4.3]).
3. The action of the closure of the Reeb flow of a $K$-contact manifold [@GNT].
4. Hyperpolar actions on symmetric spaces [@GHM].
Actions on foliated manifolds
=============================
The main algebraic ingredient of the construction of the Cartan model is the structure of a $G$-differential graded algebra on $\Omega(M)$ induced by a $G$-action on $M$. That is, the $G$-action induces contraction operators $i_X$ and Lie derivative operators $L_X$, for every $X\in {\mathfrak{g}}$, on $\Omega(M)$. It was Cartan’s original approach to abstract from the concrete geometric setting, and consider equivariant cohomology of abstract $G$-differential graded algebras, see [@Cartan1 Section 4].
In [@GT2] this was applied this to foliated manifolds, using the notion of transverse action from [@AlvarezLopez Section 2]:
A *transverse action* of a finite-dimensional Lie algebra ${\mathfrak{g}}$ on a foliated manifold $(M,{{\mathcal F}})$ is a Lie algebra homomorphism $${\mathfrak{g}}\longrightarrow l(M,{{\mathcal F}}).$$
Here, $l(M,{{\mathcal F}})= L(M,{{\mathcal F}})/\Xi({{\mathcal F}})$ is the Lie algebra of *transverse fields*: $L(M,{{\mathcal F}})$ is the Lie algebra of *foliate fields*, i.e., vector fields whose flows send leaves to leaves, which is the same as the normalizer of the subalgebra of vector fields $\Xi({{\mathcal F}})$ tangent to ${{\mathcal F}}$ in the Lie algebra $\Xi(M)$ of all vector fields on $M$. For the trivial foliation by points, a transverse action is the same as an ordinary infinitesimal action on $M$.
Recall that on a foliated manifold $(M,{{\mathcal F}})$ the ${{\mathcal F}}$-basic forms $$\Omega(M,{{\mathcal F}})=\{\omega\in \Omega(M)\mid i_X\omega={{\mathcal L}}_X\omega=0 \text{ for all }X\in \Xi({{\mathcal F}})\}$$ define, in the same way as the $G$-basic forms introduced in Definition \[defn:basicforms\], a subcomplex of the de Rham complex of $M$, thus yielding the *${{\mathcal F}}$-basic cohomology* $H^*(M,{{\mathcal F}})$. This cohomology was first considered by Reinhart [@Reinhart].
A transverse action of a finite-dimensional Lie algebra ${\mathfrak{g}}$ on a foliated manifold $(M,{{\mathcal F}})$ induces the structure of a ${\mathfrak{g}}$-differential graded algebra, thus yielding a notion of *equivariant basic cohomology* [@GT2] for transverse actions. Explicitly, one defines on $$\Omega_{{\mathfrak{g}}}(M,{{\mathcal F}}) := (S({\mathfrak{g}}^*)\otimes \Omega(M,{{\mathcal F}}))^{{\mathfrak{g}}}$$ an equivariant differential $d_{\mathfrak{g}}$ in the same way as in Definition \[def:equivariantdifferential\], and obtains $H^*_{\mathfrak{g}}(M,{{\mathcal F}})$ as the cohomology of this complex.
The main example for which this variant of equivariant cohomology was investigated was the *Molino action* of a Killing foliation [@Molino] (see [@GT2 Section 4.1] for a short summary): this is an action of an abelian Lie algebra ${\mathfrak{a}}$ whose orbits are the leaf closures of the foliation. Imitating classical results on the fixed point sets of torus actions as in Section \[sec:fixedpoints\], one can use this theory to obtain results about the set of closed leaves of a Killing foliation. For example, one obtains the following generalization of Proposition \[prop:aeqformalfixedpointset\] [@GT2]:
For any transversely oriented Killing foliation ${{\mathcal F}}$ on a compact manifold $M$, the union $C\subset M$ of closed leaves of $M$ satisfies $$\dim H^*(C,{{\mathcal F}})\leq \dim H^*(M,{{\mathcal F}}),$$ and equality holds if and only if the Molino action is equivariantly formal.
On the other hand, there are criteria for equivariant formality of the Molino action, similar to the classical setting. For example we have the following generalization of Example \[ex:morsebotteqformal\] [@GT2]:
\[thm:basiceqformal\] If ${{\mathcal F}}$ is a transversely oriented Killing foliation on a compact manifold $M$, and $f\colon M\to {{\mathbb R}}$ a basic Morse-Bott function whose critical set is the union of closed leaves of ${{\mathcal F}}$, then the Molino action is equivariantly formal.
This criterion was applied to concrete geometric situations such as contact [@GNT] or cosymplectic geometry [@BaGo] to count closed Reeb orbits. In contact geometry, the existence of a momentum map is automatic, and just as in the symplectic setting, a generic component of the momentum map is a Morse-Bott function. As its critical set is the correct one we can apply Theorem \[thm:basiceqformal\] to the foliation given by the Reeb vector field (we need $M$ to be $K$-contact in order for the foliation to be Riemannian):
Let $M$ be a compact $K$-contact manifold, and $C\subset M$ the union of closed Reeb orbits. Then $$\dim H^*(C,{{\mathcal F}}) = \dim H^*(M,{{\mathcal F}}).$$ In particular, if the number of closed Reeb orbits is finite, then it is given by $\dim H^*(M,{{\mathcal F}})$.
On a compact $K$-contact manifold $(M,\alpha)$ of dimension $2n+1$, the elements $1,[d\alpha],\ldots,[d\alpha]^n$ are nonzero in $H^*(M,{{\mathcal F}})$; in this way we obtain an alternative proof of the statement due to Rukimbira [@Rukimbira Corollary 1] that the Reeb flow of any compact $K$-contact manifold has at least $n+1$ closed Reeb orbits. Moreover, by an easy application of the Gysin sequence, we find:
Let $M$ be a compact $K$-contact manifold of dimension $2n+1$ with only finitely many closed Reeb orbits. Then the number of closed Reeb orbits is $n+1$ if and only if $M$ is a real cohomology sphere,
Similar results can be derived in other geometries where there naturally appears a Riemannian foliation, such as $K$-cosymplectic geometry (see [@BaGo Section 8]).
Spectral sequences and the module structure on equivariant cohomology
=====================================================================
We present the basics of the spectral sequence of a filtration and apply them to the Cartan model of equivariant cohomology. By also paying attention to the multiplicative structure on spectral sequences, this tool allows us to derive some fundamental properties of the $S(\mathfrak{g}^*)^G$-module structure on $H^*_G(M)$: it is finitely generated and its rank agrees with that of the final page of the spectral sequence associated to a certain filtration. Also, we use spectral sequences to prove the torus case of Remark \[rem:commutingactionprinciple\]. Finally we give an example where $E_\infty$ and $H^*_G(M)$ are not isomorphic as $S(\mathfrak{g}^*)^G$-modules, a point which is in several places unclear in the literature.
Before we start, we want to point out that the goal here is not to give a complete introduction to spectral sequences but rather to provide the reader with all the algebraic background that is needed for our (and many other topological) applications. In particular, we avoid the finer details of convergence by restricting to first-quadrant spectral sequences. For an in-depth introduction we recommend, e.g., Chapter 5 of [@Weibel].
Basic definitions
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Let $R$ be a commutative ring. When applying algebraic results to equivariant cohomology we will always take $R=\mathbb{R}$.
A *(cohomology) spectral sequence* is a sequence $\{(E_r,d_r)\}_{r\geq 0}$ of bigraded $R$-modules $E_r=\bigoplus_{p,q\in\mathbb{Z}}E^{p,q}_r$ with $R$-linear differentials $d_r^{p,q}\colon E_r^{p,q}\rightarrow E^{p+r,q-r+1}_r$ satisfying $d_r\circ d_r=0$ and isomorphisms $E_{r+1}^{p,q}\cong (\ker d_r^{p,q})/({\operatorname{im}}d_r^{p-r,q+r-1})$.
A spectral sequence is often compared to a book, where for turning the $r$th page $E_r$ one takes cohomology to arrive at the next page $E_{r+1}\cong H^*(E_r,d_r)$. The advantage of spectral sequences is that they can be used to approximate the cohomology of a cochain complex by breaking down the transition $(C^*,d)\rightsquigarrow H^*(C^*,d)$ into smaller steps. Let us now make this idea precise by defining a suitable notion of convergence.
A *first-quadrant* spectral sequence is a spectral sequence $(E_r,d_r)$ where $E_r^{p,q}=0$ whenever $p<0$ or $q<0$. Note that if we fix a bidegree $(p,q)$ and start turning through the pages, the differentials $d_r^{p,q}$ (resp. $d_r^{p-r,q+r-1}$) eventually leave (resp. come from outside) the first-quadrant and thus are trivial. This implies that $E_r^{p,q}\cong E_l^{p,q}$ for all $l\geq r$. This stable value is denoted by $E_\infty^{p,q}$ and the the bigraded $R$-module $E_\infty$ is called the final page of the spectral sequence. If for some $r$ we have $d_i=0$ for $i\geq r$, or equivalently $E_r=E_\infty$, we say that the spectral sequence *collapses* at $E_r$. While we will solely be interested in first-quadrant spectral sequences, the definition of $E_\infty$ is not limited to this special case and makes sense whenever the pointwise limit exists.
A *filtration* of a (graded) $R$-module $H$ is a sequence of (graded) submodules $$\ldots\subset F^pH\subset F^{p-1}H\subset\ldots$$ We say that the spectral sequence $(E_r,d_r)$ *converges* to a graded module $H^*$ if there is a filtration of $H^*$ such that in any degree $n$ we have $$0=F^sH^n\subset\ldots\subset F^pH^n\subset F^{p-1}H^n\subset\ldots\subset F^tH^n=H^n$$ for some $s,t\in \mathbb{Z}$ and $E_\infty^{p,q}\cong F^pH^{p+q}/F^{p+1}H^{p+q}$.
Note that when working with $\mathbb{R}$-coefficients (or over any field) there is a highly non-canonical isomorphism of vector spaces $H^n=\bigoplus_p F^pH^n/ F^{p+1}H^n=\bigoplus_{p+q=n}E_\infty^{p,q}$. In particular $H^*\cong E_\infty$ as graded vector spaces when we consider $E_\infty^{p,q}$ to be of degree $p+q$.
Spectral sequence of a filtration {#Appendix:ConstrSec}
---------------------------------
As hinted at above, the usefulness of spectral sequences stems from the fact that they can be used to break the process of taking cohomology down into several steps. Consider, e.g., the Cartan model $C_G(M)=(S(\mathfrak{g}^*)\otimes \Omega(M))^G$ with its differential $d_G=1\otimes d+\delta$ where $\delta$ is the component which raises the degree in $S(\mathfrak{g}^*)$ and $d$ is just the differential on $\Omega(M)$. Algebraically speaking, $C_G(M)$ is a huge and complicated object, but its cohomology under the differential $1\otimes d$ is much smaller (see Prop. \[prop:E1term\] below). Consequently, when analysing $H_G(M)$, it can be helpful to take cohomology with respect to $1\otimes d$ first, and then worry about the rest of $d_G$. This process of singling out the $1\otimes d$ component is achieved via a suitable filtration and the associated spectral sequence.
A filtration of a cochain complex $(C,d)$ of $R$-modules is a family $$\ldots\subset F^pC\subset F^{p-1}C\subset\ldots$$ of subcomplexes of $C$. The filtration is said to be *canonically bounded* if $F^0C=C$ and $F^{n+1}C^n=0$.
\[Appendix:Hfiltration\] A filtration of a complex $(C,d)$ induces a filtration $F^*H^*(C,d)$ of $H^*(C,d)$, where $F^pH^n(C,d)$ is the image of the map $H^n(F^pC,d)\rightarrow H^n(C,d)$.
\[Appendix:specseq\] Let $(C,d)$ be a cochain complex and $F^*C$ a canonically bounded filtration. Then the construction below gives rise to a first-quadrant spectral sequence $(E_r,d_r)$ converging to $H^*(C,d)$. More precisely we have $$E^{p,q}_\infty\cong F^{p}H^{p+q}(C,d)/F^{p+1}H^{p+q}(C,d),$$ where $F^pH^n(C,d)$ is defined as above.
In the construction we, for the moment, forget about the cohomological degree and focus purely on the filtration degree. The second component of the bidegree will be added in the end. We start by setting $$E_0^{p}=F^pC/F^{p+1}C.$$ This carries a differential induced by $d$ and $E_0=\bigoplus_p E_0^p$ is known as the associated graded chain complex. Its cohomology $E_1$ is a first approximation of the cohomology of $(C,d)$, where cocycles are represented by elements whose filtration degree increases under the differential. Note that there is a subquotient of $E_0$ that is a much better approximation of the cohomology, namely $E_\infty=\bigoplus_p E_\infty^p$ where $$E_\infty^p=\frac{\ker d\cap F^pC+F^{p+1}C}{{\operatorname{im}}d\cap F^pC+F^{p+1}C}.$$ To interpolate between the two we introduce the approximate cycles $$A_r^p=\{x\in F^pC~|~dx\in F^{p+r}C\}$$ whose filtration degree increases by $r$ under the differential. Now set $$E_r^p= \frac{A^p_r+F^{p+1}C}{d(A^{p-r+1}_{r-1})+F^{p+1}C} \cong\frac{A_r^{p}}{d\left(A_{r-1}^{p-r+1}\right)+A_{r-1}^{p+1}}.$$ The usefulness of these interpolations stems from the fact that $E_{r+1}$ can be computed from $E_r$: by definition $d$ induces a map $d_r\colon E_r^p\rightarrow E_r^{p+r}$ and one can identify $E_{r+1}$ with $H(E_r,d_r)$ (see [@Weibel Theorems 5.4.1] for details).
The bigrading in the spectral sequence arises from additionally considering the grading on $C$. We want the latter to correspond to the total degree of the bigrading so we set $A^{p,q}_r=A_r^p\cap C^{p+q}$, which naturally induces a bigrading on $E_r$. Explicitly we have $$E_r^{p,q}=\frac{A_r^{p,q}}{d\left(A_{r-1}^{p-r+1,q+r-2}\right)+A_{r-1}^{p+1,q-1}}.$$ Since $d_r$ raises the total degree by one and the filtration degree by $r$, it is of bidegree $(r,-r+1)$. To construct the isomorphism $E^{p,q}_\infty\cong F^{p}H^{p+q}(C,d)/F^{p+1}H^{p+q}(C,d)$, note that $d$ vanishes on $A^{p,q}_r$ for $r>q+1$ because the filtration is canonically bounded. Thus $E_r^{p,q}$ is represented by cocycles from $\ker(d)\cap F^pC^{p+q}$. The isomorphism is then defined by just mapping those cocycles onto their image in $F^{p}H^{p+q}(C,d)/F^{p+1}H^{p+q}(C,d)$. For further details like well-definedness of the last map we again refer to [@Weibel Theorems 5.4.1 and 5.5.1].
The spectral sequence of the Cartan model {#Appendix:Section:CartanSpeq}
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From now on let $G$ be a compact, connected group acting on a manifold $M$. Recall from the definitions in Section \[sec:CartanModel\] that the Cartan model $C_G(M)\subset S(\mathfrak{g}^*)\otimes \Omega^*(M)$ inherits a bigrading via $$\left( S(\mathfrak{g}^*)\otimes \Omega^*(M)\right)^{p,q}= S^\frac{p}{2}(\mathfrak{g}^*)\otimes\Omega^q(M),$$ whenever $p$ is even and $C_G^{p,q}(M)=0$ when $p$ is odd. In particular, $S(\mathfrak{g}^*)$ is concentrated in even degrees when considered as the subalgebra $C_G^{*,0}$. We also assign a total degree via $C_G^n(M)=\bigoplus_{p+q=n}C^{p,q}_G(M)$. The Cartan differential is $d_G=1\otimes d+\delta$ with $d$ just the regular differential in $\Omega^*(M)$ and $(\delta\omega)(X)=-i_{\overline{X}}(\omega(X))$. Note that $1\otimes d$ and $\delta$ are themselves differentials of bidegree $(0,1)$ and $(2,-1)$.
Doing a suitable degree shift one can achieve that the bidegrees of the differentials are $(0,1)$ and $(1,0)$. With this grading $C_G(M)$ becomes a double complex in the classical sense and the spectral sequence we construct below is (up to degree shifts) the spectral sequence associated to this double complex (c.f. [@GuilleminSternberg]). As the degree shift will not simplify our presentation of the material and the original bigrading is more in line with the topological conventions, we decide to stick to the original one.
In what follows we will write $C$ instead of $C_G(M)$. The filtration we consider on $C$ is defined by $$F^pC:= C^{\geq p,*}=\bigoplus_{l\geq p,q\geq 0}C^{l,q}.$$ It is canonically bounded as $$F^pC^n=\bigoplus_{l=p}^nC^{l,n-l}.$$ The differential $d_G$ restricts to the $F^pC$, so this is indeed a filtration by subcomplexes and we have an associated spectral sequence to which we just refer as the spectral sequence of $C$. Let us now explicitly compute the first pages.
We have $E_0^{p,q}=F^pC^{p+q}/F^{p+1}C^{p+q}$, which is canonically isomorphic to $C^{p,q}$ via the projection onto this summand. The differential $d_0\colon E^{p,q}_0\rightarrow E^{p,q+1}_0$ is just the one induced by $d_G$ on the quotient. The composition with the isomorphisms $$C^{p,q}\cong F^pC^{p+q}/F^{p+1}C^{p+q}\xrightarrow{d_G} F^pC^{p+q+1}/F^{p+1}C^{p+q+1}\cong C^{p,q+1}$$ is precisely the its bidegree $(0,1)$ component $1\otimes d$. Thus we see that $(E_0,d_0)$ is isomorphic to $(C,1\otimes d)$ as a cochain complex.
\[Appendix:rem:multiplicativebla\] The following observation will become relevant when discussing multiplicative aspects in Section \[Appendix:Multstrucsec\]. In fact the above isomorphism $(C,1\otimes d)\cong (E_0,d_0)$ is one of commutative differential graded algebras (cdga, see Section \[Appendix:Multstrucsec\]) with respect to the product $$F^pC/F^{p+1}C\otimes F^qC/F^{q+1}C\rightarrow F^{p+q}C/F^{p+q+1}C$$ on $E_0$ which is induced by multiplication in $C$. The cohomology of a cdga is naturally a commutative graded algebra. Morphisms between cdgas, i.e. multiplicative maps that respect the grading and commute with the differential, induce multiplicative maps in cohomology. The isomorphism in the following proposition is of this form and hence respects the algebra structure.
\[prop:E1term\] If $G$ is a compact, connected Lie group acting on a compact differentiable manifold, then the $E_1$-term in the spectral sequence associated to the Cartan complex is $$E_1 \cong S({\mathfrak{g}}^*)^G\otimes H^{*}(M).$$
We just need to compute the cohomology of $(E_0,d_0)$. Consider the inclusion of complexes $$\label{eq:a8}
(C,1\otimes d) = ((S({\mathfrak{g}}^*)\otimes \Omega(M))^G,1\otimes d) \longrightarrow (S({\mathfrak{g}}^*)\otimes \Omega(M),1\otimes d).$$ With regards to Remark \[Appendix:rem:multiplicativebla\] note that it is an inclusion of cdgas. We obtain the induced map on cohomology $$i\colon H^*(C,1\otimes d) \longrightarrow S({\mathfrak{g}}^*)\otimes H^*(M).$$ Let us show first that it is injective. Assume that $\omega \in C$ is such that $\omega=(1\otimes d)(\sigma)$ for some $\sigma\in S({\mathfrak{g}}^*)\otimes \Omega(M)$. As $\omega$ is $G$-invariant and $1\otimes d$ commutes with the diagonal $G$-action on $S({\mathfrak{g}}^*)\otimes \Omega(M)$, we have $(1\otimes d)(g^*\sigma)=\omega$ for all $g\in G$. But then also $$(1\otimes d)\left(\int_G g^*\sigma\, dg\right)= \int_G(1\otimes d)g^*\sigma\, dg = \int_G \omega\, dg = \omega.$$ Because $\int_G g^*\sigma\, dg\in C$, it follows that $[\omega]=0\in H^*(C,1\otimes d)$.
We next claim that the map $i$ takes values in $S({\mathfrak{g}}^*)^G\otimes H^*(M)$, which means that for every $[\omega]$ on the left hand side, the element $i[\omega]$ is $G$-invariant when considered as a polynomial function with values in $H^*(M)$. For $g\in G$ the diffeomorphism $g^{-1}\colon M\to M$ is homotopic to the identity, because $G$ is connected. Then, for any $X\in {\mathfrak{g}}$ we have $[\omega({\operatorname{Ad}}_gX)] = [(g^{-1})^*\omega(X)] = [\omega(X)]$.
Finally we show that $i:H^*(C,1\otimes d)\to S({\mathfrak{g}}^*)^G\otimes H^*(M)$ is surjective. For this we precompose with the inclusion $$(S({\mathfrak{g}}^*)^G\otimes \Omega(M)^G,1\otimes d) \longrightarrow (C,1\otimes d).$$ In cohomology we obtain the composition $$S({\mathfrak{g}}^*)^G\otimes H^*(\Omega(M)^G,d)\longrightarrow H^*(C,1\otimes d)\overset{i}\longrightarrow S({\mathfrak{g}}^*)^G\otimes H^*(M)$$ which, by Theorem \[thm:cohomofinvariantforms\], is an isomorphism. Thus $i$ is surjective.
Note that the proof is simpler in case of a torus action: in this case the coadjoint action on $S({\mathfrak{t}}^*)$ is trivial, so the isomorphism $E_1 = S({\mathfrak{t}}^*)\otimes H^{*}(M)$ follows directly from Theorem \[thm:cohomofinvariantforms\].
\[thm:hoddcollapse\] If the cohomology of $M$ is concentrated in even degrees, i.e., $H^n(M)=0$ whenever $n$ is odd, then the spectral sequence of the Cartan model degenerates at the $E_1$-term.
Under the hypothesis we know that $E_1^{p,q}$ vanishes whenever $p$ or $q$ is odd. Thus $d_1$ vanishes for degree reasons. The same argument applies to all subsequent pages.
\[rem:doddvanishes\] The differential $d_r$ on $E_r$ vanishes whenever $r\geq 1$ is odd, because $S(\mathfrak{g}^*)^G$ is concentrated in even degrees. In particular, the spectral sequence collapses at $E_1$ if and only if it collapses at $E_2$.
Consider the diagonal action of $S^1\subset\mathbb{C}$ on the unit sphere $S^{2n+1}\subset \mathbb{C}^{n+1}$. The Weyl-invariant polynomials are just $\mathbb{R}[u]$, where $u$ is the dual of some generator $X$ of the Lie algebra of $S^1$. The $E_1$ term of the spectral sequence is isomorphic to $\mathbb{R}[u]\otimes H^*(S^{2n+1})$, so it consists just of two copies of $\mathbb{R}[u]$, embedded as $E_1^{*,0}$ and $E_1^{*,2n+1}$. A differential can only be nonzero if it maps from the $(2n+1)^{\mathrm{st}}$ row to the $0^{\mathrm{th}}$ row. Consequently we have $d_r=0$ for $1\leq r\leq 2n+1$ and $E_1\cong E_{2n+2}$. By the same reasoning we have $d_r=0$ for $r\geq 2n+3$ and $E_{2n+3}=E_\infty$. All that remains to understand is what the differential $d_{2n+2}$ does on $E_{2n+2}$:
\(m) \[matrix of math nodes, nodes in empty cells,column sep=[0.8cm,between origins]{}, row sep=[1cm,between origins]{}\][ 2n+1 &\[+2mm\] & 0 & & & & & &\
& & & & & & & &\
0 & & 0 & & & & 0 & &\
& 0 & 1 & 2 & & 2n+2 &\
]{};
(m-1-2) to (m-3-6) node\[midway, above=3mm,right=-3mm\] [$d_{2n+2}$]{}; (m-1-4) to (m-3-8);
(m-1-2)++(-0.4,0.4)–++(0,-3.4); (m-3-1)++(-0.5,-0.4)–++(7.5,0);
Often spectral sequence arguments can work entirely without knowing the explicit definition of the differentials if one adds an extra ingredient. In this case for example, we know by Theorem \[thm:eqcohomlocallyfreeactions\] that $E_\infty$ is the cohomology of a $2n$-dimensional manifold and vanishes in degrees above $2n$. This knowledge implies that no elements of greater (total) degree must survive the transition from $E_{2n+2}$ to $E_{2n+3}$. Consequently $d_{2n+2}\colon \smash{E_{2n+2}^{p,2n+1}}\rightarrow \smash{E^{p+2n+2,0}_{2n+2}}$ has to be an isomorphism for every $p\geq 0$. All that remains on the page $E_{2n+3}=E_\infty$ is therefore $\mathbb{R}[u]/(u^{n+1})$ in the $0$th row. We have shown that $H^*(\mathbb{C}P^n)\cong H_{S^1}(S^{2n+1})\cong\mathbb{R}[u]/(u^{n+1})$ as graded vector spaces. With the help of the discussion of the $\mathbb{R}[u]$-module and algebra structures from the subsequent sections, one can deduce that this isomorphism is actually one of $\mathbb{R}[u]$-algebras. However, this is false in general and only holds because in the example, $E_\infty$ is concentrated in a single row, implying there is only one step in the filtration of $H_{S^1}(S^{2n+1})$.
Finally, let us examine explicitly the generator of $\smash{E_{2n+2}^{0,2n+1}}\cong H^{2n+1}(S^{2n+1})$. Let $\omega_0$ be a $S^1$-invariant volume form on $S^{2n+1}$. Other than suggested by the isomorphism, $\omega_0$ does not represent a generator of $\smash{E_{2n+2}^{0,2n+1}}$ because $d_{S^1}\omega_0=ui_{\overline{X}}\omega_0$ has filtration degree $2$. So $\omega_0$ is not an element of $\smash{A^{0,2n+1}_{2n+2}}$. However, we find a form $\omega_1$ such that $i_{\overline{X}}(\omega_0)=d\omega_1$ because $H^{2n}(S^{2n+1})=0$. Now $d_G(\omega_0+u\omega_1)=u^2i_{\overline{X}}\omega_1$ lies in filtration degree $4$. Inductively we construct a zigzag $\omega=\omega_0+\cdots+u^n\omega_n$ such that $d_G\omega$ is a multiple of $u^{n+1}$. So $\omega$ lies in $\smash{A^{0,2n+1}_{2n+2}}$ and induces an element of $\smash{E_{2n+2}^{0,2n+1}}$. Using the fact that the bidegree-$(0,2n+1)$ component of $\omega$, which is precisely $\omega_0$, does not lie in the the projection ${\operatorname{im}}d$ of ${\operatorname{im}}d_G$ to the $(0,2n+1)$ component, we conclude that $\omega$ descends to a generator.
Multiplicative structure {#Appendix:Multstrucsec}
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A *graded* $R$-*algebra* is an $R$-algebra $A=\bigoplus_{k\in \mathbb{Z}} A^k$ (where $A^k$ are $R$-modules) such that the multiplication map respects the grading, i.e., $A^p\cdot A^q\subset A^{p+q}$. It is called *commutative* if $xy=(-1)^{|x||y|}yx$ for homogeneous elements $x,y$ of degrees $|x|,|y|$. If $d\colon A\rightarrow A$ is an $R$-linear map which raises the degree by $1$ and satisfies $d^2=0$ as well as the graded Leibniz rule $$d(xy)=dx\cdot y+(-1)^{|x|}x\cdot dy,$$ we call $(A,d)$ a *commutative differential graded algebra* (cdga). A filtration $F^*A$ of $A$ (as a graded $R$-module) is called *multiplicative* if $F^pA\cdot F^lA\subset F^{p+l}A$.
\[Appendix:Multfiltration\] The cohomology $H^*(A,d)$ of any cdga $(A,d)$ inherits an algebra structure which turns it into a commutative graded algebra. If $F^*A$ is a multiplicative filtration of $(A,d)$ by subcomplexes, then the induced filtration on $H^*(A,d)$ (see Remark \[Appendix:Hfiltration\]) is multiplicative with respect to the induced algebra structure. In this case we have well defined product maps $$\frac{F^pH^n}{F^{p+1}H^n}\otimes \frac{F^lH^m}{F^{l+1}H^m}\longrightarrow \frac{F^{p+l}H^{n+m}}{F^{p+l+1}H^{n+m}},$$ where we write $H^k$ for $H^k(A,d)$.
The differential forms $(\Omega(M),d)$ and the Cartan model $(C_G(M),d_G)$ are cdgas with the total degree which is the sum of both components of the bidegree. The filtration of the Cartan model as defined in the previous section is a multiplicative filtration.
We have seen that for a suitably filtered complex $(C,d)$ the last page of the associated spectral sequence carries information on $H^*(C,d)$ and the two are even abstractly isomorphic as vector spaces if we use field coefficients. It is natural to ask if in case of a cdga $(A,d)$, $E_\infty$ carries information on the algebra structure on $H^*(A,d)$. While we cannot expect to have $E_\infty\cong H^*(A,d)$ as algebras, the algebra structure does indeed leave its mark on $E_\infty$ in the following manner.
\[Appendix:Thm:MultSpecsec\] Let $(A,d)$ be a cdga with a canonically bounded multiplicative filtration $F^*A$. Then the spectral sequence from Theorem \[Appendix:specseq\] carries a multiplicative structure, i.e., for any $r$ there exist multiplication maps $\mu_r\colon E^{p,q}_r\otimes E^{s,t}_r\rightarrow E_r^{p+s,q+t}$ with the following properties:
- $(E_r,d_r)$ is a cdga with respect to the total degree of the bigrading.
- The multiplication $\mu_{r+1}$ is induced by $\mu_r$ under the isomorphism $E_{r+1}\cong H(E_{r},d_{r})$.
In particular we get an induced multiplication on $E_\infty$. Under the isomorphism $$E_\infty^{p,q}= F^pH^{p+q}(A,d)/F^{p+1}H^{p+q}(A,d),$$ this product coincides with the one described in Remark \[Appendix:Multfiltration\].
Details of the proof are given e.g. in [@McCleary Section 2.3]. Let us just quickly demystify the products $\mu_r$ by giving their definition: in the explicit construction of $E_r^{p,q}$ from Section \[Appendix:ConstrSec\] one easily checks that multiplication in $A$ restricts to $A_r^{p,q}\otimes A_r^{s,t}\rightarrow A_r^{p+s,q+t}$ and that this descends to quotients inducing the map $\mu_r\colon E_r^{p,q}\otimes E_r^{s,t}\rightarrow E_r^{p+s,q+t}$ from the above theorem. Finally we want to draw the reader’s attention to Remark \[Appendix:rem:multiplicativebla\], where we argue that $$E_1\cong S(\mathfrak{g^*})^G\otimes H^*(M)$$ as algebras.
On the module structure of the equivariant cohomology {#Appendix:Section:ModuleStructure}
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One of the interesting features of equivariant cohomology is that it is not only an algebra over $\mathbb{R}$ but over $S(\mathfrak{g}^*)^G$. As we have seen, multiplicative structures carry over to the spectral sequence, so we can use the latter to analyse the $S(\mathfrak{g}^*)^G$-module structure on $H^*_G(M)$.
As the differential $d_G$ of the Cartan model vanishes on $S(\mathfrak{g}^*)^G\otimes 1$, we have $S^p(\mathfrak{g}^*)^G \subset A^{2p,0}_r$ for all $r$. The degreewise projection onto $E^{2p,0}_r$ yields a map $$S(\mathfrak{g}^*)^G\rightarrow E_r$$ whose image is the zeroth row $E_r^{*,0}$. On the page $E_1\cong S(\mathfrak{g}^*)^G\otimes H^*(M)$ (see Prop. \[prop:E1term\]) it is just the inclusion of $S(\mathfrak{g}^*)^G\otimes {{\mathbb R}}$. Note that we also obtain an induced map $S(\mathfrak{g}^*)^G\rightarrow E_\infty$. These maps are easily checked to be morphisms of algebras. Thus, the $E_r$ carry the structure of $S(\mathfrak{g}^*)^G$-modules.
For degree reasons the differentials $d_r$ vanish on $E_r^{*,0}$ for $r\geq 1$ so by the Leibniz rule we have $d_r(fx)=fd_r(x)$ for any $f\in S(\mathfrak{g}^*)^G,x\in E_r$. The module structure on $E_{r+1}$ is just the one that $H(E_r,d_r)$ inherits from the differential graded $S(\mathfrak{g}^*)^G$-module $(E_r,d_r)$.
\[Appendix:Lem:modulegenerators\] Let $x_1,\ldots,x_k\in E_\infty$ be homogeneous elements (with respect to the bigrading) that generate $E_\infty$ as an $S(\mathfrak{g}^*)^G$-module. Choose representatives $y_1,\ldots,y_k\in H^*_G(M)$ via the isomorphisms $$E_\infty^{p,q}\cong F^pH^{p+q}_G(M)/F^{p+1}H^{p+q}_G(M).$$ Then the $y_i$ generate $H^*_G(M)$ as an $S(\mathfrak{g}^*)^G$-module.
Let $c_0\in H^l_G(M)$ be any element. It is contained in some $F^pH_G^l(M)$, so we may consider its image $\overline{c_0}\in E_\infty^{p,l-p}$. We find elements $f_1,\ldots,f_k\in S(\mathfrak{g}^*)^G$ such that $$\overline{c_0}=\sum f_ix_i.$$ Recall that the multiplication in $E_\infty$ respects the bigrading. We may therefore choose the $f_i$ in such a way that they are homogeneous and if $x_i\in E_\infty^{p-m,l-p}$, we have $|f_i|=m$ (in the grading inherited from the Cartan model) or $f_i=0$. This ensures that $\sum_i f_iy_i$ lies in $F^pH^l_G(M)$. Now by the description of the multiplicative structure on $E_\infty$ from Theorem \[Appendix:Thm:MultSpecsec\] one verifies that $\sum_i f_iy_i$ projects to $\overline{c_0}$ in $E_\infty^{p,l-p}$. In particular $$c_1=c_0-\sum f_iy_i$$ projects to $0$ and thus lies in $F^{p+1}H^l_G(M)$. Now we repeat this process for $c_1$ until eventually $c_{l-p+1}\in F^{l+1}H^l_G(M)=0$. We have written $c_0$ as a linear combination of the $y_i$.
The following proposition applies in particular to compact manifolds. The proof is taken from [@AlldayPuppe Prop. 3.10.1]
If $\dim H^*(M)<\infty$, then $H^*_G(M)$ is finitely generated as an $S(\mathfrak{g}^*)^G$-module.
By Lemma \[Appendix:Lem:modulegenerators\], it suffices to show that $E_\infty$ is finitely generated. We have seen that $E_1$ is the free module $S(\mathfrak{g}^*)^G\otimes H^*(M)$. The cohomology $H^*(M)$ is finite-dimensional and in particular $E_1$ is finitely generated as an $S(\mathfrak{g}^*)^G$-module. The ring $S(\mathfrak{g}^*)^G$ is is a polynomial ring (see Section \[sec:Coadjoint\]). In particular it is Noetherian, which implies that submodules and quotients of finitely generated $S(\mathfrak{g}^*)^G$-modules are again finitely generated, see [@AtiyahMac Prop. 6.5]. Thus if $E_r$ is finitely generated, the same is true for $E_{r+1}=H(E_r,d_r)$: the differential respects the module structure so the cohomology is a quotient of the submodule $\ker d_r$. As the spectral sequence collapses after a finite number of pages (at most $\dim M$), we conclude that $E_\infty$ is finitely generated.
Note that, since $S(\mathfrak{g}^*)^G$ is concentrated in even degrees, the module structure preserves even and odd degree elements. With regard to the resulting decomposition we have the following
\[Appendix:Cor:RankGleich\] If $\dim H^*(M)<\infty$, then the ranks of the $S(\mathfrak{g}^*)^G$-modules $E^{{\operatorname{even}}}_\infty$ (resp. $E_\infty^{{\operatorname{odd}}}$) and $H^{{\operatorname{even}}}_G(M)$ (resp. $H^{{\operatorname{odd}}}_G(M)$) coincide.
For a finitely generated graded module $M$ over the polynomial ring $S(\mathfrak{g}^*)^G$, the rank is encoded in its Hilbert-Poincaré series $H_M(t)=\sum_i \dim( M^i)\, t^i$: the latter takes the form $f(t)\prod_{i=1}^r(1-t^{k_i})^{-1}$ for some $f\in \mathbb{Z}[t]$, where $r$ is the number of variables of $S(\mathfrak{g}^*)^G$ and the $k_i$ are their degrees [@AtiyahMac Thm. 11.1]. The rank is then precisely $f(1)$ (check this for a free module first and then deduce it for general $M$ via a free resolution). As we have already seen, $E_\infty$ and $H^*_G(M)$ are isomorphic as graded vector spaces, so the claim follows.
In the corollary above, it is tempting to argue that a basis of a free submodule in $H^*_G(M)$ projects down to the basis of a free submodule of $E_\infty$. However this is false in general.
Naturality and the comparison theorem
-------------------------------------
We briefly discuss maps between spectral sequences and the important comparison Theorem. The latter enables us to prove Remark \[rem:commutingactionprinciple\] in case $G$ and $H$ are tori. Also, a construction made in said proof is needed in the next and final section.
A morphism of spectral sequences $(E_r,d_r)\rightarrow (E_r',d_r')$ is a family of morphisms $f_r\colon E_r\rightarrow E'_r$, defined for large $r$, that preserve the bigrading, commute with the differentials, and have the property that $f_{r+1}$ is the map induced by $f_r$ in cohomology.
In particular, if $E_\infty$ is defined, we obtain a map $f_\infty\colon E_\infty\rightarrow E'_\infty$. Morphisms of spectral sequences associated to filtrations arise naturally via filtration-preserving maps: Suppose $(C,d)$ and $(C',d')$ are canonically bounded filtered cochain complexes and $f\colon C\rightarrow C'$ is a filtration-preserving chain map. Then $f$ maps $A_r^{p,q}$ (see the construction in Section \[Appendix:ConstrSec\]) to ${A'}_r^{p,q}$ and induces maps $f_r\colon E_r\rightarrow E_r'$ for $r\geq 0$. One checks directly via the definitions that this is a morphism of spectral sequences. For proofs of this and the theorem below we refer to [@Weibel Thm. 5.5.11].
\[Appendix:thm:Comparison\] If, in the above setting, one of the $f_r$ is an isomorphism, then so are all subsequent $f_r$ and $f$ induces an isomorphism in cohomology.
To illustrate the usefulness of the above theorem, we prove Remark \[rem:commutingactionprinciple\] in the case of tori:
\[Appendix:Prop:QuotientShenanigans\] Let a torus $T=T'\times T''$ act on $M$ in such a way that the restricted action of the $T''$-factor is free. Then there is a map $C_{T'}(M/T'')\rightarrow C_{T}(M)$ of cdgas inducing an isomorphism in cohomology.
It suffices to prove the proposition in case $T=T'\times S^1$. Then the general case $T^n=T^l\times T^r$ follows by induction. Consider now an action of $T=T'\times S^1$ on $M$ with the $S^1$ factor acting freely. Via the above product decomposition the Lie algebra of $T$ decomposes as $\mathfrak{t}\oplus\mathfrak{t}_1$. In Corollary \[cor:orbitspacedings\] it was proved that $\Omega(M/S^1)\cong \Omega_{\mathrm{bas}~S^1}(M)\rightarrow C_{S^1}(M)$ induces an isomorphism on cohomology. Note that if we restrict this map to $\Omega(M/S^1)^{T'}$, it will take values in $S(\mathfrak{t}_1^*)\otimes \Omega^{T}(M)$. We want to argue that in the diagram $$\xymatrix{
\Omega(M/S^1)^{T'}\ar[r]^(0.45){\psi_1}\ar[d]^{\psi_2}& S(\mathfrak{t}_1^*)\otimes \Omega(M)^{T}\ar[d]^{\psi_3} \\
\Omega(M/S^1)\ar[r]^{\psi_4} & C_{S^1}(M)
}$$ the map $\psi_1$ induces an isomorphism in cohomology. By Theorem \[thm:cohomofinvariantforms\] and Corollary \[cor:orbitspacedings\] (applied to the proved $S^1$ case) we know that $\psi_2$ and $\psi_4$ induce isomorphisms. Consequently, if we show that $\psi_3$ induces an isomorphism, the same will hold for $\psi_1$.
Filter both complexes, $S(\mathfrak{t}_1^*)\otimes \Omega^{T}(M)$ and $C_{S^1}(M)$, by the degree of $S(\mathfrak{t}_1^*)$ as we did for the construction of the spectral sequence for $C_{S^1}(M)$ (see Section \[Appendix:Section:CartanSpeq\]). As $\psi_3$ is $S(\mathfrak{t}_1^*)$-linear it respects the filtration and induces a morphism of spectral sequences. As argued before, the $0$th pages of the spectral sequences are isomorphic to the respective filtered complexes $ S(\mathfrak{t}_1^*)\otimes \Omega^{T}(M)$ and $C_{S^1}(M)$ and one quickly checks that the map between the $0$th pages is just $\psi_3$. On both $0$th pages, the differential $d_0$ is $1\otimes d$, with $d$ the exterior derivative on $\Omega(M)$. The inclusion $\Omega(M)^T\rightarrow \Omega(M)$ factors through $\Omega(M)^{S^1}\rightarrow\Omega(M)$ and both induce isomorphisms in cohomology by Theorem \[thm:cohomofinvariantforms\]. Consequently the inclusion $i\colon\Omega(M)^{T}\rightarrow\Omega(M)^{S^1}$ induces an isomorphism as well and we deduce that $\psi_3={{\operatorname{id}}_{S(\mathfrak{t}_1^*)}}{\otimes}\ { i}$ induces an isomorphism on $E_1=H(E_0,d_0)$. Now by the Comparison Theorem \[Appendix:thm:Comparison\], $\psi_3$ induces an isomorphism in cohomology.
The final step is to show that the map $$\varphi\colon C_{T'}(M/S^1)=S(\mathfrak{t}^*)\otimes \Omega(M/S^1)^{T'}
\longrightarrow S(\mathfrak{t}^*)\otimes \left(S(\mathfrak{t}_1^*)\otimes \Omega(M)^{T}\right)=C_{T}(M)$$ defined as ${\operatorname{id}}_{S(\mathfrak{t}_l^*)}\otimes \psi_1$ induces an isomorphism in cohomology. To see this one proceeds analogously to before: Filter both complexes by the degree of $S(\mathfrak{t}^*)$. Then the $0$th pages will be isomorphic to $C_{T'}(M/S^1)$ and $C_{T}(M)$ (the bigrading on the latter is not the usual one!) and $\varphi$ induces a morphism of spectral sequences which on $E_0$ is just $\varphi$ itself. The differentials $d_0$ are $1\otimes d$ and $1\otimes d_{S^1}$. In particular $\varphi$ induces an isomorphism on the cohomology $E_1$ because $\psi_1$ does so on the right tensor factor. Another application of Theorem \[Appendix:thm:Comparison\] yields the result.
A counterexample {#sec:counterexample}
----------------
In [@Skjelbred] it was shown that under certain topological conditions, e.g. for compact manifolds, the equivariant cohomology of a $S^1$-action and the final page of the spectral sequence are isomorphic as $S(\mathfrak{t}^*)$-modules. For tori of greater dimension this is no longer true. We construct here a $T^2$-action on a compact manifold such that the final page of the spectral sequence associated to the Cartan model is not isomorphic as a graded $S(\mathfrak{t}^*)$-module to the equivariant cohomology.
Consider the standard action of the diagonal maximal torus $T^3$ of ${\mathrm{SU}}(4)$ by left multiplication, where we identify $(s,t,u)$ with the diagonal matrix with entries $(stu,\overline{s},\overline{t},\overline{u})$. The maximal diagonal torus of ${\mathrm{SU}}(2)$ is a circle and together they yield a product action of $T^4=S^1\times T^3$ on ${\mathrm{SU}}(2)\times {\mathrm{SU}}(4)$. We pull back this action along the homomorphism $T^3\rightarrow T^4,~(s,t,u)\mapsto(s,s,t,u)$. Now we take the quotient of the first circle factor of $T^3$ and consider the action of the middle and right circle factors to obtain an action of $T^2$ on the space $$M:=({\mathrm{SU}}(2)\times{\mathrm{SU}}(4))/S^1.$$ This action has the desired properties as we will now show. In what follows the Lie algebra of the $r$-torus will be denoted $\mathfrak{t}_r$.
As it is our goal to show that $H_{T^2}^*(M)$ and $E_\infty$ are not isomorphic let us begin by pointing out the structural difference in the two modules.
In $E_\infty$ there exists a nontrivial degree $2$ element which is torsion with respect to some linear polynomial in $S(\mathfrak{t}_2^*)$. The same does not hold for $H^*_{T^2}(M)$.
To analyse $H^*_{T^2}(M)$ we will use that it is isomorphic to $H^*_{T^3}(N)$, where $N={\mathrm{SU}}(2)\times {\mathrm{SU}}(4)$ with the aforementioned $T^3$-action. The isomorphism is induced by the cdga morphism $$\varphi\colon C_{T^2}(M)=S(\mathfrak{t}_2^*)\otimes \Omega(M)^{T^2}\longrightarrow S(\mathfrak{t}_2^*)\otimes \left(S(\mathfrak{t}_1^*)\otimes\Omega(N)^{T^3}\right)=C_{T^3}(N)$$ which was constructed in the proof of Proposition \[Appendix:Prop:QuotientShenanigans\], where we decompose $\mathfrak{t}_3$ as $\mathfrak{t}_2\oplus \mathfrak{t}_1$ in such a way that $\mathfrak{t}_1$ corresponds to the subcircle of $T^3$ such that $M=N/S^1$. In the proof we also argued that $\varphi$ induces an isomorphism between the $E_\infty$-term of the spectral sequence of $C_{T^2}(M)$ and the final page $E_\infty'$ of the spectral sequence obtained by filtering $C_{T^3}(N)$ by the degree of $S(\mathfrak{t}^*_2)$. This allows us to work with the latter spectral sequence when analysing the $E_\infty$-term. Note that under the isomorphisms $H_{T^2}(M)\cong H_{T^3}(N)$ and $E_\infty\cong E_\infty'$, the $S(\mathfrak{t}_2^*)$-module structure on the left side corresponds to the pullback of the $S(\mathfrak{t}_3^*)$-module structure on the right side along the inclusion $S(\mathfrak{t}_2^*)\rightarrow S(\mathfrak{t}^*_3)$.
Now let $X,Y,Z\in \mathfrak{t}_3^*$ be the dual basis of the standard basis of $\mathfrak{t}_3$, with $X$ in the $\mathfrak{t}_1^*$ summand of the decomposition $\mathfrak{t}_3^*=\mathfrak{t}_2^*\oplus \mathfrak{t}_1^*$.
The map $S(\mathfrak{t}^*_3)\rightarrow H_{T^3}^*(N)$ is injective in degrees up to $3$ and its kernel in degree $4$ is generated by $X^2$ and $X^2+XY+Y^2+YZ+Z^2 +ZX$.
Let $(E_r,d_r)$ denote the spectral sequence of $C_{T^3}(N)$. The map $S^p(\mathfrak{t}_3^*)\rightarrow H_{T^3}^{2p}(N)$ factors as $$S^p(\mathfrak{t}_3^*)\rightarrow E_\infty^{2p,0}\cong F^{2p}H^{2p}_{T^3}(N)\subset H^{2p}_{T^3}(N),$$ where we have used that $F^{2p+1}H^{2p}_{T^3}(N)=0$ (see the definition of the isomorphism at the end of Section \[Appendix:ConstrSec\]). In particular the kernels of $S(\mathfrak{t}_3^*)\rightarrow E_\infty$ and $S(\mathfrak{t}_3^*)\rightarrow H_{T^3}^*(N)$ coincide.
We have $E_1=S(\mathfrak{t}^*_3)\otimes H^*({\mathrm{SU}}(2)\times {\mathrm{SU}}(4))$. By the Künneth formula, $H^*({\mathrm{SU}}(2)\times {\mathrm{SU}}(4))$ is trivial in degrees $1$ and $2$. For degree reasons, no elements in $E_1^{2,0}$ can be hit by a differential, and thus they live to infinity. This shows injectivity. Elements in $E_1^{4,0}$ live to $E_3^{4,0}$, where they can potentially be hit by $d_3\colon E_3^{0,3}\rightarrow E_3^{4,0}\cong H^3({\mathrm{SU}}(2)\times{\mathrm{SU}}(4))$. This is the only nonzero differential entering this position and thus the kernel in degree $4$ corresponds to the image of $d_3$ in $E_3^{4,0}$. In particular it is at most $2$-dimensional because $\dim H^3({\mathrm{SU}}(2)\times{\mathrm{SU}}(4))=2$. It remains to show that the polynomials from the lemma actually lie in the kernel in which case they will span it.
Recall that the $T^3$-action is defined as a pullback of the product $T^4$-action on $N$ along a homomorphism which on Lie algebras is given by $i\colon \mathfrak{t}_3\rightarrow\mathfrak{t}_4,~(x,y,z)\mapsto (x,x,y,z)$ where we use the standard bases. We have a commutative diagram $$\xymatrix{S(\mathfrak{t}_4^*)\ar[r]\ar[d] & H_{T^4}^*(N)\ar[d]\\
S(\mathfrak{t}_3^*)\ar[r] & H_{T^3}^*(N)
}$$ induced by the pullback map $i^*\colon C_{T^4}(N)\rightarrow C_{T^3}(N)$. Let $W,X,Y,Z$ denote the dual basis of the standard basis of $\mathfrak{t}_4$, where $W$ corresponds to the circle factor acting on ${\mathrm{SU}}(2)$ and $X,Y,Z$ correspond to the maximal torus of ${\mathrm{SU}}(4)$. Note that $N$ is actually a Lie group and that the $T^4$-action is the action of a maximal torus of $N$. By Remark \[rem:modulestructureG/H\], the kernel of $S(\mathfrak{t}_4^*)\rightarrow H_{T^4}^*(N)$ consists of the Weyl-invariant polynomials which in (cohomological) degree 4 are $p_1=W^2$ and $p_2=X^2+XY+Y^2+YZ+Z^2+ZX$. Hence the elements $i^*(p_1),i^*(p_2)$ lie in the kernel of $S(\mathfrak{t}_3^*)\rightarrow H_{T^3}^*(N)$. They are precisely the polynomials from the lemma because $i^*$ maps $W$ to $X$ and $X,Y,Z$ to themselves.
As we see from the spectral sequence of $C_{T^3}(N)$, the elements $X,Y,Z$ induce a basis of $H^2_{T^3}(N)$. No element of the degree-$4$ part of $\ker(S(\mathfrak{t}^*_3)\rightarrow H_{T^3}^*(N))$ is divisible by a linear polynomial from $S(\mathfrak{t}^*_2)$. Indeed, for an element of the form $aY+bZ$ to divide a nonzero element of the form $cX^2+d(X^2+X(Y+Z)+Y^2+YZ+Z^2)$ it is certainly necessary that $c=-d$ and $a=b$. But $Y+Z$ does not divide $X(Y+Z)+Y^2+YZ+Z^2$. This proves the claim that no nonzero element of $H_{T^2}^2(M)$ is sent to $0$ by multiplication with a linear polynomial from $S(\mathfrak{t}^*_2)$.
On the contrary, consider the element $\overline{X}\in {E'}_\infty^{0,2}$ induced by $X$ in the spectral sequence obtained by filtering $C_{T^3}(N)$ by the degree in $\mathbb{R}[Y,Z]$ (recall that $E_\infty'$ is isomorphic to the final page associated to $C_{T^2}(M)$). By the lemma, $X(Y+Z)+Y^2+YZ+Z^2$ is a coboundary. But this shows that $X(Y+Z)$ is a coboundary up to elements of filtration degree $4$ and therefore becomes trivial in ${E'}_\infty^{2,2}$. Thus $\overline{X}(Y+Z)=0$. We have shown that $E_\infty$ and $H^*_{T^2}(M)$ are not isomorphic as graded modules.
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Victor Guillemin and Catalin Zara, [*Equivariant de Rham theory and graphs*]{}, Asian J. Math [**3**]{} (1999), no. 1, 49–76. Sam Hagh Shenas Noshari, [*On the equivariant cohomology of isotropy actions*]{}, Dissertation, Philipps University of Marburg, 2018. Megumi Harada, André Henriques, and Tara S. Holm, [*Computation of generalized equivariant cohomologies of Kac–Moody flag varieties*]{}, Adv. Math [**197**]{} (2005), no. 1, 198–221. Chen He, *Localization of certain odd-dimensional manifolds with torus actions*, preprint, arXiv:1608.04392. Wu Yi Hsiang, *Cohomology Theory of Topological Transformation Groups*, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, Berlin, 1975. Richard M. Kane, [ *Reflection groups and invariant theory.*]{} CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 5. Springer-Verlag, New York, 2001. Katsuo Kawakubo, [*The theory of transformation groups*]{}, Translated from the 1987 Japanese edition. 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Graduate Studies in Mathematics, 93. American Mathematical Society, Providence, RI, 2008. Pierre Molino, *Riemannian foliations*, with appendices by G. Cairns, Y. Carrière, É. Ghys, E. Salem and V. Sergiescu, Birkhäuser Boston Inc., Boston, 1988. Liviu I. Nicolaescu, *On a theorem of Henri Cartan concerning the equivariant cohomology*, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 45 (1999), no. 1, 17–38 (2000). Arkadi L. Onishchik, [*Topology of transitive transformation groups*]{}. Johann Ambrosius Barth Verlag GmbH, Leipzig, 1994. Peter Orlik and Frank Raymond, *Actions of the torus on a $4$-manifold – II*, Topology [**13**]{} (1974), 89–112. Bruce L. Reinhart, *Harmonic integrals on foliated manifolds*, Amer. J. Math. [**81**]{} (1959), 529–536. Philippe Rukimbira, *Topology and closed characteristics of $K$-contact manifolds*, Bull. Belg. Math. Soc. Simon Stevin [**2**]{} (1995), 349–356. Hiroo Shiga, *Equivariant de Rham cohomology of homogeneous spaces*, J. Pure Appl. Algebra, [**106**]{} (2), 173–183, 1996. Hiroo Shiga and Hideo Takahashi, *Remarks on equivariant cohomology of homogeneous spaces*, Technical report 17, Technological University of Nagaoka, May 1995. Tor Skjelbred, *On the spectral sequence for the equivariant cohomology of a circle action*, preprint, Matematisk institutt, Universitetet i Oslo, 1991. Julianna S. Tymoczko, *An introduction to equivariant cohomology and homology, following Goresky, Kottwitz, and MacPherson*, Snowbird lectures in algebraic geometry, 169–188, Contemp. Math. [**388**]{}, Amer. Math. Soc., Providence, RI, 2005. Veeravalli S. Varadarajan, [*Lie groups, Lie algebras, and their representations*]{}. Reprint of the 1974 edition. Graduate Texts in Mathematics, 102. Springer-Verlag, New York, 1984. Charles Weibel, *An Introduction to Homological Algebra*, Cambridge University Press, 1994.
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[ROME prep.1249/99\
hep-th/9904200]{}
[**THE ABELIAN PROJECTION VERSUS THE HITCHIN FIBRATION OF $K(D)$ PAIRS IN FOUR-DIMENSIONAL $QCD$**]{}\
\
INFN Sezione di Roma\
Dipartimento di Fisica, Universita’ di Roma ‘La Sapienza’\
Piazzale Aldo Moro 2 , 00185 Roma\
\
We point out that the concept of Abelian projection gives us a physical interpretation of the role that the Hitchin fibration of parabolic $K(D)$ pairs plays in the large-$N$ limit of four-dimensional $QCD$.\
This physical interpretation furnishes also a simple criterium for the confinement of electric fluxes in the large-$N$ limit of $QCD$.\
There is also an alternative, compatible interpretation, based on the $QCD$ string.
April 1999
Introduction
============
Some years ago ’t Hooft introduced the concept of Abelian projection [@H1] into non-Abelian gauge theories, in order to explain the confinement of quarks in four-dimensional $QCD$ as a dual Meissner effect in a dual superconductor [@H02; @M].\
The Abelian projection allows us, by a careful choice of the gauge, to describe the physical variables of a non-Abelian $SU(N)$ gauge theory, without scalar matter fields, as a set of electric charges and magnetic monopoles interacting via a residual $U(1)^{N-1}$ Abelian gauge coupling.\
The occurrence of magnetic monopoles into a non-Abelian gauge theory without matter fields is perhaps the most crucial feature of the Abelian projection, that furnishes a precise understanding of the structure of the phases of non-Abelian gauge theories, according to the following alternatives [@H3].\
If there is a mass gap, either the electric charge condenses in the vacuum (Higgs phase) or the magnetic charge does (confinement phase). If there is no mass gap, the electric and magnetic fluxes coexist (Coulomb phase).\
Recently, in an apparently unrelated development [@MB], some mathematical control was gained over the large-$N$ limit of four-dimensional $QCD$, mapping, by means of a chain of changes of variables, the function space of the $QCD$ functional integral into an elliptic fibration of Hitchin bundles.\
Hitchin bundles [@Hi] are themselves a fibration of $U(1)$ bundles over spectral branched covers of a Riemann surface, that, in the case of [@MB], is a torus.\
In this paper, we point out that the map in [@MB] is a version, in a perhaps global algebraic-geometric setting, of the concept of Abelian projection [@H1].\
In fact, the branching points of the spectral cover are identified with the magnetic monopoles of the Abelian projection, the parabolic points of the cover with (topological) electric charges and the $U(1)$ gauge group on the cover with a global version (on the cover) of the $U(1)^{N-1}$ gauge group of the Abelian projection.\
The identifications that we have just outlined provide a physical interpretation of the mathematical construction in [@MB]. Indeed it is precisely this physical interpretation that explains naturally why the functional integral, once it is expressed as a functional measure supported over the collective field of the Hitchin fibration, is dominated by a saddle-point condition in the large-$N$ limit.\
On the other side, we may think that the mathematical proof, that the variables of the Abelian projection really capture the physics of four-dimensional $QCD$ in the large-$N$ limit, relies on the fact that those variables may be employed to dominate the functional integral in the large-$N$ limit.\
The only qualitative feature, in the treatment in [@MB], that was not already present in the concept of the Abelian projection, is the occurrence of Riemann surfaces and it is due to the global algebraic-geometric nature of the methods in [@MB]. This, however, makes contact, at least qualitatively, with another long-standing conjecture about the $QCD$ confinement, the occurrence of string world sheets [@GT] and the string program [@Po].\
Our last concluding remark is that the electric/magnetic alternative [@H3] and the physical interpretation based on the Abelian projection, applied in the mathematical framework of [@MB], give us a simple qualitative criterium to characterize the confinement phase of $QCD$ in the large-$N$ limit: confinement is equivalent to magnetic condensation, in absence of electric (parabolic) singularities of the spectral covers.\
An alternative, compatible interpretation, based on the idea that $QCD$ is equivalent, in the large-$N$ limit, to a theory of strings [@GT; @Po] is outlined in the following section. The rest of the paper is devoted to a technical explanation of the correspondence between the Abelian projection and the Hitchin fibration in four-dimensional $QCD$.
The Hitchin fibration as the Abelian projection in the gauge in which the Higgs current is a triangular matrix
==============================================================================================================
The Abelian projection, according to [@H1], is really the choice of a gauge-fixing in such a way that, after the gauge-fixing, the theory is no longer locally invariant under $SU(N)$ but only under its Cartan subgroup $U(1)^{N-1}$. The important point about this projection is that it is defined strictly locally, that is, the gauge rotation $\Omega$ performed at each point in space-time to implement the gauge-fixing condition, does not depend on the values of the physical fields in other points of space-time. This then guarantees that all observables in the new gauge frame are still locally observable. There are no propagating ghosts. But $\Omega$ is not completely defined. There is a subgroup, $U(1)^{N-1}$, of gauge rotations that may still be performed. And this is why the theory, after the Abelian projection, looks like a local $U(1)^{N-1}$ gauge theory.\
If one now tries to gauge-fix this remaining gauge freedom, one discovers that it cannot be done locally, without encountering apparent difficulties. But local gauge-fixing is not needed, since the residual gauge symmetry is the one of a familiar Abelian theory.\
There may be, however, isolated points, where the local gauge-fixing condition has coinciding eigenvalues, where the gauge symmetry is not $U(1)^{N-1}$ but a larger group. Here singularities appear, the magnetic monopoles. So we see that, topologically, the full theory can only be topologically equivalent to the $U(1)^{N-1}$ gauge theory if the latter is augmented with monopole singularities where the $U(1)$ conservation laws for the vortices are broken down into the (less restrictive) conservation laws of the $SU(N)$ vortices.\
When we try to gauge-fix completely, we hit upon the Dirac strings, whose end points are the magnetic monopoles.\
In addition to the magnetic monopoles, in the $QCD$ case, the gauge-fixed theory contains also gluon and quark fields, that are charged with respect to the residual $U(1)^{N-1}$.\
Therefore we have a set of electric charges and magnetic monopoles interacting via a residual $U(1)^{N-1}$ Abelian gauge coupling.\
We now compare this description with the one that arises in [@MB], for the pure gauge theory without quark matter fields.\
The functional integral for $QCD$ in [@MB] is defined in terms of the variables $(A_z, A_{\bar z}, \Psi_z, \Psi_{\bar z})$, obtained by means of a partial duality transformation from $(A_z, A_{\bar z}, A_u, A_{\bar u})$, where $(z, \bar z, u, \bar u)$ are the complex coordinates on the product of two two-dimensional tori, over which the theory is defined.\
$(A_z, A_{\bar z}, \Psi_z, \Psi_{\bar z})$ define the coordinates of an elliptic fibration of $T^* {\cal A}$, the cotangent bundle of unitary connections on the $(z, \bar z)$ torus, whose base is the $(u, \bar u)$ torus.\
$\Psi_z$ transforms as a field strength by gauge transformations and it is a non-hermitian matrix.\
Following Hitchin [@Hi], the gauge is chosen in which $\Psi_z$ is a triangular matrix, for example lower triangular, that leaves a $U(1)^{N-1}$ residual gauge freedom as in the Abelian projection.\
The points in space-time where $\Psi_z$ has a pair of coinciding eigenvalues, correspond to monopoles. In addition there are the charged components of $(A_z, A_{\bar z}, \Psi_z, \Psi_{\bar z})$. We have thus a set of charges and monopoles with a residual $U(1)^{N-1}$, according to the Abelian projection.\
In [@MB], however, it is found a dense set in the functional integral over (the elliptic fibration of) $T^* {\cal A}$, with the property that the quotient by the action of the gauge group exists as a Hausdorff (separable) manifold.\
This dense set is defined in [@MB] as the set of pairs $(A, \Psi)$ that are solutions of the following differential equations (elliptically fibered over the $(u, \bar u)$ torus): F\_A-i &=& \_p \^[0]{}\_[p]{} \_p i dz d|[z]{}\
|\_A &=& \_p \_[p]{} \_p dz d|[z]{}\
\_A | &=& \_p |\_[p]{} \_p d|[z]{} dz where $\delta_p$ is the two-dimensional delta-function localized at $z_p$ and $(\mu^{0}_{p},\mu_{p},\bar{\mu}_{p})$ are the set of levels for the moment maps. The moment maps are the three Hamiltonian densities generating gauge transformations on $T^* {\cal A}$ that appear in the left hand sides of Eq.(1) [@Hi1].\
$\mu^{0}_{p}$ are hermitian traceless matrices, and $\mu_{p}$ are matrices in the complexification of the Lie algebra of $SU(N)$, that determine the residues of the poles the Higgs current $\Psi$. $\psi$ and $\bar{\psi}$ are the $z$ and $\bar z$ components of the one-form $\Psi$.\
Eq.(1) defines a dense stratification of the functional integral over $T^* {\cal A}$ because the set of levels is dense everywhere in function space, in the sense of the distributions, as the divisor $D$ gets larger and larger.\
Eq.(1) defines the data of parabolic $K(D)$ pairs [@K] on a torus valued in the Lie algebra of the complexification of $SU(N)$: a holomorphic connection $\bar{\partial}_A$ of a holomorphic bundle, $E$, with a parabolic structure and a parabolic morphism $\psi$ of the parabolic bundle. The parabolic structure at a point $p$ [@MS; @K] consists in the choice of a set of ordered weights, that are positive real numbers modulo 1, and a flag structure, that is a collection of nested subspaces $ \cal{F}_{1}
\subset \cal{F}_{2} \subset...\cal{F}_{i}$ labelled by the weights $\alpha_1 \geq \alpha_2
\geq ...\alpha_k$, with the associated multiplicities defined as: $m_{i+1}=dim \cal{F}_{i+1}-dim
\cal{F}_{i}$. A parabolic morphism, $\phi$, is a holomorphic map between parabolic bundles, $E^1,E^2$, that preserves the parabolic flag structure at each parabolic point $p$ in the sense that $\alpha^{1}_{i} > \alpha^{2}_{j}$ implies $\phi( \cal{F}^{1}_{i}) \subset \phi(\cal{F}^{2}_{j+1})$. We should now explain how a parabolic structure arises from Eq.(1) and how it follows that $\psi$ is a parabolic morphism with respect to the given parabolic structure. Though we are going to choose the gauge in which $\psi$ is a lower triangular matrix in most of this paper, we start at an intermediate stage with a gauge in which $\mu^{0}_{p}$ is diagonal. The eigenvalues of $\mu^{0}_{p}$ modulo $2 \pi$ and divided by $2 \pi$ define the parabolic weights. Their multiplicities will turn out to be the multiplicities of the yet to be defined flag structure.\
Fixed $\mu^{0}_{p}$ and $\mu_p$ in Eq.(1), let $(e_k)$ be an orthonormal basis of the eigenvectors of $\mu^{0}_{p}$ in decreasing order. This basis is not necessarily unique if the eigenvalues have non-trivial multiplicities. However the corresponding flag structure will not be affected by this lack of uniqueness. Let $g$ be the gauge transformation that puts $\mu$ and $\psi$ into lower triangular form. Let $(g e_k)$ be the transformed basis and let $\cal F$ be the flag obtained by taking the unions of subspaces generated by the vectors in the transformed basis that are the images of eigenvectors of the ordered eigenvalues with the given multiplicity in such a way that the multiplicities of the resulting flag are the same as the multiplicities of the eigenvalues. In addition, by construction, $\psi$ is a parabolic morphism with respect to the flag since it is holomorphic and lower triangular in the basis $(ge_k)$.\
We have thus the data of a parabolic $K(D)$ pair from Eq.(1).\
There is also a representation theoretic interpretation of Eq.(1).\
The three equations for the moment maps are equivalent to a vanishing curvature condition for the non-hermitian connection one-form $B=A+i \Psi$ plus a harmonic condition for $\psi$ away from the parabolic divisor [@S].\
Therefore the set of solution of Eq.(1) can be figured out essentially as a collection of monodromies around the points of the divisor with values in the complexified gauge group, that form a representation of the fundamental group of the torus with the points of the parabolic divisor deleted.\
’t Hooft description of the Abelian projection previously outlined, applies to $T^* {\cal A}$ and to its dense subset defined by Eq.(1) a fortiori. In addition, we have just shown that there is an embedding of the solutions of Eq.(1) into the parabolic $K(D)$ pairs.\
However, on the parabolic $K(D)$ pairs, ’t Hooft concept of Abelian projection can be carried to its extreme consequences.\
Indeed, in the global algebraic-geometric framework of the Hitchin fibration [@Hi; @K] of parabolic $K(D)$ pairs, it is preferable to concentrate ourselves on the first eigenvalue and the first eigenstate of the lower triangular matrix $\Psi_z$, since all the information of the original parabolic bundle, up to gauge equivalence, can be reconstructed from these only data [@Hi].\
The first eigenvalue defines a spectral covering, that is a branched cover of the two-torus. The eigenspace defines a section of a line bundle, that determines a $U(1)$ connection on the cover of the torus, instead of the $U(1)^{N-1}$ bundle on the torus of the Abelian projection.\
The $U(1)$ connection on the cover, $a$, and the eigenvalue, $ \lambda$, of the Higgs current can be considered as coordinates of the cotangent bundle of unitary $U(1)$ connections on the cover, or as parabolic $K(D)$ pairs $(a, \lambda)$ on the cover, valued in the complexification of the Lie algebra of $U(1)$.\
The system is now completely abelianized. Correspondingly, not only the magnetic charges, but also the electric ones can occur only as gauge invariant topological configurations.\
The points in space-time where $\Psi_z$ has a pair of coinciding eigenvalues, that in the Abelian projection correspond to monopoles, are here, according to Hitchin, simple branching points of the spectral covers, defined by means of the characteristic equation: Det(1-\_[z]{})=0 , in which the coordinates $(u,\bar{u})$ are kept fixed.\
All the other branching points can be obtained by collision of these simple branching points, in the same way monopoles can in the Abelian projection. The branching points are the end points of string cuts on the Riemann surfaces, the Dirac strings of the Abelian projection.\
These Riemann surfaces, the only additional global ingredient with respect to the Abelian projection, are interpreted as the world sheets of strings made by electric flux lines.\
A closed string of electric flux is represented by a Wilson loop of the $U(1)$ connection $a$ on the cover, along a non-trivial generator of the fundamental group of the surface.\
In addition, the Riemann surfaces, defined by the spectral equation, may posses parabolic points, associated to poles of the eigenvalues of the Higgs current $\Psi_z$, whose origin is in the parabolic singularities of the original $su_{c}(N)$-valued $K(D)$ pair, which may be reflected into a parabolic structure for the $u_{c}(1)$-valued $K(D)$ pair on the cover.\
These poles, together with the ones of the $U(1)$ connection, are interpreted as electric charges. Indeed it is not difficult to see that they are electric sources, that appear where a boundary-electric loop shrinks to a point.\
Therefore, the electric charges occur here as topological objects associated to the parabolic degree [@MS] of the $u_{c}(1)$-valued $K(D)$ pair. On the other side, magnetic topological quantum numbers are associated, as usual, to the ordinary degree of the $U(1)$ bundle.\
We should mention however that a subtlety arises in our interpretation of the Hitchin fibration in terms of the Abelian projection. As we mentioned in the first part of this section, in the Abelian projection the gauge-fixing condition leaves a residual non-Abelian gauge symmetry where a magnetic monopole occurs. This is essentially due to the fact that ’t Hooft chooses to diagonalize a hermitian functional of the fields. On the contrary, in the case of the dense set defined by Eq.(1), since $\psi$ is a non-hermitian matrix, it can only be put in triangular form. This gauge-fixing does not leave in general a residual compact non-Abelian gauge symmetry even when the eigenvalues coincide. However this difficulty can be resolved in the following way, anticipating somehow some of the conclusions of this paper and the result of [@MB2]. Let us require for the moment that the levels of the non-hermitian moment maps be nilpotent. Since these are only $N$ conditions at each parabolic point they do not modify essentially the entropy of the functional integration in the large-$N$ limit. The true physical meaning of this choice has to do with confinement and it is explained in [@MB2]. If the residues of the Higgs field are nilpotent, Eq.(1) can be interpreted as the vanishing condition for the moment maps of the action of the compact $SU(N)$ gauge group on the pair $(A, \Psi)$ and on the cotangent space of coadjoint orbits [@Ale]: &&F\_A-i - \_p \^[0]{}\_[p]{} \_p i dz d|[z]{}=0\
&&|\_A - \_p n\_[p]{} \_p dz d|[z]{}=0\
&&\_A |- \_p |[n]{}\_[p]{} \_p d|[z]{} dz=0 In addition the quotient under the action of the compact gauge group is hyper-Kahler [@K]. By a general result of Hitchin, Karlhede, Lindström and Rocěk [@H2], the hyper-Kahler quotient under the action of the compact gauge group in Eq.(3) is the same as the quotient defined by the non-hermitian moment maps: &&|\_A - \_p n\_[p]{} \_p dz d|[z]{}=0\
&&\_A |- \_p |[n]{}\_[p]{} \_p d|[z]{} dz=0 under the action of the complexification of the gauge group. We can therefore impose a gauge condition compatible with the compact action in Eq.(3) or a gauge condition compatible with the action of the complexified group in Eq.(4) getting the same moduli space. In the second case we choose the gauge in which $\psi$ is diagonal. This condition becomes singular where two or more eigenvalues coincide. In fact it cannot be extended continuously to the points where the eigenvalues coincide. There it can only be required that $\Psi_z$ be a triangular matrix. However this condition leaves now a residual non-Abelian gauge symmetry in the complexification of the gauge group: the freedom of making triangular gauge transformations, thus confirming our analogy with ’t Hooft definition of magnetic monopoles.\
To summarize, the ingredients of the Hitchin fibration of the $su_{c}(N)$- valued $K(D)$ pairs, are the branching points, that are interpreted as magnetic monopoles, and the $U(1)$ monodromies around closed loops, that are interpreted as electric lines. In addition, the ordinary degree of the $U(1)$ bundle is interpreted as a (topological) magnetic charge, while the parabolic degree [@MS] of the $U(1)$ bundle is interpreted as a (topological) electric charge.\
The difference here, with the letter but not with the spirit of the Abelian projection, is that the system has been completely abelianized, so that both the magnetic and the electric charges are topological. We are thus given a set of charges and monopoles with a $U(1)$ gauge group on the covering, in analogy with the Abelian projection.\
We call this description a complete Abelian projection.\
The string interpretation is as follows. The spectral covers are the world sheets of strings, made by the electric flux lines. The confinement condition is equivalent to requiring that only closed string world sheet occur, since confinement requires that the flux lines can never break in absence of quarks.\
If the spectral covers posses parabolic points, the same as electric charges in the complete Abelian projection, they are, topologically, Riemann surfaces with boundaries at infinity.\
For example a sphere with two parabolic points is a topological cylinder.\
But a cylinder can occur in vacuum string world sheets (we are describing the contributions to the partition function, the vacuum to vacuum amplitude indeed) only if open strings propagate.\
In fact, a closed string that propagates through the torus breaks into an open one at the parabolic points, since the parabolic points do not belong to the world sheet.\
On the contrary, when a closed string meets a branching point, for example in a once-branched double cover of a torus, the closed string is pinched into another closed string with the form of a double loop intersecting at the (simple) branching point.\
Notice also that the branching points do belong to the world sheet.\
Thus, the string picture is consistent with the interpretation of branching points as magnetic charges, where the string electric line can self-intersect but not break, and of parabolic points as electric charges, where closed string break into open strings with the parabolic points as boundaries.
Conclusions
===========
Our conclusion is that the concept of Abelian projection in [@H1] furnishes a physical interpretation of the structures that appear in the Hitchin fibration of $K(D)$ pairs, as it is embedded in the $QCD$ functional integral in [@MB].\
In addition, there is a complementary consistent string interpretation.\
The most relevant consequence of these interpretations is a criterium for electric confinement in the framework of [@MB], that is the usual criterium of magnetic condensation of [@H1].\
Therefore, if $QCD$ confines the electric charge, the functional measure must be localized, in the large-$N$ limit, on those parabolic $K(D)$ pairs, whose image through the Hitchin map, contains monopoles but no charges, that is, in geometric language, on those spectral covers that are arbitrarily branched, but that do not posses a parabolic divisor.\
In turn, this is equivalent to the condition that only spectral covers spanned by closed strings occur as configurations in the vacuum to vacuum amplitude.\
It is amusing to notice that this condition is satisfied by the string of two-dimensional $QCD$ in the large-$N$ limit [@GT].
Acknowledgements
================
We would like to thank Gerard ’t Hooft for several clarifying remarks on the Abelian projection.
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| {
"pile_set_name": "ArXiv"
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---
abstract: 'Equivalence of convex optimization and variational inequality is well established in the literature such that the latter is formally recognized as a fixed point problem of the former. Such equivalence is also known to exist between a saddle-point problem and the variational inequality. The variational inequality is a static problem which can be further studied within the dynamical settings using a framework called the projected dynamical system whose stationary points coincide with the static solutions of the associated variational inequality. Variational inequalities have rich properties concerning the monotonicity of its vector-valued map and the uniqueness of its solution, which can be extended to the convex optimization and saddle-point problems. Moreover, these properties also extend to the representative projected dynamical system. The objective of this paper is to harness rich monotonicity properties of the representative projected dynamical system to develop the solution concepts of the convex optimization problem and the associated saddle-point problem. To this end, this paper studies a linear inequality constrained convex optimization problem and models its equivalent saddle-point problem as a variational inequality. Further, the variational inequality is studied as a projected dynamical system[@friesz1994day] which is shown to converge to the saddle-point solution. By considering the monotonicity of the gradient of Lagrangian function as a key factor, this paper establishes exponential convergence and stability results concerning the saddle-points. Our results show that the gradient of the Lagrangian function is just monotone on the Euclidean space, leading to only Lyapunov stability of stationary points of the projected dynamical system. To remedy the situation, the underlying projected dynamical system is formulated on a Riemannian manifold whose Riemannian metric is chosen such that the gradient of the Lagrangian function becomes strongly monotone. Using a suitable Lyapunov function, the stationary points of the projected dynamical system are proved to be globally exponentially stable and convergent to the unique saddle-point.'
author:
- |
P. A. Bansode[^1]\
V. Chinde[^2]\
S. R. Wagh[^3]\
R. Pasumarthy[^4]\
N. M. Singh[^5]
bibliography:
- 'dsvm\_pd.bib'
title: 'On the Exponential Stability of Projected Primal-Dual Dynamics on a Riemannian Manifold'
---
Introduction
============
The convex optimization methods have remained as the subject of substantial research for many decades. The primal-dual gradient-based method is one of such methods which dates back to late $1950s$[@arrow1958studies]. Lately, these methods (also referred to as primal-dual dynamics as its dynamical system equivalent) have found many applications in the networked systems (viz, the power networks [@zhao2014design; @mallada2017optimal; @yi2015distributed] and the wireless networks [@feijer2010stability; @chen2012convergence; @ferragut2014network]), and building automation systems [@kosaraju2018stability]. The primal-dual dynamics (PD dynamics) seek a solution to the saddle-point problem representing the original constrained convex optimization problem by taking gradient descent along the primal variable and gradient ascent along the dual variable. From the perspectives of systems and control theory, the PD dynamics have much to offer in terms of stability and convergence with respect to the saddle point solution.
During recent years the notions of stability of PD dynamics have evolved. The asymptotic stability of PD dynamics has been established as one of the most fundamental notions. Feijer [*et al.*]{}[@feijer2010stability] explores the PD dynamics with applications to network optimization problems and prove its asymptotic stability. The dual dynamics pertaining to the inequality constraints have been shown to include switching projections that restrict the dual variables to the set of nonnegative real numbers. Due to the switching projections, the PD dynamics becomes discontinuous, which is further modeled as a hybrid dynamical system. A Krasovskii-type Lyapunov function along with LaSalle invariance principle of hybrid systems [@lygeros2003dynamical] have been utilized to prove the asymptotic stability of the PD dynamics. In [@cherukuri2016asymptotic] it is proved that the PD dynamics is a special case of the projected dynamical systems. It uses the invariance principle of Carathéodory solutions to show that the saddle-point solution of PD dynamics is unique and globally asymptotically stable. Although widely established, the notion of asymptotic stability does not offer explicit convergence bounds of the PD dynamics which is an essential factor in case of the on-line optimization techniques. One must ensure that the trajectories converge to the saddle point solution in finite time. To explicitly obtain stronger convergence rates, research interests have shifted towards the notions of global exponential convergence and stability.
The pathway leading to the exponential stability of the PD dynamics is not as straightforward as it is for its asymptotic stability. The existence of right-hand side discontinuities and non-strongly monotone gradient of the associated Lagrangian function seem to prevent the saddle-point solution from being exponentially stable. The globally exponential stability has been the most desirable yet often formidable aspect of PD dynamics, which guarantees a minimum rate of convergence to the saddle point. While exhaustive literature on asymptotic stability of the PD dynamic can be encountered, its exponential stability has not been explored except for the recent studies [@cortes2018distributed; @nguyen2018contraction; @qu2019exponential; @dhingra2018proximal]. The optimization problem considered in [@cortes2018distributed] proves the exponential stability of the PD dynamic for an equality constrained optimization problem. Robustness and contraction analysis of the primal-dual dynamics establishing exponential convergence to the saddle-point solution is presented in [@nguyen2018contraction]. In [@qu2019exponential], the PD dynamics is proved to be globally exponentially stable for linear equality and inequality constrained convex optimization problem. Under assumptions on strong convexity and smoothness of the objective function and full row rank conditions of the constraint matrices, the PD dynamics is shown to have global exponential convergence to the saddle point solution. It mainly proposes the augmented Lagrangian function that results in a PD dynamics which does not have right-hand side discontinuities. By employing a quadratic Lyapunov function that has non-zero off-diagonal block matrices, it shows that the PD dynamics is globally exponentially stable. In [@dhingra2018proximal] a composite optimization problem is considered in which the objective function is represented as a sum of differentiable non-convex component and convex non-differentiable regularization component. A continuously differentiable proximal augmented Lagrangian is obtained by using a Moreau envelope of the regularization component. This results in a continuous-time PD dynamics which under the assumption of strong convexity of the objective function, is shown to be exponentially stable by employing a framework of the integral quadratic constraints (IQCs) [@587335]. By using a well-known result pertaining to linear systems with nonlinearities in feedback connection that satisfy IQCs [@hu2016exponential], it proves the global exponential stability of the PD dynamics.
Motivation and contribution
---------------------------
For a sufficiently small step size, a Euler discretized globally exponentially stable PD dynamics leads to geometric convergence to the saddle-point solution[@stuart1994numerical]. This property has been widely appreciated in recent articles such as [@dhingra2018proximal; @qu2019exponential]. The existing methods have considered the augmented Lagrangian techniques at a pivotal position for proving the globally exponential stability of the PD dynamics. This paper does not rely on augmented Lagrangian techniques to arrive at exponentially stable saddle-point solution. It presents a complementary approach that uses a combined framework of variational inequalities, projected dynamical systems [@nagurney2012projected], and the theory of Riemannian manifolds[@udriste1994convex; @da2002contributions] to derive conditions that lead to the global exponential stability of the saddle-point solution.
This paper exploits an equivalence between a constrained optimization problem and a variational inequality problem as discussed in [@kinderlehrer1980introduction; @facchinei2007finite; @nagurney2012projected]. They are shown to be equivalent when the vector-valued map of the variational inequality is the gradient of the objective function of the underlying optimization problem. Besides that, when the objective function is convex the Karush-Kuhn-Tucker (KKT) conditions of both problems reveal that the Lagrangian function associated with the variational inequality and the Lagrangian of the optimization problem have the exactly same saddle-point[@facchinei2007finite]. This further hints at formulating the saddle-point problem (of the corresponding optimization problem) as a variational inequality when both primal, as well as dual variables, are of interest.
Variational inequalities are equivalent to fixed point problems [@nagurney2012projected] in the sense that they yield only the static solutions. Thus the variational inequality formulation of the saddle-point problem would only result in the static description of the saddle-point. To understand the dynamic behavior of such variational inequality, this paper brings in the framework of projected dynamical systems[@friesz1994day; @nagurney2012projected]. The projected dynamical system combines essential features of both variational inequalities and dynamical systems such that its solution coincides with the static equilibrium of the variational inequality problem. These dynamical systems have interesting features which they derive from the underlying variational inequality problem. In [@gao2003exponential] a globally projected dynamical system [@friesz1994day] is proved to be exponential stable when the vector-valued mapping concerning the variational inequality problem is strongly monotone. This motivates to represent the saddle-point problem as a variational inequality and use the framework of the projected dynamical system for proposing a new dynamical system that is equivalent to the PD dynamics. Aiming at exponential stability of the saddle-point solution, this paper indirectly poses the saddle-point problem as a projected dynamical system (regarded hereafter as the projected primal-dual dynamics). While deriving the stability results of the proposed dynamics, our analysis reveals that the gradient of the Lagrangian function is not strongly monotone on the Euclidean space, which further deprives the proposed dynamics of being exponentially stable. Towards this end, our paper seeks a differential geometry which favors the desired properties such as strong monotonicity of the gradient of the Lagrangian and exponential stability of the proposed dynamics.
Convexity and monotonicity properties have strong connections[@karamardian1990seven]. Since Riemannian geometry is considerably the most natural framework for convexity[@udriste1994convex; @helmke2012optimization], it can also be explored for the monotonicity properties of the underlying gradient maps. There already exists a wide interest in optimization [@luenberger1972gradient; @da2002contributions; @absil2009optimization] and projected dynamical systems [@hauswirth2016projected; @hauswirth2018projected] over manifolds, which further motivates us to appreciate the Riemannian geometry for the proposed dynamics. For a linear inequality constrained convex optimization problem, our work establishes that the key to achieving an exponentially stable projected primal-dual dynamics is to choose a Riemannian metric that leads to strong monotonicity of the underlying gradient. With Lipschitz continuity of the gradient of the Lagrangian function, it is proved that the equilibrium solution of the projected primal-dual dynamics is globally exponentially stable.
The reported work envelopes following key contributions:
- The equivalence between a saddle point problem and a variational inequality problem is established and using the framework of the projected dynamical system, a projected PD dynamic is proposed.
- The restriction of the strongly monotone gradient of the Lagrangian function is overcome by proposing the projected PD dynamics constrained to a Riemannian manifold under a suitable Riemannian metric.
- The projected PD dynamics defined over a Riemannian manifold is proved to be globally exponentially stable for a linear inequality constrained convex optimization problem.
- The effectiveness of the proposed method is studied with the application of $L_2$ regularized least squares problem. The Euler discretized version of the proposed dynamics is shown to converge geometrically to the saddle-point solution.
Notations and Preliminaries {#prel}
---------------------------
The set $\mathbb{R}$ (respectively $\mathbb{R}_{\geq 0}$ or $\mathbb{R}_{> 0}$) is the set of real (respectively non-negative or positive) numbers. If $f: \mathbb{R}^n \rightarrow \mathbb{R}$ is continuously differentiable in $x \in \mathbb{R}^n$, then $\nabla_x f:\mathbb{R}^n \rightarrow \mathbb{R}^n$ is the gradient of $f$ with respect to $x$. ${\| . \|}$ denotes the Euclidean norm. For scalars $x,y$, $[x]^+_y:=x$ if $y>0$ or $x>0$, and $[x]^+_y:=0$ otherwise. For a set $X \subseteq \mathbb{R}^n$, the notation $\mathrm{relint}X$ defines the relative interior of $X$. The notation $\mathrm{P_1} \implies \mathrm{P_2}$ implies that the problem $\mathrm{P_2}$ can be derived by specializing the problem $\mathrm{P_1}$.
The following subsections provide preliminaries relevant to the main results of this paper.
### Convex Optimization, Variational Inequality, and Projected Dynamical System
(The Variational Inequality Problem, [@nagurney2012projected])\[vi\_Def\]\
For a closed convex set $X\in \mathbb{R}^n$ and vector function $F:X\rightarrow \mathbb{R}^n$, the finite dimensional variational inequality problem, $\mathrm{VI(F,X)}$, is to determine a vector $x^*\in X$ such that $$(x-x^*)^TF(x^*)\geq 0,~\forall x\in X. \label{vip}$$
The $\mathrm{VI(F,X)}$ is equivalent to solving a system of equations as given below:
(A System of Nonlinear Equations,[@nagurney2012projected])\[snle\]\
Let $F: X \rightarrow \mathbb{R}^n$ be a vector function. Then $x^* \in \mathbb{R}^n$ solves the variational inequality $\mathrm{VI(F,X)}$ if and only if $x^*$ solves the system of equations $$\begin{aligned}
F(x^*)=\mathbf{0}. \label{nle}
\end{aligned}$$
where $\mathbf{0}$ is zero vector of appropriate dimensions.
When the function $F$ is realized as a gradient of a real-valued function $f$, the following relationship between a variational inequality and an optimization problem is established.
(An Optimization Problem,[@nagurney2012projected])\[optpro\]\
Let $X\subset \mathbb{R}^n$ be closed and convex and $f:X \rightarrow \mathbb{R}$ be a continuously differentiable function. If $x^*\in X$ solves the optimization problem: $$\begin{aligned}
\min_{x\in X} f(x), \label{opt}
\end{aligned}$$ then $x^*$ solves the variational inequality problem $\mathrm{VI(\nabla f,X)}$. On the other hand, if $f(x)$ is a convex function and $x^*$ solves the $\mathrm{VI(\nabla f,X)}$, then $x^*$ is a solution to the optimization problem .
The optimization problem is denoted hereafter as $\mathrm{OPT(f,X)}$.
If convexity of $f$ holds, then the following relationship between $\mathrm{OPT(f,X)}$ and $\mathrm{VI(\nabla f,X)}$ can be derived. $$\begin{aligned}
\mathrm{VI(\nabla f,X)} \implies \mathrm{OPT(f,X)}. \label{r1}\end{aligned}$$ A variational inequality problem $\mathrm{VI(F,X)}$ is also equivalent to a fixed point problem as given below:
(A Fixed Point Problem,[@nagurney2012projected])\[fpppro\]\
$x^*$ is a solution to $\mathrm{VI(F,X)}$ if and only if for any $\alpha >0$, $x^*$ is a fixed point of the projection map: $$\begin{aligned}
x^*=P_X(x^*-\alpha F(x^*)) \label{fppprob}
\end{aligned}$$ where $$\begin{aligned}
P_{X}=\arg \min_{v\in X}{\| x-v \|}.\label{projection}
\end{aligned}$$
The fixed point problem, denoted by $\mathrm{FPP(F,X)}$ shares connection with the variational inequality problem $\mathrm{VI(F,X)}$ as described below. $$\begin{aligned}
\mathrm{FPP(F,X)} \implies \mathrm{VI(F,X)}.\label{r2}\end{aligned}$$
(Existence of Solution of Variational Inequality,[@nagurney2012projected]) \[thm2.1nagu\]\
If $X$ is compact and convex and $F(x)$ is continuous on $X$, then the variational inequality problem $\mathrm{VI(F,X)}$ admits at least one solution $x^*$.
The monotonicity properties of $F$ required in this paper are stated below:
[(Monotone Map,[@karamardian1990seven])]{}\[def1.1\]\
A mapping $F$ is monotone on $X \subseteq \mathbb{R}^n$, if for every pair of distinct points $x,y\in X$, we have $$(y-x)^T(F(y)-F(x)) \geq 0.$$
[(Strongly Monotone Map,[@karamardian1990seven])]{}\[def1.3\]\
A mapping $F$ is strongly monotone on $X \subseteq \mathbb{R}^n$, if there exists $\mu>0$ such that, for every pair of distinct points $x,y\in X$, we have $$(y-x)^T(F(y)-F(x)) \geq \mu {\| x-y \|}^2.$$
The relation between monotonicity of $F$ and positive definiteness of its Jacobian matrix $$\begin{aligned}
\nabla F(x)=\Bigg (\frac{\partial F_i(x)}{\partial x_j}\Bigg)_{i,j=1,2,\ldots,n},\end{aligned}$$ as given below.
((Strongly) Positive Definite Jacobian of $F(x)$ implies (Strongly) Monotone $F(x)$,[@nagurney2012projected])\[monoto\]\
Suppose that $F$ is continuously differentiable on $X$.
1. If the Jacobian matrix $\nabla F(x)$ is positive semidefinite, i.e., $$\begin{aligned}
y^T \nabla F(x) y \geq 0, \forall y \in \mathbb{R}^n, x \in X,
\end{aligned}$$ then $F$ is monotone on $X$.
2. If the Jacobian matrix $\nabla F(x)$ is positive definite, i.e., $$\begin{aligned}
y^T \nabla F(x) y > 0, \forall y \in \mathbb{R}^n, x \in X,
\end{aligned}$$ then $F$ is strictly monotone on $X$.
3. If $\nabla F(x)$ is strongly positive definite, i.e., $$\begin{aligned}
y^T\nabla F(x) y \geq \mu {\| z \|}^2,\forall y \in \mathbb{R}^n, x \in X,
\end{aligned}$$ then $F(x)$ is strongly monotone on $X$.
(Strongly Monotone $\nabla f$ implies Strongly Convex $f$,[[@karamardian1990seven]]{})\[prop1\]\
Let $f$ be a differentiable function on $D \subseteq \mathbb{R}^n$ that contains $X$. Then, $f$ strongly convex if and only if $\nabla f$ is strongly monotone on $X$.
(Strongly Convex Function[@karamardian1990seven])\[strong\_conv\_def\]\
A differentiable function $f$ is strongly convex on a domain $D\subseteq \mathbb{R}^n$ of $X$, if there exists $\mu>0$ such that, for all $x,y\in D$, $$\begin{aligned}
f(y)-f(x)\geq (y-x)^T\nabla f(x)+\frac{\mu}{2}{\| y-x \|}^2.\label{scvx}
\end{aligned}$$ $f$ is strongly concave, if $-f$ is strongly convex.
(Uniqueness of the Solution to Variational Inequality, [@nagurney2012projected])\[uniqueo\]\
Suppose that $F(x)$ is strongly monotone on $X$. Then there exists precisely one solution $x^*$ to $\mathrm{VI(F,X)}$.
Consider the following globally projected dynamical system proposed in [@friesz1994day], denoted here as $\mathrm{PDS(F,X)}$: $$\begin{aligned}
\dot{x}=\beta\{P_{X}[x-\alpha F(x)]-x\}\label{frz}\end{aligned}$$ where $k,\alpha$ are positive constants and $P_{X}:\mathbb{R}^{n}\rightarrow X$ is a projection operator as defined in .
(Equivalence between $\mathrm{PDS(F,X)}$ and $\mathrm{VI(F,X)}$, [@gao2003exponential])\[remark3\]\
$x^*$ is an equilibrium point of the $\mathrm{PDS(F,X)}$, if and only if $x^*$ is a solution of the variational inequality problem $\mathrm{VI(F,X)}$ defined in .
From Remark \[remark3\], $$\begin{aligned}
\dot{x}=0\implies x^* = P_X(x^*-\alpha F(x^*)).\end{aligned}$$ By using Proposition \[fpppro\], the result is immediate. The $\mathrm{PDS(F,X)}$ shares connection with the $\mathrm{VI(F,X)}$ as described below: $$\begin{aligned}
\mathrm{PDS(F,X)} \implies \mathrm{FPP(F,X)} \implies \mathrm{VI(F,X)}.\label{r3}\end{aligned}$$
\[lemma1.3\][[@xia2000stability]]{} Assume that $F$ is locally Lipschitz continuous in a domain $D$ that contains $X$. Then the solution $x(t)$ of will approach exponentially the feasible set $X$ when the initial point $x^0\notin X$. Moreover, if $x^0\in X$, then $x(t) \in X$.
[@gao2003exponential] Let $X$ be a nonempty, closed convex set, then the following holds:$$\begin{aligned}
[x-P_X(x)]^T[P_X(x)-y]\geq 0,\forall y \in X, x \in \mathbb{R}^{n}. \label{bapro}
\end{aligned}$$
Given a set $X \subset \mathbb{R}^n$ and $x \in X$, a vector $v \in \mathbb{R}^n$ is called an inward tangent vector of $X$ at $x$ if there exist a smooth curve $\gamma:[0,\xi]\rightarrow X$ such that $\xi \geq 0$, $\gamma(0)=x$, and $\gamma'_+(0)=v$.
The set of all inward tangent vectors is the tangent cone of $X$ at $x$ and denoted by $T_xX$.
Denote the boundary and interior of $X$, respectively, by $\partial X$ and $X^0$. If $x \in \partial X$, the set of inward normals to $X$ at $x$ is defined as the dual cone of $T_xX$, as follows:
$$\begin{aligned}
N_xX = \{\eta:{\| \eta \|}=1|\langle \eta^T,x-x' \rangle \leq 0, \forall x' \in T_xX\}.\end{aligned}$$
From Definition \[vi\_Def\], the necessary and sufficient condition for $x^*$ to be a solution to $\mathrm{VI(F,X)}$[@nagurney2012projected], is that $$\begin{aligned}
-F(x^*) \in N_{x^*}X.\end{aligned}$$
### Riemannian manifold
Let $\mathcal{M}$ be a differential manifold endowed with Riemannian metric $r$. Let $\mathcal{S}(\mathcal{M})$ denote the space of vector fields over $\mathcal{M}$. The tangent space of $\mathcal{M}$ at some point $x\in \mathcal{M}$ be $T_x\mathcal{M}$.
An inner product $r$ is defined as $\langle u,v \rangle_r:=r(u,v)$. In matrix form, $\langle u,v \rangle_r=u^TRv$ where $R$ is symmetric positive definite. The $2-$norm induced by $r$ is written as ${\| . \|}_r$ such that ${\| v \|}=\sqrt{\langle v,v \rangle_r}$. $r_x:T_x\mathcal{M} \times T_x\mathcal{M}\rightarrow \mathbb{R}$ is a smoothly chosen inner product on the tangent space $T_x\mathcal{M}$ of $\mathcal{M}$. For each $x \in \mathcal{M}$, $r = r_x$, satisfies the following:
1. $r(x_1,x_2)=r(x_2,x_1),\forall x_1,x_2 \in T_x\mathcal{M}$
2. $r(x,x)>0,\forall x\in T_x\mathcal{M}$
3. $r(x,x) = 0$ if and only if $x=0$.
(Riemannian Metric,[@rockafellar2009variational])\[def1.7\]\
Given a set $X\subseteq \mathbb{R}^n$, a Riemannian metric is a map $r:X\rightarrow L^n_2$ that assigns to every point $x \in X$ an inner product $\langle.,.\rangle_{r(x)}$. A metric is Lipschitz continuous if it is continuous as a map from $X$ to $L^n_2$.
If there exists a smooth function $f:\mathcal{M}\rightarrow \mathbb{R}$, and $x\in \mathcal{M}$, the differential $D_xf:T_x\mathcal{M}\rightarrow \mathbb{R}$ is defined as $$\begin{aligned}
D_xf(v)=(f \circ \gamma)'(v)\end{aligned}$$ where $\gamma(-\xi,\xi)$ is a smooth curve with $\gamma(0)=x$, and $\gamma'(0)=v$, $v\in T_x\mathcal{M}$. The gradient of $f$ at $x\in \mathcal{M}$ is defined as the unique tangent vector $\mathrm{grad}f$ such that $$\begin{aligned}
\langle \mathrm{grad}f,v\rangle_r=D_xf(v)\forall v \in T_x\mathcal{M}.\end{aligned}$$ In matrix notation, the gradient $\mathrm{grad}_rf=R^{-1}\nabla{f}^T$. A vector field $F$ is a map that assigns a vector $F(x)\in T_x\mathcal{M}$ to every point $x\in \mathcal{M}$.
Let $\nabla$ be a linear (Levi-Civita) connection and $Y$ be a $C^\infty$ vector field on $\mathcal{M}$, respectively. Then the connection $\nabla$ induces a covariant derivative with respect to $Y$, denoted by $\nabla_Y$.
The differential of $F\in \mathcal{S}(\mathcal{M})$ is a linear operator $\mathbb{H}_F:\mathcal{S}(\mathcal{M})\rightarrow\mathcal{S}(\mathcal{M})$, given by $\mathbb{H}_F(Y):=\nabla_Y(F)$. The linear map $\mathbb{H}_F(x):T_x\mathcal{M}\rightarrow\mathcal{M}$ assigned to each point $x\in \mathcal{M}$ is defined by $\mathbb{H}_F(x)v=\nabla_vF, \forall v \in T_x\mathcal{M}$.
If $F = \mathrm{grad}f$, where $f:\mathcal{M}\rightarrow \mathbb{R}$, then $\mathbb{H}_F(x)$ denotes the Hessian of $f$ at $x$.
(Generalization of Strongly Monotone Vector-valued Function on a Riemannian Manifold ,[@da2002contributions])\[prop1.5\]\
Let $\mathcal{M}$ be a Riemannian manifold and let $F$ be a vector field on $\mathcal{M}$. Then $F$ is strongly monotone if and only if there exists $\nu>0$ such that $\langle \mathbb{H}_{F(x)}v,v\rangle_r \geq \nu{\| v \|}^2_r$ for any $x\in \mathcal{M}$ and $v \in T_x\mathcal{M}$.
(Generalization of Strongly Convex Real-valued Function on a Riemannian Manifold,[@da2002contributions])\[prop1.6\]\
Let $\mathcal{M}$ be a Riemannian manifold, then the function $f:\mathcal{M}\rightarrow \mathbb{R}$ is strongly convex if and only if the gradient vector field $F=\mathrm{grad}f$ on $\mathcal{M}$ is strongly monotone.
Saddle-point problem as a variational inequality and projected primal-dual dynamics {#pf}
===================================================================================
Consider the following constrained optimization problem $$\setlength\arraycolsep{1.5pt}
\begin{array}{cc}
\mathrm{minimize} & f(x)\\
\mathrm{subject~to} & x \in X
\end{array} \label{cvx}$$ where $$\begin{aligned}
X &=\{x\in \mathbb{R}^n|g_i(x)\leq 0,\forall^m_{i=1}\},\label{constr_set} \end{aligned}$$ is the domain of the problem . The functions $f:\mathbb{R}^n\rightarrow\mathbb{R}$, $g:\mathbb{R}^n \rightarrow \mathbb{R}^m$ are assumed to be continuously differentiable $(\mathcal{C}^2)$ with respect to $x$, with the following assumptions:
\[ass1\] $\nabla f: \mathbb{R}^n\rightarrow \mathbb{R}^n$ is strongly monotone on $X$, with $\mu>0$ such that the following holds: $$(x_1-x_2)^T(\nabla f(x_1)-\nabla f(x_2)) \geq \mu {\| x_1-x_2 \|}^2.$$
As a consequence of Assumptions \[ass1\], it is derived that the objective function $f$ is strongly convex in $x$ with the modulus of convexity given by $\frac{\mu}{2}$.
\[ass2\] Constraints $g_i(x)$ are convex in $x,\forall^m_{i=1}$.
\[ass3\] There exists an $x \in \mathrm{relint}X$ such that $g_i(x)<0, \forall^{m}_{i=1}$.
Assumptions - ensure that $x$ is strictly feasible and strong duality holds for the optimization problem .
Note that the gradient function $\nabla g_i(x)$ need not be monotone.
Let $L:\mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}$ define the *Lagrangian* function of the optimization problem as given below $$\begin{aligned}
L(x,\lambda)=f(x)+\lambda^Tg(x). \label{lg}\end{aligned}$$ Let $\lambda_i$ be the Lagrange multipliers associated with $g_i(x)$, then $\lambda\in \Lambda \subseteq \mathbb{R}^m_+=\{\lambda \in \mathbb{R}^m,\lambda_i \geq 0, \forall^m_{i = 1}\}$ defines the corresponding vectors of Lagrange multipliers.
The Lagrangian function $L$ defined in is differentiable convex-concave in $x$ and $\lambda$ respectively, i.e., $L(.,\lambda)$ is convex for all $\lambda\in \Lambda$ and $L(x,.)$ is concave for all $x \in X$.
The saddle-point problem, denoted as $\mathrm{LP}(L,\Omega)$ (not to be confused with a linear programming problem usually designated as $\mathrm{LP}$), finds a pair $x^*\in X$ and $\lambda^*\in \Lambda$ such that the following holds: $$\begin{aligned}
L(x^*,\lambda)\leq L(x^*,\lambda^*)\leq L(x,\lambda^*).\label{spp}\end{aligned}$$ If $x^*$ is the unique minimizer of $L$, then it must satisfy the Karush-Kuhn-Tucker (KKT) conditions stated as follows. $$\begin{aligned}
g_i(x^*)&\leq 0,~\forall^{m}_{i=1} \label{k1}\\
\lambda^*_i&\geq 0,~\forall^{m}_{i=1} \label{k2}\\
\lambda^*_i g_i(x^*)&=0,~\forall^{m}_{i=1} \label{k3}\\
\nabla f(x^*)+\lambda^{*T}\nabla g(x^*)&=0. \label{k5}\end{aligned}$$
Let us define $z = (x,\lambda) \in \Omega = X \times \Lambda$ then $z^*=(x^*,\lambda^*)$ is the saddle point of the Lagrangian function defined in .
Since strong duality holds, the KKT conditions - are necessary and sufficient to guarantee optimality of the problem , with $x^*$ as the unique minimizer of and $z^*$ as the unique saddle-point of .
Let $G:\mathbb{R}^n \times \mathbb{R}^m \rightarrow\mathbb{R}^{n+m}$ define the gradient map of as given below: $$\begin{aligned}
G(z)&=\nabla_z L\\
&=\begin{bmatrix}
\nabla_x L(x,\lambda,\gamma)\\
-\nabla_\lambda L(x,\lambda,\gamma)
\end{bmatrix} \label{gu}\end{aligned}$$ While the primal-dual dynamics for an unconstrained optimization problem corresponds to simply $\dot{z}=-\nabla_z L = -G(z)$, the PD dynamics corresponding to the constrained optimization problem - is given by: $$\begin{aligned}
\dot{z}=\begin{bmatrix}
-\nabla_x L(x,\lambda,\gamma)\\
[\nabla_\lambda L(x,\lambda,\gamma)]^+_\lambda
\end{bmatrix} \label{pdd}\end{aligned}$$
It is well known that the solution of the PD dynamics coincides with the saddle point $z^*$ of the saddle-point problem [@arrow1958studies; @feijer2010stability]. This section describes the variational inequality based formulation of the saddle-point problem and proceeds to develop the projected primal-dual dynamics to solve the underlying variational inequality problem.
Saddle-point problem as a variational inequality
------------------------------------------------
The equivalence between variational inequality and constrained optimization problem has been well established in Proposition \[optpro\] and [@facchinei2007finite Theorem 1.3.1]\]. Using these properties, if $F = \nabla f$ holds, then the equivalence stated in proves that the optimal solution $x^*$ of the problem $\mathrm{OPT(f,X)}$ given in - can be obtained by formulating $\mathrm{OPT(f,X)}$ as a variational inequality $\mathrm{VI(\nabla f,X)}$ stated in . However, this solution concept holds only with respect to the primal optimizer $x^*$ which is necessary but not sufficient to develop the main results. The similar solution concept must be developed also for the saddle point $z^*$. To this end, this subsection establishes additional properties linking the saddle-point problem $\mathrm{LP}(L,\Omega)$ with the equivalent variational inequality problem.
To arrive at the variational inequality problem that is equivalent to the $\mathrm{LP}(L,\Omega)$, the following result from [@facchinei2007finite Proposition 1.3.4] is useful. $$\begin{aligned}
\mathbf{L}(x,\lambda)=\nabla_xL(x,\lambda) \label{dlL}\end{aligned}$$ where $\mathbf{L}(x,\lambda)$ is defined as the *vector valued* Lagrangian function of the variational inequality $\mathrm{VI(\nabla f,X)}$ stated below: $$\begin{aligned}
\mathbf{L}(x,\lambda)=F(x)+\lambda^T\nabla g(x),~\forall (x,\lambda)\in \Omega.\end{aligned}$$
From , it is obvious that for a given saddle point $z^*$, $$\begin{aligned}
\mathbf{L}(x^*,\lambda^*)=\nabla_{x^*}L(x^*,\lambda^*) \label{zstar}\end{aligned}$$
Using , an equivalence similar to can be obtained between the saddle-point problem $\mathrm{LP}(L,\Omega)$ and the corresponding variational inequality problem of the form $\mathrm{VI(\nabla L,\Omega)}$. In line with this, one can define the variational inequality problem $\mathrm{VI(\nabla L,\Omega)}$ by replacing $F$ by $\nabla L$ and $X$ by $\Omega$ in , as stated below.
\[lgvi\] For a closed convex set $\Omega \in \mathbb{R}^{n+m}$ and vector function $\nabla L:\Omega\rightarrow \mathbb{R}^{n+m}$, the finite dimensional variational inequality problem, $\mathrm{VI}(\nabla L,\Omega)$, is to determine a vector $z^*\in \Omega$ such that $$\small
(z-z^*)^T\nabla_z L(z^*)\geq 0,~\forall z\in \Omega. \label{mvi}$$ or equivalently $$\begin{aligned}
\small
(x-x^*)^T\nabla_x L(x^*,\lambda^*)\geq 0,~\forall x\in X.\\
(\lambda-\lambda^*)^T(-\nabla_\lambda L(x^*,\lambda))\geq 0,~\forall \lambda\in \Lambda.
\end{aligned}$$
Denote the boundary and interior of $\Omega$, respectively, by $\partial \Omega$ and $\Omega^0$. If $z \in \partial \Omega$, the set of inward normals to $\Omega$ at $z$ is defined as the dual cone of $T_z\Omega$, as follows:
$$\begin{aligned}
N_z\Omega = \{\eta:{\| \eta \|}=1|\langle \eta^T,z-z' \rangle \leq 0, \forall z' \in T_z\Omega\}.
\end{aligned}$$
where $T_z\Omega$ is the set of all inward tangent vectors.
From Definition \[lgvi\], the necessary and sufficient condition for $z^*$ to be a solution to $\mathrm{VI(\nabla L,\Omega)}$, is that $$\begin{aligned}
-G(z^*) \in N_{z^*}\Omega.
\end{aligned}$$
A geometric representation of the variational inequality $\mathrm{VI}(\nabla L,\Omega)$ is given in Fig. \[vipro\]. Using Proposition \[snle\], the $\mathrm{VI}(\nabla L,\Omega)$ can be proved to be equivalent to solving a system of nonlinear equations, as stated below.
\[snlez\] Let $\nabla L: \Omega \rightarrow \mathbb{R}^{n+m}$ be a vector function. Then $z^* \in \mathbb{R}^{n+m}$ solves the variational inequality $\mathrm{VI}(\nabla L,\Omega)$ if and only if $z^*$ solves the system of equations $$\begin{aligned}
\nabla L(z^*)=\mathbf{0}.\end{aligned}$$ or equivalently (from ) $$\begin{aligned}
G(z^*) = \mathbf{0}\label{nlez}.
\end{aligned}$$
![A geometric representation of the variational inequality $\mathrm{VI}(\nabla L,\Omega)$.[]{data-label="vipro"}](geometry_vip.png){width="3.0in"}
In a similar way, a fixed point problem corresponding to the $\mathrm{VI}(\nabla L,\Omega)$ can be obtained as given below.
\[lgvifpp\] $z^*$ is a solution to $\mathrm{VI(\nabla L,\Omega)}$ if and only if for any $\alpha >0$, $z^*$ is a fixed point of the projection map: $$\begin{aligned}
z^*=P_\Omega(z^*-\alpha \nabla_z L(z^*)) \label{fppprobz}
\end{aligned}$$ where $$\begin{aligned}
P_{\Omega}=\arg \min_{v\in \Omega}{\| z-v \|}.\label{projectionz}
\end{aligned}$$
Denote the fixed point problem in Proposition \[lgvifpp\] as $\mathrm{FPP}(\nabla L,\Omega)$, it follows from the equivalence stated in that $$\begin{aligned}
\mathrm{FPP}(\nabla L,\Omega) \implies \mathrm{VI}(\nabla L,\Omega).\label{same route}\end{aligned}$$ Further one can use [@konnov2002theory Corollary 1.1] to show that the problems , , and are equivalent such that the following holds: $$\begin{aligned}
\small
\mathrm{FPP}(\nabla L,\Omega) \implies \mathrm{VI}(\nabla L,\Omega) \implies \mathrm{LP}(L,\Omega) .\label{same routelp}\end{aligned}$$ The equivalence result confirms that the solution $z^*$ of the $\mathrm{VI}(\nabla L,\Omega)$ is the saddle point $z^*$ of saddle-point problem $\mathrm{LP}(L,\Omega)$.
Projected primal-dual dynamics
------------------------------
The framework of finite dimensional variational inequalities studies only the equilibrium solutions, which in a way uses to arrive at a static representation of the system at its steady state. The dynamic representation of the system which shall follow the equivalence stated in , must be developed to understand how the variable of interest, i.e., $z$ converges to the solution $z^*$ of the $\mathrm{VI}(\nabla L,\Omega)$. A dynamical model that represents the $\mathrm{VI}(\nabla L,\Omega)$ is widely known as [“the projected dynamical system”]{} (PDS). $$\begin{aligned}
\dot{z}=\beta\{P_{\Omega}[z-\alpha G(z)]-z\},\label{mpdd}\end{aligned}$$ where $\beta>0$. By invoking Remark \[remark3\] from preliminaries, the $\mathrm{VI}(\nabla L,\Omega)$ can be expressed as a $\mathrm{PDS}(\nabla L,\Omega)$ and the following equivalence can be derived. $$\begin{aligned}
\small
\mathrm{PDS}(\nabla L,\Omega) {\mathrel{\raisebox{0.3ex}{\scalebox{\@scaleFactorImplies}{\ensuremath{\Longrightarrow}}}}}\mathrm{FPP}(\nabla L,\Omega) {\mathrel{\raisebox{0.3ex}{\scalebox{\@scaleFactorImplies}{\ensuremath{\Longrightarrow}}}}}\mathrm{VI}(\nabla L,\Omega) {\mathrel{\raisebox{0.3ex}{\scalebox{\@scaleFactorImplies}{\ensuremath{\Longrightarrow}}}}}\mathrm{LP}(L,\Omega).
\label{same routelpds}\end{aligned}$$
Towards this end, the $\mathrm{PDS}(\nabla L,\Omega)$ is regarded as the projected PD dynamics whose solution converges to the saddle point solution of the saddle-point problem $\mathrm{LP}(L,\Omega)$. Since $\Omega$ is a closed and convex set, using Theorem \[thm2.1nagu\] and Theorem \[uniqueo\] from preliminaries, the existence and the uniqueness of $z^*$ are guaranteed. With this, the next section proceeds towards the stability analysis of the projected PD dynamics.
Stability analysis
------------------
In what follows, monotonicity property of $G(z)$ is explored and it is proved that the projected PD dynamics is Lyapunov stable.
\[thm1\] If Assumptions - hold, then $G(z)$ is monotone such that $$\begin{aligned}
[G(z_1)-G(z_2)]^T(z_1-z_2)\geq 0 \label{ineq1}
\end{aligned}$$ for every pair of $z_1,z_2\in \Omega$.
The Jacobian matrix of $G$ is derived as follows: $$\begin{aligned}
\small
\nabla G = \begin{bmatrix}
\nabla^2 f(x) + \lambda^T\nabla^2 g(x) & \nabla g(x)^T\\
-\nabla g(x) & \mathbf{0}
\end{bmatrix}
\end{aligned}$$ Recall from Proposition \[monoto\], $G$ is monotone if and only if the $\nabla G$ is positive semidefinite. By $\nabla G$ positive semidefinite, it is meant that $$\begin{aligned}
\frac{1}{2}\nabla G+\frac{1}{2} \nabla G^T \geq 0,\forall z\in \Omega,\forall t.\label{psd}
\end{aligned}$$
Inequality can be easily verified by checking the symmetric part of $\nabla G$: $$\begin{aligned}
\frac{\nabla G + \nabla G^T}{2} &= \begin{bmatrix}
\nabla^2 f(x) + \lambda^T\nabla^2 g(x) & \mathbf{0}_{n \times m}\\
\mathbf{0}_{m \times n} & \mathbf{0}_{m \times m}
\end{bmatrix}\\
&\geq \mathbf{0}_{(n+m) \times (n+m)},\forall z\in \Omega,\forall t\label{psd1}
\end{aligned}$$ where $\mathbf{0}$ is a zero matrix of appropriate dimensions. From it is verified that the $\nabla G$ is a $(n+m) \times (n+m)$ matrix which is rank deficient by $m$ rows. Thus it is only a positive semi-definite matrix. Thus from Proposition \[monoto\], it can be concluded that $G$ is a monotone.
\[thm3.5.1\] Let ${G}(z)$ be continuously differentiable on open convex subset of $\mathbb{R}^{n+m}$. If Assumptions \[ass1\]-\[ass3\] hold and $G$ is monotone for all $z\in \Omega$, then with $\alpha>0$, the projected PD dynamics is Lyapunov stable.
For the Lagrangian function , the following inequalities always hold: $$\begin{aligned}
{L}(x^*,\lambda^*)-{L}(x^*,\lambda)\geq 0\\
{L}(x,\lambda^*)-{L}(x^*,\lambda^*)\geq 0
\end{aligned}$$ Let us define the Lyapunov function as follows: $$\begin{aligned}
V(z) &= ({L}(x^*,\lambda^*)-{L}(x^*,\lambda))+({L}(x,\lambda^*)-{L}(x^*,\lambda^*))\nonumber\\
&~~~+\frac{1}{2}{\| z-z^* \|}^2.\label{Vz}
\end{aligned}$$ The last term in ensures that $V(z) \geq \frac{1}{2}{\| z-z^* \|}^2,\forall z \in \Omega$, thus also ensures the boundedness of the level sets of $V(z)$.
Differentiating $V(z)$ with respect to time $t$ yields: $$\begin{aligned}
\dot{V}(z)&=\nabla V(z)\dot{z}\nonumber\\
&=-[(\nabla {L}(x,\lambda^*)-\nabla {L}(x^*,\lambda))+z-z^*]^T(z-\tilde{z})\nonumber\\
&=-[G(z)+z-z^*]^T(z-\tilde{z}) \label{ff}
\end{aligned}$$ Substituting $x = z-\alpha {G}(z)$ and $y=z^*$ in [@gao2003exponential], yields $$\begin{aligned}
[z-z^*+\alpha G(z)]^T(z-\tilde{z})\geq {\| z-\tilde{z} \|}^2+\alpha (z-z^*)^T{G}(z). \label{use1}
\end{aligned}$$ Using in yields, $$\begin{aligned}
\dot{V}(z)&\leq -(z-z^*)^T{G}(z)-{\| z-\tilde{z} \|}^2\nonumber\\
&\leq -(z-z^*)^T(G(z)-G(z^*))-{\| z-\tilde{z} \|}^2
\end{aligned}$$ From , $$\begin{aligned}
\dot{V}(z)\leq 0 \label{leq1}
\end{aligned}$$ and ensure that the Lyapunov function is non-increasing along , hence the projected dynamical system is Lyapunov stable.
The Lyapunov stability result from Lemma \[thm3.5.1\] also holds for the optimization problem of the form with linear inequality constraints as defined in the set $X =\{x\in \mathbb{R}^n|Ax\leq b\}$.
From Proposition \[monoto\] and Lemma \[lemma1.3\], it is observed that the projected PD dynamics does not achieve exponential (asymptotic) stability since the gradient $G$ is not strongly (strictly) monotone on the Euclidean space. Thus to obtain desired stability results, a geometry other than the Euclidean geometry must be considered. The key to achieving a strongly monotone gradient and a globally exponentially stable PD dynamics is to formulate the problem on a Riemannian manifold as discussed in the subsection below.
Projected primal-dual dynamics over a Riemannian manifold
=========================================================
Proposition \[prop1.5\] and \[prop1.6\] establish the relation between monotonicity and convexity on Riemannian manifolds. While the gradient of the Lagrangian function is just monotone on the Euclidean space, using rich metric properties of the Riemannian manifolds, it can be modified to become strongly monotone. On a Riemannian manifold, the key to obtaining a strongly monotone gradient function is to define a Riemannian metric that yields one. Following this reasoning, in this subsection, a linear inequality constrained convex optimization problem is considered over a Riemannian manifold and the Riemannian metric is chosen such that the gradient of the Lagrangian function is strongly monotone. Using the relation between strong monotonicity and uniqueness of the solution, under the assumption of Lipschitz continuity, the projected PD dynamics is shown to have globally exponentially stable saddle-point solution.
Considers the following optimization problem $$\begin{aligned}
\min ~&f(x)\nonumber\\
\mathrm{subject~to}~&x \in X \label{lincvx}\end{aligned}$$ where $X =\{x\in \mathbb{R}^n|Ax\leq b\}$, and $A \in \mathbb{R}^{m \times n}$.
\[as4\] Let matrix $A$ have full row rank $m\leq n$ and $q_1I\leq AA^T \leq q_2I$, where $I$ is an identity matrix and $q_1,q_2$ are positive constants.
The Lagrangian function for the problem is given by $$\begin{aligned}
L(z)=f(x)+\lambda^T(Ax-b)\label{linlag}\end{aligned}$$where $z= (x,\lambda)$, $z\in \Omega=X \times \Lambda$, $x\in X$, $\lambda\in \Lambda \subseteq \mathbb{R}^m_{\geq 0}$.
The gradient of the Lagrangian is obtained as: $$\begin{aligned}
G(z)&=\nabla_z L\\
&=\begin{bmatrix}
\nabla f(x)+A^T\lambda\\
-(Ax-b)
\end{bmatrix}.\label{gud}\end{aligned}$$
Strongly monotone gradient of the Lagrangian
--------------------------------------------
Consider a smooth manifold $\mathcal{M}\subseteq\mathbb{R}^{m+n}$ and let $\mathcal{M}$ be endowed with Riemannian metric $r$. Define the Lagrangian function $L:\mathcal{M}\rightarrow \mathbb{R}$ then the gradient of $L$ at $z\in \mathcal{M}$ is the unique tangent vector $\mathrm{grad} L$ given as $$\begin{aligned}
\langle \mathrm{grad} L,v\rangle_r=D_zL(v),\forall v\in T_z\mathcal{M}. \label{lgre}\end{aligned}$$ In the matrix notation, implies the following $$\begin{aligned}
\mathrm{grad}_r L = R^{-1}\nabla L^T \label{gradL}\end{aligned}$$ where $\nabla L = G(z)$ is the gradient vector of $L$ on Euclidean space $\mathbb{R}^{m+n}$.
Denote $G_r(z)=\mathrm{grad}_rL$, the differential of $G_r(z)\in \mathcal{S}(\mathcal{M})$ is a linear operator $\mathbb{H}_{G_r}:\mathcal{S}(\mathcal{M})\rightarrow\mathcal{S}(\mathcal{M})$, given by $\mathbb{H}_{G_r}(Y):=\nabla_Y(G_r)$. The linear map $\mathbb{H}_{G_r(z)}:T_z\mathcal{M}\rightarrow\mathcal{M}$ assigned to each point $z\in \mathcal{M}$ is defined by the Hessian of $L$, denoted by $\mathbb{H}_{G_r}(z)v=\nabla_v G_r(z)=R^{-1}\nabla G(z),\forall v\in T_z\mathcal{M}$.
The projection operator $P^r_\mathcal{M}:\mathbb{R}^{n+m}\rightarrow \mathcal{M}$ defined as $$\begin{aligned}
P^r_\mathcal{M}(z)=\arg\min_{v\in T_z\mathcal{M}}{\| z-v \|}^2_r.\end{aligned}$$ Correspondingly, the projected PD dynamics on $\mathcal{M}$ is defined as follows: $$\begin{aligned}
\dot{z}=\beta\{P^r_\mathcal{M}[z-\alpha G_r(z)]-z\}.\label{mpddr}\end{aligned}$$ Let the Riemannian metric $R$ be chosen as given below: $$\begin{aligned}
R = \begin{bmatrix}
kI & A^T\\
A & kI
\end{bmatrix}^{-1},\label{P}\end{aligned}$$ where $$\begin{aligned}
k \geq \sqrt{q_2} \label{kcond0}\end{aligned}$$ meets the positive definiteness of the matrix $R$. The gradient vector $G_r(z) \in T_z\mathcal{M}$ is given by $$\begin{aligned}
G_r(z) &= R^{-1}G(z),\nonumber\\
&=\begin{bmatrix}
k\nabla f(x)-A^TAx+kA^T\lambda+A^Tb\\
A\nabla f(x)-kAx+AA^T\lambda+kb
\end{bmatrix}\label{newgrad}\end{aligned}$$
In the following section it is proved that the gradient map is strongly monotone.
\[thm3.4\] Consider the problem and let $(\mathcal{M},r)$ be a $n+m$-dimensional smooth manifold. If Assumption \[ass1\] and \[as4\] hold for the problem , then with the linear map $R^{-1}:T_z\mathcal{M}\rightarrow T_z\mathcal{M}$, the gradient vector $G_r(z)$ is strongly monotone.
Recall from Proposition \[prop1.5\] that for $G_r(z)$ to be strongly monotone, $\nabla_z {G}_r$ must be positive definite, i.e., for the symmetric part of $\nabla_z {G}_r$, i.e. $\frac{1}{2}\nabla {G}_r+\frac{1}{2}\nabla {G}^T_r$, the following must hold: $$\begin{aligned}
\small
\nabla {G}_r+\nabla {G}^T_r&=R^{-1}\nabla G+\nabla G^TR^{-1},\nonumber\\
&\geq \nu \mathrm{I},\forall z \in \mathcal{M},\forall t\label{gradg}
\end{aligned}$$ where $\nu>0$ is a constant, $\mathrm{I}$ is an identity matrix of appropriate dimensions.
The Jacobian of $G_r(z)$, denoted by $\nabla G_r(z)$ is given below: $$\begin{aligned}
\small
\nabla G_r(z) = \begin{bmatrix}
k\nabla^2 f(x) - A^TA & kA^T\\
A\nabla^2f(x)-kA & AA^T
\end{bmatrix}
\end{aligned}$$ The symmetric part of $\nabla G_r(z)$ is obtained as: $$\begin{aligned}
\small
\frac{\nabla {G}_r(z)+\nabla {G}^T_r(z)}{2} = \begin{bmatrix}
k\nabla^2 f(x) - A^TA & \frac{1}{2}(A\nabla^2f(x))^T\\
\frac{1}{2}A\nabla^2f(x) & AA^T
\end{bmatrix}\label{nablaGr}
\end{aligned}$$ Let $\mathrm{M}=\nabla {G}_r(z)+\nabla {G}^T_r(z)-q_1\mathrm{I}>0$. Then $$\begin{aligned}
\mathrm{M} &= \begin{bmatrix}
2k\nabla^2 f(x) - 2A^TA -q_1\mathrm{I} & (A\nabla^2f(x))^T\\
A\nabla^2f(x) & 2AA^T-q_1\mathrm{I}
\end{bmatrix}\nonumber\\
&\geq \begin{bmatrix}
2k\nabla^2 f(x) - 2A^TA -q_1\mathrm{I} & (A\nabla^2f(x))^T\\
A\nabla^2f(x) & AA^T
\end{bmatrix}\label{Ma}.
\end{aligned}$$ Further let $\mathrm{S}=AA^T$, then the Schur compliment of the block $\mathrm{S}$ of the matrix $\mathrm{M}$, denoted by $\mathrm{S}_{\mathrm{Schur}}$ is derived as $$\begin{aligned}
\small
\mathrm{S}_{\mathrm{Schur}} &= 2k\nabla^2 f(x) - 2A^TA-q_1I\nonumber\\
&-(A\nabla^2f(x))^T(AA^T)^{-1}A\nabla^2f(x).\label{m/s}
\end{aligned}$$ Let $\mathrm{H}=\nabla^2f(x)$ for the notational simplicity. Note that in , $2k\mathrm{H}>0,\forall k>0$, $2A^TA \geq 0$, $q_1I>0$, and $0\leq \mathrm{H}A^T(AA^T)^{-1}A\mathrm{H} \leq \mathrm{H}^2$. The last terms is a consequence of $A^T(AA^T)^{-1}A\leq I$. Rearranging as given below $$\begin{aligned}
\small
2k\mathrm{H} &> 2A^TA+q_1I+\mathrm{H}A^T(AA^T)^{-1}A\mathrm{H}\nonumber\\
2k\mathrm{H} &> 2A^TA+q_1I+\mathrm{H}^2\label{pqrs}
\end{aligned}$$ allows to choose $k$ such that $\mathrm{S}_{\mathrm{Schur}}>0$. Post multiplying by $(2H)^{-1}$ yields the following: $$\begin{aligned}
\small
2kI &> 2A^TA(2H)^{-1}+q_1(2H)^{-1}+\mathrm{H}^2(2H)^{-1}\nonumber\\
kI &> A^TAH^{-1}+0.5q_1H^{-1}+0.5\mathrm{H}\label{kcond}.\end{aligned}$$
Applying Courant-Fischer theorem [@horn1990matrix] to yields the following: $$\begin{aligned}
\small
\lambda_{max}(kI)>\lambda_{max}(A^TAH^{-1}+0.5q_1H^{-1}+0.5\mathrm{H}).\label{mkcond}
\end{aligned}$$ Since $\lambda_{max}(kI) = k$, has the following form: $$\begin{aligned}
\small
k>\lambda_{max}(A^TAH^{-1}+0.5q_1H^{-1}+0.5\mathrm{H}).\label{mkcond1}
\end{aligned}$$ By choosing $k$ as given in ensures that $\mathrm{S}_{\mathrm{Schur}}>0$. But $k$ must also satisfy , thus $k$ must be chosen such that the following holds: $$\begin{aligned}
k > \max\{\sqrt{q_2},\lambda_{max}(A^TAH^{-1}+0.5q_1H^{-1}+0.5\mathrm{H})\} \label{choosek}
\end{aligned}$$ ensures that both and are met. If $k$ is chosen according to , then $\mathrm{S}_{\mathrm{Schur}}>0$ holds such that there exists a $\nu \geq \frac{q_1}{2}$ which implies that $$\begin{aligned}
\langle \mathbb{H}_{G_r(z)}v,v\rangle_r \geq \nu{\| v \|}^2_r,\forall v \in T_z\mathcal{M}.
\end{aligned}$$ Thus by using Proposition \[prop1.5\], the following is derived $$\begin{aligned}
\small
\langle {G}_r(z_1)-{G}_r(z_2),z_1-z_2\rangle_r\geq \nu {\| z_1-z_2 \|}^2_r. \label{ffinal}
\end{aligned}$$ Hence it is proved that $G_r(z)$ is strongly monotone.
Exponential stability
---------------------
Without loss of generality, let us define $G_r(z)$ similar to as follows: $$\begin{aligned}
G_r(z)=\begin{bmatrix}
\nabla^r_x {L}(x,\lambda)\\
-\nabla^r_\lambda {L}(x,\lambda)
\end{bmatrix}, \label{gut}\end{aligned}$$ where ${L}(x,\lambda)$ would represent the modified Lagrangian function whose gradient vector field is given by $G_r(z)$. Since, $G_r(z)$ is strongly monotone on $\mathcal{M}$, will converge to a unique saddle-point solution $z^*$.
\[thm3.5\] Let $G_r(z)$ be Lipschitz continuous on $D$, then inequality and $\alpha>0$, imply that the system with $z(0) \in \mathcal{M}$ is globally exponentially stable at the unique solution $z^*$ of .
For each $z(0) \in \mathcal{M}$, there exists a unique solution $z(t)$ of , that started from $z(0)$. If $[0,t_f)$ is the maximal interval of $z(t)$, then from Lemma \[lemma1.3\], $z(t) \in \mathcal{M}$ for all $t \in [0,t_f)$. Since $G_r(z)$ is strongly monotone, the following holds: $$\begin{aligned}
L(x^*,\lambda^*)-L(x^*,\lambda)> 0\\
L(x,\lambda^*)-L(x^*,\lambda^*)> 0
\end{aligned}$$ Let us define the Lyapunov function for the dynamic as follows: $$\begin{aligned}
V_1(z) &= (L(x^*,\lambda^*)-L(x^*,\lambda))+(L(x,\lambda^*)-L(x^*,\lambda^*))\nonumber\\
&~~~+\frac{1}{2}{\| z-z^* \|}^2_r.
\end{aligned}$$ It is to be noted that $V_1(z)$ possesses a similar structure as that of $V(z)$ defined in , it is also differentiable convex on $\mathcal{M}$, with $V_1(z) \geq \frac{1}{2}{\| z-z^* \|}^2_r,\forall z \in \mathcal{M}$, thus bounding all level sets of $V_1(z)$.
Differentiating $V_1(z)$ with respect to time $t$ yields: $$\begin{aligned}
\dot{V}_1(z)&=\nabla V_1(z)\dot{z}\nonumber\\
&=-\langle\nabla L(x,\lambda^*)-\nabla L(x^*,\lambda)+z-z^*,z-\tilde{z}\rangle_r\nonumber\\
&=-\langle G_r(z)+z-z^*,z-\tilde{z}\rangle_r \label{eff}
\end{aligned}$$ Substituting $x = z-\alpha G_r(z)$ and $y=z^*$ in [@gao2003exponential], yields $$\begin{aligned}
\langle z-z^*+\alpha G_r(z),z-\tilde{z}\rangle_r\geq {\| z-\tilde{z} \|}^2_r+\langle\alpha (z-z^*),G_r(z)\rangle_r. \label{euse1}
\end{aligned}$$ Using in yields, $$\begin{aligned}
\dot{V}_1(z)&\leq -\langle\alpha (z-z^*),G_r(z)\rangle_r.\label{V1zdot}
\end{aligned}$$ If $k$ is chosen such that the condition is satisfied then $G_r(z)$ is strongly monotone. Using Proposition from the preliminary section, the strong monotonicity of $G_r(z)$ also leads to the strong convexity of the Lagrangian function $L(z),\forall z\in \mathcal{M}$, which implies that there exists a unique saddle point $z^*\in \mathcal{M}$, i.e. $\mathcal{M}^* = z^*$. Hence the following inequality can be obtained: $$\begin{aligned}
\langle z-&z^*,G_r(z)\rangle_r\geq L(x,\lambda^*)-L(x^*,\lambda)+\frac{\nu}{2}{\| z-z^* \|}^2_r, z \in \mathcal{M}.\label{strong conv}
\end{aligned}$$
Using , modifies to the following $$\begin{aligned}
\dot{V}_1(z)&\leq -\langle\alpha (z-z^*),G_r(z)\rangle_r\nonumber,\\
&\leq-\alpha \beta[L(x,\lambda^*)-L(x^*,\lambda)+\frac{\nu}{2}{\| z-z^* \|}^2_r],\nonumber\\
&\leq-\alpha \beta[(L(x^*,\lambda^*)-L(x^*,\lambda))\nonumber\\
&~~~+(L(x,\lambda^*)-L(x^*,\lambda^*))+\frac{\nu}{2}{\| z-z^* \|}^2_r].
\end{aligned}$$ With $\alpha,\beta>0$, it can be shown that, $$\begin{aligned}
\dot{V}_1(z)\leq -\beta\min\{1,\alpha\nu\}V(z).
\end{aligned}$$ Thus, it is proved that the system is exponentially stable at the unique solution $z^*$ of . Therefor, $$\begin{aligned}
{\| z-z^* \|}_r\leq ce^{-\beta\frac{\min\{1,\alpha\nu\}}{2}t}
\end{aligned}$$ where $c=\sqrt{2V_1(z(0))}$.
Further, if $G_r(z)$ is Lipschitz continuous on $\mathcal{M}$, i.e., ${\| {G}_r(z_1)-{G}_r(z_2) \|}_r\leq \ell{\| z_1-z_2 \|}_r,\forall z_1,z_2\in \mathcal{M}$. By using [@gao2003exponential Theorem 4], the global exponential stability of the projected PD dynamics can be derived: $$\begin{aligned}
{\| z(t)-z^* \|}_r\leq {\| z(0)-z^* \|}_re^{\frac{-\alpha \beta(4\nu-\alpha \ell^2)}{8}t},\forall t \geq 0. \label{conva}
\end{aligned}$$ If $\alpha<\frac{4\nu}{\ell^2}$, it follows that the projected PD dynamic is globally exponentially stable.
Simulation Results
==================
This section presents simulation studies of the projected PD dynamics . It is known that the Euler discretization of the exponentially stable dynamical system owns geometric rate of convergence [@stuart1994numerical] for sufficiently small step-sizes. The projected PD dynamics is Euler discretized with a step size $s>0$ and the following discrete-time projected PD dynamics[@nagurney2012projected] is obtained. $$\begin{aligned}
z(\tau+1)=\beta P^r_\mathcal{M}\{z(\tau)-\alpha G_r(\tau)\}.\end{aligned}$$
First example (Example 1) considers an optimization problem of the form with $m=5$ and $n=10$. The Hessian matrix is assumed to be $H = 20{I}$ with $A$ and $b$ taken as Gaussian random matrix and vector respectively. The distance to the equilibrium point i.e. $z^*=(x^*,\lambda^*)$ for different values of parameter $k$ is shown separately in Fig. \[eps1\] and \[eps12\], where $\varrho=\max\{\sqrt{q_2},\lambda_{max}(A^TAH^{-1}+0.5q_1H^{-1}+0.5\mathrm{H})\}$. It can be seen from the plots that the rate of convergence to the equilibrium points accelerates as the value of $k$ is increased. It implies that increasing the value of $k$ allows increasing the value of $\nu$, which further increases the coefficient of the negative exponential term in . The primal optimizers $x^*$ of the problem are also compared to the optimal solution of the same problem obtained using [“quadprog”]{} solver in MATLAB environment as shown in Fig. \[eps2\].
In the second example, an $L_2$ regularized least squares problem is considered with $m=30$ and $n=50$. The objective function is $f(x) = {\| Cx-d \|}^2_2+\frac{\theta}{2}{\| x \|}^2_2$ with $\theta>0$, constrained to $Ax\leq b$. Matrices $(C,A) \in \mathbb{R}^{m \times n}$, and vectors $(d,b)\in \mathbb{R}^{m \times 1}$ are Gaussian random matrices and vectors, respectively. Parameters $\alpha,\beta$ are chosen as unity and the proposed dynamics is simulated for $k = 1000\max(\varrho)$. A sketch of the error norm as a function of time is shown in Fig. \[eps3\]. It can be seen that the error norm ${\| x_i-x^*_i \|}^2$ has geometric rate of convergence.
![Distance to the primal optimizer $x^*$ for different values of $k$ (Example 1).[]{data-label="eps1"}](k_Vs_iterations_plot.eps){width="5in"}
![Distance to the dual optimizer $\lambda^*$ for different values of $k$ (Example 1).[]{data-label="eps12"}](k_Vs_iterations_plot_lambda.eps){width="5in"}
![Optimal solution compared with “[QUADPROG]{}" (MATLAB) solver.[]{data-label="eps2"}](optimizers.eps){width="5in"}
![Distance to the primal optimizer $x^*$ ($L_2$ regularized least squares problem).[]{data-label="eps3"}](fig4.eps){width="5in"}
Conclusions and discussion {#concl}
==========================
This paper has proposed a projected dynamical system based formulation of the saddle point problem to solve a constrained convex optimization problem. Monotonicity properties of the gradient of the underlying Lagrangian function are evaluated. It is found that this gradient map is just monotone on the Euclidean space which restricts the proposed dynamics from being globally exponentially stable. This has lead to a saddle-point solution of the proposed dynamics which is only Lyapunov stable. It confirmed that the desired property concerning strong monotonicity of the underlying gradient map ceases to exist on the Euclidean space. Due to which the exponential stability of the proposed dynamics cannot be obtained. It further compelled to search for a differential geometry where such properties are obtainable. To this end, our results proved that the proposed dynamics is exponentially stable on a Riemannian manifold whose Riemannian metric is chosen such that the underlying gradient is strongly monotone. Then it is shown that the exponential stability holds globally under the Lipschitz continuity of the gradient map. However, the analysis pertains to a linear inequality constrained convex optimization problem.
There are many network-based optimization problems that fall under the category of linear inequality constrained optimization, one of such problems is distributed support vector machines[@forero2010consensus; @stolpe2016distributed] which is solved in a distributed manner over a network of nodes acquiring valuable statistics. The results of this paper can be further extended to such problems.
The proposed approach can also be generalized to a convex optimization problem with convex inequality constraints under regularity conditions. However, it will not be a straightforward extension of the present work. It is expected that the underlying mathematical framework would require additional properties concerning convexity of the inequality constraints which is left as future scope of this paper.
[^1]: P. A. Bansode is with department of Instrumentation Engineering, Ramrao Adik Institute of Technology, Mumbai, 400706 India. [prashant.bansode@rait.ac.in]{}
[^2]: V. Chinde is with National Renewable Energy Laboratory, Lakewood, CO 80401, USA.
[^3]: S. R. Wagh is with department of Electrical Engineering, Veermata Jijabai Technological Institute, Mumbai, 400019 India.
[^4]: R. Pasumarthy is with department of Electrical Engineering, Indian Institute of Technology Madras, Madras, 600036 India.
[^5]: N. M. Singh is with department of Electrical Engineering, Veermata Jijabai Technological Institute, Mumbai, 400019 India.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We show that within the inverse seesaw mechanism for generating neutrino masses minimal supergravity is more likely to have a sneutrino as the lightest superparticle than the conventional neutralino. We also demonstrate that such schemes naturally reconcile the small neutrino masses with the correct relic sneutrino dark matter abundance and accessible direct detection rates in nuclear recoil experiments.'
author:
- 'C. Arina'
- 'F. Bazzocchi'
- 'N. Fornengo'
- 'J. C. Romao'
- 'J. W. F. Valle'
title: Minimal supergravity sneutrino dark matter and inverse seesaw neutrino masses
---
Introduction
============
Over the last fifteen years we have had solid experimental evidence for neutrino masses and oscillations [@Maltoni:2004ei], providing the first evidence for physics beyond the Standard Model. On the other hand, cosmological studies clearly show that a large fraction of the mass of the Universe in dark and must be non–baryonic.
The generation of neutrino masses may provide new insight on the nature of the dark matter [@Berezinsky:1993fm]. In this Letter we show that in a minimal supergravity (mSUGRA) scheme where the smallness of neutrino masses is accounted for within the inverse seesaw mechanism the lightest supersymmetric particle is likely to be represented by the corresponding neutrino superpartner (sneutrino), instead of the lightest neutralino. This opens a new window for the mSUGRA scenario. Here we consider the implications of the model for the dark matter issue. We demonstrate that such a model naturally reconciles the small neutrino masses with the correct relic abundance of sneutrino dark matter and experimentally accessible direct detection rates.
Minimal SUGRA inverse seesaw model {#Model}
==================================
Let us add to the Minimal Supersymmetric Standard Model (MSSM) three sequential pairs of 1 singlet neutrino superfields ${\widehat
\nu^c}_i$ and $\widehat{S}_i$ ($i$ is the generation index), with the following superpotential terms [@mohapatra:1986bd; @Deppisch:2004fa], $${\cal W} = {\cal W_{\rm MSSM}} +\varepsilon_{ab}\,
h_{\nu}^{ij}\widehat L_i^a\widehat \nu^c_j\widehat H_u^b
+ M_{R}^{ij}\widehat \nu^c_i\widehat S_j
+\frac{1}{2}\mu_S^{ij} \widehat S_i \widehat S_j
\label{eq:Wsuppot}$$ where ${\cal W_{\rm MSSM}}$ is the usual MSSM superpotential. In the limit $\mu_S^{ij} \to 0$ there are exactly conserved lepton numbers assigned as $(1,-1,1)$ [@mohapatra:1986bd; @Deppisch:2004fa] for $\nu$, $\nu^{c}$ and $S$, respectively.
The extra singlet superfields induce new terms in the soft–breaking Lagrangian: $$\begin{aligned}
\mathcal{-L}_{\rm soft} &=& \mathcal{-L}_{\rm soft}^{\rm MSSM} +
\tilde{\nu}^c_i\ \mathbf{M^2_{\nu^c}}_{ij} \tilde{\nu}^c_j + \tilde{S}_i\,
\mathbf{M^2_{S}}_{ij} \tilde{S}_j \\
& & + \varepsilon_{ab}\,
A_{h_{\nu}}^{ij} \tilde{L}_i^a \tilde{\nu}^c_j H_u^b
+ B_{M_{R}}^{ij} \tilde{\nu}^c_i \tilde{S}_j
+\frac{1}{2} B_{\hat{\mu_S}}^{ij} \tilde{S}_i \tilde{S}_j \nonumber
\label{eq:soft}\end{aligned}$$ where $\mathcal{L}_{\rm soft}^{\rm MSSM}$ is the MSSM SUSY–breaking Lagrangian.
Small neutrino masses are generated through the inverse seesaw mechanism [@mohapatra:1986bd; @Deppisch:2004fa; @Nunokawa:2007qh]: the effective neutrino mass matrix $m^{\rm eff}_{\nu}$ is obtained by the following relation: $$\label{eq:1}
m^{\rm eff}_{\nu}= -v_u^2 h_{\nu} \left(M_R^T\right)^{-1} {\mu_S}
M_R^{-1} h_{\nu}^T = \left(U^T\right)^{-1} m_{\mu}^{\rm diag}\ U^{-1}$$ where $h_\nu$ defines the Yukawa matrix and $v_u$ is the $H_u$ vacuum expectation value. The smallness of the neutrino mass is ascribed to the smallness of the $\mu_S$ parameter, rather than the largeness of the Majorana–type mass matrix $M_R$, as required in the standard seesaw mechanism [@Nunokawa:2007qh]. In this way light (eV scale or smaller) neutrino masses allow for a sizeable magnitude for the Dirac–type mass $m_D=v_u h_\nu$ and a TeV–scale mass for the right-handed neutrinos, features which have been shown to produce an interesting sneutrino dark matter phenomenology [@Arina:2007tm].
The main feature of our model is that the nature of the dark matter candidate, its mass and couplings all arise from the same sector responsible for the generation of neutrino masses. In order to illustrate the mechanism we consider the simplest one-generation case, for simplicity. In this case where the sneutrino mass matrix reads $$\begin{aligned}
\mathcal{M}^2 =
\begin{pmatrix}
\mathcal{M}^2_+ & \mathbf{0}\cr
\mathbf{0} & \mathcal{M}^2_-\cr
\end{pmatrix}
$$ where the two sub–matrices $\mathcal{M_\pm}^2$ are:
$$\begin{aligned}
\label{eq:snumatrix}
\mathcal{M_{\pm}}^2 =
\begin{pmatrix}
m^2_L+\frac{1}{2} m^2_Z \cos 2\beta+m^2_D & \pm (A_{h_{\nu}}v_u-\mu m_D {\rm cotg} \beta) & m_D M_R\cr
\pm (A_{h_{\nu}}v_u-\mu m_D {\rm cotg}\beta) & m^2_{\nu^c}+M_R^2+m^2_D & \mu_S M_R \pm B_{M_R}\cr
m_D M_R & \mu_S M_R \pm B_{M_R} & m^2_S+\mu^2_S+M^2_R\pm B_{\mu_S}
\end{pmatrix}
$$
in the CP eigenstates basis: $\Phi^{\dag} = (\snu_{+}^\ast
\,\tilde{\nu}_{+}^{c\ast} \, \tilde{S}_+^\ast \,\, \snu_-^\ast \,
\tilde{\nu}_-^{c\ast} \, \tilde{S}_-^\ast)$. Once diagonalized, the lightest of the six mass eigenstates is our dark matter candidate and it is stable by $R$–parity conservation.
A novel supersymmetric spectrum
===============================
![Supersymmetric particle spectrum in the standard MSUGRA scheme \[panel (a)\] and in the inverse seesaw mSUGRA model \[panel (b)\] with parameters chosen as: $m_0= 358$ GeV, $m_{1/2}= 692$ GeV, $A_0
= 0$, $\tan\beta=35$ and sign $\mu >0$. The sneutrino sector has the additional parameter $B_{\mu_S}$, fixed at 10 GeV$^2$. The squark sector is not shown. []{data-label="fig:spectrum"}](tower2_new.eps){width="\columnwidth"}
![The $m_{0}-m_{1/2}$ plane for $\tan\beta=35$, $A_0=0$ and $\mu>0$. The red and yellow areas denote the set of supersymmetric parameters where the sneutrino is the LSP in inverse seesaw models (notice that it includes all the yellow region where the $\tilde{\tau}$ is the LSP in the standard mSUGRA case). The white region has the neutralino as LSP in both standard and modified mSUGRA. For the sneutrino LSP region, the additional parameters are: $B_{\mu_S}= 10 {\, {\rm GeV}}^2$, $M_R=500 {\, {\rm GeV}}$, $m_D=50 {\, {\rm GeV}}$ and $\mu_S=1$ eV. The blue region is excluded (see text). []{data-label="fig:m0mh12"}](mhalfm0_new.eps){width="\columnwidth"}
Let us now consider the model within a minimal SUGRA scenario. In the absence of the singlet neutrino superfields, the mSUGRA framework predicts the lightest supersymmetric particle (LSP) to be either a stau or a neutralino, and only the latter case represents a viable dark matter candidate. In most of the mSUGRA parameter space, however, the neutralino relic abundance turns out to exceed the WMAP bound [@Komatsu:2008hk] and hence the cosmologically acceptable regions of parameter space are quite restricted.
In contrast, when the singlet neutrino superfields are added, a combination of sneutrinos emerges quite naturally as the LSP. Indeed, we have computed the resulting supersymmetric particle spectrum and couplings by adapting the SPheno code [@Porod:2003um] so as to include the additional singlet superfields. An illustrative example of how the minimal supergravity particle spectrum is modified by the presence of such states is given in Fig. \[fig:spectrum\]. This figure shows explicitly how a sneutrino LSP is in fact realized.
A more general analysis in the mSUGRA parameter space is shown in Fig. \[fig:m0mh12\]: the dark (blue) shaded area is excluded either by experimental bounds on supersymmetry and Higgs boson searches, or because it does not lead to electroweak symmetry breaking, while the (light) yellow region refers to stau LSP in the conventional (unextended) mSUGRA case. As expected, in all of the remaining region of the plane, the neutralino is the LSP in the standard mSUGRA case. The new phenomenological possibility which opens up thanks to the presence of the singlet neutrino superfields where the sneutrino is the LSP corresponds to the full mid-gray (red) and light (yellow) areas. In what follows we demonstrate that in this region of parameter space such a sneutrino reproduces the right amount of dark matter and is not excluded by direct detection experiments.
Sneutrino LSP as Dark Matter
============================
![Sneutrino relic abundance $\Omega h^2$ as a function of the LSP sneutrino mass $m_1$, for a full scan of the supersymmetric parameter space: $100 {\, {\rm GeV}}< m_0 < 3 {\, {\rm TeV}}$, $100 {\, {\rm GeV}}< m_{1/2}< 3
{\, {\rm TeV}}$, $1 {\, {\rm GeV}}^2 <B_\mu < 100 {\, {\rm GeV}}^2$, $A_0=0$, $3 < \tan\beta<50$, $10^{-9} {\, {\rm GeV}}<\mu_S<10^{-6} {\, {\rm GeV}}$. The yellow band delimits the WMAP [@Komatsu:2008hk] cold dark matter interval at 3 $\sigma$ of C.L.: $0.104 \leq \Omega_{\rm{CDM}} h^2 \leq 0.124$.[]{data-label="fig:omega"}](relic_inv_wmap5.eps){width="\columnwidth"}
![Sneutrino–nucleon scattering cross section $\xi \sigma^{\rm
(scalar)}_{\rm nucleon}$ vs. the sneutrino relic abundance $\Omega
h^{2}$, for the same scan of the supersymmetric parameter space given in Fig. \[fig:omega\]. The horizontal \[light blue\] band denotes the current sensitivity of direct detection experiments; the vertical \[yellow\] band delimits the 3 $\sigma$ C.L. WMAP cold dark matter range [@Komatsu:2008hk].[]{data-label="fig:direct"}](relic_direct1.eps){width="\columnwidth"}
The novelty of the spectrum implied by mSUGRA implemented with the inverse seesaw mechanism is that it may lead to a bosonic dark matter candidate, the lightest sneutrino $\tilde{\nu}_1$, instead of the fermionic neutralino. To understand the physics it suffices for us to consider the simple one sneutrino generation case [^1]. The relic density of the sneutrino candidate is shown in Fig. \[fig:omega\]. The lightest mass eigenstate is also a CP eigenstate and coannihilates with the NLSP, a corresponding heavier opposite–CP sneutrino eigenstate. We notice that this situation provides a nice realization of inelastic dark matter, a case where the dark matter possesses a suppressed scattering with the nucleon, relevant for the direct detection scattering cross section, shown in Fig. \[fig:direct\].
From Fig. \[fig:omega\] we see that a large fraction of the sneutrino configuration are compatible with the WMAP cold dark matter range, and therefore represents viable sneutrino dark matter models. Fig. \[fig:direct\] in addition shows that direct detection experiments do not exclude this possibility: instead, a large fraction of configurations are actually compatible and under exploration by current direct dark matter detection experiments. This fact is partly possible because of the inelasticity characteristics we have mentioned above, which reduces the direct detection cross section to acceptable levels [@Arina:2007tm].
We stress that all models reported in Figs. \[fig:omega\] and \[fig:direct\] have the inverse seesaw-induced neutrino masses consistent with current experimental observations for natural values of its relevant parameters. We also note that the lepton–number violating parameter $B_{\mu_S}$, which determines the lightest mass sneutrino eigenstate and its couplings, also has an impact on the neutrino sector, since it can induce one-loop corrections to the neutrino mass itself (for details, see Ref. [@Arina:2007tm] and references therein). These corrections must be small, in order not to go into conflict with the bounds on neutrino masses, and this in turn implies that the mass splitting between the sneutrino LSP and sneutrino NLSP is small (less than MeV or so) [@Arina:2007tm], implying the inelasticity of the sneutrino scattering with nuclei [@Arina:2007tm]. The parameter $\mu_s$ therefore plays a key role in controlling the neutrino mass generation, the sneutrino relic abundance and the direct detection cross section.
In conclusion, in this Letter we have presented an mSUGRA scenario in which neutrino masses and dark matter arise from the same sector of the theory. Over large portions of the parameter space the model successfully accommodates light neutrino masses and sneutrinos dark matter with the correct relic abundance indicated by WMAP as well as direct detection rates searches consistent with current dark matter searches. The neutrino mass is generated by means of an inverse seesaw mechanism, while in a large region of parameters the dark matter is represented by sneutrinos. The small superpotential mass parameter $\mu_S$ controls most of the successfull phenomenology of both the neutrino and sneutrino sector. In the absence of $\mu_S$ neutrinos become massless, Eq. (\[eq:1\]). The bilinear superpotential term $\mu_S^{ij} \widehat S_i \widehat
S_j$ could arise in a spontaneous way in a scheme with an additional lepton-number-carrying singlet superfield $\sigma$, implying the existence of a majoron [@gonzalez-garcia:1988rw]. In this case, the dominant decays of the Higgs boson are likely to be into a pair of majorons [@Joshipura:1992hp]. Such invisible mode would be “seen” experimentally as missing momentum, but the corresponding signal did not show up in the LEP data [@Abdallah:2003ry]. Although hard to catch at the LHC such decays would provide a clean signal in a future ILC facility. Similarly, the standard bilinear superpotential term $\mu H_u H_d$ present in the minimal supergravity model could also be substituted by a trilinear, in a NMSSM-like scheme [@CerdenoMunoz:2008].
Note that our proposed scheme may also have important implications for supersymmetric particle searches at the LHC, due to modified particle spectra and decay chains. Additional experimental signatures could be associated with the (quasi-Dirac) neutral heavy leptons formed by $\nu^c$ and $S$, whose couplings and masses are already restricted by LEP searches [@Dittmar:1989yg; @Abreu:1996pa].
[*Acknowledgements.*]{} We warmly thank M. Hirsch for stimulating discussions. This work was supported by MEC grant FPA2005-01269, by EC Contracts RTN network MRTN-CT-2004-503369 and ILIAS/N6 RII3-CT-2004-506222, by FCT grant POCI/FP/81919/2007 and by research grants funded jointly by the Italian Ministry of Research and by the Istituto Nazionale di Fisica Nucleare (INFN) within the [*Astroparticle Physics Project*]{}.
[10]{}
For an updated review of neutrino oscillations see archive version of M. Maltoni, T. Schwetz, M. A. Tortola and J. W. F. Valle, New J. Phys. [**6**]{}, 122 (2004), \[hep-ph/0405172\]. V. Berezinsky and J. W. F. Valle, Phys. Lett. [**B 318**]{} (1993) 360; E. K. Akhmedov, Z. G. Berezhiani, R. N. Mohapatra and G. Senjanovic,Phys. Lett. B [**299**]{} (1993) 90 \[hep-ph/9209285\]; M. Lattanzi and J. W. F. Valle, Phys. Rev. Lett. [**99**]{}, 121301 (2007), \[0705.2406\]; F. Bazzocchi, M. Lattanzi, S. Riemer-Sorensen and J. W. F. Valle, 0805.2372. R. N. Mohapatra and J. W. F. Valle, Phys. Rev. [**D34**]{}, 1642 (1986). F. Deppisch and J. W. F. Valle, Phys. Rev. [**D72**]{}, 036001 (2005), \[hep-ph/0406040\]. H. Nunokawa, S. J. Parke and J. W. F. Valle, Prog. Part. Nucl. Phys. [**60**]{}, 338 (2008), \[arXiv:0710.0554 \[hep-ph\]\], this review gives an updated discussion of the standard seesaw mechanism and its variants. C. Arina and N. Fornengo, JHEP [**11**]{}, 029 (2007), \[arXiv:0709.4477\]. WMAP collaboration, E. Komatsu [*et al.*]{}, arXiv:0803.0547. W. Porod, Comput. Phys. Commun. [**153**]{}, 275 (2003), \[hep-ph/0301101\]. M. C. Gonzalez-Garcia and J. W. F. Valle, Phys. Lett. [**B216**]{}, 360 (1989). A. S. Joshipura and J. W. F. Valle, Nucl. Phys. [**B397**]{}, 105 (1993). See, e. g. DELPHI collaboration, J. Abdallah [*et al.*]{}, Eur. Phys. J. [**C32**]{}, 475 (2004), \[hep-ex/0401022\]. D. Cerdeno, talk at DSU08, Cairo, June 2008.
M. Dittmar, A. Santamaria, M. C. Gonzalez-Garcia and J. W. F. Valle, Nucl. Phys. [**B332**]{}, 1 (1990). See, e. g. DELPHI collaboration, P. Abreu [*et al.*]{}, Z. Phys. [**C74**]{}, 57 (1997).
[^1]: We adopt the same approximation used in the relic density calculation within the standard minimal mSUGRA model, which we have checked holds in our case as well.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- Tianming Liu
- Haoyu Wang
- Li Li
- Xiapu Luo
- Feng Dong
- Yao Guo
- Liu Wang
- 'Tegawend[é]{} F. Bissyand[é]{}'
- Jacques Klein
bibliography:
- 'cite.bib'
title: 'MadDroid: Characterizing and Detecting Devious Ad Contents for Android Apps'
---
<ccs2012> <concept> <concept\_id>10002978.10003022</concept\_id> <concept\_desc>Security and privacy Software and application security</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10002951.10003260.10003272</concept\_id> <concept\_desc>Information systems Online advertising</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10003120.10003138</concept\_id> <concept\_desc>Human-centered computing Ubiquitous and mobile computing</concept\_desc> <concept\_significance>500</concept\_significance> </concept> </ccs2012>
Conclusion
==========
In this paper, we perform a large-scale characterization study of mobile ad content, which has been largely overlooked by the research community. We first create a comprehensive categorization of devious mobile ad contents, then we build [*MadDroid*]{}, a framework for automated detection of devious mobile ad contents. By applying [*MadDroid*]{} to 40,000 Android apps, we find that devious ad contents are prevalent: 6% of apps in our study are identified as delivering devious ad contents. To the best of our knowledge, [*MadDroid*]{} is the first attempt towards mitigating threats from both ad-load and ad-click introduced by mobile ad contents.
Acknowledgment {#acknowledgment .unnumbered}
==============
This work was partly supported by the National Natural Science Foundation of China (No.61702045 and No.61772042), by the Hong Kong RGC Projects (No.152223/17E, CityU C1008-16G), by the Australian Research Council (ARC) under projects DE200100016 and DP200100020, by the Fonds National de la Recherche (FNR), Luxembourg, under project CHARACTERIZE C17/IS/11693861, by the SPARTA project which has received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement No 830892.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'A short summary of constraints on the parameter space of supersymmetric models is given. Experimental limits from high energy colliders, electroweak precision data, flavor and Higgs physics, and cosmology are considered. The main focus is on the MSSM with conserved R- and CP-parity and minimal flavor violation, but more general scenarios and extended models will also be discussed briefly.'
author:
- 'A. Freitas'
title: |
\
Status of Constraints on Supersymmetry
---
INTRODUCTION
============
The purpose of this contribution is to summarize the constraints on supersymmetric models from various experimental results. Due to the large wealth of experimental searches for physics beyond the standard model (SM) and phenomenological studies on supersymmetry (SUSY) it is impossible to cover all of them in this short review. Thus the author apologizes that many valuable studies are not mentioned or cited in this report.
To set the scene, a short review of the most widely studied SUSY models is given in the next section. The following sections discuss constraints on the parameter space of these models from high energy colliders, electroweak precision data, flavor and Higgs physics, and cosmology, respectively. Finally, some qualitative comments on more general SUSY models are presented before the summary.
SUSY MODELS
===========
The most extensively studied SUSY model is the Minimal Supersymmetric Standard Model (MSSM), with the particle content listed in Table \[mssm\]. In addition the MSSM imposes R-parity, assigning $R_p = +1$ for the Higgs boson, gauge bosons, leptons, and quarks, and $R_p = -1$ for their supersymmetric partners (neutralinos, charginos, gluino, sleptons, and squarks). As a result the superpotential has the form $$W_{\rm MSSM} = y_{\rm u} \hat{Q} \cdot \hat{H}_2 \, \hat{U}^c
+ y_{\rm d} \hat{Q} \cdot \hat{H}_1 \, \hat{D}^c
+ y_{\rm e} \hat{L} \cdot \hat{H}_1 \, \hat{E}^c
- \mu \hat{H}_1 \cdot \hat{H}_2 \,.
\label{wmssm}$$ For a general introduction to the MSSM and notational definitions, see [*e.$\,$g.*]{} Ref. [@martin].
In the major part of this work, an even more minimal version of the MSSM is assumed where the CKM matrix is the only source of CP violation and flavor violation. In other words, the SUSY breaking parameters are assumed to be real and flavor blind.
**Spin 0** **Spin 1/2** **Spin 1**
----------------------------------------------------------------- ------------------------------------------------------- -----------------
Neutral Higgses Neutralinos Photon $\gamma$
$h_0,\,H_0,\,A_0$ ${\tilde{\chi}^0}_1 \dots {\tilde{\chi}^0}_4$ $Z$ boson
\[.5ex\] Charged Higgs $H^\pm$ Chargino ${\tilde{\chi}}^\pm_1, {\tilde{\chi}}^\pm_2$ $W^\pm$ bosons
\[.5ex\] Gluino $\tilde{g}$ gluon $g$
\[.5ex\] sleptons $\tilde{e}$, $\tilde{\mu}$, $\tilde{\nu}$,... leptons $e$, $\mu$, $\nu$, ...
squarks $\tilde{u}$, $\tilde{d}$, ... quarks $u$, $d$, ...
: Particle content of the MSSM
\[mssm\]
This still leaves more than one dozen [*a priori*]{} unknown SUSY breaking parameters. Many experimental searches and phenomenological analyses thus consider specific SUSY breaking scenarios:
- *mSUGRA/CMSSM:* In *minimal Supergravity* (mSUGRA) or the *constrained MSSM* (CMSSM) the scale of SUSY breaking is situated near the scale of gauge coupling unification, $M_{\rm GUT} \approx 2\times 10^{16}{{\rm \ GeV}}$. At this scale, there is one common mass parameter each for the gauginos, scalars and triple-scalar couplings ($A$-terms), respectively. At lower energies, a more complex SUSY mass spectrum emerges due to renormalization group running. As a result, the colored SUSY partners (squarks and gluino) are substantially heavier than the weakly coupled SUSY particles. The lightest SUSY particle (LSP) is typically the lightest neutralino ${\tilde{\chi}^0}_1$, with ${m_{\tilde{\chi}^0_{1}}} \sim {\cal O}(100 {{\rm \ GeV}})$.
- *GMSB:* In *gauge mediated SUSY breaking* (GMSB) the breaking of supersymmetry is transmitted by gauge interactions. The minimal version, which introduces messengers in the fundamental representation of SU(5), produces ${\cal O}(100 {{\rm \ GeV}})$ SUSY masses for a messenger scale $\Lambda_{\rm mess} \sim 100 {{\rm \ TeV}}$. Similar to mSUGRA, the gauge couplings and gaugino masses unify at $M_{\rm GUT}$, but the sfermion masses do not unify at any scale. The triple-scalar couplings ($A$-terms) are almost zero at the messenger scale $\Lambda_{\rm mess} \sim 100 {{\rm \ TeV}}$ and remain relatively small at the electroweak scale. In GSMB, the LSP is typically the gravitino, with $m_{\tilde{G}} \sim 100 {\rm \ eV} \dots 1 {{\rm \ GeV}}$.
- *AMSB:* In general, soft supersymmetry breaking terms receive contributions from the super-Weyl anomaly via loop effects. *Anomaly mediated supersymmetry breaking* (AMSB) becomes relevant only if other SUSY breaking mechanisms are suppressed or absent. AMSB predicts the gaugino mass ratios $|M_1| : |M_2| : |M_3| \approx 2.8 : 1 : 7.1$, so that the LSP is typically the lightest neutralino ${\tilde{\chi}^0}_1$ with a dominant wino component. The chargino ${\tilde{\chi}}^\pm_1$ is a almost pure wino and very close in mass to the LSP.
A shortcoming of the MSSM is the appearance of the $\mu$-term (the last term in eq. (\[wmssm\])) which must be of the order of the electroweak scale for successful electroweak symmetry breaking, leading to the unnatural hierarchy $\mu \ll M_{\rm GUT}$. One solution to this puzzle is the introduction of an additional singlet chiral superfield so that the general superpotential becomes $$W_{\rm MSSM+S} = \lambda \hat{S} \hat{H}_1 \cdot \hat{H}_2 + \kappa \hat{S}^3 +
m_{\rm S} \hat{S}^2 +
t_{\rm S} \hat{S} +
\mbox{Yukawa terms}.
\label{smssm}$$ In this general form the superpotential again has several dimensionful parameters which have to be much smaller than the GUT scale. However, the unwanted terms can be set to zero by introducing new symmetries, for example
- *Next-to-minimal MSSM (NMSSM):* A global $\mathbb{Z}_3$ symmetry mandates $m_{\rm S}=t_{\rm S}=0$, but could lead to cosmological domain walls [@domain].
- *Nearly minimal MSSM (nMSSM):* Imposing a global $\mathbb{Z}_5$ or $\mathbb{Z}_7$ symmetry forbids all singlet self-couplings at tree-level, $m_{\rm S}=t_{\rm S}=\kappa=0$. However, supergravity effects combined with SUSY breaking allow a contribution to $t_{\rm S}$ at the six- or seven-loop level, naturally generating a value $t_{\rm S} \sim {\cal O}$(TeV) as required for successful electroweak symmetry breaking [@nMSSM].
- *U(1)-extended MSSM (UMSSM):* This model introduces a U(1) gauge symmetry under which the Higgs and singlet field are charged. As a result, $m_{\rm S}=t_{\rm S}=\kappa=0$, but new D-term contributions to the Higgs potential appear which play an important role in achieving realistic electroweak symmetry breaking [@UMSSM].
HIGH ENERGY COLLIDERS
=====================
Searches for SUSY particles at $e^+e^-$ colliders are largely independent on the details of the model or scenario. Roughly speaking, results from LEP exclude sparticles up to the beam energy $E_{\rm beam} \sim 100 {{\rm \ GeV}}$. The actual exclusion bounds [@lepsusy; @pdg] are listed in Table \[lepsusy\]. The exact limits vary as a result of the different pair-production cross sections for different particles types. Furthermore, some of the searches fail if the mass difference between the pair-produced sparticle $\tilde{X}$ and the LSP becomes too small, $m_{\tilde{X}} - m_{\rm LSP} {\,\raisebox{-.3ex}{$_{\textstyle
<}\atop^{\textstyle\sim}$}\,}$ few GeV, see Refs. [@lepsusy; @pdg] for details.
**Sparticle** **lower limit** **\[GeV\]** **Sparticle** **lower limit** **\[GeV\]**
----------------------------------------------- ----------------- ------------- ---------------------- ----------------- -------------
$\;\;{\tilde{\chi}^0}_2$ 62.4 $\;\;\tilde{\nu}$ 94.0
$\;\;{\tilde{\chi}^0}_3$ 99.9 $\;\;\tilde{e}_L$ 107.0
$\;\;{\tilde{\chi}^0}_4$ 116.0 $\;\;\tilde{\mu}_R$ 91.0
$\;\;{\tilde{\chi}}^\pm_1$ 94.0 $\;\;\tilde{\tau}_1$ 81.9
$\;\;\tilde{u},\tilde{d},\tilde{c},\tilde{s}$ 97.0 $\;\;\tilde{t}_1$ 92.6
: Lower limits on SUSY particle masses from LEP searches
\[lepsusy\]
SUSY searches at hadron colliders are more intricate due to the large backgrounds. In most cases, a large signal-to-background ratio is only achievable by designing the selection strategy for some set of SUSY scenarios. For SUSY searches at the Tevatron, mSUGRA/CMSSM scenarios are usually taken as benchmark [@tevsusy]. However, when these results are expressed for more general MSSM scenarios the limits become much weaker and might drop below the LEP limits [@tevsusy; @alwall], see Table \[tevlim\]. More details about Tevatron searches and prospects for SUSY discovery at the LHC are given in the contributions by T. Adams [@adams] and O. Brandt [@brandt] at this conference.
------------------------ ------------------- -----------------------
**Sparticle** **mSUGRA/CMSSM** **more general MSSM**
**limit \[GeV\]** **limit \[GeV\]**
$\tilde{g}$ 308 $\sim 150$
$\tilde{q}$ 380 LEP limit
${\tilde{\chi}}^\pm_1$ 140 LEP limit
------------------------ ------------------- -----------------------
: Lower limits on SUSY particle masses from Tevatron searches
\[tevlim\]
The MSSM parameter space is also constrained indirectly by the lower limit on the mass of a SM-like Higgs boson from LEP searches, $m_{\rm h}^{\rm SM} >
114.4{{\rm \ GeV}}$ [@lephiggs]. In the MSSM the lightest CP-even Higgs boson mass can be calculated as a function of other parameters. The leading tree-level and one-loop contributions are given by $$m_{\rm h} {\,\raisebox{-.3ex}{$_{\textstyle
<}\atop^{\textstyle\sim}$}\,}{M_{\text{Z}}}^2 \, \cos^2 2\beta + \frac{3{m_{\text{t}}}^4}{2\pi^2v^2}
\left[ \log \frac{m_{\tilde{t}}^2}{{m_{\text{t}}}^2} + \frac{X_t^2}{m_{\tilde{t}}^2}
\right]
+ ... ,
\qquad
X_t = A_t - \frac{\mu}{\tan\beta},
\label{hfor}$$ where the dots stand for higher-order corrections. To be compatible with the LEP limit, the terms in eq. (\[hfor\]) need to be large so that at least one of the following conditions must be met:
- $\tan\beta \gg 1$ to maximize the tree-level term ${M_{\text{Z}}}^2 \, \cos^2
2\beta$,
- Large average stop mass, $m_{\tilde{t}}^2 = m_{\tilde{t}_1} m_{\tilde{t}_2}
{\,\raisebox{-.3ex}{$_{\textstyle
>}\atop^{\textstyle\sim}$}\,}(1 {{\rm \ TeV}})^2$,
- Large stop mixing to enhance the $X_t^2/m_{\tilde{t}}^2$ term.
In extended models with extra singlets (NMSSM, nMSSM) or gauge groups (UMSSM) the constraints on the SUSY parameter space from the $m_{\rm h}$ limit are much less severe due to new positive tree-level contributions to $m_{\rm h}$ [@Batra:2004vc].
ELECTROWEAK PRECISION DATA
==========================
Loop corrections from SUSY particles affect the predictions for electroweak precision observables. Much effort has been invested in calculating the radiative corrections from SUSY loops. The current state of the art in the MSSM encompasses complete one-loop corrections [@ewal] and leading two-loop corrections of order ${\cal O}(\alpha \alpha_s)$ [@ewalals] and ${\cal O}(\alpha y_{t,b}^2)$ [@ewaly]. For the NMSSM and other extensions only partial one-loop results are known.
The most important quantities that have been measured with high precision and that receive sizable corrections from new physics are:
- The $W$-boson mass ${M_{\text{W}}}$, which is determined from the muon decay width by including the relevant radiative corrections for the process $\mu^- \to e^- \bar{\nu}_e \nu_\mu$.
- The effective weak mixing angle of the $Z$ boson, defined through the effective vector and axial vector couplings of the $Z$ boson on the $Z$ resonance, $\sin \theta_{\rm eff} =
\frac{1}{4}\left(1-{\rm Re} \frac{v_{\rm eff}}{a_{\rm eff}} \right)$. The effective mixing angle can be defined for all fermion flavors although the numerical differences are small except for the $Zb\bar{b}$ vertex.
- The total $Z$-boson width, $\Gamma_{\rm Z}$, and the total $Z$ peak cross section $\sigma[e^+e^- \to Z \to f\bar{f}]$. Both of these quantities are closely related to the coupling combination $v_{\rm eff}^2+a_{\rm eff}^2$.
- The muon anomalous magnetic moment $a_\mu = (g_\mu-2)/2$.
By performing a fit of the MSSM predictions for these quantities to the experimentally determined values [@weber; @su] one finds in general good agreement for light sleptons and gauginos. The reason for this is two-fold: [*(i)*]{} in the SM the best fit to the electroweak precision observables corresponds to a Higgs mass of $m_{\rm h} \approx 87{{\rm \ GeV}}$, which creates a tension with the lower limit from direct searches, $m_{\rm h} > 114.4{{\rm \ GeV}}$. The new contributions from slepton-gaugino loops can push the best-fit Higgs mass to values $m_{\rm h} >
100{{\rm \ GeV}}$, thus improving the overall goodness-of-fit. [*(ii)*]{} SUSY loop contributions from sleptons and gauginos can account for the 3.3$\sigma$ discrepancy between the SM prediction and the measured value of the muon anomalous magnetic moment, $a_\mu^{\rm exp} - a_\mu^{\rm theo} = (27.5 \pm 8.4) \times
10^{-10}$ [@amu].
The results of a $\chi^2$ fit for the mSUGRA/CMSSM, GMSB, and AMSB scenarios are shown in Fig. \[ewsusy\]. The plots show the best fit $\chi^2$ as a function of the mass of the neutralino ${\tilde{\chi}^0}_2$, which has a dominant wino or higgsino component in these scenarios. As evident from the figure, in all three scenarios a light neutralino with ${m_{\tilde{\chi}^0_{2}}} \sim 200 \dots 700 {{\rm \ GeV}}$ is preferred, while neutralino masses above 1 TeV are strongly disfavored.
FLAVOR AND HIGGS PHYSICS
========================
Rare decays of heavy flavor mesons are very sensitive to new physics effects. For SUSY models these effects are enhanced for large values of $\tan\beta$ since the Yukawa couplings of the down-type fermions become large, $y_{\rm d} =
m_{\rm d}/v \times \tan\beta$. Schematically, the dependence of rare $B$ decays on $\tan\beta$ in the MSSM reads $$\begin{aligned}
{\rm BR}[B_s \to \mu\mu] &\sim \frac{\tan^6\beta}{M_A^4}, \\
{\rm BR}[B_u \to \tau\nu] &\sim \biggl [ 1-\frac{m_B^2}{M_A^2} \tan^2\beta \biggr ]^2, \\
{\rm BR}[b \to s\gamma] &\sim 1+ A\tan\beta+B\tan\beta/M_A^2,\end{aligned}$$ where $A$ and $B$ are coefficient that depend on other SUSY parameters in a non-trivial way.
In the region of large $\tan\beta$, the production cross section for the CP-odd Higgs boson $A_0$ at hadron colliders is also increased, $$\sigma[pp \to A \to \tau\tau] \sim \tan^2\beta,$$ establishing an intricate relationship between heavy-flavor and Higgs observables with respect to the SUSY parameter space [@menon].
The negative searches of the $A_0$ boson at the Tevatron [@a0cdf; @a0d0] exclude the parameter region of large $\tan\beta$ and small $M_A$, with very little dependence on other SUSY parameters. Similarly, the current experimental upper limit on BR$[B_s \to \mu\mu]$ is most important for large $\tan\beta$ and small $M_A$, although with some dependence on other variables, $m_{\tilde{t}_{1,2}},
m_{\tilde{b}_{1,2}}, \mu$. Depending on the value of these quantities the constraint from BR$[B_s \to \mu\mu]$ on the SUSY parameter space can be stronger or weaker than the bound from $A_0$ searches. The measurement of BR$[B_u \to \tau\nu]$, in good agreement with the SM, allows either the region of small $\tan\beta/M_A$ or of large $\tan\beta/M_A$ (although the latter is severely limited by the previous two observables), while intermediate values of $\tan\beta/M_A \sim 1/4 {{\rm \ GeV}}^{-1}$ are disfavored. Finally, BR$[b \to s\gamma]$ varies relatively mildly as a function of $M_A$, but it places both an lower and upper bound on $\tan\beta$. However, the prediction of BR$[b \to s\gamma]$ is affected by many SUSY parameters so that quantitative conclusions depend quite strongly on the scenario.
The flavor physics and Higgs constraints mentioned above are summarized in Fig. \[bh\] for the MSSM.
Note that while the $B$-physics observables seem to impose very severe limits on $M_A$ and $\tan\beta$ these bounds depend substantially on other SUSY parameters $\mu$, $m_{\tilde{g}}$, $X_t$, and $m_{\tilde{q}}$, which for the purpose of this analysis is assumed to be a common mass for all squarks. The most robust, scenario-independent constraint comes from $A_0$ searches which can be expressed roughly as $M_A/\tan\beta {\,\raisebox{-.3ex}{$_{\textstyle
>}\atop^{\textstyle\sim}$}\,}3{{\rm \ GeV}}$.
COSMOLOGY
=========
The derivation of bounds on the SUSY parameter space from cosmology depends on many details of the SUSY model as well as the history of the universe and might be impacted by theoretical uncertainties that have not been quantified so far. Nevertheless it is illustrative to study some of the constraints since even at a qualitative level they affect the parameter structure of the model.
Dark Matter
-----------
For conserved R-parity, the LSP is a stable particle and could provide a good cold dark matter candidate as long as it is neutral and weakly interacting. Within the standard cosmological model it is possible to calculate the expected relic dark matter density for a given SUSY model, although often the results depend on many model parameters. However typically only certain corners of the parameter space give good agreement with the measured value from the cosmic microwave background, $\Omega_{\rm DM} h^2 = 0.110 \pm 0.006$ [@wmap]. There are three main possibilities for LSPs as viable dark matter candidates in the MSSM:
- *Lightest neutralino ${\tilde{\chi}^0}_1$:* If the lightest neutralino has a dominant bino component, annihilation into gauge bosons is strongly suppressed. Thus, to be compatible with the observed dark matter density, one of the following enhancement mechanisms for the annihilation cross section needs to be present:
- Light sleptons, $m_{\tilde{l}} \approx 100{{\rm \ GeV}}$, together with ${m_{\tilde{\chi}^0_{1}}} < 100{{\rm \ GeV}}$ lead to a sufficiently large t-channel contribution. This parameter region is often called the *“bulk” region*.
- Co-annihilation: if the mass difference to the next-to-lightest SUSY particle (NSLP) $\tilde{X}$ is small, $m_{\tilde{X}}-{m_{\tilde{\chi}^0_{1}}} \ll {m_{\tilde{\chi}^0_{1}}}$, both particles annihilate in parallel in the early universe. A large $\tilde{X}{\tilde{\chi}^0}_1$ co-annihilation cross section can then compensate for a small ${\tilde{\chi}^0}_1{\tilde{\chi}^0}_1$ annihilation rate.
- Resonant annihilation: if the mass of the neutralino is close to half of the mass of a possible bosonic s-channel resonance, $2{m_{\tilde{\chi}^0_{1}}} \approx {M_{\text{Z}}}, \,
m_{\rm h}, \, M_A$, neutralino pair annihilation can proceed efficiently through this resonance.
Alternatively, the neutralino ${\tilde{\chi}^0}_1$ could be an admixture with sizable wino and/or higgsino components. In this case, the ${\tilde{\chi}^0}_1{\tilde{\chi}^0}_1$ annihilation rate into gauge boson naturally has the right order of magnitude for ${m_{\tilde{\chi}^0_{1}}} \sim
{\cal O}({\rm few\ 100\ GeV})$.
- *Sneutrino $\tilde{\nu}$:* It has been known for many years that the L-sneutrino $\tilde{\nu}_L$ cannot be the dominant source of dark matter since it would lead to a collision rate with ordinary matter that is much larger than the current bounds from direct detection experiments. However, as indicated by the observation of neutrino oscillations, it is likely that also L- and R-sneutrinos mix with each other. A sneutrino with a dominant R-sneutrino ($\tilde{\nu}_R$) component would constitute a good dark matter candidate in agreement with all constraints for 10 GeV ${\,\raisebox{-.3ex}{$_{\textstyle
<}\atop^{\textstyle\sim}$}\,}m_{\tilde{\nu}_R} {\,\raisebox{-.3ex}{$_{\textstyle
<}\atop^{\textstyle\sim}$}\,}$ 1 TeV [@sneu].
- *Gravitino $\tilde{G}$:* Gravitino dark matter can be produced in two ways, see [*e.$\,$g.*]{} [@steffen]:
- Gravitinos can be produced non-thermally from decays of the NLSP $\tilde{X}$. Late decays of the NLSP can lead to entropy overproduction and thus hot dark matter in disagreement with large scale structure formation. Since $$\Gamma[\tilde{X} \to X \tilde{G}] \propto
\frac{m_{\tilde{X}}^5}{m_{\tilde{G}}^2}
\biggl(1-\frac{m_{\tilde{G}}^2}{m_{\tilde{X}}^2}\biggr)^4$$ this places a lower bound $m_{\tilde{X}} > 0.5{{\rm \ TeV}}$. If this bound is satisfied the correct relic abundance can be obtained for gravitino masses in the range $1{{\rm \ GeV}}{\,\raisebox{-.3ex}{$_{\textstyle
<}\atop^{\textstyle\sim}$}\,}m_{\tilde{G}} {\,\raisebox{-.3ex}{$_{\textstyle
<}\atop^{\textstyle\sim}$}\,}700 {{\rm \ GeV}}$ [@steffen].
- Alternatively, gravitinos can be produced thermally directly from the hot plasma in the early universe. When produced from thermal equilibrium the gravitino abundance is much too large (“gravitino problem”). Therefore the reheating temperature $T_{\rm R}$ of the universe is required to be much smaller than the gravitino equilibrium temperature. In this case non-equilibrium thermal production is viable for $1{\rm \
keV} {\,\raisebox{-.3ex}{$_{\textstyle
<}\atop^{\textstyle\sim}$}\,}m_{\tilde{G}} {\,\raisebox{-.3ex}{$_{\textstyle
<}\atop^{\textstyle\sim}$}\,}1 {{\rm \ TeV}}$, depending on the exact value of $T_{\rm R}$.
Big-bang Nucleosynthesis
------------------------
If the LSP is a gravitino the energy released from NLSP decays can be problematic for successful big-bang nucleosynthesis (BBN). Hadronic and electromagnetic showers emitted by the NLSP decays can dissociate light element nuclei and thus shift the predicted ratios of element abundances. For the NLSP to disrupt BBN, the NLSP lifetime has to be $\tau_{\rm NLSP} {\,\raisebox{-.3ex}{$_{\textstyle
>}\atop^{\textstyle\sim}$}\,}100$ s. Therefore this constraint excludes small NLSP masses and large gravitino masses.
For $\tilde{\tau}_1$ as NLSP, the detailed constraints are shown in Fig. \[bbn\] [@steffen2].
Depending on assumptions in the evaluation of the primordial nuclei abundances the BBN constraints place an upper bound $m_{\tilde{G}} {\,\raisebox{-.3ex}{$_{\textstyle
<}\atop^{\textstyle\sim}$}\,}100$–500 GeV.
Baryogenesis
------------
New physics beyond the SM is needed to explain the excess of matter over antimatter in the universe. The two most well-known mechanisms are leptogenesis and electroweak baryogenesis. *Leptogenesis* generates the particle-antiparticle asymmetry through the decay of long-lived heavy neutrinos $\nu_R$ or sneutrinos $\tilde{\nu}_R$. This mechanisms imposes strong constraints on the masses, mixings and CP phases of the right-chiral (s)neutrino sector, but if $m_{\nu_R} \gg 1{{\rm \ TeV}}$ it is in general not testable by collider experiments. In *electroweak baryogenesis*, on the other hand, the matter asymmetry is created by the electroweak phase transition if it is strongly first order and involves CP-violating currents.
In the MSSM a strong first order phase transition is only realizable if one of the stops is light, $m_{\tilde{t}_1} < 140{{\rm \ GeV}}$ [@mssmbg]. The Higgs mass bound, $m_{\rm h} > 114.4 {{\rm \ GeV}}$ then requires the other stop to be much heavier, $m_{\tilde{t}_2} > 3{{\rm \ TeV}}$. In singlet extensions (NMSSM/nMSSM) the strength of the electroweak phase transition is increased by the new Higgs-singlet couplings and no special values for the stop masses are needed [@nmssmbg].
CP-violating currents can originate from the chargino/neutralino sector both in the MSSM and NMSSM/nMSSM. However, due to strong limits on electric dipole moments of the electron and neutrino, such CP phases are only allowed for very large masses of the first generation sfermions, $m_{\tilde{e}}, m_{\tilde{q}}
{\,\raisebox{-.3ex}{$_{\textstyle
>}\atop^{\textstyle\sim}$}\,}10{{\rm \ TeV}}$. In the NMSSM/nMSSM the CP phase responsible for baryogenesis can also be implemented in the Higgs sector, leading to weaker constraints from electric dipole moments [@huber].
Ultra-light neutralinos
-----------------------
Neutralinos ${\tilde{\chi}^0}_1$ that are almost exclusively bino and have negligible wino and higgsino components are not constrained by collider data. The only relevant bounds come from astrophysics and cosmology [@dreiner]:
- For conserved R-parity a lower bound ${m_{\tilde{\chi}^0_{1}}} > 3{{\rm \ GeV}}$ has to be imposed to avoid dark matter overproduction.
- Independent of R-parity conservation, very light neutralinos can contribute to supernova cooling for moderately light selectrons, $m_{\tilde{e}} {\,\raisebox{-.3ex}{$_{\textstyle
<}\atop^{\textstyle\sim}$}\,}500{{\rm \ GeV}}$. The observations from SN 1987A thus lead to a lower limit ${m_{\tilde{\chi}^0_{1}}} >
200{{\rm \ MeV}}$ in this case. However, for $m_{\tilde{e}} > 1200{{\rm \ GeV}}$ no constraint on the neutralino mass can be derived from supernova cooling.
- Very light neutralinos have a large free-streaming length and thus can jeopardize structure formation. This consideration excludes values of ${m_{\tilde{\chi}^0_{1}}}$ between 1 eV and 1 keV.
In summary, limits on light bino-like ${\tilde{\chi}^0}_1$ are very weak and ${m_{\tilde{\chi}^0_{1}}}$ is largely unconstrained.
EXTENDED MODELS
===============
In this section the assumptions of R-parity conservation, minimal flavor violation and CP conservation will be relaxed one at the time. As a result, many bounds on the MSSM parameter space become weaker or disappear altogether.
Flavor violation
----------------
The sfermion soft breaking parameters can introduce new sources of flavor violation, in particular leading to potentially large flavor changing neutral currents (FCNCs). FCNCs are strongly constrained by $K^0$, $D^0$ and $B^0$ mixing, rare $B$ decays, and limits on lepton flavor violating processes such as $\mu \to e\gamma$, $\mu \to e$ conversion, [*etc.*]{} However, if new flavor violating terms are introduced in the 2nd and 3rd generation only, flavor mixing sfermion mass terms as large as ${\cal O}(M_{\rm
SUSY})$ are still allowed by present data [@utfit].
CP violation
------------
Complex CP phases in the gaugino sector and in the parameters of the 1st generation sfermions are strongly constrained by electric dipole moments (see previous section). Sizable CP violation is however allowed in the Higgs sector and the sector of the 3rd generation sfermions.
R-parity violation
------------------
Without R-parity conservation the MSSM superpotential is extended by the following couplings: $$W_{{\not{R_p}}\rm MSSM} = W_{\rm MSSM}
+ \tfrac{1}{2} \lambda_{ijk} L_i \cdot L_j E_k^c
+ \tfrac{1}{2} \lambda'_{ijk} L_i \cdot Q_j D_k^c
+ \tfrac{1}{2} \lambda''_{ijk} U_i^c \cdot D_j^c D_k^c.$$ The product of baryon-number violating and lepton-number violating couplings is strongly constrained by proton decay, [*i.$\,$e.*]{} $|\lambda''_{ijk}\lambda_{ijk}|,
|\lambda''_{ijk}\lambda'_{ijk}| \ll 1$. Such a structure could be explained by discrete symmetries while still allowing some non-zero R-parity violating couplings. In the absence of $B$-violating terms the $L$-violating interactions are mainly constrained by data on neutrino masses, leading to $|\lambda_{ijk}|,|\lambda'_{ijk}| {\,\raisebox{-.3ex}{$_{\textstyle
<}\atop^{\textstyle\sim}$}\,}10^{-5} \dots 0.6$ [@rpl].
If R-parity is violated the LSP is not stable and the signatures for SUSY particles production at colliders are dramatically altered. As a result, experimental bounds for several SUSY particles becomes much weaker. In particular one finds $m_{\tilde{g}} > 51$ GeV [@schwartz], $m_{\tilde{b}_1}
> 7.5$ GeV, $m_{\tilde{\tau}_1} > 11$ GeV [@janot], and $m_{\rm h} > 82$ GeV [@carpenter].
SUMMARY
=======
Due to the complexity of the SUSY parameters space (even in the MSSM with R-parity conservation, minimal flavor violation and CP conservation) and the large number of experimental results it is difficult to summarize all constraints in a simple picture. In Table \[summ\] a rough overview of the main limits from direct sparticle and Higgs searches, as well as electroweak precision data and flavor physics is attempted. The three columns in the table correspond to the MSSM with and without R-parity conservation and the NMSSM, respectively. The NMSSM limits also apply for the nMSSM and UMSSM. A “—” indicates that no bound exists while “?” stands for cases where the final quantitative conclusion is not known yet.
[|c|c|c|]{} & &\
\
$m_{\tilde{g}} {\,\raisebox{-.3ex}{$_{\textstyle
>}\atop^{\textstyle\sim}$}\,}150$ GeV & $m_{\tilde{g}} {\,\raisebox{-.3ex}{$_{\textstyle
>}\atop^{\textstyle\sim}$}\,}51$ GeV & $m_{\tilde{g}} {\,\raisebox{-.3ex}{$_{\textstyle
>}\atop^{\textstyle\sim}$}\,}150$ GeV\
$m_{\tilde{\tau}_1} {\,\raisebox{-.3ex}{$_{\textstyle
>}\atop^{\textstyle\sim}$}\,}82$ GeV & $m_{\tilde{\tau}_1} {\,\raisebox{-.3ex}{$_{\textstyle
>}\atop^{\textstyle\sim}$}\,}11$ GeV & $m_{\tilde{\tau}_1} {\,\raisebox{-.3ex}{$_{\textstyle
>}\atop^{\textstyle\sim}$}\,}82$ GeV\
\
${m_{\tilde{\chi}^0_{1}}} > 3$ GeV & — & ?\
$\tan\beta {\,\raisebox{-.3ex}{$_{\textstyle
>}\atop^{\textstyle\sim}$}\,}3$ & — & —\
$\sqrt{m_{\tilde{t}_1}m_{\tilde{t}_2}} {\,\raisebox{-.3ex}{$_{\textstyle
>}\atop^{\textstyle\sim}$}\,}1$ TeV &\
and/or large $A_t$ &\
& —\
\[summ\]
The limits in the first three lines stem for direct searches at high-energy colliders, while the upper bound in the fourth line originates from electroweak precision data. The fifth line gives limits from astrophysics and cosmology on light neutralinos. Other cosmological constraints are not included in the table. Finally, the last three lines summarize bounds from flavor and Higgs physics.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The bosonic $su(n)$ Hubbard model was recently introduced. The model was shown to be integrable in one dimension by exhibiting the infinite set of conserved quantities. I derive the $R$-matrix and use it to show that the conserved charges commute among themselves. This new matrix is a non-additive solution of the Yang-Baxter equation. Some properties of this matrix are derived.'
author:
- |
[**Z. Maassarani**]{}[^1]\
\
[Département de Physique, Pav. A-Vachon]{}\
[Université Laval, Ste Foy, Qc, G1K 7P4 Canada]{}[^2]\
title: 'Exact integrability of the $su(n)$ Hubbard model'
---
23.50cm -1.7cm 0.6cm
\#1[\#1 ]{} \#1[[1 \#1]{}]{}
=msbm10 =msbm7 =msbm5 = == \#1[[\#1]{}]{}
PACS numbers: 75.10.-b, 75.10.Jm, 75.10.Lp\
Key words: Hubbard model, $su(n)$ spin-chain, integrability
October $6^{\rm th}$ 1997\
LAVAL-PHY-25/97\
Introduction
============
The two-dimensional Hubbard model was introduced to describe the effects of correlation for $d$-electrons in transition metals [@guhu]. It was then shown to be relevant to the study of high-$T_c$ superconductivity of cuprate compounds.
In one dimension the model is integrable [@liwu; @sh12; @woa]. The integrability framework of the model is the quantum inverse scattering method [@qism]. However, despite sharing many properties with discrete quantum integrable models, the model has a peculiar integrable structure which defines a class of its own.
In seeking to generalize the Hubbard model in any dimension, it was therefore natural to look for a one-dimensional generalization which is integrable. An $n$-state generalized model which contains the usual $su(2)$ model was recently introduced in [@hn]. This $su(n)$ Hubbard model was shown to possess an infinite set of conserved charges and to have an extended $su(n)$ symmetry. The model is built by coupling two copies of the recently discovered $su(n)$ XX ‘free-fermions’ model [@mm]. For $n=2$ a fermionic formulation exists, but for $n > 2$ finding an analogous framework is a tantalizing problem.
In this work I derive the $R$-matrix of the model; this provides a direct proof of the commutation of the conserved charges among themselves. Section two gives the definition of the bosonic Hamiltonian and the transfer matrix. The $R$-matrix intertwining the monodromy matrices is derived in section three. In section four some properties of this new matrix are given. I conclude with some remarks and outline some outstanding issues.
The model
=========
Let $E^{\af\be}$ be the $n\times n$ matrix with a one at row $\af$ and column $\be$ and zeros otherwise. The $su(n)$ Hubbard Hamiltonian on a ring then reads [@hn]: H\_2 &=&\_i h\_[ii+1]{} +\_i h\^[’]{}\_[ii+1]{} + U\_i h\^c\_i\[h2\]\
&=& \_i \_[< n]{} (x E\_i\^[n]{} E\_[i+1]{}\^[n]{} + x\^[-1]{} E\_i\^[n]{} E\_[i+1]{}\^[n]{} + (EE\^[’]{})) + U \_i (\_i +) (\^[’]{}\_i +)where $\rho = \sum_{\af < n} E^{\af\af} -(n-1) E^{nn}$, and primed and unprimed quantities correspond to two commuting copies of the $E$ matrices. The Hamiltonians $h$ and $h^{'}$ are $su(n)$ XX Hamiltonians [@mm]. The complex free parameter $x$ is a deformation inherited from the XX model. The Hamiltonian $H_2$ is defined in one dimension but can be evidently defined on any lattice; integrability is lost however.
For $n=2$ and $x=1$, and using Pauli matrices, the Hamiltonian is just the integrable bosonic version of the usual Hubbard Hamiltonian [@sh12]: H\_2\^[(2)]{}= \_i (\^x\_i \^x\_[i+1]{} + \^y\_i \^y\_[i+1]{}) + (\^[’]{} ) + U\_i \^z\_i\^[’z]{}\_i The Hamiltonians can be written simply in terms of $su(n)$ hermitian traceless matrices. For $|x|=1$ the Hamiltonians are hermitian.
The transfer matrix is the generator of the infinite set of conserved quantities. Its construction was given in [@hn]. We recall it here. Consider first the $R$-matrix of the $su(n)$ XX model [@mm]: R() &=& a() \[E\^[nn]{}E\^[nn]{}+\_[[, <n]{}]{} E\^E\^\]\
& & + b()\_[[<n]{}]{}(x E\^[nn]{}E\^ + x\^[-1]{} E\^E\^[nn]{})\
& & + c() \_[[<n]{}]{}(E\^[n ]{}E\^[n]{} + E\^[n]{}E\^[n]{}) where $a(\la)=\cos(\la)$, $b=\sin(\la)$ and $c(\la)=1$. The functions $a$, $b$ and $c$ satisfy the ‘free-fermion’ condition: $a^2 +b^2 = c^2$. For this set of parameters, a Jordan-Wigner transformation turns the $U=0$ Hamiltonian density for $su(2)$ into a fermionic expression for free fermions hopping on the lattice.
Consider also the matrix I\_0 (h) =() + () C\_0 C\^[’]{}\_0 =( C\_0 C\^[’]{}\_0) where $C=\sum_{\af < n} E^{\af\af}-E^{nn}$. We stress that $C$ turns out to be the fundamental matrix, not the $su(n)$ generator $\rho$. We have $\rho +\frac{n-2}{2}{\rm Id} = \frac{n}{2} C$, for $n \geq 2$. The parameter $h$ is related to the spectral parameter $\lambda$ by (2h) = (2) \[rela\] One chooses for $h(\la)$ the principal branch which vanishes for vanishing $\la$ or $U$. Then for $U=0$ the monodromy matrix becomes a tensor product of two uncoupled XX models. The Lax operator at site $i$ is given by: L\_[0i]{} () = I\_0(h) R\_[0i]{}() R\^[’]{}\_[0i]{}() I\_0(h) and the monodromy matrix is a product of Lax operators, $T(\lambda)= L_{0M}(\la)...L_{01}(\la)$, where $M$ is the number of sites on the chain. The transfer matrix is the trace of the monodromy matrix over the auxiliary space 0: $\tau (\la)= {\rm Tr}_0 \;\left[\left( L_{0M}...L_{01}\right)(\la)\right]$. One possible set of conserved quantities is given by H\_[p+1]{} = ([d\^p ()d\^p]{})\_[=0]{} The proof that $H_2$ commutes with $\tau(\lambda)$ was given in [@hn]. The derivative of the matrix $I$ gives the coupling term appearing in (\[h2\]). Note that the definition involving a logarithm has two benefits. Besides giving the most local operators, it further disentangles the two copies.
The $R$-matrix
==============
We derive the $R$-matrix intertwining two monodromy matrices at different spectral parameters. To this end we generalize the algebraic method of the Decorated Star Triangle Equation introduced by Shastry [@sh3].
The XX $R$-matrix satisfies the regularity property $\check R (0) = {\rm Id}$, the unitarity condition $\check R (\lambda) \check R (-\lambda) = {\rm Id} \;\cos^2\lambda$ and the Yang-Baxter equation \_[12]{}(-) R\_[13]{}()R\_[23]{}() = R\_[13]{}() R\_[23]{}()\_[12]{}(-)\[ybec\] where $R=P \check{R}$ and $P$ is the permutation operator on the tensor product of two $n$-dimensional spaces. It is easy to verify that it also satisfies a decorated Yang-Baxter equation \_[12]{}(+) C\_1 R\_[13]{}() R\_[23]{}() = R\_[13]{}() R\_[23]{}()C\_2\_[12]{}(+)\[dybec\]
We now look for the $R$-matrix intertwining two $L$-matrices: (\_1,\_2) L(\_1)L(\_2) = L(\_2)L(\_1) (\_1,\_2)\[rll\] The $su(2)$ case lead us to consider the following Ansatz [@sh3]: (\_1,\_2) &=& I\_[12]{}(h\_2) I\_[34]{}(h\_1) ( \_[13]{}(\_1-\_2) \_[24]{}(\_1-\_2) .\
& &.+ \_[13]{}(\_1+\_2) C\_1 \_[24]{}(\_1+\_2)C\_2 ) I\_[12]{}(-h\_1) I\_[34]{}(-h\_2) The $R$-matrix acts on the product of four auxiliary spaces labeled from 1 to 4, and $\alpha$, $\beta$ are to be determined. One then requires relation (\[rll\]) to be satisfied and uses (\[ybec\]) and (\[dybec\]) to derive the following equation: &(\_[13]{}(\_1-\_2) \_[24]{}(\_1-\_2) + C\_3\_[13]{}(\_1+\_2)C\_4 \_[24]{}(\_1+\_2) )I\_[12]{}(2h\_1) I\_[34]{}(2h\_2) =&\
&I\_[12]{}(2h\_2) I\_[34]{}(2h\_1) ( \_[13]{}(\_1-\_2) \_[24]{}(\_1-\_2) + \_[13]{}(\_1+\_2) C\_1 \_[24]{}(\_1+\_2)C\_2 )& Expanding the exponentials and taking into account all the terms yield only two equations: = (h\_1+h\_2) , = (h\_1-h\_2) where $a=\cos(\la_1-\la_2)$, $b=\sin(\la_1-\la_2)$, $A=\cos(\la_1+\la_2)$ and $B=\sin(\la_1+\la_2)$. The compatibility equation = is satisfied provided equation (\[rela\]) is satisfied for the pairs $(\la_1,h_1)$ and $(\la_2,h_2)$. One can then pull out $\alpha=\alpha(\la_1,\la_2)$ which appears as an arbitrary normalization of the $R$-matrix, to obtain: (\_1,\_2)&=& (\_1,\_2) I\_[12]{}(h\_2) I\_[34]{}(h\_1) ( \_[13]{}(\_1-\_2) \_[24]{}(\_1-\_2) +.\
& &. (h\_1+h\_2) \_[13]{}(\_1+\_2) C\_1 \_[24]{}(\_1+\_2)C\_2 ) I\_[12]{}(-h\_1) I\_[34]{}(-h\_2)\[rc\]
The monodromy matrix being a tensor product of $M$ copies of $L$ matrices, one has (\_1,\_2) T(\_1)T(\_2) = T(\_2)T(\_1) (\_1,\_2)\[rtt\] Taking the trace over the auxiliary spaces and using the cyclicity property of the trace one obtains $[\tau(\la_1),\tau(\la_2) ]=0$. We have thus proven that all the conserved charges $H_p$ mutually commute.
Note that this proof is rigorous and valid [*for all values of*]{} $n$, and for arbitrary values of the complex parameter $x$. It only involves the algebraic properties of the operators appearing in the various matrices, not the specific $n$-dependent representation. The equations (\[ybec\]) and (\[dybec\]) are the only equations of this type needed for the proof.
Properties of the $\check{R}$ matrix
====================================
I now give some properties of the $R$-matrix. At $U=0$ the two $XX$ models decouple and $h(\la,U)=0$. Expression (\[rc\]) indeed decouples as a tensor product of two $su(n)$ XX $\check{R}$-matrices.
The matrix also satisfies the regularity property (\_1,\_1) = (\_1,\_1) and the unitarity property: (\_1,\_2) (\_2,\_1)& =& \^2(\_1,\_2) \^2(\_1-\_2)\
&&(\^2(\_1-\_2) - \^2(\_1+\_2) \^2(h\_1-h\_2)) [Id]{} The derivation of the last property is straightforward and involves algebraic relation between the building blocks of the $su(n)$ XX $\check{R}$-matrix.
One can invoke the associativity of the algebra of $L$ matrices, which ultimately reduces to the associativity of usual matrix multiplication, to conclude that the intertwiner satisfies a Yang-Baxter relation of its own. The two ways of permuting a product of three $L$-matrices imply & L(\_1)L(\_2)L(\_3) \^[-1]{}= L(\_1)L(\_2) L(\_3) &\
& = ( \_[12]{}(\_2,\_3) \_[23]{}(\_1,\_3) \_[12]{}(\_1,\_2) )\^[-1]{} \_[23]{}(\_1,\_2) \_[12]{}(\_1,\_3) \_[23]{}(\_2,\_3) I am unaware of the existence of an equivalent of the Schur lemma for the algebra of $L$-matrices. This would allow to conclude that $\check{\Pi}\propto {\rm Id}$. Once proportionality is established, the regularity property ensures that the proportionality constant is one. We can argue that $\check{\Pi} = {\rm Id}$ holds because it has been explicitly verified for $n=2$ [@sw], and because the building blocks of the matrix satisfy algebraic relations which are independent of $n$. Thus the $R$-matrix satisfies the Yang-Baxter equation: \_[12]{}(\_2,\_3) \_[23]{}(\_1,\_3) \_[12]{}(\_1,\_2) = \_[23]{}(\_1,\_2) \_[12]{}(\_1,\_3) \_[23]{}(\_2,\_3)\[rybe\] where $\la$ and $h$ are related through (\[rela\]).
Using the explicit expression (\[rc\]), it should be possible and it is instructive to try to check that the above equation is satisfied for any value of $n$. The factors $I$ drop out and half of the terms on both sides of the YBE compensate each other because relations (\[ybec\]) and (\[dybec\]) hold. The eight remaining terms involve highly non-trivial relations. Each term is a product of six $R$-matrices, three for every copy, in the ordering dictated by the YBE. The $C$ factors can be dropped by changing the arguments of the $R$-matrices appropriately. However the arguments do not allow the use of (\[ybec\], \[dybec\]) because the middle argument is not the sum of the extreme ones. One is forced to expand all the products on a basis and to regroup terms and check that the resulting trigonometric constraints are satisfied. I have not verified whether all these relations hold. Although specific particularities pertaining to the XX matrix are needed, I stress again that the proof is algebraic. In this respect the proof of [@sw] for $n=2$, although following a different approach, should generalize in a straightforward way to any value of $n$.
Conclusion
==========
We have shown that all the conserved charges of the $su(n)$ Hubbard model mutually commute by exhibiting the intertwining matrix. This matrix is the $su(n)$ generalization of the $su(2)$ one obtained in [@sh12]. Some properties where then derived. A notable feature of the matrix is its non-additivity property; the $\la$ dependence cannot be reduced to a difference $(\la_1-\la_2)$.
One can now start diagonalizing the Hamiltonians by the method of the algebraic Bethe Ansatz. Preliminary results suggest an interesting structure for the Bethe eigenstates [@win].
One can also consider the extension of the $su(n)$ model to other algebras. The algebraic underlying structure of the $su(n)$ XX model should admit generalizations [@mm; @win].
[**Acknowledgement:**]{} I thank P. Mathieu for bringing to my attention reference [@sh3] and for continued support.
[**Note:**]{} While this work was being written, Martins exhibited a gauge-transformed version of the foregoing $\check{R}(\la_1,\la_2)$ matrix. The derivation is also based on a generalization of Shastry’s method. However, the proof in [@mjm] was carried out for $n=3$, and ‘extensive checks’ were made for $n=3,4$. The expression of $\check{R}$ for all values of $n$ is left as a (correct) [*conjecture*]{}. This reference also used unnecessarily complicated versions of the Yang-Baxter equations.
[30]{}
M.C. Gutzwiller, Phys. Rev. Lett. [**10**]{}, 159 (1963); J. Hubbard, Proc. Roy. Soc. [**A276**]{}, 238 (1963). E.H. Lieb and F.Y. Wu, Phys. Rev. Lett. [**20**]{}, 1445 (1968). B.S. Shastry, Phys. Rev. Lett. [**56**]{}, 1529 (1986); Phys. Rev. Lett. [**56**]{}, 2453 (1986). M. Wadati, E. Olmedilla and Y. Akutsu, J. Phys. Soc. Jpn. [**56**]{}, 1340 (1987); E. Olmedilla, M. Wadati and Y. Akutsu, J. Phys. Soc. Jpn. [**56**]{}, 2298 (1987). For reviews on QISM and the Hubbard model see [*Proceedings of the Panchgani Winter School*]{} edited by B.S. Shastry, S.S. Jha and V. Singh, Lectures Notes in Physics Vol. 242, (Springer-Verlag, Berlin 1985). Z. Maassarani, Phys. Lett. A [**239**]{} (1998) 187–190. Z. Maassarani and P. Mathieu, Nucl. Phys. B [**517**]{}, Nos. 1–3 (1998) 395–408. M. Shiroishi and M. Wadati, J. Phys. Soc. Jpn. [**64**]{} (1995) 57. Work in progress. B.S. Shastry, J. Stat. Phys. [**50**]{}, (1988) 57–79. M.J. Martins, preprint IFTA-97-36, cond-mat/9710049.
[^1]: Work supported by NSERC (Canada) and FCAR (Québec).
[^2]: email address: zmaassar@phy.ulaval.ca
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We calculate the time independent four-point function in high temperature ($T$) QCD and obtain the leading momentum dependent terms. Furthermore, we relate these derivative interactions to derivative terms in a recently proposed finite $T$ effective action based on the SU(3) Wilson Line and its trace, the Polyakov Loop. By this procedure we thus obtain a perturbative matching at finite $T$ between QCD and the effective model. In particular, we calculate the leading perturbative QCD-correction to the kinetic term for the Polyakov Loop.'
address: |
Department of Physics, Brookhaven National Laboratory, Upton, NY 11973, USA.\
email: wirstam@bnl.gov
author:
- 'J. Wirstam'
title: |
[BNL-NT-01/11]{}
One-Loop QCD Corrections to the Thermal Wilson Line Model
---
Introduction {#I}
============
At high temperatures QCD is expected to be found in a new phase, the quark-gluon plasma. While the thermal excitations are hadrons and glueballs at low $T$, the degrees of freedom in the plasma phase are the quarks and gluons. This new state of matter is believed to have existed during the first microseconds after the Big Bang, and much of the recent interest stems from the fact such conditions may be produced in heavy-ion collisions. Already the results from CERN-SPS seem to hint in that direction, and the experiments at higher energies at BNL-RHIC have provided a wealth of new interesting results after its first year of running [@rhicwebpage]. To understand and interpret the experimental signatures in terms of the evolution of the initial stage after a heavy-ion collision is clearly a very challenging theoretical task.
The most convincing theoretical results that a drastic change in the degrees of freedom takes place at a certain $T$ come from lattice studies. In the pure glue theory, there is a phase transition between the confined and deconfined phases at a critical temperature $T_c \simeq 270$ MeV [@latticesu3]. When massless quarks are added, there is similarly a phase transition to a chirally symmetric phase at $T_c \simeq 155-175$ MeV [@chiral], where the precise value depends on the number of flavors. Recent lattice simulations also suggest that the chiral transition is simultaneous with the deconfining one [@digal]. At physical quark masses the situation is not completely clear, in the sense that there may not be a true phase transition but only a rapid cross-over [@transcross]. Nevertheless, lattice simulations have shown that the pressure, divided by the ideal gas result and plotted against $T/T_c$, is almost independent of the number of flavors [@indepofnf].
Due to asymptotic freedom, the quark-gluon plasma behaves as an ideal gas at asymptotically high temperatures. Up to corrections of the order of 20 percent, this behavior holds down to temperatures $T\simeq 3T_c$, even though each higher order term in a straightforward perturbative expansion gives widely different contributions in this temperature regime [@pertpressure; @nieto]. Instead, at $T\leq 5T_c$ one needs a resummed effective theory in terms of quasi particles, the HTL effective action [@htl]. Such an effective description correctly reproduces thermodynamic quantities like the pressure, as measured by the lattice, down to approximately $2T_c$ [@blaizotetal; @peshier].
Despite this progress, it is of course highly desirable to actually have an analytical description at $T\simeq T_c$, close to the critical temperature. Since the QCD coupling constant $g
\simeq 2.5$ at $T_c$ (using a renormalization scale $\mu =2\pi T$), one is presumably forced to consider effective models that go beyond the fundamental QCD Lagrangian. In a recent paper [@robmodel], such an effective theory was constructed in terms of the thermal Wilson Line ${\bf L}$, $$\begin{aligned}
{\bf L} = {\cal P} \exp \left [ ig\int_0^{\beta} d\tau A_0 (\vec{x}, \tau ) \right ] , \label{wilsonline}\end{aligned}$$ where ${\cal P}$ denotes path ordering, $\beta$ is the inverse temperature and $A_0 = A_0^a T^a$ the time component of the gluon field, with $T^a$ the generators of the fundamental SU(3) representation, $a=1,\ldots ,8$. The trace of the Wilson Line is proportional to the Polyakov Loop $l$, $l=(1/3){\rm Tr}\,{\bf L}$. In the pure Yang-Mills theory, the Polyakov Loop is an order parameter for a global Z(3) symmetry separating the confined and deconfined phases, with $\langle l\rangle \neq 0$ ($\langle l\rangle =0$) above (below) the phase transition [@refsforl]. When dynamical quarks are introduced, $l$ ceases to be an order parameter in the strict sense, but the susceptibility of $l$ still peaks strongly at $T_c$ [@digal; @lwithquarks].
In the effective theory [@robmodel], the pressure of the quark-gluon plasma at $T>T_c$ is completely due to the condensate of $l$, and below $T_c$, where $\langle l\rangle =0$, the pressure vanishes. Moreover, the effective potential $V(l)$ changes extremely rapidly around $T_c$. Hence, as the system cools it may find itself trapped at the wrong value of $\langle l\rangle$. By coupling the effective field $l$ to hadronic degrees of freedom, e.g. the pions, hadrons can be produced as $l$ evolves from $\langle l\rangle \neq 0$ and subsequently oscillates around $\langle l\rangle =0$. This scenario is somewhat reminiscent of reheating after inflation [@linde], and much attention has lately been paid to that aspect of the model [@robadrian; @particleprod]. Remarkably, many qualitative features observed at RHIC are in accordance with the model predictions.
However, when it comes to questions related to the change of the expectation value of $l$ and particle production around $T_c$, one has to take into account the variation of $l$ in space-time. In this paper we address the question of radiative corrections to the spatial variation, by considering the leading one-loop QCD contribution to the spatial derivatives of $l$. Previous work [@robadrian; @particleprod] took into account only the classical kinetic term in the (Euclidean) effective action, $\Gamma (l) = (1/2)\{ |{\partial}_t l|^2 + |{\partial}_i l|^2\} + V(l)$. While such an approach certainly is justified at these preliminary stages, it is important to estimate how much the radiative QCD-effects can affect $\Gamma (l)$ around $T_c$, where the QCD coupling constant becomes large. As for the parameters in the potential $V(l)$, they can be fitted by comparing to QCD lattice results and so are well defined at all $T$. The kinetic term, on the other hand, has to be matched to perturbatively calculated terms in QCD, and could therefore receive large radiative corrections. The magnitude of the first one-loop QCD correction can then hopefully serve as a guideline to the importance of loop effects, and indicate how reliable the above form of $\Gamma (l)$ is at $T_c$. We want to stress that the radiative corrections to be discussed come from QCD, and not from fluctuations in $\Gamma (l)$.
To study the correction to the kinetic term $|{\partial}_i l|^2$, we first consider the one-loop induced quartic terms in QCD, that contain four powers of the external field $A_0$ and two powers of the external momenta. These terms contribute to the high $T$, dimensionally reduced QCD effective action $\Gamma (A_0)$ [@nadkarni; @landsman], and apart from providing a correction to $\Gamma (A_0)$ they can also be related to the kinetic term in $\Gamma (l)$. As a byproduct we obtain some additional derivative interactions in $\Gamma (l)$.
The paper is organized as follows. In the next section, we give the perturbative QCD calculation that corresponds to the leading derivative interactions in $\Gamma (A_0)$. In Sec. III we make the actual matching from an effective theory in terms of $A_0$ to the one in $l$, and discuss the validity of the results. We end with our conclusions and an outlook. Our conventions and some technical details can be found in the appendix.
Perturbative calculation of the four-point function {#II}
===================================================
In the high temperature regime, long distance phenomena (i.e. $|\vec{x}|\gg\beta$) are dominated by the static sector of QCD. At high $T$ it therefore makes sense to use dimensional reduction and integrate out all the nonstatic modes in the theory [@dimred]. With only the static modes left, the full QCD Lagrangian is reduced to a three-dimensional theory. In principle the integrating-out procedure gives rise to an infinite number of interaction terms, but higher dimensional operators become more suppressed by powers of the QCD coupling constant $g$ and/or $T$. In full QCD, the following terms in the resulting effective action $\Gamma (A_0)$ have been calculated [@nadkarni; @landsman], $$\begin{aligned}
\Gamma (A_0) = \beta \!\int \! d^3x \left [\frac{1}{2}{\rm Tr}\,F_{ij}^2 + {\rm Tr}\, [D_i,A_0][D_i,A_0]
+ g^2T^2\left ( 1+\frac{N_{\! f}}{6}\right ){\rm Tr}\, A_0^2 +
\frac{g^4(9-N_{\! f})}{24\pi^2} \left ( {\rm Tr}\, A_0^2\right )^2 \right ] \ , \label{knownaction}\end{aligned}$$ where $i,j=1,2,3$, $F_{ij}= F_{ij}^aT^a = ({\partial}_iA_j^a-{\partial}_jA_i^a -gf^{abc}A_i^bA_j^c)T^a$, $D_i = {\partial}_i +igA_i$ and $A_i=A_i^aT^a$. The next term in $\Gamma (A_0)$ contains two derivatives and four powers of $A_0$, and corresponds to the following part in the Euclidean effective action for $A_0(\vec{x})$, $$\begin{aligned}
\Gamma^{(4)}_{\!E} (A_0)= \frac{\beta}{24} \prod_{i=1}^4\int \!\frac{d^3k_i}{(2\pi )^3} \, \delta^{(3)}\!(k_1+\ldots +k_4)
\left [ -i\Gamma^{abcd}_{0000} (\vec{k_1},\ldots ,\vec{k_4})\right ]
A_0^a(\vec{k_1})A_0^b(\vec{k_2})A_0^c(\vec{k_3})A_0^d(\vec{k_4}) \ , \label{effactiona0}\end{aligned}$$ where $\Gamma^{abcd}_{0000}$ is the four-point function of order $O(\beta^2k^2)$, obtained by integrating out all the non-static modes. Such higher dimensional terms have been calculated in the pure Yang-Mills theory using the background field method [@chapman], as well as in QED [@landsman], but not in QCD with quarks. In this paper we will use a diagrammatic approach to the four-point function.
Perturbatively, the four-point function receives contributions from the diagrams shown in Fig. 1, together with the additional permutations of the external legs. There are five permutations adding to the graphs (a), (b) and (d), and two to the diagrams (c) and (e), where (e) has a symmetry factor $1/2$. Even though all diagrams are superficially logarithmically divergent, it is well known that both the fermion diagram and the sum of the pure Yang-Mills diagrams are ultra-violet finite. We will therefore only give explicit results for the finite $T$ part of these diagrams, where we use the imaginary time formalism [@finitetbooks] combined with the particular technique described in the appendix.
= 8 cm
The fermion contribution
------------------------
Consider first the contribution from the $N_{\! f}$ massless quarks, with the particular ordering of momenta shown in Fig. 1(a). Using the conservation of momentum, $k_4 = -(k_1+k_2+k_3)$, with $k_i = \vec{k}_i$, we find $$\begin{aligned}
\left . \Gamma^{abcd}_{0000} \right |_{(a)_1} = &&\left ( \frac{3ig^4N_{\! f}}{4\pi^{5/2}} \right ) {\rm Tr}_c (T^aT^bT^cT^d)
\int_0^1\!\!dy_1\!\int_0^{1-y_1}\!\!\!dy_2\!\int_0^{1-y_1-y_2}\!\!\!\!dy_3\! \int_{\epsilon-i\infty}^{\epsilon+i\infty}\!\!
\frac{dz}{2\pi i} (1-2^{1-z})\Gamma (z)\xi (z) \cos [\pi z/2]\beta^{-z} \times \nonumber \\
&& \left [ f_1^{-z/2}\Gamma (z/2) \left \{ \frac{5}{2}\Gamma ((1\!-\!z)/2) -6\Gamma ((3\!-\!z)/2) +\frac{2}{3}
\Gamma ((5\!-\!z)/2) \right \} +f_1^{-(2+z)/2}\Gamma ((2\!+\!z)/2) \times \right . \nonumber \\
&& \left . \left \{ f_3\Gamma ((1\!-\!z)/2)-\frac{2}{3}f_2\Gamma ((3\!-\!z)/2)\right \} +\frac{2}{3}f_4f_1^{-(4+z)/2}
\Gamma ((1\!-\!z)/2)\Gamma ((4\!+\!z)/2) \right ] \ , \label{fermionloop}\end{aligned}$$ where $\xi (z)$ is the Riemann Zeta-function and $f_i = f_i(y_1,y_2,y_3,\vec{k}_1,\ldots ,\vec{k}_3)$, given explicitly in the appendix, are functions of the Feynman parameters $y_k$ and the external momenta $\vec{k}_j$. In the high temperature limit, where $|\vec{k}_i|\ll T$, we can use the residue theorem to evaluate the $z$-integral by closing the contour on the left side in the complex $z$-plane. Although there is seemingly a logarithmic dependence on $T$ from a double-pole at $z=0$, coming from the product $\Gamma (z)\Gamma (z/2)$, the coefficient is actually proportional to $[(5/2)\Gamma (1/2)
-6\Gamma (3/2) +(2/3)\Gamma (5/2)] = 0$. This is in accordance with the fact that the fermion loop does not have any logarithmic enhancements [@nologs].
For the term of order $\beta^2$ we get, from the poles at $z=-2$, $$\begin{aligned}
\left . \Gamma^{abcd}_{0000} \right |_{(a)_1} = \left ( \frac{-7ig^4N_{\! f}\xi (3)\beta^2}{96\pi^4} \right )
{\rm Tr}_c (T^aT^bT^cT^d) \left ( k_1^2+2k_2^2+k_3^2 +2k_1k_2+2k_2k_3 \right ) \ .\end{aligned}$$ When the additional five permutations are added and the resulting four-point function inserted into Eq. (\[effactiona0\]), the contribution to the effective action from the quark loop becomes, $$\begin{aligned}
\left . \Gamma_{\!E} \right |_{(a)} (A_0)= \frac{7g^4N_{\! f}\xi (3)\beta^2}{2304\pi^4}\int_0^{\beta}\!d\tau\!\int \! d^3x \!
\left [ 2A_0^aA_0^a({\partial}_iA_0^b)^2 +({\partial}_i(A_0^aA_0^a))^2
-2f^{acm}f^{bdm} ({\partial}_iA_0^a)\!\cdot \!({\partial}_iA_0^b) A_0^cA_0^d \right ] \ , \label{fermionpart1}\end{aligned}$$ after a partial integration.
In the QED case we have $N_{\! f}{\rm Tr}_c (T^aT^bT^cT^d) \rightarrow 1$, and then our result for the two-derivative part of the QED effective action agrees with the earlier calculation in [@landsman].
The pure Yang-Mills contribution
--------------------------------
We now turn to the pure Yang-Mills contribution, i.e. the diagrams (b)-(e) in Fig. 1. To evaluate the finite $T$ part of these diagrams we will use the gauge condition ${\partial}\! \cdot \! A^a = 0$, and work in Feynman gauge. As in the fermion case, the functions $g_i = g_i(y_1,\ldots ,\vec{k}_1,\ldots ,\vec{k}_3)$ below are all functions of the relevant Feynman parameters and the external momenta. Their explicit forms can also be found in the appendix. Proceeding in a way similar to the previous section, we have for the ghost loop depicted in Fig. 1(b), $$\begin{aligned}
\left . \Gamma^{abcd}_{0000} \right |_{(b)_1} = && \frac{-ig^4}{8\pi^{5/2}}(f\!f\!f\!f) \!
\int_0^1\!\!dy_1\!\int_0^{1-y_1}\!\!\!dy_2\!\int_0^{1-y_1-y_2}\!\!\!\!dy_3\!
\int_{1+\epsilon-i\infty}^{1+\epsilon+i\infty}\!\!
\frac{dz}{2\pi i} \Gamma (z)\xi (z) \cos [\pi z/2](\beta^2g_1)^{-z/2} \Gamma ((5\!-\!z)/2)\Gamma (z/2) , \label{ym1}\end{aligned}$$ where the color structure is $f\!f\!f\!f = f^{fae}\!f^{ebg}\!f^{gch}\!f^{hdf} = \delta_{ab}\delta_{cd}\!
+\! \delta_{ad}\delta_{bc} \!+\!N(d_{abm}d_{cdm} \!-\!d_{acm}d_{bdm}\!+\!d_{adm}d_{bcm})/4$, with $N=3$ and $d_{ijk}$ the completely symmetric structure constant.
For the graph in (c) we find, $$\begin{aligned}
\left . \Gamma^{abcd}_{0000} \right |_{(c)_1} = &&\left ( \frac{3ig^4}{16\pi^{5/2}} \right ) (f\!f\!f\!f)
\int_0^1\!\!dy_1\!\int_0^{1-y_1}\!\!\!dy_2\!\int_0^{1-y_1-y_2}\!\!\!\!dy_3\!
\int_{1+\epsilon-i\infty}^{1+\epsilon+i\infty}\!\!
\frac{dz}{2\pi i} \Gamma (z)\xi (z) \cos [\pi z/2]\beta^{-z} \times \nonumber \\
&& \left [ g_2^{-z/2}\Gamma (z/2) \left \{ 5\Gamma ((1\!-\!z)/2) +16\Gamma ((3\!-\!z)/2) +32
\Gamma ((5\!-\!z)/2) \right \} -g_2^{-(2+z)/2}\Gamma ((2\!+\!z)/2) \times \right . \nonumber \\
&& \left . \left \{ g_4\Gamma ((1\!-\!z)/2)-\frac{16}{3}g_3\Gamma ((3\!-\!z)/2)\right \} -\frac{2}{3}g_5g_2^{-(4+z)/2}
\Gamma ((1\!-\!z)/2)\Gamma ((4\!+\!z)/2) \right ] \ , \label{ym2}\end{aligned}$$ where the color structure is as for the ghost loop.
For the triangle diagram (d) we get, $$\begin{aligned}
\left . \Gamma^{abcd}_{0000} \right |_{(d)_1} = &&\left ( \frac{-ig^4}{8\pi^{5/2}} \right )(f\!f\!)(f\!f\!+\!f\!f)
\int_0^1\!\!dy_1\!\int_0^{1-y_1}\!\!\!dy_2\!
\int_{1+\epsilon-i\infty}^{1+\epsilon+i\infty}\!\!
\frac{dz}{2\pi i} \Gamma (z)\xi (z) \cos [\pi z/2](\beta^2g_6)^{-z/2}
\times \nonumber \\ && \left [g_6^{-1}(g_6-3g_7)
\Gamma ((1\!-\!z)/2)\Gamma ((2\!+\!z)/2) -15\Gamma ((3\!-\!z)/2)\Gamma (z/2) \right ] \ , \label{ym3}\end{aligned}$$ where $(f\!f\!)(f\!f\!+\!f\!f) = f^{hfc}\!f^{gdf}(f^{age}\!f^{bhe}\!+\!f^{ahm}\!f^{bgm}) = -2\delta_{ab}\delta_{cd}\!
-\!\delta_{ac}\delta_{bd} \!-\!\delta_{ad}\delta_{bc}\!-\!Nd_{abm}d_{cdm}/2$.
Finally, we find for graph (e), $$\begin{aligned}
\left . \Gamma^{abcd}_{0000} \right |_{(e)_1}= &&\left ( \frac{3ig^4}{16\pi^{5/2}} \right )
(f\!f\!+\!f\!f)(f\!f\!+\!f\!f)\int_0^1\!\!dy_1\!\int_{1+\epsilon-i\infty}^{1+\epsilon+i\infty}\!\!
\frac{dz}{2\pi i} \Gamma (z)\xi (z) \cos [\pi z/2](\beta^2g_8)^{-z/2}\Gamma ((1\!-\!z)/2)\Gamma (z/2) \ , \label{ym4}\end{aligned}$$ with $(f\!f\!+\!f\!f)(f\!f\!+\!f\!f) = (f^{age}\!f^{bhe}\!+\!f^{ahe}\!f^{bge})
(f^{cgf}\!f^{dhf}\!+\!f^{chf}\!f^{dgf}) = 4\delta_{ab}\delta_{cd}\!+\!2\delta_{ac}\delta_{bd} \!+\!2
\delta_{ad}\delta_{bc}\!+\!Nd_{abm}d_{cdm}$.
Contrary to the fermion case, at order $O(\beta^0)$ each diagram contains a logarithmic dependence on $T$ and the external momenta $\vec{k}_j$. In addition, the $T=0$ part depends logarithmically on $\vec{k}_j$ and an ultraviolet cut-off $\Lambda$, that has to be introduced to regularize the loop-momentum integral. The logarithmic dependence on $\vec{k}_j$ cancels out between the $T=0$ and $T>0$ parts for each permutation of each individual diagram, whereas the terms containing $\log T$ ($\log \Lambda $) only cancel out in the total $T>0$ ($T=0$) result, i.e. when all the different diagrams are added together. This was basically noted already in [@nadkarni], and we have checked that it holds true in our calculations as well. There is also a linear divergence when $\beta k\rightarrow 0$, at order $O(1/\beta)$, in all of the Eqs. (\[ym1\])-(\[ym4\]). This divergence originates from static propagators running in the loop, i.e. the propagators with a vanishing Matsubara frequency, $\omega_n=0$. If only the non-static modes are integrated out, the static terms should be subtracted and our remaining result is then finite in the limit $\beta k\rightarrow 0$, as it should [@nadkarni]. We emphasize that the $\omega_n=0$ modes do not influence the two-derivative term.
Taking into account all the permutations of Eqs. (\[ym1\])-(\[ym4\]) and adding the different contributions, we find for the $O(\beta^2)$ term, $$\begin{aligned}
\left . \Gamma_{\!E} \right |_{(b)-(e)}(A_0) = -\frac{g^4\xi (3)\beta^2}{256\pi^4}\int_0^{\beta}\!d\tau\!\int \! d^3x \!
\left [ 2A_0^aA_0^a({\partial}_iA_0^b)^2 +({\partial}_i(A_0^aA_0^a))^2 +\frac{11}{6}
f^{acm}f^{bdm} ({\partial}_iA_0^a)\!\cdot \!({\partial}_iA_0^b) A_0^cA_0^d \right ] \ , \label{gluepart}\end{aligned}$$ where we have performed an integration by parts. This result for the pure Yang-Mills contribution disagrees slightly with the previous finding from the background field method in [@chapman], in that the first two term in Eq. (\[gluepart\]) are a factor $(2/11)$ smaller, and the last a factor $(-1/2)$[^1].
The total contribution
----------------------
By combining the results in Eqs. (\[fermionpart1\]) and (\[gluepart\]), the complete contribution to the effective action becomes, $$\begin{aligned}
\Gamma_{\!E}^{(4)} (A_0)= && \left . \Gamma_{\!E} \right |_{(a)} (A_0)+ \left . \Gamma_{\!E} \right |_{(b)-(e)} (A_0)=
\frac{g^4\xi (3)\beta^2}{256\pi^4}\left ( \frac{7N_{\! f}}{9}-1\right )\int_0^{\beta}\!d\tau\!\int \! d^3x \!
\left [ 2A_0^aA_0^a({\partial}_iA_0^b)^2 +({\partial}_i(A_0^aA_0^a))^2\right ] - \nonumber \\
&& \frac{11g^4\xi (3)\beta^2}{1536\pi^4}\left ( \frac{28N_{\! f}}{33}+1\right )\int_0^{\beta}\!d\tau\!\int \! d^3x \!
\left [ f^{acm}f^{bdm} ({\partial}_iA_0^a)\!\cdot \!({\partial}_iA_0^b) A_0^cA_0^d \right ] \ . \label{effaction}\end{aligned}$$ In the dimensionally reduced theory of QCD [@nadkarni; @landsman], Eq. (\[effaction\]) provides the leading derivative interactions between the $A_0$-fields. How large the calculated derivative term is, compared to both the ones already present in Eq. (\[knownaction\]) as well as the omitted higher dimension operators, depends on the scales of interest. For instance, at the soft scale where ${\partial}_i
\sim gT$ and $A_0 \sim T$, we have $g^4\beta^2{\partial}^2A_0^4 \sim g^6T^4$, a factor $g^2$ higher than the term $g^4A_0^4$.
In general, this effective theory is interesting on length scales $|\vec{x}|\ll \beta$ in the high temperature regime, especially when combined with nonperturbative lattice methods [@kajantie]. It is for example possible to study the non-perturbative Debye mass [@debyemass], and the 3d effective theory is also useful for calculations of the pressure in the quark-gluon phase, both perturbatively [@nieto] and nonperturbatively [@resum]. Although the derived correction in Eq. (\[effaction\]) presumably gives a minor effect only, it is somewhat interesting to note that the terms are rather sensitive to the number of quark flavors. In Eq. (\[effaction\]), the coefficient of the first term is proportional to $(7N_{\! f} -9)$, which goes from $-1$ to $4/3$ between $N_{\! f}=0$ and $N_{\! f}=3$. Similarly, the second term in Eq. (\[effaction\]) increases by more than 250% in the same range of $N_{\! f}$. This should be contrasted with the constant part of the $A_0^4$-contribution to Eq. (\[knownaction\]), that depends on $N_{\! f}$ as $(9 - N_{\! f})$ and therefore only changes by 33% when going from $N_{\! f}=0$ to $N_{\! f}=3$. Due to this strong behavior of the number of quark flavors, it is not inconceivable that the derivative interactions will make a small but noticeable difference between e.g. the pure glue theory and the three-flavor case.
Derivative terms in the Wilson Line model
=========================================
The QCD dimensionally reduced theory describes accurately static phenomena at very high $T$, but the approximations break down around a few times $T_c$ [@kajantie]. In addition, one is by construction omitting all dynamical information.
To understand the features around $T_c$ a Ginzburg-Landau type of effective theory was proposed in [@robmodel]. In this model, the potential is written in terms of $l$, $$\begin{aligned}
V(l)= a_1T^4\left [ -a_2|l|^2 -a_3\left ( l^3+{\rm c.c.} \right )+|l|^4
\right ] \ . \label{robpotential}\end{aligned}$$ The constants $a_i$ are then used to fit the pressure above $T_c$, with $a_2$ a function of temperature so that the global minimum of the potential is at $l\neq 0$ ($l=0$) above (below) $T_c$ [@robadrian]. One of the important aspects of the potential in Eq. (\[robpotential\]) is the extremely rapid change around $T_c$, due to a very sensitive dependence of $a_2$ on $T/T_c$ [@robadrian]. In a dynamical scenario one can therefore assume an instantaneous quench, where the value of $l$ suddenly no longer corresponds to the correct minimum. The $l$-field then rolls down the potential, and by coupling the $l$-field to a linear sigma model the potential energy is converted into pions [@robadrian; @particleprod]. Even though the model is of phenomenological origin, it thus makes predictions that can be compared to experimental results.
After the quench, the evolution of the Euler-Lagrange equations from the initial conditions requires, apart from the potential and the coupling to the chiral field, also a kinetic term for $l$ [@robadrian; @particleprod]. Although the time dependence is beyond the calculation presented in this paper, we can provide the first perturbative QCD-correction to the spatial derivatives. The leading coefficient is the classical contribution to the derivative term, and comes from the kinetic term of $A_0 (\vec{x})$ in Eq. (\[knownaction\]), as can be seen from the following argument [@robprivate]: decomposing the Wilson Line in Eq. (\[wilsonline\]) into an octet $\tilde{{\bf L}}$ and the singlet $l$, $$\begin{aligned}
{\bf L} = \tilde{{\bf L}} + {\bf 1}\frac{1}{N}{\rm Tr}\, {\bf L} = \tilde{{\bf L}} + {\bf 1}l \ ,\end{aligned}$$ where $\tilde{{\bf L}}$ is traceless and $N=3$, we have $$\begin{aligned}
{\rm Tr}|{\partial}_i {\bf L}|^2 = {\rm Tr}|{\partial}_i \tilde{{\bf L}}|^2 + 3|{\partial}_i l|^2 \ . \label{decomposeL}\end{aligned}$$ On the other hand, by a direct calculation in the static limit, $$\begin{aligned}
{\rm Tr}|{\partial}_i {\bf L}|^2 = g^2\beta^2 ({\partial}_iA_0^a)^2 + {\rm Tr}\, \left \{ {\rm commutator \, terms}
\right \} \ . \label{commutators}\end{aligned}$$ When the commutator terms in Eq. (\[commutators\]) are rewritten in terms of ${\bf L}$ they can only involve the adjoint field, or products of $l$ and $\tilde{{\bf L}}$, since $l$ (times the identity matrix) by itself commutes with all SU(3) matrices. Thus, by combining Eqs. (\[decomposeL\]) and (\[commutators\]), we have, $$\begin{aligned}
(1/2)({\partial}_iA_0)^2 = \frac{T^2}{2g^2}\left [ {\rm Tr}|{\partial}_i \tilde{{\bf L}}|^2 + 3|{\partial}_i l|^2 \right ] +
f(\tilde{{\bf L}}) \ , \label{leadingderivative}\end{aligned}$$ where $f(\tilde{{\bf L}})$ corresponds to the commutator terms, rewritten as a function of $\tilde{{\bf L}}$. At $T_c$, $g\simeq 2.5$ so that the leading coefficient for the kinetic term of $l$ is $3/2g^2 \simeq 0.3$, which is reasonably close to the canonical value $1/2$. Given the unknown function $f(\tilde{{\bf L}})$ it is not clear whether the kinetic term for the adjoint field actually is unique. However, $\tilde{{\bf L}}$ does not play any important role around $T_c$, and can therefore be neglected on physical grounds [@robmodel].
The procedure to obtain the kinetic term for $l$ is thus to match terms in the effective theory $\Gamma (A_0)$ to a corresponding $|{\partial}_i l|^2$ piece in $\Gamma (l)$. This means that the classical coefficient for $|{\partial}_i l|^2$ will change when radiative corrections are taken into account in $\Gamma (A_0)$. To lowest order, the kinetic term for $A_0$ can receive corrections from the polarization tensor [@nadkarni; @landsman], that would affect the $|{\partial}_i l|^2$ term via Eq. (\[leadingderivative\]). However, with the optimal choice for the counterterms [@landsman] there is in fact no $O(g^2)$-contribution, so the renormalized kinetic term in $\Gamma (A_0)$ remains $(1/2)({\partial}_i A_0)^2$.
Even though the kinetic term for $A_0$ is unchanged at one-loop, this does not mean that $|{\partial}_i l|^2$ is so. Since $l = (1/3){\rm Tr}\,{\bf L}$, the $|{\partial}_i l|^2$-term contains at least four powers of $A_0$ when ${\bf L}$ is expanded in powers of $A_0$. The two-derivative term in $\Gamma (l)$ can therefore receive a perturbative correction from the four-point function in Eq. (\[effactiona0\]). Indeed, by using the relations $$\begin{aligned}
A_0^aA_0^a({\partial}_iA_0^b)^2 = &&-\frac{12}{g^4\beta^4}\left [ |l|^2 -1\right ] \left (
{\rm Tr} |{\partial}_i \tilde{{\bf L}}|^2 +3|{\partial}_i l|^2 \right ) +O(A_0^6) \ , \nonumber \\
({\partial}_i(A_0^aA_0^a))^2 = && \frac{144}{g^4\beta^4}|{\partial}_i l|^2 +O(A_0^6)\nonumber \\
f^{acm}f^{bdm} ({\partial}_iA_0^a)\!\cdot \!({\partial}_iA_0^b) A_0^cA_0^d = && 2{\rm Tr}|
[{\partial}_i \tilde{{\bf L}}, \ \tilde{{\bf L}}] |^2 +O(A_0^6) \ ,\end{aligned}$$ we can rewrite Eq. (\[effaction\]) as $$\begin{aligned}
\Gamma_{\!E} = && \int_0^{\beta}\!d\tau\!\int \! d^3x \! \left \{ \frac{\xi (3)T^2}{16\pi^4}\left
(\frac{7N_{\! f}}{9}-1\right )
\left [ \frac{3}{2} (1-|l|^2){\rm Tr}|{\partial}_i\tilde{{\bf L}}|^2 +\frac{27}{2}|{\partial}_i l|^2
-\frac{9}{2}|l|^2 |{\partial}_i l|^2 \right ] \right . - \nonumber \\ && \left .
\frac{11\xi (3)T^2}{768\pi^4} \left (\frac{28N_{\! f}}{33}+1\right )
{\rm Tr}|[{\partial}_i \tilde{{\bf L}}, \ \tilde{{\bf L}} ]|^2 \right \} \ . \label{gammainls}\end{aligned}$$ From the calculation of the four-point function we thus find the leading perturbative correction to the kinetic term of $l$. Including the contribution from Eq. (\[gammainls\]), the kinetic term now becomes $$\begin{aligned}
\frac{3T^2}{2g^2}|{\partial}_i l|^2 \rightarrow
\frac{3T^2}{2g^2}\left [ 1+ \frac{g^2\xi (3)}{16\pi^4}(7N_{\! f}-9)\right ] |{\partial}_i l|^2 =
\frac{3T^2}{2g^2} \left [ 1+c_{{\rm corr}}\right ] |{\partial}_i l|^2 \ .\end{aligned}$$ Since $l$ is dimensionless, the correct dimension of the operator has to be supplied by some other scales. In the perturbative calculation the only scale is $T$, and hence the one-loop induced coefficient has the same $T$-dependence as the leading term in Eq. (\[leadingderivative\]). However, the perturbative correction does not depend on the QCD coupling constant and is therefore just a fixed number at this order. For example, for three flavors the coefficient is $\simeq 1.4\times 10^{-2}T^2$. Compared to the classical contribution $(3/2g^2)T^2$, the fraction of the one-loop correction is only $0.01g^2$. Even at $T_c$, this is merely of the order of 5%, and at higher $T$ even less due to the logarithmic decrease of $g$.
Having derived the first correction to the kinetic term for $l$ from perturbation theory, let us now discuss to what extent, and in what temperature range, the terms in Eq. (\[gammainls\]) can be trusted. First of all, it should be noted that the form of the effective action in Eq. (\[gammainls\]) is not completely unique. The reason is that by a partial integration, and discarding any surface terms, we can always trade factors of $[A_0^aA_0^a({\partial}_iA_0^b)^2]$ and $({\partial}_i(A_0^aA_0^a))^2$ for a term like $[A_0^aA_0^aA_0^b({\partial}_i^2\!A_0^b)]$, but this equality does not hold at the level of $l$. In fact, if a term $[A_0^aA_0^aA_0^b({\partial}_i^2\!A_0^b)]$ is kept in the action, not only do the coefficients in Eq. (\[gammainls\]) change, but there are also additional nonequivalent terms of the form $({\partial}_i|l|^2)^2$ and $|l|^2[{\partial}_i^2(l+l^{\ast})]$. Nevertheless, the action in Eq. (\[gammainls\]) is of course unique to order $O(A_0^4)$. Since higher order operators in the dimensionally reduced theory are further suppressed at high $T$, the predictions in Eq. (\[gammainls\]) should at the very least be reliable down to $T\geq 2T_c-3T_c$, i.e. when the effective 3d theory itself is applicable.
When $T\rightarrow T_c$, the question is admittedly more subtle, as higher loop effects, higher dimensional operators and possibly nonperturbative effects become important. However, Eq. (\[gammainls\]) does not have to break down completely when the 3d theory does so. The 3d theory becomes invalid because the procedure of integrating out the non-static modes is unreliable when $g^2(T)\, T \simeq \pi T$ [@kajantie]. In contrast, $\Gamma (l)$ is by construction valid near $T_c$, so the question is rather how much the coefficient for the spatial derivative term changes.
Considering first the operators of higher dimensionality, it is certainly possible to imagine that their bulk part follows from an expansion of Eq. (\[gammainls\]). The additional contributions that do not originate from these sources, e.g. terms like $T^2(|l|^2-1)^n|{\partial}_il|^2$, that are at least of order $A_0^{(2n+2)}$ (with $n\geq 2$), would then be suppressed. Not because they are unimportant a priori, but because their numerical coefficients are small. There are also higher derivative terms not accounted for in Eq. (\[gammainls\]), like $(|l|^2-1)({\partial}_i^2|l|^2)^2$, but they do not affect the kinetic term. To $O(A_0^4)$, they are in fact straightforward to obtain from Eq. (\[fermionloop\]) and Eqs. (\[ym1\])-(\[ym4\]).
When it comes to higher loop and nonperturbative effects, they will naturally induce a $T$-dependence in the radiative corrections to the kinetic term. Thus, $(3T^2/2g^2)(1+c_{\rm corr})|{\partial}_il|^2\rightarrow (3T^2/2g^2)(1+c_{\rm corr}(T/T_c, T/\Lambda ))|{\partial}_il|^2 $, which follows partly from the running of the QCD coupling constant in the higher loop effects. How much this will affect the kinetic term is difficult to estimate, but the small correction from the four-point function may indicate that the perturbative QCD-contributions are not too important for constructing $\Gamma (l)$.
Summary and Conclusions
=======================
In this paper we calculated the leading momentum dependence of the four-point function in QCD with $N_{\! f}$ massless flavors, and related this contribution to terms in both the effective action $\Gamma (A_0)$ and $\Gamma (l)$.
As for the derivative terms in $\Gamma (A_0)$, they will only have a minor influence when $T\gg T_c$. As the temperature decreases and approaches $T_c$ the 3d theory becomes less reliable, but there could very well be a temperature region where the effective theory is still valid and the derivative interactions nonnegligible. Since the contribution has a rather strong dependence on $N_{\! f}$, a difference between the pure glue theory and e.g. $N_{\! f}=3$ QCD could perhaps be noticed.
From the four-point function, we also found the lowest one-loop QCD correction to the spatial derivative term in the effective theory $\Gamma (l)$. The coefficient is independent of $g$ and much smaller than the classical term, the ratio between the two being of the order of $10^{-2}$ at $T_c$. This derivation assumes that the strange quark mass $m_s$ can be neglected even at $T_c$, which of course is an oversimplification, given that $m_s\sim T_c$. Nevertheless, the influence of $m_s$ is not likely to change the fact that the correction is small even at $T_c$.
At one-loop, there is an infinite number of terms that contribute to the coefficient of $|{\partial}_i l|^2$, to the same order in $g$ and $T$, as the four-point function. This follows from the fact the induced two-derivative interactions, with $n$ external fields $A_0$, is of the functional form $g^n{\partial}^2 A_0^n/T^{(n-2)}$, which corresponds to an expansion of $l$ to at most order $(n-2)$ in the term $T^2|{\partial}_i l|^2$. Some of these higher-dimensional contributions, maybe even the major parts, are already accounted for by rewriting the four-point function in $\Gamma (A_0)$ in terms of $l$, as in Eq. (\[gammainls\]). In any case, since the four-point contribution is very small, it is reasonable to assume that the higher $n$-point functions give even smaller corrections. In that case, Eq. (\[gammainls\]) should give the correct order of magnitude for the total one-loop correction.
As mentioned earlier, there are also higher loop effects that contribute to the kinetic term in $\Gamma (l)$. For example, taking into account the two-loop correction to the diagrams in Fig. 1 gives $c_{{\rm corr}} \rightarrow c_{{\rm corr}}[1+ag^2]$. To understand the reliability of the canonical term it is then crucial to know the magnitude of $a$. Surprisingly, studies of higher loop effects in the 3d effective theory $\Gamma (A_0)$ indicate that they only give corrections of the order of 30% at $T_c$ [@kajantie]. If these conjectures can be taken over to the Wilson Line model, one could in fact expect the derivative term $(1/2)|{\partial}_i l|^2$ to change by perhaps at most a factor two, with all QCD-corrections taken into account. Of course, this has to be regarded as a highly speculative suggestion at the present stage.
To complete the dynamical scenario one also needs the time dependence of $l$. Unfortunately, it is yet unclear how $l$ generalizes to a real time formulation [@robmodel]. Assuming that the form of the spatial derivatives can be extended to a Lorentz invariant form, the predictions for pion production and the evolution of $|l|$ will remain almost unchanged [@robadrian; @particleprod]. In particular, if the Lorentz invariant kinetic term does not change by more than a factor of two, it can easily be compensated by a difference in e.g. the expansion rate of the plasma.
Finally, to obtain a decisive estimate of the QCD-effects in $\Gamma (l)$, one has to establish either a unique mapping from $\Gamma (A_0)$ to $\Gamma (l)$, or find a way to determine $\Gamma (l)$ directly, perhaps numerically. Hopefully, the calculations presented in this paper can serve as a first step in that direction.
[**Acknowledgments**]{}
The author thanks R. D. Pisarski for discussions that initialized this project, and for reading the manuscript. D. Bödeker, A. Dumitru and R. D. Pisarski are greatfully acknowledged for useful and interesting discussions during the investigations, and K. Kajantie and M. Laine for useful comments on an early version of the manuscript. This work was supported by The Swedish Foundation for International Cooperation in Research and Higher Education (STINT) under contract 99/665, and in part by DOE grant DE-AC02-98CH10886.
Evaluation of the Feynman diagrams
==================================
In this appendix we outline our method for calculating the Feynman diagrams shown in Fig. 1. We first follow the Feynman rules given in [@peskin] for Minkowski space-time. After all contractions and traces over spinor indices have been performed, we continue to a Euclidean space compact in the imaginary time direction: $$\begin{aligned}
p_0\rightarrow i\omega_n \ , \ \ \ \ \int\frac{dp_0}{2\pi}\rightarrow
iT\sum_n \ , \end{aligned}$$ where $\omega_n$ is the Matsubara frequency, $\omega_n = (2n+1)\pi T$ ($\omega_n=2\pi T$) for fermions (bosons).
Next, we use a Feynman parametrization to combine the denominators in the loop integral, and extract the $T>0$ part from the following relations [@finitetbooks]: $$\begin{aligned}
\left . T\sum_n f(p_0=i\omega_n) \right |_{T>0} = \left \{ \matrix{ &
\int_{\delta-i\infty}^{\delta+i\infty}\frac{dp_0}{2\pi i}
\, n_B \, [f(p_0)+f(-p_0)] \ \ {\rm (bosons)} \cr & \mbox{} \cr
& \int_{\delta-i\infty}^{\delta+i\infty}\frac{dp_0}{2\pi i}
\, n_F \, [f(p_0)+f(-p_0)]
\ \ {\rm (fermions)}} \right . \ .\end{aligned}$$ where $n_B = [\exp (\beta p_0)-1]^{-1}$, $n_F =[\exp (\beta p_0)+1]^{-1}$ and $\delta \rightarrow 0^{+}$. The ghosts follow Bose statistics, despite their anticommuting properties.
By shifting the vector momentum in the loop, $\vec{p}$, to $\vec{q}=\vec{p} + h(y_i, \vec{k}_i)$, with $h(y_i, \vec{k}_i)$ a linear function of the external momenta $\vec{k}_i$, we then integrate over $\vec{q}$. Finally, by performing a Mellin transform, $$\begin{aligned}
\left ( e^x \pm 1 \right )^{-1} = \int_{c_{\pm}-i\infty}^{c_{\pm}+i\infty}\frac{dz}{2\pi i} \Gamma (z)\xi (z)v_{\pm}x^{-z} \ ,\end{aligned}$$ where $v_{+}=(1-2^{1-z})$, $v_{-}=1$, and with the contour specified by $c_{+}=\epsilon$, $c_{-}=1+\epsilon$ (where $\epsilon\rightarrow 0^{+}$), we can integrate over $p_0$ after a change of variables.
To illustrate the above procedure, consider the diagram in Fig. 1(e). Omitting the color factors for simplicity, and using the notation $|\vec{p}|=p$, $|\vec{k}_1+\vec{k}_2|=|\vec{k}_{12}|=k_{12}$, we have to calculate the following integral: $$\begin{aligned}
&& \frac{3g^4}{2}\!\int \!\frac{d^4p}{(2\pi )^4}\, \frac{1}{p_0^2-p^2}\, \frac{1}{p_0^2-(p+k_{12})^2}
\stackrel{T>0}{\rightarrow} 3ig^4 \!
\int_{\delta-i\infty}^{\delta+i\infty}\! \frac{dp_0}{2\pi i} n_B
\! \int_0^1 \! dy_1\!\int \! \frac{d^3p}{(2\pi )^3}
\frac{1}{[p^2+\{2\vec{p}\cdot \vec{k}_{12} + k_{12}^2 \}(1\!-\!y_1)-p_0^2]^2}
\nonumber \\ && \ \ \ \ = \frac{3ig^4}{2\pi^2}\int_0^1dy_1
\int_{\delta-i\infty}^{\delta+i\infty}\frac{dp_0}{2\pi i} n_B
\int_0^{\infty}dq \frac{q^2}{[q^2+g_8-p_0^2]^2} \ , \label{illustration}\end{aligned}$$ where we made the shift $\vec{q}=\vec{p}+(1-y_1)\vec{k}_{12}$ and defined $g_8 = y_1(1-y_1)k_{12}^2$ in the last integral. After performing the $q$-integral in Eq. (\[illustration\]), we are left with, $$\begin{aligned}
&& \frac{3ig^4}{8\pi}\! \int_0^1\!dy_1 \int_{\delta-i\infty}^{\delta+i\infty}\frac{dp_0}{2\pi i}\left (
\frac{1}{e^{\beta p_0}\!-\!1}\right )\frac{1}{\sqrt{g_8-p_0^2}} = \frac{3ig^4}{8\pi^2}\!\int_0^1\!dy_1
\int_{1+\epsilon-i\infty}^{1+\epsilon+i\infty}
\frac{dz}{2\pi i}\Gamma (z) \xi (z) \cos [\pi z/2]\beta^{-z} \! \int_0^{\infty}\! du
\frac{u^{-z}}{\sqrt{g_8+u^2}} \nonumber \\ && \ \ \ = \frac{3ig^4}{16\pi^{5/2}} \int_0^1dy_1
\int_{1+\epsilon -i\infty}^{1+\epsilon +i\infty}
\frac{dz}{2\pi i}\Gamma (z) \xi (z) \beta^{-z} \cos [\pi z/2] g_8^{-z/2}
\Gamma ((1-z)/2) \Gamma (z/2) \ ,\end{aligned}$$ which is Eq. (\[ym4\]), except for the color factors.
To check our method we also calculated the $O(\beta^2k_{12}^2)$ contribution to the above diagram in a different way: we first performed the $p_0$-integral, by picking up the poles in the complex $p_0$-plane, and then did the $p$-integral, without any Feynman parametrization, by expanding the integrand differently in different integration regions. The two results of course agree with each other.
For completeness, we also give the functions $f_1,\ldots ,f_4$ in Eq. (\[fermionloop\]) and $g_1,\ldots ,g_8$ in Eqs. (\[ym1\])-(\[ym4\]). Using the conservation of momentum, and the shorthand notation $\vec{k}_i+\vec{k}_j+\vec{k}_l=k_{ijl}$, their explicit forms are as follows: $$\begin{aligned}
f_1 =&& y_2k_1^2+y_3k_{12}^2+(1-y_1-y_2-y_3)k_{123}^2 - [y_2k_1+y_3k_{12}+(1-y_1-y_2-y_3)k_{123}]^2
\\
f_2 =&& 3k_1^2+4k_1k_2+k_2^2+2k_1k_3+k_2k_3-3(3k_1+2k_2+k_3)[y_2k_1+y_3k_{12}+(1-y_1-y_2-y_3)k_{123}]+\nonumber \\
&& +6[y_2k_1+y_3k_{12}+(1-y_1-y_2-y_3)k_{123}]^2 \\
f_3 =&& (1/3)\left \{ 5k_1^2y_1(2y_1-1)+k_2^2[3+10y_1^2-10y_2+10y_2^2+10y_1(2y_2-1)]+
+5k_3^2[1+2y_1^2+2y_2^2-3y_3+2y_3^2+\right . \nonumber \\ && +y_2(4y_3-3)+y_1(4y_2+4y_3-3)]+k_2k_3[8+20y_1^2+20y_2^2-10y_3+
5y_2(4y_3-5)+5y_1(8y_2+4y_3-5)] + \nonumber \\ && \left .
k_1[k_2(3+20y_1^2-5y_2+5y_1(4y_2-3))+k_3(2+20y_1^2-5y_2-5y_3+20y_1(y_2+y_3-1))] \right \} \\
f_4 =&& \left \{ [(1-y_2)k_1-y_3k_{12}-(1-y_1-y_2-y_3)k_{123}]\cdot [-y_2k_1+(1-z)k_{12}-(1-y_1-y_2-y_3)k_{123}]\right \}
\times \nonumber \\ &&
\left \{ [-y_2k_1-y_3k_{12}-(1-y_1-y_2-y_3)k_{123}]\cdot [-y_2k_1-y_3k_{12}+(y_1+y_2+y_3)k_{123}]\right \} - \nonumber \\
&& -\left \{ [-y_2k_1-y_3k_{12}-(1-y_1-y_2-y_3)k_{123}]\cdot [-y_2k_1+(1-z)k_{12}-(1-y_1-y_2-y_3)k_{123}] \right \}
\times \nonumber \\ &&
\left \{ [(1-y_2)k_1-y_3k_{12}-(1-y_1-y_2-y_3)k_{123}]\cdot [-y_2k_1-y_3k_{12}+(y_1+y_2+y_3)k_{123}]\right \}+
\nonumber \\ &&
+\left \{ [-y_2k_1-y_3k_{12}-(1-y_1-y_2-y_3)k_{123}]\cdot [(1-y_2)k_1-y_3k_{12}-(1-y_1-y_2-y_3)k_{123}]\right \} \times
\nonumber \\ &&
\left \{ [-y_2k_1+(1-z)k_{12}-(1-y_1-y_2-y_3)k_{123}]\cdot [-y_2k_1-y_3k_{12}+(y_1+y_2+y_3)k_{123}]\right \} \\
g_1 = && f_1 \\
g_2 = && f_1 \\
g_3 = && 3k_1^2+2k_2^2+2k_3^2+3k_1k_2+3k_1k_3+2k_2k_3+2[(1-y_1)k_1+(1-y_1-y_2)k_2+(1-y_1-y_2-y_3)k_3]^2 - \nonumber \\ &&
- (3k_1+2k_2+k_3)[(1-y_1)k_1+(1-y_1-y_2)k_2+(1-y_1-y_2-y_3)k_3] \\
g_4 = && (1/3)\left \{ k_1^2(10y_1-20y_1^2-7)-2k_2^2[10y_1^2-10y_2+10y_2^2+10y_1(2y_2-1)-3] - \right . \nonumber \\
&& - k_3^2[17+20y_1^2+20y_2^2-30y_3+20y_3^2+10y_2(4y_3-3)+10y_1(4y_2+4y_3-3)] - \nonumber \\
&& -2k_2k_3[2+20y_1^2+20y_2^2-10y_3+
5y_2(4y_3-5)+5y_1(8y_2+4y_3-5)] - \nonumber \\
&& \left . -2k_1[k_2(20y_1^2-5y_2+5y_1(4y_2-3)-3)+5k_3(3+4y_1^2-y_2-y_3+4y_1(y_2+y_3-1))] \right \} \\
g_5 = && \left \{ [y_1k_1+(1+y_1+y_2)k_2-(1-y_1-y_2-y_3)k_3]\cdot [(2-y_1)k_1+(1-y_1-y_2)k_2+(1-y_1-y_2-y_3)k_3] \right \}
\times \nonumber \\ && \left \{ [-y_1k_1-(y_1+y_2)k_2+(2-y_1-y_2-y_3)k_3]\cdot [(2-y_1)k_1+(2-y_1-y_2)k_2
+(2-y_1-y_2-y_3)k_3] \right \} + \nonumber \\ &&
\left \{ [(1+y_1)k_1-(1-y_1-y_2)k_2-(1-y_1-y_2-y_3)k_3]\cdot [(1+y_1)k_1+(1+y_1+y_2)k_2+(1+y_1+y_2+y_3)k_3] \right \}
\times \nonumber \\ &&
\left \{ [-y_1k_1+(2-y_1-y_2)k_2+(1-y_1-y_2-y_3)k_3]\cdot [y_1k_1+(y_1+y_2)k_2+(1+y_1+y_2+y_3)k_3] \right \} \\
g_6 = && y_1k_{12}^2+y_2k_{123}^2+(1-y_1-y_2)k_{1234}^2 -[k_1+k_2+(1-y_1)k_3+(1-y_1-y_2)k_4]^2 \label{eqg6} \\
g_7 = && [(1+y_1)k_3-(1-y_1-y_2)k_4][(2-y_1-y_2)k_4-y_1k_3] \ , \label{eqg7}\end{aligned}$$ where, in order to simplify the permutations of the triangle graph, we did not use $k_4=-(k_1+k_2+k_3)$ in Eqs. (\[eqg6\]) and (\[eqg7\]). Finally, $$\hspace{-13.9cm} \hspace{2pt}
g_8 = y_1(1-y_1)k_{12}^2 \ .$$
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[^1]: It should be noted, however, that the discrepancy is of marginal practical importance when it comes to the qualitative discussion of the influence of these QCD-terms in the Polyakov Loop action.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We derive new integral representations for objects arising in the classical theory of elliptic functions: the Eisenstein series $E_s$, and Weierstrass’ $\wp$ and $\zeta$ functions. The derivations proceed from the Laplace-Mellin transformation for multipoles, and an elementary lemma on the summation of 2D geometric series. In addition, we present new results concerning the analytic continuation of the Eisenstein series as an entire function in $s$, and the value of the conditionally convergent series, denoted by $\widetilde{E}_2$ below, as a function of summation over increasingly large rectangles with arbitrary fixed aspect ratio[^1].'
author:
- 'Andrew Dienstfrey[^2] and Jingfang Huang[^3]'
bibliography:
- '/home/andrewd/Bibliography/master.bib'
title: Integral Representations for Elliptic Functions
---
Introduction
============
In this paper we revisit the classical theory of elliptic functions as developed by Eisenstein and Weierstrass. Both of these researchers represented the meromorphic functions appearing in their theories as summations over a given lattice of elementary pole functions of a prescribed order. Our fundamental observation is that pole functions may be represented by exponentially-damped, oscillatory integrals. These representations depend on the complex half-planes in which the singularities lie, and are natural variants of the classical Mellin, or Laplace-Mellin, formulas which are valid for isolated poles lying in the right half plane, where $\Re(\w)>0$. In more recent times such integral representations have resurfaced in the development of fast multipole methods for the solution of Poisson problems; in this context they often are referred to as “plane-wave” representations ([@hr98] and, more recently, [@cgr99]). A key feature of these integral representations is that the pole centers appear in the exponents of the integrands. As a consequence the lattice summations are transformed into geometric series which may be summed explicitly underneath the integral. The result is a new class of integral representations for the Eisenstein series and other meromorphic functions of Weierstrass’ theory.
A brief summary of the paper follows. In the first section we review the definitions of the Eisenstein series $E_n$ and the Weierstrass functions ¶ and . We will analyze a generalization of Eisenstein’s series which we denote by $\tEs$, the differences being: first, we consider $s=\sig+\i t\in\C$, and second, we define $\tEs$ as a limit over lattice squares of increasing size, a significant point when $\Re(s)\le2$ and the sums are not absolutely convergent. In addition, in this preliminary section we provide elementary derivations of the requisite plane-wave formulas for general pole functions of the form $f(\w)=\w^{-s}$, and a summation identity for a two-dimensional geometric series.
In the next section we employ the plane-wave representations and the summation identity to derive an integral representation for $\tEs$ for the case $\ReSTwo$. Integral representations for Eisenstein’s $E_n$ naturally follow for $s=n\ge 3$. Subsequently, we derive an alternative representation for $\tEs$ as a contour integral from which we deduce that the sums $\tEs$, defined unambiguously for , admit an analytic continuation as an [**entire**]{} function to the whole of the complex plane. As a corollary, we prove the existence of a finite limit for $\tE_2$. We discuss $\tE_2$ and its relation to Eisenstein’s, $E_2$. As the summation processes defining these two conditionally convergent series are distinct, one expects different limiting values. We derive a closed form correction term for this difference. (Note that both Eisenstein’s convention and ours give $E_1=\tE_1=0$.) In the following section we derive analogous integral formulas for Weierstrass’ ¶ and functions. We conclude the paper with a brief discussion of these integral representations in relation to previous research in the theories of lattice sums, and elliptic functions.
In this last regard we take a moment to mention here that our formulas for $\tEs$ are the natural lattice analogues to the well-known representation for Riemann’s zeta function (which we denote with the subscript $\z_R$ so as to distinguish it from Weierstrass’ function of the same name) \[RiemannZ\] \_R(s) = \_[n=1]{}\^ = ł\^[s-1]{} \^[-ł]{} , (s)>1. For example, in the case of a square lattice we derive the following integral expression for the classical Eisenstein series E\_k(i) &=& \_[n=-]{}\^\_[m=-]{}\^\
\[EkSquare\] &=& ł\^[k-1]{} \^[-ł]{} , where $k$ is a positive integer divisible by four ($E_k(i)=0$ otherwise). The similarity between (\[RiemannZ\]) and (\[EkSquare\]) is clear. For more general lattices, we replace $\i$ by $\t$, $k\in\N$ by $s\in\C$, and the single trigonometric ratio in (\[EkSquare\]) by a sum of analogous ratios denoted by and defined in (\[fOne:def\]) and (\[fTwo:def\]). The general expression is given in theorem \[Eisenstein:thm\].
We note that a subset of the results presented below appeared previously in a slightly different form [@huang99].
Preliminaries
=============
In this section we review the definitions of the Eisenstein series and the Weierstrass ¶ and functions. Furthermore, we derive elementary lemmas concerning plane-wave representations and a geometric series identity, both of which we will use repeatedly in the subsequent sections.
The Eisenstein series and Elliptic functions
--------------------------------------------
We are given a general lattice $\Lambda\subset \C$ which we describe by generators $\mu, \nu$: $$\Lambda=\{m\cdot\mu + n\cdot\nu| m,n\in\Z\}$$ where $\mu,\nu$ are complex numbers such that the lattice ratio, $\t=\nu/\mu$, is not real. The fundamental parallelogram is the set, $\L_0=\{\al+\beta\t| |\al|\le 1/2, |\beta|\le 1/2\}$. Eisenstein began his investigations of doubly periodic, meromorphic functions with periods $\mu$ and $\nu$ through the study his eponymous series (see, for example, [@weil76] and [@schoeneberg74]) \[EisensteinEn\] E\_k = , k1; the elimination of the term $m=n=0$ is implicit here and below. For the magnitude of each term in the summand we have the bounds \[MagBound\] . Applying (\[MagBound\]) and elementary estimates one verifies that the series (\[EisensteinEn\]) are absolutely convergent for $n\ge
3$, and thus are well-defined functions of the lattice $E_n=E_n(\Lambda)$. The same estimates indicate that the series $E_n$ are absolutely divergent for $n=1$ or $2$, hence the limiting operation specified in (\[EisensteinEn\]) plays a non-trivial role in the definition of these sums. Eisenstein proved that the limiting procedure (\[EisensteinEn\]) yields finite values of $E_n$ even in these cases. We observe that, in particular, $E_1=0$. This is in keeping with the fact that, using a symmetry argument in the absolutely convergent case, one may prove that $E_n=0$ for all odd $n\ge3$. Therefore, from the point of view of convergence, the only “interesting” sum is $E_2$.
Eisenstein was cognizant that the value of $E_2$ depends on the choice of limiting procedures; and he derived many identities which connect his summation process for $E_2$ to others ([@weil76]). We choose yet a different summation convention and define $\tEs$ as the limit of partial sums over “lattice-squares” of increasing size. We generalize further in considering complex exponents. Specifically, we define $\tEs$ by \[HDEn\] = , which we consider, initially, for $s=1, s=2$ and $\Re(s)>2$. For non-integer $s$ we consider the branch of the function $\z^{s}$ with a cut along the “negative diagonal” of the lattice, $\{z=-t(\mu+\nu),\
t>0\}$. For all $z$ in the closure of this cut plane we have \[BranchW\] (z) +2,\
=(--). We further enforce the convention that points of the lattice lying along the diagonal are considered symmetrically, $$\fr{1}{(-m\mu-m\nu)^s} = \fr{1}{2(m|\mu+\nu|)^s}
\lp\fr{1}{\e^{\i\th s}}
+ \fr{1}{\e^{\i(\th+2\pi)s}}\rp$$ We will return to this point later. Finally, we note that the large $K$ limiting convention defined in (\[HDEn\]) is relevant only in the cases $s=1$ or $s=2$.
We will derive integral representations for $\tEs$. Naturally, restricting $s$ to the positive integers, our formula yields an integral representation for the classical Eisenstein series $\tE_n=E_n$, for $n\ge3$. As for the conditionally convergent series, it is straightforward to verify that $\tE_1=0$ directly from (\[HDEn\]). However, for $s=2$, it is not [*a priori*]{} obvious that (\[HDEn\]) will be finite for arbitrary $\Lambda(\mu,\nu)$. Furthermore, if finite, the relationship between its value and Eisenstein’s $E_2$ is not clear. The finite existence of the limit (\[HDEn\]) will follow as a consequence of the integral representations for $\tEs, \Re(s)>2$. In addition, we derive a formula which connects our limiting value to Eisenstein’s. Even more, we prove that $\tEs$ admits an analytic continuation to $s\in\C$ as an entire function.
Some fifteen years on the heels of Eisenstein, in 1862 Weierstrass commenced his study of doubly-periodic functions. For Weierstrass, the fundamental object was his ¶ function which he defined as \[WeierstrassP\] ¶(x,Ł) = + -. Standard estimates demonstrate that ¶ is an analytic function of $x$, and well-defined function of $\L$. From (\[WeierstrassP\]) we see that the ¶ function has a double pole at the origin and every lattice translate. Weierstrass defined his function as an indefinite integral of ¶ and developed the following summation representation \[WeierstrassZ\] (x,Ł) = -\^x ¶(s,Ł) s = + ++. Again, the correction terms inside the summation balance the asymptotics of the translated poles so that the sum converges absolutely for $x$ in a compact set containing no lattice points. As is meromorphic and has only a single pole within a fundamental parallelogram, it can not be doubly periodic (see, for example, [@ahlfors79]). Nevertheless, Weierstrass derived many properties satisfied by this function and its relatives. The absolutely convergent sums (\[WeierstrassP\]) and (\[WeierstrassZ\]) will serve as the starting points for the derivation of the integral representations for ¶ and below.
We conclude this section with a brief comment on freedom in the choice of generators for $\L$. Note that $\tEs, \P, \z$ satisfy simple rescalings with respect to $\mu$ \[Rescalings\] (,) &=&(1,),\
¶(x|,)&=& ¶.| 1,,\
(x|,)&=& .| 1,, . where $\t=\nu/\mu$.
Furthermore, one observes that two pairs of generators $(\mu,\nu)$ and $(\mu',\nu')$ will give rise to the same lattice if and only if they satisfy a linear system of the form $$\lp\ba{c} \mu' \\ \nu' \ea\rp
= \lp\ba{cc} a&b\\c&d \ea\rp \lp\ba{c} \mu \\ \nu \ea\rp$$ with $a,b,c,d\in\Z, ad-bc=\pm1$. The set of all two-by-two matrices with integral entries and determinant plus or minus one is known as the unimodular group in two variables. In this case the lattice ratios are transformed by the fractional linear transformation $$\t' = \fr{a+b\t}{c+d\t},$$ known as a unimodular substitution. One may prove that up to rescaling and unimodular substitution, any lattice ratio may be represented by a unique $\t$ chosen from the following fundamental region [@ahlfors79] \[FundamentalT\] - < () ,\
() > 0 ,\
|| 1,\
||= 1, ()0.\
.
In summary, without loss of generality, we restrict our analysis to the “inhomogeneous” functions, which are obtained from (\[HDEn\]), (\[WeierstrassP\]) and (\[WeierstrassZ\]) by fixing $\mu=1,
\nu=\t$, and consider $\L=\L(\t)$ with $\t$ satisfying (\[FundamentalT\]). For convenience we omit the variables $\mu,\nu$ below and write, for example, $\P=\P(z,\t)$.
Plane-wave representations and a 2D Geometric Series
----------------------------------------------------
To facilitate our derivations we define the truncated lattice $\LK=\{\wmn{m}{n}=m+n\t|\ |m|,|n|\!<\!K\}\setminus\{0\}$. We further group lattice points into four overlapping “quadrants” $$\LK = \LKab{+}{\bullet} \cup \LKab{\bullet}{+}
\cup \LKab{-}{\bullet} \cup \LKab{\bullet}{-}$$ defined by \[QuadrantDefn\] &=& {|& 1 && m && K, &|n|&& m}\
&=& {|& 1 && n && K, &|m|&& n}\
&=& {|&-K && m &&-1, &|n|&& -m}\
&=& {|&-K && n &&-1, &|m|&& -n}. We recall from the discussion following (\[HDEn\]) that for non-integer $s$ the shared boundary between $\LKab{-}{\bullet}$ and $\LKab{\bullet}{-}$ is identical to the branch cut (see figure \[lattice:fig\]).
We have the following elementary lemma
\[PlaneWave:lem\] Assume a complex lattice $\L(\t)$ and the quadrants defined as in (\[QuadrantDefn\]). An isolated singularity of complex order $s,
\ \Re(s)\!>\!0$ with branch cut defined as in (\[BranchW\]) may be represented by the following plane-wave integrals, each of which is valid in the appropriate quadrant determined by the location of the point $\w$. \[PlaneWave\] = { ł\^[s-1]{} \^[- ł]{} , &\
ł\^[s-1]{} \^[ ł]{} , &\
ł\^[s-1]{} \^[ił]{} , &\
ł\^[s-1]{} \^[-ił]{} , &. .
\
We have the representation of the $\G$ function $$\G(s) = \intRp \l^{s-1}\e^{-\l}\dl,\ \Re(s)>0.$$ Assume $\w\in \i\R^+$, in particular, $\w=\i t,\ t>0$. As $\t$ satisfies (\[FundamentalT\]) we observe that $-\pi<\th <-\pi/2$ hence $\arg(\w^s)=\pi s/2$. Keeping this in mind, we rescale the integration variable by $t$, factor the $-1$ in the exponential, and multiply and divide by $\exp(\i\pi s/2)$ to obtain (s) &=& t\^[s]{}ł\^[s-1]{}\^[-łt]{}\
&=& \^[-is/2]{}\^[s]{}ł\^[s-1]{}\^[ił]{}. Dividing both sides by $\G(s)\w^s$ gives the desired result for $\w=\i t$. In a similar manner we prove the formula for $\w$ lying on any of the principal coordinate rays emanating from the origin, $\w\in
\pm \R^+, \pm \i\R^+$. The full expressions (\[PlaneWave\]) then follow by analytic continuation into the appropriate quadrants. $\dagger$
We note that for integer $s$, the integral expressions may be continued further and are valid in the appropriate half-planes $\pm\Re(\w)>0$ and $\pm\Im(\w)>0$.
Next we turn to our summation convention (\[HDEn\]). From lemma \[PlaneWave:lem\], it is apparent that no single plane-wave expansion formula will be valid for all terms in the summands (\[HDEn\]), (\[WeierstrassP\]), and (\[WeierstrassZ\]); terms must be grouped with respect to quadrant. As with the our convention of splitting contributions from points $\w_{-m,-m}$ lying on the cut in (\[HDEn\]) equally between branches, we wish to treat each quadrant as symmetrically as possible. We define the symbol $\eps_{mn}$ for $m,n\in\Z$ by \[EpsilonSymbol:def\] \_[mn]{} = , & m = n\
1, & . By convention, we sum over the terms in the $\LKab{+}{\bullet}$-quadrant as \[Grouping\] f() &=& f()\
\[GroupingExplicit\] &=& \_[m=1]{}\^[K]{} f() +\_[n=-m+1]{}\^[m-1]{} f() +f() We recognize the slight abuse of notation in (\[Grouping\]) in that the left hand side is not interpreted as a standard sum over the set $\LKab{+}{\bullet}$, but rather a modified sum in which the “diagonal” terms are added with a factor of 1/2; this is made explicit in (\[GroupingExplicit\]). We choose this abuse of notation so as to not burden our summation symbols with a cascade of modifiers, and anticipate that it will not cause confusion below.
The sum over the lattice square is the sum of the quadrant sums as in (\[Grouping\]); hence the reason for the factor of $1/2$ — to avoid double-counting of the contributions from the diagonal terms — is clear. We note that for $s\in\N,\ s>2$ the numerical value of the sums is independent of any manner of grouping terms. Even so, the form of the integrands in our integral representations reflect this choice. We have found that the convention (\[Grouping\]) yields the most symmetric expressions in appearance (a different grouping for integer $s$ was employed in [@huang99]).
Next, we recognize that in sums of the form (\[Grouping\]) we may substitute the appropriate plane-wave expansion (\[PlaneWave\]) to represent the poles contained in $f$. This substitution will transform the quadrant sums into geometric series. Concerning the later, we derive the following lemma.
\[GeometricSum:lem\] For any $p,q\in\C$ and $K\in\N$, the following is an identity \[StdSum\] p\^i q\^j = - - .
\
The formula follows from iteration of the usual single variable geometric sum, and algebra.
We record the following corollary for reference as the expressions appear many times in the formulas below.
\[GeometricSum:cor\] We have the following specializations of lemma \[GeometricSum:lem\] $$\sumQuadK{m}{n} (\e^{-\l})^m(\e^{-\l\t})^n
= 2\e^{-\l}\fOne - 2\e^{-\l(K+1)}f_1^{(K)}(\t,\l)$$ and $$\sumQuadK{n}{m} (\e^{\i\t\l})^n(\e^{\i\l})^m
= 2\e^{\i\t\l}\fTwo - 2\e^{\i\t\l(K+1)}f_2^{(K)}(\t,\l)$$ where the functions $f_1, f_1^{(K)}, f_2, f_2^{(K)}$ are: \[fOne:def\] &=&\
\[fOneK:def\] f\_1\^[(K)]{}(,ł) &=& -\
\[fTwo:def\] &=&\
\[fTwoK:def\] f\_2\^[(K)]{}(,ł) &=& - .
We assume $\t$ is in the fundamental region (\[FundamentalT\]) and make several observations. First, we note that all of the functions given by (\[fOne:def\])-(\[fTwoK:def\]) have double poles at the origin, $\l=0$. Since $\Im(\t)>0$, neither nor have other poles for $\l>0$. For $\t$ strictly imaginary, the denominator $1-\exp(-\t\l)$ of $f_1^{(K)}(\t,\l)$ will have isolated simple zeros. However, these are balanced by simple zeros of the difference of bracketed terms in (\[fOneK:def\]). Thus the product, in other words $f_1^{(K)}$, has no other singularities for $\l>0$. A similar argument shows that $f_2^{(K)}$ is also finite for $\l>0$. Finally, one may show the bounds |\^[-ł]{}| &<& C\_1 \^[-ł(1-|()|)]{},\
|\^[ił]{}| &<& C\_2 \^[ił]{} for large $\l$. As $|\Re(\t)|\le1/2$ and $\Im(\t)>0$, both quantities are exponentially decreasing in $\l$. Similar reasoning shows that $\e^{-\l(K+1)}f_1^{(K)}(\t,\l)$ and $\e^{\i\t\l(K+1)}f_2^{(K)}(\t,\l)$ are exponentially decreasing in $\l$ [**and**]{} $K$.
Eisenstein series
=================
As mentioned previously, the summation $\tEs$ for $\ReSTwo$ is absolutely convergent. We begin by proving our first integral representation for this case in theorem \[Eisenstein:thm\]. As a corollary, by restricting $s=n,\ n\ge 3$ we obtain integral representations of $E_n$. Further inspection of the integral representation demonstrates the existence of $\tE_2$. Elaborating on theorem \[Eisenstein:thm\], we derive an alternative representation for $\tEs$ as a contour integral. As a consequence of this second representation, we prove that admits an analytic continuation in $s$ as an entire function. Returning to the analysis of $\tE_2$, we consider a more general limiting procedure and define as the limit over increasing “lattice rectangles” with a fixed aspect ratio defined by $\al$. We derive a closed form expression which, when added to $\tE_2=\tE_2^{(1)}$, gives . As a corollary, we derive the relationship between $\tE_2$ and the sum $E_2$ as defined by Eisenstein.
Integral representations
------------------------
For the sums $\tEs$ defined by (\[HDEn\]) we prove
\[Eisenstein:thm\] Given a lattice $\L(\t)$ with ratio $\t$ chosen from the fundamental region (\[FundamentalT\]), we have the following integral representation for $\tEs, \ReSTwo$ \[EisensteinInt\] \_[s]{}() = ł\^[s-1]{}\^[-is/2]{}\^[-ł]{} f\_1(,ł) + \^[ił]{}f\_2(,ł)where and are given by (\[fOne:def\]) and (\[fTwo:def\]).
\
We recognize that, due to the placement of branch cut (\[BranchW\]) and the symmetric summation conventions (\[Grouping\]), we have the following relations between sums over $\LKab{\pm}{\bullet}$ and $\LKab{\bullet}{\pm}$ &=& \^[-is]{}\
&=& \^[ is]{} . Therefore we may consider the positive quadrants only and scale the results by an exponential factor. Turning to the quadrant $\LKab{+}{\bullet}$, in place of the isolated singularity of degree $s$, we substitute the appropriate plane-wave expression from (\[PlaneWave\]) to obtain &=&(1+\^[-is]{})\
&=& ł\^[s-1]{}\^[-ł(m+n)]{}\
&=& ł\^[s-1]{}\^[-ł]{} -\^[-ł(K+1)]{}f\_1\^[(K)]{}(,ł)where the last line follows from corollary \[GeometricSum:cor\]. From the statements following this same corollary, we observe that the two integrands are singular at $\l=0$, and are otherwise finite and exponentially decreasing in $K$ and $\l\in \R^+$. In addition, as $\ReSTwo$, the singularity at the origin is absolutely integrable. Therefore, one may take the large $K$ limit inside the integral and compute $$\limK \sumLKab{\pm}{\bullet} \fr{1}{(m+n\t)^{s}}
=\cosPiS \fr{4}{\G(s)}
\intRp \l^{s-1}\e^{-\i s\pi/2}\e^{-\l}\fOne \dl.$$ By a similar analysis we prove that $$\limK \sumLKab{\bullet}{\pm} \fr{1}{(m+n\t)^s}
= \cosPiS \fr{4}{\G(s)}
\intRp \l^{s-1} \e^{\i\t\l}\fTwo \dl.$$ Adding these two contributions gives the theorem. $\dagger$
A few comments are in order. First, for $s=2j+1$ the term $\cosPiS=0$. This is equivalent to the well-known fact that, by symmetry, the odd Eisenstein series are zero. Furthermore, by inspection of (\[EisensteinInt\]), we see that the absolute convergence of the Eisenstein series $\tEs,
\ReSTwo$ manifests itself in the behavior of the integrand of (\[EisensteinInt\]) near the origin; the factor $\l^{s-1}$ balances the double poles of $f_1$ and $f_2$ so as to ensure the product is integrable at $\l=0$. More careful analysis reveals that the formula (\[EisensteinInt\]) is finite even for the conditionally convergent case $\tE_2$. The Laurent expansions of the integrands about the origin are \_[ł0]{} ł\^[-ł]{} &=& - + Ø[ł\^3]{}\
\_[ł0]{} ł\^[ił]{} &=& - + Ø[ł\^3]{} At $s=2$, where the exponential factor $\e^{-\i s\pi/2}=-1$, the expansions above are to be subtracted. Hence the integrand of (\[EisensteinInt\]) is finite at the origin even in this case. By a similar analysis one may show that the $K$-dependent terms also cancel at the origin. We have proved:
\[EisensteinTwo:cor\] The summation (\[HDEn\]) converges in the conditionally convergent case $s=2$, and its value, $\tE_2$, is given by the integral (\[EisensteinInt\]).
In fact, a great deal more may be said. The function \[EsIntegrand\] F(s,z) = \^[-is/2]{} \^[-z]{} + \^[iz]{} appearing as a factor in the integrand (\[EisensteinInt\]) has a singularity at the origin $z=0$, and additional simple poles in the complex plane at the points $$z \in P =
\lsb\left.\pm\fr{2\pi \i}{1\pm \t}m, \pm\fr{2\pi}{1\pm \t}n \right|
m,n\in\N\rsb.$$ We denote the minimum magnitude of all $z\in P$ by $\rho$. Next, define the contour $C$ which begins at $\infty+\i y,\ y>0$; runs parallel to real axis until it intersects the circle centered at the origin with radius $r$, where $y<r<\rho$; follows this circle counterclockwise around the origin; and runs back out to $\infty-\i y$, parallel to the real axis. We assume that $y>0$ is small enough such that $C$ encloses only the pole at $z=0$. Finally, for the function $z^{s-1}, s\not\in\Z$, situate the branch cut along the positive real axis such that ($\l\in\R$) \[BranchCut\] \_[y0]{} (ł+iy)\^[s-1]{} &=& ł\^[s-1]{}\
\_[y0]{} (ł-iy)\^[s-1]{} &=& \^[2i(s-1)]{}ł\^[s-1]{} . With these preliminaries established we prove the following theorem.
\[EisensteinCont:thm\] Given a lattice $\L(\t)$ with ratio $\t$ chosen from the fundamental region (\[FundamentalT\]), we have the following contour integral representation for $\tEs,\ \ReSTwo$ \[EisensteinCont\] = 2\_C z\^[s-1]{}F(s,z)z where $F(s,z)$ is given by (\[EsIntegrand\]).
\
Consider the contour integral $$\int_C z^{s-1}F(s,z) \d z.$$ As the integrand is analytic except for the singularity at the origin, we may deform the contour without altering the value of the integral. Specifically, shrink the radius of the circle, $r\to 0$, and take the limit as $y\to 0$ for the two components running parallel to the real axis. For $\Re(s)=2+\eps$, we estimate the contribution from the circular arc $$\lim_{r\to 0} \left| \int_{|z|=r} z^{s-1}F(s,z) \d z \right|
\le M \lim_{r\to 0} \int_{|z|=r} r^{\eps-1} |\d z| = 0.$$ Turning to the components parallel to the real axis, applying the specification of the branch cut (\[BranchCut\]) we compute \_[y0]{} \_[+iy]{}\^[\_r+iy]{} z\^[s-1]{}F(s,z)z &=& -ł\^[s-1]{}F(s,ł)\
\_[y0]{} \_[\_r-iy]{}\^[-iy]{} z\^[s-1]{}F(s,z)z &=& \^[2i(s-1)]{}ł\^[s-1]{}F(s,ł) . We recognize the integrals of theorem \[Eisenstein:thm\], make use of the identity $$\G(s)\G(1-s) = \fr{\pi}{\sin(\pi s)},$$ and obtain \_C z\^[s-1]{}F(s,z) z &=& (\^[2i(s-1)]{}-1)ł\^[s-1]{}F(s,ł)\
&=&\
&=& . The result (\[EisensteinCont\]) follows from algebra. $\dagger$
Several corollaries follow from theorem \[EisensteinCont:thm\]. We note here only the most immediate
\[EsEntire:cor\] The sums $\tEs$, defined by (\[HDEn\]) for , admit an analytic continuation to $s\in\C$ as an entire function. This continuation is given by the contour integral representation (\[EisensteinCont\]).
\
The contour integral appearing in (\[EisensteinCont\]) defines an analytic function of $s$ which is never singular. The same may be said for the cosine factor. Thus the only possible singularities would arise from the factor $\G(1-s)$ which has simple poles for $s\in\N$. From the definition (\[HDEn\]) we know that is finite for (the apparent singularities in (\[EisensteinCont\]) in this case are balanced by zeros of the cosine term, the contour integral, or both.) Therefore, we need only verify the finite existence of $\tE_1$ and $\tE_2$. We have argued above that $\tE_2$ is finite. Finally, for $s=n=1$, the pole in the Gamma function is balanced by the simple zero of the cosine factor. $\dagger$
By inspection of (\[EisensteinCont\]), we find that $\tEs$ has simple zeros for $s=1-2j,\ j\ge1$. Thus the continuation respects the symmetric limiting process (\[HDEn\]). We anticipate further results concerning the evaluation of the contour integral (\[EisensteinCont\]) via residue methods. However, we have not carried out this analysis at the time of writing this paper. Finally, as mentioned previously, the exact form of the integrands (\[EisensteinInt\]) and (\[EisensteinCont\]) reflect our summation convention with respect to grouping of summands and placement of the branch-cut. Regarding the latter, similar formulas arise if the branch-cut is situated along any of the lattice diagonals; the effect is to redistribute factors of $\exp(\i\pi s/2)$ between the two functions $f_1$ and $f_2$. There are, perhaps, additional treatments of the branch-cut that could yield relatively simple expressions. However, a simple integral expression valid for placement of the cut along an arbitrary ray in the complex plane appears to be intractable.
Aspect ratio correction
-----------------------
Although we have demonstrated that $\tE_2$ is finite, until this point its relation to Eisenstein’s definition of this series is unclear. We make this explicit in the present section.
We wish to formalize summation over lattice rectangles with a fixed aspect ratio. Given $\al\in(0,\infty)$ we define the sum \[HDEab\] = \_[|m|]{}\_[|n|K]{} , where $\lfloor x\rfloor$ denotes greatest integer less than or equal to $x$. Again, although expected, the existence of the limit (\[HDEab\]) is not [*a priori*]{} guaranteed but will follow in the course of our analysis.
Clearly, $\tE_2^{(1)}=\tE_2$ defined in (\[HDEn\]). More generally, we write \[SumSplitting\] = \_2 + (,) where the value of $\tE_2$ may be computed via the integral expression (\[EisensteinInt\]). Analysis similar to that used to prove (\[EisensteinInt\]) may be employed to compute a closed form expression for $\Delta(\al,\t)$.
\[AspectRatio:thm\] For a fixed aspect ratio determined by $\al\in(0,\infty)$, the limit $\tETwoA$ specified in (\[HDEab\]) exists. Furthermore, when written in the form (\[SumSplitting\]), the aspect ratio dependence is given by \[AspectRatio\] (,) = - i-
\
Assume $\al\ge1$. We write the limit (\[HDEab\]) &=& \_2 + (,),\
(,) &=& 2 \_[m=K+1]{}\^\_[n=-K]{}\^[n=K]{} . The contribution from the sum over lattice points $-\lfloor\al\cdot
K\rfloor \le m \le -K-1$ is accounted for by the factor of two multiplying the sum in the final line. We represent the poles using the $\LKab{+}{\bullet}$ plane-wave expansion (\[PlaneWave\]). (,) &=& 2 \_[m=K+1]{}\^ \_[n=-K]{}\^[n=K]{} ł\^[-ł(m+n)]{}\
&=& 2 ł\
&=& 2 ł\
\[AlphaInt\] &=& . Where the argument which justifies taking the large $K$ limit inside the integral runs along the same lines as in the proof of theorem \[Eisenstein:thm\]. This last integral (\[AlphaInt\]) may be evaluated in closed form using the formula \[IntegralFormula\] \^[-x]{}(x) x = , which holds for $\Re(\beta)>|\Im(\delta)|$ (see [@gr94], 3.944.5). Taking care to write (\[AlphaInt\]) as the difference of two integrals of the form (\[IntegralFormula\]), and performing algebra gives the expression for $\Delta(\al,\t),\ \al\ge 1$ in (\[AspectRatio\]).
For $\al<1$ the lattice rectangle is such that the longer side is in the $\t$-direction. As written, equations (\[HDEab\]) and (\[SumSplitting\]) suggest that this rectangle is inscribed in a lattice square of size $K$, and to compute $\Delta(\al,\t)$, one should subtract the extra contributions exterior to the rectangle but interior to the square. The problem with this approach is that, for arbitrary $\t$ and $\al$, one would have to keep track of the quadrants in which these points lie. In lieu of this, for $\al<1$ we rescale the limits in (\[HDEab\]) $$\tETwoA = \limK \sum_{|m|\le K}\sum_{|n|\le \betaK} \fr{1}{(m+n\t)^2}.$$ Informally, this has the “effect” of inscribing the square in the rectangle and motivates computing the contributions from the points in the difference using the plane-wave formulas appropriate for $\LKab{\bullet}{\pm}$. Arguing as above and using the integral identity (\[IntegralFormula\]) we compute (,) &=&-2 \_[n=K+1]{}\^ \_[m=-K]{}\^[m=K]{} ł\^[ił(m+n)]{}\
&=&\
\[BetaInt\] &=& ----.
Although perhaps not obvious at first glance, the formula (\[BetaInt\]) is the same as the formula for $\Delta(\al,\t)$ in (\[AspectRatio\]). Using standard trig identities we have \[ArcTanIdent\] \_[z’z]{} (z) - - = \_[z’z]{} = . Furthermore, for $z=\i\t$ or $z=\i\t/\al$ with $\t$ satisfying (\[FundamentalT\]) and $\al>0$, we find that we should choose the minus sign in (\[ArcTanIdent\]). Using this identity and algebra we observe that (\[BetaInt\]) is equal to (\[AspectRatio\]), thus (\[AspectRatio\]) holds for all $\al>0$. $\dagger$
We use this theorem to find the connection between our summation and Eisenstein’s. Starting from Eisenstein’s summation convention we obtain E\_2 &=&\
&=&\_[N]{}\_[n=-N]{}\^N \_ \_[m=-N]{}\^[m=-N]{}\
&=&\_ \_[N]{}\_[n=-N]{}\^N \_[m=-N]{}\^[m= N]{}\
\[EisensteinToHD\] &=&\_2 -(i). Standard estimates justify commuting the $N$ and $\al$ limits between the second and third lines, and we used (\[AspectRatio\]) to compute this limit. As $\t\not=0$, (\[EisensteinToHD\]) shows that, for finite $\t$, the value of our sum is always different from Eisenstein’s. In the limit $\t\to \i\infty,\ |\Re(\t)|\le 1/2$, both summations are equal and presumably converge to $2\z_R(2)=\pi^2/3$.
In a similar vein, we compute the difference between taking Eisenstein’s limit and “its reverse”. Arguing as above we have that &=& \_2 +\_[0]{}(,)\
\[NotEisensteinToHD\] &=& \_2 -+ Taking the difference between (\[EisensteinToHD\]) and (\[NotEisensteinToHD\]) and we obtain $$\lp \sumE - \sumNotE \rp \fr{1}{(m+n\t)^2}
= \fr{2\pi \i}{\t}.$$ For a different proof of this fact see Walker, [@walker96].
Finally, in the case of the square lattice, $\t=\i$, we observe that $f_1(\i,\l)=f_2(\i,\l)$. Collecting factors in (\[EisensteinInt\]) and performing algebra we find that $$\tEs(\i) = \e^{-\i s\pi/4}
\cosPiS \cos\lp\fr{\pi}{4}s\rp \fr{8}{\G(s)}
\intRp \l^{s-1}\e^{-\l}f_1(\i,\l) \dl.$$ Corresponding to the added $\pi/2$-symmetry of the square lattice, the product of cosines causes the sum to vanish for $s=n\ge 3$, $n$ not a multiple of four. Similarly, as the representation (\[EisensteinInt\]) is valid for $n=2$, we conclude $\tE_2(\i)=0$. Substituting this value into (\[SumSplitting\]), and taking the large $\al$ (small $\al$) limits we obtain &=&\
&=&-, well-known identities in the fast multipole community (see, for example, [@greengard88]).
Weierstrass elliptic functions
==============================
Our derivations of integral formulas for Weierstrass’ elliptic functions proceed in much the same manner as above. As with the Eisenstein series, we begin with the definitions of the functions as sums over the lattice (\[WeierstrassP\]) and (\[WeierstrassZ\]). Next, we group terms in the sum as in (\[Grouping\]), substitute the appropriate plane-wave expansions from lemma \[PlaneWave:lem\], and sum the resulting geometric series using the identities of corollary \[GeometricSum:cor\].
As a preliminary note, the integral representations for $\P(z,\t)$ and $\z(z,\t)$ which we derive in theorem \[Weierstrass:thm\] are not valid for all $z\in\C$, but rather have a finite domain of validity. This is a consequence of the way in which we group terms. More precisely, we require that $z\in D(\t)$ defined by \[FundamentalZ\] D() = {z | (-1 z )<0, (z )>0}. As $\t$ is in the region (\[FundamentalT\]), one may verify that $D(\t)$ is an open set containing the origin. However, we point out that $D(\t)$ may not contain the fundamental period parallelogram of the lattice. For example, the standard hexagonal lattice has generators $(1,\t)=(1,1/2+\i\sqrt{3}/2)$. Thus a corner of $\L_0$ is given by the point $z_0=1/2+\t/2=3/4+\i\sqrt{3}/4$. However, $\Re(-1+z_0+\tau)=1/4>0$, violating the first inequality in (\[FundamentalZ\]).
With this aside, we prove the following.
\[Weierstrass:thm\] Assume $\L=\L(\mu,\nu)$ is an arbitrary complex lattice with generators chosen such that $\t=\nu/\mu$ is in the fundamental region (\[FundamentalT\]), and that the complex number $z$ is in the domain $D(\t)$ defined by the inequalities (\[FundamentalZ\]). We have the following integral expressions for the inhomogeneous elliptic functions $\P(z,\t)$, and $\z(z,\t)$: \[WeierstrassPInt\] ¶(z,) = + 8 ł\^[-ł]{} \^2f\_1(ł,) +\^[ił]{}\^2 f\_2(ł,), and, \[WeierstrassZInt\] (z,) = + 4 \^[-ł]{} (zł-(zł))f\_1(ł,) -\^[ił]{}(zł-(zł) )f\_2(ł,), where the functions $f_1, f_2$ are defined by (\[fOne:def\]) and (\[fTwo:def\]). We may evaluate the homogeneous functions, $\P(x|\mu,\nu)$ and $\z(x|\mu,\nu)$, via the appropriate scaling relations (\[Rescalings\]) and (\[WeierstrassPInt\])-(\[WeierstrassZInt\]) with the proviso that $x/\mu=z\in D(\t)$.
\
In computing the integral representation for the ¶ function we will group terms of the sum (\[WeierstrassP\]) as in (\[Grouping\]). As in the computation of the Eisenstein sums, we wish to combine the contributions from the sums over $\LKab{\pm}{\bullet}$. We compute - &=& -\
- &=& -. Therefore the contributions from both quadrants may be expressed as a single sum over $\LKab{+}{\bullet}$ of a modified summand. Furthermore, under the assumption $z\in D$, all of the poles in this sum may be expressed using the $\Re(\w)>0$ plane-wave expansion from (\[PlaneWave\]). - &=& -+\
&=& ł(\^[łz]{}-2+\^[-łz]{}) \^[-ł(m+n)]{}\
\[WeierstrassPIntKm\] &=& 8 ł\^2\^[-ł]{}- \^[-ł(K+1)]{}f\_1\^[(K)]{}(,ł) . Arguing as before, we find the large $K$ limit of the $K$-dependent term to be zero. We compute the contribution from the terms in the quadrants $\LKab{\bullet}{\pm}$ in an analogous manner. Adding this result to (\[WeierstrassPIntKm\]) gives (\[WeierstrassPInt\]).
The derivation of the expression for the function is similar. In brief, the sum over the quadrants $\LKab{\pm}{\bullet}$ may again be expressed as a sum over the single quadrant $\LKab{+}{\bullet}$ in which we substitute the appropriate plane-wave expansion. Thus, \_ ++ &=& -++\
&=& (\^[łz]{}+2zł-\^[-łz]{}) \^[-ł(m+n)]{}\
\[WeierstrassZIntKm\] &=& 4 (zł- (zł)) \^[-ł]{}f\_1(ł,) -\^[-ł(K+1)]{}f\_1\^[(K)]{}(ł,) . As before the $K$-dependent term goes to zero in the limit. The analogous sums over $\LKab{\bullet}{\pm}$ give the other half of the expression (\[WeierstrassZInt\]). $\dagger$
Remarks:
1. As an alternative to the above derivation of theorem \[Weierstrass:thm\], we recall that the Eisenstein series appear as coefficients in the Laurent expansion for Weierstrass’ ¶function ¶(z) &=& + -\
&=& + (n-1)z\^[n-2]{}\
&=& + (n-1) E\_n z\^[n-2]{}. Substituting the integral representations (\[EisensteinInt\]) for the $E_n$, the Taylor series may be summed explicitly inside the integrand. The formula (\[WeierstrassPInt\]) above follows after algebraic simplification. Furthermore, the expression (\[WeierstrassZInt\]) for the function follows from anti-differentiation of (\[WeierstrassPInt\]).
2. As with the series $\tE_2$, and its dependence on aspect ratios derived in theorem \[AspectRatio:thm\], the slowly decaying terms of the sums defining ¶ and manifest themselves at the origin in the integral representations (equations (\[WeierstrassP\]), (\[WeierstrassZ\]) and (\[WeierstrassPInt\]), (\[WeierstrassZInt\]) respectively). In the integral representations, we observe that Weierstrass’ “correction” terms are arranged in such a way as to create third order zeros at $\l=0$, which appropriately balance the second order poles from $f_1$ and $f_2$.
Conclusion
==========
We conclude with a brief discussion of our results in relation to previous research in this field. To the best of our knowledge, there is no analog to the integral expressions for the ¶ and functions (\[WeierstrassPInt\]) and (\[WeierstrassZInt\]). The possibility of developing numerical routines for evaluation of these functions based on these representations deserves further study. We observe that the integrands are not extremely oscillatory, and decay exponentially. Thus $N$-point Gauss-Laguerre quadrature rules will converge rapidly in $N$. As one drawback, there is the perhaps awkward domain of validity in $z$. However, it may be that symmetries of the ¶ and functions imply that it is sufficient to evaluate them over domains that are smaller than the fundamental period parallelogram. Furthermore, at least for the ¶function, there exists the following closed-form Fourier-like expansion [@walker78] \[WeierstrassPFourier\] ¶(z,) = -2 + \_[n=1]{}\^ + -8\^2 \_[n=1]{}\^ (2n z). Both summands in (\[WeierstrassPFourier\]) are exponentially decreasing and the sums converge rapidly—stiff competition from a numerical perspective. Nevertheless, we have not fully explored the relative merits of this approach over the plane-wave representation (\[WeierstrassPInt\]). In addition, the integral representations may have further analytic implications.
Turning to the representations for the Eisenstein series, the existence of $\tE_2$ (corollary \[EisensteinTwo:cor\]) is not unexpected. In addition to the original finiteness proofs given by Eisenstein, many years prior to this present work, Walker derived the remarkable formula for the conditionally convergent series (see [@walker78]) $$\limK \sum_{0<m^2+n^2\le K^2} \fr{1}{(m+n\t)^2}
= \fr{-2\pi}{1-\i\t} - 4\pi \i \fr{\eta'(\t)}{\eta(\t)},$$ where the Dedekind $\eta(\t)$-function with $\Im(\t)>0$ is defined by $$\eta(\t) = \e^{\pi \i \t/12}\prod_{n=1}^\infty (1-\e^{2\pi \i\t n}).$$ We also note that a different treatment, initiated by Hecke, has become a standard approach to resolving convergence and transformation properties of $E_2$ [@schoeneberg74].
As indicated by theorem \[EisensteinCont:thm\], our expressions are quite general, and have broad implications. We note that Riemann demonstrated both the functional equation satisfied by $\z_R(s)$, and the evaluation of $\z_R(-2n+1)$ (and, via the functional equation, $\z_R(2n)$) in terms of Bernoulli numbers using the “version” of theorem \[EisensteinCont:thm\] appropriate for his zeta function. Similarly, we anticipate that a residue argument will give the evaluation of $\tE_n=E_n$ in terms of multiple Bernoulli numbers. For an alternative treatment of Eisenstein series for negative even integers using Hecke convergence factors see the recent work of Pribitkin [@pribitkin00]. The functional equation satisfied by the continuation of $\tEs$ is more elusive. We are currently pursuing this and hope to report our results in the future.
Finally, there is a possibility that representations of the form (\[EisensteinInt\]) may exist for certain Dirichlet series G(s,) &=&\
() &=& (i(m+ n)), for $\al, \beta \in \R$. (These are called “Kronecker series” in [@weil76], Chapter VIII.) A detailed discussion of the convergence of these series is given in [@bbp98]. We note that Laplace-Mellin techniques have been employed frequently in the analysis of such series. The approach up until now has been to think of $$|\w_{m,n}|^2 = |m\mu + n\nu|^2 = Q(m,n)$$ as defining a positive definite quadratic form taking $m$ and $n$ as arguments. Treating this form as “indivisible”, one may use the $\Re(\w)>0$ plane-wave formula in lemma \[PlaneWave:lem\] and obtain the integral representation $$\sumLat \fr{\chi(\w)}{|\w|^{2s}}
= \fr{1}{\G(s)} \intRp \l^{s-1} \sumLat \chi(\w)\e^{-\l Q(m,n)}\dl.$$ The analysis then proceeds via $\theta$ functions.
Our approach would be different. “Plane-wave-like” representations exist for the function $f(x,y)=\sqrt{x^2+y^2}$. Formally, one may take the true plane-wave expressions for $f(x,y,z)=\sqrt{x^2+y^2+z^2}$ derived in [@cgr99], and set $z=0$. The result is a 2D integral — as opposed to the Eisenstein case analyzed above where one complex dimension (two real) collapses into a 1D integral. However, the critical element of this representation is that the exponential function in the “plane-wave” representation is linear in $m$ and $n$. Again, as a consequence, we observe that the summation under the integrand becomes a 2D geometric series. We are considering this as a possible direction for future research.
The authors thank Peter Walker at the University of Sharjah, United Arab Emirates, and Brad Alpert at NIST/Boulder for frequent helpful discussions during the writing of this paper.
\[lattice:fig\]
{height="5in" width="5in"}
[^1]: Contribution of U.S. Government, not subject to copyright.
[^2]: National Institute of Standards and Technology, Boulder, CO 80305-3328.
[^3]: University of North Carolina, Chapel Hill, NC 27599-6011.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'B. Tuguldur and Ts. Gantsog'
title: 'Some features of the Driven Jaynes-Cummings system'
---
[School of Physics and Electronics, National University of Mongolia]{}
Introduction
============
The Jaynes-Cummings model (JCM) [@jaynes] is one of the simplest models describing the interaction of light with matter, where a single two-level atom interacts with a single mode of quantized radiation field in the electric dipole and rotating wave approximations. This model is of fundamental importance to the field of quantum optics; many interesting features have been predicted for both the atomic variables and the statistical properties of the field through the years, beginning with the well-known phenomena of collapses and revivals of the atomic population inversion oscillation [@eberly]. Numerous extensions of the JCM have been considered [@alsing; @deb; @gerry] and many experiments reported [@rempe; @diedrich; @thompson; @childs; @brune].
One of the most interesting features of the JCM is that if the atom is initially prepared in its upper state and if the cavity field is initially in the coherent state, then the quantized field evolves into an almost pure state at half of the atomic-revival time [@eiselt; @phoenix; @gea]. This approximately pure state is equal to a superposition (Schr$\ddot{\text{o}}$dinger cat) state composed of two states of light having the same amplitude, but opposite phase. The amplitude of the component states is approximately equal to the amplitude of the initial coherent state of the field mode [@buzek].
Buzek et al. [@buzek1] showed that by driving the atom with the external classical field, superposition states of the quantized cavity mode with arbitrary amplitudes and phases of component states can be produced. Recently, Gea-Banacloche and coworkers [@gea1] have verified the existence of maximally entangled state in externally driven JCM. Optical cavities with atoms have been proposed for quantum information processing [@pellizzari; @pellizzari1].
In this paper we study the dynamics of the JCM when a quantized cavity mode is pumped continuously by an external classical field. In Sec.2 we derive the eigenstates and eigenenergies of the driven Jaynes-Cummings system. Using these results we find the time dependent state vector of the system for a given initial condition. In Sec.3 the dynamics of the atomic inversion, mean photon number and the phase space distribution function $Q(\alpha,\alpha^*)$ are examined. In Sec.4 we give approximate solutions that enable us to explain the features observed. Finally in Sec.5 we summarize our results.
Driven Jaynes-Cummings System
=============================
We consider the driven Jaynes-Cummings system with the external field driving the cavity mode. The interaction Hamiltonian of the system in the interaction picture is given by \[3\] $$\hat{H}_I=g(\hat{a}\hat{\sigma}_++\hat{a}^\dagger\hat{\sigma}_-)+\mathscr{E}(\hat{a}^\dagger e^{i\phi}+\hat{a} e^{-i\phi}),\label{hhh}$$ where $g$ is the coupling constant between the atom and the cavity mode; $\mathscr{E}$ is the amplitude of the driving field; $\phi$ is the phase of the classical field; $\hat{a}^\dagger$, $\hat{a}$ are creation and annihilation operators for the cavity mode; and $\hat{\sigma}_+$ and $\hat{\sigma}_-$ are atomic pseudospin operators. We assumed that the driving field is in resonance with the atom and the cavity mode. The steady state solution of the Schr$\ddot{\text{o}}$dinger equation for the Hamiltonian (1) is possible for $\mathscr{E}<g/2$. The quasieigenenergies and the corresponding eigenstates are given by [@alsing] $$\begin{array}{l}
E_0=0,\\
\displaystyle{E_{n1}=g\sqrt{n}\left[1-\left(\frac{2\mathscr{E}}{g}\right)^2\right]^{3/4},\quad n=1,2,3,...,}\\
\displaystyle{E_{n2}=-g\sqrt{n}\left[1-\left(\frac{2\mathscr{E}}{g}\right)^2\right]^{3/4},\quad n=1,2,3,...}
\end{array}$$ and
\[steady\] $$\begin{gathered}
\vert\psi_{En1}\rangle=c_p(S(\eta)D(\beta_{n+})\vert n-1\rangle+L_+ S(\eta)D(\beta_{n+})\vert n\rangle)\vert -\rangle\\
+c_p(L_+S(\eta)D(\beta_{n+})\vert n-1\rangle+e^{2i\phi} S(\eta)D(\beta_{n+})\vert n\rangle)\vert +\rangle,\end{gathered}$$ $$\begin{gathered}
\vert\psi_{En2}\rangle=c_p(S(\eta)D(\beta_{n-})\vert n-1\rangle-L_+ S(\eta)D(\beta_{n-})\vert n\rangle)\vert -\rangle\\
+c_p(L_+S(\eta)D(\beta_{n-})\vert n-1\rangle-e^{2i\phi} S(\eta)D(\beta_{n-})\vert n\rangle)\vert +\rangle,\end{gathered}$$
where if quasienergies are positive (negative) $E_{n1}>0$ ($E_{n2}<0$) then $\displaystyle{\beta_{n+}=-\frac{2\mathscr{E}}{g}\sqrt{n}e^{i\phi}}$ ($\displaystyle{\beta_{n-}=\frac{2\mathscr{E}}{g}\sqrt{n}e^{i\phi}}$), $\eta=re^{i\theta}$, $e^{2r}=\sqrt{1-(2\mathscr{E}/g)^2}$, $\theta=2\phi$, $c_p=(1/2)\sqrt{1-\sqrt{1-(2\mathscr{E}/g)^2}}$ and $L_{+}=-(ge^{i\phi}/2\mathscr{E})(1+\sqrt{1-(2\mathscr{E}/g)^2})$. For an external field amplitude larger than $\mathscr{E}=g/2$, no normalizable steady states exist. This value is the threshold condition for spontaneous dressed-state polarization.
If the atom is initially prepared in its upper state $\vert +\rangle$ and if the cavity field is prepared initially in the coherent state $\vert \alpha_0\rangle$, then the time evolution of the system is given by $$\vert\psi(t)\rangle=\vert\psi^+(t)\rangle+\vert\psi^-(t)\rangle,\label{time}$$ where
\[timeevolution\] $$\begin{gathered}
\vert\psi^+(t)\rangle=\vert c_p\vert^2
\sum_{n=0}^{\infty}(L_+^*\langle n-1;-\beta_{n+};-\eta\vert\alpha_0\rangle+e^{-2i\phi}\langle n;-\beta_{n+};-\eta\vert\alpha_0\rangle)e^{-iE_nt}\\
\left[(L_+\vert\eta;\beta_{n+};n-1\rangle+e^{2i\phi}\vert\eta;\beta_{n+};n\rangle)\vert +\rangle
+(\vert\eta;\beta_{n+};n-1\rangle+L_+\vert\eta;\beta_{n+};n\rangle)\vert -\rangle\right],\label{df1}\end{gathered}$$ $$\begin{gathered}
\vert\psi^-(t)\rangle=\vert c_p\vert^2
\sum_{n=0}^{\infty}(L_+^*\langle n-1;-\beta_{n-};-\eta\vert\alpha_0\rangle-e^{-2i\phi}\langle n;-\beta_{n-};-\eta\vert\alpha_0\rangle)e^{iE_nt}\\
\left[(L_+\vert \eta;\beta_{n-};n-1\rangle-e^{2i\phi}\vert \eta;\beta_{n-};n\rangle)\vert +\rangle
+(\vert \eta;\beta_{n-};n-1\rangle-L_+\vert\eta;\beta_{n-};n\rangle)\vert -\rangle\right],\label{df2}\end{gathered}$$
and $E_n=g\sqrt{n}(1-(2\mathscr{E}/g)^2)^{3/4}$ and squeezed and displaced Fock states are denoted as $S(\eta)D(\beta_{n\pm})\vert n\rangle=\vert\eta;\beta_{n\pm};n\rangle$. This result is exact and no approximations are made. We shall apply it to study quantum dynamics of the system in the following section.
Quantum dynamics of the driven JCM
==================================
Atomic inversion and the mean photon number of the cavity mode
--------------------------------------------------------------
Using explicit expressions for the state vector given by equations and we can study the dynamical properties of the system under consideration. Firstly we calculate the expectation value of the atomic inversion. An illustration of the time evolution of the atomic inversion oscillation in the driven JCM is shown in figure \[fig1\] for different values of $\mathscr{E}$.
The revival of the atomic inversion oscillations is a purely quantum-mechanical effect that originates in the discreteness of quantum states of the cavity mode. It is seen from figure \[fig1\] that with the change of the amplitude $\mathscr{E}$ of the driving field the revival time of the atomic inversion as well as the overall pattern of the time evolution of the atomic inversion oscillation are changed. For larger $\mathscr{E}$ the overall revival and collapse pattern disappear due to the domination of the classical driving field on the quantized cavity field. In the standard JCM ($\mathscr{E}=0$) the total excitation number operator $\hat{R}=\hat{a}^\dagger\hat{a}+1/2(1+\hat{\sigma}_z)$ commutes with the total Hamiltonian and hence represents an integral of motion. As a consequence, the time evolution of the mean photon number of the cavity mode shows the same oscillation as the atomic inversion. However, in the case of driven JCM ($\mathscr{E}\neq 0$) we have $[\hat{R},\hat{H}]\neq 0$, and therefore the total excitation number is not conserved. Physically this corresponds to the fact that the energy is transferred from the classical field to the quantized cavity mode. Using equations and we can numerically calculate the mean photon number of the cavity mode.
In figure \[fig2\] we plot the mean photon number $\langle\hat{a}^\dagger\hat{a}\rangle$ for $\alpha_0=5$, $\phi=0$ and for different values of $\mathscr{E}$. As we can see from figure \[fig2\] the time evolution of the mean photon number in the driven JCM exhibits several interesting features. Firstly, for small values of $\mathscr{E}$ (see figure \[2a\] with $\mathscr{E}=0.01$) a typical collapse-revival pattern corresponding to the time evolution of the inversion can be observed. At the same time we can also observe the pattern of the global oscillating behavior. With the increase in the classical amplitude $\mathscr{E}$ the time evolution of the mean photon number exhibits stronger global oscillating behavior with small quantum oscillations (revivals) at times corresponding to revivals of the atomic inversion (the amplitude of these oscillations is of the order of unity and therefore they are not clearly seen in our pictures because of the scale used in the y axis).
Finishing this part of the paper we turn our attention to the figure \[fig3\], where the atomic inversion is plotted for $\alpha_0=5$, $\mathscr{E}=0.1$ and for different values of $\phi$. We can see from the pictures that the revival time of the atomic inversion as well as the overall pattern of its time evolution also depend on the phase of the driving classical field $\phi$.
$Q$-function
------------
Now we turn our attention to the dynamics of the $Q(\alpha,\alpha^*)$ function. We know from the standard JCM ($\mathscr{E}=0$) that, if the initial field is in a coherent state $\vert\alpha_0\rangle$, then during the evolution of the system the single-peaked $Q$-function of the cavity mode bifurcates into two peaks that move in opposite directions around a circular path whose radius equals the amplitude $\alpha_0$ of the initial coherent state [@eiselt]. The centre of this circular path is at the origin of the phase space. At one half of the revival time the two peaks have a maximum separation in phase space. Phoenix and Knight [@phoenix] and Gea-Banocloche [@gea] have shown that at this time the cavity mode (and the atom) returns most closely to the pure state, which is close to a macroscopic superposition composed of two field states that have the same amplitude but opposite phase. The purity of this macroscopic superposition state increases as the amplitude of the cavity mode increases. As the peaks meet at the opposite side of the circle, they produce a revival of the Rabi oscillations.
Buzek et al. [@buzek1] have shown that, by driving the atom with the external classical field, one can produce (almost pure) macroscopic superposition states of the cavity field. The amplitudes of the component states can be much larger than the amplitude of the initial cavity field mode. With a proper choice of the phase of the classical driving field, one can produce superposition states with different amplitudes and phases of the component states.
It is interesting to contrast this behavior with what occurs when the cavity quantum mode is driven by an external classical field. In figure \[fig4\] the contour plots of the $Q(\alpha,\alpha^*)$ function are shown for $\alpha_0=5$, $\mathscr{E}/g=0.1$ and for three different values of the phase $\phi=0$, $\pi$ and $\pi/2$. The most interesting result is that in the course of the time the initial one-peaked function splits into two peaks, which counterrotate on separate circles with different radii depending on the phase of the classical driving field $\phi$. It is clearly seen from the comparison of figure \[3b\] and figure \[fig5\] that atomic inversion oscillation shows revival when these two peaks collide. When $\phi=0$ and $\phi=\pi$, the two circles have different radii, and the peaks can collide only at the original site of the circle (see figures \[4a\] and \[4b\]). But when $\mathscr{E}/g$ is small enough, the peaks still do have some overlap while moving on different circles (not necessarily on the opposite sides of the circles) and as a result we observe some small revival pattern (see the first small revival in figure \[1b\]). However, with the increase of $\mathscr{E}/g$, the peaks do not overlap and the revival pattern disappears (see figure \[1c\]).
In the next section we will derive an approximate expression for the state vector that enables us to study the kinematics of the $Q(\alpha,\alpha^*)$ function in the complex plane: we will be able to determine radii and centers of two separate circles, angular velocities of rotation of the peaks around these circles, and therefore estimate the revival time of the atomic inversion oscillation.
Approximate solutions
=====================
If $\mathscr{E}$ is small compared to $g$, we can approximate the state vector - of the system to give (see Appendix A)
$$\vert \psi^+(t)\rangle\approx A^+(t)\vert (\alpha_0+\bar{\beta}_n)e^{-i\omega^+gt}-\bar{\beta}_n\rangle(\vert +\rangle+\vert -\rangle),\label{tr1}$$
$$\vert \psi^-(t)\rangle\approx A^-(t)\vert (\alpha_0-\bar{\beta}_n)e^{i\omega^-gt}+\bar{\beta}_n\rangle(\vert +\rangle-\vert -\rangle),\label{tr2}$$
where $\bar{\beta}_n=(2\mathscr{E}/g)\sqrt{\bar{n}}e^{i\phi}$, and
$$\begin{gathered}
A^+(t)=\vert c_p\vert^2 \vert L_+\vert^2\exp\left(\frac{1}{2}\vert\alpha_0+\bar{\beta}_n\vert^2\right)\\
\exp\left\{\frac{1}{2}[-\bar{\beta}_n(\alpha_0+\bar{\beta}_n)^*e^{i\omega^+gt}+\bar{\beta}_n^*(\alpha_0+\bar{\beta}_n)e^{-i\omega^+gt}]\right\}
\exp\left[\frac{1}{2}(\alpha_0^*\bar{\beta}_n-\alpha_0\bar{\beta}_n^*)\right],\label{an}\end{gathered}$$
$$\begin{gathered}
A^-(t)=\vert c_p\vert^2 \vert L_+\vert^2\exp\left(\frac{1}{2}\vert\alpha_0-\bar{\beta}_n\vert^2\right)\\
\exp\left\{\frac{1}{2}(\bar{\beta}_n(\alpha_0-\bar{\beta}_n)^*e^{-i\omega^-gt}-\bar{\beta}_n^*(\alpha_0-\bar{\beta}_n)e^{i\omega^-gt})\right\}
\exp\left[\frac{1}{2}(-\alpha_0^*\bar{\beta}_n+\alpha_0\bar{\beta}_n^*)\right],\label{ah}\end{gathered}$$
and (see Appendix B)
$$\omega^+=\frac{[1-(2\mathscr{E}/g)^2]^{3/4}}{2\vert\alpha_0+\bar{\beta}_n\vert},$$
$$\omega^-=\frac{[1-(2\mathscr{E}/g)^2]^{3/4}}{2\vert\alpha_0-\bar{\beta}_n\vert}.$$
It is clear from the above results that $\psi^+$ corresponds to the coherent state rotating along the circle of radius $\vert \alpha_0+\bar{\beta}_n\vert$ centered at $-\bar{\beta}_n$, and $\psi^-$ corresponds to the coherent state rotating along the circle of radius $\vert \alpha_0-\bar{\beta}_n\vert$ centered at $\bar{\beta}_n$. These states counterrotate in the complex plane. This approximation is valid only if time is less than revival time.
The modules of the $A^+$ and $A^-$ are not time dependent. Therefore density operator of the cavity mode takes the form $$\rho_f=Tr_{atom}(\rho)=2\vert A^+\vert^2\vert\gamma^+(t)\rangle\langle\gamma^+(t)\vert+2\vert A^-\vert^2\vert\gamma^-(t)\rangle\langle\gamma^-(t)\vert,$$ where $\rho$ is the density operator for the total system, $\gamma^+(t)=(\alpha_0+\bar{\beta}_n)e^{-i\omega^+gt}-\bar{\beta}_n$, and $\gamma^-(t)=(\alpha_0-\bar{\beta}_n)e^{i\omega^-gt}+\bar{\beta}_n$. Then $Q(\alpha,\alpha^*)$ function can be written as a sum of two counterrotating peaks in the phase space i.e., $$Q(\alpha,\alpha^*,t)\sim \vert\langle\alpha\vert\gamma^+(t)\rangle\vert^2+\vert\langle\alpha\vert\gamma^-(t)\rangle\vert^2\label{qq}.$$ In figure \[fig6\] we plotted the time evolution of $Q(\alpha,\alpha^*,t)$ using exact, as well as approximated solutions for $\phi=0$ (figures \[6a\] and \[6b\]) and for $\phi=\pi/2$ (figures \[6c\] and \[6d\]). The initial condition of the system is $\vert +\rangle\vert\alpha_0\rangle$. The left hand side pictures correspond to the exact solutions and the right hand side pictures correspond to the approximated results. The initial $Q(\alpha,\alpha^*)$ distribution is a Gaussian centered at $\alpha_0$. As we know, in the course of time the initial one-peaked function splits into two peaks. We plotted them in separate Figures. One of them ($\psi^+$) rotates counterclockwise (see figure \[6a\]) on the circle of radius $\vert\alpha_0+1\vert=6$ ($\bar{\beta}_n\approx 1$) with angular speed $\omega^+=0.0808$, and the other one ($\psi^-$) rotates clockwise (see figure \[6b\]) on the circle with smaller radius $\vert\alpha_0-1\vert=4$ at angular speed $\omega^-=0.1212$. In this case the centers of circles are shifted respectively down and up in real axis.
For $\phi=\pi/2$, $\psi^+$ and $\psi^-$ counterrotate on the different circles with the same radius $\vert\alpha_0\pm i\vert=5.099$ at angular speed $\omega^{\pm}=0.0951$. The centers of the circles are shifted horizontally. We can see from figure \[fig6\] that our approximated solutions give almost precise locations of the peaks. Therefore we can predict the kinematics of the moving peaks and find the times when those peaks collide, which enable us to calculate the revival times of the atomic inversion.
Conclusion
==========
We have calculated steady states of the generalized Jaynes-Cummings system with the external field driving the cavity mode. The steady states are the superposition of atomic states multiplied by squeezed and displaced Fock states. The calculated steady states are valid only if $2\mathscr{E}/g<1$. In other words, $2\mathscr{E}/g=1$ is a threshold value of the driving field strength and above this value normalizable steady states do not exist. But we numerically calculated the time evolution of normalized states for the arbitrary value of $2\mathscr{E}/g$. Using these results we studied the dynamics of the atomic variables and the statistical properties of the field.
Furthermore, we derived approximate solutions for the time evolution of quantum states, which allow us to understand the kinematics of $Q(\alpha,\alpha^*)$ function in the phase space. We found that as time goes on the initial single peaked $Q(\alpha,\alpha^*)$ function splits into two peaks which counterrotate on separate circles of different radii $\vert\alpha_0\pm\bar{\beta}_n\vert$ with different angular velocities $\omega^{\pm}$. Whenever these peaks collide, or even partly overlap, atomic inversion oscillation shows revival.
Approximation method
====================
If we are interested only in the kinematics of the $Q(\alpha,\alpha^*)$ function in the complex plane, but not the shapes of the peaks, then we can make some very simple approximations for the state vectors . We restrict our calculations with only first order terms of $\mathscr{E}/g$ ignoring all higher order terms to yield $r\approx 0$, and therefore $\eta\approx 0$, and $S(\eta)\approx 1$ (squeezing operator does not affect the location of the peaks in phase space, it changes only the shape of the peaks).
Doing same approximation we get $L_+\approx-(g/\mathscr{E})e^{i\phi}$. Since $\vert L_+\vert$ is much greater than 1, the terms not containing $L_+$ can be neglected in equations . Then we get $$\vert\psi^+(t)\rangle\approx\vert c_p\vert^2\vert L_+\vert^2\sum_{n=0}^{\infty}(\langle n-1;\beta_{n}\vert\alpha_0\rangle)e^{-iE_nt}
\left(\vert -\beta_{n};n-1\rangle\vert +\rangle+\vert -\beta_{n};n\rangle\vert -\rangle\right),$$ $$\vert\psi^-(t)\rangle\approx\vert c_p\vert^2\vert L_+\vert^2\sum_{n=0}^{\infty}(\langle n-1;-\beta_{n}\vert\alpha_0\rangle)e^{iE_nt}
\left(\vert \beta_{n};n-1\rangle\vert +\rangle-\vert\beta_{n};n\rangle\vert -\rangle\right),$$ where $\beta_n=(2\mathscr{E}/g)\sqrt{n}e^{i\phi}$. The difficulty to manage above equations is that $\beta_n$ depends on the photon number $n$. The multiplier $2\mathscr{E}/g$ in equation $\beta_n$ is much less than $1$. That leads to $\beta_n\neq 0$ in small interval of $n$. If we change $\beta_n$ with $$\bar{\beta}_n=(2\mathscr{E}/g)\sqrt{\bar{n}}e^{i\phi},$$ we can easily evaluate state vector $$\vert\psi^+(t)\rangle\approx\vert c_p\vert^2\vert L_+\vert^2\sum_{n=0}^{\infty}\frac{(\alpha_0+\bar{\beta}_n)^n}{\sqrt{n!}}e^{-iE_nt}
D(-\bar{\beta}_n)\left(\vert n-1\rangle\vert +\rangle+\vert n\rangle\vert -\rangle\right),\label{app1}$$ $$\vert\psi^-(t)\rangle\approx\vert c_p\vert^2\vert L_+\vert^2\sum_{n=0}^{\infty}\frac{(\alpha_0-\bar{\beta}_n)^n}{\sqrt{n!}}e^{iE_nt}
D(\bar{\beta}_n)\left(\vert n-1\rangle\vert +\rangle-\vert n\rangle\vert -\rangle\right).\label{app2}$$ If $\alpha_0$ is large enough, then $\alpha_0+\bar{\beta}_n$ and $\alpha_0-\bar{\beta}_n$ are also large and there is not much difference between $\vert n-1\rangle$ and $\vert n\rangle$. We can substitute $e^{\mp i\omega^{\pm} nt}$ instead of $e^{\mp iE_nt}$. Therefore, taking into account the relation $$D(\alpha)D(\beta)=\exp\left[\frac{1}{2}(\alpha\beta^*-\alpha^*\beta)\right]D(\alpha+\beta),$$ we get the approximate expressions for the state vectors as $$\vert\psi^+(t)\rangle\approx A^+(t)\sum_{n=0}^{\infty}\frac{[(\alpha_0+\bar{\beta}_n)e^{-i\omega^+gt}-\bar{\beta}_n]^n}{\sqrt{n!}}\vert n\rangle\left(\vert +\rangle +\vert -\rangle\right),\label{app3}$$ $$\vert\psi^-(t)\rangle\approx A^-(t)\sum_{n=0}^{\infty}\frac{[(\alpha_0-\bar{\beta}_n)e^{i\omega^-gt}+\bar{\beta}_n]^n}{\sqrt{n!}}\vert n\rangle\left(\vert +\rangle -\vert -\rangle\right).\label{app4}$$
Estimations of $\omega^+$ and $\omega^-$
========================================
Here we make a simple estimation for the rotational angular velocities $\omega^+$ and $\omega^-$ of two separate peaks of the $Q(\alpha,\alpha^*)$ function in the complex plane. First we estimate $\omega^+$. Using equation , the reduced density operator for the quantized cavity field can be written as $$\rho_f^+=\sum^{\infty}_{n,m=0}\frac{(\alpha_0+\bar{\beta}_n)^n}{\sqrt{n!}}\frac{((\alpha_0+\bar{\beta}_n)^*)^m}{\sqrt{m!}}e^{-ikgt(\sqrt{n}-\sqrt{m})}D(-\bar{\beta}_n)\vert n\rangle\langle m\vert D(\bar{\beta}_n),\label{rho}$$ where $k=(1-(2\mathscr{E}/g)^2)^{3/4}$. Using a Taylor expansion of $\sqrt{n}$ around $\sqrt{\bar{n}}$ (where $\bar{n}=\vert\alpha_0+\bar{\beta}_n\vert^2$) $$\sqrt{n}\approx \sqrt{\bar{n}}+\frac{1}{2\sqrt{\bar{n}}}(n-\bar{n})-\dots,$$ and ignoring all higher order terms we get $$\sqrt{n}-\sqrt{m}\approx (n-m)/2\sqrt{\bar{n}}.\label{fff}$$ Substituting it into the equation for $\rho_f^+$ one can find $\omega^+$ as follows $$\omega^+=\frac{[1-(2\mathscr{E}/g)^2]^{3/4}}{2\vert\alpha_0+\bar{\beta}_n\vert}.$$ Similarly, for $\omega^-$ we get $$\omega^-=\frac{[1-(2\mathscr{E}/g)^2]^{3/4}}{2\vert\alpha_0-\bar{\beta}_n\vert}.$$
[999]{}
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
We study some basic properties of the variety of characters in ${\mathrm{PSL}_2 (\C)}$ of a finitely generated group. In particular we give an interpretation of its points as characters of representations. We construct 3-manifolds whose variety of characters has arbitrarily many components that do not lift to ${\mathrm{SL}_2 (\C)}$. We also study the singular locus of the variety of characters of a free group.\
[*MSC*: 57M50, 57M05, 20C15\
*Keywords*: Representation spaces; variety of characters; ${\mathrm{PSL}_2 (\C)}$]{}
author:
- Michael Heusener and Joan Porti
title: 'The variety of characters in ${\mathrm{PSL}_2 (\C)}$'
---
Introduction
============
The varieties of representations and characters have many applications in 3-dimensional topology and geometry. The variety of ${\mathrm{SL}_2 (\C)}$-characters has been intensively studied since the seminal paper of Culler and Shalen [@CS], but for many applications it is more convenient to work with ${\mathrm{PSL}_2 (\C)}$ instead of ${\mathrm{SL}_2 (\C)}$ (see [@BZ] and [@BMP] for instance). The purpose of this note is to study some basic properties of the variety of characters in ${\mathrm{PSL}_2 (\C)}$. Most of the results of invariant theory that we use can be found in any standard reference (e.g. [@KSS], [@Kraft], [@PV]).
Throughout this paper, $\Gamma$ will denote a finitely generated group.
The set of all representations of $\Gamma$ in ${\mathrm{PSL}_2 (\C)}$ is denoted by $
R(\Gamma)
$ and it is called the *variety of representations of $\Gamma$ in ${\mathrm{PSL}_2 (\C)}$*.
The variety of representations $R(\Gamma)$ has a natural structure as an affine algebraic set over the complex numbers given as follows: the group ${\mathrm{PSL}_2 (\C)}$ is algebraic (see Section \[sec:invariants\]). Given a presentation $\Gamma=\langle
\gamma_1,\ldots,\gamma_s\mid (r_i)_{i\in I} \rangle$ we have a natural embedding: $$\begin{array}{rcl}
R(\Gamma)&\to &{\mathrm{PSL}_2 (\C)}\times\cdots\times {\mathrm{PSL}_2 (\C)}\\
\rho & \mapsto & (\rho(\gamma_1),\ldots,\rho(\gamma_s))
\end{array}$$ and the defining equations are induced by the relations. This structure can be easily seen to be independent of the presentation. In fact using the isomorphism ${\mathrm{PSL}_2 (\C)}\cong {\mathrm{SO}_{3} (\C)}$, $R(\Gamma)$ has a structure of an affine set (see Lemma \[lem:SO3\]).
The action of ${\mathrm{PSL}_2 (\C)}$ on $R(\Gamma)$ by conjugation is algebraic. The quotient $R(\Gamma)/{\mathrm{PSL}_2 (\C)}$ may be not Hausdorff and it is more convenient to consider the algebraic quotient of invariant theory, because ${\mathrm{PSL}_2 (\C)}$ is reductive.
The *variety of ${\mathrm{PSL}_2 (\C)}$-characters* $X(\Gamma)$ is the quotient $R(\Gamma)/\!/{\mathrm{PSL}_2 (\C)}$ of invariant theory.
This definition means that $X(\Gamma)$ is an affine algebraic set together with a regular map $t\co R(\Gamma) \to X(\Gamma)$ which induces an isomorphism $$t^{*}\co \C[X(\Gamma)]\to\C[R(\Gamma)]^{{\mathrm{PSL}_2 (\C)}}$$ (i.e.the regular functions on $X(\Gamma)$ are precisely the regular functions on $R(\Gamma)$ invariant by conjugation). We will use the notation $R(M)=R(\pi_1M)$ and $X(M)=X(\pi_1M)$ if $M$ is a path-connected topological space.
In this paper we study the basic properties of $X(\Gamma)$.
First we explain the name “variety of characters": given a representation $\rho\co\Gamma\to {\mathrm{PSL}_2 (\C)}$, its character is the map $$\begin{array}{rcl}
\chi_{\rho}\co\Gamma&\to&\C\\
\gamma&\mapsto &{\operatorname{tr}}^2(\rho(\gamma))
\end{array}$$
\[thm:PSL2characters\] There is a natural bijection between $X(\Gamma)$ and the set of characters of representations $\rho\in
R(\Gamma)$. This bijection maps every $t(\rho)\in X(\Gamma)$ to the character $\chi_\rho$.
In many cases the representations of $R(\Gamma)$ lift to ${\mathrm{SL}_2 (\C)}$, for instance if $\Gamma$ is a free group. In such a case, $X(\Gamma)$ is just a quotient of the usual variety of characters in ${\mathrm{SL}_2 (\C)}$ (See Proposition \[prop:naturaliso\]). This quotient is the definition already used in [@Bur90], [@HLM1],[@HLM2] and [@Ril84] for 2-bridge knot exteriors. The explicit computation for the figure eight knot exterior is found in [@GM].
There are cases where representations do not lift to ${\mathrm{SL}_2 (\C)}$, for instance the holonomy representation of an orientable hyperbolic 3-orbifold with 2 torsion. The next result proves that there are manifolds with arbitrarily many components of characters that do not lift.
\[thm:nolifts\] For every $n$, there exist a compact irreducible 3-manifold $M$ with $\partial M$ a 2-torus such that $X(M)$ has at least $n$ irreducible one dimensional components whose characters do not lift to ${\mathrm{SL}_2 (\C)}$.
In Section \[sec:invariants\] we prove Theorem \[thm:PSL2characters\]. In Section \[sec:irreducibility\] we study the fiber of the projection $t\co R(\Gamma)\to X(\Gamma)$, introducing the different notions of irreducibility. Section \[sec:lifts\] is devoted to the study of lifts of representations and the proof of Theorem \[thm:nolifts\]. In the last section we determine the singular set of $X(\Gamma)$ when $\Gamma\cong F_n$ is the free group of rank $n\geq 3$.
Invariants of ${\mathrm{PSL}_2 (\C)}$ {#sec:invariants}
=====================================
Before proving Theorem \[thm:PSL2characters\] we quickly review some basic notions of algebraic geometry and invariant theory (that the reader may prefer to skip and go directly to the proof in Subsection \[ss:ProofTheorem\]). For details see [@KSS], [@Kraft] or [@PV].
Basic notions of invariant theory
---------------------------------
A closed algebraic subset $Z\subset \C^{N}$ is called *affine*. We denote by $\C[Z]$ the ring of regular functions on $Z$. An algebraic group $G$ that acts algebraically on $Z$ acts naturally on $\C[Z]$ via $g f (z) := f(g^{-1} z) $. We denote by $\C[Z]^{G} $ the ring of invariant functions, i.e. functions $f\in\C[Z]$ for which $g
f = f$ for all $g\in G$.
The group $G$ is called *reductive* if it has the following property: for each finite dimensional rational representation $\rho\co G\to \mathrm{GL}(V)$ and every $G$-invariant subspace $W\subset V$ there exist a complementary $G$-invariant subspace $W'\subset V$, i.e. $V= W'\oplus W$.
If $Z$ is affine and $G$ is reductive, then the ring $\C[Z]^{G} $ is finitely generated. The affine set $Y$ such that $\C[Y]\cong
\C[Z]^{G} $ is called the *algebraic quotient* and it is denoted by $Z /\!/\/ G$.
We shall use the following properties of reductive groups:
- By Maschke’s theorem, finite groups are reductive.
- More generally, let $G\subset \mathrm{GL}_{n}(\C)$ be a linear algebraic group. The group $G$ is reductive if there is a Zariski-dense subgroup $K\subset G$ which is compact in the classical topology. It follows that $\mathrm{GL}_{n}(\C)$, $\mathrm{SL}_{n}(\C)$, $\textrm{O}_{n}(\C)$, $\textrm{SO}_{n}(\C)$ and $\textrm{Sp}_{n}(\C)$ are reductive.
- Let $G$ be a reductive linear algebraic group. Let $Y$ and $Z$ be varieties on which $G$ acts and let $f\co X\to Y$ be a $G$-invariant regular map. If $f^{*}\co \C[Y]\to\C[X]$ is surjective then $f^{*}(\C[Y]^{G}) = \C[X]^{G}$ holds.
Algebraic structure of ${\mathrm{PSL}_2 (\C)}$
----------------------------------------------
The group ${\mathrm{PSL}_2 (\C)}$ is algebraic, it is the quotient of ${\mathrm{SL}_2 (\C)}$ by the finite group $\{\pm {\mathrm{Id} }\}$.
It is useful to recall the isomorphism with ${\mathrm{SO}_{3} (\C)}$, that we construct next. We denote by $${\operatorname{Ad}}\co{\mathrm{PSL}_2 (\C)}\to {{\operatorname{Aut}}({\mathfrak{sl}_2 (\C)})}$$ the adjoint action of ${\mathrm{PSL}_2 (\C)}$ on its Lie algebra ${\mathfrak{sl}_2 (\C)}$. The Killing form on ${\mathfrak{sl}_2 (\C)}$ is a non degenerate symmetric bilinear form over $\C$. For each $A\in {\mathrm{PSL}_2 (\C)}$, ${\operatorname{Ad}}(A)$ preserves the Killing form and $\det ({\operatorname{Ad}}(A)) = 1$, hence ${\operatorname{Ad}}({\mathrm{PSL}_2 (\C)}) \subseteq
{\mathrm{SO}_{3} (\C)}$. The following lemma is well known from representation theory (see for instance [@FultonHarris]):
\[lem:SO3\] The action of ${\mathrm{PSL}_2 (\C)}$ on the Lie algebra induces an isomorphism ${\operatorname{Ad}}\co{\mathrm{PSL}_2 (\C)}\to {\mathrm{SO}_{3} (\C)} $.
In this paper the trace will be abbreviated by ${\operatorname{tr}}$, and ${\operatorname{tr}}^{2}(A)$ stands for $({\operatorname{tr}}(A))^{2}$. By direct computation we obtain the equality $$\label{eqn:traceAd}
{\operatorname{tr}}({\operatorname{Ad}}(A))={\operatorname{tr}}^{2}(A)-1={\operatorname{tr}}(A^2)+1
\quad\text{ for all } A\in {\mathrm{PSL}_2 (\C)}$$ that will be used later.
Given $\gamma\in\Gamma$, we have a well defined function $$\begin{array}{rcl}
\tau_{\gamma}\co R(\Gamma)&\to&\C\\
\rho&\mapsto &{\operatorname{tr}}^{2}(\rho(\gamma))
\end{array}$$ Since it is invariant by conjugation, it induces a function $$J_{\gamma}\co X(\Gamma) \to\C.$$
Proof of Theorem \[thm:PSL2characters\] {#ss:ProofTheorem}
---------------------------------------
Theorem \[thm:PSL2characters\] is a consequence of:
\[prop:generators\] The ring of invariant functions $\C[R(\Gamma)]^{{\mathrm{PSL}_2 (\C)}}$ is generated by the functions $\tau_{\gamma}$, with $\gamma\in\Gamma$.
There is a surjection $\psi\co F_{n}\to \Gamma$ where $F_n$ is a free group of rank $n\in\N$. We obtain an inclusion $\psi^{*}\co R(\Gamma) \subset R(F_{n})$. This inclusion induces a surjection $\psi_{*} \co\C[R(F_{n})]\to \C[R(\Gamma)]$. Now, ${\mathrm{PSL}_2 (\C)}$ is reductive and acts regularly by conjugation on the representation varieties. Hence we obtain a surjection $$\psi_{*}\co \C[R(F_{n})]^{{\mathrm{PSL}_2 (\C)}}\to \C[R(\Gamma)]^{{\mathrm{PSL}_2 (\C)}}$$ and it is sufficient to prove the proposition for $\Gamma =
F_{n}$ since $\psi_{*} (\tau_{\gamma}) = \tau_{\psi(\gamma)}$.
Using Lemma \[lem:SO3\] and (\[eqn:traceAd\]), we have to prove that $\C[R(F_n)]^{{\mathrm{SO}_{3} (\C)}}$ is generated by the trace functions on elements of $F_n$. Equivalently, we claim that $$\C[{\mathrm{SO}_{3} (\C)}\times\cdots\times{\mathrm{SO}_{3} (\C)}]^{{\mathrm{SO}_{3} (\C)}}$$ is generated by traces of products of matrices and their transposes.
Let ${\mathrm{M}_3(\C)}$ denote the algebra of $3\times 3$ matrices with complex coefficients. The group ${\mathrm{PSL}_2 (\C)}\cong {\mathrm{SO}_{3} (\C)}$ acts on the product ${\mathrm{M}_3(\C)}\times \cdots \times{\mathrm{M}_3(\C)}$ diagonally by conjugation. A theorem of Aslaksen, Tan and Zhu (see [@ATZ]) states that the algebra of invariant functions $$\C[{\mathrm{M}_3(\C)}\times \cdots \times {\mathrm{M}_3(\C)}]^{{\mathrm{SO}_{3} (\C)}}$$ is generated by the traces of products of matrices and their transposes. Thus the proof of the proposition reduces to show that we have a natural surjection $$\C[{\mathrm{M}_3(\C)}\times \cdots \times {\mathrm{M}_3(\C)}]^{{\mathrm{SO}_{3} (\C)}} \to
\C[{\mathrm{SO}_{3} (\C)}\times \cdots \times {\mathrm{SO}_{3} (\C)}]^{{\mathrm{SO}_{3} (\C)}}\,.$$ Since ${\mathrm{SO}_{3} (\C)}\times \cdots \times {\mathrm{SO}_{3} (\C)}\subset {\mathrm{M}_3(\C)}\times \cdots \times
{\mathrm{M}_3(\C)}$ is a closed subvariety we obtain a natural surjection $$\C[{\mathrm{M}_3(\C)}\times \cdots \times {\mathrm{M}_3(\C)}]\to \C[{\mathrm{SO}_{3} (\C)}\times \cdots\times {\mathrm{SO}_{3} (\C)}]$$ which is of course ${\mathrm{SO}_{3} (\C)}$-invariant. Using the fact that ${\mathrm{SO}_{3} (\C)}$ is reductive gives the surjection $\C[{\mathrm{M}_3(\C)}\times \cdots \times {\mathrm{M}_3(\C)}]^{{\mathrm{SO}_{3} (\C)}} \to
\C[{\mathrm{SO}_{3} (\C)}\times \cdots \times {\mathrm{SO}_{3} (\C)}]^{{\mathrm{SO}_{3} (\C)}}$.
Since $\C[X(\Gamma)]=\C[R(\Gamma)]^{{\mathrm{SO}_{3} (\C)}}$ is finitely generated, we also obtain:
\[cor:embedding\] There are finitely many elements $\gamma_{1},\ldots,\gamma_{N}$ in $\Gamma$ such that $J_{\gamma_1}\times\cdots\times J_{\gamma_N}\co X(M)\to\C^N$ is an embedding and its image is a closed algebraic set.
Other invariant functions {#ssec:others}
-------------------------
There are other natural functions to consider. Let $\Gamma^{2}$ be the subgroup of $\Gamma$ generated by the squares $\gamma^{2}$ of all elements $\gamma$ of $\Gamma$. It is well known that we have an exact sequence: $$1\to\Gamma^2\to\Gamma\to H_1(\Gamma,{C_2})\to 1,$$ where $C_2=\{\pm 1\}$ is the group with $2$ elements. For instance, if $\Gamma$ is a finite group of odd order, then $\Gamma^{2}=\Gamma$. In general, if $ \gamma,\mu \in\Gamma$ the commutator $[\gamma,\mu]=\gamma\mu\gamma^{-1}\mu^{-1}=
(\gamma\mu)^{2}(\mu^{-1}\gamma^{-1}\mu)^{2}\mu^{-2}$ is in $\Gamma^{2}$ and hence $\Gamma^{2}$ contains the commutator group $\Gamma' = [\Gamma,\Gamma]$. Notice that $$\Gamma^2=\bigcap_{\epsilon\in H^1(\Gamma,C_2)} {\operatorname{Ker}}(\epsilon)$$ where $H^1(\Gamma,C_2)={{\operatorname{Hom}}((,\Gamma)},C_{2})$. Let $R(\Gamma,{\mathrm{SL}_2 (\C)})$ denote the variety of representations of $\Gamma$ in ${\mathrm{SL}_2 (\C)}$. The cohomology group $H^1(\Gamma,{C_2})$ acts on this variety of representations as follows: an homomorphism $\epsilon\co\Gamma\to{C_2}=\{\pm 1\}$ maps the representation $\rho\in R(\Gamma,{\mathrm{SL}_2 (\C)})$ to the product of representations $\epsilon\cdot \rho$ (which maps $\gamma\in\Gamma$ to $
{\epsilon(\gamma)}\cdot \rho(\gamma)$).
### Invariant functions for the free group
Let $F$ be a finitely generated free group. For $\gamma\in F^{2}$ and $\rho\in R(F)$, ${\operatorname{tr}}(\rho(\gamma))$ is well defined since the representation $\rho\co F \to {\mathrm{PSL}_2 (\C)}$ lifts to $\tilde{\rho}\co F \to {\mathrm{SL}_2 (\C)}$ and for $\gamma\in F^{2}$ the trace ${\operatorname{tr}}(\tilde{\rho}(\gamma))$ depends only on $\gamma$. Note that two lifts $\tilde{\rho}_{1}$ and $\tilde{\rho}_{2}$ of $\rho$ differ by a homomorphism $\epsilon\in H^1(F,C_2) $ and that $F^{2}\subset {\operatorname{Ker}}(\epsilon)$ for each $\epsilon\in H^{1}(F,C_{2})$.
\[prop:sigmafunction\] Let $F$ be a free group. For every $k$-tuple $\gamma_1,\ldots,\gamma_k\in F$ such that the product $\gamma_1\cdots\gamma_k\in F^2$, the function $$\begin{array}{rcl}
\sigma_{\gamma_1,\ldots,\gamma_k}\co R(F)&\to&\C \\
\rho&\mapsto&{\operatorname{tr}}(\tilde\rho(\gamma_1))\cdots
{\operatorname{tr}}(\tilde\rho(\gamma_k))
\end{array}$$ is regular (i.e.$\sigma_{\gamma_1,\ldots,\gamma_k}\in\C[R(F)]$). Here, $\tilde\rho\co F \to{\mathrm{SL}_2 (\C)}$ denotes a lift of $\rho$.
In order to prove this proposition we shall use the following:
\[lem:RFn\] Let $F_n$ be the free group of rank $n$. We have a natural isomorphism $$R(F_n,{\mathrm{SL}_2 (\C)})/\! /H^1(F_n,{C_2})\cong R(F_n)\,.$$
Since $R(F_n,{\mathrm{SL}_2 (\C)})\cong {\mathrm{SL}_2 (\C)}^n$, $R(F_n)\cong {\mathrm{PSL}_2 (\C)}^n$ and ${\mathrm{SL}_2 (\C)}/{C_2}\cong {\mathrm{PSL}_2 (\C)} $, we have the lemma.
For a free group $F$ and $\gamma_1,\ldots,\gamma_k\in F$, the function $\tilde \sigma \co R(F,{\mathrm{SL}_2 (\C)})\to \C$ given by $\tilde\sigma(\rho)={\operatorname{tr}}(\rho(\gamma_1))\cdots {\operatorname{tr}}(\rho(\gamma_k))$ is regular. Moreover, we have $\tilde\sigma(\epsilon\cdot\rho) =
\epsilon(\gamma_1\cdots\gamma_k) \tilde\sigma(\rho)$. Since the product $\gamma_1\cdots\gamma_k\in F^{2}$ we get that $\tilde\sigma\in \C[R(F_n,{\mathrm{SL}_2 (\C)})]^{H^1(F_n,{C_2})}$ is an invariant regular function on the ${\mathrm{SL}_2 (\C)}$ representation variety. By Lemma \[lem:RFn\], this function factors through $R(F)$ and gives the regular function $\sigma_{\gamma_1,\ldots,\gamma_k}\in\C[R(F)] $.
\[ex:sigmagammaeta\] Given $\gamma,\eta\in F$, by Proposition \[prop:sigmafunction\], $\sigma_{\gamma,\eta,\gamma\eta}\in \C[R(F)]$, thus by Proposition \[prop:generators\], $\sigma_{\gamma,\eta,\gamma\eta}$ is a polynomial on the functions $\tau$.
To compute explicitly the polynomial of Example \[ex:sigmagammaeta\], we recall some identities of traces in ${\mathrm{SL}_2 (\C)}$: $${\operatorname{tr}}(AB)={\operatorname{tr}}(BA)\quad\textrm{and}\quad {\operatorname{tr}}(A)={\operatorname{tr}}(A^{-1})\qquad
\forall A,B\in {\mathrm{SL}_2 (\C)}\,.$$ In addition, we have the fundamental identity: $$\label{eqn:sl2fundamental}
{\operatorname{tr}}(AB)+{\operatorname{tr}}(A^{-1}B)={\operatorname{tr}}(A)\,{\operatorname{tr}}(B)
\qquad \forall A,B\in
{\mathrm{SL}_2 (\C)} .$$ This identity can be deduced from $A^2-({\operatorname{tr}}A)A+{\mathrm{Id} }=0$ multiplying by $A^{-1}B$ and taking traces. Taking the square of ${\operatorname{tr}}(AB^{-1})={\operatorname{tr}}(A)\,{\operatorname{tr}}(B)-{\operatorname{tr}}(AB)$ we deduce: $$2 {\operatorname{tr}}(A){\operatorname{tr}}(B){\operatorname{tr}}(AB) = {\operatorname{tr}}^{2}(A) \, {\operatorname{tr}}^{2}(B)+{\operatorname{tr}}^{2}(AB) -
{\operatorname{tr}}^{2}(AB^{-1})\,.$$ Thus $$\label{eqn:sigmaABAB}
\sigma_{\gamma,\eta,\gamma\eta}=
\frac12(\tau_{\gamma}\tau_{\eta}+\tau_{\gamma\eta}-\tau_{\gamma\eta^{-1}}).$$
For every $\gamma,\mu\in F$, the commutator $[\gamma,\mu]=\gamma\mu\gamma^{-1}\mu^{-1}$ belongs to $F^2$ and therefore $\sigma_{[\gamma,\mu]}\in\C[R(F)]$. Using the the same method as for Equation (\[eqn:sigmaABAB\]) one can find: $$\label{ex:traces}
\sigma_{[\gamma,\eta]}= \tau_{\gamma} + \tau_{\eta} + \frac 1 2
\tau_{\gamma\eta} + \frac 1 2 \tau_{\gamma\eta^{-1}} - \frac 1 2
\tau_{\gamma}\tau_{\eta} -2 \,.$$
### Invariant functions for other groups
Let $\Gamma$ be a finitely generated group, $F$ a free group and $\psi\co F\to\Gamma$ a surjection. It induces another surjection $\psi_{*}\co \C[ R(F)]\to \C[R(\Gamma)]$, $\psi_{*}f
(\rho) = f (\rho\circ \psi)$. Hence we obtain for all $\eta_{1},\ldots,\eta_{k}\in F$ such that the product $\eta_{1}\cdots\eta_{k}\in F^{2}$ a regular function $\psi_{*}\sigma_{\eta_{1},\ldots,\eta_{k}}\in \C[R(\Gamma)]$. Note that the functions $\psi_{*}\sigma_{\eta_1}$ and $\psi_{*}\sigma_{\eta_2}$ might be different even if $\psi(\eta_1)=\psi(\eta_2)$ in $\Gamma$. This reflects the fact that in general not every representation $\rho\co\Gamma\to{\mathrm{PSL}_2 (\C)}$ lifts to ${\mathrm{SL}_2 (\C)}$.
Let $\psi\co F\to\Gamma$ be the canonical projection where $F=\langle x,y \mid - \rangle$ and $\Gamma = \langle x,y \mid [x,y] =1 \rangle$. We consider the representation $\rho\co\Gamma\to{\mathrm{PSL}_2 (\C)}$ given by $ \rho (x) = \pm A_{x}$ and $\rho (y) = \pm A_{y}$ where $$A_{x} =\begin{pmatrix} i & 0 \\ 0& -i \end{pmatrix} \text{ and }
A_{y} = \begin{pmatrix} 0 & 1 \\ -1& 0 \end{pmatrix}\,.$$ We obtain ${\operatorname{tr}}([A_{x},A_{y}]) = -2$ and hence $\psi_{*}\sigma_{[x,y]}(\rho) = -2$. On the other hand we have $[x,y] =1$ in $\Gamma$ and $\psi_{*}\sigma_{1}= 2$ is a constant function.
If the representation $\rho\in R(\Gamma)$ admits a lift $\tilde\rho\co\Gamma\to{\mathrm{SL}_2 (\C)}$ then $$\label{equ:sigma}
\psi_{*}\sigma_{\eta_{1},\ldots,\eta_{k}}(\rho)=
{\operatorname{tr}}(\tilde\rho(\psi(\eta_{1})))\cdots
{\operatorname{tr}}(\tilde\rho(\psi(\eta_{k})))$$ only depends on the elements $\psi(\eta_1),\ldots,
\psi(\eta_k)\in\Gamma$.
Irreducibility {#sec:irreducibility}
==============
To study the fiber of the map $t\co R(\Gamma)\to X(\Gamma)$ we shall consider two different notions of irreducibility for $\rho\in
R(\Gamma)$, the usual one as a representation in ${\mathrm{PSL}_2 (\C)}$ and the so called ${\operatorname{Ad}}$-irreducibility, for the three dimensional representation ${\operatorname{Ad}}\circ\rho\co \Gamma\to SO_3(\C)$.
Irreducible representations
---------------------------
A representation $\rho\in R(\Gamma)$ is called *reducible* if $\rho(\Gamma)$ preserves a point of $P^1(\C)$. Otherwise it is called irreducible. A character $\chi\co\Gamma\to\C$ is called *reducible* if it is the character of a reducible representation.
\[rem:reducible\] Up to conjugation, the image of a reducible representation is contained in the set of upper-triangular matrices $\left(\begin{smallmatrix}
* & *\\ 0 & *\end{smallmatrix} \right)$.
We shall require the following well known lemma (see [@Beardon § 4.3]).
\[lem:fixedpoint\] Two non-trivial elements $g,h\in{\mathrm{PSL}_2 (\C)}$ have a common fixed point in $P^{1}(\C)$ if and only if ${\operatorname{tr}}([g,h])=2$. In addition, this fixed point is unique if $[g,h]$ is not the identity.
Irreducibility is a property that can be detected from characters:
\[lem:tracered\] A representation $\rho\in R(\Gamma)$ is reducible iff ${\operatorname{tr}}([\rho(\gamma),\rho(\eta)])=2$ for all elements $\gamma,\eta$ in $\Gamma$.
If $\rho$ is reducible then all the $\rho(\gamma)$ have a common fixed point and Lemma \[lem:fixedpoint\] gives the result.
Assume now that ${\operatorname{tr}}([\rho(\gamma),\rho(\eta)])=2$ for all elements $\gamma,\eta$ in $\Gamma$.
*Case 1:* There are two elements $\gamma$ and $\eta$ in $\Gamma$ such that $[\rho(\gamma),\rho(\eta)]$ is not the identity. Then $A=[\rho(\gamma),\rho(\eta)]$ is a non-trivial parabolic element in the image of $\Gamma$. For any $\mu\in\Gamma$, either $\rho(\mu)$ commutes with $A$ or $[\rho(\mu),A]$ is non-trivial. The former possibility implies that $\rho(\mu)$ fixes the unique fixed point of $A$, the latter too by Lemma \[lem:fixedpoint\].
*Case 2:* The image of $\rho$ is an abelian group. Abelian subgroups of ${\mathrm{PSL}_2 (\C)}$ are well-known: either they have a global fixed point in $P^{1}(\C)$ or they are conjugated to the group with four elements generated by $\pm\left(\begin{smallmatrix} 0 & 1\\ -1 &
0\end{smallmatrix} \right)$ and $\pm \left(\begin{smallmatrix} i &
0\\ 0 & -i\end{smallmatrix} \right)$. Since the commutator of these two generators is $\left(\begin{smallmatrix} -1 & 0\\ 0 &
-1\end{smallmatrix} \right)$, this possibility does not occur.
\[def:Klein\] A non-cyclic abelian subgroup of ${\mathrm{PSL}_2 (\C)}$ with four elements is called *Klein’s 4-group*. Such a group is realized by rotations about three orthogonal geodesics and it is conjugated to the one generated by $\pm\left(\begin{smallmatrix} 0 & 1\\ -1 & 0\end{smallmatrix} \right)$ and $\pm \left(\begin{smallmatrix} i & 0\\ 0 & -i\end{smallmatrix} \right)$.
Let $R^{red}(\Gamma)$ denote the set of reducible representations and $X^{red}(\Gamma)=t(R^{red}(\Gamma))$. Let $F$ be a free group and let $\psi\co F\to\Gamma$ be surjective. Lemma \[lem:tracered\] implies that $$R^{red}(\Gamma) = \{\rho\in R(\Gamma)\mid
\psi_{*}\sigma_{[\gamma,\eta]}(\rho)=2 \quad \forall \gamma,\eta\in F \}$$ is a Zariski closed subset invariant by conjugation. Thus, by invariant theory we have:
\[cor:red\] The set $X^{red}(\Gamma)$ is Zariski closed and $R^{red}(\Gamma)=t^{-1}(X^{red}(\Gamma))$.
\[rem:rhodiagonal\] Every reducible character $\chi$ is the character of a diagonal representation, because if $\rho(\gamma)=
\pm\left(\!\begin{smallmatrix}a_{\gamma}&b_{\gamma}\\
0&
c_{\gamma}\end{smallmatrix}\!\right)$ is a representation, then $\rho'(\gamma)=\pm\left(\!\begin{smallmatrix}a_{\gamma}&0\\ 0&
c_{\gamma}\end{smallmatrix}\!\right)$ is also a representation with $\chi_{\rho}=\chi_{\rho'}$.
${\operatorname{Ad}}$-irreducibility
------------------------------------
A representation $\rho\in R(\Gamma)$ is *${\operatorname{Ad}}$-reducible* if ${\mathfrak{sl}_2 (\C)}$ has a proper invariant subspace by the action of ${\operatorname{Ad}}\circ\rho$. Otherwise it is ${\operatorname{Ad}}$-irreducible.
Let $\H^3$ denote the three-dimensional hyperbolic space and $\partial_{\infty}\H^3$ its ideal boundary. We use the isomorphism $\operatorname{Isom}^+(\H^3)\cong {\mathrm{PSL}_2 (\C)}$ and the natural identification $\partial_{\infty}\H^3\cong P^1(\C)$.
A representation $\rho\co\Gamma\to{\mathrm{PSL}_2 (\C)}$ is ${\operatorname{Ad}}$-reducible if and only if $\rho(\Gamma)$ preserves either a point in $\partial_{\infty}\H^3$ or a geodesic in $\H^{3}$.
Let $V$ be a proper subspace of ${\mathfrak{sl}_2 (\C)}$ invariant by ${\operatorname{Ad}}\circ\rho(\Gamma)$. Up to taking $V^{\bot}$, we may assume $\dim V=1$, because the Killing form is not degenerate. We have then two possibilities: either the Killing form restricted to $V$ vanishes or not. In the first case $V$ consists of parabolic Killing fields, in particular the 1-parameter group $\{\exp(v)\mid
v\in V\}\cong \C$ is parabolic and fixes a unique point at infinity, that has to be fixed also by $\rho$. In the second case, when the Killing form restricted to $V$ does not vanish, the 1-parameter group $\{\exp(v)\mid v\in V\}\cong \C^{*}$ is a subgroup of index two in the group of isometries which preserve a geodesic in $\H^3$. This geodesic has to be preserved by the representation. Conversely, if a representation preserves a point in $\partial_{\infty}\H^3$ or a geodesic, the previous argument shows how to construct an invariant subspace of ${\mathfrak{sl}_2 (\C)}$.
Reducible representations are also ${\operatorname{Ad}}$-reducible.
A representation ${\operatorname{Ad}}$-reducible but not reducible is a ${C_2}$-extension of an abelian one that fixes an oriented geodesic. Thus it preserves an unoriented geodesic
We call a representation $\rho\in R(\Gamma)$ *abelian* respectively *metabelian* if its image is an abelian respectively metabelian subgroup of ${\mathrm{PSL}_2 (\C)}$
\[lem:Ad-red\] A representation $\rho\in R(\Gamma)$ is ${\operatorname{Ad}}$-reducible iff it is metabelian.
If $\rho$ is ${\operatorname{Ad}}$-reducible then its image is contained in the stabilizer of either a point in $P^1(\C)$ or a geodesic in $\mathbb
H^3$. Those stabilizers are metabelian, since they are respectively the group of affine transformations of $\C$ and the semidirect product $\C^{*}\rtimes C_{2}$.
Now assume that $\rho(\Gamma)\subset{\mathrm{PSL}_2 (\C)}$ is a metabelian subgroup. We use the fact that an abelian subgroup of ${\mathrm{PSL}_2 (\C)}$ preserves a unique point of $P^1(\C) $, a unique geodesic or it is Klein’s 4-group (Def. \[def:Klein\]). If $\rho([\Gamma,\Gamma])$ is trivial then $\rho$ is ${\operatorname{Ad}}$-reducible by this fact. If $\rho([\Gamma,\Gamma])$is not trivial, then we look at those *unique* invariant objects: the *unique* point in $P^1(\C) $, the *unique* geodesic, or the *unique* three geodesics if it is Klein’s 4-group. Since $[\Gamma,\Gamma]$ is normal in $\Gamma$, uniqueness implies that $\rho(\Gamma)$ preserves the same objects, hence $\rho$ is ${\operatorname{Ad}}$-reducible.
The set of characters of ${\operatorname{Ad}}$-reducible representations is Zariski closed.
Lemma \[lem:Ad-red\] gives that the set of ${\operatorname{Ad}}$-reducible representations is $$R^{{\operatorname{Ad}}-red}=\{ \rho\in R(\Gamma)\mid \rho(c) =
\pm{\mathrm{Id} }\quad \forall c \in \Gamma''\}$$ where $\Gamma''$ denotes the second commutator group of $\Gamma$. This is a closed subset of $R(\Gamma)$ invariant under conjugation. Hence we have $X^{{\operatorname{Ad}}-red}(\Gamma)=t(R^{{\operatorname{Ad}}-red})$ is a closed subset of $X(\Gamma)$.
The image of an ${\operatorname{Ad}}$-reducible representation is elementary, but elementary groups also include groups that fix a point in $\mathbb H^3$.
The fibers of $t\co R(\Gamma)\to X(\Gamma)$
-------------------------------------------
\[lem:fiberirred\] The fiber of an irreducible character consists of a single closed orbit (i.e. two irreducible representations have the same character iff they are conjugate).
Let $\rho_1,\rho_2\in R(\Gamma)$ be two irreducible representations with $\chi_{\rho_1}=\chi_{\rho_2}$.
We assume first that each $\rho_i$ is irreducible but ${\operatorname{Ad}}$-reducible. Thus each $\rho_i$ preserves a geodesic $l$, that we may assume to be the same after conjugation. The action of $\rho_i(\gamma)$ on $l$ is determined by the value of $\chi_{\rho_i}(\gamma)$, except in the case $\chi_{\rho_i}(\gamma)=0$, which means that $\rho_i(\gamma)$ is a rotation through angle $\pi$, but it can be either about $\gamma$ or about an axis perpendicular to $\gamma$. Thus if there exists an element $\gamma_0\in \Gamma$ with $\chi_{\rho_i}(\gamma_0)\neq 4,0$ (i.e. $\rho_i(\gamma_0)$ is either a loxodromic element or a rotation of angle $\neq\pi$) then $\forall\gamma\in\Gamma$ the action of $\rho_i(\gamma)$ on the geodesic $l$ is determined by $ \chi_{\rho_i}(\gamma)$ and $\chi_{\rho_i}(\gamma\gamma_0)$. In particular $\rho_i$ is unique up to conjugation. The exceptional case occurs when $\chi_{\rho_i}(\gamma)=0$ or $4$ for every $\gamma\in\Gamma$. In this special case, $\rho_i$ is necessarily a representation into Klein’s 4-group. The lemma is also clear in this case.
When $\rho_i$ are ${\operatorname{Ad}}$-irreducible, we can assume that $\Gamma$ is a free group. Thus we can lift $\rho_i$ to $
\tilde\rho_i\co\Gamma\to{\mathrm{SL}_2 (\C)} $. By Example \[ex:sigmagammaeta\], for every pair $\gamma,\gamma'\in \Gamma$ we obtain a regular function $\sigma_{\gamma,\gamma',\gamma\gamma'}\co X(\Gamma)\to\C$, given by $$\sigma_{\gamma,\gamma',\gamma\gamma'}(\chi_{\rho}) =
{\operatorname{tr}}\tilde\rho(\gamma\gamma')\,{\operatorname{tr}}\tilde\rho(\gamma)\,
{\operatorname{tr}}\tilde\rho(\gamma')$$ where $\tilde\rho\co\Gamma\to{\mathrm{SL}_2 (\C)}$ is any lift of $\rho$. Thus: $$\label{eqn:traceslifts}
{\operatorname{tr}}\tilde\rho_1(\gamma\gamma')\,
{\operatorname{tr}}\tilde\rho_1(\gamma)\,
{\operatorname{tr}}\tilde\rho_1(\gamma') =
{\operatorname{tr}}\tilde\rho_2(\gamma\gamma')\,
{\operatorname{tr}}\tilde\rho_2(\gamma)\,
{\operatorname{tr}}\tilde\rho_2(\gamma')\, .$$
We define $\epsilon\co\Gamma\to{C_2}=\{\pm 1\}$ by the formula: $$\phantom{aa}\qquad {\operatorname{tr}}\tilde\rho_1(\gamma)= {\epsilon(\gamma)}
{\operatorname{tr}}\tilde\rho_2(\gamma), \qquad \forall\gamma\in\Gamma\textrm{
such that } \chi_{\rho_1}(\gamma)\neq 0.$$ When $\chi_{\rho_1}(\gamma)=0$, since we assume that $\rho_i$ is ${\operatorname{Ad}}$-irreducible, we can find $\gamma_0\in \Gamma$ with $\chi_{\rho_i}(\gamma_0)\neq 0$ and $\chi_{\rho_i}(\gamma\gamma_0)\neq 0$. In this case we define $\epsilon (\gamma)=\epsilon(\gamma_0)\cdot \epsilon(\gamma
\gamma_0)$.
By (\[eqn:traceslifts\]), $\epsilon$ is a morphism. Hence $\tilde\rho_1$ and $ {\epsilon}\cdot \tilde\rho_2$ are irreducible representations in ${\mathrm{SL}_2 (\C)}$ with the same character. By [@CS] they are conjugate.
\[lem:stabilizers\]
- A character $\chi$ is irreducible iff ${\mathrm{PSL}_2 (\C)}$ acts transitively on the fiber and with finite stabilizer.
- A character is ${\operatorname{Ad}}$-irreducible iff ${\mathrm{PSL}_2 (\C)}$ acts faithfully on the fiber.
\(i) By Lemma \[lem:fiberirred\], if $\chi$ is irreducible then ${\mathrm{PSL}_2 (\C)}$ acts transitively on $t^{-1}(\chi)$. Assume now that the stabilizer is infinite: i.e. there exists nontrivial $A\in {\mathrm{PSL}_2 (\C)}$ of order $\geq 3$ (possibly infinite) and $\rho$ in the fiber such that $A$ commutes with $\rho$. If $A$ is parabolic, then it has a fixed point in $P^1(\C)$ and therefore $\rho$ fixes this point. Otherwise $A$ has an invariant geodesic; since $A$ has order $\geq 3$, $\rho$ preserves the oriented geodesic, and therefore $\rho$ is also reducible.
Assume the character is reducible, then it has a diagonal representation $\rho$ on the fiber (Rem. \[rem:rhodiagonal\]), and therefore the group of diagonal matrices stabilizes it. Thus the stabilizer is infinite.
\(ii) Assume ${\mathrm{PSL}_2 (\C)}$ does not act faithfully on the fiber, i.e. there exists nontrivial $A\in {\mathrm{PSL}_2 (\C)}$ and $\rho$ in the fiber such that $A$ commutes with $\rho$. If $A$ is parabolic, then $\rho$ fixes a point in $P^1(\C)$ by the previous argument. Otherwise $A$ has an invariant geodesic, and by commutativity, $\rho$ must preserve this geodesic. In both cases, $\rho$ is ${\operatorname{Ad}}$-reducible.
If the character is irreducible but ${\operatorname{Ad}}$-reducible, then it preserves a geodesic, and the rotation through angle $\pi$ about this geodesic commutes with $\rho$. Hence the stabilizer is nontrivial.
The projection $t\co R(\Gamma)\to X(\Gamma)$ induces a bijection between irreducible components.
A priori $R(\Gamma)$ could have more components than $X(\Gamma)$, but the number of components is the same, because ${\mathrm{PSL}_2 (\C)}$ is irreducible.
From Corollary \[cor:red\] and Proposition \[lem:stabilizers\] we deduce:
Let $\rho\in R(\Gamma)$ be an irreducible representation. Let $R_0$ denote an irreducible component of $R(\Gamma)$ that contains $\rho$ and let $X_0$ denote the corresponding irreducible component of $X(\Gamma)$. Then $$\dim R_0=\dim X_0+3.$$
Lifts of representations to ${\mathrm{SL}_2 (\C)}$ {#sec:lifts}
==================================================
Let $\overline R(\Gamma)\subset R(\Gamma)$ denote the set of representations $\rho\in R(\Gamma)$ that lift to ${\mathrm{SL}_2 (\C)}$. According to [@Culler Thm. 4.1] $\overline R(\Gamma)$ is a union of connected components of $R(\Gamma)$. In particular $\overline R
(\Gamma) $ is a Zariski-closed algebraic subset of $R(\Gamma)$, since irreducible complex varieties are connected in the $\C$-topology [@Sha VII, §2]. Moreover, $\overline R (\Gamma)$ is invariant under conjugation and hence the algebraic quotient $$\overline X(\Gamma)=\overline
R(\Gamma)/\! /{\mathrm{PSL}_2 (\C)}$$ is a well defined closed subset of $X(\Gamma)$.
In many cases, $\overline X(\Gamma)=X(\Gamma)$. For instance this is clear when $\Gamma$ is a free group. It is also true if $H^2(\Gamma,{C_2})=0$ by the following remark (see [@GM] or [@Culler]).
Let $\rho\co\Gamma\to{\mathrm{PSL}_2 (\C)}$ be a representation. There is a second Stiefel-Whitney class $w_{2}(\rho)\in H^{2}(\Gamma,{C_2})$ which is exactly the obstruction for the existence of a lift $\overline \rho\co\Gamma\to{\mathrm{SL}_2 (\C)}$.
Properties of $\overline X(\Gamma)$
-----------------------------------
Let $R(\Gamma,{\mathrm{SL}_2 (\C)})$ and $X(\Gamma,{\mathrm{SL}_2 (\C)})$ denote the variety of representations and characters in ${\mathrm{SL}_2 (\C)}$. The ring $\C[R(\Gamma,{\mathrm{SL}_2 (\C)})]^{{\mathrm{SL}_2 (\C)}}$ is generated by the trace functions $\tilde\tau_{\gamma}\co R(\Gamma,{\mathrm{SL}_2 (\C)})\to\C$, $\tilde\tau_{\gamma}(\rho)={\operatorname{tr}}(\rho(\gamma))$. The function induced by $\tilde\tau_{\gamma}$ is denoted by $I_{\gamma}\co
X(\Gamma)\to\C$, therefore $\C[X(\Gamma)]$ is finitely generated by the functions $I_{\gamma}$, $\gamma\in\Gamma$ [@CS].
Elements of the cohomology group $ H^1( \Gamma,{C_2})$ are homomorphisms $\theta\co\Gamma\to{C_2}=\{\pm 1\}$ that act on representations by multiplication. The action of $\epsilon \in H^1( \Gamma,C_2)$ on $I_{\gamma}$ is given by: $ \epsilon \cdot I_{\gamma} = \epsilon
(\gamma) I_{\gamma}$. Since $ H^1( \Gamma,{C_2})$ is finite, it is reductive and we may take the quotient of invariant theory.
Let $F$ be a finitely generated free group and $\psi\co F\to\Gamma$ be a surjection. We fix a $k$-tuple $\gamma_{1},\ldots,\gamma_{k}\in
\Gamma$ such that the product $\gamma_{1}\cdots\gamma_{k}\in\Gamma^{2}$. Moreover, we choose $\eta_{i}\in F$ such that $\psi(\eta_{i})=\gamma_{i}$ and such that the product $\eta_{1}\cdots\eta_{k}\in F^{2}$. The function $\psi^{*}\sigma_{\eta_1,\ldots,\eta_k}\in \C[\overline R(\Gamma)]$ is invariant under conjugation and gives us a function $\psi^{*}\sigma_{\eta_1,\ldots,\eta_k}\in\C[\overline X(\Gamma)]$. By Equation (\[equ:sigma\]) we have $\psi^{*}\sigma_{\eta_1,\ldots,\eta_k} (\chi) =
\tilde\chi(\gamma_{1})\cdots\tilde\chi(\gamma_{k})$ where $\tilde\chi\in X(\Gamma,{\mathrm{SL}_2 (\C)})$ is a character such that $\pi(\tilde\chi)= \chi$. Note that $\pi\co X(\Gamma,{\mathrm{SL}_2 (\C)})\to
\overline X(\Gamma)$ is surjective. The function $$\label{eqn:Sigma}
\Sigma_{\gamma_1,\ldots,\gamma_k} :=
\phi^{*}\sigma_{\eta_1,\ldots,\eta_k} \in\C[\overline X(\Gamma)]$$ depends only on the elements $\gamma_{i}\in\Gamma$.
\[prop:naturaliso\] There is a natural isomorphism: $$X( \Gamma,{\mathrm{SL}_2 (\C)})/\! /H^1( \Gamma,C_2)\cong \overline X(\Gamma).$$
Composition with the projection ${\mathrm{SL}_2 (\C)}\to{\mathrm{PSL}_2 (\C)}$ induces a surjection $$\pi\co X(\Gamma,{\mathrm{SL}_2 (\C)}) \to \overline X(\Gamma),$$ which is easily seen to be algebraic and is given by $\pi(\chi) = \chi^{2}$. At the level of function rings it induces an injection $$\pi^*\co \C[\overline X(\Gamma)]\hookrightarrow \C[X(\Gamma,{\mathrm{SL}_2 (\C)})].$$ We have $\pi^{*}f (\chi)= f(\chi^{2})$ for $f\in\C[\overline X(\Gamma)]$ and $\chi\in X(\Gamma,{\mathrm{SL}_2 (\C)})$. The image of $\pi^*$ is contained in the set of invariant functions: $${\operatorname{Im}}\pi^{*} \subseteq
\C[X(\Gamma,{\mathrm{SL}_2 (\C)})]^{H^1(\Gamma,C_2)}.$$ More precisely, we have $\pi^{*}f (\epsilon\chi)=
f(\epsilon^{2}\chi^{2}) = \pi^{*}f (\chi)$ for all $\epsilon\in H^{1}(\Gamma,C_{2})$. It remains to prove that this inclusion is an equality.
Since $ \C[X(\Gamma,{\mathrm{SL}_2 (\C)})]$ is generated as $\C$-algebra by the functions $I_{\gamma}$ with $\gamma\in\Gamma$, the monomials $$I_{\gamma_1} I_{\gamma_2}\cdots I_{\gamma_k}$$ generate $\C[X(\Gamma,{\mathrm{SL}_2 (\C)})]$ as a $\C$-vector space. Taking the average of the action of $H^1(\Gamma,C_2)$, we deduce that the subspace of invariant functions $\C[X(\Gamma,{\mathrm{SL}_2 (\C)})]^{H^1(\Gamma,C_2)}$ is generated by $$\frac 1 {2^r} \sum_{\epsilon\in H^1(\Gamma,C_2)}
\epsilon \cdot I_{\gamma_{1}}\cdots
I_{\gamma_{k}} =
\big(\frac 1 {2^r}\sum_{\epsilon\in H^1(\Gamma,C_2)}
\epsilon (\gamma_{1}\cdots\gamma_{k}) \big)
I_{\gamma_{1}}\cdots
I_{\gamma_{k}}$$ where $r$ is the rank of $H^{1}(\Gamma,C_2)$ (see [@Kraft II.3.6] for instance). Using the fact that $$\frac 1 {2^r}\sum_{\epsilon\in H^1(\Gamma,C_2)}
\epsilon(\gamma) =
\begin{cases}
1 & \text{ if } \gamma\in\Gamma^{2}\\
0 & \text{ otherwise }
\end{cases}$$ we deduce that $\C[X(\Gamma,{\mathrm{SL}_2 (\C)})]^{H^1(\Gamma,C_2)}$ is generated by the monomials $
I_{\gamma_1}\cdots I_{\gamma_k}
$ such that the product $\gamma_1\ldots\gamma_k\in\Gamma^2$.
On the other hand we have for $\chi\in X(\Gamma,{\mathrm{SL}_2 (\C)})$: $$\pi^{*}\Sigma_{\gamma_{1},\ldots,\gamma_{k}} (\chi)
= \Sigma_{\gamma_{1},\ldots,\gamma_{k}}(\chi^{2}) =
\chi(\gamma_{1})\cdots\chi(\gamma_{k}) = I_{\gamma_{1}}\cdots
I_{\gamma_{k}} (\chi)\, ,$$ where $\Sigma_{\gamma_{1},\ldots,\gamma_{k}}$ is the function defined in (\[eqn:Sigma\]). This gives that the monomials $I_{\gamma_1}\cdots I_{\gamma_k}$ such that the product $\gamma_1\ldots\gamma_k\in\Gamma^2$ is in the image of $\pi^{*}$ and therefore $\C[X(\Gamma,{\mathrm{SL}_2 (\C)})]^{H^1(\Gamma,C_2)}={\operatorname{Im}}\pi^{*}$.
Let $p\co X(\Gamma,{\mathrm{SL}_2 (\C)})\to\overline X(\Gamma)$ denote the projection. If $\chi\in \overline X(\Gamma)$ is ${\operatorname{Ad}}$-irreducible, then $p^{-1}(\chi)$ has $2^{r}$ points where $r$ is the rank of $H^{1}(\Gamma,C_{2})$. If $\chi$ is ${\operatorname{Ad}}$-reducible then the cardinality of $p^{-1}(\chi)$ is strictly less than $2^{r}$. Thus $p$ is a branched covering with branching locus the set of ${\operatorname{Ad}}$-reducible characters.
\[ex:F2\] Let $F_2$ be the free group of rank 2, with generators $\alpha$ and $\beta$. There is an isomorphism: $$(I_{\alpha},I_{\beta},I_{\alpha\beta})\co X(F_2,{\mathrm{SL}_2 (\C)})\to\C^3$$ where $I_{\gamma}$ denotes the regular function induced by $\tilde\tau_{\gamma}$. In particular $X(F_2,{\mathrm{SL}_2 (\C)})$ is smooth.
Since every representation in $R(F_2)$ lifts to ${\mathrm{SL}_2 (\C)}$, we deduce $$X(F_2)=X(F_2,{\mathrm{SL}_2 (\C)})/\!/{H^{1}(F_2,C_{2})}.$$ The group $H^{1}(F_2,C_2)\cong ({C_2})^2$ has four elements, and its action on $X(F_2,{\mathrm{SL}_2 (\C)})$ is generated by the involutions $$\begin{array}{rcl}
(I_{\alpha},I_{\beta},I_{{\alpha}{\beta}})&\mapsto&(-I_{\alpha},I_{\beta},-I_{{\alpha}{\beta}})
\\
(I_{\alpha},I_{\beta},I_{{\alpha}{\beta}})&\mapsto&(I_{\alpha},-I_{\beta},-I_{{\alpha}{\beta}}).
\end{array}$$ Thus $\C[X(F_2),{\mathrm{SL}_2 (\C)}]^{H^{1}(F_2,C_{2})}$ is generated by $X=I_{\alpha}^2$, $Y=I_{\beta}^2$, $Z=I_{{\alpha}{\beta}}^2$ and $W=I_{\alpha} I_{\beta}
I_{{\alpha}{\beta}}$. Hence $$\label{eqn:XF2}
X(F_2)\cong\{(X,Y,Z,W)\in\C^4\mid W^2=XYZ\}$$ The relationship with Corollary \[cor:embedding\] is given by the change of coordinates (cf. Equality (\[eqn:sigmaABAB\])) $$\left\{
\begin{array}{l}
J_{\alpha}=X\\
J_{\alpha}=Y\\
J_{{\alpha}{\beta}}=Z\\
J_{{\alpha}{\beta}^{-1}}=XY+Z-2W.
\end{array}
\right.$$
From Equality (\[eqn:XF2\]) we remark that the singular set of $X(F_2 )$ consists of those points such that two of $\{ X,Y,Z\}$ vanish. This is the same as the set of characters of representations generated by two rotations of angle $\pi$. This is also the set of ${\operatorname{Ad}}$-reducible but non-reducible representations.
If $M$ is a knot exterior in $S^3$, then $H_2(\pi_1M)\cong H_2(M)\cong
0$ and therefore $X(M)=\overline X(M)$. When in addition $M$ is a 2-bridge knot exterior, explicit methods of how to compute $X(M)$ are given in [@HLM1] and [@HLM2], where $X(M)$ for this particular case was already defined as $X(M,{\mathrm{SL}_2 (\C)})/\!/C_2$. The explicit computation for the figure eight knot exterior is found in [@GM], for instance.
Representations that do not lift
--------------------------------
The manifold $M$ is a bundle over $S^1$ with fiber $\dot{T}^2$ a torus minus a disk. Up to homeomorphism, $M$ is described by the action of the monodromy on $H_1(\dot{T}^2,\mathbb Z)$, which is given by the matrix $$\left(\begin{matrix} 1 & m_2 \\ m_1 & 1+m_1 m_2
\end{matrix}
\right)$$ with $m_i\in 2\mathbb Z$, $m_i>0$. We shall show that $X(M)-\overline X(M)$ has arbitrarily many components by choosing $m_i$ sufficiently large.
To have a presentation of $\pi_1M$, we use an automorphism $f$ of $\pi_1\dot T^2$ induced by the monodromy. Since $\pi_1\dot T^2$ is the free group of rank $2$ generated by $\alpha$ and $\beta$, $$\pi_1M=\langle \alpha,\beta,\mu\mid \mu\alpha\mu^{-1}=f(\alpha),
\mu\beta\mu^{-1}=f(\beta)\rangle$$ We choose $f$ such that: $$\left\{
\begin{array}{rcl}
\mu\alpha\mu^{-1}&=&\alpha\beta^{m_2}\\
\mu\beta\mu^{-1}&=&\beta(\alpha\beta^{m_2})^{m_1}
\end{array}
\right.$$ We choose odd numbers $p_1,p_2\in 2\mathbb Z+1$, with $1\leq
p_i\leq m_i/2$ and an arbitrary complex number $z\in \C$. By Example \[ex:F2\], there exist matrices $A_{z},B_{z}\in {\mathrm{SL}_2 (\C)}$ with $${\operatorname{tr}}(A_{z})=2\cos\frac{\pi p_1}{m_1},\
{\operatorname{tr}}(B_{z})=2\cos\frac{\pi p_2}{m_2} \text{ and }
{\operatorname{tr}}(A_{z} B_{z})=z\,.$$ Those trace equalities imply that $A_{z}^{m_1}=B_{z}^{m_2}=-{\mathrm{Id} }$. In particular $$\begin{array}{rcl}
A_{z} B_{z}^{m_2} &=& -A_{z},\\
B_{z} (A_{z} B_{z}^{m_2})^{m_1}&=&-B_{z}.
\end{array}$$ Let $\rho_{z}\in R(\Gamma)$ be the representation that $\rho_{z}(\alpha)=\pm A_{z} $, $\rho_z(\beta)=\pm B_{z}$ and $\rho_z(\mu)=\pm {\mathrm{Id} }$. Since $m_1$ and $m_2$ are even, this representation does not lift to ${\mathrm{SL}_2 (\C)}$. In addition, for each value of $p_1$ and $p_2$ we have defined a one parameter family of characters, with parameter $z={\operatorname{tr}}(A_{z}B_{z})\in\C$. By [@CCGLS Proposition 2.4] the dimension of each component of $X(M)$ is at most one, hence different values of $p_1$ and $p_2$ give different components.
The singular set of $X(F_n)$
============================
In this section we compute the singular set of $X(F_n)$, but before we need two preliminary subsections: in Subsection \[ss:ZariskiTS\] we recall some basic facts about the Zariski tangent space and Luna’s slice theorem, and in Subsection \[ss:cohomology\] we compute the cohomology of free groups with twisted coefficients.
The Zariski tangent space {#ss:ZariskiTS}
-------------------------
Given a representation $\rho\in R(\Gamma)$, we define the space of cocycles $$Z^1(\Gamma,{\operatorname{Ad}}\circ\rho) = \left\{
\theta\co\Gamma\to {\mathfrak{sl}_2 (\C)}\ \left\vert
\begin{array}{c}
\theta(\gamma_1\gamma_2)=\theta(\gamma_1)+{\operatorname{Ad}}_{\rho(\gamma_1)}
(\theta(\gamma_2)),
\\
\forall \gamma_1,\gamma_2\in\Gamma
\end{array}
\right.
\right\}\, .$$ Given a smooth path of representations $\rho_t$, with $t$ in a neighborhood of the origin, one can construct a cocycle as follows: $$\begin{array}{rcl}
\Gamma &\to &{\mathfrak{sl}_2 (\C)}\\
\gamma&\mapsto&
\frac{d\phantom{t}}{dt}\rho_t(\gamma)\rho_0(\gamma)^
{-1}\vert_{t=0}
\end{array} \,.$$ This construction defines an isomorphism, due to Weil [@Weil]:
\[thm:Weil\] The previous construction defines an isomorphism $${T^{\mathrm{Zar}}_{\rho}(R(\Gamma))}\cong Z^1(\Gamma,{\operatorname{Ad}}\circ\rho).$$
Here ${T^{\mathrm{Zar}}_{\rho}(R(\Gamma))}$ denotes the Zariski tangent space in the scheme sense (i.e. the defining ideals are not necessary reduced).
We also consider the space of coboundaries $$B^1(\Gamma,{\operatorname{Ad}}\circ\rho) = \left\{
\theta\co\Gamma\to{\ensuremath{R} }^2\ \left\vert
\begin{array}{c}
\textrm{ there exists } a\in {\mathfrak{sl}_2 (\C)}\textrm{ such that }\\
\theta(\gamma)={\operatorname{Ad}}_{\rho(\gamma)} (a)- a,\,
\forall \gamma\in\Gamma
\end{array}
\right.
\right\}\,.$$
The isomorphism of Theorem \[thm:Weil\] identifies the subspace of the Zariski tangent space corresponding to the orbits by conjugation with $
B^1(\Gamma,{\operatorname{Ad}}\circ\rho)$. So it seems natural that in some cases ${T^{\mathrm{Zar}}_{\chi}(X(\Gamma))}$ is isomorphic to the cohomology group $$H^1(\Gamma,{\operatorname{Ad}}\circ\rho)=
Z^1(\Gamma,{\operatorname{Ad}}\circ\rho)/B^1(\Gamma,{\operatorname{Ad}}\circ\rho)$$ as we will show next.
The stabilizer of a representation $\rho\in R(\Gamma)$ is denoted by $$Stab_{\rho}=\{A\in {\mathrm{PSL}_2 (\C)}\mid A\rho A^{-1}=\rho\}\,.$$ In particular, for and ${\operatorname{Ad}}$-irreducible representation $Stab_{\rho}$ is trivial.
\[prop:TZ\] If $\rho$ is a smooth point of $R(\Gamma)$ with closed orbit, then $${T^{\mathrm{Zar}}_{\chi_{\rho}}(X(\Gamma))}\cong
{T^{\mathrm{Zar}}_{0}(H^1(\Gamma,{\operatorname{Ad}}\circ\rho)/\!/ Stab_{\rho})}\,.$$
We use the slice theorem of Luna: there exists an algebraic subvariety $S\subset R(\Gamma)$ that contains $\rho$ and that is $ Stab_{\rho}$-invariant, such that $$\label{eqn:complement}
Z^1(\Gamma,{\operatorname{Ad}}\circ\rho)=B^1(\Gamma,{\operatorname{Ad}}\circ\rho)\oplus
{T^{\mathrm{Zar}}_{\rho}(S)}$$ and the map induced by the projection $$S/\!/Stab_{\rho}\to X(\Gamma)$$ is an étale isomorphism (in particular their tangent spaces are isomorphic). Since we assume that $\rho$ is a smooth point, Luna’s theorem shows that $S/\!/Stab_{\rho}$ and ${T^{\mathrm{Zar}}_{\rho}(S)}/\!/Stab_{\rho}$ are étale equivalent (see [@KSS p. 97 ]). Since ${T^{\mathrm{Zar}}_{\rho}(S)}$ and $H^1(\Gamma,{\operatorname{Ad}}\circ\rho)$ are isomorphic as $Stab_{\rho}$-modules (by Equation (\[eqn:complement\])), the proposition follows.
Cohomology of Free groups {#ss:cohomology}
-------------------------
We start with irreducible characters:
\[lem:cohomologyirreducible\] Let $\chi_{\rho}\in X(F_n)$ be an irreducible character. Then $$\dim H^1(F_n,{\operatorname{Ad}}\circ\rho)=3n-3.$$
Notice first that $Z^1(F_{n},{\operatorname{Ad}}\circ\rho)\cong
{\mathfrak{sl}_2 (\C)}^n\cong \C^{3n}$. Irreducibility implies that $\dim
B^1(F_{n},{\operatorname{Ad}}\circ\rho)=3$, which is maximal (even if ${\operatorname{Ad}}$-reducible representations have invariant subspaces, irreducibility implies that the eigenvalues are different from $1$).
We are interested in computing $H^1(F_n,{\operatorname{Ad}}\circ\rho)$ as a $Stab_{\rho}$-module. If $ \rho$ is ${\operatorname{Ad}}$-irreducible, then $Stab_{\rho}$ is trivial, and therefore $H^1(F_n,{\operatorname{Ad}}\circ\rho)$ is the trivial module $\C^{3n-3}$. In the reducible and ${\operatorname{Ad}}$-reducible cases we need further computations.
. Let $\chi\in X(F_n)$ be a non trivial reducible character. There exists a representation $\rho\in R(F_n)$ with character $\chi$ such that $\rho$ consists of diagonal matrices, constructed in Remark \[rem:rhodiagonal\].
We decompose the Lie algebra ${\mathfrak{sl}_2 (\C)}=h_0\oplus h_-\oplus h_+$, where $h_0$, $h_+$ and $h_-$ are the one dimensional $\C$-vector spaces generated respectively by $ \left(\!\begin{smallmatrix} 1&0\\
0&-1
\end{smallmatrix}\!\right)$, $ \left(\!\begin{smallmatrix} 0&1\\
0&0
\end{smallmatrix}\!\right)$ and $ \left(\!\begin{smallmatrix} 0&0\\
1&0
\end{smallmatrix}\!\right)$.
\[lem:splitdiagonal\] If $\rho$ is diagonal then ${\operatorname{Ad}}\circ\rho$ preserves the splitting ${\mathfrak{sl}_2 (\C)}=h_0\oplus h_-\oplus
h_+$. If in addition $\rho$ is non-trivial, then $Stab_{\rho}$ preserves the splitting ${\mathfrak{sl}_2 (\C)}=h_0\oplus (h_-\oplus h_+)$ (some elements may permute $h_+$ and $h_-$).
The first assertion is clear, because diagonal matrices preserve each factor $h_0$ and $h_{\pm}$.
When the image of $\rho$ has order $\geq 3$, the group $Stab_{\rho}$ is precisely the set of diagonal matrices. When the image has order precisely $2$, then $Stab_{\rho}$ is the group of diagonal and anti-diagonal ones $\left(\!\begin{smallmatrix} 0&*\\ *&0
\end{smallmatrix}\!
\right)$. Antidiagonal matrices preserve $h_0$ and permute $h_-$ with $h_+$, hence the second assertion is proved.
\[lem:cohmologydiagonal\] Let $\rho\in R(F_n)$ be a non-trivial diagonal representation, then $H^1(F_n,{\operatorname{Ad}}\circ\rho)\cong h_0^n\oplus (h_+\oplus h_-)^{n-1} $ as $Stab_{\rho}$-modules.
By construction, $Z^1(F_n,{\operatorname{Ad}}\circ\rho) \cong {\mathfrak{sl}_2 (\C)}^n$. We have the splitting $$H^1(F_n,{\operatorname{Ad}}\circ\rho)\cong H^1(F_n,h_0)\oplus
H^1(F_n,h_+)\oplus H^1(F_n,h_-).$$ A diagonal matrix $\pm \left(\!\begin{smallmatrix} a&0 \\ 0&a^{-1}
\end{smallmatrix}\!\right)$ acts trivially on $h_0$ and by multiplication by a factor $a^{\pm 2}$ on $h_{\pm}$. Therefore $B^1(F_n,h_0)\cong 0$ and $B^1(F_n,h_{\pm})\cong h_{\pm}$, and the lemma follows.
. Let $\rho\in R(\Gamma)$ be irreducible but ${\operatorname{Ad}}$-reducible. Up to conjugation the image of $\rho$ is contained in the group of diagonal and anti-diagonal matrices. There are two possibilities for the stabilizer $Stab_{\rho}$. If the image of $\rho$ has more than four elements, then $Stab_{\rho}$ has two elements: the identity and $\pm\left(\!\begin{smallmatrix} i&0\\ 0&-i
\end{smallmatrix}\!\right)$. Otherwise the image of $\rho$ is Klein’s 4-group (i.e. the group generated by $\pm\left(\!\begin{smallmatrix} i&0\\ 0&-i
\end{smallmatrix}\!\right)$ and $\pm\left(\!\begin{smallmatrix} 0&1\\
-1&0
\end{smallmatrix}\!\right)$ ). In this case $Stab_{\rho}$ equals the image of $\rho$. With the same argument as in Lemma \[lem:splitdiagonal\], one can prove:
Let $\rho$ be as above. Then both ${\operatorname{Ad}}\circ\rho$ and $Stab_{\rho}$ preserve the splitting ${\mathfrak{sl}_2 (\C)}=h_0\oplus
(h_+\oplus h_-)$.
\[lem:cohomologyantidiagonal\] Let $\rho\in R(F_n)$ be an irreducible but ${\operatorname{Ad}}$-reducible representation, then $H^1(F_n,{\operatorname{Ad}}\circ\rho)\cong {\mathfrak{sl}_2 (\C)}^{n-1}$ as $Stab_{\rho}$-modules.
Again $Z^1(F_n,{\operatorname{Ad}}\circ\rho) \cong {\mathfrak{sl}_2 (\C)}^n$, and we have the decomposition $$H^1(F_n,{\operatorname{Ad}}\circ\rho)\cong
H^1(F_n,h_0)\oplus H^1(F_n,h_+\oplus h_-).$$ The group $B^1(F_n,h_0)$ has dimension one, because the antidiagonal matrices act on $h_0$ by change of sign. In addition, $\dim(B^1(F_n,h_+\oplus h_-))=2$ is also maximal, because this is the case when we restrict it to diagonal representations (see the proof of Lemma \[lem:cohmologydiagonal\]).
Singular locus for free groups
------------------------------
We saw above that $X(F_2,{\mathrm{SL}_2 (\C)})\cong\C^3$ is smooth. We also showed that the singular points of $X(F_2 )$ are ${\operatorname{Ad}}$-reducible but irreducible characters.
For $n\geq 3$ the singular set of $X(F_n )$ is precisely the set of ${\operatorname{Ad}}$-reducible characters.
Since $R(F_n)\cong {\mathrm{PSL}_2 (\C)}^{n}$, $X(F_n)$ is irreducible and of dimension $3 n-3$. Thus $\chi\in X(F_n)$ is singular if and only if $$\dim T^{Zar}_{\chi} X(F_n)> 3 n -3.$$ This dimension is computed by means of Proposition \[prop:TZ\]: if the orbit of $\rho\in t^{-1}(\chi)$ is closed then $$\dim T^{Zar}_{\chi} X(F_n)=\dim
T_0^{Zar}(H^1(F_n,{\operatorname{Ad}}\circ\rho)/\!/Stab_{\rho}).$$ If $\rho\in R(F_n)$ is irreducible, by Lemma \[lem:cohomologyirreducible\] $\dim H^1(F_n,{\operatorname{Ad}}\circ\rho)=3n-3$. If in addition $\rho$ is ${\operatorname{Ad}}$-irreducible, then $Stab_{\rho}$ is trivial and therefore $\chi_{\rho}$ is smooth.
If $\rho$ is irreducible but ${\operatorname{Ad}}$-reducible, then $H^1(F_n,{\operatorname{Ad}}\circ\rho)\cong {\mathfrak{sl}_2 (\C)}^{n-1} $ as $Stab_{\rho}$ modules, by Lemma \[lem:cohomologyantidiagonal\]. We may assume that the image of $\rho$ has more than 4 elements, because the adherence set of such characters is the whole set of irreducible but ${\operatorname{Ad}}$-reducible characters, and the singular set is closed. Hence $Stab_{\rho}$ is the group generated by the involution $\pm
\left(\!\begin{smallmatrix} i&0\\ 0&-i
\end{smallmatrix}\!
\right)$, that acts trivially on $h_0$ but as a change of sign on $h_+\oplus h_-$. Thus the action of $Stab_{\rho}$ on $H^1(F_n,{\operatorname{Ad}}\circ\rho)$ is equivalent to the involution on $\C^{3n-3}$ that fixes $(n-1)$ coordinates and changes the sign of the remaining $(2n-2)$ coordinates. The quotient of $\C^{3n-3}$ by this involution is not smooth, hence $\dim
T_0^{Zar}(H^1(F_n,{\operatorname{Ad}}\circ\rho)/\!/Stab_{\rho})>3n-3$.
When $\chi_{\rho}$ is reducible but non trivial, we may assume that $\rho$ is diagonal and its image has more that three elements (again the adherence set of those characters is the whole set of reducible ones). Thus $Stab_{\rho}$ is the group of diagonal matrices, and by Lemma \[lem:cohmologydiagonal\], $H^1(F_n,{\operatorname{Ad}}\circ\rho)\cong h_0^n\oplus (h_+\oplus h_-)^{n-1}$ as $Stab_{\rho}$-module. We have an isomorphism $Stab_{\rho}\cong
\C^*$ and $t\in \C^*$ acts on $h_{0}$ trivially and on $h_{\pm}$ by multiplication by $t^{\pm 1}$. An elementary computation shows that $(h_+\oplus
h_-)^{n-1}/\!/\C^*$ has dimension $2n-3$ and it is not smooth for $n>2$.
A similar argument yields that for $n\geq 3$ the singular part of $X(F_n,{\mathrm{SL}_2 (\C)})$ is precisely the set of reducible characters.
**Acknowledgment** The second author was partially supported by MCYT through grant BFM2000-0007.
<span style="font-variant:small-caps;">Laboratoire de Mathématiques Pures, Université Blaise Pascal, F-63177, Aubière Cedex, France</span>, heusener@math.univ-bpclermont.fr.
<span style="font-variant:small-caps;">Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain</span>, porti@mat.uab.es.
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| {
"pile_set_name": "ArXiv"
} |
=1
Introduction
============
The aim of this article is to study rational parallelisms of algebraic varieties by means of the transcendence of their symmetries. Our original motivation was to understand the possible obstructions to the third Lie theorem for algebraic Lie pseudogroups. This article is concerned with the simply transitive case. These obstructions should appear in the Galois group of certain connection associated to a Lie algebroid. However, we have written the article in the language of regular and rational parallelisms of algebraic varieties and their symmetries.
A theorem of P. Deligne says that any Lie algebra can be realized as a parallelism of an algebraic variety. This is a sort of algebraic version of the third Lie theorem. Notwithstanding, there is one main problem: given an algebraic variety with a parallelism, how far is it from being an algebraic group? Is it possible to conjugate this parallelism with the canonical parallelism of invariant vector fields on an algebraic group?
In the analytic context, from the Darboux–Cartan theorem [@sharpe p. 212], a $\mathfrak{g}$-parallelized complex manifold $M$ has a natural $(G,G)$ structure where $G$ is a Lie group with $\mathfrak{lie}(G) = \mathfrak{g}$. The obstruction to be a covering of $G$, as manifold with a $(G,G)$ structure, is contained in a monodromy group [@sharpe p. 130]. In [@Wang], Wang proved that parallelized compact complex manifolds are biholomorphic to quotients of complex Lie groups by discrete cocompact subgroups. This result has been extended by Winkelmann in [@Winkelmann1; @Winkelmann2] for some open complex manifolds.
In this article we address the problem of classification of rational parallelisms on algebraic varieties up to birational transformations. Such a classification seems impossible in the algebraic category but we prove a criterion to ensure that a parallelized algebraic variety is isogenous to an algebraic group. Summarizing, we pursue the following plan: We regard infinitesimal symmetries of a rational parallelism as horizontal sections of a connection that we call the reciprocal Lie connection. This connection has a Galois group which is represented as a group of internal automorphisms of a Lie algebra. The obstruction to the algebraic conjugation to an algebraic group, under some assumptions, appear in the Lie algebra of this Galois group.
In Section \[section\_parallelisms\] we introduce the basic definitions; several examples of parallelisms are given here. In Section \[section\_lie\] we study the properties of connections on the tangent bundle whose local analytic horizontal sections form a sheaf of Lie algebras of vector fields. We call them [*Lie connections*]{}. They always come by pairs, and they are characterized by having vanishing curvature and constant torsion (Proposition \[prop\_Lie\_char\]). We see that each rational parallelism has an attached pair of Lie connections, one of them with trivial Galois group. We compute the Galois groups of some parallelisms given in examples (Proposition \[prop\_example\]), and prove that any algebraic subgroup of ${\rm PSL}_2(\mathbf C)$ appears as the differential Galois group of a $\mathfrak{sl}_2(\mathbf C)$-parallelism (Theorem \[thm\_SL2\]). Section \[section\_DC\] is devoted to the construction of the isogeny between a $\mathfrak g$-parallelized variety and an algebraic group $G$ whose Lie algebra is $\mathfrak g$. In order to do this, we introduce the Darboux–Cartan connection, a $G$-connection whose horizontal sections are parallelism conjugations. We prove that if $\mathfrak g$ is centerless then the Darboux–Cartan connection and the reciprocal Lie connection have isogenous Galois groups. We prove that the only centerless Lie algebras $\mathfrak{g}$ such that there exists a $\mathfrak{g}$-parallelism with a trivial Galois group are algebraic Lie algebras, i.e., Lie algebras of algebraic groups. In particular this allows us to give a criterion for a parallelized variety to be isogenous to an algebraic group. The vanishing of the Lie algebra of the Galois group of the reciprocal connection is a necessary and sufficient condition for a parallelized variety to be isogenous to an algebraic group:
Let $\mathfrak g$ be a centerless Lie algebra. An algebraic variety $(M,\omega)$ with a rational parallelism of type $\mathfrak g$ is isogenous to an algebraic group if and only if $\mathfrak{gal}(\nabla^{\rm rec}) = \{0\}$.
The notion of [*isogeny*]{} can be extended beyond the simply-transitive case. Let us consider a complex Lie algebra $\mathfrak g$. An [*infinitesimally homogeneous variety*]{} of type $\mathfrak g$ is a pair $(M,\mathfrak s)$ consisting of a complex smooth irreducible variety $M$ and a finite-dimensional Lie algebra isomorphic to $\mathfrak g$ that spans the tangent bundle of $M$ on the generic point.
We are interested in conjugation by rational or by algebraic maps, so that, whenever necessary, we replace $M$ by a suitable Zariski open subset. In this context, we say that a dominant rational map $f\colon M_1 \dasharrow M_2$ between varieties of the same dimension conjugates the infinitesimally homogeneous varieties $(M_1,\mathfrak s_1)$ and $(M_2,\mathfrak s_2)$ if $f^*(\mathfrak s_2) = \mathfrak s_1$. We say that $(M_1,\mathfrak s_1)$ and $(M_2,\mathfrak s_2)$ are [*isogenous*]{} if they are conjugated to the same infinitesimally homogeneous space of type $\mathfrak g$.
Under some hypothesis on the Lie algebra $\mathfrak s\subset \mathfrak X(M)$ one can prove that $(M,\mathfrak s)$ is isogenous to a homogeneous space $(G/H,\mathfrak{lie}(G)^{\rm rec})$ with the action of right invariant vector fields. These hypothesis are satisfied by transitive actions of $\mathfrak{sl}_{n+1}(\mathbf C)$ on $n$-dimensional varieties. As a particular case of Theorem \[homogeneous\] one has
Let $(M,\mathfrak s)$ be an infinitesimally homogeneous variety of complex dimension $n$ such that $\mathfrak s$ is isomorphic to $\mathfrak{sl}_{n+1}(\mathbf C)$. Then there exists a dominant rational map $M \dasharrow \mathbf{CP}_n$ conjugating $\mathfrak s$ with the Lie algebra $\mathfrak{sl}_{n+1}(\mathbf C)$ of projective vector fields in $\mathbf{CP}_n$.
Appendix \[ApA\] is devoted to a geometrical presentation of Picard–Vessiot theory for linear and principal connections. Finally, Appendix \[apB\] contains a detailed proof of Deligne’s theorem of the realization of a regular parallelism modeled over any finite-dimensional Lie algebra. This includes also a computation of the Galois group that turns out to be, for this particular construction, an algebraic torus.
Parallelisms {#section_parallelisms}
============
Let $M$ be a smooth connected affine variety over $\mathbf C$ of dimension $r$. We denote by $\mathbf C[M]$ its ring of regular functions and by $\mathbf C(M)$ its field of rational functions. Analogously, we denote by $\mathfrak X[M]$ and $\mathfrak X(M)$ respectively the Lie algebras of regular and rational vector fields in $M$, and so on.
Let $\mathfrak g$ be a Lie algebra of dimension $r$. We fix a basis $A_1,\ldots,A_r$ of $\mathfrak g$, and the following notation for the associated structure constants $[A_i,A_j] = \sum_{k}\lambda_{ij}^kA_k$.
A parallelism of type $\mathfrak g$ of $M$ is a realization of the Lie algebra $\mathfrak g$ as a Lie algebra of pointwise linearly independent vector fields in $M$. More precisely:
A regular parallelism of type $\mathfrak g$ in $M$ is a Lie algebra morphism, $\rho\colon \mathfrak g \to \mathfrak X[M]$ such that $\rho A_1(x), \ldots, \rho A_r(x)$ form a basis of $T_xM$ for any point $x$ of $M$.
\[ex:AGP\]Let $G$ be an algebraic group and $\mathfrak g$ be its Lie algebra of left invariant vector fields. Then the natural inclusion $\mathfrak g\subset \mathfrak X[G]$ is a regular parallelism of $G$. The Lie algebra $\mathfrak g^{\rm rec}$ of right invariant vector fields is another regular parallelism of the same type. Let invariant and right invariant vector fields commute, hence, an algebraic group is naturally endowed with a pair of commuting parallelisms of the same type.
From Example \[ex:AGP\], it is clear that any *algebraic* Lie algebra is realized as a parallelism of some algebraic variety. On the other hand, Theorem \[TDeligne\] due to P. Deligne and published in [@Malgrange], ensures that any Lie algebra is realized as a regular parallelism of an algebraic variety. Analogously, we have the definitions of rational and local analytic parallelism. Note that a rational parallelism in $M$ is a regular parallelism in a Zariski open subset $M^\star \subseteq M$.
There is dual definition, equivalent to that of parallelism. This is more suitable for calculations.
A regular parallelism form (or coparallelism) of type $\mathfrak g$ in $M$ is a $\mathfrak g$-valued $1$-form $\omega\in\Omega^1[M]\otimes_{\mathbb C} \mathfrak g$ such that:
- For any $x\in M$, $\omega_x\colon T_x M \to \mathfrak g$ is a linear isomorphism.
- If $A$ and $B$ are in $\mathfrak g$ and $X$, $Y$ are vector fields such that $\omega(X) = A$ and $\omega (Y) = B$ then $\omega[X,Y] = [A,B]$.
Analogously, we define local analytic and rational coparallelism of type $\mathfrak g$ in $M$. It is clear that each coparallelism induces a parallelism, and reciprocally, by the relation $\omega (\rho (A)) = A$. Thus, there is a natural equivalence between the notions of parallelism and coparallelism. From now on we fix $\rho$ and $\omega$ equivalent parallelism and coparallelism of type $\mathfrak g$ on $M$.
The Lie algebra structure of $\mathfrak g$ forces $\omega$ to satisfy Maurer–Cartan structure equations $$\begin{gathered}
{\rm d}\omega + \frac{1}{2}[\omega,\omega] = 0.\end{gathered}$$ Taking components $\omega = \sum_{i}\omega_iA_i$ we have $$\begin{gathered}
{\rm d}\omega_i +\sum_{j,k=1}^r \frac{1}{2}\lambda_{jk}^i\omega_j\wedge \omega_k = 0.\end{gathered}$$
\[ex:group\] Let $G$ be an algebraic group and $\mathfrak g$ be the Lie algebra of left invariant vector fields in $G$. Then the structure form $\omega$ is the coparallelism corresponding to the parallelism of Example \[ex:AGP\].
\[ex:B\]Let $\mathfrak g = \langle A_1,A_2\rangle$ be the 2-dimensional Lie algebra with commutation relation $$\begin{gathered}
= A_1.\end{gathered}$$ The vector fields $$\begin{gathered}
X_1 = \frac{\partial}{\partial x},\qquad X_2 = x\frac{\partial}{\partial x}+ \frac{\partial}{\partial y},\end{gathered}$$ define a regular parallelism via $\rho (A_i)= X_i$ of $\mathbf C^2$. The associated parallelism form is $$\begin{gathered}
\omega = A_1{\rm d}x + (A_2 - x A_1){\rm d}y.\end{gathered}$$
\[ex:MD\] Let $\mathfrak g = \langle A_1,A_2,A_3 \rangle$ be the 3-dimensional Lie algebra with commutation relations $$\begin{gathered}
= \alpha A_2,\qquad [A_1,A_3]=\beta A_3, \qquad [A_2,A_3] = 0,\end{gathered}$$ with $\alpha$, $\beta$, non zero complex numbers. In particular, if $\alpha/\beta$ is not rational then $\mathfrak g$ is not the Lie algebra of an algebraic group. The vector fields $$\begin{gathered}
X_1 = \frac{\partial}{\partial x} + \alpha y \frac{\partial}{\partial y} + \beta z\frac{\partial}{\partial z},\qquad
X_2 = \frac{\partial}{\partial y},\qquad X_3 = \frac{\partial}{\partial z},\end{gathered}$$ define a regular parallelism via $\rho (A_i)= X_i$ of $\mathbf C^3$. The associated parallelism form is $$\begin{gathered}
\omega = (A_1 - A_2\alpha y - A_3\beta z){\rm d}x + A_2{\rm d}y + A_3{\rm d}z.\end{gathered}$$
\[isogenous\] Let $(M,\omega)$ and $(N,\theta)$ be algebraic manifolds with coparallelisms of type $\mathfrak g$. We say that they are isogenous if there is an algebraic manifold $(P,\eta)$ with a coparallelism of type $\mathfrak g$ and dominant maps $f\colon P\to M$ and $g\colon P \to N$ such that $f^*(\omega) = g^*(\theta) = \eta$.
Clearly, the notion of isogeny of parallelized varieties extends that of isogeny of algebraic groups.
\[ex:cover\] Let $f\colon M\dasharrow G$ be a dominant rational map with values in an algebraic group with $\dim_{\mathbf{C}}M =\dim_{\mathbf{C}}G$. Then $\theta =f^*(\omega)$ is a rational parallelism form in $M$.
\[ex:finite\]Let $H$ be a finite subgroup of the algebraic group $G$ and $$\begin{gathered}
\pi\colon \ G\to M = H\setminus G = \{Hg \colon g\in G\}\end{gathered}$$ be the quotient by the action of $H$ on the left side. The structure form $\omega$ in $G$ is left-invariant and then it is projectable by $\pi$. Then, $\theta = \pi_*(\omega)$ is a regular parallelism form in $M$.
\[ex:coverfinite\] Combining Examples \[ex:cover\] and \[ex:finite\], let $H\subset G$ be a finite subgroup and $f\colon M\to H \setminus G$ be a dominant rational map between manifolds of the same dimension. Then $\theta = f^*(\pi_*(\omega))$ is a rational parallelism form in $M$.
By application of Example \[ex:coverfinite\] to the case of the multiplicative group we obtain rational multiples of logarithmic forms in $\mathbf{CP}_1$, $\frac{p}{q}\frac{{\rm d}f}{f}$ where $f\in\mathbf C(z)$. Thus, rational multiples of logarithmic forms in $\mathbf{CP}_1$ are the rational coparallelisms isogenous to that of the multiplicative group.
By application of Example \[ex:coverfinite\] to the case of the additive group we obtain the exact forms in $\mathbf{CP}_1$, ${\rm d}F$ where $F\in\mathbf C(z)$. Thus, the exact forms in $\mathbf{CP}_1$ are the rational coparallelisms isogenous to that of the additive group.
\[ex:quotient\_c\]Let $H$ be a subgroup of the algebraic group $G$, with Lie algebra $\mathfrak h\subset \mathfrak g$. Let us assume that $\mathfrak h$ admits a supplementary Lie algebra $\mathfrak h'$ $$\begin{gathered}
\mathfrak g = \mathfrak h \oplus \mathfrak h' \qquad \mbox{(as vector spaces).}\end{gathered}$$ We consider the left quotient $M= H \setminus G$ of $G$ by the action of $H$ and the quotient map $\pi\colon G\to M$. It turns out that $\mathfrak h'$ is a Lie algebra of vector fields in $G$ projectable by $\pi$, and thus $\pi_*|_{\mathfrak h'} \colon \mathfrak h' \to \mathfrak X[M]$ gives a parallelism of $M$ that is regular in the open subset $$\begin{gathered}
\{Hg\in M \colon \operatorname{Adj}_g(\mathfrak h) \cap \mathfrak h' = \{0\} \}.\end{gathered}$$ It turns out to be regular in $M$ if $H\lhd G$. Examples \[ex:B\] and \[ex:MD\] are particular cases where $G$ is $\operatorname{Af\/f}(2,\mathbf C)$ and $\operatorname{Af\/f}(3,\mathbf C)$ respectively.
We can see also Example \[ex:quotient\_c\] as a coparallelism. Let $\pi'\colon\mathfrak g\to \mathfrak h'$ be the projection given by the vector space decomposition $\mathfrak g = \mathfrak h \oplus \mathfrak h'$. Since $\pi'\circ \omega$ is left invariant form in $G$, it is projectable by $\pi$. Hence, there is a form $\omega'$ in $M$ such that $\pi^*\omega' = \pi'\circ \omega$. This form $\omega'$ is the corresponding coparallelism.
Associated Lie connection {#section_lie}
=========================
Reciprocal connections
----------------------
Let $\nabla$ be a linear connection (rational or regular) on $TM$. The reciprocal connection is defined as $$\begin{gathered}
\nabla^{\rm rec}_{\vec X}\vec Y = \nabla_{\vec Y} \vec X + \big[\vec X,\vec Y\big].\end{gathered}$$ From this definition it is clear that the difference $\nabla - \nabla^{\rm rec} = \operatorname{Tor}_{\nabla}$ is the torsion tensor, $\operatorname{Tor}_{\nabla} = -\operatorname{Tor}_{\nabla^{\rm rec}}$ and $(\nabla^{\rm rec})^{\rm rec} = \nabla$.
Connections and parallelisms
----------------------------
Let $\omega$ be a coparallelism of type $\mathfrak g$ in $M$ and $\rho$ its equivalent parallelism. Denote by $\vec X_i$ the basis of vector fields in $M$ such that $\omega(\vec X_i)=A_i$ is a basis of $\mathfrak g$.
The connection $\nabla$ associated to the parallelism $\omega$ is the only linear connection in $M$ for which $\omega$ is a $\nabla$-horizontal form.
Clearly $\nabla$ is a flat connection and the basis $\{\vec X_i\}$ is a basis of the space of $\nabla$-horizontal vector fields. In this basis $\nabla$ has vanishing Christoffel symbols $$\begin{gathered}
\nabla_{\vec X_i} \vec X_j = 0.\end{gathered}$$
Let us compute some infinitesimal symmetries of $\omega$. A vector field $\vec Y$ is an infinitesimal symmetry of $\omega$ if ${\rm Lie}_{\vec Y}\omega = 0$, or equivalently, if it commutes with all the vector fields of the parallelism $$\begin{gathered}
\big[\vec X_i, \vec Y\big] = 0, \qquad i = 1,\ldots, r.\end{gathered}$$
\[Lemma1\]Let $\nabla$ be the connection associated to the parallelism $\omega$. Then for any vector field $\vec Y$ and any $j=1,\ldots,r$ $$\begin{gathered}
\big[\vec X_j, \vec Y\big] = \nabla^{\rm rec}_{\vec X_j}{\vec Y}.\end{gathered}$$ Thus, $\vec Y$ is an infinitesimal symmetry of $\omega$ if and only if it is a horizontal vector field for the reciprocal connection $\nabla^{\rm rec}$.
A direct computation yields the result. Take $\vec Y = \sum\limits_{k=1}^r f_k\vec X_k$, for each $j$ we have $$\begin{gathered}
\nabla^{\rm rec}_{\vec X_j} \vec Y = \sum_{k=1}^r\big( \big(\vec X_jf_k\big)\vec X_k + f_k\big[\vec X_j, \vec X_k\big]\big) = \big[\vec X_j,\vec Y\big]. \tag*{\qed}\end{gathered}$$
The above considerations also give us the Christoffel symbols for $\nabla^{\rm rec}$ in the basis $\{\vec X_i\}$ $$\begin{gathered}
\nabla^{\rm rec}_{\vec X_i}\vec X_j = \big[\vec X_i, \vec X_j\big] = \sum_{k=1}^r \lambda_{ij}^k \vec X_k,\end{gathered}$$ i.e., the Christoffel symbols of $\nabla^{\rm rec}$ are the structure constants of the Lie algebra $\mathfrak g$.
\[Lemma2\] Let $\nabla$ be the connection associated to a coparallelism in $M$. Then, $\nabla^{\rm rec}$ is flat, and the Lie bracket of two $\nabla^{\rm rec}$-horizontal vector fields is a $\nabla^{\rm rec}$-horizontal vector field.
The flatness and the preservation of the Lie bracket by $\nabla^{\rm rec}$ are direct consequences of the Jacobi identity. Let us compute the curvature $$\begin{gathered}
R\big(\vec X_i,\vec X_j,\vec X_k\big) = \nabla^{\rm rec}_{\vec X_i}\big(\nabla^{\rm rec}_{\vec X_j} X_k\big)
- \nabla^{\rm rec}_{\vec X_j}\big(\nabla^{\rm rec}_{\vec X_i} \vec X_k\big)
- \nabla^{\rm rec}_{[\vec X_i,\vec X_j]}\vec X_k\\
\hphantom{R\big(\vec X_i,\vec X_j,\vec X_k\big)}{} = \rho ([A_i,[A_j,A_k]] - [A_j,[A_i, A_k]] - [[A_i,A_j],A_k]) = 0.\end{gathered}$$ Let us compute the Lie bracket for $\vec Y$ and $\vec Z$ $\nabla^{\rm rec}$-horizontal vector fields $$\begin{gathered}
\nabla^{\rm rec}_{\vec X_i}\big[\vec Y, \vec Z\big]\! = \big[\vec X_i,\big[\vec Y, \vec Z\big]\big]\! =
\big[\big[\vec X_i, \vec Y\big], \vec Z\big]\! + \big[ \vec Y, \big[\vec X_i, \vec Z \big]\big]\! =
\big[\nabla^{\rm rec}_{\vec X_i}\vec Y, \vec Z\big]\! +\big[\vec Y, \nabla^{\rm rec}_{\vec X_i}\vec Z\big]\! = 0.\!\!\!\!\!\!\tag*{{}}\end{gathered}$$
\[Lemma3\] Let $x\in M$ be a regular point of the parallelism form $\omega$. The space of germs at $x$ of horizontal vector fields for $\nabla^{\rm rec}$ is a Lie algebra isomorphic to $\mathfrak g$. Moreover, let $\vec Y_1,\ldots, \vec Y_r$ be horizontal vector fields with initial conditions $\vec Y_i(x) = \vec X_i(x)$, then $\big[\vec Y_i, \vec Y_j\big] = - \sum\limits_{k=1}^r \lambda_{ij}^k \vec Y_k$, where the $\lambda_{i,j}$ are the structure constants of the Lie algebra generated by the $\vec X_i$.
We can write the vector fields $\vec Y_i$ as linear combinations of the vector fields $\vec X_i$: $\vec Y_i = \sum\limits_{j=1}^r a_{ji}\vec X_j$. The matrix $(a_{ij})$ satisfies the differential equation $$\begin{gathered}
\vec X_{k}a_{ij} = - \sum_{\alpha = 1}^r \lambda_{k \alpha}^i a_{\alpha j}, \qquad a_{ij}(x) = \delta_{ij}.\end{gathered}$$ On the other hand, we have $\big[\vec Y_i, \vec Y_j\big](x) = \sum\limits_{k=1}^r \hat\lambda_{ij}^k \vec Y_k (x)$, for certain unknown structure constants $\hat\lambda_{ij}^k$. Let us check that $\hat\lambda_{ij}^k = \lambda_{ji}^k = -\lambda_{ij}^k$, $$\begin{gathered}
\big[\vec Y_i,\vec Y_j\big] = \left[\sum_{\alpha=1}^r a_{\alpha i} \vec X_\alpha, \sum_{\beta = 1}^r a_{\beta j} \vec X_{\beta} \right] \\
\hphantom{\big[\vec Y_i,\vec Y_j\big]}{} = \sum_{\alpha, \beta, \gamma =1}^r -a_{\alpha i}\lambda_{\alpha \gamma}^\beta a_{\gamma j}\vec X_\beta
+ \sum_{\alpha, \beta, \gamma =1}^r a_{\beta j}\lambda_{\beta\gamma}^\alpha a_{\gamma i} \vec X_{\alpha}
+ \sum_{\alpha, \beta, \gamma =1}^r a_{\beta j} a_{\alpha i} \lambda_{\alpha \beta}^\gamma \vec X_\gamma.\end{gathered}$$ Taking values at $x$, we obtain $$\begin{gathered}
\big[\vec Y_i, \vec Y_j\big](x) = \sum_{\beta =1}^r - \lambda_{i j}^\beta \vec Y_\beta(x)
+ \sum_{\alpha =1}^r \lambda_{j i}^\alpha \vec Y_{\alpha}(x) + \sum_{\gamma =1}^r \lambda_{i j}^\gamma \vec Y_\gamma(x) =
\sum_{\alpha =1}^r \lambda_{j i}^k \vec Y_{k}(x).\tag*{{}}\end{gathered}$$
Let $G$ be an algebraic group with Lie algebra $\mathfrak g$. As seen in Example \[ex:group\] the Maurer–Cartan structure form $\omega$ is a coparallelism in $G$. Let $\nabla$ be the connection associated to this coparallelism. There is another canonical coparallelism, the right invariant Maurer–Cartan structure form $\omega_{\rm rec}$, let us consider ${\bf i}\colon G\to G$ the inversion map, $$\begin{gathered}
\omega_{\rm rec} = - {\bf i}^*(\omega).\end{gathered}$$ As may be expected, the connection associated to the coparallelism $\omega_{\rm rec}$ is $\nabla^{\rm rec}$. Right invariant vector fields in $G$ are infinitesimal symmetries of left invariant vector fields and vice versa. In this case, the horizontal vector fields of $\nabla$ and $\nabla^{\rm rec}$ are regular vector fields.
As shown in the next three examples, symmetries of a rational parallelism are not in general rational vector fields.
Let us consider the Lie algebra $\mathfrak g$ and the coparallelism $\omega = A_1{\rm d}x + (A_2 - x A_1){\rm d}y$, of Example \[ex:B\]. Let $\nabla$ be its associated connection. In cartesian coordinates, the only non-vanishing Christoffel symbol of the reciprocal connection is $\Gamma_{21}^1 = -1$. A basis of $\nabla^{\rm rec}$-horizontal vector fields is $$\begin{gathered}
\vec Y_1 = e^y\frac{\partial}{\partial x}, \qquad \vec Y_2 = \frac{\partial}{\partial y}.\end{gathered}$$ Note that they coincide with $\vec X_1$, $\vec X_2$ at the origin point and $\big[\vec Y_1,\vec Y_2\big] = - Y_1$.
Let us consider the Lie algebra $\mathfrak g$ and the coparallelism $\omega = (A_1 - \alpha y A_2 - \beta z A_3){\rm d}x + A_2{\rm d}y + A_3{\rm d}z$ of Example \[ex:MD\]. Let $\nabla$ be its associated connection. In cartesian coordinates, the only non-vanishing Christoffel symbols of the reciprocal connection are $$\begin{gathered}
\Gamma_{11}^2 = -\alpha,\qquad \Gamma_{11}^3 = -\beta.\end{gathered}$$ A basis of $\nabla^{\rm rec}$-horizontal vector fields is $$\begin{gathered}
\vec Y_1 = \frac{\partial}{\partial x}, \qquad \vec Y_2 = e^{\alpha x}\frac{\partial}{\partial y},\qquad \vec Y_3 = e^{\beta x}\frac{\partial}{\partial z}.\end{gathered}$$ Note that they coincide with $\vec X_1$, $\vec X_2$, $\vec X_3$ at the origin point and $$\begin{gathered}
\big[\vec Y_1,\vec Y_2\big] = - \alpha Y_2,\qquad \big[\vec Y_1,\vec Y_3\big] = -\beta \vec Y_3.\end{gathered}$$
Let us consider the Lie algebra $\mathfrak g$ of Example \[ex:MD\] and the coparallelism $$\begin{gathered}
\omega = (A_1 - \alpha y A_2 - \beta z A_3)\frac{{\rm d}x}{x} + A_2{\rm d}y + A_3{\rm d}z.\end{gathered}$$ Let $\nabla$ be its associated connection. In cartesian coordinates, the only non-vanishing Christoffel symbols of the reciprocal connection are $$\begin{gathered}
\Gamma_{11}^2 = -\alpha,\qquad \Gamma_{11}^3 = -\beta.\end{gathered}$$ A basis of $\nabla^{\rm rec}$-horizontal vector fields on a simply connected open subspace $U\subset \mathbb C^\ast \times \mathbb C^2$ is $$\begin{gathered}
\vec Y_1 = x\frac{\partial}{\partial x}, \qquad \vec Y_2 = x^{\alpha}\frac{\partial}{\partial y}, \qquad \vec Y_3 = x^{\beta}\frac{\partial}{\partial z}.\end{gathered}$$
Lie connections
---------------
The connections $\nabla$ and $\nabla^{\rm rec}$ associated to a coparallelism $\omega$ of type $\mathfrak g$ are particular cases of the following definition.
A Lie connection (regular or rational) in $M$ is a flat connection $\nabla$ in $TM$ such that the Lie bracket of any two horizontal vector fields is a horizontal vector field.
Given a Lie connection $\nabla$ in $M$, there is a $r$-dimensional Lie algebra $\mathfrak g$ such that the space of germs of horizontal vector fields at a regular point $x$ is a Lie algebra isomorphic to $\mathfrak g$. We will say that $\nabla$ is a Lie connection of type $\mathfrak g$. The following result gives several algebraic characterizations of Lie connections:
\[prop\_Lie\_char\] Let $\nabla$ be a linear connection in $TM$, the following statements are equivalent:
- $\nabla$ is a Lie connection;
- $\nabla^{\rm rec}$ is a Lie connection;
- $\nabla$ is flat and has constant torsion, $\nabla \operatorname{Tor}_{\nabla} = 0$;
- $\nabla$ and $\nabla^{\rm rec}$ are flat.
Let us first see (1)$\Leftrightarrow$(2). Let $\nabla$ be a Lie connection. Around each point of the domain of $\nabla$ there is a parallelism, by possibly transcendental vector fields, such that $\nabla$ is its associated connection. Then, Lemma \[Lemma2\] states (1)$\Rightarrow$(2). Taking into account that $(\nabla^{\rm rec})^{\rm rec} = \nabla$ we have the desired equivalence.
Let us see now that (1)$\Leftrightarrow$(3). Let us assume that $\nabla$ is a flat connection. For any three vector fields $X$, $Y$, $Z$ in $M$ we have $$\begin{gathered}
(\nabla_X \operatorname{Tor}_{\nabla})(Y,Z) = - \operatorname{Tor}_{\nabla}(\nabla_X Y,Z ) - \operatorname{Tor}_{\nabla}(Y,\nabla_X Z) + \nabla_X \operatorname{Tor}_{\nabla}(Y,Z).\end{gathered}$$ Let us assume that $Y$ and $Z$ are $\nabla$-horizontal vector fields. Then, we have $$\begin{gathered}
\operatorname{Tor}_{\nabla}(Y,Z) = \nabla_Y Z - \nabla_Z Y - [Y,Z] = - [Y,Z]\end{gathered}$$ and the previous equality yields $$\begin{gathered}
(\nabla_X \operatorname{Tor}_{\nabla})(Y,Z) = - \nabla_X[Y,Z].\end{gathered}$$ Thus, we have that $\nabla \operatorname{Tor}_{\nabla}$ vanishes if and only if the Lie bracket of any two $\nabla$-horizontal vector fields is also $\nabla$-horizontal. This proves (1)$\Leftrightarrow$(3).
Finally, let us see (1)$\Leftrightarrow$(4). It is clear that (1) implies (4) so we only need to see (4)$\Rightarrow$(1). Assume $\nabla$ and $\nabla^{\rm rec}$ are flat. Then, locally, there exist a basis $\{\vec X_i\}$ of $\nabla$-horizontal vector fields and a basis $\{\vec Y_i\}$ of $\nabla^{\rm rec}$-horizontal vector fields. By the definition of the reciprocal connection, we have that a vector field $\vec X$ is $\nabla$-horizontal if and only if it satisfies $[\vec X, \vec Y_i]=0$ for $i=1,\ldots,r$. By the Jacobi identity we have $$\begin{gathered}
\big[\big[\vec X_i,\vec X_j\big],\vec Y_k\big]= 0.\end{gathered}$$ The Lie brackets $\big[\vec X_i,\vec X_j\big]$ are also $\nabla$-horizontal and $\nabla$ is a Lie connection.
Let $\nabla$ be a Lie connection on $M$. Let $x$ be a regular point and $\vec X_1,\ldots, \vec X_r$ and $\vec Y_1,\ldots, \vec Y_r$ be basis of horizontal vector field germs on $M$ for $\nabla$ and $\nabla^{\rm rec}$ respectively with same initial conditions $\vec X_i(x) = \vec Y_i(x)$. Then $$\begin{gathered}
\big[\vec X_i,\vec X_j\big](x) = - \big[\vec Y_i,\vec Y_j\big](x).\end{gathered}$$ It follows that $\nabla$ and $\nabla^{\rm rec}$ are of the same type $\mathfrak g$.
By definition $\nabla$ is the connection associated to the local analytic parallelism given by the basis $\{\vec X_i\}$ of horizontal vector fields. Then we apply Lemma \[Lemma3\] in order to obtain the desired conclusion.
Some results on Lie connections by means of Picard–Vessiot theory
-----------------------------------------------------------------
Definitions and general results concerning the Picard–Vessiot theory of connections are given in Appendix \[ApA\].
Let $\nabla$ be a rational Lie connection in $TM$. The $\nabla$-horizontal vector fields are the symmetries of a rational parallelism of $M$ if and only if $\operatorname{Gal}(\nabla^{\rm rec}) = \{1\}$.
We will use the notations of Section \[A6\]: $R^1(TM)$ is the ${\rm GL}_n(\mathbb{C})$-principal bundle associated to $TM$ and $\mathcal F'$ is the ${\rm GL}_n(\mathbb{C})$-invariant foliation on $R^1(TM)$ given by graphs of local basis of $\nabla$-horizontal sections. The Galois group $\operatorname{Gal}(\nabla^{\rm rec})$ can be computed as soon as we know the Zariski closure $\overline{{\mathscr{L}}}$ of a leaf ${\mathscr{L}}$ of the induced foliation $\mathcal F'$ on $R^1(TM)$. $\operatorname{Gal}(\nabla^{\rm rec})$ is finite is and only if $\overline{{\mathscr{L}}} = {\mathscr{L}}$ and is $\{1\}$ if and only if ${\mathscr{L}}$ is the graph of a rational section $M \to R^1(TM)$. This means that there exists a basis of rational $\nabla^{\rm rec}$-horizontal sections. These sections give the desired parallelism.
For any Lie connection $\nabla$, $\operatorname{Gal}(\nabla)\subseteq \operatorname{Aut}(\mathfrak g)$.
Let us choose a point $x \in M$ regular for $\nabla$ and a basis $A_1,\ldots, A_r$ of $\mathfrak{g}$, i.e., a basis $Y_1,\ldots, Y_r$ of local $\nabla$-horizontal section of $TM$ at $x$.
Using notation of Section \[A6\], we will identify $R^1(T_xM)$ with the set of isomorphisms of linear spaces $ \sigma\colon \mathfrak g \to T_xM$; now $\operatorname{Gal}(\nabla)\subseteq {\rm GL}(\mathfrak g)$. Because of the construction of $\mathfrak g$, we have a canonical point in $R^1(TM)$ corresponding to the identity $ \sigma_o\colon \mathfrak g \to T_xM$.
For $m \in M$, if $\sigma$ is an isomorphism from $\mathfrak g$ to $T_{m}M$ then one defines $H^k_{{i,j}}(\sigma)$ to be $$\begin{gathered}
\frac{[X_{i},X_{j}] \wedge X_{1}\wedge \cdots \wedge \widehat{X_{k}}\wedge \cdots \wedge X_{r}}{X_{k} \wedge X_{1}\wedge \cdots \wedge \widehat{X_{k}} \wedge \cdots \wedge X_{r}} \Big|_m, \end{gathered}$$ where $X_i$ is the horizontal section such that $X_i(m)= \sigma A_i$. These functions are regular functions on $R^1(TM)$. Moreover they are constant and equal to the constant structures on the Zariski closure of the leaf passing through $\sigma_o$. The Galois group is the stabilizer of this leaf then the functions $H^k_{{i,j}}$ are invariant under the action of the Galois group, i.e., the Galois group preserves the Lie bracket.
\[prop\_example\] Let $\mathfrak h'$ be a Lie sub-algebra of the Lie algebra of some algebraic group and let $G$ be the smallest algebraic subgroup such that ${\mathfrak{lie}}(G) = \mathfrak g \supset \mathfrak h'$. Assume the existence of an algebraic subgroup $H$ of $G$ whose Lie algebra $\mathfrak h$ is supplementary to $\mathfrak h'$ in $\mathfrak g$, $\mathfrak g = \mathfrak h \oplus \mathfrak h'$. Let us consider the following objects:
- the quotient map $\pi\colon G \to M$ where $M$ is the variety of cosets $H\setminus G$, and $\nabla$ the Lie connection associated to the parallelism $\pi_*\colon \mathfrak h' \to \mathfrak X[M]$ in $M$ $($as given in Example [\[ex:quotient\_c\])]{};
- its reciprocal Lie connection $\nabla^{\rm rec}$ on $M$;
- the Lie algebras of right invariant vector fields $$\begin{gathered}
\mathfrak g^{\rm rec} = {\bf i}_*(\mathfrak g), \qquad \mathfrak h'^{\rm rec} = {\bf i}_*(\mathfrak h'),\end{gathered}$$ where ${\bf i}$ is the inverse map on $G$.
Then, the following statements are true:
- $\mathfrak h'$ is an ideal of $\mathfrak g$ $($equivalently $\mathfrak h'^{\rm rec}$ is an ideal of $\mathfrak g^{\rm rec})$;
- $\mathfrak h$ is commutative $($equivalently $H$ is virtually abelian$)$;
- the adjoint action of $G$ on $\mathfrak g^{\rm rec}$ preserves $\mathfrak h'^{\rm rec}$ and thus gives, by restriction, a morphism $\overline{\operatorname{Adj}}\colon G \to \operatorname{Aut}(\mathfrak h'^{\rm rec})$;
- The Galois group of the connection $\nabla^{\rm rec}$ is ${\overline{\operatorname{Adj}}}(H) \subseteq \operatorname{Aut}(\mathfrak h'^{\rm rec})$ and thus virtually abelian.
We have that $\mathfrak g$ is the algebraic hull of $\mathfrak h'$. From Lemma \[ap2\_2\] in Appendix \[apB\] we obtain $[\mathfrak g,\mathfrak g] \subseteq \mathfrak h'$. Statement (i) follows straightforwardly. Let us consider $A$ and $B$ in $\mathfrak h$. Then $[A,B]$ is in $\mathfrak h$ and also in $\mathfrak h'$ by the previous argument. Thus, $[A,B]=0$ and this finishes the proof of statement (ii). Let us denote by $H'$ the subgroup of $G$ spanned by the image of $\mathfrak h'$ by the exponential map. For each element $h\in H'$, the adjoint action of $h$ preserves the Lie algebra $\mathfrak h'$. By continuity of the adjoint action in the Zariski topology, we have that $\mathfrak h'$ is preserved by the adjoint action of all elements of $G$. This proves statement (iii). In order to prove the last statement in the proposition we have to construct a Picard–Vessiot extension for the connection $\nabla^{\rm rec}$. Let us consider a basis $\{A_1,\ldots, A_m\}$ of $\mathfrak h'$ and let $\bar A_i$ be the projection $\pi_*(A_i)$. We have an extension of differential fields $$\begin{gathered}
(\mathbf C(M), \bar{\mathcal D}) \subseteq (\mathbf C(G), \mathcal D),\end{gathered}$$ where $\bar{\mathcal D}$ stands for the $\mathbf C(M)$-vector space of derivations spanned by $\bar A_1,\ldots, \bar A_m$ and $\mathcal D$ stands for the $\mathbf C(G)$-vector space of derivations spanned by $A_1,\ldots,A_m$ (see Appendix \[ApA\] for our conventions on differential fields).
The projection $\pi$ is a principal $H$-bundle. Any rational first integral of $\{A_1,\ldots,A_m\}$ is constant along $H'$ and thus it is necessarily a complex number. Thus, the above extension has no new constants and it is strongly normal in the sense of Kolchin, with Galois group $H$. Note that the differential field automorphism corresponding to an element $h\in H$ is the pullback of functions by the left translation $L_{h}^{-1}$, that is, $(hf)(g) = f\big(h^{-1}g\big)$.
The horizontal sections for the connection $\nabla^{\rm rec}$ are characterized by the differential equations $$\begin{gathered}
\label{eq_proof_H}
[\bar A_i, X] = 0.\end{gathered}$$ Let us consider $\{B_1,\ldots,B_m\}$ a basis of $\mathfrak h'^{\rm rec}$. From the Zariski closedness of $H$ in $G$ it follows that there are regular functions $f_{ij}\in\mathbf C[G]$ such that $B_i = \sum\limits_{j=1}^m f_{ij} A_j$. Thus let us define $\bar B_i = \sum\limits_{j=1}^m f_{ij} \bar A_j$. Those objects are vector fields in $M$ with coefficients in $\mathbf C[G]$, and clearly satisfy equation . Thus, the Picard–Vessiot extension of $\nabla^{\rm rec}$ is spanned by the functions $f_{ij}$ and it is embedded, as a differential field, in $\mathbf C(G)$. Let us denote such extension by $\mathbf L$. We have a chain of extensions $$\begin{gathered}
\mathbf C(M) \subseteq \mathbf L \subseteq \mathbf C(G).\end{gathered}$$ By Galois correspondence, the Galois group of $\nabla^{\rm rec}$ is a quotient $H/K$ where $K$ is the subgroup of elements of $H$ that fix, by left translation, the functions $f_{ij}$. In order to prove statement (iv) we need to check that this group $K$ is the kernel of the morphism $\overline{\operatorname{Adj}}$.
Let us note that the image under the adjoint action by $g\in G$ of an element $B\in \mathfrak h'^{\rm rec}$ is given by the left translation, $\overline{\operatorname{Adj}}(g)(B) = L_{g*}(B)$. This transformation makes sense for any derivation of $\mathbf C[G]$, and thus we have an action of $G$ on $\mathfrak X(G)$. Let us take $h$ in the kernel of $\overline{\operatorname{Adj}}$, thus, $\overline{\operatorname{Adj}}(h)(B_j) = B_j$ for any index $j$. Applying the transformation $L_{h*}$ to the expression of $B_i$ as linear combination of the left invariant vector fields $A_j$ we obtain $B_i = \sum\limits_{j=1}^m L_{h*}(f_{ij} A_j) = \sum\limits_{j=1}^m h(f_{ij})A_j$. The coefficients of $B_i$ as linear combination of the $A_j$ are unique, and thus, $h(f_{ij}) = f_{ij}$ we conclude that $h$ is an automorphism fixing $\mathbf L$. On the other hand, let us take $h\in H$ fixing $\mathbf L$. Then $L_{h*}\big(\sum f_{ij}A_j\big) = \sum f_{ij} A_j$ thus $\overline{\operatorname{Adj}}(h)(B_i) = B_i$ and then $h$ is in the kernel of $\overline{\operatorname{Adj}}$.
Some examples of $\boldsymbol{\mathfrak{sl}_2}$-parallelisms {#sl2}
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We will construct some parallelized varieties as subvarieties of the arc space of the affine line ${\mathbb A}^1_\mathbf C$. This family of examples show how to realize every subgroup of ${\rm PSL}_2(\mathbf C)$ as the Galois group of the reciprocal Lie connection.
### The arc space of the affine line and its Cartan 1-form {#sl2parallelisms}
In our special case, the arc space of the affine line ${\mathbb A}^1_\mathbf C$ with affine coordinate $z$, is the space of all formal power series $\widehat{z} = \sum z^{(i)} \frac{x^i}{i !}$. It will be denoted by ${\mathscr{L}}$, its ring of regular functions is $\mathbf C[ {\mathscr{L}}] = \mathbf C\big[z^{(0)},z^{(1)}, z^{(2)},\ldots \big]$. For an open subset $U\subset \mathbf C$ one denotes by ${\mathscr{L}}U$ the set of power series $\widehat z$ with $z^{(0)} \in U$.
A biholomorphism $f\colon U \to V$ between open sets of $\mathbf C$ can be lift to a biholomorphism $ {\mathscr{L}}f\colon {\mathscr{L}}U \to {\mathscr{L}}V$ by composition $\widehat{z} \to f\circ\widehat{z}$.
Let $\widehat{\mathfrak X}$ be the Lie algebra of formal vector fields $\mathbf C[[x]]\frac{\partial}{\partial x}$. One can build a rational form $\sigma\colon T{\mathscr{L}} \to \widehat{\mathfrak X}$ in following way (see [@guillemin-sternberg Section 2]). Let $v = \sum a_i \frac{\partial}{\partial z^{(i)}}$ be a tangent vector at the formal coordinate $\widehat{p}$, i.e., an arc in the Zariski open subset $\{z^{(1)} \not = 0\}$. The local coordinate $\widehat{p}$ can be used to have formal coordinates $p_0,p_{1}, p_{2}, \ldots $, on $\mathscr{L}$ and $v$ can be written $v = \sum b_i \frac{\partial}{\partial p_{i}}$. The form $\sigma$ is defined by $\sigma(v) = \sum b_i \frac{x^i}{i !} \frac{\partial}{\partial x} $. This form is rational and is an isomorphism between $T_p {\mathscr{L}}$ and $\widehat{\mathfrak X}$ satisfying $d\sigma = - \frac{1}{2}[\sigma, \sigma]$ and $({\mathscr{L}}f)^\ast \sigma = \sigma$ for any biholomorphism $f$.
This means that $\sigma$ provides an action of $\widehat{\mathfrak X}$ commuting with the lift of biholomorphisms. This form seems to be a coparallelism but it is not compatible with the natural structure of pro-variety of ${\mathscr{L}}$ and $\widehat{\mathfrak X}$: $\sigma^{-1}\big(\frac{\partial}{\partial x}\big) = \sum\limits_{i \geq 0} z^{(i+1)}\frac{\partial}{\partial z^{(i)}}$ is a derivation of degree $+1$ with respect to the pro-variety structure of ${\mathscr{L}}$. The total derivation above will be denoted by $E_{-1}$. This gives a differential structure to the ring $\mathbf C[ {\mathscr{L}}]$.
### The parallelized varieties
Let $\nu\in \mathbf C(z)$ be a rational function, $f$ be the rational function on the arc space given by the Schwarzian derivative $$\begin{gathered}
f\big(z^{(0)},z^{(1)},z^{(2)},z^{(3)}\big) = \frac{z^{(3)}}{z^{(1)}}-\frac{3}{2} \left(\frac{z^{(2)}}{z^{(1)}}\right) ^2+ \nu\big(z^{(0)}\big)\big(z^{(1)}\big)^2,\end{gathered}$$ and $I \subset \mathbf C[{\mathscr L}]$ be the $E_{-1}$-invariant ideal generated by $p\big(z^{(0)}\big) {z^{(1)}}^2f\big(z^{(0)},z^{(1)},z^{(2)},z^{(3)}\big)$ where $p$ is a minimal denominator of $\nu$.
The zero set $V$ of $I$ is a dimension $3$ subvariety of ${\mathscr{L}}$ and $\omega (TV) = {\mathfrak{sl}}_2(\mathbf C) \subset \widehat{\mathfrak X}$. This provides a ${\mathfrak{sl}}_2$-parallelism on $V$.
One can compute explicitly this parallelism using $z^{(0)}$, $z^{(1)}$ and $z^{(2)}$ as étale coordinates on a Zariski open subset of $V$. Let us first compute the $\mathfrak{sl}_2$ action on ${\mathscr{L}}$. The standard inclusion of $\mathfrak{sl}_2$ in $\widehat{\mathfrak X}$ is given by $E_{-1} = \frac{\partial}{\partial x}$, $E_0 = x\frac{\partial}{\partial x}$ and $E_{1} = x^2\frac{\partial}{\partial x}$. Their actions on ${\mathscr{L}}$ are given by $E_{-1} = \sum z^{(i+1)}\frac{\partial}{\partial z^{(i)}}$, $E_0 = \sum iz^{(i)}\frac{\partial}{\partial z^{(i)}}$ and $E_1= \sum i(i-1) z^{(i-1)}\frac{\partial}{\partial z^{(i)}}$. The ideal $I$ is generated by the functions $E_{-1}^n\cdot f$. By definition $E_{-1}\cdot f \in I$, a direct computation gives that $E_0\cdot f = 2f \in I$, $E_1\cdot f = 0 \in I$. The relations in $\mathfrak{sl}_{2}$ give that $E_{-1}\cdot I \subset I$, $E_0\cdot I \subset I$ and $E_1\cdot I \subset I$, i.e., the vector fields $E_{-1}$, $E_0$ and $E_1$ are tangent to $V$.
Now parameterizing $V$ by $z^{(0)}$, $z^{(1)}$ and $z^{(2)}$ one gets $$\begin{gathered}
E_{-1}|_{\mathbf C ^3} = z^{(1)}\frac{\partial}{\partial z^{(0)}} + z^{(2)}\frac{\partial}{\partial z^{(1)}} + \left(-\nu\big(z^{(0)}\big)\big(z^{(1)}\big)^3 + \frac{3}{2}\frac{(z^{(2)})^2}{z^{(1)}}\right) \frac{\partial}{\partial z^{(2)}},\nonumber\\
E_0|_{\mathbf C^3} = z^{(1)}\frac{\partial}{\partial z^{(1)}} + 2 z^{(2)}\frac{\partial}{\partial z^{(2)}}, \qquad E_1|_{\mathbf C^3} = 2 z^{(1)}\frac{\partial}{\partial z^{(2)}}.\label{parrallel}\end{gathered}$$ They form a rational $\mathfrak{sl}_{2}$-parallelism on $\mathbf C^3$ depending on the choice of a rational function in one variable.
### Symmetries and the Galois group of the reciprocal connection
\[thm\_SL2\]Any algebraic subgroup of ${\rm PSL}_{2}(\mathbf C)$ can be realized as the Galois group of the reciprocal connection of a parallelism of $\mathbf C^3$.
A direct computation shows that $z\mapsto \varphi(z)$ is an holomorphic function satisfying $$\begin{gathered}
\frac{\varphi'''}{\varphi'}-\frac{3}{2} \left(\frac{\varphi''}{\varphi'}\right)^2 + \nu(\varphi)(\varphi')^2 = \nu(z),\end{gathered}$$ if and only if its prolongation ${\mathscr{L}}\varphi\colon \widehat{z} \mapsto \varphi (\widehat{z})$ on the space ${\mathscr{L}}$ preserves $V$ and preserves each of the vector fields $E_{-1}$, $E_0$ and $E_1$.
Taking infinitesimal generators of this pseudogroup, one gets for any local analytic solution of the linear equation $$\begin{gathered}
\label{lin}
a''' + 2 \nu a' +\nu 'a =0,\end{gathered}$$ a vector field $X = a(z)\frac{\partial}{\partial z}$ whose prolongation on ${\mathscr{L}}$ is $$\begin{gathered}
{\mathscr{L}}X = a\big(z^{(0)}\big)\frac{\partial}{\partial z^{(0)}} + a'\big(z^{(0)}\big)z^{(1)}\frac{\partial}{\partial z^{(1)}} + \big( a''\big(z^{(0)}\big)\big(z^{(1)}\big)^2 + a'\big(z^{(0)}\big) z^{(2)}\big)\frac{\partial}{\partial z^{(2)}} + \cdots.\end{gathered}$$ The equation (\[lin\]) ensures that ${\mathscr{L}}{X}$ is tangent to $V$. The invariance of $\sigma$, $({\mathscr{L}}X)_{\ast}\sigma =0$, ensures that ${\mathscr{L}}X$ commutes with the ${\mathfrak{sl}}_{2}$-parallelism given above. This means that for any solution $a$ of (\[lin\]) the vector field $$\begin{gathered}
a\big(z^{(0)}\big)\frac{\partial}{\partial z^{(0)}} + a'\big(z^{(0)}\big)z^{(1)}\frac{\partial}{\partial z^{(1)}}+ \big( a''\big(z^{(0)}\big)\big(z^{(1)}\big)^2 + a'\big(z^{(0)}\big) z^{(2)}\big)\frac{\partial}{\partial z^{(2)}}, \end{gathered}$$ commutes with $E_{-1}|_{\mathbf C ^3}$, $E_0|_{\mathbf C ^3}$ and $E_{1}|_{\mathbf C ^3}$.
Then the linear differential system of flat section for the reciprocal connection reduces to the linear equation (\[lin\]). This equation is the second symmetric power of $y'' = \nu(z)y$. If $G \subset {\rm SL}_2(\mathbf C)$ is the Galois group of $y'' = \nu(z)y$ then the image of its second symmetric power representation $s^2\colon G \to \operatorname{Sym}^2(\mathbf C^2)$ is the Galois group of (\[lin\]). The kernel of this representation is $\{{\rm Id}, -{\rm Id}\}$ then the Galois group of (\[lin\]) is an algebraic subgroup of ${\rm PSL}_{2}(\mathbf C)$.
Let us remark that, as it follows from its definition, the Galois group of an equation contains the monodromy group. Moreover one can determine the monodromy group of classical differential equations. Hypergeometric equations depend on three complex numbers $(a,b,c)$ $$\begin{gathered}
z(1-z) F'' + (c-(a+b+1)z)F' - abF = 0, $$ and is equivalent to $$\begin{gathered}
y'' = \nu(\ell, n,m ; z) y,\end{gathered}$$ with $$\begin{gathered}
\nu(\ell,m,n ;z) = \frac{\big(1-\ell^2\big)}{4z^2} +\frac{1-m^2}{4(1-z)^2} +\frac{1-\ell^2-m^2+n^2}{4 z(1-z)},\end{gathered}$$ and $$\begin{gathered}
F = z^{-c/2}(1-z)^{(c-a-b-1)/2} y, \qquad \ell = 1-c,\qquad m =c-a-b,\qquad n=a-b.\end{gathered}$$ These two equations have the same projectivized Galois group in ${\rm PGL}_2(\mathbf C)$. Any algebraic subgroup of ${\rm PGL}_2(\mathbf C)$ will be realized by an appropriate choice of $(a,b,c)$.
### The whole group
For $a=b=1/2$, $c = 1$, the hypergeometric equation is the Picard–Fuchs equation of Legendre family. Its monodromy group is $\Gamma(2) \subset {\rm SL}_2(\mathbf Z)$ and is Zariski dense in ${\rm SL}_2(\mathbf C)$.
### The triangular subgroups
For $b=0$ and $a=-1$ one can compute a basis of solutions of the equation: $1$ and $\int \big( \frac{1-z}{z} \big)^c {\rm d}z$. If $c$ is not rational, the Galois group is the group of invertible matrices $\left[\begin{smallmatrix} u & v\\ 0 & 1 \end{smallmatrix}\right]$. When $c$ is rational then $u$ must be a root of the unity of order the denominators of $c$. When $c\in \mathbf Z$, the Galois group is the group of matrices $\left[\begin{smallmatrix} 1 & v\\ 0 & 1 \end{smallmatrix}\right]$.
For $b=0 $ and $c= a+1$ a basis of solution is given by $z^{-a}$ and $1$. Its Galois group is a subgroup of the group of matrices $\left[\begin{smallmatrix} u& 0 \\ 0 & 1 \end{smallmatrix}\right]$. The parameter $a$ is rational if and only if it is a finite subgroup.
### The dihedral subgroups
For $c=1/2$ and $a+b=0$, a basis of solution is given by $\big(\sqrt{z}+\sqrt{1-z}\big)^a$ and $\big(\sqrt{z}-\sqrt{1-z}\big)^a$. The monodromy group is a dihedral group in ${\rm GL}_2(\mathbf C)$ whose quotients give dihedral subgroups of ${\rm PGL}_2(\mathbf C)$.
### The tetrahedral subgroup
This group is the monodromy group of hypergeometric equation for $\ell = 1/3$, $m= 1/2$ and $n=1/3$. A basis of solution is given by $$\begin{gathered}
(z-1)^{-1/12}\Big(\sqrt{3}\big(z^{1/3}+1\big) \pm 2\sqrt{z^{2/3} + z^{1/3} +1} \Big)^{1/4}. \end{gathered}$$
### The octahedral subgroup
This group is the monodromy group of hypergeometric equation for $\ell = 1/2$, $m= 1/3$ and $n=1/4$. A basis of solution is given by $$\begin{gathered}
(z-1)^{-1/24}\Big[\sqrt{3} \big( \big(\sqrt{z}-1\big)^{1/3} +\big(\sqrt{z}+1\big)^{1/3}\big)^{1/3}\\
\qquad{} \pm 2\sqrt{\big(\sqrt{z}-1\big)^{2/3} + (z-1)^{1/3} +\big(\sqrt{z}+1\big)^{2/3}}\Big]^{1/4}.\\\end{gathered}$$
### The icosaedral subgroup
This group is the monodromy group of hypergeometric equation for $\ell = 1/2$, $m= 1/3$ and $n=1/5$. As icosahedral group is not solvable, the solution space is not described using formulas as simple as in preceding examples.
Darboux–Cartan connections {#section_DC}
==========================
Connection of parallelism conjugations {#DC conjugation}
--------------------------------------
Let $\omega$ be a rational coparallelism on $M$ of type $\mathfrak g$ and $G$ an algebraic group with Lie algebra of left invariant vector fields $\mathfrak g$ and Maurer–Cartan form $\theta$. Denote by $M^\star$ the open subset of $M$ in wich $\omega$ is regular. We will study the contruction of conjugating maps between the parallelisms $(M,\omega)$ and $(G,\theta)$.
Let us consider the trivial principal bundle $\pi\colon P = G\times M \to M$. In this bundle we consider the action of $G$ by right translations $(g, x) * g' = (gg', x)$. Let $\Theta$ be the $\mathfrak g$-valued form $\Theta = \theta - \omega$ in $P$.
\[DCconnection\] The kernel of $\Theta$ is a rational flat invariant connection on the principal bundle $\pi\colon P \to M$. We call it the Darboux–Cartan connection of parallelism conjugations from $(M,\omega)$ to $(G,\theta)$.
The equation $\Theta = 0$ defines a foliation on $P$ transversal to the fibers at regular points of $\omega$. The leaves of the foliation are the graphs of analytic parallelism conjugations from $(M,\omega)$ to $(G,\theta)$. By means of differential Galois theory the Darboux–Cartan connection has a Galois group $\operatorname{Gal}(\Theta)$ with Lie algebra $\mathfrak{gal}(\Theta)$. The following facts are direct consequences of the definition of the Galois group:
- there is a regular covering map $c\colon (M^\star, \omega) \to (U,\theta)$ with $U$ an open subset of $G$, and $c^*(\theta) = \omega$ if and only if $\operatorname{Gal}(\Theta) = \{1\}$;
- there is a regular covering map $c\colon (M^\star, \omega) \to (U,q_*\theta)$ with $U$ an open subset of $G/H$, $H$ a group of finite index, and $c^*(q_*\theta|_U) = \omega$ if and only if $\mathfrak{gal}(\Theta) = \{0\}$.
In any case, the necessary and sufficient condition for $(M,\omega)$ and $(G,\theta)$ to be isogenous parallelized varieties is that $\mathfrak{gal}(\Theta) = \{0\}$.
Darboux–Cartan connection and Picard–Vessiot
--------------------------------------------
Note that the coparallelism $\omega$ gives a rational trivialization of $TM$ as the trivial bundle of fiber $\mathfrak g$. In $TM$ we have defined the connection $\nabla^{\rm rec}$ whose horizontal vector fields are the symmetries of the parallelism. On the other hand, $G$ acts in $\mathfrak g$ by means of the adjoint action. The Cartan-Darboux connection induces then a connection $\nabla^{\rm adj}$ in the associated trivial bundle $\mathfrak g\times M$ of fiber $\mathfrak g$.
\[adj\_conjugation\]The map $$\begin{gathered}
\tilde\omega\colon \ \big(TM, \nabla^{\rm rec}\big) \to \big(\mathfrak g \times M, \nabla^{\rm adj}\big), \qquad X_x \mapsto (\omega_x(X_x), x)\end{gathered}$$ is a birational conjugation of the linear connections $\nabla^{\rm rec}$ and $\nabla^{\rm adj}$.
It is clear that the map $\tilde\omega$ is birational. Let us consider $\{A_1,\ldots,A_m\}$ a basis of $\mathfrak g$. Let $\rho\colon \mathfrak g\to \mathfrak X(M)$ be the parallelism associated to the parallelism $\omega$ and let us define $X_i = \rho(A_i)$. Then $\{X_1,\ldots, X_n\}$ is a rational frame in $M$ and the map $\tilde\omega$ conjugates the vector field $X_i$ with the constant section $A_i$ of the trivial bundle of fiber $\mathfrak g$. By definition of the reciprocal connection $$\begin{gathered}
\nabla^{\rm rec}_{X_i}X_j = [X_i,X_j].\end{gathered}$$ On the other hand, by definition of the adjoint action and application of the covariant derivative as in equation of Appendix \[ap7\_associated\] we obtain $$\begin{gathered}
\nabla^{\rm adj}_{X_i}A_j = [A_i,A_j].\end{gathered}$$ Therefore we have that $\tilde\omega$ is a rational morphism of linear connections that conjugates $\nabla^{\rm rec}$ with $\nabla^{\rm adj}$.
The following facts follow directly from Proposition \[adj\_conjugation\], and basic properties of the Galois group.
\[cor:rec\] Let us consider the adjoint action $\rm{ Adj}\colon G \to {\rm GL}(\mathfrak g)$ and its derivative ${\rm adj}\colon \mathfrak{g}\to \operatorname{End}(\mathfrak g)$. The following facts hold:
- $\operatorname{Gal}(\nabla^{\rm rec}) = \operatorname{Adj}({\rm Gal(\Theta)})$;
- $\mathfrak{gal}(\nabla^{\rm rec}) = {\rm adj}(\mathfrak{gal}(\Theta))$;
- if $\mathfrak g$ is centerless then $\mathfrak{gal}(\nabla^{\rm rec})$ is isomorphic to $\mathfrak{gal}(\Theta)$;
- assume $\mathfrak g$ is centerless, then the necessary and sufficient condition for $(M,\omega)$ and $(G,\theta)$ to be isogenous is that $\mathfrak{gal}(\nabla^{\rm rec})=\{0\}$.
\(a) and (b). First, by Proposition \[adj\_conjugation\] we have that $\operatorname{Gal}(\nabla^{\rm rec}) = \operatorname{Gal}(\nabla^{\rm adj})$ and so $\mathfrak{gal}(\nabla^{\rm rec}) = \mathfrak{gal}(\nabla^{\rm adj})$. By definition $\nabla^{\rm adj}$ is the associated connection induced by $\Theta$ in the associated bundle $\mathfrak g\times M$. This trivial bundle is the associated bundle induced by the adjoint representation $\operatorname{Adj}\colon G\to \operatorname{End}(\mathfrak g)$. Then, by Theorem \[associated galois\], we have $\operatorname{Gal}(\nabla^{\rm rec}) = \operatorname{Adj}({\rm Gal(\Theta)})$ and $\operatorname{Gal}(\nabla^{\rm rec}) = \operatorname{Adj}({\rm Gal(\Theta)})$.
\(c) It is a direct consequence of (b). The kernel of ${\rm adj}\colon \mathfrak g \to \operatorname{End}(\mathfrak g)$ is the center of $\mathfrak g$.
\(d) It follows from the definition of Darboux–Cartan connection (see remarks after Definition \[DCconnection\]) that the necessary and sufficient condition for $(M,\omega)$ and $(G,\theta)$ to be isogenous is that $\mathfrak{gal}(\nabla^{\rm rec})=\{0\}$. By point (b) we conclude.
Algebraic Lie algebras
----------------------
Let us consider $(M,\omega)$ a rational coparallelism of type $\mathfrak g$ with $\mathfrak g$ a centerless Lie algebra. We do not assume *a priori* that $\mathfrak g$ is an algebraic Lie algebra. The connection $\nabla^{\rm rec}$ is, as said in Proposition \[adj\_conjugation\], conjugated to the connection in $\mathfrak g\times M$ induced by the adjoint action. Note that, in order to define this connection we do not need the group operation but just the Lie bracket in $\mathfrak g$. We have an exact sequence $$\begin{gathered}
0\to \mathfrak g' \to \mathfrak g \to \mathfrak g^{ab}\to 0,\end{gathered}$$ where $\mathfrak g'$ is the derived algebra $[\mathfrak g, \mathfrak g]$. Since the Galois group acts by adjoint action, we have that $\mathfrak g'\times M$ is stabilized by the connection $\nabla^{\rm rec}$ and thus we have an exact sequence of connections $$\begin{gathered}
0\to (\mathfrak g'\times M, \nabla')\to \big(\mathfrak g \times M,\nabla^{\rm rec}\big)\to \big(\mathfrak g^{ab}\times M, \nabla^{ab}\big)\to 0.\end{gathered}$$
The Galois group of $\nabla^{ab}$ is the identity, therefore $\nabla^{ab}$ has a basis of rational horizontal sections.
By definition, the action of $\mathfrak g$ in $\mathfrak g^{ab}$ vanishes. Thus, the constant functions $M\to \mathfrak g^{ab}$ are rational horizontal sections.
\[th:algebraic\] Let $\omega$ be a rational coparallelism of $M$ of type $\mathfrak g$ with $\mathfrak g$ a centerless Lie algebra. If $\mathfrak{gal}(\nabla^{\rm rec}) = \{0\}$ then $\mathfrak g$ is an algebraic Lie algebra.
Assume $\mathfrak g$ is a linear Lie algebra and et $E$ be the smallest algebraic subgroup such that $ \operatorname{Lie} (E) = \mathfrak e\supset \mathfrak g$. We may assume that $E$ is also centerless. Let $A_{1}, \ldots, A_{r}$ be a basis of $\mathfrak g$, for $i=1,\ldots,r$, $X_{i} = \omega^{-1}(A_{i})$. Complete with $B_{1}, \ldots, B_{p}$ in such way that $A_1,\ldots,A_r,B_1,\ldots,B_p$ is a basis of $\mathfrak e$. We consider in $E \times M$ the distribution spanned by the vector fields $A_{i}+X_{i}$. This is a $E$-principal connection called $\nabla$.
Let $\overline{\nabla}$ be the induced connection via the adjoint representation on $\mathfrak e \times M$ then
1. $\overline{\nabla}$ preserves $\mathfrak g$ and $\overline{\nabla}|_{\mathfrak g} = \nabla^{\rm rec}$, by hypothesis $\mathfrak{gal}(\overline{\nabla}|_{\mathfrak g}) = \{0\}$;
2. if $\widetilde{\nabla}$ is the quotient connection on ${\mathfrak e}/ \mathfrak g$ then $\mathfrak{gal}(\widetilde{\nabla}) =\{0\}$.
If $\varphi \in \mathfrak{gal}(\overline{\nabla})$ then for any $X \in \mathfrak g$, $[X,B_{i}] \in \mathfrak g$ thus $0 = \varphi [X,B_{i}] = [ X, \varphi B_{i}]$ and $\varphi B_i$ commute with $\mathfrak g$. From the second point above $\varphi B \in \mathfrak g$. By hypothesis $\varphi B_i =0$ and $\mathfrak{gal}(\overline{\nabla}) =\{0\}$. The projection on $E$ of an algebraic leaf of $\nabla$ gives an algebraic leaf for the foliation of $E$ by the left translation by $\mathfrak g$. This proves the lemma.
\[th\_criteria\] Let $\mathfrak g$ be a centerless Lie algebra. An algebraic variety $(M,\omega)$ with a rational parallelism of type $\mathfrak g$ is isogenous to an algebraic group if and only if $\mathfrak{gal}(\nabla^{\rm rec}) = \{0\}$.
It follows directly from Lemma \[th:algebraic\] and Corollary \[cor:rec\].
\[th:pair\]Let $\mathfrak g$ be a centerless Lie algebra. Any algebraic variety endowed with a pair of commuting rational parallelisms of type $\mathfrak g$ is isogenous to an algebraic group endowed with its two canonical parallelisms of left and right invariant vector fields.
Just note that to have a pair of commuting parallelism is a more restrictive condition than having a parallelism with vanishing Lie algebra of the Galois group of its reciprocal connection.
This result can be seen as an algebraic version of Wang result in [@Wang]. It gives the classification of algebraic varieties endowed with pairs of commuting parallelisms. Assuming that the Lie algebra is centerless is not a superfluous hypothesis, note that the result clearly does not hold for abelian Lie algebras. There are rational $1$-forms in $\mathbf{CP}_1$ that are not exact (isogenous to $(\mathbf C, {\rm d}z)$) nor logarithmic (isogenous to $(\mathbf C^*,{\rm d}\log(z))$). In these examples, the pair of commuting parallelisms is given by twice the same parallelism.
Let $(M,\omega,\omega')$ be a manifold endowed with a pair of commuting parallelism forms of type $\mathfrak g$, a centerless Lie algebra. From Lemma \[th:algebraic\] we have that $\mathfrak g$ is an algebraic Lie algebra. We can construct the algebraic group enveloping $\mathfrak g$ as follows. We consider the adjoint action $$\begin{gathered}
{\rm adj}\colon \ \mathfrak g\hookrightarrow \operatorname{End}(\mathfrak g).\end{gathered}$$ The algebraic group enveloping $\mathfrak g$ is identified with the algebraic subgroup $G$ of $\operatorname{Aut}(\mathfrak g)$ whose Lie algebra is ${\rm adj}(\mathfrak g)$. From Corollary \[cor:rec\](a), we have that $\operatorname{Gal}(\Theta)=\{e\}$. Thus, there is a rational map $f\colon M\to G$ such that $f^*(\theta) = \omega$, where $\theta$ is the Maurer–Cartan form of $G$. We can express explicitly this map in terms of the commuting parallelism forms. For each $x\in M$ in the domain of regularity of the parallelisms, $\omega(x)$ and $\omega'(x)$ are isomorphisms of $T_xM$ with $\mathfrak g$. We define $$\begin{gathered}
f(x) = - \omega(x)\circ \omega'(x)^{-1}.\end{gathered}$$
In virtue of Corollary \[th:pair\], if $\mathfrak g$ is a non-algebraic centerless Lie algebra, there is no algebraic variety endowed with a pair of regular commuting parallelisms of type $\mathfrak g$. This limits the possible generalizations of Theorem \[TDeligne\].
B. Malgrange has given in [@malgrange-P] another criterion: If $(M, \omega)$ is a parallelized variety and $\mathcal F$ is the foliation on $M \times M$ given by $\operatorname{pr}_1^\ast \omega - \operatorname{pr}_2^\ast \omega = 0$. Then $(M,\omega)$ is birational to an algebraic group if and only if leaves of $\mathcal F$ are graphs of rational maps. The relations with Theorem \[th\_criteria\] and Corollary \[th:pair\] are the following. One can identify $TM$ with the vertical tangent (i.e., the kernel of ${\rm d} \operatorname{pr}_2$) along the diagonal in $M\times M$. The diagonal is a leaf of $\mathcal{F}$ and the linearization of $\mathcal F$ along the diagonal defines a connection $\nabla_{\mathcal F}$ on $TM$. By construction:
- $\nabla_{\mathcal F}$-horizontal sections commute with the parallelism, it is the reciprocal Lie connection;
- if leaves of $\mathcal F$ are algebraic then $\nabla_{\mathcal F}$-horizontal section are algebraic.
Some homogeneous varieties
==========================
The notion of [*isogeny*]{} can be extended beyond the simply-transitive case. Let us consider a complex Lie algebra $\mathfrak g$. An [*infinitesimally homogeneous variety*]{} of type $\mathfrak g$ is a pair $(M,\mathfrak s)$ consisting of a complex smooth irreducible variety $M$ and a finite-dimensional Lie algebra isomorphic to $\mathfrak g$.
As before, we are interested in conjugation by rational and algebraic maps so that, whenever necessary, we replace $M$ by a suitable Zariski open subset. In this context, we say that a dominant rational map $f\colon M_1 \dasharrow M_2$ between varieties of the same dimension conjugates the infinitesimally homogeneous varieties $(M_1,\mathfrak s_1)$ and $(M_2,\mathfrak s_2)$ if $f^*(\mathfrak s_2) = \mathfrak s_1$. We say that $(M_1,\mathfrak s_1)$ and $(M_2,\mathfrak s_2)$ are [*isogenous*]{} if they are conjugated to the same infinitesimally homogeneous space of type $\mathfrak g$.
Let $G$ be an algebraic group over $\mathbf C$, $K$ an algebraic subgroup, $\mathfrak{lie}(G)$ its Lie algebra of left invariant vector fields and $\mathfrak{lie}(G)^{\rm rec}$ its Lie algebra of right invariant vector fields. A natural example of infinitesimally homogeneous space are the homogeneous spaces $G/H$ endowed with the induced action of the Lie algebra $\mathfrak{lie}(G)^{\rm rec}$. We want to recognize when a infinitesimally homogeneous space is isogenous to an homogeneous space. We prove that if $\mathfrak s \subset \mathfrak X(M)$ is a [*normal*]{} Lie algebra of vector fields then $(M,\mathfrak s)$ is isogenous to a homogeneous space. In particular, we prove that any $n$-dimensional infinitesimally homogeneous space of type $\mathfrak{sl}_{n+1}(\mathbf C)$ is isogenous to the projective space. Our answer is based on a generalization of the computations done in Section \[sl2\].
The $\boldsymbol{\mathfrak{sl}_2}$ case
---------------------------------------
Let ${\mathscr{C}}$ be a curve with $X$, $Y$, $H$ three rational vector fields such that $[X,Y] = H$, $[H,X] = -X$ and $[H,Y] = Y$. Then there exists a rational dominant map $h \colon {\mathscr{C}}\dasharrow \mathbf {CP}_1$ such that $X = h^\ast\big(\frac{\partial}{\partial z}\big)$, $H = h^\ast \big(z\frac{\partial}{\partial z}\big)$ and $Y = h^\ast \big(z^2\frac{\partial}{\partial z}\big)$.
Their proof is elementary. We outline here a more sophisticated proof in the case ${\mathscr{C}}= A^1_{\mathbf C}$ that will be generalized in the next section.
Notations are the ones introduced in Section \[sl2\]. ${\mathscr{L}}$ is the space of parameterized arcs $\widehat{z} = \sum_i z^{(i)} \frac{x^i}{i!}$ on ${\mathscr{C}}$. The vector space $\mathbf C X + \mathbf C H + \mathbf C Y$ is denoted by $\mathfrak g$. Let $r_o\colon (\mathbf C, 0) \to A^1_\mathbf C$ be an arc with $r_o'(0) \not = 0$ and consider $V \subset {\mathscr{L}}$ defined by $$\begin{gathered}
V =\{ \widehat{z} \in {\mathscr{L}}\, | \, \widehat{z}^\ast \mathfrak g =r_o^\ast \mathfrak g \}.\end{gathered}$$
This is a $3$-dimensional algebraic variety.
The prolongations ${\mathscr{L}}X$, ${\mathscr{L}}Y$ and ${\mathscr{L}}H$ define a $\mathfrak{sl}_2$-parallelism on $V$.
Let us describe the canonical structure of ${\mathscr{L}}$ (see [@guillemin-sternberg pp. 11–12] or next section for a different presentation). For $k$ an integer greater or equal to $-1$, let us consider the vector field on ${\mathscr{L}}$ $$\begin{gathered}
E_k = \sum_{i\geq k} \frac{i !}{(i-k-1)!} z^{(i-k)}\frac{\partial}{\partial z^{(i)}}.\end{gathered}$$ We define a morphism of Lie algebra $\rho\colon \widehat{\mathfrak X} \to \mathfrak X({\mathscr{L}})$ by $x^{k+1}\frac{\partial}{\partial x} \mapsto E_k$ and the adic continuity.
The Cartan form $\sigma$ $($as defined in Section [\[sl2parallelisms\])]{} restricted to $V$ takes values in the Lie algebra $r_0^*(\mathfrak g)$. It is the parallelism form reciprocal to the parallelism ${\mathscr{L}}X$, ${\mathscr{L}}H$ and ${\mathscr{L}}Y$ of $V$.
Using Corollary \[th:pair\], $V$ is isogeneous to ${\rm PSL}_2(\mathbf C)$ as defined in Definition \[isogenous\]. For $p\in M$, $V_p = \{ \widehat{z} \in V \ |\ \widehat{z}(0)=p\}$ are homogeneous spaces for the action of $\widetilde{K} = \{\varphi \colon (\mathbf C,0) \to (\mathbf C,0) \, |\, r_o\circ \varphi \in V\}$, i.e., ${\mathscr{C}}= V/\widetilde{K}$. Let $K$ be the subgroup of ${\rm PSL}_2(\mathbf C)$ of upper triangular matrices.
The actions of $\widetilde{K}$ on $V$ and the right action of $K$ on ${\rm PSL}_2(\mathbf C)$ are conjugated by the isogeny.
This induces an isogeny between ${\mathscr{C}}$ and $\mathbf {CP}_1$. Let $\pi_1$ and $\pi_2$ be the two maps of the isogeny. A local transformation $\varphi$ such that $\pi_1\circ \varphi = \pi_1$ satisfies $\varphi^\ast \pi_1^\ast(X,H,Y) = \pi_1^\ast(X,H,Y)$ and the same is true for the push-forward $(\pi_2)_\ast \varphi$ of $\varphi$ on $\mathbf {CP}_1$. Then $(\pi_2)_\ast \varphi$ preserves $\frac{\partial}{\partial z}$ and $z\frac{\partial}{\partial z}$. It is the identity. This finishes the proof.
Some jet spaces {#some_jet_spaces}
---------------
Let $M$ be a $n$-dimensional affine variety. The space of parameterized subspaces of $M$ is the set of formal maps: $ M^{[n]} = \{ r\colon (\mathbf C^n,0) \to M \}$. Like the arc space, it has a natural structure of pro-algebraic variety. We will give the construction of its coordinate ring following [@beilinson-drinfeld Section 2.3.2, p. 80]. Let $\mathbf C[\partial_1, \ldots, \partial_n]$ be the $\mathbf C$-vector space of linear partial differential operators with constant coefficients. The coordinate ring of $M^{[n]}$ is $\operatorname{Sym}(\mathbf C[M]\otimes \mathbf C[\partial_1,\ldots,\partial_n]) / \mathcal L$ where
- the tensor product is a tensor product of $\mathbf C$-vector spaces;
- $\operatorname{Sym}( V )$ is the $\mathbf C$-algebra generated by the vector space $V$;
- $\mathbf C[M]\otimes \mathbf C[\partial_1,\ldots, \partial_n]$ has a structure of $\mathbf C[\partial_1,\ldots,\partial_n]$-module [*via*]{} the right composition of differential operators;
- $\operatorname{Sym}(\mathbf C[M]\otimes \mathbf C[\partial_1,\ldots, \partial_n])$ has the induced structure of $\mathbf C[\partial_1,\ldots,\partial_n]$-algebra;
- the Leibniz ideal $\mathcal L$ is the $\mathbf C[\partial_1,\ldots, \partial_n]$-ideal generated by $fg\otimes 1 - (f\otimes1)(g\otimes1)$ for all $(f,g) \in \mathbf C[M]^2$ and by $1 - 1\otimes 1$.
Local coordinates $(z_1, \ldots, z_n)$ on $M$ induce local coordinates on $M^{[n]}$ [*via*]{} the Taylor expansion of maps $r$ at $0$ $$\begin{gathered}
r(x_1\ldots, x_n) = \left( \sum_{\alpha \in \mathbf N^n} r_1^{\alpha} \frac{x^\alpha}{\alpha!}, \ldots,\sum_{\alpha \in \mathbf N^n} r_n^{\alpha} \frac{x^\alpha}{\alpha!} \right).\end{gathered}$$ One denotes by $z_i^{\alpha}\colon M^{[n]} \to \mathbf C$ the function defined by $z_i^{\alpha}(r) = r_i^{\alpha}$. This function is the element $z_i\otimes \partial^\alpha$ in $\mathbf C[M^{[n]}]$.
### Prolongation of vector fields
Any derivation $Y$ of $\mathbf C[M]$ can be trivially extended to a derivation of $\operatorname{Sym}(\mathbf C[M]\otimes \mathbf C[\partial_1, \ldots, \partial_n])$. It preserves the ideal generated by $fg\otimes 1 - (f\otimes1)(g\otimes1)$ for all $(f,g) \in \mathbf C[M]^2$ and by $1 - 1\otimes 1$ and commutes with the action of $\mathbf C[\partial_1, \ldots, \partial_n]$ then it preserves the Leibniz ideal and defines a derivation of $\mathbf C[M^{[n]}]$. This derivation is called the prolongation of $Y$, and it is denoted by $Y^{[n]}$.
The same procedure can be used to define the prolongation of analytic or formal vector fields on $M$ to $M^{[n]}$.
### The canonical structure {#canonicalst}
The jet space $M^{[n]}$ is endowed with a differential structure on its coordinate ring and with a group action by “reparameterizations”. The compatibility condition between these two structures is well-known (see [@guillemin-sternberg pp. 11–23]) and is easily obtained using the construction above.
The action of $\partial_j\colon \mathbf C[M^{[n]}] \to \mathbf C[M^{[n]}]$ can be written in local coordinates and gives the total derivative operator $\sum_{i,\alpha} z_i^{\alpha + 1_j} \frac{\partial}{\partial z_i^{\alpha}}$. It is the differential structure of the jet space. The pro-algebraic group $$\begin{gathered}
\Gamma = \big\{ \gamma\colon (\mathbf C^n,0) \overset{\sim}{\rightarrow} (\mathbf C^n,0); \text{ formal invertible}\big\}\end{gathered}$$ acts on $M^{[n]}$.This action is denoted by $S \gamma (r) = r\circ \gamma$.
These two actions arise from the action of the Lie algebra $\widehat{\mathfrak{X}} = \bigoplus \mathbf C[[x_1,\ldots,x_n]]\partial_i$ on $M^{[n]}$. This action is described on the coordinate ring in the following way. For $\xi \in \widehat{\mathfrak{X}}$, $f\in \mathbf C[M]$ and $P \in \mathbf C[\partial_1,\ldots,\partial_n]$, we define $\xi \cdot ( f \otimes P) = f\otimes (P\circ \xi)|_0$ where the composition is evaluated in $0$ in order to get an element of $\mathbf C[\partial_1,\ldots,\partial_n]$. The action of $\bigoplus \mathbf C \partial_i$ is the differential structure. The action of $\widehat{\mathfrak{X}}^0 = \mathfrak{lie}(\Gamma)$, the Lie subalgebra of vector fields vanishing at $0$ is the infinitesimal part of the action of $\Gamma$.
\[canonique\] Let $M^{[n]\ast}$ be the open subset of submersions. The action above gives a canonical form $\sigma\colon T M^{[n]\ast} \to \widehat{\mathfrak{X}}$ satisfying:
- for any $r \in M^{[n]\ast}$, $\sigma$ is a isomorphism from $T_r M^{[n]\ast}$ to $\widehat{\mathfrak{X}}$;
- for any $\gamma \in \Gamma$, $(S\gamma)^\ast \sigma = \gamma^\ast \circ \sigma$;
- $d\sigma = -\frac{1}{2}[\sigma, \sigma]$.
These equalities are [*not*]{} compatible with the projective systems.
Normal Lie algebras of vectors fields
-------------------------------------
Without lost of generality, we should
1. identify $\mathfrak g$ with its image in $\mathfrak X(M)$;
2. replace $M$ by a Zariski open subvariety on which $\mathfrak g$ is defined and of maximal rank at any point.
If $p\in M$ one can identify $\mathfrak g$ with a Lie subalgebra of $\widehat{\mathfrak X}(M,p)$, the Lie algebra of formal vector fields on $M$ at $p$.
For a Lie subalgebra $\mathfrak g \subset \mathfrak X[M]$, its normalizer at $p\in M$ is $$\begin{gathered}
\widehat{N}(\mathfrak g,p) = \big\{ Y \in \widehat{\mathfrak X}(M,p) \, |\, Y,\mathfrak g] \subset \mathfrak g\big\}.\end{gathered}$$
A Lie subalgebra $\mathfrak g \subset \mathfrak X[M]$ is said to be normal if for generic $p \in M$ on has $ \widehat{N}(\mathfrak g,p) = \mathfrak g$.
If $\mathfrak g$ is transitive then the Lie algebra $ \widehat{N}(\mathfrak g,p) $ is finite-dimensional.
Let $k$ be an integer large enough so that the only element of $\mathfrak g$ vanishing at order $k$ at $p$ is $0$. If $\widehat{N}(\mathfrak g,p) $ is not finite-dimensional then there exists a non-zero $Y \in \widehat{N}(\mathfrak g,p)$ vanishing at order $k+1$ at $p$. For $X \in \mathfrak g$, the Lie bracket $[Y,X]$ is an element of $\mathfrak g$ vanishing at order $k$ at $p$. It is zero meaning that $Y$ is invariant under the flows of vector fields in $\mathfrak g$. The transitivity hypothesis together with $Y(p)=0$ proves the lemma.
If there exists a point $p\in M$ such that $\mathfrak g$ is maximal among finite-dimensional Lie subalgebra of $\widehat{\mathfrak X}(M,p)$ then $\mathfrak g$ is normal.
Because of the preceding lemma, if such a point exists then $\mathfrak g = \widehat{N}(\mathfrak g,p)$ in $\widehat{\mathfrak X}(M,p)$. By transitivity, for any couple of points $(p_1,p_2) \in M^2$ there is a composition of flows of elements of $\mathfrak g$ sending $p_1$ on $p_2$. These flows preserve $\mathfrak g$ thus the equality holds at any $p$.
Let $M$ be $n$-dimensional and $\mathfrak g$ be a transitive Lie subalgebra of rational vector fields isomorphic to $\mathfrak{sl}_{n+1}(\mathbf C)$. Then $\mathfrak g$ is normal (see [@cartan]).
Centerless, transitive and normal $\boldsymbol{\Rightarrow}$ isogenous to a homogeneous space
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\[homogeneous\] Let $M$ be a smooth irreducible algebraic variety over $\mathbf C$ and $\mathfrak g$ be a transitive, centerless, normal, finite-dimensional Lie subalgebra of $\mathfrak X(M)$. Then there exists an algebraic group $G$, an algebraic subgroup $H \subset G$ and an isogeny between $(M,\mathfrak g)$ and $(G/H, \mathfrak{lie}(G))$. Moreover, if $N_G(\mathfrak{lie}(H)) = H$ then the isogeny is a dominant rational map $M \dasharrow G/H$.
Because of the finiteness and the transitivity, there exists an integer $k$ such that at any $p \in M$ and for any $Y \in \widehat{N}(\mathfrak g,p)$, $j_k(Y)(p) \not = 0$, unless $Y=0$.
Let $r_o\colon (\mathbf C^n,0) \to M$ be an invertible formal map with $r_o(0) =p$ a regular point. Let us consider the subspace of $M^{[n]}$ defined by $$\begin{gathered}
V = \{ r \colon (\mathbf C^n,0) \to M \, |\, r^\ast \mathfrak g = r_o^\ast \mathfrak g\}.\end{gathered}$$
$V$ is finite-dimensional.
If $r_o^{-1}\circ r$ is tangent to the identity at order $k$ then the induced automorphism of $\mathfrak g$ is the identity. The map $r_o^{-1}\circ r$ fixes $p$, thus it is the identity. This proves the lemma.
Using $r_o$ one can identify the Lie algebra $\widehat{N}(\mathfrak g,p)$ with a Lie subalgebra of $\widehat{\mathfrak{X}}$. The latter acts on $M^{[n]}$ as described in Section \[canonicalst\]. As an application of the Theorem \[canonique\], one gets:
The restriction of the canonical structure of $M^{[n]}$ gives an parallelism $$\begin{gathered}
TV = r_o^\ast(\widehat{N}(\mathfrak g,p)) \times V,\end{gathered}$$ called the canonical parallelism.
The horizontal sections of the reciprocal Lie connection of the canonical parallelism are $Y^{[n]}$ for $Y \in \widehat{N}(\mathfrak g,q)$ for $q\in M$.
Under the hypothesis of normality of $\mathfrak g$, $V$ has two commuting parallelisms of type $\mathfrak g$.
Using Corollary \[th:pair\], $\mathfrak g$ is the Lie algebra of an algebraic group $G$ isogeneous to $V$. $V$ is foliated by the orbits of the subgroup $K$ of $\Gamma$ stabilizing $V$. This group is algebraic with Lie algebra $\mathfrak k = r_o^\ast (\mathfrak g) \cap \widehat{\mathfrak{X}}^0$. Let $\mathfrak h \subset \mathfrak{lie}(G)$ be the Lie algebra corresponding to $\mathfrak k$ by the isogeny. Then the orbits of $\mathfrak h$ are algebraic. This means that $\mathfrak h$ is the Lie algebra of an algebraic subgroup $H$ of $G$, and that $V/K$ and $G/H$ are isogenous.
Assume that $N_{G}(\mathfrak{lie}(H)) = H$. If $W$ is the isogeny between $V$ and $G$. The push-forward of a local analytic deck transformation of $W \to V$ is a transformation of $G$ preserving each element of $\mathfrak g$, it is a right translation. A deck transformation preserves the orbits of the pull-back of $\mathfrak k$ on $W$. Its push-forward preserves the orbits of a group containing $H$ with the same Lie algebra. By hypothesis the push-forward is in $H$ and then the isogony obtained by taking the quotient under $K$ and $H$ is the graph of a dominant rational map.
Picard–Vessiot theory of a principal connection {#ApA}
===============================================
In the previous reasoning we have used the concept of differential Galois group of a connection. Here, we present a dictionary between invariant connection and strongly normal differential field extension (in the sense of Kolchin). In our setting a differential field is a pair $(\mathcal K, \mathcal D)$ where $\mathcal K$ is a finitely generated field over $\mathbf C$ and $\mathcal D$ is a $\mathcal K$ vector space of derivations of $\mathcal K$ stable by Lie bracket. The dimension of $\mathcal D$ is called the rank of the differential field. Note that we can adapt this notion easily to that of a finite number of commuting derivations by taking a suitable basis of $\mathcal D$. However we prefer to consider the whole space of derivations. With our definition a differential field extension $(\mathcal K, \mathcal D) \to (\mathcal K', \mathcal D')$ is a field extension $\mathcal K \subset \mathcal K'$ such that each element of $\mathcal D$ extends to a unique element of $\mathcal D'$ and such extensions span the space $\mathcal D'$ as $\mathcal K'$-vector space.
Differential field extensions and foliated varieties
----------------------------------------------------
First, let us see that there is a natural dictionary between finitely generated differential fields over $\mathbf C$ and irreducible foliated varieties over $\mathbf C$ modulo birational equivalence. Let $(M,\mathcal F)$ be an irreducible foliated variety of dimension $n$. The distribution $T\mathcal F \subset TM$ is of rank $r\leq n$. We denote by $\mathfrak X_{\mathcal F}$ the space of rational vector fields in $T\mathcal F$; it is a $\mathbf C(M)$-Lie algebra of dimension $r$. Hence, the pair $(\mathbf C(M),\mathfrak X_{\mathcal F})$ is a differential field. The field of constants is the field $\mathbf C(M)^{\mathcal F}$ of rational first integrals of the foliation.
Let $(M,\mathcal F)$ and $(M',\mathcal F')$ be foliated varieties. A regular (rational) map $\phi\colon (M',\mathcal F')\dasharrow (M,\mathcal F)$ is a regular (rational) morphism of foliated varieties if ${\rm d}\phi$ induces an isomorphism between $T_x\mathcal F'$ and $T_{\phi(x)}\mathcal F$ for (generic values of) $x\in M'$. It is clear that $\mathcal F'$ and $\mathcal F$ have the same rank.
A differential field extension, correspond here to a dominant rational map of irreducible foliated varieties $\phi\colon (M',\mathcal F')\dasharrow (M,\mathcal F)$. It induces the extension $\phi^*\colon (\mathbf C(M),\mathfrak X_{\mathcal F})\to (\mathbf C(M'),\mathfrak X_{\mathcal F'})$ by composition with $\phi$.
Let $\mathcal F$ the foliation of $\mathbf C^2$ defined by $\{{\rm d}y-y{\rm d}x = 0\}$. It corresponds to the differential field $\big(\mathbf C(x,e^x), \big\langle\frac{{\rm d}}{{\rm d}x}\big\rangle\big) $.
Throughout this appendix “connection” means “flat connection”.
Invariant $\boldsymbol{\mathcal F}$-connections
-----------------------------------------------
Let us consider from now a foliated manifold of dimension $n$ and rank $r$ without rational first integrals $(M,\mathcal F)$, an algebraic group $G$ and a principal irreducible $G$-bundle $\pi\colon P\to M$. A $G$-invariant connection in the direction of $\mathcal F$ is a foliation $\mathcal F'$ of rank $r$ in $P$ such that:
- $\pi\colon (P,\mathcal F')\to (M,\mathcal F)$ is a dominant regular map of foliated varieties;
- The foliation $\mathcal F'$ is invariant by the action of $G$ in $P$.
With this definition $(\mathbf C(M), \mathfrak X_{\mathcal F})\to (\mathbf C(P), \mathfrak X_{\mathcal F'})$ is a differential field extension. Also, each element $g\in G$ induces a differential field automorphism of $(\mathbf C(P), \mathfrak X_{F'})$ that fixes $(\mathbf C(M), \mathfrak X_F)$ by setting $(g\cdot f)(x) = f(x\cdot g)$.
Let $\mathfrak g$ be the Lie algebra of $G$. There is a way of defining a $G$-equivariant form $\Theta_{\mathcal F'}$ with values in $\mathfrak g$, and defined in ${\rm d}\pi^{-1}(T\mathcal F)$ in such way that $T\mathcal F'$ is the kernel of $\Theta_{\mathcal F'}$. First, there is a canonical form $\Theta_0$ defined in $\ker(d\pi)$ that sends each vertical vector $X_p\in \ker d_p\pi\subset T_pP$ to the element $\mathfrak g$ that verifies, $$\begin{gathered}
\left.\frac{{\rm d}}{{\rm d}\varepsilon}\right|_{\varepsilon = 0} p \cdot \exp{\varepsilon A} = X_p.\end{gathered}$$ This form is $G$-equivariant in the sense that $R_g^*(\Theta_0) = \operatorname{Adj}_{g^{-1}} \circ \omega$. We have a decomposition of the vector bundle ${\rm d}\pi^{-1}(T\mathcal F) = \ker({\rm d}\pi) \oplus T\mathcal F'$. This decomposition allows to extend $\Theta_0$ to a form $\Theta_{\mathcal F'}$ defined for vectors in ${\rm d}\pi^{-1}(T\mathcal F)$ whose kernel is precisely $T\mathcal F$. We call *horizontal frames* to those sections $s$ of $\pi$ such that $s^*(\Theta_{\mathcal F}) = 0$.
Picard–Vessiot bundle
---------------------
We say that the principal $G$-bundle with invariant $\mathcal F$-connection $\pi\colon (P,\mathcal F')\to (M,\mathcal F)$ is a *Picard–Vessiot* bundle if there are no rational first integrals of $\mathcal F'$. The notion of Picard–Vessiot bundle corresponds exactly to that of primitive extension of Kolchin. In such case $G$ is the group of differential field automorphisms of $(\mathbf C(P), \mathfrak X_{F'})$ that fix $(\mathbf C(M), \mathfrak X_F)$ and $(\mathbf C(M), \mathfrak X_F)\to (\mathbf C(P), \mathfrak X_{F'})$ is a strongly normal extension. Moreover, any strongly normal extension with constant field $\mathbf C$ can be constructed in this way (see [@Kolchin Chapter VI, Section 10, Theorem 9]).
One of the most remarkable properties of strongly normal extensions is the Galois correspondence (from [@Kolchin Chapter VI, Section 4]).
Assume that $(\mathbf C(M),\mathfrak X_{\mathcal F})\to (\mathbf C(P),\mathfrak X_{\mathcal F'})$ is strongly normal with group of automorphisms $G$. Then, there is a bijection between the set of intermediate differential field extensions and algebraic subgroups of $G$. To each intermediate differential field extension, it corresponds the group of automorphisms that fix such an extension point-wise. To each subgroup of automorphisms it corresponds its subfield of fixed elements.
The Picard–Vessiot bundle of an invariant $\boldsymbol{\mathcal F}$-connection
------------------------------------------------------------------------------
Let us consider an irreducible principal $G$-bundle $\pi\colon (P,\mathcal F') \to (M,\mathcal F)$ endowed with an invariant $\mathcal F$-connection $\mathcal F'$. We assume that $\mathcal F$ has no rational first integrals. A result of Bonnet (see [@Bonnet Theorem 1.1]) ensures that for a very generic point in $M$ the leaf passing through such point is Zariski dense in $M$. Let us consider such a Zariski-dense leaf $\mathcal L$ of $\mathcal F$ in $M$. Let us consider any leaf $\mathcal L'$ of $\mathcal F'$ in $P$ that projects by $\pi$ onto $\mathcal L$. Its Zariski closure is unique in the following sense:
\[uniqueness\]Let $\mathcal L'$ and $\mathcal L''$ two leaves of $\mathcal F'$ whose projections by $\pi$ are Zariski dense in $M$. Then, there exist an element $g\in G$ such that $\overline{\mathcal L'} \cdot g = \overline{\mathcal L''}$.
By construction, there is some $x\in\pi(\mathcal L')\cap\pi(\overline{\mathcal L''})$. Let us consider $p\in \pi^{-1}(\{x\})\cap \mathcal L'$ and $q\in\pi^{-1}(\{x\})\cap \overline{\mathcal L''}$. Since $p$ and $q$ are in the same fiber, there is a unique element $g\in G$ such that $p\cdot g = q$. By the $G$-invariance of the connection $\mathcal L'\cdot g$ is the leaf of $\mathcal F'$ that passes through $q$. The set $\overline{\mathcal L''}$ is, by construction, union of leaves of $\mathcal F'$ and contains the point $q$. Thus, $\overline{\mathcal L' \cdot g} \subseteq \overline{\mathcal L''}$, and $\overline{\mathcal L'}\cdot g \subseteq \overline{\mathcal L''}$. Now, by exchanging the roles of $\mathcal L'$ and $\mathcal L''$, we prove that there is an element $h$ such that $\overline{\mathcal L''}\cdot h\subseteq \overline{\mathcal L'}$. It follows $h = g^{-1}$. This finishes the proof.
Let $L$ be the Zariski closure of $\mathcal L'$. Let us consider the algebraic subgroup $$\begin{gathered}
H = \{g\in G \colon L \cdot g = L\}\end{gathered}$$ stabilizing $L$. The projection $\pi$ restricted to $L$ is dominant, thus there is a Zariski open subset $M^\star$ such that $\pi^\star \colon L^\star \to M^\star$ is surjective. Let us call $\mathcal F^\star$ the restriction of $\mathcal F'$ to $L^\star$. It follows that the bundle: $\pi^\star \colon (L^\star,\mathcal F^\star) \to (M^\star, \mathcal F|_{M^\star})$ is a principal bundle of structure group $H$ called Picard–Vessiot bundle. The differential field extension $(\mathbf C(M),\mathfrak X_{\mathcal F}) \to (\mathbf C(L^\star),\mathfrak X_{\mathcal F^\star})$ is the so-called Picard–Vessiot extension associated to the connection. The algebraic group $H$ is the differential Galois group of the connection.
Split of a connection
---------------------
Let us consider a pair of morphisms of foliated varieties $$\begin{gathered}
\phi_j\colon \ (M_j,\mathcal F_j)\to(M,\mathcal F),\qquad \mbox{for} \quad j=1,2.\end{gathered}$$ Then, we can define in $M_1\times_M M_2$ a foliation $\mathcal F_1\times_{\mathcal F} \mathcal F_2$ in the following way. A vector $X = (X_1,X_2)$ is in $T(\mathcal F_1\times_{\mathcal F}\mathcal F_2)$ if and only if ${\rm d}\phi_1(X_1)= {\rm d}\phi_2(X_2)\in T\mathcal F$. Let us consider $(P,\mathcal F')$ a principal $\mathcal F$ connection. Note that the projection $$\begin{gathered}
\pi_1 \colon \ (M_1\times_M P, \mathcal F_1\times_{\mathcal F} \mathcal F')\to (M_1,\mathcal F_1)\end{gathered}$$ is a principal $G$-bundle endowed of a $\mathcal F_1$-connection. We call this bundle the pullback of $(P,\mathcal F')$ by $\phi_1$.
We also may consider the trivial $G$-invariant connection $\mathcal F_0$ in the trivial principal $G$-bundle $$\begin{gathered}
\pi_0\colon \ (M\times G,\mathcal F_0) \to (M,\mathcal F),\end{gathered}$$ for what the leaves of $\mathcal F_0$ are of the form $(\mathcal L, g)$ where $\mathcal L$ is a leaf of $\mathcal F$ and $g$ a fixed element of $G$. We say that the $G$-invariant connection $(P,\mathcal F')$ is rationally trivial if there is a birational $G$-equivariant isomorphism of foliated manifolds between $(P,\mathcal F)$ and $(M\times G, \mathcal F_0)$.
Invariant connections are always trivialized after pullback; there is a universal $G$-equivariant isomorphism defined over $P$ $$\begin{gathered}
(P \times G, \mathcal F'\times_{\mathcal F} \mathcal F_0) \to (P\times_M P, \mathcal F'\times_{\mathcal F}\mathcal F'), \qquad (p, g) \mapsto (p, p\cdot g),\end{gathered}$$ that trivializes any $G$-invariant connection. However, the differential field $(\mathbf C(P),\mathfrak X_{\mathcal F'})$ may have new constant elements. To avoid this, we replace the pullback to $P$ by a pullback to the Picard–Vessiot bundle $L^\star$ $$\begin{gathered}
(L^\star \times G, \mathcal F^\star\times_{\mathcal F} \mathcal F_0) \to (L^\star\times_M P, \mathcal F^\star\times_{\mathcal F}\mathcal F'), \qquad (p, g) \mapsto (p, p\cdot g).\end{gathered}$$
The Picard–Vessiot bundle has some minimality property. It is the smallest bundle on $M$ that trivializes the connection. We have the following result.
\[uniqueness2\] Let us consider $\pi\colon (P,\mathcal F') \to (M,\mathcal F)$ be as above, $\pi^\star \colon (L^\star,\mathcal F^\star)\to (M,\mathcal F)$ the Picard–Vessiot bundle, and and $\phi\colon \big(\tilde M, \tilde{\mathcal F}\big) \to (M, \mathcal F)$ any dominant rational map of foliated varieties such that:
- $\tilde{\mathcal F}$ has no rational first integrals in $\tilde M$;
- the pullback $\big(\tilde M \times_M P,\tilde F \times_{\mathcal F}\mathcal F'\big)\to \big(\tilde M, \tilde{\mathcal F}\big)$ is rationally trivial.
There is a dominant rational map of foliated varieties $\psi\colon \tilde M \dasharrow L^\star$ such that $\pi^\star\circ\psi = \phi$ in their common domain.
Let us take $\tau \colon \tilde M \times G \dasharrow \tilde M \times_M P$ a birational trivialization, $\pi_2 \colon \tilde M \times_M P \to P$ be the projection in the second factor, and $\iota \colon \tilde M \to \tilde M \times G$ the inclusion $p \mapsto (p,e)$. Then, $\tilde\psi = \pi_2 \circ \tau \circ \iota$ is a rational map from $\tilde M$ to $P$ whose differential sends $T\tilde{\mathcal F}$ to $T\mathcal F$. By Bonnet theorem, $\tilde M$ is the Zariski closure of a leaf of $\tilde{\mathcal F}$ that projects by $\phi$ into a Zariski dense leaf of $\mathcal F$. From this, $\tilde\psi$ contains a dense leaf of $\mathcal F'$ in $P$. By applying a suitable right translation in $P$ and the uniqueness Theorem \[uniqueness\], we obtain the desired conclusion.
Linear connections {#A6}
------------------
Let $(M,\mathcal F)$ be as above, of dimension $n$ and rank $r$. Let $\xi\colon E\to M$ be a vector bundle of rank $k$. A linear integrable $\mathcal F$-connection is a foliation $\mathcal F_E$ of rank $r$ which is compatible with the structure of vector bundle in the following sense: the point-wise addition of two leaves of any dilation of a leaf is also a leaf. This can also be stated in terms of a covariant derivative operator $\nabla$ wich is defined only in the direction of $\mathcal F$. First, the kernel of ${\rm d}\xi$ is naturally projected onto $E$ itself $$\begin{gathered}
{\rm vert}_0 \colon \ \ker({\rm d}\xi) \to E, \qquad X_v \mapsto w,\end{gathered}$$ where $\left.\frac{{\rm d}}{{\rm d}\varepsilon}\right|_{\varepsilon = 0} v + \varepsilon w = X_v$. Then, the decomposition of ${\rm d}\xi^{-1}(T\mathcal F)$ as $\ker({\rm d}\xi)\oplus T\mathcal F_E$ allows us to extend ${\rm vert_0}$ to a projection $$\begin{gathered}
{\rm vert}\colon \ {\rm d}\xi^{-1}(T\mathcal F) \to E.\end{gathered}$$ Thus, we define for each section $s$ its covariant derivative $\nabla s = s^*({\rm vert}\circ {\rm d}s|_{T\mathcal F})$. This is a $1$-form on $M$ defined only for vectors in $T \mathcal F$. This covariant derivative has the desired properties, it is additive and satisfies the Leibniz formula $$\begin{gathered}
\nabla (fs) = {\rm d}f|_{T\mathcal F}\otimes s + f \nabla s.\end{gathered}$$ In general, we write for $X$ a vector in $T\mathcal F$, $\nabla_X s$ for the contraction of $\nabla s$ with the vector $X$. It is an element of $E$ over the same base point in $M$ that the vector $X$. We call *horizontal sections* to those sections $s$ of $\xi$ such that $\nabla s = 0$.
Let $\pi\colon R^1(E)\to M$ be the bundle of linear frames in $E$. It is a principal linear ${\rm GL}_k(\mathbf C)$-bundle. The foliation $\mathcal F_E$ induces a foliation $\mathcal F'$ in $R^1(E)$ that is a $G$-invariant $\mathcal F$-connection. Let us consider the Picard–Vessiot bundle, $(L^\star,\mathcal F^\star)$. The uniqueness Theorem \[uniqueness2\] on the Picard–Vessiot bundle, can be rephrased algebraically in the following way. The Picard–Vessiot extension $(\mathbf C(M), \mathfrak X_{\mathcal F}) \to (\mathbf C(L^\star), \mathfrak X_{\mathcal F^\star})$ is characterized by the following properties (cf. [@SingerVanderput Section 1.3]):
- there are no new constants, $\mathbf C(L^\star) = \mathbf C$;
- it is spanned, as a field extension of $\mathbf C(M)$, by the coefficients of a fundamental matrix of solutions of the differential equation of the horizontal sections.
Associated connections {#ap7_associated}
----------------------
Let $\pi\colon (P,\mathcal F')\to (M,\mathcal F)$ be a $G$-invariant connection, as before, where $\mathcal F$ is a foliation in $M$ without rational first integrals. Let us consider $\phi\colon G\to {\rm GL}(V)$ a finite-dimensional linear representation of $G$. It is well known that the associated bundle $\pi_P \colon V_P \to M$, $$\begin{gathered}
V_P = P\times_G V = (P\times V)/G, \qquad (p\cdot g,v) \sim (p,g\cdot v),\end{gathered}$$ is a vector bundle with fiber $V$. Here we represent the action of $G$ in $V$ by the same operation symbol than before. The $G$-invariant connection $\mathcal F'$ rises to a foliation in $P\times G$ and then it is projected to a foliation $\mathcal F_V$ in $V_P$. In this way, the projection $$\begin{gathered}
\pi_P\colon \ (V_P,\mathcal F_V)\to (M,\mathcal F),\end{gathered}$$ turns out to be a linear $\mathcal F$-connection. It is called the *Lie–Vessiot* connection induced in the associated bundle. The Galois group of the principal and the associated Lie–Vessiot connection are linked in the following way.
\[associated galois\]Let $H\subset G$ be the Galois group of the principal connection $\mathcal F'$. Then, the Galois group of the associated Lie–Vessiot connection $\mathcal F_V$ is $\phi(H)\subseteq {\rm GL}(V)$.
Let us consider the bundle of frames $R^1(V_P)$, with its induced invariant connection $\mathcal F''$. Let us fix a basis $\{v_1,\ldots,v_r\}$ of $V$. Then, we have a map $$\begin{gathered}
\tilde \pi \colon \ P \to R^{1}(V_P), \qquad p\mapsto ([p, v_1],\ldots, [p, v_r]),\end{gathered}$$ where the pair $[p, v]$ represents the class of the pair $(p,v)\in P\times V$. By construction, $\tilde\pi$ sends $T\mathcal F'$ to $T\mathcal F''$. It implies that, if $\mathcal L^{\star}$ is a Picard–Vessiot bundle for $\mathcal F'$ then $\tilde\pi(L^\star)$ is a Picard–Vessiot bundle for $\mathcal F''$. Second, if $\mathcal L^\star$ is a principal $H$ bundle, then $\tilde\pi(L^\star)$ is a principal $H/K$ bundle where $K$ is the subgroup of $H$ that stabilizes the basis $\{v_1,\ldots,v_r\}$.
Let us discuss how the covariant derivative operator in $\nabla$ is defined in terms of $\Theta_{\mathcal F'}$ and the action of $G$ in $V$. Let us denote by $\phi'\colon \mathfrak g\to \mathfrak{gl}(V)$ the induced Lie algebra morphism. Let $s$ be a local section of $\xi$. Let us consider the canonical projection $\bar\pi \colon P \times V \to V(P)$. This turns out to be also a principal bundle, here the action on pairs is $(p,v)\cdot g = \big(p \cdot g, g^{-1} \cdot v\big)$. Now we can take any section $r$ of this bundle, and define $\tilde s = r\circ s$. As $r$ takes values in a cartesian product, we obtain $\tilde s = (s_1, s_2)$ where $s_1$ is a section of $\pi$ and $s_2$ is a function with values in $V$. Finally we obtain $$\begin{gathered}
\label{covariant_associated}
\nabla s = {\rm d}s_2|_{T\mathcal F} - \phi'(s_1^*(\Theta_{\mathcal F'}))(s_2).\end{gathered}$$ A calculation shows that it does not depend of the choice of $r$ and it is the covariant derivative operator associated to $\mathcal F_V$. In particular, if $s_2$ is already an horizontal frame, then the covariant differential is given by the first term ${\rm d}s_s|_{T\mathcal F}$.
Deligne’s realization of Lie algebra {#apB}
====================================
The proof of the existence of a regular parallelism for any complex Lie algebra $\mathfrak g$ is written in a set of two letters from P. Deligne to B. Malgrange (dated from November of 2005 and February of 2010 respectively) that are published verbatim in [@Malgrange]. We reproduce here the proof with some extra details.
\[TDeligne\]Given any complex Lie algebra $\mathfrak g$ there exist an algebraic variety endowed with a regular parallelism of type $\mathfrak g$.
\[ap2\_1\]Let $T$ be an algebraic torus acting regularly by automophisms in some algebraic group $H$ and let $\mathfrak t$ be the Lie algebra of $T$. Let us consider the semidirect product $$\begin{gathered}
\mathfrak t \ltimes H, \qquad (t,h)(t',h') = (t+t', (\exp(t')\cdot h)h')\end{gathered}$$ as an algebraic variety and analytic Lie group. Its left invariant vector fields form a regular parallelism of $\mathfrak t \ltimes H$. The Galois group of this parallelism is a torus.
Let us denote by $\alpha$ the action of $T$ in $H$ and $\alpha'\colon \mathfrak t \mapsto \mathfrak X[H]$ the Lie algebra isomorphism given by the infinitesimal generators $$\begin{gathered}
(\alpha'X)_h= \left.\frac{{\rm d}}{{\rm d}\varepsilon}\right|_{\varepsilon=0} \alpha_{\exp(\varepsilon t)}(h).\end{gathered}$$ Let $X$ be an invariant vector field in $\mathfrak t$. Let us compute the left invariant vector field in $t\ltimes H$ whose value at the identity is $(X_0,0)$. In order to perform the computation we write the vector as an infinitesimally near point to $(0,e)$. $$\begin{gathered}
L_{(t,h)}(0 + \varepsilon X_0, e) = (t + \varepsilon X_t, \alpha_{\exp(\varepsilon X)}(h) ) = (t + \varepsilon X_t, h + \varepsilon (\alpha'X)_h).\end{gathered}$$ And thus $dL_{(t,h)}(X_0,0) = (X_t, (\alpha'X)_h)$. We conclude that $(X,\alpha'X)\in\mathfrak X[\mathfrak t\ltimes H]$ is the left invariant vector field whose value at $(0,e)$ is $(X_0,0)$. Let us consider now $Y$ a left invariant vector field in $H$. Let us compute, as before, the left invariant vector field whose value at $(t,h)$ is $(0,Y_h)$ $$\begin{gathered}
L_{(t,h)}(0,e+\varepsilon Y_e) = (t, L_h(e + \varepsilon Y_e)) = (t, h + \varepsilon Y_h).\end{gathered}$$ And thus $(0,Y)$ is the left invariant vector field whose value at $(0,e)$ is $(0,Y_e)$. These vector fields of the form $(X,\alpha'X)$ and $(0,Y)$ are regular and span the Lie algebra of left invariant vector fields in $\mathfrak t\ltimes H$. Hence, they form a regular parallelism.
In order to compute the Galois group of the parallelism, let us compute its reciprocal parallelism. It is formed by the right invariant vector fields in the analytic Lie group $\mathfrak t \ltimes H$. A similar computation proves that if $X$ is an invariant vector field in $\mathfrak t$ then $(X,0)$ is right invariant in $\mathfrak t \ltimes H$. For each element $\tau\in T$, $\alpha_\tau$ is a group automorphism of $H$. Thus, it induces a derived automorphism $\alpha_{\tau*}$ of the Lie algebra of regular vector fields in $H$. Let $Y$ be now a right invariant vector field in $H$. Let us compute the right invariant vector field $Z$ in $\mathfrak t\ltimes H$ whose value at $(0,e)$ is $(0,Y_e)$: $$\begin{gathered}
R_{(t,h)}(0,e+ \varepsilon Y_e) = (t,\alpha_{\exp(t)}(e+ \varepsilon Y_e)h) = (t, h + \varepsilon (\alpha_{\exp(t)*}Y)_h)\end{gathered}$$ and $Z_{t,h} = (0, (\alpha_{\exp(t)*} Y)_h)$. Those analytic vector fields depend on the exponential function in a torus thus we can conclude, by a standard argument of differential Galois theory, that the associated differential Galois group is a torus.
Let us consider $\mathfrak g$ an arbitrary, non algebraic, finite-dimensional complex Lie algebra. We consider an embedding of $\mathfrak g$ in the Lie algebra of general linear group and $E$ the smallest algebraic subgroup whose Lie algebra $\mathfrak e$ contains $\mathfrak g$. $E$ is a connected linear algebraic group.
\[ap2\_2\] With the above definitions and notation $[\mathfrak e, \mathfrak e] = [\mathfrak g, \mathfrak g]$.
Let $H$ be the group of matrices that stabilizes $\mathfrak g$ and acts trivially on $\mathfrak g/[\mathfrak g, \mathfrak g]$. Its Lie algebra $\mathfrak h$ contains $\mathfrak g$ and thus $H\supseteq E$ and $\mathfrak h \supseteq \mathfrak e$. By definition of $H$ we have $[\mathfrak h, \mathfrak g] = [\mathfrak g, \mathfrak g]$, therefore $[\mathfrak e, \mathfrak g]\subseteq [\mathfrak g,\mathfrak g]$. Let us now consider the group $H_1$ that stabilizes $\mathfrak e$ and $\mathfrak g$ and that acts trivially in $\mathfrak e/[\mathfrak g, \mathfrak g]$. This is again an algebraic group containing $E$, and its Lie algebra $\mathfrak h_1$ satisfies $[\mathfrak h_1,\mathfrak e] \subseteq [\mathfrak g, \mathfrak g]$. Taking into account $\mathfrak e \subseteq \mathfrak h_1$ we have $[\mathfrak e, \mathfrak e] \subseteq [\mathfrak g, \mathfrak g]$. The other inclusion is trivial.
Because of Lemma \[ap2\_2\], the abelianized Lie algebra $\mathfrak g^{ab} = \mathfrak g / [\mathfrak g, \mathfrak g]$ is a subspace of $\mathfrak e^{ab} = \mathfrak e/ [\mathfrak e, \mathfrak e]$. Moreover, if we consider the quotient map, $\pi\colon \mathfrak e \to \mathfrak e^{ab}$, then $\mathfrak g = \pi^{-1}(\mathfrak g^{ab})$.
Let us consider an algebraic Levy decomposition $E\simeq L \ltimes U$ (see [@Onishchik Chapter 6]). Here, $L$ is reductive and $U$ is the unipotent radical, consisting in all the unipotent elements of $E$. The semidirect product structure is produced by an action of $L$ in $U$, so that, $(l_1,u_1)(l_2,u_2) = (l_1l_2, (l_2\cdot u_1)u_2)$.
Since $L$ is reductive, its commutator subgroup $L'$ is semisimple. Let $T$ be the center of $L$, which is a torus, the map $$\begin{gathered}
\varphi\colon \ T\times L' \to L, \qquad (t,l) \mapsto tl,\end{gathered}$$ is an isogeny. The isogeny defines an action of $T\times L'$ in $U$ by $(t,l)\cdot u = tl\cdot u$. We have found an isogeny $$\begin{gathered}
(T\times L')\ltimes U \to E.\end{gathered}$$
The Lie algebra $\mathfrak u$ of $U$ is a nilpotent Lie algebra, so that the exponential map $\exp\colon \mathfrak u \to U$ is regular and bijective. In general, if $V$ is an abelian quotient of $U$ with Lie algebra $\mathfrak v$ then the exponential map conjugates the addition law in $\mathfrak v$ with the group law in $V$.
\[ap2\_3\] With the above definitions and notation, let $\bar{\mathfrak u}$ be the biggest quotient of $\mathfrak u^{ab}$ in which $L$ acts by the identity. We have a Lie algebra isomorphism $\mathfrak e^{ab} \simeq \mathfrak t \times \bar{\mathfrak u}$.
Let us compute $\mathfrak e^{ab}$. We compute the commutators $\mathfrak e$ by means of the isomorphism $\mathfrak e \simeq (\mathfrak t \times \mathfrak l') \ltimes \mathfrak u$. We obtain $$\begin{gathered}
= (0,[l_1,l_2], a(t_2,l_2)u_1 +[u_1,u_2]),\end{gathered}$$ where $a$ represents the derivative at $(e,e)$ of the action of $L$ in $U$. From this we obtain that $[\mathfrak e, \mathfrak e]$ is spanned by $(\{0\}\times \mathfrak l')\ltimes \mathfrak \{0\}$, $\{0\}\ltimes [\mathfrak u, \mathfrak u]$ and $\{0\}\times \langle a(\mathfrak l)\mathfrak u \rangle$. Taking into account that $\bar{\mathfrak u}/ ( \langle a(\mathfrak l)\mathfrak u \rangle + [\mathfrak u, \mathfrak u] )$ is the biggest quotient of $\mathfrak u^{ab}$ in which $L$ acts trivially, we obtain the result of the lemma.
Let $\mathfrak t$ be the Lie algebra of $T$. Its exponential map is an analytic group morphism and thus we may consider the analytic action of $\mathfrak t \times L'$ in $U$ given by $(t,l)\cdot u = (\exp(t) l )\cdot u$. Let $\tilde E$ be the algebraic variety and analytic Lie group $(\mathfrak t \times L') \ltimes U$. By application of Lemma \[ap2\_1\], and taking into account that $\tilde E \simeq \mathfrak t \ltimes H$, where $H$ is the group $L'\cdot U$, we have that the left invariant vector fields in $\tilde E$ are regular. Let us consider the projection $$\begin{gathered}
\pi_1\colon \ \tilde E \to \mathfrak e^{ab} = \mathfrak t \times \bar{\mathfrak u}, \qquad (t,l,u) \mapsto (t,[\log(u)]),\end{gathered}$$ this projection is algebraic by construction, and also a morphism of Lie groups. By Lemmas \[ap2\_2\] and \[ap2\_3\], $\mathfrak g^{ab}$ is a vector subspace of the image. Then, let us take $\tilde G$ the fiber $\pi^{-1}_1(\mathfrak g^{ab})$. It is an algebraic submanifold of $\tilde E$ and an analytic Lie group. The derivative at the identity of $\pi_1$ is precisely the abelianization $\pi$ and it follows that the Lie algebra of $\tilde G$ is precisely $\mathfrak g$. Finally $\tilde G$ is an algebraic variety with a regular $\mathfrak g$-parallelism. This finishes the proof of Theorem \[TDeligne\].
The right invariant vector fields in $\tilde G$ are constructed as in Lemma \[ap2\_1\] by means of the exponential function in the torus. Hence, Galois groups of the parallelisms obtained via this construction are always tori.
Acknowledgements {#acknowledgements .unnumbered}
----------------
The authors thank the ECOS-Nord program C12M01 and the project “IsoGalois” ANR-13-JS01-0002-01. They also thank the “Universidad Nacional de Colombia”(project HERMES code 37243) and the “Université de Rennes 1” (Actions Internationales 2016) for supporting this reseach, and also the Centre Henri Lebesgue ANR-11-LABX-0020-01 for creating an attractive mathematical environment.
The authors thank Juan Diego Vélez for his help with the final redaction of the manuscript and the anonymous referees who gave relevant contributions to improve the paper.
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| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'Taiya <span style="font-variant:small-caps;">Munenaka</span> and Hirohiko <span style="font-variant:small-caps;">Sato</span>[^1]'
title: ' A Novel Pyrochlore Ruthenate: Ca$_{2}$Ru$_{2}$O$_{7}$ '
---
Frustration results in many types of unexpected phenomena. Among three-dimensional frustrated systems, the pyrochlore lattice is particularly interesting for its strong frustration originating from a purely geometric reason. A typcial pyrochlore oxide has the composition A$_{2}$B$_{2}$O$_{7}$. In this system, the B sites (and also the A sites) form a three-dimensional network based on the B$_{4}$ tetrahedron. From another point of view, we can regard the pyrochlore lattice as a three-dimensional version of a Kagomé lattice. Therefore, perfect geometric frustration is inherent in this structure, and many interesting phenomena emerge. For example, Y$_{2}$Mo$_{2}$O$_{7}$ exhibits spin-glass behavior,[@gingras97; @gardner99; @miyoshi00] demonstrating that the geometric frustration due to the antiferromagnetic pyrochlore lattice itself is responsible for the glassy state, even if there is no structural disorder. On the other hand, the nearest-neighbor interaction is ferromagnetic in Ho$_{2}$Ti$_{2}$O$_{7}$ and Dy$_{2}$Ti$_{2}$O$_{7}$. In this case, single-ion magnetic anisotropy causes another type of frustration and consequently, “spin ice” behavior appears.[@harris97; @ramirez99; @higashinaka05]
Conductive pyrochlores are also remarkable systems. For Nd$_{2}$Mo$_{2}$O$_{7}$, there was the epoch-making interpretation that the anomalous Hall effect detects the chirality of Nd magnetic moments.[@taguchi03] A theoretical study proved that the Berry phase plays an important role in systems with a chiral spin arrangement.[@onoda03] Tl$_{2}$Mn$_{2}$O$_{7}$ has metallic conductivity and undergoes a ferromagnetic transition. Near the transition temperature, a giant magnetoresistance appears[@shimakawa96]. In Cd$_{2}$Re$_{2}$O$_{7}$, a superconducting transition was discovered at 1.5 K.[@sakai01; @hanawa01] Furthermore, $\beta$-type pyrochlore osmates, AOs$_{2}$O$_{6}$ (A = K, Rb, Cs), also exhibit superconductivity with a relatively high $T_{c}$ (9.7 K for A = K).[@yonezawa04; @yonezawa04b; @hiroi04; @hiroi05]
While searching for new materials with exotic electronic states, we have become interested in pyrochlore ruthenates. Because ruthenium 4$d$-orbitals have a character intermediate between localized and itinerant orbitals, a variety of electronic phases appear. In particular, the discovery of spin-triplet superconductivity in Sr$_{2}$RuO$_{4}$[@maeno94; @ishida98] has aroused the interest of many material scientists.
Ruthenates with pyrochlore structures have also been actively investigated. Bi$_{2}$Ru$_{2}$O$_{7}$ and Pb$_{2}$Ru$_{2}$O$_{6.5}$ are metallic with Pauli paramagnetism,[@longo69; @cox83; @hsu88] whereas Ln$_{2}$Ru$_{2}$O$_{7}$ and Y$_{2}$Ru$_{2}$O$_{7}$ are insulators with localized magnetic moments.[@aleonard62; @subramanian83; @yoshii99; @ito00] Tl$_{2}$Ru$_{2}$O$_{7}$ undergoes a metal-insulator transition.[@takeda98] These observations reveal that pyrochlore ruthenates display a variety of electronic phases, widely distributed over the Mott boundary. Their electronic states are very sensitive to the Ru-O distance or the Ru-O-Ru bond angle, which is related to the radius of the cations on the A site. In addition to controlling the band width by changing the cation radius, filling control of the $4d$-band also seems important in searching for novel electronic phases. However, there have been few trials[@yoshii99] on controlling the band filling of pyrochlore ruthenates. This is probably because the cation on the A site is trivalent in most stable pyrochlore ruthenates. Apart from Cd$_{2}$Ru$_{2}$O$_{7}$[@wang98], there are no reports on stoichiometric pyrochlores composed of only Ru$^{5+}$. In the present study, we succeeded in synthesizing a new pyrochlore ruthenate with Ru$^{5+}$, Ca$_{2}$Ru$_{2}$O$_{7}$, by maintaining a high-oxidization atmosphere.
Single crystals of Ca$_{2}$Ru$_{2}$O$_{7}$ were synthesized by a hydrothermal method. A mixture of RuO$_{2}$ (40 mg), obtained by oxidizing Ru metal (Furuya Metals, 99.99% purity), CaO (34 mg, Soekawa Chemical, 99.99% purity), and 0.3 ml of 30% H$_{2}$O$_{2}$ solution was encapsulated in a gold tube. Then, it was kept in an autoclave under 150 MPa hydrostatic pressure at 600$^{\circ}$C for 3 days. The chemical composition was determined using an energy dispersive X-ray spectrometer (EDS) installed on a scanning electron microscope. The crystal structure was analyzed using a single crystal and an imaging-plate X-ray diffractometer (Rigaku, R-Axis RAPID), in which Mo-K$\alpha$ radiation was generated using an X-ray tube and monochromized using graphite. We also used a powder X-ray diffractometer to check whether there was any contamination due to impurity phases. The magnetic susceptibility between 2 and 400 K was measured using a superconducting quantum-interference-device magnetometer. In the measurement, approximately 10 mg of nonoriented single crystals were wrapped in a piece of aluminum foil. The resistivity was measured by a DC four-wire method on an array of single crystals, connected with each other, in a closed-cycle helium refrigerator whose temperature range was between 5.5 and 300 K. The array was composed of four single crystals, and we attached the voltage leads to the same crystal located at the center. Therefore, we consider that the observed resistivity approximately reflects the behavior of a single crystal.
The obtained materials were black crystals with an octahedral shape. Observations using a microscope did not detect any other type of crystal as shown in Fig. \[fig1\](a). An EDS analysis showed that the atomic compositional ratio of Ca and Ru is almost 1:1. The single-crystal X-ray diffraction revealed an F-type cubic unit cell with $a = 10.197$ Å, which is very close to 10.143 Å for Y$_{2}$Ru$_{2}$O$_{7}$[@kennedy95]. This strongly suggested that the our material has a pyrochlore structure. A further structural refinement was carried out by observing about 2000 reflections at room temperature. The results are summarized in Table \[table1\],
------- ---------- ----------- ------- ------- ----------
Atom Position $x$ $y$ $z$ $B_{eq}$
Ru(1) 16$c$ 0 0 0 0.602(4)
Ca(1) 16$d$ 0.5 0.5 0.5 2.587(9)
O(1) 48$f$ 0.3219(1) 0.125 0.125 1.48(2)
O(2) 8$b$ 0.375 0.375 0.375 3.65(3)
------- ---------- ----------- ------- ------- ----------
: Fractional atomic coordinates and equivalent isotropic displacement parameters (Å$^{2}$) for Ca$_{2}$Ru$_{2}$O$_{7}$. The lattice symmetry and the space group are *cubic* and $Fd\bar{3}m$ (\#227), respectively. The lattice parameters are $a = 10.197(2)$ Å, $V = 1060.4(3)$ Å$^3$ and $Z=8$. The final reliability factor is $R(F) = 2.7 \%$ for 1888 observed reflections.[]{data-label="table1"}
![(a) Micrograph of single crystals of Ca$_{2}$Ru$_{2}$O$_{7}$. The typical dimensions of the crystals are $0.1 \times 0.1 \times 0.1$ mm$^{3}$. (b) X-ray powder pattern of batch used for magnetic measurement. The whole batch was ground before measurement. The weak peaks at 28.1$^{\circ}$ and at 54.3$^{\circ}$ are from traces of RuO$_{2}$.[]{data-label="fig1"}](fig1.eps){width="1.0\linewidth"}
and they coincide with those for a pyrochlore structure. The analysis did not detect any clear evidence that the composition deviates from the ideal pyrochlore, although the temperature factors of Ca(1) and O(2) seem unexpectedly large. We also analyzed the powder X-ray diffraction pattern. As shown in Fig. \[fig1\](b), almost all of the peaks are in agreement with those of a pyrochlore lattice, except for two weak peaks from the traces of RuO$_{2}$. To our best knowledge, there have been no reports on the existence of the Ca$_{2}$Ru$_{2}$O$_{7}$ phase despite many studies on pyrochlore ruthenates. There has only been a study on the mixed crystal Y$_{2-x}$Ca$_{x}$$_{2}$Ru$_{2}$O$_{7}$,[@yoshii99] although the upper limit of $x$ was 0.6. The reason why Ca$_{2}$Ru$_{2}$O$_{7}$ has not been successfully obtained is probably that a strong oxidization atmosphere is necessary for maintaining the Ru$^{5+}$ valence at high temperatures. We suppose that a hydrothermal reaction using a strong oxidant, H$_{2}$O$_{2}$, is advantageous for realizing this condition.
Figure \[fig2\](a) shows the temperature dependence of the magnetic susceptibility.
![(a) Magnetic susceptibility of Ca$_{2}$Ru$_{2}$O$_{7}$ at 0.1 T. The magnetization becomes irreversible below 23 K. The inset shows a comparison of the low-temperature susceptibilities under several magnetic fields. (b) Time dependence of magnetization. The sample was cooled under the ZFC condition down to 15 K. After waiting for 1800 s, a magnetic field of 0.1 T was applied. Then, the magnetization was recorded as a function of time.[]{data-label="fig2"}](fig2.eps){width="0.8\linewidth"}
The susceptibility above 30 K is almost perfectly reproduced using the function $$\label{eq1}
\chi=\frac{C}{T-\Theta} + \chi_{0},$$ where $C$, $\Theta$ and $\chi_{0}$ are the Curie constant, the Weiss temperature and a constant term independent of temperature, respectively. The best-fitted values are $C = 1.65 \times 10^{-2}$ emu$\cdot$K/mol Ru, $\Theta = -4.3$ K and $\chi_{0} = 5.95 \times 10^{-4} $ emu/mol Ru. Assuming a simple ionic model, the oxidation number of ruthenium is 5+. Therefore, there are three 4$d$ electrons per Ru atom, and each Ru atom has a $S=3/2$ spin. In this case, we can expect $\mu_{\mbox{\scriptsize eff}}= 3.87$ $\mu_{\mbox{\scriptsize B}}$. However, the observed $\mu_{\mbox{\scriptsize eff}}$ is only 0.36 $\mu_{\mbox{\scriptsize B}}$, smaller by one order of magnitude. On the other hand, the value of $\chi_{0}$ is comparable to that of typical Pauli paramagnetism in highly correlated oxides, which is consistent with the resistivity result, as shown below.
The most striking feature is the appearance of a sharp transition with an irreversibility between the field-cooled (FC) curve and the zero-field-cooled (ZFC) curve below 23 K. The ZFC curve exhibits a sharp cusp and decreases with decreasing temperature. On the other hand, the FC curve below 23 K is almost constant. This is a typical behavior of a spin glass. Under a higher magnetic field, the cusp becomes less sharp, and the onset temperature of the irreversibility becomes lower as shown in the inset. The time dependence of the magnetization below the glass-transition temperature is shown in Fig. \[fig2\](b). The magnetization exhibits an aging phenomenon, clearly demonstrating another characteristic of a spin-glass state.
Figure \[fig3\] shows the temperature dependence of the resistivity of Ca$_{2}$Ru$_{2}$O$_{7}$. The resistivity at room temperature is 2$\times$$10^{-3}$ $\Omega$cm, as large as a typical value for metallic, highly correlated oxides.
![Temperature dependence of resistivity of Ca$_{2}$Ru$_{2}$O$_{7}$ single crystal. The resistivity is also plotted as a function of $\log T$ in the inset.[]{data-label="fig3"}](fig3.eps){width="1.0\linewidth"}
The behavior is not that of a simple metal, because the temperature coefficient is negative over the whole temperature range measured between 5.5 and 300 K. In the lowest temperature region, the resistivity does not appear to saturate but continues to increase with decreasing temperature. However, the rise in the resistivity at low temperatures is less steep; the ratio, $\rho_{\mbox{\scriptsize 5.5K}} / \rho_{\mbox{\scriptsize 295K}}$, is only about 2. Clearly, an activation function, $\rho \propto \exp (E_{\mbox{\scriptsize A}}/k_{\mbox{\scriptsize B}}T)$, does not reproduce the observation, indicating that no gap is formed at the Fermi level. Therefore, the electronic state is basically metallic, and the $\chi_{0}$ term in the susceptibility can be interpreted in terms of Pauli paramagnetism. We also attempted to fit the data with a variable-range hopping funcition, $\rho \propto \exp \{ (E_{\mbox{\scriptsize A}}/k_{\mbox{\scriptsize B}}T)^{(1/(d+1))} \}$, but were unsuccessful. On the other hand, it seems that a $\rho$ vs $\log T$ plot begins to saturate at low temperatures, as shown in the inset. The resistivity shows no anomaly at the glass-transition temperature, $T_{g}$. This is not unusual for a conductive spin-glass system.[@taniguchi04]
It is known that Y$_{2}$Ru$_{2}$O$_{7}$ is an insulator but becomes metallic when Bi is substituted for Y.[@yoshii99] This is explained in terms of a Mott transition. The Ru-O-Ru bond angles are 129$^{\circ}$ and 139$^{\circ}$ in Y$_{2}$Ru$_{2}$O$_{7}$ and Bi$_{2}$Ru$_{2}$O$_{7}$, respectively.[@ishii00] The larger angle gives a broader band width, resulting in a metallic state. Ca substitution was also attempted by the same author. However, the upper limit of $x$ was 0.6 in Y$_{2-x}$Ca$_{x}$$_{2}$Ru$_{2}$O$_{7}$, and the electronic state was still that of an insulator. In Ca$_{2}$Ru$_{2}$O$_{7}$, the Ru-O-Ru bond angle is 135.72$^{\circ}$, larger than that in Y$_{2}$Ru$_{2}$O$_{7}$. This may be one of the reasons why Ca$_{2}$Ru$_{2}$O$_{7}$ is metallic. Furthermore, the change in the band filling is also important. Note that the isovalent Cd$_{2}$Ru$_{2}$O$_{7}$ is also metallic.[@wang98]
We now explain the origin of the localized spins, all of which undergo the spin-glass transition. One might interpret the data to conclude that Ca$_{2}$Ru$_{2}$O$_{7}$ itself exhibits only Pauli paramagnetism, and that the coexistence of a small amount of another phase contributes to the spin-glass behavior. However, our sample is an ensemble of small single crystals with well-defined shapes, and the powder X-ray diffraction demonstrated that there is no contamination, as shown in Fig. \[fig1\], except for a very small amount of RuO$_{2}$, which is known to exhibit Pauli paramagnetism.[@ryden70] We measured the magnetic susceptibility for three different batches. All the results quantitatively agreed; the ratio of the Curie-Weiss contribution and the Pauli paramagnetism is always the same. If the Curie-Weiss contribution were from an extrinsic origin, this ratio would vary from batch to batch. Therefore, it is most likely that the spin-glass behavior is an intrinsic feature of our phase.
Next, we clarify whether our pyrochlore includes some magnetic impurity atoms in the crystal lattice, because an inclusion of only 0.4 % of Fe$^{3+}$ ions may cause the observed magnetic moments. We can exclude this possibility because we used reagents of more than 99.99 % purity. In the Ru powder used in this study, the content of magnetic elements such as Fe was less than 14 ppm. In the CaO powder, it was less than 1 ppm. The H$_{2}$O$_{2}$ solution used was almost free of magnetic impurities.
Whatever the origin of the magnetic moments, it is unlikely that only nearest-neighbor interactions cause the spin-glass state, because the glass temperature is as high as 23 K despite the very low spin concentration. Therefore, there should be Ruderman-Kittel-Kasuya-Yoshida interactions mediated by the conduction electrons. In such a mechanism, the exchange interactions can be ferromagnetic or antiferromagnetic. The coexistence of these interactions reduces the absolute value of the Weiss temperature. In our crystal, the factor $T_{\mbox{\scriptsize g}} / |\Theta|$ is as large as 5.4. This is in contrast with the insulating spin-glass pyrochlore, Y$_{2}$Mo$_{2}$O$_{7}$, in which $T_{\mbox{\scriptsize g}} / |\Theta|$ is as small as 0.11, showing that geometric frustration is predominant in this molybdate.
Although our X-ray structural analysis did not detect any clear evidence, we cannot yet exclude the possibility of a deviation from ideal Ca$_{2}$Ru$_{2}$O$_{7}$ composition. If there are oxygen defects, for example, some of the Ru atoms are deoxidized into Ru$^{4+}$, possessing an excess electron. If these excess electrons behave as localized moments, they can cause a Curie-Weiss magnetism in addition to the Pauli paramagnetism coming from the conduction electrons. Another possibility is that the conduction electrons themselves have weak magnetic polarizations delocalized over many atoms, similar to those in stoner glass[@hertz79] and spin-density glass.[@sachdev95] In any case, there remains a fundamental question: why did the Ru 4$d$ electrons become partially magnetic despite the metallic conduction? Note that many perovskite-structure metallic ruthenates exhibit anomolous magnetism instead of ordinary Pauli paramagnetism. For example, CaRuO$_{3}$, which has metallic conduction, exhibits Curie-Weiss magnetic susceptibility with weak irreversibility.[@felner00] Therefore, the metallic state in Ca$_{2}$Ru$_{2}$O$_{7}$ might be so fragile that a small purturbation causes magnetic moments. This is in contrast with the study on Bi$_{2-x}$$M_{x}$Ru$_{2}$O$_{7}$ ($M=$ Mn, Fe, Co, Ni and Cu), where the substitution with magnetic atoms $M$ does not affect the Pauli paramagnetic state of the Ru-O sublattice of Bi$_{2}$Ru$_{2}$O$_{7}$.[@haas02]
In conclusion, we have discovered a novel pyrochlore ruthenate, Ca$_{2}$Ru$_{2}$O$_{7}$. The magnetic susceptibility exhibits the behavior of a spin glass in addition to Pauli paramagnetism, although the effective magnetic moment of the spin glass was smaller than expected by one order of magnitude. The resistivity of $2 \times 10^{-3}$ $\Omega$cm at room temperature suggests a metallic electronic state, but the $\rho$ vs $T$ curve has negative slope over the whole temperature region, indicating that the conduction electrons are strongly scattered by the magnetic moments. In future studies, we intended to first verify whether the spin-glass behavior is intrinsic in Ca$_{2}$Ru$_{2}$O$_{7}$. Detailed measurements of the transport properties under a magnetic field will be useful, because they may detect chirality in the spin-glass state.[@taniguchi04] NMR or neutron-scattering measurements are necessary to clarify the origin of the magnetic moments and the mechanism of the spin glass. Investigation of the electronic phase diagram by doping with Y will also be important.
This study was supported by Grants-in-Aid for Scientific Research No. 15750127 and No. 17540341 from the Ministry of Education, Culture, Sports, Science and Technology.
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[^1]: E-mail address: hirohiko@phys.chuo-u.ac.jp
| {
"pile_set_name": "ArXiv"
} |
LYCEN 2001-38\
****
First Results of the EDELWEISS WIMP Search using a 320 g Heat-and-Ionization Ge Detector
[The EDELWEISS Collaboration:]{}\
A. Benoit$^{1}$, L. Bergé$^{2}$, A. Broniatowski$^{2}$, B. Chambon$^{3}$, M. Chapellier$^{4}$, G. Chardin$^{5}$, P. Charvin$^{5,6}$, M. De Jésus$^{3}$, P. Di Stefano$^{5}$, D. Drain$^{3}$, L. Dumoulin$^{2}$, J. Gascon$^{3}$, G. Gerbier$^{5}$, C. Goldbach$^{7}$, M. Goyot$^{3}$, M. Gros$^{5}$, J.P. Hadjout$^{3}$, A. Juillard$^{2,5}$, A. de Lesquen$^{5}$, M. Loidl$^{5}$, J. Mallet$^{5}$, S. Marnieros$^{2}$, O. Martineau$^{3}$, N. Mirabolfathi$^{2}$, L. Mosca$^{5}$, L. Miramonti$^{5}$, X.-F. Navick$^{5}$, G. Nollez$^{7}$, P. Pari$^{4}$, M. Stern$^{3}$, L. Vagneron$^{3}$
[$^{1}$Centre de Recherche sur les Très Basses Températures, SPM-CNRS, BP 166, 38042 Grenoble, France\
$^{2}$Centre de Spectroscopie Nucléaire et de Spectroscopie de Masse, IN2P3-CNRS, Université Paris XI, bat 108, 91405 Orsay, France\
$^{3}$Institut de Physique Nucléaire de Lyon-UCBL, IN2P3-CNRS, 4 rue Enrico Fermi, 69622 Villeurbanne Cedex, France\
$^{4}$CEA, Centre d’Études Nucléaires de Saclay, DSM/DRECAM, 91191 Gif-sur-Yvette Cedex, France\
$^{5}$CEA, Centre d’Études Nucléaires de Saclay, DSM/DAPNIA, 91191 Gif-sur-Yvette Cedex, France\
$^{6}$Laboratoire Souterrain de Modane, CEA-CNRS, 90 rue Polset, 73500 Modane, France\
$^{7}$Institut d’Astrophysique de Paris, INSU-CNRS, 98 bis Bd Arago, 75014 Paris, France ]{}
[**Abstract**]{}
The EDELWEISS collaboration has performed a direct search for WIMP dark matter using a 320 g heat-and-ionization cryogenic Ge detector operated in a low-background environment in the Laboratoire Souterrain de Modane. No nuclear recoils are observed in the fiducial volume in the 30-200 keV energy range during an effective exposure of [4.53]{} kg$\cdot$days. Limits for the cross-section for the spin-independent interaction of WIMPs and nucleons are set in the framework of the Minimal Supersymmetric Standard Model (MSSM). The central value of the signal reported by the experiment DAMA is excluded at 90% CL.
[**Introduction**]{}
A general picture of matter and energy in the Universe is now emerging (see e.g. Ref. [@bib-review] for a review), suggesting that our Galaxy could be immersed in a halo of Dark Matter made of Weakly Interacting Massive Particles (WIMPs). The collision of a WIMP with an atomic nucleus would produce a nuclear recoil with a kinetic energy of the order of ten keV [@bib-sandl]. In the event that WIMPs are the neutralinos of the Minimal Supersymmetric extension of the Standard Model (MSSM), interaction rates per kilogram of matter would vary between 1 event per day to one per decade, depending on model parameters [@bib-mssm].
Experimental searches for these recoils in germanium ionization detectors [@bib-ge] and NaI scintillators [@bib-nai] have yielded upper limits on their rate per kilogram of detector material, which are interpreted in the framework of the MSSM in terms of limits on the WIMP-nucleon interaction cross-section. These searches are limited by the interaction rate due to natural radioactivity, which is at best limited to approximately 1 count/kg/day in the low energy range where recoils are expected.
Recently, the DAMA experiment has reported an annual modulation of the low-energy rate recorded in their $~100$ kg NaI detector array over a period of four years [@bib-dama]. This was attributed [@bib-dama] to the modulation of the WIMP flux impinging on the detector due to the Earth rotation around the Sun corresponding to a WIMP mass of 52$\pm^{10}_{8}$ GeV/c$^2$ and a WIMP-nucleon interaction cross-section of (7.2$\pm^{0.4}_{0.9}$) $\times$10$^{-6}$pb. In contrast, the CDMS collaboration [@bib-cdms] observed no excess of nuclear recoils above the rate expected from the scattering of cosmic-ray induced neutrons after accumulating an exposure of 10.6 kg$\cdot$days in the fiducial volume of their heat-and-ionization cryogenic germanium detectors. The two experimental results are not compatible if one applies the standard procedure to scale rates in different detectors described in ref. [@bib-sandl].
More data are needed to resolve definitely this discrepancy. The most exciting developments here are to be expected from the rapidly evolving domain of heat-and-ionization (or heat-and-scintillation [@bib-cresst]) cryogenic detector technology, which provides excellent event-by-event rejection of the dominating $\gamma$-ray background. The EDELWEISS collaboration has recently commissioned a massive (320 g) heat-and-ionization Ge detector [@bib-navick]. We report here on the first physics results obtained with this detector in a low-background environment in the underground site of the Laboratoire Souterrain de Modane (LSM). In this deep-underground experiment, the cosmic-ray induced neutron background that limited the recent CDMS results (one nuclear scattering per kg$\cdot$day above 10 keV recoil energy [@bib-cdms]) should be reduced by orders of magnitude. The CDMS and EDELWEISS detectors differ by their mass, geometry and electrode implantation scheme, and a comparison of their performance will benefit the development of this novel technology. The results presented in this letter represent a significant improvement relative to our previous results [@bib-distefano; @bib-benoit], obtained with a 70 g detector and with higher background levels.
[**Experimental Setup**]{} The experimental site is the Laboratoire Souterrain de Modane in the Fréjus Tunnel under the French-Italian Alps. The 1780 m rock overburden (4800 m water equivalent) results in a muon flux of about 4 m$^{-2}$day$^{-1}$ in the experimental hall and the flux of neutrons in the 2-10 MeV range is .
The detector is mounted in a dilution cryostat shielded from the radioactive environment by 10 cm of copper and 15 cm of lead [@bib-cryostat]. Pure nitrogen gas is circulated around the cryostat in order to reduce radon accumulation. The radioactivity of all material in the close vicinity of the detectors was measured using a dedicated low-background germanium $\gamma$-ray detector, also at the LSM. All electronic components were moved away from the detector and hidden behind a 7 cm thick archeological lead shield. The entire setup is surrounded by a 30 cm thick paraffin shielding against neutrons. According to Monte Carlo simulations of the various shields based on the measured neutron flux in the experimental hall, the rate of neutron scattering events producing nuclear recoils above 30 keV is expected to be of the order of 0.03 per kg and per day.
The detector [@bib-navick] is a cylindrical Ge single crystal with a diameter of 70 mm and a thickness of 20 mm. The edges have been beveled at an angle of 45$^o$. The plane surfaces and chamfers have been metalized for ionization measurement. The electrodes are made of 100 nm Al layers sputtered on the surfaces after etching. The top electrode is divided in a central part and a guard ring, electrically decoupled for radial localization of the charge deposition. During data taking, a voltage $V_o =$ 6.37 V was applied to the top electrodes. The electrodes were regularly shorted in order to prevent charge accumulation due to trapping of carriers in the detector volume.
The cross-talk between the centre and guard ring electrode signal is approximately 10%. This does not affect the ionization energy resolution since the cross-talk amplitude is a fixed fraction of the signal amplitude on the other electrode. Its shape is constant in time and this effect is easily taken into account by the simultaneous analysis of the signals recorded on both electrodes.
The thermal sensor consists of a Neutron Transmutation Doped germanium crystal (NTD) of 4 mm$^3$ glued to the beveled part of the surface of the detector. The residual radioactivity of the activated NTD sensor should thus be mostly contained in the region covered by the guard electrode. The resistance of the DC-polarized sensor was approximately 3 M$\Omega$ for a base temperature of 27 mK, stabilized to within $\pm$10 $\mu$K.
The signals from all channels are sent to digitizers triggered by either of the two ionization channels. To generate the trigger, the ionization signals are processed by shaping amplifiers and sent to two discriminators. The trigger is the “or” of the output of the two discriminators. The rise time of the heat signal is of the order of 10 ms, much slower than the $\mu$s-fast ionization channel, and was not used for triggering.
[**Detector Calibration**]{} The heat and ionization responses to $\gamma$ rays were calibrated using $^{57}$Co sources. Over the entire data-taking period, the baseline resolutions on the centre and guard ring ionization signals are better than 2.0 keV FWHM, and it varied between 1.9 and 3.5 keV FWHM for the heat signal. The corresponding resolutions measured at 122 keV for the centre, guard ring and heat signals are approximately 3, 2 and 3.5 keV FWHM, respectively.
The ionization resolution was limited by microphonic noise at frequencies varying with time. This also constrained the trigger level of the data acquisition, which was set on the relatively faster ionization signals. The individual trigger levels on the two ionization channels were repeatedly measured during the data taking period using the low-energy part of the Compton plateau recorded with a $^{60}$Co source. The low-energy edge of this plateau corresponds to the decrease of efficiency due to the trigger threshold. Its shape is well described by an error function corresponding to a 50% efficiency at 5.7$\pm$0.3 keV ionization and reaching full efficiency at approximately 8 keV ionization.
The calibration of the response to photons and nuclear recoils was performed using $^{57}$Co and $^{60}$Co $\gamma$-ray sources and a $^{252}$Cf neutron source. The variable used to discriminate electron and nuclear recoils is the ratio of the ionization signal to the recoil energy. To obtain the recoil energy $E_R$, the heat signal is corrected for the Joule heating proportional to the charge signal amplitude [@bib-luke]. For this, the ionization and heat signals calibrated using $\gamma$-ray sources (the electron-equivalent energies $E_I$ and $E_H$, respectively) are combined event-by-event to get the true recoil energy $E_R$ $$\begin{aligned}
E_R & = & (1+\frac{V_o}{V_{pair}})E_H
- \frac{V_o}{V_{pair}} E_I \nonumber\end{aligned}$$ where $V_{pair}=3~V$ is the electron-hole pair creation potential in Ge. By construction, the ratio of the ionization energy to the recoil energy, $Q = E_I / E_R$, is equal to 1 for energy deposits coming from $\gamma$-rays. For neutrons, $Q$ is a function of $E_R$ determined using the data recorded with a $^{252}$Cf source shown in Fig. \[fig-neutron\]. The average value of $Q(E_R)$ is well described by $Q=0.16(E_R)^{0.18}$, with $E_R$ in keV, a parameterization similar to that obtained elsewhere [@bib-qn]. From the dispersion of these data is obtained the [*nuclear recoil band*]{}, defined as the region in the ($Q$,$E_R$) plane where 90% of the nuclear recoils are expected. Since neutrons have a significant probability for multiple scattering inside the detector volume, the $Q(E_R)$ distribution for an actual WIMP signal would differ slightly from that measured with the $^{252}$Cf source. However, Monte Carlo simulations indicate that multiple scattering shifts down the average measured Q values for neutrons by approximately 0.01 units relative to a WIMP signal. This value is small compared to the width of the adopted nuclear recoil band shown in Fig. \[fig-neutron\] and has been neglected. Furthermore, the small decrease of the WIMP detection efficiency would be largely compensated by the narrowing of the $Q$ distribution at a given recoil energy due to the absence of multiple scattering.
An important feature of the detector is the ability to use the guard electrode to tag interactions occurring near the perimeter of the detector. This part of the surface is the most exposed to elements of the detector environment that are known to have perceptible levels of radioactivity (such as the NTD and Cu-Be support springs) and its surface-to-volume ratio makes it more susceptible to surface contaminants. Interactions in this region can also suffer from electric field inhomogeneities leading to incomplete charge collection, and thus mimic the ionization deficit expected for nuclear recoils. For this reason, the fiducial volume of the detector is chosen to correspond to events for which more than 75% of the charge is collected in the centre electrode.
The fiducial volume fraction $f_V$ corresponding to this selection was measured using nuclear recoil events recorded with the $^{252}$Cf source, as neutron interactions are expected to be more evenly spread throughout the detector than low-energy $\gamma$-ray interactions. A clean sample of neutron events is obtained from the $^{252}$Cf data by requiring Q$<0.5$ and a recoil energy between 30 and 200 keV. These energies correspond to an ionization well above the trigger threshold, and the loose requirement on $Q$ makes the selection insensitive to difference in resolution between the two electrode signals. In this sample where the selection efficiency does not depend on the relative strength of the signal on the two electrodes, the fiducial volume cut selects $f_{event}$ $=$ 53$\pm$2% of the events. Some sharing of the charge between the two electrodes is expected to arise because of multiple scattering events occurring both inside and outside the fiducial volume and of interactions occurring close to the boundary between the two electrodes. Multiple scattering must thus be taken into account in the derivation of the volume fraction from the measured fraction of events passing the fiducial cut. To evaluate this correction, the response of the detector, the cryostat and the shielding to neutrons from the $^{252}$Cf source was simulated using GEANT [@bib-geant]. According to the simulations, the volume fraction $f_V$ corresponding to the fiducial selection is 54$\pm$2%. This number differs from the event fraction $f_{event}$ by only 1% because the fiducial selection partially compensates the loss of “pure centre” events due to multiple scattering by allowing that up to 25% of the charge be collected on the guard ring electrode.
Simple electrostatic simulations of the bending of the field lines inside the detector can reproduce the volume fractions to within 5%. This discrepancy is taken as a systematic error on the measured fraction, which is then $f_V$ $=$ 54 $\pm$2 (stat.) $\pm$5 (syst.) %.
[**Results and Discussion**]{} The low-background data were accumulated over two consecutive months. Over these two months, no physics runs have been excluded from the data sample. In two series of runs, an increase of the microphonic noise level on the centre electrode channel lead to unacceptably high trigger rates. These were brought under control by attenuating the centre electrode signal by a calibrated amount, increasing the 50% efficiency level for these runs from 5.7 to 9 and 11 keV. Using the value of $f_V$ obtained in the previous section, the exposure recorded with 5.7, 9 and 11 keV ionization thresholds are [3.80]{}, [0.63]{} and [0.60]{} kg$\cdot$day, respectively, for a total of [5.03]{} kg$\cdot$day (fiducial volume). The principal source of down-time were the regular interruptions for calibrations with $\gamma$ sources and cryostat operations.
The data taking was interrupted by a series of power cuts that were followed by a significant deterioration of charge collection, as attested by the rate of events below the nuclear recoil band. No such events were observed between 30 and 200 keV recoil energy before the incident (corresponding to a rate inferior to 0.5 /kg/day at 90% confidence level), while the observed rate is 1.8$\pm$0.6 /kg/day afterwards. Work is in progress to understand the exact cause of this deterioration and to restore the original charge collection properties of the detector.
Fig. \[fig-ion\]a shows the distribution of the total (centre+guard ring) ionization energy recorded for events passing the fiducial volume cut, before applying the nuclear recoil selection. Fig. \[fig-ion\]b shows the corresponding distribution for events rejected by the fiducial volume cut. The 46.5 keV $^{210}$Pb line provides a convincing illustration of the ability of the fiducial volume selection to reject localized sources of background radioactivity. Indeed, its yield in the rejected sample is 3.4 $\pm$ 0.5 counts/day while it is less than 0.3 counts/day in the fiducial volume (at 90% CL), proving that the source of this contamination is located towards the outside perimeter of the detector.
The 10.4 keV line observed in both spectra and corresponding to the Ga K-shell energy originates from the cosmogenic activation of the detector leading to the creation of $^{68}$Ge with a half-life of T$_{1/2}$=271 days [@bib-nuclide]. No significant variation of its intensity is observed as a function of time, indicating that the contribution from the decay of the $^{71}$Ge nucleus (T$_{1/2}$=11.2 days) created by thermal neutron capture during the exposure to the $^{252}$Cf source is small. In both cases, these decays should be evenly distributed in the detector volume and thus provide an alternative tool for the measurement of the fiducial volume fraction. This method is statistically less precise, and the effect of the finite trigger threshold must be taken into account. Nevertheless, the value of $f_V$ obtained with this method (50 $\pm$ 4 %, where only the statistical error is quoted) is compatible with the neutron data result.
Another illustration of the usefulness of the fiducial volume selection is the reduction in the overall count rates between 15 and 40 keV ionization energy observed in fig. \[fig-ion\], from 4.5$\pm$0.2 to 1.8$\pm$0.1 counts/kg/day/keV.
Fig. \[fig-data\] shows the distribution of $Q$ versus $E_R$ for the entire data set. Most events are within the 99.9% efficiency photon band. A few events lie between this region and the nuclear recoil band. They are interpreted as surface events with reduced charge collection [@bib-benoit]. The present results are significantly better than those obtained previously [@bib-distefano; @bib-benoit] with smaller detectors. Part of the improvements comes from the decrease of radioactive backgrounds in the close vicinity of the detector, as well as the mass of the detector (320 g, the largest mass achieved so far for heat-and-ionization cryogenic Ge detectors) which allows the definition of a large fiducial volume with a relatively uniform electrostatic field surrounded by a thick protective guard region. The sputtered Al electrodes which equip the present detector also appear to have reduced the charge collection problems for surface events that severely limited the previous detector performances [@bib-distefano].
The limit on the WIMP rate is taken from the total number of counts in the nuclear recoil band for recoil energies between 30 and 200 keV. The lower limit corresponds to the recoil energy for which the efficiency is close to 90% and excludes the region where the $\gamma$-rays rejection is worse than 99.9%, and in particular the low-$Q$ tail of the 10.4 keV line. The experimental limit on the rate between 30 and 200 keV recoil energy is less than [0.51]{} counts per kg$\cdot$day at 90% CL.
These results are interpreted in terms of an upper limit at 90% CL on the WIMP-nucleon scattering cross-section using the prescriptions of Ref. [@bib-sandl]. The limits are shown in Fig. \[fig-exclus\]. In the calculation of the WIMP flux, a galactic halo WIMP density of 0.3 $GeV/cm^3$ is assumed, together with an r.m.s. velocity of 270 km/s, an escape velocity of 650 km/s and a relative Earth-halo velocity of 230 km/s. The interaction rates are calculated using cross-sections scaled by the square of the target mass number and the Helm parameterization of the form factor [@bib-sandl; @bib-helm]. The expected number of WIMPs as a function of their mass and scattering cross-section takes into account the experimental efficiency for nuclear recoils as a function of the recoil energy and uses the ionization threshold values relevant to each data sample.
To ensure the stability of this result within the systematic uncertainties of the measurement, the analysis has been repeated using an increased fiducial volume. It corresponds to the selection of all events where the signal on the centre electrode is larger than that on the guard. The fiducial volume evaluated from the neutron source data is 63$\pm$2%, corresponding to a 17% increase of the fiducial volume. No nuclear recoil candidates are observed in the increased-acceptance sample. This increase in acceptance is larger than the variations corresponding to the uncertainties on the trigger threshold and on the measurement of $f_V$. A further increase in efficiency can be achieved by increasing the width of the nuclear recoil band to 95% efficiency with still no events entering the band. It is thus believed that the limits shown in Fig. \[fig-exclus\] are conservative.
For WIMP masses above 30 GeV/c$^2$, the present limits are better than those obtained by Ge diode experiments without heat measurement [@bib-ge]. Although the effective exposure in this experiment is approximately half that accumulated by the CDMS collaboration [@bib-cdms], the limits obtained for WIMP masses above 200 GeV/c$^2$ are very similar, as seen in Fig. \[fig-exclus\]. This is due to the absence of any event in the EDELWEISS acceptance while the CMDS results relies on a statistical subtraction of its neutron background. At lower WIMP masses, the present results suffer from the relatively poorer energy resolution and high ionization trigger level. This should be solved with planned improvements in the wiring of the detector to reduce microphonic noise and adjustments of the NTD sensor excitation and readout.
Based on the usual assumptions for the comparison of direct WIMP searches described in Ref. [@bib-sandl] such as target mass scaling, and using the same standard galactic halo model [@bib-dama; @bib-cdms], the EDELWEISS results [^1] exclude at more than 90% CL the central value for the WIMP signal reported by the annual modulation measurement of the DAMA collaboration [@bib-dama] (WIMP mass $M_W$ = 52 GeV/c$^2$ and interaction cross-section $\sigma_n$ = 7.2 $\times$10$^{-6}$pb). It does not exclude at 90% CL the other central value obtained by DAMA when the annual modulation data is combined with their own exclusion data based on pulse shape discrimination ($M_W$ = 44 GeV/c$^2$ and $\sigma_n$ = 5.4 $\times$10$^{-6}$pb).
[**Conclusion**]{} The EDELWEISS collaboration has searched for nuclear recoils due to the scattering of WIMP dark matter using a 320 g heat-and-ionization Ge detector operated in a low-background environment in the Laboratoire Souterrain de Modane. After an effective exposure of [4.53]{} kg$\cdot$day, the rate of Ge recoils with kinetic energies between 30 and 200 keV is measured to be less than [0.51]{} per kg$\cdot$day at 90% CL. This is the most stringent limit based on the observation of zero event and not relying on any statistical background subtraction. Within the usual assumption for the comparison of direct WIMP searches [@bib-sandl] and using the same standard galactic halo model as in Ref. [@bib-dama; @bib-cdms], the EDELWEISS results exclude at more than 90% CL a 52 GeV/c$^2$ WIMP with an interaction cross-section of 7.2 $\times$10$^{-6}$ pb. With a four-fold increase of exposure time or with some improvements in the detector resolution, the sensitivity of the present detector should be able to test the whole parameter space of the DAMA candidate, without requiring a statistical subtraction of nuclear recoils due to neutron scattering interactions. Later this year it is planned to operate at the LSM this detector together with two other similar 320 g detectors presently under construction.
Acknowledgements {#acknowledgements .unnumbered}
================
The help of the technical staff of the Laboratoire Souterrain de Modane and the participating laboratories is gratefully acknowledged. This work has been partially funded by the EEC Network program under contract ERBFMRXCT980167.
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[^1]: The EDELWEISS 90% CL cross-section limit for a WIMP mass of 52 GeV/c$^2$ is $\sigma_n$ = 6.3 $\times$10$^{-6}$pb.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Traditional 3D convolutions are computationally expensive, memory intensive, and due to large number of parameters, they often tend to overfit. On the other hand, 2D CNNs are less computationally expensive and less memory intensive than 3D CNNs and have shown remarkable results in applications like image classification and object recognition. However, in previous works, it has been observed that they are inferior to 3D CNNs when applied on a spatio-temporal input. In this work, we propose a convolutional block which extracts the spatial information by performing a 2D convolution and extracts the temporal information by exploiting temporal differences, i.e., the change in the spatial information at different time instances, using simple operations of shift, subtract and add without utilizing any trainable parameters. The proposed convolutional block has same number of parameters as of a 2D convolution kernel of size $n\times n$, i.e. $n^2$, and has $n$ times lesser parameters than an $n\times n \times n$ 3D convolution kernel. We show that the 3D CNNs perform better when the 3D convolution kernels are replaced by the proposed convolutional blocks. We evaluate the proposed convolutional block on UCF101 and ModelNet datasets. All the codes and pretrained models will be publicly available at \_.'
author:
- Gagan Kanojia
- Sudhakar Kumawat
- Shanmuganathan Raman
title: Exploring Temporal Differences in 3D Convolutional Neural Networks
---
Introduction {#sec:intro}
============
Lately, 3D convolutional neural networks are gaining popularity over the 2D CNNs when the task is to deal with 3D data representations which could be videos, shapes or other formats [@hara2018can; @tran2017convnet]. This is because 2D CNN lack in exploiting the temporal information. 3D CNNs are more proficient than 2D CNNs in extracting temporal information and utilizing it to perform specific tasks. It has been shown that a 3D CNN of same depth as that of a 2D CNN performs better on tasks like action recognition [@hara2018can; @tran2018closer]. However, this proficiency comes with a cost in terms of the number of learnable parameters, memory requirements, and risks of overfitting. For example, 3D ResNet (18 layers) [@hara2018can] has around 3 times more parameters than the 2D ResNet (18 layers) [@he2016deep].\
In this work, our focus is on acquiring both spatial and temporal structure of the 3D data while reducing the cost in terms of trainable parameters. We propose a convolutional block which exploits both the spatial information and the temporal information by utilizing a 2D convolution and temporal differences, i.e., the change in the spatial information at different time instances, using simple operations of shift, subtract and add. We have also incorporated temporal max pooling in order to downsample the temporal depth of the feature maps along the depth of the network. None of the operations other than 2D convolution require trainable parameters which makes the number of trainable parameters of the proposed convolutional block equal to the 2D convolution kernel with same kernel size. The major contributions of the work are as follows. **(a)** We propose a novel convolutional block which captures spatial information by performing a 2D convolution and captures temporal information using simple operations of shift, subtract and add. **(b)** We reduce the number of parameters by $n$ times by replacing the 3D convolution kernel of size $n\times n\times n$ with the proposed convolution block comprising a 2D convolution kernel of size $1\times n\times n$. **(c)** We show that the proposed convolutional block helps the 3D CNNs to perform better while utilizing lesser parameters than the 3D convolution kernels.
Related work {#sec:related}
============
In recent years, 2D CNNs have been dominating several applications of computer vision like object detection [@he2016deep] and image classification[@he2016deep]. However, they lack in extracting the temporal information present in the spatio-temporal data [@tran2018closer]. There are works which extend the 2D CNNs on videos by processing the video frames individually and then combining the extracted information along the temporal dimension to obtain the output [@xu2015discriminative; @girdhar2017actionvlad]. Recently, 3D CNNs have shown great potential in dealing with the spatio-temporal data or 3D CAD models as inputs [@tran2015learning; @zhi2017lightnet; @maturana2015voxnet]. It has been observed that 3D CNNs are much better in exploiting the temporal information than 2D CNNs[@tran2018closer]. However, 3D CNNs are computationally expensive and they are prone to overfit due to their large number of parameters. Hence, the researchers moved on to find better and more efficient ways of mimicking 3D convolutions. There has been notable advances in the separable convolutions in 2D CNNs to reduce the space-time complexity [@sandler2018mobilenetv2; @chollet2017xception; @xie2017aggregated]. In many works, the idea of separable convolutions has been extended to 3D CNNs [@sun2015human; @xie2018rethinking; @qiu2017learning; @tran2018closer]. In [@qiu2017learning], the authors proposed the idea of replacing the 3D convolution kernel by a 2D convolution kernel to capture the spatial information followed by a 1D convolution kernel to convolve along the temporal direction. They showed that the proposed technique has several advantages, like parameter reduction and better performance, over the 3D convolutions, which has been further explored in [@tran2018closer]. Temporal differences has been explored in few recent works [@wang2016temporal; @lee2018motion]. Wang *et al.* [@wang2016temporal] use difference in two frames as the approximation of motion information. Similarly, Lee *et al.* [@lee2018motion] propose a motion block which extracts features using spatial and temporal shifts. In this work, we only rely on the temporal differences. Instead of relying on only the adjacent frames, we compute aggregated temporal differences over several frames. The proposed SSA Layer does not involve any trainable parameter to extract temporal information via temporal differences. Our focus is to propose an efficient alternative to the 3D convolution filters which utilizes lesser parameters without compromising the performance.
Proposed Approach {#sec:conv_block}
=================
In this section, we discuss the proposed convolutional block which extracts both spatial and temporal information. The proposed convolutional block has three parts: 2D convolution kernel, SSA layer, and temporal pooling layer. Here, SSA stands for Shift, Subtract and Add. Let the input to the proposed convolutional block be $\mathcal{X} \in \mathbb{R}^{c\times f \times h \times w}$. Here, $\mathcal{X}$ is the output feature maps of the previous convolutional block or layer, $c$ is number of channels, $f$ corresponds to the temporal depth, and $h$ and $w$ are the height and width of $\mathcal{X}$, respectively.\
**2D convolution.** In traditional 3D CNNs, the feature maps are convolved with a 3D filter $\hat{g} \in \mathbb{R}^{c\times k \times k \times k}$ with $c$ channels and kernel size $k \times k \times k$ [@hara2018can]. In the proposed framework, first we obtain $\mathcal{X}_c = \mathcal{X}\star g$. Here, $\star$ stands for convolution, and $g$ is a 2D filter of kernel size $1\times k \times k$ and $c$ channels. The purpose of the 2D convolution is to extract the spatial information present in the input feature maps[@zeiler2014visualizing]. We, then, pass $\mathcal{X}_c$ through the proposed SSA layer to obtain the temporal structure of the feature maps.\
**SSA Layer.** SSA stands for Shift, Subtract and Add operations performed in the SSA layer. The purpose of the SSA layer is to extract the temporal information present in the spatio-temporal data. For example, in action recognition, motion features extracted from the videos can hold important information. In order to capture the motion information, optical flow techniques can be used [@dosovitskiy2015flownet]. However, capturing optical flow is in itself a computationally expensive task which can require a dedicated network [@dosovitskiy2015flownet]. In the proposed SSA layer, we rely on temporal differences, i.e., the change in the spatial information at different time instances, to extract the necessary temporal information present in the spatio-temporal data.In the case of action recognition, temporal differences can provide the rough extimate of the location of moving objects or non-rigid bodies [@park2013exploring]. However, there is a possibility that there has not been enough change occurred in the adjacent frames. Hence, we take multiple frames into the consideration. The difference could be due to motion like in the case of action recognition or due to the structure of the input, like in the case of shapes. This makes the SSA layer to be used in a more general sense.\
Let the input to the SSA layer be $\mathcal{X}_c \in \mathbb{R}^{c\times f \times h \times w}$. Here, $c$ is the number of channels, $f$ is the temporal depth, and $h$ and $w$ are the height and width of $\mathcal{X}_c$, respectively. We obtain the temporal differences between the volumes of the feature map $\mathcal{X}_c$ along the temporal depth. Let $\{\mathcal{X}_c^1,\mathcal{X}_c^2, \mathcal{X}_c^3,\ldots,\mathcal{X}_c^f\} \in \mathbb{R}^{c\times 1 \times h \times w}$ be the volumes of the feature map $\mathcal{X}_c$ along the temporal depth. $\mathcal{X}_c$ is passed through the SSA layer to obtain $\mathcal{X}_s \in \mathbb{R}^{c\times f \times h \times w} $ as shown in Eq. \[eq:ssa\_layer2\]. Here, $\mathcal{X}_s$ is obtained by concatenating $\{\mathcal{X}_s^1,\mathcal{X}_s^2, \mathcal{X}_s^3,\ldots,\mathcal{X}_s^f \}\in \mathbb{R}^{c\times 1 \times h \times w}$ along the temporal dimension. $$\mathcal{X}_s^i = \mathcal{X}_c^i+ \frac{1}{f}\sum\limits_{k=1}^{i-1} \frac{f-(i-k)}{f}(\mathcal{X}_c^{i}-\mathcal{X}_c^{k}), \ \ \forall i = 2, \ldots, f
\label{eq:ssa_layer2}$$ Here, $k$ is the shift and $i$ is a location along the temporal direction. If $i= 1$, $\mathcal{X}_s^i =\mathcal{X}_c^i$. Since, the nearby frames can have more contextual relation, the term $\frac{f-(i-k)}{f}$ is to ensure that the larger shifts get smaller weights than smaller shifts.\
Instead of computing for each temporal volume separately, $X_s$ can be computed in a cumulative manner as illustrated in Fig. \[fig:ssa\_layer\]. However, the mathematical formulation and the illustration shown in Fig. \[fig:ssa\_layer\] lead to the same output. Since, $\mathcal{X}_c$ is a four dimensional volume, it would be hard to provide a clean illustration. We have also omitted the multiplicative constants from the illustration to keep it clean. Hence, we have used 1-D representation to illustrate its operations visually. In Fig. \[fig:ssa\_layer\], each column refers to a single shift. It can be seen that the input feature map $\mathcal{X}_c$ is subtracted from its shifted version and then, the difference is added to it in the corresponding locations to obtain $\hat{\mathcal{X}_s}$. Then, we again shift the feature map $\mathcal{X}_c$ by one more step, subtract it from its original version and then add the difference to the corresponding locations of $\hat{\mathcal{X}_s}$. At the end of $f-1$ steps, we obtain $\mathcal{X}_s$.\
**Temporal pooling.** As, we move along the depth of the 3D convolution networks, the temporal depth of the feature maps keeps reducing as we perform 3D convolutions of stride more than one. In our case, we are not performing convolution along the temporal depth. Hence, to reduce the temporal depth, we perform max pooling along the temporal direction whenever we want to reduce the temporal depth of the feature maps.\
**Parameter Analysis.** A standard 3D convolutional kernel of size $n\times n\times n$ and $c$ channels contains $cn^3$ parameters. The proposed convolution block comprises a standard 2D-convolution kernel, an SSA layer and temporal max pooling. A standard 2D-convolution kernel of size $n\times n$ and $c$ channels contains $cn^2$ parameters and an SSA layer consists of shift, subtract and add operations which do not require any trainable parameters. Also, temporal max pooling does not require any trainable parameters. Hence, the overall number of trainable parameters used in the proposed convolution block is $cn^2$ which is $n$ times less than the standard 3D convolution kernel.
Experiments and Discussions {#sec:experiments}
===========================
\[fig:blocks\]
In this section, we show that the 3D CNNs perform better when the standard 3D convolution kernels are replaced by the proposed convolutional block. Our focus is mostly on the residual networks. We evaluate their performances on two types of 3D data: spatio-temporal image sequences and 3D CAD models.
Spatio-Temporal Image Sequences
-------------------------------
**Dataset.** UCF101 [@soomro2012ucf101] is a benchmark action recognition dataset containing complex real world videos which has been used in several works[@tran2015learning; @tran2017convnet; @diba2018spatio]. The videos of the dataset cover 101 action categories. We use UCF101 split-1 for all our experiments regarding spatio-temporal image sequences.
### Network Architectures {#sec:network_arch}
We employ deep 3D residual networks to evaluate the proposed convolutional block [@hara2018can]. Fig. \[fig:blocks\] (a) and (b) show the basic and bottleneck block used in 3D ResNet architecture [@hara2018can]. Fig. \[fig:blocks\] (c) shows the bottleneck block used in ResNeXT architecture [@hara2018can]. We replace the 3D convolution kernel of the residual blocks by the proposed convolutional block as shown in Fig. \[fig:blocks\] (d), (e) and (f). In Fig. \[fig:blocks\] (d), (e) and (f), it can be seen that we preserve the overall structure of the blocks while replacing the 3D convolution kernel by the proposed convolutional block. This is done to show the true effect of the proposed block on the existing networks. We have also experimented with the WideResNet architecture with a widening factor of 2 [@hara2018can]. The structure of bottleneck block of WideResNet is same as the bottleneck block of ResNet. The only difference is the number of channels of the feature maps in the layers. To show that the proposed approach is not constrained to the residual networks, we have also done experiments with C3D network proposed in [@tran2015learning]. Similar to the residual networks, we replace the 3D convolution kernel with the proposed convolutional block. The training details are provided in the supplementary material.
Network Layers Accuracy(%)
------------------------------------------------------ -------- --------------- -- -- -------------
3D ResNet[@hara2018can; @tran2017convnet] (baseline) 18 $\approx$ 33 45.6
SSA-ResNet (ours) 18 $\approx$ 11 52.8
SSA-ResNet (ours) 18 $\approx$ 11 **55.7**
3D ResNeXT[@hara2018can] (baseline) 50 $\approx$ 26 49.3
SSA-ResNeXT (ours) 50 $\approx$ 23 54.9
SSA-ResNeXT (ours) 50 $\approx$ 23 **56.9**
3D WideResNet[@hara2018can] (baseline) 50 $\approx$ 157 46.8
SSA-WideResNet(ours) 50 $\approx$ 67 50.7
SSA-WideResNet(ours) 50 $\approx$ 67 **52.9**
C3D[@tran2015learning] (baseline) 5 $\approx$ 18 44
SSA-C3D (ours) 5 $\approx$ 14 50
SSA-C3D (ours) 5 $\approx$ 14 **51.6**
3D ResNet[@diba2018spatio; @hara2018can] (baseline) 101 $\approx$ 88 46.7
SSA-ResNet (ours) 101 $\approx$ 43 52.1
SSA-ResNet (ours) 101 $\approx$ 43 **54.4**
: **Comparisons with baselines.** The comparison of the test accuracies obtained by the baseline 3D models with the networks obtained by replacing the 3D convolution kernel by the proposed convolution block in the baseline 3D models on UCF101 split-1 when trained from scratch.
\[tab:ablation\_baseline\]
### Comparisons with baselines
We perform our experiments by training the networks from scratch on UCF101 split-1. The test accuracies of 3D ResNeXT and 3D WideResNet when trained from scratch on UCF101 split-1 are not available in the previous works[@hara2018can]. So, we train these networks on UCF101 from scratch to obtain them. For the other baseline networks, we mention the accuracies reported in [@hara2018can; @diba2018spatio; @tran2017convnet]. SSA-ResNet, SSA-WideResNet, SSA-ResNeXT, and SSA-C3D are obtained by replacing the 3D convolution kernels in ResNet, WideResNet, ResNeXT, and C3D[@tran2015learning] by the proposed convolutional block. We train these networks from scratch on UCF101 with same hyperparameter settings. Table \[tab:ablation\_baseline\] shows that the accuracies obtained by the baseline 3D models when trained from scratch on the split-1 of UCF101 dataset. It also shows the accuracies obtained by replacing the 3D convolution kernel by the proposed convolution block. It can be seen that the networks perform significantly better with the proposed convolutional block while utilizing lesser trainable parameters.
Network Layers Accuracy
------------------------------------------ -------- ---------------- ------- ----------
2D-ResNet[@he2016deep; @tran2017convnet] 18 $\approx$11.2 - 42.2
2D-ResNet[@he2016deep; @tran2017convnet] 34 $\approx$21.5 - 42.2
3D-ResNet[@tran2017convnet] 18 $\approx$33.2 254 45.6
3D-ResNet[@tran2017convnet] 34 $\approx$63.5 485 45.9
3D-ResNet[@diba2018spatio] 101 $\approx$86.06 657 46.7
3D STC-ResNet[@diba2018spatio] 18 - - 42.8
3D STC-ResNet[@diba2018spatio] 50 - - 46.2
3D STC-ResNet[@diba2018spatio] 101 - - 47.9
C3D[@tran2015learning] 5 $\approx$18 139.6 44
R(2+1)D[@tran2018closer] 18 $\approx$33.3 128 48.37
SSA-ResNet (ours) 18 $\approx$11 88.5 55.7
SSA-ResNeXt (ours) 50 $\approx$23 185.9 **56.9**
: **Comparisons with the state-of-the-art.** The comparison of the proposed approach with the state-of-the-art methods when trained from scratch on UCF101 dataset.
\[tab:results\]
### Comparisons with the state-of-the-art
Table \[tab:results\] compares the proposed approach with the state-of-the-art methods when trained from scratch on UCF101 dataset. The test accuracy of R(2+1)D[@tran2018closer] when trained from scratch on UCF101 is not available in the previous works. So, we trained the network on UCF101 split-1 from scratch to obtain it using the same hyperparameter settings as ours. It can be observed that SSA-ResNeXT performs significantly better than the previous approaches. SSA-ResNet (18 layers) utilizes approximately 11 million parameters which is roughly equal to the parameters used in 2D-ResNet [@he2016deep] (18 layers). Inspite of having almost equal parameters, SSA-ResNet (18 layers) outperforms 2D-ResNet (18 layers) by 13.5 % in terms of classification accuracy. Also, SSA-ResNet (18 layers) utilizes approximately 3 times less parameters than 3D-ResNet (18 layers)[@tran2017convnet], 3D STC-ResNet (18 layers)[@diba2018spatio], and R(2+1)D (18 layers)[@tran2018closer] and still outperforms them by 10.1%, 12.9 %, and 7.33%, respectively.
\#Shift Temporal pooling Accuracy
--------- ------------------ ----------
0 46.3
0 52.8
1 52.6
2 53.4
3 53.9
f-1 51.3
f-1 **55.7**
: **Analysis of different shifts and temporal pooling**. The comparison of test accuracies obtained on UCF101 split-1 using SSA-ResNet (18 layers) (when trained-from-scratch) with varying number of shifts along with the effect of temporal pooling.
\[tab:ablation\_shift\]
### Analysis
In the proposed convolutional block, apart from a standard 2D convolution kernel, there are two components: SSA layer and Temporal pooling.\
**SSA Layer.** As shown in Fig.\[fig:ssa\_layer\], we perform the shift operation $f-1$ times, where $f$ is the temporal depth of the input feature map. We perform the experiments on SSA-ResNet (18 layers) with different values of shifts. The results are shown in Table \[tab:ablation\_shift\]. It can be seen that as we increase the fixed number of shifts from 1 to 3, the test accuracy increases and we obtain the highest accuracy when we perform $f-1$ shifts.\
**Temporal Pooling.** In Table \[tab:ablation\_shift\], it can be observed that by using 2D-convolution kernel and only max temporal pooling, the network outperforms the baseline case, i.e. with only 2D convolution kernels. The same pattern can be observed in Table \[tab:ablation\_baseline\], in which the baseline 3D models are replaced with the proposed convolution block without SSA layer (second row for each network) and the networks performed significantly better than the baseline 3D CNNs.
Network Framework Augmentation ModelNet40 (%) ModelNet10 (%)
--------------------------------- ------------ ---------------- --------------- ---------------- ----------------
3D ShapeNets[@wu20153d] Volumetric Az $\times$ 12 $\approx$38 77 83.5
Beam Search[@xu2016beam] Volumetric Az $\times$ 12 $\approx$0.08 81.26 88
3D-GAN[@wu2016learning] Volumetric Az $\times$ 12 $\approx$11 83.3 91
VoxNet[@maturana2015voxnet] Volumetric Az $\times$ 12 $\approx$0.92 83 92
LightNet[@zhi2017lightnet] Volumetric Az $\times$ 12 $\approx$0.30 86.90 93.39
ORION[@sedaghat2017orientation] Volumetric Az $\times$ 12 $\approx$.91 - **93.8**
SSA-ResNeXT8 (ours) Volumetric Az $\times$ 12 $\approx$3.38 **89.5** 93.3
: **Comparisons with the state-of-the-art.** The comparison of the SSA-ResNeXT8 with the state-of-the-art methods on the voxelized version of ModelNet40 and ModelNet10 datasets.
\[tab:modelnet\]
3D CAD Models
-------------
**Dataset.** ModelNet[@wu20153d] is a collection of 3D CAD models of objects. It has two subsets: ModelNet10 and ModelNet40. ModelNet10 and ModelNet40 contains 10 and 40 classes of objects, respectively, which are manually aligned to a canonical frame. In our experiments, we use the voxelized version of size $32\times 32 \times 32$ and augmentation with 12 orientations [@maturana2015voxnet]. Similar to [@maturana2015voxnet; @brock2016generative], we add noise, random translations, and horizontal flips for data augmentation to the training data. Similar to [@brock2016generative], we scale the binary voxel range from $\{0,1\}$ to $\{-1,5\}$.\
**Network Architecture.** To avoid overfitting on ModelNet40 and ModelNet10, we use a smal network SSA-ResNext8 to evaluate our approach on 3D CAD models. We use the SSA-ResNeXT bottleneck block in the architecture of the network. Let us denote the SSA-ResNeXT bottleneck block with $SSAR(k,F,s)$, where $1\times k\times k$ is the kernel size of the 2D convolution filter, $F$ is the number of channels in the input feature map and $s$ is the value of stride passed to the block. The architecture of SSA-ResNext8 is as follows: $Conv2D(3,1)\rightarrow MP(3,2)\rightarrow SSAR(3,64,1)\rightarrow SSAR(3,256,1)\rightarrow SSAR(3,256,2)\rightarrow SSAR(3,512,1)\rightarrow SSAR(3,512,2)\rightarrow SSAR(3,1024,1)$ $\rightarrow GP\rightarrow FC$. Here, $Conv2D(3,1)$ is a 2D convolution kernel of size $1\times 3\times 3$ and stride of 1, followed by a batch normalization layer and ReLU, and $MP(3,2)$ is the max-pooling layer with kernel size of $3\times 3\times 3$ and stride of 2. $GP$ and $FC$ stands for global average pooling and fully connected layer, respectively. The training details are provided in the supplementary material.\
**Comparisons with the state of the art.** Table \[tab:modelnet\] shows the comparison of the SSA-ResNeXT8 with the state-of-the-art methods that use voxelized/volumetric ModelNet datasets as input. For fair comparison, we only consider volumetric frameworks. It can be observed that the network with the proposed convolutional block performs better than the state-of-the-art on ModelNet40 and comparable on ModelNet10 in the case when the networks are trained with shapes augmented with 12 orientations. This shows that the proposed convolution block is not restricted to videos and can be further exploited in shapes.
Conclusion
==========
We propose a novel convolutional block which is proficient in capturing both spatial and temporal structure of the 3D data while utilizing lesser parameters than the 3D convolution kernel. It comprises three components: a 2D-convolution kernel to capture the spatial information, a novel SSA layer to capture the temporal structure, and a temporal pooling layer to reduce the temporal depth of the input feature map. We show that the 3D CNNs perform better when the 3D convolution kernels are replaced by the proposed convolutional block. SSA-ResNet (18 layers) outperforms the state-of-the-art accuracy on the UCF101 dataset split-1 while utilizing lesser parameters when networks are trained-from-scratch. We have also evaluated the proposed convolutional block on 3D CAD models and we outperform the state-of-the-art on ModelNet40 among the volumetric framework, when the training data is augmented with 12 rotations.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We report the results from a new, highly sensitive ($\Delta T_{mb} \sim 3 $mK) survey for thermal OH emission at 1665 and 1667 MHz over a dense, 9 x 9-pixel grid covering a $1\degr \times 1\degr $ patch of sky in the direction of $l = 105\fdg00, b = +2\fdg50$ towards the Perseus spiral arm of our Galaxy. We compare our Green Bank Telescope (GBT) 1667 MHz OH results with archival (1-0) observations from the Five College Radio Astronomy Observatory (FCRAO) Outer Galaxy Survey within the velocity range of the Perseus Arm at these galactic coordinates. Out of the 81 statistically-independent pointings in our survey area, 86% show detectable OH emission at 1667 MHz, and 19% of them show detectable CO emission. We explore the possible physical conditions of the observed features using a set of diffuse molecular cloud models. In the context of these models, both OH and CO disappear at current sensitivity limits below an A$_{\rm v}$ of 0.2, but the CO emission does not appear until the volume density exceeds 100-200 . These results demonstrate that a combination of low column density A$_{\rm v}$ and low volume density $n_{H}$ can explain the lack of CO emission along sight lines exhibiting OH emission. The 18-cm OH main lines, with their low critical density of $n^{*}$ $ \sim 1 $ , are collisionally excited over a large fraction of the quiescent galactic environment and, for observations of sufficient sensitivity, provide an optically-thin radio tracer for diffuse H$_2$.'
author:
- 'Michael P. Busch'
- 'Ronald J. Allen'
- 'Philip D. Engelke'
- 'David E. Hogg'
- 'David A. Neufeld'
- 'Mark G. Wolfire'
bibliography:
- 'Mendeley.bib'
title: |
The Structure of Dark Molecular Gas in the Galaxy - II.\
Physical State of ”CO-Dark” Gas in the Perseus Arm
---
[^1]
Introduction
============
The formation of molecular hydrogen gas from atomic hydrogen in galaxies is widely considered to be a critical step for the formation of new stars from the interstellar gas, and hence it is one of the most important processes that occurs in the interstellar medium (ISM). Unfortunately, as a symmetric molecule without a dipole moment, is practically invisible in emission at the temperature range of 10-100K expected for the bulk of the ISM in the Galaxy, and indirect estimates are required that make use of surrogate tracers. The most universally-accepted surrogate tracer for in the ISM is the lowest-energy rotational spectral line of (1-0) at $\lambda = 3 $mm. This line is relatively bright and easily observed, often with instrumentation designed specifically for that purpose. CO observations are commonly used jointly with an empirically-derived conversion factor, usually called the “X-factor” [see @Bolatto2013TheFactor for a review]. The strength of the 3-mm CO line emission is measured in units of K $\kmps$ and, by multiplying this line strength by the X-factor, one directly obtains an estimate of the column density.
A growing body of observational evidence points to an extra component of the ISM not traced by either the 21-cm line or (1-0) emission. This excess component is usually referred to as “dark gas” [@Grenier2005UnveilingNeighborhood; @Wolfire2010TheGas]. This dark gas may be a large fraction of the total molecular gas content in the Galaxy [@Pineda2013AComponents; @Li2015QuantifyingGas]. Quantifying how much dark gas exists is therefore of great interest in an effort to calibrate the X-factor and work towards a single prescription for scaling between a molecular tracer line emission and an accurate total molecular mass of . @Wolfire2010TheGas constructed models of molecular cloud surfaces and determined that dark gas in molecular cloud surfaces can amount to $\sim 30\%$ of the total molecular mass of the cloud. Here, the column density is large enough so that has sufficient column to remain self-shielded against the ambient UV flux, but CO is photodissociated and the carbon is the form of C or C$^{+}$. However, a large fraction of the faint CO gas might also arise in the diffuse ISM [@Papadopoulos2002MolecularDistances; @Liszt2010TheGas; @Allen2012Faint5circ; @Allen2015The+1deg; @Xu2016EvolutionTaurusb].
Initially discovered in absorption at centimeter radio wavelengths in the diffuse ISM[^2], OH has also recently been detected in the far-IR [@Wiesemeyer2016Far-infraredClouds], also in absorption. The work reported here mainly concerns new 18-cm emission observations in the outer Galaxy. [@Allen2012Faint5circ] reported faint and widespread 18-cm OH emission in a blind survey of a small region in the second quadrant of the Galactic plane using the 25-m radio telescope at Onsala, Sweden. However, owing to spectrometer limitations and radio interference in the spectra, the Onsala blind survey was limited to the 1667 MHz OH line and to within 2 kpc of the Sun. Similar results were presented by [@Dawson2014SPLASH:Region] with the SPLASH survey using the Parkes Telescope; however, their survey of OH encountered high levels of background synchrotron continuum emission from the inner Galaxy at levels approaching the typical excitation temperatures of the 18-cm OH lines, effectively suppressing the extended OH emission outside of the ‘CO-bright’ clouds. In the outer Galaxy, we can typically avoid this issue as the integrated background synchrotron emission is much weaker. More recent observations in the outer Galaxy by @Allen2015The+1deg [hereafter Paper 1] using the GBT have shown main-line (1665 and 1667 MHz) OH emission to be present in regions largely devoid of CO emission, strongly suggesting that OH may be a good tracer of the dark gas. The main lines of OH emission in these regions are observed to be in the 5:9 ratio characteristic of optically-thin emission lines with level populations in local thermodynamic equilibrium (LTE). This makes calculating a column density of OH relatively straightforward as long as a good estimate for the excitation temperatures can be found. With an OH column density known from emission observations, the column density can be directly estimated using the N()/N(OH) ratio of approximately $10^{-7}$ measured from UV absorption data [@Weselak2010TheMolecules; @Nguyen2018Dust-GasISM; @Engelke2018OHW5].
In this paper we present the results of a densely-sampled, blind, highly-sensitive GBT emission survey of the two main OH lines in a one-square-degree field of the Perseus Arm. The purpose of this survey is to further explore the apparent connection of extended OH emission to the so-called ’dark molecular gas’, and attempt to resolve structures of diffuse molecular gas on a large scale that are otherwise invisible to CO observations.
In Paper 1, we were able to compare sparse OH survey measurements taken with the GBT (FWHM $\sim$ 7.6’) with spectra provided directly from archives of the CFA CO survey [@Dame2001TheSurvey], since both data sets were observed at closely similar angular resolution ($\sim 7.6'$ for the GBT vs $\sim 8.4'$ for the CfA telescopes). However, the Five College Radio Astronomy Observatory (FCRAO) “Outer Galaxy Survey” [@Heyer1998TheGalaxy] has significantly higher angular resolution ($\sim 50\arcsec$) than the CFA survey, and hence significantly better sensitivity when smoothed to the GBT beam. We have therefore compared our OH data with a smoothed version of the FCRAO data. This allows for a direct, dense, observational comparison between these two important molecular tracers with comparable sensitivities for the first time.
It was observed in Paper 1 that OH emission was widespread, and that there was varying structure to the emission profiles on scales of $\sim$ 30 pc at the distance of the Perseus Arm feature. In an attempt to resolve the structure of this emission a new observing program was undertaken to increase the coverage of the original 2015 sparse survey. In this paper we therefore chose to restrict our analysis to the Perseus Arm feature in particular in an attempt to map the OH emission spatially and compare it to the CO emission at the $\sim$ 7 pc scale, as we show in Sect. \[morphology\]. Due to the absence of the kinematic distance ambiguity in the outer Galaxy, we are able to differentiate components of the OH spectra (see Fig. \[fig:exampleSpectrum\]). The distance of the Perseus Arm offered us the best consistently apparent option to measure this structure. The availability of accurate parallax distances in this direction (see Sect. \[distancetoarm\]) made this spatial comparison possible. Statistical and spatial comparisons of OH and CO emission between the local and inter-arm features would also be interesting and will be the subject of future papers.
Observations and Data {#Observations}
=====================
We carried out highly-sensitive observations of main-line OH emission at 18 cm with the Robert C. Byrd Green Bank Telescope (GBT) in project AGBT14B\_031 at the 81 locations indicated with small circles in Fig.\[fig:osdArea\]. For comparison with the OH data, we used the (1-0) data from the FCRAO survey, as explained above.
![The blind survey areas discussed in this work. The “X” markers indicate the original 3x9 ‘sparse’ survey carried out with the GBT ACS spectrometer in program AGBT13B\_044 as reported in Paper 1. The set of 9 “X” sightlines in the range of $+2\degr$ to $+3\degr$ were later re-observed using the newer VEGAS backend for consistency with the remaining sightlines. The “+” markers indicate the next 5x6 survey carried out in program AGBT14B\_031. As unexpected but significant L band GBT observing time became available, we changed the observing goal to sample an entire square degree. The “O” markers indicate the final “One-Square-Degree” 9x9 dense 81-point grid, also carried out under AGBT14B\_031. These data are the subject of this paper.[]{data-label="fig:osdArea"}](OSD-Area.png){width="45.00000%"}
OH Observations {#data}
---------------
Paper 1 reported the results of a ’sparse’ survey with the GBT at $0.5^{\circ}$ spacing interval at sightlines indicated with “X” in Fig.\[fig:osdArea\]. The observations discussed in this paper were made between September 11, 2015 and January 31, 2016, also with the GBT. The receivers selected are situated at the Gregorian Focus and operate in the frequency range 1.15 to 1.73 GHz (L-Band). There is one beam on the sky, with dual polarizations. The amplifiers are cooled Field Effect Transistors (FET) with an effective system temperature of 20 K or less in good weather and for intermediate elevations of the GBT. These observations were made with linear polarization.
This new OSD survey covers a $9 \times 9$ grid of GBT sightlines, which roughly corresponds to $60 \times 60$ pc region at a 3.2 kpc distance to the Perseus Arm, centered on l = 105.0, b = 2.5 and at intervals of 0.125 = $7.5'$, closely corresponding to the angular resolution of the GBT at the frequency of the OH main lines (FWHM $\sim 7.6'$). At each position 12 individual scans, each of duration 10 minutes, were made. For an individual scan, in one polarization, the effective integration time was approximately 292 seconds, and the expected rms in a single channel in the corresponding spectrum is approximately 34 mK. The effective integration time of the final spectrum after combination of the data from all scans, in both polarizations, is just under two hours.
{width="\textwidth"} {width="\textwidth"}
Archival CO Data
----------------
The (1-0) observations were made between 1995 and 1998 using the 14m telescope of the FCRAO [@Heyer1998TheGalaxy]. This FCRAO “Outer Galaxy Survey” observed a 336 square degree region of the second quadrant of the Galaxy sampled at $50\arcsec$ intervals with a FWHM beamwidth resolution of $45\arcsec$. For the purposes of our direct comparison, this survey was first smoothed to the GBT resolution. The final CO spectra have a sensitivity of 60 to 100 mK rms, and a velocity resolution of 0.8 .
Data Reduction
--------------
The GBT OH data were reduced using the *GBTIDL* software [@Garwood2006GBTIDL:Data]. The quality of this data is generally very high, and each polarization of each 10 minute scan was reviewed for the presence of radio frequency interference and problems from instrumental effects. It was quickly evident that the spectra would be heavily influenced by the baseline ripple discovered during commissioning of the GBT (Fisher, J. R., Norrod, R.D., and Balser, D.S. 2003, Electronic Division Internal Report No. 312). The dominant feature is a broad ripple with an approximate periodicity of 9 MHz. Fisher et al. deduce that this ripple is caused by multipath reflections in which a part of the system noise enters the receiver system directly and a part is returned from the subreflector. Typical values of the amplitude are 0.4 K in Y and 0.1 K in X, the orthogonal polarized channel. Because the periodicity of this ripple is much larger than the frequency range expected for the OH signal, the ripple can in general be satisfactorily removed by fitting and subtracting a polynomial of low order to the baseline.
The residual spectrum shows an additional ripple feature that is more problematic. This ripple is comprised of several components having frequencies in a range between 1.3 and 1.8 MHz according to Fisher et al. The ripple is stronger in the Y polarization, is variable with a dependence on the configuration of the main reflector and the subreflector, and is presumed to also arise in multipath reflections from the circumferential gaps between the surface panels. The amplitude of the ripple is larger in the Y linear polarization whose E-vector is parallel to the plane of symmetry of the telescope. Typical worst case amplitudes are about $\pm 1.5$ mK in the Y polarization. The placement of the pattern in the bandpass is a strong function of the position of the subreflector. An attempt was made to reduce the amplitude of the ripple by observing the same position with the subreflector set at focus positions differing by $\lambda/8$. Some improvement was noted, but the ripple was never completely canceled out.
All of the data for a given position were assembled and reviewed for quality. Where necessary, data contaminated by (infrequent) interference or equipment instability were edited out. The data for each polarization were averaged, and the 9 MHz ripple was removed by fitting a baseline polynomial of order 5 to the spectrum in the frequency range 1661.4 MHz to 1671.4 MHz. Each spectrum was then used to compare the 1667 and 1665 MHz lines to judge if the emission is in LTE line ratio. The relative intensities of the four lines are 1:5:9:1 for the 1612-, 1665-, 1667-, and 1720-MHz lines respectively. Instances where the ratio differed from the LTE value proved to be rare. The comparison of the two spectra also provided a useful validation of faint spectral features.
The next step was to average the two polarizations in order to improve S/N, and to smooth the data by a Gaussian function to an effective spectral resolution of 1.0 . For these spectra the rms of an individual point is 2.2 mK, but there is variation from spectrum to spectrum because of the remaining uncertainty in the determination of the baseline. Including baseline uncertainties, the resultant RMS uncertainty in the OH spectra is on the order of 3mK in $T_{mb}$ units. The final spectrum was shifted to the rest frame velocity of the 1665 or 1667 MHz line for further detailed parameterization. An example spectrum from the OSD is shown in Fig. \[fig:exampleSpectrum\], the OH emission from the local gas, inter-arm, and Perseus spiral arm are observed at $V_{LSR}$ $\sim$ 0, -20 and -65 km s$^{-1}$ respectively.
The final step was to make the best estimate of OH emission from the Perseus Arm Region. We assumed that emission from the Perseus Arm would be limited to a spectral region of width 25 km/s centered on the expected velocity. For the expected velocity we were guided by the profile of the neutral hydrogen as mapped in the DRAO “LRDS” survey, in the vicinity of -75 to -50 [@Higgs2000ThePlane; @Higgs2005TheII.]
To remove the residual effects of the 1.6 MHz ripple we defined two regions proximate to the OH window, each of width 10 km/s, and fitted a linear baseline to these regions. At the same time the CO emission from the region is computed from the archival data from the FCRAO [@Heyer1998TheGalaxy] over the same velocity range. The resulting parameters have been collected into a table, a fragment of which is shown in Table \[table:data\]. The complete version of Table \[table:data\] is provided in the electronic version of this paper.
The finished products from the data sometimes suffer from a long-wavelength ripple in the spectra, which has been removed by a similar polynomial fit. However, negative values are sometimes recovered owing to the variation in spectra where there is no CO signal (as is the case in most of the sightlines in the area studied here), see e.g. Figure 1 in [@Heyer1998TheGalaxy], where there is a demonstrative negative feature, a probable artifact of this baseline ripple, at roughly the Perseus Arm velocity range.
{width="80.00000%"}
Distance to the Perseus Arm {#distancetoarm}
---------------------------
The physical size of the Perseus Arm structures that we are observing are known from precise VLBI distance measurements [@Reid2009TRIGONOMETRICMOTIONS; @Reid2014TRIGONOMETRICWAY; @Choi2014TRIGONOMETRICARM]. In the direction of $l = 105\fdg00, b = +2\fdg50$ we use the Bayesian Distance Calculator[^3] from The Bar and Spiral Structure Legacy Survey (BeSSeL) [@Reid2016ASources] to estimate the distance to our observed Perseus Arm sources. We note that the galactic rotation model utilized by the BeSSeL project does not incorporate the so-called “rolling motions” of the Perseus Arm [@Yuan1973TheArms; @Foster2010StructurePicture], and that the Bayesian Distance Calculator treats the Perseus Arm $V_{LSR}$ as constant with galactic latitude. While many of our observed features at high latitudes contain LSR radial velocities inconsistent with the Perseus Arm $V_{LSR}$ range provided by the BeSSeL model, we believe that rolling motions account for this variation in $V_{LSR}$ and that our observed features do indeed fall within the Perseus Arm. We therefore use the parallax distances from the calculator as opposed to the kinematic distance PDF; we adopt a distance estimate of 3.2 kpc to the Perseus Arm. As noted in the previous section, any feature we observe in the vicinity of -75 to -50 we associate with the Perseus Arm.
Determining Line Strengths
--------------------------
The line strengths of all features corresponding to the Perseus Arm velocity interval ($V_{LSR} \sim -65$ , see Fig. \[fig:exampleSpectrum\]) of the 81 1667 MHz OH spectra in the survey were calculated. The line profile strengths of the CO and OH lines are plotted against each other in Fig. \[fig:profileIntegrals\]. All 81 Perseus features in the OH spectra were manually identified and the velocity range’s integration limits were chosen from visual inspection of the OH spectra. The corresponding line strengths from the CO spectra were obtained over the same velocity range. The expression used to calculate the line strengths is:
$$S = \sum T_{mb} \times \Delta V,$$
in units of K , where $T_{mb}$ is in units of main-beam brightness temperature, and $\Delta V$ is the channel spacing of the data. Note that, for the GBT, $T_a$ and $T_{mb}$ are the same to within 5%. The summation is done numerically over the channels containing measurable signal for each Perseus Arm feature in the 81 OH spectra. After this process, the (1-0) spectrum is extracted from a spatially-smoothed version of the FCRAO data cube and integrated over the same velocity range. This process was repeated for all 81 Perseus Arm features (and lack of features around $V_{LSR} \sim -65$ ) until all of the data in the OSD was processed. The error analysis is the same procedure as outlined in [@Allen2015The+1deg] section 4.2. We also computed the scaled difference between the 1665 and 1667 MHz lines with each sightline and find that most emission is the LTE ratio of 5:9, as is expected if these lines are excited chiefly by collisions.
{width="49.55000%"} {width="49.55000%"}
Estimating the Continuum and Excitation Temperatures
----------------------------------------------------
In order to fit our OH observations and the corresponding CO data to models of diffuse clouds in the ISM, we need estimates of the column density of OH based on our line strength data. OH column densities are calculated from OH 1667 MHz emission lines using the following equation [see, e.g. @Liszt1996GalacticSources]: $$N({\rm OH}) = C\frac{T_{ex}}{T_{ex} - T_C}\int{T_b(\nu) d\nu}\textbf,
\label{eq:emission}$$ which in turn means that values for the continuum temperature in the background of the OH and of the excitation temperature of the 1667 MHz line are both needed as inputs. The coefficient $C = 2.3 x 10^{14}$ for the 1667 MHz line yields N(OH) in terms of cm$^{-2}$. However, determining the exact value of the continuum temperature is difficult, as observations have been made at 408 MHz, 1420 MHz, and 2695 MHz, but not at 1667 MHz; moreover, there are different components to the continuum, each of which have different spectral indices, making interpolation complicated. In order to estimate $T_C$ at 1667 MHz, we use the “diffuse component” of the continuum at 1420 MHz as reported by @Reich1997The240, and at 2695 MHz as reported by @Furst1969A240DEG, and we estimate the diffuse component from [@Haslam1969AMaps] as reported in [@Taylor2003TheSurvey]. We then made a quadratic interpolation on a logarithmic plot among the continuum temperature at these three frequencies to estimate the diffuse component of the continuum at 1667 MHz. Uncertainty still remains, however, since we do not know how much of the Galactic background portion of the continuum exists in the foreground versus the background of the OH. As such, we choose two possible values representative of the range of plausible $T_C$ values of 4.0 K and 5.0 K for this work. The value $T_C$ = 5.0 K assumes that the continuum is mainly in the background, and $T_C$ = 4.0 K assumes that half of the continuum is in the background, using @Xu2016TheWay Figure 2 as a rough basis.
The excitation temperature is another unknown quantity, but reasonable estimates are possible. Given that at the coordinate $104.75^{\circ}, 2.75^{\circ}$ CO is detected while the corresponding OH signal is very faint, and also that the source component of the continuum contains a small elevated patch in this vicinity which has been catalogued as the probable HII region GB6 B2214+5950 [@Gregory1996TheSources], it seems plausible that at this coordinate the higher value of $T_C$ is close to the value of $T_{ex}$. That would put $T_{ex}$ at approximately 1 K above the surrounding value of $T_C$. Although this is the most likely scenario, since a $T_{ex}$ value closer to $T_C$ would have a significant effect on the resulting column densities, we also try a value of $T_{ex}$ only 0.5 K above the surrounding $T_C$ in our analysis.
Results
=======
Here we present the main results of this survey, beginning with a scatter plot of OH and CO profile integrals to demonstrate the presence of extended OH emission outside of CO-bright clouds in Fig. \[fig:profileIntegrals\]. In addition, we display a 9 $\times$ 9 color-coded *heatmap* based on profile integral strengths, which shows spatially the ubiquitous OH emission in the Perseus Arm in contrast to the relatively compact CO emission in Fig. \[fig:heatMap\]. Each square in the heatmap represents a GBT beam which, at the assumed distance of the Perseus Arm, corresponds to roughly 7 pc in spatial extent. In a following subsection we present and discuss theoretical models of diffuse clouds which allow us to explore the physical environments of this gas in Fig. \[fig:cloudModel\].
Cloud Model Predictions {#cloudModelSection}
-----------------------
We have compared the observed (1-0) and OH 1667 MHz profile integrals with the predictions of a set of diffuse cloud models. Here, we used the model described by [@Hollenbach2012TheIons], with the modifications discussed by [@Neufeld2016TheClouds], to obtain predicted CO profile integrals and OH column densities. These were obtained as a function of the thickness of the cloud – measured in magnitudes of visual extinction, $A_V({\rm tot})$ – and the volume density of H nuclei, $n_{\rm H}$. Results are shown in Fig. \[fig:cloudModel\], where colored contours representing fixed values of $n_{\rm H}$ and black contours representing fixed values of $A_V({\rm tot})$ are plotted in the plane of observable quantities (i.e. the velocity-integrated brightness temperatures for the OH and CO transitions with the OH converted to column density.) All of the predictions shown here were obtained for an assumed cosmic-ray ionization rate of $2\times 10^{-16}\,\rm s^{-1}$ (primary ionizations per H atom) – the mean Galactic value favored by [@Neufeld2017TheIons]–and an assumed interstellar ultraviolet radiation field equal to that given by [@Draine1978PhotoelectricGas].
The resulting values of gas volume density predicted by the cloud model for three representative values of the factor F = $T^{67}_{ex}$/($T^{67}_{ex} - T_C$) are displayed in Table \[table:densities\]. Note that for all three cases of input values of $T_C$ and $T^{67}_{ex}$, the predicted average volume density is greater for the CO-bright gas than it is for the CO-dark gas. As the CO-dark gas volume density predictions are upper limits, this result is further strengthened.
The diffuse cloud models suggest that the portion of the survey that is OH-bright, CO-dark gas is mainly molecular. According to the model, in the representative case where $A_V({\rm tot})$ = 0.3 mag and $n_{H}$ = 166 cm$^{-3}$, the molecular fraction of the gas is found to be 0.7. We also find that the predicted $A_V({\rm tot})$ is greater than $\sim 0.2, 0.25, 0.3$ mag for F = 5, 6, 11 respectively. This lower limit to $A_V({\rm tot})$ most likely results from the sensitivity of the observations and the required OH line strength to be detected above the noise.
$T_C$ (K) $T^{67}_{ex}$ (K) F CO-Bright Mean Volume Density (cm$^{-3}$) CO-Dark Mean Volume Density (cm$^{-3}$)
----------- ------------------- ----- ------------------------------------------- -----------------------------------------
4.0 5.0 5.0 $400 \pm 70$ < $210 \pm 20$
5.0 6.0 6.0 280 $\pm 40$ < $200 \pm 20$
5.0 5.5 11 160 $\pm 15$ < $120 \pm 10$
{width=".46\textwidth"} {width=".46\textwidth"} {width=".46\textwidth"}
Discussion
==========
Comparing OH and CO Line Strengths
----------------------------------
Fig. \[fig:profileIntegrals\] shows the 81 data points resulting from a profile analysis of the OH and CO features, over the same velocity ranges, in the Perseus Arm. OH emission appears widespread in this region and, if there is a CO signal detected, it is roughly correlated with the OH signal. It is apparent that a substantially larger region of the ISM contains molecular gas than is indicated by the line.
It is noteworthy to mention that the area of this survey (densely covering one square degree) and that of @Allen2015The+1deg [sparsely covering four square degrees] are quite different, and yet they both statistically show very similar results: $40\%$ of the pointings in Paper I. showed 3$\sigma$ CO emission, while all of the pointings showed OH emission. In the present survey, $19\%$ of the analyzed pointings showed CO emission and $86\%$ showed OH emission. Both these results reveal large amounts of ’CO-dark’ molecular gas in the ISM. The present survey is dense, showing structure on the scale of the GBT beam itself is 7 pc at the 3.2 kpc distance to the Perseus Arm in this direction, whereas the survey of [@Allen2015The+1deg] was not restricted to certain features in velocity space and was a sparse, blind OH survey that showed ’CO-Dark’ gas structures on larger scales with large gaps between sight lines.
Dark Gas Morphology on $0.125^{\circ}$ Intervals {#morphology}
------------------------------------------------
The structure of dark gas is of interest for many reasons. Indirect observational evidence suggests that the diffuse molecular gas surrounds the ’CO-bright’, molecular gas. Direct mapping of OH emission is extremely difficult due to sensitivity requirements but also shows observationally consistent results: the OH is extended beyond the CO-bright clouds [@Wannier1993WarmObservations; @Allen2012Faint5circ; @Allen2015The+1deg; @Xu2016EvolutionTaurusb]. If one assumes that the OH molecule can trace all of the diffuse molecular gas, then the extent of the OH emission can tell us how extended this ’CO-faint’ component of the molecular gas actually is. In this regard, we visually identified three interesting regions of note in Fig. \[fig:heatMap\]:
- Feature A: A CO-bright region centered on l = 104.7, b = 2.75, on the order of 12x12 pc, surrounded by extended OH emission;
- Feature B: A CO-bright region centered on l = 105.25, b = 2.25. A dimmer and smaller structure than feature A, surrounded by extended OH emission, and;
- Feature C: A CO-dark region around l = 105.25, b = 2.875 which shows no CO emission but significant amounts of OH emission. This latter feature extends beyond feature A and B by at least 10-12 pc.
In the case of Feature C, we note that there is clearly a CO-dark region with corresponding OH emission. Note also that our survey is also not fully sampled, that is: the intervals are beamwidths and not half-beamwidths. Finally, while the structure of the CO-bright regions are compact clouds which appear to be localized in our survey area, the structure of the OH emission is more diffuse and we cannot reliably conclude from our limited survey just how extended the dark gas actually is in relation to the CO-bright regions. [@Grenier2005UnveilingNeighborhood] have described the ‘dark gas’ as being ‘similar in extent to that of atomic HI; it appears to surround all CO-bright regions and bridges the dense cores to the more extended atomic distribution’. Our results show that OH emission is morphologically and quantitatively consistent with this description of the dark gas.
The Physical Environment of the Dark Molecular Gas
--------------------------------------------------
The physical state of the dark molecular gas is inferred to be diffuse clouds of , without the necessary volume density to collisionally excite the 3mm CO(J=1-0) line. The Planck satellite detected an overabundance of IR emission in the column density ranges corresponding to $A_{v}$ < 2 [@PlanckCollaboration2011PlanckGalaxyb]. [@Wolfire2010TheGas] theoretically studied the dark gas phenomenon at the surface of molecular cloud surfaces and concluded that in this density range $H_{2}$ self-shields while CO can be photo-dissociated. As noted before, the OH-bright, CO-dark gas maintains a high molecular fraction ($\sim$ 0.7) in the models and remains a good candidate tracer of the dark molecular gas. However, for lower column and volume density regimes than studied in this paper, this may not be the case and further study is warranted.
Fig. \[fig:cloudModel\] shows the diffuse cloud model predictions for the CO emission line strength and OH column densities, along with our Perseus Arm observations overplotted (see \[cloudModelSection\]). We observe a density effect that is responsible for the CO-faint gas at the current sensitivities of the observations. In the context of these models, variations in the gas density are the most likely explanation for why some positions with detectable OH emission show CO emission above our detection threshold whereas other such positions do not. A density dependence arises because the CO J = 1 state is subthermally populated over much of the density range of interest, and thus the rate of CO emission per molecule is roughly proportional to the density. The OH lambda-doubling transitions, by contrast, become thermally-populated at much lower densities, owing to their much smaller spontaneous radiative decay rates. Thus, the OH emission rate per molecule is independent of density, and the CO/OH line ratio is an increasing function of density in the regime of relevance. Given the line sensitivities achieved in these observations, this behavior results in OH detections that are unaccompanied by CO detections when the gas density is low.
The results presented in Table \[table:densities\] for the mean volume density of the CO-bright and CO-dark gas imply that, for the current level of sensitivity discussed in this paper, molecular gas becomes CO-dark below a volume density of H nuclei of $\sim 100-200$ in the survey region, which corresponds to a gas volume density of $\sim 50 - 100$ if the gas is primarily molecular hydrogen. The result that molecular gas becomes CO-dark below $\sim 50 - 100$ is consistent with the fact that the critical density for the (1-0) line is 1000 times larger than that of OH 18 cm transitions [see e.g. @Wilson2013ToolsEdition Chap. 16.2.1 and Fig. 16.1]. Since the OH emission lines are observed to be in the LTE ratio of 5:9, collisions are expected to be the dominant source of excitation in this environment. The measured fraction of CO-dark molecular gas will depend on the detection threshold and velocity resolution of the comparison CO data set used [@Donate2017SensitiveTwo; @Li2018WhereDMG].
Conclusions
===========
The results presented in this paper demonstrate that a substantial fraction of molecular gas is invisible in (1-0) emission at the current sensitivity level of the FCRAO survey, and this component of the molecular ISM can be effectively traced by sensitive 18-cm OH observations.
- We presented a dense OH survey in the outer Galaxy with particular attention to the Perseus Arm molecular cloud velocity range ($V_{LSR}\sim -65$ ). We showed that OH emission is widespread, which reaffirms previous observational results (see [@Allen2012Faint5circ; @Allen2013ERRATUM:97; @Allen2015The+1deg]), and can be spatially decoupled from CO emission.
- We presented diffuse cloud models to predict the physical conditions of the molecular clouds studied in this paper. We demonstrated that the data, with reasonable assumptions for $T^{67}_{ex}$ and $T_C$, prefer theoretical densities that are well below the optically-thin critical density approximation for the (1-0) line. This offers a simple physical explanation for the simultaneous lack of observed CO emission, and yet the presence of extended OH emission in this region. The reason for this is because in the radio regime of low density molecular gas spectral line emission, stimulated emission is the dominant radiation mechanism [@Wilson2013ToolsEdition].
- The 18-cm OH lines appear to trace the behavior of the dark molecular gas, the CO-dark component of the molecular ISM that has been revealed from analysis of Planck data [@PlanckCollaboration2011PlanckGalaxyb], gamma-ray [@Grenier2005UnveilingNeighborhood; @Abdo2010FermiGalaxy; @Ackermann2011TheTelescope], and \[CII\] data [@Velusamy2010CIIClouds; @Glover2016CO-darkGas].
We also showed how CO and OH data, examined jointly with simulations of the diffuse ISM, can reveal the physical conditions of the dark gas. The large-scale distribution of 18-cm OH emission in the galaxy could shed light on the features of the galactic molecular ISM not revealed by conventional CO mapping observations. Future observations by the 500m FAST telescope [@Li2016TheProject], and the SKA [@Dickey2012GASKAPSurvey] may help to reveal a portion of this dark molecular ISM content through OH absorption studies; however, the low excitation temperature of the 18-cm OH lines [@Engelke2018OHW5] remains an impediment to the straightforward interpretation of OH observations in the inner Galaxy in the presence of substantial ambient nonthermal continuum emission. An all-sky survey for OH emission in the outer Galaxy would not be hindered by the much weaker continuum of the galactic background there; however, an effective OH emission survey may be presently out of reach of current instruments owing to the long integration times required to obtain the necessary sensitivity. Nevertheless, such a survey could be critical for an understanding the true extent of diffuse molecular gas in the outer Galaxy.
We are grateful to the staff at the Green Bank Observatory for their advice and assistance with the operation of the GBT, in particular Karen O’Neil, Jay Lockman, Toney Minter, Ron Maddalena, and Amanda Kepley, and for the development and support of the GBTIDL data analysis software, especially Jim Braatz and Bob Garwood. We would like thank Joanne Dawson, Claire Murray, Anita Petzler, Katie Jameson and Bruce Draine for helpful and insightful discussions, and Mark Heyer for providing an updated digital copy of the FCRAO data. We also want to thank the anonymous referee for constructive comments that helped improve this manuscript. M. P. B. is supported by a National Science Foundation Graduate Research Fellowship under grant No. 1746891. The work of D. A. N. and M. G. W. was supported by grant No. 120364 from NASA’s Astrophysical Data Analysis Program (ADAP). This research has been supported by the Director’s Research Fund at STScI.
[cccccccccc]{}
104.5 & 3 & 0.104 & 0.0033 & -64.019 & 0.092 & 2.56 & 0.146 & -64.21 & 0.4\
104.625 & 3 & 0.0738 & 0.0029 & -63.49 & 0.185 & 2.012 & 0.137 & -64.22 & 0.406\
104.75 & 3 & 0.0185 & 0.0022 & -61.11 & 0.185 & 0.085 & 0.084 & -61.78 & 1.21\
104.875 & 3 & 0.018 & 0.0025 & -61.29 & 0.64 & -0.28 & 0.11 & -61.37 & 4.06\
105 & 3 & 0.102 & 0.003 & -64.0192 & 0.09 & 2.563 & 0.146 & -64.2199 & 0.406\
105.125 & 3 & 0.0334 & 0.003 & -63.162 & 0.55 & -0.0294 & 0.137 & -63.81 & 5.68\
105.25 & 3 & 0.0196 & 0.002 & -61.67 & 0.277 & -0.0596 & 0.109 & -60.9 & 3.25\
105.375 & 3 & 0.0312 & 0.0023 & -66.137 & 0.277 & 0.3428 & 0.119 & -67.47 & 0.812\
105.5 & 3 & 0.093 & 0.0033 & -65.59 & 0.185 & 0.367 & 0.1547 & -55.65 & 0.812\
104.5 & 2.875 & 0.0428 & 0.0024 & -62.998 & 0.185 & 0.00791 & 0.119 & -63.41 & 2.03\
104.625 & 2.875 & 0.052 & 0.0025 & -62.26 & 0.185 & 1.74 & 0.109 & -61.78 & 0.406\
104.75 & 2.875 & 0.024 & 0.0024 & -62.47 & 0.277 & 0.377 & 0.109 & -60.96 & 0.4\
104.875 & 2.875 & 0.0069 & 0.0017 & -62.66 & 0.55 & -0.187 & 0.084 & -62.59 & 1.62\
105 & 2.875 & 0.0156 & 0.00259 & -61.082 & 0.55 & -0.101 & 0.109 & -31.29 & 62.59\
105.125 & 2.875 & 0.021 & 0.0026 & -61.86 & 0.46 & -0.0239 & 0.119 & -60.56 & 4.06\
105.25 & 2.875 & 0.035 & 0.0035 & -62.43 & 0.277 & -0.154 & 0.154 & -63.81 & 7.31\
105.375 & 2.875 & 0.0317 & 0.002 & -63.05 & 0.185 & -0.049 & 0.109 & -32.1 & 64.21\
105.5 & 2.875 & 0.036 & 0.0029 & -51.9 & 0.37 & -0.094 & 0.128 & -52.0314 & 4.87\
... & ... & ... & ... & ... & ... & ... & ... & ... & ...\
[^1]: To whom correspondence should be addressed: mpbusch@jhu.edu\
MPB is a National Science Foundation Graduate Fellow
[^2]: Additional historical information on the discovery and early observations of the 18-cm radio lines of OH both in emission and absorption can be found in [@Allen2012Faint5circ] and [@Allen2015The+1deg].
[^3]: http://bessel.vlbi-astrometry.org/bayesian
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Cluster structure detection is a fundamental task for the analysis of graphs, in order to understand and to visualize their functional characteristics. Among the different cluster structure detection methods, spectral clustering is currently one of the most widely used due to its speed and simplicity. Yet, there are few theoretical guarantee to recover the underlying partitions of the graph for general models. This paper therefore presents a variant of spectral clustering, called $\ell_1$-spectral clustering, performed on a new random model closely related to stochastic block model. Its goal is to promote a sparse eigenbasis solution of a $\ell_1$ minimization problem revealing the natural structure of the graph. The effectiveness and the robustness to small noise perturbations of our technique is confirmed through a collection of simulated and real data examples.'
author:
- |
[Champion Camille$^{1}$, Blazère Mélanie$^1$, Burcelin Rémy$^2$, Loubes Jean-Michel$^1$, Risser Laurent$^3$]{}\
[$^1$ Toulouse Mathematics Institute (UMR 5219)]{}\
[University of Toulouse F-31062 Toulouse, France]{}\
[$^2$ Metabolic and Cardiovascular Diseases Institute (UMR 1048)]{}\
[University of Toulouse F-31432 Toulouse, France]{}\
[$^3$ Toulouse Mathematics Institute (UMR 5219)]{}\
[CNRS F-31062 Toulouse, France]{}
bibliography:
- 'example\_paper.bib'
title: Robust spectral clustering using LASSO regularization
---
[*Keywords:*]{} Spectral clustering, community detection, eigenvectors basis, $\ell_1$-penalty.
Introduction {#section1}
============
Graphs play a central role in complex systems as they can conveniently model interactions between the variables of a system. Finding variable sets with similar attributes can then help understanding the mechanisms underlying a complex system. Graphs are commonly used in a wide range of applications, ranging from Mathematics (graph theory) to Physics [@Hopfield82], Social Networks [@Handcock10], Informatics [@Pastor07] or Biology [@Jeong00; @Meunier09]. For instance, in genetics, groups of genes with high interactions are likely to be involved in a same function that drives a specific biological process.
One of the most relevant features when analyzing graphs is cluster structures. Clusters are generally defined as connected subsets of nodes that are more densely connected to each other than to the rest of the graph. Different strategies make it possible to define more specifically variable clusters depending on whether this property of vertices is considered locally (on a connected subset of vertices) or globally (on the whole network). First, cliques (subset of vertices such that every two distinct vertices in the clique are adjacent)[@Wasserman94], n-clique (maximal subgraph such that the distance of each pair of its vertices is not larger than n) [@Wasserman94] and k-core (maximal connected subgraph of G in which all vertices have degree at least k) [@Seidman83] characterize local cluster structure. Secondly, one of the global cluster structure definition is based on the notion of modularity [@Newman04; @Newman06] that quantifies the extent to which the fraction of the edges that fall within the given groups differs from the expected fraction if edges were distributed at random. The most popular random model is proposed by [@Newman04], where edges are reconnected randomly, under the constraint that the expected degree of each vertex corresponds to the degree of the vertex in the original graph. The last definition of cluster structure, and the most natural is related to similarity between each pair of vertices, that includes local or global definitions of a cluster structure. It is really natural to assume that cluster structures are groups of vertices that are close to each other. Similarity measures are the foundations of traditional methods as detailed below. These include traditional distance measures such as Manhattan or Euclidean distances or computing correlations between rows of the adjacency matrix or random walk based similarities [@Pons05].
Once the definition of cluster structure is fixed, it is crucial to build efficient procedures and algorithms for the identification of such structures in the network. The ability to find and to analyze such groups can provide an invaluable help in understanding and visualizing the functional components of the whole graph [@Girvan02; @Newman04]. Classical techniques for data clustering, like hierarchical clustering, partitioning clustering and spectral clustering, detailed below, are sometimes adopted for graph clustering too. Hierarchical clustering [@Hastie01] builds a hierarchy of nested clusters organized as a tree. partitioning clustering [@Pothen97] decomposes the graph into a set of disjoint clusters. Given $N$ variables/nodes, it builds $k$ partitions of the data by satisfying: (i) each group contains at least one point (ii) each point belongs to exactly one group. In recent years, spectral clustering has become one of the most widely used methods due to its speed and simplicity [@Luxburg07; @Chung97; @Ng02; @Fortunato10]. This method extracts the geometry and local information of the dataset by computing the top or bottom eigenvectors of specially constructed matrices. The observations are projected into this eigenspace to reduce the dimensionality of the problem and $k$-means procedure is then applied in an easier subspace to detect clusters.
$k$-means, that belongs to partitional clustering methods, aims to find a set of $k$ cluster centers of a dataset such that the sum squared of distances of each point to its closest cluster center is minimized. Lloyd’s 1957 procedure [@Lloyd82] remains one of the widely used because of its speed and simplicity. It has been studied for several decades [@Lloyd82; @Wu08] and many versions of this technique has recently been developed. [@Xu19] proposed alternatives to Lloyd’s algorithm that preserves its simplicity, makes it more robust to initialization and relieves its tendency to get trapped by local minima. [@Lattanzi19] developed a new variant of $k$-means++ seeding algorithm [@Arthur07] to achieve a constant approximation guarantee.
**Our contribution.** Observed real networks differ from random graphs from their edge distribution and from their underlying structures. Erdös Renyi random graphs models [@Erdos59], where all the pairs of nodes have equal probability of being connected by an edge, independently of all other pairs fail to model real observed graphs. Additionnally, stochastic block models are not always relevant to infer their structures. To remedy this problem, we developed a new random model, closely related to stochastic block model, but better suited to model graphs that have been inferred from the observations. In practice, graphs that are studied are not known beforehand but often estimated.To achieve a good clustering recovery, random graph models are often associated to their similarity matrix to maintain the clustering structure of the graph. [@Wang16] developed a model to learn a doubly stochastic matrix which encodes the probability of each pair of data points to be connected, used to normalize the affinity matrix such that the data graph is more suitable for clustering tasks. [@Peng15] has shown that for a wide class of graphs, spectral clustering gives a good approximation of the optimal cluster. In our model, we assume that a group does not emerge by chance but because there exists an underlying structure. This randomized version of the deterministic graph with exact cluster structure, is used to check whether it displays the original cluster structure. [@Sussman12; @Lei15; @Rohe11] proved consistency of spectral clustering applied to stochastic block models for some specific adjacency type matrices. Even if the consistency of spectral clustering has been proved for stochastic block models, there is no convergence guarantee for general models. Thus, $k$-means can fail to reach the true underlying partitions of the graph. Moreover, spectral clustering technique fails to recover the original clusters when it comes to a higher randomization coefficient. This is mostly due to the computed eigenbasis that is not equally informative.
In order to tackle this issue, we develop an alternative method to the spectral clustering that promote a sparse eigenvectors basis solution of an $\ell_0$ optimization problem, corresponding to the indicator vectors of each cluster. Since the natural constrained $\ell_0$ is a NP-hard problem, it was then replaced by its convex relaxation $\ell_1$ [@Ramirez13]. Actually we can show that the solution of the $\ell_0$ optimization problem is still the same when replacing the $\ell_0$-norm by the $\ell_1$-norm if we add a constraint on the maximum of the coefficients. Hence, the algorithm turns out to solve an $\ell_1$-penalty optimization problem that is feasible and easy to implement, even for very large graphs. In a wider scope, research papers have explored differently regularized spectral clustering to robustly identify clusters in large networks. Although [@Zhang18] and [@Joseph16] show the effect of regularization on spectral clustering through graph conductance and respectively through stochastic block models. Equally, [@Lara19], shows on a simple block model that the spectral regularization separates the underlying blocks of the graph.
In this paper, we introduce in Section \[section2\] a new random graph model, used to solve spectral clustering (Section \[section3\]) and its new variant (Section \[section4\]) objective function. We prove the efficiency and accuracy of the variant algorithm in Section \[section5\] through experiments on simulated and real medical dataset (Section \[section6\]).
New random graph model {#section2}
======================
Notations
---------
This work considers the framework of an unweighted undirected graph $G(V,E)$ with no self-loops consisting of vertices $V=\left\{ 1,\dots,n \right\}$ and $p$ edges connecting each pair of vertices. An edge $e \in E$ that connects a node $i$ and a node $j$ is denoted by $e = (i,j)$. In this paper, we consider that the existence of a link between two nodes in an interaction network is already inferred from the estimation of a statistical dependance measure. The graph $G$ is represented hereusing an adjacency matrix $A = (A_{ij})_{(i,j) \in V^2}$ defined by
$A_{i,j} = \left\{
\begin{array}{ll}
1 \ \ \mbox{if there is an edge between } i \mbox{ and } j,\\
0 \ \ \mbox{otherwise.}
\end{array}
\right.$
Since the graph is undirected and with no self-loops, $A \in \mathbb{M}_n(\mathbb{R})$ is a symmetric matrix with coefficients zero on the diagonal.
For each node $i$, the degree $d_i$ is defined as the number of edges incident to $i$ and is equal to : $d_i =\sum\limits_{j=1}^n A_{ij}$. We denote by $D$ the diagonal degree matrix containing $(d_1,\dots ,d_n)$ on the diagonal and zero elsewhere.
A subset $C \in V$ of a graph is said to be connected if any two vertices in $C$ are connected by a path in $C$. Non empty sets $ C_1, \dots, C_k $ form a partition of the graph $G(V,E)$ if $C_i \cap C_j =\emptyset$ and $C_1\cup \cdots \cup C_k=V$. In addition, $C_i$ are called connected components if there are no connections between vertices in $C_i$ and $\overline{C_i}$ for all i in $ \left\{1, \dots, k \right\}$.
We define the indicators of connected components $\textbf{1}_{C_i}$ whose entries are defined by:
$(\textbf{1}_{C_i})_j = \left\{
\begin{array}{ll}
1 \ \ \mbox{if vertex } j \mbox{ belong to } C_i,\\
0 \ \ \mbox{otherwise.}
\end{array}
\right.$
Graph models
------------
As mentioned in Section \[section1\], random graph models, in general, are not always relevant to represent the structure of a graph that has been inferred from observations. To tackle this issue, we create a new random model with an underlying structure that is a randomized version of a deterministic graph with exact cluster structure.
### Ideal model
We consider that the graph $G_*(V,E)$ is the union of $k$ complete graphs that are disconnected from each other. We denote by $C_1, \dots, C_k$ the $k$ connected components of the graph, that match the $k$ clusters. We allow the number of vertices in each subgraph to be different. We denote by $c_1,\cdots,c_k $ $(\geq 2)$ their respective size ($\sum_{i=1}^k c_i=n$). To simplify, we assume that the nodes, labeled from $1$ to $n$, are ordered with respect to their block membership and in increasing order with respect to the size of the blocks.
From a matricial point of view, the associated adjacency matrix $A_*$ is a $k$-block diagonal matrix of size $n$ of the form:
$$A_*=
\left[
\begin{array}{c@{}c@{}c}
C_1 & & \mathbf{0} \\
& \ddots & \\
\mathbf{0} & & C_k \\
\end{array}\right]$$
where $C_1, \cdots, C_k$ are symmetric matrices of size $c_1 \times c_1, \cdots, c_k \times c_k$.
### Perturbed model
The reality is that we consider the graph $G_*$ but we observe a randomized version of this graph, denoted by $\tilde{G}$.
We introduce the Erdös–Rényi model of a graph [@Erdos59; @Stewart90], one of the oldest and best studied random graph model.
Given a set of $n$ vertices, we consider the variable $X_{ij}$ that indicates the presence/absence of an edge between vertices $i$ and $j$. Then, for all $ \left\{ X_{ij} \right\} i.i.d.$, we have $ X_{ij} \sim B(p)$. Some edges have been added between the clusters and others have been removed within the clusters independently with respect to the same probability $p$. The adjacency matrix $B$ of the Erdos-Renyi graph of size $n$, whose upper entries are realizations of independent Bernoulli variables, can be written as
$$\left\{
\begin{array}{r c l}
B_{ij} &\sim& \mathcal{B}(p) \ \ i.i.d, \ i<j \\
B_{ii}&=&0 \\
B_{ij}&=&B{ji}
\end{array}
\right.$$
The graph $\tilde{G}$ of the new model, is derived from a deterministic graph with an exact cluster structure, whose edges have been disturbed Erdös Renyi random graph. For instance, this perturbation may arise because of a partial knowledge of the graph.
Let $\tilde{A}$ be the adjacency matrix associated to $\tilde{G}$. $\tilde{A}$ is defined as follows:
$\tilde{A}$ $ =A_* \bigoplus\limits^2 B$
where $\tilde{A}_{ij} = A_{ij*} + B_{ij} \ [2]$, namely,
$\tilde{A}_{ij}= \left\{
\begin{array}{ll}
0 \ \ \mbox{ if } B_{ij}=1 \mbox{ and } A_{ij}=1 \\
\mbox{ or } B_{ij}=0 \mbox{ and } A_{ij}=0,\\
1 \ \ \mbox{ if } B_{ij}=1 \mbox{ and } A_{ij}=0 \\
\mbox{ or } B_{ij}=0 \mbox{ and } A_{ij}=1.
\end{array}
\right. $
Graph clustering through spectral clustering {#section3}
=============================================
Spectral clustering algorithm
-----------------------------
Looking first at the ideal graph model, let $A_*$ and $D_*$ the adjacency and degree matrices associated to the graph $G_*$.
Spectral clustering algorithm is based on graph Laplacian matrices. Among them, three different variants are used:
- the Unormalized Laplacian:
$L_*=D_*-A_*$,
- the Symmetric Laplacian:
$L_{sym*}=D_*^{-\frac{1}{2}}A_*D_*^{-\frac{1}{2}}$,
- the Random Walk Laplacian:
$L_{rw*}=I-D_*^{-1} A_*$.
The original spectral clustering method has been proposed by [@Luxburg07] to cluster the nodes of the graph into $k$ connected components. The idea behind spectral clustering is to use the first $k$ eigenvectors (corresponding the $k$ smallest eigenvalues) a normalized or unormalized version of the Laplacian matrix (derived from the adjacency one) to recover the connected components of the graph. If these matrices are so appealing in graph clustering, it is because of the following proposition:
\[prop1\](Number of connected components): The multiplicity $k$ of the eigenvalue $0$ of $L_*$ and $L_{rw*}$ and the multiplicity of the generalized eigenvalue $0$ of $L_{sym*}$ are equal the number of connected components $C_1 , \dots, C_k $ in the graph. For $L_*$ and $L_{rw*}$, the eigenspace associated to $0$ is spanned by the indicators of connected components $\left\{ 1_{C_i} \right\}_{1 \leq i \leq n}$. For $L_{sym*}$, the eigenspace associated to $0$ is spanned by $\left\{ D_*^{1/2} 1_{C_i} \right\}_{1 \leq i \leq n}$.
We deduce from **Proposition 1** that a particular basis of the associated eigenspace is spanned by the connected components indicators. In addition, the rows of the matrix resulting from the concatenation of the $k$ first eigenvectors, are equal for indices corresponding to nodes in the same component. Therefore, it is natural to apply $k$-means to these rows to provide, by the same way the blocks. Moreover, as the graph is made of exactly $k$ connected components, the computation of the eigenvectors of $L_*$, $L_{sym*}$, $L_{rw*}$ enables to recover these components.
Limits
------
Secondly, we consider the perturbed version $\tilde{G}$ of the graph $G_*$. Thus, $\tilde{G}$ is no longer made of connected components, but of densely connected subgraphs that are sparsely connected to each other. These densely connected subgraphs represent somehow a perturbed version of the initial connected components that form the clusters. As our model is closely related to stochatic block models and if the perturbation is not too high, we can still hope that the rows of the $k$ concatenated eigenvectors are still closed for indices corresponding to nodes in the same cluster. But there is no theoritical guarantee that it still contains enough information on the graph structure to detect these clusters using $k$-means procedure.
To overcome this issue, we developed an alternative to the standard spectral clustering, called $\ell_1$-spectral clustering, that aims at finding the $k$ underlying connected components of a graph $G_*$ with an exact cluster structure from its perturbed version $\tilde{G}$.
$\ell_1$-spectral clustering, a new method for connected component detection {#section4}
=============================================================================
To ensure a good recovery of the connected components, the way the eigenvectors basis is built is of the highest importance. The key is to replace the $k$-means procedure by the selection of relevant eigenvectors that provide useful information about the structure of the data. $\ell_1$-spectral clustering focused on the graph adjacency matrix instead of the Laplacian matrix or its normalized version, and its good properties. The idea remains the same if we replace the adjacency matrix by the Laplacian or its normalized version as proved by [@Sussman12; @Rohe11].
We consider the ideal adjacency matrix $A_*$ associated to the graph $G_*(V,E)$, we assume in what follows that the eigenvalues of $A_*$ are sorted increasing order. And the same goes for the associated eigenvectors.
The indicators ${\{\textbf{1}_{C_i}\}}_{n-k+1 \leq i\leq n}$ of the connected components $C_{n-k+1}, \dots, C_{n}$ are the eigenvectors associated this time to the largest eigenvalues. Theses eigenvalues are equal to the degree coefficients of the connected components $d_{n-k+1}, \dots, d_n$. The $k$ first eigenvectors of $A_*$ (associated to the $k$ largest eigenvalues) are thus denoted $v_{n-k+1}, \dots, v_n$. Let $V_{1,k}$ the matrix that contains $v_{n-k+1}, \cdots, v_n$ in columns and by $V_2,n$ the one that contains $v_{1}, \dots, v_{n-k}$. We define $\mathcal{V}_{1,k}^0 =\text{Span}\{v_{n-k+1},\dots,v_n\}$.
Unlike the traditional spectral clustering method, $\ell_1$-spectral clustering does not directly use the subspace spanned by the eigenvectors associated to the largest eigenvalues to recover the connected components but computes another eigenbasis that promotes sparse solutions for the eigenvectors.
General $\ell_0$ minimization problem
-------------------------------------
Proposition 2 and 3 below show that the connected components indicators are solution of some specific problem.
\[prop2\] The minimization problem ($\mathcal{P}_0$)
$\underset{v\in \mathcal{V}_{1,k}^0 \backslash\{0\}}{\arg\min} {\|v\|}_0 $
has a unique solution (up to a constant) given by $\textbf{1}_{C_{n-k+1}}$.
In other words, $\textbf{1}_{C_{n-k+1}}$ is the sparsest non-zero eigenvector in the space spanned by the eigenvectors associated to the $k$ largest eigenvectors.
We recall that $\|v\|_0=| \left\{ j : v_j \ne 0 \right\}|$. Let $v \in \mathcal{V}_{1,k}^0 \backslash \left\{ 0 \right\}$. Therefore, as $\textbf{1}_{C_{n-k+1}} \in \mathcal{V}_{1,k}^0$, $v$ can be decomposed as $v=\sum\limits_{j=n-k+1}^n \alpha_j \textbf{1}_{C_j}$ where $\alpha=(\alpha_{n-k+1},\dots ,\alpha_n)\in \mathbb{R}^k \ $ and $ \exists j, \ \alpha_j \ne 0$.
The connected components of sizes $c_{n-k+1},\dots, c_n$ are sorted in increasing order of size. Therefore, by Proposition \[prop1\], $\|v\|_0=\textbf{1}_{\alpha_{n-k+1} \ne 0}c_{n-k+1}+\dots +\textbf{1}_{\alpha_n \ne 0} c_n$. The solution of ($\mathcal{P}_0$) is given by the vector in $\mathcal{V}_{1,k}\backslash\{0\}$ with the smallest $\ell_0$-norm such that $\alpha=(\alpha_{n-k+1} , 0, \dots, 0)$ where $\alpha_{n-k+1} \ne 0$.
We can generalize Proposition \[prop2\] to find, iteratively and with sparsity constraint, the other following indicators of connected components.
For $i=n-k+2,\dots,n$, let $\mathcal{V}_{1,k}^i =\left\{v\in \mathcal{V}_{1,k} : v \perp \textbf{1}_{C_l}, l=n-k+1, \dots ,i-1 \right\}$.
\[prop3\] The minimization problem ($\mathcal{P}_i$)
$\underset{v\in \mathcal{V}_{1,k}^i \backslash\{0\}}{\arg\min} {\|v\|}_0 $
has a unique solution (up to a constant) given by $\textbf{1}_{C_i}$.
Solving ($\mathcal{P}_0$) (Proposition \[prop2\]) is a NP-hard problem. In order to tackle this issue, we replace the ${\ell_0}$-norm by its convex relaxation ${\ell_1}$-norm. We can show that the solution of the $\ell_0$ optimization problem is still the same by replacing the $\ell_0$-norm by the $\ell_1$-norm, if we add the constraint on the maximum of the coefficients.
General $\ell_1$ minimization problem to promote sparsity {#l1}
---------------------------------------------------------
In addition to the number of connected components, we assume that we know one representative of each component i.e. a node belonging to this component. This assumption is not so restrictive compared to traditional spectral clustering where the number of clusters is assumed to be known. If we do not exactly know a representative for each component, we can estimate them by first applying a rough partitioning algorithm or just an algorithm that aims to find hubs of very densely connected parts of the graph.
Let $I_{n-k+1},\dots, I_n$ be the row indices of the representative element of each component and let $\tilde{\mathcal{V}}_{1,k}^1 =\{v\in \mathcal{V}_{1,k}^0 : v_{I_j}=1\}$ for all $j\in \left\{n-k+1, \dots,n \right\}$. This is straightfoward to see that the indicator vector of the smallest component is solution of the following optimization problem.
\[prop4\] The minimization problem ($\mathcal{P}_1$)
$\underset{v\in \tilde{\mathcal{V}}_{1,k}^1 }{\arg\min} {\|v\|}_1 $
has a unique solution given by $\textbf{1}_{C_{n-k+1}}$.
\[proof2\] We recall that $\|v\|_1=\sum\limits_{i=1}^n |v_i|$. Let $v \in \tilde{\mathcal{V}}_k^1$. Therefore, as $\textbf{1}_{C_{n-k+1}} \in \mathcal{V}_{1,k}^0$, $v$ can be decomposed as $v=\sum\limits_{j=n-k+1}^n \alpha_j \textbf{1}_{C_j}$ where $\alpha=(\alpha_{n-k+1},\dots ,\alpha_n)\in \mathbb{R}^k \ $ and there exists $ j \in \left\{n-k+1, \dots, n \right\} , \ \alpha_j \ne 0$.
The connected components of sizes $c_{n-k+1},\dots, c_n$ are sorted in increasing order of size. Therefore, $\|v\|_1=\alpha_{n-k+1} c_{n-k+1}+\dots +\alpha_n c_n$. The solution of ($\mathcal{P}_1$) is given by the vector in $\tilde{\mathcal{V}}_{1,k}^1$ that satisfies $\|v\|_{+\infty}$ and with the smallest $\ell_1$-norm such that $\alpha=(\alpha_{n-k+1} , 0, \dots, 0)$ where $\alpha_{n-k+1}=1$.
To simplify and without loss of generality, we assume that $I_{n-k+1}$ corresponds to the first index (up to a permutation). We can rewrite ($\mathcal{P}_1$) (Proposition \[prop4\]) as:
$\underset{(1,v) \in \mathcal{V}_{1,k}}{\underset{v \in \mathbb{R}^{n-1}}{\arg\min}} {\|v\|}_1 $
Constraints in ($\mathcal{P}_1$) can be moved to the following equality contraints:
\[prop5\] Let $w$ be the first column of $V_{2,n}^T$. We define $W$ as the matrix $V_{2,n}^T$ whose first column $w$ has been deleted.
The minimization problem ($\tilde{\mathcal{P}}_1$)
$\underset{W v=-w}{\underset{v \in \mathbb{R}^{n-1}}{\arg\min}} {\|v\|}_1 $
with $w=V_{2,n}^{(1)}$ and $W=V_{2,n}^{(n-1)}$ has a unique solution $\tilde{v}$ equals to $\textbf{1}_{C_{n-k+1}}$ such that $\tilde{v}=(1,v)$.
We recall that $V_{2,n}$ is the restriction of the eigenvectors matrix to the $n-k$ first columns. Because the columns of this matrix form an orthogonal basis, $v\in \mathcal{V}_{1,k}$ is equivalent to $V_{2,n}^T v=0$. Thus, $\tilde{v}=(1,v)$ satisfies the equation: $V_{2,n}^{(1)} + V_{2,n}^{(n-k-1)} v$, where $V_{2,n}^{(1)}$ is the first row of $V_{2,n}$ and $V_{2,n}^{(n-k-1)}$ the matrix $V_{2,n}$ whose first row has been deleted.
For all $ \tilde{v}=(1,v), $ $$\begin{aligned}
V_{2,n}^T\tilde{v} & =V_{2,n}^{(1)} + V_{2,n}^{(n-k-1)} v\\
&=0 \\
\Leftrightarrow \qquad V_{2,n}^{(n-k-1)} v &= -V_{2,n}^{(1)}\end{aligned}$$
Note that in Proposition \[prop5\], $V_{2,n}^{(1)}$ and $V_{2,n}^{(n-k-1)}$ are denoted $w$ and $W$.
**Remark:** Constraint problem ($\tilde{\mathcal{P}}_1$) (Proposition \[prop5\]) can be equivalently written as the following penalized problem:
$\underset{v \in \mathbb{R}^{n-1}}{\arg\min} \ {\|W v+w\|}_2^2 + \lambda {\| v \|}_1$.
where $\lambda >0$ is the regularization parameter that controls the balance between the constraint and the sparsity norm, $W \in \mathbb{M}_{n-k,n-1}$ is the matrix $V_{2,n}^T$ whose first column $w$ has been deleted.
In the following, we will provide an algorithm based on the contraint problem ($\tilde{\mathcal{P}}_1$) introduced in Proposition \[prop5\].
$\ell_1$-spectral clustering algorithm {#section5}
======================================
Now, we only consider graphs with an exact cluster structure whose edges have been perturbed by a coefficient $p \in [0,1]$.
$\ell_1$-spectral clustering algorithm is developed in a Matlab software. Starting with the number of blocks $k$ of an adjacency matrix $A$ and the column index of one representative element of each block $I_{n-k+1}, \dots, I_n$, the pseudo-code for the algorithm is presented in Algorithm \[algo\].
Steps from $3$ to $14$ are dedicated to the recovery of the indicators of connected components. The minimization problem introduced in Section \[l1\] is solved using the $\ell_1$-eq function of the Matlab optimization package $\ell_1$-magic [@Candes05]. Vector $\tilde{v}_j$ contains the solution of the minimization problem (step 11).
To find the other connected components indicators, we add the constraint of being orthogonal to the previous computed vectors by deflating the matrix $A$ (step 13) and we do the same to estimate the other connected component indicators.
Let $F$ be the concatenation of the vectors $\tilde{v}_j$. As the algorithm is applied on a perturbed adjacency matrix, the elements in $F$ are not exactly equal to one or zero but are very close to one for the indices associated to edges belonging to a same cluster and to zero for the remaining ones. Therefore, we shrink the solution (steps 16 to 20):
For all $j=1,\dots,n$, for all $i=1,\dots k$,
$F_{ij}=\left\{
\begin{array}{ll}
1 \ \ \mbox{if} \ F_{ij} >\frac{1}{2}, \\
0 \ \ \mbox{if} \ F_{ij} \leq \frac{1}{2}.
\end{array}
\right.$
The indicators of the clusters are given by the $k$ column vectors of $F$.
number of clusters $k$ , adjacency matrix $A$, indices of representative elements $Index=[I_{n-k+1},\dots, I_n]$. Initialize $F=[]$. Eigen decomposition $[V,U]$ of $A$: $A=VU^tV$. Sort in ascending order the eigenvalues and the associated eigenvectors of $A$. Form the matrix $V_{2,n}$ by stacking the $n-k+j-1$ eigenvectors associated to the smallest eigenvalues. Computation of constraints of the $l_1$ minimization problem Compute $T=V_{2,n}^t$ Compute $W=T^{-Index[j]}$ and $w=T^{Index[j]}$(where $T^{Index[j]}$ is the column $Index[j]$ of $T$ and $V^{-I_j}$ the matrix $T$ wihtout the column $Index[j]$) Compute the solution $v^j$ of the following problem
$\underset{Wv=-w}{\underset{v \in \mathbb{R}^{n-1}}{\arg\min}} {\|v\|}_1 $.
Recovery of the $j_{th}$ cluster:
$\tilde{v}_j=[v^j_{1} \ v^j_{2} \ \dots \ v^j_{Index[j] -1} \ 1 \ v^j_{Index[j]+1} \ \dots \ v^j_{n}] $ $F$ concatenation of the $j_{th}$ clusters and deflation of $A=A - \tilde{v}_j ^t \tilde{v}_j$ to recover the other indicator vectors $F((F>0.5))=1$ $F((F \leq 0.5))=0$ $k$ column vectors $F$.
$\ell_1$-spectral clustering applications {#section6}
==========================================
Spectral clustering and $\ell_1$-spectral clustering on simulated dataset
-------------------------------------------------------------------------
### Performances
In Section \[section5\], we introduced a new algorithm (called $\ell_1$-spectral clustering) that aims to detect cluster structures in complex graphs. To illustrate graphically the performances of this method, we simulated a perturbed version of a graph with an exact group structure. The associated adjacency matrix is composed of $k \in [5,10]$ blocks of size $c_{n-k+1},\dots, c_n \in [10,20]$. Let $p$ be the level of Bernoulli noise applied on the adjacency matrix.
Once the matrix is disturbed by a strictly positive coefficient, we no longer have exact block structures. To recover it, we applied the traditional spectral clustering algorithm and the new $\ell_1$-spectral clustering algorithm. Figure \[recovery\] gives the performances of both algorithm with a perturbation coefficient of $p=2$.
0.2in
![Input Adjacency matrix: Adjacency matrix with exact community structure. Perturbed Adjacency matrix $p=0.2,0.3$: Adjacency matrix after perturbation. L1 Adjacency matrix: Adjacency matrix recovery after $\ell_1$-spectral clustering application. Spectral Adjacency matrix: Adjacency matrix recovery after spectral clustering application.[]{data-label="recovery"}](p2bis.pdf){width="\columnwidth"}
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We can notice that our model performs well in this task as both methods effectively recovers the clustering structure, which indicates the robustness of our model.
### Robustness to perturbations
Then, we tested the robustness under perturbations of the spectral clustering ang $l_1$-spectral clustering algorithms. Let $p$ be the level of Bernoulli noise, discretized in this section between $0$ and $0.4$. In this experiment, we simulate $100$ graphs with $k \in [5,10]$ clusters of size $c_{n-k+1},\dots, c_n \in [10,20]$.
We introduce the block membership function: for all node $i \in \left\{1,\dots, n \right\}$ of a graph $G(V,E)$ made of block structures of size $c_{n-k+1},\dots, c_n$, $$\begin{aligned}
\tau \colon & V \to \left\{n-k+1,\dots, n \right\}\\
&i \mapsto c.\end{aligned}$$
For each value of $p$, we test the perfomances of both algorithms to recover the clusters of the graphs. The performances of the algorithms were evaluated by computing the percentage of missassigned nodes in average defined as $\frac{1}{100} \sum\limits_{j=1}^{100} |\left\{i\in V : \tau(i) \ne \hat{\tau}_j(i)\right\}|$, where $\tau_j$ is the block membership function and $\hat{\tau}_j$ is the estimated membership function for the $j$-th model. The results are plotted in Figure \[comparaison\].
0.2in
![Fraction of nodes correctly classified using spectral (red line) and $\ell_1$-spectral clustering (blue line) under increasing perturbation coefficient. []{data-label="comparaison"}](compl1spec3.pdf){width="\columnwidth"}
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Figure \[comparaison\] captures the fraction of nodes correctly classified and the associated region of confidence when $\ell_1$-spectral clustering (blue) and spectral clustering (red) are applied under increasing perturbation coefficient.
### Results
Both simulations show that the perturbation coefficient has an impact on the performance of spectral clustering and $\ell_1$-spectral clustering. Moreover, we observe that $\ell_1$-spectral clustering works better on simulated data for small perturbations (up to $30\%$ Bernoulli noise) than spectral clustering. Thus, the new technique provides powerful results on small perturbations (rate of exact assignment is equal or very close to one).
$\ell_1$-spectral clustering on real dataset
--------------------------------------------
### FLORINASH dataset
This section is dedicated to experimental studies to assess the performances of our method through real dataset. Experiments have been performed on R using the packages igraph, PLNmodels [@Chiquet19]. The dataset we used belongs to the project FLORINASH that proposes an innovative research concept to address the role of intestinal microfloral activity in Non-Alcoholic Fatty Liver Disease (NAFLD).
Hepatic steatosis is often observed in obese patients and is a preliminary stage to non-alcoholic fatty liver disease. The studied cohort [@Hoyles18] is made of obese patients featured with hepatic steatosis. It has been deeply studied and numerous clinical and biochemical data sets are available. We ran an ancillary study on $51$ control and $6$ diabetic patients with a median age of $42,50$ years, and characterized by a median body weight of $124,125$ kg and a glycemia of $5.8,6.5$ mM.
The underlying dataset includes the output of sequencing 16S rDNA gene from liver biopsies to study microbial composition and diversity of obese patients. The standard approach to analyzing 16S rDNA sequence data relies on clustering reads by sequence similarity into Operational Taxonomic Units (OTUs). All OTUs are assigned to a taxonomic rank (phylum, class, order, family, genus and species). The standard way of representing the community structure inferred from microbial data is by means of an abundance table, where the rows correspond to samples ($57$ patients) and columns to features ($831$ microbial taxa). The goal of this analysis is to detect clusters of OTUs at their family taxonomy level according to their abundance by patients ($53$ OTUs at this taxonomy). Our aim is to identify the associations between the different microbial families by reconstructing the ecological network and make a direct comparisons between the two groups of patients.
### Results
Microbiome data is compositional because the information that abundance tables contain is relative, the total number of counts per sample being highly variable. Few universal multivariate models are available for compositional data and existing models often impose undesired constraints on the dependency structure. To tackle this issue, we use the Poisson Log Normal model [@Chiquet19]. We use the framework of graphical models to model the dependency structure of the dataset.
From the graph modeled, we deduce the adjacency matrix and the score of each underlying hubs [@Kleinberg98]. A hub is a node with a number of links that greatly exceeds the average, also called a high degree node. A node is given a high hub score by linking to a large number of nodes.The number of hubs selected give us the total number of clusters do be detected. $l_1$-spectral clustering applied on the adjacency matrix outputs $4$ (respectively $5$) clusters in control and diabetic patients (Figure \[comparaison\]).
0.2in
![Application of $\ell_1$-spectral clustering on the cohort composed of patients with and without diabetes. (a) Graph representing hubs and clusters related to diabetic patients (b) Graph representing hubs and clusters related to healthy patients.[]{data-label="comparaison"}](Graphs.pdf){width="\columnwidth"}
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0.2in
![(a) Proportion of Phyla in control and diabetic patients (b) Proportion of Families in control and diabetic patients.[]{data-label="phyla"}](barplot_der.pdf){width="\columnwidth"}
-0.2in
Figure \[phyla\] shows that the gut microbiota from control and diabetic patients is characterized by two dominating phyla Firmicutes and Bacteroidetes ([@Ley05; @Gill06]). We also added to the knowledge that in diabetic patients there was a disappearance of the frequency of the variable Verrucomicrobiaceae. This is also in agreement with the data from literature which demonstrate that this bacteria could be considered as a probiotic controlling metabolic diseases [@Everard13]. Eventually, from $\ell_1$-spectral clustering, we identified that a novel variable i.e. the Fusobacteriaceae are important discriminant signatures of the non diabetic group while that of the diabetic group is related to the Firmicutes variable. Altogether, our algorithm was validated by the previously published findings from the FLORINASH cohort and even add to the knowledge that some specific variables could be associated with the diabetic or non diabetic signatures.
Conclusion
==========
We present $\ell_1$-spectral clustering, a novel variation of spectral clustering algorithm based on promoting a sparse eigenvectors basis that provides information about the structure of the system observed. The associated graph is assumed to contain connected subnetworks. We characterized the indicators of these subnetworks as the ones that have the minimal $\ell_1$-norm with respect to a specific restricted space. $\ell_1$-spectral clustering benefits from this feasible objective function as a substitution of the $k$-means step of the traditional spectral clustering. Its effectiveness and robustness to small noise perturbations is confirmed by simulated and real examples.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We theoretically investigate heat transport in temperature-biased Josephson tunnel junctions in the presence of an in-plane magnetic field. In full analogy with the Josephson critical current, the phase-dependent component of the heat flux through the junction displays *coherent diffraction*. Thermal transport is analyzed in three prototypical junction geometries highlighting their main differences. Notably, minimization of the Josephson coupling energy requires the quantum phase difference across the junction to undergo $\pi$ *slips* in suitable intervals of magnetic flux. An experimental setup suited to detect thermal diffraction is proposed and analyzed.'
author:
- 'F. Giazotto'
- 'M. J. Martínez-Pérez'
- 'P. Solinas'
title: Coherent diffraction of thermal currents in Josephson tunnel junctions
---
Introduction {#intro}
============
The impressive advances achieved in nanoscience and technology are nowadays enabling the understanding of one central topic in science, i.e., *thermal flow* in solid-state nanostructures [@Giazotto2006; @Dubi2011]. Control and manipulation [@heattransistor; @ser] of thermal currents in combination with the investigation of the origin of dissipative phenomena are of particular relevance at such scale where heat deeply affects the properties of the systems, for instance, from *coherent caloritronic* circuits, which allow enhanced operation thanks to the quantum phase [@Meschke2006; @Vinokur2003; @Eom1998; @Chandrasekhar2009; @Ryazanov1982; @Panaitov1984; @virtanen2007; @Martinez2013], to more developed research fields such as ultrasensitive radiation detectors [@Giazotto2006; @Giazotto2008] or cooling applications [@Giazotto2006; @Giazotto2002]. In this context it has been known for more than $40$ years that heat transport in Josephson junctions can be, in principle, phase-dependent [@Maki1965; @Guttman97; @Guttman98; @Zhao2003; @Zhao2004; @Golubev2013]. The first ever Josephson thermal interferometer has been, however, demonstrated only very recently [@giazotto2012; @martinez2012; @giazottoexp2012; @simmonds2012], therefore proving that phase coherence extends to thermal currents as well. The heat interferometer of Ref. [@giazottoexp2012] might represent a prototypical circuit to implement novel-concept coherent caloritronic devices such as heat transistors [@martinez2012], thermal splitters and rectifiers [@Martinez2013].
In the present work we theoretically analyze heat transport in temperature-biased extended Josephson tunnel junctions showing that the phase-dependent component of thermal flux through the weak-link interferes in the presence of an in-plane magnetic field leading to *heat diffraction*, in analogy to what occurs for the Josephson critical current. In particular, thermal transport is investigated in three prototypical *electrically-open* junctions geometries showing that the quantum phase difference across the junction undergoes $\pi$ *slips* in order to minimize the Josephson coupling energy. These phase slips have energetic origin and are not related to fluctuations as conventional phase slips in low-dimensional superconducting systems [@Langer1967; @Zaikin1997; @astafiev2012]. We finally propose how to demonstrate thermal diffraction in a realistic microstructure, and to prove such $\pi$ slips exploiting an uncommon observable such as the heat current.
The paper is organized as follows: In Sec. \[model\] we describe the general model used to derive the behavior of the heat current in a temperature-biased extended Josephson tunnel junction. In Sec. \[results\] we obtain the conditions for the quantum phase difference across an electrically-open short Josephson junction in the presence of an in-plane magnetic field, and the resulting behavior of the phase-dependent thermal current. In particular, we shall demonstrate the occurrence of phase-slips of $\pi$, independently of the junction geometry, in order to minimize the Josephson coupling energy. The phase-dependent heat current in three specific junction geometries is further analyzed in Sec. \[differentgeo\], where we highlight their main differences. In Sec. \[experiment\] we suggest and analyze a possible experimental setup suited to detect heat diffraction through electronic temperature measurements in a microstructure based on an extended Josephson junction, and to demonstrate the existence of $\pi$ slips. Finally, our results are summarized in Sec. \[summary\].
![(Color online) (a) Cross section of a temperature-biased extended S$_1$IS$_2$ Josephson tunnel junction in the presence of an in-plane magnetic field $H$. The heat current $J_{S_1\rightarrow S_2}$ flows along the $z$ direction whereas $H$ is applied in the $x$ direction, i.e., parallel to a symmetry axis of the junction. Dashed line indicates the closed integration contour, $T_i$, $t_i$ and $\lambda_i$ represent the temperature, thickness and London penetration depth of superconductor S$_i$, respectively, and $d$ is the insulator thickness. $\Phi$ denotes the magnetic flux piercing the junction. Prototypical junctions with rectangular, circular, and annular geometry are shown in panel (b), (c) and (d), respectively. $L$, $W$, $R$ and $r$ represent the junctions geometrical parameters. []{data-label="fig1"}](fig1.pdf){width="\columnwidth"}
Model
=====
Our system is schematized in Fig. \[fig1\](a), and consists of an extended Josephson tunnel junction composed of two superconducting electrodes S$_1$ and S$_2$ in thermal steady-state residing at different temperatures $T_1$ and $T_2$, respectively. We shall focus mainly on symmetric Josephson junctions in the *short* limit, i.e., with lateral dimensions much smaller than the Josephson penetration depth \[see Fig. \[fig1\](b,c,d)\], $L,W,R,r\ll \lambda_J=\sqrt{\frac{\pi \Phi_0}{\mu_0i_ct_H}}$, where $\Phi_0=2.067\times 10^{-15}$ Wb is the flux quantum, $\mu_0$ is vacuum permeability, $i_c$ is the critical current areal density of the junction, and $t_H$ is the junction effective magnetic thickness to be defined below. In such a case the self-field generated by the Josephson current in the weak-link can be neglected with respect to the externally applied magnetic field, and no traveling solitons can be originated. $t_i$ and $\lambda_i$ denote the thickness and London penetration depth of superconductor S$_i$, respectively, whereas $d$ labels the insulator thickness. We choose a coordinate system such that the applied magnetic field ($H$) lies parallel to a symmetry axis of the junction and along $x$, and that the junction electrodes planes are parallel to the $xy$ plane. Furthermore, the junction lateral dimensions are assumed to be much larger than $d$ so that we can neglect the effects of the edges, and each superconducting layer is assumed to be thicker than its London penetration depth (i.e., $t_i>\lambda_i$) so that $H$ will penetrate the junction in the $z$ direction within a thickness $t_H=\lambda_1+\lambda_2+d$ [@magneticlength]. For definiteness, we assume $T_1\geq T_2$ so that the Josephson junction is temperature biased only, and no electric current flows through it. If $T_1\neq T_2$ there is a finite electronic heat current $J_{S_1\rightarrow S_2}$ flowing through the junction from S$_1$ to S$_2$ \[see Fig. \[fig1\](a)\] which is given by [@Maki1965; @Guttman97; @Guttman98; @Zhao2003; @Zhao2004; @Golubev2013] $$J_{S_1\rightarrow S_2}(T_1,T_2,\varphi)=J_{qp}(T_1,T_2)-J_{int}(T_1,T_2)\textrm{cos}\varphi.
\label{heatcurrent}$$ Equation (\[heatcurrent\]) describes the oscillatory behavior of the thermal current flowing through a Josephson tunnel junction as a function of $\varphi$ predicted by Maki and Griffin [@Maki1965], and experimentally verified in Ref. [@giazottoexp2012]. In Eq. (\[heatcurrent\]), $J_{qp}$ is the usual heat flux carried by quasiparticles [@Giazotto2006; @Frank1997], $J_{int}$ is the phase-dependent part of the heat current which is peculiar of Josephson tunnel junctions, and $\varphi$ is the macroscopic quantum phase difference between the superconductors. By contrast, the Cooper pair condensate does not contribute to heat transport in a static situation [@Maki1965; @Golubev2013; @giazottoexp2012].
The two terms appearing in Eq. (\[heatcurrent\]) read [@Maki1965; @Guttman97; @Guttman98; @Zhao2003; @Zhao2004; @Golubev2013] $$J_{qp}=\frac{1}{e^2 R_J} \int^{\infty}_{0} d\varepsilon \varepsilon \mathcal{N}_1 (\varepsilon,T_1)\mathcal{N}_2 (\varepsilon,T_2)[f(\varepsilon,T_2)-f(\varepsilon,T_1)],$$ and $$J_{int}=\frac{1}{e^2 R_J}\int^{\infty}_{0}d\varepsilon \varepsilon \mathcal{M}_{1}(\varepsilon,T_1)\mathcal{M}_{2}(\varepsilon,T_2)[f(\varepsilon,T_2)-f(\varepsilon,T_1)],$$ where $\mathcal{N}_{i}(\varepsilon,T_i)=|\varepsilon|/\sqrt{\varepsilon^2-\Delta_{i}(T_i)^2}\Theta[\varepsilon^2-\Delta_{i}(T_i)^2]$ is the BCS normalized density of states in S$_{_i}$ at temperature $T_i$ ($i=1,2$), $\mathcal{M}_{i}(\varepsilon,T_i)=\Delta_{i}(T_i)/\sqrt{\varepsilon^2-\Delta_{i}(T_i)^2}\Theta[\varepsilon^2-\Delta_{i}(T_i)^2]$, and $\varepsilon$ is the energy measured from the condensate chemical potential. Furthermore, $\Delta_i(T_i)$ is the temperature-dependent superconducting energy gap, $f(\varepsilon,T_i)=\text{tanh}(\varepsilon/2 k_BT_i)$, $\Theta(x)$ is the Heaviside step function, $k_B$ is the Boltzmann constant, $R_J$ is the junction normal-state resistance, and $e$ is the electron charge. In the following analysis we neglect any contribution to thermal transport through the Josephson junction arising from lattice phonons.
Results
=======
In order to discuss the effect of the applied magnetic field on the heat current we shall focus first of all onto the phase-dependent component. To this end we need to determine the phase gradient $\varphi (x,y)$ induced by the application of the external magnetic flux. By choosing the closed integration contour indicated by the dashed line depicted in Fig. \[fig1\](a) it can be shown [@Tinkham; @Barone] that, neglecting screening induced by the Josephson current, $\varphi (x,y)$ obeys the equations $\partial \varphi/\partial x=0$ and $\partial \varphi/\partial y=2\pi \mu_0 t_H H/\Phi_0$. The latter equation can be easily integrated to yield $$\varphi (y)=\kappa y+\varphi_0,$$ where $\kappa \equiv 2\pi \mu_0 t_H H/\Phi_0$ and $\varphi_0$ is the phase difference at $y=0$. The phase-dependent component of the heat current can then be written as $$J_{H}(T_1,T_2,H)=\int\int dxdy J_A(x,y,T_1,T_2)\textrm{cos}(\kappa y+\varphi_0),
\label{phasea}$$ where the integration is performed over the junction area, and $J_A(x,y,T_1,T_2)$ is the heat current density per unit area. We note that the integrand of Eq. (\[phasea\]) oscillates sinusoidally along the $y$ direction with period given by $\Phi_0(\mu_0t_H H)^{-1}$. After integration over $x$ we can write Eq. (\[phasea\]) as $$\begin{aligned}
J_{H}(T_1,T_2,H)&=&\int dy \mathcal{J}(y,T_1,T_2)\textrm{cos}(\kappa y+\varphi_0)\nonumber\\
&=&\textrm{Re}\left\{e^{i\varphi_0}\int^{\infty}_{-\infty} dy \mathcal{J}(y,T_1,T_2)e^{i\kappa y}\right\},
\label{phaseb}\end{aligned}$$ where $\mathcal{J}(y,T_1,T_2)\equiv \int dx J_A(x,y,T_1,T_2)$ is the heat current density per unit length along $y$. In writing second equality in Eq. (\[phaseb\]) we have replaced the integration limits by $\pm \infty$ since the thermal current is zero outside the junction. Equation (\[phaseb\]) for $J_{H}(T_1,T_2,H)$ resembles the expression for the Josephson current, $I_{H}(T_1,T_2,H)$, which is given by [@Tinkham; @Barone] $$\begin{aligned}
I_{H}(T_1,T_2,H)&=&\textrm{Im}\left\{e^{i\varphi_0}\int^{\infty}_{-\infty} dy \mathcal{I}(y,T_1,T_2)e^{i\kappa y}\right\} \nonumber\\
&=&\textrm{sin}\varphi_0\int^{\infty}_{-\infty} dy\mathcal{I}(y)\textrm{cos}\kappa y,
\label{critical}\end{aligned}$$ where $\mathcal{I}(y,T_1,T_2)$ is the supercurrent density integrated along $x$, and second equality in Eq. (\[critical\]) follows from the assumed junctions *symmetry*, i.e., $\mathcal{I}(y,T_1,T_2)=\mathcal{I}(-y,T_1,T_2)$. In the actual configuration of electrically-open junction, the condition of *zero* Josephson current for any given value of $H$ yields the solution $\varphi_0=m\pi$, with $m=0,\pm 1,\pm2\ldots$. On the other hand, the Josephson coupling energy of the junction ($E_J$) can be expressed as $$\begin{aligned}
E_J(T_1,T_2,H)&=&E_{J,0}-\frac{\Phi_0}{2\pi}\textrm{Re}\left\{e^{i\varphi_0}\int_{-\infty}^{\infty} dy \mathcal{I}(y,T_1,T_2)e^{i\kappa y}\right\}\nonumber\\
&=&E_{J,0}-\frac{\Phi_0}{2\pi}\textrm{cos}\varphi_0\int_{-\infty}^{\infty} dy\mathcal{I}(y)\textrm{cos}\kappa y
\label{energy}\end{aligned}$$ where $E_{J,0}=\Phi_0I_c/2\pi$, $I_c$ is the zero-field critical supercurrent, and in writing the second equality we have used the symmetry property of $\mathcal{I}(y,T_1,T_2)$. Minimization of $E_J$ for any applied $H$ imposes the second term on rhs of Eq. (\[energy\]) to be always negative, so that $\varphi_0$ will undergo a $\pi$ *slip* whenever the integral does contribute to $E_J$ with negative sign. As a result, the Josephson coupling energy turns out to be written as $$E_J(T_1,T_2,H)=E_{J,0}-\frac{\Phi_0}{2\pi}\left|\int_{-\infty}^{\infty} dy\mathcal{I}(y,T_1,T_2)\textrm{cos}\kappa y\right|.
\label{energybis}$$ We also assume that the symmetry of the junction and of the electric current density are reflected in an analogous symmetry in the heat current, i.e., $\mathcal{J}(y,T_1,T_2) = \mathcal{J}(-y,T_1,T_2)$. It therefore follows from Eq. (\[phaseb\]) that $J_H$ can be written as $$J_{H}(T_1,T_2,H)=\left|\int^{\infty}_{-\infty} dy \mathcal{J}(y,T_1,T_2)\textrm{cos}\kappa y\right|.
\label{heatcurrfin}$$ Equation (\[heatcurrfin\]) is the main result of the paper.
The above results hold for symmetric Josephson junctions under in-plane magnetic field parallel to a symmetry axis, and only occur without any electrical bias. The discussed phase slips, however, exists for any arbitrary junction geometry. If the junction has not a symmetric geometry with respect to the magnetic field direction, the constraints on vanishing Josephson current and minimization of coupling energy are translated in a more complex condition for the phase $\varphi_0$. As we shall demonstrate below, the phase undergoes nevertheless a $\pi$ slip as well. We choose the $x$ axis of the coordinate system parallel to the magnetic field \[as in Fig. \[fig1\](a)\]. For a junction with arbitrary symmetry, we split $\mathcal{I}(y,T_1,T_2)$ in its symmetric, $ \mathcal{I}_s(y)$, and antisymmetric part, $ \mathcal{I}_a(y)$. We thus have $$\begin{aligned}
I_H&=&\textrm{Im}\left\{e^{i\varphi_0}\int^{\infty}_{-\infty} dy \mathcal{I}(y)e^{i\kappa y}\right\} \nonumber\\
&=& \cos \varphi_0 \int_{-\infty}^{\infty} dy \mathcal{I}_a (y) \sin \kappa y + \sin \varphi_0 \int_{-\infty}^{\infty} dy \mathcal{I}_s (y) \cos \kappa y\nonumber\\
&=& \cos \varphi_0 I_a + \sin \varphi_0 I_s,
\label{eq:zero_current}\end{aligned}$$ where we have denoted the symmetric and antisymmetric integrals as $I_s$ and $I_a$, respectively. The case of symmetric junctions has already been discussed above so that in the following we shall focus on junctions with no symmetry, i.e., with $I_s\neq0$ [*and*]{} $I_a\neq0$. Notice that this already has consequences on the values of $\varphi_0$. In fact, if $I_s\neq0$ [*and*]{} $I_a\neq0$ we must have $\cos \varphi_0 \neq 0$ [*and*]{} $\sin \varphi_0 \neq 0$ to satisfy the zero-current condition.
The Josephson coupling energy can be written as $$\begin{aligned}
E_J&=&E_{J,0} - \frac{\Phi_0}{2 \pi} \Big [\cos \varphi_0 \int_{-\infty}^{\infty} dy \mathcal{I}_s (y) \cos \kappa y\nonumber\\
&-& \sin \varphi_0 \int_{-\infty}^{\infty} dy \mathcal{I}_a (y) \sin \kappa y \Big ]\nonumber\\
&=&E_{J,0} + \frac{\Phi_0}{2 \pi} \Big [ -\cos \varphi_0 I_s + \sin \varphi_0 I_a \Big].\end{aligned}$$ To find the energy minima, we differentiate twice with respect to $\varphi_0$ and impose the condition $\partial^2 E_J / \partial \varphi_0^2 > 0$. We therefore obtain $$\frac{\partial^2 E_J}{ \partial \varphi_0^2} = \frac{\Phi_0}{2 \pi} \Big [ \cos \varphi_0 I_s - \sin \varphi_0 I_a \Big] >0.
\label{eq:energy_min}$$ Assuming that that $I_s \neq 0$, the condition of vanishing $I_H$ for any applied $H$ from Eq. (\[eq:zero\_current\]) reads $$\sin \varphi_0 = - \cos \varphi_0 \frac{I_a}{I_s} ~ {\rm or }~ \tan \varphi_0 = - \frac{I_a}{I_s}.
\label{eq:sin_varphi_0}$$ By using the first of Eqs. (\[eq:sin\_varphi\_0\]), the condition to have minima in Eq. (\[eq:energy\_min\]) gives $$\frac{\partial^2 E_J}{ \partial \varphi_0^2} = \frac{\Phi_0}{2 \pi} \cos \varphi_0 \Big(I_s + \frac{I_a^2}{I_s} \Big) = \frac{\Phi_0}{2 \pi} \frac{\cos \varphi_0}{I_s} \left(I_s^2 + I_a^2 \right) > 0
\label{eq:min_condition}$$ that depends on the signs of $I_s$ and $\cos \varphi_0$.
Now we turn to equations (\[eq:sin\_varphi\_0\]) that impose a constraint on $\varphi_0$ as a function of $I_s$ and $I_a$. To simplify the discussion we denote as $\phi_0 = - \arctan (I_a/I_s)$, and consider only solutions within a $2\pi$ variation from the latter. We have two solutions of Eqs. (\[eq:sin\_varphi\_0\]): $\varphi_{0,1} = \pi + \phi_0$ and $\varphi_{0,2} = \phi_0$ which correspond to cosine function $$\begin{aligned}
\cos \varphi_{0,1} &=& - \cos \phi_0 = - \frac{1}{\sqrt{1+ \left(\frac{I_a}{I_s} \right)^2}} \nonumber \\
\cos \varphi_{0,2} &=& \cos \phi_0 = \frac{1}{\sqrt{1+ \left(\frac{I_a}{I_s} \right)^2}},\end{aligned}$$ where we have used the relation $\cos( \arctan x)= 1/\sqrt{1+x^2}$. As we can see, the first solution gives a negative $\cos \varphi_{0,1}$ while the second one corresponds to positive $\cos \varphi_{0,2}$. Going back to the inequality (\[eq:min\_condition\]), if $I_s>0$ we need to choose the solution $\varphi_{0,2} = \phi_0$ (for which $ \cos \varphi_{0,2}>0$) to minimize the Josephson coupling energy. By contrast, if $I_s<0$, we must choose the solution $\varphi_{0,1} = \pi + \phi_0$ (for which $ \cos \varphi_{0,1}<0$). Therefore, we get that the superconducting phase must undergo a $\pi$ slip to minimize the Josephson coupling energy whenever the integral $I_s$ changes sign as a function of the magnetic field.
We shall conclude by discussing the pure *antisymmetric* junction case, i.e., $I_a\neq0$ and $I_s =0$. Because of the zero current condition the only values that the phase $\varphi_0$ can assume are $\pi/2$ or $3\pi/2 $. Equation (\[eq:energy\_min\]) implies that, if $I_a >0$, $\varphi_0=3\pi /2$ and, if $I_a <0$, $\varphi_0=\pi/2$. Therefore, also in this case, the phase $\varphi_0$ undergoes a $\pi$ slip when $I_a$ changes sign.
We remark that the discussed phase slips differ from those present in low dimensional superconductors, caused by thermal [@Langer1967] and quantum [@Zaikin1997; @astafiev2012] fluctuations. In those cases, the phase slips are generated when, because of fluctuations, the modulus of the complex order parameter goes to zero, the phase becomes unrestricted and jumps of $2 \pi$ [@arutyunov08].
By contrast, the phase slips discussed above have an energetic origin and they occur when the system passes from one energetically stable configuration to another one [@kuplevakhsky06]. This transition takes place when the magnetic flux crosses one of the critical points and therefore can be experimentally induced by changing the magnetic flux. The different origin of the slips is exemplified by the fact that the fluctuation-induced phase slips are always of $2 \pi$ while in the present case we have slips of $\pi$. The identification of this effect is possible only in the electrically-open junctions. In fact, the presence of an electric current or a voltage bias would destroy or hide the original effect.
![(Color online) Normalized phase-dependent component of the heat current $J_{H}$ versus magnetic flux $\Phi$ calculated for a rectangular \[(a)\], circular \[(b)\], and annular \[(c)\] Josephson tunnel junction. In the curves of panel (c) we set $\alpha=0.9$, and $n$ indicates the number of fluxons trapped in the junction barrier. []{data-label="fig2"}](fig2.pdf){width="\columnwidth"}
Heat current in Josephson junctions with different geometries {#differentgeo}
=============================================================
With the help of Eq. (\[heatcurrfin\]) we can now determine the behavior of $J_{H}(T_1,T_2,H)$ for the three prototypical junction geometries sketched in Fig. (\[fig1\]). In particular, we shall consider two well-known examples such as the *rectangular* \[see Fig. \[fig1\](b)\] and *circular* \[see Fig. \[fig1\](c)\] junction, and the more exotic *annular* one \[see Fig. \[fig1\](d)\]. Annular junctions offer the possibility to investigate fluxons dynamics due to the absence of collisions with boundaries; yet, they provide fluxoid quantization thanks to their geometry which allows fluxons trapping. We assume that the total phase-dependent heat current is characterized by a uniform distribution, i.e., by a constant thermal current areal density $J_A(x,y,T_1,T_2)$ in Eq. (\[phasea\]). $J_{H}$ can therefore be calculated for the three considered geometries by following, for instance, Refs. [@Tinkham; @Barone]. In particular, for the rectangular junction, the absolute value of the sine cardinal function is obtained, $$J_{H}^{rect}(T_1,T_2,\Phi)=J_{int}(T_1,T_2)\left|\frac{\textrm{sin}(\pi\Phi/\Phi_0)}{(\pi\Phi/\Phi_0)}\right|,$$ where $J_{int}(T_1,T_2)=WLJ_A(T_1,T_2)$, $\Phi=\mu_0 HLt_H$, $L$ is the junction length and $W$ its width. For the circular geometry one gets the Airy diffraction pattern, $$J_{H}^{circ}(T_1,T_2,\Phi)=J_{int}(T_1,T_2)\left|\frac{J_1(\pi\Phi/\Phi_0)}{(\pi\Phi/2\Phi_0)}\right|,$$ where $J_{int}(T_1,T_2)=\pi R^2 J_A(T_1,T_2)$, $J_1(y)$ is the Bessel function of the first kind, $\Phi=2\mu_0 HRt_H$, and $R$ is the junction radius. Finally, for the annular junction [@Martucciello1996; @Nappi1997], the phase-dependent component of the heat current takes the form $$J_{H}^{ann}(T_1,T_2,\Phi)=\frac{2J_{int}(T_1,T_2)}{1-\alpha^2}\left|\int^1_\alpha dxxJ_n(x\pi \Phi/\Phi_0)\right|,$$ where $J_{int}(T_1,T_2)=\pi(R^2-r^2)J_A(T_1,T_2)$, $\Phi=2\mu_0 HRt_H$, $\alpha=r/R$, $J_n(y)$ is the $n$th Bessel function of integer order, $R$ ($r$) is the external (internal) radius, and $n=0,1,2,...$ is the number of $n$ trapped fluxons in the junction barrier.
Figure \[fig2\] illustrates the behavior of $J_{H}$ for the three geometries. In particular, the curve displayed in Fig. \[fig2\](a) for the rectangular case shows the well-known Fraunhofer diffraction pattern analogous to that produced by light diffraction through a rectangular slit. In such a case, the heat current $J_{H}$ vanishes when the applied magnetic flux through the junction equals integer multiples of $\Phi_0$. Furthermore, the heat current is rapidly damped by increasing the magnetic field falling asymptotically as $\Phi^{-1}$.[@Tinkham] The behavior for a circular junction is displayed in Fig. \[fig2\](b). Here, the flux values where $J_{H}$ vanishes do not coincide anymore with multiples of $\Phi_0$, and $J_H$ falls more rapidly than in the rectangular junction case, i.e., as $\Phi^{-3/2}$.[@Tinkham] Figure \[fig2\](c) shows $J_H$ for an annular junction. In particular, the heat current diffraction pattern is strongly $n$-dependent and, differently from the rectangular and circular case, $J_H$ decays in general more slowly. It is apparent that annular junctions may provide, in principle, enhanced flexibility to tailor the heat current response.
![(Color online) (a) Possible experimental setup to demonstrate heat diffraction in a temperature-biased rectangular Josephson junction. Source and drain normal-metal electrodes are tunnel-coupled to one of the junction electrodes (S$_1$). Superconducting tunnel junctions operated as heaters and thermometers are connected to source and drain. A static in-plane magnetic field $H$ is applied perpendicular to the S$_1$IS$_2$ junction. (b) Thermal model describing the main heat exchange mechanisms existing in the structure shown in (a).[]{data-label="fig3"}](fig3.pdf){width="\columnwidth"}
Proposed experimental setup {#experiment}
===========================
Demonstration of diffraction of thermal currents could be achieved in the setup shown in Fig. \[fig3\](a). It consists of two normal-metal source and drain electrodes tunnel-coupled via resistances $R_t$ to one electrode (S$_1$) of a Josephson junction which, for the sake of clarity, is assumed to be *rectangular*. An in-plane static magnetic field $H$ is applied perpendicular to the Josephson weak-link. Furthermore, superconducting probes tunnel-coupled to both source and drain electrodes either implement heaters or allow accurate measurement of the electronic temperature in the leads [@Giazotto2006]. By intentionally heating electrons in the source up to $T_{src}$ yields a quasiparticle temperature $T_1>T_2$ in S$_1$, therefore leading to a finite heat current $J_{S_1\rightarrow S_2}$. Yet, the latter can be modulated by the applied magnetic field. Measurement of the drain electron temperature ($T_{dr}$) would thus allow to assess heat diffraction.
Drain temperature can be predicted by solving a couple of thermal balance equations accounting for the main heat exchange mechanisms existing in the structure, according to the model shown in Fig. \[fig3\](b). In particular, S$_1$ exchanges heat with source electrons at power $J_{src\rightarrow S_1}$, with drain at power $J_{S_1\rightarrow drain}$, and with quasiparticles in S$_2$ at power $J_{S_1 \rightarrow S_2}$. Furthermore, electrons in the structure exchange heat with lattice phonons residing at bath temperature $T_{bath}$, in particular, at power $J_{e-ph, S_1}$ in S$_1$, and at power $J_{e-ph,src}$ and $J_{e-ph,dr}$ in source and drain electrodes, respectively. Finally, we assume S$_2$ to be large enough to provide substantial electron-phonon coupling $J_{e-ph, S_2}$ so that its quasiparticles will reside at $T_{bath}$. The electronic temperatures $T_1$ and $T_{dr}$ can therefore be determined under given conditions by solving the following system of thermal balance equations $$\begin{aligned}
-J_{src\rightarrow S_1}+J_{S_1 \rightarrow S_2}+J_{S_1\rightarrow drain}+J_{e-ph, S_1}&=&0\\
-J_{S_1\rightarrow drain}+J_{e-ph,dr}&=&0\nonumber\end{aligned}$$ for S$_1$ and drain, respectively. In the above expressions, $J_{S_1 \rightarrow S_2}(T_1,T_{bath},\Phi)=J_{qp}(T_1,T_{bath})-J_{H}^{rect}(T_1,T_{bath},\Phi)$, $J_{src\rightarrow S_1}(T_{src},T_1)=\frac{1}{e^2 R_t} \int^{\infty}_{0} d\varepsilon \varepsilon \mathcal{N}_1 (\varepsilon,T_1)[f(\varepsilon,T_1)-f(\varepsilon,T_{src})]$, $J_{S_1\rightarrow drain}(T_1,T_{dr})=\frac{1}{e^2 R_t} \int^{\infty}_{0} d\varepsilon \varepsilon \mathcal{N}_1 (\varepsilon,T_1)[f(\varepsilon,T_{dr})-f(\varepsilon,T_{1})]$, and $J_{e-ph,dr}=\Sigma_{dr}\mathcal{V}_{dr}(T^5_{dr}-T^5_{bath})$ [@Giazotto2006], $\Sigma_{dr}$ and $\mathcal{V}_{dr}$ being the electron-phonon coupling constant and the volume of drain, respectively. Furthermore [@Timofeev2009], $$\begin{aligned}
J_{e-ph,S_1}&=&-\frac{\Sigma_{S_1} \mathcal{V}_{S_1}}{96\zeta (5)k_B^5}\int^{\infty}_{-\infty}dEE\int^{\infty}_{-\infty}d\varepsilon \varepsilon^2\text{sign}(\varepsilon)M_{E,E+\varepsilon}\nonumber\\
&\times& [\text{coth}(\frac{\varepsilon}{2k_B T_{bath}})(f_E-f_{E+\varepsilon})-f_Ef_{E+\varepsilon}+1],
\label{eph}\end{aligned}$$ where $f_E(T_1)=\text{tanh}(E/2k_B T_1)$, $M_{E,E'}(T_1)=\mathcal{N}_1(E,T_1)\mathcal{N}_1(E',T_1)[1-\Delta_1^2(T_1)/EE']$, $\Sigma_{S_1}$ is the electron-phonon coupling constant, and $\mathcal{V}_{S_1}$ is the volume of S$_1$. As a set of parameters representative for a realistic microstructure we choose $R_t=2\,\text{k}\Omega$, $R_J=500\,\Omega$, $\mathcal{V}_{dr}=10^{-20}$ m$^{-3}$, $\Sigma_{dr}=3\times 10^9$ WK$^{-5}$m$^{-3}$ (typical of Cu) [@Giazotto2006], $\mathcal{V}_{S_1}=10^{-18}$ m$^{-3}$, $\Sigma_{S_1}=3\times 10^8$ WK$^{-5}$m$^{-3}$ and $\Delta_1(0)=\Delta_2(0)=200\,\mu$eV, the last two parameters typical of aluminum (Al) [@Giazotto2006]. Finally, our thermal model neglects both heat exchange with photons, owing to poor matching impedance [@schmidt; @Meschke2006], and pure phononic heat conduction [@Maki1965; @giazottoexp2012].
![(Color online) (a) Drain temperature $T_{dr}$ vs $\Phi$ calculated at $T_{bath}=250$ mK for several values of source temperature $T_{src}$ for a structure based on a *rectangular* Josephson junction. (b) Flux-to temperature transfer function $\mathcal{T}$ vs $\Phi$ calculated at 250 mK for a few selected values of $T_{src}$. (c) $T_{dr}$ vs $\Phi$ calculated for a few values of $T_{bath}$ at $T_{src}=1$ K. (d) $\mathcal{T}$ vs $\Phi$ at a few selected $T_{bath}$ calculated for $T_{src}=1$ K.[]{data-label="fig4"}](fig4.pdf){width="\columnwidth"}
The results of thermal balance equations for drain temperature are shown in Fig. \[fig4\] [@Gamma]. In particular, panel (a) displays $T_{dr}$ vs $\Phi$ for different values of $T_{src}$ at $T_{bath}=250$ mK. As expected, $T_{dr}$ shows a response to magnetic flux resembling a Fraunhofer-like diffraction pattern. The minima appearing at integer values of $\Phi_0$ are the inequivocal manifestation of the above-described phase-slips. Increasing $T_{src}$ leads to a monotonic enhancement of the maximum of $T_{dr}$ at $\Phi=0$ which stems from an increased heat current flowing into drain electrode. Furthermore, the amplitude of $T_{dr}$ lobes follows a non-monotonic beahavior, initially increasing with source temperature, being maximized at intermediate temperatures, and finally decreasing at higher $T_{src}$ values. With the above-given structure parameters one would obtain a maximum peak-to-valley amplitude exceeding $\sim 60$ mK at $T_{src}\sim 700$ mK. By defining a figure of merit in the form of flux-to-temperature transfer coefficient, $\mathcal{T}=\partial T_{dr}/\partial \Phi$, we get that $\mathcal{T}$ as large as $\sim 90$ mK$/\Phi_0$ could be achieved at $T_{src}=600$ mK in the present structure \[see Fig. \[fig4\](b)\]. Moreover, the transfer coefficient clearly demonstrates the non-monotonicity of the amplitude of drain temperature lobes as a function of $T_{src}$.
The impact of bath temperature on the structure response is shown in Fig. \[fig4\](c) where $T_{dr}$ is plotted against $\Phi$ for a few $T_{bath}$ values at fixed $T_{src}=1$ K. In particular, by increasing $T_{bath}$ leads to a smearing of drain temperature joined with a reduction of the lobes amplitude. This originates from both reduced temperature drop across the Josephson junction and enhanced electron-phonon relaxation in S$_1$ and drain at higher $T_{bath}$. We notice that already at 550 mK the temperature diffraction pattern is somewhat suppressed for a structure realized according to the chosen parameters. The drain temperature behavior as a function of $T_{bath}$ directly reflects on the transfer coefficient $\mathcal{T}(\Phi)$ \[see Fig. \[fig4\](d)\] which is calculated for a few selected values of $T_{bath}$.
We finally notice that the temperature diffraction patterns shown in Figs. \[fig4\] implicitly assume the presence of the $\pi$ slips and, therefore, the same heat diffraction measure can be considered as a proof of the existence of such phase slips.
Summary
=======
In summary, we have investigated thermal transport in temperature-biased extended Josephson tunnel junctions under the influence of an in-plane magnetic field. We have shown, in particular, that the heat current through the junction displays *coherent diffraction*, in full analogy with the Josephson critical current. In an electrically-open junction configuration, minimization of the Josephson coupling energy imposes the quantum phase difference across the junction to undergo $\pi$ slips in suitable magnetic flux intervals, the latter depending on the specific junction geometry. Finally, we have proposed and analyzed a hybrid superconducting microstructure, easily implementable with current technology, which would allow to demonstrate diffraction of thermal currents. We wish further to stress that the described temperature detection is uniquely suited to reveal the hidden physical properties of the quantum phase in electrically-open tunnel junctions of whatever geometry otherwise more difficult to access with electric-type transport measurement. The effects here predicted could serve to enhance the flexibility to master thermal currents in emerging coherent caloritronic nanocircuitry.
We would like to thanks C. Altimiras for useful discussions. F.G. and M.J.M.-P. acknowledges the FP7 program No. 228464 “MICROKELVIN”, the Italian Ministry of Defense through the PNRM project “TERASUPER”, and the Marie Curie Initial Training Action (ITN) Q-NET 264034 for partial financial support. P.S. acknowledges financial support from FIRB - Futuro in Ricerca 2012 under Grant No. RBFR1236VV HybridNanoDev.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We investigate the influence of the additional third level on the dynamic evolution of a Two-Level system interacting with a coherent field in the strong coupling regime where Rotating Wave Approximation is not valid. We find that the additional level has great influence on the evolution of the system population. Our results show that the Two-Level model is not a good approximation in this strong light-matter coupling regime. We further investigate the parameter space where the Two-Level model can still be justified.'
author:
- Faheel Ather Hashmi
- 'Shi-Yao Zhu'
title: 'The justification of the ‘Two-Level Approximation’ in strong light-matter interaction'
---
Introduction
============
‘Two-level approximation’ is a very convenient tool to study light-matter interaction because of its simplicity and because of its applicability to a large number of real light matter interaction scenarios. This approximation together with Rotating Wave Approximation (RWA) can be easily justified when two distinct levels of a system weakly interact with the light field at resonance or quasi-resonance. The other levels of the system are far apart in energy and can be ignored along with the ‘Counter Rotating Terms (CRT)’. However, RWA becomes invalid in strong light-matter coupling regime. This strong coupling regime is becoming more accessible for experiments [@niemczyk10; @forn10; @you11; @you13], and hence there is a surge of interest in going beyond RWA (taking into account the effects of CRT) in the description of the interaction [@casanova10; @albert12]. The breakdown of RWA and the role of CRT has also been reported [@larson12; @wang09]. In the strong coupling regime, the two-level system without RWA has been very extensively studied [@irish07; @larson07; @werlang08; @Braak11; @zubairy88; @Milonni83; @Spreeuw90]. In these studies the counter-rotating terms are taken into account, but the two-level approximation is still adopted. The counter-rotating terms involve the contribution proportional to $1/(\omega_{ab}+\omega_0)$ where $\omega_{ab}$ is the transition frequency of the system and $\omega_0$ is the frequency of the field. If there is an additional level of the system (say ${\left|c\right\rangle}$ with transition frequency $\omega_{ac}$), the rotating wave terms on this transition will result in the contribution proportional to $1/(\omega_{ac}-\omega_0)$, which can be larger than the contribution proportional to $1/(\omega_{ab}+\omega_0)$. Consequently, neglecting the additional levels is questionable. Therefore, in the strong coupling regime where CRT effects have to be taken into account, the question ‘whether the effects due to additional levels can be neglected?’ needs to be treated carefully. In this paper, we consider the effects of the third level on the population dynamics of a two-level system in strong coupling regime, and focus on when the third level can be neglected.
The three level system interacting with resonant or quasi-resonant fields beyond RWA has also been studied, mainly in the context of trapping dark state in $\Lambda$ configuration [@unanyan00; @ho85; @matisov95; @xiaohong12; @sanchez04]. However, these studies concerned with the effects of CRT terms on the three level system, and did not discuss the consequence of the third level on the evolution. In the present work we consider the three level ‘V-system’, where the third level is non-resonant and is far away in energy from the two-level transition, and study the effects of the third level on the population dynamics. Our focus is on when the effects of the third level will diminish in the strong coupling regime.
Model
=====
![A three level system in ‘V’ configuration. We are interested in the effects of the additional level ${\left|c\right\rangle}$ on the population dynamics of level ${\left|b\right\rangle}$.[]{data-label="fig:1"}](system)
Consider a three level system in ‘V’ configuration as shown in FIG. \[fig:1\] interacting with a single mode quantized field. The Hamiltonian of the system can be written as (with $\hbar=1$) $$\label{eqHamil}
\begin{aligned}
{\hat{H}}=& {\omega_{ab}}{{\left|b\right\rangle}{\left\langleb\right|}} + {\omega_{ac}}{{\left|c\right\rangle}{\left\langlec\right|}} + {\omega_{0}}{\hat{b}^{\dagger}}{\hat{b}}\\ & +{g_{ab}}{\left( {\hat{b}^{\dagger}}+ {\hat{b}}\right)} {\left[{{\left|a\right\rangle}{\left\langleb\right|}} + {{\left|b\right\rangle}{\left\langlea\right|}} \right]}
\\ & +{g_{ac}}{\left( {\hat{b}^{\dagger}}+ {\hat{b}}\right)} {\left[{{\left|a\right\rangle}{\left\langlec\right|}} + {{\left|c\right\rangle}{\left\langlea\right|}} \right]},
\end{aligned}$$ where ${\omega_{ab}}$ and ${\omega_{ac}}$ are the transition frequencies for the excited states and ${\hat{b}^{\dagger}}$, ${\hat{b}}$, and ${\omega_{0}}$ are the creation and annihilation operators and the frequency for the field. The coupling constants ${g_{ab}}$ and ${g_{ac}}$ are real. We choose ${\left|n,l\right\rangle}$ as the basis to write the system ket where $n$ is the number of photon and $l=\{a,b,c\}$ denotes the level of the system. The system ket can be written as $$\begin{aligned}
{\left|\Psi\right\rangle}= \sum_n^{\infty}\sum_{p=0}^{1} e^{-i{\hat{H_0}}t} a_{2n+p} {\left|2n+p,a\right\rangle} + b_{2n+p}{\left|2n+p,b\right\rangle} + c_{2n+p}{\left|2n+p,c\right\rangle}
\label{eqKet}\end{aligned}$$ In this expression ${\hat{H_0}}={\omega_{ab}}{{\left|b\right\rangle}{\left\langleb\right|}} + {\omega_{ac}}{{\left|c\right\rangle}{\left\langlec\right|}} + {\omega_{0}}{\hat{b}^{\dagger}}{\hat{b}}$ is the non-interacting part of the Hamiltonian in eq(\[eqHamil\]). The system ket eq(\[eqKet\]) actually consists of two independent kets that separately satisfy the Schrödinger equation. This is because of the fact that the Hamiltonian admits even and odd parity chains of evolution like the JCM Hamiltonian with CRT [@casanova10]. One such chain consists of the states with even number of photons in ${\left|a\right\rangle}$ and odd number of photons in ${\left|b\right\rangle}$ and ${\left|c\right\rangle}$, and the other chain has states with odd number of photons in ${\left|a\right\rangle}$ and even number of photons in ${\left|b\right\rangle}$ and ${\left|c\right\rangle}$. The time evolution of the amplitudes is given by $$\label{eqMaster}
\begin{aligned}
i \dot{a}_{2n+p} &= {g_{ab}}\left[\sqrt{2n+p} \, b_{2n+p-1} {e^{-i \Delta_{ab}t}}+ \sqrt{2n+p+1} \,b_{2n+p+1}{e^{-i \Delta_{ab}^{\prime}t}}\right]
\\ &+ {g_{ac}}\left[\sqrt{2n+p} \, c_{2n+p-1} {e^{-i \Delta_{ac}t}}+ \sqrt{2n+p+1} \,c_{2n+p+1}{e^{-i \Delta_{ac}^{\prime}t}}\right]
\\
i \dot{b}_{2n+p} &= {g_{ab}}\left[\sqrt{2n+p+1} \,a_{2n+p+1} {e^{i \Delta_{ab}t}}+ \sqrt{2n+p} \,a_{2n+p-1}{e^{i \Delta_{ab}^{\prime}t}}\right]
\\
i \dot{c}_{2n+p} &= {g_{ac}}\left[\sqrt{2n+p+1} \,a_{2n+p+1} {e^{i \Delta_{ac}t}}+ \sqrt{2n+p} \,a_{2n+p-1}{e^{i \Delta_{ac}^{\prime}t}}\right]
\end{aligned}$$ Here $\Delta_{ab}= \omega_{ab}-\omega_0$, $\Delta_{ab}^{\prime}=\omega_{ab}+\omega_0$, $\Delta_{ac}= \omega_{ac}-\omega_0$, and $\Delta_{ac}^{\prime}=\omega_{ac}+\omega_0$. $p$ in each equation is either $0$ for state ${\left|a\right\rangle}$ and $1$ for the states $\{{\left|b\right\rangle}, {\left|c\right\rangle} \}$, or $1$ for the state ${\left|a\right\rangle}$ and $0$ for excited states. In the following we solve these coupled differential equations numerically for a system that is initially in state ${\left|b\right\rangle}$, and for the field that is in a coherent state with average photon number $\bar n=10$. The population in state ${\left|b\right\rangle}$ $=\sum_n {\left|b_n\right|}^2$ is calculated in the presence (${g_{ac}}\ne 0$) and the absence (${g_{ac}}=0$) of the additional level ${\left|c\right\rangle}$. The summation in eq (\[eqKet\]) is truncated at $n=200$.
Results & Discussion
====================
![(color online) The population dynamics of state ${\left|b\right\rangle}$ in the absence (dashed-green) and presence (solid-red) of the additional level ${\left|c\right\rangle}$. The difference of the two (dotted-blue) is also shown. The parameters are ${g_{ab}}=0.02 \omega_{ab}$, $\omega_0=\omega_{ab}$, and $\bar n=10$.[]{data-label="fig:2"}](g02g002o13 "fig:"){width="40.00000%"} ![(color online) The population dynamics of state ${\left|b\right\rangle}$ in the absence (dashed-green) and presence (solid-red) of the additional level ${\left|c\right\rangle}$. The difference of the two (dotted-blue) is also shown. The parameters are ${g_{ab}}=0.02 \omega_{ab}$, $\omega_0=\omega_{ab}$, and $\bar n=10$.[]{data-label="fig:2"}](g02g002o17 "fig:"){width="40.00000%"}\
![(color online) The population dynamics of state ${\left|b\right\rangle}$ in the absence (dashed-green) and presence (solid-red) of the additional level ${\left|c\right\rangle}$. The difference of the two (dotted-blue) is also shown. The parameters are ${g_{ab}}=0.02 \omega_{ab}$, $\omega_0=\omega_{ab}$, and $\bar n=10$.[]{data-label="fig:2"}](g02g01o13 "fig:"){width="40.00000%"} ![(color online) The population dynamics of state ${\left|b\right\rangle}$ in the absence (dashed-green) and presence (solid-red) of the additional level ${\left|c\right\rangle}$. The difference of the two (dotted-blue) is also shown. The parameters are ${g_{ab}}=0.02 \omega_{ab}$, $\omega_0=\omega_{ab}$, and $\bar n=10$.[]{data-label="fig:2"}](g02g01o17 "fig:"){width="40.00000%"}\
![(color online) The population dynamics of state ${\left|b\right\rangle}$ in the absence (dashed-green) and presence (solid-red) of the additional level ${\left|c\right\rangle}$. The difference of the two (dotted-blue) is also shown. The parameters are ${g_{ab}}=0.02 \omega_{ab}$, $\omega_0=\omega_{ab}$, and $\bar n=10$.[]{data-label="fig:2"}](g02g02o13 "fig:"){width="40.00000%"} ![(color online) The population dynamics of state ${\left|b\right\rangle}$ in the absence (dashed-green) and presence (solid-red) of the additional level ${\left|c\right\rangle}$. The difference of the two (dotted-blue) is also shown. The parameters are ${g_{ab}}=0.02 \omega_{ab}$, $\omega_0=\omega_{ab}$, and $\bar n=10$.[]{data-label="fig:2"}](g02g02o17 "fig:"){width="40.00000%"}
In FIG. \[fig:2\] we show the population dynamics in the state ${\left|b\right\rangle}$ in the absence (dashed-green), and presence (solid-red) of the additional level ${\left|c\right\rangle}$ for a moderately strong coupling ${g_{ab}}=0.02\omega_{ab}$. The difference of the two signifies the importance of additional level effects, and is shown in dotted (blue) curve. The dashed (green) curves exhibit traditional collapse and revival of the dynamics in the two-level model [@eberly80] superimposed by strong oscillations due to counter-rotating terms [@zubairy88]. This behavior is modified by the additional level ${\left|c\right\rangle}$. With the additional level present, and for short times, the dynamics follow the two-level case but at longer times strong differences appear in the dynamics. These differences can give us the criteria for the justification of ‘Two-Level Approximation’ in the strong coupling regime. As we can see in the figure that for weak additional level coupling ${g_{ac}}/{g_{ab}}\rightarrow 0$, the differences are small, and hence the approximation can be justified. However, for ${g_{ab}}/{g_{ac}}=0.5$, a level placed at ${\omega_{ac}}/{\omega_{ab}}=1.3$ strongly modifies the dynamics, and raises concerns on the two-level model. For still higher coupling ${g_{ac}}={g_{ab}}$, even a level placed at ${\omega_{ac}}/{\omega_{ab}}=1.7$ can not be ignored.
![(color online) Additional level effects when the two-level transition ${\left|a\right\rangle} \leftrightarrow {\left|b\right\rangle}$ is detuned from resonance. Parameters are ${g_{ab}}=0.02 {\omega_{ab}}$, ${\omega_{0}}= 0.95{\omega_{ab}}$, and $\bar n=10$[]{data-label="fig:3"}](g02g01o17D05){width="50.00000%"}
The additional level effects become more prominent when the two-level system is detuned from the resonance as shown in FIG. \[fig:3\]. Here we make the field non-resonant on ${\left|a\right\rangle}\leftrightarrow {\left|b\right\rangle}$ transition by taking ${\omega_{0}}=0.95{\omega_{ab}}$. The resulting difference in dynamics are more pronounced than the corresponding case in FIG. \[fig:2\] (with parameters ${g_{ac}}/{g_{ab}}=0.5$ and ${\omega_{ac}}/{\omega_{ab}}=1.7$).
![(color online) The average of the absolute population difference in state ${\left|b\right\rangle}$ in the presence and absence of the additional level. The parameters are ${g_{ab}}=0.02 \omega_{ab}$, and $\bar n=10$.[]{data-label="fig:5"}](g02tab){width=".8\columnwidth"}
Finally, in FIG. \[fig:5\] we plot an estimate of the error in population due to the additional level. This error is calculated by taking the average of the absolute difference in population dynamics in the presence and absence of the additional level, and is plotted as the function of the placement of the additional level ${\omega_{ac}}/{\omega_{ab}}$ for various coupling strengths ${g_{ac}}/{g_{ab}}$ with ${g_{ab}}=0.02{\omega_{ab}}$. Here, we again see that the error due to the additional level can be significant even if it is placed as far as ${\omega_{ac}}/{\omega_{ab}}=1.3$ for non-vanishing coupling ${g_{ac}}$. However, the error decreases as the additional level goes further away in energy, and as the coupling of the ground state to this additional level ${g_{ac}}$ decreases in comparison with ${g_{ab}}$.
Conclusion
==========
We have shown that the additional level can have important effects on the population dynamics of the two-level model in the strong coupling regime. However, the ‘Two-Level approximation’ can still be valid when the additional level is far away (few units) from resonance, and the coupling with the additional level is weaker compared to the coupling with the two-level model. The effects of the additional level may become very crucial in experiments in the strong coupling regime where precise control of the quantum dynamics is required. Our results suggest that in such strong light-matter interaction regime, the ‘Two-Level approximation’ should be used with care.
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| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'Jan-e Alam'
title: In search of quark gluon plasma in nuclear collisions
---
[**Abstract**]{}\
At high temperatures and densities the nuclear matter undergoes a phase transition to a new state of matter called quark gluon plasma (QGP). This new state of matter which existed in the universe after a few microsecond of the big bang can be created in the laboratory by colliding two nuclei at relativistic energies. In this presentation we will discuss how the the properties of QGP can be extracted by analyzing the spectra of photons, dileptons and heavy flavours produced in nuclear collisions at Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) energies.
Introduction
============
The theory of strong interaction - Quantum Chromodynamics (QCD) has a unique feature - it possess the property of asymptotic freedom which implies that at very high temperatures and/or densities nuclear matter will convert to a deconfined state of quarks and gluons [@collins]. Recent lattice QCD based calculations [@lqcd] indicate that the value of the temperature for the nuclear matter to QGP transition $\sim 175$ MeV. It is expected that such high temperature can be achieved in the laboratory by colliding nuclei at RHIC and LHC energies.
A high multiplicity system of deconfined quarks and gluons with power law type of momentum distributions can created just after the nuclear collisions at high energies. Interactions among these constituents may alter the momentum distribution of quarks and gluons from a power law to an exponential one - resulting in a thermalized state of quarks and gluons with initial temperature, $T_i$. This thermalized system with high internal pressure expands very fast as a consequence it cools and reverts to hadronic matter at a temperature, $T_c\sim 175$ MeV. The hadrons formed after the hadronization of quarks may maintain thermal equilibrium among themselves until the expanding system becomes too dilute to support collectivity at a temperature, $T_F (\sim 120$ MeV) called freeze-out temperature from where the hadrons fly freely from the interaction zone to the detector.
The electromagnetic (EM) probes [@mclerran] (see [@rapp; @alam1; @alam2] for review) [*i.e.*]{} real photons and dileptons can be used to follow the evolution of the system from the pristine partonic stage to the final hadronic stage through an intermediary phase transition or cross over. In the state of QGP some of the symmetries of the physical vacuum may either be restored or broken - albeit transiently. The electromagnetic probes, especially the lepton pairs can be used very effectively to investigate whether these symmetries in the system are restored/broken at any stage of the evolving matter. Results from theoretical calculations will be shown in the presentation to demonstrate this aspect of the electromagnetically interacting probes. We will demonstrate that lepton pairs can be used very effectively to probe the collective motion (radial and elliptic) of the system.
The other promising probe of the QGP that will be discussed here - is the depletion of the transverse momentum spectra of energetic quarks (and gluons) in QGP. The magnitude of the depletion can be used to estimate the transport coefficients of QGP which is turn can be used to understand the fluidity of the matter.
The transport coefficients of QGP and hot hadrons calculated by using perturbative QCD and effective field theory respectively have been applied to evaluate the nuclear suppression ($R_{\mathrm AA}$) of heavy flavours. Theoretical results on $R_{\mathrm AA}$ will be compared with the experimental data available from RHIC and LHC energies. The azimuthal asymmetry of the system estimated through the single leptons originating from the decays of open heavy flavours produced from the fragmentation of heavy quarks will also be discussed.
The electromagnetic probes
==========================
The dilepton production per unit four-volume from a thermal medium produced in heavy ion collisions is well known to be given by: &=&- L(M\^2)f\_[BE]{}(p\_0) g\^\
&&W\_(p\_0,[p]{}) \[eq1\] where the factor $L(M^2)=(1+{2m_l^2}/{M^2})~
(1-4m_l^2/M^2)^{1/2}~$ is of the order of unity for electrons, $M(=\sqrt{p^2})$ being the invariant mass of the pair and the hadronic tensor $W_{\mu\nu}$ is defined by W\_(p\_0,[p]{})=d\^4xe\^[ipx]{}\[eq2\] where $J^{em}_\mu(x)$ is the electromagnetic current and $\langle.\rangle$ indicates ensemble average. For a deconfined thermal medium such as the QGP, Eq. (\[eq1\]) leads to the standard rate for lepton pair productions from $q\bar q$ annihilation at lowest order.
The production of low mass dileptons from the decays of light vector mesons in the hadronic matter can be obtained as (see [@sabya] for details): &=&- f\_[BE]{}(p\_0) g\^\
&&\_[R=,,]{}K\_R \^R\_(p\_0,[p]{}) \[eq2\] where $f_{BE}$ is the thermal distributions for bosons, $\rho^R_{{\mu\nu}}(q_0,{\vec q})$ is the spectral function of the vector meson $R (=\rho,\omega,\phi)$ in the medium, $K_R=F_R^2 m_R^2$, $m_R$ is the mass of $R$ and $F_R$ is related to the decay of $R$ to lepton pairs.
The interaction of the vector mesons with the hadrons in the thermal bath will shift the location of both the pole and the branch cuts of the spectral function - resulting in mass modification or broadening - which can be detected through the dilepton measurements and may be connected with the restoration of chiral symmetry in the thermal bath. In the present work the interaction of $\rho$ with thermal $\pi$,$\omega$, $a_1$, $h_1$ [@sabya; @sabya2] and nucleons [@ellis] have been considered to evaluate the in-medium spectral function of $\rho$. The finite temperature width of the $\omega$ spectral function has been taken from [@Weise].
To evaluate the dilepton yield from a dynamically evolving system produced in heavy ion collisions (HIC) one needs to integrate the fixed temperature production rate given by Eq. \[eq2\] over the space time evolution of the system - from the initial QGP phase to the final hadronic freeze-out state through a phase transition in the intermediate stage. We assume that the matter is formed in QGP phase with zero net baryon density at temperature $T_i$ in HIC. Ideal relativistic hydrodynamics with boost invariance [@bjorken] has been applied to study the evolution of the system. The EoS required to close the hydrodynamic equations is constructed by taking results from lattice QCD for high $T$ [@lqcd] and hadron resonance gas comprising of all the hadronic resonances up to mass of $2.5$ GeV [@victor; @bmja] for lower $T$. The system is assumed to get out of chemical equilibrium at $T=T_{ch}=170$ MeV [@tsuda]. The kinetic freeze-out temperature $T_{F}=120$ MeV fixed from the $p_T$ spectra of the produced hadrons.
Invariant mass spectra of lepton pairs
--------------------------------------
The $M$ distribution of lepton pairs originating from quark matter (QM) and hadronic matter (HM) with and without medium effects on the spectral functions of $\rho$ and $\omega$ are displayed in Fig. \[fig1\]. We observe that for $M\,>\,M_{\phi}$ the QM contributions dominate. For $M_{\rho}\lesssim M\lesssim M_{\phi}$ the HM shines brighter than QM. For $M\,<M_{\rho}$, the HM (solid line) over shines the QM due to the enhanced contributions primarily from the medium induced broadening of $\rho$ spectral function. However, the contributions from QM and HM become comparable in this $M$ region if the medium effects on $\rho$ spectral function is ignored (dotted line). Therefore, the results depicted in Fig. \[fig1\] indicate that a suitable choice of $M$ window will enable us to unravel the contributions from a particular phase (QM or HM). An appropriate choice of $M$ window will also allow us to extract the medium induced effects.
To further quantify these points we evaluate the following [@payalv2]: $$\begin{aligned}
&&F=
\frac{\int^\prime
\left(\frac{dN}{d^4xd^2p_TdM^2dy}\right)dxdyd\eta\tau d\tau d^2p_TdM^2}
{\int\left(\frac{dN}{d^4xd^2p_TdM^2dy}\right)dxdyd\eta\tau d\tau d^2p_TdM^2}\nn
\label{eq3}\end{aligned}$$ where the $M$ integration in both the numerator and denominator are performed for selective windows from $M_1$ to $M_2$ with mean $M$ defined as $\langle M\rangle = (M_1+M_2)/2$. While in the denominator the integration is done over the entire lifetime, the prime in $\int^\prime$ in the numerator indicates that the $\tau$ integration in the numerator is done from $\tau_1=\tau_i$ to $\tau_2=\tau_i+\Delta\tau$ with incremental $\Delta\tau$ until $\tau_2$ attains the life time of the system. In the inset of Fig. \[fig1\] $F$ is plotted against $\tau_{\mathrm av} (=(\tau_1+\tau_2)/2)$. The results substantiate that pairs with high $\langle M\rangle\sim 2.5$ GeV originate from early time ($\tau_{\mathrm av}\lesssim 3$ fm/c, QGP phase) and pairs with $\langle M\rangle\sim 0.77$ GeV mostly emanate from late hadronic phase ($\tau_{\mathrm av}\geq 4$ fm/c). The change in the properties of $\rho$ due to its interaction with thermal hadrons in the bath is also visible through $F$ evaluated for $\langle M\rangle\sim 0.3$ GeV with and without medium effects.
![Invariant mass distribution of dileptons from hadronic matter (HM) for modified and unmodified $\rho$ meson. []{data-label="fig1"}](fig1.eps)
![\[a\] and \[b\] indicate elliptic flow of lepton pairs as a function of $p_T$ for various $M$ windows. \[c\] displays the effect of the broadening of $\rho$ spectral function on the elliptic flow for $\langle M\rangle = 300$ MeV. \[d\] shows the variation of $R$ (see text) with $p_T$ for $\langle M\rangle=0.3$ GeV, 0.77 GeV and 2.5 GeV. []{data-label="fig2"}](fig2.eps)
![Variation of dilepton elliptic flow as function of $\langle M\rangle$ for QM, HM (with and without medium effects) and for the entire evolution. The inset shows the variation of momentum space anisotropy with proper time.[]{data-label="fig3"}](fig3.eps)
![The dilepton yield plotted against $M_T-M_{av}$ for different $M$ windows for LHC initial condition.[]{data-label="fig4"}](fig4.eps)
Elliptic flow of lepton pairs
-----------------------------
The elliptic flow of dilepton, $v_2(p_T,M)$ can be defined as: $$\begin{aligned}
&&v_2=
\frac{\sum\int cos(2\phi)
\left(\frac{dN}{d^2p_TdM^2dy}\arrowvert_{y=0}\right) d\phi}
{\sum\int\left(\frac{dN}{d^2p_TdM^2dy}\arrowvert_{y=0}\right)d\phi }
\label{eqv2}\end{aligned}$$ where the $\sum$ stands for summation over QM and HM phases.
Fig. \[fig2\] (\[a\] and \[b\]) show the differential elliptic flow, $v_2(p_T)$ of dileptons arising from various $\langle M\rangle$ domains. We observe that for $\langle M\rangle=2.5$ GeV $v_2$ is small for the entire $p_T$ range because these pairs arise from the early epoch (see inset of Fig. \[fig1\]) when the flow is not developed entirely. However, the $v_2$ is large for $\langle M\rangle=0.77$ GeV as these pairs originate predominantly from the late hadronic phase when the flow is fully developed. It is also interesting to note that the medium induced enhancement of $\rho$ spectral function provides a visible modification in $v_2$ for dileptons below $\rho$ peak (Fig. \[fig2\] \[c\]). The medium-induced effects lead to an enhancement of $v_2$ of lepton pairs which is culminating from the ‘extra’ interaction (absent when a vacuum $\rho$ is considered) of the $\rho$ with other thermal hadrons in the bath. In Fig. \[fig2\] \[d\] we depict the variation of $R$ with $p_T$ for $\langle M\rangle=0.3$ GeV (solid circle) 0.77 GeV (solid line) and 2.5 GeV (open circle), the quantity $R$ is defined as $R=v_2^{\mathrm QM}/(v_2^{\mathrm QM}+v_2^{\mathrm HM})$ where $v_2^{\mathrm i}$ is the elliptic flow of the phase $i(=QM+HM$. The results clearly illustrate that $v_2$ of lepton pairs in the large $\langle M\rangle$ domain originate from QM.
Fig. \[fig3\] shows $p_T$ integrated elliptic flow, $v_2(\langle M\rangle)$ evaluated for different $\langle M\rangle$ windows defined above. The $v_2$ (which is proportional to momentum space anisotropy, $\epsilon_p$) of QM is small because the pressure gradient is not fully developed in the QGP phase as evident from the inset plot of $\epsilon_p$ with $\tau$. The hadronic phase $v_2$ has a peak around $\rho$ pole indicating large flow at late times. For $\langle M\rangle\>>\, m_\phi$ the $v_2$ obtained from the combined phases approach the value corresponding to the $v_2$ for QGP. Therefore, measurement of $v_2$ for large $\langle M\rangle$ will bring information of the properties of the QGP. It is important to note that the $p_T$ integrated $v_2(\langle M\rangle)$ of lepton pairs with $\langle M\rangle\,\sim m_\pi, m_K$ is close to the hadronic $v_2^\pi$ and $v_2^K$ if the thermal effects on $\rho$ properties are included. Exclusion of medium effects give lower $v_2$ for lepton pairs compared to hadrons. We also observe that the variation of $v_2(\langle M\rangle)$ with $\langle M\rangle$ has a structure similar to $dN/dM$ vs $M$. As indicated by Eq. \[eq1\] we can write $v_2(\langle M\rangle)\sim \sum v_2^{\mathrm i}\times f_{\mathrm i}$. The structure of $dN/dM$ is reflected in $v_2(\langle M\rangle)$ through $f_i$.
Radial flow of dileptons
------------------------
The transverse mass distributions of the lepton pairs at LHC is displayed in Fig. \[fig4\]. The variation of inverse slope (deduced from the from the transverse mass distributions of lepton pairs, Fig. \[fig4\]) with $\langle M\rangle$ for LHC is depicted in Fig. \[fig5\]. The radial flow in the system is responsible for the rise and fall of $T_{\mathrm eff}$ with $\langle M\rangle$ (solid line) in the mass region ($0.5<$ M(GeV)$<1.3$), for $v_T=0$ (dashed line) a completely different behaviour is obtained.
![$T_{eff}$ for different values of the $M$-bins for LHC conditions. The dashed line is obtained by setting $v_T=0$.[]{data-label="fig5"}](fig5.eps)
![$R_{side}$ and $R_{out}$ as a function of $\langle M \rangle$. The dashed, dotted and the solid line (with asterisk) indicate the HBT radii for the QGP, hadronic and total dilepton contributions from all the phases respectively. The solid circles are obtained by switching off the contributions from $\rho$ and $\omega$. []{data-label="fig6"}](fig6.eps)
Radial flow from HBT interferometry of lepton pairs
---------------------------------------------------
It was shown in Ref. [@payal] that the variation of HBT radii ($R_{side}$ and $R_{out}$) extracted from the correlation of dilepton pairs with $\langle M\rangle$ can used to extract collective properties of the evolving QGP. While the radius ($R_{\mathrm side}$) corresponding to $q_{side}$ is closely related to the transverse size of the system and considerably affected by the collectivity, the radius ($R_{\mathrm out}$) corresponding to $q_{out}$ measures both the transverse size and duration of particle emission. The extracted $R_{\mathrm side}$ and $R_{\mathrm out}$ for different $\langle M\rangle$ are shown in Fig. \[fig6\]. The $R_{\mathrm side}$ shows non-monotonic dependence on $M$, starting from a value close to QGP value (indicated by the dashed line) it drops with increase in $M$ finally again approaching the QGP value for $\langle M\rangle \,>\,m_\phi$. It can be shown that $R_{side}\sim 1/(1+E_{\mathrm collective}/E_{\mathrm thermal})$. In the absence of radial flow, $R_{\mathrm side}$ is independent of $q_{\mathrm side}$. With the radial expansion of the system a rarefaction wave moves toward the center of the cylindrical geometry as a consequence the radial size of the emission zone decreases with time. Therefore, the size of the emission zone is larger at early times and smaller at late time. The high $\langle M\rangle$ regions are dominated by the early partonic phase where the collective flow has not been developed fully [*i.e.*]{} the ratio of collective to thermal energy is small hence show larger $R_{\mathrm side}$ for the source. In contrast, the lepton pairs with $M\sim m_\rho$ are emitted from the late hadronic phase where the size of the emission zone is smaller due to larger collective flow giving rise to a smaller $R_{\mathrm side}$. The ratio of collective to thermal energy for such cases is quite large, which is reflected as a dip in the variation of $R_{\mathrm side}$ with $\langle M\rangle$ around the $\rho$-mass region (Fig. \[fig6\] upper panel). Thus the variation of $R_{\mathrm side}$ with $M$ can be used as an efficient tool to measure the collectivity in various phases of matter. The dip in $R_{\mathrm side}$ at $\langle M\rangle\sim
m_\rho$ is due to the contribution dominantly from the hadronic phase. The dip, in fact vanishes if the contributions from $\rho$ and $\omega$ is switched off (circle in Fig. \[fig6\]). We observe that by keeping the $\rho$ and $\omega$ contributions and setting radial velocity, $v_r=0$, the dip in $R_{\mathrm side}$ vanishes, confirming the fact that the dip is caused by the radial flow of the hadronic matter. Therefore, the value of $R_{\mathrm side}$ at $\langle M\rangle\sim m_\rho$ may be used to estimate the average $v_r$ in the hadronic phase.
The $R_{\mathrm out}$ probes both the transverse dimension and the duration of emission as a consequence unlike $R_{\mathrm side}$ it does not remain constant even in the absence of radial flow and its variation with $M$ is complicated. The large $M$ regions are populated by lepton pairs from early partonic phase where the effect of flow is small and the duration of emission is also small - resulting in smaller values of $R_{\mathrm out}$. For lepton pair from $M\sim m_\rho$ the flow is large which could have resulted in a dip as in $R_{\mathrm side}$ in this $M$ region. However, $R_{\mathrm out}$ probes the duration of emission too which is large for hadronic phase. The larger duration compensates the reduction of $R_{\mathrm out}$ due to flow in the hadronic phase resulting in a bump in $R_{\mathrm out}$ for $M\sim m_\rho$ (Fig. \[fig6\] lower panel). Both $R_{\mathrm side}$ and $R_{\mathrm out}$ approach QGP values for $\langle M\rangle\sim 2.5$ GeV implying dominant contributions from partonic phase.
![$R_{AA}$ as a function of $p_T$ for $D$ and $B$ mesons at LHC. Experimental data taken from [@ALICE].[]{data-label="fig7"}](fig7.eps)
Suppression of heavy flavours in QGP
====================================
The depletion of hadrons with high transverse momentum ($p_T$) produced in Nucleus + Nucleus collisions with respect to those produced in proton + proton (pp) collisions has been considered as a signature of QGP formation. The two main processes which cause the depletion are (i) the elastic collisions and (ii) the radiative loss or the inelastic collisions of the high energy partons with the quarks, anti-quarks and gluons in the thermal bath.
In the present work we focus on the energy loss of heavy quarks in QGP in deducing the properties of the medium. Because (i) the abundance of charm and bottom quarks in the partonic plasma for the expected range of temperature to be attained in the experiments is small, consequently the bulk properties of the plasma is not decided by them and (ii) they produce early and therefore, can witness the entire evolution history. Hence heavy quarks may act as an efficient probe for the diagnosis of QGP. The depletion of heavy quarks in QGP has gained importance recently in view of the measured nuclear suppression in the $p_T$ spectra of non-photonic single electrons [@stare; @phenixe].
We assume here that the light quarks and gluons thermalize before heavy quarks. The charm and bottom quarks execute Brownian motion [@we1] (see references therein) in the heat bath of QGP. Therefore, the interaction of the heavy quarks with QGP may be treated as the interactions between equilibrium and non-equilibrium degrees of freedom. The Fokker-Planck (FP) equation provide an appropriate framework for the evolution of the heavy quark in the expanding QGP heat bath which can be written as [@we1]: &=&C\_[1]{}(p\_[x]{},p\_[y]{},t) +C\_[2]{}(p\_[x]{},p\_[y]{},t)\
&+& C\_[3]{}(p\_[x]{},p\_[y]{},t) +C\_[4]{}(p\_[x]{},p\_[y]{},t)\
&+& C\_[5]{}(p\_[x]{},p\_[y]{},t)f +C\_[6]{}(p\_[x]{},p\_[y]{},t). \[fpeqcartesian\] . where, C\_[1]{}&=& D\
C\_[2]{}&=& D\
C\_[3]{}&=& p\_[x]{} +2 \
C\_[4]{}&=& p\_[y]{} +2 \
C\_[5]{}&=& 2 + + \
C\_[6]{}&=& 0 . where the momentum, $\textbf{p}=(\textbf{p}_T,p_z)=(p_x,p_y,p_z)$, $\gamma$ is the drag coefficient and $D$ is the diffusion coefficient. We numerically solve Eq. \[fpeqcartesian\] [@antia] with the boundary conditions: $f(p_x,p_y,t)\ra 0$ for $p_x$,$p_y\ra \infty$ and the initial (at time $t=\tau_i$) momentum distribution of charm and bottom quarks are taken MNR code [@MNR].
The system under study has two components. The equilibrium component, the QGP comprising of the light quarks and the gluons. The non-equilibrium component, the heavy quarks produced due to the collision of partons of the colliding nuclei has momentum distribution determined by the perturbative QCD (pQCD), which evolves due to their interaction with the expanding QGP background. The evolution of the heavy quark momentum distribution is governed by the FP equation. The interaction of the heavy quarks with the QGP is contained in the drag and diffusion coefficients. The drag and diffusion coefficients are provided as inputs, which are, in general, dependent on both temperature and momentum. The evolution of the temperature of the background QGP with time is governed by relativistic hydrodynamics. The solution of the FP equation at the (phase) transition point for the charm and bottom quarks gives the (quenched) momentum distribution of hadrons ($B$ and $D$ mesons) through fragmentation. The fragmentation of the initial momentum distribution of the heavy quarks results in the unquenched momentum distribution of the $B$ and $D$ mesons. The ratio of the quenched to the unquenched $p_T$ distribution is the nuclear suppression factor which is experimentally measured. The quenching is due to the dragging of the heavy quark by QGP. Hence the properties of the QGP can be extracted from the suppression factor.
Nuclear suppression factor
--------------------------
The variation of the nuclear suppression factor, $R_{AA}$ [@we1] with $p_T$ of the electron originating from the decays of $D$ and $B$ mesons have been displayed in Fig. \[fig7\] for RHIC initial condition ($T_i=300$ MeV). A less suppression of $B$ is observed compared to $D$. The theoretical results show a slight upward trend for $p_T$ above 10 GeV both for mesons containing charm and bottom quarks. Similar trend has recently been experimentally observed for light mesons at LHC energy [@CMS]. This may originate from the fact that the drag (and hence the quenching) for charm and bottom quarks are less at higher momentum.
The same formalism is extended to evaluate the nuclear suppression factor, $R_{AA}$ both for charm and bottom at LHC energy. Result has been compared with the recent ALICE data(Ref. [@ALICE]) in Fig. \[fig8\]. The data is reproduced well by assuming formation of QGP at an initial temperature $\sim 550$ MeV after Pb+Pb collisions at $\sqrt{s_{\mathrm NN}}=2.76$ TeV.
Elliptic flow of heavy flavours
-------------------------------
In Fig. \[fig9\] we compare the experimental data obtained by the PHENIX [@phenixemb] collaborations for Au + Au minimum bias collisions at $\sqrt{s_{\mathrm NN}}=200$ GeV with theoretical results obtain in the present work. We observe that the data can be reproduced by including both radiative and collisional loss with $c_s=1/\sqrt{4}$. In this case $v_2^{HF}$ first increases and reaches a maximum of about 7% then saturates for $p_T>2$ GeV. However, with ideal equation of state ($c_s^2=1/3$) we fail to reproduce data. This is because with larger value of $c_s$ the system expands faster as a result has shorter life time for fixed $T_i$ and $T_c$. Consequently the heavy quarks get lesser time to interact with the expanding thermal system and fails to generate enough flow. From the energy dissipation we have evaluated the shear viscosity ($\eta$) to entropy ($s$) density ratio using the relation [@mmw]: $\eta/s\sim 1.25T^3/\hat{q}$, where $\hat{q}=
\langle p^2_T\rangle/L$ and $dE/dx\sim \alpha_s\langle p^2_T\rangle$ [@RB], $L$ is the length traversed by the heavy quark. The average value of $\eta/s\sim 0.1-0.2$, close to the AdS/CFT bound [@KSS].
![$R_{AA}$ as a function of $p_T$ for $D$ and $B$ mesons at LHC. Experimental data taken from [@ALICE].[]{data-label="fig8"}](fig8.eps)
![Elliptic flow of single electrons originating from the heavy mesons decays. []{data-label="fig9"}](fig9.eps)
Summary
=======
In this work we have discussed the productions of lepton pairs from nuclear collisions at relativistic energies and shown that lepton pairs can trace the evolution of collectivity of the system. The elliptic flow and the nuclear suppression factor of the electrons originating from the heavy flavour decays have been studied by including both the radiative and the collisional processes of energy loss in evaluating the effective drag and diffusion coefficients of the heavy quarks. The results have been compared with the available experimental data and properties of QGP expected to be formed at RHIC collisions have been extracted.
[**Acknowledgment:**]{} The author is grateful to Trambak Bhattacharyya, Santosh K Das, Sabyasachi Ghosh, Surasree Mazumder, Bedangadas Mohanty, Payal Mohanty and Sourav Sarkar for collaboration and to Tetsufumi Hirano for providing hadronic chemical potentials.
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| {
"pile_set_name": "ArXiv"
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---
abstract: 'Vision-to-Language tasks aim to integrate computer vision and natural language processing together, which has attracted the attention of many researchers. For typical approaches, they encode image into feature representations and decode it into natural language sentences. While they neglect high-level semantic concepts and subtle relationships between image regions and natural language elements. To make full use of these information, this paper attempt to exploit the text-guided attention and semantic-guided attention to find the more correlated spatial information and reduce the semantic gap between vision and language. Our method includes two level attention networks. One is the text-guided attention network which is used to select the text-related regions. The other is semantic-guided attention network which is used to highlight the concept-related regions and the region-related concepts. At last, all these information are incorporated to generate captions or answers. Practically, image captioning and visual question answering experiments have been carried out, and the experimental results have shown the excellent performance of the proposed approach.'
author:
- 'Xuelong Li, Aihong Yuan, and Xiaoqiang Lu, [^1] [^2] [^3] [^4] [^5] [^6]'
bibliography:
- 'egbib.bib'
title: 'Vision-to-Language Tasks Based on Attributes and Attention Mechanism'
---
[Shell : 3G Structure for Image Caption Generation]{}
Image captioning, visual question answering, deep learning, multi-modal.
Introduction
============
ision-to-Language (V2L) tasks aim to integrate natural language processing and computer vision together. Typical V2L tasks are image captioning [@ATT2U; @DBLP:conf/aaai/LiuMSY17; @DBLP:conf/cits/QuLTL16; @Binqiang], visual question answering (VQA) [@DBLP:conf/icml/XiongMS16; @DBLP:conf/cvpr/ShihSH16; @TGIF-QA] and video description [@DBLP:conf/cvpr/DonahueHGRVDS15; @HVC; @TDVD; @DBLP:conf/ijcai/LiZL17; @DBLP:conf/mm/ZhaoLL17]. Recently, due to the advent of artificial intelligence (AI), V2L tasks have attracted extensive attention. Practically, V2L tasks enable many important applications, including early childhood education, human-robot interaction, visually impaired people assistance and so on.
Many recent approaches for V2L tasks have achieved a lot of gratifying results through combining Convolutional Neural Networks (CNNs) and Recurrent Neural networks (RNNs) for image encoding and text generating, respectively [@DBLP:conf/cvpr/KarpathyL15; @DBLP:conf/cvpr/VinyalsTBE15; @DBLP:journals/corr/MaoXYWY14a; @DBLP:conf/nips/RenKZ15]. Concretely, a CNN pre-trained on ImageNet [@DBLP:journals/ijcv/RussakovskyDSKS15] is used to extract global image feature while a RNN is used to encode the language information. Most of the recently approaches are belong to the “CNN-RNN” paradigm and these approaches have attained some promising results, further improvements should be got over some limitations.
Motivation and Overview
-----------------------
![Overview of our scheme for vision-to-language tasks. The proposed approach is composed of two level attention modules. The first attention module is text-guided attention (*i.e.*, word-guided attention for image captioning and question-guided attention for VQA) which is used to dig up the mapping relation between word and image region. Semantic-guided attention is the second attention module, which is used for explore the relationship between image and semantic concept. At last, the two parts of representations which output from the two attention modules are jointly embedded into multi-modal space to generate the corresponding captions or answers.[]{data-label="introduction"}](intro_1.pdf "fig:"){width="0.90\linewidth"}\
Image high-level semantic concepts (also called image attributes, *i.e.*, objects, actions, scenes and object’s attributes of images) are very important information for V2L tasks. In previous works, the spatial attention based methods (to distinguish the semantic-guided attention network, the spatial attention network is called text-guided attention network in this paper) are the most popular scheme for V2L tasks. Namely, the relationships between natural language elements with image regions via computing the attention weights between words/questions and image regions. The core idea of the spatial attention mechanism is that every word in captions or every question should only correspond to one or several regions of an image. Although the spatial attention based methods can dig up the subtle relationships between image and text elements, high-level semantic concepts have not been fully utilized for V2L tasks in most of the previous works, while these concepts are important for humans when understanding a scene [@MLAN; @DBLP:conf/cvpr/WuSLDH16]. Actually, the semantic concepts bridge images and natural language information together and can contribute significantly to eliminate the well-known semantic gap. That is because the semantic concepts are not only the important high-level visual information for images, but also the important component of captions. For example, image in Fig. \[introduction\] shows an brown cat. According to the semantic concepts in this paper, the words “*brown*” and “*cat*” are high-level information of the image, and they provide very important information for understanding the image. Meanwhile, the two words also the components of the corresponding captions. To eliminate the semantic gap between images and natural language, the high-level semantic information is an extra input of the proposed method.
Moreover, every high-level semantic concept should correlated to a specific image region. Some previous methods, such as [@DBLP:conf/cvpr/WuSLDH16; @TPAMI/Q], attempt to use the image semantic information to complete the V2L tasks, the semantic information is encoded into vectors and directly input into the language generating model. However, this process is not the optimal one because it cannot dig up the relationships between the semantic concepts and image regions. In this paper, a semantic-guided attention network is designed to explore the relationships between image semantic concepts and image regions. Namely, the image semantic concepts information is used to attend the corresponding regions. Other works like [@MLAN] use the attention mechanism for image semantic information, but the guidance information is natural language. In other words, the high-level semantic information is attended the by the corresponding caption or the question. Actually, every semantic concept is more correlated to a specific image regions. For instance, image in Fig. \[introduction\] shows a brown cat. Both the concept words “*cat*” and “*brown*” are important elements of the captions, while they correspond to the “cat” regions. So, exploring the relationships between image regions and image semantic concepts are more effective than the relationships between image semantic concepts and natural language elements.
Motivated by the aforementioned two reasons, we propose a methods with a semantic-guided attention network. The semantic-guided attention network contains two sub-parts which are used to highlight the concept-related regions and the region-related concepts, respectively. In addition, text-guided attention network is also reserved to explore the subtle relationships between image regions and natural language parts. For example, when describing the content of the image in Fig. \[introduction\], the phrase “*brown cat*” should map with the “*cat*” region of the image. For VQA task, the question is “*What is the cat on?*”. When answering this question, the “*cat*” and its surrounding regions should be focused on because these regions are most related to the question. So, to simultaneously learn the relationships between high-level semantic concepts and image regions and the correlations between natural language elements and image regions, we unify two sub-attention networks (semantic-guided attention network and tex-guided attention network) into a framework.
Fig. \[introduction\] shows the overall scheme of the proposed approach. The approach mainly includes two level attention networks. One is the text-guided attention network which is used to select text-related regions. The text-guided attention network has two variants for image captioning and VQA, respectively. In image captioning task, the text-guided attention network is called word-guided attention which is used to explore the relationships between words and image regions. In VQA task, the text-guided attention network is called question-guided attention which is used to select the image regions corresponding to the question. The other is the semantic-guided attention network which is used to dig up the relationship between image regions and high-level semantic concepts. The outputs of these two networks are projected into the same multi-modal space to generate captions or answers.
Contributions
-------------
The core contributions can be summarized as follows:
1\) An approach based on image high-level semantic attributes and local image features is proposed to address the challenges of V2L tasks. Specially, the high-level semantic attributes information is used to reduce the semantic gap between images and text.
2\) An novel semantic-guided attention network is designed to explore the mapping relationships between semantic attributes and image regions. The semantic-guided attention network highlights the concept-related regions and selects the region-related concepts.
3\) Two special V2L tasks (*i.e.*, image captioning and visual question answering) are addressed by the proposed approach. Taking into account their characteristics, two sub-models was designed for image captioning and VQA, respectively. Experimental results show that our models are effective for V2L tasks.
Organization
------------
The rest of this paper is organized as follows. In Section \[Related Work\], some previous works are briefly introduced. Section \[our model\] presents our approach for V2L tasks. To validate the proposed method, the experimental results are shown in Section \[Experiments\]. At last, Section \[Conclusion\] makes a brief conclusion for this paper.
Related Work {#Related Work}
============
With the development of deep learning, some related and recent work on deep learning has been researched for visual content analysis [@DBLP:journals/tcsv/HanCLZ18; @DBLP:journals/tcyb/HanCLYL18; @DBLP:journals/tip/ChengHZX19]. In this section, some typical methods for V2L tasks, *i.e.*, image captioning and VQA, are introduced.
Image caption generation
------------------------
Using a natural language sentence to describe the content of the given image has long been researched in artificial intelligence. A traditional approach is to use predefined visual templates to generate sentences by filling detected visual concepts. Kuznetsova *et al.* [@DBLP:journals/pami/KulkarniPODLCBB13] pose the image caption generation task as a retrieval problem. They first retrieval a similar image and the corresponding descriptions from the training set, and compose a new sentence based on the retrieval descriptions. Sentences generated by these methods are less variety and very limited, which cannot describe the contents of the test image very well.
Recent works using the deep neural networks has gained many encouraging results on image caption generating task. Mao *et al.* [@DBLP:journals/corr/MaoXYWY14a] proposed a *multi-modal recurrent neural network* (m-RNN) to explore the relationships between vision and text information. This model predicts the next word by computing the probability distribution of the next word conditioned on the previous words and visual features at each time-step. Karpathy *et al.* [@DBLP:conf/cvpr/KarpathyL15] also proposed a multi-modal RNN model to generate sentences to describe the content of a given image. But in contrast to m-RNN, the image features are input into the multi-modal RNN only at the first time-step. Vinyals *et al.* [@DBLP:conf/cvpr/VinyalsTBE15] proposed a similar method, which combined deep CNN for image feature extracting with an LSTM for sentence generating. Donahue *et al.* [@DBLP:conf/cvpr/DonahueHGRVDS15] proposed an unified model for activity recognition, image captioning and video description. To generate captions for image, this model use multiple layers of LSTM. Wu *et al.* [@DBLP:conf/cvpr/WuSLDH16; @TPAMI/Q] proposed a caption generation model based on attributes. They use the most common words as the semantic attributes. At the sentence generating step, not only the global image feature is input into RNN, but the semantic attribute vector also be used as one input of RNN.
Attention-based model becomes a hot topic on image caption generation. Xu *et al.* [@DBLP:conf/icml/XuBKCCSZB15] proposed an attention model to solve the image caption generation problem. In contrast to the previous models, it uses the output of last convolutional layer as the image features. Through flattening the feature map into 196 vectors, each vector denotes one region of the image. At each time-step, only one or several regions are selected by the attention mechanism. [@DBLP:conf/cvpr/YouJWFL16] proposed an image captioning model with semantic attention. It uses a set of attribute detectors to get some semantic concepts and the attention mechanism can select specific items form these concepts. Fu *et al.* [@Kun_Fu] proposed a model based on spatial attention and scene-specific contexts.
Visual question answering
-------------------------
Malinowski *et al.* [@DBLP:conf/nips/MalinowskiF14] may be the first researchers to study the “open-world” visual question answering problem. They proposed a method with two important parts. One for semantic text parsing and the other is image segmentation with a Bayesian formulation to sample from nearest neighbors in the training set. This approach is very dependent on the human defined predicates and the accuracy of the image segmentation. Tu *et al.* [@DBLP:journals/ieeemm/TuMLCZ14] proposed a question answering based on joint parse graph from text and videos. All these early approaches have a common shortage: the answer is limited on the form of question.
Recently, deep neural network models have gained many encouraging results in the field of computer vision and natural language processing. Inspired by these encouraging results, an architecture based on “CNN-RNN” has become the most popular trend. Gao *et al.* [@DBLP:conf/nips/GaoMZHWX15] used CNN to encode the image. Another two RNNs are used to encode the question and generate the answer, respectively. Similar to [@DBLP:conf/nips/GaoMZHWX15], Malinowski *et al.* also proposed a method based on “CNN-RNN” architecture. However, [@DBLP:conf/nips/GaoMZHWX15] only used one RNN as question encoder and decoding the image and question into answer. In [@DBLP:conf/nips/RenKZ15], Ren *et al.* took the visual question answering as classification problem. Their method used the LSTM as the question encoder and the image was treated as the first world. The answer was generated from an classifier which is a softmax layer. The input of the softmax layer was the output of the last time-step of the LSTM. Wu *et al.* [@DBLP:conf/cvpr/WuSLDH16] proposed a method which contains two different LSTMs to encode the question together with decode it and image information into answer with multiple words. It is worth noting that this model used the global image feature and image attribute vector output from the attribute detector as image information. Their team also did another work. This work was more complicate than ever before because on the basis of [@DBLP:conf/cvpr/WuSLDH16], they added external knowledge and caption vector as another two inputs to the encoder LSTM. They encoded five descriptions into vectors and pooled these vectors into one vector as caption vector. Noh *et al.* [@DBLP:conf/cvpr/NohSH16] used CNN with dynamic parameter prediction to solve the image question answering problem. To reduce the complexity of the problem, they incorporated a hashing technique to select the weights. [@DBLP:conf/aaai/MaLL16] proposed a model with CNN architectures for learning not only for image and question, but also their inter-modal relationships to produce the answering.
A limitation of the most aforementioned methods is that they only use global image feature to represent the input image. This may lead to some irrelevant or noisy information input into the answering module. To address the aforementioned problem, attention mechanism is widely used in question answering system. A typical model is SANs [@DBLP:conf/cvpr/YangHGDS16] which is short for stacked attention networks. This model used semantic representation of a question to search for the corresponding regions in an image which related to the question and the answer. It also stacked the attention network because the authors argued that visual question answering needs to multiple steps of reasoning. Shih *et al.* [@DBLP:conf/cvpr/ShihSH16] presented a method which learns to answer questions by selecting image regions relevant to the questions. Unlike to [@DBLP:conf/cvpr/YangHGDS16], which used one layer neural network to compute the attention distribution, this model mapped question queries and image features from various regions into a shared space through an inner product manuscription. Xu *et al.* [@DBLP:conf/eccv/XuS16] proposed a spatial memory network to the visual question answering task. Their memory networks were recurrent neural networks with attention mechanism that choose relevant regions stored in memory. [@DBLP:conf/cvpr/NamHK17] presented an dual attention model which jointly used visual and textual attention to capture the fine-grained relationship between vision and language. Lu *et al.* [@DBLP:conf/nips/LuYBP16] proposed a co-attention for both image and question. Different from the most above models, this model used an hierarchical question encoding. Kazemi *et al.* [@DBLP:journals/corr/KazemiE17] proposed a strong baseline for visual question answering. Their model used two-layer convolutional neural network to realize the stacking attention and produce probabilities over answer classes. Yu *et al.* [@MLAN] presented a multi-level attention model which contained context-aware visual attention and semantic attention modules. The context-aware module used a question to select relevant regions and the semantic attention module aimed to find important concepts. Xiong *et al.* [@DBLP:conf/icml/XiongMS16] proposed a model named dynamic memory network which mainly contained two important parts: input module and episodic memory module. The core component of the input module is the bidirectional gated recurrent unit which was used to explore the relationship between local regional image features. In fact, the episodic memory module is also an attention module, which extracted a contextual vector based upon the current focus.
{width="0.90\linewidth"}\
Proposed Approach {#our model}
=================
Fig. \[V2L\_fig\] shows the overall framework of our approach for image captioning and VQA, two typical V2L tasks. Both the two sub-frameworks consist of six part: 1) a multi-label CNN, 2) an attribute layer, 3) a bidirectional GRU module, 4) semantic-guided attention network, 5) text-guided attention network and 6) joint embedding layer. The first two modules are used to extract image attributes and local features. The bidirectional GRU [@DBLP:conf/emnlp/ChoMGBBSB14] module is used to explore the relationships among the local image features. The concept vectors output from the attribute layer and the proposed local image feature output from the bidirectional GRU are input into semantic-guided attention network. This is designed to highlight the concept-related regions and select the region-related concepts. The text-guided attention network explores the fine-grained mapping relationship between language elements and image regions. All the information is fused in the joint embedding layer. At last, the multi-modal information is used to generate the caption or the answer. There are two little differences between the two sub-frameworks. First, image captioning is considered as a generating problem, which means the captions are generated word by word. While VQA is treated as a classification problem, which all the answers are processed as class labels. Second, the text-guided attention network in image captioning and VQA are called word-guided attention network and question-guided attention network, respectively.
Image Concepts Predicting {#image_concept}
-------------------------
To train an image concepts predictor, the concepts vocabulary should be built at first. Similar to [@MLAN], we collect all words from the MS COCO image captioning dataset [@lin2014coco]. All words are reverted to the prototype (*i.e.*, the form of nouns and the tense of verbs are not sensitive). To select the concept words, the word frequencies are counted at first. And then the meaningless words (*e.g.*, “*a*”, “*is*”, “*on*” and so on) are abandoned. After the rough screening, we select the $c$ most frequent words as the image semantic concepts candidate.
After constructing the concept vocabulary, we label each image with a $c$-dimensional vector through comparing the captions with the concept vocabulary. We then train the concept detector (*i.e.*, Multi-label CNN in Fig. \[V2L\_fig\]). As a results, each image $I$ is represented as a concept vector ${{v}_I} = {\left[ {\begin{array}{*{20}{c}}
{{v_{I1}}}&{{v_{I2}}}& \cdots &{{v_{Ic}}}
\end{array}} \right]^T} \in {\mathbb{R}^{c}}$ and each element denotes the probability of the corresponding concept. To train the concept predictor, the last softmax layer of the single label CNN is replaced by the sigmoid cross entropy loss layer. Suppose that there are $N$ training samples and ${y_i} = {\left[ {\begin{array}{*{20}{c}}
{{y_{i1}}}&{{y_{i2}}}& \cdots &{{y_{ic}}}
\end{array}} \right]^T}$ (where $y_{ij}=0\ or\ 1, i=1,2,\cdots,N, j=1,2,\cdots,c$) is the attribute label of $i$-th image. And the loss function is defined as follows: $${L_M} = - \frac{1}{N}\sum\limits_{i = 1}^N {\sum\limits_{j = 1}^c {\left[ {{y_{ij}}\log v_{Ij}^{(i)} + \left( {1 - {y_{ij}}} \right)\log \left( {1 - v_{Ij}^{(i)}} \right)} \right]} } .$$
According to $v_I$, the concept set $v_c$ of image $I$ is defined as follows: $$\label{v_c}
{{v}_c} = \left\{ {{v_{ci}}} \right\},{v_{ci}} = {e_i} \cdot \delta \left( {{v_{Ii}} \ge \varepsilon } \right),i \in \left\{ {1,2, \cdots ,c} \right\},$$ where $\delta \left( \cdot \right)$ denotes the indicator function, $\varepsilon$ is a threshold and we set it as 0.6 in this paper, ${e_i} \in {\mathbb{R}^c}$ is a vector where the $i$-th element equal to 1 and the other elements equal to 0.
Local Image Feature Processing {#local_image_representation}
------------------------------
As illustrated in [@DBLP:conf/icml/XiongMS16], we use a pre-trained CNN (*i.e.*, VGG-19 in this paper) to extract local image features. When a raw image $I$ is input into VGG-19, we flat the feature map output from the CONV5-4 layer. The process can be written as follows: $$\label{v_l}
{{v}_l} = \left\{ {{v_{l1}},{v_{l2}},\cdots,{v_{lC}}} \right\} = flatten\left( {Conv(I)} \right),$$ where ${v}_{li}\in \mathbb{R}^{D},\ i\in \{0,1,\cdots ,C\}$ denotes the feature of $i$-th location of image $I$. In other words, each image $I$ is divided into $C$ locations and every ${v}_{li}$ represents one location. So, we call ${v}_{li}$ is the location feature representation.
The local image feature extracted from above do not yet have global information available for them. Without global information, their representational power is quite limited because it suffers from the simple issues like locational variance causing accuracy problems or object scaling. According to [@DBLP:conf/icml/XiongMS16], the bidirectional RNN can solve the aforementioned problem. Following this idea, we use a bidirectional GRU to explore the relationship among regions (As illustrated in Fig. \[local\_image\_feature\]). The formulas are shown as follows: $$\label{v_l'}
\begin{array}{l}
{{\vec v}_{li}} = GR{U_f}\left( {{v_{li}},{{\vec v}_{li - 1}}} \right)\\
{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftarrow$}}
\over v} }_{li}} = GR{U_b}( {{v_{li}},{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftarrow$}}
\over v} }_{li + 1}}} )\\
{v'_{li}} = {{\vec v}_{li}} + {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftarrow$}}
\over v} }_{li}}
\end{array},$$ where ${\vec v}_{li}$ and ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftarrow$}} \over v} }_{li}$ are the hidden states of forward and backward GRU at time-step $i$, respectively. At last, the sum of ${\vec v}_{li}$ and ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftarrow$}} \over v} }_{li}$ denotes the context-aware visual representation of the $i$-th image region ${v'_{li}}$. We denote ${v'_l} = \left[ {\begin{array}{*{20}{c}}
{{v'_{l1}}}&{{v'_{l2}}}& \cdots &{{v'_{lC}}}
\end{array}} \right]$.
![An illustration of local image processing. The local image feature output from convolutional layer are encoded into bi-directional GRU. The sum of hidden states of the forward and backward GRU is denoted as the context-aware visual representation.[]{data-label="local_image_feature"}](local_image_feature.pdf "fig:"){width="0.8\linewidth"}\
Semantic-guided Attention {#SGA}
-------------------------
To find the fine-grained relationship between image regions and semantic concepts, we propose a semantic-attention network. We connect the visual representation $v'_{l}$ and semantic concepts representation ${{v}_c}$ by similarity between them at all image-concepts and concept-regions. Specifically, given an image representation $v'_{l}\in \mathbb{R}^{D \times C}$, and the concepts representation ${{v}_c}\in \mathbb{R}^{c \times c}$, the similarity matrices are calculated as follows: $$\label{sim}
\begin{array}{l}
P = {\left( {{W_l}{v'_l} \oplus {{b}_l}} \right)^T} \cdot \left( {{W_c}{{v}_c} \oplus {{b}_c}} \right)\\
P' = {\left( {{W'_c}{{v}_c} \oplus {b'_c}} \right)^T} \cdot \left( {{W'_l}{v'_l} \oplus {b'_l}} \right)
\end{array},$$ where $P\in \mathbb{R}^{C \times c}$ and $P'\in \mathbb{R}^{c \times C}$ represent the similarity matrices between image regions and concepts. Concretely, ${P_{ij}}$ and ${P'_{i'j}}$ are scores which represent the similarity of the $j$-th concept with $i$-th region representation and the similarity of the $j$-th region representation with $i'$-th concept, respectively. ${W_l} \in {\mathbb{R}^{d \times D}}$, ${W_c} \in {\mathbb{R}^{d \times c}}$, ${W'_c} \in {\mathbb{R}^{d' \times c}}$, ${W'_l} \in {\mathbb{R}^{d' \times D}}$, $b_l \in {\mathbb{R}^d }$, $b_c \in {\mathbb{R}^d }$, $b'_c \in {\mathbb{R}^{d'} }$, and $b'_l \in {\mathbb{R}^{d'} }$ denote weights parameters. Note that “$\oplus$” represents the addition of a matrix and a vector. The addition between a matrix and a vector is performed by adding each column of the matrix by the vector.
After calculating the similarity matrices, the formula of attention weights is as follows: $$\label{alp}
\begin{array}{l}
\alpha _i^l = \frac{{\exp \left( {{{\max }_j}\left( {{P_{ij}}} \right)} \right)}}{{\exp \left( {\sum\nolimits_k {{P_{ik}}} } \right)}},\left( {i \in \{ 1,2, \cdots ,C\} } \right)\\
\alpha _{i'}^c = \frac{{\exp \left( {{{\max }_j}\left( {{P'_{i'j}}} \right)} \right)}}{{\exp \left( {\sum\nolimits_k {{P'_{i'k}}} } \right)}},\left( {i' \in \{ 1,2, \cdots ,c\} } \right)
\end{array}.$$
Based on the above attention weights, the concept-based image region representation and the region-based concept representation are calculated as follows: $$\label{v}
\begin{array}{l}
{{\hat v}_l} = \sum\limits_{i = 1}^C {\alpha _i^l{{v'}_{li}}} ,\\
{{\hat v}_c} = \sum\limits_{i' = 1}^c {\alpha _{i'}^c{v_{ci'}}} .
\end{array}$$
After computing the weighted image and concept representations, we concatenate them into a vector which contains the image feature and semantic concepts representation. The formula is: $$\label{v_II}
{v'_I} = \left[ {{{\hat v}_l};{{\hat v}_c}} \right].$$
Image Captioning {#IC}
----------------
The proposed model for image captioning is summarized in Fig. \[V2L\_fig\] (a). Similar to [@DBLP:conf/cvpr/VinyalsTBE15], our language generation model is trained by maximizing the probability of the correct description conditioned on the given image. Combined with our model, the log-likelihood function can be written as follows: $$\label{log}
\log P\left( {S\left| {{{v}_l}, {v'_I}} \right.} \right) = \sum\limits_{t = 1}^T {\log P\left( {{w_t}\left| {{w_{1:t - 1}},{{v}_l},{v'_I}} \right.} \right)} ,$$ where $S = \left\{ {{w_0},{w_1}, \cdots ,{w_T}} \right\}$ is the description of image $I$, $w_t$ is the $t$-th word of the sentence $S$, and $T$ is the length of the sentence. Based on Eq. (\[log\]), the probability of generating the word $w_t$ (*i.e.*, $P\left( {{w_t}\left| {{w_{1:t - 1}},{v'_I}} \right.} \right)$) is determined by the output of the semantic-guided attention network ${v'_I}$ and the previous words ${w_{0:t - 1}}$. We exploit GRU to model this.
### Sentence Representation {#Sentence Representation}
In our model, we encode words into one-hot vectors. For example, the benchmark dataset has $N_{0}$ different words, and every word is encoded into a $N_{0}$-dimension vector in which only one value equals to 1 and others equal to 0. When a raw image is input into our model, a corresponding sentence $S$ is generated which is encoded as a sequence of one-hot vectors. We denote $S=({w_1}, {w_2},\cdots ,{w_T})$, where ${w}_i\in \mathbb{R}^{N_0}$ represents the $i$-th word in the sentence. We project these words into embedding space. The concrete formula is as follows: $$\label{sentence representation}
{{\bf{s}}_t} = {W_s}\cdot {w_t},\ t\in \{1,2,\cdots ,N\},$$ where $W_s$ is the embedding matrix of sentences which projects the word vector into the embedding space. So the projection matrix $W_s$ is a $N_0\times h$ matrix where $N_0$ is the size of the dictionary and $h$ is the dimensionality of the embedding space.
### Word-guided Attention {#Word-guided Attention}
Similar to [@DBLP:conf/icml/XuBKCCSZB15], we use word-guided attention mechanism for local feature. At each time-step, the attention mechanism uses the previous hidden state $h_{t - 1}$ which concludes the previous words information to decide the local feature. The attention model is defined as follows: $$\label{att}
\begin{array}{l}
{{{\bf{\tilde \alpha }}}_t} = \tanh \left[ {{{\left( {{\bf{w}}_a^T{v'_l}} \right)}^T} + {U_a}h_{t - 1} + {b_a}} \right]\\
{{\bf{\alpha }}_t} = {\rm{softmax}}\left( {{{{\bf{\tilde \alpha }}}_t}} \right) \buildrel \Delta \over = {\left[ {\begin{array}{*{20}{c}}
{{\alpha _{t1}}}& \cdots &{{\alpha _{tC}}}
\end{array}} \right]^T}
\end{array},$$ where ${{\bf{w}}_a}\in {\mathbb{R}^{D}}$ and ${U_a}\in {\mathbb{R}^{C\times h}}$ are weights, ${b_a}\in {\mathbb{R}^{C}}$ is bias. ${{\alpha }_t}\in{\mathbb{R}^C}$ is a probability vector whose each value denotes the probability of the corresponding local image feature. In our algorithm, we use the soft attention model. Therefore, ${{\bf{\hat z}}_t}$, the word-related region representation at time-step $t$, is calculated as follows: $$\label{lv}
{{\hat z}_t} = {v'_l}{{\bf{\alpha }}_t} = \sum\limits_{i = 1}^C {{\alpha _{ti}}{v'_{li}}} .$$
Through Eq. (\[lv\]) we know that ${{\bf{\alpha }}_t}$ decides which locals should be emphasized at the current time-step.
### Gate for $v'_I$ {#Gate for ${v'}_I$}
To control when and how much $v'_I$ should be input into sentence generation GRU, we design a gate to achieve it. The gate is defined as follow: $$\label{gate-global}
{g_t} = \sigma ({\bf{w}}_g^T{h}_{t - 1} + {b_g}),$$ where ${\bf{w}}_g \in {\mathbb{R}^{h}}$ is weight vector and $b_g \in {\mathbb{R}}$ is bias. After calculating the gate, the $v'_I$ is controlled as following formula: $$\label{gg}
{v'_t} = {g_t}{v'_I}.$$
### Sentence Generating
After getting ${{\hat z}_t}$ and $v'_t$, we use GRU to generate description for the given image. The formulae are as follows: $${h_t} = GRU\left[ {{{\hat z}_t},{v'_t},{h_{t - 1}}} \right],$$ $$\label{p}
{p_{t + 1}}{\rm{ }} = {\rm{softmax}} ({h_t}).$$ The loss function is written as follows: $$\begin{array}{c}
L_C = - \frac{1}{N}\sum\limits_{i = 1}^N {\log P\left( {{S^{(i)}}\left| {v_l^{(i)},{v'_I}^{(i)}} \right.} \right)} \\
= - \frac{1}{N}\sum\limits_{i = 1}^N {\sum\limits_{t = 1}^{{T^{(i)}}} {\log {p_t}\left( {w_t^{(i)}} \right)} }
\end{array},$$ where $N$ is the number of training images and $T^{(i)}$ is the length of the sentence for the $i$-th training image. ${p_t}\left( {w_t^{(i)}} \right)$ equals to $p_{t+1}$ in Eq. (\[p\]).
Visual Question Answering {#VQA}
-------------------------
The model for VQA is illustrated in Fig. \[V2L\_fig\] (b). Similar to [@DBLP:conf/nips/RenKZ15], we take VQA as a classification task. So, all the information should be jointly embedded into a classifier. Given the image $I$ and corresponding question $Q$, we expect the probability of the correct answer to reach maximum. The object function can be written as follows: $$\hat A = \mathop {\arg \max }\limits_A P\left( {A\left| {{v_q},{v'_I}} \right.} \right),$$ where $v_q \in \mathbb{R}^{q}$ denotes the representation of question $Q$, $v'_I$ represents the image feature and attribute representation of image $I$. After encoding each question into a vector $v_q$, we calculate the question-guided region representation.
### Question-guided Attention Network
Similar to word-guided attention in Section \[Word-guided Attention\], we design a question-guided attention network to select the question-related regions which can improve the accuracy of the answer. The formula of the attention weight is as follows: $${\alpha _i} = \frac{{\exp \left( {\sigma \left( {\left\langle {{W_q}{v_q},{W_l}{v'_{li}}} \right\rangle } \right)} \right)}}{{\sum\nolimits_k {\exp \left( {\sigma \left( {\left\langle {{W_q}{v_q},{W_l}{v'_{lk}}} \right\rangle } \right)} \right)} }},$$ where $\left\langle { \cdot , \cdot } \right\rangle $ is the inner product operation symbol, ${W_q} \in {\mathbb{R}^{h' \times q}}$ and ${W_l} \in {\mathbb{R}^{h' \times D}}$ are projection matrices which project the question and region representation into the $h'$-dimensional multi-modal space. After that, the question-related region can be represented as follow: $$\label{v_lq}
{v_{lq}} = \sum\limits_{i = 1}^C {{\alpha _i}{v'_{li}}} ,{v'_{lq}} = \left[ {{v_q};{v_{lq}}} \right].$$
### Joint Embedding
Finally, we feed all the vectors (*i.e.*, $v'_I$ and $v'_{lq}$) into classifier with an joint embedding layer to generate the answer. This can be represented as the following formulae: $$\begin{array}{l}
u = \tanh \left( {W{v'_I} + U{v_{lq}} + b} \right)\\
{P_a} = {\rm{softmax}}(u)
\end{array},$$ where $W$, $U$ and $b$ are the parameters of the last parameter layer, the input of the classifier, $v'_I$ and $v_{lq}$ are calculated in Eq. (\[v\_II\]) and Eq. (\[v\_lq\]), respectively, $P_a$ is the distribution of probability of answer candidates. The answer is the maximum probability of the candidates.
The loss function can be written as follows: $$L_A = - \frac{1}{N}\sum\limits_{i = 1}^N {\log P_a^{(i)}} ,$$ where $N$ is the number of the train examples.
Experiment {#Experiments}
==========
Train Details and Experimental Setup
------------------------------------
This section mainly shows the training details and the parameter setting. For both image captioning and VQA tasks, the variants of our models are trained with stochastic gradient descent [@DBLP:conf/icml/Zhang04] within adaptive learning rates. Specially, for the Flickr30K [@DBLP:journals/tacl/YoungLHH14], MS COCO [@lin2014coco], VQA [@DBLP:conf/iccv/AntolALMBZP15] and COCO-QA [@DBLP:conf/nips/RenKZ15], Adam algorithm is used. For Flickr8K [@DBLP:journals/jair/HodoshYH(A); @DBLP:conf/ijcai/HodoshYH(B)], RMSProp is used to train the models. The parameter setting is shown in the following subsections.
### Local Image Feature
In the proposed model, deep features generated from the CONV5-4 layer of VGG-19 are used to represent the images. The dimensionality of the feature map output from the Conv5-4 layer is $14 \times 14 \times 512$. Through flattening operation, the feature map is transformed into $196 \times 512$. So, in Section \[local\_image\_representation\], the parameters $C = 196$ and $D = 512$.
### Image Concepts Encoding
In the proposed method, $512$ concepts word are collected from the MS COCO image captioning dataset. So, each image is encoded into a $512$-dimensional vector. In other words, $c = 512$ in Section \[image\_concept\].
### Word Encoding
In our model, we encode words into one-hot vectors. For example, the benchmark dataset has $M$ different words, every word is encoded into a $M$-dimension vector, in which only one value is equal to 1 and others are equal to 0. So the location of 1 in the vector denotes the corresponding word in the dictionary. It implies that $N_{0}$ in Section \[Sentence Representation\] equals $M$. Specially, after filtering words less than 5 times in the training set, the value of $N_0$ equals 2538, 7414 and 8791 words for Flickr8K, Flickr30K and MS COCO, respectively.
### Question Encoding
To encode the questions, we first cast all question words which appear at least twice in the training and validation sets into lowercase. After collecting the question words vocabulary, each word is represented as one-hot vector. We use one layer Gated Recurrent Unit (GRU) with $512$-dimensionality hidden state to encode the question, and the last hidden state of the GRU as the question representation. So the parameter $q=512$ in Section \[VQA\].
### Other Parameters
In this paper, to calculate the relationship between the concept word and image regions, the parameter $d$ and $d'$ are introduced (In Section \[SGA\]). In the experiments, we set $d'=d=512$. For image caption, the hidden state’s dimensionality of the captioning GRU is set as 512 (*i.e.* $h = 512$ in Section \[IC\]).
Image Captioning {#image-captioning}
----------------
### Dataset and Evaluation Metrics
**Dataset.** We report results on the most popular three datasets: Flickr8K, Flickr30K and MS COCO. Among them, Flickr8K and Flickr30K have 8,092 and 31,783 images respectively, and each image has 5 reference sentences. MS COCO dataset has 123,287 images and the most images has 5 reference sentences. Before the experiment, we preprocess the datasets as [@DBLP:conf/cvpr/KarpathyL15] did. First, we convert all letters of sentences to lowercase, remove non-alphanumeric characters and get rid of words that occur less than five times on the training set. Second, we discard these data which have more than 5 corresponding sentences to guarantee that every image has the same number of describing sentences. For MS COCO, we evaluate our model with the widely used publicly available splits in [@DBLP:conf/cvpr/KarpathyL15].
**Evaluation Metrics.** We report results with the BLEU [@papineni2002bleu], METEOR [@banerjee2005meteor] and CIDEr [@DBLP:conf/cvpr/VedantamZP15] metrics which are the most frequently used in the caption generation literature. The first two metrics are originally designed for evaluating the quality of the automatically machine translation. BLEU score represents the precision ratio of the generated sentence compared with the reference sentences. METEOR score reflects the precision and recall ratio of the generated sentence. It is based on the harmonic mean of uniform precision and recall. CIDEr measures consistency between n-gram occurrences in generated and reference sentences, where this consistency is weighted by n-gram saliency and rarity. For BLEU, we report the scores from BLEU-1 to BLEU-4, which denote the precision of N-gram (N equals to 1, 2, 3 and 4). For both metrics, the higher score they are, the higher quality of the generated sentences they have.
### Results on Flickr8K and Flickr30K
We compare our method with several state-of-the-art methods on the Flickr8K and Flickr30K datasets. The contrast models can be roughly divided into three categories. The first category, such as NeralTalk [@DBLP:conf/cvpr/KarpathyL15], Google-NIC [@DBLP:conf/cvpr/VinyalsTBE15] and m-RNN [@DBLP:journals/corr/MaoXYWY14a] in Table \[results\_on \_Flickr\], only uses the global image feature extracted by CNN, and only the feature is input into the sentence generator RNN. The second category is attribute-based models, which the global image feature and attribute vector are used to sentence generating. In Table \[results\_on \_Flickr\], Att-SVM + LSTM [@TPAMI/Q] and Att-GlobalCNN + LSTM [@TPAMI/Q] belong to this category. The third category is attention-based models. The attention-based model try to explore the relationship between image regions and words. NIC-VA [@DBLP:conf/icml/XuBKCCSZB15], ATT [@DBLP:conf/cvpr/YouJWFL16] and RA [@Kun_Fu] *et al.* in Table \[results\_on \_Flickr\] are all attention-based models. Table \[results\_on \_Flickr\] reports the image captioning results on the Flickr8K and Flickr30K. Between the contrast models, the attribute-based models show better performance than attention-based models. Specially for the Flikr8K, the Att-GlobalCNN + LSTM brings significant improvements nearly $5\%$ for B-1, $8\%$ for B-2, $8\%$ for B-3 and $7\%$ for B-4 on average. And the similarity improvements on the Flickr30K dataset. The phenomenon implies that the high-level semantic information (*i.e.*, attributes) is very important for image captioning task. Compared with the basic models (*i.e.*, none attribute and none attention are used), the attention-based models show much better performance for image captioning. The main reason is that the attention-based models can dig up the relationship between the image regions and sentence elements.
Although the state-of-the-art attribute-based models and attention-based models show good performance on the Flickr8K and Flickr30K datasets, Table \[results\_on \_Flickr\] shows that our model gains a much better results on these datasets (only the B-1 score less than Att-SVM + LSTM on the Flickr8K dataset). The main reason is our model combines the semantic information and attention mechanism masterly. Specially, both the semantic information vector output from the semantic-guided attention network and the image region feature are selected by the word-guided attention network are used to generate the description sentence.
### Results on MS COCO
{width="0.98\linewidth"}\
Table \[results\_on \_MSCOCO\] shows image captioning results on the MS COCO dataset. Similar to the experiment on the Flickr8K and Flickr30K, the contrast models also be classified into three categories, *i.e.*, none attention and none attribute models (such as NeuralTalk, Google-NIC, LRCN and m-RNN), only attribute-based models (such as Att-CNN + LSTM) and only attention-based models (such as NIC-VA and ATT-FCN). Among the contrast models, Att-CNN +LSTM gets the highest scores both on BLEU and METEOR metrics. It shows that the high-level semantic information is important to transform an image into natural language sentence. That is to say, the high-level semantic information can contribute significantly to eliminate the semantic gap between vision and language. Compared with the proposed model in this paper, no attention mechanism has been used in Att-CNN + LSTM. In other words, the attributes information is encoded into one vector and imported into the language model. Table \[results\_on \_MSCOCO\] shows that our model gets higher scores on most of the metrics. It implies that our model with attention mechanism is more effective than Att. In addition, the results of attention-based models are obviously better than the none attribute-based and attention-based models. The fact indicates that the attention mechanism can find fine-grained relationship between image region and sentence element, and this relationship is effective for image captioning.
Although the attention-based and attribute-based models show the powerful ability on image captioning task, our model further improves the performance. There are two main reasons: 1) the word-guided attention network can find the fine-grained relationship between image regions and words; 2) the semantic-guided attention network adds high-level semantic information which contributes to eliminating the semantic gap between vision and language.
Fig. \[visualization\] shows the visualization of generated captions, attributes and image attention maps on the MS COCO dataset. According to the Fig. \[visualization\], we see that our model successfully learns to align the local image regions, image attributes and words. For instance, when generate captions for the first image in the third row, the attribute layer predicts four attributes (i.e., “*man*”, “*next*”, “*motorcycle*” and “*build*”) of this image. When generate the word “riding”, the proposed model attends the most related region (i.e., the man’s region of the image). The histogram shows the attention weights of the four attributes, and the attention weights are computed by Eq. (\[v\]). All the instances prove that the model can explore the relationships among the attributes, local image regions and captions very well.
### Ablation Study
{width="0.98\linewidth"}\
To verify the effectiveness of each component in our model, we perform ablation studies by ablating certain components:
- None attention is used for image captioning (None-Att). The word-guided and semantic-guided attention networks are abandoned. Only the global image feature output from FC7 layer of VGG-19 are used to generate descriptions.
- Word-guided attention (WA). Only the word-guided attention network is used.
- Word and semantic-guided attention (WSA). The two attention networks are both used while the gate for $v'_I$ are abandoned.
- Word and semantic-guided attention with gating controlling (FULL). Our full model.
[l||ccccc]{}
------------------------------------------------------------------------
height 1.5pt @a xhline
Model & B-1 & B-2 & B-3 & B-4 & METEOR\
None-Att & 63.1 & 46.3 & 31.9 & 23.1 & 20.2\
WA & 70.8 & 50.2 & 37.4 & 25.8 & 23.8\
WSA & 72.9 & 53.4 & 41.2 & 30.7 & 27.9\
FULL &73.9 & 56.4 & 41.7 & 30.9 & 27.1\
------------------------------------------------------------------------
height 1.5pt @a xhline
Table \[basline\_1\] shows the performance of the ablation models. The results confirm the truth that both the word-guided and semantic-guided attention networks. First, the WA model improves the performance on the bias of the Non-Att model. That is because the word-guided attention network can automatically focus on the most word-related regions. Second, due to introducing the concept information and semantic-guided attention network, the WSA model further improves the performance on the bias of the WA model. The concept information is an important supplement for image information and the semantic-guided attention network explores the relationship between the concepts and regions. Last, our full model, which adds a gate to control the vector output from the semantic-guided attention model, is an improved version of the WSA model. As can be seen in Tabel \[basline\_1\], the gate is necessary, because the gate automatically controls whether and how much the concept-region representation should be input to the RNN module at each time-step.
Fig. \[example\] shows some examples of image captioning on the validation set of the MS COCO dataset. Generally speaking, our full model shows best performance among the ablation models. The None-Att model may loss the attribute information and some important object information when generates description for image. For example, the second instance of the first row in Fig. \[example\], the None-Att model can describe the main object in the scene (such as “*cat* ” and “*laptop*”), but the attribute of the “*cat*” (*i.e.*, the color of the cat—yellow) and some important objects (such as “*desk*” and “*textbook*”) are not been described. Two main reasons may led to such a result: 1) the attribute information (*i.e.*, the color of the cat) is not be used to generate the caption; and 2) the None-Att model does not have the word-guided attention network which can exploit the attention-transfer mechanism. In other words, the None-Att model only focuses on the main region (the “*cat*” and “*laptop*” region) and describes it, but it cannot transfer the attention into other regions (such as the “*desk*” and “*textbook*” region). The WA model tries to dig all important object in an image, but it may make some mistakes. For instance, no “*book*” in the second image of the third row in Fig. \[example\], but the WA model identifies some object as “*book*”. Simultaneously, the “*posters*” are missed. However, the WSA model has expressed the information of the “*posters*”. This is mainly because the “*poster*” is an attribute word and the WSA has the semantic-guided attention network which can make full use of the attribute information. Our FULL model not only considers both the semantic information and the relationship between the word and image region, but also uses an gate to control when an how much the semantic information output from the semantic-guided attention network should be used to generated the description. This structure can correct some mistakes by the WSA. For example, when the WSA model describe the first image of the third row in Fig. \[example\], two “*coffee table*” are generated, but our FULL model correct this mistake and generate the right caption—“*dining-room*”.
Visual Question Answering {#visual-question-answering-1}
-------------------------
[c||cccc|ccc]{}
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height 1.5pt @a xhline
Model & Object & Number & Color & Location &Accuracy & WUPS@0.9 & WUPS@0.0\
VIS+BOW [@DBLP:conf/nips/RenKZ15] &60.17 &43.99 &52.97 &51.53 & 57.27 & 68.03 & 89.58\
VIS+LSTM [@DBLP:conf/nips/RenKZ15] &56.54 &45.12 &45.58 &45.76 & 53.21 & 63.82 & 88.31\
2-VIS+BLSTM [@DBLP:conf/nips/RenKZ15] &58.49 &44.57 &50.76 &47.70 & 55.53 & 65.88 & 88.91\
DPP-Net [@DBLP:conf/cvpr/NohSH16] &- &- &- &- & 61.19 & 70.84 & 90.61\
Att-LSTM [@DBLP:conf/cvpr/WuSLDH16] &63.92 &51.83 &57.29 &54.84 & 61.38 & 71.15 & 91.58\
SAN [@DBLP:conf/cvpr/YangHGDS16] &64.5 &48.6 &57.9 &54.0 & 61.6 & 71.6 & 90.9\
ABC-CNN [@DBLP:journals/corr/ChenWCGXN15] &62.46 &45.7 &46.81 &53.67 &58.10 &68.44 & 89.85\
CoATT + VGG [@DBLP:conf/nips/LuYBP16] &65.6 &49.6 &61.5 &56.8 &63.3 &73.0 & 91.3\
Ours &67.51 &51.55 &62.10 &56.78 & 64.32 & 74.89 & 92.02\
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[c||cccc|cccc]{}
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& &\
&
--------
Yes/No
--------
&
--------
Number
--------
&
--------
Others
--------
&
-----
All
-----
&
--------
Yes/No
--------
&
--------
Number
--------
&
--------
Others
--------
&
-----
All
-----
\
LSTM Q + I [@DBLP:conf/iccv/AntolALMBZP15] & 78.9 & 35.2 & 36.4 & 53.7 & 79.0 & 35.6 & 36.8 & 54.1\
DPP-Net [@DBLP:conf/cvpr/NohSH16] & 80.7 & 37.2 & 41.7 & 57.2 & 80.3 & 36.9 & 42.2 & 57.4\
SAN [@DBLP:conf/cvpr/YangHGDS16] & 79.3 & 36.6 & 46.1 & 58.7 & - & - & - & 58.9\
CoATT [@DBLP:conf/nips/LuYBP16] & 79.5 & 38.7 & 48.3 & 60.1 & - & - & - & -\
SMem-VQA Two-Hop [@DBLP:conf/eccv/XuS16] & 80.87 & 37.32 & 43.12 & 57.99 & 80.8 & 37.53 & 43.48 & 58.24\
DAN (VGG) [@DBLP:conf/cvpr/NamHK17] & 82.1 & 38.2 & 50.2 & 62.0 & - & - & - & -\
DAN (ResNet) [@DBLP:conf/cvpr/NamHK17] & 83.0 & 39.1 & 53.9 & 64.3 & 82.8 & 38.1 & 54.0 & 64.2\
Att-LSTM [@DBLP:conf/cvpr/WuSLDH16] & 78.90 & 36.11 & 40.07 & 55.57 & 78.73 & 36.08 & 40.60 & 55.84\
MLAN (ResNet) [@MLAN] & 82.9 & 39.2 & 52.8 & 63.7 & - & - & - & -\
MLAN (ResNet, train + val) [@MLAN] & 83.8 & 40.2 & 53.7 & 64.6 & 83.7 & 40.9 & 53.7 & 64.8\
MLAN (ResNet, train + val + VG) [@MLAN] & 81.8 & 41.2 & 56.7 & 65.3 & 81.3 & 41.9 & 56.5 & 65.2\
Ours (VGG) & 83.5 & 41.5 & 55.8 & 65.2 & 83.2 & 42.0 & 55.1 & 65.5\
Ours (ResNet) & 83.7 &41.3 & 56.5 & 66.0 & 83.4 & 42.1 & 56.0 & 65.8\
------------------------------------------------------------------------
height 1.5pt @a xhline
### Dataset and Evaluation Metrics
**Dataset.** We report VQA results on Toronto COCO-QA, VQA dataset which are most popular publicly available visual question answering datasets based on MS COCO. Toronto COCO-QA dataset contains 8,000 images with 79,000 question/answer pairs for training and 4,000 images with 39,171 question/answer pairs for testing. The questions have four types (*i.e.*, object, number, color and location). The answers are all single-word. VQA dataset is a much larger dataset which contains 614,163 questions. The training and testing split follows COCO official split, which contains 82,783 training images, 40,504 validation images and 81,434 test images, each has 3 questions and 10 answers. We use the official test split for our testing. The dataset has two different tasks : open-ended and multiple-choice tasks. We only report the experiment result on open-ended task.
**Evaluation Metrics.** We formulate VQA as a classification problem. The proposed model is evaluated with classification accuracy. The WUPS score [@wu1994verbs] is also reported. The WUPS calculates the similarity between two words based on the similarity between their common subsequence in the taxonomy.
### Results on COCO-QA Dataset
Table \[results 2\] shows the results on the COCO-QA dataset. We categorize the contrast models as [1]{}) none attribute and attention models, [2]{}) only attribute-based models and [3]{}) only attention-based models (every category is separated with double horizontal line in Table \[results 2\]). From the Table \[results 2\], we can easily draw a conclusion that both the attention-based models and the attribute-based models significantly improved accuracy (about $5\%$ increase on both the four types of questions) on the COCO-QA dataset. The Att-LSTM model shows more powerful performance on the COCO-QA dataset than the none-att based (none-attribute and none-attention based) models and the attention-based models (except the CoATT + VGG model [@DBLP:conf/nips/LuYBP16] which include image attention and question attention). It confirms that the high-level semantic information is important to solve the VQA problem. In addition, the results of attention-based models are obviously better than the none att-based and attention-based models. It main because the attention model can focus on the important region which is very correlation with the question.
Through the results in Table \[results 2\], we find that our model improves the state-of-the-art from $63.3\%$ (CoATT + VGG [@DBLP:conf/nips/LuYBP16]) to $64.32\%$. For the different types of questions, all the models in Table \[results 2\] show less powerful performance on the Number and Location questions than the Object and Color Question. That mainly caused by unbalanced data: the COCO-QA dataset contains $70\%$ Object questions, $7\%$ Number questions, $17\%$ Color questions and $6\%$ Location questions. However, our model and the Att-LSTM model increase the accuracy much greater than the attention-based models on the Number questions. Furthermore, the proposed method is further improved the performance for the Object questions. Two main reasons cause the result: 1) the question-guided attention network focuses on the important region which relates to the question; 2) the semantic-guided-attention provide the attribute information and the object noun is the main element for the attribute set. All the results show that the proposed method outperforms almost all the contrast model on all types of questions.
### Results on VQA Dataset
{width="0.98\linewidth"}\
We compare our method with several state-of-the-art methods on the VQA dataset. For equality, the per-answer category accuracy and overall accuracy of the models are shown in Table \[results on VQA\]. The compared models are divided into four categories: [1]{}) none attribute and attention models, [2]{}) only attribute-based models, [3]{}) only attention-based models and [4]{}) both the semantic-guided attention and attention-guided attention models (every category is separated with double horizontal line in Table \[results on VQA\]). It is observed that our model get almost the best performance on all category questions. Only the MLAN model [@MLAN] achieves a comparable results with ours. Two reasons for such a result. Firstly, the MLAN model uses two level attention networks, the visual attention which is benefit for fine-grained spatial inference and the semantic attention which reduces the semantic gap. Secondly, the MLAN model use the ResNet [@DBLP:conf/cvpr/HeZRS16] as image information extractor, which is more powerful than the VGG-Net. However, our model is not inferior to it. To get a more fair comparison (*i.e.*, to eliminate the influence of the visual features), ResNet-152 is also used as image feature extractor (Ours (ResNet) in Table \[results on VQA\]). Specially, feature maps output from the last convolutional layer of the ResNet-152 are used as visual features. So each image is cropped into 49 regions and each region is represented as a 2048-dimension vector. Compared with MLAN, almost for all types of questions our gets the highest scores. It shows that with the same image feature representation, our model shows a better performance than MLAN. Furthermore, there are two differences between MLAN and ours: (a) our approach is designed for both image captioning and visual questioning answering, while MLAN only used for VQA; (b) the semantic-guided attention in our approach is a dual structure, *i.e.*, the semantic-guided attention network is used not only to find the most attribute-related image regions, but also to find the most region-related attributes. While the semantic attention network in MLAN is used to find the attributes coressponding to the question. The experimental results shown in Table \[results on VQA\] demonstrates that the dual structure of the semantic-guided attention is efficiency for VQA task. All results in Table \[results 2\] and \[results on VQA\] confirmed that the semantic information and the attention network are vital to VQA task.
Fig. \[vqa\_example\] shows some typical examples on the VQA validation subset. The letter Q denotes the corresponding questions, “Ours” and “LSTM” represent the answers generated by our model and the VIS+LSTM proposed in reference [@DBLP:conf/nips/RenKZ15], respectively. According to the instances shown in Fig. \[vqa\_example\], we can find that our model shows a much better performance than the VIS+LSTM, especially on the number questions. For example, the second instance of the second row, two cats are in the photo, but the VIS+LSTM only finds one. It mainly because the proposed model contains two level attention networks which helps the model focus on the most related regions for the question. Moreover, when answering the object attribute’s question, the proposed model gets more accurate answers than VIS + LSTM. For example, the forth image in the first row show a big football field. Obviously, the football field is green, while the VIS+LSTM judges the grass is white. The examples show that our model is effective for the VQA task.
[c||cccc|ccc]{}
------------------------------------------------------------------------
height 1.5pt @a xhline
Model & Object & Number & Color & Location &Accuracy & WUPS@0.9 & WUPS@0.0\
None-Att &56.52 &45.22 &48.35 &46.24 & 53.72 & 64.02 & 88.51\
QA &64.27 &49.13 &57.72 &54.33 & 61.50 &70.87 & 90.43\
SA &64.38 &51.22 &59.24 &54.86 & 62.01 & 71.42 & 91.34\
FULL &67.51 &51.55 &62.10 &56.78 & 64.32 & 74.89 & 92.02\
------------------------------------------------------------------------
height 1.5pt @a xhline
### Ablation Study
To further verify the effectiveness of each component in our model, we perform ablation studies by ablating certain components:
- None-attention (None-Att). Only the image representation $v_I$ and question representation $v_q$ are used for VQA problem ,without the semantic-guided attention network and question-guided attention network.
- Question-guided attention (QA). Only the question-guided attention network are used.
- Semantic-guided attention (SA). Only the semantic-guided attention network are used.
- Question and semantic-guided attention (FULL). All attention networks are used for VQA.
Table \[basline\_2\] shows the performance of the ablation models on the COCO-QA dataset. The results are similar to those in Table \[basline\_1\], so we can get a similar conclusion: 1) the question-guided attention network is effective in finding question-related regions and 2) the semantic-guided attention network provide more important supplement information which helps the model get the correct answers.
Conclusion {#Conclusion}
==========
We propose a novel model based on attributes and attention mechanism for V2L problems. The model concludes two level attention networks. The text-guided attention network enables subtle understanding between vision and language, and the semantic-guided attention network provides high-level concepts information and explores the subtle relationships between concepts and regions which reduces the gap between language and visual information. Our model makes full use of the complementarity of the different level visual representations. The extensive experiments both for image captioning and visual question answering show that our model outperforms any single visual attention or attribute model. The semantic attention network is an important supplement for text-guided attention.
[Xuelong Li]{} is a full professor with School of Computer Science and Center for OPTical IMagery Analysis and Learning (OPTIMAL), Northwestern Polytechnical University, Xi’an 710072, P.R. China.
[Aihong Yuan]{} is currently pursuing the Ph.D. degree with the Key Laboratory of Spectral Imaging Technology CAS, Xi’an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences, Xi’an 710119, Shaanxi, P. R. China. His research interests include image/video content understanding and deep learning.
[Xiaoqiang Lu]{} is a Full Professor with the Key Laboratory of Spectral Imaging Technology CAS, Xi’an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences, Xi’an 710119, Shaanxi, P. R. China.
[^1]: This work was supported in part by the National Natural Science Foundation of China under Grant no. 61772510 and in part by the Young Top-Notch Talent Program of Chinese Academy of Sciences under Grant no. QYZDB-SSWJSC015. *(Corresponding author: Xiaoqiang Lu)*
[^2]: Xuelong Li is with School of Computer Science and Center for OPTical IMagery Analysis and Learning (OPTIMAL), Northwestern Polytechnical University, Xi’an 710072, P.R. China.
[^3]: Aihong Yuan is with the Key Laboratory of Spectral Imaging Technology CAS, Xi’an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences, Xi’an 710119, Shaanxi, P. R. China, and also with the University of Chinese Academy of Sciences, Beijing 100049, China.
[^4]: Xiaoqiang Lu is with the Key Laboratory of Spectral Imaging Technology CAS, Xi’an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences, Xi’an 710119, Shaanxi, P. R. China.
[^5]: Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.
[^6]: 20XX IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'Y. Shimajiri, Ph. Andr$\acute{\rm e}$, P. Palmeirim , D. Arzoumanian , A. Bracco , V. K$\ddot{\rm o}$nyves , E. Ntormousi , and B. Ladjelate'
bibliography:
- 'B211\_accretion.bib'
date: 'Received ; accepted '
title: |
Probing accretion of ambient cloud material\
into the Taurus B211/B213 filament
---
[$Herschel$ observations have emphasized the role of molecular filaments in star formation. However, the origin and evolution of these filaments are not yet well understood, partly because of the lack of kinematic information.]{} [We aim to confirm that Taurus B211/B213 filament is accreting background cloud material from a kinematic viewpoint and to investigate the potential influence of large-scale external effects on the formation of the filament. ]{} [To examine whether the B211/B213 filament is accreting background gas due to its gravitational potential, we produced a toy accretion model and compared its predictions to the velocity patterns observed in $^{12}$CO (1–0) and $^{13}$CO (1–0). We also examined the spatial distributions of H${\alpha}$, $Planck$ 857 GHz dust continuum, and HI emission to search for evidence of large-scale external effects.]{} [We estimated the depth of the Taurus cloud around the B211/B213 filament to be $\sim$0.3–0.7 pc under the assumption that the density of the gas is the same as the critical density of $^{13}$CO (1–0). Compared to a linear extent of > 10 pc in the plane of the sky, this suggests that the 3D morphology of the cloud surrounding the B211/B213 filament is sheet-like. Position-velocity ($PV$) diagrams observed in $^{12}$CO (1–0) and $^{13}$CO (1–0) perpendicular to the filament axis show that the emission from the gas surrounding B211/B213 is redshifted to the northeast of the filament and blueshifted to the southwest, respectively, and that the velocities of both components approach the velocity of the B211/B213 filament as the line of sight approaches the crest of the filament. The $PV$ diagrams predicted by our accretion model are in good agreement with the observed $^{12}$CO (1–0) and $^{13}$CO (1–0) $PV$ diagrams, supporting the scenario of mass accretion into the filament proposed by Palmeirim et al. Moreover, inspection of the spatial distribution of the H$\alpha$ and $Planck$ 857 GHz emission in the Taurus-California-Perseus region on scales up to >200 pc suggests that the B211/B213 filament as a result of an expanding supershell generated by the Per OB2 association. ]{} [Based on these results, we propose a scenario in which the B211/B213 filament was initially formed by large-scale compression of HI gas and then is now growing in mass due to the gravitational accretion of ambient cloud molecular gas.]{}
Introduction {#Sect1}
============
{width="170mm"}
\[fig:3color\]
![Map of $^{12}$CO (1–0) optical depth derived from the @Goldsmith08 $^{12}$CO (1–0) and $^{13}$CO (1–0) data.[]{data-label="fig:tau"}](tau.jpg){width="90mm"}
The observations of the [*Herschel*]{} Gould Belt survey (HGBS) have revealed an omnipresence of parsec-scale filaments in molecular clouds and emphasized their importance for solar-type star formation [e.g. @Andre10; @Menshchikov10; @Arzoumanian11; @Palmeirim13]. In particular, most $Herschel$ prestellar cores are found to lie in dense (thermally supercritical) filaments, suggesting that cores generally form by filament fragmentation [eg. @Konyves15; @Marsh16; @Benedettini18]. Molecular line observations of the velocity field around cores and filaments further support this view [@Tafalla15]. Based on the HGBS results, @Andre14 proposed a filament paradigm for star formation, whereby large-scale compression of interstellar material in supersonic flows generates a quasi-universal web of $\sim$0.1-pc wide filaments in the cold interstellar medium (ISM) and then the denser filaments fragment into prestellar cores by gravitational instability. Recently, @Shimajiri17 found that the star formation efficiency in dense molecular gas ($A_{\rm v}$ > 8), where filamentary structures dominate the mass budget, is remarkably uniform over a wide range of scales from 1-10 pc to >10 kpc [see also, @Gao04; @Lada10; @Lada12; @Chen15]. Furthermore, @Shimajiri17 proposed that this common star formation efficiency in dense gas results from the microphysics of star formation in filaments [see also @Andre14]. This result suggests the existence of a universal “star formation law” converting dense molecular gas into stars along filaments. Therefore, unveiling how molecular filaments grow in mass and fragment is crucial to understanding star formation in filaments.
The B211/B213 filament system is located in the Taurus molecular cloud, which is one of the nearest star-forming regions [$d$$\sim$140 pc, @Elias78]. Wide-field mapping observations in $^{12}$CO, $^{13}$CO, C$^{18}$O, N$_2$H$^+$, and SO emission revealed a whole network of filamentary structures in the B211/B213 area [@Goldsmith08; @Hacar13; @Panopoulou14; @Tafalla15]. @Goldsmith08 and @Palmeirim13 found that many low-density striations are elongated parallel to the magnetic field, and that blueshifted and redshifted components in both $^{12}$CO (1–0) and $^{13}$CO (1–0) emission are distributed to the southwest and the northeast of the B211/B213 filament, respectively, as shown in Fig. \[fig:3color\]. This morphology was suggestive of mass accretion along magnetic field lines into the B211/B213 filament. To quantify mass accretion, @Palmeirim13 assumed cylindrical geometry and used the observed mass per unit length $M_{\rm line}$ to estimate the gravitational acceleration $\phi$($R$) = 2$GM_{\rm line}/R$ of a piece of gas in free-fall toward the filament (where $R$ and $G$ denote radius from filament center and the gravitational constant, respectively). The free-fall velocity of gas initially at rest at a cylindrical radius $R_{\rm init}\sim$2 pc was estimated to reach 1.1 km s$^{-1}$ when the material reached the outer radius $R_{\rm out}\sim$0.4 pc of the B211/B213 filament. This estimation was consistent with the velocity observed in CO, suggesting that the background gas accretes into the B211/B213 filament owing to the gravitational potential of the B211/B213 filament. However, the velocity structure was not investigated in detail. Investigation of the velocity structure is crucial to confirm this suggested scenario from the kinematic viewpoint. This is the topic of the present paper.
The paper is organized as follows: in Sect. \[Sect2\], we describe the $^{12}$CO (1–0) and $^{13}$CO (1–0) data, as well as complementary H${\alpha}$, 857 GHz, and HI data. In Sect. \[Sect3\], we estimate the optical depth of the $^{12}$CO (1–0) line and present the $^{12}$CO (1–0) and $^{13}$CO (1–0) velocity structures observed in the B211/B213 cloud. In Sect. \[Sect4\], we discuss the cloud structure, whether the surrounding material accretes onto the B211/B213 filament from the kinematic viewpoint, and whether the filament is formed by large-scale compression. In Sect. \[Sect5\], we summarize our results.
{width="155mm"} {width="155mm"}
Observational data {#Sect2}
==================
In this paper, we used the $^{12}$CO (1–0) and $^{13}$CO (1–0) data obtained by @Goldsmith08 [@Narayanan08] with the 14 m diameter millimeter-wave telescope of the Five College Radio Astronomy Observatory (FCRAO). The half-power beam width of the telescope was 45$\arcsec$ for $^{12}$CO (1–0) and 47$\arcsec$ for $^{13}$CO (1–0). We applied Gaussian spatial smoothing to improve the signal to noise ratio, resulting in an effective beam resolution of $\sim$76$\arcsec$, corresponding to $\sim$0.05 pc at a distance of 140 pc. The velocity resolution of the data is 0.26 km s$^{-1}$ for $^{12}$CO (1–0) and 0.27 km s$^{-1}$ for $^{13}$CO (1–0). The rms noise level is 0.1 K ($T_{\rm A}^*$) for $^{12}$CO (1–0) and 0.05 K ($T_{\rm A}^*$) for $^{13}$CO (1–0), respectively.
As complementary observations of the Taurus cloud region and its large-scale environment, we also used the H${\alpha}$ data[^1] of @Finkbeiner03, as well as $Planck$ 857 GHz[^2] [@Planck14] and HI data[^3] [@Kalberla17] from the archive.
{width="190mm"}
\[fig:model\]
Analysis and results {#Sect3}
====================
$^{12}$CO (1–0) and $^{13}$CO (1–0) optical depths \[optical\_depth\]
---------------------------------------------------------------------
The optical depth of the $^{12}$CO (1–0) line was estimated from the FCRAO $^{12}$CO and $^{13}$CO data. Assuming the same excitation temperature for the $^{12}$CO (1–0) and $^{13}$CO (1–0) lines, an isotopic ratio, $R_{\rm i}$ = 62 for $^{12}$C/$^{13}$C [@Langer93], and the same beam filling factor in both lines, we evaluated the optical depth of $^{12}$CO (1–0) using the following equation:
$$\frac{T({\rm ^{13}CO)}}{T({\rm ^{12}CO})}=\frac{1-e^{-\tau({\rm ^{12}CO})/R_{\rm i}}}{1-e^{-\tau(\rm{^{12}CO})}}.$$
Here, $T(\rm{^{12}CO})$ and $\tau(\rm{^{12}CO})$ denote the peak intensity and the optical depth of $^{12}$CO (1–0), respectively. While the $^{12}$CO (1–0) emission preferentially traces the diffuse extended cloud, the $^{13}$CO (1–0) emission traces the central B211/B213 filament (see Fig. \[fig:3color\]). The typical inner width of the filaments observed with [*Herschel*]{} is $\sim$0.1 pc [@Arzoumanian11; @Arzoumanian18; @Palmeirim13], which is larger than the 0.05 pc effective beam size of the FCRAO data. Thus, assuming the same beam filling factor in $^{12}$CO (1–0) and $^{13}$CO (1–0) is reasonable. Figure \[fig:tau\] displays the resulting map of $^{12}$CO (1–0) optical depth. The optical depth in this map ranges from $\sim$3 to $\sim$300, showing that the $^{12}$CO (1–0) emission is optically thick. In particular, the $^{12}$CO (1–0) optical depth toward the B211/B213 filament itself ($\tau(\rm ^{12}CO)$$\sim$100) is much larger than that found for the surrounding lower density material ($\tau(\rm ^{12}CO)$$\sim$20).
$^{12}$CO (1–0) and $^{13}$CO (1–0) velocity channel maps {#section:channel}
---------------------------------------------------------
Figure \[fig:channel\_co\] shows the velocity channel maps observed in $^{12}$CO (1–0) and $^{13}$CO (1–0). In the maps for $V_{\rm LSR}$ < 3.7 km s$^{-1}$, both $^{12}$CO (1–0) and $^{13}$CO (1–0) emission is seen in the northeastern part of the maps (RA, DEC = 4:24:00, 28:15:00). In the channel maps for 4.0 < $V_{\rm LSR}$ < 7.3 km s$^{-1}$, enhanced emission is seen toward the B211/B213 filament in both $^{12}$CO (1–0) and $^{13}$CO (1–0). The emission at these velocities is likely to be directly associated with the B211/B213 filament. Furthermore, while the emission at 4 km s$^{-1}$ < $V_{\rm LSR}$ < 6 km s$^{-1}$ is distributed to the southwest of the B211/B213 filament, the emission at 6 km s$^{-1}$ < $V_{\rm LSR}$ < 7 km s$^{-1}$ is distributed to the northeast of the filament. In the channel maps for $V_{\rm LSR}$ > 7.3 km s$^{-1}$, the distribution of the $^{12}$CO (1–0) and $^{13}$CO (1–0) emission is suggestive of an arc-like structure around L1495. Figure \[fig:3color\] ($right$) is a sketch showing the location of each velocity component.
Modeling of the data and discussion {#Sect4}
===================================
3D morphology of the B211/B213 ambient cloud {#sect:cloud}
--------------------------------------------
Here, we discuss the 3D morphology of the material surrounding the B211/B213 filament by comparing the extent of the gas in the plane of the sky and its depth along the line of sight. Hereafter, we refer to the system consisting of the B211/B213 filament and its surrounding gas as the B211/B213 cloud (i.e. red, green, and dark blue areas in Fig. \[fig:3color\] ($right$)).
The projected extent of the B211/B213 cloud in the plane of the sky is more than $\sim$10 pc. Taking the viewing angle into account, the real extent of the cloud may be larger. At the same time, we can estimate the depth of the cloud along the line of sight under the assumption that the surrounding material is filled by gas with density exceeding the critical density of the $^{13}$CO (1–0) line, since $^{13}$CO (1–0) emission is observed over the entire mapped area. The critical density of $^{13}$CO (1–0), $n_{\rm critical}^{\rm ^{13}CO}$, may be estimated as follows:
$$n_{\rm critical}^{\rm ^{13}CO} = \frac{A_{\rm ul}}{\sigma_{\rm cross} \nu} = \frac{A_{\rm ul}}{10^{-15}{\rm cm}^{-2} \times 10^4 \sqrt{T_{\rm ex}}},$$
where $A_{\rm ul}$, $\sigma_{\rm cross}$, $\nu$, and $T_{\rm ex}$ are the Einstein spontaneous emission coefficient, collision cross section, collision velocity, and line excitation temperature. The values of $A_{10}$ and $\sigma_{\rm cross}$ in the LAMDA database[^4] are 6.294$\times$10$^{-8}$ s$^{-1}$ and 10$^{-15}$ cm$^{-2}$. The collision velocity can be calculated as $v$=$\sqrt{3k_{\rm B}T_{\rm ex}/m}$ = $10^{4}\sqrt{T_{\rm ex}}$ cm s$^{-1}$, where $k_{\rm B}$ is the Boltzmann constant and $m$ is hydrogen molecular mass. This leads to a value of 1.7 $\times$ 10$^3$ cm$^{-3}$ for the critical density of $^{13}$CO (1–0) assuming $T_{\rm ex}$ $\simeq$ 14 K. Here, we assumed that the excitation temperature $T_{\rm ex}$ of the $^{13}$CO (1–0) line is the same as the dust temperature $T_{\rm dust}$ $\sim$ 14K derived by @Palmeirim13 from ${\it Herschel}$ data in the ambient cloud around B211/B213 (red and dark blue area in Fig. \[fig1\] ($right$)). @Palmeirim13 also estimated the mean [*Herschel*]{} column density in the material surrounding the B211/B213 filament to be $N_{{\rm H}_2}$ $\simeq$ 1.4 $\times$ 10$^{21}$ cm$^{-2}$. Thus, the depth of the cloud (=$N_{{\rm H}_2}/n_{\rm critical}^{\rm ^{13}CO}$) is estimated to be 0.3 pc.
Recently, @Qian15 independently estimated the depth of the whole Taurus molecular cloud and found a value of $\sim$0.7 pc using the core velocity dispersion (CVD) method. With a projected extent of more than 10 pc and a depth of $\sim$0.3 – 0.7 pc, we conclude that the 3D morphology of the cloud resembles a sheet-like structure (see Fig. \[fig:cloud\_structure\]).
![Schematic picture of the definition of velocity components associated with the B211/B213 filament. The spectrum in each panel is the $^{13}$CO (1-0) spectrum averaged over a 15$\arcmin$ $\times$ 15$\arcmin$ area with a center position indicated in the top-left corner. The velocity components with a velocity of < 4.0 km s$^{-1}$ or > 7.0 km s$^{-1}$ are regarded as components not associated with the B211/B213 filament. These components are subtracted from the data cube.[]{data-label="fig:subt"}](sample_spectrum_A.jpg "fig:"){width="80mm"} ![Schematic picture of the definition of velocity components associated with the B211/B213 filament. The spectrum in each panel is the $^{13}$CO (1-0) spectrum averaged over a 15$\arcmin$ $\times$ 15$\arcmin$ area with a center position indicated in the top-left corner. The velocity components with a velocity of < 4.0 km s$^{-1}$ or > 7.0 km s$^{-1}$ are regarded as components not associated with the B211/B213 filament. These components are subtracted from the data cube.[]{data-label="fig:subt"}](sample_spectrum_B.jpg "fig:"){width="80mm"}
{width="190mm"}
Accretion of background gas into the B211/B213 filament {#sect:accretion}
-------------------------------------------------------
Here, we compare the velocity pattern seen in $^{12}$CO (1–0) and $^{13}$CO (1–0) emission with the prediction of an accretion gas model, in order to investigate whether the B211/B213 filament accretes ambient cloud gas from a kinematic viewpoint.
### Observed position-velocity diagrams {#section:pv_discription}
As mentioned in Sect. \[section:channel\], the highly blueshifted and redshifted components at $V_{\rm LSR}$ < 3.7 km s$^{-1}$ and $V_{\rm LSR}$ > 7.3 km s$^{-1}$ do not seem to be directly connected to the B211/B213 cloud/filament. To focus on the velocity field of the gas associated with the B211/B213 filament, we subtracted these two components as follows. We applied Gaussian fitting with $N$ Gaussian components to each pixel, where $N$=1, 2, 3, 4, or 5. Wherever the signal to noise (S/N) ratio of the residual peak intensity was less than 5, the fit was deemed to be acceptable and the corresponding spectrum was assumed to consist of $N$ Gaussian components. Then, if the peak LSR velocity of a Gaussian component was lower than 4.0 km s$^{-1}$ or higher than 7.0 km s$^{-1}$, the component was not considered to be associated with the B211/B213 filament or cloud and was subtracted from the data cube (see also Fig. \[fig:subt\] and Fig. \[fig:fitting\]). Figure \[fig:channel\_subt\] displays the $^{12}$CO (1–0) and $^{13}$CO (1–0) velocity channel maps after subtracting these components. Hereafter, we used these subtracted data cubes.
Figure \[fig:pv\] shows the resulting position-velocity ($PV$) diagrams in $^{12}$CO (1–0) and $^{13}$CO (1–0) along a direction perpendicular to the B211/B213 filament as indicated in Fig. \[fig:3color\]. On these $PV$ diagrams, distinct velocity pattern can be recognized in $^{12}$CO (1–0) and $^{13}$CO (1–0) toward the filament (|offset| < 10$\arcmin$ $\sim$ 0.4 pc). This is probably due to differing optical depths in the two lines. As described in Sect. \[optical\_depth\], the $^{12}$CO (1–0) optical depth toward the filament is $>$ 50 and much larger than the optical depth toward the outskirts of the filament, suggesting that the $^{12}$CO (1–0) emission only traces the surface of the filament. In the outskirts of the B211/B213 filament (|offset| > 10$\arcmin$), the blueshifted emission is distributed to the southwest (offset > 0$\arcmin$) and the redshifted emission is distributed to the northeast (offset < 0$\arcmin$) of the filament. It can be seen that the velocities of the blueshifted and redshifted components approach the velocity of the B211/B213 filament as the offset approaches 0 (i.e. the crest of the filament).
### Gas accretion model {#section:model}
Component Parameter
----------- ------------------------------ ----------------------------------------- --
$M_{\rm line}$ 54 $M_{\odot}$ $^{\dag}$
$n_{\rm H_2}^0$ 4.5$\times$10$^{4}$ cm$^{-3}$ $^{\dag}$
$p$ 2 $^{\dag}$
$R_{\rm flat}$ 0.03 pc $^{\dag}$
$R_{\rm out}$ 0.4 pc $^{\dag}$
$\mathcal{V}_{\rm filament}$ 6.2 km s$^{-1}$ $^{\ddag}$
$\mathcal{V}_{\rm init,N}$ 6.8 km s$^{-1}$ $^{\ddag}$
$R_{\rm init,N}'$ 10 pc
$\theta_{\rm N}$ 70 deg
$\mathcal{V}_{\rm init,S}$ 4.4 km s$^{-1}$ $^{\ddag}$
$R_{\rm init,S}'$ 10 pc
$\theta_{\rm S}$ 20 deg
: []{data-label="Table1"}
The $PV$ diagrams in Fig. \[fig:pv\] show an asymmetric velocity distribution on either side of the 0 position (filament crest), suggesting that the sheet-like ambient cloud surrounding the B211/B213 filament has a different inclination to the plane of the sky to the northeast and the southwest of the filament. To investigate whether the B211/B213 filament accretes gas from the ambient cloud, we thus constructed a 3-component toy model (one filament component and two components for the northeastern and southwestern sheets) under the assumption that the sheet components to the northeast (red-shifted) and the southwest (blues-shifted) lie on the near and far sides of the B211/B213 filament, respectively, as shown in Fig. \[fig:model\]. Our modeling procedure is summarized in the schematic picture shown in Fig. \[fig:modeling\_flow\].
### $\bullet$ Model for the central filament component {#bullet-model-for-the-central-filament-component .unnumbered}
First, we produced a model for the filament. [*Herschel*]{} observations of nearby clouds have shown that the radial column density profiles of molecular filaments in the radial direction $R'$ (i.e. perpendicular to the filament crest) can be well described by the following “Plummer-like” function [@Arzoumanian11; @Palmeirim13]:
$$\begin{aligned}
\begin{tiny}
\begin{split}
\Sigma_p(R')/\mu m_{\rm H} & = \frac{N_{\rm H_2}^0}{[1+(R'/R_{\rm flat})^2]^\frac{p-1}{2}}
&\to
\rho_{p}(R') = \frac{\rho_{\rm c}}{[1+(R'/R_{\rm flat})^2]^\frac{p}{2}} ,
\end{split}
\end{tiny}\end{aligned}$$
where $\rho_{\rm c}$, $\Sigma_p$, $\mu$, $m_{\rm H}$, $N_{\rm H_2}^0$, $p$, and $R_{\rm flat}$ are the central density of the filament, the column density as a function of radius $R'$, the mean molecular mass, the hydrogen atom mass, the central column density, the index of the power-law density profile at large radii ($R'$ $\gg$ $R_{\rm flat}$), and the radius of the flat inner region, respectively. For the B211/B213 filament, we adopted $N_{\rm H_2}^0$=1.4$\times$10$^{21}$ cm$^{-2}$, $p$=2.0, and $R_{\rm flat}$=0.03 pc from the fitting results of @Palmeirim13. We assumed that the filament itself lies in the plane of the sky and that the shape of the intensity profile of the B211/B213 filament as traced in $^{12}$CO (1–0) and $^{13}$CO (1–0) emission is the same as that found in the [*Herschel*]{} column density map. Then, we rescaled the peak integrated intensity to be 2 K km s$^{-1}$ as observed in $^{13}$CO (1–0).
Approximating the Plummer density profile of the filament by a broken power-law, the gravitational potential in the radial direction $R'$ can be expressed as follows[^5] [cf. @Hennebelle13]:
$$\begin{tiny}
\phi(R') =
\begin{cases}
G \rho_{\rm flat} \pi R'^2 & \text{for $R'$ $\le$ $R_{\rm flat}$} \\
G M_{\rm line,flat} \left[1 + 2 \ln\left(\frac{R'}{R_{\rm flat}}\right) + 2 \left(\ln\frac{R'}{R_{\rm flat}}\right)^2 \right] & \text{for $R_{\rm flat}$ < $R'$ $\le$ $R_{\rm out}$} \\
G M_{\rm line,flat} \left[1 + 2 \ln \left(\frac{R_{\rm out}}{R_{\rm flat}}\right) + 2 \left(\ln\frac{R_{\rm out}}{R_{\rm flat}} \right)^2 \right] \\
\qquad +2G M_{\rm line} \ln \left(\frac{R'}{R_{\rm out}}\right) & \text{for $R_{\rm out}$ < $R'$}
\end{cases}
\end{tiny}$$
where $\rho_{\rm flat}$ and $R_{\rm out}$ are the density of the filament at $R'$ $\le$ $R_{\rm flat}$ and outer radius of the filament, respectively. We adopted $n_{\rm H_2}^0$=$\rho_{\rm flat}$/$\mu m_{\rm H}$ = $4.5 \times 10^4 \ {\rm cm}^{-3}$, $R_{\rm flat}$ = 0.03 pc, and $R_{\rm out}$ = 0.4 pc from @Palmeirim13 as summarized in Table \[Table1\]. In the above equation, $M_{\rm line,flat}$ and $M_{\rm line}$ represent the inner and total masses per unit length of the filament and are given by:
$$\begin{aligned}
M_{\rm line,flat} = \rho_{\rm flat} \pi R_{\rm flat}^2 \end{aligned}$$
$$\begin{aligned}
M_{\rm line} = \rho_{\rm flat} \pi R_{\rm flat}^2 \left[1+2 \ln\left( \frac{R_{\rm out}}{R_{\rm flat}}\right)\right]\end{aligned}$$
### $\bullet$ Models for the northeastern and southwestern sheet components {#bullet-models-for-the-northeastern-and-southwestern-sheet-components .unnumbered}
Second, we produced models for the northeastern and southwestern sheet components assuming that the B211/B213 filament accretes the gas of the sheets as a result of its gravitational potential.
Taking into account the pressure gradient force, conservation of energy for a parcel of unit mass of the ambient cloud falling onto the central filament may be expressed as follows [cf. @Smith94; @Smith12]:,
$$\begin{aligned}
\begin{split}
\scriptsize
\frac{1}{2}(\mathcal{V}_{\rm init,N/S}'^0)^2 \mathalpha{+}
& \phi(R_{\rm init,N/S}') \mathalpha{+} C_{\rm s,eff}^2 \ln(\rho_{\rm init}) \\
&\mathalpha{=}
\frac{1}{2}\mathcal{V}(R')^2 \mathalpha{+} \phi(R') \mathalpha{+} C_{\rm s,eff}^2 \ln(\rho(R')),
\end{split}\end{aligned}$$
where $\mathcal{V}(R)$ is the projected velocity and $C_{\rm s,eff}$ is the effective sound speed. The projected velocity $\mathcal{V}(R)$ of the gas flow can thus be expressed as follows:
$$\begin{aligned}
\scriptsize
\mathcal{V}(R)\mathalpha{=}\mathcal{V}_{\rm filament} \mathalpha{\pm} \sqrt{2\left[\frac{1}{2}(\mathcal{V}_{\rm init,N/S}'^0)^2 \mathalpha{+} \phi(R_{\rm init,N/S}')\mathalpha{-}\phi(R') \mathalpha{+} C_{\rm s,eff}^2 \ln \left(\frac{\rho_{\rm init}}{\rho(R')}\right)\right]} \mathalpha{\times} \cos(\theta_{\rm N/S}),\end{aligned}$$
where $\mathcal{V}_{\rm filament}$, $\mathcal{V}_{\rm init,N/S}'^0$, $R_{\rm init,N/S}'$, and $\rho_{\rm init}$ are the systemic velocity of the filament, the velocity of the accreting gas at the initial point corrected for inclination, the initial radius of the accreting gas corrected for inclination, and the volume density at the initial point, respectively. Here, we define ${\mathcal{V}_{\rm init,N/S}'^0}$ as $(\mathcal{V}_{\rm init,N/S} - \mathcal{V}_{\rm filament} )/\cos(\theta_{\rm N/S})$, where $\mathcal{V}_{\rm init,N/S}$ is the projected velocity of the northeastern/southwestern sheet component at $R_{\rm init,N/S}'$. We adopted $C_{\rm s,eff}$ = 0.9 km s$^{-1}$ from $C_{\rm s,eff} \simeq \delta V_{\rm FWHM}(^{12}{\rm CO})/\sqrt{8\ln2}$, where $\delta V_{\rm FWHM}(^{12}{\rm CO})$ is the $^{12}$CO (1-0) line width (=2.1 km s$^{-1}$) observed toward the B211/B213 filament. Wherever the value of $C_{\rm s,eff}^2 \ln \left(\frac{\rho_{\rm init}}{\rho(R')}\right)$ was larger than $\frac{1}{2}{(\mathcal{V}_{\rm init,N/S}'^0)^2} + \phi(R_{\rm init,N/S}')-\phi(R')$, we adopted $\mathcal{V}(R) = \mathcal{V}_{\rm filament}$. The $Herschel$ observations show that the density profile of the B211/B213 filament is proportional to $R'^{-2}$ at $R'$ $\le$ $R'_{\rm out}$ and has a shallower slope at $R'$ $\ge$ $R'_{\rm out}$ [@Palmeirim13]. Furthermore, the slope for the southwestern sheet component is slightly steeper than the slope for the northeastern sheet component. At $R'$ > $R'_{\rm init}$, the gas density in the model was assumed to be constant.
To summarize, we assumed the following density distribution as a function of radial direction $R'$ (see Fig. \[fig:density\]):
For the northeastern sheet component,
$$\begin{aligned}
\footnotesize
\rho(R') =
\begin{cases}
\rho_{\rm flat} &\text{for $R'$ $\le$ $R_{\rm flat}$} \\
\rho_{\rm flat} \left(\frac{R'}{R_{\rm flat}}\right)^{-2} &\text{for $R_{\rm flat}$ < $R'$ $\le$ $R'_{\rm out}$} \\
\rho_{\rm flat} \left(\frac{R_{\rm out}}{R_{\rm flat}}\right)^{-2}\left(\frac{R'}{R_{\rm out}}\right)^{-1.0} &\text{for $R'_{\rm out}$ < $R'$ $\le$ $R'_{\rm init}$} \\
{\rm constant} = \rho_{\rm flat} \left(\frac{R_{\rm out}}{R_{\rm flat}}\right)^{-2}\left(\frac{R'_{\rm init}}{R_{\rm out}}\right)^{-1.0} & \text{for $R'_{\rm init}$ < $R'$} \\
\end{cases}\end{aligned}$$
For the southwestern sheet component,
$$\begin{aligned}
\footnotesize
\rho(R') =
\begin{cases}
\rho_{\rm flat} &\text{for $R'$ $\le$ $R_{\rm flat}$} \\
\rho_{\rm flat} \left(\frac{R'}{R_{\rm flat}}\right)^{-2} &\text{for $R_{\rm flat}$ < $R'$ $\le$ $R'_{\rm out}$} \\
\rho_{\rm flat} \left(\frac{R_{\rm out}}{R_{\rm flat}}\right)^{-2}\left(\frac{R'}{R_{\rm out}}\right)^{-1.5} &\text{for $R'_{\rm out}$ < $R'$ $\le$ $R'_{\rm init}$} \\
{\rm constant} = \rho_{\rm flat} \left(\frac{R_{\rm out}}{R_{\rm flat}}\right)^{-2}\left(\frac{R'_{\rm init}}{R_{\rm out}}\right)^{-1.5} & \text{for $R'_{\rm init}$ < $R'$} \\
\end{cases}\end{aligned}$$
![Assumed density profile for the 3-component model. Red and blue lines indicate the densities for the northeastern and southwestern sheet components, respectively.[]{data-label="fig:density"}](density_profile.jpg){width="95mm"}
We also assumed that both sheet components have integrated intensities of $\sim$1 K km s$^{-1}$ as observed in $^{13}$CO (1-0). To get a good agreement between the models and the observations (see Appendix \[appendix:inclination\]), we adopted $\theta_{\rm N}$ = 70$^{\circ}$, $\mathcal{V}_{\rm init,N}$ = 6.8 km s$^{-1}$, and $R_{\rm init,N}'$ = 10 pc for the northeastern sheet component and $\theta_{\rm S}$ = 20$^{\circ}$, $\mathcal{V}_{\rm init,S}$ = 4.4 km s$^{-1}$, and $R_{\rm init,S}'$ = 10 pc for the southwestern sheet component. The parameters of our model are summarized in Table \[Table1\].
### $\bullet$ Combined 3-component model {#bullet-combined-3-component-model .unnumbered}
We first generated an integrated intensity distribution and a peak velocity field for each of the three components with IDL (Interactive Data Language). Using the MIRIAD task $velimage$[^6], we then produced individual data cube components for the filament and the two sheet components assuming uniform velocity dispersions of 1.3 km s$^{-1}$ for the filament and 0.9 km s$^{-1}$ for the sheet components. The velocity dispersions were obtained from fitting the observed $^{13}$CO (1–0) spectra.
Finally, we used IDL to co-add the three individual data-cube components and produce a combined model data cube.
### $\bullet$ Large-scale kinematic model {#bullet-large-scale-kinematic-model .unnumbered}
We adopted initial velocities (corrected for inclination) of [$\mathcal{V}_{\rm init,N}'^0$ = 1.8 km s$^{-1}$ $(=[\mathcal{V}_{\rm init,N} - \mathcal{V}_{\rm filament}]/\cos(\theta_{\rm N}))$]{} for the northeastern sheet component and [$\mathcal{V}_{\rm init,S}'^0$ = $-$1.9 km s$^{-1}$ $(=[\mathcal{V}_{\rm init,S} - \mathcal{V}_{\rm filament}]/\cos(\theta_{\rm S}))$]{} for the southwestern sheet component. This almost symmetric initial velocity pattern after correction for inclination is suggestive of gravitational accretion. If the accreting gas comes from far away positions $R_{\rm far,N/S}'$ ($\gg$ $R_{\rm init}'$) and is accelerated by the gravitational potential of the B211/B213 filament/cloud, the line of sight velocity at $R_{\rm far,N/S}'$ is likely to be similar to $\mathcal{V}_{\rm filament}$. The positions $R_{\rm far,N/S}'$ for the northeastern and southwestern sheet components can be estimated from the equation of $\mathcal{V}_{\rm init,N/S}'=\sqrt{2[\phi(R_{\rm far,N/S}')-\phi(R_{\rm init,N/S}')]} \times \cos(\theta_{\rm N/S})$ since the pressure density gradient is probably small and can be neglected. We adopted $R_{\rm init,N}'$ = 10 pc, and $R_{\rm init,S}'$ = 10 pc, respectively. Thus, assuming that the initial velocities are entirely generated by gravitational acceleration, the surrounding gas for the northeastern and southwestern sheet components would have to come from $R_{\rm far,N}'$ ($R_{\rm far,N}$) = 270 (260) pc and $R_{\rm far,S}'$ ($R_{\rm far,S}$) = 520 (180) pc (see Fig. \[fig:pv\_large\]). Here, for simplification, we did not include the mass of the sheets when estimating the gravitational potential. Thus, these $R_{\rm far,N/S}'$ values should be considered upper limits. The HI emission observed at $V_{\rm LSR}$ $\sim$ 6 km s$^{-1}$, which corresponds to the systemic velocity of the B211/B213 filament, has an extended emission with an extent of several $\times$ 100 pc which is consistent with the above value of $R_{\rm far,N}'$ (see also Sect. \[Sect:compress\] and Fig. \[fig:channel\_hi\]). Thus, one of the reasons why the initial velocities at $R_{\rm init,N/S}'$ in the northeastern and southwestern sheet components differ from the velocity of the filament may be the large-scale effect of the gravitational potential of the B211/B213 cloud/filament. We will discuss another possible explanation in Sect. \[Sect:compress\].
### Comparing the combined model with the observations {#section:comp_model}
The synthetic $PV$ diagram predicted by the model is shown in Fig. \[fig:pv\] ($c)$ for comparison with the $PV$ diagrams observed in $^{12}$CO (1–0) and $^{13}$CO(1–0) (Figs. \[fig:pv\]($a$) and \[fig:pv\]($b$)). A good quantitative agreement especially with the $^{13}$CO (1–0) diagram can be seen. In particular, in both the model and the observed $PV$ diagrams, the velocity of the gas surrounding the B211/B213 filament (red and dark blue area in Fig. \[fig:3color\]($right$)) approaches the systemic velocity of the B211/B213 filament as the positional offset approaches 0 (i.e. the filament crest). While the gas is accelerated by the gravitational potential of the filament/cloud at large scales (several$\times$10 pc), it is decelerated by the pressure gradient force of the dense filament at small scales (several pc) (see Fig. \[fig:pv\_large\] and Fig. \[fig:ob2\]). The good agreement between the model and the data indicates that observational kinematic constraints are consistent with the B211/B213 filament accreting background cloud material as a result of its gravitational potential. This provides strong support to the scenario of mass accretion along magnetic field lines into the filament proposed by @Palmeirim13. The mass accretion rate onto the B211/B213 filament was estimated to be 27-50 $M_{\odot}$ pc$^{-1}$ Myr$^{-1}$ by @Palmeirim13, suggesting that it took $\sim$1–2 Myr to form the filament. Thus, accretion of gas from the ambient cloud in B211/B213 likely plays a key role in the evolution of the filament.
![Position-velocity diagram of the model for the large scale. $\theta_{\rm N}$=70$^{\circ}$ and $\theta_{\rm S}$=20$^{\circ}$ are assumed. Fig. \[fig:pv\] corresponds to -2.4 pc < offset < 1.6 pc in this figure. The vertical dashed lines mark $R_{\rm init,N}$ = -9.4 pc ($R'_{\rm init,N}$ = 10 pc) and $R_{\rm init,S}$ = 3.4 pc ($R'_{\rm init,S}$ = 10 pc).[]{data-label="fig:pv_large"}](large_scale_pv.jpg){width="80mm"}
Formation of the B211/B213 filament by large-scale compression {#Sect:compress}
--------------------------------------------------------------
![ Schematic picture of the relation between the B211/B213 cloud and Per OB2 association. The black arrows indicate the line of sight. The horizontal line indicate the sky plane. Red and blue arrows indicate the direction of the gas accretion in the northeastern and southwestern sheet components, respectively. Green arrows indicate the direction of the compression by Per OB2 association. $\theta_{\rm N}$ and $\theta_{\rm S}$ are the inclination angles of the northeastern and southwestern sheet components to the line of sight. Red and blue arrows of length scaling quantitatively with the magnitude velocity field indicate the direction of the acceleration flow of ambient cloud material. []{data-label="fig:ob2"}](B211-OB_ver2.png){width="80mm"}
![Distributions of the H${\alpha}$ [color, @Finkbeiner03] and 857 GHz dust [grey, @Planck14] emission. The units of the H${\alpha}$ and 857 GHz maps are R (Rayleigh, 4$\pi$ $\times$ 10$^{-4}$ photons cm$^{-2}$ s$^{-1}$ sr$^{-1}$) and MJy str$^{-1}$, respectively. The magenta dashed circle indicates a HI supershell [@Lim13]. The diameter of the HI supershell might be $>$ 200 pc since the distances to the Taurus and Perseus clouds are 140 pc and 340 pc, respectively. The distribution of HI emission is shown in Figs. \[fig:channel\_hi\] and \[fig:3color\_hi\]. See also \[fig:halpha\_appendix\].[]{data-label="fig:halpha"}](planck_halpha_b2.jpg){width="90mm"}
As described in Sect. \[sect:accretion\], we adopted different inclinations for the northeastern sheet component ($i$=70$^{\circ}$) and for the southwestern sheet component ($i$=20$^{\circ}$) in our model to get a good agreement with the observations. This suggests that the B211/B213 cloud is actually shaped like a shell (see Fig. \[fig:ob2\]). One possibility is that this shell-like structure was produced by large-scale compression.
In this section, we try to investigate whether the cloud surrounding the B211/B213 filament is affected by large-scale flow phenomena using wide-field H${\alpha}$ maps tracing gas ionized by massive stars [@Finkbeiner03], the $Planck$ 857 GHz dust continuum map tracing cold dust [@Planck14], and HI map tracing lower density atomic gas [@HI4PI16].
Figure \[fig:halpha\] (see also Figs \[fig:channel\_hi\] and \[fig:3color\_hi\]) compare the spatial distributions of the H${\alpha}$ and 857 GHz emission in the Taurus-California-Perseus region (e.g. Taurus, Auriga, California, and Perseus). The 857 GHz dust emission traces each molecular cloud and exhibits a hole-like structure. This hole-like structure can also be seen in HI emission as shown in Fig. \[fig:channel\_hi\] and Fig. \[fig:3color\_hi\]. The H${\alpha}$ emission fills the hole-like structure seen in the 857 GHz dust emission near the center of the field. The Taurus, California, and Perseus molecular complexes traced by the 857 GHz dust emission are distributed at the edge of the hole-like structure. @Lim13 also found evidence of a shell-like structure using dust extinction and $^{12}$CO (1–0) maps. The hole-like structure may result from the expansion of a large-scale supershell produced by a supernova in the Per OB2 association that compresses the Taurus cloud from the far side [@Olano87; @Bally08]. An H${\alpha}$ absorption feature is detected toward the Taurus cloud (see Fig. \[fig:halpha\] and Fig. \[fig:halpha\_appendix\]), suggesting that the Taurus cloud lies at the front surface of the large-scale supershell produced by the Per OB2 association. The distance to the Per OB2 association is estimated to be 340 pc from the Sun [@Cernis93], while the distance to the Taurus cloud is $\sim$140 pc [@Elias78]. These distances are consistent with the Taurus cloud lying in front of the Per OB2 association. The B211/B213 filament also appears to be in front of the HI shell [see Fig. 10 in @Chapman11]. This morphology suggests that the B211/B213 filament may have formed as a result of an expanding supershell. This may provide another reason for the different initial gas velocities differed for the northeastern and southwestern sheet components besides large-scale acceleration by the gravitational potential of the B211/B213 cloud (see Sect. \[section:comp\_model\]). The Local Bubble surrounding the Sun might also compress the Taurus cloud from the opposite direction. The Local Bubble surrounding the Sun was produced by supernovae [@Snowden98; @Sfeir99] and the wall of the Local Bubble is located close to the Taurus cloud [@Konyves07; @Lallement14].
Interestingly, the $Planck$ 353 GHz data show variations in the polarization fraction (i.e., polarized intensity/total intensity) across the B211/B213 filament, with lower and higher polarization fractions in the southwestern and northeastern parts of the filament, respectively [see Fig. 10 in @Planck16]. If the gas surrounding the filament is shaped as a shell-like structure with an ordered magnetic field in the plane of each sheet component and if the southwestern sheet component is oriented closer to the line of the sight compared to the northeastern sheet component (cf. Fig. \[fig:model\]($right$)), the polarization fraction is expected to be lower in the southwestern area (dark blue in Fig. \[fig1\]($right$)) than in the northeastern area (red in Fig. \[fig1\]($right$)) assuming uniform dust grain properties. Moreover, both the polarization fraction and the polarization angle show smooth variations across the filament, which is consistent with the northeastern and southwestern sheets being curved (i.e., shell-like). The $Planck$ polarization results are therefore support the present model.
Using magnetic magnetohydrodynamic (MHD) numerical simulations, @Inutsuka15 and @Inoue18 have argued that multiple compressions associated with expanding bubbles can create star-forming filamentary structures within sheet-like molecular cloud. A similar model of anisotropic filament formation in shock compressed layers has been proposed by @Chen14, also based on MHD simulations. Such anisotropic filament formation model naturally account for transverse velocity gradients across the B211/B213 filament (see Fig. \[fig:3color\]) and other dense molecular filaments [@Dhabal18], and are good agreement with the observational picture presented.
Based on these considerations, we propose the following scenario for the formation and evolution of the B211/B213 filamentary system:
1. A large-scale flow associated with the Per OB2 supershell compressed and deformed the cloud centered on the B211/B213 filament and created a bent shell-like structure.
2. Owing to its strong gravitational potential, the B211/B213 filament is growing in mass due to accretion of background gas from the surrounding shell-like structure.
Conclusions {#Sect5}
===========
To examine whether the B211/B213 filament is accreting gas from the surrounding cloud, we investigated the velocity patterns observed in the $^{12}$CO (1–0) and $^{13}$CO (1–0) lines. Our main findings may be summarized as follows:
1. The optical depth of the $^{12}$CO (1–0) line was estimated to be $\sim$3–300. The $^{12}$CO optical depth toward the B211/B213 filament is much larger than that toward the outskirts of the filament. The position-velocity diagrams observed in $^{12}$CO (1–0) and $^{13}$CO (1–0) exhibit different velocity patterns close to the filament, which is likely due to different optical depths.
2. The $^{12}$CO (1–0) and $^{13}$CO (1–0) emission from the B211/B213 filament are seen at an LSR velocity of $\sim$6 km s$^{-1}$. In the northeastern and southwestern parts of the B211/B213 filament, the $^{12}$CO (1–0) and $^{13}$CO (1–0) emission are redshifted and blueshifted, respectively. The line of sight velocities are gradually approaching the systematic velocity of the filament as one gets closer to the filament.
3. The linear extent of the cloud around the B211/B213 filament is more than 10 pc in the plane of the sky. In contrast, the depth of the cloud along the line of sight is estimated to be $\sim$0.3–0.7 pc (=$N_{\rm H_2}$/$n_{\rm critical}^{\rm ^{13}CO}$) under the assumption that the density of the surrounding material is the same as the critical density of $^{13}$CO (1–0). These results suggest that the 3D morphology of the gas cloud surrounding the B211/B213 filament is sheet-like.
4. To investigate whether the B211/B213 filament is in the process of accreting the surrounding gas material, we compared the velocity patterns observed in $^{12}$CO (1–0) and $^{13}$CO (1–0) with our 3-component model. The predictions of the model were found to be in good agreement with the distribution of $^{12}$CO (1–0) and $^{13}$CO (1–0) emission in the observed position-velocity diagrams, supporting the scenario of mass accretion along magnetic field lines into the B211/B213 filament proposed by @Palmeirim13.
5. From an inspection of the wide-field spatial distributions of H${\alpha}$ and 857 GHz dust emission in the Taurus-California-Perseus region, we concluded that the B211/B213 filament was probably formed as a result of the expansion of a large-scale supershell originated in the Per OB2 association. This scenario provides a simple explanation for the different inclinations of the northeastern and southwestern sheet components inferred from our modeling analysis.
6. Based on these results, we propose that a) large-scale compression(s) generated by the Per OB2 association initially formed the B211/B213 filament system, and b) accretion of ambient gas material due to the gravitational potential of the filament is now responsible for the growth of the filament.
This work was supported by the ANR-11-BS56-010 project “STARFICH“ and the European Research Council under the European Union’s Seventh Framework Programme (ERC Advanced Grant Agreement no. 291294 – ‘ORISTARS’). YS also received support from the ANR (project NIKA2SKY, grant agreement ANR-15-CE31-0017). P. P. acknowledges support from the Fundação para a Ciência e a Tecnologia of Portugal (FCT) through national funds (UID/FIS/04434/2013) and by FEDER through COMPETE2020 (POCI-01-0145-FEDER-007672) and also by the fellowship SFRH/BPD/110176/2015 funded by FCT (Portugal) and POPH/FSE (EC). This research has made use of ”Aladin sky atlas" developed at CDS, Strasbourg Observatory, France [@Bonnarel00; @Boch14].
Complementary figures
=====================
{width="140mm"}
{width="150mm"} {width="150mm"}
{width="150mm"}
{width="180mm"}
![Large-scale spatial distribution of HI emission in the Taurus-Auriga-California-Perseus region from @Kalberla17. Red, green,and blue are the HI components at the velocity of -29.7–3.35, 4.93–8.09, and 8.88–14.41 km s$^{-1}.$ Contour indicates a level of 30 MJy str$^{-1}$ at the $Planck$ 857 GHz emission.[]{data-label="fig:3color_hi"}](HI_shell_3color.jpg){width="80mm"}
{width="60mm"} {width="60mm"}
Inclinations of the two sheet components in the model {#appendix:inclination}
=====================================================
To investigate the effect of the assumed inclinations for the two sheet components in our model, we expected a range of inclinations (10$^\circ$, 20$^\circ$, 30$^\circ$, 40$^\circ$, 50$^\circ$, 60$^\circ$, 70$^\circ$, and 80$^\circ$) and compared, for each inclination, the (peak) velocities predicted by the model with the $^{12}$CO/$^{13}$CO observations. The northeastern and southwestern sheet components were examined separately. We assumed the same parameters as listed in Table \[Table1\], except for the inclination angles ($\theta_{\rm N/S}$). Figure \[fig:variation\] shows the velocity offsets between models and observations. The velocity offset for the southwestern sheet component (offset $>$ 0) increases as the inclination angle increases, while the velocity offset for the northeastern sheet component (offset $<$ 0) increases as the inclination angle decreases. The velocity offset at |offset| $<$ $R_{\rm out}$ tends to be larger than that at |offset| $>$ $R_{\rm out}$. One possible reasons is that the $^{12}$CO (1–0) and $^{13}$CO (1–0) emissions do not trace the inner part of the filament (|offset| $<$ $R_{\rm out}$) since the $^{12}$CO (1–0) and $^{13}$CO (1–0) optical depths are much larger than a unity (see Sect. \[optical\_depth\]). In order to further investigate the velocity fields of the accreting gas, observations in optical thin dense gas tracers such as N$_2$H$^{+}$(1-0) and H$^{13}$CO$^+$(1-0) which trace the filament well [cf. @Shimajiri17 for H$^{13}$CO$^{+}$ (1–0)] are required. Table \[TableA1\] summarizes the mean values of the velocity offsets for the northeastern and southwestern sheet components for each model. Inclinations of 70$^{\circ}$ for the northeastern sheet component and 20$^{\circ}$ for the southwestern sheet component provide the minimum velocity offset. We therefore adopted these inclination values in the model presented in Sect. \[section:model\].
{width="160mm"}
Inclination$^{\dag}$ North/South line Mean$\pm$Stddev$^{\ddag}$
---------------------- ------------- ----------- ---------------------------
North $^{12}$CO 0.48$\pm$ 0.44
South $^{12}$CO 0.29$\pm$ 0.25
North $^{13}$CO 0.41$\pm$ 0.49
South $^{13}$CO 0.25$\pm$ 0.19
North $^{12}$CO 0.48$\pm$ 0.17
South $^{12}$CO 0.26$\pm$ 0.24
North $^{13}$CO 0.43$\pm$ 0.17
South $^{13}$CO 0.17$\pm$ 0.15
North $^{12}$CO 0.50$\pm$ 0.14
South $^{12}$CO 0.27$\pm$ 0.24
North $^{13}$CO 0.45$\pm$ 0.17
South $^{13}$CO 0.21$\pm$ 0.17
North $^{12}$CO 0.50$\pm$ 0.14
South $^{12}$CO 0.26$\pm$ 0.19
North $^{13}$CO 0.45$\pm$ 0.17
South $^{13}$CO 0.25$\pm$ 0.19
North $^{12}$CO 0.50$\pm$ 0.14
South $^{12}$CO 0.31$\pm$ 0.29
North $^{13}$CO 0.45$\pm$ 0.17
South $^{13}$CO 0.37$\pm$ 0.26
North $^{12}$CO 0.38$\pm$ 0.23
South $^{12}$CO 0.47$\pm$ 0.41
North $^{13}$CO 0.33$\pm$ 0.14
South $^{13}$CO 0.54$\pm$ 0.44
North $^{12}$CO 0.26$\pm$ 0.28
South $^{12}$CO 0.62$\pm$ 0.45
North $^{13}$CO 0.20$\pm$ 0.18
South $^{13}$CO 0.63$\pm$ 0.52
North $^{12}$CO 0.29$\pm$ 0.35
South $^{12}$CO 0.73$\pm$ 0.51
North $^{13}$CO 0.21$\pm$ 0.29
South $^{13}$CO 0.74$\pm$ 0.60
: Mean values of the velocity offset between $^{12}$CO (1–0)/$^{13}$CO (1–0) observations and model[]{data-label="TableA1"}
[^1]: <https://faun.rc.fas.harvard.edu/dfink/skymaps/halpha/data/v1_1/index.html>
[^2]: <https://irsa.ipac.caltech.edu/data/Planck/release_1/all-sky-maps/previews/HFI_SkyMap_857_2048_R1.10_survey_2_ZodiCorrected/>
[^3]: <https://www.astro.uni-bonn.de/hisurvey/AllSky_profiles/index.php>
[^4]: <http://home.strw.leidenuniv.nl/~moldata/datafiles/13co.dat>
[^5]: Here, $R'$ denotes the radius corrected for inclination to the line of sight, where the relation between the corrected radius $R'$ and radius in the sky plane $R$ is $R' = R/\sin(\theta_{\rm N/S})$ and $\theta_{\rm N/S}$ is the inclination angle of the northeastern/southwestern sheet component to the line of sight.
[^6]: $velimage$ makes a data cube $output\_cube(x,y,v_{\rm centroid})$ from an input integrated intensity image $input\_intensity(x,y)$, input centroid velocity image $input\_velocity(x,y)$, and dispersion $\sigma$. The $v_{\rm centroid}$-values are the centroid velocity and are input as an ($x,y$) image. The output cube image is produced as $output\_cube(x,y,v_{\rm centroid}) = input\_intensity(x,y) \times \exp(-(v_{\rm centroid}-input\_velocity(x,y))^2/(2\sigma^2)))$.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We formulate axisymmetric general relativity in terms of real Ashtekar–Barbero variables. We study the constraints and equations of motion and show how the Kerr, Schwarzschild and Minkowski solutions arise. We also discuss boundary conditions. This opens the possibility of a midisuperspace quantization using loop quantum gravity techniques for spacetimes with axial symmetry and time dependence.'
author:
- 'Rodolfo Gambini$^{1}$, Esteban Mato$^{1}$, Javier Olmedo$^{2,3}$, Jorge Pullin$^{3}$'
title: Classical axisymmetric gravity in real Ashtekar variables
---
Introduction
============
Due to the complexities of the quantized version of the Einstein equations in loop quantum gravity, the study of mini and midisuperspaces has proved a valuable tool to gain insights into the physics of the theory. The study first started with homogeneous cosmologies, giving rise to loop quantum cosmology (see [@Ashtekar:2011ni] and references therein). It was later expanded to include spherically symmetric spacetimes (see [@Gambini:2013hna] and references therein), including charged black holes [@Gambini:2014qta]. In both cases interesting physical insights, like the elimination of singularities due to quantum effects, were found. It is natural to try to extend these studies to situations with less symmetry, like the case of axisymmetric spacetimes, which include physically important situations, like the Kerr geometry. There is virtually no literature on the subject. An exception is the work on isolated horizons and black hole entropy [@krasnov:1998; @bojowald:2000; @perez:2011; @bianchi:2011; @frodden:2014; @achour:2016; @croken:2017]. An early study of spacetimes with one Killing vector field made some progress, partially addressing the situation of axial symmetry in complex connection variables [@Husain:1989mp]. Some progress was also made in planar space-times ([@Neville:2013xba] and references therein, [@Hinterleitner:2011rb]) and the case of two spatial Killing vector fields was also discussed for the Gowdy models [@Husain:1989qq], including the use of hybrid quantizations (see [@ElizagaNavascues:2016vqw] for a review). Some of these studies were in terms of the early form of the Ashtekar variables which were complex.
Here we would like to discuss the case of axisymmetric space-time using real Ashtekar–Barbero variables. We introduce a suitable Killing vector field and coordinates adapted to it. We will also show how the Kerr, Schwarzschild and Minkowski solutions arise. Besides, we will make some remarks on boundary conditions. This completes a classical setup suitable to perform a loop quantization, which we will discuss in a subsequent paper. This is the first example of a system with only one Killing vector field to be formulated with the real Ashtekar–Barbero variables.
The organization of this paper is as follows. In section 2 we discuss a set of symmetry adapted variables and set up the kinematics of the problem. In section 3 we introduce the constraints of general relativity in terms of the reduced axisymmetric variables introduced. In section 4 we work out the equations of motion. In section 5 we check that some particular solutions of interest including the Kerr, Schwarzschild and Minkowski space-times solve the equations we present. In section 6 we discuss boundary conditions. We end with a discussion.
Kinematics: symmetry adapted variables
======================================
Here we will impose a symmetry reduction due to a spatial Killing field with orbits tangent to $S^1$. Let us consider a choice of fiducial coordinates $\{x,y,\phi\}$, where $\phi\in S^1$ and $x,y\in \mathbb{R}$. The Killing field will be then $$K^a=\left(\partial_\phi\right)^a.$$
We will be following the typical reduction procedure adopted for connection variables. Namely, a connection $ A=A_a^i\tau_i \mathrm{d}x^a$ will be invariant under the Killing symmetries if it satisfies the condition $${\cal L}_{\tilde K}A^i_a=\epsilon_{ijk}\lambda^jA^k_a,$$ where $\tilde K = \lambda_i \partial^i=\lambda_3\partial_\phi$ and $\lambda_1=0=\lambda_2$. The previous equation amounts to $$\label{eq:axi-sym}
\partial_\phi A^i_a=\epsilon_{i3k}A^k_a.$$ Notice that we are imposing that the Lie derivative be proportional to a constant $O(2)$ gauge transformation [@cordero]. We have found this to be the simplest choice that is general enough to recover all solutions with axisymmetry. In other situations one may need to consider $\lambda^i$ that are more general, perhaps including spatial dependence.
The same equation is valid for the densitized triad $E^a_i$. The most general solution (see Appendix \[app:red\]) to these equations are $$\begin{aligned}
\label{eq:red-A}
A&=&A_a^i\tau_i \mathrm{d}x^a= \left((\cos(\phi)\tau_1 + \sin(\phi) \tau_2) \a_a^1 + (-\sin(\phi) \tau_1 + \cos(\phi) \tau_2) \a_a^2 + \a_a^3 \tau_3 \right)\mathrm{d}x^a \\ \label{eq:red-E}
E&=&E^a_i \tau^i \partial_a= \left((\cos(\phi)\tau^1 + \sin(\phi) \tau^2) \e^a_1 + (-\sin(\phi) \tau^1 + \cos(\phi) \tau^2) \e^a_2 + \e^a_3 \tau^3 \right)\partial_a,\end{aligned}$$ where the symmetry adapted variables $(\a_a^i,\e^b_j)$ do not depend on the angular coordinate $\phi$, i.e. only on $(x,y)$, and are canonically conjugate. In order to prove this, it is very easy to verify that $$\Omega=\frac{1}{8\pi G \beta}\int dx dy d\phi \;\delta E^a_i\wedge \delta A^i_a = \frac{1}{4 G \beta}\int dx dy \;\delta \e^a_i\wedge \delta\a^i_a,$$ with $\beta$ the Immirzi parameter. In other words, $$\{\a^i_a(\vec x),\e^b_j(\vec x')\}=4 G\beta\,\delta^i_j\delta^b_a\delta^{(2)}(\vec x-\vec x').$$ Another geometrical quantity that will be useful and can be computed now is the determinant of the symmetry-reduced densitized triad $$E = \det(E) =\frac{1}{3!}\varepsilon_{abc}\varepsilon^{ijk}E^a_iE^b_jE^c_k=\frac{1}{3!}\varepsilon_{abc}\varepsilon^{ijk}\e^a_i\e^b_j\e^c_k=\det (\e)=\e.$$ The inverse of the densitized triad, $E^i_a$, takes a similar form as $E^a_i$, but replacing $\e^a_i$ by $\e_a^i$. One can easily see that $\e^i_a$ fulfills $\e^i_a\e^a_j=\delta^i_j$ and $\e^i_a\e^b_i=\delta^a_b$, i.e. it is the inverse of $\e^a_i$ (and therefore it can be written in terms of $\e^a_i$). Then, the symmetry-reduced spatial metric can be written as $$q_{ab}=EE^i_aE^i_b=\e\,\e^i_a\e^i_b,$$ and it only depends on $\e^a_i$.
Similarly, the same reduction process can be applied to the extrinsic curvature $ K=K_a^i\tau_i \mathrm{d}x^a$, in its triadic form, and the spin connection $ \Gamma=\Gamma_a^i\tau_i \mathrm{d}x^a$, namely $$\begin{aligned}
K&&=\left((\cos(\phi)\tau_1 + \sin(\phi) \tau_2) \k_a^1 + (-\sin(\phi) \tau_1 + \cos(\phi) \tau_2) \k_a^2 + \k_a^3 \tau_3 \right)\mathrm{d}x^a,\\
\Gamma&&=\left((\cos(\phi)\tau_1 + \sin(\phi) \tau_2) {\gamma}_a^1 + (-\sin(\phi) \tau_1 + \cos(\phi) \tau_2) \gamma_a^2 + \gamma_a^3 \tau_3 \right)\mathrm{d}x^a.\end{aligned}$$ Actually, the components of the symmetry-reduced spin connection can be written in terms of the components of the symmetry-reduced triads as $$\gamma^i_a=\frac{1}{2}\epsilon_{ijk}\e^b_j\left(\e_{a,b}^k-\e_{b,a}^k+\e^c_k\e_a^l\e^l_{c,b}+\e_a^k\e^l_{c}\e_{l,b}^{c}\right)-\delta^i_3\delta^\phi_a= \g^i_a-\delta^i_3\delta^\phi_a,$$ where $\g^i_a$ is the spin connection compatible with $\e^a_i$. This means that $\{\e^a_i,\g^j_b\}=0$, and therefore $\{\e^a_i,\gamma^j_b\}=0$.
The relation between the components of the symmetry-reduced extrinsic curvature with the ones of the symmetry-reduced Ashtekar-Barbero and the spin connections is $$\beta\k^i_a=\a^i_a-\gamma^i_a = \a^i_a-\g^i_a+\delta^i_3\delta^\phi_a.$$
Finally, we will conclude this section by introducing some identities that can be useful for the calculations in the next sections. The first identity is the Poisson bracket of the connection with the inverse triad, $$\{\a^i_a(\vec x),\e^j_b(\vec x')\}=-4 G\beta\delta^{(2)}(\vec x-\vec x')\, \e^i_b\e^j_a.$$ and it is easy to prove. The second identity, $$\{\a^i_a(\vec x),\e(\vec x')\}=4 G\beta\delta^{(2)}(\vec x-\vec x')\, \e\,\e^i_a,$$ is based on $\delta\e = \e \,\e^i_b \delta \e_i^b$, for any given variation $\delta\e$ (in particular it is therefore valid for $\partial_a\e$). For the next identity, we should first notice that, given any phase space tensor of the form $T^a_i$ with density weight one, we can define $$\g(T)=\int dx dy \,T_i^a\g^i_a,$$ at least formally (we are not taking into account boundary terms neither fall-off conditions). These types of expressions appear, for instance, in the Lorentzian part of the symmetry-reduced Hamiltonian constraint. After a lengthy but simple calculation, one can prove that $$\label{eq:PB-ident}
\{\a^i_a(\vec x),\g(T)\}=4 G \beta \left.{P^{ij}}{}^b_{ac}{}^{\e}\!D_bT^c_{j}\right|_{\vec x}+\int dx' dy' \,\{\a^i_a(\vec x),T_j^b(\vec x')\}\g^j_b(\vec x'),$$ where $$\label{eq:PB-P-def}
{P^{ij}}{}^b_{ac}=\frac{1}{2}\left[\epsilon_{i j k}\e^{l}_c\left(\e^{l}_a \e_{k}^b + \e_{l}^b \e^{k}_a\right) + \epsilon_{ljk} \e_{k}^b\left(\e^i_c \e^{l}_a - \e^i_a \e^{l}_c \right)\right]$$ and ${}^{\e}\!D_b$ is the covariant derivative compatible with $\e^a_i$, namely ${}^{\e}\!D_b\e^a_i=0$.
Similar identities also hold in the full theory, namely, for the original phase space variables $A^i_a$ and $E_j^b$, and the spin connection $\Gamma^i_a$.
The constraints
===============
The total Hamiltonian of the full theory is a combination of 7 constraints: 3 Gauss constraints, 3 vector constraints and the Hamiltonian constraint. Concretely, $$H_T=\frac{1}{16\pi G}\left[G(\vec\Lambda)+ D(\vec N)+C(N)\right],$$ where $$\begin{aligned}
G(\vec\Lambda)&=\frac{2}{\beta}\int d^3x \Lambda^i\left(\partial_aE^a_i+\varepsilon_{ijk}A^j_aE^a_k\right),\\
D(\vec N)&=\frac{2}{\beta}\int d^3x N^a\left(E^b_i\partial_aA^i_b-\partial_b(E^b_iA^i_a)\right), \\
C(N) &= H_E(N)+H_L(N),\end{aligned}$$ and where $H_E(N)$ and $H_L(N)$ are the Euclidean and Lorentzian parts of the Hamiltonian constraint, given, respectively, by $$\begin{aligned}
H_E(N)&=-\int d^3x N e^{-1}(A^{i}_{b,a} - A^{i}_{a,b} + \epsilon_{ilm}A^{l}_{a} A^{m}_{b})\epsilon_{ijk}E_{j}^{a} E_{k}^{b} ,\\
H_L(N)&=\int d^3x N (1+\beta^2)e^{-1}\epsilon_{ijk} \epsilon_{ilm} E_{j}^{a} E_{k}^{b} K^{l}_{a} K^{m}_{b}.\end{aligned}$$
Now, we replace the symmetry-reduced connection $A^i_a$ and the densitized triad $E^a_i$ in the previous expressions. The symmetry-reduced Hamiltonian will be $$h_T=\frac{1}{8 G}\left[g(\vec\Lambda)+ d(\vec N)+c(N)\right],$$ and the (smeared) constraints are, $$\begin{aligned}
g(\vec\lambda)&=\frac{2}{\beta}\int dxdy \lambda^i\left(\partial_a\e^a_i+\varepsilon_{ijk}\a^j_a\e^a_k+\varepsilon_{ijk}\delta^j_3\delta^\phi_a \e^a_k\right),\\
d(\vec N)&=\frac{2}{\beta}\int dxdy N^a\left(\e^b_i\partial_a\a^i_b-\partial_b(\e^b_i\a^i_a)+\delta_a^\phi\delta^i_3\varepsilon_{ijk}\a^j_b\e^b_k%-\delta_a^\phi\partial_b(\e^b_i\a^i_c\delta^c_\phi)
\right), \\
h_E(N)&=-\int dxdy \frac{N}{\sqrt{\e}}\left[(\a^{i}_{b,a} - \a^{i}_{a,b} + \epsilon_{ilm}\a^{l}_{a} \a^{m}_{b})\epsilon_{ijk}\e_{j}^{a} \e_{k}^{b} +2\delta^j_3 \delta_b^\phi\left(\a^i_a \e_i^a\e_j^b - \a^i_a\e_i^b \e_j^a\right)\right] ,\\
h_L(N)&=\int dxdy \frac{N}{\sqrt{\e}} (1+\beta^2)\epsilon_{ijk} \epsilon_{ilm} \e_{j}^{a} \e_{k}^{b} \k^{l}_{a} \k^{m}_{b},\end{aligned}$$ where, as before, we have written the scalar constraint as $c(N)=h_E(N) + h_L(N)$.
Equations of motion
===================
In this section we will provide the Poisson brackets of the components of the symmetry-reduced connection and densitized triad with the constraints. We will perform a local analysis, assuming suitable boundary terms have been chosen. We will discuss the issue of boundary terms for the asymptotically flat case in section 5, similar analyses can be carried out for other asymptotic behaviors. Let us start with the Gauss constraint. One can easily see that $$\begin{aligned}
\{\e^a_i(\vec x),g(\vec\lambda)\} &= \left. \left(\epsilon_{ijk} \lambda^j \e_k^a\right)\right|_{\vec x} \\
\{\a_a^i(\vec x),g(\vec\lambda)\} &= \left. \left(-\lambda^i_{,a}+\epsilon_{ijk} \lambda^j \a^k_a+\varepsilon_{ijk}\lambda^j\delta^k_3\delta^\phi_a \right)\right|_{\vec x}.\end{aligned}$$ We see that $\e^a_i$ transforms as a tensor and that $\a^k_a+\delta^k_3\delta^\phi_a$ transforms as a connection.
The vector constraint yields $$\begin{aligned}
\{\e^a_i(\vec x),d(\vec N)\} &= \left. \left(\left(N^b \e^a_{i}\right)_{,b}-\e_i^b N^a_{,b}
-\varepsilon_{ijk}N^d\delta_d^\phi\delta^k_3\e^a_j
%- N^d_{,b}\delta_d^\phi\delta^a_\phi\e^b_i
\right)\right|_{\vec x} \\
\{\a_a^i(\vec x),d(\vec N)\} &= \left. \left(\a^i_b N^b_{,a} + N^b \a^i_{a,b}
+\varepsilon_{ijk}N^d\delta_d^\phi\delta^j_3\a^k_a
%+ N^d_{,a}\delta_d^\phi\delta^c_\phi\a^i_c
\right)\right|_{\vec x}.\end{aligned}$$ The Poisson brackets with the Euclidean and Lorentzian parts of the Hamiltonian constraint are, $$\begin{aligned}
\{\e^a_i(\vec x),h_E(N)\} &= - \frac{\beta}{2}\left. \left[2\left(\frac{N}{\sqrt{\e}}\varepsilon_{ijk}\e_{j}^{b} \e_{k}^{a}\right)_{,b}-2 \frac{N}{\sqrt{\e}} \varepsilon_{ilm} \varepsilon_{mjk} \a^{l}_b \e_{j}^{a} \e_k^b - 2\frac{N}{\sqrt{\e}}\delta^j_3 \delta_b^\phi\left(\e_i^a\e_j^b - \e_i^b \e_j^a\right)\right]\right|_{\vec x} \\
\{\e^a_i(\vec x),h_L(N)\} &= \frac{\beta}{2}\left. \left(
\frac{2}{\beta} (1+\beta^2)\frac{N}{\sqrt{\e}} \varepsilon_{ilm} \varepsilon_{jkm} \k^{l}_{b} \e_{j}^{b} \e_k^a
\right)\right|_{\vec x} \end{aligned}$$
Finally, we will provide the Poisson brackets of the components of the symmetry-reduced connection with the symmetry-reduced Hamiltonian constraint. The Euclidean part is simply given by $$\begin{aligned}
\nonumber
\{\a_a^i(\vec x),h_E(N)\} &= -\frac{\beta}{2}\left[-\frac{N}{2}\C_E\e^i_a+2\frac{N}{\sqrt{\e}}\epsilon_{ijk}\F^k_{ab}\e^b_j\right.\\
&\left.\left.+\frac{2N}{\sqrt{\e}}\left(\delta^j_3 \delta_b^\phi\a^i_a \e_j^b+\delta^i_3 \delta_a^\phi\a^j_b \e_j^b-\delta^j_3 \delta_a^\phi\a^i_b \e_j^b-\delta^i_3 \delta_b^\phi\a^j_a \e_j^b
\right)\right]\right|_{\vec x},\end{aligned}$$ where $\F_{ab}^i= \a^{i}_{b,a} - \a^{i}_{a,b} + \epsilon_{ilm}\a^{l}_{a} \a^{m}_{b}$. Here we have introduced $$\C_E=\frac{1}{\sqrt{\e}}\left[\F_{ab}^i\epsilon_{ijk}\e_{j}^{a} \e_{k}^{b} +2\delta^j_3 \delta_b^\phi\left(\a^i_a \e_i^a\e_j^b - \a^i_a\e_i^b \e_j^a\right)\right],$$ that is nothing but the Euclidean part of the local Hamiltonian constraint $\C$. On the other hand, for the Lorentzian part we must notice that $\{\a^i_a,\k^j_b\}=-\beta^{-1}\{\a^i_a,\g^j_b\}$ and also remember the identity in and the definition . After some manipulations, one gets, $$\begin{aligned}
\nonumber
\{\a_a^i(\vec x),h_L(N)\} &= -\frac{\beta}{2} \left\{-\frac{N}{2}\C_L\e^i_a-2(1+\beta^2)\frac{N}{\sqrt{\e}} \varepsilon_{ijk} \varepsilon_{lmk} \k^{l}_{a} \k^{m}_{b} \e_j^b\right.\\
&+\left.\left.\frac{1}{\beta}(1+\beta^2) \left(\varepsilon_{ijk}\e^m_a-\frac{1}{2}\varepsilon_{mjk}\e^i_a\right)\e_j^b \e_k^c\left[{{}^{\e}\!D_b}\left(\frac{N \k^m_{c}}{\sqrt{\e}}\right)-{{}^{\e}\!D_c}\left(\frac{N \k^m_{b}}{\sqrt{\e}}\right)\right]
\right\}\right|_{\vec x}. \end{aligned}$$ Similarly, $$\C_L=\frac{1}{\sqrt{\e}} (1+\beta^2)\epsilon_{ijk} \epsilon_{ilm} \e_{j}^{a} \e_{k}^{b} \k^{l}_{a} \k^{m}_{b},$$ is the Lorentzian part of the Hamiltonian constraint $C$.
The Kerr, Schwarzschild and Minkowski solutions
===============================================
We have explicitly checked that the symmetry-reduced model admits the well-known solution of the full theory given by the Kerr metric.
In the usual spherical coordinates $(r,\theta,\phi)$, the densitized triad for Kerr in a diagonal gauge, identifying the internal directions with the coordinates, takes the following form: $$\begin{aligned}
\nonumber
\e^r_3 &&= \sin\theta \sqrt{(r^2+a^2)(r^2+a^2 \cos^2\theta)+a^2 ~ r ~ r_s \sin^2\theta},\\\nonumber
\e^\theta_1&&=\frac{\sin\theta \sqrt{(r^2+a^2)(r^2+a^2 \cos^2\theta)+a^2 ~ r ~ r_s \sin^2\theta}}{\sqrt{a^2+r^2-r~r_s}},\\\label{eq:triad}
\e^\phi_2&&=\frac{r^2+a^2\cos^2\theta}{a^2+r^2-r~r_s},\end{aligned}$$ where $a=J/r_s$, while the rest of its components vanish. Together with the well-known choices of lapse and shift (e.g. [@visser; @chandrasekhar]) $$\begin{aligned}
N&&=\sqrt{\frac{(a^2+r(r-r_s))(a^2+2r^2+a^2 \cos(2\theta))}{2(a^2+r^2)(r^2+a^2\cos^2\theta+2 a^2 ~r ~r_s \sin^2\theta)}},\\
N_\phi&&=-\frac{r r_s a \sin^2\theta}{r^2+a^2\cos^2\theta},\quad N_r=0,\quad N_\theta = 0,\end{aligned}$$ and one can easily verify that this solution corresponds to the Kerr metric in Boyer–Lindquist coordinates. Here, $r_s$ is the Schwarzschild radius and $a$ the angular momentum per unit mass. The spin connection can be computed out of the densitized triad. Finally, the connection components can be easily computed provided the extrinsic curvature in triadic form. Concretely, $$\begin{aligned}
k_a^i&=\delta^{ij}\frac{\e^{b}_j}{\sqrt{\e}}K_{ab}\end{aligned}$$ where we obtain the extrinsic curvature from $$\begin{aligned}
K_{ab}&=\frac{1}{2N}\left( -\dot h_{ab}+\nabla_a N_b + \nabla_b N_a \right),\end{aligned}$$ keeping in mind that for our stationary solution $\dot h_{ab}=0$. The expressions of the components of the connection are rather lengthy, as well as those of the Langrange multipliers $\lambda^i$ of the Gauss constraint. We give them in the appendix. To determine the Lagrange multipliers we insert the above expressions for the triad, connection, lapse and shift in the equations of motion.
It is also very easy to check that in the limit $a\to 0$ we recover the Schwarzschild (static) solution. Concretely, the densitized triad reduces to $$\begin{aligned}
\nonumber
\e^r_3&&=r^2 \sin\theta, \\\nonumber
\e^\theta_1&&=\frac{r\sin\theta}{\sqrt{1-\frac{r_s}{r}}}, \\
\e^\phi_2&&=\frac{r}{\sqrt{1-\frac{r_s}{r}}}.\end{aligned}$$ the lapse now takes the familiar form $$\begin{aligned}
N&=\sqrt{1-\frac{r_s}{r}},\end{aligned}$$ while the shift vanishes, namely $N_\phi=0$. Besides, we also have that $K_{ab}=0$. Therefore, the connection is completely determined by the spin connection.
The Lagrange multipliers for the Gauss constraint take the simple form (for the Schwarzschild case, for the Kerr case see the appendix): $$\begin{aligned}
\nonumber
\lambda^1&=0 \\\nonumber
\lambda^2&=0 \label{lambda_schwarzschild} \\
\lambda^3&=-\beta \frac{r_s}{r^2}\end{aligned}$$
Finally, the Minkowski solution can be recovered from the limit $a\to 0$ and $r_s\to 0$. The densitized triad reduces to $$\begin{aligned}
\e^r_3&&=r^2 \sin\theta, \\
\e^\theta_1&&=r\sin\theta, \\
\e^\phi_2&&=r.\end{aligned}$$ The lapse function $N=1$ becomes the usual one in flat spacetimes. As in the previous case, the spin connection completely determines the connection, and any other Lagrange multiplier (shift and $\lambda^i$) vanish.
Boundary terms
==============
Up to now the analysis we made has been local. When one is in asymptotically flat space-times one needs to be mindful about falloff rates and integrations by parts. In particular, in addition to the constraints, one has a true Hamiltonian associated to the generators of the Lorentz group at infinity. In this section we will identify the boundary contributions needed to make the action differentiable in the asymptotically flat case for the Ashtekar–Barbero variables with axial symmetry. We will review individually each set of constraints to see if boundary terms are needed. We will follow closely [@thiemann_boundary; @campiglia_asymptotic].
Diffeomorphism constraint
-------------------------
We start this section by writing the portion of the action that corresponds to the diffeomorphism constraints in ADM-variables: $$\begin{aligned}
D[\vec{N}] = - 2 \int_{\Sigma} \mathrm{d}^3 x\,N^a\, \nabla_b P_{~a}^b,\end{aligned}$$ $\nabla$ being the covariant derivative compatible with the metric $g_{ab}$ and $P^{ab}$ the ADM-momentum, given by: $$\begin{aligned}
P_{ab}=-\frac{1}{16 \pi G}\sqrt{q}\left( K_{ab}-q^{ab} K \right),\end{aligned}$$ where $K_{ab}$ is the extrinsic curvature $$\begin{aligned}
K_{ab}=\frac{1}{2N}\left( -\dot{q}_{ab}+\nabla_b N_a + \nabla_a N_b \right).\end{aligned}$$ Taking variations of that term of the action with respect to the canonical variables yields the following boundary contribution, $$\begin{aligned}
2 \int_{\Sigma} \mathrm{d}^3 x \nabla_b \left( N_a \delta P^{ab} \right),
\end{aligned}$$ which must be canceled at infinity. We must thus add to the action the following surface term: $$\begin{aligned}
\label{eq:P-bound}
{\cal P}=- 2 \oint_{\delta \Sigma} \mathrm{d} S^b N^a P_{ab}.\end{aligned}$$ As we see, this boundary term not only depends on the phase space variables, but also on the Lagrange multipliers (the shift functions $N^a$ in this case). In the following, we will assume that the latter will be prescribed functions at spatial infinity (determined by the asymptotic form of the Kerr metric given below in Eq. ). Therefore, we will not consider variations of these functions on the boundary.
The boundary term in Eq. can be easily evaluated for the Kerr metric at spatial infinity. In spherical coordinates, its asymptotic form is given by $$\label{eq:kerr-aymp}
\mathrm{d}S^2 = -\left[ 1-\frac{2m}{r} \right] \mathrm{d}t^2 - \frac{2J \sin^2 \theta}{r}\left(\mathrm{d}t \mathrm{d}\phi + \mathrm{d}\phi \mathrm{d}t \right) + \left[1 + \frac{2m}{r} \right] \mathrm{d}r^2 + r^2 (\mathrm{d}\theta^2+ \sin^2 \theta \mathrm{d}\phi^2).$$ Moreover, $$\begin{aligned}
\mathrm{d}S^b=\mathrm{d}S n^b &= \mathrm{d}S \delta_r^b,
%N^a &= N^\phi \delta^a_\phi\end{aligned}$$ and $\mathrm{d}S=r^2\sin\theta {\rm d}\theta {\rm d}\phi$. From the metric in Eq. , the only contributions to the integral are $$\begin{aligned}
{\cal P} =- 2 \oint_{\delta\Sigma} \mathrm{d}S N^\phi P_{\phi r}=-2 \oint_{\delta\Sigma} \mathrm{d}\theta \mathrm{d}\phi r^2 \sin\theta N^\phi \left( \frac{\sqrt{q}}{16\pi G} (K_{\phi r} - q_{\phi r} K ) \right),\end{aligned}$$ where $q_{ab}$ is the spatial metric and $K^{ab}$ the extrinsic curvature. At the boundary $\delta \Sigma$, we have $$\begin{aligned}
q_{\phi r}&=0, \\
\sqrt{q}&=r^{3/2} \sqrt{r+r_s} \sin\theta, \\
N_\phi &= -\frac{2J \sin^2\theta}{r} ~~ \rightarrow ~~ N^\phi=-\frac{2J}{r^3}, \\
K_{\phi r}&=\frac{3 J \sin^2 \theta}{\sqrt{r^3 (r-r_s)+4J^2 \sin^2 \theta}}, \\
K&=0.\end{aligned}$$ Performing the integral, the leading term in the expansion in $1/r$ is $$\begin{aligned}
{\cal P} = \lim_{r\to\infty}\frac{9\pi}{64 G} \frac{J^2}{r}=0.\label{6.9}\end{aligned}$$
Now, in Ashtekar-Barbero variables, the diffeomorphism constraint of the full theory takes the form $$\begin{aligned}
\frac{1}{16\pi G}D(\vec N)&=\frac{1}{8\pi G\beta}\int_\Sigma N^a\left(E^b_i\partial_aA^i_b-\partial_b(E^b_iA^i_a)
\right).\end{aligned}$$ Its boundary term (for asymptotically flat spacetimes) takes the form $$\begin{aligned}
{\cal P}= \frac{1}{8\pi G\beta} \oint_{\delta\Sigma} N^a E^b_i ~ A_a^i ~ \mathrm{d}S_b.\end{aligned}$$ On the other hand, in our reduced theory, the reduced diffeomorphism constraint is given by $$\begin{aligned}
\frac{1}{8\pi G}d(\vec N)&=\frac{1}{4\pi G\beta}\int_\sigma N^a\left(\e^b_i\partial_a\a^i_b-\partial_b(\e^b_i\a^i_a)+\delta_a^\phi\delta^i_3\varepsilon_{ijk}\a^j_b\e^b_k
\right),\end{aligned}$$ where $\sigma$ are the $r,\theta$ 2D spatial sections of our reduced theory, with topology $\mathbb R^2$. The corresponding boundary term is $$\begin{aligned}
{\mathfrak p}= \frac{1}{4G\beta} \oint_{\delta\sigma} N^a \e^b_i ~ \a_a^i ~ \mathrm{d}s_b,\end{aligned}$$ where $$\begin{aligned}
\mathrm{d}s^b=\mathrm{d}s n^b &= \mathrm{d}s \delta_r^b, \\
%N^a &= N^\phi \delta^a_\phi\end{aligned}$$ with ${\rm d}s=r^2\sin\theta {\rm d}\theta$. Since the only non-vanishing component of the shift is $N^\phi$, the required boundary term is $$\begin{aligned}
{\mathfrak p}= \frac{1}{4 G\beta} \oint_{\delta\sigma} N^\phi ~ \e^r_3 \a^3_\phi ~ q_{rr} ~ r^2 \sin\theta ~ \mathrm{d}\theta .\end{aligned}$$ Where we have chosen the triad as in . Again, we have at infinity: $$\begin{aligned}
N^\phi&=-\frac{2J}{r^3}, \\
e^r_3 &= r^2 \sin\theta, \\
q_{rr} &= \frac{1}{1+\frac{r_s}{r}}, \\
a_\phi^3 &= \cos\theta + \gamma \frac{3J r \sin^2 \theta}{\sqrt{r(r+r_s)(r^3(r-r_s)+ 4 J^2 \sin^2\theta)}}.\end{aligned}$$ The term proportional to $\cos\theta$ integrates to zero (since it contains the integral of $\cos\theta \sin\theta$ between $0$ and $\pi$) while the other term is easily seen to yield the same result as in (\[6.9\]).
Hamiltonian constraint
----------------------
Now we turn to the portion of the action involving the Hamiltonian constraint. In ADM variables: $$\begin{aligned}
\frac{1}{16 \pi G}C[N]=\frac{1}{16 \pi G}\int_\Sigma N \left( q^{-1/2} \left( P_{ab} P^{ab} - \frac{1}{2} P \right)- q^{1/2} R \right).\end{aligned}$$ Where $R$ is the Ricci scalar. In order for the variations with respect to the dynamical variables to be well defined, it is necessary to add to the action the surface term (see e.g. [@thiemann_boundary]): $$\begin{aligned}
\label{boundary_adm_0}
{\cal E}=\frac{1}{16 \pi G} 2 \oint_{\delta \Sigma}\mathrm{d}S_d N \sqrt{q} q^{ac} q^{bd} \bar{\nabla}_{\left[c\right.} q_{b\left. \right]a}.\end{aligned}$$ Where $\bar{\nabla}$ is the covariant derivative compatible with the order zero of expansion in $1/r$ of the spatial metric at infinity. This term corresponds to time translations at infinity. The surface term actually has another contribution coming from the fact that the Ricci tensor has second derivatives, requiring two integration by parts. That contribution corresponds to boosts at infinity, but due to our choice of adapted coordinates we do not allow such boosts.
We will evaluate the previous boundary term at spatial infinity in order to show that it is finite. For convenience, we will introduce an asymptotically Cartesian coordinate system with coordinates $\left\lbrace x^a \right\rbrace$. We then expand our metric asymptotically as $g_{ab}=\eta_{ab}+h_{ab}$, with $\eta_{ab}$ the flat space metric and $h_{ab}$ a small perturbation around $\eta_{ab}$. We also expand the lapse as $N=1+{\cal O}(1/r)$. Then, in the limit $r\to\infty$, the leading contribution to the boundary term takes the form $$\begin{aligned}
\label{boundary_adm}
{\cal E} = \frac{1}{16\pi G} \oint_{\delta \Sigma} \left( \frac{\partial h_a^b}{\partial x^b} - \frac{\partial h_b^b}{\partial x^a} \right) \mathrm{d}S^a=\frac{r_s}{2G}.\end{aligned}$$
Now, a direct calculation shows that, in Ashtekar variables, the equivalent boundary term to takes the following form: $$\begin{aligned}
{\cal E}= -\frac{1}{8 \pi G\beta}\oint_{\delta \Sigma} \mathrm{d}S_a \frac{N}{\sqrt{E}} \left(E^a_i \bar{D}_b E^b_i + E^b_i \bar{D}_b E^a_i \right),\end{aligned}$$ where, similar to the derivative $\bar{\nabla}$ defined earlier, $\bar{D}$ is the covariant derivative compatible with the order zero component of the triad at infinity.
In our reduced theory, the boundary term is given by $$\begin{aligned}
{\mathfrak e}= -\frac{1}{4 G\beta}\oint_{\delta \sigma} \mathrm{d}s_a \frac{N}{\sqrt{e}} \left(\e^a_i \bar{D}_b \e^b_i + \e^b_i \bar{D}_b \e^a_i \right).\end{aligned}$$ Its evaluation in Cartesian coordinates agrees with the result given in .
Gauss constraint
----------------
The contribution to the action of the Gauss constraint in the full theory is given by $$\begin{aligned}
\frac{1}{16\pi G}G(\vec\lambda)&=\frac{1}{8\pi G\beta}\int_\Sigma \lambda^i\left(\partial_aE^a_i+\varepsilon_{ijk}A^j_aE^a_k\right).\end{aligned}$$ The variation of this contribution also requires another boundary term in order to make the full variational problem well defined. It is given by $$\label{eq:Q}
{\cal Q}=\frac{1}{8\pi G\beta} \oint_{\delta \Sigma} dS_a (E^a_i-\bar{E}^a_i) \lambda^i,$$ where $\bar{E}^a_i$ is the densitized triad at spatial infinity. Without this term, inserting the asymptotic form of the triad, the result is divergent. Since at spatial infinity the metric is flat, $\bar{E}^a_i$ is independent of $M$ and $J$. Therefore, any variational derivative of this term will be zero. Similar boundary terms have been suggested in previous treatments [@thiemann_boundary; @campiglia_asymptotic].
In our symmetry reduced theory, the reduced Gauss constraint is given by: $$\begin{aligned}
\frac{1}{8G}g(\vec\lambda)&=\frac{1}{4G\beta}\int_\sigma \lambda^i\left(\partial_a\e^a_i+\varepsilon_{ijk}\a^j_a\e^a_k+\varepsilon_{ijk}\delta^j_3\delta^\phi_a \e^a_k\right).\end{aligned}$$ The boundary term takes the same form in terms of the reduced densitized triad, namely, $${\mathfrak q}=\frac{1}{4G\beta} \oint_{\delta \sigma} dS_a (\e^a_i-\bar{\e}^a_i) \lambda^i,$$ where $\bar{\e}^a_i$ is the reduced triad at spatial infinity. After evaluation at spatial infinity, one gets $$\begin{aligned}
{\mathfrak q} = \frac{5\pi^2}{64 G} \frac{J^2}{r_{S}} .\end{aligned}$$ We note that it only depends on $M$ and $J$. Therefore, no new observables appear. This is due to our choice of diagonal triads, which ties spatial rotations generated by $J$ to internal rotations.
Conclusions
===========
We have developed the Ashtekar–Barbero framework for axisymmetric spacetimes. We found triads and connections adapted to the symmetry and wrote the Gauss law, vector and Hamiltonian constraints. We showed that the Kerr solution indeed solves the constraints and the evolution equations. We also discussed the boundary terms needed to make the action differentiable in a canonical treatment. This lays out a framework to attempt a loop quantization of axially symmetric spacetimes. These represent the most complex midisuperspaces considered up to date. The strategy we intend to follow for quantization is similar to the one we pursued in spherical symmetry ([@Gambini:2013hna; @Gambini:2014qta]). We will build spin network states based on the reduced connection. The third component of the connection will be represented by a point holonomy and the first two with genuine holonomies in the two dimensional reduced space adapted to the symmetry (for instance $r,\theta$ if one were to consider spherical coordinates). On such states the fluxes of the triads will act naturally. We will use these basic operators to construct the Hamiltonian of the theory. We will discuss details in a future publication.
Acknowledgments {#acknowledgments .unnumbered}
===============
We wish to thank Miguel Campiglia for discussions. This work was supported in part by grant No. NSF-PHY-1603630, funds of the Hearne Institute for Theoretical Physics, CCT-LSU, Pedeciba, grant ANII FCE-1-2014-1-103974, grant No. NSF-PHY-1505411, the Eberly research funds of Penn State, Project. No. MINECO FIS2014-54800-C2-2-P from Spain and its continuation Project. No. MINECO FIS2017-86497-C2-2-P.
Axisymmetric classical reduction {#app:red}
================================
From the components of Eq. we get the set of differential equations $$\partial_\phi A^1_a = -A^2_a, \quad \partial_\phi A^2_a = A^1_a, \quad \partial_\phi A^3_a = 0.$$ From $\partial_\phi A^3_a = 0$ we conclude that $A^3_a$ must be equal to a function $\a^3_a$ independent of $\phi$. On the other hand, the most general solutions for the differential equations of $A^1_a$ and $A^2_a$ are $$A^1_a = \cos\phi \,\a^1_a-\sin\phi\, \a^2_a,\quad A^2_a=\sin\phi\, \a^1_a+\cos\phi\, \a^2_a,$$ with $\a^1_a$ and $\a^2_a$ independent of $\phi$. These solutions yield to Eq. . The symmetry reduction of the densitized triad $E^a_i$ (despite being a tensor density of weight one) is similar to the one of $A^i_a$ for axisymmetric spacetimes (for $\lambda_3={\rm const}$). The reduced components of $E^a_i$ take a similar form as those of $A^i_a$ but replacing $\a^i_a$ by $\e^i_a$. One then obtains Eq. .
Kerr solution: Lagrange multipliers and connexion components
============================================================
Gauss’ law Lagrange multipliers: $$\begin{aligned}
\lambda^1&=0,
\\
\lambda^2&=\frac{\sqrt{2} a \sqrt{a^2+r(r-r_s)} r_s ( a^4 - 3 a^2 r^2 - 6 r^4 + 4 a r (a^2+r^2) \beta \cos(\theta)+a^2(a^2-r^2)\cos(2\theta) ) \sin(\theta) %
}{ ( ( a^2+2 r^2 + a^2 \cos(2 \theta) ) ( a^4 + 2 r^4 + a^2 r (3r + r_s)+a^2 (a^2+r(r-r_s) ) \cos(2\theta))^{3/2} )},\end{aligned}$$ $$\begin{aligned}
&\lambda^3=\left(\frac{a^2 + r (r - r_s)}{(a^2 + r^2) (r^2 + a^2 \cos^2(\theta)) +
a^2 r r_s \sin^2(\theta)}\right)^{1/2} \left\lbrace\frac{(2 r - r_s) \sin(\theta)}{(a^2 + r (r - r_s))^{1/2}} \right.
\\
&+ \frac{8 a^3 r (a^2 + r (r -
r_s))^{1/2} r_s (1 + \beta^2) \cos(\theta) \sin^3(\theta)}{\beta \
(a^2 + 2 r^2 + a^2 \cos(2 \theta)) (a^4 + 2 r^4 + a^2 r (3 r + r_s) + \
a^2 (a^2 + r (r - r_s)) \cos(2 \theta))}
\\
&- \left[ \frac{4 (r^2 + a^2
\cos^2(\theta)) ((a^2 + r^2) (r^2 + a^2 \cos^2(\theta)) + a^2 r
r_s (\sin(\theta))^2)^{1/2}}{{a^4 + 2 r^4 + a^2 r (3 r +
r_s) + a^2 (a^2 + r (r - r_s)) \cos(2 \theta)}} \right.
\\
& \left( \frac{(16 r^5 + 4 a^2 r^2 (4 r - r_s) +
a^4 (6 r + r_s) + 4 a^2 r (2 a^2 + r (4 r + r_s)) \cos(2 \theta) + a^4
(2 r - r_s) \cos(4 \theta)) \sin(\theta)}{8 \sqrt{2} (r^2 + a^2
\cos^2(\theta))^2 \left(\frac{a^4 + 2 r^4 + a^2 r (3 r + r_s)}{a^2 + r (r
- r_s)} + a^2 \cos(2 \theta)\right)^{1/2}} \right.
\\
&\left. \left. \left.
+\frac{8 a^3 r r_s \beta \cos(\theta) \sin^3(\theta) ((a^2 + r (r - r_s)) ((a^2 + r^2) (r^2 +
a^2 \cos^2(\theta)) + a^2 r r_s \sin^2(\theta)))^{1/2}}{(a^2 + 2
r^2 + a^2 \cos(2 \theta))^2 (a^4 + 2 r^4 + a^2 r (3 r + r_s) + a^2
(a^2 + r (r - r_s)) \cos(2 \theta))}\right)\right]\right\rbrace.\end{aligned}$$
Connection components: $$\begin{aligned}
&a_r^1=\frac{-2 a r_s \beta (r^2 + a^2 \cos^2(\theta)) (a^4 -3 a^2 r^2 -6 r^4 + a^2 (a - r) (a + r) \cos(2 \theta)) \sin(\theta)}{(a^2 + r (r - r_s))^{1/2} (a^2 + 2 r^2 + a^2 \cos(2 \theta))^2 (a^4 + 2 r^4 + a^2 r (3 r + r_s) + a^2 (a^2 + r (r - r_s)) \cos(2 \theta))}
\\
&+ \frac{a^2 \sin(2 \theta)}{(a^2 + r (r - r_s))^{1/2} (a^2 + 2 r^2 + a^2 \cos(2 \theta))}, \\
&a_r^2=0, \\
&a_r^3=0, \\
&a_\theta^1=\frac{r (a^2 + r (r - r_s))^{1/2}}{r^2 + a^2 \cos^2(\theta)}
\\
&- \frac{8 a^3 r (a^2 + r (r - r_s))^{1/2} r_s \beta \cos(\theta) (r^2 + a^2 \cos^2(\theta)) \sin^2(\theta)}{(a^2 + 2 r^2 + a^2 \cos(2 \theta))^2 (a^4 + 2 r^4 + a^2 r (3 r + r_s) + a^2 (a^2 + r (r - r_s)) \cos(2 \theta))}, \\
&a_\theta^2=0, \\
&a_\theta^3=0, \\
&a_\phi^1= 0, \\
&a_\phi^2\!=\! \frac{(16 r^5 + 4 a^2 r^2 (4 r - r_s) + a^4 (6 r + r_s) + 4 a^2 r (2 a^2 + r (4 r + r_s)) \cos(2 \theta) + a^4 (2 r - r_s) \cos(4 \theta)) \sin(\theta)}{8 \sqrt{2} (r^2 + a^2 \cos^2(\theta))^2 (\frac{a^4 + 2 r^4 + a^2 r (3 r + r_s)}{a^2 + r (r - r_s)} + a^2 \cos(2 \theta))^{1/2}}
\\
&+ \frac{8 a^3 r r_s \beta \cos(\theta) \sin^3(\theta) ((a^2 + r (r - r_s)) ((a^2 + r^2) (r^2 + a^2 \cos^2(\theta)) + a^2 r r_s \sin^2(\theta)))^{1/2}}{(a^2 + 2 r^2 + a^2 \cos(2 \theta))^2 (a^4 + 2 r^4 + a^2 r (3 r + r_s) + a^2 (a^2 + r (r - r_s)) \cos(2 \theta))},\end{aligned}$$ $$\begin{aligned}
&a_\phi^3=\frac{2 (a^2 (a^2 + r (r - r_s)) ((5 a^2 + 8 r^2)
\cos(3 \theta) + a^2 \cos(5 \theta))}{2 \sqrt{2} (a^2 + 2 r^2 + a^2
\cos(2 \theta))^2 (a^4 + 2 r^4 + a^2 r (3 r + r_s) + a^2 (a^2 + r (r
- r_s)) \cos(2 \theta))^{1/2}} \\
&+\frac{2 (5 a^6 + 8 r^6 + 4 a^2 r^3 (5 r + r_s) + a^4 r (17 r + 3 r_s)) \cos(\theta) }{2 \sqrt{2} (a^2 + 2 r^2 + a^2 \cos(2 \theta))^2 (a^4 + 2 r^4 + a^2 r (3 r + r_s) + a^2 (a^2 + r (r - r_s)) \cos(2 \theta))^{1/2}} \\
&+\frac{\sqrt{2} a r_s \beta (- a^4 + 3 a^2 r^2 + 6 r^4 + a^2 (- a^2 + r^2) \cos(2 \theta)) \sin^2(\theta)}{(a^2 + 2 r^2 + a^2 \cos(2 \theta))^2 (a^4 + 2 r^4 + a^2 r (3 r + r_s) + a^2 (a^2 + r (r - r_s)) \cos(2 \theta))^{1/2}}.\end{aligned}$$
[99]{} ref
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The past few years have seen remarkable progress in the theory and phenomenology of QCD, bringing perturbative and nonperturbative methods into closer contact with each other and with experiment.'
title:
---
ø Ł
\#1 [\[\#1\]]{}
ITP-SB-99-22
[^1]
George Sterman
*Institute for Theoretical Physics, SUNY at Stony Brook, Stony Brook, NY 11794-3840*
Introduction: QCD a Group Portrait
==================================
In this talk, I will summarize some recent developments in QCD, concentrating on, but not limited to, topics discussed at this meeting. Details, of course, can be found in the talks themselves, presented in lively sessions organized by Lance Dixon and Joey Huston for perturbative QCD, and by Paul Mackenzie and Claude Bernard for nonperturbative QCD. The future of the field is in the already-advanced convergence of these topics, sometimes thought of as nearly independent. Progress on heavy quark physics is described in the proceedings from separate parallel sessions and plenary talks. I must necessarily pass over some of the most interesting recent advances in other closely related fields as well for lack of space.
I have tried in the following to interleave perturbative and nonperturbative treatments of QCD dynamics. Let me begin with a few general comments on the place of QCD studies in high energy physics.
[*Why QCD?*]{} By now it is clear that QCD is a “correct", or phenomenologically relevant theory, at least the way classical Maxwell theory is “correct". Classical electromagnetism is an effective theory appropriate to the limit of many photons; quantum chromodynamics might be the long-distance limit of some more fundamental, underlying theory. Its self-consistency, however, leaves us to free to study QCD in its own terms.
This is a fascinating, and dauting, task, despite the fact that QCD is defined through a single dimensionless parameter, $\alpha_s$. The overall dimensional scale is determined by comparison with other interactions, for example, by measuring the strong coupling at the mass of the Z: $\alpha_s(M_{\rm Z})$. Its intrinsic interest aside, QCD has important “practical" applications, in the calculation of backgrounds to new physics and, for hadron colliders particularly, in predictions for new particle cross sections. From a theoretical point of view, however, what makes QCD so attractive is that it is a quantum field theory that requires all orders in perturbation theory and nonperturbative analysis to confront data at available energies.
[*QCD is the exemplary quantum field theory.*]{} QCD exhibits most of the classic quantum field-theoretic phenomena discovered in the sixties and seventies, including asymptotic freedom, confinement, spontaneous symmetry breaking and instantons. The problems of strong interactions that gave rise to QCD were also, in the same time period, the original inspiration for concepts of duality and strings. As we saw at this conference, the strong interactions, now understood as QCD, are once more the meeting ground for field theory, duality and string theory.
Again, QCD is evidently the correct theory of the strong interactions. Given its depth, however, “tests" of QCD should be thought of as tests, or perhaps better, “explorations", of quantum field theory itself. Complementary to the extraordinary accuracy of selected perturbative predictions in quantum electrodynamics are the broad predictions of QCD, interweaving nonperturbative and perturbative scales and phenomena.
QCD at the Shortest Distances
=============================
It was the asymptotic freedom of QCD that first drew attention to gauge field theory as the unique description of the strong interactions at short distances. This theme continues to unfold in current experiments, and to provide a basis for extrapolations between energy scales and the detection of signals for new physics. Let us begin with a run-through of the underlying methods [^2].
Methods
-------
[*Infrared safety.*]{} Infrared safe quantities are insensitive to long-distance effects, and may be calculated in perturbation theory [@tasi95; @glover]. An infrared safe quantity may or may not be directly observable. The classic examples include the total cross section for $\e^+\e^-$ annihilation to hadrons, and jet and event shape cross sections in $\e^+\e^-$. These can be written in the general form, Q\^2 \_[phys]{}(Q\^2) = \_n c\_n(Q\^2/\^2) \^n() , \[IRS\] with the $c_n$ dimensionless functions of the ratio of the hard scale to the renormalization scale $\mu$.
[*Factorization.*]{} Cross sections for deep-inelastic scattering (DIS) and for jet or heavy quark production in hadron-hadron scattering, are not purely perturbative, but appear as convolutions in parton momentum fractions of distributions $f_{a/h}(\xi,\m)$ of partons $a$ in hadrons $h$, with perturbative hard-scattering functions $\hat\sigma^{\rm PT}$, Q\^2\_[phys]{}(Q,x) = \_[[partons]{} a]{}\_a\^[PT]{}(Q/,())f\_[a/h]{}() = \_[[partons]{} a]{} \_x\^1 d \_a\^[PT]{}(x/,Q/,()) f\_[a/h]{}(,) , \[fact\] where in the second equality we have exhibited the convolution appropriate to deeply inelastic scattering (with $x=Q^2/2p\cdot q$). Corrections to Eq. (\[fact\]) are suppressed by ${\cal O}(1/Q^2)$ [@tasi95]. In this formula, there is a complementarity between the roles of parton distributions and the hard scattering. The distributions $f_{a/h}$ are universal among hard-scattering processes, but particular to hadron $h$, while the functions $\hat \sigma^{\rm PT}$ are particular to the process, but universal among external hadrons. This last feature enables us to calculate realistic $\hat\sigma^{\rm PT}$ in “unrealistic", but technically manageable (infrared regulated) scattering processes, in which the initial state hadrons are partons.
[*Evolution.*]{} The physical cross sections of Eqs. (\[IRS\]) and (\[fact\]) above must both be independent of the scales $\m$ that define the parton distributions: $\m d\sigma_{\rm phys}/d\m=0$. This self-consistency requirement is readily translated into the “DGLAP" equation for the evolution of parton densities, =\_b P\_[ab]{}(/,())f\_[b/h]{}(,) . \[evol\] Here, the dimensionless kernel $P$ depends only on variables that are in common between the hard scattering functions and the parton distributions: $\alpha_s$ and the momentum fractions. The scale-independence of physical quantities and their relations can be studied systematically [@commensurate].
The idealized pattern for determining and applying the distributions may be summarized as follows. We measure one cross section, $\sigma_{\rm phys}$, at momentum transfer $Q_0$. Given a “next-to-leading" order calculation of its hard scattering functions $\hat\sigma_a^{\rm (NLO)}$, we determine NLO parton distributions $f_{a/h}^{\rm (NLO)}$ at $\m=Q_0$. Using evolution, we can then predict $\sigma_{\rm phys}$ for any hard process at all $Q$.
The coefficients $c_n$ in Eq. (\[IRS\]) are known for many processes to NLO [@glover]. They have been determined at NNLO for inclusive DIS and Drell-Yan [@nnlo], and even to three loops [@3loop] for selected quantities. Generally, two loop corrections are available only for single-scale processes, and the calculation of two-loop corrections for genuine scattering diagrams is an as-yet unsolved, but actively studied, problem in QCD [@kaufmann; @bern].
Perturbative QCD is most successful for inclusive processes, and/or single-scale semi-inclusive. Evolution in DIS is perhaps still the best illustration (see below). The current experimental situation in hadronic hard-scattering is reviewed in Ref. [@huston]. In multiscale problems, logarithms of ratios of distinct but perturbative scales often require resummation to all orders. Formally beyond perturbative resummation, but not always less important numerically, are corrections suppressed by powers of the hard scale(s). In DIS, and a few other cases, these corrections can be described by the operator product expansion. We shall encounter below “generalizations" of this famous technique, usually in the form of effective field theories.
Prime Examples
--------------
[*Tevatron Jets.*]{} The most impressive success in orders of magnitude continues to be the Tevatron inclusive single-jet and dijet cross sections [@huston; @bhatta; @hauser; @krane], as illustrated in Fig. \[cteqSinglejet\], which shows a plot from Ref. [@cteq5]. According to Eq. (\[fact\]), these cross sections are of the form \_[p|pJ+X]{}=\_[ab]{} f\_[a/p]{}\^[(NLO)]{}\_[abJ+X]{}f\_[b/|p]{} . \[1jet\] We find a consistency between NLO theory and experiment at a few tens of percent, well within the overall systematic errors, over a range in which the cross section decreases by seven or so factors of ten. As the figure shows, reasonable choices of parton distributions (in this case CTEQ5HJ) can account even for the highest-momentum data, although a slight difference remains between the D0 and CDF data at the high end.
[*DIS Scaling violations.*]{} Next, we should cite measurements of DIS structure functions [@doyle; @erdmann; @cross] for $\ell^\pm N\to \ell^\pm N$, through the cross sections = [2\^2xQ\^4]{} , at HERA, the Tevatron, SLAC and elsewhere, with $x=Q^2/2p\cdot q \quad q^2=-Q^2 \quad y=Q^2/xs$. Because the evolution is universal in the factorized forms, F\_[h]{}(Q\^2) &=& \_a C\_[a]{}(Q\^2/\^2) f\_[a/h]{}(\^2) |\_[Q=]{}\
&=& \_b P\_[ab]{}(())f\_[b/h]{} , \[factevol\] we may think of evolution as in, but not of, the nucleon, except perhaps in the smallest-$x$ region, where target-dependent shadowing, in which partons begin to interfere with each other, comes into play.
The quality of the data, and of QCD fits to it, is illustrated by recent results on $F_2(x,Q^2)$ from H1 [@H1data], shown in Fig. [\[H12\]]{} for various fixed $x$ as a function of $Q^2$. Notice in particular the sharp rise of $F_2$ with $Q^2$ for the smaller values of $x$. This is a prediction of the evolution through parton branching described by Eq. (\[factevol\]). Deviations from such DGLAP evolution have been surprisingly difficult to find. The interesting results of a next-to-next to leading order analysis of DIS has been described in Ref. [@nnlodis]. At the other end of the spectrum, the excess of events reported two years ago at the largest $x$ and $Q^2$ has all but disappeared [@highq].
[*Multijets, and event shapes.*]{} In $\e^+\e^-$ annihilation, the Born level process $\e^+\e^-
\rightarrow q\bar q$ produces two jets, as is the case as well for hadron colliders, while in DIS, the Born scattering produces a single jet. In all three “canonical" scattering processes, the cross sections for the minimal reactions are known at NLO. Beyond this, NLO cross sections are available for two jets in DIS [@zeppen], for four in $\e^+\e^-$ [@dixon], and NLO three-jet cross sections are just now becoming available for hadron-hadron scattering [@kilgore]. Recent years have seen an explosion of data on these processes, and new results were discussed at this conference in [@bhatta; @hauser; @krane; @zeppen; @gary], along with jet fragmentation properties [@gary; @klapp; @abe; @dong; @schyns; @lamsa]. The production of vector bosons associated with jets [@croninh] is another important source of information about short-distance dynamics, and a reanalysis of W+1 jet cross sections has brought them back into agreement with theory over the past year.
Jets are theoretical-phenomenological constructs [@tkachov]. Their value is not that they are an exact reflection of short-distance reactions, but that, if they are defined properly, they are related to them in a calculable fashion. Jets are generally defined in terms of energy flow into some angular region, or in terms of interative clustering schemes for the momenta of observed particles. Any (sufficiently smooth [@irshapes]) quantity that is insensitive to the emission of zero-momentum particles, or to the collinear branching of finite-energy massless particles, can be used to define an infrared-safe cross section. Event shapes in $\e^+\e^-$ annihilation or DIS are chosen for this property. The best-known is the thrust, defined for individual events as the maximum fractional projection of the momenta of observed particles on an axis, as that axis is varied about the unit sphere. Although event shape cross sections are infrared safe, they generally receive nonperturbative corrections that decrease rather slowly with energy, typically as a single power [@webberlect], \_[phys]{}=\^[PT]{} (1+ [O]{}(1/Q)) . \[1overQ\] The theory of these power corrections is a passageway between the short-distance perturbative, and the long-distance nonperturbative aspects of QCD. We shall have more to say about them below.
[*The $\as$ lineup.*]{} Any of the short-distance cross sections above allows a determination of the strong coupling. For example, in a factorized cross section, Eq. (\[fact\]), given parton distributions $f$, we compare $\hat\sigma^{\rm PT}(\as)$ to experiment, and solve for $\alpha_s$, typically evaluated at a renormalization scale equal to the factorization scale. Other, more refined, choices [@commensurate] are related by perturbative corrections. For cross sections that are infrared safe, as jet or event shape cross sections at LEP [@duchesneau], the comparison is even more direct, although power corrections, as in Eq. (\[1overQ\]), must be taken into account for any precise determination [@gary; @dokshitzer]. Yet another way of determining $\as$ is from lattice QCD calculations of energy level differences in heavy quark systems. These may be related, on the one hand to the strong coupling, and on the other to experiment, which then determines the size of the coupling at a scale that grows with the heavy quark mass.
For ease of comparison, the couplings can be evolved to the mass of the Z, at which a “world average" of $\alpha_s(M_Z)=0.119 \pm 0.004$ has been quoted in Ref. [@Betheke98]. Since varied experiments measure the coupling over a wide range of scales, Fig. \[alphafig\], which shows the variation of measured $\alpha_s(\mu)$ with $\m$ illustrates one of the great successes of renormalized field theory.
Resummation
-----------
In the presence of multiple scales, high order contributions to hard scattering functions $\hat \sigma^{\rm PT}$ can become important. For example, in Eq. (\[fact\]), the limit $x/\xi\to 1$ is associated with integrable singularities that usually enhance the cross section. These “threshold" singularities were resummed long ago for the Drell-Yan cross section, and have been applied for some time to top production at leading-log accuracy [@topresum]. In QCD hard-scatterings, such as heavy quark and jet production, color exchange makes the resummation somewhat more complex beyond leading log. This problem has now been solved, and threshold resummation is understood in principle for a wide variety of hard scattering cross sections [@colorresum1; @colorresum2; @colorresum3]. As a practical matter, differing approaches to the inversion of certain Mellin transforms can lead to differing numerical predictions. From a broad perspective, however, the main lesson is that the theory stays relatively close to NLO for cross sections like top production, even in the presence of superficially large corrections at higher order. An important observation [@topresum; @colorresum2], made particularly clear in the very recent publication [@colorresum3], is a marked decrease in factorization-scale dependence for resummed cross sections.
Another example, currently being discussed widely, is related to the data of Fig. \[E706fig\], single-photon and pion inclusive cross sections measured by the fixed-target Fermilab experiment E706 [@begel; @e706]. So far, this data can be fit only by supplementing NLO with the old method of $k_T$ smearing for the initial partons [@706cteq]. At the same time, it has been noted that the full range of fixed-target direct photon data may not be consistent among themselves [@aurenche]. It should also be noted that at collider energies, experiment and NLO agree, at least at the higher transverse momenta [@gordon].
Can the resummation of higher orders in QCD lead to an effective $k_T$ smearing? For W or Z production at low transverse momentum, the answer is yes, and the formalism has existed for some time. In this case, fitting the cross section [@d0wpt] requires the introduction of nonperturbative parameters that are accessible to experiment. In the past year or two, there has been some discussion on the best way of going about this [@ellisvesl; @yuanmor], but the underlying theory is relatively well-understood. The same cannot be said for W or photon cross sections at high transverse momenta, because the logarithms whose resummation requires the nonperturbative input at $Q_T\sim 0$ cancel order-by-order in perturbation theory in the calculation of $\hat \sigma$ at high $Q_T$. Interesting observations on the relationship between the two regimes have been made, however [@likt], and futher progress in this direction can be anticipated in the coming year.
One of the major challenges in perturbative QCD is the development of a theory of these and related higher-order effects, and a method for estimating their importance. We shall return to developments on resummation below.
The Long and Short of Hadron Structure
======================================
Parton Distributions 1999
-------------------------
The parton distribution functions (PDFs) in Eq. (\[fact\]) summarize the structure of hadrons as seen at short distances, one parton at a time. As the hadron is probed at ever shorter distances, the distributions evolve perturbatively according to Eq. (\[evol\]), but the boundary condition for this evolution stands as a truely nonperturbative reflection of the dynamics that holds the hadron together. These parton distributions have been the subject of intense study since the recognition of approximate scaling in DIS structure functions thirty years ago.
Over the past decade, two groups, CTEQ and MRS, whose memberships have themselves evolved somewhat, have undertaken coordinated “global" approaches to the determination of parton distributions, taking into account data from a variety of processes and momentum scales. The past year has seen the development of the latest, best fits of these two groups, MRST [@mrst] and CTEQ5 [@cteq5], which were discussed and compared at the conference in [@wkttalk].
Global fits test the self-consistency of factorized cross sections in the sense that the fits are overconstrained, and because they can be checked against experiments not incorporated into the fits. Nevertheless, the fits must be improved as the data improves, and as it extends to extreme values of fractional momenta. Surprises can occur, especially in regions where a particular parton distribution does not contribute at leading order. Examples of such refinements from 1998 involved the ratio of $d$ to $u$ quarks in the proton, as tested by the W asymmetry [@Wasy] and Drell-Yan [@DYdtou]. Generally speaking, indirect constraints on parton distribution functions are tentative.
Other cases where the results of global fits have been rethought involve higher-order or power (“higher-twist") corrections to the cross section. The extraction of PDFs requires a parameterization of such effects, assuming that they can be brought under control. Examples include the role of higher twist in the extraction of neutron PDFs from deuterium data [@yangbodek].
Up to this cycle of global fits, the primary processes employed were DIS, Drell-Yan and direct photons, the latter thought to be especially valuable for constraining the gluon distribution, which does not appear at leading order in the other two. The data of E706 [@begel; @e706], however, a sample of which was shown above in Fig. \[E706fig\], has thrown this neat picture into disarray, as it disagrees decisively with the predictions of Eq. (\[fact\]) at NLO. A $k_T$ smearing approach has been used in the MRST fits of last year [@mrst]. The CTEQ5 fits abandon direct photons in favor of jet cross sections to constrain the gluon at large $x$ [@cteq5].
Yet another interesting problem is the treatment of heavy quarks. For energies near a heavy-quark mass, $m_Q$, their production may be calculated directly as a hard process, part of $\hat \sigma^{\rm PT}$ in Eq. (\[fact\]). For energies much above $m_Q$, however, it may be advantageous to treat the heavy quark as a parton, thus automatically resumming logs of $m_Q^2/s$. A number of schemes to make this transition have been proposed [@cteq5; @mrst; @svnhq; @thornerv], to treat charmed quark production at HERA.
The sophistication of these considerations of global analysis, and the need to make accurate predictions at high energy raises the question of how to estimate uncertainties in the distributions [@ctequncer]. In part to explore these issues, new sets of distributions have been produced based on DIS data only [@disPDF], and methods have been introduced to quantify uncertainties systematically through statistical analysis [@bayseanpdfs].
Spin and Off-diagonal Distributions
-----------------------------------
The past few years has seen a rebirth of interest in the high-energy physics of spin, which, with advances in the technology of polarized beams and targets, has made possible the systematic study of spin at the parton level [@doyle]. The DIS cross section for a nucleon with spin $s$ may be represented in the standard form ${d^2\sigma/d\Omega dE}
=
(\alpha_{\rm EM}^2/2mQ^4)(E_e/ E'_e)
L^{\mu\nu}W_{\mu\nu}$, in terms of spin structure functions defined as W\_ = W\_\^[unpol]{} + [iE\_e-E’\_e]{}\_q\^s\^ g\_1(x,Q) + [i(E\_e-E’\_e)\^2]{}\_q\^g\_2(x,Q) . \[spinstructure\] The function $g_1$ has a particularly transparent interpretation in terms of quark helicity: $g_1(x,Q) = {1\over 2}\sum_f e_f^2\Delta q_f(x,Q) + O(\alpha_s)$, where $\Delta q_f$ is the difference between the distributions of quarks with helicity parallel to the hadron’s helicity and against it. At this conference, precision data for $g_1^p$ and $g_1^n$ were presented by E155 [@mitchell].
One result of these measurements is a test of of the benchmark Bjorken sum rule, \_0\^1 ( g\_1\^p(x)-g\_1\^n(x))dx = [16]{} |[g\_Ag\_V]{}| (1+c\_n\_s\^n) , \[bjsr\] which is now verified to a new level of accuracy.
The “spin" distributions of the nucleon, such as $\Delta q_f$, do not necessarily describe its full spin content, and the possibility of orbital angular momentum must also be taken into account. At the same time, for gluons the distinction between these two types of angular momentum is not gauge invariant. This problem notwithstanding, an attractive formalism for the description of orbital angular momentum has been proposed [@xji], in terms of form factors, $J_{q,g} = {1\over 2}\; \left[A_{q,g}(0)+B_{q,g}(0)\right]$, which arise in matrix elements of the energy-momentum tensor, p+| T\_[q,g]{}\^ |p= |u(p+)u(p) , \[offdiagonal\] with $\bar p=p+\Delta/2$. At zero momentum transfer, the $J_{q,g}$ become expectation values of the angular momentum operators \_q = d\^3x , \_g = d\^3x([**E**]{}) . The measurement of these off-diagonal matrix elements, unlike the diagonal matrix elements that define classic PDFs (see Eq. (\[msdist\]) below), require the measurement of exclusive, or semi-exclusive amplitudes, such as “deeply-virtual Compton scattering", $\gamma^*(Q^2)+N(p)\rightarrow \gamma+N(p')$, with $p^2=p'{}^2=m_N^2$. The measurement of such amplitudes is a challenge, but one that is of great interest for the Jefferson Laboratory facility, and there is a correspondingly vigorous theoretical program to study the factorization and evolution properties of off-diagonal distributions [@radyod; @collinsod; @balitskyod]. Off-diagonal matrix elements interpolate, in some ways, between inclusive and exclusive processes, and between parton distributions and hadronic light-cone wave functions [@brodsky].
The consideration of orbital angular momentum leads us to the doorstep of the long-distance, low-energy properties of hadrons, where progress has continued in lattice QCD and instanton studies.
Lattice Hadron Spectra, Quark Masses
------------------------------------
Lattice methods [@sharpe] approach QCD from a limit complementary to perturbation theory, and make possible direct calculation of long-distance properties of hadrons. Typically, lattice computations involve the evaluation of expectations of nonlocal products of operators, such as C\_J = 0 |J(x) J(0)| 0 , J = \_i |q\_i(x+) O\_[ij]{} q\_j(x-) , \[lattvev\] with the $q_i$ quark fields, where $O_{ij}$ projects a set of quantum numbers. For $x\rightarrow \infty$, the $x$-dependence is dominated by the energy of the lowest-lying state(s) of the relevant quantum numbers. As noted above, such studies can be used to set the scale for $\alpha_s$ by studying the hyperfine splitting in heavy quarkonia.
The numerical evaluations of expectations like (\[lattvev\]) are said to be quenched, partially quenched, or fully unquenched, depending on how many species light fermions are allowed to fluctuate out of the vacuum. At the conference, progress was reported in quenched and unquenched lattice QCD, on the calculation of realistic spectra for hadrons, and of matrix elements of hadrons involving both heavy and light quarks [@kuramashi; @gottlieb]. Variations on this theme are now making possible the calculation of realistic decay matrix elements [@gottlieb; @simone; @pukurovsky]. In addition, they allow the exploration of ideas on the mechanisms of confinement [@cornwall]. In alternative formulations, lattice methods may be applied to light-cone formulations of hadronic structure [@brodsky; @vandesande].
The current sophistication of lattice techniques, and the growing power of new machines, including Teraflop computers at the U. of Tsukuba and at Brookhaven(RIKEN)-Columbia, are making possible a new generation of investigations of chiral symmetry breaking.
Roughly speaking, chiral symmetry in QCD (also discussed at the conference in [@tandean]) reflects the observation that gluon emission doesn’t change helicities. As a result, in the absence of quark masses in the propagator, left- and right-handed quarks decouple altogether in perturbation theory. This means that a massless quark stays massless in perturbation theory. Chiral symmetry, however, is broken nonperturbatively at zero temperature, but restored at finite termperatures, a transition which can also be studied on the lattice [@negele; @vranas]. At the same time, current algebra requires that the square of the pion mass would vanish linearly with the lightest quark masses, m\_\^2 \~m\_[light]{} . \[pionqkmass\] Historically, this kind of relation has been difficult to realize on the lattice because of corrections that vanish as a low power of the lattice spacing in the continuum limit.
With the advent of ever more powerful machines, the method of “domain wall fermions" [@kaplan] makes possible new approaches to relations like Eq. (\[pionqkmass\]). In the domain wall technique, the chiral symmetry is manifest, up to [*exponential*]{} corrections, at the price of introducing a 5th dimension, which we label by coordinate $s$, in the five-dimensional Dirac Lagrange density = | \[D\[A\] + m(s)\] , \[wallL\] where the mass parameter depends on $s$, and vanishes at endpoints 0 and $s_0$: $m(0) = m(s_0)=0$. The light quarks are zero modes that propogate in “our world" at $s=0$ and $s_0$, one wall for each helicity. These new methods have shown early success [@vranas; @blum] in preserving chiral symmetry, in terms of relations such as Eq. (\[pionqkmass\]). In the lattice domain wall construction, the extra dimension is just a convenience, but it cannot be denied that the technique bears an eerie resemblance to the brane constructions of modern string theory [@terning].
Hadrons and Instantons
----------------------
Successes in the treatment of chiral symmetry in lattice QCD lead us naturally to the reemergence of instanton studies of hadron dynamics [@Shuryak]. In the language of spontaneous symmetry breaking, the masses of hadrons are related to the generation of quark “condensates" $\langle \bar q q\rangle
\sim \langle \bar q_R q_L\rangle+\langle \bar q_L q_R\rangle$ in the QCD vacuum, which couple (the perturbatively decoupled) left- and right-handed components of the quark field.
Instantons may be thought of as tunneling events between the inequivalent QCD vacuum configurations that are distinuished by different phases of the nonabelian gauge fields at infinity (even at zero energy). The instantons couple to the quarks, and the process of tunneling produces an effective $2N_f$-point interaction between their left- and right-handed components, of the general form =G , \[fourquark\] even though the quarks are massless.
In the “instanton liquid" model, the nonperturbative soft gluon field is replaced by an ensemble of instantons and anti-instantons that couple left- and right-handed quark-antiquark pairs. Meson masses are produced as pairs “hop" between instantons, changing helicities as they go. In this model, mesons and baryons, with realistic spectra and matrix elements emerge, and provide a cross-check for lattice calculations [@negele] at zero and at high temperatures.
Factorizations, Evolutions and Effective Theories
=================================================
Every hard-scattering experiment includes a complete evolution all the way from short distance to long distance dynamics. Factorization allows us to organize the long distance dynamics, and thus to calculate perturbative short-distance dependence, and compare the results to experiment. The essence of factorization is to interpret long-distance information in terms of matrix elements in the underlying theory. For example, in the classic factorization in Eq. (\[factevol\]) for DIS structure functions of hadron $h$, $F^{(h)}(Q)= \sum_a C_a(Q/\mu)\otimes f_{a/h}(\mu)$, the quark ($a=q$) distributions may be interpreted as f\_[q/h]{}(x,) = \_[-]{}\^ \^[-ixPny]{} h(P)| |q(yn)n \_n(y,0) q(0) |h(P) , \[msdist\] where $n$ is a lightlike vector not in the direction of the hadron momentum $P$, and $\m$ enters as the renormalization scale for the matrix element, which is ultraviolet divergent for $n^2=0$. The function $\Phi_n(y,0)$ is a path-ordered exponential of the gauge field, \_n(y,0)= P , \[oexp\] which makes the matrix element gauge invariant. The evolution of Eq. (\[evol\]) may be thought of as a consequence of the renormalization properties of the nonlocal operators in $f_{a/h}$, which summarize an infinite set of twist-two matrix elements in the light-cone expansion. The same formalism is at the basis of heavy quark effective theory and of nonrelativistic QCD (NRQCD) [@benekerv], although generally with a finite sum over operators rather than a convolution. In a sense, the operator $\Phi_n$ in Eq. (\[msdist\]) plays the role of the heavy quark field in heavy quark effective theory, as a nonrecoiling source of gluons.
Over the past few years, factorization, analyzed in terms of effective operators, has been applied to multiscale hard scattering processes, with Sudakov [@cls; @sotir] and Regge high-energy limits. The former refers to cross sections with large momentum transfer and low QCD radiation, the latter to low momentum transfer and essentially unlimited radiation. The Regge limit is related to the total cross section, while the Sudakov limit highlights its short-distance components.
One of the simplest examples of the Sudakov limit is the dijet cross section in $\e^+\e^-$ at fixed jet masses $m_i^2$, $i=1,2$. In the limit of light jets, $m_i^2\ll Q^2$, the dijet cross section is related to the integrated thrust cross section by \_T\^1 dT’ = dm\_1\^2dm\_2\^2 ( 1-T-[m\_1\^2+m\_2\^2Q\^2]{}) . The cross section for “nearly-lightlike" jets in $\e^+\e^-$ satisfies a factorization [@cls] = [C(Q/,\_i)Q\^6]{} \_[i=1]{}\^2 J\_i(,\_i) S(,\_i) , \[dijetfact\] up to corrections suppressed by powers of $m_i^2/Q^2$, with $\beta_i$ the 4-velocity of jet $i$. In (\[dijetfact\]), there is a double factorization, separating the dynamics of the jets, included in the functions $J_i$, from both the truely short-distance “coefficient" function $C$ and from the dynamics of relatively low-energy partons emitted coherently by the jets and included in the function $S$. The “soft" function $S$ is associated with a particularly interesting composite operator in QCD. $S$ describes the emission of gluons whose wavelenths are so long that they cannot resolve the internal structure of the jets, and are thus generated by the product W(0)=\_[\_1]{}(,0) \^\_[\_2]{}(,0) , \[Wdef\] where, in the notation of Eq. (\[oexp\]), the $\Phi$’s are ordered exponentials, and $\beta_1$ is the velocity of the quark jet, and $\beta_2$ of the antiquark jet.
We need not dwell on the nature of the convolutions denoted by $\otimes$ in Eq. (\[dijetfact\]), but the double factorization itself is adequate to imply a resummation [@cls] of double logarithms in $1-T$ [@cttw], \_T\^1 dT’[d(T’)dT’]{} = , \[thrustresum\] where corrections include fewer logarithms of $1-T$ in the exponent. These results are also related to the renormalization properties of the operators $W$ in Eq. (\[Wdef\]).
Another application of factorization and effective operators is to resummation in the “Regge" limit $s\to \infty$, $t$ fixed, the “BFKL" regime for QCD. The BFKL equation may be derived from a “multiperipheral" reexpression of DIS factorization, Eq. (\[factevol\]) [@tasi95; @balitsky; @kw_lievol]: F(x,Q\^2)&=& \_x\^1 [d]{} C([x]{},[\^2Q\^2]{})G(,Q\^2)+[O]{}([1/Q\^2]{})\
&=& d\^2k\_T c([x’]{},Q,k\_T)(’,k\_T)+[O]{}([1/(1/x)]{}) , \[xrefact\] where the $k_T$-dependent distribution $\psi$ is related to the gluon PDF by [@ccfm]: G(,Q\^2)=\^Qd\^2k\_T (,k\_T) . \[psiktdef\] In the second form of Eq. (\[xrefact\]), the roles of longitudinal and transverse momenta have been reversed, and corrections are suppressed only by logarithmics of $x$, rather than powers of $Q$. In these terms, the BFKL equation, = d\^2k’\_T(k\_T,k’\_T)(,k’\_T) = -[N\^2]{} , \[bfkl\] with $\tilde\psi\equiv(1/k_T^2)\psi$, describes the evolution of $\psi(\xi,k_T)$ in $\xi$. The same equation may also be derived from the renormalization of ordered exponentials, like Eq. (\[oexp\]). Indeed, distributions of exponentials integrated over a transverse density [@balitsky] d\^2x\_T (x\_T) P \[distribution\] are being studied to develop “unified" effective theories that describe the variety of evolution equations, including DGLAP, BFKL and others [@kw_lievol].
BFKL, Diffraction and Color Dynamics
====================================
BFKL at NLO
-----------
What’s so special about the BFKL equation? It addresses the total cross section in gauge theory in terms of its fundamental quanta, the quarks and gluons. Now it is not entirely obvious that such a project will work, ultimately, in an asymptotically free theory with confinement, but if it does, it will say something fundamental about field theory. In an older language, this would be a theory of the “pomeron" [@levin1]. Also, in the language of the parton model, BFKL appears to predict that, as we evolve to low $x\ \leftrightarrow$ high $s_{\gamma^*N}$ at fixed $Q$, we reach a region of high parton density at nearly fixed (actually slowly diffusing [@muellerdiff]) virtuality. This is a new “intermediate" regime of QCD, between perturbative and hadronic phases [@mueller98]. It is relevant to the screened “plasma" state, which we hope to encounter at RHIC.
The solutions to the lowest order BFKL equation are of the form \~x\^[-ø]{} ([k\_T\^2\^2]{})\^[-i-1/2]{} . The largest permissible value of $\omega$ gives the dominant low-$x$ behavior, which is found to be =0, ø= ø\_0 4N2(/) . \[losoln\] From the kinematic relation in DIS, $s_{\gamma^* N}= Q^2(1-x)/x$, the low-$x$ behavior of $\psi(x,k_T)$ determines the large-$s$ behavior of the $\gamma^* N$ total cross section, for which the lowest-order BFKL result (\[losoln\]) gives \~“\_[tot]{}"\~s\^[4N2(/)]{} . \[sigtot\] This is a derivation of Regge-like behavior for the total cross section from perturbative QCD, and, because it involves the exchange of no overall quantum numbers, may be considered as a perturbative model for the pomeron.
Where should one look for BFKL behavior in experiment? Suggestions include correlations in dijet and rapidity-gap cross sections in DIS and ${\rm p}\bar{\rm p}$, and in non-DGLAP evolution in DIS at low $x$ and moderate $Q^2$. In the first case, there may be hints in the dijet data from HERA and in the comparison of jet correlations at 630 and 1800 GeV [@krane]. In the later case, the strong rise in $x$ of the structure functions $F_i(x,Q^2)$ cannot be sustained indefinitely, since, by (\[sigtot\]), this would eventually violate unitarity bounds for the cross section. Before this happens, interference between partons, or “shadowing", which is absent in both DGLAP and BFKL evolutions, must begin to set in [@gotsman]. One of the much-discussed data presentations of the past year, Fig. \[slopeplot\] from the ZEUS collaboration, shows the transition between perturbative and nonperturbative behavior in a particularly suggestive form [@zeusv].
1998 was the year of the NLO BFKL kernel, the year in which the decade-long project of computing the next-to-leading order kernel in Eq. (\[bfkl\]) bore fruit [@nlo2loop]. At NLO, $\cal K$ is fairly complicated, but the effect on $\o$ in Eq. (\[losoln\]) is simple enough, ø\^[(NLO)]{}= ø\_0 . \[omeganlo\] Now 6 and a half is not by itself a large number, but the size of this correction nevertheless presents a challenge, because unless $\alpha_s$ is quite small, the second term may overwhelm the first and lead to an unrealistic falling cross section. In addition, it has been observed that the NLO kernel even implies a non-Regge behavior at high orders [@nlorun]. Interesting responses to these challenges were discussed in [@schmidt]. Evidently, the NLO fruit of BFKL will be an acquired taste, but the coming year surely promises intensive work and further clarification
Diffraction and Diffractive PDFs
--------------------------------
Through the optical theorem, the total cross section is closely related to elastic and diffractive scattering amplitudes [@erdmann]. In diffractive DIS, we can relax inclusivity and thus probe QCD dynamics in the final state, while retaining a large momentum transfer. Convenient variables to describe diffraction are x\_[P]{}=[M\_X\^2+Q\^2W\^2+Q\^2]{} =[xx\_[P]{}]{} , \[diffx\] where $M_X$ is the mass of an observed system $X$ moving in the “current" ([*i.e.*]{} photon) direction, and $W^2=s_{\gamma^*s}$. $x_{\cal P}$ is the fractional longitudinal momentum transfer from the nucleon to $X$, and $\beta$ is the equivalent fractional momentum of a parton in the (hypothetical) exchanged “pomeron".
Diffractive events are typically defined by a large gap in rapidity between $X$ and the elastically-scattered, or diffractively-excited, proton (or its low-mass fragments), which has experienced invariant momentum transfer $t$. In these terms, a fully differential cross section is = [2\_[EM]{}\^2Q\^4]{} ( 1+(1-y)\^2) F\^[D(4)]{}\_2 . \[diffcrossx\] In the simplest diffractive processes $X$ consists of a single vector boson. In this case, the relevant amplitude is factorized in terms of off-diagonal PDF’s, related for small $x$ to the gluon distribution, which behaves as $G(x)\sim x^{-\lambda}$. The resulting cross section is proportional to $G^2(x)$, and hence increases with $W$, as \~[1M\_X\^4]{} W\^[4(M\_X)]{} . There is compelling evidence for this behavior, with a value of $\lambda$ increasing with $M_X$, suggesting once again a Regge-like behavior reminiscent of BFKL. This connection has yet to be completely explored.
High-$Q^2$ DIS diffractive cross sections may be factored [@difffactprf] using specifically diffractive PDFs, also referred to as fracture functions [@fracture], F\^D=\_a C\_a\^Df\_a\^D . \[diffact\] Phenomenological fits to $f_{q,g}^D$’s have been carried out [@diffpdf], and lead to predictions whenever a factorization like Eq. (\[diffact\]) applies [@hautmann]. It is important to realize, however, that because diffractive PDFs are not fully inclusive, they depend on the details of evolution into the final state, in particular the likelyhood of the target proton staying together. This probability cannot be expected to be the same in ${\rm p}\bar{\rm p}$ cross sections, where the fragments of two initial-state hadrons pass through each other, as in DIS, where only a single hadron is involved. And indeed, studies [@alves] have shown that diffraction at the Tevatron is much less likely than would be suggested by a direct generalization of Eq. (\[diffact\]) to this case with universal diffractive PDFs.
Nevertheless, double-diffraction (double rapidity gap) jet production is seen at the Tevatron, and the jets show a standard parton-parton $E_T$-dependence [@alves], indicating that the short-distance process is independent of the long-distance evolution. This suggests that another factorization is possible in this case, and may shed light on diffractive dynamics.
Color Dynamics
--------------
Diffractive processes are naturally interpreted in terms of color-singlet exchange in the $t$-channel. Large momentum transfer processes, however, are described perturbatively in terms of single-gluon exchange, carrying octet quantum numbers. Over the past few years, there has been progress in understanding the relation between these two pictures.
Although color is not observable, representations ($1,q=3,g=8\dots$) are, at least in principle. In NRQCD, a factorization that includes the mixing of operators with differing color content has already led to valuable insights and phenomenological successes [@benekerv]. For high-energy processes, rapidity gaps were long ago suggested by Bjorken as an ideal arena to study color exchange, with singlet exchange expected to produce an excess of events with very low interjet multiplicity.
It is clear, however, that it is not possible to separate short- from long-distance color exchange uniquely, since gluons of all momenta carry the same color content, and models in which the color content of the final state is determined at the longest distances have had success [@halzenzepp]. At the same time, energy flow into regions between two high-$p_T$ jets is senstive to all time scales between $1/|p_T|$ and $1/\Lambda_{\rm QCD}$. Energy flow at the shorter time scales is both perturbative and sensitive to the color content in the $t$-channel. This observation led [@OdSt] to an analysis of dijet cross sections in terms of energy flow $Q_c$ into the interjet region, in the range $\Lambda_{\rm QCD}\ll Q_c\ll \sqrt{-t}$. The cross section at fixed $Q_c$ may be factorized, and logarithms of $Q_c^2/t$ resummed. This behavior is found from the renormalization properties of composite operators that are products of ordered exponentials, which generalize Eq. (\[Wdef\]) to the $2\to 2$ scattering of partons with color exchange (labelled here by f), W\^[ f]{}\_[\_j\_j]{}(0) = \_[i=3]{}\^4 \_[n\_i]{}(,0)\_[\_i,\_i]{} T\^[ f]{}\_[\_4…\_1]{} \_[i=1]{}\^2\_[n\_i]{}(0,-)\_[\_i,\_i]{} \[Wfdef\] where $T^{\; \rm f}$ is a matrix that couples the color of the incoming and outgoing ordered exponentials $\Phi_{n_i}$, representing the active partons of the hard scattering. These operators mix under renormalization and induce an evolution that tracks mixing in the color space as the scattering particles (including $q\bar q,\, qg,\, gg$) evolve from short to long distances. This analysis offers a new set of predictions for $p_T$, energy and rapidity dependence of gap events, which can be tested at Run II of the Tevatron, and at the LHC.
Power corrections
=================
There has been considerable interest in power corrections to infrared safe quantities. As noted in Sec. 2 above, such power corrections are quite important in the phenomenology of jet cross sections and event shapes in $\e^+\e^-$ annihilation [@gary; @duchesneau]. Behind this work is a hypothesis, that it is not necessary to model the details of hadronization to parameterize leading corrections to perturbation theory, and a hope, that plausible parameterizations inspired by perturbation theory will lead to useful insights at the perturbative-nonperturbative interface [@webberlect]. The hypothesis seems to be correct; whether the hope will be realized remains to be seen, but there are preliminary indications that it might be [@irr].
How does perturbation theory imply nonperturbation corrections? In the calculation of any IR safe quantity at NLO, we always encounter integrals of the general form I(,p)=f(x) \_s(Q\^2) \_0\^Q dk k\^p,\[Ias\] with $p>-1$, where $k$ may be thought of as a gluon momentum scale, and $f(x)$ represents the remaining (IR finite) dependence. Now in many cases (see Eq. (\[thrustresum\]) for example) we can argue (or derive) that higher order corrections modify (\[Ias\]) to I\^[(resum)]{}(,p) =f(x)\_0\^Q dk (k\^2) k\^p … . \[Irunas\] This result shows that the perturbative reexpansion in terms of $\alpha_s(Q)$ is asymptotic, with high orders that grow as $n!$ at large $n$. This information is encoded in the singularity of the perturbative expression for $\alpha_s(k^2)$ at $k=\Lambda_{\rm QCD}$. We can further reinterpret this behavior by means of a Borel transform, but the inverse transform will not be unique in any case. Taking a more practical approach, we retain the perturbative factor $f(x)$ and the perturbative range in the $k$ integral in Eq. (\[Irunas\]), and simply replace the lower end of the integral, $k<\mu_1$, for some fixed $\mu_1$, by a parameter $\alpha_p(\mu_1)$, \_0\^[\_1]{} dk (k\^2) k\^p \_1\^[p+1]{} \_p(\_1) , \[irreplace\] Since the overall integral in Eq. (\[Ias\]) behaves as $Q^{p+1}$ this is automatically a power correction. Clearly the value of this approach depends on the assumption that $\alpha_p(\mu_1)$ is in some sense universal [@irr]. Evidently, this is true approximately, and this method has found applications in models for DIS higher twist [@disht]. As the notation suggests, the parameter $\alpha_p$ is often thought of as a reflection of a universal, nonperturbative low-scale running coupling. This is suggested by (\[Irunas\]) above, where it is a moment of the lowest-order running $\alpha_s(k^2)$. It has been argued that the relation is more general, and that higher orders of $\alpha(k^2)$ may be incorporated into a reconstructed effective coupling, defined through dispersion relations [@dispersive; @milano].
In interpreting these developments, it is important to keep in mind that the values of higher-twist parameters cannot be defined independently of perturbation theory [@irr], and that they will change as new orders, or resummations, are computed. A striking example of this effect was illustrated by the NNLO analysis of [@nnlodis], which reduced the size of higher twist contributions, relative to those found in fits based on NLO.
Another interesting application of these ideas is to resummed event shapes, as in Eq. (\[thrustresum\]), where a replacement like Eq. (\[irreplace\]) leads to a simple shift [@shift] in the thrust ($T=1-t$) distribution, which vanishes as $1/Q$, +[O]{}([1(tQ)\^2]{}) , \[thrustshift\] with $\lambda$ a constant related to $\alpha_0$ in (\[irreplace\]). In a somewhat more general approach, we may once again factorize soft gluon emission into the region between the two jets, and derive a convolution expression for the cross section [@shape], \_0\^[tQ]{} d f() +[O]{}([1tQ\^2]{}) , \[shapeeq\] where $f(\epsilon)$, a “shape function", has a field-theoretic interpretation [@shape; @ftshape], which involves the composite operator of Eq. (\[Wdef\]). $f(\epsilon)$ is $Q$-independent and summarizes all $1/(tQ)^n$ corrections, while Eq. (\[shapeeq\]) reduces to (\[thrustshift\]) with the replacement $f(\epsilon)=\delta(\epsilon-\lambda)$. A fit [@shape] to $f(\epsilon)$ using the extensive thrust data at $Q=M_Z$, and the perturbative resummation of [@cttw], faithfully predicts $d\sigma/dT$ for a wide range of $Q$, as shown in Fig. \[thrustfig\]. Given the discussion of the foregoing section, by following this line of reasoning we may hope to relate event shapes in $\e^+\e^-$ annihilation to energy flow in hadronic hard-scattering cross sections. This relation remains unexplored, although it is certainly related to multiplicity and correlation studies of the final states in jet events [@giacomelli].
. \[thrustfig\]
QCD at High Temperature and Baryon Number
=========================================
I have already mentioned the path from perturbative QCD to high parton density through BFKL evolution. Certainly, these considerations are made more interesting by the pending turn-on of the RHIC accelerator at Brookhaven, at which nuclei will be collided at unprecented energies. This development has led to a fresh look at QCD in “extreme" conditions, long of relevance to studies of the early universe.
[*Color Superconductivity.*]{} At high enough density and temperature, the long-distance interactions that lead to confinement in the normal, hadronic phase of QCD are screened, and in some sense nonsinglet degrees of freedom are freed. In fact, QCD is expected to have a possibly quite rich phase structure in the plane of temperature (T) and baryon density (B).
An exciting exploration of these features of the theory is the fresh look at the long-standing conjecture of color superconductivity at large B and low T, in the light of the instanton liquid model referred to in Sec. IIID. It was realized in Ref. [@supercond] that the four-fermion effective Lagrange density in Eq. (\[fourquark\]) produces an attractive potential between quarks, which can lead to a condensate of quark Cooper pairs at the Fermi surface, in a manner analogous to superconductors of electric current. In fact, the condensate can be driven by gluon exchange, but the energy gaps produced by instantons are much larger in most, but not all [@son], of parameter space. Although more likely to be relevant for neutron stars than nuclear collisions, these interesting considerations were clearly inspired by RHIC physics, which has led to an efflorescence of studies of the QCD B/T plane [@tricrit; @contin].
[*Energy Loss.*]{} As a final example, I will very briefly refer to some recent considerations on a topic of direct interest to RHIC and hadron-nucleus scattering, energy loss of fast partons in dense media. High-energy partons travelling through matter (partons or hadrons) will scatter and radiate, and their evolution into the final state will be modified in some way. In sufficiently inclusive processes, these effects are (perhaps surprisingly) higher twist [@guo]. It is of interest, however, also to look at changes in radiation at transverse momentum scales set by the medium, rather than a hard scattering. Along these lines, recent work on the energy loss [@energyloss] experienced by a quark travelling through a medium over length $(L)$ has identified a QCD analog of the famous Landau-Pomeranchuk effect in QED. This work analyzes “induced" gluon radiation at $\L\ll k_T\ll \langle Q\rangle$, where $\langle Q\rangle$ is the typical momentum transfer in a projectile-medium scattering. If the Debye length of the medium is short enough in a hot, dense medium, $Q$ could be perturbative.
The amplitude for an emission $q\rightarrow q+g$ at impact parameter $b$ is denoted $f(b,t)$. The effect of the scatterings is a diffusion in $b$, f(b,t) \~b Q\^2[ddQ\^2]{} , \[fbt\] with known corrections to the Gaussion. For large $\langle Q\rangle$, the cross section involves the interference between amplitudes where the radiation occurs at different positions along the path through the medium, leading to a power spectrum in path length and frequency given by \~\_s \_0\^L dt (1-[tL]{}) d\^2b f(b,t) f(b,0) . This results in an energy loss per unit length that grows with length (!): -[dEdz]{} \~[\_s N\_cL4]{} Q\^2[ddQ\^2]{} , at mean free path $\lambda$, which is a diagnostic for $\left\langle Q^2 {d\sigma\over dQ^2}\right\rangle$, potentially able to distinguish the composition of the medium, whether hadrons, Debye-screened plasma or something else. Here again, the transition from energy loss at relatively low transverse momentum, to more inclusive cross sections, should be an interesting one [@jetloss].
Conclusion
==========
The central conclusion of this little review is the variety and vitality of the work itself. Beyond this, the nature of our knowledge of QCD is such that important ideas and techniques only require the possibility that they can be tested to receive further theoretical development. The ongoing experiments at LEP, HERA and the Tevatron have already transformed perturbative QCD into a truely quantitative discipline. The pending RHIC accelerator has inspired creative theoretical developments. A strong QCD component to future high-energy projects is sure to be richly rewarded by insights into quantum field theory .
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I would like to thank the organizers of DPF 99 for the invitation to participate in a vibrant meeting. I am indebted to Werner Vogelsang for invaluable help. This work was supported in part by the National Science Foundation, under grant PHY9722101.
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[^1]: Plenary talk presented at the meeting of the American Physical Society Division of Particles and Fields (DPF 99), UCLA, Los Angeles, CA, 5-9 Jan 1999.
[^2]: I have described some of the technical background in Ref [[@tasi95]]{}.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'A resolvent-based reduced-order representation is used to capture time-averaged second-order statistics in turbulent channel flow. The recently-proposed decomposition of the resolvent operator into two distinct families related to the Orr-Sommerfeld and Squire operators \[K. Rosenberg and B. J. McKeon, Efficient representation of exact coherent states of the Navier-Stokes equations using resolvent analysis, Fluid Dynamics Research 51, 011401 (2019)\] results in dramatic improvement in the ability to match all components of the energy spectra and the $uv$ cospectrum. The success of the new representation relies on the ability of the Squire modes to compete with the vorticity generated by Orr-Sommerfeld modes, which is demonstrated by decomposing the statistics into contributions from each family. It is then shown that this competition can be used to infer a phase relationship between the two sets of modes. Additionally, the relative Reynolds number scalings for the two families of resolvent weights are derived for the universal classes of resolvent modes presented by Moarref *et al.* \[R. Moarref, A. S. Sharma, J. A. Tropp, and B. J. McKeon, Model-based scaling of the streamwise energy density in high-Reynolds-number turbulent channels, Journal of Fluid Mechanics 734, 275 (2013)\]. [These developments can be viewed as a starting point for further modeling efforts to quantify nonlinear interactions in wall-bounded turbulence.]{}'
author:
- 'Ryan M. McMullen'
- Kevin Rosenberg
- 'Beverley J. McKeon'
bibliography:
- 'references.bib'
title: |
Interaction of forced Orr-Sommerfeld and Squire modes\
in a low-order representation of turbulent channel flow
---
\[sec:intro\]Introduction and background
========================================
Techniques from linear systems theory applied to wall-bounded turbulent shear flows have met with much success. For example, analyses of the Navier-Stokes equations (NSE) linearized about the turbulent mean velocity predict the spanwise length scales associated with the near-wall cycle and large-scale structures in the outer region of the flow from both a transient growth [@delAlamo2006; @cossu2009; @willis2010] and energy amplification of harmonic and stochastic forcing perspective [@hwang2010; @willis2010]. More recently, the linearized equations have been used to develop linear estimators [@illingworth2018; @madhusudanan2019; @towne2020] and compute impulse responses [@vadarevu2019] that qualitatively reproduce the coherence and self-similarity of large-scale motions.
As turbulence is an inherently nonlinear phenomenon, a complete model must account for nonlinear interactions. A common approach to incorporate the effects of nonlinearity into linear models is to augment the linearized equations with an eddy viscosity, such that the turbulent mean profile is fixed as an equilibrium solution of the modified mean momentum equation [@delAlamo2006; @cossu2009; @willis2010; @hwang2010; @illingworth2018; @madhusudanan2019; @vadarevu2019]. While this approach justifies linearization about the turbulent mean profile, it precludes the study of finite-amplitude fluctuations, since their nonlinear interactions would feed back on and further alter the mean. Instead of using an eddy viscosity, @zare2017 considered colored-in-time stochastic forcing of the linearized NSE in the problem of completing partially-known second-order statistics. Notably, they demonstrated that their approach can be equivalently represented as a low-rank modification of the original equations.
In a different approach to dealing with nonlinearity, the resolvent analysis framework introduced by @mckeon2010 retains the nonlinear term and interprets it as endogenous forcing of the linear dynamics through triadic interactions with the velocity fluctuations at other wavenumber-frequency combinations. This framework eliminates the need to incorporate an eddy viscosity for self-consistency, as no linearization is performed. [@landahl1967 arrived at a similar formulation, deriving a forced Orr-Sommerfeld equation in the study of wall-pressure fluctuations, but focused on obtaining approximate solutions in eigenfunction expansions.]{}
Closure of the loop requires determination of the forcing such that it yields the correct velocity Fourier modes, as well as the mean velocity profile, which is assumed known. The forcing can be expanded as a sum over a set of basis functions such that the unknowns are the complex amplitudes, called the resolvent weights. An exact equation for the weights can be formulated [@mckeon2013], though it is intractable to solve for most complex flows of interest. Consequently, there have been previous attempts to estimate the weights from data, e.g. by using either a single time series or power spectral density of the velocity fluctuations [@gomez2016; @beneddine2016]. Alternatively, @moarref2014 used convex optimization to compute the weights for a resolvent-based low-order representation of time-averaged velocity spectra that minimize the deviation from spectra obtained from a direct numerical simulation (DNS) of $\Rey=2003$ channel flow [@hoyas2006]. @towne2018 established a link between resolvent analysis and spectral proper orthogonal decomposition (SPOD) and showed that if the resolvent weights are treated as stochastic quantities, their covariance matrix can be calculated from the SPOD modes, which inherently rely on statistical data.
[In special cases where full information of the nonlinear forcing is available, such as for exact coherent states (ECS), the resolvent weights can be computed exactly by projecting the forcing onto the aforementioned set of basis functions [@sharma2016]. For ECS families in channel and pipe flow, which come in pairs of upper and lower branch solutions, the lower branch ones are typically well-represented by only a few resolvent modes, whereas many of the upper branch solutions are not captured as efficiently. Furthermore, the wall-normal and spanwise velocity components converge much more slowly than the streamwise velocity. However, an alternative decomposition of the resolvent operator recently proposed by @rosenberg2019a yields two families of modes related to the Orr-Sommerfeld and Squire operators from classical linear stability theory. By projecting the same channel ECS, they demonstrated that the new sets of basis functions enable a much more compact representation of both branches of solutions, and, notably, all three velocity components converge at roughly the same rate. Subsequent analysis attributed the improved efficacy of the alternative decomposition to the isolation of the wall-normal velocity response into the Orr-Sommerfeld modes, such that the Squire wall-normal vorticity is free to interact with that generated by the Orr-Sommerfeld modes [@rosenberg2018].]{}
[While the utility of the decomposition into Orr-Sommerfeld and Squire modes for highly simplified flows like ECS has been established, an open question is whether or not it remains relevant for high Reynolds number turbulence. In the present work, it is shown that the second-order statistics of turbulent channel flow can be accurately represented using a low-order approximation based on this framework. It is additionally shown that the vorticity produced by the Orr-Sommerfeld and Squire modes act to oppose each other, and this observation reveals information about how the resolvent weights for the two families scale relative to each other with Reynolds number. Altogether, these insights point to a mechanism in turbulent channel flow that is important for low-order modeling efforts.]{}
\[sec:form\]Formulation
=======================
\[sec:res\]Resolvent analysis of turbulent channel flow
-------------------------------------------------------
The approach is based on the resolvent analysis framework of @mckeon2010, in which the incompressible NSE,
\[eq:NSE\] $$\begin{aligned}
\partial_t \tilde{\vect{u}} + \left( \tilde{\vect{u}} \boldsymbol{\cdot} \bnab \right) \tilde{\vect{u}} = - \bnab \tilde{p} + \Rey^\inv\bnab^2 \tilde{\vect{u}}&,
\\
\bnab \boldsymbol{\cdot} \tilde{\vect{u}} = 0&,\end{aligned}$$
here nondimensionalized using the friction velocity $u_\tau$ and channel half-height $h$, are first Reynolds decomposed as $\tilde{\vect{u}} = \vect{U} + \vect{u}$, where $\vect{U} = \begin{pmatrix}U(y) & 0 & 0\end{pmatrix}^\trsp$ is the turbulent mean velocity profile and $\vect{u}$ are the fluctuations about the mean, and then Fourier transformed in the homogeneous wall-parallel and temporal directions $x$, $z$, and $t$, resulting in equations for the Fourier coefficients, denoted by $\hat{\blankop}\,$. For each wavenumber-frequency triplet $\begin{pmatrix}k_x & k_z & \omega\end{pmatrix}^\trsp \neq \mathbf{0}$, we have
$$\begin{aligned}
i\omega \hat{\vect{u}} + \left( \vect{U} \boldsymbol{\cdot} \bnab \right) \hat{\vect{u}} + \left( \hat{\vect{u}} \boldsymbol{\cdot} \bnab \right) \vect{U} + \bnab \hat{p} - \Rey^\inv\bnab^2 \hat{\vect{u}} =& \hat{\vect{f}}
\\
\bnab \boldsymbol{\cdot} \hat{\vect{u}} =& 0,\end{aligned}$$
\[eq:NSEf\]
where $\vect{f} = -\left( \vect{u} \boldsymbol{\cdot} \bnab \right) \vect{u} + \langle \left( \vect{u} \boldsymbol{\cdot} \bnab \right) \vect{u} \rangle$ and $\langle\,\cdot\,\rangle$ denotes an averaged quantity, is interpreted as a forcing that drives the dynamics linear in $\hat{\vect{u}}$. The pressure can be projected out of \[eq:NSEf\] using the standard mapping to wall-normal velocity $\hat{v}$ and wall-normal vorticity $\hat{\eta} = ik_z\hat{u} - ik_x \hat{w}$. The equations are then concisely written as $$\begin{pmatrix}
\hat{v}\\
\hat{\eta}
\end{pmatrix}
= \res\!\left( k_x,\, k_z,\, \omega \right)
\hat{\vect{g}},
\label{eq:NSEres}$$ where $$\mathcal{H} =
\begin{pmatrix}
-i \omega - \lapinv \mathcal{L}^\OS & 0\\
-i k_z U' & -i \omega - \mathcal{L}^\SQ
\end{pmatrix}^{-1}
\label{eq:resop}$$ is the resolvent operator, $\lap = \Dop^2-k^2$, $\Dop=\diff/\diff y$, $k^2=k_x^2+k_z^2$, and $U'=\Dop U$. Additionally,
\[eq:Lossq\] $$\begin{aligned}
\mathcal{L}^\OS &= i k_x\!\left(U'' - U\lap\right) + \Rey^\inv\lap^2,
\label{eq:Los}\\
\mathcal{L}^\SQ &= -i k_x U + \Rey^\inv\lap
\label{eq:Lsq}
\end{aligned}$$
are the Orr-Sommerfeld (OS) and Squire (SQ) operators, respectively. The forcing term $\hat{\vect{g}} = \begin{pmatrix} \hat{g}_v & \hat{g}_\eta \end{pmatrix}^\trsp$ in \[eq:NSEres\] is related to $\hat{\vect{f}}$ via $$\hat{\vect{g}}
=
\underbrace{
\begin{pmatrix}
-i k_x \lapinv \Dop & -k^2\lapinv & -i k_z \lapinv \Dop\\
i k_z & 0 & -i k_x
\end{pmatrix}
}_{\textstyle \mathcal{B}}
\hat{\vect{f}}.
\label{eq:forcing}$$ Note that $\hat{\vect{g}}$ is solenoidal, since the irrotational component of $\hat{\vect{f}}$ lies in the null space of $\mathcal{B}$ [@rosenberg2019a].
With the aim of obtaining a low-order representation of \[eq:NSEres\], we compute the Schmidt decomposition of $\res$: $$\res = \sum_{j=1}^\infty \vect{\psi}_j \sigma_j \langle \,\cdot\,,\vect{\phi}_j\rangle,
\label{eq:svd}$$ where $\sigma_j \geq \sigma_{j+1}\geq 0$ $\forall j$ are the singular values, and $\vect{\psi}_j$ and $\vect{\phi}_j$ are the left and right singular vectors, respectively. Since the $\vect{\psi}_j$ are a basis for the output space, i.e., the space to which the response belongs, they are referred to as response modes; similarly, the $\vect{\phi}_j$ are referred to as the forcing modes. The Schmidt decomposition applies to linear operators on infinite-dimensional vector spaces. For the finite-dimensional matrix approximation obtained from numerically discretizing the operator, this becomes the singular value decomposition (SVD), which we refer to hereafter for simplicity. As is evident from \[eq:svd\], the SVD depends on the choice of inner product $\langle \,\cdot\, , \,\cdot\, \rangle$. On both the input and output spaces we adopt the standard kinetic energy inner product [@schmid2001]: $$\langle \vect{x}_1, \vect{x}_2 \rangle = \int_{-1}^1 \vect{x}_2^* \mathcal{Q} \vect{x}_1 \diff y,
\label{eq:IP}$$ where $\blankop^*$ denotes the conjugate transpose and $$\mathcal{Q} = \frac{1}{k^2}
\begin{pmatrix}
-\lap & 0\\
0 & 1
\end{pmatrix}.
\label{eq:Q}$$ The left and right singular vectors are orthonormal with respect to this inner product, i.e., $\langle \vect{\psi}_j, \vect{\psi}_k \rangle = \langle \vect{\phi}_j, \vect{\phi}_k \rangle = \delta_{jk}$, where $\delta_{jk}$ is the Kronecker delta.
The desired low-order approximation of \[eq:NSEres\] is obtained by truncating the sum in \[eq:svd\]: $$\begin{pmatrix}
\hat{v}\\
\hat{\eta}
\end{pmatrix}
\approx \sum_{j=1}^N \vect{\psi}_j \sigma_j \chi_j,
\label{eq:approx}$$ for some $N\geq 1$. We refer to this as the rank-$N$ approximation. The $\chi_j = \langle \hat{\vect{g}},\vect{\phi}_j\rangle$ are called the resolvent weights and quantify how much of the forcing $\hat{\vect{g}}$ is in the direction $\vect{\phi}_j$. [For broadband forcing in $y$, i.e., $\chi_j = \chi$ $\forall j$, \[eq:approx\] is optimal in the norm induced by the inner product in \[eq:IP\]. Furthermore, if $\sum_{j=1}^N \sigma_j^2 \approx \sum_{j=1}^\infty \sigma_j^2$ for relatively small $N$, $\res$ is said to be effectively low-rank.]{} It has been shown that this property holds for a large portion of spectral space that is energetically significant [@moarref2013], and this low-rank behavior has previously been exploited to model salient features in wall-bounded turbulence [@mckeon2010; @sharma2013; @moarref2013]. [However, the assumption of broadband forcing is in general not valid. For the case of structured forcing, \[eq:approx\] is an accurate approximation of the full system, provided the forcing is not too aligned in any of the truncated directions.]{}
Finally, in order to compute the second-order velocity statistics, the velocity $\hat{\vect{u}}$ is recovered from the response via $$\hat{\vect{u}}
=
%\underbrace{
\frac{1}{k^2}
\begin{pmatrix}
i k_x \Dop & -i k_z\\
k^2 & 0\\
i k_z \Dop & i k_x
\end{pmatrix}
%}_{\textstyle \mathcal{C}}
\begin{pmatrix}
\hat{v}\\
\hat{\eta}
\end{pmatrix}.
\label{eq:uvw}$$
\[sec:OSSQ\]Orr-Sommerfeld and Squire decomposition of the resolvent
--------------------------------------------------------------------
As discussed in \[sec:res\], the decomposition of $\res$ given in \[eq:svd\], hereafter referred to as the standard resolvent decomposition, is optimal in the kinetic energy norm induced by the inner product in \[eq:IP\]. However, in wall-bounded turbulence the kinetic energy is often dominated by the the streamwise velocity, which means that all three velocity components may not be approximated uniformly well [@moarref2014; @sharma2016]. In such situations, an alternative decomposition that more faithfully represents the underlying dynamics may be desirable. This idea has been explored previously by @juttijudata2005, who transformed near-wall data from turbulent channel flow into Squire’s coordinate system and then performed POD on modes associated with the streamwise streaks and rolls separately. While the resulting basis functions are energetically suboptimal compared to those from standard POD, they demonstrate that the reconstruction of wall-normal, spanwise, and Reynolds shear stress statistics improve substantially. In a similar spirit, @rosenberg2019a, proposed the following alternative decomposition of $\res$. Note that \[eq:NSEres\] can be rewritten as $$\begin{pmatrix}
\hat{v}\\
\hat{\eta}
\end{pmatrix}
=
\begin{pmatrix}
\res_{vv} & 0\\
\res_{\eta v} & \res_{\eta\eta}
\end{pmatrix}
\begin{pmatrix}
\hat{g}_v\\
\hat{g}_\eta
\end{pmatrix},
\label{eq:NSEOS}$$ where
\[eq:Hops\] $$\begin{aligned}
\res_{vv} &= \left( -i\omega - \lapinv \mathcal{L}^\OS \right)^{-1},
\label{eq:Hvv}\\
\res_{\eta\eta} &= \left( -i\omega - \mathcal{L}^\SQ \right)^{-1},
\label{eq:Hetaeta}\\
\res_{\eta v} &= -i k_z \res_{\eta \eta} U' \res_{vv}.
\label{eq:Hetav}
\end{aligned}$$
Apparently, $\res_{vv}$ and $\res_{\eta v}$ are forced by $\hat{g}_v$ only, while $\res_{\eta \eta}$ is forced by $\hat{g}_\eta$ only. This motivates the separation of the response into two distinct families:
\[eq:OSSQ\] $$\begin{aligned}
\begin{pmatrix}
\hat{v}\\
\hat{\eta}^\OS
\end{pmatrix}
&=
\begin{pmatrix}
\res_{vv}\\
\res_{\eta v}
\end{pmatrix}
\hat{g}_v,
\label{eq:resOS}\\
\hat{\eta}^\SQ &= \res_{\eta\eta}\,
\hat{g}_\eta.
\label{eq:resSQ}
\end{aligned}$$
In the following, we refer to the family of modes in \[eq:resOS\] as Orr-Sommerfeld (OS) modes and the family in \[eq:resSQ\] as Squire (SQ) modes. [The separation of $\hat{\eta}$ into two distinct families is common practice in linear stability analysis, where the SQ and OS modes are, respectively, the homogeneous and particular solutions of the Squire equation: $(-i\omega - \op{L}^\SQ)\hat{\eta}=-ik_zU'\hat{v}$ [@schmid2001]. That is, the OS modes can be interpreted as a response to the wall-normal velocity. This interpretation still holds in the nonlinear setting, since the second component of \[eq:resOS\] can be written as $\hat{\eta}^\OS = -ik_z\res_{\eta \eta}U'\hat{v}$. However, the SQ modes are no longer the homogeneous solutions, but are now interpreted as the response to forcing by $\hat{g}_\eta$.]{}
[Note that only the OS modes contain a $\hat{v}$ response, such that the SQ modes contribute only to the $\hat{\eta}$ response, i.e., to the wall-parallel velocity components. There is thus the potential for interaction between the OS and SQ vorticity in ways that are not admitted by the standard resolvent decomposition. This fact is of central importance for the OS-SQ resolvent decomposition, and it will be demonstrated in \[sec:kxspectra\] that this drastically improves the accuracy of a low-order resolvent-based representation of the second-order statistics for turbulent channel flow.]{}
An SVD of each operator in \[eq:OSSQ\] is performed separately, and the resulting decomposition is referred to as the OS-SQ decomposition of the resolvent. The approximation of the response becomes $$\begin{pmatrix}
\hat{v}\\
\hat{\eta}
\end{pmatrix}
\approx \sum_{j=1}^{N^\OS} \vect{\psi}^\OS_j \sigma^\OS_j \chi^\OS_j + \sum_{k=1}^{N^\SQ} \vect{\psi}^\SQ_k \sigma^\SQ_k \chi^\SQ_k.
\label{eq:seriesOSSQ}$$ Note that \[eq:seriesOSSQ\] is now a sum of $N^\OS + N^\SQ$ terms. Furthermore, while the left and right singular vectors of each family still comprise orthonormal sets with respect to the inner product given in \[eq:IP\], it is not guaranteed that modes belonging to different families are orthonormal, e.g. $\langle \vect{\psi}^\OS_j, \vect{\psi}^\SQ_k \rangle \neq \delta_{jk}$ in general.
\[sec:opt\]Emprical determination of the resolvent weights via convex optimization
----------------------------------------------------------------------------------
The singular values and vectors are computed directly from the resolvent operator, which depends only on the (assumed known) mean velocity profile $U$, whereas computation of the weights requires solution of a nonlinear programming problem [@mckeon2013]. This can be done exactly in special cases, such as for exact coherent states (ECS) [@rosenberg2018]. However, it rapidly becomes intractable with an increasing number of degrees of freedom, and, to our knowledge, fully turbulent flows remain out of reach.
Consequently, several attempts have been made to determine the weights empirically [@moarref2014; @gomez2016; @beneddine2016; @zare2017; @towne2018]. In particular, @moarref2014, used convex optimization to compute the weights that minimize the deviation between a resolvent-based representation of the energy spectra and DNS data. We take the same approach here and largely adopt their formulation, with the major exception that we employ the OS-SQ decomposition discussed in \[sec:OSSQ\]. That is, for given $N^\OS$ and $N^\SQ$, we attempt to approximate the DNS statistics using the approximation given in \[eq:seriesOSSQ\].
As introduced by @moarref2014, the resolvent three-dimensional streamwise energy spectra are $$E_r(y, k_x, k_z, c) = \Re\!\left\{ \tr\! \left( \tens{A}_r\tens{X} \right) \right\},
\label{eq:E}$$ with $r\in\{uu, vv, ww, uv\}$, and where $\Re\{\,\cdot\,\}$ is the real part of a complex number and $\tr\blankop$ is the matrix trace. Note that we have chosen to parameterize the spectra in terms of the wavespeed $c=\omega/k_x$ since resolvent modes tend to be localized about the critical layers $y_c$, where $U(y_c) = c$ [@mckeon2010], and it has been observed experimentally that the range of energetic wavespeeds is relatively compact, with the most energetic motions typically being confined to the range $8 \lesssim c \lesssim U_{cl}$ [@lehew2011], where $U_{cl}$ is the mean centerline velocity. In \[eq:E\], the matrix $\tens{A}_{uu}$, for example, with entries $$\tenscomp{A}_{uu,ij} = \sigma_i\sigma_j \hat{u}_i \hat{u}_j^*,
\label{eq:A}$$ represents the contributions of the singular values and response modes and can be determined *a priori* from the SVD of the resolvent. The matrix $\tens{X}$, with entries $$\tenscomp{X}_{ij} = \chi_i^*\chi_j,
\label{eq:X}$$ is the weights matrix. Apparent from this definition is that $\tens{X}^{\,\trsp} = \vect{\chi}\vect{\chi}^* \succeq \vect{0}$, where $\vect{\chi}$ is the vector of weights and $\succeq$ denotes the Löwner order, i.e., $\tens{X}$ is a rank-1 positive-semidefinite matrix. The OS-SQ decomposition is incorporated into this framework simply by partitioning the $\tens{A}_r$ and $\tens{X}$ matrices as $$\begin{aligned}
&\tens{A}_r =
\begin{pmatrix}
\tens{A}_r^{\OS/\OS} & \tens{A}_r^{\OS/\SQ}\\
\tens{A}_r^{\SQ/\OS} & \tens{A}_r^{\SQ/\SQ}
\end{pmatrix},
&\tens{X} =
\begin{pmatrix}
\tens{X}^{\OS/\OS} & \tens{X}^{\OS/\SQ}\\
\tens{X}^{\OS/\SQ\,*} & \tens{X}^{\SQ/\SQ}
\end{pmatrix},
\label{eq:partition}\end{aligned}$$ where the superscript $\text{X}/\text{Y}$ denotes the family of the $i$th and $j$th mode, respectively, in \[eq:A,eq:X\]. The goal is to compute the weights matrix such that the deviation between the wavespeed-integrated resolvent spectra in \[eq:E\] and time-averaged DNS spectra is minimized. After discretization of the wavespeed range $c\in [0,U_{cl}]$, this can be formally cast as the following optimization problem: For fixed $k_x$ and $k_z$, $$\begin{aligned}
&\underset{\{\tens{X}_l\}_{l = 1,2,\dots,N_c}, \, e}{\text{minimize}} &&e
\\
&\text{subject to} &&\frac{\lVert E^{\text{DNS}}_r - \sum_{l=1}^{N_c} k_x\,\diff c\, \Re\!\left\{ \tr\!\left( \tens{A}_{r,l}\tens{X}_l\right) \right\} \rVert^2}{\lVert E^{\text{DNS}}_r \rVert^2} \leq e
\label{eq:optprob}\\
&&&\tens{X}_l \succeq \vect{0},\, l = 1,2,\dots,N_c,
\end{aligned}$$ where the subscript $l$ denotes a quantity evaluated at $c=c_l$. Note that the norm $\lVert \,\cdot\, \rVert$ is not the one induced by \[eq:IP\]. It is defined as $$\left\Vert f \right\Vert^2 = \int_{y_{\min}^+}^{y_{\max}^+} \left\vert f(\log y^+) \right\vert^2 \diff \log y^+$$ and is designed to penalize deviations across the channel equally [@moarref2014]. Thus deviations from the DNS spectra are enforced for $5\leq y_{\min}^+ \leq y^+ \leq y_{\max}^+<\Rey$.
\[eq:optprob\] is a semidefinite program for the weights matrices $\tens{X}_l$ and can therefore be solved efficiently using a convex optimization software package. Note that imposing the rank-1 constraint on the $\tens{X}_l$ would make \[eq:optprob\] non-convex. @moarref2014 employed an iterative rank-reduction procedure to recover rank-1 matrices from the full-rank solution [@huang2009]. However, we do not employ this algorithm here and instead choose to work with the full-rank weights matrices. In this case, the $\tens{X}_l$ can be interpreted as the covariance matrices of the weights, similar to @towne2018. [Finally, since the optimization is performed for second-order statistics, the present approach does not provide phase information about modes with different wavenumbers. This means that the computed weights do not yield a closed, self-consistent system, as such information is necessary to recover the mean velocity profile [as well as the fluctuations]{}. Extension of the method to incorporate phase is a direction for future work.]{}
\[sec:numer\]Numerical details
------------------------------
The resolvent operators are discretized in MATLAB using a Chebyshev pseudospectral method [@weideman2000]; all results presented here use 203 Chebyshev polynomials. The SVDs of the discretized operators are performed with a random matrix algorithm, which is faster than MATLAB’s built-in `svd()` [@halko2011]. The time-averaged two-dimensional DNS spectra for $\Rey=934$ and $\Rey=2003$ are obtained from @hoyas2006, and for $\Rey=4219$ from @lozano2014. Additionally, spectra were generated for $\Rey=185$ using Channelflow [@channelflow]. The resolutions of all the DNS considered are given in \[tab:DNS\].
For the results presented in \[sec:kxspectra\], which focus on $\Rey=2003$, the DNS spectra are interpolated onto a grid of $N_{k_x}=30$ by $N_{k_z}=31$ logarithmically spaced wavenumbers, [which is sufficient to reproduce the general shape of the spectra. Furthermore, it has recently been shown that statistics such as the $uv$ Reynolds stress can be accurately reproduced even when retaining only about 2% of the wavenumbers from DNS [@toedtli2019]]{}. Both the spectra and resolvent modes are interpolated onto a common grid of $N_y=60$ logarithmically spaced points in the wall-normal direction, and the wavespeed range $c \in [0, U_{cl}]$ is discretized into $N_c=100$ linearly spaced wavespeeds. The optimization problem \[eq:optprob\] is then solved with CVX [@cvx]. The results are insensitive to further increases in $N_y$ and $N_c$ [@moarref2014].
[lrrr]{} $\Rey$ & $N_x$ & $N_y$ & $N_z$\
185 & 384 & 129 & 128\
934 [@hoyas2006] & 1024 & 385 & 768\
2003 [@hoyas2006] & 2048 & 635 & 1536\
4219 [@lozano2014] & 1024 & 1081 & 1024\
Analysis of the optimized spectra
=================================
\[sec:kxspectra\]Reconstruction of time-averaged statistics
-----------------------------------------------------------
The accuracy of the optimized spectra is evaluated by comparing them to the time-averaged statistics from the DNS for $\Rey=2003$. The premultiplied 1D $k_x$ spectra, $$k_xE_r(y,\, k_x) = \int_{k_{z,\mathrm{min}}}^{k_{z,\mathrm{max}}} \int_{0}^{U_{\mathit{cl}}} k_x^2 \, E_r(y, k_x, k_z, c) \,\diff c \, \diff k_z,$$ using $N^{\OS}=N^{\SQ}=3$ modes, i.e., six modes per wavenumber-frequency triplet, are compared to the DNS in the right column of \[fig:kxspect\], which is plotted in terms of $\lambda_x^+ = 2\pi/k_x^+$. Clearly, $N^{\OS}=N^{\SQ}=3$ modes is sufficient to accurately reproduce the spectra since the overall agreement between the resolvent and DNS spectra is very good, and in particular, the peaks are captured almost exactly. The only significant discrepancies are in $k_xE_{uu}$ at large $\lambda_x^+$ and $y^+ \lesssim 100$ and $-k_xE_{uv}$ at large $\lambda_x^+$ and $y^+ \lesssim 50$. Further discussion of these discrepancies, as well as the accuracy of the optimized spectra using different numbers of modes is given in \[sec:err\]. Also shown in the left column are the spectra obtained using the standard decomposition with the same total number of modes. The performance is significantly worse, with $k_xE_{uu}$ and $k_xE_{ww}$ being greatly over-predicted, and $k_xE_{vv}$ and $-k_xE_{uv}$ being under-predicted. In fact, the standard resolvent decomposition fails to capture the 90% energy level (darkest blue contours) for $-k_xE_{uv}$.
![Premultiplied one-dimensional spectra from the resolvent (filled contours) and DNS (black contours) for $\Rey=2003$. (a,c,e,g) Standard resolvent decomposition using $N=6$ modes per wavenumber-frequency triplet; (b,d,f,h) OS-SQ resolvent decomposition using $N^{\OS} = N^{\SQ} = 3$ modes per wavenumber-frequency triplet. (a,b) $k_xE_{uu}$, (c,d) $k_xE_{vv}$, (e,f) $k_xE_{ww}$, (g,h) $-k_xE_{uv}$. Contour levels are from 10% to 90% of the DNS maximum in 20% increments.\[fig:kxspect\] ](spect_N3.pdf){width="62.00000%"}
Subsequent integration over $k_x$ gives the intensities, which are shown in figure \[fig:ints\]. The deviation errors are 4.3%, 0.95%, 0.66%, and 3.8% for $\langle u^2\rangle$, $\langle v^2\rangle$, $\langle w^2\rangle$, and $\langle -uv \rangle$, respectively. These should be compared with errors of 30%, 14%, 12%, and 31% using the standard resolvent decomposition, shown in the dashed curves.
[As the goal of the optimized spectra is to obtain a *low-order* representation of the spectra, it is worth comparing the number of degrees of freedom of the resolvent spectra to the original DNS. For a given $k_x,k_z$, the $\Rey=2003$ DNS spectra were computed using $N_y=665$ wall-normal grid points and $N_t=7730$ snapshots [@hoyas2006]. For the results shown in \[fig:kxspect,fig:ints\], the resolvent representation was computed using $N=6$ resolvent modes and $N_c=100$ wavespeeds, about 0.01% of the degrees of freedom in the DNS.]{}
![Intensities from the resolvent with $N^{\OS}=N^{\SQ}=3$ modes per wavenumber-frequency triplet (blue) and DNS (black) for $\Rey=2003$. Also shown in dashed lines are the intensities obtained from the standard resolvent decomposition approach using the same total number of modes.\[fig:ints\]](ints_N3.pdf){width="65.00000%"}
\[sec:pwrspect\]Prediction of additional statistics
---------------------------------------------------
Since the optimization only attempts to match time-averaged spectra, the distribution of energetic content in $c$ is not directly constrained. To assess this, the power spectra computed from the resolvent representation using the optimized weights with $N^\OS=N^\SQ=3$ are compared to those computed from DNS for $\Rey=185$ using Welch’s method with 8491 snapshots divided into 10 segments having 50% overlap. \[fig:pwrspect\](a) and \[fig:pwrspect\](b) show the 2D premultiplied streamwise power spectra in the $k_x-\omega$ and $k_z-\omega$ planes, respectively, at $y^+\approx15$.
![2D premultiplied streamwise velocity power spectra at $y^+\approx15$ for $\Rey=185$. (a) $\omega k_x E_{uu}$, (b) $\omega k_z E_{uu}$. Filled contours: Optimized weights with $N^\OS=N^\SQ=3$. Line contours: DNS; levels are 10% and 50% of the maximum value. The slope of the dashed line indicates the the local mean velocity $U(y^+=15)$. \[fig:pwrspect\]](pwrspect.pdf){width="0.7\linewidth"}
The distribution in the $k_x-\omega$ plane is fairly good, with most of the energetic content of the resolvent spectrum falling within the 10% DNS contour. As discussed above, the distribution in $c$ is not directly constrained. However, the localization of the leading resolvent modes at the critical layer implies that the energetic content at a given wall-normal location is largely contributed by modes with a wavespeed matching the local mean velocity. This is evident in \[fig:pwrspect\], where both the DNS and resolvent spectra are concentrated around the dashed line representing a constant wavespeed $c=U(y^+=15)$. There is no such localization mechanism in the $k_z-\omega$ plane. Nonetheless, the resolvent representation still reproduces the general shape of the DNS spectrum quite well. Note that to produce \[fig:pwrspect\](b), the resolvent spectrum was interpolated onto a common $\omega$ grid prior to integration over $k_x$. The optimized weights can also be used to compute an approximation of the forcing spectra in a manner that is directly analogous to the velocity spectra in \[eq:E\]:
\[eq:Efrc\] $$\begin{aligned}
E_{g_vg_v}(y, k_x, k_z, c) &= \Re\!\left\{ \tr\! \left( \tens{B}_{vv}\tens{X}^{\OS/\OS} \right) \right\},
\\
E_{g_\eta g_\eta}(y, k_x, k_z, c) &= \Re\!\left\{ \tr\! \left( \tens{B}_{\eta\eta}\tens{X}^{\SQ/\SQ} \right) \right\},
\end{aligned}$$
where $\tenscomp{B}_{vv,ij} = \phi_{v,i} \phi_{v,j}^*$, and $\tenscomp{B}_{\eta\eta,ij} = \phi_{\eta,i} \phi_{\eta,j}^*$. The estimates of the 2D forcing spectra with $N^\OS=N^\SQ=3$ in the $k_x - k_z$ plane at $y^+\approx15$ for $\Rey=185$ are compared to the full forcing spectra computed from DNS [@rosenberg2018] in \[fig:frcspect\]. The resolvent estimate reasonably predicts the general shape of the full spectra with only a few modes; this is consistent with results indicating that the OS-SQ decomposition yields not only an efficient response basis, but also a forcing basis that is more efficient than the one obtained from the standard resolvent approach [@rosenberg2019a]. Furthermore, this estimate was obtained using only information about the velocity statistics, an interesting implication of which is that potentially much can be be learned about the nonlinear forcing directly from commonly-computed flow quantities. We note that similar observations have been made by @towne2020, who use a limited set of flow statistics to infer forcing statistics, which are in turn used to estimate the unknown flow statistics.
![2D forcing spectra at $y^+\approx15$ for $\Rey=185$. (a) $E_{g_v g_v}$, (b) $E_{g_\eta g_\eta}$. Filled contours: Optimized weights with $N^\OS=N^\SQ=3$. Line contours, reproduced from @rosenberg2018: DNS; levels are 10% and 50% of the maximum value. \[fig:frcspect\]](frcspect.pdf){width="0.7\linewidth"}
Finally, to give additional insight into how energetic modes are distributed across spectral space, the magnitudes of the total mode coefficients, i.e., the weight multiplied by the singular value, are plotted for the leading OS and SQ modes in \[fig:pointcloud\](a) and \[fig:pointcloud\](b), respectively, for $\Rey=2003$; for ease of visualization, only coefficients larger than 1% of the maximum value over all spectral space are plotted. Interestingly, they are largely concentrated at large $\lambda_x$ and $c$ close to $U_{cl}$. [In addition, there are large coefficients for very low $c$ and large $\lambda_x^+$, which are likely related to near-wall over-compensation, discussed in \[sec:err\].]{} This observed clustering may have implications for further model reduction by highlighting important regions of spectral space. [Finally, with the exception of some SQ coefficients at small $\lambda_x^+$ and $c\approx U_{cl}$, the large OS and SQ coefficients occupy the essentially the same regions of spectral space which is a reflection of the interactions between the two families of modes; this is discussed in detail in the next section.]{}
![Magnitudes of the total leading mode coefficients (a) $\sigma_1^\OS\sqrt{\tenscomp{X}_{11}^{\OS/\OS}}$, (b) $\sigma_1^\SQ\sqrt{\tenscomp{X}_{11}^{\SQ/\SQ}}$ [larger than 1% of the maximum value over all of spectral space]{} for $\Rey=2003$. Marker sizes are proportional to the magnitude and are normalized by the maximum. [The axes show the full range of wave parameters included in the optimization.]{} \[fig:pointcloud\]](pointcloud.pdf){width="0.7\linewidth"}
Interpretation of the OS-SQ decomposition: a competition mechanism
------------------------------------------------------------------
It has been demonstrated that the performance of the optimization is greatly improved by employing the OS-SQ decomposition of the resolvent. Previous work reported similar results for channel ECS [@rosenberg2018; @rosenberg2019a]. In that case, the relatively poor performance of the traditional resolvent method was attributed to the fact that the $\eta$ response dominates under the kinetic energy norm. Thus, matching the statistics for $u$ (or $w$) results in under-prediction of the $v$ statistics, as observed in \[fig:kxspect,fig:ints\]. However, in the OS-SQ decomposition, isolating the $v$ response in only the OS modes allows $v$ and $\eta$, to be ‘tuned’ somewhat independently, with the role of the SQ modes then being to saturate the OS vorticity. The improved matching of all components in \[fig:ints\] indicates that this is also the case for fully-turbulent channel flow, where the dynamics are significantly more complex than for the aforementioned equilibria.
[To understand why the OS and SQ modes comprise a much more efficient basis, note that in certain cases the response modes of the standard resolvent, \[eq:NSEOS\], coincide with those of the OS resolvent, \[eq:resOS\]. A detailed description of the regions of parameter space where this holds is beyond the present scope, [but we note that since $\res_{\eta v}$ contains the coupling term $-ik_zU'$, it is expected to hold whenever the lift-up mechanism is dominant, one such example being for highly streamwise-elongated modes.]{} Further discussion can be found in @dawson2019. Here, we simply illustrate by example]{} for a particular wavenumber-frequency triplet in \[fig:modes\], which compares the singular values and magnitudes of the $\hat{\eta}$ response for the standard, OS, and SQ resolvents. Due to the symmetry of the channel geometry about $y=0$, the modes come in symmetric-antisymmetric pairs with correspondingly paired singular values. [The singular values of the OS and standard resolvents are almost equal, with the separation between them growing slowly with increasing mode index. Looking now at the $\hat{\eta}$ response modes, those from OS and standard resolvents are almost indistinguishable. Though not shown, the same is true for the $\hat{v}$ responses.]{}
[The SQ singular values are significantly smaller than those for the standard or OS resolvent – by more than an order of magnitude for the first pair. Interestingly, the SQ singular values do not demonstrate clear pairing beyond this. The SQ $\hat{\eta}$ modes are distinct from the other two, in particular having slightly narrower wall-normal support. However, the shapes are still largely similar. Importantly, there is still a significant region of overlap with the OS modes in the wall-normal direction, which is a necessary condition for the SQ modes to interact with the the OS modes.]{}
![(a) First ten singular values of the OS (blue), SQ (red), and standard (black) resolvent operators for $(k_x,k_z,c)=(0.25, 2.5, 24)$ and $\Rey=2003$. (b)-(d) Magnitudes of the vorticity responses from the first, second, and third mode pairs (same color scheme as in (a)); modes having the same wall-normal symmetry have been selected from each pair. The standard resolvent (black) and OS (blue) modes are visually indistinguishable. The gray lines in (b)-(d) are the locations of the critical layers, $y_c = \pm 0.194$.\[fig:modes\]](modes_compare.pdf){width="65.00000%"}
[It is also instructive to look at the corresponding forcing modes, which are shown in the top row of \[fig:forcing\]. As with the response modes, $\phi_v$ for the OS and standard resolvents are virtually identical. This is at first surprising since the standard resolvent has $\phi_\eta$ with comparable amplitude to $\phi_v$. However, its contribution to the norm is $\lesssim 1\%$. The bottom row of \[fig:forcing\] shows $\phi_\eta$ for the SQ and standard resolvents normalized by their maximum amplitude for ease of comparison. Despite some differences that become more pronounced for the higher-order modes, their shapes are overall quite similar. Therefore, though there may be traces of the SQ modes in the leading standard resolvent modes, they are clearly dominated by the OS ones. This implies that using the standard resolvent operator to generate a low-order representation is effectively equivalent to using only the OS family of modes, and the linear mechanisms encoded in the SQ operator are thus not accounted for.]{}
![(a)-(c) Magnitudes of the forcing modes corresponding to the response modes shown in \[fig:modes\], using the same color scheme. $\phi_v$ for the standard resolvent (solid black) is indistinguishable from OS (blue). Dotted lines are the standard resolvent $\phi_\eta$. (d)-(f) SQ and standard resolvent $\phi_\eta$ normalized by their maximum values. The gray lines in (a)-(c) are the locations of the critical layers, $y_c = \pm 0.194$.\[fig:forcing\]](forcing_compare.pdf){width="\textwidth"}
To further examine the relationship between the OS and SQ modes, we decompose the intensities shown in \[fig:ints\] into contributions from OS modes only, SQ modes only, and a cross term (C) that represents the interaction of OS and SQ modes, e.g. $\langle u^2 \rangle$ becomes $${\left\langle u^2 \right\rangle =
\underbrace{\left\langle \left(u^\OS\right)^2 \right\rangle}_{\OS}
+
\underbrace{\left\langle \left(u^\SQ\right)^2 \right\rangle}_{\SQ}
+
\underbrace{2\left\langle u^\OS u^\SQ \right\rangle}_{\mathrm{C}}.}
\label{eq:udecomp}$$ The results with $N=3$ for $\langle u^2 \rangle$, $\langle w^2 \rangle$, and $\langle -uv \rangle$ are shown in \[fig:intsdecomp\]. The decomposition for $\langle v^2 \rangle$ is not shown since, as seen from \[eq:resSQ\], the SQ modes have no $v$ response, and hence $\langle v^2 \rangle = \langle (v^\OS)^2 \rangle$. Similarly, there is no SQ-only contribution to $\langle -uv \rangle$.
![(a) $\langle u^2 \rangle$, (b) $\langle w^2 \rangle$, (c) $\langle -uv \rangle$ decomposed into OS (blue), SQ (red), and C (green) terms for $N=3$. The totals are plotted in black.\[fig:intsdecomp\]](ints_decomp_N3.pdf){width="\textwidth"}
For $\langle u^2 \rangle$ and $\langle w^2 \rangle$, the OS and SQ terms are similar, with the OS term having slightly larger magnitude. However, for all three components the C term is negative, which supports the claim that [the SQ vorticity acts to saturate the OS vorticity.]{} In fact, information about the phase relationship between the OS and SQ modes can be deduced from this observation. Note that the third term in \[eq:udecomp\] is simply twice the covariance of $u^\OS$ and $u^\SQ$.
For simplicity, express each as a Fourier sine series in one variable:
\[eq:fourser\] $$\begin{aligned}
u^\OS &= \sum_{i=1}^{\infty} A^\OS_i \sin(k_i x + \theta^\OS_i),
\\
u^\SQ &= \sum_{i=1}^{\infty} A^\SQ_i \sin(k_i x + \theta^\SQ_i),\end{aligned}$$
where $A_i^{\mathrm{X}}$ and $\theta_i^{\mathrm{X}}$ are the amplitude and phase of the $i$th mode, respectively, and $0<k_1<k_2<\cdots$. Then, $$\begin{aligned}
2\left\langle u^{\text{OS}}u^{\text{SQ}} \right\rangle &= 2 \sum_i \sum_j A^\OS_i A^\SQ_j \left\langle \sin(k_i x + \theta^\OS_i) \sin(k_j x + \theta^\SQ_j) \right\rangle
\nonumber\\
&=\sum_i \sum_j A^\OS_i A^\SQ_j \left( \left\langle \cos\!\left[ (k_i-k_j) x + \theta^\OS_i-\theta^\SQ_j \right]\right\rangle - \left\langle \cos\!\left[ (k_i+k_j) x + \theta^\OS_i+\theta^\SQ_j \right]\right\rangle \right)
\label{eq:dtheta}\\
&= \sum_i A^\OS_i A^\SQ_i \cos(\Delta\theta_i),
\nonumber\end{aligned}$$ where $\Delta\theta_i=\theta^\OS_i-\theta^\SQ_i$. The term labeled C in \[eq:udecomp\] can thus be interpreted as a weighted (by the amplitudes) sum of the cosines of the phase difference between the OS and SQ modes.
Therefore, $\langle u^\OS u^\SQ \rangle < 0$ implies $\pi/2 < \Delta\theta_i < 3\pi/2$ on average. Furthermore, the relative magnitudes of the three terms suggests that for the majority of modes the phase difference is relatively close to $\pi$, i.e, the OS and SQ modes are close to being exactly out of phase. Finally, we note that while the individual terms in \[eq:udecomp\] depend on $N$, the trends discussed above, namely the similarity of the OS and SQ terms and the C term being negative, do not. Furthermore, performing the decomposition in \[eq:udecomp\] for $\langle \eta^2 \rangle$ from $\Rey=185$ DNS data reveals the same features [@rosenberg2018]. This provides strong evidence that they are not merely consequences of the particular optimization procedure, but are in fact robust features of turbulent channel flow.
[Since the SQ modes are exclusively wall-parallel motions, there is a passing resemblance to the notion of the ‘inactive’ motions proposed by @townsend1961. It is supposed that, at first order, the inactive motions to not interact with the ‘active’ shear stress-carrying motions. However, \[fig:intsdecomp\](c) shows that the interaction of the SQ vorticity with $v$ produced by the OS modes, the C term, contributes significantly to the overall Reynolds shear stress profile, suggesting that there is not an exact correspondence between the SQ modes and inactive motions.]{} [In this section it was shown that the OS-SQ decomposition of the resolvent provides an improved basis for efficiently representing the statistics of turbulent channel flow, and that this provides insight into the complex physics at play, namely a competition mechanism, interpreted as a phase difference, between OS and SQ modes that results in saturation of the wall-normal vorticity. In the next section, we use this insight to derive simple scalings for the relative magnitudes of the OS and SQ weights of modes belonging to several special classes.]{}
\[sec:scaling\]Weights scaling for the universal classes of resolvent modes
===========================================================================
@moarref2013 leveraged universal scaling regimes of the mean velocity profile to derive the $\Rey$ scaling for several universal classes of resolvent modes. Here, we extend this to the OS-SQ resolvent decomposition and show that for the outer and geometrically self-similar classes, each family of modes has a distinct scaling for the singular values. From this, the scalings of each submatrix of the energy density matrices $\tens{A}_r$ given in \[eq:partition\] can be determined. Combining these scalings with the hypothesis that competition of the OS and SQ modes remains relevant at different Reynolds numbers and in different regions of the flow enables the relative scaling of the OS and SQ weights belonging to the universal classes to be deduced. The universal classes investigated here are the inner, outer, and geometrically self-similar classes. These consist of resolvent modes that are localized within the near-wall, wake, and logarithmic regions of the flow, respectively, and rely on universality of the mean velocity profile under the appropriate scaling in these regions. \[fig:means\](a) demonstrates the universality of $U$ for $y^+\lesssim100$, and \[fig:means\](b) shows that the velocity defect $U_{cl}-U$ is universal for $y\gtrsim0.1$; these approximate boundaries are indicated by the vertical dashed lines in \[fig:means\]. [Between these regions, there exists an intermediate region of the mean velocity profile in which both scalings hold. In this overlap region, it is widely accepted that the mean varies logarithmically with distance from the wall.]{} Classical estimates put the beginning of the logarithmic region at $y^+=O(100)$. However, there is recent evidence that this lower boundary moves outward as $\Rey^{1/2}$ [@klewicki2009; @marusic2013]. In the logarithmic region, the resolvent operator admits self-similar modes localized about their critical layers [@moarref2013]. [Furthermore, the scaling of these modes reduces to the inner and outer scalings when $y^+$ or $y$, respectively, is held fixed, reflecting their mutual validity in the logarithmic region.]{}
![(a) Mean velocity profile $U(y^+)$ and (b) velocity defect $U_{cl}-U(y)$ for $\Rey=934$ (blue), $\Rey=2003$ (red), and $\Rey=4219$ (green). The gray boxes indicate the regions where the profiles are $\Rey$-invariant in the respective coordinates. \[fig:means\]](means.pdf){width="70.00000%"}
In each of the next subsections, we briefly summarize the scaling of the wave parameters for each of the three aforementioned universal classes derived by @moarref2013, as well as the distinct scalings for the OS and SQ singular values. Finally, the relative scaling of the OS and SQ weights are presented and tested against the computed optimal weights.
\[sec:inner\]Inner class
------------------------
Following @moarref2013, the relevant length scale for the inner class is the viscous unit $\nu/u_\tau$, so that the corresponding inner-scaled parameters are $$k_x^+ = \Rey^{-1}k_x, \quad k_z^+ = \Rey^{-1}k_z, \quad y^+ = \Rey y,
\label{eq:li}$$ and the inner class wave parameters are $$\mathscr{S}_i: \quad 0 \leq c \lesssim 16.4.
\label{eq:Si}$$ The upper wavespeed limit is obtained from the critical layer at the top of the inner region, i.e., $U(y^+=100)=16.4$; this is indicated by the horizontal dashed line in \[fig:means\]a. Note that this bound is slightly different from the one given in @moarref2013 since the mean velocity profiles they used were obtained from an eddy viscosity model, whereas the ones used here are taken directly from the DNS that the spectra are obtained from. Using \[eq:li\] and continuity, it follows that all three velocity components scale in the same way. Furthermore, the orthonormality constraint on the resolvent modes imposes $$\hat{\vect{u}} = \Rey^{1/2}\hat{\vect{u}}^+,
\label{eq:ui}$$ where a superscript $\blankop^+$ indicates a quantity that is $\Rey$-invariant for modes belonging to the inner class. \[eq:li\] can be used to obtain the inner-scaled versions of the weighted resolvent operators:
\[eq:Hopsi\] $$\begin{aligned}
\begin{pmatrix}
\op{F}_v \res_{vv} \op{F}_v^\inv \\
\op{F}_\eta \res_{\eta v} \op{F}_v^\inv
\end{pmatrix}
&= \Rey^\inv
\begin{pmatrix}
\op{F}_v^+ \res_{vv}^+ \op{F}_v^{+\, \inv} \\
\op{F}_\eta^+ \res_{\eta v}^+ \op{F}_v^{+\, \inv}
\end{pmatrix}
\label{eq:Hosi}
\\
\op{F}_\eta \res_{\eta\eta} \op{F}_\eta^\inv &= \Rey^\inv \op{F}_\eta^+ \res_{\eta\eta}^+ \op{F}_\eta^{+\, \inv},
\label{eq:Hsqi}
\end{aligned}$$
where $\op{F} = \diag( \op{F}_v, \, \op{F}_\eta)$ is the square root of the positive-definite operator $\op{Q}$ defined in \[eq:Q\], i.e., $\op{Q} = \op{F}^{\adj}\op{F}$, and the superscript $\blankop^\adj$ denotes the adjoint with respect to the inner product \[eq:IP\]. Computing the SVDs of \[eq:Hosi,eq:Hsqi\], it is clear that both the OS and SQ singular values have the same scaling: $$\sigma_j^\OS = \Rey^\inv \sigma_j^{\OS\, +}, \quad \sigma_j^\SQ = \Rey^\inv \sigma_j^{\SQ\, +}.
\label{eq:sigi}$$ [For a particular $k_x^+$, $k_z^+$, and $c\in\mathscr{S}_i$, the magnitudes of the leading vorticity response modes of the OS and SQ resolvent operators and their corresponding forcing modes for the three $\Rey$ in \[fig:means\] are shown in \[fig:modesinner\](a) and \[fig:modesinner\](b), respectively, using the scalings derived in . For comparison, the leading response and forcing mode for the standard resolvent operator are also shown; as discussed in \[sec:kxspectra\], $\psi_\eta$ and $\phi_v$ are indistinguishable from the OS modes.]{}
![(a) Magnitudes of the scaled OS, SQ, and standard resolvent (color scheme as in \[fig:modes\]) inner-class leading vorticity response modes for the three $\Rey$ shown in \[fig:means\], with $k_x^+ = 1/934$, $k_z^+ = 10 k_x^+$, and $c=10$. (b) Corresponding scaled leading forcing mode magnitudes, with the relevant axes indicated by the arrows. $\psi_\eta$ and $\phi_v$ for the standard resolvent are indistinguishable from the OS modes. The gray line indicates the location of the critical layer. \[fig:modesinner\]](inner_compare.pdf){width="70.00000%"}
Substituting \[eq:ui,eq:sigi\] into \[eq:A\], we obtain the inner-scaled energy density matrices: $$\begin{array}{c c c}
\tenscomp{A}^{\OS/\OS}_{r,ij} = \Rey^\inv \tenscomp{A}^{\OS/\OS\, +}_{r,ij}, &
\tenscomp{A}^{\OS/\SQ}_{r,ij} = \Rey^\inv \tenscomp{A}^{\OS/\SQ\, +}_{r,ij}, &
\tenscomp{A}^{\SQ/\SQ}_{r,ij} = \Rey^\inv \tenscomp{A}^{\SQ/\SQ\, +}_{r,ij}.
\end{array}
\label{eq:Ai}$$ With this, the decomposed version of the three-dimesional streamwise energy spectrum becomes $$E_{uu} = \Rey^\inv \Re\!\left\{ \tr\! \left( \tens{A}^{\OS/\OS\, +}_{uu}\tens{X}^{\OS/\OS} \right) \right\} + 2\Rey^\inv \Re\!\left\{ \tr\! \left( \tens{A}^{\SQ/\OS\, +}_{uu}\tens{X}^{\OS/\SQ} \right) \right\} + \Rey^\inv \Re\!\left\{ \tr\! \left( \tens{A}^{\SQ/\SQ\, +}_{uu}\tens{X}^{\SQ/\SQ} \right) \right\},
\label{eq:Euui}$$ where the Reynolds number dependence of the right-hand side is made explicit, save for the unscaled weights matrices $\tens{X}^{\text{X}/\text{Y}}$. \[eq:Ai\] can be used to write similar expressions for the other components of the spectra.
Since the overall scaling of $E_{uu}$ for the inner class is not known, the absolute scaling of the weights cannot be determined directly from \[eq:Euui\]. However, in \[sec:kxspectra\] it was shown that the vorticity generated by OS and SQ modes compete. That is, the vorticity generated by the SQ modes acts to ‘saturate’ the OS vorticity. We hypothesize that this mechanism is not specific to $\Rey=2003$ for which the optimization results were presented, but instead holds for arbitrary $\Rey$. This is only possible if all three terms remain of the same order in \[eq:Euui\], which is satisfied if the inner class OS and SQ weights have the same scaling, i.e., if the ratio $$\left\vert \frac{\chi_j^\SQ}{\chi_j^\OS} \right\vert \neq \mathrm{fn}(\Rey)
\label{eq:rati}$$ for modes belonging to the universal inner class.
This scaling is tested by computing the weights matrices for the three Reynolds numbers depicted in \[fig:means\] for fixed inner-scaled wavenumber combinations $(k_x^+,k_z^+)$. As discussed in \[sec:opt\], the individual weights are not recovered from the full-rank solutions. However, $\tenscomp{X}_{jj}^{\OS/\OS} \sim \vert \chi^\OS_j \vert^2$ and $\tenscomp{X}_{jj}^{\SQ/\SQ} \sim \vert \chi^\SQ_j \vert^2$, so that $\vert\chi_j^\SQ/\chi_j^\OS\vert \sim \sqrt{\tenscomp{X}_{jj}^{\SQ/\SQ}/\tenscomp{X}_{jj}^{\OS/\OS}}$. This ratio with $j=1$ is computed for the three Reynolds numbers, and the results for several wavenumber combinations spanning a large range of scales are shown in \[fig:win\]. Overall, the agreement is quite good, showing a reasonable collapse despite some scatter. The main exception is for $(k_x^+, k_z^+)=(2\pi/10^3,2\pi/10^2)$ in \[fig:win\]b, where there is clearly some dependence on $\Rey$ for $c\lesssim10$. This trend is consistent for modes of relatively small scale ($k_x^+ \gtrsim O(2\pi/10^3)$) and large aspect ratio ($k_z^+/k_x^+ \gtrsim O(10)$). The reason for the failure of the scaling for these modes is unclear. However, it is observed that in such cases the profiles of the time-averaged energy spectra are localized very near the wall. Additionally, the upper limit on the wavespeed range for inner class modes given in \[eq:Si\], $c=16.4$ is shown in each panel of \[fig:win\] as the vertical gray line. [The fact that the scaling given by \[eq:rati\] holds reasonably well for $c > 16.4$ is a reflection of the fact that the inner scaling remains valid in the logarithmic region, as seen in \[fig:means\](a).]{}
![Leading weights ratio for $\Rey=934$ (blue), $\Rey=2003$ (red), and $\Rey=4219$ (green) with $N=3$ for several different wavelengths: (a) $(k_x^+, k_z^+)=(2\pi/10^2,2\pi/10^2)$; (b) $(k_x^+, k_z^+)=(2\pi/10^3,2\pi/10^2)$; (c) $(k_x^+, k_z^+)=(2\pi/10^3,2\pi/10^3)$; (d) $(k_x^+, k_z^+)=(2\pi/10^4,2\pi/10^3)$. Filled circles denote modes belonging to the universal inner class, and the highest inner class wavespeed, $c = 16.4$, is indicated by the gray line.\[fig:win\]](weights_inner.pdf){width="\textwidth"}
\[sec:outer\]Outer class
------------------------
The outer class length scales are $$\out{k}_x = \Rey k_x, \quad \out{k}_z = k_z, \quad \out{y} = y,
\label{eq:lo}$$ and the outer class wave parameters are $$\mathscr{S}_{o} : \left\{
\begin{array}{l}
0 \leq U_{c l}-c \lesssim 6.17
\\
k_z / k_x \gtrsim \gamma \Rey / \mathit{Re}_{\tau, \min}
\end{array} \right.,
\label{eq:So}$$ where The upper bound on the wavespeed defect $U_{cl}-c=6.17$ is obtained from the setting the minimum critical layer location at the bottom of the outer region, i.e., $U_{cl}-U(y=0.1)=6.17$; this is indicated by the horizontal dashed line in \[fig:means\]b. Again, this value is slightly different from the one given in @moarref2013 due to the different source for the mean profiles. As \[eq:So\] indicates, the outer class modes must satisfy an aspect ratio constraint for all $\Rey$ considered, where the minimum aspect ratio is $\gamma$ when $\Rey=\mathit{Re}_{\tau,\min}$ [@moarref2013]. Here, $\mathit{Re}_{\tau,\min}=934$. From \[eq:lo\] and continuity, it follows that $$\hat{\vect{u}} =
\begin{pmatrix}
\out{u}\\
\Rey^\inv\out{v}\\
\Rey^\inv\out{w}
\end{pmatrix}.
\label{eq:uo}$$ where $\out{\blankop}$ indicates a quantity that is approximately $\Rey$-invariant for modes belonging to the outer class. [See also @sharma2017 and @moarref2014b for the scaling of each velocity component, as well as for the components of the forcing modes.]{}
![(a) Magnitudes of the scaled OS, SQ, and standard resolvent (color scheme as in \[fig:modes\]) outer-class leading vorticity response modes for the three $\Rey$ shown in \[fig:means\], with $\out{k}_x = 934$, $\out{k}_z = \gamma=1.5\sqrt{10}$, and $U_{cl}-c=1$. (b) Corresponding scaled leading forcing mode magnitudes. $\psi_\eta$ and $\phi_v$ for the standard resolvent are indistinguishable from the OS modes. The gray line indicates the location of the critical layer. \[fig:modesouter\]](outer_compare.pdf){width="70.00000%"}
The outer-scaled versions of the weighted resolvent operators are
\[eq:Hopso\] $$\begin{aligned}
\begin{pmatrix}
\op{F}_v \res_{vv} \op{F}_v^\inv \\
\op{F}_\eta \res_{\eta v} \op{F}_v^\inv
\end{pmatrix}
&=
\begin{pmatrix}
\Rey \out{\op{F}}_v \out{\res}_{vv} \out{\op{F}}_v^\inv \\
\Rey^2 \out{\op{F}}_\eta \out{\res}_{\eta v} \out{\op{F}}_v^\inv
\end{pmatrix}
\label{eq:Hoso}
\\
\op{F}_\eta \res_{\eta\eta} \op{F}_\eta^\inv &= \Rey \out{\op{F}}_\eta \out{\res}_{\eta\eta} \out{\op{F}}_\eta^\inv.
\label{eq:Hsqo}
\end{aligned}$$
Computing the SVDs of \[eq:Hoso,eq:Hsqo\], we have for the leading singular values, $$\sigma_j^\OS = \Rey^2 \out{\sigma}_j^\OS, \quad \sigma_j^\SQ = \Rey \out{\sigma}_j^\SQ.
\label{eq:sigo}$$ Note that because the components of \[eq:Hoso\] do not scale uniformly for outer class modes, the scaling of the OS singular values is only expected to hold for the first several modes. However, since good agreement between the resolvent and DNS spectra is achieved using only a small number of modes, it is reasonable to adopt the scalings in what follows.
[For a particular $(k_x, k_z, c)\in\mathscr{S}_o$, the magnitudes of the leading vorticity response modes of the OS and SQ resolvent operators and their corresponding forcing modes for the three $\Rey$ in \[fig:means\] are shown in \[fig:modesouter\](a) and \[fig:modesouter\](b), respectively, using the scalings derived in . For comparison, the leading response and forcing mode for the standard resolvent operator are also shown; again, $\psi_\eta$ and $\phi_v$ are indistinguishable from the OS modes. Apparent from \[fig:modesouter\] is that the scaling of the outer class modes is only approximate. Recalling that the derivation of such universal classes relies on universal behavior of the mean profile, this is not surprising, since it is clear from \[fig:means\](b) that the mean profiles for the three $\Rey$ do not collapse perfectly for $y>0.1$. Additionally, $\phi_\eta$ for the standard resolvent (black dotted line in \[fig:modesouter\](b)) does not obey the same scaling as $\phi_\eta^\SQ$. Indeed, using the scaling of the standard resolvent in primitive variables presented in @sharma2017, it can be shown that $\phi_\eta = O(\Rey^\inv)$.]{}
Substituting \[eq:uo,eq:sigo\] into \[eq:A\], we obtain the outer-scaled energy density matrices:
$$\begin{aligned}
&\begin{array}{c c c}
\tenscomp{A}^{\OS/\OS}_{uu,ij} = \Rey^4 \out{\tenscomp{A}}^{\OS/\OS}_{uu,ij}, &
\tenscomp{A}^{\OS/\SQ}_{uu,ij} = \Rey^3 \out{\tenscomp{A}}^{\OS/\SQ}_{uu,ij}, &
\tenscomp{A}^{\SQ/\SQ}_{uu,ij} = \Rey^2 \out{\tenscomp{A}}^{\SQ/\SQ}_{uu,ij},
\end{array}
\\
&\begin{array}{c}
\tenscomp{A}^{\OS/\OS}_{vv,ij} = \Rey^2 \out{\tenscomp{A}}^{\OS/\OS}_{vv,ij},
\end{array}
\\
&\begin{array}{c c c}
\tenscomp{A}^{\OS/\OS}_{ww,ij} = \Rey^2 \out{\tenscomp{A}}^{\OS/\OS}_{ww,ij}, &
\tenscomp{A}^{\OS/\SQ}_{ww,ij} = \Rey \out{\tenscomp{A}}^{\OS/\SQ}_{ww,ij}, &
\tenscomp{A}^{\SQ/\SQ}_{ww,ij} = \out{\tenscomp{A}}^{\SQ/\SQ}_{ww,ij},
\end{array}
\\
&\begin{array}{c c}
\tenscomp{A}^{\OS/\OS}_{uv,ij} = \Rey^3 \out{\tenscomp{A}}^{\OS/\OS}_{uv,ij}, &
\tenscomp{A}^{\OS/\SQ}_{uv,ij} = \Rey^2 \out{\tenscomp{A}}^{\OS/\SQ}_{uv,ij}.
\end{array}
\end{aligned}$$
\[eq:Ao\]
The streamwise energy spectrum is thus $$E_{uu} = \Rey^4 \,\Re\!\left\{ \tr\! \left( \out{\tens{A}}^{\OS/\OS}_{uu}\tens{X}^{\OS/\OS} \right) \right\} + 2\Rey^3 \,\Re\!\left\{ \tr\! \left( \out{\tens{A}}^{\SQ/\OS}_{uu}\tens{X}^{\OS/\SQ} \right) \right\} + \Rey^2 \,\Re\!\left\{ \tr\! \left( \out{\tens{A}}^{\SQ/\SQ}_{uu}\tens{X}^{\SQ/\SQ} \right) \right\},
\label{eq:Euuo}$$ where again the Reynolds number dependence of the right-hand side is explicit, save for the unscaled weights matrices $\tens{X}^{\text{X}/\text{Y}}$, and \[eq:Ao\] can be used to write similar expressions for the other energy spectra.
As for the inner class modes, competition between the OS and SQ modes requires that all three terms are of the same order for arbitrary $\Rey$, which is satisfied if $$\left\vert \frac{\chi_j^\SQ}{\chi_j^\OS} \right\vert \sim \Rey
\label{eq:rato}$$ for modes belonging to the universal outer class.
The weights ratio $\sqrt{\tenscomp{X}_{11}^{\SQ/\SQ}/\tenscomp{X}_{11}^{\OS/\OS}}$ for the three Reynolds numbers is shown in the top row of \[fig:wout\] for several values of the outer-scaled wavenumber combinations $(\out{k}_x,\out{k}_x)$ and a minimum aspect ratio $\gamma=\sqrt{10}$. In agreement with \[eq:rato\], the data from all three Reynolds numbers show reasonable collapse onto a single curve for $U_{cl}-c\lesssim 6.17$ when scaled by $\Rey^\inv$, as seen in the bottom row of \[fig:wout\].
![(a)-(d) Leading weights ratio for $\Rey=934$ (blue), $\Rey=2003$ (red), and $\Rey=4219$ (green) with $N=2$. (e)-(h) Weights ratio scaled according to \[eq:rato\]. Each column represents a different scaled streamwise wavenumber $\out{k}_x$: (a),(e) $\out{k}_x=4219$; (b),(f) $\out{k}_x=8438$; (c),(g) $\out{k}_x=12657$; (d),(h) $\out{k}_x=16876$. In all cases the spanwise wavenumber is $k_z=\gamma \out{k}_x/\mathit{Re}_{\tau, \min}$, with $\gamma=\sqrt{10}$. Filled circles denote modes belonging to the universal outer class, and the largest outer class wavespeed defect, $U_{cl}-c = 6.17$, is indicated by the gray line.\[fig:wout\]](weights_outer.pdf){width="\textwidth"}
\[sec:selfssim\]Geometrically self-similar class
------------------------------------------------
The self-similar resolvent modes in the logarithmic region of the mean velocity profile belong to hierarchies parameterized by the critical layer location $\yc$ [@moarref2013]. The corresponding length scales along a hierarchy are $$\ssim{k}_x = \ycp\yc k_x, \quad \ssim{k}_z = \yc k_z, \quad \ssim{y} = y/\yc,
\label{eq:lss}$$ and the self-similar class wave parameters are $$\mathscr{S}_{h} : \left\{
\begin{array}{l}
16.4 \lesssim c \lesssim U_{c l}-6.17
\\
c = U(\ycp) = \kappa^\inv \log \ycp + B
\\
k_z / k_x \gtrsim \gamma
\end{array} \right.,
\label{eq:Sss}$$ where $\kappa$ is the Kármán constant. Note that the lower wavespeed bound for the self-similar class, $c=16.4$, is the same as the upper limit for the inner class modes, i.e., the beginning of the logarithmic region is taken to be $y^+=100$. As discussed above, recent evidence suggests that this lower limit is $\Rey$-dependent [@klewicki2009; @marusic2013]. However, @moarref2013 demonstrated successful scaling of the self-similar modes using the fixed lower limit in \[eq:Sss\], so we continue to use it here. For reference, the beginning of the logarithmic region according to the balance of terms in the mean momentum equation, $y^+\approx2.6\Rey^{1/2}$ [@klewicki2009], is indicated by the gray dashed line in \[fig:wss\]. The self-similar modes must also satisfy an aspect ratio constraint. Since the aspect ratio increases like $\ycp$ along a given hierarchy, it is sufficient that the lowest member on the hierarchy with critical layer $y_{c,l}$ and wavenumbers $k_{x,l},k_{z,l}$ satifies $k_{z,l} / k_{x,l} \gtrsim \gamma$, where a conservative lower bound is $\gamma\approx\sqrt{10}$ [@moarref2013].
Using \[eq:lss\] and continuity, as well as the orthonormality constraint on the resolvent modes, it follows that $$\hat{\vect{u}} = \yc^{-1/2}
\begin{pmatrix}
\ssim{u}\\
\yc^{+\, \inv} \ssim{v}\\
\yc^{+\, \inv} \ssim{w}
\end{pmatrix}.
\label{eq:uss}$$ where $\ssim{\blankop}$ indicates a quantity that is approximately $\yc$- and $\Rey$-invariant for modes belonging to the self-similar class.
![(a) Magnitudes of the scaled OS, SQ, and standard resolvent (color scheme as in \[fig:modes\]) self-similar leading vorticity response modes for five members of a hierarchy at $\Rey=2003$, with $k_{x,l} = 10$ and $k_{z,l} = 10^{3/2}$. (b) Corresponding scaled leading forcing mode magnitudes. $\psi_\eta$ and $\phi_v$ for the standard resolvent (black) are indistinguishable from the OS modes (blue). \[fig:modesss\]](ss_compare.pdf){width="70.00000%"}
The $\yc$-scaled versions of the weighted resolvent operators are
\[eq:Hopsss\] $$\begin{aligned}
\begin{pmatrix}
\op{F}_v \res_{vv} \op{F}_v^\inv \\
\op{F}_\eta \res_{\eta v} \op{F}_v^\inv
\end{pmatrix}
&=
\begin{pmatrix}
\yc\ycp \ssim{\op{F}}_v \ssim{\res}_{vv} \ssim{\op{F}}_v^\inv \\
\yc\yc^{+\,2} \ssim{\op{F}}_\eta \ssim{\res}_{\eta v} \ssim{\op{F}}_v^\inv
\end{pmatrix}
\label{eq:Hosss}
\\
\op{F}_\eta \res_{\eta\eta} \op{F}_\eta^\inv &= \yc\ycp \ssim{\op{F}}_\eta \ssim{\res}_{\eta\eta} \ssim{\op{F}}_\eta^\inv,
\label{eq:Hsqss}
\end{aligned}$$
so that their leading singular values scale as $$\sigma_j^\OS = \yc\yc^{+\,2} \ssim{\sigma}_j^\OS, \quad \sigma_j^\SQ = \yc\ycp \ssim{\sigma}_j^\SQ.
\label{eq:sigss}$$ As with the outer class modes, the scaling of the OS singular values are only expected to hold for the first several modes.
[The magnitudes of the leading vorticity response modes of the OS and SQ resolvent operators and their corresponding forcing modes for five members of a particular hierarchy at $\Rey=2003$ are shown in \[fig:modesss\](a) and \[fig:modesss\](b), respectively, using the scalings derived in . For comparison, the leading response and forcing mode for the standard resolvent operator are also shown; again, $\psi_\eta$ and $\phi_v$ are indistinguishable from the OS modes. Here again, $\phi_\eta$ for the standard resolvent (black dotted line in \[fig:modesss\](b)) and $\phi_\eta^\SQ$ exhibit different scalings, with the standard resolvent $\phi_\eta = O(\yc^{-3/2}\yc^{+\,-1})$.]{}
Substituting \[eq:uss,eq:sigss\] into \[eq:A\], we obtain the scaled energy density matrices:
$$\begin{aligned}
&\begin{array}{c c c}
\tenscomp{A}^{\OS/\OS}_{uu,ij} = \yc\yc^{+\,4} \ssim{\tenscomp{A}}^{\OS/\OS}_{uu,ij}, &
\tenscomp{A}^{\OS/\SQ}_{uu,ij} = \yc\yc^{+\,3} \ssim{\tenscomp{A}}^{\OS/\SQ}_{uu,ij}, &
\tenscomp{A}^{\SQ/\SQ}_{uu,ij} = \yc\yc^{+\,2} \ssim{\tenscomp{A}}^{\SQ/\SQ}_{uu,ij},
\end{array}
\\
&\begin{array}{c}
\tenscomp{A}^{\OS/\OS}_{vv,ij} = \yc\yc^{+\,2} \ssim{\tenscomp{A}}^{\OS/\OS}_{vv,ij},
\end{array}
\\
&\begin{array}{c c c}
\tenscomp{A}^{\OS/\OS}_{ww,ij} = \yc\yc^{+\,2} \ssim{\tenscomp{A}}^{\OS/\OS}_{ww,ij}, &
\tenscomp{A}^{\OS/\SQ}_{ww,ij} = \yc\ycp \ssim{\tenscomp{A}}^{\OS/\SQ}_{ww,ij}, &
\tenscomp{A}^{\SQ/\SQ}_{ww,ij} = \yc \ssim{\tenscomp{A}}^{\SQ/\SQ}_{ww,ij},
\end{array}
\\
&\begin{array}{c c}
\tenscomp{A}^{\OS/\OS}_{uv,ij} = \yc\yc^{+\,3} \ssim{\tenscomp{A}}^{\OS/\OS}_{uv,ij}, &
\tenscomp{A}^{\OS/\SQ}_{uv,ij} = \yc\yc^{+\,2} \ssim{\tenscomp{A}}^{\OS/\SQ}_{uv,ij},
\end{array}
\end{aligned}$$
and the streamwise energy spectrum is $$E_{uu} = \yc\yc^{+\,4} \,\Re\!\left\{ \tr\! \left( \ssim{\tens{A}}^{\OS/\OS}_{uu}\tens{X}^{\OS/\OS} \right) \right\} + 2\yc\yc^{+\,3} \,\Re\!\left\{ \tr\! \left( \ssim{\tens{A}}^{\SQ/\OS}_{uu}\tens{X}^{\OS/\SQ} \right) \right\} + \yc\yc^{+\,2} \,\Re\!\left\{ \tr\! \left( \ssim{\tens{A}}^{\SQ/\SQ}_{uu}\tens{X}^{\SQ/\SQ} \right) \right\},
\label{eq:Euuss}$$ where the $\yc$ dependence of the right-hand side is explicit, except for the unscaled weights matrices $\tens{X}^{\text{X}/\text{Y}}$. Balancing all three terms requires $$\left\vert \frac{\chi_j^\SQ}{\chi_j^\OS} \right\vert \sim \ycp.
\label{eq:ratss}$$ The ratio $\sqrt{\tenscomp{X}_{11}^{\SQ/\SQ}/\tenscomp{X}_{11}^{\OS/\OS}}$ is plotted along several hierarchies with different $k_{x,l},k_{z,l}$ for $\Rey=2003$ in \[fig:wss\]. [In this case, to have a sufficient number of wavespeeds belonging to $\mathscr{S}_h$ while keeping the size of the optimization problem manageable, the matching of the DNS spectra is only enforced for $y^+_{\min}=100\leq y^+\leq 0.1\Rey=y^+_{\max}$, with $N_c=25$.]{} The scaling given by \[eq:ratss\] is clearly demonstrated, as the data in all cases exhibit a linear dependence on $\ycp$ to within a good approximation. In all cases shown the aspect ratio at the bottom of the hierarchies is $\gamma=5$. Similar results are obtained using different aspect ratios, provided that $\gamma \gtrsim \sqrt{10}$ [@moarref2013]. [The slopes of the lines are observed to decrease with increasing $k_{x,l}$. Although not shown here, the slopes tend to increase with increasing $\gamma$.]{}
![Leading weights ratio along hierarchies for $\Rey=2003$ with $N=2$. Each panel is a different hierarchy, represented by the streamwise wavenumber at the bottom of the hierarchy: (a) $k_{x,l}=1$; (b) $k_{x,l}=5$; (c) $k_{x,l}=10$; (d) $k_{x,l}=20$. In all cases the spanwise wavenumber at the bottom of the hierarchy is $k_{z,l}=\gamma k_{x,l}$, with $\gamma=5$. [The solid gray lines are the least squares linear fits, with slopes (a) 0.158, (b) 0.145, (c) 0.107, and (d) 0.069.]{} The dashed gray lines are $y^+=2.6\Rey^{1/2}\approx116$.\[fig:wss\]](weights_ss.pdf){width="\textwidth"}
\[sec:conclusions\]Discussion and Conclusions
=============================================
A low-order [representation of]{} the time-averaged energy spectra of turbulent channel flow based on the resolvent analysis framework was presented. The resolvent mode weights, which encode information about the nonlinear interactions in the flow were determined empirically by computing the weights that minimize the deviation between the resolvent spectra and spectra obtained from DNS using a convex optimization scheme. The present approach is a modification of previous work [@moarref2014], with the major difference being the incorporation of a recently-proposed alternative decomposition of the resolvent operator into two distinct families of modes, referred to as the Orr-Sommerfeld and Squire families [@rosenberg2019a].
It was demonstrated that the alternative OS-SQ decomposition results in a dramatic improvement in the performance of the representation. This improvement is attributed to the isolation of the $v$ response in the OS family, which enables the $\eta$ response of the SQ family to compete with the large $\eta$ response generated by the OS modes. Furthermore, for certain values of wave the parameters, the leading modes of the standard resolvent operator are almost identical to the leading modes of the OS resolvent, so that the mechanisms encoded in the SQ operator are essentially neglected; this helps explain the relatively poor performance of the representation obtained using the standard resolvent. A decomposition of the statistics into contributions from the OS modes, SQ modes, and an interaction between the two families supports this claim and is in agreement with results from DNS at $\Rey=185$ [@rosenberg2018]. It was further shown that the competition between the OS and SQ modes can be interpreted as a phase difference, and that this phase difference is speculated to be close to $\pi$ over large portions of spectral space.
Next, the scaling of the leading singular values for the OS and SQ families were derived for the inner, outer, and geometrically self-similar universal classes of resolvent modes [@moarref2013]. For the inner class, both sets of singular values scale as $\Rey^\inv$. For the outer and self-similar classes, the OS singular values are larger than the SQ ones by a factor of $\Rey$ and $\ycp$, respectively. Interestingly, this [large difference in amplification]{} suggests that modes in these classes are likely to be among those for which the OS and standard resolvent modes are nearly identical. Indeed, the scaling of the leading OS singular values is the same as that for the leading singular values of the standard resolvent for the outer and self-similar classes. For the inner class, the scaling for both the OS and SQ singular values match the standard resolvent. Combining the scalings with the hypothesis that the competition between SQ and OS modes discussed above remains relevant for arbitrary $\Rey$ and throughout the flow domain was used to derive the relative scalings of the OS and SQ weights in each of the universal classes. The scaling predictions were tested against the optimized weights, and, with the exception of high aspect ratio modes of the inner class localized very near the wall, good agreement with the computed optimal weights was found for each of the universal classes.
[The results presented herein have several important implications for equation-driven modeling of turbulent channel flow. The first is that partitioning the resolvent operator into Orr-Sommerfeld and Squire subsystems, originally presented in the context of ECS [@rosenberg2019a], is also advantageous in terms of its ability to develop compact representations of fully turbulent channel flow at high Reynolds number. Furthermore, it provides valuable insight into the complex dynamics by identifying the competition mechanism between the OS and SQ modes, which has ramifications for modeling nonlinear interactions. Specifically,]{} considering that for large $\Rey$, the OS singular values in the logarithmic and outer regions of the flow are much larger than the SQ ones, it may be tempting from a modeling perspective to neglect the SQ family of modes. However, doing so does not take into account the relative scaling of the forcing terms $\hat{g}_v$ and $\hat{g}_\eta$ in \[eq:NSEOS\] – it implicitly assumes they remain of the same order. The present results indicate that this is not the case. In fact, the scaling results of the weights for all of the classes can be summarized as $\vert \chi^\SQ_j/\chi^\OS_j \vert \sim \sigma^\OS_j/\sigma^\SQ_j$.
Though the absolute scalings of the weights were not determined, the present work can be considered a starting point to guide further modeling efforts toward quantifying nonlinear interactions in turbulent channel flow. [For instance, it is particularly intriguing that, as discussed in \[sec:kxspectra\], the $v$ statistics depend only on the OS modes. Consequently, if the scaling of the OS weights can be determined from these, empirically or otherwise, then the results given in \[sec:scaling\] can be used to determine the scaling of the SQ weights, effectively reducing the number of unknowns by half. Then a single computation at a relatively low Reynolds number could be combined with the scalings to make predictions of the spectra at Reynolds numbers that are currently unattainable by DNS.]{}
Taken together, the results point to the competition between the OS and SQ modes being an important mechanism in turbulent channel flow that should be respected in order to accurately model the statistics. We hypothesize that if this mechanism could be interrupted, the dynamics, and consequently the statistics, of the system would be significantly different. This line of inquiry is the subject of ongoing work.
The support of AFOSR under grant FA 9550-16-1-0361 and ONR under N00014-17-1-3022 is gratefully acknowledged. Additionally, the authors would like to thank Javier Jiménez for making the spectra for the $\Rey=934$ and $\Rey=2003$ simulations publicly available, as well as Adrián Lozano-Durán for sharing the spectra for $\Rey=4219$.
\[sec:err\]Behavior at large ${\lambda}_x$
==========================================
As discussed in \[sec:kxspectra\], the most significant discrepancies between the DNS and OS-SQ representation of 1D the spectra shown in \[fig:kxspect\] occur in $k_xE_{uu}$ at large $\lambda_x^+$ and $y^+ \lesssim 100$ and $-k_xE_{uv}$ at large $\lambda_x^+$ and $y^+ \lesssim 50$. Furthermore, these errors do not improve considerably with an increasing number of modes, as demonstrated in \[fig:intserr\], which shows a slow decrease in the error for $N^{\OS}=N^{\SQ}=N > 4$.
![Relative errors in $\langle u^2 \rangle$ (blue), $\langle v^2 \rangle$ (red), $\langle w^2 \rangle$ (green), and $\langle -uv \rangle$ (orange), as a function of $N^{\OS}=N^{\SQ}=N$.\[fig:intserr\]](ints_err.pdf){width="46.00000%"}
![Comparison of DNS (black) and the optimization results with $y^+_{\text{min}}=5$ (red) and $y^+_{\text{min}}=50$ (green) using $N^{\OS}=N^{\SQ}=6$ modes for $(\lambda_x^+, \lambda_z^+) \approx (3.83\times10^4, 2.78\times10^3)$. \[fig:ypi\]](ypi_compare.pdf){width="42.00000%"}
The reason for the persisting error is that for large $\lambda_x^+$, there is significant energetic content below the peaks of the [lowest wavespeed modes, which typically sit around $y^+\approx 40-50$ for $\Rey=2003$ [@moarref2013a]. Thus, trying to match near the wall results in overcompensation at larger $y^+$. This is illustrated for the representative wavelenghts $(\lambda_x^+, \lambda_z^+) \approx (3.83\times10^4, 2.78\times10^3)$ in \[fig:ypi\]. To confirm that this is indeed the cause, \[fig:ypi\] also shows the result of the optimization with $y^+_{\text{min}}=50$, in which case the large oscillations disappear.]{} [The near-wall errors eventually diminish as the number of modes tends to infinity.]{} However, the fact that the spectra for these wavenumbers are not well-represented by a low-rank approximation suggests that the response modes may not be the most efficient basis. Indeed, @rosenberg2019b outline conditions under which the flowfields around a cylinder and for channel ECS are more compactly represented by the response to the forcing generated by the leading response at a different wavenumber triplet; it is possible that the present case is a similar situation, but the number of triadic interactions that would have to be accounted for in fully turbulent flow significantly complicates matters. It has also recently been shown that augmentation of \[eq:resop\] with an eddy viscosity improves the representation of large-scale structures [@hwang2016; @illingworth2018; @madhusudanan2019]. [Both approaches attempt to constrain the forcing, the former by using triadic interactions to identify which scales are most important, and the latter by choosing to only directly model the large-scale coherent motions. However, as pointed out in \[sec:intro\], such nonlinear interactions are incompatible with the turbulent mean velocity profile when an eddy viscosity is included.]{}
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We consider the problems of error-correcting codes and image restoration with multiple stages of dynamics. Information extracted from the former stage can be used selectively to improve the performance of the latter one. Analytic results were derived for the mean-field systems using the cavity method. We find that it has the advantage of being tolerant to uncertainties in hyperparameter estimation, as confirmed by simulations.'
address:
- 'Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong'
- 'Department of Physics, Tokyo Institute of Technology, Oh-Okayama, Meguro-ku, Tokyo 152-8551, Japan'
author:
- 'K. Y. Michael Wong'
- Hidetoshi Nishimori
title: |
Error-correcting codes and image restoration\
with multiple stages of dynamics
---
INTRODUCTION
============
The corruption of signals by noise is a common problem encountered in information processing. To retrieve signals from messages corrupted during the transmission through noisy channels, various error-correcting codes have been proposed [@eliece]. In particular, the error-correction mechanism of a class of parity-checking codes can be considered as the search for thermodynamically stable states of a Hamiltonian constructed in terms of the message bits [@sourlas]. These codes have been demonstrated to saturate the Shannon information bound in the limit that each encoded bit checks the parity of an infinitely large number of message bits [@sourlas; @kabasaad]. While in practice, each encoded bit can only check the parity of a finite number of message bits, these codes still maintain a very low bit error probability.
The need to retrieve signals from corrupted messages is also inherent in image restoration [@geman]. Although parity-checking bits may not be explicitly introduced for the task, prior knowledge about the images plays a similar role. For example, the smoothness of real-world images provides a mechanism for checking the pixel values in comparison with those of their neighbors. A corresponding Hamiltonian, consisting of a ferromagnetic bias to reflect the smoothening tendency, can be constructed in terms of the image pixels. Modern techniques of image restoration based on Markov random fields correspond to the search for thermodynamically stable states of the Hamiltonian system, using methods such as simulated annealing [@geman].
In a recent paper, we have shown that the problems of error-correcting codes and image restoration can be formulated in a unified framework [@nishiwong]. In both tasks, the choice of the so-called hyperparameters is an important factor in determining their performances. Hyperparameters refer to the coefficients of the various interactions appearing in the Hamiltonian of the tasks. In error correction, they determine the statistical significance given to the parity-checking terms and the received bits. Similarly in image restoration, they determine the statistical weights given to the prior knowledge and the received data. It was shown, by the use of inequalities, that the optimal choice of the hyperparameters correspond to the Maximum Posterior Marginal (MPM) method, where there is a match between the source and model priors. The choice of these values correspond to the Nishimori point in the space of hyperparameters [@nishimori]. It is equivalent to a thermodynamic process at finite temperature, and the task performance is better than the Maximum A Posteriori probability (MAP) method, where the values of the hyperparameters are taken to infinity, equivalent to a zero temperature process. Furthermore, from the analytic solution of the infinite-range model and the Monte Carlo simulation of finite-dimensional models, it was shown that an inappropriate choice of the hyperparameters can lead to a rapid degradation of the tasks.
In fact, hyperparameter estimation has been the subject of many previous studies [@zhou], a recently popular one using the “evidence framework” [@mackay]. However, if the prior models the source poorly, no hyperparameters can be reliable [@pryce]. Even if they can be estimated accurately through steady-state statistical measurements, they may fluctuate when interfered by bursty noise sources in communication channels. Hence it is important to devise decoding or restoration procedures which are robust against the uncertainties in hyperparameter estimation.
In this paper we propose the technique of selective freezing as a method to increase the tolerance to uncertainties in hyperparameter estimation. The technique has been studied for pattern reconstruction in neural networks, where it led to an improvement in the retrieval precision, a widening of the basin of attraction, and a boost in the storage capacity [@wong]. The idea is best illustrated for Ising bits or pixels with binary states $\pm 1$, though it can be easily generalized to other cases. In a finite temperature thermodynamic process, the Ising variables keep moving under thermal agitation. Some of them have smaller thermal fluctuations than the others, implying that they are more certain to stay in one state than the other. This stability implies that they have a higher probability to stay in the correct state for error-correction or image restoration tasks, even when the hyperparameters are not optimally tuned. It may thus be interesting to separate the thermodynamic process into two stages. In the first stage we select those relatively stable bits or pixels whose time-averaged states have a magnitude exceeding a certain threshold. In the second stage we subsequently fix (or freeze) them in the most probable thermodynamic states (for Ising variables this corresponds to the sign of the time-averaged state). Thus these selectively frozen bits or pixels are able to provide a more robust assistance to the less stable bits or pixels in their search for the most probable states. The selective freezing procedure reduces to the usual finite-temperature decoding or restoration process if all bits or pixels are frozen (since nothing happens in the second stage), or no bits or pixels are frozen (since the second stage is merely a continuation of the equilibration process of the first stage).
The two-stage thermodynamic process can be studied analytically in the mean-field model, which provides a qualitative guide to the behavior of more realistic cases of lower dimensions. However, it is necessary to give a remark about the theoretical approach. That is, as far as we have tried, the analytical solution has been inaccessible by the more conventional replica method. Rather, we have to use the cavity method to obtain the equations for the order parameters. In particular, the cavity method leads to the appearance of a term called the trans-susceptibility, which correctly describes the effects of the thermodynamics of the first stage on that of the second.
The paper is organized as follows. In Section II we briefly review the formulation of error-correcting codes and image restoration in a unified framework. In Sections III and IV, we consider the mean-field model for error-correcting codes and image restoration respectively. We derive the equations for the order parameters of the two-stage thermodynamics using the cavity method, and present numerical results illustrating the robustness of selective freezing against uncertainties in hyperparameter estimation. We further demonstrate that even when the noise model changes without the receiver/restoration agent realizing the change (i.e. it makes a wrong estimation of the prior), the task performance is still robust. For the more realistic cases of lower dimensions, simulation results illustrate the relevance of the infinite-range model in providing qualitative guidance. The conclusion is given in Section V.
FORMULATION
===========
Consider an information source which generates data represented by a set of Ising spins $\{\xi_i\}$, where $\xi_i=\pm 1$ and $i=1, \cdots, N$. The data is generated according to the source prior $P_s(\{\xi_i\})$. For error-correcting codes transmitting unbiased messages, all sequences are equally probable and $P_s(\{\xi\})=2^{-N}$. For images with smooth structures, the prior consists of ferromagnetic Boltzmann factors, which increase the tendencies of the neighboring spins to stay at the same spin states, that is, $$P_s(\{\xi\})=\frac{1}{Z(\beta_s)}
\exp \left({\beta_s\over z}\sum_{\langle ij\rangle}
\xi_i \xi_j \right).
\label{prior}$$ Here $\langle ij\rangle$ represents pairs of neighboring spins, $z$ is the valency of each site, and the partition function $Z(\beta_s)$ is given by $$Z(\beta_s)={\rm Tr}_\xi
\exp \left({\beta_s\over z}\sum_{\langle ij\rangle}
\xi_i \xi_j \right).
\label{partition}$$ The data is coded by constructing the codewords, which are the products of $p$ spins $J^0_{i_1 \cdots i_p}=\xi_{i_1}\cdots\xi_{i_p}$ for appropriately chosen sets of of indices $\{i_1,\cdots,i_p\}$, the choice of which determines the type of code. Each spin may appear in a number of $p$-spin codewords; the number of times of appearance is called the valency $z_p$. The Sourlas code [@sourlas] is equivalent to the infinite-range model in which all possible codewords of $p$ spins are chosen from $N$ spins. On the other hand, the Kabashima-Saad code [@kabasaad] consists of combinations in which each spin appears in a random pre-selection of $z_p$ codewords. For conventional image restoration, codewords with only $p=1$ are transmitted, corresponding to the pixels in the image; the inclusion of terms with $p>1$, and their positive effects on restoring the original image, have also been discussed in [@nishiwong]. For simplicity, we restrict ourselves to the case of a single non-vanishing value of $p$ with $p\ge 2$, and $p=1$.
When the signal is transmitted through a noisy channel, the output consists of the sets $\{ J_{i_1\cdots i_p}\}$ and $\{\tau_i\}$, which are the corrupted versions of $\{J^0_{i_1 \cdots i_p}\}$ and $\{\xi_i\}$ respectively. In the binary symmetric channel, the outputs $J_{i_1 \cdots i_p}$ are equal to $\mp J^0_{i_1 \cdots i_p}$ with probabilities $p_J$ and $1-p_J$ respectively, and $\tau_i$ equal to $\mp\xi_i$ with probabilities $p_\tau$ and $1-p_\tau$ respectively. Thus $$P_{\rm out}(\{J\},\{\tau\}|\{\xi\})
\propto\exp\left(
\beta_J\sum J_{i_1 \cdots i_p}\xi_{i_1}\cdots\xi_{i_p}
+\beta_\tau\sum\tau_i\xi_i
\right),
\label{BSC}$$ where $$\beta_J={1\over 2}\ln\frac{1-p_J}{p_J}
\quad{\rm and}\quad
\beta_\tau={1\over 2}\ln\frac{1-p_\tau}{p_\tau}.$$ The first summation in the exponent of Eq. (\[BSC\]) extends over an appropriate set of the indices $(i_1, \cdots ,i_p)$.
The Gaussian channel is defined by, for a given sequence $\{\xi_i\}$, $$P_{\rm out}(\{ J\},\{\tau \}|\{\xi\})
\propto\exp\left(
-\frac{1}{2J^2}\sum
(J_{i_1 \cdots i_p}-J_0\xi_{i_1}\cdots\xi_{i_p})^2
-\frac{1}{2\tau^2}\sum
(\tau_i-a\xi_i)^2\right).
\label{gauss}$$ $J_0$ and $a$ are the strengths of the signals to be fed into the channel, and $J^2$ and $\tau^2$ are the variances of the noise. We note that by letting $\beta_J$ and $\beta_\tau$ to be $J_0/J^2$ and $a/\tau^2$ respectively, the input-dependent terms of Eq. (\[gauss\]) reduce to those of Eq. (\[BSC\]), which therefore can be regarded as the noise model for both binary symmetric and Gaussian channels.
According to Bayesian statistics, the posterior probability that the source sequence is $\{\sigma\}$, given the outputs $\{ J\}$ and $\{\tau\}$, takes the form $$P(\{\sigma\}|\{ J\},\{\tau\})
\propto P_{\rm out}(\{J\},\{\tau\}|\{\sigma\})P_s(\{\sigma\}).$$ Using Eq. (\[BSC\]) and (\[prior\]), we have $$P(\{\sigma\}|\{ J\},\{\tau\})
\propto\exp\left(
\beta_J\sum J_{i_1\cdots i_p}\sigma_{i_1}\cdots\sigma_{i_p}
+\beta_\tau\sum\tau_i\sigma_i
+{\beta_s\over z}\sum_{\langle ij\rangle}\sigma_i\sigma_j
\right).
\label{Bayes0}$$ It often happens that the receiver at the end of the noisy channel does not have precise information on $\beta_J$, $\beta_\tau$ or $\beta_s$. One then has to estimate these parameters. If the receiver estimates $\beta$, $h$ and $\beta_m$ for $\beta_J$, $\beta_\tau$ and $\beta_s$ respectively, then the mean of the posterior distribution of $\sigma_i$ is equal to the thermal average $$\langle\sigma_i\rangle
={{\rm Tr}\sigma_i e^{-H\{\sigma\}}
\over{\rm Tr} e^{-H\{\sigma\}}},
\label{sigmai}$$ where the Hamiltonian is given by $$H\{\sigma\}
=-\beta\sum J_{i_1 \cdots i_p}\sigma_{i_1}\cdots\sigma_{i_p}
-h\sum\tau_i\sigma_i
-{\beta_m\over z}\sum_{\langle ij\rangle}\sigma_i\sigma_j.
\label{Hamiltonian}$$ One then regards ${\rm sgn}\langle\sigma_i\rangle$ as the $i$th bit of the decoded/restored information.
To reduce the sensitivity of the decoding/restoration process to the uncertainties in parameter estimation, we propose a two-stage process of selective freezing instead of the one-stage thermodynamic process implied by Eq. (\[sigmai\]). In the first stage the spins evolve thermodynamically as prescribed in Eq. (\[sigmai\]), and the thermal averages $\langle\sigma_i\rangle$ of the spins are monitored. We may relate $\langle\sigma_i\rangle$ to an effective field $H_i$ by $\langle\sigma_i\rangle=\tanh H_i$. Spins with larger magnitudes of $\langle\sigma_i\rangle$ correspond to larger magnitudes of $H_i$. They are more likely to agree with the correct message or image bit, and are less likely to change signs even when the hyperparameters vary. Their relative stability can be used to assist the less stable spins to boost their robustness against hyperparameter uncertainties. Hence we select those spins with $|\langle\sigma_i\rangle|$ exceeding a given threshold $\theta$, and freeze them in the second stage of the thermodynamics. The average of the spin $\tilde\sigma_i$ in the second stage is then given by $$\langle\tilde\sigma_i\rangle
={{\rm Tr}\tilde\sigma_i
\prod_j\left[\Theta\left(\langle\sigma_j\rangle^2-\theta^2\right)
\delta_{\tilde\sigma_j,{\rm sgn}\langle\sigma_j\rangle}
+\Theta\left(\theta^2-\langle\sigma_j\rangle^2\right)\right]
e^{-\tilde H\{\tilde\sigma\}}
\over{\rm Tr}
\prod_j\left[\Theta\left(\langle\sigma_j\rangle^2-\theta^2\right)
\delta_{\tilde\sigma_j,{\rm sgn}\langle\sigma_j\rangle}
+\Theta\left(\theta^2-\langle\sigma_j\rangle^2\right)\right]
e^{-\tilde H\{\tilde\sigma\}}},$$ where $\Theta$ is the step function, $\tilde H\{\tilde\sigma\}$ is the Hamiltonian for the second stage, and has the same form as Eq. (\[Hamiltonian\]) in the first stage. To increase the flexibility in the process, the parameters $\beta$, $h$ and $\beta_m$ can be replaced by $\tilde\beta$, $\tilde h$ and $\tilde\beta_m$ respectively in the second stage. One then regards ${\rm sgn}\langle\tilde\sigma_i\rangle$ as the $i$th spin of the decoding/restoration process.
The most important quantity in selective freezing is the overlap of the decoded/restored bit ${\rm sgn}\langle\tilde\sigma_i\rangle$ and the original bit $\xi_i$ averaged over the output probability and the spin distribution. This is given by $$M_{\rm sf}
=\sum_\xi\prod\int dJ\prod\int d\tau
P_s(\{\xi\})P_{\rm out}(\{J\},\{\tau\}|\{\xi\})
\xi_i{\rm sgn}\langle\tilde\sigma_i\rangle.
\label{M}$$ Following Appendix A of [@nishiwong], we can prove the following inequality $$M_{\rm sf}
\le M(\beta=\beta_J, h=\beta_\tau, \beta_m=\beta_s),
\label{bound1}$$ where the right hand side is the overlap of the [*single-stage*]{} dynamics when the model parameters $\beta$, $h$ and $\beta_m$ match the source parameters $\beta_J$, $\beta_\tau$ and $\beta_s$ respectively. Hence selective freezing cannot outperform the single-stage process if the hyperparameters can be estimated precisely. However, we remark that the purpose of selective freezing is rather to provide a relatively stable performance when the hyperparameters cannot be estimated precisely. This cannot be revealed from the inequality, but will be confirmed by the analytic and simulation results in Sections III and IV.
THE INFINITE-RANGE MODEL FOR ERROR-CORRECTING CODES
===================================================
Let us now suppose that the output of the transmission channel consists of only the set of $p$-spin interactions $\{J_{i_1\cdots i_p}\}$. The Hamiltonian (\[Hamiltonian\]) then becomes $$H\{\sigma\}
=-\beta\sum_{i_1<\cdots<i_p}
J_{i_1\cdots i_p}\sigma_{i_1}\cdots\sigma_{i_p},
\label{p-hamiltonian}$$ where we have set $\beta_m=0$ for the case that all messages are equally probable.
Analytical solutions for the overlap are in general unavailable. We therefore consider the infinite-range model in which the exchange interactions are present for all possible pairs of sites in the Hamiltonian of Eq. (\[p-hamiltonian\]).
To consider the transition between error-free and errored regimes, we are interested in the noise model in which $J_{i_1\cdots i_p}$ is Gaussian with mean $p!j_0\xi_{i_1}\cdots\xi_{i_p}/N^{p-1}$ and variance $p!J^2/2N^{p-1}$. Since all messages are equally probable, we can apply a gauge transformation $\sigma_i\to\sigma_i\xi_i$ and $J_{i_1\cdots i_p}\to J_{i_1\cdots i_p}\xi_{i_1}\cdots\xi_{i_p}$ to (\[p-hamiltonian\]), and arrive at an equivalent $p$-spin model with a ferromagnetic bias, where $$P(J_{i_1\cdots i_p})
=\left(N^{p-1}\over\pi J^2p!\right)^{1/2}
\exp\left[-{N^{p-1}\over J^2p!}
\left(J_{i_1\cdots i_p}-{p!\over N^{p-1}}j_0\right)^2\right].
\label{jdis}$$ The Nishimori point for this model is located at $\beta=2j_0/J^2$.
The infinite-range model is exactly solvable using mean-field theoretical techniques for disordered systems such as the replica or cavity method [@mezard]. Here we use the cavity method because of its more transparent physical interpretation, and some obstacles encountered in the use of the replica method.
The cavity method uses a self-consistency argument to consider what happens when a spin is added or removed from the system. The central quantity in this method is the [*cavity field*]{}, which is the local field of a spin when it is added to the system, assuming that the exchange couplings act only one-way from the system to the new spin (but not from the spin back to the system). Since the exchange couplings feeding the new spin have no correlations with the system, the cavity field becomes a Gaussian variable in the limit of large valency.
Average spin in the first stage
-------------------------------
We start with the so-called “clustering property” for mean-field systems [@mezard], $$\langle\sigma_{i_1}\cdots\sigma_{i_p}\rangle
=\langle\sigma_{i_1}\rangle\cdots\langle\sigma_{i_p}\rangle,
\label{cluster}$$ where $\langle\quad\rangle$ represents thermodynamic averages. As shown in Appendix A, the clustering property enables us to express the thermal averages of a spin in terms of the cavity field, say, for spin 1, $$\langle\sigma_1\rangle
=\tanh\beta h_1;\quad
h_1=\sum_{1<j_2<\cdots<j_p}J_{1j_2\cdots j_p}
\langle\sigma_{j_2}\rangle^{\backslash 1}\cdots
\langle\sigma_{j_p}\rangle^{\backslash 1},
\label{cavity}$$ where the superscript $\backslash 1$ denotes the thermal averages for a Hamiltonian in which $\sigma_1$ and the associated exchange interactions are absent, but otherwise identical to Eq. (\[p-hamiltonian\]). Thus $h_1$ is the cavity field obeying a Gaussian distribution, whose mean and variance are $pj_0m^{p-1}$ and $pJ^2q^{p-1}/2$ respectively, where $m$ and $q$ are the magnetization and Edwards-Anderson order parameter respectively, given by $$m\equiv{1\over N}\sum_i\langle\sigma_i\rangle
\quad{\rm and}\quad
q\equiv{1\over N}\sum_i\langle\sigma_i\rangle^2.
\label{mq}$$ It is convenient to write $$\beta h_i=\hat m+\sqrt{\hat q}u_i,$$ where $$\hat m=p\beta j_0m^{p-1}
\quad{\rm and}\quad
\hat q={p\over 2}\beta^2J^2q^{p-1},
\label{mqhat}$$ and $u_i$ is a Gaussian variable with mean 0 and variance 1.
Order parameters in the first stage
-----------------------------------
Applying self-consistently the cavity argument to all terms in Eq. (\[mq\]), we can obtain self-consistent equations for $m$ and $q$: $$\begin{aligned}
m&=&\int Du\tanh G,
\label{morder}\\
q&=&\int Du\tanh^2 G,
\label{qorder}\end{aligned}$$ where $Du\equiv du e^{-u^2/2}/\sqrt{2\pi}$ is the Gaussian measure, $G=\hat m+\sqrt{\hat q}u$. The overlap for the one-stage decoding process is given by $$M\equiv{1\over N}\sum_i{\rm sgn}\langle\sigma_i\rangle
={\rm erf}{\hat m\over\sqrt{2\hat q}}.$$ Now we consider selective freezing. If we introduce a freezing threshold $\theta$ so that all spins with $\langle\sigma_i\rangle^2>\theta^2$ are frozen, then the freezing fraction $f$ is given by $$f\equiv{1\over N}\sum_i\Theta\left(
\langle\sigma_i\rangle^2-\theta^2\right)
=1-{1\over 2}{\rm erf}{u_+\over\sqrt{2}}
+{1\over 2}{\rm erf}{u_-\over\sqrt{2}},$$ where $u_\pm=(\pm u_0-\hat m)/\sqrt{\hat q}$ with $\tanh u_0=\theta$.
Average spin in the second stage
--------------------------------
Assuming that the spin $\tilde\sigma_1$ is dynamic in the second stage, we can write $$H\{\tilde\sigma\}\approx
H\{\tilde\sigma\}^{\backslash 1}
-\tilde\beta\sum_{1<j_1\cdots<j_{p-1}}
\tilde\sigma_1J_{1j_1\cdots j_{p-1}}
\prod_{s=1}^{p-1}\left[
\tilde\sigma_{j_s}
\Theta\left(\theta^2-\langle\sigma_{j_s}\rangle^2\right)
+{\rm sgn}\langle\sigma_{j_s}\rangle
\Theta\left(\langle\sigma_{j_s}\rangle^2-\theta^2\right)
\right],
\label{2-hamiltonian}$$ where $H\{\tilde\sigma\}^{\backslash 1}$ is the Hamiltonian when spin 1 is completely removed from the system in both stages of the thermodynamic process. Removing spin 1 may cause the thermal averages of other spins to adjust slightly in the first stage. Hence some dynamic spins (with $\langle\sigma_k\rangle^2<\theta^2$) may become frozen ones (with $\langle\sigma_k\rangle^2>\theta^2$) and vice versa, so that strictly speaking, further terms should be considered in Eq. (\[2-hamiltonian\]) to account for these secondary effects. For example, if spin $k$ is induced to switch from dynamic to frozen (or vice versa) on removal of spin 1, then the Taylor expansion of $H\{\tilde\sigma\}$ implies that an extra term $$\begin{aligned}
&&-\tilde\beta\left(
{\rm sgn}\langle\sigma_k\rangle^{\backslash 1}
-\tilde\sigma_k\right)\left[
\delta(\langle\sigma_k\rangle^{\backslash 1}-\theta)
-\delta(\langle\sigma_k\rangle^{\backslash 1}+\theta)\right]
(\langle\sigma_k\rangle-\langle\sigma_k\rangle^{\backslash 1})
\nonumber\\
&&\sum_{1<j_1\cdots<j_{p-1}\ne k}
J_{kj_1\cdots j_{p-1}}
\prod_{s=1}^{p-1}\left\{
\tilde\sigma_{j_s}\Theta\left[
\theta^2-(\langle\sigma_{j_s}\rangle^{\backslash 1})^2\right]
+{\rm sgn}\langle\sigma_{j_s}\rangle^{\backslash 1}\Theta\left[
(\langle\sigma_{j_s}\rangle^{\backslash 1})^2-\theta^2\right]
\right\}\end{aligned}$$ should be incorporated in Eq. (\[2-hamiltonian\]). Here, we have neglected these terms for clarity. Nevertheless, justification a posteriori can be provided for their deletion.
Using a cavity argument similar to Appendix A, we can show that $$\langle\tilde\sigma_1\rangle
=\tanh\tilde\beta\left\{\sum_{1<j_1\cdots<j_{p-1}}
J_{1j_1\cdots j_{p-1}}
\prod_{s=1}^{p-1}\left[
\langle\tilde\sigma_{j_s}\rangle^{\backslash 1}
\Theta\left(\theta^2-\langle\sigma_{j_s}\rangle^2\right)
+{\rm sgn}\langle\sigma_{j_s}\rangle^{\backslash 1}
\Theta\left(\langle\sigma_{j_s}\rangle^2-\theta^2\right)
\right]\right\}.
\label{2-average}$$ However, the effective field on the right hand side of Eq. (\[2-average\]) is still not a cavity field because $\langle\sigma_{j_s}\rangle$, which are used in the step functions to decide whether the spin $j_s$ is dynamic or frozen in the second stage, is different from $\langle\sigma_{j_s}\rangle^{\backslash 1}$. Hence it may have correlations with spin 1. Taylor expansion of $\langle\sigma_{j_s}\rangle$ about $\langle\sigma_{j_s}\rangle^{\backslash 1}$ yields $$\begin{aligned}
&&\langle\tilde\sigma_1\rangle
=\tanh\tilde\beta\Biggl\{\tilde h_1
+\sum_{1j\ne j_1\cdots<j_{p-2}}
J_{1jj_1\cdots j_{p-2}}\nonumber\\
&&\prod_{s=1}^{p-2}\left[
\langle\tilde\sigma_{j_s}\rangle^{\backslash 1}
\Theta\left[\theta^2
-(\langle\sigma_{j_s}\rangle^{\backslash 1})^2\right]
+{\rm sgn}\langle\sigma_{j_s}\rangle^{\backslash 1}
\Theta\left[
(\langle\sigma_{j_s}\rangle^{\backslash 1})^2-\theta^2\right]\right]
\nonumber\\
&&\left[{\rm sgn}\langle\sigma_j\rangle^{\backslash 1}
-\langle\tilde\sigma_j\rangle^{\backslash 1}\right]
\left[\delta(\langle\sigma_j\rangle^{\backslash 1}-\theta)
-\delta(\langle\sigma_j\rangle^{\backslash 1}+\theta)\right]
(\langle\sigma_j\rangle-\langle\sigma_j\rangle^{\backslash 1})
\Biggr\},
\label{2-without1}\end{aligned}$$ where $\tilde h_1$ is the generic cavity field which is now completely uncorrelated with spin 1. It is given by $$\tilde h_1
=\sum_{1<j_1\cdots<j_{p-1}}
J_{1j_1\cdots j_{p-1}}
\prod_{s=1}^{p-1}\left\{
\langle\tilde\sigma_{j_s}\rangle^{\backslash 1}
\Theta\left[\theta^2
-(\langle\sigma_{j_s}\rangle^{\backslash 1})^2\right]
+{\rm sgn}\langle\sigma_{j_s}\rangle^{\backslash 1}
\Theta\left[(\langle\sigma_{j_s}\rangle^{\backslash 1})^2
-\theta^2\right]
\right\}.
\label{2-cavity}$$ To evaluate the difference $\langle\sigma_j\rangle-\langle\sigma_j\rangle^{\backslash 1}$ appearing in Eq. (\[2-without1\]), we have to apply the cavity method a second time, by comparing the changes when both spins 1 and $j$ are removed. This is done in Appendix B and the result is $$\langle\sigma_j\rangle-\langle\sigma_j\rangle^{\backslash 1}
=\left(\beta{\rm sech}^2\beta h_j^{\backslash 1}\right)
\left(h_{j1}\tanh\beta h_1^{\backslash j}\right),
\label{2-change}$$ where $$h_{1j}=h_{j1}
=\sum_{1j\ne k_1\cdots<k_{p-2}}
J_{1jk_1\cdots k_{p-2}}
\langle\sigma_{k_1}\rangle^{\backslash 1j}\cdots
\langle\sigma_{k_{p-2}}\rangle^{\backslash 1j}.
\label{h_1j}$$ When Eqs. (\[2-change\]-\[h\_1j\]) are substituted into Eq. (\[2-without1\]), the significant contribution comes from the terms which pair up $J_{1jj_1\cdots j_{p-2}}$ and $J_{1jk_1\cdots k_{p-2}}$. The various terms appearing in the summation over $j\ne j_1<\cdots<j_{p-2}$ involve thermal averages in the absence of spins 1 or $j$. We assume that the effects of removing a spin is negligible (which can be shown to be equivalent to the replica symmetric approximation in the replica method [@wong2]). Then replacing the components of the terms by their mean values, and counting that $N^{p-2}/(p-2)!$ terms appearing in the summation over $j_1<\cdots<j_{p-2}$, we arrive at $$\begin{aligned}
&&\langle\tilde\sigma_1\rangle
=\tanh\tilde\beta\Biggl\{\tilde h_1
+{p\over 2}(p-1) J^2 {1\over N}\sum_j
\left[\delta(\langle\sigma_j\rangle-\theta)
-\delta(\langle\sigma_j\rangle+\theta)\right]
\left[{\rm sgn}\langle\sigma_j\rangle
-\langle\tilde\sigma_j\rangle\right]\nonumber\\
&&\left(\beta{\rm sech}^2\beta h_j\right)
\left(r^{p-2}\tanh\beta h_1\right)\Biggr\},
\label{2-paired}\end{aligned}$$ where $r$ is the order parameter describing the spin correlations of the two thermodynamic stages: $$r\equiv
{1\over N}\sum_i
\langle\sigma_i\rangle\left\{
\langle\tilde\sigma_i\rangle
\Theta\left[\theta^2-\langle\sigma_i\rangle^2\right]
+{\rm sgn}\langle\sigma_i\rangle
\Theta\left[\langle\sigma_i\rangle^2-\theta^2\right]
\right\}.
\label{r}$$ Eq. (\[2-paired\]) can be simplified by introducing the trans-susceptibility $\chi_{tr}$, which describes the response of a spin in the second stage to variations of the cavity field in the first stage, namely $$\chi_{tr}
\equiv{1\over N}\sum_i
{\partial\langle\tilde\sigma_i\rangle\over
\partial h_i}.$$ Since $\langle\tilde\sigma_i\rangle$ equals ${\rm sgn}h_i$ for $\tanh^2\beta h_i>\theta^2$, and $\tanh\beta\tilde h_i$ otherwise, we get $$\chi_{tr}
={1\over N}\sum_i
\left[\delta(\langle\sigma_i\rangle-\theta)
-\delta(\langle\sigma_i\rangle+\theta)\right]
\left[{\rm sgn}\langle\sigma_i\rangle
-\langle\tilde\sigma_i\rangle\right]
\beta{\rm sech}^2\beta h_i.
\label{trans}$$ Eq. (\[2-paired\]) can thus be simplified to $$\langle\tilde\sigma_1\rangle
=\tanh\tilde\beta\left\{\tilde h_1
+{p\over 2}(p-1) J^2 r^{p-2} \chi_{tr}
\tanh\beta h_1\right\}.
\label{2-result}$$
Order parameters in the second stage
------------------------------------
The cavity field $\tilde h_1$ in the second stage is a Gaussian variable. Its mean and variance are $pj_0\tilde m^{p-1}$ and $pJ^2\tilde q^{p-1}/2$ respectively, where $\tilde m$ and $\tilde q$ are the magnetization and Edwards-Anderson order parameter respectively, given by $$\begin{aligned}
\tilde m
&\equiv&{1\over N}\sum_i
\left[\Theta(\theta^2-\langle\sigma_i\rangle^2)
\langle\tilde\sigma_i\rangle
+\Theta(\langle\sigma_i\rangle^2-\theta^2)
{\rm sgn}\langle\sigma_i\rangle\right],
\label{2-m}\\
\tilde q
&\equiv&{1\over N}\sum_i
\left[\Theta(\theta^2-\langle\sigma_i\rangle^2)
\langle\tilde\sigma_i\rangle^2
+\Theta(\langle\sigma_i\rangle^2-\theta^2)
\right].
\label{2-q}\end{aligned}$$ Furthermore, the covariance between $h_1$ and $\tilde h_1$ is $pJ^2 r^{p-1}/2$, where $r$ is given in Eq. (\[r\]).
Algebraic manipulations can be simplified if we write, for $i=1$, $$\begin{aligned}
\beta h_i
&=&\hat m+\sqrt{\hat q}u_i,\\
\tilde\beta\tilde h_i
&=&\hat{\tilde m}+\sqrt{\hat{\tilde q}}
(\eta u_i+\sqrt{1-\eta^2}v_i),\end{aligned}$$ where $u_i$ and $v_i$ are independent Gaussian variables with mean 0 and variance 1, $\hat m$, $\hat q$ are given in Eq. (\[mqhat\]), and $$\begin{aligned}
&&\hat{\tilde m}=p\tilde\beta j_0\tilde m^{p-1},
\quad{\rm and}\quad
\hat{\tilde q}={p\over 2}\tilde\beta^2J^2\tilde q^{p-1},\\
&&\hat r={p\over 2}\beta\tilde\beta J^2 r^{p-1},
\quad{\rm and}\quad
\eta={\hat r\over\sqrt{\hat q\hat{\tilde q}}}.
\label{2-mqhat}\end{aligned}$$ Applying self-consistently the same cavity argument to all terms in Eqs. (\[2-m\]), (\[2-q\]), (\[r\]) and (\[trans\]) and performing the Gaussian average over $u_i$ and $v_i$, we arrive at the following self-consistent equations for $\tilde m$, $\tilde q$, $r$ and $\chi_{tr}$: $$\begin{aligned}
\tilde m&=&
-{1\over 2}{\rm erf}{u_+\over\sqrt 2}
-{1\over 2}{\rm erf}{u_-\over\sqrt 2}
+\int_{u_-}^{u_+}Du\int Dv\tanh L,
\label{2-morder}\\
\tilde q&=&
1-{1\over 2}{\rm erf}{u_+\over\sqrt 2}
+{1\over 2}{\rm erf}{u_-\over\sqrt 2}
+\int_{u_-}^{u_+}Du\int Dv\tanh^2 L,
\label{2-qorder}\\
r&=&
\left(\int_{-\infty}^{u_-}+\int_{u_+}^\infty\right)Du
\left|\tanh G\right|
+\int_{u_-}^{u_+}Du\int Dv\tanh G\tanh L,
\label{rorder}\\
\chi_{tr}&=&
{\exp(-u_+^2/2)\over J\sqrt{\pi pq^{p-1}}}
\int Dv(1-\tanh L_v^{(+)})+
{\exp(-u_-^2/2)\over J\sqrt{\pi pq^{p-1}}}
\int Dv(1+\tanh L_v^{(-)}),
\label{transorder}\end{aligned}$$ where $$\begin{aligned}
L&=&
\hat{\tilde m}+\sqrt{\hat{\tilde q}}(\eta u+\sqrt{1-\eta^2}v)
+{p\over 2}(p-1)\tilde\beta J^2 r^{p-2} \chi_{tr} \tanh G,\\
L_v^{(\pm)}&=&
\hat{\tilde m}+\sqrt{\hat{\tilde q}}(\eta u_\pm+\sqrt{1-\eta^2}v)
\pm{p\over 2}(p-1)\tilde\beta J^2 r^{p-2} \chi_{tr} \theta.\end{aligned}$$ Eqs. (\[morder\]-\[qorder\]), (\[2-morder\]-\[transorder\]) for the order parameters $m$, $q$, $\tilde m$, $\tilde q$, $r$ and $\chi_{tr}$ form a close set of equations. The performance of selective freezing is measured by $$M_{\rm sf}
\equiv{1\over N}\sum_i
\left[\Theta(\theta^2-\langle\sigma_i\rangle^2)
{\rm sgn}\langle\tilde\sigma_i\rangle
+\Theta(\langle\sigma_i\rangle^2-\theta^2)
{\rm sgn}\langle\sigma_i\rangle\right].$$ From the above parameters, $M_{\rm sf}$ can be derived as: $$M_{\rm sf}
=-{1\over 2}{\rm erf}{u_+\over\sqrt 2}
-{1\over 2}{\rm erf}{u_-\over\sqrt 2}
+\int_{u_-}^{u_+}Du
{\rm erf}{L_u\over\sqrt{2\tilde q(1-\eta^2)}},$$ where $L_u=\hat{\tilde m}+\sqrt{\hat{\tilde q}}\eta u
+[p(p-1)/2]\tilde\beta J^2 r^{p-2} \chi_{tr} \tanh G$.
We have also tried to derive the above equations using the replica method. However, in the nearest results that we could find, terms involving the trans-susceptbility are absent, which we believe to be unphysical. Therefore the replica approach to the order parameter equations remain an open question.
We show an example of the case $p=2$ and $j_0=J=1$ in Fig. \[bif.vgr\], where the overlap $M_{\rm sf}$ is plotted as a function of the decoding temperature $T (=\beta^{-1}=\tilde\beta^{-1})$ for various given values of freezing fraction $f$. When $f=0$ (no spins frozen) and $f=1$ (all spins frozen), the dynamics is equivalent to one with single stage, and the overlap reaches its maximum at the Nishimori point $T=J^2/2j_0$ as expected. We observe that the tolerance against variations in $T$ is enhanced by selective freezing for certain values of $f$.
It is therefore interesting to consider the appropriate values of $f$ for the best overlap at a given decoding temperature. Figs. \[bif2.vgr\](a-f) shows that at high temperatures such as in Figs. \[bif2.vgr\](a-c), there is a single maximum and its position is fairly independent of temperature, lying around $f=0.9$ in the present case. At intermediate temperatures such as in Figs. \[bif2.vgr\](d-e), there appear two maxima and as temperature changes, there is a discontinuous jump in the maximum position. Fig. \[bif2.vgr\](f) shows that when the temperature is lower than the Nishimori point ($T_N=0.5$), the overlap cannot be improved by selective freezing.
Figure \[bit3.vgr\] compares the overlap of the one-stage dynamics with that of the best of selective freezing. It shows that when the decoding temperature is mis-determined to be higher than its optimal value at the Nishimori point, selective freezing can provide a fairly robust performance. Furthermore, the choice of the freezing fraction for such robust performance appears to be quite independent of the temperature. The solid line in Fig. \[bif4.vgr\] locates the position for the best overlap and, as observed from Figs. \[bif2.vgr\](a-f), lies in the vicinity of $f\approx 0.9$ for a large range of temperature. The unshaded region in the same figure also indicates that selective freezing leads to an improvement in the overlap over a wide range of the parameter space.
We have also studied the dependence of the overlap on varying the freezing threshold $\theta$ rather than the freezing fraction $f$. However, Fig. \[bit4.vgr\] shows that the optimal value of $\theta$ has a much larger dependence on the temperature. This is due to the sensitive dependence of the thermal averages of the spins on temperature. At high temperatures, most spins are thermally agitated, and the freezing threshold has to be set to a very low value in order to freeze a given fraction of spins. On the other hand, at low temperatures, most spins are relatively stable, and the freezing threshold has to be set to a very high value in order to keep a given fraction of spins dynamic in the second stage. We conclude that the freezing fraction is a better controlling parameter for the decoding performance.
The advantages of selective freezing are confirmed by Monte Carlo simulations shown in Fig. \[sff2.vgr\]. For one-stage dynamics, the overlap is maximum at the Nishimori point ($T_N=0.5$) as expected. However, it deterriorates rather rapidly when the decoding temperature increases. In contrast, selective freezing maintains a more steady performance, especially when $f=0.9$.
THE MEAN-FIELD MODEL FOR IMAGE RESTORATION
==========================================
In conventional image restoration problems, a given degraded image consists of the set of pixels $\{\tau_i\}$, but not the set of exchange interactions $\{J_{i_1,\cdots,i_p}\}$. On the other hand, effective restoration requires the introduction of a model prior distribution of the pixels for smooth images. In this case the Hamiltonian corresponds to that of a random field Ising model, $$H\{\sigma\}=-h\sum_i\tau_i\sigma_i
-{\beta_m\over z}\sum_{\langle ij\rangle}\sigma_i\sigma_j.$$ In mean-field systems, each pixel $i$ has an extensive valency. The pixels $\tau_i$ are the degraded versions of the source pixels $\xi_i$, corrupted by noise which, for convenience, is assumed to be Gaussian with mean $a\xi_i$ and variance $\tau^2$, i.e. $$P(\tau_i|\xi_i)
={\exp\left[-{1\over 2\tau^2}(\tau_i-a\xi_i)^2\right]
\over\sqrt{2\pi\tau^2}}.$$ In turn, the source pixels satisfy the prior distribution in Eq. (\[prior\]). Applying the cavity argument for mean-field systems, the prior distribution becomes factorizable, $$P(\xi_i)={\exp(\beta_s m_0 \xi_i)
\over 2\cosh\beta_sm_0},$$ where $m_0=\tanh\beta_sm_0$. The order parameter in the first stage is given by $$m\equiv{1\over N}\sum_i\langle\sigma_i\rangle
={1\over 2\cosh\beta_sm_0}\sum_{\xi=\pm 1}
\exp(\beta_sm_0\xi)\int Dx\tanh U,$$ where $U=\beta_mm+ha\xi+h\tau x$. The overlap for the one-stage restoration process is given by $$M\equiv{1\over N}\sum_i\xi_i{\rm sgn}\langle\sigma_i\rangle
={1\over 2\cosh\beta_sm_0}\sum_{\xi=\pm 1}
\exp(\beta_sm_0\xi)\xi
{\rm erf}{\beta_mm+ha\xi\over\sqrt 2h\tau}.
\label{m_ca}$$ Next we consider selective freezing in the second stage with a freezing threshold $\theta$. The freezing fraction is given by $$f\equiv{1\over N}\sum_i
\Theta\left(\langle\sigma_i\rangle^2-\theta^2\right)
={1\over 2\cosh\beta_sm_0}\sum_{\xi=\pm 1}
\exp(\beta_sm_0\xi)\left[1
-{1\over 2}{\rm erf}{u_+(\xi)\over\sqrt 2}
+{1\over 2}{\rm erf}{u_-(\xi)\over\sqrt 2}\right],$$ where $u_\pm(\xi)=(\pm u_0-\beta_mm-ha\xi)/h\tau$ with $\tanh u_0=\theta$. The order parameter of the second stage is given by $$\begin{aligned}
\tilde m
&\equiv&{1\over N}\sum_i
\left[\Theta(\theta^2-\langle\sigma_i\rangle^2)
\langle\tilde\sigma_i\rangle
+\Theta(\langle\sigma_i\rangle^2-\theta^2)
{\rm sgn}\langle\sigma_i\rangle\right]\nonumber\\
&=&{1\over 2\cosh\beta_sm_0}\sum_{\xi=\pm 1}
\exp(\beta_sm_0\xi)\left[
-{1\over 2}{\rm erf}{u_+(\xi)\over\sqrt 2}
-{1\over 2}{\rm erf}{u_-(\xi)\over\sqrt 2}
+\int_{u_-(\xi)}^{u_+(\xi)}Dx\tanh L\right],\end{aligned}$$ where $L=\beta_m\tilde m+ha\xi+h\tau x$. The overlap for selective freezing is given by $$\begin{aligned}
M_{\rm sf}
&\equiv&{1\over N}\sum_i\xi_i
\left[\Theta(\theta^2-\langle\sigma_i\rangle^2)
{\rm sgn}\langle\tilde\sigma_i\rangle
+\Theta(\langle\sigma_i\rangle^2-\theta^2)
{\rm sgn}\langle\sigma_i\rangle\right]\nonumber\\
&=&{1\over 2\cosh\beta_sm_0}\sum_{\xi=\pm 1}
\exp(\beta_sm_0\xi)\xi{\rm erf}
{g(\beta_m\tilde m)+ha\xi\over\sqrt 2h\tau},
\label{m_sf}\end{aligned}$$ where $$g(\beta_m\tilde m)=\left\{\matrix{
\beta_mm-u_0\hfill &&\hfill && \beta_m\tilde m<\beta_mm-u_0,\hfill\cr
\beta_m\tilde m\hfill &&\hfill &&
\beta_mm-u_0<\beta_m\tilde m<\beta_mm+u_0,\hfill\cr
\beta_mm+u_0\hfill &&\hfill &&
\beta_m\tilde m>\beta_mm+u_0.\hfill}\right.
\label{gfunc}$$ We note that since the spin-glass interaction is absent in this case, there are no trans-susceptibility effects. This is unlike the case of error-correcting codes, in which $\chi_{tr}$ is nonzero when $J$ is nonzero.
The three cases of the function $g(\beta_m\tilde m)$ in Eq. (\[gfunc\]) correspond to three situations. When $\beta_m\tilde m<\beta_mm-u_0$, all the dynamic spins in the second stage have negative thermodynamic averages and therefore take the value $-1$ in the two-stage restoration process. This is equivalent to a one-stage restoration process in which all spins with thermodynamic averages above the threshold $+\theta$ are frozen to $+1$, and to $-1$ otherwise. Similarly, when $\beta_m\tilde m>\beta_mm+u_0$, all the dynamic spins in the second stage have positive thermodynamic averages. Only when $\beta_mm-u_0<\beta_m\tilde m<\beta_mm+u_0$, do we have the dynamic spins frozen to partly $+1$ and partly $-1$.
We can consider the condition for the optimal performance $M_{\rm sf}$ of selective freezing. For a given distribution of data and noise, $g(\beta_m\tilde m)$ is the only adjustable parameter in Eq. (\[m\_sf\]), playing the same role as the adjustable parameter $\beta_mm$ for one-stage dynamics in Eq. (\[m\_ca\]). In the space of $h$ and $\beta_m$, the performance is optimal along the line $h/\beta_\tau=\beta_mm/\beta_sm_0$ for one-stage dynamics [@nishiwong] ($\beta_\tau=a/\tau^2$ for Gaussian noise). Analogously, there exists a line of optimal performance defined by $h/\beta_\tau=g(\beta_m\tilde m)/\beta_sm_0$ for selective freezing.
An example of the lines of optimal performance is shown in Fig. \[ilsf.vgr\]. It is interesting to note the kinks for certain freezing fractions. They correspond to transitions of cases in which the dynamic spins are partially or completely frozen to $\pm 1$.
A comparison of Eqs. (\[m\_ca\]) and (\[m\_sf\]) shows that selective freezing performs as well as one-stage dynamics, but cannot outperform it. Nevertheless, selective freezing provides a rather stable performance when the hyperparameters cannot be estimated precisely. In image restoration, the usual practice is to choose a fixed ratio of $\beta_m/h$. Fig. \[irb10.vgr\] confirms this stability along the line of operation with $\beta_m/h$ set to the optimal ratio $\beta_s/\beta_\tau$. Note especially that the lines with $f=0.7$ and $0.9$ attain a nearly optimal value of $M_{\rm sf}$ over a wide range of parameters. The kink at $f=0.9$ is, again, due to the appearance of the $-1$ frozen dynamic spins (to the right of the kink).
The stable performance of selective freezing can be partly explained by the proximity of the lines of optimal performance with the line of operation which, as discussed in [@nishiwong], is an important factor in hyperparameter estimation. This is illustrated by the optimal lines for small values of $f$ near the Nishimori point $(T_m,h)=(1.05^{-1},1)$ in Fig. \[ilsf.vgr\].
However, the advantage of selective freezing does not only rely on the fortuitous combination of parameters. Even when the parameters are not chosen optimally, selective freezing still maintains a rather robust performance. For example, along the line of optimal performance for $f=0.9$ in Fig. \[ilsf.vgr\], the bending at the kink only causes a modest reduction in the overlap $M_{\rm sf}$ in Fig. \[irb10.vgr\].
To study the robustness of the performance of selective freezing, we model a situation common in modern communication channels carrying multimedia traffic, which are often bursty in nature. Since burstiness results in intermittent interferences, we consider a noise with two Gaussian components, each with its own characteristics. A random fraction $f_1$ of the pixels are influenced by Gaussian noise with signal strength $a_1$ and noise variance $\tau_1^2$. The rest of the pixels have strength $a_2$ and noise variance $\tau_2^2$. Hence the distribution of the degraded pixels are $$P(\tau_i|\xi_i)
=f_1{\exp\left[-{1\over 2\tau_1^2}(\tau_i-a_1\xi_i)^2\right]
\over\sqrt{2\pi\tau_1^2}}
+f_2{\exp\left[-{1\over 2\tau_2^2}(\tau_i-a_2\xi_i)^2\right]
\over\sqrt{2\pi\tau_2^2}},$$ where $f_2=1-f_1$. The equations for the order parameters can be generalized from the single component case in a straightforward manner.
A case of interest is that the restoration agent operates on the assumption of the characteristics of the majority component of the channel, say the first component. Hence it operates at the ratio $\beta_m/h=\beta_s\tau_1^2/a_1$. Suppose the Gaussian noise is partly interrupted to take the characteristics of the second component, but the operation parameters cannot be adjusted soon enough, then there will be a degradation of the quality of the restored images. In the example in Fig. \[ird8.vgr\], the reduction of the overlap $M_{\rm sf}$ for selective freezing is much more modest than the one-stage process ($f=0$).
An alternative situation is that the restoration agent is able to detect the changes in the average signal strengths and noise variance, but still operates on the assumption of a single-component Gaussian channel. Suppose that such simple statistics as $\langle{\rm sgn}\tau_i\rangle$, $\langle\tau_i\rangle$ and $\langle\tau_i^2\rangle$ are accessible. Then the parameters $m_0^*$, $a^*$ and $\tau^*$ estimated by the restoration agent are obtained, for $\tau_1=\tau_2=\tau$, from the solutions of $$\begin{aligned}
m_0^*{\rm erf}{a^*\over\sqrt 2\tau^*}
&=&\langle{\rm sgn}\tau_i\rangle=m_0\left[
f_1{\rm erf}{a_1\over\sqrt 2\tau_1}+
f_2{\rm erf}{a_2\over\sqrt 2\tau_2}\right],
\label{estimate1}\\
m_0^*a^*
&=&\langle\tau_i\rangle=m_0\left[f_1a_1+f_2a_2\right],
\label{estimate2}\\
a^{*2}+\tau^{*2}
&=&\langle\tau_i^2\rangle
=f_1(a_1^2+\tau_1^2)+f_2(a_2^2+\tau_2^2),
\label{estimate3}\end{aligned}$$ and $\beta_s^*=\tanh^{-1}m_0^*/m_0^*$. Using these estimated parameters, the performances in Fig. \[irc8.vgr\] improve over their counterparts based on only the majority component in Fig. \[ird8.vgr\]. Still, one-stage restoration cannot avoid the performance drop when $h$ vanishes, whereas correspondingly, selective freezing has a much more gentle drop in performance.
It is interesting to study the more realistic case of two-dimensional images, since we have so far presented analytical results for the mean field model only. As confirmed by the results for Monte carlo simulations in Fig. \[imsf.eps\], the overlaps of selective freezing are much more steadier than that of the one-stage dynamics when the decoding temperature changes. This steadiness is most remarkable for a freezing fraction of $f=0.9$.
DISCUSSIONS
===========
We have introduced a multistage technique for error-correcting codes and image restoration, in which the information extracted from the former stage can be used selectively to improve the performance of the latter one. While the overlap $M_{\rm sf}$ of the selective freezing is bounded by the optimal performance of the one-stage dynamics derived in [@nishiwong], it has the advantage of being tolerant to uncertainties in hyperparameter estimation. The performance is especially steady when the fraction of frozen spins, rather than the threshold of their thermodynamic averages, is fixed in the process. This is confirmed by both analytical and simulational results for mean-field and finite-dimensional models. As an example, we have illustrated its advantage of robustness when the noise distribution is composed of more than one Gaussian components, such as in the case of modern communication channels supporting multimedia applications.
We found that selective freezing is most useful when more than one hyperparameters have to be estimated, as illustrated by the example of image restoration, where both $\beta_m$ and $h$ have to be estimated. In the example of error-correcting codes discussed in Section III, there is only one hyperparameter $T_m$, and it is found that selective freezing has performance advantages only when $T_m$ is chosen above the Nishimori point. However, more than one hyperparameters are often present in practical applications.
Selective freezing can be generalized to more than two stages, in which spins that remain relatively stable in one stage are progressively frozen in the following one. It is expected that the performance can be even more robust.
While the multistage process described here has a robust performance, it does not raise the critical temperature or the critical noise level for the existence of the ordered phase. Nor can it widen the basin of attraction for the ordered phase. Other multistage processes, proposed in [@wong] for neural networks, may be able to achieve this. This remains an area for further research.
We have made progress in the theoretical treatment of multistage processes using the cavity method. It allows the thermal averages of spins to be expressed in terms of the cavity fields. Since a cavity field is uncorrelated with the spin in consideration, it can in turn be expressed in terms of the means and covariances of the spin averages, thereby arriving at a set of self-consistent equations for the order parameters. In particular, there appears a trans-susceptibility term since variations of the cavity field in the first stage are correlated with the spin average in the second stage due to the selective nature of the freezing process in the second stage. However, for the ordered phase considered in this paper, the effects of the trans-susceptibility term is not too large except near the phase boundary.
On the other hand, we have a remark about the basic assumption of the cavity method, namely that the addition or removal of a spin causes a small change in the system describable by a perturbative approach. In fact, adding or removing a spin may cause the thermal averages of other spins to change from below to above the thresholds $\pm\theta$ (or vice versa). This change, though often small, induces a non-negligible change of the thermal averages from fractional values to the frozen values of $\pm 1$ (or vice versa) in the second stage. The perturbative analysis of these changes is only approximate. The situation is reminiscent of similar instabilities in other disordered systems such as the perceptron, and are equivalent to Almeida-Thouless instabilities in the replica method [@wong3]. A full treatment of the problem would require the introduction of a rough energy landscape [@wong3], or the replica symmetry breaking ansatz in the replica method [@mezard]. Nevertheless, previous experiences on disordered systems showed that the corrections made by a more complete treatment may not be too large in the ordered phase. For example, corresponding analytical and simulational results in Figs. \[bif.vgr\] and \[sff2.vgr\] respectively are close to each other.
In practical implementations of error-correcting codes, algorithms based on belief-propagation methods, rather than Monte Carlo methods, are often employed [@frey]. It has recently been shown that such decoded messages converge to the solutions of the TAP equations in the corresponding thermodynamic system [@kabasaad2]. Again, the performance of these algorithms are sensitive to the estimation of hyperparameters. We propose that the selective freezing procedure has the potential to make these algorithms more robust.
Incidentally, multistage dynamics has also been applied in the recently popular turbo codes [@turbo]. Messages are coded in sequences with two possible permutations and at each iterative stage, the information derived from decoding one sequence is fed to the other in the form of external fields for each bit. The techniques developed in the present context can be used to study this iterative process.
KYMW wishes to thank Tokyo Institute of Technology for hospitality. HN is grateful to Hong Kong University of Science and Technology for hospitality. This work was partially supported by research grant HKUST6157/99P of the Research Grant Council of Hong Kong.
Thermal averages of spins
=========================
In this appendix we derive Eq. (\[cavity\]) starting from the clustering property Eq. (\[cluster\]). For convenience we illustrate the derivation for $p=2$. We separate the Hamiltonian into two parts, one does not contain $\sigma_1$ and the other does. Hence $$H=H^{\backslash 1}-\beta\sum_{j>1}J_{1j}\sigma_1\sigma_j.$$ Thus the thermal average can be written as $$\langle\sigma_1\rangle={
{\rm Tr}^{\backslash 1}e^{-H^{\backslash 1}}
{\rm Tr}_1\sigma_1
\exp\left(\beta\sigma_1\sum_jJ_{1j}\sigma_j\right)
/{\rm Tr}^{\backslash 1}e^{-H^{\backslash 1}}\over
{\rm Tr}^{\backslash 1}e^{-H^{\backslash 1}}
{\rm Tr}_1
\exp\left(\beta\sigma_1\sum_jJ_{1j}\sigma_j\right)
/{\rm Tr}^{\backslash 1}e^{-H^{\backslash 1}}}.
\label{thermal}$$ Expanding the exponential function in the denominator and tracing over $\sigma$, we get $${\rm Den.}=
2\sum_{n\ \rm even}{\beta^n\over n!}
\sum_{j_1\cdots j_n}J_{1j_1}\cdots J_{1j_n}
\langle\sigma_{j_1}\cdots\sigma_{j_n}\rangle^{\backslash 1}.$$ Next, we use the clustering property to factorize the thermal average $\langle\sigma_{j_1}\cdots\sigma_{j_n}\rangle^{\backslash 1}$. For the coupling distribution specified by Eq. (\[jdis\]), only two kinds of contributions are significant in the summation over the indices $j_1\cdots j_n$. In the first kind, an index $j$ remains distinct from the rest, contributing a factor of $J_{1j}\langle\sigma_j\rangle^{\backslash 1}$. In the second kind, two indices become paired up. However, when $j$ and $k$ pair up, the thermal average $\langle\sigma_j\sigma_k\rangle^{\backslash 1}$ becomes 1 instead of $(\langle\sigma_j\rangle^{\backslash 1})^2$. Hence the additional contribution due to the pairing is $J_{1j}^2[1-(\langle\sigma_j\rangle^{\backslash 1})^2]$. Other than these, the contributions due to the pairing of three or more indices are smaller by factors of $N$.
The denominator can now be considered a summation over $n$ and $m$, which are respectively the total number of indices and the number of pairs of paired indices appearing in a term. The number of such terms is $n!/m!2^m(n-2m)!$. Hence $${\rm Den.}=
2\sum_{n\ \rm even}\sum_{m=0}^{n/2}
{\beta^n\over n!}{n!\over m!2^m(n-2m)!}
\left[\sum_jJ_{1j}\langle\sigma_j\rangle^{\backslash 1}\right]^{n-2m}
\left\{\sum_jJ_{1j}^2\left[1-
(\langle\sigma_j\rangle^{\backslash 1})^2\right]\right\}^m,$$ which can be simplified to $${\rm Den.}=
2\exp\left\{{1\over 2}\beta^2\sum_j
J_{1j}^2\left[1-(\langle\sigma_j\rangle^{\backslash 1})^2
\right]\right\}\cosh\left\{\beta\sum_j
J_{1j}\langle\sigma_j\rangle^{\backslash 1}\right\}.
\label{den}$$ Similarly, the numerator can be written as $${\rm Num.}=
2\exp\left\{{1\over 2}\beta^2\sum_j
J_{1j}^2\left[1-(\langle\sigma_j\rangle^{\backslash 1})^2
\right]\right\}\sinh\left\{\beta\sum_j
J_{1j}\langle\sigma_j\rangle^{\backslash 1}\right\}.
\label{num}$$ Substituting Eq. (\[den\]) and (\[num\]) into Eq. (\[thermal\]), we arrive at Eq. (\[cavity\]).
Change in thermal averages on removal of a spin
===============================================
In this appendix we derive Eq. (\[2-change\]). For convenience we illustrate the derivation for $p=2$. We separate the Hamiltonian into four parts: (a) does not contain spins 1 and $j$, (b) contains only spins 1 and $j$, (c) contains spin 1 but not $j$, (d) contains spin $j$ but not 1. This yields $$H=H^{\backslash 1j}
-\beta J_{1j}\sigma_1\sigma_j
-\beta\sum_{k\ne 1j}J_{k1}\sigma_k\sigma_1
-\beta\sum_{k\ne 1j}J_{kj}\sigma_k\sigma_j.$$ The thermal average of $\sigma_j$ can then be written as $$\langle\sigma_j\rangle={
{\rm Tr}_{1j}
{\rm Tr}^{\backslash 1j}e^{-H}\sigma_j/
{\rm Tr}^{\backslash 1j}e^{-H^{\backslash 1j}}\over
{\rm Tr}_{1j}
{\rm Tr}^{\backslash 1j}e^{-H}/
{\rm Tr}^{\backslash 1j}e^{-H^{\backslash 1j}}}.$$ Using the mean-field technique developed in Appendix A, the denominator can be written as $$\begin{aligned}
{\rm Den.}=
{\rm Tr}_{1j}\exp\biggl\{
\beta J_{1j}\sigma_1\sigma_j
&+&\beta\sum_{k\ne 1j}\langle\sigma_k\rangle^{\backslash 1j}
(J_{k1}\sigma_1+J_{kj}\sigma_j)\nonumber\\
&+&{1\over 2}\beta^2\sum_{k\ne 1j}
\left[1-(\langle\sigma_k\rangle^{\backslash 1j})^2\right]
(J_{k1}\sigma_1+J_{kj}\sigma_j)^2\biggr\}.\end{aligned}$$ After collecting terms and discarding negligible ones, $${\rm Den.}=
{\rm Tr}_{1j}\exp\left\{
\beta\sigma_1\sum_{k\ne 1j}J_{1k}
\langle\sigma_k\rangle^{\backslash 1j}+
\beta\sigma_j\sum_{k\ne 1j}J_{jk}
\langle\sigma_k\rangle^{\backslash 1j}
+\beta J_{1j}\sigma_1\sigma_j
+\beta^2(1-q)J^2\right\}.$$ Together with a similar manipulation of the numerator, we obtain $$\langle\sigma_j\rangle=
\tanh\beta\left(h_j^{\backslash 1}
+J_{j1}\tanh\beta h_1^{\backslash j}\right),$$ whose Taylor expansion yields $$\langle\sigma_j\rangle=
\langle\sigma_j\rangle^{\backslash 1}
+\left(\beta{\rm sech}^2\beta h_j^{\backslash 1}\right)
\left(J_{j1}\tanh\beta h_1^{\backslash j}\right),$$ which becomes Eq. (\[2-change\]) for the case $p=2$.
[99]{}
R. J. Eliece, [*The Theory of Information and Coding*]{}, Encyclopedia of Mathematics and its Applications (Addison-Wesley, Reading, MA 1977). N. Sourlas, Nature [**339**]{}, 693 (1989). Y. Kabashima and D. Saad, Europhys. Lett. [**45**]{}, 97 (1999). S. Geman and D. Geman, IEEE Trans. PAMI [**6**]{}, 721 (1984). H. Nishimori and K. Y. M. Wong, Phys. Rev. E [**60**]{}, 132 (1999). H. Nishimori, J. Phys. Soc. Jpn. [**62**]{}, 2973 (1993). Z. Zhou, R. M. Leathy, and J. Qi, IEEE Trans. Image Proc. [**6**]{}, 844 (1997). D. J. C. Mackay, Neural Computation [**4**]{}, 415 (1992). J. M. Pryce and A. D. Bruce, J. Phys. A [**28**]{}, 511 (1995). K. Y. M. Wong, Europhys. Lett. [**36**]{}, 631 (1996). M. Mézard, G. Parisi, and V.A. Virasoro, [*Spin Glass Theory and Beyond*]{} (World Scientific, Singapore 1987). K. Y. M. Wong, Europhys. Lett. [**30**]{}, 245 (1995). K. Y. M. Wong, Advances in Neural Information Processing Systems [**9**]{}, 302 (1997). B. J. Frey, [*Graphical Models for Machine Learning and Digital Communication*]{} (MIT Press, 1998). Y. Kabashima and D. Saad, Europhys. Lett. [**44**]{}, 668 (1998). C. Berrou, A. Glavieux, and P. Thitimajshima, Proc. IEEE Int. Conf. Comm. ’93, 1064 (1993).
| {
"pile_set_name": "ArXiv"
} |
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LPT-ENS 02/42\
Bernard Julia$^a$
0.5cm
$^a$Laboratoire de Physique Th[é]{}orique CNRS-ENS\
24 rue Lhomond, F-75231 Paris Cedex 05, France[^1]\
0.5cm
[**ABSTRACT**]{}
It seems to me at this time that two recent developments may permit fast progress on our way to understand the symmetry structure of toroidally (compactified and) reduced M-theory. The first indication of a (possibly) thin spot in the wall that prevents us from deriving a priori the U-duality symmetries of these models is to be found in the analysis of the hyperbolic billiards that control the chaotic time evolution of (quasi)homogeneous anisotropic String, Supergravity or Einstein cosmologies near a spacelike singularity. What happens is that U-duality symmetry controls chaos via negative constant curvature. On the other hand it was noticed in 1982 that (symmetrizable) ”hyperbolic” Kac-Moody algebras have maximal rank ten, exactly like superstring models and that two of these four rank ten algebras matched physical theories. My second reason for optimism actually predates also the previous breakthrough, it was the discovery in 1998 of surprising superalgebras extending U-dualities to all (p+1)-forms (associated to p-branes). They have a super-nonlinear sigma model structure similar to the symmetric space structure associated to 0-forms and they obey a universal self-duality field equation. As the set of forms is doubled to implement electric-magnetic duality, they obey a set of first order equations. More remains to be discovered but the beauty and subtlety of the structure cannot be a random emergence from chaos. In fact we shall explain how a third maximal rank hyperbolic algebra $BE_{10}$ controls heterotic cosmological chaos and how as predicted Einstein’s General Relativity fits into the general picture.
Classifications
===============
It is well known to conformal field theorists but a much more general and venerable fact that positive definite symmetric matrices with integer entries tend to appear in many classification problems. More precisely the ADE philosophy is to list such occurrences and try to relate them to each other. Let us be a bit more specific, this set of problems corresponds to the emergence in various contexts of symmetric matrices with diagonal entries equal to 2 and negative integral off-diagonal entries. The prototype of a successful correspondence is the work of Brieskorn (with some help from Grothendieck) realising the simple complex Kleinian singularities related to discrete subgroups of SU(2) as rational singularities in the set of unipotent elements of the corresponding complex Lie group of type ADE. The singularity is at the conjugacy class of subregular elements in other words the elements whose centraliser has 2 more dimensions than regular ones namely $r+2$ instead of $r$, the group rank. The resolution of the singularity introduces exceptional divisors whose intersection matrix is the opposite of the group Cartan matrix, there the ADE matrices arise in two different disguises but there is a relation to the Lie group in both cases.
Halfway (in time) between the Brieskorn results and the classification of modular invariant partition functions by Cappelli et al. $N=8$ supergravity was constructed in 4 dimensions as well as its toroidal decompactification family up to 11 spacetime dimensions. It was quickly remarked to me by Y. Manin that the $E_r$ internal symmetry groups appearing upon compactification on a r-torus suggested a role for (regular) del Pezzo surfaces, these are variants of $CP_2$ or $CP_1 \times CP_1$ which admit a canonical projective embedding . However the other pure supergravities in 4 dimensions with fewer supersymmetries also belonged to families which together formed a magic triangle of theories. They led also to ADE groups yet not all in split form, this meant that real geometry was to be tackled rather than the simpler complex algebraic one.
It also rapidly became clear if not rigorously established that the $E_r$ family included the infinite dimensional $E_9 \equiv E_8^{(1)}$ in 2 dimensions ie. after compactification on a 9-torus. Our partial understanding of these duality groups includes their $A_{r-1} \times \R$ subgroups related to the r ignorable coordinates. The infinite dimensional hidden symmetry is related to the existence of a Lax pair in 2 residual dimensions leading to a quasi integrable situation yet it was known that chaos remained prevalent in some particular flows.
In 1982 I noticed that a naive extrapolation to 1 dimension would suggest a role for $E_{10}$ as one would expect its subalgebra $A_9$ to appear there, however the implementation was problematic. Nevertheless a similar analysis of so-called type I supergravity in other words pure (without matter multiplet) supergravity in 10 dimensions led to the suggestion of a corresponding role for the other hyperbolic Kac-Moody algebra $DE_{10}$ the “overextended” $D_8$ in split form (namely $SO(8,8)$ affinized with one more $SL(2,\R)$ generating subgroup right next to the affinizing $SL(2,\R)$ in the Dynkin diagram as implied by the $A_{r-1}$ argument). Now according to Bourbaki (actually Chein) $E_{10}$ and $DE_{10}$ are exactly the two simply laced hyperbolic Kac-Moody algebras of maximal rank. There are only two more (non simply laced) hyperbolic Kac-Moody algebras of maximal rank: $BE_{10}$ (see section 3) and $CE_{10}$ (to be seen).
Let us recall that hyperbolicity here means that the Dynkin diagram becomes a product of finite or affine Dynkin diagrams after removal of any node (this gives nice arithmetic properties as well). The rank ten here is directly related to the rank eight of $E_8$, the largest exceptional simple Lie group which itself comes about also from some subtle classification analysis. On the other hand ten is the critical dimension of superstring models and as such it is related to quantum conformal invariance. How could the two derivations be related? They must be as the rank of the Lie group is closely related to the dimension of the compactification torus!
Still a puzzling fact emerged from the subsequent construction of heterotic strings namely the possibility of duality groups of rank higher than 8 for instance 16 in dimension 3 ($SO(8,24)$) corresponding putatively to rank 18 in one dimension. This paradoxical apparent violation of the bound ten on the rank will be clarified in section 3. Let us also remark that we are talking of superstrings and (bosonic) Kac-Moody algebras, hence fermionic structures should emerge in an a priori bosonic context. The upper limit 26 does not arise so simply yet, although it can be argued to be the sum of ten and sixteen the latter being the rank of the two even euclidean unimodular lattices dictated again by quantum anomaly considerations.
Let us add some examples of ADE objects. The basic one is the set of integral positive definite matrices occuring as Cartan matrices of simple simply laced complex Lie groups. The positive definiteness, resp positive semi-definiteness, resp hyperbolicity guarantees a relatively simple classification, for instance in the positive definite case (that of finite dimensional simply laced Lie algebras) there is a finite number of objects for each rank; on the other hand the three cases of $E_6, E_7, E_8$ look exceptional in this context. In the semi-definite case also called affine Kac-Moody situation very much the same is true; in the hyperbolic case however as we have seen the rank is bounded and there are finitely many instances of a given rank except when it is equal to two.
The finite dimensional (irreducible) Coxeter groups generated by reflections are closely related objects, their list encompasses that of the Weyl groups of the simple Lie groups but one gets 3 extra Coxeter diagrams with rotation angle $2\pi/5$ and an infinite family of dihedral groups of rotation angles $2\pi/k$ for non cristallographic k’s integers at least 7. These are all nonsimply laced cases. It is important to note that non simply laced Lie groups have a symmetrisable Cartan matrix: a basic assumption for most of Kac-Moody theory and Borcherds algebras. The reflections preserve a symmetric form that is a symmetrisation of the non symmetric Cartan matrix.
By definition Coxeter groups admit a finite presentation by involutions $S_i
\, , \, i=1,\dots,r$ satisfying $$( S_i S_j)^{m_{ij}} = I \, , \, i\neq j.$$ In the simply laced case we consider only exponents $ m_{ij} =2$ for commuting involutions or 3 for dihedral subgroups of order 6. The matrix is encoded by a Dynkin diagram with r vertices and simple bonds for exponent 3; it turns out that no loop is allowed, that at most three legs occur and finally that the sum $1/m+1/n+1/p$ of the inverses of the number of vertices (including the potentially trivalent vertex) on each leg must be strictly larger than one. One recognises two infinite families $A_k, D_k$ and three exceptions $E_6, E_7, E_8$ with numbers of vertices respectively $(m,n,p)=(2,3,3), (2,3,4), (2,3,5)$, let us notice that $E_5\equiv D_5$ corresponds to $(2,3,2)$ and $E_4 \equiv A_4$ to $(2,3,1)$. $E_3\equiv A_2 \times A_1$ which is semi-simple but not simple is the next group in the $E_r$ family.
The list of affine Kac-Moody algebras is very closely related to the list of finite dimensional simple Lie algebras. It permits one loop for the $A_k^{(1)}$ Dynkin diagrams, sums of inverse numbers of vertices at one three-valent vertex equal to one, two three valent vertices for $D_k^{(1)} \, k\geq 5$ resp. one 4-valent vertex for $D_4^{(1)}$. The list of hyperbolic diagrams can be found in [@Og; @Sa]. Their defining property given above guarantees the Lorentzian signature of the invariant bilinear form on the Cartan subalgebra. The hyperbolic algebras one meets in supergravity theories are overextensions of finite Lie algebras, the construction was discussed in [@J82; @F83]. Not all overextensions are hyperbolic but all hyperbolic algebras of rank at least 7 are overextensions (called superaffine in [@Og]). The derivation of the signature is straightforward, if $A$ is a kxk affine Cartan matrix it has null determinant and signature $(+^{k-1}, 0)$ , the (k+1)st line and column contain a 2 at their intersection and a $-1$ at the affine column or row, the corresponding quadratic form after completing the square has manifest Lorentzian signature. Another characteristic property we shall use in section 3 is the fact that the Weyl chamber on the unit hyperboloid has finite volume (it may actually be non compact, for instance when the rank is strictly larger than 5 [@Bo; @Vi]).
Let us now discuss in more detail the case of overextended $A_{k-1}$. The affine $A_{k-1}^{(1)}$ has a circular Dynkin diagram, its (over)extension has just one line and one extra vertex attached to it, now the criterion of hyperbolicity prevents overextended $A_8 $ to be hyperbolic, overextended $A_7$ is the last hyperbolic $HA_9$ in the family, and this can be viewed again as a consequence of the fact that there are only three Lie groups of type E. So one may say that the exceptionality of the E family is related to the bound on the rank of the hyperbolic $HA_{k+1}$. We shall see in section 3 the dramatic difference between pure gravity dynamics at a singularity in ten or eleven dimensions as a consequence of this algebraic fact.
Let us recall that $E_{10}$ and $HD_{10}$ were identified in [@J82], the Weyl chamber of $HB_{10}$ (in other words overextended $B_8$) was suggestively recognised by [@DH] as controlling the classical chaos of heterotic string theory yet following Narain and Sen one expects the U duality group to have rank 16 in 3 dimensions and not 8. The answer lies in the simple observation that the classical action is a real functional and the real equations are invariant under a real Lie group, the precise real form of which is critically important. But as an expert (M. Reid) puts it, real algebraic geometry is $2^N$ times more difficult than complex algebraic geometry with N large.
Real forms of Lie algebras
==========================
We are familiar with the classification of complex simple Lie algebras as a monument of group theory. Its relative simplicity is permitted by the algebraic closure of the field of complex numbers, indeed the main tool is the simultaneous diagonalisation of commuting Cartan generators (observables) and the analysis of the root spectrum (quantum numbers). Up to central elements (non simple-connectedness) and up to isomorphism there is a unique compact form of the associated Lie group. The theory of non compact forms and their representation theory is much richer and even the split (also called maximally non-compact forms) have complicated representations. The existence of the split form follows from the observation that the structure constants of a complex Lie algebra can be taken to be integers in the appropriate basis. The restriction of the field of coefficients from the complex to the real numbers or even the natural integers is possible, the choice of real ”angles of rotations" in the Cartan-Chevalley basis defines the split form. For $SL(2,C)$ the split form is $SL(2,\R)$ whereas the compact form is $SU(2)$ and the arithmetic form $SL(2,\Z)$.
Over the complex numbers or in the compact case the Cartan subalgebras are all conjugate to each other, not so for other real forms $G$ even the split one, still there is a finite collection of inequivalent ones. The classification of real forms of simple Lie groups has been given by E. Cartan together with the classification of maximally symmetric spaces. Since then two strategies have been applied. Either one selects a maximally non-compact Cartan torus in $G$ and diagonalises as many observables as possible over the reals, there appears a maximal split subalgebra $S$ ([@BT] p.116) inside our noncompact real form $G$ and the roots project onto roots or twice the roots of $S$, this is the Tits-Satake theory with bicoloured diagrams as developped for instance in [@Ar]. One key result is that the roots restricted to the noncompact Cartan generators form a not necessarily reduced root system. In particular this implies that after choosing the compact real form that contains the Cartan (compact) generators resp. their multiples by the imaginary unit i (for the non-compact ones), all “imaginary roots” ie. those that vanish on the (maximal) set of non-compact Cartan generators must be associated to compact eigengenerators.
The other frequent choice of maximal torus leaves more freedom, given a non-compact real form one chooses a maximally compact Cartan subalgebra. Vogan introduced other bicoloured diagrams for that situation. For instance the case of fully compact tori arises sometimes for non-compact Lie algebras and is very interesting. Now the arbitrariness with the Vogan strategy stems from the fact that the choice of simple roots is not unique there. The bicoloured diagrams may be chosen to have all or all but one compact vertices. Clearly this choice permits an easier identification of compact subgroups whereas the Tits-Satake strategy is best for split subgroups. It is natural to expect a dual result to the previous one namely that now no root can vanish on the (maximally) compact part of the Cartan torus, in other words one departs maximally from the complex root analysis. We refer to [@Kn] for an introduction to the Borel-de Siebenthal-Murakami-Vogan theory.
The real rank $l$ of a simple real Lie algebra of full rank $r$ is the maximal dimension of an abelian subalgebra of diagonalisable (called semi-simple) generators of non-compact type (whose Killing norm is positive). $0\leq l \leq r$, with $r=l$ in the split case and $l=0$ in the compact case. If one starts from the split form the compact form is obtained by multiplying some generators by $\sqrt{-1}$. For instance given the standard basis $e,f,h$ of $SL(2,\R)$ the compact $SU(2)$ is generated by $(e-f) $ which is already compact as well as $i(e+f)$ and $ih$. We see that the diagonalisability of $ih$ has been lost over the real numbers.
Turning now to the applications we may read off the tables of [@Ar; @He] that the restricted root system associated to the non-split $E_7(-5)$ of real rank 4 (the U-duality symmetry of N=6 4d supergravity reduced to three dimensions) is that of its maximal split subalgebra $F_4$ (one speaks of Freudenthal-Tits geometry of type $F_4$) and this is a common feature of all Maxwell-Einstein N=2, d=5 supergravities constructed by G" unaydin et al. in 1983 after reduction to three dimensions. We refer to [@He] p.534 for the Tits Satake diagram of type EVI symmetric space or real form of $E_7$. On this diagram the white dots denote non compact $SL(2,\R)$ subalgebras, three of them building up a compactification $SL(4,\R)$ symmetry. The black dots refer to compact $SU(2)$’s.
In the tables one finds also the multiplicity of the restricted roots; for instance for the Lorentz group $SO(1,3)$ of real rank 1 the Tits-Satake diagram is composed of two disconnected white dots with one double arrow between them. This can be understood as follows, the root analysis over the real numbers goes through for the noncompact (white) Cartan generator(s). Upon projection of the full root system on its restriction to linear forms over the noncompact part of the Cartan subalgebra this may imply multiple occurrence of the same restricted roots, when this happens for two simple roots one joins the corresponding (white) dots by a double arrow. The real Lie algebra admits a complex structure precisely when all restricted roots have multiplicity two. On the other hand black dots project to zero. When the restricted root system is not reduced, ie for some geometries of type $B_k$ one must also give the multiplicity of the doubles of the restricted roots.
Our present interest in the Tits-Satake analysis stems from the observation that for the orthogonal groups of the form $SO(8,8+p)$ with p between 1 and 16 the real rank is 8 and the geometry is of type $SO(8,9)$. Now such groups arise as U duality groups in three dimensions and the sigma model Lagrangians that control chaos in the quasi homogeneous situation must lead to the overextension $BE_{10}$ if one is to recover the experimental discovery of [@DH].
Chaos controlled by symmetry
============================
In a remarkable analysis [@DH] it was found that near a cosmological singularity the chaotic behaviour of essentially one dimensional (homogeneous) string theories was well approximated by an Anosov flow in the hyperbolic billiard defined by the Weyl cell of hyperbolic Kac-Moody algebras. For M-theory (alias in this approximation 11d SUGRA or type II String theory) $E_{10}$ emerged, and for type I SUGRA $DE_{10}$ replaces it. As we mentioned above both Lie algebras had surfaced as tantalising candidates for hidden symmetries in exactly these situations 20 years ago. Symmetry controls chaos which is not so surprising when negative curvature non-compact symmetric spaces are the arena. What remains more puzzling is the precise sense in which the 2 dimensional reductions of these models can be called integrable and still allow in their midst chaotic islands: they are well known to admit Lax pairs...
The chaotic solutions are even older and this tension between order and ergodicity is one of the most striking features of these Lagrangian theories. Another surprise of this work was the emergence of $BE_{10}$’s Weyl cell as the billiard relevant for heterotic and type I cosmological singularities. The relevant U-duality group in three dimensions is $SO(8,24)$ and its overextension is not hyperbolic. How should the overextension of $SO(8,9)$ take its place. On the other hand it was also remarked in [@J82] that $BE_{10}$ and $CE_{10}$ were nonsimply laced analogues of the previous two maximal rank algebras and ought to appear somewhere, half this prediction has been now fulfilled. We shall explain momentarily the reconciliation of $SO(8,24)$ and $SO(8,9)$.
In [@J82] it was also predicted that the overextension $HA_3$ of $SL(2,\R)$, the Ehlers group of stationary General Relativity, was the more conservative candidate for a hidden symmetry but now in a well tested theory and again in the homogeneous situation. It was a very powerful experience to meet Alex Feingold and Igor Frenkel in Chicago that summer who independently and for mathematical reasons were working on [@F83] and had developed the theory of superaffine algebras (better called hyperaffine maybe) as a handle for hyperbolic algebras while I had been concentrating on the $A_{k-1}$ concept for physical reasons. We learned a lot from each other then and it is a good place to express my thanks to the organisers I. Singer, P. Sally, G. Zuckerman, H. Garland and M. Flato for their invitation. Now this experience immediately suggested that the corresponding billiard should appear in the celebrated Belinsky-Khalatnikov-Lifschitz chaos. This time it appeared only in the quasi-homogeneous situation ie not quite in the 1-dimensional setting, yet again the three dimensional U-duality controlled chaos by the same mechanism [@DHJN]. Furthermore the earlier observation by the belgian team of the absence of BKL chaos for pure gravity beyond 10 dimensions exactly matched the above remark about $HA_9$ as the largest hyperbolic overextension of a group of A type. It was known that $A_k$ is precisely the U-duality group of the three dimensional reduction of pure gravity in k+3 dimensions [@CJLP].
Let us now see how the dimensionally reduced action of such a gravity theory (on a torus by homogeneity) implies the dynamical mechanism that is approximated by a hard walled billiard for time evolution near a singularity. We shall begin with the case of split U-dualities ie the better studied examples of [@CJLP]. As is well known in three dimensions all propagating fields can be dualised to scalars and form a noncompact symmetric space. There is a choice between two descriptions here. Either as it is the case when one actually discovers the system upon dimensional reduction one works in a fixed (or partially fixed) gauge and one uses coordinates on the symmetric space; for instance by using the Iwasawa decomposition one may use affine coordinates on the Borel subgroup $AN$ of the noncompact group $G=KAN$ where $K$ is the maximal compact subgroup of $G$. Or else one may restore a local $K$ gauge invariance and use $G$ valued scalar fields, this is the better way to discover and restore symmetries but the formalism developed in [@CJLP] and references therein is best suited for the fixed gauge approach.
The general action distinguishes the dilaton fields ie the coordinates along the (fully) non-compact Cartan subalgebra $Lie(A)$ from the other scalar fields that correspond to positive root generators. It is simply a sum of quadratic kinetic terms for the latter weighted by appropriate exponentials of their respective roots ie. linear forms in the dilatons plus free kinetic terms for the dilatons themselves. These exponential factors are responsible for the walls of the effective potential of the piecewise Kasner metric evolving in time [@DH; @DHJN] after suitable overextension, in particular one must include the so-called symmetry wall.
Let us now recall the Iwasawa decomposition in the general case. It can be studied in [@He] for instance. We are now considering a real Lie algebra with a maximally non-compact Cartan subalgebra whose noncompact part we denote by $a$. The maximal compact subgroup has a higher dimension than in the split case and the coset space has lower dimension. The decomposition reads now $$Lie(G)=Lie(K)+a+n$$ where $n$ is the nilpotent subalgebra of positive restricted root vectors.
Clearly in the non-split case the Cartan subalgebra is to be replaced by its non compact part and the nilpotent subalgebra of positive root vectors by that of positive restricted root vectors. Again the restricted roots form a possibly nonreduced root system. This brings two possible complications: firstly the multiplicities but they do not change the walls of the billiards and hence are irrelevant but also the nonreduced roots for some $SO(2k+1)$ restricted root systems and geometries.
Applications will appear soon, firstly in [@HJK] we shall examine the replacement of $SO(8,24)$ by $SO(8,9)$; this paper contains also anomaly free string realisations of the $DE_{10}$ theory as well as that of a theory with U-duality $SO(8,9)$ (this work was started before [@DH] and independently). Other instances of non split forms of U-dualities occuring in pure supergravities in 4d the so-called magic triangle will be analysed in [@HJ] in order to check the precise control of chaos by symmetry in this more general situation. In many cases the simple rule of overextension of 3d U-duality is sufficient to analyse the chaotic or non chaotic behaviour of the flow. It is important to reach a more rigorous level of characterisation of the chaos and work is being done in this direction whereas theorems are available on the non-chaotic side [@AR].
Acknowledgements: I am grateful to J. McKay and M. Henneaux for references and discussions.
[99]{} A.P. Ogg, Can. J. Math. 36 (1984) 800
C. Sacioglu, J. Phys. A 22 (1989) 3753
B. Julia, Proc AMS-SIAM Chicago meeting July 1982, Lectures in Applied Mathematics 21 (1985) 355
A. Feingold and I. Frenkel, Math. Ann. 263 (1983) 87
Bourbaki, Groupes et alg\` ebres de Lie 4,5,6 (1968) Hermann
E.B. Vinberg, Geometry II Encyclopedia of Math. Sciences 29 (1993) Springer
T. Damour and M. Henneaux, Phys.Rev.Lett. 86 (2001) 4749
S. Araki, J. of Math. Osaka City Un. 13 (1962) 1
A.W. Knapp, Lie groups beyond an introduction, Birkh" auser (1996)
S. Helgason, Differential geometry, Lie groups and symmetric spaces, Academic Press (1978)
T. Damour, M. Henneaux, B. Julia and H. Nicolai, Phys. Lett. B509 (2001) 323
E. Cremmer, B. Julia, H. Lu and C. Pope, hep-th/9909099 to be submitted to Com. Math. Phys.
A. Hanany, B. Julia and A. Keurentjes, LPTENS 02-24, hep-th/0210xxx
M. Henneaux and B. Julia, Hyperbolic billiards of pure d=4 SUGRAS, in preparation, hep-th/0210xxy
L. Andersson and A.D. Rendall, gr-qc/0001047, see also gr-qc/0202069
A. Borel and J. Tits, Pub. IHES 27 (1965) 55
[^1]: UMR 8549 du CNRS et de l’[É]{}cole Normale Sup[é]{}rieure.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Given a partial action $\alpha$ of a connected groupoid ${\mathcal{G}}$ on an associative ring $A$ we investigate under what conditions the partial skew groupoid ring $A\star_{\alpha}{\mathcal{G}}$ can be realized as a partial skew group ring. In such a case applications concerning to the separability, semisimplicity and Frobenius property of the ring extension $A\subset A\star_{\alpha}{\mathcal{G}}$ as well as to the artinianity of ${{A}}\star_{{\alpha}}{\mathcal{G}}$ are given.'
address:
- |
Departamento de Matemática, Universidade Federal de Santa Maria, 97105-900\
Santa Maria-RS, Brasil
- |
Instituto de Matemática e Estatística, Universidade federal de Porto Alegre, 91509-900\
Porto Alegre-RS, Brazil
- 'Escuela de Matematicas, Universidad Industrial de Santander, Cra. 27 Calle 9 ´ UIS Edificio 45, Bucaramanga, Colombia'
author:
- Dirceu Bagio
- Antonio Paques
- Héctor Pinedo
title: On partial skew groupoids rings
---
\#1[[**(\*\*\* \#1 \*\*\*)**]{} ]{}
\#1
[^1] [^2]
Introduction
============
In this work we will consider partial actions of groupoids on rings. We are interested in studying the structure of the corresponding partial skew groupoids rings.
Partial groupoids actions on rings were introduced in [@BP] and they are a natural generalization of partial group actions. It is well known that every groupoid is a direct sum of its connected component. A partial action of a groupoid on a ring $A$ is completely determined by the partial actions of its connected components on $A$. Thence, we can reduce the study of groupoid partial action on rings to the context of connected groupoids.
The structure of a connected groupoid is also well known. If ${\mathcal{G}}$ is a connected groupoid then ${\mathcal{G}}\simeq {\mathcal{G}}_0^2\times {\mathcal{G}}(x)$, where ${\mathcal{G}}_0^2$ is the coarse groupoid associated to the set ${\mathcal{G}}_0$ of the objects of ${\mathcal{G}}$ and ${\mathcal{G}}(x)$ is the isotropy group of an object $x$ of ${\mathcal{G}}$.
For a partial action $\alpha$ of a connected groupoid ${\mathcal{G}}$ on a ring $A$ we can construct the partial skew groupoid ring $A\star_{\alpha}{\mathcal{G}}$. If ${\mathcal{G}}_0$ is finite and $\alpha$ is unital then $A\star_{\alpha}{\mathcal{G}}$ is an associative and unital ring which is an extension of $A$.
The partial skew groupoid rings have an important role in the partial Galois theory for groupoids as it is explicit in Theorem 5.3 of [@BP]. They also are examples of Leavitt path algebras, which are important in the theory of $C^{\ast}$-algebras (see Theorem 3.11 of [@GY]). In the last years, algebraic properties to the extension $A\subset A\star_{\alpha}{\mathcal{G}}$ have been studied. For example, the separability and semisimplicity properties of the extension $A\subset A\star_{\alpha}{\mathcal{G}}$ were studied in [@BPi] whereas in [@NOP] the authors investigate chain conditions between $A$ and $A\star_{\alpha}{\mathcal{G}}$.
Our purpose in this work is to study the following problem. Let ${\mathcal{G}}$ be a connected groupoid such that ${\mathcal{G}}_0$ is finite and $\alpha$ a unital partial action of ${\mathcal{G}}$ on a ring $A$. Does the factorization ${\mathcal{G}}\simeq {\mathcal{G}}_0^2\times {\mathcal{G}}(x)$ induce a factorization of $A\star_{\alpha}{\mathcal{G}}$? Theorem 4.1 provides sufficient conditions for the answer to this question to be affirmative. Precisely, when $\alpha$ is a group-type partial action, we construct a groupoid action $\beta$ of ${\mathcal{G}}_0^2$ on $A$ and a partial group action $\gamma$ of ${\mathcal{G}}(x)$ on $A\star_{\beta} {\mathcal{G}}_0^2$ and we prove that $A\star_{\alpha}{\mathcal{G}}\simeq (A\star_{\beta} {\mathcal{G}}_0^2)\star_{\gamma}{\mathcal{G}}(x)$.
We organize our work as follows. The background about groupoids is presented in Section 2. The topics of partial groupoid actions that will be used are in Section 3. In Section 4, we construct the actions $\beta$ and $\gamma$ which allow us to prove the factorization of ${{A}}\star_{{\alpha}}{\mathcal{G}}$ mentioned in the previous paragraph. Applications of this result concerning to the separability, semisimplicity and Frobenius property of the extension $A\subset {{A}}\star_{{\alpha}}{\mathcal{G}}$ as well as to the artinianity of ${{A}}\star_{{\alpha}}{\mathcal{G}}$ are given in Section 5.
Conventions {#subsec:conv .unnumbered}
-----------
Throughout this work, by ring we mean an associative and not necessarily unital ring. The center of a ring $A$ will be denoted by $C(A)$. We will denote the cardinality of a finite set $X$ by $|X|$.
Groupoids
=========
We recall that a [*groupoid*]{} is a small category in which every morphism is an isomorphism. The set of the objects of a groupoid ${\mathcal{G}}$ will be denoted by ${\mathcal{G}}_0$. If $g:x\to y$ is a morphism of ${\mathcal{G}}$ then $s(g)=x$ and $t(g)=y$ are called the [*source*]{} and the [*target*]{} of $g$ respectively. We will identify any object $x$ of ${\mathcal{G}}$ with its identity morphism, that is, $x={{\rm id}}_x$. The [*isotropy group*]{} associated to an object $x$ of ${\mathcal{G}}$ is the group ${\mathcal{G}}(x)=\{g\in {\mathcal{G}}:\,s(g)=t(g)=x\}$.
The composition of morphisms of a groupoid ${\mathcal{G}}$ will be denoted via concatenation. Hence, for $g,h\in {\mathcal{G}}$, there exists $gh$ if and only if $t(h)=s(g)$. Notice that, if $g\in {\mathcal{G}}$ then $s(g)=g^{-1}g$ and $t(g)=gg{{}^{-1}}$. Also, $s(gh)=s(h)$ and $t(gh)=t(g)$ for all $g,h\in {\mathcal{G}}$ with $t(h)=s(g)$.
A groupoid ${\mathcal{G}}$ is said to be [*connected*]{} if for any $x,y\in {\mathcal{G}}_0$ there exists a morphism $g\in {\mathcal{G}}$ such that $s(g)=x$ and $t(g)=y$, that is, the morphism $g$ connects the objects $x$ and $y$. It is well-known that any groupoid is a disjoint union of connected subgroupoids. In order to justify this fact, we consider the following equivalence relation on ${\mathcal{G}}_0$: for any $x,y\in{\mathcal{G}}_0$, $x\sim y$ if and only if there exists $ g\in{\mathcal{G}}$ such that $s(g)=x$ and $t(g)=y$. Every equivalence class $X\in{\mathcal{G}}_0/\!\!\sim$ determines a full connected subgroupoid ${\mathcal{G}}_X$ of ${\mathcal{G}}$. The set of objects of ${\mathcal{G}}_X$ is $X$. The set ${{\mathcal{G}}_X}(x,y)$ of morphisms of ${\mathcal{G}}_X$ from $x$ to $y$ is equal to ${{\mathcal{G}}}(x,y)$, for all $x,y\in X$. By construction, ${\mathcal{G}}$ is the disjoint union of the subgroupoids ${\mathcal{G}}_X$, i. e. $$\label{direct-sum}
{\mathcal{G}}=\dot\cup_{X\in {\mathcal{G}}_0/\!\sim}{\mathcal{G}}_X.$$
For the convenience of the reader, we will prove a well-known result about the structure of connected groupoids. In order to do this, we need to introduce some extra notation. Let $X$ be a nonempty set and $X^2=X\times X$. Then $X^2$ is a groupoid. The source and target maps of $X^2$ are, respectively, $s(x,y)=x$ and $t(x,y)=y$, for all $x,y\in X$. The rule of composition is given by: $(y,z)(x,y)=(x,z)$, for all $x,y,z\in X$. The groupoid $X^2$ is called the [*coarse groupoid associated to $X$*]{}.
\[group:connec\] Let ${\mathcal{G}}$ be a connected groupoid. Then ${\mathcal{G}}\simeq {\mathcal{G}}_0^2\times {\mathcal{G}}(x)$ as groupoids.
Let $x\in {\mathcal{G}}_0$ a fixed object of ${\mathcal{G}}$. For each $y\in {\mathcal{G}}_0$, we choose a morphism $\tau_y:x\to y$ of ${\mathcal{G}}$. We also choose $\tau_x=x$. Define $\varphi:{\mathcal{G}}\to {\mathcal{G}}_0^2\times{\mathcal{G}}(x)$ by $\varphi(g)= ((s(g),t(g)), g_x)$, where $g_x={\tau}^{-1}_{t(g)}g{\tau}_{s(g)}$, for all $g\in {\mathcal{G}}$. It is straightforward to prove that $\varphi$ in a groupoid morphism. Suppose that $\varphi(g)$ is an identity of ${\mathcal{G}}_0^2\times{\mathcal{G}}(x)$. Then, $\varphi(g)=((y,y),x)$ for some $y\in{\mathcal{G}}_0$. Hence, $s(g)=t(g)=y$ and $x=g_x={\tau}_y^{-1}g{\tau}_y$ which implies that $g={\tau}_y{\tau}_y^{-1}=x$. This ensures that $\varphi$ is injective. Given an element $((y,z),h)\in {\mathcal{G}}_0^2\times {\mathcal{G}}(x)$, consider $g=\tau_zh\tau{{}^{-1}}_y\in {\mathcal{G}}$. Notice that $g_x=h$ and whence $\varphi(g)=((y,z),h)$, that is, $\varphi$ is surjective, so an isomorphism of groupoids.
Partial actions
===============
In this section we recall the notion of partial actions of groupoids. Some properties related to partial actions, that will be used later, are presented. The definition of group-type partial groupoid actions, which has a central role for our purposes, will be introduced.
Partial groupoid action
-----------------------
We recall from [@BP] that a *partial action* of a groupoid ${\mathcal{G}}$ on a ring ${{A}}$ is a pair ${\alpha}=({{A}}_g,{\alpha}_g)_{g\in {\mathcal{G}}}$ such that
1. ${{A}}_g$ is an ideal of ${{A}}_{t(g)}$ and ${{A}}_{t(g)}$ is an ideal of ${{A}}$, for all $g\in {\mathcal{G}}$,
2. ${\alpha}_g:{{A}}_{g{{}^{-1}}}\to {{A}}_g$ is an isomorphism of rings, for all $g\in {\mathcal{G}}$,
3. $\alpha_x={{\rm id}}_{A_x}$, for all $x\in {\mathcal{G}}_0$,
4. ${\alpha}_g{\alpha}_h\le {\alpha}_{gh}$, for all $g,h\in {\mathcal{G}}$ such that $t(h)=s(g)$.
The condition (iv) means that $\alpha_{gh}$ is an extension of ${\alpha}_g{\alpha}_h$. Since the domain of ${\alpha}_g{\alpha}_h$ is ${\alpha}_h^{-1}({{A}}_{g^{-1}}\cap{{A}}_h)$, it follows that (iv) is equivalent to $$\quad{\rm (v)}\quad
{\alpha}_h^{-1}({{A}}_{g^{-1}}\cap{{A}}_h)\subset {{A}}_{(gh)^{-1}}\,\text{ and }\,{\alpha}_{gh}(a)={\alpha}_g{\alpha}_h(a), \, \text{ for all }\, a\in {\alpha}_h^{-1}({{A}}_{g^{-1}}\cap{{A}}_h).$$
The partial action ${\alpha}$ is said to be [*global*]{} if ${\alpha}_g{\alpha}_h={\alpha}_{gh}$, for all $g,h\in {\mathcal{G}}$ such that $t(h)=s(g)$. Also, ${\alpha}$ is called [*unital*]{} if each $A_g$ is a unital ring, i. e., there exists a central element $1_g$ of $A$ such that $A_g=A1_g$, for all $g\in {\mathcal{G}}$.
Now we recall Lemma 1.1 of [@BP] which give us some useful properties of partial actions that will be used in what follows.
\[lem:BP\] Let ${\alpha}=({{A}}_g,{\alpha}_g)_{g\in {\mathcal{G}}}$ be a partial action of a groupoid ${\mathcal{G}}$ on a ring $A$. Then:
1. $\alpha$ is global if and only if $A_g=A_{t(g)}$, for all $g\in {\mathcal{G}}$;
2. ${\alpha}_{g{{}^{-1}}}={\alpha}{{}^{-1}}_g$, for all $g\in {\mathcal{G}}$;
3. ${\alpha}_g({{A}}_{g{{}^{-1}}}\cap {{A}}_h)={{A}}_{g}\cap {{A}}_{gh}$, for all $g,h\in {\mathcal{G}}$ such that $t(h)=s(g)$.
\[obs-pag\][Let ${\alpha}=({{A}}_g,{\alpha}_g)_{g\in {\mathcal{G}}}$ be a partial action of a groupoid ${\mathcal{G}}$ on a ring $A$. Notice that ${\alpha}$ induces by restriction a partial action ${\alpha}_{(x)}=({{A}}_h,{\alpha}_h)_{h\in{\mathcal{G}}(x)}$ of the isotropy group ${\mathcal{G}}(x)$ on the ring ${{A}}_x$, for each $x\in{\mathcal{G}}_0$.]{}
[Let ${\mathcal{G}}$ be a groupoid. Using the decomposition of ${\mathcal{G}}$ given in , it is straightforward to check that partial actions of ${\mathcal{G}}$ on a ring ${{A}}$ induce by restriction partial actions of ${\mathcal{G}}_X$ on ${{A}}$, for all $X\in {\mathcal{G}}_0/\!\!\sim$. Conversely, partial actions of ${\mathcal{G}}$ on $A$ are uniquely determined by partial actions of ${\mathcal{G}}_X$, $X\in {\mathcal{G}}_0/\!\!\sim$, on $A$ . Hence, we can reduce the study of partial groupoid actions to the connected case.]{}
Group-type partial groupoid action
----------------------------------
Let ${\mathcal{G}}$ be a connected groupoid, $x\in {\mathcal{G}}_0$ and $\mathcal{S}_x=\{h\in {\mathcal{G}}\,:\,s(h)=x\}$. Consider the following equivalence relation on $\mathcal{S}_x$: $$g\sim_x l\,\,\ \text{ if and only if }\,\, t(g)=t(l), \qquad g,l\in \mathcal{S}_x.$$ A transversal $\tau(x)=\{\tau_{y}:y\in {\mathcal{G}}_0\}$ for $\sim_x$ such that $\tau_x=x$ will be called a *transversal for $x$*. Hence, ${\tau}_y:x\to y$ is a chosen morphism of ${\mathcal{G}}$, for each $y\in{\mathcal{G}}_0$ and ${\tau}_x=x$.
A partial action ${\alpha}=(A_g,{\alpha}_g)_{g\in {\mathcal{G}}}$ of a connected groupoid ${\mathcal{G}}$ on $A$ will be called [*group-type*]{} if there exist $x\in {\mathcal{G}}_0$ and a transversal $\tau(x)=\{\tau_{y}:y\in {\mathcal{G}}_0\}$ for $x$ such that $$\begin{aligned}
\label{cond1} A_{\tau{{}^{-1}}_y}=A_x \ \ \text{and} \ \ A_{\tau_y}=A_y, \ \ \text{ for all } \ y\in{\mathcal{G}}_0.\end{aligned}$$
\(i) Notice that the notion of group-type partial action not depend on the choice of object $x$. Indeed, for another object $z$ of ${\mathcal{G}}$, consider $\tilde{\tau}_y:=\tau_{y}\tau{{}^{-1}}_{z}$, for all $y\in {\mathcal{G}}_0$. Clearly, $\tilde{\tau}(z)=\{\tilde{\tau}_y:y\in {\mathcal{G}}_0\}$ is a transversal for $z$. From follows that ${\alpha}_{\tau_{y}}{\alpha}_{\tau{{}^{-1}}_{z}}={\alpha}_{\tau_{y}\tau{{}^{-1}}_{z}}={\alpha}_{\tilde{\tau}_y}$. Thus, $A_{(\tilde{\tau}_y){{}^{-1}}}=A_z$ and $A_{\tilde{\tau}_y}=A_y$, for all $y\in {\mathcal{G}}_0$.
\(ii) We use the term “group-type partial actions" since by Theorem \[teo-decomp\], proved in the next section for this kind of partial actions, the corresponding partial skew groupoid ring is indeed a partial skew group ring.
By Lemma \[lem:BP\] (i), any global groupoid action is group-type. The converse is not true as we can see in the next example.
\[57\] [Let ${\mathcal{G}}=\{g,h,l,m,l^{-1},m^{-1}\}$ be the groupoid with objects ${\mathcal{G}}_0=\{x,y\}$ and the following composition rules $$g^2=x,\quad h^{2}=y,\quad lg=m=hl,\quad g\in{\mathcal{G}}(x),\quad h\in{\mathcal{G}}(y)\,\,\text{ and }\,\, l,m:x\to y.$$ The diagram bellow illustrates the structure of ${\mathcal{G}}$: $$\xymatrix{& x\ar[r]^{l} &y \ar[d]^{h}\\
& x\ar[u]^{g} \ar[r]^{m} & y}$$ Consider $A=\mathbb{C}e_1\oplus\mathbb{C}e_2\oplus\mathbb{C}e_3\oplus\mathbb{C}e_4$, where $\mathbb{C}$ denotes the complex number field, $e_ie_j=\delta_{i,j}e_i$ and $e_1+\ldots+e_4=1$. We define the following partial action $\alpha=\big(A_p,{\alpha}_p\big)_{p\in{\mathcal{G}}}$ of ${\mathcal{G}}$ on $A$: $$\begin{aligned}
&A_x=\mathbb{C}e_1\oplus\mathbb{C}e_2=A_{l{{}^{-1}}},& & A_y=\mathbb{C}e_3\oplus\mathbb{C}e_4=A_l, &\\[.2em]
& A_g=\mathbb{C}e_1=A_{g{{}^{-1}}}=A_{m{{}^{-1}}},& & A_m=A_h=\mathbb{C}e_3=A_{h{{}^{-1}}}, &\
\end{aligned}$$ and $$\begin{aligned}
&{\alpha}_x=id_{A_x},\ \ \ {\alpha}_y= id_{A_y},\ \ \ {\alpha}_g:ae_1\mapsto \overline{a}e_1, \ \ \ {\alpha}_h:ae_3\mapsto \overline{a}e_3,\ \ \ {\alpha}_m:ae_1\mapsto \overline{a}e_3, \\[.3em]
&\ \ {\alpha}_{m{{}^{-1}}}: ae_3\mapsto\overline{a}e_1,\ \ \ {\alpha}_l:ae_1+be_2\mapsto ae_3+be_4, \ \ \ {\alpha}_{l{{}^{-1}}}:ae_3+be_4\mapsto ae_1+be_2,
\end{aligned}$$ where $\overline{a}$ denotes the complex conjugate of $a$, for all $a\in\mathbb{C}$. Notice that ${\alpha}$ is a group-type (not global) partial action. Indeed, to see this it is enough to take the transversal ${\tau}(x)=\{{\tau}_x=x, {\tau}_y=l\}$ for $x$.]{}
The partial skew groupoid ring
===============================
In this section, we will assume that ${\mathcal{G}}$ is a connected groupoid such that ${\mathcal{G}}_0$ is finite, $x\in{\mathcal{G}}_0$ is a fixed object of ${\mathcal{G}}$ and ${\alpha}=(A_g,{\alpha}_g)_{g\in {\mathcal{G}}}$ is a unital partial action of ${\mathcal{G}}$ on a ring $A$ with $A_g=A1_g$, where $1_g$ is a central idempotent of $A$, for all $g\in{\mathcal{G}}$. We will also assume that $\alpha$ is group-type and $\tau(x)=\{\tau_{y}:y\in {\mathcal{G}}_0\}$ is a transversal for $x$ such that is satisfied.
The [*partial skew groupoid ring*]{} $A\star_{\alpha}{\mathcal{G}}$ associated to $\alpha$ is the set of all formal sums $\sum_{g\in {\mathcal{G}}}a_g\delta_g$, where $a_g\in A_g$, with the usual addition and multiplication induced by the following rule $$(a_g\delta_g)(a_h\delta_h)=
\begin{cases}
a_g\alpha_g(a_h1_{g^{-1}})\delta_{gh} &\text{if $s(g)=t(h),$}\\
0 &\text{otherwise},
\end{cases}$$ for all $g, h\in {\mathcal{G}}$, $a_g\in A_g$ and $a_h\in A_h$. The partial skew groupoid ring $A\star_{\alpha}{\mathcal{G}}$ is an associative ring. Since by assumption ${\mathcal{G}}_0$ is finite, $A\star_{\alpha}{\mathcal{G}}$ is unital with identity $1_{A\star_{\alpha}{\mathcal{G}}}=\sum_{y\in{\mathcal{G}}_0}1_y\delta_y$ (see $\S\,3$ of [@BFP] for more details).
As it was mentioned in the introduction section of [@NOP], the ring structure of $A\star_\alpha {\mathcal{G}}$ only depends on the choice of the ideals $A_y$, $y\in {\mathcal{G}}_0$. Hence, we can choose $A$ to be any ring having the ideals as above described. In this sense, we will assume for the rest of this paper that $$A=\oplus_{y\in {\mathcal{G}}_0} A_y.$$
The main theorem
----------------
In this subsection we will prove that the factorization of ${\mathcal{G}}$, given by Proposition \[group:connec\], induces a factorization of $A\star_{\alpha}{\mathcal{G}}$. Particularly, we obtain that $A\star_{\alpha}{\mathcal{G}}$ is a partial skew group ring. In order to prove this result we will use some lemmas that will be proved in the sequel.
\[xx\] The pair ${\beta}=(B_u,{\beta}_u)_{u\in{\mathcal{G}}_0^2}$, where $B_u=A_{t(u)}$ and ${\beta}_u={\alpha}_{{\tau}_{t(u)}}{\alpha}_{{\tau}{{}^{-1}}_{s(u)}}$, is a global action of ${\mathcal{G}}_0^2$ on $A$.
For any identity $e=(y,y)$ of ${\mathcal{G}}_0^2$ we have that $B_e=A_y$ and ${\beta}_e={\alpha}_{{\tau}_{y}}{\alpha}_{{\tau}{{}^{-1}}_{y}}$ is the identity map of $A_{{\tau}_y}=A_y$. Also, if $u=(y,z)$ and $v=(r,y)$ are elements in ${\mathcal{G}}_0^2$ then $uv=(r,z)$ and $ {\beta}_u {\beta}_v={\alpha}_{{\tau}_z}{\alpha}_{{\tau}{{}^{-1}}_y}{\alpha}_{{\tau}_y}{\alpha}_{{\tau}{{}^{-1}}_r}={\alpha}_{{\tau}_z}{\alpha}_{{\tau}{{}^{-1}}_r}= {\beta}_{uv}$.
Thanks to Lemma \[xx\] we can consider the skew groupoid ring $C:=A\star_ {{\beta}}{\mathcal{G}}_0^2$. In the sequel we will see that the group ${\mathcal{G}}(x)$ acts partially on $C$. Let $z\in {\mathcal{G}}_0$. Since ${\alpha}$ is group-type, it follows from that $A_{{\tau}{{}^{-1}}_z}=A_x$. Then, for all $h\in {\mathcal{G}}(x)$, $A_h\subset A_x$ and $$\begin{aligned}
\label{Czh}
C_{z,h}:={\alpha}_{{\tau}_z}(A_h),\end{aligned}$$ is well-defined. Hence, we can set $$\begin{aligned}
\label{Ch}
C_h:=\oplus_{u\in{\mathcal{G}}_0^2}C_{t(u),h}\delta_u.\end{aligned}$$
\[63\] $C_h$ is a unital ideal of $C$, for all $h\in G(x)$. Moreover, $C_x=C$.
Note that $$\begin{aligned}
C &=\oplus_{u\in {\mathcal{G}}_0^2}B_{u}\delta_u\\
&=\oplus_{u\in {\mathcal{G}}_0^2}A_{{t(u)}}\delta_u\\
&\overset{\mathclap{\eqref{cond1}}}{=} \oplus_{u\in {\mathcal{G}}_0^2}A_{{\tau}_{t(u)}}\delta_u \\
&= \oplus_{u\in {\mathcal{G}}_0^2}{\alpha}_{{\tau}_{t(u)}}(A_{{\tau}{{}^{-1}}_{t(u)}})\delta_u \\
&\overset{\mathclap{\eqref{cond1}}}{=} \oplus_{u\in {\mathcal{G}}_0^2}{\alpha}_{{\tau}_{t(u)}}(A_x)\delta_u \\
&\overset{\mathclap{\eqref{Czh}}}{=} \oplus_{u\in {\mathcal{G}}_0^2}C_{t(u),x}\delta_u \\
&\overset{\mathclap{\eqref{Ch}}}{=}C_x.
\end{aligned}$$ Observe also that $1'_h=\sum_{z\in{\mathcal{G}}_0}{\alpha}_{{\tau}_z}(1_h)\delta_{(z,z)}$ is the identity element of $C_h$, for all $h\in{\mathcal{G}}(x)$. Indeed, let $u=(y,w)\in {\mathcal{G}}_0^2$ and $a\in C_{t(u),h}\delta_u=C_{w,h}\delta_u$. By , there exists $a_h\in A_h$ such that $a={\alpha}_{{\tau}_w}(a_h)\delta_{(y,w)}$ and consequently $$\begin{aligned}
a1'_h &=\sum_{z\in{\mathcal{G}}_0}{\alpha}_{{\tau}_w}(a_h)\delta_{(y,w)}{\alpha}_{{\tau}_z}(1_h)\delta_{(z,z)}\\
&={\alpha}_{{\tau}_w}(a_h)\delta_{(y,w)}{\alpha}_{{\tau}_y}(1_h)\delta_{(y,y)}\\
&={\alpha}_{{\tau}_w}(a_h) {\beta}_{(y,w)}({\alpha}_{{\tau}_y}(1_h))\delta_{(y,w)}\\
&={\alpha}_{{\tau}_w}(a_h){\alpha}_{{\tau}_w}{\alpha}_{{\tau}{{}^{-1}}_y}{\alpha}_{{\tau}_y}(1_h)\delta_{(y,w)} \qquad \text{(see Lemma \ref{xx}))} \\
&={\alpha}_{{\tau}_w}(a_h){\alpha}_{{\tau}_w}(1_h)\delta_{(y,w)}\\
&={\alpha}_{{\tau}_w}(a_h)\delta_{(y,w)}\\
&=a.\end{aligned}$$ Similarly, $1'_ha=a$. Hence $1'_h$ is a central idempotent of $C$. A straightforward calculation shows that $C_h=1'_hC$. Thus, $C_h$ is an ideal of $C$.
Let $(z,h)\in {\mathcal{G}}_0\times {\mathcal{G}}(x)$. We define $\gamma_{z,h}:C_{z,h{{}^{-1}}}\to C_{z,h}$, ${\alpha}_{{\tau}_z}(a)\mapsto {\alpha}_{{\tau}_z}({\alpha}_h(a))$, for all $a\in A_{h{{}^{-1}}}$. Clearly $ \theta_{z,h}$ is a bijection. Moreover, these maps induce the following bijective map $$\gamma_h:C_{h{{}^{-1}}}\to C_h,\quad \gamma_h({\alpha}_{{\tau}_{t(u)}}(a)\delta_u)= \gamma_{t(u),h}(a)\delta_u, \, \text{ for all } a\in A_{h{{}^{-1}}} \text{ and } u\in {\mathcal{G}}_0^2.$$
\[64\] The pair $\gamma=(C_h, \gamma_h)_{h\in {\mathcal{G}}(x)}$ is a unital partial action of ${\mathcal{G}}(x)$ on $C$.
By definition, $\gamma_x$ is the identity map of $C_x=C$. Note that $\gamma_h$ preserves the operation of multiplication. In fact, for all $a,b\in A_{h{{}^{-1}}}$, $$\begin{aligned}
\hspace*{-1.5cm} \gamma_h(({\alpha}_{{\tau}_z}(a)\delta_{(y,z)})({\alpha}_{{\tau}_y}(b)\delta_{(w,y)}))&= \gamma_h({\alpha}_{{\tau}_z}(a) {{\alpha}^\ast}_{(y,z)}({\alpha}_{{\tau}_y}(b))\delta_{(w,z)})\\
&= \gamma_h({\alpha}_{{\tau}_z}(a){\alpha}_{{\tau}_z}({\alpha}_{{\tau}{{}^{-1}}_y}({\alpha}_{{\tau}_y}(b)))\delta_{(w,z)})\\
&= \gamma_h({\alpha}_{{\tau}_z}(ab)\delta_{(w,z)})\\
&={\alpha}_{{\tau}_z}({\alpha}_h(ab))\delta_{(w,z)}\\
&={\alpha}_{{\tau}_z}({\alpha}_h(a)){\alpha}_{{\tau}_z}({\alpha}_h(b))\delta_{(w,z)}.\end{aligned}$$ On the other hand, $$\begin{aligned}
\hspace*{-1.2cm} \gamma_h({\alpha}_{{\tau}_z}(a)\delta_{(y,z)}) \gamma_h({\alpha}_{{\tau}_y}(b)\delta_{(w,y)})&={\alpha}_{{\tau}_z}({\alpha}_h(a))\delta_{(y,z)}{\alpha}_{{\tau}_y}({\alpha}_h(b))\delta_{(w,y)}\\
&={\alpha}_{{\tau}_z}({\alpha}_h(a)){\alpha}_{{\tau}_z}({\alpha}_{{\tau}{{}^{-1}}_y}({\alpha}_{{\tau}_y}({\alpha}_h(b))))\delta_{(w,z)}\\
&={\alpha}_{{\tau}_z}({\alpha}_h(a)){\alpha}_{{\tau}_z}({\alpha}_h(b))\delta_{(w,z)}.\end{aligned}$$ Hence, $\gamma_h$ is a ring isomorphism. It remains to show that $\gamma$ satisfies the condition (v) given in $\S\,3.1$.
Firstly, note that $\gamma_{l{{}^{-1}}}(C_l\cap C_{h{{}^{-1}}})\subset C_{(hl){{}^{-1}}}$, for all $h,l\in {\mathcal{G}}(x)$. Indeed, by definition, $\gamma_{l{{}^{-1}}}$ is additive and whence $$\begin{aligned}
\hspace*{-1cm} \gamma_{l{{}^{-1}}}(C_l\cap C_{h{{}^{-1}}})&=\oplus_{u\in {\mathcal{G}}_0^2} \gamma_{l{{}^{-1}}}({\alpha}_{{\tau}_{t(u)}}(A_l\cap A_{h{{}^{-1}}})\delta_u)\\
&=\oplus_{u\in {\mathcal{G}}_0^2}{\alpha}_{{\tau}_{t(u)}}({\alpha}_{l{{}^{-1}}}(A_l\cap A_{h{{}^{-1}}}))\delta_u\\
&\subset \oplus_{u\in {\mathcal{G}}_0^2}{\alpha}_{{\tau}_{t(u)}}(A_{l{{}^{-1}}}\cap A_{l{{}^{-1}}h{{}^{-1}}})\delta_u\\
&=C_{(hl){{}^{-1}}},\end{aligned}$$ where the last inclusion above holds because ${\alpha}_{(x)}$ is a partial action of ${\mathcal{G}}(x)$ on $A_x$ as defined in Remark \[obs-pag\]. Finally, let $c={\alpha}_{{\tau}_z}({\alpha}_{l{{}^{-1}}}(a))\delta_{(y,z)}\in \gamma_{l{{}^{-1}}}(C_l\cap C_{h{{}^{-1}}})$. Then $$\begin{aligned}
\gamma_h( \gamma_l(c))&= \gamma_h( \gamma_l({\alpha}_{{\tau}_z}({\alpha}_{l{{}^{-1}}}(a))\delta_{(y,z)}))\\
&= \gamma_h({\alpha}_{{\tau}_z}(a)\delta_{(y,z)})\\
&={\alpha}_{{\tau}_z}({\alpha}_h(a))\delta_{(y,z)}.\end{aligned}$$ On the other hand, since ${\alpha}_{hl}={\alpha}_h{\alpha}_l$ in ${\alpha}_{l{{}^{-1}}}(A_l\cap A_{h{{}^{-1}}})$ we have $$\gamma_{hl}(c)={\alpha}_{{\tau}_z}({\alpha}_{hl}({\alpha}_{l{{}^{-1}}}(a)))\delta_{(y,z)}={\alpha}_{{\tau}_z}({\alpha}_h(a))\delta_{(y,z)},$$ and consequently $\gamma$ satisfies (v) of $\S\,3.1$.
By Lemma \[64\] we can consider the partial skew group ring $(A\star_{{\beta}}{\mathcal{G}}_0^2)\star_ \gamma{\mathcal{G}}(x)$ and thus present the main result of this section which give us a factorization of the ring $A\star_{\alpha}{\mathcal{G}}$.
\[teo-decomp\] $\,\,A\star_{\alpha}{\mathcal{G}}\simeq (A\star_{{\beta}}{\mathcal{G}}_0^2)\star_ \gamma{\mathcal{G}}(x).$
Consider the map $\varphi:A\star_{\alpha}{\mathcal{G}}\to (A\star_{{\beta}}{\mathcal{G}}_0^2)\star_ \gamma{\mathcal{G}}(x)$ given by $ a\delta_g\mapsto a\delta_{(s(g),t(g))}\delta_{g_x}$, where $g_x=\tau{{}^{-1}}_{t(g)}g\tau_{s(g)}$. In order to prove that $\varphi$ is a ring isomorphism we proceed by a series of steps.
*Step 1: $\varphi$ is well defined.*
By Lemma \[xx\], $A_g\subseteq A_{{t(g)}}=B_{(s(g),t(g))}$, for all $g\in{\mathcal{G}}$. Hence, we only need to show that $a\delta_{(s(g),t(g))}\in C_{g_x}$, for all $a\in A_g$. Notice that $$\begin{aligned}
{\alpha}_{{\tau}_{t(g)}}(A_{g_x})&={\alpha}_{{\tau}_{t(g)}}(A_{{\tau}{{}^{-1}}_{t(g)}g{\tau}_{s(g)}})\\
&={\alpha}_{{\tau}_{t(g)}}(A_{{\tau}{{}^{-1}}_{t(g)}g{\tau}_{s(g)}}\cap A_x)\\
&=A_{g{\tau}_{s(g)}}\cap A_{{\tau}_{t(g)}x} \qquad (\text{by Lemma \ref{lem:BP} {\rm (iii)}})\\
&=A_{g{\tau}_{s(g)}}\cap A_{{\tau}_{t(g)}}\\
&\overset{\mathclap{\eqref{cond1}}}{=}A_{g{\tau}_{s(g)}}\cap A_{t(g)}\\
&=A_{g{\tau}_{s(g)}} \qquad \qquad\qquad (A_{g{\tau}_{s(g)}}\subseteq A_{t(g{\tau}_{s(g)})}=A_{t(g)} ).
\end{aligned}$$ Since $A_{g{{}^{-1}}}=A_{g{{}^{-1}}}\cap A_{s(g)}\overset{\eqref{cond1}}{=} A_{g{{}^{-1}}}\cap A_{{\tau}_{s(g)}}$ we have $A_g={\alpha}_g(A_{g{{}^{-1}}}\cap A_{{\tau}_{s(g)}})=A_g\cap A_{g{\tau}_{s(g)}}$. Hence $a\in A_g\subseteq A_{g{\tau}_{s(g)}}={\alpha}_{{\tau}_{t(g)}}(A_{g_x})$ which implies $a\delta_{(s(g),t(g))}\in C_{g_x}$ by and .
*Step 2: $\varphi$ is a ring homomorphism.*
It is enough to prove that $\varphi$ preserves the operation of multiplication. Let $g,h\in{\mathcal{G}}$ such that $s(g)=t(h)$. It is easy to see that $(gh)_x=g_xh_x$. Hence $$\begin{aligned}
\varphi((a\delta_g)(b\delta_h))&=\varphi(a{\alpha}_g(b1_{g{{}^{-1}}})\delta_{gh})\\
&=a{\alpha}_g(b1_{g{{}^{-1}}})\delta_{(s(gh),t(gh))}\delta_{(gh)_x}\\
&=a{\alpha}_g(b1_{g{{}^{-1}}})\delta_{(s(h),t(g))}\delta_{g_xh_x},\end{aligned}$$ for all $a\in A_g$ and $b\in A_h$. On the other hand $$\begin{aligned}
\varphi(a\delta_g)\varphi(b\delta_h)&=(a\delta_{(s(g),t(g))}\delta_{g_x})(b\delta_{(s(h),t(h))}\delta_{h_x})\\
&=(a\delta_{(s(g),t(g))})(\gamma_{g_x}(b\delta_{(s(h),t(h))}1'_{g{{}^{-1}}_x}))\delta_{g_xh_x}.\end{aligned}$$ As in Step 1, we have that $A_h\subseteq A_{h{\tau}_{s(h)}}={\alpha}_{{\tau}_{t(h)}}(A_{h_x})$. Since $b\in A_h$, there is $b'\in A_{h_x}$ such that $b={\alpha}_{{\tau}_{t(h)}}(b')$ and $$\begin{aligned}
b\delta_{(s(h),t(h))}1'_{g{{}^{-1}}_x}&={\alpha}_{{\tau}_{t(h)}}(b')\delta_{(s(h),t(h))}\sum_{z\in{\mathcal{G}}_0}{\alpha}_{{\tau}_z}(1_{g{{}^{-1}}_x})\delta_{(z,z)}\\
&={\alpha}_{{\tau}_{t(h)}}(b')\delta_{(s(h),t(h))}{\alpha}_{{\tau}_{s(h)}}(1_{g{{}^{-1}}_x})\delta_{(s(h),s(h))}\\
&={\alpha}_{{\tau}_{t(h)}}(b'){\alpha}_{{\tau}_{t(h)}}(1_{g{{}^{-1}}_x})\delta_{(s(h),t(h))}\\
&={\alpha}_{{\tau}_{t(h)}}(b'1_{g{{}^{-1}}_x})\delta_{(s(h),t(h))}.\end{aligned}$$ Hence, $$\begin{aligned}
\varphi(a\delta_g)\varphi(b\delta_h)&=(a\delta_{(s(g),t(g))})(\gamma_{g_x}(b\delta_{(s(h),t(h))}1'_{g{{}^{-1}}_x}))\delta_{g_xh_x}\\
&=(a\delta_{(s(g),t(g))})(\gamma_{g_x}({\alpha}_{{\tau}_{t(h)}}(b'1_{g{{}^{-1}}_x})\delta_{(s(h),t(h))}))\delta_{g_xh_x}\\
&=(a\delta_{(s(g),t(g))})({\alpha}_{{\tau}_{t(h)}}({\alpha}_{g_x}(b'1_{g{{}^{-1}}_x}))\delta_{(s(h),t(h))})\delta_{g_xh_x}\\
&=a{\alpha}_{{\tau}_{t(g)}}{\alpha}_{{\tau}{{}^{-1}}_{s(g)}}{\alpha}_{{\tau}_{t(h)}}{\alpha}_{g_x}(b'1_{g{{}^{-1}}_x})\delta_{(s(h),t(g))}\delta_{g_xh_x}\\
&=a{\alpha}_{{\tau}_{t(g)}}{\alpha}_{g_x}(b'1_{g{{}^{-1}}_x})\delta_{(s(h),t(g))}\delta_{g_xh_x}\\
&\overset{\mathclap{\eqref{cond1}}}{=}a{\alpha}_{{\tau}_{t(g)}}({\alpha}_{g_x}(b'1_{g{{}^{-1}}_x})1_{{\tau}{{}^{-1}}_{t(g)}})\delta_{(s(h),t(g))}\delta_{g_xh_x}\\
&=a{\alpha}_{{\tau}_{t(g)}g_x}(b'1_{({\tau}_{t(g)}g_x){{}^{-1}}})1_{{\tau}_{t(g)}}\delta_{(s(h),t(g))}\delta_{g_xh_x}\qquad (\text{by (v) of $\S\,$3.1})\\
&=a{\alpha}_{g{\tau}_{s(g)}}(b'1_{(g{\tau}_{s(g)}){{}^{-1}}})1_{{\tau}_{t(g)}}\delta_{(s(h),t(g))}\delta_{g_xh_x}\\
&\overset{\mathclap{\eqref{cond1}}}{=}a{\alpha}_{g{\tau}_{s(g)}}(b'1_{(g{\tau}_{s(g)}){{}^{-1}}})1_g\delta_{(s(h),t(g))}\delta_{g_xh_x}\\
&=a{\alpha}_g({\alpha}_{{\tau}_{s(g)}}(b'1_{{\tau}{{}^{-1}}_{s(g)}})1_{g{{}^{-1}}})\delta_{(s(h),t(g))}\delta_{g_xh_x}\qquad (\text{by (v) of $\S\,$3.1})\\
&\overset{\mathclap{\eqref{cond1}}}{=}a{\alpha}_g({\alpha}_{{\tau}_{t(h)}}(b')1_{g{{}^{-1}}})\delta_{(s(h),t(g))}\delta_{g_xh_x}\\
&=a{\alpha}_g(b1_{g{{}^{-1}}})\delta_{(s(h),t(g))}\delta_{g_xh_x}\\
&=\varphi((a\delta_g)(b\delta_h)).\end{aligned}$$
*Step 3: $\varphi$ is injective.*
Let $v=\sum_{g\in{\mathcal{G}}}a_g\delta_g\in A\star_{\alpha}{\mathcal{G}}$ such that $\varphi(v)=0$. Then $$0=\sum_{g\in{\mathcal{G}}}a_g\delta_{(s(g),t(g))}\delta_{g_x}=\sum_{h\in{\mathcal{G}}(x)}\sum_{\substack{g\in {\mathcal{G}}\\ g_x=h}} a_g\delta_{(s(g),t(g))}\delta_h$$ Since $C\star_ \theta{\mathcal{G}}(x)$ is a direct sum, it follows that $$\begin{aligned}
\label{injective}
\sum_{\substack{g\in {\mathcal{G}}\\ g_x=h}}a_g\delta_{(s(g),t(g))}=0,\quad \text{for all } h\in {\mathcal{G}}(x).\end{aligned}$$ Consider $h\in{\mathcal{G}}(x)$ and $g,g'\in{\mathcal{G}}$ such that $g_x=g'_x=h$. It is straightforward to check that $(s(g),t(g))=(s(g'),t(g'))$ if and only if $g=g'$. Therefore holds if and only if $a_g=0$, for all $g\in {\mathcal{G}}$. Thus $v=0$.
*Step 4: $\varphi$ is surjective.*
It is enough to check that given any element of the type ${\alpha}_{{\tau}_z}(a)\delta_{(y,z)}\delta_h$, with $h\in{\mathcal{G}}(x)$ and $a\in A_h$, there exists an element $w\in A\star_{\alpha}{\mathcal{G}}$ such that $\varphi(w)={\alpha}_{{\tau}_z}(a)\delta_{(y,z)}\delta_h$. To do that observe that $$\begin{aligned}
{\alpha}_{{\tau}_z}(a)\in{\alpha}_{{\tau}_z}(A_h)&={\alpha}_{{\tau}_z}(A_h\cap A_x)\\
&\overset{\mathclap{\eqref{cond1}}}{=}{\alpha}_{{\tau}_z}(A_h\cap A_{{\tau}{{}^{-1}}_z})\\
&=A_{{\tau}_zh}\cap A_{{\tau}_z}\qquad\quad (\text{by Lemma \ref{lem:BP} (iii)})\\
&\overset{\mathclap{\eqref{cond1}}}{=}A_{{\tau}_zh}\cap A_z\\
&=A_{{\tau}_zh}\qquad\quad (\text{because}\,\ A_{{\tau}_zh}\subseteq A_{t({\tau}_zh)}=A_{t({\tau}_z)}=A_z).\end{aligned}$$ Therefore, for $g={\tau}_zh{\tau}{{}^{-1}}_y$ we have $t(g)=t({\tau}_z)=z$, $s(g)=s({\tau}{{}^{-1}}_y)=y$ and $${\alpha}_{{\tau}_z}(a)\in A_{{\tau}_zh}= A_{{\tau}_{t(g)}h}\overset{(\ast)}{=}{\alpha}_{{\tau}_{t(g)}}(A_h)\overset{(\ast\ast)}{\subseteq}{\alpha}_{{\tau}_{t(g)}}(A_{h{\tau}{{}^{-1}}_{s(g)}})\overset{(\ast)}{=} A_{{\tau}_{t(g)}h{\tau}{{}^{-1}}_{s(g)}}=A_{{\tau}_zh{\tau}{{}^{-1}}_y}=A_g,$$ where $(\ast)$ is ensured by $${\alpha}_{{\tau}_{t(g)}}(A_h)={\alpha}_{{\tau}_{t(g)}}(A_h\cap A_x)\overset{\eqref{cond1}}{=}{\alpha}_{{\tau}_{t(g)}}(A_h\cap A_{{\tau}{{}^{-1}}_{s(g)}})=A_{{\tau}_{t(g)}h},$$ and $(\ast\ast)$ by $$A_h={\alpha}_h(A_{h{{}^{-1}}}\cap A_x)\overset{\eqref{cond1}}{=}{\alpha}_h(A_{h{{}^{-1}}}\cap A_{{\tau}{{}^{-1}}_{s(g)}})=A_h\cap A_{h{\tau}{{}^{-1}}_{s(g)}}\subseteq A_{h{\tau}{{}^{-1}}_{s(g)}}.$$
Now, taking $w={\alpha}_{{\tau}_z}(a)\delta_g$ we are done.
\[ob-global\] [Since global actions are group-type actions, the factorization given in Theorem \[teo-decomp\] holds for all unital global action $\alpha$ of ${\mathcal{G}}$ on $A$. In such a case, the partial group action $\gamma$ of ${\mathcal{G}}(x)$ on $A_x$ is indeed a global action and consequently $A\star_{{\alpha}}{\mathcal{G}}$ is a skew group ring.]{}
Applications
============
The aim of this section is to present some applications of Theorem \[teo-decomp\]. In what follows, ${\mathcal{G}}$ is connected and ${\mathcal{G}}_0$ is finite. The partial action $\alpha$, the ring $A$ and the transversal $\tau(x)$ are assumed as in the previous section. Also, $\beta$ is the global action of ${\mathcal{G}}_0^2$ on $A$ given in Lemma \[xx\] and $\gamma$ is the partial action of ${\mathcal{G}}(x)$ on $A\star_{{\beta}}{\mathcal{G}}_0^2$ given in Lemma \[64\].
Note that $\varphi\colon {{A}}\to {{A}}\star_{{\alpha}}{\mathcal{G}}$, $\ a\mapsto \sum_{y\in {\mathcal{G}}_0}(a1_y)\delta_y$, is a monomorphism of rings and whence ${{A}}\star_{{\alpha}}{\mathcal{G}}$ is a ring extension of $A$. By Theorem \[teo-decomp\], $A\subset {{A}}\star_{\beta}{\mathcal{G}}_0^2\subset ({{A}}\star_{\beta}{\mathcal{G}}_0^2)\star_{\gamma}{\mathcal{G}}(x)\simeq {{A}}\star_{{\alpha}}{\mathcal{G}}$. Therefore, we will investigate some properties of the extension $A\subset {{A}}\star_{{\alpha}}{\mathcal{G}}$ using the intermediate extensions and the results known for partial group actions.
Separability
------------
In this subsection we will study the separability property to the ring extension $A\subset A\star_{{\alpha}}{\mathcal{G}}$. We recall that a unital ring extension $R\subset S$ is called [*separable*]{} if the multiplication map $m:S\otimes_R S\to S$ is a splitting epimorphism of $S$-bimodules. This is equivalent to saying that there exists an element $x\in S\otimes_RS$ such that $sx=xs$, for all $s\in S$, and $m(x)=1_S$. Such a element $x$ is usually called [*an idempotent of separability*]{} of $S$ over $R$.
Throughout this subsection, we will assume that ${\mathcal{G}}$ is [*finite*]{}. As in [@BPi], consider the maps $t_{y,z}:A\to A$ and $t_{z}:A\to A$ given by $$t_{y,z}(a)=\sum_{g\in {\mathcal{G}}(y,z)} {\alpha}_g(a1_{g{{}^{-1}}}), \quad t_z(a)=\sum_{y\in {\mathcal{G}}_0} t_{y,z}(a),\,\,\quad y,z\in {\mathcal{G}}_0,\,\,a\in A.$$ Particularly, if ${\mathcal{G}}$ is a group then ${\mathcal{G}}_0=\{x\}$ and $t_x:A\to A$ is the trace map for partial group actions as defined in Section 2 of [@DFP].
Now, we recall Theorem 4.1 of [@BPi] which will be useful for our purposes.
\[teo:BPin\] The ring extension $A\subset A\star_{{\alpha}}{\mathcal{G}}$ is separable if and only if there is an element $a$ in the center $C(A)$ of $A$ such that $t_{z}(a)=1_z$, for all $z\in {\mathcal{G}}_0$.
In order to apply Theorem \[teo-decomp\] to determine when the extension $A\subset A\star_{{\alpha}}{\mathcal{G}}$ is separable, we consider the separability problem for the extensions $A\subset A\star_{\beta}{\mathcal{G}}_0^2$ and $A\star_{\beta}{\mathcal{G}}_0^2\subset (A\star_{\beta}{\mathcal{G}}_0^2)\star_{\gamma}{\mathcal{G}}(x)$.
\[lem:sep1\] The extension $A\subset A\star_\beta{\mathcal{G}}_0^2$ is separable if and only if there exists $a\in C(A)$ such that $\sum_{z\in{\mathcal{G}}_0}{\alpha}_{{\tau}{{}^{-1}}_{z}}(a1_z)=1_x$.
Let $z\in {\mathcal{G}}_0$ and $a\in A$. Then $$\begin{aligned}
t_z(a)&=\sum_{y\in {\mathcal{G}}_0}t_{y,z}(a)=\sum_{y\in {\mathcal{G}}_0}\sum_{u\in {\mathcal{G}}_0^2(y,z)}\beta_u(a1_{u^{-1}})\\
&=\sum_{y\in {\mathcal{G}}_0}\beta_{(y,z)}(a1_{y})=\sum_{y\in {\mathcal{G}}_0}\alpha_{\tau_z}\alpha_{\tau{{}^{-1}}_y}(a1_{y})\\
&=\alpha_{\tau_z}(\sum_{y\in {\mathcal{G}}_0}\alpha_{\tau{{}^{-1}}_y}(a1_{y})).\end{aligned}$$ Consequently $t_z(a)=1_z$ if and only if $\sum_{y\in {\mathcal{G}}_0}\alpha_{\tau{{}^{-1}}_y}(a1_{y})=1_x$ and the result follows by Theorem \[teo:BPin\].
\[lem:centro\] $C(A\star_\beta{\mathcal{G}}_0^2)=\left\{\sum_{z\in{\mathcal{G}}_0} \alpha_{\tau_z}(a_x)\delta_{(z,z)}\,:\,a_x\in C(A_x)\right\}.$
Let $a_x\in C(A_x)$ and $\Lambda=\sum_{z\in{\mathcal{G}}_0} \alpha_{\tau_z}(a_x)\delta_{(z,z)}$. Given $(y,w)\in {\mathcal{G}}_0^2$ and $a_w\in A_w$ we have that $$\begin{aligned}
a_w\delta_{(y,w)}\cdot \Lambda=a_w\delta_{(y,w)}\cdot\alpha_{\tau_y}(a_x)\delta_{(y,y)}=a_w\alpha_{\tau_w}(a_x)\delta_{(y,w)}
\end{aligned}$$ and $$\begin{aligned}
\Lambda\cdot a_w\delta_{(y,w)}=\alpha_{\tau_w}(a_x)\delta_{(w,w)}\cdot a_w\delta_{(y,w)}=\alpha_{\tau_w}(a_x)a_w\delta_{(y,w)}.
\end{aligned}$$ Since $a_x\in C(A_x)$ and $\alpha_{\tau_w}$ is an isomorphism it is clear that $\alpha_{\tau_w}(a_x)\in C(A_w)$. Thus, $\Lambda\in C(A\star_\beta{\mathcal{G}}_0^2)$. Conversely, consider $\Lambda=\sum_{y,z\in{\mathcal{G}}_0} a_{(y,z)}\delta_{(y,z)}\in C(A\star_\beta{\mathcal{G}}_0^2)$, with $a_{(y,z)}\in A_{(y,z)}=A_z$, for all $y\in {\mathcal{G}}_0$. From $\Lambda\cdot 1_w\delta_{(w,w)}= 1_w\delta_{(w,w)}\cdot \Lambda$, it follows that $$\sum_{z\in{\mathcal{G}}_0}a_{(w,z)}\delta_{(w,z)}=\sum_{y\in{\mathcal{G}}_0}a_{(y,w)}\delta_{(y,w)},\,\, \text{ for all } w\in {\mathcal{G}}_0.$$ Hence $a_{(y,z)}=0$ if $y\neq z$ and whence $\Lambda=\sum_{z\in{\mathcal{G}}_0} a_{(z,z)}\delta_{(z,z)}$. Moreover, for all $y,w\in {\mathcal{G}}_0$ and $a\in A_w$, $$\Lambda\cdot a\delta_{(y,w)}=a_{(w,w)}a\delta_{(y,w)}\,\,\,\text{ and }\,\,\, a\delta_{(y,w)}\cdot \Lambda=a\alpha_{\tau_w}(\alpha_{\tau{{}^{-1}}_y}(a_{(y,y)}))\delta_{(y,w)}.$$ Therefore $$\begin{aligned}
\label{equ:comu}
a_{(w,w)}a=a\alpha_{\tau_w}(\alpha_{\tau{{}^{-1}}_y}(a_{(y,y)})), \,\,\,\text{ for all }\,\, y,w\in {\mathcal{G}}_0,\,\,a\in A_w.
\end{aligned}$$ When $a=1_w$ we obtain that $a_{(w,w)}=\alpha_{\tau_w}(\alpha_{\tau{{}^{-1}}_y}(a_{(y,y)}))$, for all $y,w\in {\mathcal{G}}_0$. Particularly, $a_{(w,w)}=\alpha_{\tau_w}(a_{(x,x)})$ for all $w\in {\mathcal{G}}_0$. Given $b\in A_x$, consider $a:=\alpha_{\tau_w}(b)\in A_w$. By , $\alpha_{\tau_w}(a_{(x,x)}b)=\alpha_{\tau_w}(ba_{(x,x)})$. Thus $a_{(x,x)}b=ba_{(x,x)}$ and consequently $a_{(x,x)}\in C(A_x)$.
\[lem:sep2\] The extension $A\star_\beta{\mathcal{G}}_0^2\subset (A\star_\beta{\mathcal{G}}_0^2)\star_{\gamma} {\mathcal{G}}(x)$ is separable if and only if the extension $A_x\subset A_x\star_{\alpha_{(x)}} {\mathcal{G}}(x)$ is separable.
Let $\Lambda\in C(A\star_\beta{\mathcal{G}}_0^2)$. By Lemma \[lem:centro\], $\Lambda=\sum_{z\in {\mathcal{G}}_0}\alpha_{\tau_z}(a_x)\delta_{(z,z)}$, with $a_x\in C(A_x)$. Notice that $$\begin{aligned}
t_x(\Lambda)&=\sum_{h\in {\mathcal{G}}(x)}\gamma_h(\sum_{z\in {\mathcal{G}}_0}\alpha_{\tau_z}(a_x)\delta_{(z,z)}\cdot \sum_{w\in {\mathcal{G}}_0}\alpha_{\tau_w}(1_{h{{}^{-1}}})\delta_{(w,w)} )\\
&=\sum_{h\in {\mathcal{G}}(x)}\sum_{z\in {\mathcal{G}}_0}\gamma_h(\alpha_{\tau_z}(a_x)\delta_{(z,z)}\cdot \alpha_{\tau_z}(1_{h{{}^{-1}}})\delta_{(z,z)} )\\
&=\sum_{h\in {\mathcal{G}}(x)}\sum_{z\in {\mathcal{G}}_0}\gamma_h(\alpha_{\tau_z}(a_x1_{h{{}^{-1}}})\delta_{(z,z)} )\\
&=\sum_{h\in {\mathcal{G}}(x)}\sum_{z\in {\mathcal{G}}_0}\alpha_{\tau_z}(\alpha_h(a_x1_{h{{}^{-1}}}))\delta_{(z,z)}\\
&=\sum_{z\in {\mathcal{G}}_0}\alpha_{\tau_z}(\sum_{h\in {\mathcal{G}}(x)}\alpha_h(a_x1_{h{{}^{-1}}}))\delta_{(z,z)}\\
&=\sum_{z\in {\mathcal{G}}_0}\alpha_{\tau_z}(t_x(a_x))\delta_{(z,z)}.\\\end{aligned}$$ Hence, there is $\Lambda\in C(A\star_\beta{\mathcal{G}}_0^2)$ such that $t_x(\Lambda)=1=\sum_{z\in {\mathcal{G}}_0} 1_z\delta_{(z,z)}$ if and only if there is $a_x\in C(A_x)$ such that $t_x(a_x)=1_x$. it Then the result follows from Theorem \[teo:BPin\] and Theorem 3.1 of [@BLP].
\[teo:sep\] The following statements hold:
1. if $A_x\subset A_x\star_{{\alpha}_{(x)}}{\mathcal{G}}(x)$ is a separable extension and there exists $a\in C(A)$ such that $\sum_{z\in{\mathcal{G}}_0}{\alpha}_{{\tau}{{}^{-1}}_{z}}(a1_z)=1_x$ then $A\subset A\star_{{\alpha}}{\mathcal{G}}$ is a separable extension;
2. if $A\subset A\star_{{\alpha}}{\mathcal{G}}$ is a separable extension then $A_x\subset A_x\star_{{\alpha}_{(x)}}{\mathcal{G}}(x)$ also is.
\(i) By Lemmas \[lem:sep1\] and \[lem:sep2\], $A\subset A\star_\beta{\mathcal{G}}_0^2$ and $A\star_\beta{\mathcal{G}}_0^2\subset (A\star_\beta{\mathcal{G}}_0^2)\star_{\gamma} {\mathcal{G}}(x)$ are separable extensions. By the transitivity of the separability property (see Proposition 2.5 of [@HS]), $A\subset (A\star_\beta{\mathcal{G}}_0^2)\star_{\gamma} {\mathcal{G}}(x)$ is separable. Thus the result follows from Theorem \[teo-decomp\].
\(ii) From Theorem \[teo-decomp\] and Proposition 2.5 of [@HS] it follows that $A\star_\beta{\mathcal{G}}_0^2\subset (A\star_\beta{\mathcal{G}}_0^2)\star_{\gamma} {\mathcal{G}}(x)$ is a separable extension. Hence, Lemma \[lem:sep2\] implies that $A_x\subset A_x\star_{{\alpha}_{(x)}}{\mathcal{G}}(x)$ is separable.
\[cor-1\]Suppose that $|{\mathcal{G}}_0|1_A$ is invertible in $A$ and $(|{\mathcal{G}}_0|1_A)^{-1}=n1_A$, with $n\in \mathbb{N}$. If $A_x\subset A_x\star_{{\alpha}_{(x)}}{\mathcal{G}}(x)$ is separable then $A\subset A\star_{{\alpha}}{\mathcal{G}}$ is separable.
Let $a=n1_A\in C(A)$. Then $$\begin{aligned}
\sum_{z\in{\mathcal{G}}_0}{\alpha}_{{\tau}{{}^{-1}}_{z}}(a1_z)&=\sum_{z\in{\mathcal{G}}_0}{\alpha}_{{\tau}{{}^{-1}}_{z}}(n1_z)=n\sum_{z\in{\mathcal{G}}_0}{\alpha}_{{\tau}{{}^{-1}}_{z}}(1_z)\\
&=n|{\mathcal{G}}_0|1_x=1_x,
\end{aligned}$$ and whence the result follows by Theorem \[teo:sep\] (1).
[Let ${\mathcal{G}}$, $A$, $\alpha$ and ${\tau}(x)$ be as in Example \[57\]. Consider $a_x=\frac{1}{2}e_1+e_2\in A_x$. Observe that $t_x(a_x)={\alpha}_x(a_x)+{\alpha}_g(a_x1_{g{{}^{-1}}})=a_x+\frac{1}{2}e_1=e_1+e_2=1_x$. It follows from Theorem 3.1 of [@BLP] that $A_x\subset A_x\star_{{\alpha}_{(x)}} {\mathcal{G}}(x)$ is a separable extension. Moreover, if $a=\frac{1}{2}(e_1+e_2+e_3+e_4)\in A$ then $$\alpha_{\tau{{}^{-1}}_x}(a1_x)+\alpha_{\tau{{}^{-1}}_y}(a1_y)=\alpha_x(a1_x)+\alpha_{l{{}^{-1}}}(a1_l)=e_1+e_2=1_x.$$ By Theorem \[teo:sep\] (1), the extension $A\subset A\star_{\alpha}{\mathcal{G}}$ is separable. ]{}
Semisimple extension
--------------------
In this subsection we investigate the semisimplicity for the ring extension $A\subset A\star_{{\alpha}}{\mathcal{G}}$. Recall from [@HS] that a ring extension $R\subset S$ is called left (right) [*semisimple*]{} if any left (right) $S$-submodule $N$ of a left (right) $S$-module $M$ having an $R$-complement in $M$, has an $S$-complement in $M$.
For the convenience of the reader, we recall Proposition 2.6 of [@HS].
\[sep-ss\] If $R\subset S$ is a separable ring extension then $R\subset S$ is a left (right) semisimple ring extension.
\[teo-ss\] If ${\mathcal{G}}$ is [*finite*]{} and
1. there exists $a_x\in C(A_x)$ such that $t_x(a_x)=1_x$ and
2. there exists $a\in C(A)$ such that $\sum_{z\in{\mathcal{G}}_0}{\alpha}_{{\tau}{{}^{-1}}_{z}}(a1_z)=1_x$,
then $A\subset A\star_{{\alpha}}{\mathcal{G}}$ is a left (right) semisimple extension.
Using (i), we obtain from Theorem 3.1 of [@BLP] that $A_x\subset A_x\star_{{\alpha}_{(x)}}{\mathcal{G}}(x)$ is separable. Then, by Theorem \[teo:sep\] (i) the extension $A\subset A\star_{{\alpha}}{\mathcal{G}}$ is separable and so the result follows by Proposition \[sep-ss\].
Frobenius extension
-------------------
In this subsection we will prove that $A\subset A\star_{{\alpha}}{\mathcal{G}}$ is a Frobenius extension. We recall that a ring extension $R\subset S$ is called [*Frobenius*]{} if there exist an element $\Delta=\sum_{i=1}^{n}s_{i1}\otimes s_{i2}\in S\otimes_R S$ and an $R$-bimodule map $\varepsilon:S\to R$ such that $\Delta s=s\Delta$, for all $s\in S$, and $\sum_{i=1}^{n}\varepsilon(s_{i1})s_{i2}=\sum_{i=1}^{n}s_{i1}\varepsilon(s_{i2})=1$. More details on Frobenius extensions can be seen, for example, in [@CIM] or [@K].
Firstly, we note that the natural inclusion $A\subset A\star_{\beta} {\mathcal{G}}_0^2 $, given by $a\mapsto \sum_{z\in{\mathcal{G}}_0}(a1_z)\delta_{(z,z)}$, induces the following $({{A}},{{A}})$-bimodule structure on $A\star_{\beta} {\mathcal{G}}_0^2$: $$\begin{aligned}
\label{aabim}
&a\cdot (a_z\delta_{(y,z)})=aa_z\delta_{(y,z)},\quad\quad
(a_z\delta_{(y,z)})\cdot a=a_z{\beta}_{(y,z)}(a1_y)\delta_{(y,z)},&\end{aligned}$$ for all $(y,z)\in{\mathcal{G}}_0^2$, $a_z\in A_z$ and $a\in {{A}}$.
\[teo-frobenius\] If ${\mathcal{G}}$ is finite then $A\subset A\star_{{\alpha}}{\mathcal{G}}$ is a Frobenius extension.
Let $\Delta=\sum_{z\in {\mathcal{G}}_0}1_z\delta_{(z,z)}\otimes 1_z\delta_{(z,z)}\in (A\star_{\beta}{\mathcal{G}}_0^2)\otimes_{A} (A\star_{\beta}{\mathcal{G}}_0^2)$ and $\varepsilon:A\star_{\beta}{\mathcal{G}}_0^2\to A$ given by $$\varepsilon(a_z\delta_{(y,z)})=\left\{\begin{array}{lc}
a_z, & \text{ if } y=z\\
0,& \text{ otherwise.}
\end{array}
\right.$$ It is straightforward to check that $\varepsilon$ is an $(A,A)$-bimodule map. Also, note that $$(a_w\delta_{(y,w)}) \Delta=a_w\delta_{(y,w)}=\Delta(a_w\delta_{(y,w)}),$$ for all $(y,w)\in {\mathcal{G}}_0^2$ and $a_w\in A_w$. Since $$\sum_{z\in {\mathcal{G}}_0}\varepsilon(1_z\delta_{(z,z)})(1_z\delta_{(z,z)})=\sum_{z\in {\mathcal{G}}_0}1_z\delta_{(z,z)}\varepsilon(1_z\delta_{(z,z)})
=\sum_{z\in {\mathcal{G}}_0}1_z\delta_{(z,z)}=1_{A\star_{\beta}{\mathcal{G}}_0^2},$$ it follows that $A\subset A\star_{\beta}{\mathcal{G}}_0^2$ is a Frobenius extension. Notice that ${\mathcal{G}}(x)$ is finite because ${\mathcal{G}}$ is finite. Then, by Theorem 3.6 of [@BLP], $A\star_{\beta}{\mathcal{G}}_0^2\subset (A\star_{\beta}{\mathcal{G}}_0^2)\star_{\gamma}{\mathcal{G}}(x)$ is a Frobenius extension. As Frobenius extension is a transitive notion (see e. g. [@K pg. 6]), we obtain from Theorem \[teo-decomp\] that $A\subset A\star_{{\alpha}}{\mathcal{G}}$ is Frobenius.
Artinianity
-----------
The artinian property for partial skew groupoids rings was studied in [@NOP]. In our context, using Theorem \[teo-decomp\], we obtain the following refinement of Theorem 1.3 of [@NOP].
The partial skew groupoid ring $A\star_{{\alpha}}{\mathcal{G}}$ is artinian if and only if $A$ is artinian and $A_h=\{0\}$, for all but finitely many $h\in {\mathcal{G}}(x)$.
Assume that $A\star_{{\alpha}}{\mathcal{G}}$ is artinian. By Theorem 1.3 of [@NOP], $A$ is artinian and $A_g=\{0\}$ for all but finitely many $g\in {\mathcal{G}}$. Particularly, $A_h=\{0\}$ for all but finitely many $h\in {\mathcal{G}}(x)$.
For the converse, consider $h\in {\mathcal{G}}(x)$. By $\eqref{Czh}$ and , $C_h\neq \{0\}$ if and only if there is $z\in {\mathcal{G}}_0$ such that ${\alpha}_{\tau_z}(A_h)\neq \{0\}$. Consequently, $C_h=\{0\}$, for all but finitely many $h\in {\mathcal{G}}(x)$. Also, since ${\mathcal{G}}_0^2$ is finite and ${{A}}$ is artinian it follows from Theorem 1.3 of [@NOP] that ${{A}}\star_{\beta}{\mathcal{G}}_0^2$ is artinian. Using again the Theorem 1.3 of [@NOP], we conclude that $({{A}}\star_{\beta}{\mathcal{G}}_0^2)\star_{\gamma}{\mathcal{G}}(x)$ is artinian and the result follows by Theorem \[teo-decomp\].
[99]{}
D. Bagio, D. Flôres and A. Paques, [*Partial actions of ordered groupoids on rings.*]{} J. Algebra Appl. **9**, 501–517 (2010).
D. Bagio, J. R. Lazzarin and A. Paques, [*Crossed products by twisted partial actions: separability, semisimplicity and Frobenius properties.*]{} Comm. Alg. **38**, 496–508 (2010).
D. Bagio and A. Paques, Partial Groupoid Actions: Globalization, Morita theory, and Galois theory. *Comm. Algebra* **40**, 3658-3678 (2012).
D. Bagio and H. Pinedo [*On the separability of the partial skew groupoid ring.*]{} São Paulo J. Math. Sci. **11**(2), 370-384 (2017).
S. Caenepeel, B. Ion and G. Militaru [*The structure of Frobenius algebras and separable algebras.*]{} K-Theory. **19**, 365-402 (2000).
M. Dokuchaev, M. Ferrero and A. Paques, Partial actions and Galois theory. [*J. Pure Appl. Algebra*]{} **208**, 77-87 (2007).
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[^1]: [**Mathematics Subject Classification**]{}: Primary 20L05, 16W22, 16S99. Secondary 18B40, 20N02.
[^2]: [**Key words and phrases:**]{} Groupoid, partial groupoid action, partial skew groupoid ring, separability, semisimplicity, Frobenius property, artinianity.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'A successful state transfer (or teleportation) experiment must perform better than the benchmark set by the ‘best’ measure and prepare procedure. We consider the benchmark problem for the following families of states: (i) displaced thermal equilibrium states of given temperature; (ii) independent identically prepared qubits with completely unknown state. For the first family we show that the optimal procedure is heterodyne measurement followed by the preparation of a coherent state. This procedure was known to be optimal for coherent states and for squeezed states with the ‘overlap fidelity’ as figure of merit. Here we prove its optimality with respect to the trace norm distance and supremum risk. For the second problem we consider $n$ i.i.d. spin-$\frac12$ systems in an arbitrary unknown state $\rho$ and look for the measurement-preparation pair $(M_{n},P_{n})$ for which the reconstructed state $\omega_{n}:=P_{n}\circ M_{n} (\rho^{\otimes n})$ is as close as possible to the input state, i.e. $\|\omega_{n}- \rho^{\otimes n}\|_{1}$ is small. The figure of merit is based on the trace norm distance between input and output states. We show that asymptotically with $n$ the this problem is equivalent to the first one. The proof and construction of $(M_{n},P_{n})$ uses the theory of [*local asymptotic normality*]{} developed for state estimation which shows that i.i.d. quantum models can be approximated in a strong sense by quantum Gaussian models. The measurement part is identical with ‘optimal estimation’, showing that ‘benchmarking’ and estimation are closely related problems in the asymptotic set-up.'
author:
- Mădălin Guţă
- Peter Bowles
- Gerardo Adesso
date: 'September 17, 2010'
title: |
Quantum teleportation benchmarks for independent and identically-distributed\
spin states and displaced thermal states
---
[^1]
Introduction
============
Quantum teleportation [@Telep] and quantum state storage [@Storage] are by now well-established protocols in quantum information science. In both cases the procedure amounts to mapping one quantum state onto another (at a remote location in the case of teleportation), by making use of quantum correlations in the form of entanglement or interaction between systems. However, approximate transformations could also be accomplished without any use of quantum correlations, by means of classical ‘measure and prepare’ (MAP) schemes. Whilst in the ideal case, the entanglement resource gives quantum teleportation a clear advantage in terms of performance, there exists inevitable degradation of the quantum channel in realistic implementations. This has led to a number of investigations into the existence of optimal MAP schemes to locate classical-quantum boundaries and assign precise benchmarks for proving the presence of quantum effects [@bfkjmo]. Any experimental demonstration of quantum teleportation and state storage has to perform better than the optimal MAP scheme, to be certified as a truly quantum demonstration. A review of the quantum benchmarks for completely unknown pure input states of $d$-dimensional systems can be found in [@BrussMac]. More recent research has largely focused on benchmarks originating in the context of teleportation and quantum memory for continuous variable (CV) systems [@COVAQIAL], with notable results obtained for transmission of pure and mixed coherent input states, and squeezed states [@Hammerer; @AdessoC; @Namiki; @Owari; @Calsamiglia]. Beautiful experiments [@Furusawa98; @memorypolzik; @telepolzik; @polzikalessio] involving light (Gaussian modes) and matter (coherent and spin-squeezed atomic ensembles) have demonstrated unambiguous quantum teleportation, storage and retrieval of these infinite-dimensional quantum states with a measured ‘fidelity’ between input and output exceeding the benchmark set by the optimal MAP strategy (see also [@noteluk; @luk]).
In each of the above cases, the benchmarks deal with the case of teleportation or storage of [*single*]{} input states drawn from a set, in a Bayesian or pointwise set-up. To date, there exist no nontrivial benchmarks for the transmission of multiple copies of quantum states – a ‘quantum register’ – in particular for an ensemble of $n$ independent and identically-distributed (i.i.d.) qubits. Such a task comes as a primitive in distributed quantum communication. Quantum registers can be locally initialised and then transferred to remote processing units where a quantum computation is going to take place. Also, in hybrid interfaces between light and matter [@qinternet], storage and retrieval of e.g. coherent states, involves mapping the state of $n$ i.i.d. atoms onto a light mode (back and forth). Therefore, strictly speaking, a quantum benchmark for this precise input ensemble would be needed to assess the success of the experiment. In the current practice [@memorypolzik; @telepolzik] the problem is circumvented by noting that the collective spin components of the atomic ensemble (with $n \sim 10^{12}$ [@PolzikNAT]) approximately satisfy canonical commutation relations, henceforth the atomic system is treated [*a priori*]{} as a CV system, and the corresponding benchmarks are used.
In this paper we put this procedure on firm grounds, by proving rigorously that the optimal MAP scheme for teleportation and storage of $n$ i.i.d. unknown [*mixed*]{} qubits converges when $n \rightarrow \infty$ to the optimal MAP scheme for a single-mode displaced thermal state. Additionally, we also prove that the heterodyne measurement followed by the preparation of a coherent state is optimal MAP scheme for displaced thermal states when the figure of merit is the trace norm distance. The same scheme is known to be optimal for thermal and squeezed states, but for a figure of merit based on overlap fidelity [@Owari].
The key tool in deriving our results is the theory of local asymptotic normality (LAN) for quantum states which is the quantum extension of a fundamental concept in mathematical statistics introduced by Le Cam [@LeCam]. In the classical context this roughly means that a large i.i.d. sample $X_{1},\dots , X_{n}$ from an unknown distribution contains approximately the same amount of [*statistical*]{} information as a single sample from a Gaussian (normal) distribution with unknown mean and known variance.
In the quantum case, LAN means that the joint (mixed) state $\rho_{\theta}^{\otimes n}$ of $n$ identically prepared (finite dimensional) quantum systems can be transferred by means of a quantum channel to a quantum-classical Gaussian state, with asymptotically vanishing loss of statistical information. More precisely, for any fixed $\theta_{0}$ there exist quantum channels $T_{n}, S_{n}$ such that $$\begin{aligned}
& & \quad \left\|T_{n} \left(\rho_{\theta_{0}+u/\sqrt{n}}^{\otimes n}\right) -
N_{{u}}\otimes \Phi_{{u}} \right \|_{1} \nonumber \\
&\mbox{and}& \quad
\left\|\rho_{\theta_{0}+u/\sqrt{n}}^{\otimes n} - S_{n} \left(
N_{{u}}\otimes \Phi_{{u}}\right) \right\|_{1} \nonumber\end{aligned}$$ converge to zero as $n\to\infty$, [*uniformly*]{} over a $n^{-1/2+\epsilon}$ local neighbourhood of the state $\rho_{\theta_{0}}$. Here $N_{{u}}$ is a classical normal distribution and $\Phi_{{u}}$ is a Gaussian state on an ensemble of oscillators whose means are linear transformations of ${u}$ and the covariance matrices depend only on $\theta_{0}$. The qubit case is described in detail in Section \[sec.lan\] and the precise result is formulated in Theorem \[th.qlan.qubits\]. The LAN theory has been used to find asymptotically optimal estimation procedures for qubits and qudits and to show that the Holevo bound for state estimation is achievable . Here we use it to solve the benchmark problem for qubits by casting it into the corresponding one for displaced thermal states. The following diagram illustrates the asymptotically optimal MAP scheme: the measurement $M_{n}$ consists in composing the channel $T_{n}$ with the heterodyne measurement $H$. The preparation procedure consists in creating the coherent state $|\alpha_{\hat{\vec{u}}}\rangle$ and mapping it back to the qubits space by the channel $S_{n}$. The optimality of the scheme is proved in Theorem \[th.main\]. $$\label{comm.diagram}
\begin{CD}
\rho^{n}_{\vec{u}} @>M_n>> X_n @>P_n>>\omega(X_{n}) \\
@V{T_n}VV @. @AA {S_n} A \\
\Phi_{\vec{u}} \otimes N_{\vec{u}}
@> H >> \hat{\vec{u}} @>P>> |\alpha_{\hat{\vec{u}}} \rangle\langle\alpha_{\hat{\vec{u}}} |\otimes \delta_{\hat{u}_{3}} \\
\end{CD}$$
The paper is organised as follows. In Section \[sec.stat.formulation\] we give a precise statistical formulation of the benchmark problem, and we explain in some detail the definition of the asymptotic risk. A brief overview of the necessary classical and quantum concepts from the LAN theory is given in Section \[sec.lan\]. In Section \[sec.thermal\] we then revisit the benchmark problem for displaced thermal states. When the figure of merit is the overlap fidelity, the solution was found in [@Owari] to be the heterodyne detection followed by the repreparation of a coherent state. We solve the same problem using the trace norm loss function, and again find this MAP scheme to be optimal. Interestingly, the optimality proof is based on the concept of stochastic ordering which was previously used for finding optimal cloning maps for thermal states . This result allows us to exploit LAN and solve the benchmark problem for $n$ i.i.d. qubits in the asymptotic limit $n\rightarrow\infty$. A constructive solution along with a proof of optimality is given in Section \[sec.qubits\]. We conclude with discussions in Section \[sec.discussion\]. The Appendix contains additional mathematical details on the LAN theory for qubits.
Statistical formulation of the benchmark problem {#sec.stat.formulation}
================================================
The general statistical formulation of quantum benchmarking is as follows. Let $$\mathcal{Q}:= \{\rho_\theta: \theta\in\Theta\}$$ be a quantum model, i.e. a family of quantum states on a Hilbert space $\mathcal{H}$, indexed by a parameter $\theta\in \Theta$. In this paper $\Theta$ is always an open subset of $\mathbb{R}^{k}$, i.e. we are in a parametric set-up. The model encodes our prior information about the state and the parameter $\theta$ is considered to be unknown. We are given a quantum system prepared in the state $\rho_{\theta}$ and we would like to find the MAP (or classical) channel $T :\mathcal{T}_{1}(\mathcal{H})\to \mathcal{T}_{1}(\mathcal{H})$ for which $T(\rho_{\theta})$ is ‘close’ to $\rho_\theta$. A MAP channel is the composition $T= P\circ M$ of a measurement $M$ with outcomes in a measure space $(\mathcal{X},\Sigma)$, and a repreparation $P$ which assigns to every result $x\in\mathcal{X}$ a state $\rho_x \in \mathcal{T}_1(\mathcal{H})$. There are several natural distance functions on the state space, such as the Bures distance [@Bures] with its associated Uhlmann fidelity [@Uhlmann], or the trace-norm distance considered in this paper. Since $\theta$ is unknown we choose the maximum risk as the overall figure of merit of a scheme $L:=(M,P)$ $$\label{eq.max.risk}
R_{\max}(L):= \sup_{\theta\in\Theta} \|T(\rho_{\theta}) -\rho_{\theta}\|_{1}, \qquad T=P\circ M.$$ Alternatively one may use a Bayesian risk where the performance at different parameters is weighted by a prior distribution over $\Theta$. The goal is to find a MAP scheme with the lowest risk $R_{\max}(L)$, which will be called a [*minmax scheme*]{}.
The above formulation is particularly suitable in the case of covariant models such as the displaced thermal state treated in Section \[sec.thermal\] or the single qubit case of [@Calsamiglia]. Let us consider now the scenario where we want to teleport $n$ identically prepared systems, such as in the case of atomic clouds. The relevant model is then $$\mathcal{Q}^{n}:= \{\rho^{\otimes n}_{\theta} :\theta\in\Theta\},$$ and we would like to find the optimal MAP scheme $L_{n}:= (M_{n},P_{n})$ for a given $n$. The experience accumulated in the related domain of state estimation indicates that this problem can rarely be solved explicitly , but becomes tractable in an asymptotic framework . We adopt this set-up in our benchmark problem and define the local risk of a procedure $ L_{n}:= (M_{n},P_{n})$, around $\theta_{0}\in \Theta$ by $$R_{\max}(L_{n}; \theta_{0}):=
\sup_{ \| \rho_{\theta} - \rho_{\theta_{0}} \|_{1}\leq n^{-1/2+\epsilon}}
\|T_{n}(\rho_{\theta}^{\otimes n}) -\rho_{\theta}^{\otimes n}\|_{1}.$$ where unlike formula we take the maximum over a neighbourhood of $\theta_{0}$ of size $n^{-1/2+\epsilon}$ with $0<\epsilon \ll 1$ arbitrary.
The [*asymptotic local risk*]{} of the sequence $\{L_{n}:= (M_{n},P_{n}):n\in \mathbb{N}\}$ is defined by $$\label{eq.local.asymptotic.risk}
R(\{L_{n} : n\in N\}; \theta_{0}):= \limsup_{n\to\infty} R_{\max}(L_{n};\theta_{0}),$$ and by definition a MAP sequence $\{ L_{n} : n\in \mathbb{N}\}$ is optimal if it achieves the lowest possible asymptotic local risk at any point $\theta_{0}\in \Theta$. The latter is called the [*asymptotic minmax risk*]{} $$R_{\min\!\max}(\theta_{0})= \limsup_{n\to\infty} \inf_{L_{n}}
R_{\max}(L_{n};\theta_{0}).$$
Since the above formula may look rather ad-hoc to the reader who is not familiar with statistical methodology, we would like to explain its meaning in some detail. First of all, note that we compare the input and output states globally rather than locally on each system. This means that even though as $n\to \infty$ we get more and more information about the parameter $\theta$, and we can estimate it with accuracy $O(n^{-1/2})$, in the same time the task of repreparing the state $\rho_{\theta}^{\otimes n}$ becomes more and more difficult! This can be easily understood by looking at fidelity in the case of pure states. Let $\hat\theta_{n}$ be an estimator of $\theta$ obtained by measuring $\psi_{\theta}^{\otimes n}$, so that $\hat\theta_{n}-\theta =O(n^{-1/2})$, and suppose that we reprepare the state $\psi_{\hat{\theta}_{n}}$. Then
$$\left|\left\langle \psi_{\theta}^{\otimes n}| \psi_{\hat{\theta}_{n}}^{\otimes n}
\right \rangle\right|^{2} =
(\cos \alpha_{n})^{2n} = 1-\alpha_{n}^{2}/2+ o(n^{-1})$$ where $\alpha_{n}$ is an angle of order $n^{-1/2}$. Since $$\left(1- \frac{c}{n} +o(n^{-1})\right)^{2n} \longrightarrow \exp(-2c),$$ we see that the input-output fidelity cannot converge to $1$. We will show that this is the case for arbitrary states, and also when we allow for other MAP schemes.
The second remark concerns the supremum over the small ball $\|\rho_{\theta}-\rho_{\theta}\|_{1}\leq n^{-1/2+\epsilon}$ in definition . Why not consider the supremum over all $\theta$ as we did in the non-asymptotic case ? The reason is that the global supremum would be overly pessimistic and would be dominated by the region in the parameter space which is hardest for the benchmark problem. Restricting to a ball whose size is roughly that of the uncertainty in the parameter captures the local behaviour of MAP scheme at each point and is more informative than the global maximum. The ball should have size $n^{-1/2+\epsilon}$ because even if $\theta$ is unknown beforehand, it can be localised within such a region by measuring a small proportion $n^{1-\epsilon}\ll n$ of the systems, so that effectively [*we know*]{} that we are in the local ball, and this should be reflected in the definition of the risk. The localisation argument is standard in statistics and its application in quantum statistics is detailed in .
Finally, the relation between our figure of merit and the Bayes risk can be sketched as follows. If $R_{\pi}$ denotes the asymptotically optimal Bayes risk for the prior $\pi$, $$R_{\pi} := \limsup_{n\to\infty} \inf_{L_{n}} \int \pi(d\theta)
\| T_{n}(\rho_{\theta}^{\otimes n}) -\rho_{\theta}^{\otimes n}\|_{1}$$ then under suitable conditions on $\pi$ and the model $\mathcal{Q}$ one obtains $$R_{\pi}= \int \pi(d\theta) R_{\min\!\max}(\theta).$$ The intuitive explanation is that when $n\to\infty$ the features of the prior $\pi$ are washed out and the posterior distribution concentrates in a local neighbourhood of the true parameter, where the behaviour of the MAP procedures is governed by the local minmax risk. A full proof of this relation is beyond the scope of this paper (and will be presented elsewhere) but the interested reader may consult for the proof of the corresponding statement in the case of state estimation.
Local asymptotic normality {#sec.lan}
==========================
In this Section we will give a brief, self-contained introduction to the theory of LAN in as much detail as it is necessary for this paper and we refer to for proofs and more analysis.
LAN is a fundamental concept in mathematical statistics introduced by the French statistician Le Cam [@LeCam]. It roughly means that a large i.i.d. sample $X_{1},\dots , X_{n}$ from an unknown distribution contains approximately the same amount of statistical information as a single sample from a Gaussian distribution with unknown mean and known variance. More precisely if $X_{i}$ has distribution $\mathbb{P}_{\theta}$ depending ‘smoothly’ on a finite dimensional parameter $\theta\in \Theta\subset\mathbb{R}^{k}$, then in a $n^{-1/2}$-size neighbourhood of any point $\theta_{0}$, the statistical model $$\mathcal{P}^{n}: =
\left\{
\mathbb{P}_{\theta_{0}+u/\sqrt{n}}^{n} :u\in \mathbb{R}^{k}
\right\}$$ is well approximated by a simpler Gaussian shift model $$\mathcal{N}:= \left\{ N\left(u, I^{-1}(\theta_{0})\right) :u\in \mathbb{R}^{k}\right\}$$ where $u\in\mathbb{R}^{k}$ is the local unknown parameter of the distribution, and $I^{-1}(\theta_{0})$ is the inverse Fisher information matrix at $\theta_{0}$. Note that this approximation holds only locally, reflecting the intrinsic uncertainty in the unknown parameter for the sample size $n$, which should not be seen as an additional assumption about $\theta$. Indeed, the parameter can always be localised in such a region using a preliminary estimator, with vanishing probability of failure. The point of this approximation is to reduce a statistical problems (e.g. estimation) about the more complex model $\mathcal{P}^{n}$ to a simpler problem about the Gaussian model $\mathcal{N}$.
Local asymptotic normality also provides a convenient description of quantum statistical models involving i.i.d. quantum systems. Here the idea is: when the quantum ‘sample’ is large, the model can be approximated by a simpler quantum Gaussian model. If this approximation holds in a sufficiently strong sense, then statistical problems about the qubits model can be reformulated in terms of the Gaussian one, without any loss of optimality.
In quantum statistics this technique has been used for optimal state estimation with completely unknown finite dimensional quantum states , for optimal classification (learning) of spin states , for state transfer between matter and light . In the physics literature, LAN is used in an informal way to describe the dynamics of atomic gases in a simplified Gaussian approximation, in quantum memories and quantum metrology with spin coherent and squeezed states.
LAN for qubit systems
---------------------
We are given $n$ independent identically prepared spin-$\frac12$ particles (qubits) in a state $$\rho_{\vec{r}} = \frac{1}{2}(\mathbf{1} + \vec{r}\vec{\sigma})$$ where $\vec{r}$ is the Bloch vector of the state and $\vec{\sigma} = (\sigma_{x}, \sigma_{y}, \sigma_{z})$ are the Pauli matrices in $M(\mathbb{C}^{2})$.
Although a priori the state $\rho_{\vec{r}}$ is completely unknown, the following argument shows that that without loss of generality we can assume it to be [*localised*]{} within a small ball of size $n^{-1/2+\epsilon}$ where $\epsilon>0$ is arbitrarily small. Indeed by measuring a small proportion $n^{1-\epsilon}\ll n$ of the systems we can devise an initial rough estimator $\rho_{0}:=\rho_{\vec{r}_{0}}$ so that with high probability the state is in a ball of size $n^{-1/2+\epsilon}$ around $\rho_{0}$ (see Lemma 2.1 in ). We label the states in this ball by the local parameter $\vec{u}$ $$\rho_{\vec{u}/\sqrt{n}} =
\frac{1}{2} \left(
\mathbf{1} + (\vec{r}_{0} + \vec{u}/\sqrt{n})\vec{\sigma}\right)$$ and define the local statistical model by $$\label{eq.q.n}
\mathcal{Q}_{n}:= \left\{\rho^{n}_{\vec{u}} : \| \vec{u}\|\leq n^{\epsilon} \right\} ,\qquad \rho^{n}_{\vec{u}} :=\rho^{\otimes n}_{\vec{u}/\sqrt{n}}.$$
By choosing the reference frame with its $z$ axis along $\vec{r}_{0}$ we find that up to $O(n^{-1})$ terms the state $\rho_{\vec{u}/\sqrt{n}}$ is obtained by perturbing the eigenvalues of $\rho_{0}$ and rotating it with a ‘small unitary’
$$\rho_{\vec{u}/\sqrt{n}}
=
U_{\vec{u}/\sqrt{n}}
\left(
\begin{array}{cc}
\mu_{0}+\frac{u_{z}}{2\sqrt{n}} & 0\\
0 & 1-\mu_{0}-\frac{u_{z}}{2\sqrt{n}}
\end{array}
\right)
U_{\vec{u}/\sqrt{n}}^{\dagger},$$ where $$U_{\vec{u}/\sqrt{n}}:=
\exp(i(- u_{y} \sigma_{1}+u_{x}\sigma_{2} )/2r_{0}\sqrt{n}),\qquad
r_{0}:=\|\vec{r}_{0}\|.$$
The big Bloch ball picture
--------------------------
The asymptotic behaviour of the multiple spins state can be intuitively explained through the ‘big Bloch sphere’ picture commonly used to describe spin coherent [@Radcliffe] and spin squeezed states . Let $$L_{a}:= \sum_{i=1}^{n} \sigma^{(i)}_{a} ,\qquad a=x,y,z$$ be the collective spin components along the reference frame directions. By the Central Limit Theorem, the distributions of $L_{a}$ with respect to $\rho_{0}^{\otimes n}$ converge as $$\begin{aligned}
\frac{1}{\sqrt{n}}(L_{z} - n r_{0}) &\overset{\mathcal{D}}{\longrightarrow}&
N(0, 1-r_{0}^{2}),\\
\frac{1}{\sqrt{n}}L_{x,y} & \overset{\mathcal{D}}{\longrightarrow} & N(0, 1),\end{aligned}$$ so that the joint spins state can be pictured as a vector of length $nr_{0}$ whose tip has a Gaussian blob of size $\sqrt{n}$ representing the uncertainty in the collective variables (see Figure \[fig.big.ball\]). Further more, by a law of large numbers argument we evaluate the commutators $$\begin{aligned}
&&\left[\frac{1}{\sqrt{n}} L_{x}, \frac{1}{\sqrt{n}} L_{y}\right]=
2i \frac{1}{n} L_{z} \approx
2i r_{0} \mathbf{1},\\
&&\left[\frac{1}{\sqrt{n}} L_{x,y}, \frac{1}{\sqrt{n}} L_{z}\right]
\approx 0.
%\left[\frac{1}{\sqrt{n}} L_{2}, \frac{1}{\sqrt{n}} L_{3}] &=&
%2iL_{1} \approx 0\end{aligned}$$ Thus the rescaled observables $ L_{x}/\sqrt{2r_{0}n}$ and $L_{y}/\sqrt{2r_{0}n}$ converge to the canonical coordinates $Q$ and $P$ of a quantum harmonic oscillator. Moreover, the variances correspond to that of a thermal equilibrium state $$\Phi:= (1-s) \sum_{k=0}^{\infty} s^{k} | k\rangle\langle k|,\qquad
s= \frac{1-r_{0}}{1+r_{0}},$$ where $\{|k\rangle :k\geq 0\}$ represents the Fock basis.
![(Color online) Big Bloch ball picture for $n$-qubit i.i.d. mixed states.[]{data-label="fig.big.ball"}](bigball.pdf){width="7cm"}
As for the third component, $(L_{z} - n r_{0})/\sqrt{n}$ converges to a [*classical*]{} Gaussian variable $X\sim N:= N(0,1-r_{0}^{2})$ which is independent of the quantum state.
How does the limit change when we perturb the state of the spins ? By the same argument we find that the variables $Q,P,X$ pick up expectations which (in the first order in $n^{-1/2}$) are proportional to the local parameters $(u_{x},u_{y},u_{z})$ while the variances remain unchanged. More precisely the oscillator is in a displaced thermal equilibrium state $
\Phi_{\vec{u}} := D(\vec{u}) \Phi D(\vec{u})^{\dagger} ,
$ where $D(\vec{u})$ is the displacement operator $$D(\vec{u}):=\exp \left( i (- u_{y} Q+u_{x}P )/\sqrt{2r_{0}} \right),$$ and the classical part has distribution $N_{\vec{u}}:=N(u_{z}, 1-r_{0}^{2})$.
We have thus identified the limit Gaussian model which is a tensor product of a classical distribution and a quantum state, which together can be seen as a state (positive normal functional) on the von Neumann algebra $\mathcal{B}(\ell^{2}(\mathbb{N})) \otimes L^{\infty}(\mathbb{R})$.
\[def.quantum.gaussian.shift\] The quantum Gaussian shift model $\mathcal{G}$ is defined by the family of quantum-classical states $$\label{eq.q.Gaussian.shift}
\mathcal{G}:= \{ \Phi_{\vec{u}}\otimes N_{\vec{u}} : \vec{u}\in\mathbb{R}^{3} \}$$ on $\mathcal{B}(\ell^{2}(\mathbb{N})) \otimes L^{\infty}(\mathbb{R})$.
In the next subsection we formulate a precise statement about the convergence to the Gaussian model which goes beyond the Central Limit type argument presented above.
Strong convergence to Gaussian shift model
------------------------------------------
The notion of strong convergence of classical statistical models was introduced by Le Cam and is based on defining a natural distance between statistical models with the same parameter space, so that models models at zero distance are statistically equivalent and models which are close, have similar behaviour for ‘regular’ statistical decision problems. The existing results on quantum sufficiency and quantum LAN indicate the existence of a theory of quantum decision and convergence of models.
In the classical set-up the distance between models is defined operationally, in terms of randomisations which can be seen as the classical counterpart of quantum channels.
A positive linear map $$T:L^{1}(\mathcal{X},\mathcal{A},\mathbb{P}) \to
L^{1}(\mathcal{Y},\mathcal{B},\mathbb{Q})$$ is called a stochastic operator (or randomisation) if $\|T(p)\|_{1}= \|p\|_{1}$ for every $p\in L^{1}_{+}(\mathcal{X})$.
Since we work with models which may contain both classical and quantum ‘states’ we will call a channel, a completely positive, normalised map between preduals of von Neumann algebras. In finite dimensions, this reduces to the familiar notion of a channel, this time with [*block-diagonal*]{} input and output density matrices.
\[def.quantum.LeCam.distance\] Let $\mathcal{P} := \{ \rho_{\theta}: \theta\in \Theta\}$ and $\mathcal{Q}:= \{\sigma_{\theta}: \theta\in \Theta\} $ be two quantum statistical models over $\Theta$ with $\rho_{\theta}$ and $\sigma_{\theta}$ normal states of von Neumann algebras $\mathcal{A}$ and respectively $\mathcal{B}$.
The deficiencies $\delta(\mathcal{P},\mathcal{Q}) $ and $\delta(\mathcal{Q},\mathcal{P})$ are defined as $$\begin{aligned}
\delta(\mathcal{P},\mathcal{Q}) &:=&
\inf_{T} \sup_{\theta\in \Theta}\| T(\rho_{\theta}) -\sigma_{\theta} \|_{1}\\
\delta(\mathcal{Q},\mathcal{P}) &:=&
\inf_{S} \sup_{\theta\in \Theta}
\| S(\sigma_{\theta}) -\rho_{\theta}\|_{1}\end{aligned}$$ where the infimum is taken over all channels $T,S$ and $\|\cdot\|_{1}$ denotes the $L_{1}$-norm on the preduals.
The Le Cam distance between $\mathcal{P}$ and $\mathcal{Q}$ is $$\Delta(\mathcal{P},\mathcal{Q}):=
{\rm max}(\delta(\mathcal{Q},\mathcal{P}) ,\, \delta(\mathcal{P} ,\mathcal{Q})).$$
With this definition we can formulate the strong convergence of the sequence $\mathcal{Q}_{n}$ of i.i.d. qubit models to the Gaussian limit.
\[th.qlan.qubits\] Let $\mathcal{Q}_{n}$ be the sequence of statistical models for $n$ i.i.d. local spin-$\frac12$ states. and let $\mathcal{G}_{n}$ be the restriction of the Gaussian shift model to the range of parameters $\|\vec{u}\|\leq n^{\epsilon}$. Then $$\lim_{n\to\infty}\Delta(\mathcal{Q}_{n}, \mathcal{G}_{n}) =0,$$ i.e. there exist sequences of channels $T_{n}$ and $S_{n}$ such that
$$\label{eq.channel.conv.}
\begin{split}
\lim_{n\to \infty}\,
\sup_{\| \vec{u}\|\leq n^{\epsilon}}
\| \Phi_{ \vec{u}}\otimes N_{\vec{u}} -
T_{n} \left( \rho_{ \vec{u}}^{n}\right) \|_{1} =0, \\
\lim_{n\to \infty} \,
\sup_{\| \vec{u}\|\leq n^{\epsilon}} \| \rho_{\vec{u}}^{n} - S_{n} \left( \Phi_{ \vec{u}}\otimes N_{\vec{u}} \right) \|_{1} =0. \\
\end{split}$$
Let us make a few comments on the significance of the above result. The first point is that LAN provides a stronger characterisation of the ‘Gaussian approximation’ than the usual Central Limit Theorem arguments. Indeed the convergence in Theorem \[th.qlan.qubits\] is strong (in $L_{1}$) rather than weak (in distribution), it is [*uniform*]{} over a range of local parameters rather than at a single point, and has an operational meaning based on quantum channels.
Secondly, one can exploit these features to devise asymptotically optimal measurement strategies for state estimation which can be implemented in practice by coupling with a bosonic bath and performing continuous time measurements in the bath .
Thirdly, the result is not restricted to state estimation but can be applied to a range of quantum statistical problems involving i.i.d. qubit states such as cloning, teleportation benchmarks, quantum learning, and can serve as a mathematical framework for analysing quantum state transfer protocols.
For completeness, we give a brief review of the main ideas involved in the proof of Theorem \[th.qlan.qubits\] and the description of channels $T_{n},S_{n}$ in the Appendix.
Quantum benchmark for displaced thermal states with trace norm distance {#sec.thermal}
=======================================================================
In this Section we address the problem of finding the best MAP scheme for the Gaussian family of displaced thermal states with unknown mean and given variance $$\mathcal{T}:= \{\Phi_{z} : z\in \mathbb{C} \}.$$ To our knowledge this problem has only been solved in the case when the figure of merit is ‘overlap fidelity’ [@Owari]. Here we show that the same procedure is optimal when the loss function is the trace norm distance.
Let $L:=(M,P)$ be a MAP procedure and define the maximum risk as $$R_{\max}(L) = \sup_{z\in \mathbb{C}} \| P\circ M(\Phi_{z}) - \Phi_{z}\|_{1}.$$
The main result of this Section is the following.
\[th.benchmark.Gaussian\] Let $L^{*}:=(H,P)$ be given by heterodyne measurement followed by preparation of a coherent state centred at the outcome of the measurement. Then $L^{*}$ is minmax i.e. for any $L$ $$R_{\max}(L^{*})\leq R_{\max}(L),$$ and its risk $R^{*}(s)$ is given in Lemma \[lemma.risk.calculation\].
Before proceeding with the proof let us recall some basic definitions. The quantum particle or ‘one mode’ continuous variables system is characterised by the Weyl (or CCR) algebra generated by the operators $W_{\xi}$ with $\xi\in \mathbb{C}$ satisfying the commutation relations $$W_{\xi} W_{\zeta} = W_{\zeta}W_{\xi} \exp( -i {\rm Im}\langle \xi, \zeta\rangle ).$$ The algebra is represented on the Hilbert space $L^{2}(\mathbb{R})$ as $ W_{\xi}:=\exp(\xi a^{\dagger}-\bar{\xi} a)$ with $a,a^{\dagger}$ the creation and annihilation operators. The latter are defined by their action on the Fock basis $$|k\rangle := H_{k}(x)e^{-x^{2}/2}/\sqrt{k! 2^{k} \sqrt{\pi}} \quad k=0,1,2...$$ such that $a^{\dagger}|k\rangle= \sqrt{k+1}|k+1\rangle$ and $a |k\rangle= \sqrt{k}|k-1\rangle$.
A thermal state is a mixed state defined as $$\Phi := \sum_{k=0}^{\infty} (1-s) s^{k} |k\rangle\langle k|,$$ where $0<s<1$ is a parameter related to the temperature by $s=e^{-\beta}$, which will be considered fixed and known. The displaced thermal states are $\Phi_{z}:= W_{z} \Phi W_{z}^{\dagger}$. The heterodyne (or coherent) measurement is defined by its POVM $$H(d\zeta):= |\zeta\rangle \langle \zeta | d\zeta/2\pi$$ where $|\zeta\rangle:=W_{\zeta}|0\rangle$ are the coherent states. For any state $\rho$ the probability density of the heterodyne outcomes is the $Q$-function [@Leonhardt] $P_{\rho}(\zeta)= \langle \zeta |\rho |\zeta \rangle/ 2\pi $. Now if we heterodyne the state $\Phi_{z}$ we obtain $\zeta\sim N(z, \mathbf{1} (1-s)^{-1})$ and by preparing the coherent state $|\zeta\rangle$ we get the average output state $$T(\Phi_{z}):= P\circ H(\Phi_{z}) = \tilde{\Phi}_{z},$$ where $\tilde{\Phi} $ is the thermal state with $\tilde{s}=(2-s)^{-1}$.
\[lemma.risk.calculation\] The risk of the measure and prepare strategy $L^{*}:= (H,P)$ is $$\begin{aligned}
R_{\max}(L^{*})&:=& \sup_{z\in \mathbb{C}} \| P\circ H (\Phi_{z}) - \Phi_{z}\|_{1}\\
&=& R^{*}(s):=2 (2-s)^{-m_{0}-1}-2s^{m_{0}+1},\end{aligned}$$ where $m_{0}$ is the integer part of $-\log(2-s)/\log s(2-s) $.
By covariance we have $R_{\max}(L^{*})= \|\Phi -\tilde\Phi\|_{1}$. Both states are diagonal and we denote their elements $q_{i}:= (1-s)s^{i}$ and $p_{i}:=(1-\tilde{s})\tilde{s}^{i}$ so that $$\|\Phi -\tilde\Phi\|_{1}=\sum_{i=0}^{\infty}| q_{i}-p_{i}|.$$ For such geometric distributions there exists an integer $m_{0}$ such that $p_{l} \leq q_{l}$ for $m\leq m_{0}$ and $p_{l}> q_{l}$ for $m>m_{0}$, more precisely $$m_{0}
%= [ \log ( (1-\tilde{s})/(1-s)) /\log(s/\tilde{s}) ]
=\lfloor-\log(2-s)/\log s(2-s)\rfloor.$$ In conclusion $$\| p- q\|_1 = 2\sum_{i=0}^{m_{0}}
q_{i}-p_{i}= 2 \tilde{s}^{m_{0}+1}-2s^{m_{0}+1}.$$
Figure \[fig.riskheterodyne\] shows the decay of the risk $R(L^{*})$ as a function of the parameter $s=e^{-\beta}$ of the input state $\Phi$.
![(Color online) Risk of the optimal measure and prepare scheme as function of $s=e^{-\beta}$. All the quantities plotted are dimensionless.[]{data-label="fig.riskheterodyne"}](risk.pdf){width="8cm"}
We start the proof of Theorem \[th.benchmark.Gaussian\] by following a standard argument [@Owari] which shows that we can first restrict to phase space (or displacement) covariant, entanglement breaking channels, and then that it is enough to show that $L^{*}$ is optimal in a larger class of ‘time-reversible’ channels which can be easily characterised by their action on the Weyl operators $W_{\xi}:= \exp(\xi a^{\dagger}-\bar{\xi} a)$: $$\label{eq.covariant.channels}
T^{\dagger}(W_{\xi})= f(\xi \sqrt{2}) W_{\xi},$$ where $f$ is a quantum characteristic function $f(\xi)= {\rm Tr}(\tau W_{\xi})$ for some state $\tau$ with positive Wigner function, i.e. $f$ is also a classical characteristic function.
For such channels $T=P\circ M$, the risk is independent of $z$ and is equal to $$R_{\max}(L)= \| T(\Phi)-\Phi\|_{1}.$$ Now, since $\Phi$ is invariant under phase rotations, $\Phi= \exp(i\theta N)\Phi \exp(-i\theta N)$, we can apply the covariance argument again to conclude that we can restrict to channels which are covariant under phase rotation, which amounts to taking $\tau$ to be diagonal in the Fock basis. Since for diagonal states $f(\xi)=f(| \xi |)$ we obtain the following Schrödinger version of $$\begin{aligned}
{\rm Tr}(T(\Phi) W_{\xi})&=& {\rm Tr}(\tau W_{\sqrt{2}\xi})
{\rm Tr}(\Phi W_{\xi}) \\ &=& {\rm Tr}(\tau W_{\sqrt{2}\xi})
{\rm Tr}(\Phi W_{-\bar{\xi}})\\
&=&
{\rm Tr}\left(\tau\otimes \Phi \exp ( \xi c -\bar{\xi}c^{\dagger} )\right)\end{aligned}$$ where $$\label{eq.mode.c}
c:=\sqrt{2}b+a^{\dagger}$$ is the annihilation operator of a new mode. In other words, the output state is the same as the state of the ‘amplified’ mode $c$ when $b$ and $a$ are prepared in states $\tau$ and respectively $\Phi$. From this point we follow closely the arguments used in which were originally devised for finding the optimal amplification channel for displaced thermal states. The candidate channels are labelled by diagonal matrices $\tau$ and we denote by $p^{\tau}$ the probability distribution consisting of the elements of the output state $p^{\tau}_{i}= T_{\tau}(\Phi)_{i,i}$. We further denote $q_{i}:= \Phi_{ii}=(1-s)s^{i}$ the geometric distribution of the input thermal state and by $p^{\omega}$ the distribution of the output state corresponding to $\omega= |0\rangle\langle 0|$. It is easy to verify that $T_{\omega}$ is the channel associated to $L^{*}:=(H,P)$, hence we would like to show that for any $\tau$ $$\|p^{\tau}- q\|_{1} \geq \|p^{\omega} - q\|_{1}.$$ The proof is split into two parts and relies on the notion of stochastic ordering as the key ingredient.
Let $p=\{p_{l} : l\in \mathbb{N}\}$ and $q= \{q_{l} : l\in \mathbb{N}\}$ be two probability distributions over $\mathbb{N}$. We say that $p$ is stochastically smaller than $q$ ($p\preceq q$) if $$\sum_{l=0}^{m} p_{l} \geq \sum_{l=0}^{m} q_{l}, \quad \forall m\geq 0.$$
\[lemma.stoch.ordering\] For any state $\tau$ the following stochastic ordering holds: $$p^{\omega}\preceq p^{\tau}.$$
The first step is to ‘purify’ $\Phi$ by writing the mode $a$ as one of the outputs of a degenerate parametric amplifier $$a = \cosh(t) a_{1} + \sinh(t) a_{2}^{\dagger}$$ with $a_{1,2}$ in the vacuum. If $\tanh (t)^{2} =s$ then the state of $a$ is $\Phi$. Plugging into we get $$c = \sinh(\tilde{t}) a_{1}^{\dagger}+ \cosh(\tilde{t}) ( T a_{2} + R b )$$ where $ \sinh(\tilde{t})=\cosh(t), T= \sinh(t) /\cosh(\tilde{t})$ and $R= \sqrt{2}/\cosh(\tilde{t})$ with $T^{2}+R^{2}=1$. Physically this means that the modes $a_{2}$ and $b$ are mixed with a beamsplitter and then a different degenerate parametric amplifier is applied together with mode $a_{1}$ which is in the vacuum. Note that $b$ is in the vacuum state if and only if $\tilde{b}:=T a_{2} + R b$ is also in the vacuum. For general diagonal states $\tau$ the mode $\tilde{b}$ is in the state $\tilde{\tau}$ given by the binomial formula [@Leonhardt] $$\tilde{\tau}=\sum_{k=0}^{\infty} \tau_{k} \sum_{p=0}^{k} \binom{k}{p} T^{2(p-k)} R^{2k} |p\rangle \langle p|
= \sum_{p=0}^{\infty} \tilde{\tau}_{p} |p\rangle \langle p| .$$
The result of the purification argument is that we have reduced the problem of proving stochastic ordering for states of $c = a^{\dagger} + \sqrt{2}b$ when $a$ is in a [*thermal*]{} state, to the analogue problem for $c= \sinh(\tilde{t}) \tilde{a}^{\dagger} + \cosh(\tilde{t})\tilde{b}$ with $\tilde{a}:=a_{1}$ in the vacuum. Since stochastic ordering is preserved under convex combinations, it suffices to prove the statement when $\tilde\tau=|k\rangle\langle k|$ for any $k\neq 0$.
The following formula gives a computable expression of the output two-modes vector state of the amplifier $$\begin{aligned}
\psi &=&
e^{\Gamma \tilde{a}^{\dagger} \tilde{b}^{\dagger}} e^{-g (\tilde{a}^{\dagger}\tilde{a}+ \tilde{b}^{\dagger}\tilde{b}+{\bf 1})}
e^{-\Gamma \tilde{a}\tilde{b}} |0,k\rangle
\\ &=&
e^{-g(k+1)} \sum_{l=0}^{\infty} \Gamma^{l} \binom{l+k}{k}^{1/2} |l,l+k\rangle,\end{aligned}$$ where $\Gamma = \tanh(\tilde{t})$ and $e^{g}=\cosh(\tilde{t})$. By tracing over the mode $\tilde{a}$ we obtain the desired state of $c$ $$\sum_{l=0}^{\infty} d^{k}_{l} |l \rangle \langle l|= e^{-2g(k+1)}
\sum_{l=k}^{\infty} \Gamma^{2(l-k)} \binom{l}{l-k} |l\rangle
\langle l|.$$ The relation $p^{\omega}\preceq p^{\tau}$ reduces to showing that $$\sum_{l=0}^{m} d^{0}_{l} \geq \sum_{l=0}^{m} d^{k}_{l},$$ for all $m$. If $m<k$ the right side is equal to zero and the inequality is trivial. With the notation $\gamma=\Gamma^{2}$ we get $$\begin{aligned}
\sum_{l=0}^{p+k} d^{k}_{l}&=&
(1-\gamma)^{k+1} \sum_{l=0}^{p}\gamma^{l} \binom{l+k}{k}\\
&=&\frac{(1-\gamma)^{k+1}}{k!} \left(\frac{1-\gamma^{k+p+1}}{1-\gamma} \right)^{(k)}\\
&=&1- \gamma^{p+1} \sum_{r=0}^{k} (1-\gamma)^{r} \gamma^{k-r} \binom{k+p+1}{r}\\
&\leq&
1-\gamma^{p+1} \sum_{r=0}^{k} (1-\gamma)^{r} \gamma^{k-r} \binom{k}{r}\\
&=&1-\gamma^{p+1} = \sum_{l=0}^{p} d^{0}_{l} \leq \sum_{l=0}^{p+k} d^{0}_{l} .\end{aligned}$$
Stochastic ordering can be transformed into the desired optimality result by a standard argument which has some interest in its own and is summarised in the following lemma. The key property needed here is that $p^{\omega}_{l} \leq q_{l}$ if and only if $l\leq m_{0}$ (see Lemma \[lemma.risk.calculation\]).
The following inequality holds for any $\tau$ $$\|p^{\tau}- q\|_{1} \geq \|p^{\omega} - q\|_{1}.$$
Define $$m_{a} := \max (m: \sum_{l=0}^{m} q_{l}\leq a)$$ and $
\mathcal{D}(a, \tau) = \{ D\subset \mathbb{N}: \sum_{l\in D}\tau_{l}\leq a\},
$ for all $a\geq 0$. Note that by Lemma \[lemma.stoch.ordering\] we have $
\sum_{l=0}^{m_{a}} p^{\tau} \leq a$ for all $\tau$, and thus $\{0,1,\dots ,m_{a}\} \in \mathcal{D}(a, \tau)$. Using the relation $\|p-q\|_{1} = 2\sup_{D} \sum_{l\in D}(p_{l}- q_{l}) $ we obtain the chain of inequalities $$\begin{aligned}
\|q- p^{\tau}\|_{1} &=& 2 \sup_{a\geq 0}\, \sup_{D \in \mathcal{D}(a,\tau)}\, \sum_{l\in D}
(q_{l}- p^{\tau}_{l})\\
&
\geq&
2\sup_{a\geq 0} \sum_{l=0}^{m_{a}} (q_{l}- p^{\tau}_{l})
\geq
2\sup_{a\geq 0} \sum_{l=0}^{m_{a}} (q_{l} - p^{\omega}_{l})\\
&
=&2\sup_{m\geq 0} \sum_{l=0}^{m} (q_{l}-p^{\omega}_{l})=
\|q-p^{\omega}\|_{1}.\end{aligned}$$ The first equality follows directly form the definition of $\mathcal{D}(a,\tau)$. The subsequent inequality restricts the supremum over all $D\in\mathcal{D}(a,\tau)$ to one element $\{0,1,\dots, m_{a}\}$. In the second inequality we replace the distribution $p^{\tau}$ by $p^{\omega}$ using the stochastic ordering proved in Lemma \[lemma.stoch.ordering\]. In the subsequent equality we used the fact that both distributions $p$ and $q^{\omega}$ are geometric.
We have completed the proof of Theorem \[th.benchmark.Gaussian\], showing that, also with respect to the trace norm figure of merit, a heterodyne measurement followed by the preparation of a coherent state centred at the measurement outcome yields the optimal MAP scheme for the transmission of displaced thermal states.
Asymptotic quantum benchmark for independent qubits {#sec.qubits}
===================================================
Here we tackle the benchmark problem for $n$ i.i.d. qubits in the asymptotic limit $n\rightarrow\infty$, we reformulate it in terms of the local coordinates according to the recipe described in Section \[sec.stat.formulation\], and we obtain a constructive solution to it, by proving that this benchmark problem is equivalent to the one for displaced thermal states discussed and solved in Section \[sec.thermal\]. The rigorous link between the two settings is provided by the LAN theory, whose main concepts are reviewed in Section \[sec.lan\], and which will constitute an essential mathematical tool for the findings of this Section.
Given $n$ i.i.d. spin-$\frac12$ particles in state $\rho_{\vec{r}}^{\otimes n}$, we perform a measurement $M_{n}$ with outcome $X_{n}\sim \mathbb{P}^{M_{n}}_{\vec{r}}$ in a measure space $(\mathcal{X}_{n},\Sigma_{n})$ and reprepare a quantum state $\omega(X_{n})$ on $\left(\mathbb{C}^{2}\right)^{\otimes n}$. The MAP channel is $L_{n}:=P_{n}\circ M_{n}$ and only such channels will be considered in our optimisation. A full MAP protocol will be denoted by $L:=\{L_{n}:n\in\mathbb{N}\}$.
The risk (figure of merit) of the protocol at $\vec{r}$ for a given $n$ is $$\begin{aligned}
R(\vec{r},L_{n})
&:=&
\| \rho^{\otimes n}_{\vec{r}} - \mathbb{E}(\omega(X_{n})) \|_{1}
\\ &=&
\left\| \rho^{n}_{\vec{r}} -\int_{\mathcal{X}}\omega(x) \mathbb{P}^{M_{n}}_{\vec{r}} (dx)\right\|_{1} \nonumber\end{aligned}$$
As a measure of the overall performance of $L_{n}$ we consider the [*local maximum risk*]{} a neighbourhood of $\vec{r}_{0}$ $$\begin{aligned}
R_{\max}(\vec{r}_{0},L_{n}) :=
\sup_{\|\vec{r}-\vec{r}_{0}\|\leq n^{-1/2+\epsilon}} R(\vec{r},L_{n})\end{aligned}$$ whose asymptotic behaviour is $$R(\vec{r}_{0}, L) :=
\underset{n\to\infty}{\lim\sup} \,
R_{\max}(\vec{r}_{0},L_{n})$$
The local minmax risk at $\vec{r}_{0}$ is defined by $$R_{\min\!\max}(\vec{r}_{0}):=\underset{n\to\infty}{\lim\sup} \, \inf_{L_{n}} R_{\max}(\vec{r}_{0},L_{n}).$$ A protocol $L$ is called locally mimimax at $\vec{r}_{0}$ if $
R(\vec{r}_{0}, L)=R_{\min\!\max}(\vec{r}_{0}).
$
In Theorem \[th.main\] we show that the following sequence of MAP maps is locally minmax.
[*Adaptive measure and prepare protocol:*]{}
1. The state is first localised in a neighbourhood $\|\vec{r}-\vec{r}_{0}\|\leq n^{-1/2+\epsilon}$ of $\vec{r}_{0}$ by using a small proportion of the systems (see Section \[sec.lan\]).
2. The remaining spins are mapped by the channel $T_{n}$ close to the classical-quantum Gaussian state $\Phi_{\vec{u}} \otimes N_{\vec{u}}$ as in Theorem \[th.qlan.qubits\].
3. A heterodyne measurement together with an observation of the classical component give an estimator $\hat{\vec{u}}$ of $\vec{u}$.
4. A coherent state $|\alpha_{\hat{\vec{u}}}\rangle$ is prepared with the mean equal to the outcome of the heterodyne measurement. The inverse channel $S_{n}$ is applied to the coherent state and the classical part.
The procedure is illustrated in the commutative diagram drawn in the introduction: the upper line represents the MAP steps which are realised through the alternative route described in the lower part. Thus $M_{n}= H \circ T_{n}$ and $P_{n}:= S_{n}\circ P$.
\[th.main\] The sequence of MAP maps $$L^{*}_{n}:= (M_{n}:= H \circ T_{n}, \,P_{n}:= S_{n}\circ P)$$ is locally asymptotically minmax. The minmax risk at $\vec{r}_{0}$ is equal to the benchmark for the MAP problem of a displaced thermal equilibrium state with parameter $s=(1-r_{0})/(1+r_{0})$ (see Theorem \[th.benchmark.Gaussian\]) $$R_{\min\!\max}(\vec{r}_{0})= R (\vec{r}_{0},L^{*})= R_{\min\!\max}(s)$$
The idea is that the spins problem can be transferred the Gaussian one with vanishing difference in the risks. We first show that $R_{\max} (\vec{r}_{0},L^{*})\leq R_{\min\!\max}(s)$ and then argue that a strict inequality would be in contradiction with the optimality of $R_{\min\!\max}(s)$.
By Lemma 2.1 in the first step of the procedure has $o(1)$ failure probability, in which case the output state may be very different from the desired one. However since the norm distance between states is bounded by $2$, an application of the triangle inequality shows that this has no influence on the asymptotic risk. Thus we can consider that $\vec{r}$ is in the local neighbourhood $\|\vec{r}-\vec{r}_{0}\|\leq n^{-1/2+\epsilon}$ and we can apply the LAN machinery of Section \[sec.lan\].
The (average) output state is $$\omega_{n}:=\mathbb{E} (\omega(X_{n}))=
S_{n} \left(
\mathbb{E} \left[
|\alpha_{\hat{\vec{u}}} \rangle\langle\alpha_{\hat{\vec{u}}}| \right] \otimes N_{\vec{u}} \right)$$ where the expectation on the right side is over the heterodyne measurement outcomes. Since measurements and preparations are contractive we have $$\label{eq.bound1}
\|
\mathbb{E} \left[
|\alpha_{\hat{\vec{u}}} \rangle\langle\alpha_{\hat{\vec{u}}}| \right] -
\tilde{\Phi}_{\vec{u}} \|_{1}\leq \|T_{n}(\rho^{n}_{\vec{u}}) - \Phi_{\vec{u}} \otimes N_{\vec{u}}\|_{1} =
o(1)$$ where $\tilde{\Phi}_{\vec{u}}$ is the displaced thermal equilibrium state of variance $1+1/2r_{0}$ obtained by heterdyning $\Phi_{\vec{u}}$, preparing the coherent state $|\alpha_{\hat{\vec{u}}}\rangle \langle \alpha_{\hat{\vec{u}}}|$ and averaging over $(\hat{u}_{1},\hat{u}_{2})$.
On the other hand, by using contractivity of $S_{n}$ $$\begin{aligned}
\label{eq.bound2}
& &\| \rho^{n}_{\vec{u}}- \omega_{n} \|_{1} \nonumber
\\ & & \quad \leq \| \rho^{n}_{\vec{u}}-S_{n}(\Phi_{\vec{u}} \otimes N_{\vec{u}}) \|_{1}
+
\| \Phi_{\vec{u}} - \mathbb{E} \left[
|\alpha_{\hat{\vec{u}}} \rangle\langle\alpha_{\hat{\vec{u}}}| \right] \|_{1}\nonumber\\
& & \quad =
\| \Phi_{\vec{u}} - \mathbb{E} \left[
|\alpha_{\hat{\vec{u}}} \rangle\langle\alpha_{\hat{\vec{u}}}| \right] \|_{1}+o(1).\end{aligned}$$
From and we get $$R_{\max}(\vec{r}_{0},L^{*}) \leq \|\Phi -\tilde{\Phi}\|_{1}= R_{\min\!\max}(s).$$ As shown in Theorem \[th.benchmark.Gaussian\], the right side is the MAP benchmark for mixed Gaussian states with optimal procedure consisting of heterodyne measurement followed by the preparation of a coherent state with mean equal to the outcome of the measurement.
Now we show that no MAP strategy can achieve a lower asymptotic risk at $\vec{r}_{0}$ than $R_{\min\!\max}(s)$. Indeed suppose that $\tilde{L}:= (\tilde{M}_{n},\tilde{P}_{n})$ is a sequence of procedures for spins which satisfies $R_{\max}(\vec{r}_{0},\tilde{L}) < R_{\min\!\max}(s)$. Then as shown below, we could construct a MAP procedure for $\Phi_{\vec{u}}$ which is strictly better that the optimal one, which is impossible.
Let $\delta>0$ be a small number to be fixed later. We mix $\Phi_{\vec{u}}$ with the thermal state of the same temperature $\Phi_{0}$, through a beamsplitter with small reflectivity $r=\delta$ and transmissivity $t= \sqrt{1-\delta^{2}}$ and obtain the output $$\Phi_{t \vec{u}}\otimes \Phi_{r\vec{u}}.$$ By heterodyning $\Phi_{r \vec{u}}$ we obtain an estimator $\vec{u}_{0}$ such that $$\mathbb{P} \left[ \|\vec{u} - \vec{u}_{0}\| \geq L \right] \leq \epsilon_{2}/2$$ for some (large) $L$ which increases when $\delta,\epsilon_{2}\downarrow 0$.
We displace the unmeasured component $\Phi_{t \vec{u}}$ by $-t\vec{u}_{0}$ so that from now on we can assume that $\|\vec{u}\|\leq L$ (with an $\epsilon_{2}$ loss of risk).
Now we choose $n$ large enough so that $L \leq n^{\epsilon}$. Thermal states with such displacements are in the range of applicability of the inverse map $S_{n}$ in LAN Theorem \[th.qlan.qubits\]. We apply the channel $S_{n}$ mapping the Gaussian state to an i.i.d. spins state, with the small difference that the classical parameter is now fixed to zero, i.e. the spins will be prepared in a state with Bloch vector of length $r_{0}$. By Theorem \[th.qlan.qubits\] we have uniformly in $\|\vec{u}\|<L$ $$\|S_{n}(\Phi_{t\vec{u}}\otimes N_{0}) - \rho^{n}_{t\vec{u}}\|_{1} = o(1).$$
Next we measure and re-prepare the spins using the procedure $\tilde{L}$ and obtain a state $\omega_{n}:= \mathbb{E}\left[\omega(X_{n})\right]$ such that $$\|\omega_{n} - \rho^{n}_{t\vec{u}}\|_{1} \leq
R_{\max}(\vec{r}_{0},\tilde{L}) +o(1).$$
Finally, we apply the map $T_{n}$ to the output state $\omega(X_{n})$ and keep only the quantum part $T^{(q)}(\omega_{n})$. By the same contractivity argument as before we have $$\begin{aligned}
\|\Phi_{t\vec{u}} - T^{(q)}_{n} (\omega_{n})\|_{1} &\leq&
\|\Phi_{t\vec{u}} - T^{(q)}_{n} (\rho^{n}_{t\vec{u}})\|_1+
\|\omega_{n} - \rho^{n}_{t\vec{u}}\|_{1}\nonumber\\
&\leq& R_{\max}(\vec{r}_{0},\tilde{L}) + o(1).\end{aligned}$$
At this point we can directly compare out state $T^{(q)}_{n} (\omega_{n})$ with the target $\Phi_{\vec{u}}$, or use a quantum amplifier to make up for the loss in amplitude induced by the initial use of a beam splitter. We follow the second ‘unbiased’ line. The amplifier can be described by a linear transformation on the mode $a$ of the oscillator together with an ancillary mode $b$ prepared in the vacuum. The output modes are $$\begin{aligned}
c_{1}&:=& \sqrt{t^{-1}} a+ \sqrt{t^{-1}-1} b^{\dagger}\\
c_{2}&:=&\sqrt{ t^{-1}} b+ \sqrt{t^{-1}-1} a^{\dagger}.\end{aligned}$$ Let $A$ denote the channel mapping the state of the mode $a$ to that of the amplified mode $c_{1}$. Then $$A\Phi_{t\vec{u}}= \Phi^{\prime}_{\vec{u}}$$ where $\Phi^{\prime}$ is a thermal equilibrium state with variance $$\tilde{V}= t^{-2}/2r_{0} +(t^{-1}-1)/2.$$
The final distance estimate (conditional on successful localisation of $\Phi_{\vec{u}}$) is $$\begin{aligned}
\| \Phi_{u} - A\circ T^{(q)}_{n}(\omega_{n})\|_{1} \!\!&\leq&\!
\|\Phi_{\vec{u}}- \Phi^{\prime}_{\vec{u}}\|_{1} +
\| \Phi_{t\vec{u}} - T_{n}(\omega_{n})\|_{1}\nonumber\\
\!\!&\leq&\!\|\Phi-\Phi^{\prime}\|_{1} + R_{\max}(\vec{r}_{0},\tilde{L}) +o(1). \nonumber \\ & &\end{aligned}$$
Since we assumed that $R_{\max}(\vec{r}_{0},\tilde{L})< R_{\min\!\max}(s)$, it is enough to choose $\epsilon_{2}$ and $\|\Phi-\Phi^{\prime}\|_{1}$ small enough to obtain a contradiction with the optimality of $R_{\min\!\max}(s)$. This can be done by choosing $\delta, \epsilon_{2}$ small enough and $n$ large enough.
Discussion and concluding remarks {#sec.discussion}
=================================
The problem of finding an optimal MAP reconstruction scheme for a family of quantum states has attracted significant attention due to its relevance in establishing fidelity benchmarks for teleportation and state storage. In the case of a family of displaced thermal (and coherent) states with unknown displacement the problem was solved in [@Hammerer; @Owari] and the optimal procedure is the heterodyne measurement followed by the preparation of a coherent state. We showed that the same MAP procedure is again optimal for this family of states, with a more natural figure of merit - the trace norm distance. Moreover, in the case of i.i.d. mixed qubit states, the benchmark problem can be solved asymptotically by mapping it to the previous problem, using LAN theory.
Interestingly, the same heterodyne measurement is also optimal from the point of view of state estimation [@Holevo]. On the other hand, in [@Calsamiglia] it was shown that for a particular family of states in $\mathbb{C}^{2}$, the optimal MAP scheme involves a measurement which is different from the optimal one for state estimation. One of our motivations was to see whether this peculiarity survives in the asymptotic limit, and the conclusion is that for large $n$ the estimation and benchmarking can be performed optimally [*simultaneously*]{}, i.e. their measurement parts are identical.
Another difference between benchmarking and estimation pointed out in [@Calsamiglia] refers to the preparation part of the protocols. More exactly, it turns out that in the case of a special family of one qubit pure states, the optimal re-prepared state [*does not*]{} belong to the family. However, our asymptotic benchmark for qubits can be easily extended to treat the case of pure rather than mixed states and the result is that [*asymptotically*]{}, the optimal re-prepared state [*is*]{} in the original model. Thus the effect pointed out in [@Calsamiglia] is due to the particular geometry of the states space in the one sample situation, which ‘linearises’ asymptotically. To briefly explain our claim, note that asymptotically, the parameter space reduces effectively to a small interval and the tangent space approximation kicks in. Then the problem is transferred to that of finding the benchmark for a family of coherent states [*on a line*]{}, where the solution is a homodyne measurement followed by the preparation of a coherent state [*in the family*]{}.
Oh the other hand, we have seen that in the case of mixed states, the re-prepared state is pure, hence not in the family, even in the asymptotic framework. However this fact is not surprising as it happens already in the classical case: if $X$ is a random variable with distribution $\mathbb{P}$ then the best ‘re-preparation’ is the ‘pure state’ represented by the $\delta$ measure $\delta_{X}$ leading to a trivial benchmark. If $\mathbb{P}$ is not a $\delta$ measure itself then the re-prepared ‘state’ $\delta_{X}$ is outside the model.
Explicit construction of the channels $T_{n},S_{n}$ {#secApp}
===================================================
We give here a brief review of the main ideas involved in the proof of Theorem \[th.qlan.qubits\] and the description of channels $T_{n},S_{n}$.
On $\left(\mathbb{C}^{2}\right)^{\otimes n}$ we have two [*commuting*]{} unitary group representations $$\begin{aligned}
\pi_{n}(U)&:&\psi_{1}\otimes \dots\otimes \psi_{n} \mapsto
U \psi_{1}\otimes \dots \otimes U\psi_{n} ,\\
\tilde{\pi}(t) &:& \psi_{1}\otimes \dots\otimes \psi_{n}\mapsto
\psi_{t^{-1}(1)}\otimes \psi_{t^{-1}(n)}\end{aligned}$$ where $U\in SU(2)$ and $t\in S(n)$, with $S(n)$ denoting the symmetric group. By Weyl’s Theorem, the representation space decomposes into a direct sum of tensor products $$\label{eq.decomposition}
\left( \mathbb{C}^{2}\right)^{\otimes n} = \bigoplus_{j=0, 1/2}^{n/2} \mathcal{H}_{j} \otimes \mathcal{H}^{j}_{n},$$ where $j$ is half-integer, $\mathcal{H}_{j} \cong \mathbb{C}^{2j+1}$ is an irreducible representation of $SU(2)$, and $\mathcal{H}^{j}_{n}\cong \mathbb{C}^{n_{j} }$ is the irreducible representation of $S(n)$. Since the density matrix $\rho^{n}_{\vec{u}}$ is invariant under permutations, it has a block diagonal form $$\label{blocks}
\rho^{n}_{\vec{u}} = \bigoplus_{j=0, 1}^{n} p^{n}_{\vec{u}}(j) \rho^{n}_{j,\vec{u}} \otimes
\frac{\mathbf{1}}{n_{j}} .$$
This can be interpreted as being given a random variable $J$ with distribution $p^{n}_{\vec{u}}(j)$ and conditionally on $J=j$, a quantum state $\rho^{n}_{j,\vec{u}}$. The classical and quantum components of the ‘data’ will be ‘processed’ by randomising $J$ and mapping $\rho^{n}_{j,\vec{u}}$ through a quantum channel.
[*Classical component.*]{}— Each block is an eigenspace of the total spin operator $L^{2}= L_{x}^{2}+L_{y}^{2}+L_{z}^{2}$ with eigenvalue $4j(j+1)\approx (2j)^{2}$. As the big Bloch ball argument suggests, the main contribution to $L^{2}$ comes from $L_{z}^{2}$ so that the distribution $p^{n}_{\vec{u}}(j)$ can be approximated by the binomial $$p^{n}_{\vec{u}}(j) \approx {n\choose j+n/2}
\left(\frac{1+r}{2}\right)^{j+n/2} \left(\frac{1-r}{2}\right)^{n/2-j}
%B(n, (1+r)/2)[j+n/2],$$ with $r=r_{0}+u_{z}$. The fact that $p^{n}_{\vec{u}}(j)$ converges to $N_{\vec{u}}$ follows now from the classical version of LAN [@LeCam] for i.i.d. samples of binary variables. We first constructs the rescaled variable $$G_{n}:=\sqrt{n} \left(\frac{L}{n} - r_{0} \right)
\overset{\mathcal{D}}{\longrightarrow} N(u_{z}, 1-r_{0}),$$ but since $G_{n}$ has a discrete probability distribution, we need to ‘smooth’ it by randomising with e.g. a Gaussian Markov kernel of variance $1/(2\sqrt{n})$ $$K_{n,j}(x) := (n^{1/4}/\sqrt{\pi})\exp\left(-\sqrt{n} (x-G_n(j))^2\right).$$
In this way the convergence in distribution is converted into strong ($L_{1}$) convergence.
[*Quantum component.*]{}— Conditionally on obtaining $J=j$ in the which-block measurement, we are left with a quantum state $\rho^{n}_{j,\vec{u}}$ on $\mathcal{H}_{j}$. The action of $T_{n}$ will be to imbed it into the quantum oscillator space by the isometry $V_{j}:\mathcal{H}_{j}\to \ell^{2}(\mathbb{Z}) $ define below.
Let $\pi_{j}$ denote the irreducible representation of $SU(2)$ on $\mathcal{H}_{j}$ and denote its generators by $L_{j,a}= \pi_{j}(\sigma_{a})$. The space has an orthonormal basis $\left\{|j,m\rangle , m=-2j, \dots, 2j\right\}$ such that $$L_{j,z} |j,m \rangle = m |j,m\rangle.$$
The Quantum Central Limit Theorem suggests that the properly normalised operators $L_{j, \pm} := L_{j,x} \pm i L_{j,y}$ converge to the annihilation and creation operators $a,a^{\dagger}$ when $j\approx n r_{0}/2\to\infty$. Since they act as ladder operators $$\begin{aligned}
&&
J_{j, +} |j,m\rangle = \sqrt{j-m} \sqrt{ j+m+1} \, |j,m+1\rangle ,\\
&&
J_{j, -} |j,m\rangle = \sqrt{j-m+1} \sqrt{ j+m} \, |j,m-1 \rangle .\end{aligned}$$ it is natural to define the isometry $$V_{j}:|m,j\rangle \mapsto |j-m\rangle$$ defining the embedding (channel) $$T_{j}: \rho^{n}_{j,\vec{u}}\longmapsto V_{j} \rho^{n}_{j,\vec{u}} V_{j}^{j}.$$
Putting everything together we obtain the channel $T_{n}$ $$T_{n} : \rho^{n}_{\vec{u}}
\longmapsto
\sum_{j} p^{n}_{\vec{u}} (j) K_{n,j} \otimes V_{j} \rho^{n}_{j,\vec{u}} V_{j}^{j}$$ which implements the convergence to Gaussian in Theorem \[th.qlan.qubits\].
The channel $S_{n}$ is basically an inverse of $T_{n}$: the normal distribution is discretised to produce the distribution $p^{n}_{\vec{j}}$ and the quantum Gaussian state is compressed to the first $2j+1$ levels and mapped to $\mathcal{H}_{j}$ with the co-isometry $V_{j}^{\dagger}$. For more details on the proof we refer to .
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R. Namiki, Phys. Rev. A [**78**]{}, 032333 (2008).
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Let us mention, however, that an experimental implementation of the conventional benchmark criteria for CV systems can be highly demanding. Alternative approaches to the derivation of benchmarks for quantum channels, such as memory or teleportation channels, have been then pursued [@luk]. These benchmarks, using entanglement-verification tools such as the ‘expectation value matrix’, directly assess the ability of quantum transmission channels (as realized in an experiment) to preserve entangled states, as opposed to MAP channels which are entanglement-breaking. Such testing procedures, encompassing also the realistic possibility of using mixed states to probe the quantum channels, might be verified with very few experimental resources.
H. Häseler and N. Lütkenhaus, Phys. Rev. A [**80**]{}, 042304 (2009); [*ibid.*]{} [**81**]{}, 060306(R) (2010).
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, in **, edited by (, ), pp. .
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[^1]: Corresponding author.\
Electronic address: [madalin.guta@nottingham.ac.uk]{}
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The aim of this paper is that of discussing Closed Graph Theorems for bornological vector spaces in a way which is accessible to non-experts. We will see how to easily adapt classical arguments of functional analysis over $\R$ and $\C$ to deduce Closed Graph Theorems for bornological vector spaces over any complete, non-trivially valued field, hence encompassing the non-Archimedean case too. We will end this survey by discussing some applications. In particular, we will prove de Wilde’s Theorem for non-Archimedean locally convex spaces and then deduce some results about the automatic boundedness of algebra morphisms for a class of bornological algebras of interest in analytic geometry, both Archimedean (complex analytic geometry) and non-Archimedean.'
author:
- Federico Bambozzi
title: Closed graph theorems for bornological spaces
---
Introduction {#introduction .unnumbered}
============
This paper aims to discuss the Closed Graph Theorems for bornological vector spaces in a way which is accessible to non-specialists and to fill a gap in literature about the non-Archimedean side of the theory at the same time. In functional analysis over $\R$ or $\C$ bornological vector spaces have been used since long time ago, without becoming a mainstream tool. It is probably for this reason that bornological vector spaces over non-Archimedean valued fields were rarely considered. Over the last years, for several reasons, bornological vector spaces have drawn new attentions: see for example [@Bam], [@BaBe], [@BaBeKr], [@CAM] and [@M]. These new developments involve the non-Archimedean side of the theory too and that is why it needs adequate foundations. Among the classical books on the theory of bornological vector spaces, the only one considering also non-Archimedean base fields, in a unified fashion with the Archimedean case, is [@H2]. But that book cannot cover all the theory, and in particular it lacks of a unified treatment of the Closed Graph Theorems. This work comes out from the author’s need for a reference for these theorems and also in the hope that in the future bornological vector spaces will gain more popularity and that this work may be useful for others.
The Closed Graph Theorem for Banach spaces over $\R$ and $\C$ is one of the most celebrated classical theorems of functional analysis. Over the years it has been generalized in many ways, for example to Fréchet and LF spaces as a consequence of the Open Mapping Theorems. Although it is a classical argument that the Closed Graph Theorem can be deduced from the Open Mapping Theorem, people have understood that the Closed Graph Theorem can be proved in an independent way, with argumentations that also work when the Open Mapping Theorem fails. The two most famous examples of this kind of proofs are given in [@POPA] by Popa and [@DW] by de Wilde. In particular, de Wilde’s Theorem is probably the most general Closed Graph Theorem for locally convex spaces, and states the following:
(De Wilde’s Closed Graph Theorem)\
If $E$ is an ultrabornological locally convex space and $F$ is a webbed locally convex space over $\R$ or $\C$, then every linear map $f: E \to F$ which has bornologically closed graph with respect to the convex bornologies on $E$ and $F$ that are generated by all bounded Banach disks in $E$ and in $F$, respectively, is continuous even if regarded as a mapping into the ultrabornologification $F_\uborn$ of $F$.
The terminology of the theorem will be explained in the course of this work. What is interesting to notice is that, although we would like to prove a theorem for locally convex spaces we are naturally led to talk about bornologies and bounded maps. Popa’s Theorem, on the contrary, is an explicit bornological statement which is the Archimedean case of our Theorem \[thm:net\].
The content of the paper is the following: in the first section we give an overview of the theory of bornological vector spaces. In particular, since we adopt the unusual attitude of discussing the Archimedean and non-Archimedean case of the theory at the same time, we spend some time in recalling basic definitions and discuss in details the notions from the theory bornological vector spaces that will be used. In the second section we will introduce the notion of bornological nets and then give the main examples of bornological vector spaces endowed with nets. We will then deduce our first Closed Graph Theorem, which is the unified version of Popa’s Theorem (cf. [@POPA]), stated as follows:
Let $E$ and $F$ be separated convex bornological vector spaces, where $E$ is complete and $F$ has a net compatible with its bornology. Then, every linear map $f: E \to F$ with bornologically closed graph is bounded.
In the subsequent section the notion of bornological net is generalized by the notion of bornological web and the analogous Closed Graph Theorem for webbed bornological vector spaces is proved quite easily as a consequence of the previous discussion. In this case our theorem is the direct generalization, for all base fields, of the Closed Graph Theorem proved by Gach in [@G], Theorem 4.3. In the last section we discuss some applications of the theorems we proved. We will see how one can deduce Isomorphism Theorems from Closed Graph Theorems and following [@G] we will see how de Wilde’s Closed Graph Theorem can be deduced. We would like to remark that for non-Archimedean base fields we need to add some restrictions, that do not affect the Archimedean side of the theory. Our generalization of de Wilde’s Theorem is the following:
If $E$ is an ultrabornological locally convex space and $F$ is a polar webbed locally convex space defined over a spherically complete field $K$, then every linear map $f: E \to F$ which has bornologically closed graph with respect to the convex bornologies on $E$ and $F$ that are generated by all bounded Banach disks in $E$ and in $F$, respectively, is continuous even if regarded as a mapping into the ultrabornologification $F_\uborn$ of $F$.
Therefore, in order to deduce de Wilde’s Theorem for non-Archimedean base fields we needed to suppose that the base field $K$ is spherically complete and that $F$ is a polar locally convex space (cf. Definition \[defn:polar\]), conditions which are always satisfied when $K$ is Archimedean. We remark that these hypothesis on $K$ and $F$ are only used in Lemma \[lemma:polar\] and in Lemma \[lemma:web\_ultraborn\]; one might ask if it is possible to prove that lemmata without these restrictions. We do not address this problem in this work. Finally, in the last part of the paper we show how to use the Closed Graph Theorems for bornological spaces to deduce that all algebra morphisms between dagger affinoid algebras, as defined in [@Bam], are bounded. This application, and others coming in [@BaBeKr] and planned in future works, are our main motivations for this study.
Bornological spaces and Closed Graph Theorems
=============================================
Closed graph {#closed-graph .unnumbered}
------------
Let $u: E \to F$ be a map of sets, then the set $$\Gamma(u) = \{ (x, y) \in E \times F \ | y = u(x) \ \}$$ is called the graph of $u$. If $E$ and $F$ are Hausdorff topological spaces and $u$ a continuous map, then $\Gamma(u)$ is a closed subspace of $E \times F$ endowed with the product topology. This is a basic property of Hausdorff topological spaces. If $E$ and $F$ are vector spaces over a field $K$ and $u$ is a linear map, then $\Gamma(u)$ is a vector subspace of $E \times F$. If $E$ and $F$ are separated bornological vector spaces over a complete, non-trivially valued field $K$ and $u$ is linear and bounded, then $\Gamma(u)$ is bornologically closed in $E \times F$ endowed with the product bornology (see below for what it means for a subset of a bornological vector space to be bornologically closed). This assertion is pretty easy to check. Let $(x_n, u(x_n))$ be a sequence of elements of $\Gamma(u)$ which converges bornologically to $( x, y)$ in $E \times F$. Then, by definition of product bornology, $x_n \to x$ in $E$ and $u(x_n) \to y$ in $Y$. Since $u$ is bounded, the sequence $(u(x_n))$ converges bornologically to $u(x)$ and since $Y$ is separated, we must have $y = u(x)$. Therefore $(x, y) \in \Gamma(u)$ and $\Gamma(u)$ is bornologically closed in $E \times F$.
The Closed Graph Theorems are converses of the above statements for some special class of bornological or topological $K$-vector spaces. Here we pursue the main ideas of [@G] for which the bornological Closed Graph Theorems are of more fundamental nature and extend it for non-Archimedean base fields. The statements of our Closed Graph Theorems assert that if $u: E \to F$ is a linear map which has bornologically closed graph then it is bounded when $E$ and $F$ belong to some particular classes of bornological vector spaces: we will prove it when $E$ is a complete bornological vector space and $F$ a separated bornological vector space endowed with a net, in Section \[sec:closed\_nets\], and when $F$ is endowed with a web in Section \[sec:closed\_webs\]. Both the proofs of these sections are adaptations of results from [@H] and [@G] on bornological vector spaces over Archimedean base fields to any non-trivially valued, complete base field $K$ Archimedean or not.
Bornologies {#bornologies .unnumbered}
-----------
Bornological vector spaces are well studied objects in functional analysis over $\R$ and $\C$. They are not mainstream, as the theory of topological or locally convex vector spaces, but they are often useful in addressing problems for locally convex spaces and they found a good amount of applications. Thus, during the years, a good amount of work has been done to study the properties of bornological vector spaces and algebras over $\R$ and $\C$: for examples [@H], [@Hog2], [@Hog3], [@WAEL], [@M] discuss various aspects and applications of the theory. On the other hand, for non-Archimedean base fields the theory has never got much attention. The only works known to the author which deal with bornological vector spaces over non-Archimedean base fields date back to many years ago and they often discuss the non-Archimedean base field case as a mere example, interesting for working out general theories and general theorems, pursuing a “Bourbaki" study of the subject; however, this side of the theory were seldom thought to have applications to “real" mathematics. An example of this kind of work is [@H2]. In recent years a renewed interest in the theory of bornological vector spaces may challenge this attitude: for example, on the Archimedean side of the theory, one has the work [@M], [@M2], [@M3] showing the usefulness of bornological vector spaces in non-commutative geometry and representation theory and [@Bam], [@CAM], [@BaBe] where also the theory of non-Archimedean bornological vector spaces and algebras is used in analytic geometry and higher order local field theory. We will recall the basis of the theory of bornoloogical vector spaces as far as needed for our scopes.
Let $X$ be a set. A *bornology* on $X$ is a collection $\cB$ of subsets of $X$ such that
1. $\cB$ is a covering of $X$, $\forall x \in X, \exists B \in \cB$ such that $x \in \cB$;
2. $\cB$ is stable under inclusions, $A \subset B \in \cB \then A \in \cB$;
3. $\cB$ is stable under finite unions, for each $n \in \N$ and $B_1, ..., B_n \in \cB$, $\underset{i = 1}{\overset{n}\bigcup} B_i \in \cB$.
A pair $(X, \cB)$ is called a *bornological set*, and the elements of $\cB$ are called *bounded subsets* of $X$ (with respect to $\cB$, if it is needed to specify). A family of subsets $\cA \subset \cB$ is called a *basis* for $\cB$ if for any $B \in \cB$ there exist $A_1, \dots, A_n \in \cA$ such that $B \subset A_1 \cup \dots \cup A_n$. A *morphism* of bornological sets $\varphi: (X, \cB_X) \to (Y, \cB_Y)$ is defined to be a bounded map $\varphi: X \to Y$, a map of sets such that $\varphi(B) \in \cB_Y$ for all $B \in \cB_X$.
From now on let’s fix a complete, non-trivially valued field $K$. We will use the notation $K^\circ = \{ x \in K | |x| \le 1 \}$. Clearly $K$ has a natural structure of bornological field.
A *bornological vector space* over $K$ is a $K$-vector space $E$ along with a bornology on the underlying set of $E$ for which the maps $(\la, x) \mapsto \la x$ and $(x, y) \mapsto x + y$ are bounded.
We will only work with bornological vector spaces whose bounded subsets can be described using convex subsets, in the following way.
Let $E$ be a $K$-vector space. A subset $B \subset E$ is called *balanced* if for every $\la \in K^\circ$ one has that $\la B \subset B$. A subset $B \subset E$ is called *absolutely convex* (or *disk*) if
1. for $K$ Archimedean, it is convex and balanced, where convex means that for every $x, y \in B$ and $t \in [0, 1]$ then $(1 -t) x + t y \in B$;
2. for $K$ non-Archimedean, it is a $K^\circ$-submodule of $E$.
The definition of absolutely convex subset of $E$ is posed in two different ways, depending on $K$ being Archimedean of non-Archimedean, although the formal properties are essentially the same in both cases. Moreover, using the theory of generalized rings (in the sense of Durov [@DUR]) one can put the two situations on equal footing, but we are not interested to this issue in this work.
A bornological $K$-vector space is said to be *of convex type* if its bornology has a basis made of absolutely convex subsets.
We denote by $\bBorn_K$ the category of bornological vector spaces of convex type over $K$. Then, every semi-normed space over $K$ can be endowed with the bornology induced by the bounded subsets for the semi-norm and is manifestly of convex type. This association permits to see the category of semi-normed spaces over $K$ as a full sub-category of $\bBorn_K$.
For every bornological vector space of convex type $E$ there is an isomorphism $$E \cong \limind_{B \in \cB_E} E_B$$ where $\cB_E$ denotes the family of bounded absolutely convex subsets of $E$ and $E_B$ is the vector subspace of $E$ spanned by the elements of $B$ equipped with the *gauge semi-norm* (also called *Minkowski functional*) defined by $B$. Notice that all morphism of this inductive system are monomorphisms.
\[defn:separated\_born\] A bornological vector space over $K$ is said to be *separated* if its only bounded vector subspace is the trivial subspace $\{0\}$.
A bornological vector space of convex type over $K$ is separated if and only if for each $B\in \cB_E$, the gauge semi-norm on $E_{B}$ is actually a norm.
\[defn:banach\_disk\] A disk $B \subset E$ is said to be a *Banach disk* if $E_B$ is a Banach space.
\[defn:complete\_born\] A bornological space $E$ over $k$ is said to be *complete* if $$E \cong \limind_{i \in I} E_i$$ for a filtered colimit of Banach spaces over $K$ for which the system morphisms are all injective.
It can be shown that the definition of complete bornological vector space just given is equivalent to the request that the family of bounded disks $\cB_E$ of $E$ admits a final subfamily made of Banach disks, cf. [@H2] Proposition 7 page 96.
Bornological convergence and bornologically closed subsets {#bornological-convergence-and-bornologically-closed-subsets .unnumbered}
----------------------------------------------------------
The data of a bornology is enough for endowing a vector space of a notion of convergence.
\[defn:born\_conv\] Let $E$ be a bornological $k$-vector space and $\{x_n\}_{n \in \N}$ a sequence of elements of $E$. We say that $\{x_n\}_{n \in \N}$ *converges (bornologically) to $0$ in the sense of Mackey* if there exists a bounded subset $B \subset E$ such that for every $\la \in K^\times$ there exists an $n = n(\la)$ for which $$x_m \in \la B, \forall m > n.$$ We say that $\{ x_n \}_{n \in \N}$ converges (bornologically) to $a \in E$ if the sequence $\{x_n - a\}_{n \in \N}$ converges (bornologically) to zero.
An analogous definition can be given for the convergence of filters of subsets of $E$. We omit the details of this definition since is not important for our scope.
The notion of bornological convergence on a bornological vector space of convex type $E = \underset{B \in \cB_E}\limind E_B$ can be restated in the following way: $\{ x_n \}_{n \in \N}$ is convergent to zero in the sense of Mackey if and only if there exists a $B \in \cB_E$ and $N \in \N$ such that for all $n > N$, $x_n \in E_B$ and $x_n \to 0$ in $E_B$ for the semi-norm of $E_B$.
Let $E$ be a bornological vector space over $K$.
- a sequence $\{x_n\}_{n \in \N} \subset E$ is called *Cauchy-Mackey* if the double sequence $\{x_n - x_m\}_{n,m \in \N}$ converges to zero;
- a subset $U \subset E$ is called *(bornologically) closed* if every Mackey convergent sequence of elements of $U$ converges (bornologically) to an element of $U$.
A bornological vector space is called *semi-complete* if every Cauchy-Mackey sequence is convergent.
The notion of semi-completeness is not as useful as the notion of completeness in the theory of topological vector spaces. We remark that any complete bornological vector space is semi-complete, but the converse is false.
It can be shown that the notion of bornologically closed subset induces a topology on $E$, but this topology is neither a vector space topology, nor a group topology in general. Therefore, an arbitrary intersection of bornological closed subsets of a bornological vector space is bornologically closed. So, the following definition is well posed.
Let $U \subset E$ be a subset of a bornolgical vector space. The closure of $U$ is the smallest bornologically closed subset of $E$ in which $U$ is contained. We denote the closure of $U$ by $\ol{U}$.
The concept of bornologically closed subspace fits nicely in the theory. For example a bornological vector space is separated, in the sense of Definition \[defn:separated\_born\], if and only if $\{0 \}$ is a bornologically closed subset. We have to warn the reader that the closure of a subset $X \subset E$ of a bornological subset is not always equal to the limit points of convergent sequences of elements of $X$ but strictly contains it.
Duality between bornologies and topologies {#duality-between-bornologies-and-topologies .unnumbered}
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Let $\bLoc_K$ denote the category of locally convex vector spaces over $K$. We recall the definitions of two functors from [@H2], ${}^t: \bBorn_K \to \bLoc_K$ and ${}^b: \bLoc_K \to \bBorn_K$. To a bornological vector space of convex type $E$ we associate the topological vector space $E^t$ in the following way: we equip the underlying vector space of $E$ with a topology for which a basis of neighborhoods of $0$ is given by *bornivorous subsets*, subsets that absorb all bounded subsets of $E$. The association $E \mapsto E^t$ is well defined and functorial. Then, if $E$ is a locally convex space, $E^b$ is defined to be the bornological vector space obtained by equipping the underlying vector space of $E$ with the *von Neumann* (also called canonical) bornology, whose bounded subsets are the subsets of $E$ absorbed by all neighborhood of $0 \in E$. Also, this association is well defined and functorial. In chapter 1 of [@H2] one can find many properties of these constructions of which the main one is that they form an adjoint pair of functors.
We conclude this review of the theory by recalling some relations between bornological and topological vector spaces that we will use later in our proofs.
\[prop:born\_convergence\] Let $E$ be a locally convex space and consider the bornological vector space $E^b$. Then, if a sequence $\{x_n\}_{n \in \N}$ converges to $0$ in the sense of Mackey in $E^b$ it converges topologically in $E$.
Let $B \subset E$ be a von Neumann bounded subset such that the condition of Mackey convergence to $0$ for $\{x_n\}_{n \in \N}$ is satisfied. Given any neighborhood of zero $U \subset E$, then there must exist a $\la \in K^\times$ such that $$\la B \subset U$$ and by Definition \[defn:born\_conv\] there exists an $n = n(\la)$ such that $$x_i \in \la B \subset U, \ \ \forall i > n,$$ hence the sequence converge topologically.
\[def:born\_metrizable\] Let $E$ be a bornological vector space of convex type. We say that $E$ is metrizable if $E \cong F^b$ for a metrizable locally convex space.
The following result is a well-known statement of the theory of bornological vector spaces over $\R$ or $\C$, see for example proposition (3) of section 1.4.3 of [@H]. But as far as we know the non-Archimedean version of the result is hard to find in literature. The last lines of page 108 of [@H2], essentially affirm, without proof, the statement of next proposition. Here we offer a detailed proof.
\[prop:metric\] Let $E$ be metrizable bornological vector space of convex type. Then, a sequence $\{ x_n \}_{n \in \N}$ converges bornologically in $E$ if and only if it converges topologically in $E^t$.
Thanks to Proposition \[prop:born\_convergence\] we only need to check that topological convergence implies bornological convergence. So, let $\{V_n \}_{n \in \N}$ be a countable base of absolutely convex neighbourhoods of $0$ in $E$ such that $V_{n + 1} \subset V_n$ for every $n \in \N$. Let $A = \{ x_n \}_{n \in \N}$ be a sequence in $E$ which converges to $0$ topologically.
Since the sequence $A$ converges topologically to zero, then it is absorbed by every neighbourhood of zero. Therefore, for every $n \in \N$, there exists a $\a_n \in |K^\times|$ such that $$A \subset \la_n V_n$$ for $\la_n \in K^\times$ with $|\la_n| = \a_n$. It follows that $$A \subset \bigcap_{n = 1}^\infty \la_n V_n.$$ Let $\{\be_n \}_{n \in \N}$ be a sequence of strictly positive real numbers such that $\be_n \in |K^\times|$ and $\be_n \to 0$ as $n \to \infty$. Let $\gamma_n = \frac{\a_n}{\be_n}$ and $\epsilon \in |K^\times|$ be given. Define $$B = \bigcap_{n = 1}^\infty \mu_n V_n$$ with $\mu_n \in K^\times$ and $|\mu_n| = \gamma_n$. $B$ is clearly a bounded subset of $E$, because it is absorbed by all $V_n$. We are going to prove the following assertion: for every $\epsilon \in |K^\times|$ there is an integer $m \in \N$ for which $A \cap V_m \subset \la B$, for $\la \in K^\times$ with $|\la| = \epsilon$, from which the proposition will be then proved.
Since the sequence $\frac{\gamma_n}{\a_n} = \frac{1}{\be_n}$ tends to $\infty$ for $n \to \infty$, there is an integer $p \in \N$ such that $$\forall n > p, \ \frac{\gamma_n}{\a_n} > \frac{1}{\epsilon}.$$ Therefore $$A \subset \bigcap_{n = 1}^\infty \la_n V_n \then A \subset \rho_n V_n,$$ for $\rho_n = \la \mu_n \in K^\times$ with $|\la| = \epsilon$ and $n > p$. But the set $$\bigcap_{n = 1}^p \rho_n V_n,$$ with again $\rho_n = \la \mu_n \in K^\times$ and $|\la| = \epsilon$, is a neighbourhood of $0$. Hence there exists an integer $m \in \N$ such that $$V_m \subset \bigcap_{n = 1}^p \rho_n V_n$$ therefore $$A \cap V_m \subset \bigcap_{n = 1}^\infty \rho_n V_n = \bigcap_{n = 1}^\infty \la \mu_n V_n = \la B,$$ proving the claim.
The last proposition has the following interesting consequence.
\[cor:metric\] Let $E$ be a metrizable locally convex space. Then $E$ is complete topologically, if and only if $E^b$ is complete bornologically if and only if $E^b$ is semi-complete.
That the completeness of $E$ implies the completeness of $E^b$ is proved in Proposition 15 of page 101 of [@H2], and as we have already remarked the completeness of $E^b$ implies its semi-completeness. Then, Proposition \[prop:metric\] directly implies that the semi-completeness of $E^b$ implies the completeness of $E$ proving the corollary.
The closed graph theorem for bornological spaces with nets {#sec:closed_nets}
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In this section we will prove the Closed Graph Theorem for bornological vector spaces endowed with a net. Essentially we will adapt the proof of Popa’s Theorem that one can find in [@H] and we will make it work over any complete, non-trivially valued field $K$.
\[defn:net\] Let $F$ be a $K$-vector space. A *net* on $F$ is a map $\cN: \underset{k \in \N}\bigcup \N^k \to \sP(F)$ such that
1. each $\cN(n_1, \ldots, n_k)$ is a disk;
2. $\cN(\void) = F$;
3. for every finite sequence $(n_0, \ldots, n_k)$ one has $$\cN(n_0, \ldots, n_k) = \bigcup_{n \in \N} \cN(n_0, \ldots, n_k, n).$$
Notice that condition $(2)$ of the previous definition is used to give meaning to the formula $$F = \cN(\void) = \bigcup_{n \in \N} \cN(n).$$ If $s: \N \to \N$ is a sequence we will use the notation $$\cN_{s, k} = \cN(s(0), \ldots, s(k)).$$
\[defn:net\_compatible\] Let $F$ be a separated bornological $K$-vector space of convex type. Then we can say that a net $\cN$ on $F$ is compatible with its bornology if
1. for every sequence $s: \N \to \N$ there is a sequence of positive real numbers $b(s): \N \to \R_{> 0}$ such that for all $x_k \in \cN_{s,k}$ and $a_k \in K$ with $|a_k| \le b(s)_k$ the series $$\sum_{k \in \N} a_k x_k$$ converges bornologically in $F$ and $\underset{k \ge n}\sum a_k x_k \in \cN_{s,n}$ for every $n \in \N$.
2. For every sequences $\{ \la_k\}_{k \in \N}$ of elements of $K$ and $s: \N \to \N$ the subsets $$\bigcap_{k \in \N} \la_k \cN_{s, k}$$ are bounded in $F$.
We say that a separated bornological vector space *has a net* if there exists a net which is compatible with its bornology.
The most common bornological vector spaces used in functional analysis have nets.
\[exa:nets\]
1. Let $F$ be a bornological Fréchet space. By this we mean that $F \cong E^b$ for a Fréchet space $E$. Then $F$ has a net compatible with its bornology. To show this, consider a base of neighborhoods $\{ V_n \}_{n \in \N}$ of $0 \in F$. If the base field $K$ is Archimedean it is well known that one can define a net in the following way. For any $k$-tuple of integers $(n_1, \ldots, n_k)$ define $$\cN(n_1, \ldots, n_k) = n_1 V_{n_1} \cap \ldots \cap n_k V_{n_k}.$$ See section 4.4.4 of [@H] for a detailed proof of this fact. This definition does not work for non-Archimedean base fields since $|n_k| \le 1$ and so it does not satisfy condition (3) of Definition \[defn:net\]. So, let’s pick an element $\a \in K$ such that $|\a| > 1$. This is always possible since $K$ is non-trivially valued. Then, let’s define $$\cN(n_1, \ldots, n_k) = \a^{n_1} V_{n_1} \cap \ldots \cap \a^{n_k} V_{n_k}$$ and check that this is a net compatible with the bornology of $F$. Since every neighborhood of $0$ is absorbent then the condition of $\cN$ to be a net is clearly satisfied. Let $\{ n_k \}_{k \in \N}$ be a sequence of integers, define $v_k = |\a^{ -2^k n_k}|$ and choose $\la_k \in K$ with $|\la_k| \le v_k$. Then, the series $$\sum_{k = 0}^\infty \la_k x_k$$ converges bornologically in $F$ for every $x_k \in \cN(n_1, \ldots, n_k)$ because it converges for the metric of $F$, and by Proposition \[prop:metric\] the convergence for the metric of $F$ is equivalent to the bornological convergence in $F$. Moreover, for every $k_0$ $$\sum_{k = k_0}^\infty \la_k x_k \in \la_{k_0} \cN(n_1, \ldots, n_k) \subset \cN(n_1, \ldots, n_k),$$ because $|\la_k| < 1$ and for non-Archimedean base fields disks are additive sub-groups of $F$. Then, we need to check the second condition on compatibility of the net with the bornology, which is pretty easy to check since given any sequence $\{\la_k\}_{k \in \N}$ of elements of $K$, we can consider the set $$A = \bigcap_{k = 0}^\infty \la_k \cN(n_1, \ldots, n_k) = \bigcap_{k = 0}^\infty \la_k \a^{n_1} V_{n_1} \cap \ldots \cap \a^{n_k} V_{n_k}.$$ Thus, given any $V_{n_r}$, we have that $$A \subset \la_r \a^{n_r} V_{n_r}$$ hence $A$ is absorbed by any neighborhood of $0 \in F$, hence it is bounded for the von Neumann bornology of the Fréchet space.
2. Let $F = \underset{n \in \N}\limind F_n$ be a monomorphic inductive limit such that $F_n$ are separated bornological vector spaces which have nets compatible with the bornology, then we claim that $F$ is a separated bornological vector space which has a compatible net. To show this, first we note that since the inductive limit is monomorphic $$F = \bigcup_{n \in \N} F_n$$ and $F$ is separated when equipped with the direct limit bornology, see Proposition 6 at page 49 of [@H2] for a proof of this fact. Then, let $\cN_n$ be a net for $F_n$. Let’s define $$\cE(n) = \cN_1(n_1), \ \ \cE(n_1, \ldots, n_k) = \cN_{n_1}(n_2, \ldots, n_k).$$ One can check directly that $\cE$ is a net on $F$ compatible with the bornology of $F$.
3. From the previous example it follows that every complete bornological vector space with countable base for its bornology has a net compatible with the bornology. In particular regular LB spaces and regular LF spaces have nets for their von Neumann bornology.
4. As an example of bornological vector space which cannot be endowed with a net, one can consider an infinite dimensional Banach space $E$ endowed with the bornology of pre-compact subsets, if the base field is locally compact, or the bornology of compactoid subsets if the base field is not locally compact. Let’s denote $E^c$ the vector space $E$ equipped with this bornology, which is of convex type and complete (see examples 1.3 (9) and (10) for a proof of this fact for Archimedean base fields. The same argument works for any base field). It is also well known that the identity map $E^c \to E^b$ is bounded, but the two bornologies do not coincide in general. This also means that the bornology of $E^c$ cannot have a net because the fact that the identity map is bounded implies that, if $E^c$ would have a net, we could apply the Isomorphism Theorem \[thm:iso\] to deduce that the identity map is bounded also in the other direction $E^b \to E^c$. Notice that, by previous examples, this also shows that the bornology of $E^c$ has not a countable base.
Let’s move towards the proof of the Closed Graph Theorem, and before proving it let’s prove some lemmata.
\[lemma:absorbing\] Let $E$ be and $F$ be $K$-vector spaces. Let $B \subset E$ be bounded, $C \subset F$ any subset and $f: E \to F$ a linear map. Then, $C$ absorbs $f(B)$ if and only if $f^{-1}(C)$ absorbs $B$.
This is a very basic property of linear maps. $C$ absorbs $f(B)$ means that there exists a $\la \in K^\times$ such that $$f(B) \subset \la C$$ hence $$B \subset f^{-1}(f(B)) \subset f^{-1}(\la C) = \la f^{-1}(C).$$ On the other hand, if $B \subset \la f^{-1}(C)$ then $$f(B) \subset f( \la f^{-1}(C)) \subset \la f( f^{-1}(C)) \subset \la C.$$
\[lemma:baire\] Let $E$ be a $K$-Banach space and $F$ be a separated convex bornological vector space endowed with a net $\cN$, not necessarily supposed to be compatible with the bornology. Let $f: E \to F$ be a linear map, then there exists a sequence $\{ n_k \}_{k \in \N}$ of integers such that $f^{-1}(\cN(n_1, \ldots, n_k))$ is not meagre in $E$ for each $k \in \N$.
By the definition of net $F = \underset{n_1 \in \N}\bigcup \cN(n_1)$, so $E = \underset{n_1 \in \N}\bigcup f^{-1}(\cN(n_1))$ and since $E$ is a Baire space, it follows that there must exist a $n_1$ such that $f^{-1}(\cN(n_1))$ is not meagre in $E$. Then, we can apply the same reasoning to the relation $\cN(n_1) = \underset{n_2 \in \N}\bigcup \cN(n_1, n_2)$ obtaining a $f^{-1}(\cN(n_1, n_2))$, for some $n_2 \in \N$, which is not meagre in $E$, and inductively for any $k$ we get a $f^{-1}(\cN(n_1, \ldots, n_k))$ which is not meagre in $E$, for suitable $n_1, \ldots, n_k \in \N$.
The next lemma is the key technical lemma to prove the Closed Graph Theorem for bornological spaces equipped with a net.
\[lemma:net\] Let $E$ be a $K$-Banach space and $F$ be a separated convex bornological vector space endowed with a net $\cN$ compatible with its bornology. Let $B \subset E$ denotes the open unit ball of $E$. If $f: E \to F$ is a morphism with bornologically closed graph, then there exist a sequence $(n_k)$ of integers such that $f(B)$ is absorbed in each $\cN(n_1, \ldots , n_k)$.
By Lemma \[lemma:baire\] we can produce a sequence $\{ n_k \}_{k \in \N}$ of integers such that $f^{-1}(\cN(n_1, \ldots, n_{k_0}))$ is not meagre in $E$ for each $k_0 \in \N$. It is sufficient to show that for each fixed $k_0$ the set $f^{-1}(\cN(n_1, \ldots, n_{k_0}))$ absorbs $B$ by Lemma \[lemma:absorbing\].
Let’s denote by $s: \N \to \N$ the sequence obtained by applying Lemma \[lemma:baire\]. By the compatibility of $\cN$ with the bornology on $F$ there exists a sequence $b(s)$ of positive real numbers such that for each sequence $a_k$ of elements of $K$ with $|a_k| \le b(s)_k$ and for all $x_k \in \cN_{s, k}$, the series $\underset{k \ge k_0}\sum a_k x_k$ converges bornologically to an element in $\cN_{s, k_0}$. Let $\epsilon > 0$, we can choose $a_k$ such that $$\sum_{k \in \N} |a_k| \le \epsilon.$$ Let’s denote $A_k = a_k f^{-1}(\cN_{s, k})$. Since $A_k$ is not meagre, then there is a point $b_k$ in the interior of $\ol{A_k}$ and a radius $\rho_k > 0$ such that the open ball $$D(b_k, \rho_k) = b_k + \mu_k B, \ \ \ \mu_k \in K, \ \ \ |\mu_k| = \rho_k,$$ of radius $\rho_k$ and centred in $b_k$ is contained in $\ol{A_k}$. We can assume that $b_k \in A_k$. In fact, since $b_k \in \ol{A_k}$, then there exists $b_k' \in A_k$ such that $|b_k - b_k'|_E < \frac{\rho_k}{2}$. So, $$b_k' + D(0, \frac{\rho_k}{2}) = (b_k' - b_k) + (b_k + D(0, \frac{\rho_k}{2})) \subset \ol{A_k}.$$ We may also suppose that $\rho_k \le \frac{1}{k}$ without loss of generality. So for a fixed $k_0$, we have $$D(0, \rho_{k_0}) \subset \ol{A_{k_0}} - b_{k_0} \subset \ol{A_{k_0}} - \ol{A_{k_0}}.$$ Thus, if $K$ is Archimedean we can deduce that $$D(0, \rho_{k_0}) \subset 2 \ol{A_{k_0}}$$ and if $K$ is non-Archimedean that $$D(0, \rho_{k_0}) \subset \ol{A_{k_0}},$$ hence if we redefine $\rho_{k_0}$ being half its value when $K$ is Archimedean we can always suppose that $$D(0, \rho_{k_0}) \subset \ol{A_{k_0}}.$$ We will conclude the proof by showing that there exists $\gamma \in K^\times$, such that $$\ol{f^{-1}(\cN_{s, k_0})} \subset \gamma f^{-1}(\cN_{s, k_0}))$$ because then we can deduce that $$\mu_{k_0} B = D(0, \rho_{k_0}) \subset \gamma A_{k_0} \then B \subset \mu_{k_0}^{-1} \gamma a_{k_0} \cN(n_1, \ldots, n_{k_0}), \ \ \mu_{k_0} \in K, \ \ |\mu_{k_0}| = \rho_{k_0}.$$ Pick an element $x \in \ol{f^{-1}(\cN_{s, k_0})}$. There exists an element $y_{k_0} \in f^{-1}(\cN_{s, k_0})$ such that $|x - y_{k_0}|_E \le \rho_{k_0 + 1}$. Therefore $$(y_{k_0} - x) + b_{k_0 + 1} \in D(b_{k_0 + 1}, \rho_{k_0 + 1}) \subset \ol{A_{k_0 + 1}}.$$ Then, we can find $y_{k_0 + 1} \in A_{k_0 + 1}$ such that $|x - y_{k_0} - y_{k_0 + 1} + b_{k_0 + 1}|_E \le \rho_{k_0 + 2}$. So, by induction for every $N > k_0$ we can find elements $y_k \in \ol{A_k}$ such that $$|x - \sum_{k \ge k_0}^N y_k + \sum_{k \ge k_0 + 1}^N b_k |_E \le \rho_{N + 1}$$ Since $\rho_N \to 0$, the left-hand side converges to $0$. Let’s show that the series $$\sum_{k \ge k_0}^N z_k + \sum_{k \ge k_0 + 1}^N c_k$$ where $z_k = f(y_k)$ and $c_k = f(b_k)$ converges to $f(x)$. By hypothesis $z_{k_0} \in \cN_{s, k_0}$ and $z_k, c_k \in a_k \cN_{s, k}$ for $k > k_0$, so by the compatibility of the net with the bornology of $F$ the series $$\sum_{k \ge k_0}^\infty z_k , \ \ \sum_{k \ge k_0 + 1}^\infty c_k$$ converge bornologically in $F$. Moreover, since $\cN_{s, k} \subset \cN_{s, k_0}$ for each $k > k_0$, one has that $$\sum_{k \ge k_0}^\infty z_k \in \cN_{s, k_0} + \gamma' \cN_{s, k_0}$$ $$\sum_{k \ge k_0 + 1}^\infty c_k \in \gamma' \cN_{s, k_0}$$ for a $\gamma' \in K$ such that $|\gamma'| \le \epsilon$. So $$y = \sum_{k \ge k_0}^\infty z_k - \sum_{k \ge k_0 + 1}^\infty c_k \in \gamma \cN_{s, k_0}$$ where $\gamma \in K^\times$ can be chosen to have absolute value $1 + 2 \epsilon$ if $K$ is Archimedean and $1$ if $K$ is non-Archimedean, because $\cN_{s, k_0} + \gamma' \cN_{s, k_0} = \cN_{s, k_0}$ for $|\gamma'| < 1$ in this case. Since the graph of $f$ is bornologically closed in $E \times F$, then we have that $$0 = f(0) = f(x - \sum_{k \ge k_0}^\infty y_k + \sum_{k \ge k_0 + 1}^\infty b_k) = f(x) - f(\sum_{k \ge k_0}^\infty y_k + \sum_{k \ge k_0 + 1}^\infty b_k) = f(x) - y$$ so $f(x) \in \gamma \cN_{s, k_0}$ which implies that $x \in \gamma f^{-1}(\cN_{s, k_0})$, thus proving the lemma.
\[thm:net\] Let $E$ and $F$ be separated convex bornological vector spaces, where $E$ is complete and $F$ has a net $\cN$ compatible with its bornology. Then every linear map $f: E \to F$ with bornologically closed graph is bounded.
Notice that $E \cong \underset{B \in \cB_E}\limind E_B$, where $B$ runs through all bounded Banach disks, i.e. $E$ can be described as a monomorphic inductive limit of a family of Banach spaces. In order to prove that $f$ is bounded we only have to show that the compositions of $f$ with the canonical maps $i_B: E_B \to E$ are bounded. Indeed, the graph of $f \circ i_B$ is bornologically closed, so we just need to show the theorem for $E$ supposed to a Banach space. To see that the graph of $f \circ i_B$ is closed one can consider a sequence $\{x_n\} \subset E_B$, converging to $x \in E$. Since $i_B$ is bounded then $i_B(x_n) \to i(x)$, in the sense of Makey, so $f(i_B(x_n)) \to f(i(x))$ in the sense of Mackey in $F$, because the graph of $f$ is bornologically closed.
Therefore, suppose that $E$ is a $K$-Banach space with unit ball $B \subset E$. By Lemma \[lemma:net\] there exists a sequence $\{ n_k \}_{k \in \N}$ of integers such that $f(B)$ is absorbed in each $\cN(n_1, \ldots , n_k)$. It follows that there exists a sequence $\{ a_k \}_{k \in \N}$ of elements of $k$ such that $$f(B) \subset \bigcap_{k \in \N} a_k \cN(n_1, \ldots, n_k)$$ and the latter subset is bounded in $F$, by the request of the net to be compatible with the bornology of $F$. So, we can conclude that $f(B)$ is bounded in $F$.
We will discuss some applications of this theorem in the last section. Let’s now see some stability properties of bornological nets, for which there is not much in literature known to the author.
\[prop:stability\] Bornological nets have the following stability properties:
1. if $E = \underset{i \in \N}\limind E_i$ is a monomorphic direct limit of bornological vector spaces with nets then $E$ has a net;
2. if $(E, \cB)$ has a net and $\cB'$ is another bornology on $E$ such that the identity $(E, \cB) \to (E, \cB')$ is bounded, then $(E, \cB')$ has a net;
3. every closed subspace $F \subset E$ of a bornological vector space with a net has a net;
4. if $E = \underset{i \in \N}\limpro E_i$ is the countable projective limits of bornological spaces with nets, then $E$ has a net;
5. if $E = \underset{i \in \N}\bigoplus E_i$ is the countable coproduct of bornological spaces with nets, then $E$ has a net.
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1. This claim has already been discussed in Example \[exa:nets\] (3).
2. Indeed, one can use the same given net on $(E, \cB)$ on the space $(E, \cB')$, which is easily seen to be compatible with $\cB'$ too.
3. If $E$ is equipped with a net $\cN$ compatible with its bornology and $F \subset E$ is bornologically closed, then the association $$\cN_F(n_1, \ldots, n_k) = \cN(n_1, \ldots, n_k) \cap F$$ defines a net on $F$ which is compatible with the bornololgy induced by $E$ on $F$ because $F$ is bornologically closed.
4. The projective limit of a diagram $I \to \bBorn_K$ is the linear subspace $$P = \left \{ (x_i) \in \prod_{i \in I} E_i | f_{i, j}(x_j) = x_i \text{ for all } f_{i,j} \right \},$$ endowed with the induced bornology. It is easy to check that if all spaces $E_i$ are separated then $P$ is bornologically closed in $\underset{i \in I}\prod E_i$. Thus, by the previous point it is enough to show that the product of any countable family of bornological vector spaces with nets is a bornological vector space with a net. So, let $\{ (E_i, \cN^{(i)}) \}_{i \in \N}$ be a countable family of bornological vector spaces endowed with nets.
The product bornology on $\underset{i \in \N}\prod E_i$ is separated, see Proposition 5 on page 48 of [@H2]. For any $n$ let’s fix bijections $f_n: \N^{n+1} \to \N$ and let $s: \N \to \N$ be any given sequence. For any $k \in \N$ there exists $(a_0^{(k)}, \ldots, a_{k}^{(k)}) \in \N^{k + 1}$ such that $$s(k) = f_k(a_0^{(k)}, \ldots, a_{k}^{(k)})$$ So, we define a family of sequences $\{ s_n : \N \to \N \}_{n \in \N}$ by $$s_n(k) = a_n^{(k + n)}$$ for each $k \in \N$. We set $$\cN_{s, k} \doteq \cN_{s_0, k}^{(0)} \times \cN_{s_1, k - 1}^{(1)} \times \ldots \times \cN_{s_k, 0}^{(k)} \times \prod_{i > k} E_i$$ to get a well-defined map $\cN: \underset{i \in \N}\bigcup \N^i \to \sP( \underset{i \in \N}\prod E_i)$, by setting $\cN(\void) = \underset{i \in \N}\prod E_i$ and $\cN(n_1, \ldots, n_k) = \cN_{s, k}$ when we choose $s: \N \to \N$ such that $s(i) = n_i$ for each $i \in \N$.
Let’s check that $\cN$ is a net on $\underset{i \in \N}\prod E_i$. The properties (1) and (2) of Definition \[defn:net\] are obvious. So let’s check the last condition.
First, let’s fix for any $n$ a sequence $t^{(n)}: \N \to \N$ such that $t^{(n)}(i) = s(i) = n_i$ for all $i \le k$, $t^{(n)}(k + 1) = n$, and the other values of $t^{(n)}$ can be freely chosen. Then, consider $$x = (x_k)_{k \in \N} \in \cN_{s, k} \times \prod_{i > k} E_i.$$ We have to show that there exists an $n$ such that $$x \in \cN_{t^{(n)}, k + 1} \times \prod_{i > k + 1} E_i.$$ Then, for any $n$ $$\cN_{t^{(n)}, k + 1} = \cN_{s_0, k + 1}^{(0)} \times \cN_{s_1, k }^{(1)} \times \ldots \times \cN_{s_k, 1}^{(k)} \times \cN_{t^{(n)}_k, 0}^{(k + 1)}$$ so that the first $k$ components do not change changing $n$, whereas $$\cN_{t^{(n)}_k, 0}^{(k + 1)} = \cN^{(k + 1)} ((a_n)_k^{(k)})$$ for the $(k + 1)$-tuple such $f_k((a_n)_0^{(k)}, \ldots, (a_n)_k^{(k)}) = n$. Thus, since $f_k$ is a bijection then $\underset{n \to \infty}\limsup (a_n)_k^{(k)} \to \infty$ and since $\cN^{(k + 1)}$ is a net we have that $$\bigcup_{n \in \N} \cN^{(k + 1)} ((a_n)_k^{(k)}) = E_{k + 1},$$ which shows (3) of Definition \[defn:net\].
Then, let’s prove that $\cN$ is compatible with the bornology of $\underset{i \in \N}\prod E_i$. For any $s: \N \to \N$ let’s define $$b(s)_k \doteq \min \{ b^{(i)} (s_i)_{k - i} | 0 \le i \le k \}$$ where $b^{(i)}$ is as in Definition \[defn:net\_compatible\] for $\cN^{(i)}$. Let’s fix a sequence $\mu_k$ of elements of $K$ such that $|\mu_k| \le b(s)_k$. It is an easy consequence of the definition of product bornology that a series $\underset{k \in \N}\sum \mu_k x_k$, with $x_k \in \cN_{s, k}$, converges in $\underset{i \in \N}\prod E_i$ if and only if its components converge in $E_i$ for each $i \in \N$. Let’s denote with $\pi_n: \underset{i \in \N}\prod E_i \to E_n$ the canonical projection, so for $l > n$ $$\pi_n \left ( \sum_{k = 0}^l \mu_k x_k \right ) =$$ $$= \sum_{k = 0}^{n - 1} \mu_k \pi_n(x_k) + \sum_{k = n}^{l} \mu_{(k - n)}^{(n)} \frac{\mu_k}{\mu^{(n)}_{(k - n)}} \pi_n(x_k)$$ where $\mu_{(k - n)}^{(n)} \in K$ is such that $|\mu_{(k - n)}^{(n)}| \le b^{(n)}(s_n)_{(k - n)}$. And notice that we can always have that $$\left | \frac{\mu_k}{\mu^{(n)}_{(k - n)}} \right | \le 1, \ \ \forall k \ge n.$$ Hence the series $$\sum_{k = n}^{\infty} \mu^{(n)}_{(k - n)} \frac{\mu_k}{\mu^{(n)}_{(k - n)}} \pi_n(x_k)$$ converges because $\cN^{(n)}$ is a net on $E_n$.
Then, we need to check that for any given $m \in \N$ we have that $$\sum_{k \ge m} \mu_k x_k \in \cN_{s, m}$$ and since $$\cN_{s, m} = \cN_{s_0, m}^{(0)} \times \cN_{s_1, m - 1}^{(1)} \times \ldots \times \cN_{s_m, 0}^{(m)} \times \prod_{i > m} E_i$$ we need to check only the first $m + 1$ components. So, we need to check that $$\pi_n \left ( \sum_{k = m}^\infty \mu_k x_k \right ) \in \cN_{s_n, m - n}^{(n)}$$ for $0 \le n \le m$. But for the above formula $$\pi_n \left ( \sum_{k = m}^\infty \mu_k x_k \right ) = \sum_{k = m}^{\infty} \mu_{(k - n)}^{(n)} \frac{\mu_k}{\mu^{(n)}_{(k - n)}} \pi_n(x_k)$$ which belongs to $\cN_{s_n, m - n}^{(n)}$ because $\cN^{(n)}$ is a net on $E_n$ compatible with its bornology.
Last condition of Definition \[defn:net\_compatible\] is easy to check because a subset $B \subset \underset{i \in \N}\prod E_ni$ is bounded if and only if $\pi_i(B)$ is bounded in $E_i$.
5. Finally the last assertion of the proposition is a consequence of (1) and (4), because for any family of bornological vector spaces $\{ E_n \}_{n \in \N}$ one has the isomorphism $$\bigoplus_{n \in \N} E_n \cong \limind_{n \in \N} \prod_{i = 1}^n E_i$$ and the inductive system is monomorphic.
It is easy to see that the quotient of a bornological vector space endowed with a net is not necessarily endowed with a net. One can consider a Fréchet-Montel space $E$ and a quotient $E/F$, for a closed subspace $F \subset E$ which is not Fréchet-Montel. Examples of these kind of spaces are well known both for Archimedean and non-Archimedean $K$. So, the von Neumann bornology of $E$ coincides with the compact(oid) bornology of $E$, but the quotient bornology of $E/F$ is the compact(oid) one and does not coincide with the von Neumann one. Example \[exa:nets\] (4) implies that $E/F$ does not admit a net compatible with its bornology.
The closed graph theorem for bornological spaces with webs {#sec:closed_webs}
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In this section we will prove the most general version of the Closed Graph Theorem for bornological vector space of the paper.
\[defn:born\_web\] Let $E$ be a separated bornological vector space of convex type over $K$. A pair $(\cV, b)$ consisting of mappings $\cV : \underset{k \in \N}\bigcup \N^k \to \mathcal{P}(E)$ and $b : \N^\N \to (|K^\times|)^\N$ is called a *bornological web* if all the conditions below hold:
1. The image of $\cV$ consists of disks.
2. $\cV(\void) = E$.
3. Given a finite sequence $(n_0, \dots, n_k)$, then $\cV(n_0, \dots, n_k)$ is absorbed by $$\bigcup_{n \in \N} \cV(n_0, \dots, n_k, n).$$
4. For every $s: \N \to \N$ the series $\underset{k \in \N}\sum \la(s)_k x_k$, with $\la(s) \in K$, converges bornologically in $E$, whenever we choose $x_k \in \cV(s(0), \dots ,s(k))$ and $|\la(s)_k| = b(s)_k$.
As we did in previous section we will use the shorthand notation $\cV_{s,k} = \cV(s(0), \ldots , s(k))$. We define the following sets, which depend on $b$: $$\forall s: \N \to \N, \forall n \in \N : \wtilde{\cV}_{s,n} = \Gamma( \cV_{s,n} \cup \l \{ \sum_{k \ge n +1} \la(s)_k x_k | \forall k \ge n + 1 : x_k \in \cV_{s,k}, |\la(s)_k| = b(s)_k \r \} )$$ where $\Gamma$ denotes the absolutely convex hull. Furthermore, let $\cB_{(\cV,b)}$ denote the convex linear bornology on $E$ which is generated by all subsets of the form $$\bigcap_{k \in \N} \mu_k \wtilde{\cV}_{s,k},$$ where the $\{\mu_k \}_{k \in \N}$ is an arbitrary $K$-valued sequence.
A separated bornological vector space of convex type $E$ which is endowed with a bornology of the form $\cB_{(\cV, b)}$ for a bornological web $(\cV, b)$ on $E$ is called a *webbed convex bornological space*.
The following is the Closed Graph Theorem for webbed bornological vector spaces.
\[thm:web\] Let $E$ and $F$ be separated convex bornological vector spaces, where $E$ is complete and $F$ is endowed with a bornological web $(\cV, b)$. Then, every linear map $f: E \to F$ with bornologically closed graph is bounded for the bornology $\cB_{(\cV, b)}$.
First, as in Theorem \[thm:net\] we can reduce the proof to the case in which $E$ is a Banach space, because $f: E = \limind E_i \to F$ is bounded if and only if $f \circ \a_i$ is bounded for every $i$, where $\a_i: E_i \to \limind E_i$ are the canonical maps.
By condition (3) of Definition \[defn:born\_web\] we can use a reasoning similar to the one given in Lemma \[lemma:baire\] to produce a sequence $s: \N \to \N$ such that $f^{-1}(\cV_{s, k})$ is not meagre in $E$ for any $k$. Moreover let’s put $b_k = b(s)_k$, for all $k \in \N$. Since $(\cV, b)$ satisfies condition (4) of Definition \[defn:born\_web\], the series $\underset{k \in \N}\sum \la_k x_k$ converges bornologically in $F$, whenever we choose $x_k \in \cV_k$ and $\la_k \in K$ with $|\la_k| = b_k$.
Next, let $D(r)$ denote the ball of radius $r$ in $E$ centred in zero. If we can show that $f (D(1))$ is absorbed by $\wtilde{\cV}_{s, k}$ , or equivalently by Lemma \[lemma:absorbing\], that $D(1)$ is absorbed by $f^{-1} ( \wtilde{\cV}_{s, k} )$, for all $k \in \N$, then $f(D(1)) \in \cB_{(\cV,b)}$, and we are done.
Define $A_k = \la_k f^{-1} (\cV_{s, k} )$, for all $k \in \N$ and pick $\la_k \in K$ with $|\la_k| = b_k$. Since $A_k$ is not meagre and consequently not nowhere dense, the interior of $\ol{A}_k$ is not empty. Hence there exist $\ol{y}_k \in \ol{A}_k$ and $\rho_k < \frac{1}{k+1}$ such that $\ol{y}_k + D(2 \rho_k) \subset \ol{A}_k$. Since $\ol{y}_k \in \ol{A}_k$, there is a $y_k \in A_k$ such that $y_k \in \ol{y}_k + D(\rho_k)$, thus $$y_k + D(\rho_k) = (y_k - \ol{y}_k) + (\ol{y}_k + D(\rho_k)) \subset \ol{y}_k + D(2 \rho_k) \subset \ol{A}_k.$$
So, $D(\rho_k) \subset \ol{A}_k - y_k$ which implies $B(\rho_k) \subset 2 \ol{A}_k$ if $K$ is Archimedean and $D(\rho_k) \subset \ol{A}_k$ if $K$ is non-Archimedean.
So fix $n \in \N$ and let $x \in \ol{f^{-1} ( \wtilde{\cV}_{s,n} )}.$ Then there is a $u_n \in f^{-1} ( \wtilde{\cV}_{s,n} )$ with $x - u_n \in D(\rho_{n +1}).$ $$x - u_n + y_{n+1} \in D(\rho_{n +1}) + y_{n+1} \subset \ol{A}_{n+1}.$$ So, there is a $u_{n+1} \in A_{n+1}$ with $(x - u_n + y_{n+1}) - u_{n+1} \in D(\rho_{n +2})$ and inductively we find $u_k \in A_k$, $k > n$, such that we have $$x - \sum_{k=n}^l u_k + \sum_{k=n+1}^l y_k \in D(\rho_{l + 1}),$$ for $l > n$. Hence, the series $x - \underset{k = n}{\overset{\infty}\sum} u_k + \underset{k = n + 1}{\overset{\infty}\sum} y_k$ converges to $0$, since $\rho_{l +1} \to 0$. Define $v_k = f(u_k )$ and $z_k = f (y_k )$. Then $v_n \in \cV_{s, n}$, $z_n \in \la_n \cV_{s,n}$, and $\forall k > n$ one has that $v_k , z_k \in \la_k \cV_{s,k}$. It follows from (4) of Definition \[defn:born\_web\] that $\underset{k \in \N}\sum v_k$ and $\underset{k \in \N}\sum z_k$ converge bornologically in $F$ and moreover $$y = \sum_{k \ge n} v_k - \sum_{k \ge n + 1} z_k = v_n + \sum_{k \ge n + 1} v_k - \sum_{k \ge n + 1} z_k \in \wtilde{\cV}_{s, n} + \wtilde{\cV}_{s, n} - \wtilde{\cV}_{s, n}.$$ Then, since $f$ has bornologically closed graph, we infer $0 = f (0) = f (x) - y$, i.e. $f (x) = y$ which shows that $$f(x) \in f(\wtilde{\cV}_{s,n})$$ if $K$ is non-Archimedean and $$f(x) \in 3 f(\wtilde{\cV}_{s,n})$$ if $K$ is Archimedean.
Therefore, we can deduce that $$D(\rho_k) \subset 2 \ol{A}_k = 2 \ol{\la_k f^{-1} (\cV_{s,k} )} \subset 2 \ol{\la_k f^{-1} (\wtilde{\cV}_{s,k} )} \subset 6 \la_k f^{-1} (\wtilde{\cV}_{s,k} )$$ if $K$ is Archimedean or $$D(\rho_k) \subset \ol{A}_k = \ol{\la_k f^{-1} (\cV_k )} \subset \ol{\la_k f^{-1} (\wtilde{\cV}_{s,k} )} \subset \la_k f^{-1} (\wtilde{\cV}_{s,k} )$$ if $K$ is non-Archimedean, completing the proof.
We conclude this section by proving some stability properties of bornological webs.
Bornological webs have the following stability properties:
1. if $E = \underset{i \in \N}\limind E_i$ is a monomorphic direct limit of webbed bornological vector spaces then $E$ is webbed;
2. if $(E, \cB)$ is webbed and $\cB'$ is another bornology on $E$ such that the identity $(E, \cB) \to (E, \cB')$ is bounded, then $(E, \cB')$ is webbed;
3. every closed subspace of a webbed bornological vector space is a webbed bornological vector space;
4. every countable projective limit of webbed bornological vector spaces is a webbed bornological space;
5. every countable coproduct of webbed bornological vector spaces is a webbed bornological vector space.
The proof of this proposition is similar to the proofs of Proposition \[prop:stability\]. For details we refer to [@G], Theorem 4.11 where a full proof is given when the base field is Archimedean.
Applications
============
In this last section we deduce some consequences from the theorems we have proved so far. We start by discussing the more classical ones: various forms of Isomorphisms Theorems and then we deduce de Wilde’s Theorem for arbitrary base field. We conclude showing some applications to the theory of bornological algebras from [@Bam] and [@BaBeKr]. We remark, that one of the main differences in this exposition with respect to other works in literature, is that we work over any complete non-trivially valued field $K$, treating on the same footing, for as much as it is possibile, the Archimedean and the non-Archimedean sides of the theory.
Isomorphism theorems {#isomorphism-theorems .unnumbered}
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\[thm:iso\] Let $f: E \to F$ be a bijective bounded morphism between separated bornological vector spaces with $F$ complete and $E$ with a net compatible with the bornology (resp. is webbed), then $F$ is an isomorphism of bornological vector spaces.
The map $f^{-1}: F \to E$ is a linear map between bornological vector spaces whose domain is complete and whose codomain has a net compatible with the bornology (resp. is webbed). Then, the graph of $f^{-1}$ coincides with the graph of $f$, up to swap domain with the codomain, thus is a closed subset of $F \times E$. So, by Theorem \[thm:net\] (resp. Theorem \[thm:web\]) $f^{-1}$ is a bounded map.
For any locally convex space $E$ let’s denote with $(E, \cB_\Ban)$ the vector space $E$ endowed with the vector space bornology of convex type on $E$ generated by all bounded Banach disks of $E$ and $E_\uborn = (E, \cB_\Ban)^t$, where ${}^t$ is the functor which associate to every bornological vector space the topological vector space identified by the bornivorous subset. $E_\uborn$ is called the *ultrabornologification* of $E$.
\[defn:born\_ultra\] Let $E$ be a locally convex space over $K$. $E$ is called *bornological* if $E \cong (E^b)^t$. $E$ is called *ultrabornological* if $E \cong E_\uborn$.
Let $f: E \to F$ be a bijective continuous morphism between locally convex spaces. Suppose that $E$ ultrabornological with $E^b$ endowed with a net of webbed, $F$ is bornological and $F^b$ complete. Then, $f^{-1}$ is continuous.
Direct consequence of Theorem \[thm:iso\], Definition \[defn:born\_ultra\] and the fact that the functors ${}^b$ and ${}^t$ are adjoints.
Let $f: E \to F$ be a bijective continuous morphism $E$ and $F$ Fréchet then $f$ is isomorphism.
Fréchet spaces are bornological and ultrabornological.
Although one can use the Closed Graph Theorem to deduce the Isomorphism Theorems, it cannot be used to deduce Open Mapping Theorems for bornological spaces, that under some hypothesis a surjective bounded map must be a quotient map.
We conclude this section discussing a result that is not a consequence of the Bornological Closed Graph Theorems we are discussing, but for which we think it is important to have a proof that extends the classical one given over $\R$ and $\C$ to any valued base field. This result is Buchwalter’s theorem, which is an analogous of the Open Mapping Theorem for bornological space. The interest for this result rely on the fact that such kind of results are very rare for bornological spaces. We need a definition and a couple of lemmata, which are adaptations of [@WAEL2], section 1.5.
\[defn:compatible\_completant\] Let $E$ be a vector space, and $B_1$, $B_2$ be two Banach disks of $E$. We say that $B_1$ and $B_2$ are *compatible* if their intersection is a Banach disk.
\[lemma:banach\_disks\] Two Banach disks $B_1$ and $B_2$ of a vector space $E$ are compatible if $B_1 + B_2$ does not contain a non-zero vector subspace.
We shall write $E_1 = E_{B_1}$ and $E_2 = E_{B_2}$. We let also $E_1 + E_2$ be the semi-normed space absorbed by $B_1 + B_2$ with the Minkowski functional associated to $B_1 + B_2$. We have the short exact sequence $$0 \to E_1 \cap E_2 \to E_1 \oplus E_2 \to E_1 + E_2 \to 0,$$ where the first morphism maps $x \in E_1 \cap E_2$ to $(x, -x) \in E_1 \oplus E_2$ and the second morphism maps $(x, y) \in E_1 \oplus E_2$ to $x + y \in E_1 + E_2$. The space $E_1 + E_2$ is normed if and only if the kernel of the map $E_1 \oplus E_2 \to E_1 + E_2$ is closed and the kernel is closed if and only if its unit ball is a Banach disk. The unit ball of the kernel is the image of the unit ball of $E_1 \cap E_2$ by an injective map.
(Grothendieck’s lemma) Let $E$ be a vector space, let $B$ be a Banach disk in $E$ and $ \{ B_n \}_{n \in \N}$ be an increasing sequence of Banach disks of $E$ such that $B = \underset{n = 0}{\overset{\infty}\bigcup} B_n$. Then $B$ is absorbed by some $B_n$.
It is an immediate consequence of Lemma \[lemma:banach\_disks\] that the disk $B$ is compatible with all the $B_n$ for all $n \in \N$, since one has that $B_n \subset B$. The space $E_B$ is a Banach space therefore it is a Baire space. According to the Baire’s Theorem, for some $n \in \N$, $\ol{B_n}$ has a non-empty interior, where the closure is taken in $E_B$. As $B_n$ is absolutely convex, $\ol{B_n}$ contains a ball $D(\a)$ of radius $\a > 0$ in the Banach space $E_B$. Let $x_0 \in D(\a) \subset \ol{B_n}$. We can find $y_0 \in B_n$ and $x_1 \in D(\a) \subset \ol{B_n}$ such that $x_0 = y_0 + \la x_1$, with $\la \in K^\times$ and $|\la| \le \frac{1}{2}$. Next we choose $y_1 \in B_n$ and $x_2 \in D(\a) \subset \ol{B_n}$ such that $x_1 = y_1 + \la x_2$, etc. For each $j \in \N$, we see that $$x_0 = \sum_{l = 0}^l \la^j y_l + \la^{ j-1} x_j.$$ The series $\underset{l = 0}{\overset{\infty}\sum} \la^j y_l$ converges in the Banach space $E_{B_n}$ and in that space, the norm of the sum is at most equal to $2$. So the sum $\underset{l = 0}{\overset{\infty}\sum} \la^j y_l$ belongs to $\la^{-1} B_n$. In the Banach space $E_B$, $\la^{ -j} x_k \to 0$. Thus $x_0 \in \la^{-1}B_n$ and $D(\a) \subset \la^{-1} B_n$.
(Buchwalter’s theorem) Let $E$ be a complete bornological vector space whose bornology has a countable basis and $F$ be a complete bornological vector space. Let $f: E \to F$ be a surjective bounded linear map. Then $f$ is a strict epimorphism.
Let $\{ B_n \}_{n \in \N}$ be a basis of the bornology of $E$. We assume that the $B_n$ are Banach disks and that for all $n \in \N$, $B_n \subset B_{n+1}$. Let $C$ be a bounded Banach disk in $F$. Since $f$ is bounded, the subsets $f(B_n)$ are bounded Banach disks and $F = \underset{n = 0}{\overset{\infty}\bigcup} f(B_n)$. Lemma \[lemma:banach\_disks\] implies that for all $n \in \N$, the subset $f(B_n) \cap C$ is a Banach disk as $f(B_n) + C$, being bounded in $F$, does not contain any non-zero subspace. Moreover $C = \underset{n = 0}{\overset{\infty}\bigcup} (f(B_n ) \cap C)$. Then Grothendieck’s lemma shows that $C$ is absorbed by one of the sets $f(B_n)$. It follows that there exist $\la \in K^\times$ and $n \in \N$ such that $C \subset \la f(B_n )$, which yields $C \subset f(\la B_n )$. Thus, we showed that the map $f$ is a strict epimorphism.
De Wilde’s Theorem {#de-wildes-theorem .unnumbered}
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Before proving our generalization of the de Wilde’s Theorem, let’s see how Theorem \[thm:web\] generalizes Theorem \[thm:net\].
\[prop:net\_web\] Let $(E, \cB)$ be a separated convex bornological vector space and $\cN$ a net on $E$ which is compatible with $\cB$. Then, for every $b : \N^\N \to |K^\times|^\N$ satisfying $(1)$ of Definition \[defn:net\_compatible\], the couple $(\cN, b)$ is a bornological web on $E$ such that $$\cB_{(\cN,b)} \subset \cB$$
The first three conditions of the definition of bornological web are direct consequences of the definition of net. The last one is imposed by hypothesis and by condition $(1)$ of Definition \[defn:net\_compatible\]. So, we need only to check that $\cB_{(\cN,b)} \subset \cB$. And this follows directly from condition $(2)$ of Definition \[defn:net\_compatible\].
Theorem \[thm:net\] is consequence of \[thm:web\].
Direct consequence of Proposition \[prop:net\_web\].
Then, we need to introduce the topological version of the notion of web.
\[defn:top\_webs\] Let $E$ be a Hausdorff locally convex space. A map $\cW: \underset{k \in \N}\bigcup \N^k \to \sP(E)$ is called a *topological web* if
1. the image of $\cW$ consists of absolutely convex sets;
2. $\cW(\void) = E$;
3. Given a finite sequence $(n_0, \ldots, n_k)$, then $\cW(n_0, \ldots , n_k)$ is absorbed by $$\bigcup_{ n \in \N} \cW(n_0, \ldots, n_k, n).$$
4. for every finite sequence $(n_0, \ldots , n_k, n_{k+1})$ one has $$\cW(n_0, \ldots , n_k, n_{k+1}) + \cW(n_0, \ldots , n_k, n_{k+1}) \subset \cW(n_0, \ldots , n_k).$$
A separated locally convex space $E$ that carries a topological web is called *webbed locally convex space*. Moreover, we say that $\cW$ is *completing* if the following condition is satisfied: For every $s : \N \to \N$ and for every choice of $y_k \in \cW(s(1), \ldots, s(k))$ the series $$\sum_{k \in \N} y_k$$ converges topologically in $E$.
\[rmk:la\] Notice that condition (4) of the last definition when $K$ is non-Archimedean reduces to $$\cW(n_0, \ldots , n_k, n_{k+1}) \subset \cW(n_0, \ldots , n_k)$$ because in this case $\cW(n_0, \ldots , n_k, n_{k+1}) + \cW(n_0, \ldots , n_k, n_{k+1}) = \cW(n_0, \ldots , n_k, n_{k+1})$, and reduces to the condition $$2 \cW(n_0, \ldots , n_k, n_{k+1}) \subset \cW(n_0, \ldots , n_k)$$ when $K$ is Archimedean. Hence, from here on we define the constant $$\la = \begin{cases} 2 & \text{ if $K$ is Archimedean } \\ 1 & \text{ if $K$ is non-Archimedean }
\end{cases}.$$
Also for topological webs we will use the notation introduced in previous sections:
$$\cW_{s,k} \doteq \cW(s(0), \ldots , s(k)), \text{ where } s: \N \to \N.$$
We need some technical lemmata. The following is the generalization of Proposition 5.2.1 of [@J].
\[lemma:top\_equiv\] Let $E$ be a locally convex space and $\cW$ a topological web on $E$, then $\cW$ is completing if and only if for any open $0$-neighborhood $U \subset E$ and any $s: \N \to \N$ there exists a $k \in \N$ such that $\cW_{s, k} \subset U$.
Suppose that $\cW$ is completing. Then every sequence of elements $y_k \in \cW_{s, k}$ must be a zero sequence. Suppose that there exists a $0$-neighborhood such that for every $k \in \N$ one has $\cW_{s, k} \not\subset U$. In this way we can construct a sequence $y_k \in \cW_{s, k} - U$ which cannot be a zero sequence, thus $\cW$ cannot be completing.
For the reverse implication, consider $s: \N \to \N$ and $y_k \in \cW_{s, k}$ for each $k \in \N$. So, for each $0$-neighborhood $U \subset E$ we can find a $k_0 \in \N$ such that $\cW_{s, k_0} \subset U$, therefore $\cW_{s, k} \subset U$ for all $k \ge k_0$. Applying inductively (4) of Definition \[defn:top\_webs\] we get that for any $m, k \in \N$, with $k \ge k_0$ $$\sum_{n = 1}^p y_{k + n} \in \cW_{s, k + 1} + \ldots + \cW_{s, k + p} \subset \cW_{s, k} \subset U.$$ This shows that the sequence of partial sums $\underset{n = 1}{\overset{p}\sum} y_{k + n}$ for $p \to \infty$, is a zero sequence for the topology of $E$.
\[lemma:new\_web\] Let $E$ be a Hausdorff locally convex space which is endowed with a topological web $\cW$. Then the map $\cV: \bigcup_{k \in \N} \N^k \to \sP(E)$, defined by $$\cV (n_0 , \ldots , n_k ) := \frac{1}{\la^k} \cW (n_0 , \ldots , n_k ),$$ is again a topological web on $E$, where $\la \in K$ is as in Remark \[rmk:la\]. Moreover, if $\cW$ is completing then also $\cV$ is.
$\cV$ clearly satisfies the first three conditions of Definition \[defn:top\_webs\], so let’s check the fourth. $$\la V (n_0 , \ldots , n_k , n_{k+1}) = \frac{1}{\la^{k -1}} \cW (n_0 , \ldots , n_k , n_{k+1} ) \subset \frac{1}{\la^k}W (n_0 , \ldots , n_k ) = \cV (n_0 , \ldots , n_k ).$$ Finally, let’s suppose that $\cW$ is completing. Then, since the sets $\cW(n_0, \ldots , n_k)$ are absolutely convex they are in particular balanced, so $$\cV (n_0, \ldots , n_k) = \frac{1}{\la^k} \cW(n_0, \ldots , n_k) \subset \cW(n_0, \ldots , n_k),$$ therefore the completing condition for $\cW$ implies the completing condition for $\cV$.
From now on we will consider only completing topological webs.
\[defn:polar\] Let $E$ be a $K$-vector space. A semi-norm $p$ on $E$ is called *polar* if $$p = \sup \{ |f| | f \in E^*, |f| \le p \}$$ where $E^*$ denotes the algebraic dual of $E$. A locally convex space $E$ is said to be polar if its topology can be defined by polar semi-norms.
If $K$ is spherically complete (hence also for $\R$ and $\C$), all locally convex spaces are polar, cf. Theorem 4.4.3 of [@PGS].
Let $E$ be a locally convex space and $X \subset E$ any subset we define $$X^\circ \doteq \{ f \in E' | |f(x)| \le 1, \forall x \in E \}$$ $$X^{\circ \circ} \doteq \{ x \in E | |f(x)| \le 1, \forall f \in X^\circ \}$$ where $E'$ is the continuous dual of $E$.
In next lemma we use the notation $S_K = l^1_K$ if $K$ is Archimedean and $S_K = c_K^0$ if $K$ is non-Archimedean, where $l^1_K$ is the $K$-Banach space of summable sequences and $c_K^0$ is the $K$-Banach space of zero sequences.
\[lemma:polar\] Let $E$ be a polar locally convex space over a spherically complete field $K$. Let $\{y_k\}_{k \in \N}$ be a sequence of elements of $E$ such that $\underset{k \in \N}\sum \mu_k y_k$ converges for each possible choice of $\mu_k \in K^{\circ}$. Then, $B = (\{y_k\}_{k \in \N})^{\circ \circ}$ is a Banach disk in $E$.
By hypothesis the series $\underset{k \in \N}\sum \mu_k y_k$ is convergent when we choose $\{ \mu_k\}_{k \in \N} \in S_K$. Let’s denote by $D \subset S_K$ the unit ball and with $e_k \in S_K$, for each $k \in \N$, the elements of the canonical Schauder base.
The map $T: S_K \to E$ defined $\{ \mu_k\} \mapsto \underset{k \in \N}\sum \mu_k y_k$ sends $D \subset S_K$ into $B = (\{y_k\}_{k \in \N})^{\circ \circ}$. Moreover, $T$ is adjoint to the map $$E' \to c_K^0: u \mapsto \{ \lt u , y_k \gt\}_{k \in \N}$$ so $T$ is $(\sigma(S_K, c_K^0), \sigma(E, E'))$-continuous, where $\sigma$ stands for the usual notation for weak topologies. The fact that $T(D)$ is $\sigma(E, E')$-bounded implies that $T(D)$ is $\sigma(E, E')$-compact for $K$ Archimedean (this follows from the Bourbaki-Alaoglu Theorem) and $\sigma(E, E')$-c-compact for $K$ non-Archimedean, because we are supposing $K$ spherically complete (for $K$ non-Archimedean and spherically complete we can apply Theorem 5.4.2 and 6.1.13 of [@PGS] to deduce weak-c-compactness from weak-boundedness). Hence $T(D)$ is weakly closed also for $K$ non-Archimedean because we can apply Theorem 6.1.2 (iii) of [@PGS]. Thus, since $T e_k = y_k$ for each $k$, and $T(D)$ is absolutely convex the Bipolar Theorem implies $T(D) = B$ (see Theorem 5.2.7 for the non-Archimedean version of the Bipolar Theorem). Moreover, $B$ is a bounded (because $E$ is polar and we can apply Theorem 5.4.5 of [@PGS]) Banach disk.
So, from now on until the end of this section $K$ will be supposed to be spherically complete.
\[lemma:web\_ultraborn\] If $(E, \cW)$ is a polar webbed locally convex space, then also $E_\uborn$ is a webbed locally convex space.
We shall show that given a topological web $\cW$ on $E$, then the topological web $\cV$ associated to $\cW$ as in Lemma \[lemma:new\_web\] is a topological web for $E_\uborn$. The only non-trivial thing to check is that $\cV$ is completing for the topology of $E_\uborn$. So, we reproduce here the argument of Theorem 13.3.3 of [@J] adapting it for any base field.
Let’s consider $s: \N \to \N$ and $x_k \in \cV_{s, k}$ for each $k \in \N$. Then, by condition (4) of Definition \[defn:born\_web\] we know that $$\la \cV_{s, k + 1} \subset \cV_{s, k} \then \la^{k -1} \cV_{s, k + 1} \subset \cW_{s, k}.$$ Thus, for each $x_k$ we can write $$x_k = \frac{y_k}{\la^{k-1}}, \ \ \text{ with } y_k \in \cW_{s, k}$$ and since $\cW_{s, k}$ are balanced, the assumption that $\cW$ is completing implies that the series $\underset{k \in \N}\sum \mu_k y_k$ converges for the topology of $E$ for each possible choice of $\mu_k \in K^\circ$. Thus we can apply Lemma \[lemma:polar\] to deduce that $B = (\{y_k\}_{k \in \N})^{\circ \circ}$ is a bounded Banach disk of $E$. This implies that $\underset{k \in \N}\sum x_k$ converges in $E_B$ and so also in $E_\uborn$.
\[lemma:web\_top\_born\] Let $E$ be a polar locally convex space. If $E$ is webbed, then $(E, \cB_\Ban)$ is a webbed convex bornological space with a bornological web $(\cV, b)$ that may be chosen in such a way that $\cB_{(\cV,b)}$ is finer than the von Neumann bornology of $E_\uborn$.
Let $\cW$ be a topological web on $E$. By Lemma \[lemma:new\_web\] $$\cV (n_0, \ldots , n_k) \doteq \frac{1}{\la^k} \cW(n_0, \ldots , n_k)$$ is another topological web of $E$. For $s: \N \to \N$ define $b(s): \N \to |K^\times|$ to be constant with value $1$. We claim that $(\cV, b)$ is a bornological web for $(E, \cB_\Ban)$. The first three conditions of bornological web are clear, so only the last one need to be checked. Given a sequence of elements $x_k \in \cV_{s, k}$ then we can define $$x_k = \frac{y_k}{\la^{k-1}}, \ \ \text{ with } y_k \in \cW_{s, k}$$ and apply Lemma \[lemma:polar\] to the sequence $\{y_k\}_{k \in \N}$ for obtaining that $\underset{k \in \N}\sum x_k$ converges in $E_B$, where $B = (\{y_k\}_{k \in \N})^{\circ \circ}$. So $\{y_k\}_{k \in \N}$ is a zero sequence in $E_B$ which implies that $\underset{k \in \N}\sum x_k$ converges for the topology of $E_\uborn$.
In order to prove that the $(\cV, b)$ is finer than the von Neumann bornoloy of $E_\uborn$, first let’s notice that $(\cV, b)$ is also a bornological web for $(E_\uborn)^b$, since the bounded Banach disks that generate $\cB_\Ban$ are all bounded subsets for $E_\uborn$. Next, let $\{\mu_k\}_{k \in \N}$ be a sequence with values in $|K^\times|$. For every choice of $x_k \in \cV_{s,k}$, $k \in \N$, the value of $$\sum_{k > n + 1}^\infty x_k$$ belongs to $\ol{\cV_{s,n}}$. Hence $\tilde{\cV}_{s,n} \subset \ol{\cV_{s,n}}$, but $\underset{k \in \N}\bigcap \mu_k \ol{\cV_{s,n}} $ is bounded, since for any given closed and absolutely convex $0$-neighbourhood $U$ there is an index $n \in \N$ such that $\cV_{s,n} \subset U$, by Lemma \[lemma:top\_equiv\]. Hence, $\mu_n \ol{\cV_{s,n}}$, and consequently $\underset{k \in \N}\bigcap \mu_k \ol{V_{s,k}}$, is absorbed by $U$. Thus, we proved that $\cB_{(\cV,b)}$ is finer with respect to the canonical bornology of $E_\uborn$.
Since $\la = 1$ for $K$ non-Archimedean, last lemma proves that a topological web for a locally convex space $E$ is automatically a topological web for $E_\uborn$ in this case, and this essentially follows from the fact that for non-Archimedean base fields a sequence is summable if and only if it is a zero sequence.
\[lemma:compatible\_neumann\] Let $E$ be a polar webbed locally convex space. Then $(E_\uborn, \cB_\Ban)$ carries a bornological web $(\cV, b)$ such that the corresponding convex bornology $\cB_{(\cV,b)}$ is contained in the von Neumann bornology of $E_\uborn$.
By Lemma \[lemma:web\_ultraborn\] the ultrabornologification $E_\uborn$ of a polar webbed locally convex space $E$ is webbed. Then, Lemma \[lemma:web\_top\_born\] applied to $E$ yields the assertion.
Finally we are able to deduce the main result of this sectiom. Here we propose again the full statement of the theorem as already stated in the introduction.
If $E$ is an ultrabornological locally convex space and $F$ is a polar webbed locally convex space defined over a sphericaly complete field $K$, then every linear map $f: E \to F$ which has bornologically closed graph with respect to $(E, \cB_\Ban)$ and $(F, \cB_\Ban)$, is continuous even if regarded as a map $f: E \to F_\uborn$.
We consider $(E, \cB_\Ban)$ as domain space of $f$ and $(F_\uborn, \cB_\Ban)$ as the codomain of $f$. Notice that the family of bounded Banach disks of $F$ and of $F_\uborn$ coincide and that, by Lemma \[lemma:web\_ultraborn\], $F_\uborn$ carries a bornological web $(\cV, b)$ such that $\cB_{(\cV,b)}$ is finer than the canonical bornology of $F_\uborn$. By hypothesis $f$ has bornologically closed graph with respect to $(E, \cB_\Ban)$ and $(F, \cB_\Ban)$, therefore we may apply Theorem \[thm:web\] in order to see that $f$ is bounded, which also implies that $f$ is bounded if regarded as a map from $(E, \cB_\Ban)$ to $(F_\uborn)^b$. Since $E$ is ultrabornological, we get that $f: E \to F_\uborn$ is continuous.
The proof of de Wilde’s Theorem presented here closely follow the proof given in [@G], adapting it in order to treat on the same footing both the Archimedean and the non-Archimedean base fields case. The main difference is the need of polarity assumption on $E$ and spherically completeness assumption on $K$ which are automatic in [@G].
Applications to bornological algebras {#applications-to-bornological-algebras .unnumbered}
-------------------------------------
In this last section we show the applications which gave us the main motivations for writing down the proofs of the theorems discussed so far. The material of this section is mainly taken from one of the key technical point of author’s Ph.D. thesis, [@Bam].
\[defn:born\_algebra\] A bornological $K$-vector space $A$ equipped with a bilinear associative function $A \times A \to A$, called *multiplication map*, is said to be a *bornological algebra* if the multiplication map is bounded. We always suppose that $A$ has an identity and that the multiplication is commutative. A morphism of bornological algebras is a bounded linear map that preserves multiplication and maps $1$ to $1$.
Our next proposition is a generalization of Proposition 3.7.5/1 of [@BGR], which holds for Banach algebras.
\[prop:algebra\_morphism\] Let $A, B$ be bornological algebras over $K$ for which the underlying bornological vector space of $A$ is complete and the one of $B$ is a webbed bornological vector space and let $\phi: A \to B$ be an algebra morphism. Suppose that in $B$ there is a family of ideals $\cI$ such that
each $I \in \cI$ is bornologically closed in $B$ and each $\phi^{-1}(I)$ is bornologically closed in $A$;
for each $I \in \cI$ one has $\dim_K B/I < \infty$;
$\underset{I \in \cI}\bigcup I = (0)$. Then, $\phi$ is bounded.
Let $I \in \cI$ and let’s denote $\be: B \to B/I$ the qutient epimorphism and $\psi = \be \circ \phi$. Let $\ol{\psi}: A/\Ker(\psi) \to B/I$ denote the canonical injection, which give us the following commutative diagram $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=2.6em, column sep=2.8em,
text height=1.5ex, text depth=0.25ex]
{ A & B \\
A/\Ker(\psi) & B/I \\};
\path[->,font=\scriptsize]
(m-1-1) edge node[auto] {$\phi$} (m-1-2);
\path[->,font=\scriptsize]
(m-1-1) edge node[auto] {} (m-2-1);
\path[->,font=\scriptsize]
(m-1-2) edge node[auto] {$\be$} (m-2-2);
\path[->,font=\scriptsize]
(m-2-1) edge node[auto] {$\ol{\psi}$} (m-2-2);
\path[->,font=\scriptsize]
(m-1-1) edge node[auto] {$\psi$} (m-2-2);
\end{tikzpicture}.$$ We have that $\Ker(\psi) = \phi^{-1}(I)$, therefore, since by hypothesis $B/I$ is finite dimensional, also $A/\Ker(\psi)$ is. Thus, both $B/I$ and $A/\Ker(\psi)$ are finite dimensional separated bornological algebras, when they are equipped with the quotient bornology. Therefore, their underlying bornological vector spaces are isomorphic to the direct product of a finite number of copies of $K$. So, $\ol{\psi}$ is bounded and this implies the boundedness of $\psi$.
Let’s consider a sequence $\{a_n\}_{n \in \N} \subset A$ such that $\underset{n \to \infty}\lim a_n = 0$, bornologically. Then $$\be(\lim_{n \to \infty} \phi(a_n)) = \lim_{n \to \infty} (\be \circ \phi)(a_n)$$ since $\be$ is bounded, and therefore $$\lim_{n \to \infty} (\be \circ \phi)(a_n) = \lim_{n \to \infty} \psi(a_n) = \psi(\lim_{n \to \infty} a_n) = 0$$ which implies that $\phi(\underset{n \to \infty}\lim a_n) \in I$. Since this must be true for any $I \in \cI$ and $\underset{I \in \cI}\bigcup I = (0)$ we deduce that $\phi(\underset{n \to \infty}\lim a_n) = 0$. This implies that the graph of $\phi$ is bornologically closed, because then for any sequence $\{ a_n \}_{n \in \N}$ in $A$ such that $\underset{n \to \infty}\lim a_n = a$ one has that $$\underset{n \to \infty}\lim ( a_n, \phi(a_n)) = (a, \phi(a)) \in \Gamma(\phi).$$ Now we can apply Theorem \[thm:web\] to infer that $\phi$ is bounded.
Let $\rho = (\rho_1, ..., \rho_n) \in \R_+^n$ be a polyradius. We denote by $W_K^n(\rho)$ the algebra of overconvergent (also called germs) analytic functions on the polycylinder of polyradius $\rho$. One can check that there is a bijection $$W_K^n(\rho) \cong \limind_{r > \rho} T_K^n(r)$$ where $T_K^n(r)$ denotes the algebra of strictly convergent analytic function on the polycylinder of polyradius $r$. Since $T_K^n(r)$ are $K$-Banach algebras and by the Identity Theorem for analytic functions the system morphism of the inductive system are monomorphism, then $W_K^n(\rho)$ has a canonical structure of complete bornological algebra. When $\rho = (1, \ldots, 1)$ we will simply write $W_K^n$. For a detailed discussion of the algebras $W_K^n(\rho)$, their properties and their relations with the classical affinoid algebras and the algebras of germ of analytic functions on compact Stein subsets of complex analytic spaces the reader can refer to chapter 3 of [@Bam].
\[defn:dagger\_algebra\] A *strict $K$-dagger affinoid algebra* is a complete bornological algebra which is isomorphic to a quotient $W_K^n/I$, for an ideal $I \subset W_K^n$.
A (non-strict) *$K$-dagger affinoid algebra* is a bornological algebra which is isomorphic to a quotient $$\frac{W_K^n(\rho)}{I}$$ for an arbitrary polyradius $\rho$.
It is easy to check that the underlying bornological vector space of a $K$-dagger affinoid algebra is an LB-space, hence in particular it is a webbed bornological vector space.
From Proposition \[prop:algebra\_morphism\] we can deduce the following result.
Every morphism between dagger affinoid algebras is bounded.
If $\phi: A \to B$ is an algebra morphism between strict $K$-dagger affinoid algebras then we can apply Proposition \[prop:algebra\_morphism\] choosing as family $\cI$ the family of all powers of maximal ideals of $B$. The only non-trivial fact to check for applying Proposition \[prop:algebra\_morphism\] is the requirement that all the elements of $\cI$ must be bornologically closed in $B$ and their preimages bornologically closed in $A$. But in Section 3.2 of [@Bam] is proved that all ideals of dagger affinoid algebras are bornologically closed, hence it follows that $\phi$ is bounded.
The non-strict case can be reduced to the strict case noticing that any non-strict dagger affinoid algebra can be written as a direct limit of strict ones and that every algebra morphism can be written as a morphism of direct systems of algebras, as explained in Section 3.2 of [@Bam]. Therefore, every morphism between non-strict dagger affinoid algebras can be written as a direct limit of bounded ones, hence it is bounded.
We conclude this overview of applications of the bornological Closed Graph Theorem by saying that the last proposition can be generalized to encompass a more general class of bornological algebras used in analytic geometry: Stein algebras and (at least a big subclass of) quasi-Stein algebras, both dagger and non-dagger. The arguments for showing the boundedness of algebra morphisms for this class of bornological algebras become more involved and do not fit in this discussion. The reader can refer to [@BaBeKr] for such a study.
[A]{}
Bambozzi, F., “On a generalization of affinoid varieties”, Ph.D. thesis, University of Padova, 2013, available at http://arxiv.org/pdf/1401.5702.pdf. Bambozzi, F., Ben-Bassat, O., “Dagger Geometry as Banach Algebraic Geometry”, preprint. Bambozzi, F., Ben-Bassat, O., Kremnizer, K., “Stein Domains in Banach Algebraic Geometry”, preprint. Bosch, S., Güntzer, U., Remmert, R., [Non-Archimedean analysis. A systematic approach to rigid analytic geometry,]{} Springer, 1984. Cámara, A., “Interaction of Topology and Algebra in Arithmetic Geometry.” (2013). Durov, N. “New approach to Arakelov geometry.”, arXiv preprint arXiv:0704.2030 (2007). De Wilde, M. “Ultrabornological spaces and the closed graph theorem.” Bull. Soc. Roy. Sci. Liege 40 (1971): 116-118. Gach, F., “A note on closed graph theorems.”, Acta Math. Univ. Comenianae 75.2 (2006): 209-218. Gilsdorf, T., E., and Jerzy K. “On some non-archimedean closed graph theorems.” LECTURE NOTES IN PURE AND APPLIED MATHEMATICS (1997): 153-158. Schiffmann, Jacquet, Ferrier, Gruson, Houzel, “Seminaire Banach”, Lecture Notes in Mathematics 277, Edited by C. Houzel, Springer-Verlag, 1972. Hogbe-Nlend, H., “Bornologies and functional analysis: introductory course on the theory of duality topology-bornology and its use in functional analysis”. Vol. 26. Elsevier, 1977. Hogbe-Nlend, H., “Théorie des bornologies et applications.”, Springer, 1971. Hogbe-Nlend, H., and Moscatelli, V. B., “Nuclear and conuclear spaces.”, Elsevier, 2011. Jarchow, H. “Locally convex spaces.”, Stuttgart: BG Teubner, 1981. Meyer, R., “Local and Analytic Cyclic Homology”, European Mathematical Society, 2007. Meyer, R., “Smooth group representations on bornological vector spaces.”, Bulletin des sciences mathematiques 128.2 (2004): 127-166. Meyer, R. “Bornological versus topological analysis in metrizable spaces.”, Contemporary Mathematics 363 (2004): 249-278. Perez-Garcia, C., and Schikhof, W. H., “Locally convex spaces over non-Archimedean valued fields.” Cambridge University Press, 2010. Popa, N. “Le théoreme du graphe (b)-fermé.” CR Acad. Sci. Paris Sér. A–B 273 (1971): A294-A297. Waelbroeck, L. “Topological vector spaces and algebras.”, Berlin: Springer-Verlag, 1971. Waelbroeck, L., Guy N. “Bornological quotients.”, Classe des sciences, Académie Royale de Belgique, 2005.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
We develop a new approach for building cryptographic implementations. Our approach goes the last mile and delivers assembly code that is provably functionally correct, protected against side-channels, and as efficient as hand-written assembly. We illustrate our approach using -, one of the mandatory ciphersuites in TLS 1.3, and deliver formally verified vectorized implementations which outperform the fastest non-verified code.
We realize our approach by combining the framework, which offers in a single language features of high-level and low-level programming, and the proof assistant, which offers a versatile verification infrastructure that supports proofs of functional correctness and equivalence checking. Neither of these tools had been used for functional correctness before. Taken together, these infrastructures empower programmers to develop efficient and verified implementations by game hopping, starting from reference implementations that are proved functionally correct against a specification, and gradually introducing program optimizations that are proved correct by equivalence checking.
We also make several contributions of independent interest, including a new and extensible verified compiler for , with a richer memory model and support for vectorized instructions, and a new embedding of in .
author:
-
bibliography:
- 'dblp.bib'
- 'abbrev3.bib'
- 'crypto.bib'
title: 'The Last Mile: High-Assurance and High-Speed Cryptographic Implementations'
---
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Given a rational homology 3–sphere $M$ with $|H_1(M,\Z)|=b$ and a link $L$ inside $M$, colored by odd numbers, we construct a unified invariant $I_{M,L}$ belonging to a modification of the Habiro ring where $b$ is inverted. Our unified invariant dominates the whole set of the $SO(3)$ Witten–Reshetikhin–Turaev invariants of the pair $(M,L)$. If $b=1$ and $L=\emptyset$, $I_M$ coincides with Habiro’s invariant of integral homology 3–spheres. For $b>1$, the unified invariant defined by the third author is determined by $I_M$. One of the applications are the new Ohtsuki series (perturbative expansions of $I_M$ at roots of unity) dominating all quantum $SO(3)$ invariants.'
address:
- 'Institut für Mathematik, Universität Zurich, Winterthurerstrasse 190, 8057 Zürich, Switzerland'
- 'Department of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332–0160, USA '
author:
- Anna Beliakova
- Irmgard Bühler
- Thang Le
title: |
A unified quantum SO(3) invariant\
for rational homology 3–spheres
---
Introduction {#introduction .unnumbered}
============
Background {#background .unnumbered}
----------
The $SU(2)$ Witten–Reshetikhin–Turaev (WRT) invariant is defined for any closed oriented 3–manifold $M$ and any root of unity $\xi$ [@Tu]. Kirby and Melvin [@KM] introduced the $SO(3)$ version of the invariant $\tau_M(\xi)\in \Q(\xi)$ for roots of unity $\xi$ of odd order. If the order of $\xi$ is prime, then by the results of Murakami [@Mu] (also Masbaum–Roberts [@MR]), $\tau_M(\xi)$ is an algebraic integer. This integrality result was the starting point for the construction of finite type 3–manifold invariants, Ohtsuki series [@Ohtsukibook], integral TQFTs, representations of the mapping class group over $\Z[\xi]$ [@GM], and categorification of quantum 3–manifold invariants [@Kho]. The proofs in [@Mu] and [@MR] depend heavily on the arithmetic of $\Z[\xi]$ for a root of unity $\xi$ of [*prime*]{} order and do not extend to other roots of unity.
Is it true that $\tau_M(\xi)$ is always an algebraic integer (belongs to $\Z[\xi]$), even when the order of $\xi$ is not a prime? The positive answer to this question was given first for [*integral homology spheres*]{} by Habiro [@Ha], and then for arbitrary 3–manifolds by the first and third author [@BL], in connection with the study of “strong integrality”.
What Habiro proved for integral homology 3–spheres is actually much stronger than integrality. For any integral homology 3–sphere $M$, Habiro [@Ha] constructed a [*unified invariant*]{} $J_M$ whose evaluation at any root of unity coincides with the value of the Witten–Reshetikhin–Turaev invariant at that root. Habiro’s unified invariant $J_M$ is an element of the following ring (Habiro’s ring) $$\Habiro:=\lim_{\overleftarrow{\hspace{2mm}k\hspace{2mm}}}
\frac{\Z[q]}{
((q;q)_k)}, \qquad \text{ where} \quad (q;q)_k = \prod_{j=1}^k (1-q^j).$$ Every element $f(q)\in \Habiro$ can be written as an infinite sum $$f(q)= \sum_{k\ge 0} f_k(q)\, (1-q)(1-q^2)...(1-q^k),$$ with $f_k(q)\in \Z[q]$. When $q=\xi$, a root of unity, only a finite number of terms on the right hand side are not zero, hence the right hand side gives a well–defined value, called the evaluation $\ev_\xi(f(q))$. Since $f_k(q)\in \Z[q]$, $\ev_\xi(f(q))\in \Z[\xi]$ is an algebraic integer. The fact that the unified invariant belongs to $\Habiro$ is stronger than just integrality of $\tau_M(\xi)$. We will refer to it as “strong” integrality.
The Habiro ring has beautiful arithmetic properties. Every element $f(q) \in \Habiro$ can be considered as a function whose domain is the set of roots of unity. Moreover, there is a natural Taylor series for $f$ at every root of unity. Two elements $f,g \in \Habiro$ are the same if and only if their Taylor series at a root of unity coincide. In addition, each function $f(q) \in \Habiro$ is totally determined by its values at, say, infinitely many roots of order $3^n,\, n\in \N$. Due to these properties the Habiro ring is also called a ring of “analytic functions at roots of unity” [@Ha]. Thus belonging to $\Habiro$ means that the collection of the $SO(3)$ WRT invariants is far from a random collection of algebraic integers; together they form a nice function.
Perturbative expansion at 1 of WRT invariants for rational homology 3–spheres was first constructed by Ohtsuki in the case when the order of the quantum parameter $\xi$ is prime [@Oh]. General properties of the Habiro ring imply that for any integral homology 3–sphere $M$, the Taylor expansion of the unified invariant $J_M$ at $q= 1$ coincides with the Ohtsuki series and dominates WRT invariants of $M$ at all roots of unity (not only of prime order).
To generalize Habiro’s results to rational homology 3–spheres, new ideas and techniques are required. Strong integrality of quantum invariants for rational homology 3–spheres was studied in [@Le] and [@BL]. Among other things, in [@Le] a unified invariant was constructed for the case when the order $r$ of the quantum parameter $\xi$ is coprime with $b$. In [@BL], it was proved that for any 3–manifold $M$ (not necessary a rational homology 3–sphere), the $SO(3)$ WRT invariant $\tau_M(\xi)$ is always an algebraic integer, i.e. $\tau_M(\xi)\in \Z[\xi]$ with no restriction on the order of $\xi$ at all. There we used a (2nd order) Laplace transform method [@BBL] and a difficult identity of Andrews [@And] in $q$–calculus, generalizing those of Rogers–Ramanujan.
Thus, although we have had integrality of all SO(3) WRT invariants, we still lacked a “strong integrality” for the case when $(r,b) \neq 1$. This is the main object of this paper.
In this paper we will generalize Habiro’s construction of the unified invariant to all rational homology 3–spheres. Our new unified invariant $I_M$ dominates SO(3) WRT invariants also in the case when the order $r$ of the quantum parameter is not coprime with $b=|H_1(M,\Z)|$. Although this includes the case $(r,b)=1$ of [@Le], the ring our invariant belongs to is simpler than the one obtained in [@Le] and [@BL]. In particular, we don’t need any fractional power of $q$. We show that the Taylor expansion of our unified invariant at a root of unity of order $c$ (new Ohtsuki series) dominates all WRT invariants with $r=cl$ and $(l,b)=1$.
For rational homology 3–spheres the universal finite type invariant was constructed by Le, Murakami and Ohtsuki [@LMO]. It determines Ohtsuki series and, hence, $\{\tau_M(\xi)\,|\,({\operatorname{ord}}(\xi),b)=1\}$ [@Le]. An interesting open question is whether the Le–Murakami–Ohtsuki invariant determines $I_M$.
Results {#results .unnumbered}
-------
The WRT or quantum $SO(3)$ invariant $\tau_{M,L}(\xi)$ is defined for a pair of a closed 3–manifold $M$ and a link $L$ in it, with link components colored by integers. Here $\xi$ is a root of unity of odd order. We will recall the definitions in Section \[defs\].
Suppose $M$ is a rational homology 3–sphere, i.e. $|H_1(M,\Z)|:={\rm card}\,H_1 (M,\Z) < \infty$. There is a unique decomposition $ H_1(M,\Z)=\bigoplus_{i}
\Z/{b_{i}\Z}$, where each $b_i$ is a prime power. We renormalize the $SO(3)$ WRT invariant of the pair $(M, L)$ as follows: $$\tau'_{M, L}(\xi)=\frac{\tau_{M,L}(\xi)}
{\prod\limits_{i}\;\,
\tau_{L(b_{i},1)}(\xi)}\; ,
\label{0910}$$ where $L(b,a)$ denotes the $(b,a)$ lens space. We will see that $\tau_{L(b,1)}(\xi)$ is always nonzero.
For any positive integer $b$, we define the cyclotomic completion ring $\R_b$ to be \[ab\] \_b:=\_ , (q;q\^2)\_k = (1-q)(1-q\^3) …(1-q\^[2k-1]{}). For any $f(q)\in \R_b$ and a root of unity $\xi$ of [*odd*]{} order, the evaluation $\ev_\xi (f(q)):= f(\xi)$ is well–defined. Similarly, we put $$\calS_b :=\lim_{\overleftarrow{\hspace{2mm}k\hspace{2mm}}}
\frac{\Z[1/b][q]}
{((q;q)_k)}\; .$$ Here the evaluation at any root of unity is well–defined. For odd $b$, there is a natural embedding $\calS_b\to \R_b$, see Section \[cyc\].
Let us denote by $\M_b$ the set of rational homology 3–spheres such that $|H_1(M,\Z)|$ divides $b^n$ for some $n$. The main result of this paper is the following.
\[main\] Suppose the components of a framed oriented link $L \subset M$ have odd colors, and $M\in \cM_b$. Then there exists an invariant $I_{M,L} \in \R_b$, such that for any root of unity $\xi$ of odd order $$\ev_\xi(I_{M,L})=\tau'_{M,L}(\xi)\, .$$ In addition, if $b$ is odd, then $I_{M,L}\in \calS_b$.
If $b=1$ and $L$ is the empty link, $I_{M}$ coincides with Habiro’s unified invariant $J_M$.
The proof of Theorem \[main\] uses the Laplace transform method and Andrew’s identity. We also construct a Frobenius type isomorphism to get rid of the formal fractional power of $q$ that appeared in [@Le], [@BL]. As a byproduct, we generalize the deep integrality result of Habiro (Theorem 8.2 in [@Ha]), underlying the construction of the unified invariants, to a union of an algebraically split link with any odd colored one.
The rings $\R_b$ and $\calS_b$ have properties similar to those of the Habiro ring. An element $f(q) \in \R_b$ is totally determined by the values at many infinite sets of roots of unity (see Section \[cyc\]), one special case is the following.
\[main-cor\] Let $p$ be an odd prime not dividing $b$ and $T$ the set of all integers of the form $p^k b'$ with $k\in \N$ and $b'$ any odd divisor of $b^n$ for some $n$. Any element $f(q) \in \R_b$, and hence also $\{\tau_M(\xi)\}$, is totally determined by the values at roots of unity with orders in $T$.
The Ohtsuki series [@Oh; @Le3], originally defined through some arithmetic congruence property of the $SO(3)$ invariant, can be identified with the Taylor expansion of $I_M$ at $q=1$ [@Ha; @Le]. We will also investigate the Taylor expansions of $I_M$ at roots of unity and show that these Taylor expansions satisfy congruence relations similar to the original definition of the Ohtsuki series, see Section \[Oh-strategy\].
Plan of the paper {#plan-of-the-paper .unnumbered}
-----------------
In Section \[defs\] we recall known results and definitions. In the next section we explain the strategy of our proof of Theorem \[main\]. In Sections \[cyc\] and \[map-frob\], we develop properties of cyclotomic completions of polynomial rings. New Ohtsuki series are discussed in Section \[Oh-strategy\]. The unified invariant of lens spaces, needed for the diagonalization, is defined in Section \[diag\]. The main technical result of the paper based on Andrew’s identity is proved in Section \[laplace\]. The Appendix is devoted to the proof of the generalization of Habiro’s integrality theorem.
Acknowledgments {#acknowledgments .unnumbered}
---------------
The authors would like to thank Kazuo Habiro and Christian Krattenthaler for helpful remarks and stimulating conversations.
Quantum (WRT) invariants {#defs}
========================
Notations and conventions
-------------------------
We will consider $q^{1/4}$ as a free parameter. Let $$\{n\} = q^{n/2}-q^{-n/2},
\quad \{n\}!=
\prod_{i=1}^n \{i\} ,\quad [n] =\frac{\{n\}}{\{1\}}, \quad
{\text{$\left[\begin{array}{c}n\\ k\end{array}
\right]$}} = \frac{\{n\}!}{\{k\}!\{n-k\}!}.$$ We denote the set $\{1,2,3,\ldots\}$ by $\N$. We also use the following notation from $q$–calculus: $$(x;q)_n := \prod_{j=1}^n (1-x q^{j-1}).$$ Throughout this paper, $\xi$ will be a primitive root of unity of [*odd*]{} order $r$ and $e_n:=\exp(2\pi I/n)$.
All 3–manifolds in this paper are supposed to be closed and oriented. Every link in a 3–manifold is framed, oriented, and has components ordered.
In this paper, $L\sqcup L'$ denotes a framed link in $S^3$ with disjoint sublinks $L$ and $L'$, with $m$ and $l$ components, respectively. Surgery along the framed link $L$ transforms $(S^3,L')$ into $(M,
L')$. We use the same notation $L'$ to denote the link in $S^3$ and the corresponding one in $M$.
The colored Jones polynomial
----------------------------
Suppose $L$ is a framed, oriented link in $S^3$ with $m$ ordered components. For positive integers $n_1,\dots,n_m$, called the colors of $L$, one can define the quantum invariant $J_L(n_1,\dots,n_m)\in \Z[q^{\pm 1/4}]$, known as the colored Jones polynomial of $L$ (see e.g. [@Tu; @MM]). Let us recall here a few well–known formulas. For the unknot $U$ with 0 framing one has $$J_U(n) = [n]. \label{unknot}$$ If $L_1$ is obtained from $L$ by increasing the framing of the $i$th component by 1, then $$\label{framing}
J_{L_1}(n_1,\dots,n_m) = q^{(n_i^2-1)/4} J_{L}(n_1,\dots,n_m).$$ If all the colors $n_i$ are odd, then $J_{L}(n_1,\dots,n_m) \in \Z[q^{\pm 1}]$.
Evaluation and Gauss sums
-------------------------
For each root of unity $\xi$ of odd order $r$, we define the evaluation map $\ev_\xi$ by replacing $q$ with $\xi$. Suppose $f(q;n_1,\dots,n_m)$ is a function of variables $q^{\pm 1}$ and integers $n_1,\dots,n_m$. In quantum topology, the following sum plays an important role $${\sum_{n_i}}^\xi f := \sum_{\substack{0< n_i< 2r\\ n_i \text{ odd}}}
\ev_\xi f(q; n_1,\dots, n_m)$$ where in the sum all the $n_i$ run over the set of [*odd*]{} numbers between $0$ and $2r$.
In particular, the following variation of the Gauss sum $$\gamma_b(\xi):= {\sum_{n}}^\xi q^{b\frac{n^2-1}{4}}$$ is well–defined, since for odd $n$, $4\mid n^2-1$. It is known that, for odd $r$, $|\gamma_b(\xi)|$ is never 0.
Definition of the WRT invariant
-------------------------------
Suppose the components of $L'$ are colored by fixed integers $j_1,\dots,j_l$. Let $$ F_{L\sqcup L'}(\xi):= {\sum_{n_i}}^\xi\;
\left \{ J_{L\sqcup L'}(n_1,\dots,n_m, j_1,\dots,j_l)\prod_{i=1}^m [n_i]\right \}.$$
An important special case is when $L=U^b$, the unknot with framing $b \neq 0$, and $L'=\emptyset$. In that case $F_{U^{b}}(\xi)$ can be calculated using the Gauss sum and is nonzero, see Section \[diag\] below.
Let $\sigma_+ $ (respectively $\sigma_-$) be the number of positive (negative) eigenvalues of the linking matrix of $L$. Then the quantum $SO(3)$ invariant of the pair $(M, L')$ is defined by (see e.g. [@KM; @Tu]) $$\tau_{M,L'}(\xi) =
\frac{F_{L\sqcup L'}(\xi)}{(F_{U^{+1}}(\xi))^{\sigma_+}\,
(F_{U^{-1}}(\xi))^{\sigma_-} }\, .
\label{def_qi}$$ The invariant $\tau_{M,L'}(\xi)$ is multiplicative with respect to the connected sum.
For example, the $SO(3)$ invariant of the lens space $L(b,1)$, obtained by surgery along $U^b$, is $$\tau_{L(b,1)} (\xi)= \frac{ F_{U^b}(\xi)}{F_{U^{{\operatorname{sn}}(b)}}(\xi) },
\label{2005}$$ were ${\operatorname{sn}}(b)$ is the sign of the integer $b$.
Let us focus on the special case when the linking matrix of $L$ is diagonal, with $b_1, b_2, \dots, b_m$ on the diagonal. Assume each $b_i$ is a power of a prime up to sign. Then $H_1(M,\Z) = \oplus_{i=1}^m \Z/|b_i|$, and $$\sigma_+ = {\rm card}\, \{ i\mid b_i >0\}, \quad \sigma_- =
{\rm card}\, \{ i \mid b_i < 0\}.$$ Thus from the definitions , and we have $$\tau'_{M,L'}(\xi) = \left( \prod_{i=1}^m \tau'_{L(b_i,1)}(\xi)
\right)\,
\frac{F_{L\sqcup L'}(\xi)}
{\prod_{i=1}^m F_{U^{b_i}}(\xi) }
\, ,
\label{0077}$$ with $$\tau'_{L(b_i,1)}(\xi)= \frac{\tau_{L(b_i,1)}(\xi)}{\tau_{L(|b_i|,1)}(\xi)}\, .$$
Habiro’s cyclotomic expansion of the colored Jones polynomial
-------------------------------------------------------------
Recall that $L$ and $L'$ have $m$ and $l$ components, respectively. Let us color $L'$ by fixed $\bj=(j_1,\dots,j_l)$ and vary the colors $\bn=(n_1,\dots,n_m)$ of $L$.
For non–negative integers $n,k$ we define $$A(n,k) := \frac{\prod^{k}_{i=0}
\left(q^{n}+q^{-n}-q^i -q^{-i}\right)}{(1-q) \, (q^{k+1};q)_{k+1}}.$$ For $\bk=(k_1,\dots,k_m)$ let $$A(\bn,\bk):= \prod_{j=1}^m \; A(n_j,k_j).$$ Note that $A(\bn,\bk)=0$ if $k_j \ge n_j$ for some index $j$. Also
$$A(n,0)= q^{-1} J_U(n)^2.$$
The colored Jones polynomial $J_{L\sqcup L'}
(\n, \bj)$, when $\bj$ is fixed, can be repackaged into the invariant $C_{L\sqcup L'}
(\bk, \bj)$ as stated in the following theorem.
\[GeneralizedHabiro\] Suppose $L\sqcup L'$ is a link in $S^3$, with $L$ having zero linking matrix. Assume the components of $L'$ have fixed [*odd*]{} colors $\bj = (j_1,\dotsm j_l)$. Then there are invariants $$\label{Jones2}
C_{L\sqcup L'}(\bk,\bj) \in \frac{(q^{k+1};q)_{k+1}}{(1-q)}
\,\,\BZ[q^{\pm 1}] ,\quad \text{where $k=\max\{k_1,\dots, k_m\}$}$$ such that for every $\bn =(n_1,\dots, n_m)$ $$\label{Jones}
J_{L\sqcup L'}
(\n, \bj) \, \prod^m_{i=1}\; [n_i] = \sum_{0\le k_i \le n_i-1}
C_{L\sqcup L'}(\bk,\bj)\;
A(\bn, \bk).$$
When $L'=\emptyset$, this is Theorem 8.2 in [@Ha]. This generalization, essentially also due to Habiro, can be proved similarly as in [@Ha]. For completeness we give a proof in the Appendix. Note that the existence of $C_{L\sqcup L'}(\bk,\bj)$ as rational functions in $q$ satisfying is easy to establish. The difficulty here is to show the integrality of .
Since $A(\bn, \bk) =0$ unless $ \bk < \bn$, in the sum on the right hand side of one can assume that $\bk$ runs over the set of all $m$–tuples $\bk$ with non–negative integer components. We will use this fact later.
Strategy of the proof of the main theorem {#strategy}
==========================================
Here we give the proof of Theorem \[main\] using technical results that will be proved later.
As before, $ L\sqcup L'$ is a framed link in $S^3$ with disjoint sublinks $L$ and $L'$, with $m$ and $l$ components, respectively. Assume that $L'$ is colored by fixed $\bj=(j_1,\dots, j_l)$, with $j_i$’s odd. Surgery along the framed link $L$ transforms $(S^3,L')$ into $(M,
L')$. We will define $I_{M,L'}\in \cR_{b}$, such that $$\tau'_{M,L'}(\xi)\;=\; \ev_{\xi}\left(I_{M,L'}\right)
\label{0080}$$ for any root of unity $\xi$ of odd order. This unified invariant is multiplicative with respect to the connected sum.
The following observation is important. By Proposition \[main-cor\], there is [*at most one*]{} element $f(q)\in \R_b$ such that for every root $\xi$ of odd order one has $$\tau'_{M,L} (\xi) = \ev_\xi\left( f(q)\right).$$ That is, if we can find such an element, it is unique, and we put $I_{M,L'} := f(q)$.
Laplace transform
-----------------
The following is the main technical result of the paper. A proof will be given in Section \[laplace\].
Suppose $b=\pm 1$ or $b= \pm p^l$ where $p$ is a prime and $l$ is positive. For any non–negative integer $k$, there exists an element $Q_{b,k} \in \R_b $ such that for every root $\xi$ of odd order $r$ one has $$\frac{{\sum\limits_n}^\xi \, q^{b\frac{n^2-1}{4}} A(n,k) }{F_{U^b}(\xi)}
= \ev_\xi (Q_{b,k}).$$ \[0078\] In addition, if $b$ is odd, $Q_{b,k} \in \calS_b $.
Definition of the unified invariant: diagonal case {#2501}
--------------------------------------------------
Suppose that the linking number between any two components of $L$ is 0, and the framing on components of $L$ are $b_i=\pm p_i^{k_i}$ for $i=1,\dots, m$, where each $p_i$ is prime or 1. Let us denote the link $L$ with all framings switched to zero by $L_0$.
Using , taking into account the framings $b_i$’s, we have $$J_{L\sqcup L'}(\bn,\bj)\prod_{i=1}^m [n_i] = \sum_{\bk\ge 0} C_{L_0 \sqcup L'}
(\bk,\bj) \, \prod_{i=1}^m q^{b_i \frac{n_i^2-1}{4}} A(n_i,k_i).$$ By the definition of $F_{L\sqcup L'}$, we have $$F_{L\sqcup L'}(\xi)= \sum_{\bk \ge 0}
\ev_\xi(C_{L_0 \sqcup L'}(\bk,\bj)) \, \prod_{i=1}^m
{\sum_{n_i}}^\xi \, q^{b_i \frac{n_i^2-1}{4}} A(n_i,k_i).$$ From and Theorem \[0078\], we get $$\tau'_{M,L'}(\xi) = \ev_\xi \left \{ \prod_{i=1}^m I_{L(b_i,1)} \,
\sum_{\bk} C_{L_0 \sqcup L'}(\bk,\bj) \, \prod_{i=1}^m Q_{b_i,k_i} \right \},$$ where the unified invariant of the lens space $I_{L(b_i,1)}\in \R_b$, with $\ev_\xi(I_{L(b_i,1)})=\tau'_{L(b_i,1)}(\xi)$, exists by Lemma \[0079\] below. Thus if we define $$I_{(M,L')}:= \prod_{i=1}^m I_{L(b_i,1)} \, \sum_{\bk}
C_{L_0 \sqcup L'}(\bk,\bj) \, \prod_{i=1}^m Q_{b_i,k_i}\, ,$$ then is satisfied. By Theorem \[GeneralizedHabiro\], $C_{L_0 \sqcup L'}(\bk,\bj)$ is divisible by $(q^{k+1};q)_{k+1}/(1-q)$, which is divisible by $(q;q)_k$, where $k = \max k_i$. It follows that $I_{(M,L')} \in \R_b$. In addition, if $b$ is odd, then $I_{(M,L')} \in \calS_b$.
Diagonalization using lens spaces
---------------------------------
The general case reduces to the diagonal case by the well–known trick of diagonalization using lens spaces. We say that $M$ is [*diagonal*]{} if it can be obtained from $S^3$ by surgery along a framed link $L$ with diagonal linking matrix, where the diagonal entries are of the form $\pm p^k$ with $p=0,1$ or a prime. The following lemma was proved in [@Le Proposition 3.2 (a)].
For every rational homology sphere $M$, there are lens spaces $L(b_i,a_i)$ such that the connected sum of $M$ and these lens spaces is diagonal. Moreover, each $b_i$ is a prime power divisor of $|H_1(M,\Z)|$. \[diagonalization\]
To define the unified invariant for a general rational homology sphere $M$, one first adds to $M$ lens spaces to get a diagonal $M'$, for which the unified invariant $I_{M'}$ had been defined in Subsection \[2501\]. Then $I_M$ is the quotient of $I_{M'}$ by the unified invariants of the lens spaces. But unlike the simpler case of [@Le], the unified invariant of lens spaces are [*not*]{} invertible in general. To overcome this difficulty we insert knots in lens spaces and split the unified invariant into different components. This will be explained in the remaining part of this section.
Splitting of the invariant
--------------------------
Suppose $p$ is a prime divisor of $b$, then it’s clear that $\R_p \subset \R_b$.
In Section \[cyc\] we will see that there is a decomposition
$$\R_b = \R_{b}^{p,0} \times \R_{b}^{p,{{\bar 0}}},$$ with canonical projections $\pi^p_0: \R_b \to \R_{b}^{p,0}$ and $\pi^p_{{\bar 0}}:\R_b \to \R_{b}^{p,{{\bar 0}}}$. If $f\in \R_{b}^{p,0}$ then $\ev_\xi(f)$ can be defined when the order of $\xi$ is coprime with $p$; and in this case $ \ev_\xi(g) = \ev_\xi(\pi^p_0(g))$ for every $g\in \R_b$.
On the other hand, if $f\in \R_{b}^{p,{{\bar 0}}}$ then $\ev_\xi(f)$ can be defined when the order of $\xi$ is divisible by $p$, and one has $ \ev_\xi(g) = \ev_\xi(\pi^p_{{\bar 0}}(g))$ for every $g\in \R_b$.
It also follows from the definition that $\R_p^{p,\ve}
\subset \R_b^{p,\ve}$ for $\ve = 0$ or ${{\bar 0}}$.
For $\calS_b$, there exists a completely analogous decomposition. For any odd divisor $p$ of $b$, an element $ x \in \R_b$ (or $\calS_b$) determines and is totally determined by the pair $(\pi^p_0(x), \pi^p_{{\bar 0}}(x))$. If $p=2$ divides $b$, then for any $x\in \R_b$, $x=\pi^p_0(x)$.
Hence, to define $I_M$ it is enough to fix $I^0_M= \pi^p_0(I_M)$ and $I^{{\bar 0}}_M=\pi^p_{{\bar 0}}(I_M)$. The first part $I^0_M= \pi^p_0(I_M)$, when $b=p$, was defined in [@Le] (up to normalization), where the third author considered the case when the order of roots of unity is coprime with $b$. We will give a self–contained definition of $I^0_M$, and show that it is coincident (up to normalization) with the one introduced in [@Le].
Lens spaces
-----------
Suppose $b,a,d$ are integers with $(b,a)=1$ and $b\neq 0$. Let $M(b,a;d)$ be the pair of a lens space $L(b,a)$ and a knot $K\subset L(b,a)$, colored by $d$, as described in Figure \[figure:LensSpaceKnot\].
Among these pairs we want to single out some whose quantum invariants are invertible. For $\ve\in \{0,{{\bar 0}}\}$, let $M^\ve(b,a) := M(b,a;d(\ve))$, where $d(0):=1$ and $d({{\bar 0}})$ is the smallest odd positive integer such that $|a|d({{\bar 0}}) \equiv 1 \pmod {b}$. Note that if $|a|=1$, $d(0)=d({{\bar 0}})=1$.
It is known that if the color of a link component is $1$, then the component can be removed from the link without affecting the value of quantum invariants. Hence $$\tau_{M(b,a;1)} = \tau_{L(b,a)}.$$
Suppose $b=\pm p^{l}$ is a prime power. For $\ve \in \{0,{{\bar 0}}\}$, there exists an invertible invariant $I^\ve_{M^\ve(b,a)} \in \R^{p,\ve}_p$ such that $$\tau'_{M^\ve(b,a)}(\xi)\;=\; \ev_{\xi}\left(I^\ve_{M^\ve(b,a)}\right)$$ where $\ve=0$ if the order of $\xi$ is not divisible by $p$, and $\ve={{\bar 0}}$ otherwise. Moreover, if $p$ is odd, then $I^\ve_{M^\ve(b,a)}$ belongs to and is invertible in $\calS^{p,\ve}_p$. \[0079\]
A proof of Lemma \[0079\] will be given in Section \[diag\].
Definition of the unified invariant: general case
-------------------------------------------------
Now suppose $(M,L')$ is an arbitrary pair of a rational homology 3–sphere with a link $L'$ in it colored by odd numbers $j_1,\dots, j_l$. Let $L(b_i,a_i)$ for $i=1,\dots, m$ be the lens spaces of Lemma \[diagonalization\]. We use induction on $m$. If $m=0$, then $M$ is diagonal and $I_{M,L'}$ has been defined in Subsection \[2501\].
Since $(M,L') \# M(b_1,a_1;d)$ becomes diagonal after adding $m-1$ lens spaces, the unified invariant of $(M,L') \# M(b_1,a_1;d)$ can be defined by induction, for any odd integer $d$. In particular, one can define $I_{M^\ve}$, where $M^\ve := (M,L') \# M^\ve(b_1,a_1)$. Here $\ve = 0$ or $\ve ={{\bar 0}}$ and $b_1$ is a power of a prime $p$ dividing $b$. It follows that the components $\pi^p_\ve(I_{M^\ve}) \in \R^{p,\ve}_b$ are defined.
By Lemma \[0079\], $I^\ve_{M^\ve(b_1,a_1)}$ is defined and invertible. Now we put $$I^\ve_{M,L'} := I^\ve_{M^\ve} \cdot (I^\ve_{M^\ve(b_1,a_1)})^{-1}.$$ It is easy to see that $I_{M,L'}:= (I^0_{M,L'}, I^{{\bar 0}}_{M,L'})$ satisfies . This completes the construction of $I_{M,L'}$. It remains to prove Lemma \[0079\] and Theorem \[0078\].
Cyclotomic completions of polynomial rings {#cyc}
==========================================
In this section we adapt the results of Habiro on cyclotomic completions of polynomial rings [@Ha1] to our rings.
On cyclotomic polynomial
------------------------
Recall that $e_n := \exp(2\pi I/n)$ and denote by $\Phi_n(q)$ the cyclotomic polynomial $$\Phi_n(q) = \prod_{\substack{(j,n)=1\\0<j<n}} (q - e_n^j).$$ The degree of $\Phi_n(q)\in \Z[q]$ is given by the Euler function $\varphi(n)$. Suppose $p$ is a prime and $n$ an integer. Then (see e.g. [@Na]) $$\Phi_n(q^p)= \begin{cases} \Phi_{np}(q) & \text{ if } p \mid n \\
\Phi_{np}(q) \Phi_n(q) & \text{ if } p \nmid n.
\end{cases}$$ It follows that $\Phi_n(q^p)$ is always divisible by $\Phi_{np}(q)$.
The ideal of $\Z[q]$ generated by $\Phi_n(q)$ and $\Phi_m(q)$ is well–known, see e.g. [@Le Lemma 5.4]:
$\text{ }$
- If $\frac{m}{n} \neq p^e$ for any prime $p$ and any integer $e\neq 0$, then $(\Phi_n)+ (\Phi_m)=(1)$ in $\Z[q]$.
- If $\frac{m}{n} = p^e$ for a prime $p$ and some integer $e \neq 0$, then $(\Phi_n)+ (\Phi_m)=(1)$ in $\Z[1/p][q]$. \[0911\]
Note that in a commutative ring $R$, $(x) + (y) =(1)$ if and only if $x$ is invertible in $R/(y)$. Also $(x) + (y) =(1)$ implies $(x^k) +
(y^l) =(1)$ for any integers $k,l \ge 1$.
Habiro’s results
----------------
Let us summarize some of Habiro’s results on cyclotomic completions of polynomial rings [@Ha1]. Let $R$ be a commutative integral domain of characteristic zero and $R[q]$ the polynomial ring over $R$. For any $S\subset \N$, Habiro defined the $S$–cyclotomic completion ring $R[q]^S$ as follows: \[rs\] R\[q\]\^S:=\_ where $\Phi^*_S$ denotes the multiplicative set in $\Z[q]$ generated by $\Phi_S=\{\Phi_n(q)\mid n\in S\}$ and directed with respect to the divisibility relation.
For example, since the sequence $(q;q)_n$, $n\in \N$, is cofinal to $\Phi^*_\N$, we have \[cofinal\] \^.
Note that if $S$ is finite, then $R[q]^S$ is identified with the $(\prod \Phi_S)$–adic completion of $R[q]$. In particular, $$R[q]^{\{1\}}\simeq R[[q-1]], \quad
R[q]^{\{2\}}\simeq R[[q+1]].$$ Suppose $S' \subset S$, then $\Phi^*_{S'}\subset \Phi^*_S$, hence there is a natural map $$\rho^R_{S, S'}: R[q]^S \to R[q]^{S'}.$$
Recall important results concerning $R[q]^S$ from [@Ha1]. Two positive integers $n, n'$ are called [*adjacent*]{} if $n'/n=p^e$ with a nonzero $e\in \Z$ and a prime $p$, such that the ring $R$ is $p$–adically separated, i.e. $\bigcap_{n=1}^\infty (p^n) =0$ in $R$. A set of positive integers is [*$R$–connected*]{} if for any two distinct elements $n,n'$ there is a sequence $n=n_1, \,n_2, \dots,\, n_{k-1},\, n_k= n'$ in the set, such that any two consecutive numbers of this sequence are adjacent. Theorem 4.1 of [@Ha1] says that if $S$ is $R$–connected, then for any subset $S'\subset S$ the natural map $ \rho^R_{S,S'}: R[q]^S \hookrightarrow R[q]^{S'}$ is an embedding.
If $\zeta$ is a root of unity of order in $S$, then for every $f(q)\in R[q]^S$ the evaluation $\ev_\zeta(f(q))\in R[\zeta]$ can be defined by sending $q\to\zeta$. For a set $\Xi$ of roots of unity whose orders form a subset $\cT\subset S$, one defines the evaluation $$\ev_\Xi: R[q]^S \to \prod_{\zeta \in \Xi} R[\zeta].$$ Theorem 6.1 of [@Ha1] shows that if $R\subset \Q$, $S$ is $R$–connected, and there exists $n\in S$ that is adjacent to infinitely many elements in $\cT$, then $\ev_\Xi$ is injective.
Taylor expansion
----------------
Fix a natural number $n$, then we have $$R[q]^{\{n\}} =
\lim_{\overleftarrow{\hspace{2mm}k\hspace{2mm}}}
\frac{R[q]}{(\Phi^k_n(q))}\; .$$ Suppose $ \Z \subset R \subset \Q$, then the natural algebra homomorphism
$$h: \frac{R[q]}{(\Phi^k_n(q))} \to \frac{R[e_n][q]}{((q-e_n)^k)}$$ is injective, by Proposition \[h\_kInjektive\] below. Taking the inverse limit, we see that there is a natural injective algebra homomorphism
$$h : R[q]^{\{n\}} \to R[e_n][[q-e_n]].$$
Suppose $n \in S$. Combining $h$ and $\rho_{S, \{n\}}: R[q]^S \to R[q]^{\{n\}}$, we get an algebra map
$${{\mathfrak t}}_n: R[q]^S \to R[e_n][[q-e_n]].$$ If $f\in R[q]^S$, then ${{\mathfrak t}}_n(f)$ is called the Taylor expansion of $f$ at $e_n$.
Splitting of $\calS_p$ and evaluation
-------------------------------------
For every integer $a$, we put $\N_a := \{ n \in \N \mid (a,n)=1\}$.
Suppose $p$ is a prime. Analogously to , we have $$\calS_p \simeq \Z[1/p][q]^\N\,.$$
Observe that $\N$ is not $\Z[1/p]$–connected. In fact one has $\N =\amalg_{j=0}^\infty \; p^j \N_p$, where each $p^j\N_p$ is $\Z[1/p]$–connected. Let us define $$\calS_{p,j}:= \Z[1/p][q]^{p^j \N_p}.$$ Note that for every $f \in \calS_p$, the evaluation $\ev_\xi(f)$ can be defined for every root $\xi$ of unity. For $f\in \calS_{p,j}$, the evaluation $\ev_\xi(f)$ can be defined when $\xi$ is a root of unity of order in $p^j\N_p$.
For every prime $p$ one has $$\calS_p\simeq \prod_{j=0}^\infty \calS_{p,j}.
\label{0912}$$
Suppose $n_i \in p^{j_i}\N_p$ for $ i=1,\dots, m$, with distinct $j_i$’s. Then $n_{i}/n_{s}$, with $i \neq s$, is either not a power of a prime or a non–zero power of $p$, hence by Lemma \[0911\] (and the remark right after Lemma \[0911\]), for any positive integers $k_1,\dots, k_m$, we have $$(\Phi_{n_i}^{k_i})+ (\Phi_{n_s}^{k_s})=(1) \quad \text{ in }\Z[1/p][q].$$ By the Chinese remainder theorem, we have $$\frac{ \Z[1/p][q]}{\left(\prod_{i=1}^m \Phi_{n_i}^{k_i}\right)}\;\simeq\;
\prod_{i=1}^m
\frac{\Z[1/p][q]}{\left(\Phi_{n_i}^{k_i}\right)} .$$ Taking the inverse limit, we get .
Let $\pi_j: \calS_p \to \calS_{p,j}$ denote the projection onto the $j$th component in the above decomposition.
Suppose $\xi$ is a root of unity of order $r= p^j r'$, with $(r',p)=1$. Then for any $x\in \calS_p$, one has $$\ev_\xi(x) = \ev_\xi(\pi_j(x)).$$ If $i\neq j$ then $\ev_\xi(\pi_i(x))=0$.
\[1100\]
Note that $\ev_\xi(x)$ is the image of $x$ under the projection $\calS_p \to \calS_p/(\Phi_r(q))= \Z[1/p][\xi]$. It remains to notice that $\calS_{p,i}/(\Phi_r(q))=0$ if $i\neq j$.
Splitting of $\calS_b$
----------------------
Suppose $p$ is a prime divisor of $b$. Let $$\calS_{b}^{p,0} := \Z[1/b][q]^{\N_p} \qquad \text {and } \quad
\calS_{b}^{p,{{\bar 0}}} := \Z[1/b][q]^{p\N}\simeq \prod_{j>0}\Z[1/b][q]^{p^j\N_p}$$
We have similarly $$\calS_b = \calS_{b}^{p,0} \times \calS_{b}^{p,{{\bar 0}}}$$ with canonical projections $\pi^p_0 : \calS_b \to \calS_b^{p,0}$ and $\pi^p_{{\bar 0}}: \calS_b \to \calS_b^{p,{{\bar 0}}}$. Note that if $b=p$, then $\calS_{p}^{p,0}=\calS_{p,0}$ and $\calS_{p}^{p,{{\bar 0}}}=\prod_{j>0} \calS_{p,j}$. As before we set $\calS_{b,0}:=\Z[1/b][q]^{\N_b}$ and $\pi_0:\calS_b\to\calS_{b,0}$.
Suppose $f \in \calS_b$. If $\xi$ is a root of unity of order coprime with $p$, then $\ev_\xi(f) = \ev_\xi(\pi^p_0(f))$. Similarly, if the order of $\xi$ is divisible by $p$, then $\ev_\xi(f) = \ev_\xi(\pi^p_{{\bar 0}}(f))$.
Properties of the ring $\cR_b$
------------------------------
For any $b\in \N$, we have $$\R_b\simeq\Z[1/b][q]^{\N_2}$$ since the sequence $(q;q^2)_k,\, k\in \N$, is cofinal to $\Phi^*_{\N_2}$. Here $\N_2$ is the set of all odd numbers. Let $\{p_i\,|\, i=1,\dots,m\}$ be the set of all distinct [*odd*]{} prime divisors of $b$. For $\bn=(n_1,\dots,n_m)$, a tuple of numbers $n_i\in \N$, let $\bp^{\bn}=\prod_{i}p_i^{n_i}$. Let $S_{\bn}:=\bp^{\bn}\N_{2b}$. Then $\N_2=\amalg_{\bn}\, S_\bn$. Moreover, for $a\in S_{\bn},$ $a'\in S_{\bn'}$, we have $(\Phi_{a}(q),\Phi_{a'}(q))=(1)$ in $\Z[1/b]$ if $\bn\neq\bn'$. In addition, each $S_\bn$ is $\Z[1/b]$–connected. An argument similar to that for Equation gives $$\cR_b\simeq\prod_{\bn}\Z[1/b][q]^{S_\bn}.$$ In particular, $\R^{p_i,0}_b:=\Z[1/b][q]^{\N_{2p_i}}$ and $\R^{p_i,{{\bar 0}}}_b:=\Z[1/b][q]^{p_i\N_2}$ for any $1\leq i\leq m$. If $2\mid b$, then $\R^{2,0}_b$ coincides with $\R_b$.
Let $T$ be an infinite set of powers of an odd prime not dividing $b$ and let $P$ be an infinite set of odd primes not dividing $b$.
\[main-corProof\] With the above notations, one has the following.
- For any $l\in S_\bn$, the Taylor map ${{\mathfrak t}}_l : \Z[1/b][q]^{S_\bn}\;\to \;\Z[1/b][e_{l}][[q-e_{l}]]$ is injective.
- Suppose $f, g\in \Z[1/b][q]^{S_{\bn}}$ such that $\ev_\xi(f)=\ev_\xi(g)$ for any root of unity $\xi$ with $\text{ord}(\xi)\in \bp^\n T$, then $f=g$. The same holds true if $\bp^\n T$ is replaced by $\bp^\n P$.
- For odd $b$, the natural homomorphism $\rho_{\N,\N_2}:\calS_b\to \R_b$ is injective. If $2\mid b$, then the natural homomorphism $\calS_{b}^{2,0}\to \R_{b}$ is an isomorphism.
\(a) Since each $S_\bn$ is $\Z[1/b]$–connected in Habiro sense, by [@Ha1 Theorem 4.1], for any $l\in S_\bn$ \[inj\] \_[S, {l}]{}: \^[S\_]{}\^[{ l}]{} is injective. Hence ${{\mathfrak t}}_l = h \circ \rho_{S, \{l\}}$ is injective too.
\(b) Since both sets contain infinitely many numbers adjacent to $\bp^\n$, the claim follows from Theorem 6.1 in [@Ha1].
\(c) Note that for odd $b$ $$\calS_b\simeq \prod_{\bn}\Z[1/b][q]^{S'_\bn}$$ where $S'_{\bn}:=\bp^{\bn}\N_{b}$. Further observe that $S'_{\bn}$ is $\Z[1/b]$–connected if $b$ is odd. Then by [@Ha1 Theorem 4.1] the map $$\Z[1/b][q]^{S'_\bn}\hookrightarrow \Z[1/b][q]^{S_{\bn}}$$ is an embedding. If $2\mid b$, then $\calS^{2,0}_b:=\Z[1/b][q]^{\N_2}\simeq\R_b$.
Assuming Theorem \[main\], Proposition \[main-corProof\] (b) implies Proposition \[main-cor\].
On the Ohtsuki series at roots of unity {#Oh-strategy}
=======================================
The Ohtsuki series was defined for $SO(3)$ invariants by Ohtsuki [@Oh] and extended to all other Lie algebras by the third author [@Le2; @Le3].
In the works [@Oh; @Le2; @Le3], it was proved that the sequence of quantum invariants at $e_{p}$, where $p$ runs through the set of primes, obeys some congruence properties that allow to define uniquely the coefficients of the Ohtsuki series. The proof of the existence of such congruence relations is difficult. In [@Ha], Habiro proved that Ohtsuki series coincide with the Taylor expansion of the unified invariant at $q=1$ in the case of integral homology spheres; this result was generalized to rational homology spheres by the third author [@Le].
Here, we prove that the sequence of $SO(3)$ invariants at the $pr$th roots $e_{r}e_p$, where $r$ is a fixed odd number and $p$ runs through the set of primes, obeys some congruence properties that allow to define uniquely the coefficients of the “Ohtsuki series” at $e_r$, which is coincident with the Taylor expansion at $e_r$.
Extension of $\Z[1/b][e_r]$
---------------------------
Fix an odd positive integer $r$. Assume $p$ is a prime bigger than $b$ and $r$. The cyclotomic rings $\Z[1/b][e_{pr}]$ and $\Z[1/b][e_r]$ are extensions of $\Z[1/b]$ of degree $\varphi(rp)=\varphi(r) \varphi(p)$ and $\varphi(r)$, respectively. Hence $\Z[1/b][e_{pr}]$ is an extension of $\Z[1/b][e_r]$ of degree $\varphi(p)=p-1$. Actually, it is easy to see that for $$f_p(q):= \frac{q^p - e_r^p}{q-e_r},$$ the map $$\frac{\Z[1/b,e_r][q]}{(f_p(q))} \;\to\; \Z[1/b][e_{pr}], \hspace{5mm}
q\mapsto e_p e_r,$$ is an isomorphism. We put $x=q-e_r$ and get $$\Z[1/b][e_{pr}]\simeq \frac{\Z[1/b,e_r][x]}{(f_p(x+e_r))}\; .
\label{5500}$$ Note that $$f_p(x+ e_r) = \sum_{n=0}^{p-1} \binom{p}{n+1} x^n e_r^{p-n-1}$$ is a monic polynomial in $x$ of degree $p-1$, and the coefficient of $x^n$ in $f_p(x+e_r)$ is divisible by $p$ if $n \le p-2$.
Arithmetic expansion of $\tau'_M$
---------------------------------
Suppose $M$ is a rational homology 3–sphere with $|H_1(M,\Z)|=b$. By Theorem \[main\], for any root of unity $\xi$ of order $pr$ $$\tau'_M(\xi)\in \Z[1/b][e_{pr}] \simeq \frac{\Z[1/b,e_r][x]}{(f_p(x+e_r))}\; .$$ Hence we can write \[Ohtsuki\] ’\_M(e\_[r]{}e\_[p]{})= \_[n=0]{}\^[p-2]{} a\_[p,n]{} x\^n where $a_{p,n}\in \Z[1/b,e_r]$. The following proposition shows that the coefficients $a_{p,n}$ stabilize as $p\to \infty$.
\[main-cor1\] Suppose $M$ is a rational homology 3–sphere with $|H_1(M,\Z)|=b$, and $r$ is an odd positive integer. For every non–negative integer $n$, there exists a unique invariant $a_n= a_n(M) \in \Z[1/b,e_r]$ such that for every prime $p > \max (b,r)$, we have $$a_n\equiv a_{p,n} \pmod p\;\;\; \text{in $ \Z[1/b,e_r]$ for} \;\;\;
0\le n \le p-2.
\label{5501}$$ Moreover, the formal series $\sum_{n} a_n (q-e_r)^n$ is equal to the Taylor expansion of the unified invariant $I_M$ at $e_r$.
The uniqueness of $a_n$ follows from the easy fact that if $a \in \Z[1/b,e_r]$ is divisible by infinitely many rational primes $p$, then $a=0$.
Assume Theorem \[main\] holds. We define $a_n$ to be the coefficient of $(q-e_r)^n$ in the Taylor series of $I_M$ at $e_r$, and will show that Equation holds true.
Recall that $x= q-e_r$. The diagram $$\begin{CD} \Z[\frac{1}{b}][q]^{\N_2} @>>> \Z[\frac{1}{b}, e_r][q]^{r\N_2} @>>> \Z[\frac{1}{b},e_r][[x]] \\
@VV q \to e_re_pV @VV /(f_p(q))V @VV /(f_p(x+e_r))V \\
\Z[\frac{1}{b}][e_{rp}] @> e_re_p \to q >> \frac{\Z[\frac{1}{b}, e_r][q]}{(f_p(q))} @>>>\frac{ \Z[\frac{1}{b},e_r][[x]]}{(f_p(x+e_r))}
\end{CD}$$ is commutative. Here the middle and the right vertical maps are the quotient maps by the corresponding ideals. Note that $I_M$ belongs to the upper left corner ring, its Taylor series is the image in the upper right corner ring, while the evaluation is in the lower middle ring. Using the commutativity at the lower right corner ring, we see that
$$\sum_{n=0}^{p-2} a_{p,n}x^n = \sum_{n=0}^{\infty} a_{n}x^n \pmod{f_p(x+e_r)}
\quad \text{in} \quad \Z[1/b,e_r][[x]].$$ Since the coefficients of $f_p(x+e_r)$ up to degree $p-2$ are divisible by $p$, we get the congruence .
Proposition \[main-cor1\], when $r=1$, was the main result of Ohtsuki [@Oh], which leads to the development of the theory of finite type invariant and the LMO invariant.
When $(r,b)=1$, then Taylor series at $e_r$ determines and is determined by the Ohtsuki series. But when, say, $r$ is a divisor of $b$, a priori the two Taylor series, one at $e_r$ and the other at $1$, are independent. We suspect that the Taylor series at $e_r$, with $r\mid b$, corresponds to a new type of LMO invariant.
Frobenius maps {#map-frob}
==============
The proof of Theorem \[0078\], and hence of the main theorem, uses the Laplace transform method. The aim of this section is to show that the image of the Laplace transform, defined in Section \[laplace\], belongs to $\R_b$, i.e. that certain roots of $q$ exist in $\R_b$.
On the module $\Z[q]/(\Phi^k_{n}(q))$
-------------------------------------
Since cyclotomic completions are built from modules like $\Z[q]/(\Phi^k_{n}(q))$, we first consider these modules. Fix $n,k\ge 1$. Let $$E := \frac{\Z[q]}{(\Phi^k_{n}(q))}, \quad \text { and } \quad G:=
\frac{\Z[e_{n}][x]}{(x^k)}\; .$$
The following is probably well–known.
\[h\_kInjektive\]$\text{ }$
- Both $E$ and $G$ are free $\Z$–modules of the same rank $k \varphi(n)$.
- The algebra map $h: \Z[q] \to \Z[e_n][x]$ defined by
$$h (q) = e_n + x$$ descends to a well–defined algebra homomorphism, also denoted by $h$, from $E$ to $G$. Moreover, the algebra homomorphism $h: E \to G$ is injective.
\(a) Since $\Phi^k_n(q)$ is a monic polynomial in $q$ of degree $k \varphi(n)$, it is clear that $$E= \Z[q]/(\Phi^k_n(q))$$ is a free $\Z$–module of rank $k\varphi(n)$. Since $G = \Z[e_n]\otimes_\Z \Z[x]/(x^k)$, we see that $G$ is free over $\Z$ of rank $k\varphi(n)$.
\(b) To prove that $h$ descends to a map $E \to G$, one needs to verify that $h(\Phi^k_n(q))=0$. Note that $$h(\Phi^k_n(q))= \Phi^k_{n}(x + e_n) =
\prod_{(j,n)=1} (x+e_n - e_n^j)^k.$$ When $j=1$, the factor is $x^k$, which is 0 in $\Z[e_n][x]/(x^k)$. Hence $h(\Phi^k_n(q))=0$.
Now we prove that $h$ is injective. Let $f(q)\in\Z[q]$. Suppose $h(f(q))=0$, or $f(x+e_n)=0$ in $\Z[e_n][x]/(x^k)$. It follows that $f(x+e_n)$ is divisible by $x^k$; or that $f(x)$ is divisible by $(x-e_n)^k$. Since $f$ is a polynomial with coefficients in $\Z$, it follows that $f(x)$ is divisible by all Galois conjugates $(x-e_n^j)^k$ with $(j,n)=1$. Then $f$ is divisible by $\Phi^k_n(q)$. In other words, $f=0$ in $E= \Z[q]/(\Phi^k_n(q))$.
A Frobenius homomorphism
------------------------
We use $E$ and $G$ of the previous subsection. Suppose $b$ is a positive integer coprime with $n$. If $\xi$ is a primitive $n$th root of 1, i.e. $\Phi_n(\xi)=0$, then $\xi^b$ is also a primitive $n$th root of $1$, i.e. $\Phi_n(\xi^b)=0$. It follows that $\Phi_n(q^b)$ is divisible by $\Phi_n(q)$.
Therefore the algebra map $F_b: \Z[q] \to \Z[q]$, defined by $F_b(q)=q^b$, descends to a well–defined algebra map, also denoted by $F_b$, from $E$ to $E$. We want to understand the image $F_b(E)$.
The image $F_b(E)$ is a free $\Z$–submodule of $E$ of maximal rank, i.e. ${\operatorname{rk}}(F_b(E)) = {\operatorname{rk}}(E)$. Moreover, the index of $F_b(E)$ in $E$ is $b^{k(k-1)\varphi(n)/2}$.
Using Proposition \[h\_kInjektive\] we identify $E$ with its image $h(E)$ in $G$.
Let $\tilde F_b: G \to G$ be the $\Z$–algebra homomorphism defined by $\tilde F_b(e_n) =e_n^b, \tilde F_b(x)=
(x+e_n)^b - e_n^b$.
Note that $\tilde F_b(x) = b e_n^{b-1} x + O(x^2)$, hence $\tilde F_b(x^k)=0$. It is easy to see that $\tilde F_b$ is a well–defined algebra homomorphism, and that $\tilde F_b$ restricted to $E$ is exactly $F_b$. Since $E$ is a lattice of maximal rank in $G\otimes \Q$, it follows that the index of $F_b$ is exactly the determinant of $\tilde F_b$, acting on $G\otimes \Q$.
A basis of $G$ is $e_n^j x^l$, with $(j,n)=1, 0<j<n$ and $j=0$, and $0\le l < k$. Note that $$\tilde F_b(e_n^j x^l) = b^le_n^{jb} e_n^{(b-1)l} x^l + O(x^{l+1}).$$ Since $(b,n)=1$, the set $e_n^{jb}$, with $(j,n)=1$ is the same as the set $e_n^{j}$, with $(j,n)=1$. Let $f_1: G\to G$ be the $\Z$–linear map defined by $f_1(e_n^{jb} x^l) =
e_n^{j} x^l$. Since $f_1$ permutes the basis elements, its determinant is $\pm 1$. Let $f_2: G\to G$ be the $\Z$–linear map defined by $f_2(e_n^{j} x^l) = e_n^{j} (e_n^{1-b}x)^l$. The determinant of $f_2$ is again $\pm 1$. This is because, for any fixed $l$, $f_2$ restricts to the automorphism of $\Z[e_n]$ sending $a$ to $ e^s_n a$, each of these maps has a well–defined inverse: $a \mapsto e^{-s}_n a$. Now $$f_1 f_2 \tilde F_b(e_n^j x^l) = b^l e_n^j x^l + O(x^{l+1})$$ can be described by an upper triangular matrix with $b^l$’s on the diagonal; its determinant is equal to $b^{k(k-1)\varphi(n)/2}$.
From the proposition we see that if $b$ is invertible, then the index is equal to 1, hence we have
\[onen\] For any $n$ coprime with $b$ and $k\in \N$, the Frobenius homomorphism $F_b: \Z[1/b][q]/\left(\Phi^k_n(q)\right) \to \Z[1/b][q]/\left(\Phi^k_n(q)\right)$, defined by $F_b(q)= q^b$, is an isomorphism.
Frobenius endomorphism of $\calS_{b,0}$
---------------------------------------
For finitely many $n_i\in\N_b$ and $k_i\in \N$, the Frobenius endomorphism
$$F_b :\frac{\Z[1/b][q]}{\left(\prod_i\Phi^{k_i}_{n_i}(q)\right)}\to
\frac{\Z[1/b][q]}{\left(\prod_i\Phi^{k_i}_{n_i}(q)\right)}$$ sending $q$ to $q^b$, is again well–defined. Taking the inverse limit, we get an algebra endomorphism $$F_b: \Z[1/b][q]^{\N_b} \to \Z[1/b][q]^{\N_b}.$$
\[frob\] The Frobenius endomorphism $F_b: \Z[1/b][q]^{\N_b} \to \Z[1/b][q]^{\N_b}$, sending $q$ to $q^b$, is an isomorphism.
For finitely many $n_i\in\N_b$ and $k_i\in \N$, consider the natural algebra homomorphism $$J:\frac{\Z[1/b][q]}{\left(\prod_i\Phi^{k_i}_{n_i}(q)\right)}
\to
\prod_i \frac{ \Z[1/b][q]}{\left(\Phi^{k_i}_{n_i}(q)\right)} .$$ This map is injective, because in the unique factorization domain $\Z[1/b][q]$, one has $$(\Phi_{n_1}(q)^{k_1} \dots \Phi_{n_s}(q)^{k_s})
= \bigcap_{j=1}^s \Phi_{n_j}(q)^{k_j} \, .$$ Since the Frobenius homomorphism commutes with $J$ and is an isomorphism on the target of $J$ by Proposition \[onen\], it is an isomorphism on the domain of $J$. Taking the inverse limit, we get the claim.
Existence of $b$th root of $q$ in $\calS_{b,0}$ {#qthroot}
-----------------------------------------------
Suppose $n$ and $b$ are coprime positive integers and $y \in \Q[e_n]$ such that $y^b=1$. Then $y =\pm 1$. If $b$ is odd then $y=1$. \[0923\]
Let $d\mid b$ be the order of $y$, i.e. $y$ is a primitive $d$th root of $1$. Then $\Q[e_n]$ contains $y$, and hence $e_d$. Since $(n,d)=1$, one has $\Q[e_n]\cap \Q[e_d]=\Q$ (see e.g. [@Lang Corollary of IV.3.2]). Hence if $e_d \in \Q[e_n]$, then $e_d\in\Q$, it follows that $d=1$ or $2$. Thus $y=1$ or $y=-1$. If $b$ is odd, then $y$ cannot be $-1$.
Let $b$ be a positive integer, $T\subset \N_b$, and $y \in \Q[q]^T$ satisfying $y^b=1$. Then $y =\pm 1$. If $b$ is odd then $y=1$. \[0913\]
It suffices to show that for any $n_1, n_2 \dots n_m \in T$, the ring $\Q[q]/(\Phi^{k_1}_{n_1}\dots\Phi^{k_m}_{n_m})$ does not contains a $b$th root of $1$ except possibly for $\pm 1$. Using the Chinese remainder theorem, it suffices to consider the case where $m=1$. The ring $\Q[q]/(\Phi_{n}^k(q))$ is isomorphic to $\Q[e_n][x]/(x^k)$, by Proposition \[h\_kInjektive\]. If
$$y= \sum_{j=0}^{k-1} a_j x^j, \quad a_j \in \Q[e_n]$$ satisfies $y^b=1$, then it follows that $a_0^b=1$. By Lemma \[0923\] we have have $a_0=\pm1$. One can easily see that $a_1=\dots =a_{k-1}=0$. Thus $y=\pm 1$.
In contrast with Lemma \[0913\], we have
\[cor9\] For any odd positive $b$, and any subset $T\subset \N_b$, the ring $\Z[1/b][q]^{T}$ contains a unique $b$th root of $q$, which is invertible in $\Z[1/b][q]^T$.
For any even positive $b$, and any subset $T\subset \N_b$, the ring $\Z[1/b][q]^{T}$ contains two $b$th roots of $q$, which are invertible in $\Z[1/b][q]^T$; one is the negative of the other.
Let us first consider the case $T=\N_b$. Since $F_b$ is an isomorphism by Theorem \[frob\], we can define a $b$th root of $q$ by $$q^{1/b}:=F^{-1}_b (q) \in \calS_{b,0}\,.$$ If $y_1$ and $y_2$ are two $b$th root of the same element, then their ratio $y_1/y_2$ is a $b$th root of 1. From Lemma \[0913\] it follows that if $b$ is odd, there is only one $b$th root of $q$ in $\Z[1/b][q]^{\N_b}$, and if $b$ is even, there are 2 such roots, one is the minus of the other. We will denote them $\pm q^{1/b}$.
Further it is known that $q$ is invertible in $\Z[q]^\N$ (see [@Ha1]). Actually, there is an explicit expression $q^{-1}=\sum_n q^n (q;q)_n $. Hence $q^{-1}\in \Z[1/b][q]^{\N_b}$, since the natural homomorphism from $\Z[q]^\N$ to $\Z[1/b][q]^{\N_b}$ maps $q$ to $q$. In a commutative ring, if $x\mid y$ and $y$ is invertible, then so is $x$. Hence any root of $q$ is invertible.
In the general case of $T \subset \N_b$, we use the natural map $\Z[1/b][q]^{\N_b}\hookrightarrow \Z[1/b][q]^T$.
Relation with [@Le] {#relation-with .unnumbered}
--------------------
By Proposition \[cor9\], $\calS_{b,0}$ is isomorphic to the ring $\Lambda^{\N_b}_b:=\Z[1/b][q^{1/b}]^{\N_b}$ used in [@Le]. Furthermore, our invariant $\pi_0 I_M$ and the one defined in [@Le] belong to $\calS_{b,0}$. This follows from Theorem \[main\] for $b$ odd, and from Proposition \[main-corProof\](c) for $b$ even. Finally, the invariant defined in [@Le] for $M$ divided by the invariant of $\#_i L(b^{k_i}_i,1)$ (which is invertible in $\calS_{b,0}$ [@Le Subsection 4.1]) coincides with $\pi_0 I_M$ up to factor $q^{\frac{1-b}{4}}$ by Theorem \[main\], [@Le Theorem 3] and Proposition \[main-corProof\](b).
Another Frobenius homomorphism
------------------------------
We define another Frobenius type algebra homomorphism. The difference of the two types of Frobenius homomorphisms is in the target spaces of these homomorphisms.
Suppose $m$ is a positive integer. Define the algebra homomorphism $$G_m : R[q]^T \to R[q]^{mT} \quad \text{ by } \qquad G_m(q) = q^m.$$ Since $\Phi_{mr}(q)$ always divides $\Phi_{r}(q^m)$, $G_m$ is well–defined.
Realization of $q^{a^2/b}$ in $\calS_p$ {#defxb}
---------------------------------------
Throughout this subsection, let $p$ be a prime or $1$. Suppose $b= \pm p^l$ for an $l\in \N$ and let $a$ be an integer. Let $B_{p,j} = G_{p^j}(\calS_{p,0})$. Note that $B_{p,j}\subset \calS_{p,j}$. If $b$ is odd, by Proposition \[cor9\] there is a unique $b$th root of $q$ in $\calS_{p,0}$; we denote it by $x_{b;0}$. If $b$ is even, by Proposition \[cor9\] there are exactly two $b$th root of $q$, namely $\pm q^{1/b}$. We put $x_{b;0}=q^{1/b}$. We define an element $z_{b,a} \in \calS_p$ as follows.
If $b\mid a$, let $z_{b,a} := q^{a^2/b} \in \calS_p$.
If $ b=\pm p^l \nmid a$, then $z_{b,a}\in \calS_p$ is defined by specifying its projections $\pi_j(z_{b,a}):=z_{b,a;j}\in \calS_{p,j}$ as follows. Suppose $a = p^s e$, with $(e,p)=1$. Then $s < l$. For $j >s$ let $z_{b,a;j}:=0$. For $ 0\le j \le s$ let $$z_{b,a;j} := [G_{p^j}(x_{b;0})]^{a^2/p^j} = [G_{p^j}(x_{b;0})]^{e^2 \, p^{2s-j}}
\in B_{p,j} \subset \calS_{p,j}.$$
Similarly, for $b=\pm p^l$ we define an element $x_b \in \calS_p$ as follows. We put $\pi_0(x_b): =x_{b;0}$. For $j<l$, $\pi_j(x_b):= [G_{p^j}(x_{b;0})]^{p^j}$. If $j\geq l$, $\pi_j(x_b):=q^{b}$. Notice that for $c=(b,p^j)$ we have $$\pi_j(x_b)=z_{b,c;j}.$$
\[eval\_z\] Suppose $\xi$ is a root of unity of order $r = c r'$, where $c= (r,b)$. Then $$\ev_{\xi}(z_{b,a}) = \begin{cases} 0 & \text{ if } c \nmid a \\
(\xi^c)^{a_1^2 b'_*} & \text{ if } a=ca_1,
\end{cases}$$ where $b'_*$ is the unique element in $\Z/r'\Z$ such that $ b'_\ast (b/c) \equiv 1 \pmod{r'}$. Moreover, $$\ev_\xi(x_b)=(\xi^c)^{b'_\ast}\; .$$
Let us compute $\ev_\xi(z_{b,a})$. The case of $\ev_\xi(x_b)$ is completely analogous.
If $b\mid a$, then $c\mid a$, and the proof is obvious.
Suppose $b\nmid a$. Let $a=p^s e$ and $c= p^i$. Then $s <l$. Recall that $z_{b,a} = \prod_{j=0}^\infty z_{b,a;j}$. By Lemma \[1100\], $$\ev_\xi(z_{b,a}) = \ev_\xi(z_{b,a;i}).$$
If $c\nmid a$, then $i > s$. By definition, $z_{b,a;i}=0$, hence the statement holds true.
It remains the case $c\mid a$, or $ i \le s$. Note that $\zeta = \xi^c$ is a primitive root of order $r'$ and $(p, r')=1$. Since $z_{b,a;i} \in B_{p,i}$, $$\ev_\xi(z_{b,a;i}) \in \Z[1/p][\zeta].$$ From the definition of $z_{b,a;i}$ it follows that $(z_{b,a;i})^{b/c} =
(q^c)^{a^2/c^2}$, hence after evaluation we have $$[\ev_\xi(z_{b,a;i})]^{b/c} = (\zeta)^{a^2_1}.$$ Note also that $$[(\xi^c)^{a^2_1 b'_*}]^{b/c} = (\zeta)^{a^2_1}.$$ Using Lemma \[0923\] we conclude $\ev_\xi(z_{b,a;i}) =
(\xi^c)^{a^2_1 b'_*}$ if $b$ is odd, and $\ev_\xi(z_{b,a;i}) =(\xi^c)^{a^2_1 b'_*}$ or $\ev_\xi(z_{b,a;i}) =-(\xi^c)^{a^2_1 b'_*}$ if $b$ is even. Since $\ev_{1}(q^{1/b})=1$ and therefore $\ev_{\xi}(q^{1/b})=\xi^{b_*}$ (and not $-\xi^{b*}$) we get the claim.
Invariant of lens spaces {#diag}
========================
The purpose of this section is to prove Lemma \[0079\]. Throughout this section we will use the following notations.
Let $a$ and $b$ be coprime integers. Choose $\hat{a}$ and $\hat{b}$ such that $b\hat{b}+a\hat{a}=1$ with $0<{\operatorname{sn}}(a)\hat{a}<|b|$. Notice that for $a=1$ we have $\hat{1}=1$ and $\hat{b}=0$. Let $r$ be a fixed odd integer (the order of $\xi$). For $l\in\Z$ coprime to $r$, let $l_*$ denote the inverse of $l$ modulo $r$. If $(b,r)=c$, let $b'_*$ denote the inverse of $b':=\frac{b}{c}$ modulo $r':=\frac{r}{c}$. Notice that for $c=1$, we have $b_*=b'_*$.
Further, we denote by $\left(\frac{x}{y}\right)$ the Jacobi symbol and by $s(a,b)$ the Dedekind sum (see e.g. [@KM-Dedekind]).
Invariants of lens spaces
-------------------------
Let us compute the $SO(3)$ invariant of the lens space $M(b,a;d)$. Recall that $M(b,a;d)$ is the lens space $L(b,a)$ together with a knot $K$ inside colored by $d$ (see Figure 1).
\[1409\] Suppose $c=(b,r)$ divides $d-{\operatorname{sn}}(a)\hat{a}$. Then $$\begin{aligned}
\tau'_{M(b,a;d)}(\xi)&= (-1)^{\frac{c+1}{2}\,\frac{{\operatorname{sn}}(ab)-1}{2}}
\left(\frac{|a|}{c}\right)
\left(\frac{1-\xi^{-{\operatorname{sn}}(a)db'_*}}{1-\xi^{-{\operatorname{sn}}(b)b'_*}}\right)^{\chi(c)}
\xi^{
4_*u-4_*b'_* \frac{a(\hat{a}-{\operatorname{sn}}(a)d)^2}{c}
}\end{aligned}$$ where $$u=12s(1,b)-12 {\operatorname{sn}}(b)s(a,b)+
\frac{1}{b}\left(a(1-d^2)+2({\operatorname{sn}}(a)d-{\operatorname{sn}}(b))
+a(\hat{a}-{\operatorname{sn}}(a)d)^2\right) \in \Z$$ and $\chi(c)=1$ if $c=1$ and is zero otherwise. If $c\nmid (\hat{a}\pm d)$, $\tau_{M(b,a;d)}(\xi)=0\; .$ \[lens\]
In particular, it follows that $\tau_{L(b,a)}(\xi)=0$ if $c\nmid \hat{a}\pm 1$.
We consider first the case where $b,a>0$. Since two lens spaces $L(b,a_1)$ and $L(b,a_2)$ are homeomorphic if $a_1 \equiv a_2 \pmod{b}$, we can assume $a<b$. Let $b/a$ be given by a continued fraction $$\frac{b}{a}=m_n-\frac{1}{\displaystyle m_{n-1}-\frac{1}{\displaystyle
m_{n-2}-\dots \frac{1}{\displaystyle m_2-\frac{1}{\displaystyle
m_1}}}}.$$ Using the Lagrange identity $$a-\frac{1}{b}=(a-1)+\frac{1}{\displaystyle 1+\frac{1}{\displaystyle
(b-1)}}$$ we can assume $m_i\geq 2$ for all $i$.
The $\tau_{M(b,a;d)}(\xi)$ can be computed in the same way as the invariant $\xi_r(L(b,a),A)$ in [@LiLi], after replacing $A^2$ (respectively $A$) by $\xi^{2_*}$ (respectively $\xi^{4_*}$). Representing the $b/a$–framed unknot in Figure \[figure:LensSpaceKnot\] by a Hopf chain (as e.g. in Lemma 3.1 of [@BL]), we have $$F_{L\sqcup K}(\xi,d)={\sum_{j_i}}^{\xi}\prod_{i=1}^{n}
q^{m_i \frac{j_i^2-1}{4}}
\prod_{i=1}^{n-1} [j_i j_{i+1}]\cdot[j_n d][j_1]
=\frac{S_n(d)}{(\xi^{2_*}-\xi^{-2_*})^{n+1}}\cdot \xi^{-4_*\sum_{i=1}^{n}m_i}$$ where $$S_n(d)=
{\sum_{\substack{j_i=1 \\ \text{odd}}}^{2r}}
\xi^{\sum m_ij^2_i}
(\xi^{2_*j_1}-\xi^{-2_*j_1})
(\xi^{2_*j_1j_2}-\xi^{-2_*j_1j_2})\dots
(\xi^{2_*j_{n-1}j_n}-\xi^{-2_*j_{n-1}j_n})
(\xi^{2_*j_n d}-\xi^{-2_*j_n d})\, .$$
Using Lemmas 4.11, 4.12 and 4.20 of [@LiLi][^1] (and replacing $e_r$ by $\xi^{4_*}$, $c_n$ by $c$, $N_{n,1}=p$ by $b$, $N_{n-1,1}=q$ by $a$, $N_{n,2}=q^*$ by $\hat{a}$ and $-N_{n-1,2}=p^*$ by $\hat{b}$), we get $$S_n(d)=(-2)^n (\sqrt{r} \epsilon(r))^n \sqrt{c} \epsilon(c)
\left(\frac{\frac{b}{c}}{\frac{r}{c}}\right)\left(\frac{a}{c}\right)
(-1)^{\frac{r-1}{2}\frac{c-1}{2}}
\cdot\sum_{\pm}\chi^{\pm}(d)
\xi^{-ca4_*b'_*
\left(\frac{d\mp \hat{a}}{c}\right)^2
\pm 2_*\hat{b}(d\mp \hat{a}) +4_*\hat{a} \hat{b}}$$ where $\chi^{\pm}(d)=\pm 1$ if $c\mid d\mp \hat{a}$ and is zero otherwise. Further $\epsilon(x)=1$ if $x \equiv 1 \pmod 4$ and $ \epsilon(x)=I$ if $x \equiv 3 \pmod 4$. This implies the second claim of the lemma.
Note that when $c=1$, both $\chi^\pm(d)$ are nonzero. If $c>1$ and $c \mid (d-\hat{a})$, $\chi^+(d)=1$, but $\chi^- (d)=0$. Indeed, for $c$ dividing $d-\hat{a}$, $c \mid (d+\hat{a})$ if and only if $c\mid \hat{a}$ which is impossible, because $c\mid b$ but $(b,\hat{a})=1$.
Inserting the last formula into the Definition we get $$\tau_{M(b,a;d)}(\xi)=
\frac{S_n(d)}{\xi^{2_*}-\xi^{-2_*}}
\left(-2 \xi^{-3\cdot4_*} \sum_{j=1}^{r} \xi^{4_*j^2}\right)^{-n}
\xi^{-4_* \sum_{i=1}^{n}m_i}$$ where we used that $\sigma_+=n$ and $\sigma_-=0$ (compare [@KM-Dedekind p. 243]). From $\sum_{j=1}^{r}\xi^{4_*j^2}=\epsilon(r)\sqrt{r}$, we obtain $$\tau_{M(b,a;d)}(\xi)=(-1)^{\frac{(c-1)(r-1)}{4}}\epsilon(c)
\left(\frac{b'}{r'}\right)
\left(\frac{a}{c}\right)
\sqrt{c}
\; \frac{(1-\xi^{- db'_*})^{\chi(c)}}{\xi^{2_*}-\xi^{-2_*}}
\; \xi^{4_*(3n-\sum_i m_i)
-4_*\hat{b}(\hat{a}-2d)
-4_*b'_*\frac{a(d-\hat{a})^2}{c}}\,.$$ Applying the following formulas for the Dedekind sum (compare [@KM-Dedekind Theorem 1.12]) \[Dedekind\] 3n- \_i m\_i=-12 s(a,b)+, -3+b=12 s(1,b)- and dividing the formula for $\tau_{M(b,a;d)}(\xi)$ by the formula for $\tau_{L(b,1)}(\xi)$ we get $$\tau'_{M(b,a;d)}(\xi)=
\left(\frac{a}{c}\right)
\left(\frac{1-\xi^{-db'_*}}{1-\xi^{-b'_*}}\right)^{\chi(c)}
\xi^{4_*u-4_*b'_*\frac{a(d-\hat{a})^2}{c}}$$ where $$u=
-12s(a,b)+12s(1,b)+
\frac{1}{b}\left(a+\hat{a}
-2
-\hat{b}b(\hat{a}-2d)\right).$$ Notice, that $u\in\Z$. Further observe, that by using $a\hat{a}+b\hat{b}=1$, we get $$a+\hat{a}-2-\hat{b}b(\hat{a}-2d)=
2(d-1)+a(1-d^2)+a(\hat{a}-d)^2.$$ This implies the result for $0<a<b$.
To compute $\tau_{M(-b,a;d)}(\xi)$, observe that $\tau_{M(b,-a;d)}=\tau_{M(-b,a;d)}$ is equal to the complex conjugate of $\tau_{M(b,a;d)}$. The ratio $$\tau'_{M(-b,a;d)}(\xi)=\frac{\;\;\overline{\tau_{M(b,a;d)}(\xi)}\;\;}
{\tau_{L(b,1)}(\xi)}$$ can be computed analogously. Using $\overline{\epsilon(c)}=(-1)^{\frac{c-1}{2}}\epsilon(c)$, we have for $a,b>0$ $$\tau'_{M(-b,a,d)}(\xi) =(-1)^{\frac{c+1}{2}}
\left(\frac{a}{c}\right)
\left(\frac{1- \xi^{db'_*}}{1-\xi^{-b'_*}}\right)^{\chi(c)}
\xi^{4_*\tilde{u}+4_*b'_*\frac{a(d-\hat{a})^2}{c}}$$ where $$\tilde{u}=12s(a,b)+12s(1,b)+\frac{1}{b}
\left(
-a-\hat{a}-2+\hat{b}b(\hat{a}-2d)
\right)$$ Using $s(a,b)=s(a,-b)=-s(-a,b)$, we get the result.
For $b>0$, we have $$\tau'_{L(-b,1)}(\xi)
=
(-1)^{\frac{c+1}{2}-{\chi(c)}}\,
\xi^{2_*(b-3)+b_*\chi(c)}
\; .$$
Proof of Lemma \[0079\]
-----------------------
Assume $b=\pm p^l$ and $p$ is prime. We have to define the unified invariant of $M^\ve(b,a) := M(b,a;d(\ve))$, where $d(0)=1$ and $d({{\bar 0}})$ is the smallest odd positive integer such that ${\operatorname{sn}}(a)ad({{\bar 0}}) \equiv 1 \pmod {b}$. First observe that such $d({{\bar 0}})$ always exists. Indeed, if $p$ is odd, we can achieve this by adding $b$, otherwise the inverse of any odd number modulo $2^l$ is odd.
Recall that we denote the unique positive $b$th root of $q$ in $S_{p,0}$ by $q^{\frac{1}{b}}$. We define the unified invariant $I_{M^\ve(b,a)}\in \R_b$ as follows. If $p\neq 2$, then $I_{M^\ve(b,a)}\in \calS_p$ is defined by specifying its projections $$\pi _j I_{M^\ve(b,a)}
:=
\,
\begin{cases}
q^{3 s(1,b)-3{\operatorname{sn}}(b)\, s(a,b)}
&\text{if } j=0, \;\ve =0
\\&\\
(-1)^{\frac{p^j+1}{2}\,\frac{{\operatorname{sn}}(ab)-1}{2}}
\left(\frac{|a|}{p}\right)^j \,
q^{\frac{u'}{4}}
&\text{if } 0<j<l, \;\ve={{\bar 0}}\\&\\
(-1)^{\frac{p^l+1}{2}\,\frac{{\operatorname{sn}}(ab)-1}{2}}
\left(\frac{|a|}{p}\right)^l q^{\frac{u'}{4}}
&\text{if } j\geq l, \;\ve={{\bar 0}}\end{cases}$$ where $u':=u-\frac{a(\hat{a}-{\operatorname{sn}}(a)d({{\bar 0}}))^2}{b}$ and $u$ is defined in Proposition \[1409\]. If $p=2$, then only $\pi_0 I_{M(b,a)}\in \calS_{2,0}=\R_2$ is non–zero and it is defined to be $q^{3s(1,b)-3 {\operatorname{sn}}(b)\; s(a,b)}$.
The $I_{M^{\varepsilon}(b,a)}$ is well–defined due to Lemma \[0622\] below, i.e. all powers of $q$ in $I_{M^{\ve}(b,a)}$ are integers for $j>0$ or lie in $\frac{1}{b}\Z$ for $j=0$. Further, for b odd (respectively even) $I_{M^\ve(b,a)}$ is invertible in $\calS_{p}^{p,\varepsilon}$ (respectively $\R_p^{p,\varepsilon}$) since $q$ and $q^{\frac{1}{b}}$ are invertible in these rings.
In particular, for odd $b=p^l$, we have $I_{L(b,1)}=1$, and $$\pi _j I_{L(-b,1)}
=
\,
\begin{cases}
q^{\frac{b-3}{2}+\frac{1}{b}}
&\text{if } j=0
\\&\\
(-1)^{\frac{p^j+1}{2}}q^{\frac{b-3}{2}}
&\text{if } 0<j<l, \; p \text{ odd} \\&\\
(-1)^{\frac{p^l+1}{2}}q^{\frac{b-3}{2}}
&\text{if } j\geq l, \; p \text{ odd} \, .
\end{cases}$$
It is left to show, that for any $\xi$ of order $r$ coprime with $p$, we have $$\ev_\xi ( I_{M^0(b,a)})=\tau'_{M^0(b,a)}(\xi)\,$$ and if $r=p^j k$ with $j>0$, then $$\ev_\xi ( I_{M^{{{\bar 0}}}(b,a)})=\tau'_{M^{{\bar 0}}(b,a)}(\xi)\,.$$ For $\ve=0$, this follows directly from Propositions \[eval\_z\] and \[1409\] with $c=d=1$. For $\ve={{\bar 0}}$, we have $c=(p^j,b)>1$ and we get the claim by using Proposition \[1409\] and $$\label{0623}
\xi^{\frac{a(\hat{a}-{\operatorname{sn}}(a)d({{\bar 0}}))^2}{b}}=
\xi^{c\,\frac{a(\hat{a}-{\operatorname{sn}}(a)d({{\bar 0}}))^2}{bc}}=
\xi^{bb'_*\,\frac{a(\hat{a}-{\operatorname{sn}}(a)d({{\bar 0}}))^2}{bc}}=
\xi^{b'_*\,\frac{a(\hat{a}-{\operatorname{sn}}(a)d({{\bar 0}}))^2}{c}},$$ where for the second equality we use $c\equiv bb'_*\pmod{r}$. Notice that due to part (2) of Lemma \[0622\] below, $b$ and $c$ divide $\hat{a}-{\operatorname{sn}}(a)d({{\bar 0}})$ and therefore all powers of $\xi$ in (\[0623\]) are integers.
The following Lemma is used in the proof of Lemma \[0079\].
\[0622\] We have
- $ 3 s(1,b) - 3{\operatorname{sn}}(b)\, s(a,b) \in \frac{1}{b}\Z$,
- $b\mid \hat{a}-{\operatorname{sn}}(a)d({{\bar 0}})$ and therefore $u'\in\Z$, and
- $4\mid u'$ for $d=d({{\bar 0}})$.
The first claim follows from the formulas (\[Dedekind\]) for the Dedekind sum. The second claim follows from the fact that $(a,b)=1$ and $$a(\hat{a}-{\operatorname{sn}}(a)d)=1-{\operatorname{sn}}(a)ad-b\hat{b}\equiv 0\pmod{b},$$ since $d$ is chosen such that ${\operatorname{sn}}(a)ad\equiv 1\pmod{b}$. For the third claim, notice that for odd $d$ we have $$4\mid (1-d^2) \;\;\text{ and }\;\;
4\mid 2({\operatorname{sn}}(a)d-{\operatorname{sn}}(b)).$$
Laplace transform {#laplace}
=================
This section is devoted to the proof of Theorem \[0078\] by using Andrew’s identity. Throughout this section, let $p$ be a prime or $p=1$, and $b= \pm p^l$ for an $l\in \N$.
Definition
----------
The Laplace transform is a $\Z[q^{\pm 1}]$–linear map defined by $$\begin{aligned}
\cL_{b}: \Z[z^{\pm 1}, q^{\pm 1}] &\to& \calS_p \\
z^a &\mapsto& z_{b,a}.\end{aligned}$$ In particular, we put $\cL_{b;j}:=\pi_j \circ \cL_{b}$ and have $\cL_{b;j}(z^a)=z_{b,a;j} \in \calS_{p,j}$.
Further, for any $f \in \BZ[z^{\pm 1},q^{\pm 1}]$ and $n\in \Z$, we define $$\hat f:=f|_{z=q^n}\in \Z[q^{\pm n}, q^{\pm 1}]\,.$$
Suppose $f \in \BZ[z^{\pm 1},q^{\pm 1}]$. Then for a root of unity $\xi$ of odd order $r$, $${\sum_{n}}^\xi q^{b\frac{n^2-1}{4}} \hat f = \gamma_{b}(\xi) \,
\ev_\xi(\cL_{-b}(f)).$$ \[1001\]
It is sufficient to consider the case $f=z^a$. Then, by the same arguments as in the proof of [@BL Lemma 1.3], we have \[four\] [\_[n]{}]{}\^q\^[b]{} q\^[ na]{} =
0 &\
(\^c)\^[-[a\_1\^2 b’\_\*]{}]{} \_b() & .
The result follows now from Proposition \[eval\_z\].
Proof of Theorem \[0078\]
-------------------------
Recall that $$A(n,k) = \frac{\prod^{k}_{i=0}
\left(q^{n}+q^{-n}-q^i -q^{-i}\right)}{(1-q) \, (q^{k+1};q)_{k+1}}.$$ We have to show that there exists an element $Q_{b,k} \in \R_b $ such that for every root of unity $\xi$ of odd order $r$ one has $$\frac{{\sum_n}^{\hspace{-1.8mm}\xi } \; q^{b\frac{n^2-1}{4}} A(n,k) }{F_{U^b}(\xi)}
= \ev_\xi (Q_{b,k}).$$ Applying Lemma \[1001\] to $F_{U^b}(\xi)={\sum\limits_ n}^\xi q^{b\frac{n^2-1}{4}}
[n]^2$, we get for $c=(b,r)$ \[1122\] F\_[U\^b]{}()=2\_b() \_() ,where as usual, $\chi(c)=1$ if $c=1$ and is zero otherwise. We will prove that for an odd prime $p$ and any number $j\geq 0$ there exists an element $Q_k(q,x_b,j) \in \calS_{p,j}$ such that \[imp\] \_[b;j]{}( \_[i=0]{}\^k (z+z\^[-1]{} - q\^i -q\^[-i]{}) ) = 2 Q\_k(q\^[(b)]{}, x\_[b]{},j). If $p=2$ we will prove the claim for $j=0$ only, since $\calS_{2,0}\simeq\cR_{2}$. The case $p=\pm 1$ was done e.g. in [@BBL]. Theorem \[0078\] follows then from Lemma \[1001\] and where $Q_{b,k}$ is defined by its projections $$\pi_j Q_{b,k}:=\;
\frac{1-q^{-1}}{(1-x_{-b})^{\chi(p^j)}}\; Q_k(q^{-{\operatorname{sn}}(b)},x_{-b},j).$$ We split the proof of (\[imp\]) into two parts. In the first part we will show that there exists an element $Q_{k}(q,x_b,j)$ such that Equality holds. In the second part we show that $Q_k(q,x_b,j)$ lies in $\calS_{p,j}$.
Part 1, $b$ odd case {#part-1-b-odd-case .unnumbered}
--------------------
Assume $b=\pm p^l$ with $p\not=2$. We split the proof into several lemmas.
\[S\_[b;j]{}(k,q)\] For $x_{b;j}:=\pi_j(x_b)$ and $c=(b,p^j)$, $$\cL_{b;j}\left( \prod_{i=0}^k (z+z^{-1} - q^i -q^{-i}) \right)=
2\, (-1)^{k+1} \, {\text{$\left[\begin{array}{c}2k+1\\ k\end{array}
\right]$}} \, S_{b;j}(k,q),$$ where $$\label{unif-s}
S_{b;j}(k,q):=1+\sum_{n=1}^{\infty}\frac{q^{(k+1)cn}(q^{-k-1};q)_{cn}}
{(q^{k+2};q)_{cn}}
(1+q^{cn}) x_{b;j}^{n^2}.$$
Observe that for $n> \frac{k+1}{c}$ the term $(q^{-k-1};q)_{cn}$ is zero and therefore the sum in is finite.
Since $\cL_{b}$ is invariant under $z \to z^{-1}$ one has $$\cL_b\left(\prod_{i=0}^k (z+z^{-1} - q^i -q^{-i})\right) =
-2\cL_{b} (z^{-k}(zq^{-k};q)_{2k+1}),$$ and the $q$–binomial theorem (e.g. see [@GR], II.3) gives $$\label{qbinomial}
z^{-k}(zq^{-k};q)_{2k+1}=
(-1)^k \sum_{i=-k}^{k+1}(-1)^i{\text{$\left[\begin{array}{c}2k+1\\ k+i\end{array}
\right]$}}z^i.$$ Notice that $\cL_{b;j}(z^a)\not=0$ if and only if $c \mid a$. Applying $\cL_{b;j}$ to the RHS of (\[qbinomial\]), only the terms with $c \mid i$ survive and therefore $$\cL_{b;j}\left(z^{-k}(zq^{-k};q)_{2k+1}\right)=
(-1)^{k} \sum_{n=-\lfloor k/c \rfloor}^{\lfloor (k+1)/c \rfloor} (-1)^{cn}
{\text{$\left[\begin{array}{c}2k+1\\ k+cn\end{array}
\right]$}} z_{b,cn;j}.$$ Separating the case $n=0$ and combining positive and negative $n$ this is equal to $$(-1)^{k}{\text{$\left[\begin{array}{c}2k+1\\ k\end{array}
\right]$}}
+(-1)^k \sum_{n=1}^{\lfloor (k+1)/c \rfloor} (-1)^{cn}\left({\text{$\left[\begin{array}{c}2k+1\\ k+cn\end{array}
\right]$}}+{\text{$\left[\begin{array}{c}2k+1\\ k-cn\end{array}
\right]$}}\right) z_{b,cn;j},$$ where we use the convention that ${\text{$\left[\begin{array}{c}x\\ -1\end{array}
\right]$}}$ is put to be zero for positive $x$. Further, $${\text{$\left[\begin{array}{c}2k+1\\ k+cn\end{array}
\right]$}} +{\text{$\left[\begin{array}{c}2k+1\\ k-cn\end{array}
\right]$}} =\frac{\{k+1\}}{\{2k+2\}}{\text{$\left[\begin{array}{c}2k+2\\ k+cn+1\end{array}
\right]$}}(q^{cn/2}+q^{-cn/2})$$ and $$\frac{\{k+1\}}{\{2k+2\}}{\text{$\left[\begin{array}{c}2k+2\\ k+cn+1\end{array}
\right]$}}{\text{$\left[\begin{array}{c}2k+1\\ k\end{array}
\right]$}}^{-1}=
(-1)^{cn}q^{(k+1)cn+\frac{cn}{2}}\frac{(q^{-k-1};q)_{cn}}{(q^{k+2};q)_{cn}}
.$$ Using $z_{b,cn;j}=(z_{b,c;j})^{n^2}=x_{b;j}^{n^2}$ we get the result.
To define $Q_k(q,x_b,j)$ we will need Andrew’s identity (3.43) of [@And]: $$\begin{aligned}
&&\hspace{-6mm}\sum_{n\geq 0} (-1)^n\alpha_n t^{-\frac{n(n-1)}{2}+sn+Nn} \frac{(t^{-N})_n}{(t^{N+1})_n}
\prod_{i=1}^{s}\frac{(b_i)_n (c_i)_n}{b_i^nc_i^n(\frac{t}{b_i})_n(\frac{t}{c_i})_n}
=\\
&&
\hspace{-4mm}
\frac{(t)_N(\frac{q}{b_sc_s})_N}{(\frac{t}{b_s})_N(\frac{t}{c_s})_N}
\sum_{n_s\geq \cdots\geq n_2\geq n_1 \geq 0}
\beta_{n_1}
\frac{t^{n_s}(t^{-N})_{n_s}(b_s)_{n_s}(c_s)_{n_s}}{(t^{-N} b_s c_s)_{n_s}}
\prod_{i=1}^{s-1}
\frac{t^{n_i}}{b_i^{n_{i}} c_i^{n_{i}}}
\frac{(b_i)_{n_{i}}(c_i)_{n_{i}}}{(\frac{t}{b_i})_{n_{i+1}}(\frac{t}{c_i})_{n_{i+1}}}
\frac{(\frac{t}{b_i c_i})_{n_{i+1}-n_i}}{(t)_{n_{i+1}-n_i}}\;.\end{aligned}$$ Here and in what follows we use the notation $(a)_n:=(a;t)_n$ . The special Bailey pair $(\alpha_n,\beta_n)$ is chosen as follows $$\begin{array}{rclcrcl}
\alpha_0&=&1, &&\alpha_n&=&(-1)^nt^{\frac{n(n-1)}{2}}(1+t^n)\\
\beta_0&=&1, &&\beta_n&=&0 \;\;\;\text{ for } n\geq 1.
\end{array}$$
\[LHS\] $S_{b;j}(k,q)$ is equal to the LHS of Andrew’s identity with the parameters fixed below.
Since $$S_{b;j}(k,q)=S_{-b;j}(k,q^{-1}),$$ it is enough to look at the case when $b>0$. Define $b':=\frac{b}{c}$ and let $\omega$ be a $b'$th primitive root of unity. For simplicity, put $N:=k+1$ and $t:=x_{b;j}$. Using the following identities $$\begin{aligned}
(q^{y};q)_{cn}&=&\prod_{l=0}^{c-1}(q^{y+l};q^c)_n\\
(q^{yc};q^c)_n&=&\prod_{i=0}^{b'-1}(\omega^i t^y;t)_n,\end{aligned}$$ where the later is true due to $t^{b'}=x_{b;j}^{b'}=q^c$ for all $j$, and choosing a $c$th root of $t$ denoted by $t^{\frac{1}{c}}$ we can see that $$S_{b;j}(k,q)=1+\sum_{n=1}^{\infty}
\prod_{i=0}^{b'-1}\prod_{l=0}^{c-1}
\frac{
(\omega^i t^{\frac{-N+l}{c}})_{n}
}{
(\omega^i t^{\frac{N+1+l}{c}})_{n}
}
(1+t^{b'n})t^{n^2+b'Nn}.$$
Now we choose the parameters for Andrew’s identity as follows. We put $a:=\frac{c-1}{2}$, $d:=\frac{b'-1}{2}$ and $m:=\lfloor \frac{N}{c}\rfloor$. For $l\in\{1,\ldots,c-1\}$ there exist unique $u_l,v_l\in\{0,\ldots,c-1\}$ such that $u_l\equiv N+l\pmod{c}$ and $v_l\equiv N-l\pmod{c}$. Note that $v_l=u_{c-l}$. We define $U_l:=\frac{-N+u_l}{c}$ and $V_l:=\frac{-N+v_l}{c}$. Then $U_l, V_l \in \frac{1}{c}\Z$ but $U_l+V_l \in \Z$. We define $$\begin{array}{rclrcll}
b_l&:=&t^{U_l}, &c_l&:=&t^{V_l} & \text{for }\; l=1,\ldots, a, \\
b_{a+i}&:=&\omega^it^{-m},
&c_{a+i}&:=&\omega^{-i}t^{-m}
& \text{for }\;i=1,\ldots,d, \\
b_{a+ld+i}&:=&\omega^it^{U_l},
&c_{a+ld+i}&:=&\omega^{-i}t^{V_l}
& \text{for }\;i=1,\ldots,d \text{ and } l=1,\ldots, c-1, \\
b_{g+i}&:=&-\omega^{i}t, &c_{g+i}&:=&-\omega^{-i}t &\text{for }\;i=1,\ldots,d,\\
b_{s-1}&:=&t^{-m}, &c_{s-1}&:=&t^{N+1}, &\\
b_s&\rightarrow&\infty,&c_s&\rightarrow&\infty.&\\
\end{array}$$ where $g=a+cd$ and $s=(c+1)\frac{b'}{2}+1$.
We now calculate the LHS of Andrew’s identity. Using the notation $$(\omega^{\pm 1} t^x)_n=(\omega t^x)_n(\omega^{-1} t^x)_n$$ and the identities $$\lim_{c\to \infty} \frac{(c)_n}{c^n}=(-1)^n t^{\frac{n(n-1)}{2}} \;\;\text{ and }\;\; \lim_{c\to\infty} \left(\frac{t}{c}\right)_n=1$$ we get $$\begin{aligned}
LHS&=&1+
\sum_{n\geq 1}
t^{n(n-1+s+N-y)}\;
(1+t^n)\frac{(t^{-N})_n}{(t^{N+1})_n}
\cdot
\prod_{l=1}^{a}
\frac{
(t^{U_l})_n (t^{V_l})_n
}{
(t^{1-U_l})_n (t^{1-V_l})_n
}
\cdot
\prod_{i=1}^{d}
\frac{
(\omega^{\pm i} t^{-m})_n
}{
(\omega^{\pm i} t^{1+m})_n
}
\\ && \hspace{3cm}
\cdot\prod_{i=1}^{d}\prod_{l=1}^{c-1}
\frac{
(\omega^i t^{U_l})_n
(\omega^{-i}t^{V_l})_n
}{
(\omega^{-i} t^{1-U_l})_n
(\omega^{i}t^{1-V_l})_n
}
\cdot
\prod_{i=1}^{d}
\frac{
(-\omega^{\pm i}t)_n
}{
(-\omega^{\pm i})_n
}
\cdot
\frac{
(t^{-m})_n (t^{N+1})_n
}{
(t^{1+m})_n (t^{-N})_n
}\end{aligned}$$ where $$y:=
\sum_{l=1}^{a} (U_l+V_l)+\sum_{i=1}^{d}\sum_{l=1}^{c-1}(U_l+V_l)-m(2d+1)+2d+1+N.$$ Since $\sum_{l=1}^{c-1}(U_l+V_l)=2\sum_{l=1}^{a}(U_l+V_l)=2(-N+m+\frac{c-1}{2})$ and $2d+1=b'$, we have $$n-1+s+N-y=n+Nb'.$$ Further, $$\prod_{i=1}^{d} \frac{(-\omega^{\pm i}t)_n}{(-\omega^{\pm i})_n}
=
\prod_{i=1}^{b'-1}\frac{1+\omega^i t^n}{1+\omega^i}=\frac{1+t^{b'n}}{1+t^n}$$ and $$\begin{aligned}
&&
\prod_{l=1}^{a}
\frac{
(t^{U_l})_n (t^{V_l})_n
}{
(t^{1-U_l})_n (t^{1-V_l})_n
}
\cdot
\prod_{i=1}^{d}
\frac{
(\omega^{\pm i} t^{-m})_n
}{
(\omega^{\pm i} t^{1+m})_n
}
\cdot
\prod_{i=1}^{d}\prod_{l=1}^{c-1}
\frac{
(\omega^i t^{U_l})_n
(\omega^{-i}t^{V_l})_n
}{
(\omega^{-i} t^{1-U_l})_n
(\omega^{i}t^{1-V_l})_n
}
\cdot
\frac{
(t^{-m})_n
}{
(t^{1+m})_n
}
\\
&&\hspace{105mm}=
\prod_{i=0}^{b'-1}\prod_{l=0}^{c-1}
\frac{
(\omega^i t^{\frac{-N+l}{c}})_{n}
}{
(\omega^i t^{\frac{N+1+l}{c}})_{n}
}.\end{aligned}$$ Taking all the results together, we see that the LHS is equal to $S_{b;j}(k,q)$.
Let us now calculate the RHS of Andrew’s identity with parameters chosen as above. For simplicity, we put $\delta_j:=n_{j+1}-n_j$. Then the RHS is given by $$\begin{aligned}
RHS
&=&
(t)_N\sum_{n_s\geq \cdots\geq n_2 \geq n_1=0}
\frac{
t^{x}\cdot (t^{-N})_{n_s}(b_s)_{n_s}(c_s)_{n_s}
}{
\prod_{i=1}^{s-1} (t)_{\delta_{i}}(t^{-N}b_sc_s)_{n_s}
}
\cdot
\frac{
(t^{-m})_{n_{s-1}}(t^{N+1})_{n_{s-1}}
(t^{m-N})_{\delta_{s-1}}
}{
(t^{m+1})_{n_s}(t^{-N})_{n_s}
}
\\
&&\hspace{7mm}
\cdot
\prod_{l=1}^{a}
\frac{
(t^{U_l})_{n_l}(t^{V_l})_{n_l}
(t^{1-U_l-V_l})_{\delta_{l}}
}{
(t^{1-U_l})_{n_{l+1}}(t^{1-V_l})_{n_{l+1}}
}
\cdot
\prod_{i=1}^{d}
\frac{
(\omega^{\pm i}t^{-m})_{n_{a+i}}
(t^{2m+1})_{\delta_{a+i}}
}{
(\omega^{\pm i}t^{m+1})_{n_{a+i+1}}
}
\frac{
(-\omega^{\pm i}t)_{n_{g+i}}
(t^{-1})_{\delta_{g+i}}
}{
(-\omega^{\pm i})_{n_{g+i+1}}
}
\\
&&\hspace{7mm}
\cdot
\prod_{i=1}^{d}\prod_{l=1}^{c-1}
\frac{
(\omega^it^{U_l})_{n_{a+ld+i}}
(\omega^{-i}t^{V_l})_{n_{a+ld+i}}
(t^{1-U_l-V_l})_{\delta_{a+ld+i}}
}{
(\omega^{-i}t^{1-U_l})_{n_{a+ld+i+1}}
(\omega^{i}t^{1-V_l})_{n_{a+ld+i+1}}
}\end{aligned}$$ where $$\begin{aligned}
x
&=&\sum_{l=1}^{a}(1-U_l-V_l)\,n_l
+\sum_{i=1}^{d}(2m+1)\,n_{a+i}
\\&&\hspace{2cm}
+\sum_{i=1}^{d}\sum_{l=1}^{c-1}(1-U_l-V_l)\, n_{a+ld+i}
-\sum_{i=1}^{d}n_{g+i}+(m-N)\,n_{s-1}+n_s.\end{aligned}$$ For $c=1$ or $d=0$, we use the convention that empty products are set to be 1 and empty sums are set to be zero.
Let us now have a closer look at the RHS. Notice, that $$\lim_{b_s,c_s\to\infty}\frac{
(b_s)_{n_s}(c_s)_{n_s}
}{
(t^{-N}b_sc_s)_{n_s}
}
=
(-1)^{n_s}t^{\frac{n_s(n_s-1)}{2}}t^{Nn_s}.$$ The term $(t^{-1})_{\delta_{g+i}}$ is zero unless $\delta_{g+i}\in \{0,1\}$. Therefore, we get $$\prod_{i=1}^{d}\frac{
(-\omega^{\pm i}t)_{n_{g+i}}
}{
(-\omega^{\pm i})_{n_{g+i+1}}
}
=
\prod_{i=1}^{d}
(1+\omega^{\pm i}t^{n_{g+i}})^{1-\delta_{g+i}}.$$ Due to the term $(t^{-m})_{n_s}$, we have $n_s\leq m$ and therefore $n_i\leq m$ for all $i$. Multiplying the numerator and denominator of each term of the RHS by $$\begin{aligned}
&&
\prod_{l=1}^{a} (t^{1-U_l+n_{l+1}})_{m-n_{l+1}}(t^{1-V_l+n_{l+1}})_{m-n_{l+1}}
\prod_{i=1}^{d} (\omega^{\pm i}t^{m+1+n_{a+i+1}})_{m-n_{a+i+1}}
\\
&&\hspace{35mm}\cdot\prod_{i=1}^{d}\prod_{l=1}^{c-1}
(\omega^{-i}t^{1-U_l+n_{a+ld+i+1}})_{m-n_{a+ld+i+1}}
(\omega^{i}t^{1-V_l+n_{a+ld+i+1}})_{m-n_{a+ld+i+1}}\end{aligned}$$ gives in the denominator $\prod_{i=0}^{b'-1}\prod_{l=1}^{c-1}(\omega^it^{1-U_l})_m
\cdot
\prod_{i=1}^{b'-1}(\omega^it^{m+1})_m$. This is equal to $$\prod_{l=1}^{c-1}(t^{b'(1-U_l)};t^{b'})_m
\cdot \frac{(t^{b'(m+1)};t^{b'})_m}{(t^{m+1};t)_m}
=
\frac{(q^{N+1};q)_{cm}}{(t^{m+1};t)_m}.$$ Further, $$(t)_N(t^{N+1})_{n_{s-1}}=(t)_{N+n_{s-1}}=(t)_{m}(t^{m+1})_{N-m+n_{s-1}}.$$ The term $(t^{-N+m})_{\delta_{s-1}}$ is zero unless $\delta_{s-1}\leq N-m$ and therefore $$\frac{(t^{m+1})_{N-m+n_{s-1}}}{(t^{m+1})_{n_s}}
=
(t^{m+1+n_s})_{N-m-\delta_{s-1}}.$$ Taking the above calculations into account, we get $$\label{RHS}
RHS=
\frac{(t;t)_{2m}}{(q^{N+1};q)_{cm}}
\cdot T_k(q,t)$$ where $$\begin{aligned}
T_k(q,t)&:=&
\sum_{n_s\geq \cdots\geq n_2 \geq n_1=0}
(-1)^{n_s}t^{x'}\cdot
(t^{-m})_{n_{s-1}}\cdot
(t^{m+1+n_s})_{N-m-\delta_{s-1}}
\cdot\frac{
(t^{-N+m})_{\delta_{s-1}}
}{
\prod_{i=1}^{s-1} (t)_{\delta_i}
}
\\
&&\hspace{3mm}
\cdot
\prod_{l=1}^{a}(t^{1-U_l-V_l})_{\delta_l}
\cdot
\prod_{i=1}^{d} (t^{2m+1})_{\delta_{a+i}}(t^{-1})_{\delta_{g+i}}
\cdot
\prod_{i=1}^{d}\prod_{l=1}^{c-1}(t^{1-U_l-V_l})_{\delta_{a+ld+i}}
\\
&&\hspace{3mm}
\cdot\prod_{l=1}^{a}
(t^{U_l})_{n_l}(t^{V_l})_{n_l}
(t^{1-U_l+n_{l+1}})_{m-n_{l+1}}
(t^{1-V_l+n_{l+1}})_{m-n_{l+1}}
\cdot
\prod_{i=1}^{d}
(1+\omega^{\pm i}t^{n_{g+i}})^{1-\delta_{g+i}}
\\
&&\hspace{3mm}
\cdot
\prod_{i=1}^{d}
(\omega^{\pm i}t^{-m})_{n_{a+i}}
(\omega^{\pm i}t^{m+1+n_{a+i+1}})_{m-n_{a+i+1}}
\cdot
\prod_{i=1}^{d}
\prod_{l=1}^{c-1}
(\omega^{i}t^{U_l})_{n_{a+ld+i}}
(\omega^{-i}t^{V_l})_{n_{a+ld+i}}
\\
&&\hspace{3mm}
\cdot
\prod_{i=1}^{d}
\prod_{l=1}^{c-1}
(\omega^{-i}t^{1-U_l+n_{a+ld+i+1}})_{m-n_{a+ld+i+1}}
(\omega^{i}t^{1-V_l+n_{a+ld+i+1}})_{m-n_{a+ld+i+1}}\end{aligned}$$ and $x':=x+\frac{n_s(n_s-1)}{2}+Nn_s$.
We now define the element $Q_{k}(q,x_b,j)$ by $$Q_{k}(q,x_b,j):=
\left((-1)^{k+1}q^{-\frac{k(k+1)}{2}}\right)
^{\frac{1+{\operatorname{sn}}(b)}{2}}
\left(q^{(k+1)^2}\right)^{\frac{1-{\operatorname{sn}}(b)}{2}}
\frac{(x_{b;j};x_{b;j})_{2m}}{(q;q)_{N+cm}}
\;T_k(q,x_{b;j}).$$
By Lemmas \[S\_[b;j]{}(k,q)\] and \[LHS\], Equation (\[RHS\]) and the following Lemma \[bPosbNeg\], we see that this element satisfies Equation (\[imp\]).
\[bPosbNeg\] The following formula holds. $$(-1)^{k+1}{\text{$\left[\begin{array}{c}2k+1\\ k\end{array}
\right]$}}(q^{k+1};q)_{k+1}^{-1}=
(-1)^{k+1}\frac{q^{-k(k+1)/2}}{(q;q)_{k+1}}
=
\frac{q^{-(k+1)^2}}{(q^{-1};q^{-1})_{k+1}}$$
This is an easy calculation using $$(q^{k+1};q)_{k+1}=(-1)^{k+1} q^{(3k^2+5k+2)/4}\frac{\{2k+1\}!}{\{k\}!}.$$
Part 1, $b$ even case. {#part-1-b-even-case. .unnumbered}
----------------------
Let $b=\pm 2^l$. We have to prove Equality (\[imp\]) only for $j=0$, i.e. we have to show $$\frac{1}{(q^{k+1};q)_{k+1}}
\,\cL_{b;0}\left( \prod_{i=0}^k (z+z^{-1} - q^i -q^{-i}) \right)
=
2\, Q_k(q^{{\operatorname{sn}}(b)}, x_{b},0).$$ The calculation works similar to the odd case. Note that we have $c=1$ here. This case was already done in [@BL] and [@Le]. Since their approaches are slightly different and for the sake of completeness, we will give the parameters for Andrew’s identity and the formula for $Q_{k}(q,x_{b},0)$ nevertheless.
We put $t:=x_{b;0}$, $d:=\frac{b}{2}-1$, $\omega$ a $b$th root of unity and choose a primitive square root $\nu$ of $\omega$. Define the parameters of Andrew’s identity by $$\begin{array}{rclrcll}
b_{i}&:=&\omega^it^{-N},
&c_{i}&:=&\omega^{-i}t^{-N}
& \;\text{for }\;i=1,\ldots,d, \\
b_{d+i}&:=&-\nu^{2i-1}t, &c_{d+i}&:=&-\nu^{-(2i-1)}t &\;\text{for }\;i=1,\ldots,d+1,\\
b_{b}&:=&-t^{-N},
&c_{b}&:=&-t^0=-1, &\\
b_{s-1}&:=&t^{-N}, &c_{s-1}&:=&t^{N+1}, &\\
b_s&\rightarrow&\infty,&c_s&\rightarrow&\infty,&
\end{array}$$ where $s=b+2$. Now we can define the element $$Q_{k}(q,x_b,0):=
\left((-1)^{k+1}q^{-\frac{k(k+1)}{2}}\right)
^{\frac{1+{\operatorname{sn}}(b)}{2}}
\left(q^{(k+1)^2}\right)^{\frac{1-{\operatorname{sn}}(b)}{2}}
\frac{(x_{b;0};x_{b;0})_{2N}}{(q;q)_{2N}}
\frac{1}{(-x_{b;0};x_{b;0})_N} T_k(q,x_{b;0})$$ where $$\begin{aligned}
T_k(q,t)
&:=&
\sum_{n_{s-1}\geq \cdots \geq n_1=0}
(-1)^{n_{s-1}}t^{x''}
\cdot
\frac{
\prod_{i=1}^{d}
(t^{2N+1})_{\delta_{i}}
\cdot\prod_{i=1}^{d+1}(t^{-1})_{\delta_{d+i}}
\cdot
(t^{N+1})_{\delta_b}
}{
\prod_{i=1}^{s-2}(t)_{\delta_i}
}
\\
&&\hspace{13mm}
\cdot
(t^{-N})_{n_{s-1}}
\cdot (-t^{N+1+n_{s-1}})_{N-n_{s-1}}
\cdot(-t^{-N})_{n_{b}}
\cdot (-t)_{n_{b}-1}
\cdot (-t^{n_{s-1}+1})_{N-n_{s-1}}
\\
&&\hspace{13mm}
\cdot
\prod_{i=1}^{d}
(\omega^{\pm i}t^{-N})_{n_{i}}
(\omega^{\pm i}t^{N+1+n_{i+1}})_{N-n_{i+1}}
\cdot
\prod_{i=1}^{d+1}(1+\nu^{\pm(2i-1)}t^{n_{d+i}})^{1-\delta_{d+i}}\end{aligned}$$ and $
x'':=\sum_{i=1}^{d}(2N+1)n_i-\sum_{i=1}^{d+1}n_{d+i}+\frac{n_{s-1}(n_{s-1}-1)}{2}+(N+1)(n_b+n_{s-1})$. We use the notation $(a;b)_{-1}=\frac{1}{1-ab^{-1}}$.
Part 2 {#part-2 .unnumbered}
------
We have to show that $Q_{k}(q,x_b,j) \in\calS_{p,j}$, where $j \in \N\cup\{0\}$ if $p$ is odd, and $j=0$ for $p=2$. The following two lemmas do the proof.
For $t=x_{b;j}$, $$T_k(q,t)\in \Z[q^{\pm 1}, t^{\pm1}].$$
Let us first look at the case $b$ odd and positive. Since for $a\not=0$, $(t^a)_n$ is always divisible by $(t)_n$, it is easy to see that the denominator of each term of $T_k(q,t)$ divides its numerator. Therefore we proved that $T_k(q,t)\in\Z[t^{\pm 1/c},\omega]$. Since \[Sbj\] S\_[b;j]{}(k,q)= T\_k(q,t), there are $f_0,g_0\in\Z[q^{\pm 1},t^{\pm 1}]$ such that $T_k(q,t)= \frac{f_0}{g_0}$. This implies that $T_k(q,t)\in \Z[q^{\pm 1}, t^{\pm 1}]$ since $f_0$ and $g_0$ do not depend on $\omega$ and the $c$th root of $t$.
The proofs for the even and the negative case work similar.
For $t=x_{b;j}$, $$\frac{(t;t)_{2m}}{(q;q)_{N+cm}}
\frac{1}{((-t;t)_{N})^{\lambda}}
\in\calS_{p,j}$$ where $\lambda=1$ and $j=0$ if $p=2$, and $\lambda=0$ and $j\in \N\cup \{0\}$ otherwise.
Notice that $$(q;q)_{N+cm}=\widetilde{(q;q)}_{N+cm}(q^c;q^c)_{2m},$$ where we use the notation $$\widetilde{(q^a;q)}_{n}:=
\prod_{\substack{j=0\\c\nmid (a+j)}}^{n-1}(1-q^{a+j}).$$ We have to show that $$\frac{(q^c;q^c)_{2m}}{(t;t)_{2m}}
\cdot
\widetilde{(q;q)}_{N+cm}\cdot
((-t;t)_{N})^{\lambda}$$ is invertible in $\Z[1/p][q]$ modulo any ideal $(f)=(\prod_{n} \Phi^{k_n}_n (q))$ where $n$ runs through a subset of $p^j\N_p$. Recall that in a commutative ring $A$, an element $a$ is invertible in $A/(d)$ if and only if $(a)+(d)=(1)$. If $(a)+(d)=(1)$ and $(a)+(e)=(1)$, multiplying together we get $(a)+(de)=(1)$. Hence, it is enough to consider $f=\Phi_{p^j n}(q)$ with $(n,p)=1$. For any $X\in\N$, we have $$\begin{aligned}
\label{denomin1}
{\widetilde{(q;q)}_X}&=&\prod_{\text{\scriptsize{$\begin{array}{c}i=1 \\ c\nmid i \end{array}$}}}^{X} \prod_{d\mid i} \Phi_d(q),
\\\label{denomin2}
(-t;t)_X&=&\frac{(t^2;t^2)_X}{(t;t)_X}=\prod_{i=1}^{X}\prod_{d\mid i}\Phi_{2d}(t)
\\\label{denomin3}
\frac{(q^c;q^c)_X}{(t;t)_X}
&=&\frac{(t^{b'};t^{b'})_X}{(t;t)_X}=
\frac{
\prod_{i=1}^{X}\prod_{d\mid ib'}\Phi_d(t)
}{
\prod_{i=1}^{X}\prod_{d\mid i}\Phi_d(t)
}\end{aligned}$$ for $b'=b/c$. Recall that $(\Phi_r(q),\Phi_a(q))=(1)$ in $\Z[1/p][q]$ if either $r/a$ is not a power of a prime or a power of $p$. For $r=p^j n$ odd and $a$ such that $c\nmid a$, one of the conditions is always satisfied. Hence is invertible in $\calS_{p,j}$. If $b=c$ or $b'=1$, and do not contribute. For $c<b$, notice that $q$ is a $cn$th primitive root of unity in $\Z[1/p][q]/(\Phi_{cn}(q)) = \Z[1/p][e_{cn}]$. Therefore $t^{b'}=q^{c}$ is an $n$th primitive root of unity. Since $(n,b')=1$, $t$ must be a primitive $n$th root of unity in $\Z[1/p][e_{cn}]$, too, and hence $\Phi_n(t) = 0$ in that ring. Since for $j$ with $(j, p) > 1$, $(\Phi_j(t),\Phi_n(t)) = (1)$ in $\Z[1/p][t]$, we have $\Phi_j(t)$ is invertible in $\Z[1/p][e_{cn}]$, and therefore and are invertible, too.
Proof of Theorem \[GeneralizedHabiro\]
======================================
The appendix is devoted to the proof of Theorem \[GeneralizedHabiro\], a generalization of the deep integrality result of Habiro, namely Theorem 8.2 of [@Ha]. The existence of this generalization and some ideas of the proof were kindly communicated to us by Habiro.
Reduction to a result on values of the colored Jones polynomial {#reduction-to-a-result-on-values-of-the-colored-jones-polynomial .unnumbered}
---------------------------------------------------------------
We will use the notations of [@Ha]. We put $q=e^{h}$, and $v=e^{h/2}$, where $h$ is a free parameter. The quantum algebra $U_h=U_h(sl_2)$, generated by $E, F$ and $H$, subject to some relations, is the quantum deformation of the universal enveloping algebra $U(sl_2)$.
Let $V_n$ be the unique $(n+1)$–dimensional irreducible $U_h$–module. In [@Ha], Habiro defined a new basis $\tilde{P}'_k$, $k=0,1,2,\dots$, for the Grothendieck ring of finite–dimensional $U_h(sl_2)$–modules with $$\tilde{P}_k':=\frac{v^{\frac{1}{2}k(1-k)}}{\{k\}!} \, \prod_{i=0}^{k-1}
(V_1-v^{2i+1}-v^{-2i-1}).$$
Put $\tilde{P}'_{\bk}=\{\tilde{P}'_{k_1},\ldots,\tilde{P}'_{k_m}\}$. It follows from Lemma 6.1 of [@Ha] that we will have identity of Theorem \[GeneralizedHabiro\] if we put
$$C_{L\sqcup L'}(\bk,\bj)=
J_{L\sqcup L'}\,(\tilde{P}'_{\bk},\bj)\,
\prod_{i}(-1)^{k_i}q^{k^2_i+k_i+1}\, .$$
Hence to prove Theorem \[GeneralizedHabiro\] it is enough to show the following.
\[main-integrality\] Suppose $L\sqcup L'$ is a colored framed link in $S^3$ such that $L$ has zero linking matrix and $L'$ has odd colors. Then for $k=\max\{k_1,\dots, k_m\}$ we have $$J_{L\sqcup L'}(\tilde{P}'_{\bk},\bj) \in \frac{(q^{k+1};q)_{k+1}}{1-q}
\,\,\BZ[q^{\pm 1}].$$
In the case $L'=\emptyset$, this statement was proved in [@Ha Theorem 8.2]. Since our proof is a modification of the original one, we first sketch Habiro’s original proof for the reader’s convenience.
Sketch of the proof of Habiro’s integrality theorem
---------------------------------------------------
Geometric part {#geometric-part .unnumbered}
--------------
Let us first recall the notion of a bottom tangle, introduced by Habiro in [@Ha-b].
An $n$–component bottom tangle $T=T_1\sqcup \dots \sqcup
T_n$ is a framed tangle consisting of $n$ arcs $T_1,\ldots ,T_n$ in a cube such that all the endpoints of $T$ are on a line at the bottom square of the cube, and for each $i=1,\ldots ,n$ the component $T_i$ runs from the $2i$th endpoint on the bottom to the $(2i-1)$st endpoint on the bottom, where the endpoints are counted from the left. An example, the Borromean bottom tangle $B$, is given in Figure \[borro\].
In [@Ha-b], Habiro defined a braided subcategory $\modB$ of the category of framed, oriented tangles which acts on the bottom tangles by composition (vertical pasting). The objects of $\modB $ are the symbols $\modb ^{\otimes n}$, $n\ge 0$, where $\modb:=\downarrow \uparrow $. For $m,n\ge 0$, a morphism $X$ of $\modB$ from $\modb ^{\otimes m}$ to $\modb ^{\otimes n}$ is the isotopy class of a framed, oriented tangle $X$ which we can compose with $m$–component bottom tangles to get $n$–component bottom tangles. Let $\modB (m,n)$ be the set of morphisms from $\modb ^{\otimes m}$ to $\modb ^{\otimes n}$. The composite $YX$ of two morphisms is the gluing of $Y$ to the bottom of $X$, and the identity morphism $1_{\modb ^{\otimes m}}=\downarrow \uparrow \cdots\downarrow \uparrow $ is a tangle consisting of $2m$ vertical arcs. The monoidal structure is given by pasting tangles side by side. The braiding for the generating object $\modb$ with itself is given by $\psi _{\modb ,\modb }=\incl{1.5em}{phibb+2}$.
Corollary 9.13 in [@Ha-b] states the following.
\[thmASL\][*(Habiro)*]{} If the linking matrix of a bottom tangle $T$ is zero then $T$ can be presented as $T=W B^{\otimes k}$, where $k\geq 0$ and $W\in \modB(3k,n)$ is obtained by horizontal and vertical pasting of finitely many copies of $1_\modb$, $\psi_{\modb,\modb}$, $\psi^{-1}_{\modb,\modb}$, and
$$\def\sss{1.5em}
\eta _\modb =\incl{\sss}{bot0},\quad
\def\sss{2em}
\mu _\modb =\incl{\sss}{bomu},\quad
\gamma_+=\incl{2.5em}{gamma+},\quad
\gamma_-=\incl{2.5em}{gamma-}.
$$
Algebraic part {#algebraic-part .unnumbered}
--------------
Let $K=v^{H}=e^\frac{hH}{2}$. Habiro introduced the integral version ${\mathcal{U}}_q$, which is the $\Z[q, q^{- 1}]$–subalgebra of $U_h$ freely spanned over $\Z[q, q^{- 1}]$ by $\tilde{F}^{(i)}K^{j}e^k$ for $i,k\geq 0, j\in \Z$, where $$\tilde{F}^{(n)}=\frac{F^nK^n}{v^{\frac{n(n-1)}{2}}[n]!}\;\;\;\;\text{ and
}\;\;\;\; e=(v-v^{-1})E.$$
There is $\Z/2\Z$–grading, ${\mathcal{U}}_q={\mathcal{U}}^0_q\oplus {\mathcal{U}}^1_q$, where ${\mathcal{U}}^{0}_q $ (resp. ${\mathcal{U}}_q^{1}$) is spanned by $\tilde{F}^{(i)}K^{2j}e^k$ (resp. $\tilde{F}^{(i)}K^{2j+1}e^k$). We call this the $\ve$–grading, and ${\mathcal{U}}^{0}_q $ (resp. ${\mathcal{U}}_q^{1}$) the even (resp. odd) part.
The two–sided ideal $\F_p$ in ${\mathcal{U}}_q$ generated by $e^p$ induces a filtration on $({\mathcal{U}}_q)^{\otimes n}$, $n\geq 1$, by $$\F_p(({\mathcal{U}}_q^{})^{\otimes n})=
\sum^n_{i=1} ({\mathcal{U}}_q^{})^{\otimes i-1}\otimes
\F_p({\mathcal{U}}_q^{})\otimes ({\mathcal{U}}_q^{})^{\otimes n-i}
\subset ({\mathcal{U}}_q^{})^{\otimes n}\, .$$ Let $(\tilde{{\mathcal{U}}}_q)^{\tilde\otimes n}$ be the image of the homomorphism $$\lim_{\overleftarrow{\hspace{1mm}{p\geq 0}\hspace{1mm}}} \;\;
\frac{({\mathcal{U}}_q)^{\otimes n}}{\F_p(({\mathcal{U}}_q)^{\otimes n})} \to
U_h^{\hat \otimes n}$$ where $\hat\otimes$ is the $h$–adically completed tensor product. By using $\F_p({\mathcal{U}}_q^\ve):= \F_p({\mathcal{U}}_q)\cap {\mathcal{U}}^{\ve}_q$ one defines $(\tilde{{\mathcal{U}}}^{\ve}_q)^{\tilde\otimes n}$ for $\ve\in\{0,1\}$ in a similar fashion.
By definition (Section 4.2 of [@Ha]), the universal $sl_2$ invariant $J_T$ of an $n$–component bottom tangle $T$ is an element of $U_h^{\hat\otimes n}$. Theorem 4.1 in [@Ha] states that, in fact, for any bottom tangle $T$ with zero linking matrix, $J_T$ is even, i.e. \[even\] J\_T(\^[0]{}\_q)\^[n]{} .
Further, using the fact that $J_K$ of a $0$–framed bottom knot $K$ (i.e. a 1–component bottom tangle) belongs to the center of $\tilde {\mathcal{U}}^0_q$, Habiro showed that $$J_K=\sum_{n\geq 0} (-1)^n q^{n(n+1)}\frac{(1-q)}{(q^{n+1};q)_{n+1}}\, J_K(\tilde P'_n)\, \sigma_n$$ where $$\sigma_n=\prod^n_{i=0}(C^2-(q^i+2+ q^{-i}))\, \quad {\rm with}\quad
C=(v-v^{-1}){\tilde{F}}^{(1)}K^{-1}e+vK+v^{-1}K^{-1}\, ,$$ the quantum Casimir operator. The $\sigma_n$ provide a basis for the even part of the center. From this, Habiro deduced that $J_K(\tilde P'_n)\in \frac{ (q^{n+1};q)_{n+1}}{(1-q)} \Z[q,q^{-1}]$.
The case of $n$–component bottom tangles reduces to the 1–component case by partial trace, using certain integrality of traces of even element (Lemma 8.5 of [@Ha]) and the fact that $J_T$ is invariant under the adjoint action.
Algebro–geometric part {#algebrogeometric-part .unnumbered}
----------------------
The proof of uses Proposition \[thmASL\], which allows to build any bottom tangle $T$ with zero linking matrix from simple parts, i.e. $T=W(B^{\otimes k})$.
On the other hand, the construction of the universal invariant $J_T$ extends to the braided functor $J:\modB\to \operatorname{ Mod}_{U_h}$ from $\modB$ to the category of $U_h$–modules. This means that $J_{W(B^{\otimes k})}=J_W(J_{B^{\otimes k}})$. Therefore, in order to show , we need to check that $J_B \in (\tilde {\mathcal{U}}^0_q)^{\tilde \otimes 3} $, and then verify that $J_W$ maps the even part to itself. The first check can be done by a direct computation [@Ha Section 4.3]. The last verification is the content of Corollary 3.2 in [@Ha].
Proof of Theorem \[main-integrality\]
--------------------------------------
Generalization of Equation {#generalization-of-equation .unnumbered}
---------------------------
To prove Theorem \[main-integrality\] we need a generalization of Equation or Theorem 4.1 in [@Ha] to tangles with closed components. To state the result let us first introduce two new gradings.
Suppose $T$ is an $n$–component bottom tangle in a cube, homeomorphic to the 3–ball $D^3$. Let ${\tilde {\mathcal S}}(D^3\setminus T)$ be the $\Z[q^{\pm 1/4}]$–module freely generated by the isotopy classes of framed unoriented colored links in $D^3\setminus T$, including the empty link. For such a link $L \subset D^3\setminus T$ with $m$–components colored by $n_1,\dots, n_m$, we define our new gradings as follows. First provide the components of $L$ with arbitrary orientations. Let $l_{ij}$ be the linking number between the $i$th component of $T$ and the $j$th component of $L$, and $p_{ij}$ be the linking number between the $i$th and the $j$th components of $L$. For $X=T\sqcup L$ we put $${\operatorname{gr}}_\ve(X):= (\ve_1,\dots, \ve_n)\in (\Z/2\Z)^n, \quad \text{where}\quad
\ve_i := \sum_{j} l_{ij} n'_{j} \pmod 2, \quad \text{ and}
\label{3306}$$
$${\operatorname{gr}}_q(L) := \sum_{1\le i,j\le m }p_{ij} n_i'n_j' + 2\sum_{1\le j\le m}(p_{jj}+1) n_j' \pmod 4,
\quad \text{where} \quad n_i':= n_i-1.$$ It is easy to see that the definitions do not depend on the orientation of $L$.
The meaning of ${\operatorname{gr}}_q(L)$ is the following: The colored Jones polynomial of $L$, a priori a Laurent polynomial of $q^{1/4}$, is actually a Laurent polynomial of $q$ after dividing by $q^{{\operatorname{gr}}_q(L)/4}$; see [@LeDuke] for this result and its generalization to other Lie algebras.
We further extend both gradings to ${\tilde {\mathcal S}}(D^3\setminus T)$ by $${\operatorname{gr}}_\ve(q^{1/4})=0, \quad {\operatorname{gr}}_q(q^{1/4}) =1 \pmod 4.$$
Recall that the universal invariant $J_X$ can also be defined when $X$ is the union of a bottom tangle and a colored link (see [@Ha-b Section 7.3]). In [@Ha-b], it is proved that $J_X$ is adjoint invariant. The generalization of Theorem 4.1 of [@Ha] is the following.
\[MMM\] Suppose $X=T \sqcup L$, where $T$ is a $n$–component bottom tangle with zero linking matrix and $L$ is a framed unoriented colored link with ${\operatorname{gr}}_\ve(X)= (\ve_1,\dots,\ve_n)$. Then $$J_X \in q^{{{\operatorname{gr}}_q(L)/4}}\, \left ( \tilde {\mathcal{U}}_q^{\ve_1}\, \tilde \otimes
\dots \tilde \otimes\; \tilde {\mathcal{U}}_q^{\ve_n}\right ).$$ \[3310\]
\[MMM1\] Suppose $L$ is colored by a tuple of odd numbers, then $$J_X\in (\tilde {\mathcal{U}}_q^{0})^{\tilde\otimes n}\, .$$
Since $J_X$ is invariant under the adjoint action, Theorem \[main-integrality\] follows from Corollary \[MMM1\] by repeating Habiro’s arguments.
Hence it remains to prove Theorem \[MMM\]. In the proof we will need a notion of a [*good morphism*]{}.
### Good morphisms {#good-morphisms .unnumbered}
Let $I_m:=1_{\modb^{\otimes m}}\in \modB(m,m)$ be the identity morphism of $\modb ^{\otimes m}$ in the cube $C$. A framed link $L$ in the complement $C \setminus I_m$ is [*good*]{} if $L$ is geometrically disjoint from all the up arrows of $\modb ^{\otimes m}$, i.e. there is a plane dividing the cube into two halves, such that all the up arrows are in one half, and all the down arrows and $L$ are in the other. Equivalently, there is a diagram in which all the up arrows are above all components of $L$. The union $W$ of $I_m$ and a colored framed good link $L$ is called a [*good morphism*]{}. If $Y$ is any bottom tangle so that we can compose $X =WY$, then it is easy to see that ${\operatorname{gr}}_\ve(X)$ does not depend on $Y$, and we define ${\operatorname{gr}}_\ve(W):={\operatorname{gr}}_\ve(X)$. Also define ${\operatorname{gr}}_q(W):= {\operatorname{gr}}_q(L)$.
As in the case with $L=\emptyset$, the universal invariant extends to a map $J_W: {\mathcal{U}}_h^{\otimes m} \to {\mathcal{U}}_h^{\otimes m}$.
Proof of Theorem \[MMM\] {#proof-of-theorem-mmm .unnumbered}
------------------------
The strategy here is again analogous to the Habiro case: In Proposition \[TWWB\] we will decompose $X$ into simple parts: the top is a bottom tangle with zero linking matrix, the next is a good morphism, and the bottom is a morphism obtained by pasting copies of $\mu_\modb$. Since, any bottom tangle with zero linking matrix satisfies Theorem \[MMM\] and $\mu_\modb$ is the product in ${\mathcal{U}}_q$, which preserves the gradings, it remains to show that any good morphism preserves the gradings. This is done in Proposition \[MM2\] below.
\[TWWB\] Assume $X=T\sqcup L$ where $T$ is a $n$–component bottom tangle with zero linking matrix and $L$ is a link. Then there is a presentation $X =W_2 W_1 W_0$, where $W_0$ is a bottom tangle with zero linking matrix, $W_1$ is a good morphism, and $W_2$ is obtained by pasting copies of $\mu_\modb$.
Let us first define $\tilde \gamma_\pm\in \modB(i, i+1)$ for any $i\in {\mathbb N}$ as follows. $$\def\sss{3em}
\tilde\gamma_+ =\incl{\sss}{tildegamma+}\quad
\def\sss{3em}
\tilde\gamma_- =\incl{\sss}{tildegamma-}.$$
If a copy of $\mu_\modb$ is directly above $\psi^{\pm 1}_{\modb, \modb}$ or $ \gamma_\pm$, one can move $\mu_\modb$ down by isotopy and represent the result by pasting copies of $\psi^{\pm 1}_{\modb, \modb}$ and $\tilde \gamma_\pm$. It is easy to see that after the isotopy $\gamma_\pm$ gets replaced by $\tilde\gamma_\pm$ and $\psi^{\pm 1}_{\modb, \modb}$ by two copies of $\psi^{\pm 1}_{\modb, \modb}$.
Using Proposition \[thmASL\] and reordering the basic morphisms so that the $\mu$’s are at the bottom, one can see that $T$ admits the following presentation: $$T= W_2 \tilde W_1 (B^{\otimes k})$$ where $B$ is the Borromean tangle, $W_2$ is obtained by pasting copies of $\mu_\modb$ and $\tilde{W}_1$ is obtained by pasting copies of $\psi^{\pm 1}_{\modb,\modb}$, $\tilde\gamma_{\pm}$ and $\eta_\modb$.
Let $P$ be the horizontal plane separating $\tilde W_1$ from $W_2$. Let $P_+$ ($P_-$) be the upper (lower, respectively) half–space. Note that $W_0=\tilde W_1(B^{\otimes k})$ is a bottom tangle with zero linking matrix lying in $P_+$ and does not have any minimum points. Hence the pair $(P_+,W_0)$ is homeomorphic to the pair $(P_+,\; l \text{ trivial arcs})$. Similarly, $W_2$ does not have any maximum points; hence $L$ can be isotoped off $P_-$ into $P_+$. Since the pair $(P_+,W_0)$ is homeomorphic to the pair $(P_+,\; l\text{ trivial arcs})$ one can isotope $L$ in $P_+$ to the bottom end points of down arrows. We then obtain the desired presentation.
\[MM2\] For every good morphism $W$, the operator $J_{W}$ preserves gradings in the following sense. If $x\in {\mathcal{U}}_q^{\ve_1} \otimes \dots \otimes {\mathcal{U}}_q^{\ve_m}$, then $$J_{W}(x) \in q^{{\operatorname{gr}}_q(W)/4} \left( {\mathcal{U}}_q^{\ve_1'} \otimes \dots \otimes
{\mathcal{U}}_q^{\ve_m'}\right), \quad \text{where} \quad (\ve'_1,\dots, \ve'_m) =
(\ve_1,\dots, \ve_m) + {\operatorname{gr}}_{\ve}(W).$$
The rest of the appendix is devoted to the proof of Proposition \[MM2\].
Proof of Proposition \[MM2\] {#ProofMM2 .unnumbered}
----------------------------
We proceed as follows. Since $J_X$ is invariant under cabling and skein relations, and by Lemma \[A8\] below, both relations preserve ${\operatorname{gr}}_\ve$ and ${\operatorname{gr}}_q$, we consider the quotient of $\tilde S(D^3\setminus T)$ by these relations known as a skein module of $D^3\setminus T$. For $T=I_n$, this module has a natural algebra structure, with good morphisms forming a subalgebra. By Lemma \[A9\] (see also Figure \[Wgamma\]), the basis elements $W_\gamma$ of this subalgebra are labeled by $n$–tuples $\gamma=(\gamma_1,\dots, \gamma_n)\in (\Z/2\Z)^n$. It’s clear that if the proposition holds for $W_{\gamma_1}$ and $W_{\gamma_2}$, then it holds for $W_{\gamma_1} W_{\gamma_2}$. Hence it remains to check the claim for $W_\gamma$’s. This is done in Corollary \[corA10\] for basic good morphisms corresponding to $\gamma$ whose non–zero $\gamma_j$’s are consecutive. Finally, any $W_\gamma$ can be obtained by pasting a basic good morphism with few copies of $\psi^{\pm}_{\modb, \modb}$. Since $J_{\psi^\pm}$ preserves gradings (compare (3.15), (3.16) in [@Ha]), the claim follows from Lemmas \[A9\], \[A8\] and Proposition \[3308\] below.
### Cabling and skein relations {#cabling-and-skein-relations .unnumbered}
Let us introduce the following relations in ${\tilde {\mathcal S}}(D^3\setminus T)$.
Cabling relations:
- Suppose $n_i=1$ for some $i$. The first cabling relation is $L=\tilde L$, where $\tilde L$ is obtained from $L$ by removing the $i$th component.
- Suppose $n_i \ge 3$ for some $i$. The second cabling relation is $L= L'' -L'$, where $L'$ is the link $L$ with the color of the $i$th component switched to $n_i-2$, and $L''$ is obtained from $L$ by replacing the $i$th component with two of its parallels, which are colored with $n_i-1$ and $2$.
Skein relations:
- The first skein relation is $U=q^{\frac 12}+q^{-\frac 12}$, where $U$ denotes the unknot with framing zero and color 2.
- Let $L_R$, $L_V$ and $L_H$ be unoriented framed links with colors 2 which are identical except in a disc where they are as shown in Figure \[Skein\]. Then the second skein relation is $L_R = q^{\frac 14}L_V + q^{-\frac 14}L_H$ if the two strands in the crossing come from different components of $L_R$, and $L_R = \epsilon(q^{\frac 14}L_V - q^{-\frac 14}L_H)$ if the two strands come from the same component of $L_R$, producing a crossing of sign $\epsilon=\pm 1$ (i.e. appearing as in $L_{\epsilon}$ of Figure \[Skein\] if $L_R$ is oriented).
We denote by $S(D^3\setminus T)$ the quotient of ${\tilde {\mathcal S}}(D^3\setminus T)$ by these relations. It is known as the [*skein module*]{} of $D^3\setminus T$ (compare [@Pr], [@SikPr] and [@Bullock]). Recall that the ground ring is $\Z[q^{\pm 1/4}]$.
Using the cabling relations, we can reduce all colors of $L$ in $S(D^3\setminus T)$ to be 2. Note that the skein module $S(C\setminus I_n)$ has a natural algebra structure, given by putting one cube on the top of the other. Let us denote by $A_n$ the subalgebra of this skein algebra generated by good morphisms.
For a set $\gamma=(\gamma_1,\dots, \gamma_n)\in (\Z/2\Z)^n$ let $W_\gamma$ be a simple closed curve encircling the end points of those downward arrows with $\gamma_i=1$. See Figure \[Wgamma\] for an example.
Similarly to the case of Kauffman bracket skein module [@Bullock], one can easily prove the following.
The algebra $\mathcal A_n$ is generated by $2^n$ curves $W_\gamma$. \[A9\]
Using linearity, we can extend the definition of $J_X$ to $X =T \sqcup L$, where $L$ is [*any element*]{} of ${\tilde {\mathcal S}}(D^3\setminus T)$. It is known that $J_X$ is invariant under the cablings and skein relations (Theorem 4.3 of [@KM]), hence $J_X$ is defined for $L \in S(D^3\setminus T)$. Moreover, we have
\[A8\] Both gradings ${\operatorname{gr}}_\ve$ and ${\operatorname{gr}}_q$ are preserved under the cabling and skein relations.
The statement is obvious for the $\ve$–grading. For the $q$–grading, notice that $${\operatorname{gr}}_q(L)=2\sum_{1\leq i<j\leq m}p_{ij}n_i' n_j' +
\sum_{1\leq j \leq m}p_{jj}n_j'^2+2\sum_{1\leq j\leq m}(p_{jj}+1)n_j',$$ and therefore ${\operatorname{gr}}_q(L'')\equiv {\operatorname{gr}}_q(L')\equiv{\operatorname{gr}}_q(L)\pmod{4}$. This takes care of the cabling relations.
Let us now assume that all colors of $L$ are equal to 2 and therefore $${\operatorname{gr}}_q(L)=2\sum_{1\leq i< j\leq m} p_{ij}+3\sum_{i=1}^{m} p_{ii}+2m.$$ The statement is obvious for the first skein relation. For the second skein relation, choose an arbitrary orientation on L. Let us first assume that the two strands in the crossing depicted in Figure \[Skein\] come from the same component of $L_R$ and that the crossing is positive. Then, $L_V$ and $L_H$ have one positive self–crossing less, and $L_V$ has one link component more than $L_R$. Therefore $$\begin{aligned}
{\operatorname{gr}}_q(q^{\frac 14}L_V)&=&{\operatorname{gr}}_q(L_R)-3+2+1 \equiv {\operatorname{gr}}_q{L_R} \pmod{4}, \\
{\operatorname{gr}}_q(q^{-\frac 14}L_H)&=&{\operatorname{gr}}_q(L_R)-3-1 \equiv {\operatorname{gr}}_q{L_R} \pmod{4}.\end{aligned}$$ It is obvious, that this does not depend on the orientation of $L_R$. If the crossing of $L_R$ is negative or the two strands do not belong to the same component of $L_R$, the proof works similar.
### Basic good morphisms {#basic-good-morphisms .unnumbered}
Let $ Z_n$ be $W_\gamma$ for $\gamma=(1,1,\dots, 1)\in (\Z/2\Z)^n$. $$\label{Zn}
\def\sss{3em}
Z_n =\incl{\sss}{ZnVers2}\quad\quad
$$
One has a presentation $$J_{ Z_n} =
\sum z^{(n)}_{i_1} \otimes \sum z^{(n)}_{i_2} \otimes \dots \otimes
\sum z^{(n)}_{i_{2n}},$$ such that $z^{(n)}_{i_{2j-1}}\, z^{(n)}_{i_{2j}}\in v\; {\mathcal{U}}^1_q$ for every $j=1,\dots, n$. \[3308\]
\[corA10\] $J_{ Z_n}$ satisfies Proposition \[MM2\].
Assume $x\in {\mathcal{U}}_q^{\ve_1} \otimes \dots \otimes {\mathcal{U}}_q^{\ve_n}$, then we have $$J_{ Z_n}(x) =
\sum z^{(n)}_{i_1} x_1 z^{(n)}_{i_2} \otimes \dots \otimes
\sum z^{(n)}_{i_{2n-1}} x_n z^{(n)}_{i_{2n}}.$$ Hence, by Proposition \[3308\], we get $$J_{ Z_n}(x) \in q^{1/2} \left( {\mathcal{U}}_q^{\ve_1'} \otimes \dots \otimes
{\mathcal{U}}_q^{\ve_n'}\right), \quad \text{where} \quad (\ve'_1,\dots, \ve'_m) =
(\ve_1,\dots, \ve_n) + (1,1,\dots, 1).$$ The claim follows now from the fact that ${\operatorname{gr}}_\ve( Z_n)=(1,1,\dots,1)$ and ${\operatorname{gr}}_q(L)=2$.
Proof of Proposition \[3308\]
-----------------------------
The statement holds true for $J_{ Z_1}= C\otimes {\operatorname{id}}_\uparrow$. Now Lemma 7.4 in [@Ha-b] states that applying $\Delta$ to the $i$th component of the universal quantum invariant of a tangle is the same as duplicating the $i$th component. Using this fact we represent \[Jn\] J\_[Z\_[n+1]{}]{} = (1\_[\^[n-1]{}]{} \_) ( J\_[Z\_[n]{}]{} ),where $\Phi$ is defined as follows. For $x \in {\mathcal{U}}_q$ with $\Delta (x)=\sum x_{(1)}\otimes x_{(2)}$, we put $$\Phi(x) := \sum_{(x), m,n} x_{(1)} \otimes \beta_m S(\beta_n)
\otimes \alpha_n \, x_{(2)} \alpha_m$$ where the $R$–matrix is given by $R =\sum_l {\alpha_l \otimes \beta_l}$. See Figure below for a picture.
We are left with the computation of the $\ve$–grading of each component of $\Phi(x)$.
In ${\mathcal{U}}_q$, in addition to the $\ve$–grading, there is also the $K$–grading, defined by $|K|=|K^{-1}|=0, |e|=1, |F|=-1$. In general, the co–product $\Delta$ does not preserve the $\ve$–grading. However, we have the following.
Suppose $x\in {\mathcal{U}}_q$ is homogeneous in both $\ve$–grading and $K$–grading. Then we have a presentation $$\Delta(x) = \sum_{(x)} x_{(1)} \otimes x_{(2)},$$ where each $x_{(1)}$, $x_{(2)}$ are homogeneous with respect to the $\ve$–grading and $K$–grading. In addition, for $x\in {\mathcal{U}}_q^\ve$, we have $x_{(2)}\in {\mathcal{U}}_q^\ve$ and $x_{(1)} \, K^{-|x_{(2)}|}\in {\mathcal{U}}_q^\ve$. \[3304\]
If the statements hold true for $x, y \in {\mathcal{U}}_q$, then they hold true for $xy$. Therefore, it is enough to check the statements for the generators $e, {\tilde{F}}^{(1)},$ and $K$, for which they follow from explicit formulas of the co–product.
Suppose $x \in {\mathcal{U}}_q$ is homogeneous in both $\ve$–grading and $K$–grading. There is a presentation $$\Phi(x) =\sum x_{i_0}\otimes x_{i_1}\otimes x_{i_2}$$ such that each $x_{i_j}$ is homogeneous in both $\ve$–grading and $K$–grading, and for $x \in {\mathcal{U}}^\ve_q$, $x_{i_2}$ and $ x_{i_0}\, x_{i_1}$ belong to ${\mathcal{U}}^\ve_q$. \[3303\]
We put $D=\sum D' \otimes D'':=v^{\frac 12 H\otimes H}$. Using (see e.g. [@Ha]) $$R=D\left(\sum_{n} q^{\frac 12 n(n-1)}{\tilde{F}}^{(n)}K^{-n}\otimes e^n\right),$$ we get $$\begin{aligned}
\Phi(x)&=&\sum_{(x),n,m}
q^{\frac{1}{2}\left(m(m-1)+n(n-1)\right)}
x_{(1)}\otimes
D''_2 e^m S(D''_1e^{n})\otimes
D'_1{\tilde{F}}^{(n)}K^{-n}x_{(2)}D'_2{\tilde{F}}^{(m)}K^{-m}\\
&=&
\sum_{(x),n,m}(-1)^n q^{-\frac 12 m(m+1)-n(|x_{(2)}|+1)}
x_{(1)}\otimes e^{m}e^n K^{-|x_{(2)}|}\otimes {\tilde{F}}^{(n)}x_{(2)}{\tilde{F}}^{(m)},\end{aligned}$$ where we used $({\operatorname{id}}\otimes S)D=D^{-1}$ and $D^{\pm1}(1\otimes x)=(K^{\pm|x|}\otimes x)D^{\pm 1}$ for homogeneous $x\in {\mathcal{U}}_q$ with respect to the $K$–grading. Now, the claim follows from Lemma \[3304\].
By induction on $n$ in , given that $C\in v \;{\mathcal{U}}^1_q$, Lemma \[3303\] implies Proposition \[3308\].
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[^1]: There are misprints in Lemma 4.21: $q^*\pm n$ should be replaced by $q^*\mp n$ for $n=1,2$.
| {
"pile_set_name": "ArXiv"
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abstract: 'We use resistive magnetohydrodynamical simulations with the nested grid technique to study the formation of protoplanetary disks around protostars from molecular cloud cores that provide the realistic environments for planet formation. We find that gaseous planetary-mass objects are formed in the early evolutionary phase by gravitational instability in regions that are decoupled from the magnetic field and surrounded by the injection points of the magnetohydrodynamical outflows during the formation phase of protoplanetary disks. Magnetic decoupling enables massive disks to form and these are subject to gravitational instability, even at $\sim$ 10 AU. The frequent formation of planetary-mass objects in the disk suggests the possibility of constructing a hybrid planet formation scenario, where the rocky planets form later under the influence of the giant planets in the protoplanetary disk.'
author:
- 'Shu-ichiro Inutsuka, Masahiro N. Machida, and Tomoaki Matsumoto'
title: Emergence of Protoplanetary Disks and Successive Formation of Gaseous Planets by Gravitational Instability
---
Introduction
============
Recent direct imaging of outer planets in extra-solar planetary systems [@Kalas+2008; @Marois+2008; @Lagrange+2008; @Greaves+2008; @Thalmann+2009] provides a challenging question to the theory of planet formation: how are giant planets formed in the distant regions far from the central star? Almost all the dynamical timescales for the various important processes in planet formation essentially scale with the Kepler rotation timescale [e.g., @KokuboIda2002]. In the standard core-accretion scenario, there are severe timescale constraints on the in situ formation of Jovian planets, even at 5 AU from the central star [e.g., @Pollack+1996; @IkomaNakazawaEmori2000; @HubickyjBodenheimerLissauer2005; @HoriIkoma2008; @Lissauer+2009; @Machida+2010]. Consequently, the formation of planets at greater distances (up to $\sim 10^2$ AU) seems to be impossible within an observationally reasonable timescale in the core-accretion scenario of planet formation [e.g., @IdaLin2004]. On the other hand, the formation of giant planets due to gravitational instability of massive protoplanetary disks has been extensively investigated by various authors [e.g., see review by @Durisen+2007]. However, most analyses focused only on the evolution of hypothetical disks and cannot be applied directly to actual systems. The result of gravitational instability or absence of it does depend on how the protoplanetary disks are formed, but the formation of the protoplanetary disk is closely related to the formation of the central star. In other words, the initial conditions of planet formation by gravitational instability should be provided by the star formation process.
The last decade has seen dramatic progress in our understanding of protostar formation [e.g., @AndreBasuInutsuka2008] and now provides us with the opportunity to study the formation phase of protoplanetary disks. A highlight of the recent non-ideal magnetohydrodynamical (MHD) calculations of protostellar collapse from molecular cloud cores is the driving of outflows from the first cores and well-collimated fast jets from the protostars; these may be regarded as proof of the importance of various physical processes such as the ohmic dissipation of magnetic fields due to the low degree of ionization at higher density phases. In this Letter, we study the formation phase of protoplanetary disks in a self-consistent, non-ideal MHD system. Our protostellar collapse calculations start from a molecular cloud core, include (practically) all the realistic physical processes, and show how knowledge of the protostar formation process provides convincing evidence for the formation of giant planets in the early phase.
Non-Ideal MHD Simulations
=========================
The initial condition of our resistive MHD simulation is the critical Bonner-Ebert Sphere (an isothermal sphere at gravitational equilibrium) with a temperature of 10 K and a radius of 4750 AU. In order to initiate gravitational collapse, we increase the density uniformly by a factor of 3 to an initial central density of $3~\times~10^6~{\rm cm}^3$. The mass is 1.6 times the solar mass. In a typical simulation, we use a uniform rotation of the angular velocity of $\Omega_{\rm init}=1.1\times10^{-13}{\rm s}^{-1}$ and a uniform magnetic field strength of $B_{\rm init}=37~\mu$ G in the initial state. The initial ratios of thermal, rotational, and magnetic energy to the negated gravitational energy are 0.3, 0.005, 0.014, respectively [cf., @Machida+2008c]. We adopt the nested-grid scheme in order to increase the spatial resolution of the central region [@Machida+2005]; the number of nest-grids is typically 12 and each grid has $n_x~\times~n_y~\times n_z = 128~\times~128\times~32$ cells [@Machida+2006c]. As a result, spatial resolution of the innermost grid is 0.58AU and that of the outermost grid is 1200AU.
To describe a realistic evolution of the magnetic field in protostar formation, we should take into account the non-ideal MHD effects of weakly ionized molecular gas. In general, the ambipolar diffusion is important in the low-density phase, but it is slow and not critical in the dynamically collapsing state, In the intermediate-density phase, the Hall term effect can produce a modest effect depending on the size distribution of dust grains [@WardleNg1999]. In contrast, ohmic dissipation dominates in the high-density phase, and is shown to be the most efficient mechanism for the dissipation of magnetic field in the magnetically supercritical cloud core [e.g., @NakanoNishiUmebayashi2002]. Therefore, we model the non-ideal effects of the magnetic field by the effective resistivity in the induction equation. We adopt the resistivity evolution of the fiducial model of @MachidaInutsukaMatsumoto2007 that corresponds to the ionization equilibrium in standard molecular clouds. As the cloud core collapses, the degree of ionization decreases with increasing density. The gas becomes magnetically decoupled at densities of around $10^{10} {\rm cm}^{-3}$ but couples again when the temperature exceeds about 1000 K. The adopted equation of state is the same as that used in @Machida+2009 that follows the radiation hydrodynamical calculations of protostellar collapse [@MasunagaInutsuka2000].
Results
=======
![ Bird’s eye-view of the result of non-ideal MHD simulation with nested grid technique, covering the evolution of the molecular cloud core to the protostar. The left panel shows the structure in the grid, level $l=8$, where the high-density region ($n=10^{10}{\rm cm}^{-3}$; blue isodensity surface) and magnetic field lines are plotted. Two cocoon-like structures (brown) above and below the flattened core denote the zero-velocity surface inside of which the gas is outflowing from the center. The density contours (color and contour lines) and velocity vectors (thin white arrows) are projected in each wall surface. The right upper panel shows the structure in part of the 10th grid, where we can clearly see the central cavity in the outflowing region. The right lower panel (12th grid) shows the protoplanetary disk in the formation phase, and two newly formed planetary-mass objects in the disk. []{data-label="fig:1"}](inutsuka_fig1.eps){width="170mm"}
Figure 1 shows a typical “bird’s eye-view” snapshot of our simulations. The timescale of the gravitational collapse (i.e., free-fall time) is a decreasing function of density; therefore, the dense central region shrinks faster than the less-dense surrounding regions. This property of gravitational collapse almost always leads to the successive decrease of mass inside the faster shrinking region in a run-away manner, This gravitational “run-away collapse” is decelerated by a gradual increase in the temperature of the central region, and eventually a quasi-steady object called “the first core” is formed [@Larson1969; @WinklerNewman1980a; @WinklerNewman1980b; @MasunagaMiyamaInutsuka1998]. It consists mainly of hydrogen molecules and has a radial extent exceeding 10 AU. The molecular gas surrounding the first core continues to accrete onto it, resulting in a slow but monotonic increase in density and temperature at the center of the first core. If the initial angular momentum of the molecular cloud core is on the order of the value suggested by observation, the resultant first core rotates significantly fast, and its formation corresponds to the onset of bipolar outflows driven by magnetic fields . In Figure 1, the outflow regions are shown by the two brown cocoon-like structure that correspond to the zero vertical velocity surfaces ($v_z=0$): gas inside the cocoons has a significant vertical velocity. Most of the angular momenta in gravitationally collapsing objects are removed by the Maxwell stress of the field, which is called magnetic braking [@MachidaInutsukaMatsumoto2007]. In addition the outflowing gas carries away angular momentum during this phase. When the central temperature becomes sufficiently high ($\sim 2\times 10^3$K), the dissociation of hydrogen molecules becomes significant, providing effective cooling that makes the core gravitationally unstable, triggering “the second collapse” .
The upper right panel of Figure 1 shows an enlarged view of the region inside the outflow-launching regions, where the resistivity is so significant that the magnetic field is decoupled from the gas. The outflow region envelopes the “dead zone” for magnetic field [@Machida+2008a] where magnetic braking is not operating. Therefore, the infalling gas in the first core maintains angular momentum and reaches the radius of the centrifugal barrier to form a circumstellar disk-like structure. The formation of this disk-like structure corresponds to the birth of the protoplanetary disk, which happens inside the outflow-launching region.
The infalling from the envelope to the central region continues, and the radius of the outflow-lauching region increases over time, as does the outer boundary of the magnetic dead zone. Likewise, both the mass and the outer radius of the circumstellar disk increase over time. Eventually, the radius of the disk extends beyond the initial radius of the first core and the first core disappears; in other words, the first core transforms itself into a protoplanetary disk.
![ Schematic diagram for the evolution of protostellar objects, in terms of mass. The vertical axis denotes mass (in units of solar mass) and the horizontal axis denotes time (in years). The red curve on the left-hand side depicts the mass of the fast collapsing region in the center of the molecular cloud core in the collapsing phase, which essentially defines the gravitationally unstable mass, and therefore corresponds to the Jeans mass. Note that the mass of the first core is much larger than the mass of the central protostar at its birth. The right-hand side describes the evolution in the main accretion phase, where gas in the envelope of the molecular cloud core accretes onto the central region and the protostar gains its mass. As the first core gradually changes into the protoplanetary disk, the mass of the protoplanetary disk remains larger than the mass of the protostar for a while. This configuration is gravitationally unstable and creates self-gravitating objects in the disk. The protostar mass increases monotonically and overwhelms the mass of the disk later in the accretion phase. []{data-label="fig:2"}](inutsuka_fig2.eps){width="170mm"}
![ Left panel shows actual evolution of protostar mass and disk mass, where the horizontal axis denotes elapsed time in years after the formation of the protostar. The right panel shows time evolution of the gravitational instability indicator $Q \equiv \kappa C_{\rm S}/(\pi G \Sigma)$. $Q_{\rm max}$ is the maximum value in the disk and $Q_{\rm ave}$ is the average value. The curve labeled $R$ denotes the ratio of the most unstable wavelength to the disk radius. At $t\approx~800$yr, $R$ becomes smaller than unity so that the gravitationally unstable mode is possible in the disk, and planetary-mass objects are formed. []{data-label="fig:3"}](inutsuka_fig3.eps){width="170mm"}
An important consequence of the early formation of the protoplanetary disk is the gravitational fragmentation of the disk and the formation of planetary-mass objects, as shown in the lower right panel of Figure 1. The reason for the gravitational instability can be understood in terms of the ratio of the disk mass to the central object. Figure 2 shows a schematic evolution of protostellar objects, in terms of mass. The vertical axis denotes mass (in units of solar mass) and the horizontal axis denotes time (in units of year). The red curve on the left-hand side depicts the mass of the fast collapsing region in the center of the molecular cloud core in the first collapse phase, which essentially defines the gravitationally unstable mass, and thus, corresponds to the Jeans mass. The decrease of this mass in the isothermal phase describes the run-away collapse of central region, a characteristic of gravitational collapse of cooling gas. The slight increase of the central mass (the Jeans mass) corresponds to the formation of the first core and its gradual increase in a quasi-steady state. The second fall of the central mass corresponds to “the second collapse” that is triggered by the endothermic dissociation of hydrogen molecules. After most of the hydrogen molecules are dissociated, the adiabatic increase in the pressure of the atomic gas eventually overcomes the gravitational collapse and the second core forms. Therefore, the decrease of the central mass in the second collapse stage is closed by the formation of a protostar. We should note that the mass of the first core is much larger than the mass of the central protostar at its birth.
The right-hand side of Figure 2 describes the evolution in the main accretion phase, where gas in the envelope of the molecular cloud core accretes onto the central region and the protostar gains mass. As the first core gradually changes into the protoplanetary disk, the mass of the protoplanetary disk remains large (and even larger than the protostar) for a while. This configuration is gravitationally unstable and the spiral arms are excited, which promotes gas accretion in the disk, but the accretion is not efficient enough to avoid gravitational fragmentation of the disk, creating self-gravitating objects in the disk. The typical mass of the formed objects is a fraction of the disk mass (i.e., smaller than $10^{-2}$ solar masses) which corresponds to the range of Jovian planets to brown dwarfs. The protostar mass increases monotonically and eventually overwhelms the mass of the disk. In effect Figure 2 describes why and how the gravitationally unstable protoplanetary disk is created.
The left panel of Figure 3 shows the actual evolution of the protostar mass and disk mass, where the horizontal axis denotes elapsed time (in years after the formation of the protostar). The right panel shows the time evolution of the indicator of gravitational instability $Q \equiv \kappa C_{\rm S}/(\pi G \Sigma)$, where $\kappa$, $C_{\rm S}$, $G$, and $\Sigma$ are epicyclic frequency, sound speed, gravitational constant, and surface density, respectively. $Q_{\rm max}$ is the maximum value in the disk and $Q_{\rm ave}$ is the average value. During the first several hundred years, $Q$ remains smaller than unity and the disk satisfies the instability criterion obtained by local linear analysis, although the size of the disk remains too small for the unstable mode to appear, as shown by the curve labeled $R$ that denotes the ratio of the most unstable wavelength to the disk radius. At $t\approx~800$ yr, $R$ becomes smaller than unity so that the gravitationally unstable mode becomes feasible in the disk; this epoch does indeed correspond to the fragmentation of the disk. Note that the fragmentation has occurred in the region where the effective equation of state is almost adiabatic. If additional radiative cooling operates, it will become even more unstable, and thus, the fragmentation is definitely expected. In other words, the gravitational fragmentation shown in this early formation phase of the massive disk does not require efficient radiative cooling that is supposed to be important in the later phase of the evolution where the disk is less-massive and less-unstable [e.g., @Gammie2002; @Rice+2003; @RiceLodatoArmitage2005]. Here we emphasize that we should expect the growth of gravitational instability in the disks that excessively satisfy the local instability criterion ($Q~<~1$) for the mode whose wavelength is sufficiently smaller than the disk. Nevertheless, the consequence of the growth and the resultant fragmentation should be studied in more detail including the effect of irradiation from the central star that is not taken into account in the present calculations. The formation of planetary-mass objects and the excited spiral arm structure provide an efficient transfer of angular momentum in the disk and promote mass accretion onto the protostar, as shown by the rapid increase in the protostar mass in the left panel. This property will be further investigated in our next paper.
Discussion
==========
#### Cooling Efficiency
Since we are not solving realistic radiative transfer equation in our dynamical simulations, one might wonder whether each fragment can sufficiently shrink by radiating an excess energy obtained by compressional heating or not. Here we estimate the amount of energy that should be radiated away from a fragment and compare with the cooling capability of the fragments. In order for the self-gravity to dominate over gas pressure the effective ratio of specific heats ($\gamma_{\rm eff}$) should be less than 4/3, while the ratio of specific heats ($\gamma$) for the molecular gas in adiabatic evolution is mostly 7/5 [e.g., @MasunagaInutsuka2000]. Thus, we can roughly calculate the amount of energy $\Delta E$ that should be radiated during the compressional motion of the planet with mass $M_{\rm p}$ from the density $\rho_{\rm d}$, pressure $P_{\rm d}$, and temperature $T_{\rm d}$ in the gas disk to the average density $\rho_{\rm p}$ and pressure $P_{\rm p}$ of the planet as, $$\Delta E = \frac{M_{\rm p}}{\gamma-1}
\left( \frac{P_{\rm d}}{\rho_{\rm p}} \right)
\left[
\left(\frac{\rho_{\rm p}}{\rho_{\rm d}}\right)^{\gamma}
- \left(\frac{\rho_{\rm p}}{\rho_{\rm d}}\right)^{\gamma_{\rm eff}}
\right] .$$ If we take the density enhancement factor of $10^5$ (e.g., $\rho_{\rm d} = 10^{-11} {\rm g~cm}^{-3}$ and $\rho_{\rm p} = 10^{-6} {\rm g~cm}^{-3}$) the second term is negligible compared to the first term on the right-hand side of the above equation. Thus, the expected time-averaged luminosity of the fragment that cools with the timescale $\Delta t$ can be calculated as, $$\langle L \rangle \equiv
\frac{\Delta E}{\Delta t} \approx
1.5 \times 10^{-1}
\left( \frac{M_{\rm p}}{10^{-3} M_{\odot}} \right)
\left( \frac{T_{\rm d}}{10^2{\rm K}} \right)
\left( \frac{10^2 {\rm yr}}{\Delta t} \right)
\left( \frac{\rho_{\rm p}}{10^5\rho_{\rm d}} \right)^{2/5}
L_{\odot} ,$$ where we assumed that the mean molecular weight is 2.3. On the other hand, the luminosity $L_{\rm p}$ of the planet with radius $R_{\rm p}$ and the surface temperature $T_{\rm p}$ is $$L_{\rm p} = 4 \pi R_{\rm p}^2 \sigma_{\rm SB} T_{\rm p}^4
= 1.9 \times 10^{-1}
\left( \frac{R_{\rm p}}{10^{12}{\rm cm}} \right)^2
\left( \frac{T_{\rm p}}{10^{3}{\rm K}} \right)^4
L_{\odot} ,$$ where $\sigma_{\rm SB}$ is Stephan-Boltzmann constant. Since $(T/T_{\rm d})=(\rho/\rho_{\rm d})^{\gamma_{\rm eff}-1}$, compression by a factor of $\rho_{\rm p}/\rho_{\rm d} = 10^5$ with $\gamma_{\rm eff}=4/3$ results in the increase of temperature by a factor of about 46. Thus, the fragment can radiate efficiently enough to collapse gravitationally by a factor of at least $10^5$ in density within 100 years (i.e., the resultant surface temperature of the planet will be somewhat smaller than $10^3$K). Thus, the objects formed by the gravitational instability of the disk as in our simulation are expected to be dense enough to avoid being washed out in the dynamical environment.
#### Further Evolution
What is the fate of these gravitationally formed planetary-mass objects? Does gravitational interaction with the gaseous disk lead to the migration of these objects? In fact, some of our simulations showed the objects falling into the central star, while in other cases they migrated outward but remained in the disk. Their chaotic behavior is not surprising, since eccentricities of the orbits of planetary-mass objects in the disk are, in general, not small. Their large eccentricities are due to the fact that they absorb infalling gas. In addition, the gravitational fragmentation and formation of planetary-mass objects repeat many times during the main accretion phase, where the gas accretion from the envelope of the molecular cloud core continues. Some of the planetary-mass objects fall onto the central star while some remain in the disk. Therefore, a statistical method seems to be required to predict the distribution of objects that survive the gas dispersal, which will be the scope of our subsequent paper.
#### Implication for Observations
Astronomical observation of the early formation phase of protoplanetary disks would enable us to directly test these expectation. Although our numerical simulations enable us to frequently observe the gravitational fragmentation of protoplanetary disks, in reality this process is hidden in dense gas and may remain invisible in optical and near-infrared wavelengths. However, radio interferometers are potentially capable of observing the interiors of protostellar cloud cores. The Atacama Large Millimeter Array (ALMA) would be an ideal tool for this type of observation. In order to directly observe the process described in this Letter, it is necessary to observe the early evolutionary phase within about 1000 years from the birth of a protostar. As the average lifetime of a protostar is estimated to be $10^5$ year, the required time span for this observation is only 1 % of the protostar’s lifetime, meaning that we should observe, on average, about 100 protostars to find one protostar at an early evolutionary phase. In our simulations, we almost always observe very time-dependent mass accretion through the gravitationally unstable disk. Therefore, another possible observational signature is the time variability of the protostellar source. We hope that the gravitational fragmentation reported in this Letter will be analyzed observationally and a realistic planet formation scenario will be refined in the near future.
The frequent formation of planetary-mass objects in the disks during the formation phase suggests that we should consider their influence on the evolution of the protoplanetary disk. In particular, it suggests a new possibility for the core-accretion scenario in the region outside the orbit of the planetary-mass object where the shepherding effect from a massive object prevents dust grains from raining out onto the central star [@MutoInutsuka2009], which corresponds to the far right-side of Figure 2. Therefore, it might be interesting to search for a hybrid scenario of planet formation where the rocky planets form after the gravitational formation of the gaseous giant planets.
This work was supported in part by Grants-in-Aid for Scientific Research from the MEXT of Japan (15740118, 16077202, 18540238, 18740104, 20540238, and 21740136).
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Vehicular networks, an enabling technology for Intelligent Transportation System (ITS), smart cities, and autonomous driving, can deliver numerous on-board data services, e.g., road-safety, easy navigation, traffic efficiency, comfort driving, infotainment, etc. Providing satisfactory quality of service (QoS) in vehicular networks, however, is a challenging task due to a number of limiting factors such as hostile wireless channels (e.g., high mobility or asynchronous transmissions), increasingly fragmented and congested spectrum, hardware imperfections, and explosive growth of vehicular communication devices. Therefore, it is highly desirable to allocate and utilize the available wireless network resources in an ultra-efficient manner. In this paper, we present a comprehensive survey on resource allocation (RA) schemes for a range of vehicular network technologies including dedicated short range communications (DSRC) and cellular based vehicular networks. We discuss the challenges and opportunities for resource allocations in modern vehicular networks and outline a number of promising future research directions.'
author:
- 'Md. Noor-A-Rahim, Zilong Liu, Haeyoung Lee, G. G. Md. Nawaz Ali, Dirk Pesch, Pei Xiao [^1]'
bibliography:
- 'refs.bib'
title: A Survey on Resource Allocation in Vehicular Networks
---
Intelligent Transportation System, Vehicular network, Autonomous Driving, DSRC V2X, Cellular V2X, Resource Allocation, Network Slicing, Machine Learning.
Introduction
============
The prevalent vision is that vehicles (e.g., cars, trucks, trains, etc.) will in the future be highly connected with the aid of ubiquitous wireless networks, anytime and anywhere, to provide unprecedented travel experiences and offer a series of far-reaching benefits such as significantly improved road safety, enhanced situational awareness, less traffic congestion, reduced pollution emission, and lower capital expenditure. Central to this vision is a scalable and intelligent vehicular network which is responsible for efficient information exchange among vehicles and/or between vehicles and infrastructure. As an instrumental enabler for ITS, smart cities, and autonomous driving, vehicular networks have attracted significant research interests in recent years both from the academic and industrial communities [@Liu2017CodingAssisted; @Wang2017Centrality; @Nguyen2018; @Cheng2015; @Ali2018_tvt]. In particular, the concept of connected vehicles, also known as vehicle-to-everything (V2X) communications, has gained substantial momentum by bringing in increased data throughput and enhanced road safety along with novel onboard computing and sensing technologies. So far, there are two major approaches for V2X communications: DSRC and cellular based vehicular communication [@Seo2016; @Bazzi2017]. DSRC is supported by a family of standards including the IEEE 802.11p amendment for Wireless Access in Vehicular Environments (WAVE), the IEEE 1609.1$\sim$0.4 standards for resource management, security, network service, and multi-channel operation [@Kenney2011DSRC]. On the other hand, cellular based vehicular communication, also called C-V2X, designed over cellular networks such as Long-Term Evolution (LTE) and 5G new radio (5G NR), allows every vehicle to communicate with different types of communication entities, such as pedestrians, roadside units (RSU), satellites, internet/cloud, and other vehicles. Both V2X techniques have their respective advantages and limitations when they are adopted in vehicular environments. As a result, an integration of such heterogeneous vehicular networks has been suggested to exploit their unique benefits, while addressing their individual drawbacks.
Wireless networks suffer from a wide range of impairments like shadowing, path loss, time- and/or frequency- selectivity of wireless channels, jamming and/or multi-user interference, etc. To deal with these impairments, radio resources (such as time slots, frequency bands, transmit power levels, etc.) should be allocated in an optimized manner to cater for instantaneous channels and network conditions. Dynamic Resource Allocation (RA) schemes are preferred as they give rise to significantly improved performance (compared to the Static RA schemes) by efficiently exploiting diversities from various dimensions [@Georgiadis2006; @Zhang2010_cog; @Wang2011_mul]. For instance, authors in [@Botsov2014; @Ren2015; @Sun2016; @Sun2016b; @Cheng2017] studied RA schemes for device-to-device (D2D) V2X networks by taking into account fast vehicular channel variations. Nevertheless, RA in vehicular networks are far more challenging due to the following reasons:
1. Highly dynamic mobility from low-speed vehicles (e.g., less than 60 km/h) to high-speed cars/trains (e.g., 500 km/h or higher) [@Zhang2011; @ZijunZhao2013]. The air interface design for high mobility communication, for instance, may require more time-frequency resources in order to combat the impairments incurred by Doppler spread/shifts and multi-path channels.
2. Vast range of data services (e.g., in-car multimedia entertaining, video gaming/conferencing, ultra-reliable and low-latency delivery of safety messages, high-precision map downloading, etc) with different QoS requirements in terms of reliability, latency, and data rates. In particular, some requirements (e.g., high data throughput against ultra-reliability) may be conflicting and hence it may be difficult to support them simultaneously.
3. Explosive growth of vehicular communication devices in the midst of increasingly fragmented and congested spectrum. Moreover, these devices usually have different hardware parameters and therefore may display a wide variation in their communication capabilities under different channel and network conditions. For example, a vehicular sensor device aiming for long battery life (e.g., more than 10 years) is unlikely to use sophisticated signal processing algorithms for power saving purposes whereas more system resources and more signal processing capabilities may be required for ultra-reliable transmission of safety messages.
Driven by these challenges, over the past decade, numerous disruptive ideas and techniques have been emerging aiming for optimizing/addressing various aspects/challenges of vehicular networks. In the existing literature, however, a survey with an extensive high-level overview as well as detailed up-to-date advances on RA in vehicular networks is still lacking to the best of our knowledge. To fill this gap and to stimulate more innovations in this area, we provide a comprehensive survey on the state-of-the-art of RA in vehicular networks and suggest a number of promising research directions.
This article is organized as follows. We start our discourse in Section II by a high-level overview of vehicular networks which include DSRC network, C-V2X network and heterogeneous network. Detailed literature surveys on these three types of vehicular networks are presented in Sections III-V, respectively. As machine learning is gaining ever-increasing research attention in numerous areas such as data-driven decision making, we provide a dedicated survey in Section VI on applications of machine learning for RA in vehicular networks. In Section VII, we summarize three important future directions of the RA research by taking advantage of network slicing, machine learning, and context awareness. Finally, this article is concluded in Section VIII.
Overview of Vehicular Networks
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DSRC Vehicular Network
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DSRC is a wide-consensus wireless technology that is designed to support ITS applications in vehicular networks. The underlying standard for DSRC is 802.11p, which is derived from IEEE 802.11e with small modifications in the QoS aspects. DSRC supports communications between vehicles and RSUs. The US Department of Transportation estimates that vehicle-to-vehicle (V2V) communications based on DSRC can eliminate up to 82% of all crashes involving unimpaired drivers in the US, and about 40% of all crashes occurred at intersections [@Kenney2011DSRC]. These statistics imply a significant potential for DSRC technology to reduce crashes and to improve road safety.
DSRC technology supports two classes of devices [@Morgan2010; @Hartenstein2010]: on-board unit (OBU) and road side unit (RSU), which are equivalent to the mobile station (MS) and base station (BS) in traditional cellular systems, respectively. An overview of a typical DSRC vehicular network in shown in Fig. \[fig:DSRC\]. The Federal Communications Commission in the United States has allocated 75 MHz licensed spectrum for DSRC communications in the 5.9 GHz frequency band [@Noor-A-Rahim2018_acc]. Out of the 75 MHz spectrum, 5 MHz is reserved as the guard band and seven 10-MHz channels are defined for DSRC communications. The available spectrum is configured into one control channel (CCH) and six service channels (SCHs). The CCH is reserved for carrying high-priority short messages or control data, while other data are transmitted over the SCHs. Several modulation and coding schemes (MCS) are supported by DSRC with the transmitter (TX) power ranging from 0 dBm to 28.8 dBm. Based on the communication environments, the coverage distance may range from 10m to 1km.
A fundamental mechanism for medium/channel access in DSRC technology is known as distributed coordination function (DCF). With this DCF, vehicles contend for the transmission channels using a carrier-sense multiple access (CSMA) with collision avoidance (CA) technique. To transmit a packet from a vehicle, the channel must be sensed idle for a guard period. This guard period is known as the distributed interframe space (DIFS). If the channel is sensed busy, the vehicle initiates a slotted backoff process and vehicles are only permitted to start transmissions at the beginning of slots. Vehicles randomly choose their individual backoff time from the range $[0, CW-1]$, where $CW$ is known as the contention window. The backoff time counter is decreased by $1$, when the channel is sensed idle for a time slot. The counter is frozen when the channel is sensed occupied and reactivated after the channel is sensed idle again for a DIFS time interval period. A vehicle transmits when its backoff counter reaches zero. A packet collision occurs when two or more vehicles choose the same time slot for transmission. Along with the above channel access mechanism, IEEE 802.11p adopts Enhanced Distributed Channel Access (EDCA) mechanism, which allows four access categories in a vehicle with different priorities.
Cellular based Vehicular Network (C-V2X)
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Despite of the fact that DSRC is generally considered as the de facto technique for vehicular networks, cellular/LTE based vehicular communications (also known as C-V2X) have recently attracted significant attention due to its large coverage, high capacity, superior quality of services, and multicast/broadcast support. An overview of cellular based vehicular network in shown in Fig. \[fig:Cellular\]. LTE-V2V communication exploits LTE uplink resources while utilizing single carrier frequency division multiple access (SC-FDMA) at the PHY and MAC layers [@Cecchini2018]. According to the LTE specifications, the available bandwidth is subdivided into equally-spaced (spacing of 15 kHz) orthogonal subcarriers. A resource block (RB) in LTE is formed by 12 consecutive subcarriers (i.e., 180 kHz) and one time slot (i.e., 0.5 ms). The number of data bits carried by each RB depends on specific Modulation and Coding Schemes (MCS).
To utilize the available radio resources, two side-link modes are defined by 3GPP standard release 14: Mode 3 and Mode 4. In Mode 3, it is assumed that the vehicles are fully covered by one or more evolved NodeBs (eNBs) who dynamically assign the resources being used for V2V communications through control signalling. This type of resource assignment is called dynamic scheduling. An eNodeB may also reserve a set of resources for a vehicle for its periodic transmissions. In this case, the eNodeB defines how long resources will be reserved for the vehicle. In Sidelink Mode 4, vehicles are assumed to be in areas without cellular coverage and hence, resources are allocated in a distributed manner. A sensing based semi-persistent transmission mechanism is introduced in Sidelink Mode 4 to enable distributed resource allocation.
The distributed algorithm is implemented among vehicles, which optimizes the use of the available channels by increasing the resource reuse distance between vehicles that are using the same resources. A distributed congestion control mechanism is also applied which calculates the channel busy ratio and the channel occupancy ratio. Then, a vehicle reserves resource for a random interval and sends a reservation messages using Side link Control Information (SCI). The reservation message is also called Scheduling Assignment (SA). Using SA, other vehicles which sense and listen to medium find out the list of busy resources and avoid selection of those resources. To increase the reliability, a vehicle may send a data message more than once in this mode. In Release 14, 3GPP mentioned that D2D communications included in Releases 12 and 13 can also be applied to vehicular networks as the localization characteristics of vehicular networks are similar to D2D networks [@Lin2014; @Sun2016].
Heterogeneous Vehicular Network
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Despite the potential of DSRC vehicular networks, DSRC technology suffers from several drawbacks [@HameedMir2014; @Seo2016; @Araniti2013] such as limited coverage, low data rate, and limited QoS guarantee, and unbounded channel access delay. As a matter of fact, the PHY and MAC layers of DSRC are inherited from IEEE 802.11 standards which have been originally optimized for wireless local area networks with low mobility. As concluded in [@Araniti2013], although the current DSRC technology is shown to be effective in supporting vehicular safety applications in many field trials, significant challenges remain for employing DSRC technology in some hostile vehicular environments.
While cellular based vehicular networks can provide wide coverage and high data rate services, they may not be able to support decentralized communication as the networks may become easily overloaded in situation with very high vehicle density, e.g. traffic jams, etc. Thus, both DSRC and cellular based vehicular networks have their respective advantages and limitations when used in vehicular environments. A depiction of a heterogeneous vehicular network in shown in Fig. \[fig:DSRCandCellular\]. A range of efforts [@Zheng2015a; @Dressler2014; @Atat2012; @Huang2010; @LiuFuqiang2010; @Zheng2015; @Dai2018; @Cespedes2015; @Shafiee2011; @He2016] have been made towards the integration of both DSRC and cellular based vehicular networks (e.g., LTE) for enhanced vehicular communications. Besides the integration of DSRC and cellular based vehicular networks, emerging V2X applications require efficient utilization of heterogeneous access technologies, such as Wi-Fi and TV broadcasting networks.
Resource Allocation in DSRC Networks
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In this section, we review a number of resource allocation approaches for DSRC based vehicular networks. Previous works on the resource allocation strategies for DSRC network are mainly focused on MAC parameter allocation, channel allocation and rate allocation techniques. Hence, in the following, we classify all the resource allocation approaches for DSRC network in those three categories.
![Comparison between default DSRC and the scheme proposed in [@Harigovindan2012] in terms of data transfer ratio (for fast and slow vehicles) versus mean velocity of slow vehicles.[]{data-label="fig:Harigovindan2012"}](Figures/Harigovindan2012.pdf "fig:"){width="\figwidth"}\
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MAC Parameter Allocation
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In a traditional DSRC network, all vehicles adopt identical MAC parameters by default and hence have equal opportunity to access the network resources. However, this setting may be unfair for high-mobility vehicles which in turn could significantly degrade the network performance. For example, the data throughput of a vehicle with high velocity may degrade severely compared to that of a slowly moving vehicle because the latter is expected to have a better chance to communicate with its RSU (due to its long residence time in the coverage area of the RSU). Several studies have been carried out on MAC parameter allocation in DSRC networks to enhance reliability, throughput, and fairness. [@Harigovindan2012] presented a contention window allocation strategy to resolve the aforementioned unfairness problem. Specifically, an optimal selection on the minimum contention window (required for any vehicle) has been derived by taking into consideration the mean speed of vehicles in the network. Fig. \[fig:Harigovindan2012\] compares the DSRC default scheme and the scheme proposed in [@Harigovindan2012] in terms of the data transfer ratio (for fast and slow vehicles) versus mean velocity of slow vehicles. It is observed that for the DSRC default scheme, the data transfer ratio increases as the mean velocity of slow vehicles increases. In fact, in this case, the residence time of slowly moving vehicles decreases within RSU’s coverage and hence the data transfer decreases correspondingly. On the other hand, a relatively flat data transfer ratio is maintained with their proposed contention window allocation scheme which ensures equal chances of communication with the RSU for both slow and fast vehicles[^2] A modified MAC scheme was proposed in [@Harigovindan2012] to dynamically adapt the MAC parameters based on the residence time of vehicles.
To maximize the throughput among neighboring vehicles, a stochastic model was proposed in [@Rossi2015; @Rossi2017] to find the optimal maximum contention window using the surrounding vehicle density. Fig. \[fig:Rossi2017\] shows that the proposed protocol in [@Rossi2015; @Rossi2017] offers much lower average transmission delay as well as significantly improved throughput (compared to the standard DSRC protocols) due to reduced packet collision with optimized contention window size.
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In [@Alasmary2012], two dynamic contention window allocation schemes have been proposed to improve the network performance in high mobility environments. The first scheme is the p-persistent based approach [@Cali2000] which dynamically assign the contention window based on the number of neighboring vehicles, while the second scheme performs contention window adaptation based on the vehicle’s relative velocity. Fig. \[fig:Alasmary2012\] compares their proposed schemes in terms of the packet delivery ratios and network throughput. It is observed that both schemes provide enhanced performance (compared to the default DSRC one) as they give rise to reduced packet collisions. Moreover, each scheme provides better performance than the other in certain scenarios. For example, the first scheme exhibits better packet delivery ratio when the number of vehicles in the network is large. In terms of network throughput, the second scheme outperforms the first when the number of vehicles is higher than 80.
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Channel Allocation for Emergency Messages
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DSRC/WAVE uses orthogonal frequency bands to support multi-channel operation while considering equal share of available channels to all messages. Emergency messages (e.g., mission critical messages that carry safety-related information) in vehicular networks need to be processed with high priority, ultra reliability, and low latency. Ryu et al. [@Ryu2011] proposed a multi-channel allocation strategy called DSRC-based Multi-channel Allocation for Emergency message dissemination (DMAE) by first identifying the available bandwidth of channels and then allocating the channel with the largest bandwidth to the emergency message while maintaining QoS between RSU and OBU through periodic channel switching. Fig. \[fig:Ryu2011\] compares the packet delivery ratio (PDR) and end-to-end delay between DMAE and the traditional allocation scheme adopted by WAVE. It is observed that the emergency PDR of DMAE is higher than the PDR of WAVE as DMAE assigns available SCH with maximum bandwidth to the emergency messages. Moreover, DMAE outperforms WAVE in terms of delay performance as it can assign emergency messages to reserved channels in the event of heavy traffic scenario.
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**MCS Index** **Modulation** **Code rate** **Data rate (Mbps)** **Communication range (m)**
\[0.5ex\] 1 BPSK $\frac{1}{2}$ 3 1000
2 BPSK $\frac{3}{4}$ 4.5 900
3 QPSK $\frac{1}{2}$ 6 800
4 QPSK $\frac{3}{4}$ 9 700
5 16-QAM $\frac{1}{2}$ 12 600
6 16-QAM $\frac{3}{4}$ 18 500
7 64-QAM $\frac{2}{3} $ 24 400
8 64-QAM $\frac{3}{4} $ 27 300
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Rate Allocation
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IEEE 802.11p based communication supports multiple MCS to allow a wide range of data transmission rates ranging from 3 Mbps to 27 Mbps. The data rates and transmission ranges for different MCS are shown in Table \[Table:MCS\]. For the sake of simplicity, a constant MCS is often assumed in previous works on vehicular communications. This strategy may deteriorate the communication performance as constant MCS may not be suitable for diverse traffic environments in different roadway scenarios. As a solution, [@Sheu2010] proposed a new vehicular channel access scheme (VCAS) to maintain a trade-off between overall throughput and fairness. In this scheme, a number of vehicles with similar transmission rates are grouped into one channel to achieve the overall throughput requirement, while the fairness requirement is achieved by controlling the group sizes. By adopting a marginal utility model to allocate appropriate transmission rate per SCH (determined by predefined transmission distance thresholds), it is shown in [@Sheu2010] that their proposed scheme can simultaneously achieve enhanced fairness and overall system throughput over the existing scheme adopted in DSRC system. More recently, [@Ali2018; @Ali2018_ICC] presented allocation of variable MCS (i.e., variable data rates) in network coding-assisted heterogeneous on-demand data access, in which the MCS for disseminating data items were assigned based on the distance of the requested vehicles from the RSU. Simulation results show that the schemes proposed in [@Ali2018; @Ali2018_ICC] are capable of improving the on-demand requests serving capability and reducing the system response time.
Resource Allocation in C-V2X
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The capability of supporting diverse vertical industries/applications is a major feature of 5G communication systems and beyond. Examples of vertical industries include smart homes/cites, e-health, factories of the future, intelligent refineries and chemical plants, and Cellular V2X (C-V2X). A strong catalyst for deeper and wider integration of wireless communications into our lives, C-V2X has been advocated by many mobile operators under the evolution of 3GPP’s LTE and 5G NR [@Huawei2017]. Compared to DSRC, C-V2X acts as a “long-range sensor" (aided by sophisticated cameras, radar, lidar, RSUs, cellular infrastructure and network) to allow vehicles to see/predict various traffic situations, road conditions, and emergent hazards several miles away.
From a network point of view, there are three major 5G use cases to be supported: enhanced mobile broadband (eMBB) communications, massive machine-type communications (mMTC), ultra-reliable and low-latency communications (URLLC). As far as C-V2X is concerned with, eMBB, aiming to provide data rates of at least 10 Gbps for the uplink and 20 Gbps for the downlink channels, plays a pivotal role for in-car video conferencing/gaming, various multimedia services, or high-precision map downloading, etc; mMTC will allow future driverless vehicles to constantly sense and learn the instantaneous driving environments using massive number of connected sensors deployed in-car or attached to the infrastructure; URLLC, targeting to achieve 1 ms over-the-air round-trip time for a single transmission with reliability of at least $99.999\%$ is instrumental for autonomous emergency braking and hazard prevention.
That being said, C-V2X has to share and compete with other vertical applications for system resources (e.g., spectrum/network bandwidth, storage and computing, etc) under a common physical infrastructure. RA for C-V2X therefore is a trade-off with a variety of data requirements from different vertical applications. A central question is how to design an efficient network to provide guaranteed quality of service (QoS) for C-V2X while balancing the data services to other vertical applications.
RA for Traditional Cellular System
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![Sum-rate comparison with traditional and optimal schemes [@Zhang2013].[]{data-label="fig:Zhang2013"}](Figures/Zhang2013.pdf "fig:"){width="\figwidth"}\
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Graph based interference aware RA strategies have been proposed in [@Zhang2013; @Meng2018], where the weights of the edges are assigned according to the interference terms between the related vertices. The scheme in [@Zhang2013] formulates an optimization problem with the objective of maximizing the network sum rate with low computational complexity. It is shown in Fig. \[fig:Zhang2013\] that their proposed scheme exhibits higher network sum rate than traditional orthogonal communication mode. In contrast, the work in [@Meng2018] aims at improving the connectivity of vehicular communications by introducing a metric called *connectivity index*, which is obtained from the percentage of vehicles in the network being assigned with resources while satisfying the interference constraints. With the aid of the minimum spanning tree approach [@West1996], Meng et al. [@Meng2018] proposed a RA algorithm to improve the connectivity of the network. Fig. \[fig:Meng2018\] shows the performance of the RA scheme proposed in [@Meng2018]. The connectivity index performance is presented in Fig.\[fig:Meng2018a\] with varying number of vehicles, whilst the performance of brute force search algorithm is shown as a benchmark. We observe that the connectivity index of [@Meng2018]’s algorithm is only 17.1% away from the optimum solution obtained from the brute force search algorithm. In Fig. \[fig:Meng2018b\], we present the full connectivity performance of the algorithm proposed in [@Meng2018] and compare with the greedy graph coloring algorithm [@Etzion1998]. We observe a similar full connectivity performance for both algorithms, while graph coloring algorithm exhibits high computational complexity. As expected, the full connectivity percentage decays with the increase of vehicle arrival rate (i.e., denser vehicular network).
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By exploiting geographical information, [@Liang2017] proposed a joint RA and power control scheme for reliable D2D-enabled vehicular communications by considering slow fading channel information. Queuing dynamics was also considered in [@Liang2017] in order to meet the requirements of different QoS in vehicular networks. [@Sun2016] developed a heuristic algorithm, named Separate resOurce bLockand powEr allocatioN (SOLEN), under large-scale vehicular fading channels to maximize the sum rate of cellular users while satisfying the vehicular users’ requirements on latency and reliability. Similar to [@Sun2016], [@Mei2018] incorporated dynamic MCS in the process of RBs and transmit power allocation for guaranteed reliability and latency. A latency performance comparison between the works of [@Sun2016] and [@Mei2018] is shown in Fig. \[fig:Mei2018\]. By adopting dynamic MCS in the allocation algorithm, the algorithm proposed in [@Mei2018] outperforms that of [@Sun2016] in terms of average outage probability and packet latency. To support D2D-based safety-critical vehicular communication, a cluster-based RA scheme was proposed in [@Sun2016b] by maximizing the cellular users’ sum rate. This is achieved by a three-step heuristic algorithm with the knowledge of the slowly varying channel state information of uplink channel.
The work in [@Zhang2016] proposed a centralized RA algorithm by utilizing the spectral radius estimation theory. Their proposed algorithm maximizes the number of concurrent reuses of resources by multiple vehicles instead of maximizing the sum rate (a method often used in traditional allocation algorithms). With eNodeB centrally deciding the resource reuse for the vehicles in the network, the scheme proposed in [@Zhang2016] exhibits significant improvement in the spectrum efficiency and demonstrates the capability of maintaining the required QoS when the vehicle density is high.
[@Guo2019] proposed a RA scheme to support V2X communications in a D2D-enabled cellular system, where the V2I communication is supported by a traditional cellular uplink strategy and the V2V communication is enabled by the D2D communications in the reuse mode. [@Guo2019] formulated an optimization problem to maximize the sum ergodic capacity of the vehicle-to-infrastructure (V2I) links while satisfying the delay requirements of V2V links. The optimization problem was solved by combining the bipartite matching algorithm and the effective capacity theory.
RA for Vehicular Computing System
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In recent years, integration of vehicular network with cloud computing, also known as vehicular computing system, has attracted increasing attention for its capability of providing real-time services to on-board users [@Gerla2012; @Bitam2015]. RA for vehicular computing systems has been investigated in [@Lin2018a; @Lin2018]. A semi-Markov decision process based RA scheme was proposed in [@Lin2018a] for a vehicular cloud computing system while considering heterogeneous vehicles, i.e., vehicles with different amounts of computing resources. In particular, [@Lin2018a] integrated the computational resources of vehicles and RSUs in the vehicular cloud computing system to provide optimum services. [@Lin2018] aimed to reduce the serving time by optimally allocating the available bandwidth in a vehicular fog computing system. The optimization problem of [@Lin2018], formulated based on the requirements of the serving methods, was solved in the following two steps: 1) finding the sub-optimal solutions by applying the Lagrangian algorithm; 2) performing selection process to obtain the optimum solution.
RA for Secured Vehicular Network
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RA may also be exploited to enhance the secrecy of cellular vehicular networks. By observing that LTE-based V2X communication cannot properly preserve the privacy, [@Ahmed2018] evaluated the message delivery with specified security. A joint channel and security key assignment policy was presented in [@Ahmed2018] to enable a robust and secure V2X message dissemination. In [@Yang2017], a RA scheme was proposed to enhance the physical layer security in cellular vehicular communication. A max-min secrecy rate based problem was formulated to allocate power and sub-carrier while taking into account the outdated channel state information (CSI) due to the high mobility. The problem was solved in two stages: (i) with fixed sub-carrier assignment, allocating the power level by using a bisection method allocation problem; (ii) finding suboptimal sub-carrier allocation by using greedy algorithm.
RA for Vehicle Platooning
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In recent years, vehicle platooning networks have been gaining growing research interest as they can lead to significant road capacity increase. In [@Wang2018], the authors proposed a RA scheme for D2D based vehicle platooning to share control information efficiently and timely. A time-division based intra-platoon and minimum rate guaranteed inter-platoon RA scheme was proposed to allocate the resources within the platoon, while ensuring optimized cellular users’ rate. Moreover, to obtain a stable platoon, a formation algorithm was proposed in [@Wang2018] based on a leader evaluation method. Authors in [@Meng2018a] presented a RA strategy to reduce the re-allocation rate that enhances the number of guaranteed services in vehicle platooning network. A time dynamic optimization problem was formulated in [@Meng2018a] under the constraint of a network re-allocation rate. To further reduce the computational complexity, their proposed optimization problem was converted into a deterministic optimization problem using the Lyapunov optimization theory [@Georgiadis2006]. Joint optimization of communication and control in vehicle platooning was proposed in [@Mei2018a]. An improved platooning system model was developed by taking into account both control and communication factors in a vehicle platooning. A safety message dissemination scenario was considered under an LTE based vehicular network, where the platoon leader vehicle coordinates the allocation of available communication and control resources. A joint optimization problem of RB allocation and control parameter assignment was formulated with the constraints of communication reliability and platoon stability. Through simulation results, it was shown that their proposed RA algorithm reduces the tracking error while maintaining the stability of the platoon.
RA for Out-of-Coverage Scenario
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A two-step distributed RA scheme was proposed in [@Yang2016] for out-of-coverage (i.e., out of eNodeB coverage) LTE V2V communication. In the first step, RBs are assigned based on the heading directions of vehicles. In other words, the same set of RBs are assigned to the vehicles moving in the same direction. In the second step, a channel sensing based strategy is utilized to avoid the packet collision between the vehicles which travel in parallel on the road. Recently, authors in [@Sahin2018] studied RA scheme for delimited out-of-coverage scenario, where the network infrastructure assigns the resources to vehicles based on the estimated location of vehicles. More recently, authors in [@our_TVT_2019] analyzed and evaluated the safety message broadcasting performance of LTE-V2V out-of-coverage mode in an urban intersection scenario, where two resource allocation strategies were presented to improve the broadcasting performance through vehicle assisted relaying.
RA for Heterogeneous Vehicular Networks
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A graph based resource scheduling approach was proposed in [@Zheng2013] for cooperative relaying in heterogeneous vehicular networks. In LTE, vehicles close to the base station usually enjoy high data rates due to favourable radio links, while vehicles far away from the base station suffer from lower data rates due to poor channel conditions. To tackle this problem, cooperative relaying may be adopted to establish V2V communications for distant vehicles through DSRC. [@Zheng2013] proposed a bipartite graph based scheduling scheme to determine the transmission strategy for each vehicle user from base station (i.e., cooperative or non-cooperative) and the selection of relaying vehicles. The scheme proposed in [@Zheng2013] consists of the following three steps: 1) construct a weighted bipartite graph, where the weight of each edge is determined based on the capacity of the corresponding V2V link, 2) solve the maximum weighted matching problem using the Kuhn–Munkres algorithm, and 3) optimize the number of messages that need to be relayed, where binary search was utilized to find the optimal solution. The proposed approach guarantees fairness among vehicle users and can improve the data rates for the vehicles far away from the base station.
![Comparison of overall throughput of V2I links for Guo’s [@Guo2019] and Fang’s [@Fang2017] methods with respect to reliability of the V2V link $(p_v)$ and cellular user link $(p_f)$ [@Guo2019].[]{data-label="fig:Guo2019a"}](Figures/Guo2019a.pdf "fig:"){width="\figwidth"}\
![Software defined network (SDN) based heterogeneous vehicular network.[]{data-label="fig:Huang2018a"}](Figures/SDN_controller.jpg "fig:"){width="\figwidth"}\
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Very recently, a cascaded Hungarian channel allocation algorithm was presented in [@Guo2019] for non-orthogonal multiple access (NOMA) based heterogeneous vehicular networks. [@Guo2019] addressed the channel assignment problem in high-mobility environments with different user QoS requirements and imperfect CSI by formulating a chance constrained throughput optimization problem. In Fig. \[fig:Guo2019a\], the overall throughput is compared with that of the RA method reported in [@Fang2017]. Enhanced performance is observed for the allocation scheme of [@Guo2019], thanks to an efficient user scheduling algorithm which fully utilizes the transmit power to maximize the throughput. It is also observed that the method proposed in [@Guo2019] provides more benefits with increasing transmit powers.
Xiao et al. [@Xiao2018] investigated the spectrum sharing for vehicle users in heterogeneous vehicular networks by exploiting available white space spectrum such as TV white space spectrum. A non-cooperative game theoretic approach was proposed with correlated equilibrium. Their proposed approach allows macrocell base stations to share the available spectrum with the vehicle users and improves the spectrum utilization by reusing the white space spectrum without degrading the macrocell performance. By sharing the available spectrum from the LTE and Wi-Fi networks, [@Huang2018] presented a quality of experience (QoE) based RA scheme for a software defined heterogeneous vehicular network. The system model considered in [@Huang2018] is shown in Fig. \[fig:Huang2018a\]. To maximize the QoE of all vehicles, the proposed scheme exploits the CSI of vehicles to extract transmission qualities of the vehicles with different access points. A heuristic solution was proposed to allocate the available resources (in LTE and Wi-Fi networks), which can be used in both centralized and hybrid software defined network systems. Fig. \[fig:Huang2018b\] presents the performance comparison between the proposed SDN based scenario and non-SDN based scenario. In the non-SDN based scenario, the optimization for the allocation of LTE and Wi-Fi resource is carried out separately. Due to the joint optimization of RA, the proposed method effectively allocate the resources and hence outperforms its non-SDN counterpart. An allocation approach for joint LTE and DSRC network was proposed in [@Cao2016]. The proposed approach allocates the LTE resources to minimize the number of vehicles that compete for channel access in DSRC based communication. The LTE resources are optimally allocated from the eNodeB, which jointly pairs one vehicle to another and allocates the resources to the pair considering a guaranteed signal strength for all communication links.
Machine Learning based RA for Vehicular Communications {#Sect:MLRA}
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In vehicular networks, whilst vehicles are expected to employ various facilities such as advanced on-board sensors including radar and cameras and even high-performance computing and storage facilities, massive amounts of data will be generated, processed and transmitted. Machine Learning (ML) is envisaged to be an effective tool to analyse such a huge amount of data and to make more data-driven decisions to enhance vehicular network performance [@Liang2019]. For details on machine learning, readers can refer to [@Jiang2017; @Sutton1998; @Arulkumaran2017].
For resource allocation, the traditional approach is to formulate an optimisation problem and then obtain an optimal or suboptimal solution depending on the trade-off between target performance and complexity. However, in vehicular networks where the channel quality and network topology can vary continuously, the conventional optimization approach would need to be rerun whenever a small change happens, thus incurring huge overhead [@Ye2018]. While the ML approach could be an alternative to the prevalent optimisation method, research on applying ML in vehicular networks is still at an early stage [@Liang2019].
In the existing literature [@Ye2018ICC; @Li2017; @Xu2014; @Salah2016; @Zheng2016], machine learning considering the dynamic characteristics of vehicular networks has been applied to channel and power allocation, user association and handoff for load balancing, and virtual resource management for V2V and V2I communications.
In [@Ye2018ICC], for V2V communications in a cellular network, a distributed channel and power allocation algorithm employing deep reinforcement learning (RL) [@Arulkumaran2017] has been proposed. With the assumption that an orthogonal resource is allocated for V2I links beforehand, the study focuses on resource allocation for V2V links under the constraints of V2V link latency and minimized interference impact to V2I links.
The structure of reinforcement learning for V2V links is shown in Fig. \[fig:sysModelML\]. While the agent corresponds to each V2V link, it interacts with the environment which includes various components outside the V2V links. The state for characterising the environment is defined as a set of the instantaneous channel information of the V2V link and V2I link, the remaining amounts of traffic, the remaining time to meet the latency constraints, and the interference level and selected channels of neighbours in the previous time slot. At time epoch $t$, each V2V link, as an agent, observes a state $s_t$ from the state set $\mathcal{S}$, and depending on its policy $\pi$, takes an action $a_t$ among the action set $\mathcal{A}$. The action is to select the sub-band and transmission power. Following the action, the agent receives a reward $r_t$ calculated by the capacity of V2I links and the V2V latency. The decision policy $\pi$ is determined by deep learning. At the beginning, the agent tends to select actions randomly. However, with an exploration and exploitation strategy, the agent prefers to exploit the effective actions yielding good rewards in the past and it also explorers new actions that may produce higher rewards in the future. Since the proposed approach can adjust the power and channel dynamically considering the latency constraints, the proposed algorithm is shown to outperform reference schemes in terms of the probability to satisfy the latency constraint of V2V links.
In [@Pressas2017], a contention-based MAC protocol for V2V broadcast transmission using IEEE 802.11p standard for DSRC is investigated. In a scenario with fewer than 50 vehicles, IEEE 802.11p can exhibit better performance than LTE in terms of lower latency and higher packet delivery ratio than LTE. However, as vehicle density gets high, the standard becomes unable to accommodate the increased traffic. In [@Pressas2017], with the aim of overcoming the scalability issue with the vehicular density, the ML based approach is proposed to find the optimal contention window to enable efficient data packet exchanges with strict reliability requirements. As a independent learning agent, each vehicle employs learning to decide contention window size. The result of each packet transmission, either success or fail, is feedback and utilized for the window size decision. Through simulation results, it is shown that the proposed ML based approach achieves more reliable packet delivery and higher system throughput performance.
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In [@Li2017], the ML approach is exploited to develop the user association algorithm for load balancing in heterogeneous vehicular networks. Considering the regularity characteristics of the data flow (generated from vehicular networks) in the spatial-temporal dimension, a two-step association algorithm is proposed. The initial association decision is made by a single-step reinforcement learning (RL) [@Sutton1998]. Subsequently, base station (i.e., macro, pico and femto cells) uses historical association patterns to make decisions for association. In addition, a base station, as an agent of learning, keeps accumulating feedback information and update the association results adaptively. While each base station runs the proposed algorithm in a distributed manner, in the long run, it is shown that both the real-time feedback and the regular traffic association patterns help the algorithm deal with the network changes.
In [@Xu2014], a vertical handoff strategy has been devised by using a fuzzy Q-learning approach [@Sutton1998] for heterogeneous vehicular networks consisting of a globally covered cellular network complemented by the V2I mode. From the OBU side, various information including average received signal strength (RSS) level, vehicle velocity and the amount of data is sent to the RSU side. Then, the RSU side considers the delivered information as well as the traffic load (i.e., the number of users associated with the target network) and makes handoff decisions by using the fuzzy Q-learning method. With the simulation results, it is shown that the proposed algorithm, which has a real-time learning capability, can determine the network connectivity to ensure seamless mobility management without prior knowledge about handoff behaviour.
In [@Salah2016; @Zheng2016], a machine learning approach is exploited to devise the virtual resource allocation in vehicular networks. Vertical clouds [@Lee2014] consisting of various OBUs, RSUs, and remote cloud servers can provide a pool of processing, sensing, storage, and communication resources that can be dynamically provisioned for vehicular services. The importance of resource allocation in the vehicular cloud is highlighted in [@Salah2016]. Poorly designed resource allocation mechanisms could result in QoS violation or under-utilisation of resources, whereas dynamic resource provisioning techniques are crucial for meeting the dynamically changing QoS demands of vehicular services. Against this background, a reinforcement learning framework has been proposed for resource provisioning to cater for dynamic demands of resources with stringent QoS requirements. In [@Zheng2016], a two-stage delay-optimal dynamic virtualisation radio scheduling scheme has been developed. Based on the time-scale, the proposed algorithm is divided into two stages, macro allocation for large time-scale variables (traffic density) and micro allocation with short time-scale variables (channel state and queue state). The dynamic delay-optimal problem is formulated as a partially observed Markov decision process (POMDP) [@Jiang2017] and is then solved by an online distributed learning approach.
Future Research Directions {#Sect:Future_Dir}
==========================
\
Network Slicing based Resource Allocation for C-V2X
---------------------------------------------------
Network slicing (NS) is a new paradigm that has arisen in recent years which helps to create multiple logical networks tailored to different types of data services and business operators [@NGMN2015]. NS offers an effective way to meet the requirements of all use cases and enables individual design, deployment, customization, and optimization of different network slices on a common infrastructure [@Foukas2017]. In addition to providing vertical slices (for vertical industries), NS may be used to generate horizontal slices which aim to improve the performance of user equipment (UE) and enhance the user experience [@Intel2016]. Although initially proposed for the partition of core networks (CN) using techniques such as network function virtualization (NFV) and software defined networking (SDN), the concept of NS has been extended to provide efficient end-to-end data services by slicing radio resources in radio access networks (RANs). The slicing of radio resources has mainly involves dynamic allocations of time and frequency resources based on the characteristics of multiple data services. This is achieved by providing multiple numerologies, each of which constitutes a set of data frame parameters such as multi-carrier waveforms, sub-carrier spacings, sampling rates, and frame and symbol durations. For example, an mMTC slice in C-V2X is allocated with relatively small subcarrier spacing (i.e., for massive connectivity) and hence large symbol duration. In contrast, URLLC requires large subcarrier spacing to meet the requirements of ultra-low latency and stringent reliability. Fig. \[NS\] depicts the NS for a C-V2X network consisting of RSUs, high-speed trains, railway stations and vehicles.
A step-wise approach for designing and applying function decomposition for NS in a 5G CN has been proposed in [@Sama2016]. Their main idea is to identify those functions which could be merged in different network elements as well as their corresponding implications for procedure and information storage. [@Soenen2017] presented a concrete NS example in the vehicular network domain on efficient distribution of unexpected road conditions among cars within a certain range. By properly configuring the SDN switch and controller, it is shown in [@Soenen2017] that a network slice for such inter-car communication can be readily created. In [@Silva2016], the impact of NS on a 5G RAN, such as the CN/RAN interface, the QoS framework, and the management framework, has been discussed. It is pointed out in [@Silva2016] that dynamic NS is preferred in order to cater for rapid change of traffic patterns. A comprehensive work on applications of NS to support a diverse range of C-V2X use cases has been proposed in [@Campolo2017]. Major C-V2X slices identified in [@Campolo2017] are: autonomous driving, tele-operated driving, vehicular infotainment, and vehicular remote diagnostics and management. For example, the slice for supporting tele-operated driving enables URLLC and the slice for vehicular infotainment may use multiple random access technologies (RATs) to support higher throughput. Moreover, slicing may be carried out in different vehicular devices according to their storage and computing capacities as well as the nature of the data services, a scenario similar to mobile edge computing [@Campolo2017].
It is noted that NS can be carried out not only at higher levels of wireless networks, but also in the physical layer (PHY). In 2017, a multi-service system framework implemented in both time and frequency domains has been proposed in [@Zhang2017; @Zhang-VTC2017]. A major issue here is how to select and design multicarrier waveforms with good time-frequency localization, low out-of-band power emission, low inter-carrier interference (ICI) among different sub-bands using different numerologies, and capability to support multi-rate implementation. Multicarrier waveform design for PHY NS such as filtered orthogonal frequency-multiple access (F-OFDM), windowed-OFDM, and universal filtered multi-carrier (UFMC) have been studied in [@Zhang2017; @Zhang2018; @Zhang-TVT2018]. In the context of C-V2X, the design of multiple numerologies for modest and high mobility environments is an interesting and pressing research issue. In this case, one needs to deal with doubly selective fading channels which could lead to severe ICI and inter-symbol interference. Another interesting research is direction is how to design and optimize network slices to provide guaranteed quality of services (QoS) for C-V2X while balancing the services of other vertical applications under the constraint of limited radio resources.
Machine Learning Perspective in Resource Allocation
---------------------------------------------------
Whilst the strong potentials of applying ML in vehicular networks have been discussed with the initial efforts in Section \[Sect:MLRA\], how to adapt and exploit ML to account for the peculiar characteristics of vehicular networks and services still remains as challenges and represents a promising research direction [@Ye2018]. Vehicular networks significantly differ from the scenarios where machine learning has been conventionally exploited in terms of strong dynamics in wireless networks, network topologies, traffic flow, etc. How to efficiently learn and predict such dynamics based on historical data for the benefit or reliable communications is still an open issue [@Liang2019]. In addition, data is supposed to be generated and stored across various units in vehicular networks, e.g., OBUs, RSUs, and remote clouds. It could be interesting to investigate whether traditional centralised ML approaches can be exploited to work efficiently in a distributed manner. For collective intelligent decision making in learning-capable vehicular networks, the overhead for information sharing and complexity of learning algorithms need to be taken into account [@Peng2019].
Context Aware Resource Allocation for Vehicular Communications
--------------------------------------------------------------
Existing work on resource allocation for vehicular networks mostly deals with efficient allocation of resource blocks such as frequency carriers or time-slots. However, most of the prior work on resource allocation did not consider context-aware/on-demand data transfer applications in vehicular networks. Since on-demand data transfer applications need to meet constraints such as deadline of the requested data items or priority of data items, to ensure a reliable service, there is a need for research to consider those more thoroughly. Although there is a lot of prior work [@Zhan2011; @Wang2014; @Chen2017] on performance evaluation of on-demand data dissemination scenarios in terms of the above constraints, they do not deal with the allocation of resource blocks, which is important for 5G networks.
Conclusions {#Sect:Conclusions}
===========
In this paper, we have surveyed radio resource allocation schemes in vehicular networks. We have categorized these schemes into three categories based on the types of vehicular networks, i.e., DSRC vehicular network, cellular vehicular network, and heterogeneous vehicular network. For each category, the available literature is reviewed and summarized while highlighting the pros and cons of the resource allocation schemes. We have also discussed several open and challenging future research directions for radio resource allocation in vehicular networks. It is anticipated that this paper will provide a quick and comprehensive understanding of the current state of the radio resource allocation strategies in vehicular networks while attracting and motivating more researchers into this challenging area.
[^1]: Md. Noor-A-Rahim and Dirk Pesch are with the School of Computer Science & IT, University College Cork, Ireland (E-mail: [m.rahim@cs.ucc.ie d.pesch@cs.ucc.ie]{}).
Zilong Liu, Haeyoung Lee, and Pei Xiao are with Institute for Communication Systems, 5G Innovation Centre, University of Surrey, United Kingdom (E-mail: [zilong.liu@surrey.ac.uk, haeyoung.lee@surrey.ac.uk, p.xiao@surrey.ac.uk]{}).
G. G. Md. Nawaz Ali is with the Department of Automotive Engineering, Clemson University, USA (E-mail: [gga@clemson.edu]{}).
This publication has emanated from research conducted with the financial support of Science Foundation Ireland (SFI) and is co-funded under the European Regional Development Fund under Grant Number 13/RC/2077. It has also received funding from the European Union’s Horizon 2020 Research and Innovation Programme under the EDGE CO-FUND Marie Skłodowska Curie grant agreement No. 713567. The work of Z. Liu and P. Xiao was supported by the U.K. Engineering and Physical Sciences Research Council under Grant EP/P03456X/1.
[^2]: A contention window allocation approach similar to that in [@Harigovindan2012] can be found in [@Karamad2008].
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Shi and Xie \[8\] predicted the N-shape current-voltage characteristic $(CVC)$ for a 2DEG in a magnetic field where the zero-resistance state has been observed in recent experiments. However it is known that in the absence of a magnetic field the zero resistance state (zero differential resistance state) is achieved in a system with the S-shape $CVC$. The difference in the behaviour of systems with N- and S-shape $CVC$ was studied more than three decades ago (see for example the review \[9\] and references therein) and is briefly explained in this Comment. At present it is not clear whether the N-shape $CVC$ may lead to the zero resistance state in a 2DEG in a magnetic field.'
address: |
$^{1}$ Theoretische Physik III,\
Ruhr-Universität Bochum, D-44780 Bochum, Germany\
$^{2}$Institute of Radioengineering and Electronics of the Russian\
Academy of Sciencies, Moscow 103907, Russia.
author:
- 'A.F. Volkov$^{1,2}$'
title: 'Comment on: ”Radiation-Induced ”Zero-Resistance State” and the Photon Assisted Transport”'
---
An interesting effect has been observed in recent papers [@Klitzing; @Zudov]: the resistance of a 2-DEG subjected to microwave irradiation drops to zero in some interval of an applied magnetic field $B.$ Possible mechanisms for this phenomenon are discussed in a number of papers [@Philips; @Yale; @Andreev; @Anderson; @Shrivastava; @Shi; @Raikh; @Mikhailov]. It was shown in Refs. [@Yale; @Shi; @Raikh] that in the presence of irradiation and a sufficiently strong magnetic field the conductivity $\sigma _{xx}$ may become negative in a weak electric field $E_{x}$. In addition, using a simple model, the authors of Ref. [@Shi] have calculated the current-voltage characteristic $I(V)$ of an irradiated system with a density-of-states periodic in energy $\varepsilon $. They have demonstrated that not only regions with negative conductance $G$, but also regions with positive $G$, may have negative differential conductance $G_{d}=dI/dV$ on the current-voltage characteristic ($CVC$) curve.
In the present Comment I would like to note that effects in systems with negative differential conductance (or resistance) depend in a crucial way on the type of $CVC$, i.e. whether it has the N- or S-shape form (see, for example, the review [@KoganVolkovUs] and references therein). It was established long ago that the states corresponding to regions with negative $%
G_{d}$ on $CVC$ are unstable with respect to nonhomogeneous fluctuations. However the type of instability and final nonhomogeneous state which is formed as a result of the instability depend on the type of $CVC.$
In the case of the N-shape $CVC$ (three voltages $V_{i}$ correspond to one current $I$) a domain with a strong electric field (as happens, for example, in the Gunn effect) arises as a result of an instability in a homogeneous initial state (it is assumed that the total voltage is fixed). This domain moves in the direction of the applied electric field $E=V/L$ ($L$ is the length of the sample). In the presence of the high field domain the $CVC$ changes drastically: an almost flat region (plateau) appears on the $I(V)$ curve. The differential conductance $G_{d}$ (but not the resistance $%
R_{d}=dV/dI$) is zero on this plateau. The current $I_{pl}$ corresponding to this flat part of $CVC$ depends on the particular form of the $CVC$. In the case of a ”symmetric” $I(V)$ curve with absolute negative conductance (i.e. min{$dI/dV$} corresponds to $V=0$ ) this current is equal to zero (that is the total conductance is zero). This means that when the bias voltage varies in some limits, the current remains equal to zero. The $CVC$ obtained in Ref. [@Shi] on the basis of a toy model can be regarded as a chain of N-shape $I(V)$ curves.
In the case of the S-shape $CVC$ (three currents $I_{i}$ correspond to one voltage $V$) small perturbations increase in the direction transverse to the current $I$ if this current corresponds to the part of $CVC$ with negative $%
R_{d}$. As a result of this instability, a domain of a strong current density arises in the sample (it is assumed that total current is fixed, otherwise the system goes over to a state on the stable part of the $CVC$). In the presence of this current domain (or a filament with a higher current density) the $CVC$ is modified so that an almost vertical part appears on the $I(V)$ curve ($R_{d}$ is close to zero). The electric field $E_{ver}$ corresponding to this part of the $CVC$ is again determined by the shape of $%
CVC$. All these statements concerning the S-type of $CVC$ are based on the study of the so called superheating mechanism of the S-shape $I(V)$ curve [@KoganVolkovZh] . In the case of a ”symmetric” S-shape $CVC$ with negative $G$ (i.e. min{$dV/dI$} corresponds to $I=0$) the field $E_{ver}$ is zero (when the bias current varies in some limits, the voltage across the sample is zero ). Similar conclusions were made in a recent paper on a simple phenomenological model [@Andreev].
Of course the behaviour of a system with the N- or S-shape $CVC$ in the absence or presence of a magnetic field $B$ is different. In the latter case the Hall field $E_{y}$ arises for example if the transverse current $%
I_{y}$ is zero. The form of the $CVC$ of a homogeneous 2DEG (in the absence of domains of the electric field or current) depends on whether the Hall current $I_{y}$ or the Hall field $E_{y}$ is zero. For instance, if the $%
I_{x}(V_{x})$ dependence corresponds to the N-shape $CVC$ in the absence of the Hall field $E_{y}$ $(I_{y}\neq 0)$, this correspondence changes completely in the absence of the Hall current $I_{y}$ ($E_{y}\neq 0$). It acquires a complicate form and can not be assigned to the N- or S-shape type of $CVC$ (several values of $I_{x}$ correspond to one $E_{x}$ and several values of $E_{x}$ correspond to one $I_{x}$). Therefore one has to analyse the type of instability and the form of the final state of the system separately for these two cases ($I_{y}=0$ or $E_{y}=0$).
To summarize, one can conclude that the absolute negative conductance obtained in Refs. [@Yale; @Shi; @Raikh] is not sufficient to explain the observed zero resistance (or differential resistance) states. In a system with negative conductance at low dc fields $E,$ the $CVC$ may have either the N- or S-shape form, and the behaviour of the system will be different in these both cases. I also would like draw attention to the fact that negative conductance was predicted long ago in Refs.[@Ryzhii; @Suris; @Elesin; @V'yurkov] (2DEG and 3DEG in a quantizing magnetic field in the presence of microwave irradiation) and in Refs.[@Epstein] (superlattices under ac irradiation).
I would like to thank Sh.M.Kogan as well as B.Huckestein, N.Rick and J.Shi for useful discussions and comments.
R.G.Mani, J.H. Smet, K. von Klitzing, V. Narayanmurti, W.B. Jonson, V. Umansky, Nature [**420**]{}, 646 (2002).
M.A. Zudov, R.R. Du, L.N. Pfeiffer, and K.W. West, Phys.Rev.Lett. [**90**]{}, 046807 (2003); M.A. Zudov, R.R. Du, J.A. Simmons, and L. Reno, Phys.Rev. [**B 64**]{}, R201311 (2001).
J.C.Phillips, cond-mat/0212416, 0303181; 0303184.
A.C. Durst, S. Sachdev, N. Read, and S.M. Girvin, cond-mat/0301569.
A.V.Andreev, I.L. Aleiner, and A.J.Millis, cond-mat/0302063.
P.W. Anderson and W.F. Brinkman, cond-mat/0302129.
K. Shrivastava, cond-mat/0302320.
J.Shi and X.C. Xie, cond-mat/0302393.
A.A.Koulakov and M.E. Raikh, cond-mat/0302465.
J.Shi and X.C. Xie, cond-mat/0303141.
S.A.Mikhailov, cond-mat/0303130.
A.F. Volkov and Sh.M. Kogan, Sov.Phys. Uspekhi [**11**]{}, 881 (1969).
A.F. Volkov and Sh.M. Kogan, Sov.Phys. JETP [**25**]{}, 1095 (1967).
V.I.Ryzhii, Sov.Phys.Solid State [**11**]{}, 2078 (1970).
V.I.Ryzhii, R.A.Suris and B.S.Shchamkhalova, Sov.Phys.Semicond. [**20**]{}, 1299 (1987).
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| {
"pile_set_name": "ArXiv"
} |
---
author:
- |
M. Vivekanand[^1]\
National Center for Radio Astrophysics, TIFR,\
Pune University Campus, P. O. Box 3,\
Ganeshkhind, Pune 411007, India.
title: |
The issue of aliasing in 0943\
II Signal processing arguments
---
[**Abstract**]{}: [@DR1999; @DR2001] claim that the frequency of the very narrow feature, in the spectrum of radio flux variations of 0943, is an alias of its actual value. They also claim to have detected an amplitude modulation on the above phase modulation. This paper argues that both these claims are unjustified. [**Keywords**]{}: pulsars: general – pulsars: individual (0943, 0031): stars – neutron — fluctuation spectrum — aliased feature — signal processing — drifting sub pulses.
Introduction
============
The rotation powered radio pulsar 0943 exhibits sub pulses that are drifting systematically from period to period within the observable pulse window. This pulsar has a very narrow feature in the longitude resolved spectrum of intensity fluctuations; its $Q$, defined as its central frequency divided by its width, is relatively high (@TH1971; @BRC1975; @SO1975). Recently @DR1999 ([-@DR1999], [-@DR2001]; henceforth DR1999 and DR2001) put its $Q$ at $\ge$ 500. It has been speculated that this very narrow spectral feature, occurring at 0.465 cycles per pulsar period \[cpp\], could be an alias of the actual value 0.535 \[cpp\] (@SO1975). [@DB1973] noted such a general possibility in radio pulsars and claimed, based on the phase information in the fluctuation spectrum, that PSR B2303+30 has an aliased spectral feature. This was contested by [@SO1975], who state on their page 326 that “even a phase analysis, contrary to Backer’s statements (@DB1973), is unable to decide between the two possibilities”, viz., whether the spectral feature is aliased or not. Indeed, in a later paper Backer did not repeat such a claim for 0943, whose fluctuation spectrum is similar to that of PSR B2303+30 (@BRC1975); he quoted the the true value as 0.465 \[cpp\]. However, DR1999 and DR2001 recently claimed that the spectral feature is indeed an alias; that the actual value is 0.535 \[cpp\]. In their view [@SO1975] “came to the wrong conclusion” (DR2001). They also claim that the weak, symmetrically spaced sidebands, at 0.027 \[cpp\] away from the above spectral feature “strongly suggest ... a regular, highly periodic amplitude modulation of the ... drifting sub pulse sequences”.
This paper argues that DR1999 and DR2001 are unjustified in drawing these two conclusions. The question, whether 0943 has an aliased spectral feature or not, is still unresolved; and the latter observation is as likely, if not more likely, due to an additional phase modulation of the drifting sub pulses.
A review of the relevant signal processing
==========================================
The signal from an ideal pulsar with drifting sub pulses falls under the topic “pulse position modulation” (PPM); it consists of periodically occurring narrow pulses whose positions are modulated by another periodicity. Its general principles can be found in books on electronic communication engineering (see @BPL1998). The spectrum of a PPM signal when the position modulation is due to a pure “tone” (a single frequency) is given by [@PP1965] on his page 541 and by @SBS1966 ([-@SBS1966]; henceforth SBS1966) on their pages 252 – 253.
Frequency domain discussion
---------------------------
To begin with let us assume that the drifting sub pulse pattern is PPM due to a pure tone.
The abscissa in fig. 6-2-3 on page 251 of SBS1966 represents the phase of the sampling signal at which a pulse is observed; in our case the sampling signal has the pulsar period $P$ (sec), or frequency $1 / P$ Hz or 1 \[cpp\]. The ordinate represents the corresponding phase of the modulating signal; in our case it has period $P3$ pulsar periods (frequency $1 / P3$ \[cpp\]) which is the repetition time of the drifting sub pulse pattern. The motion in time of the drifting sub pulses in this figure is along a straight line of slope $1 / P3$. In the last para on their page 254, SBS1966 state “the average pulse repetition frequency should be at least twice the highest signal frequency in order to obtain the minimum number of samples necessary for satisfactory signal recovery”; i.e., $1 > 2 / P3$ ($\Rightarrow 1 / P3 < 0.5$) to avoid aliasing. This can be verified by considering two straight lines of slopes $< 0.5$ and $> 0.5$ in fig. 6-2-3 of SBS1966.
Thus, the Nyquist sampling criterion is the same for a PPM signal and a canonical pulse amplitude modulation (PAM) signal (amplitude modulation of periodic pulses). DR2001 are wrong when they claim in their conclusion that “A harmonic resolved fluctuation spectrum uses the information within the finite width of the pulse to achieve a Nyquist frequency of 1 \[cpp\], showing clearly that the primary feature is aliased”. Their average pulse repetition frequency is obviously $P$ (sec); so their Nyquist frequency is only 0.5 \[cpp\]. Consequently, they are also wrong in concluding that “the primary feature is aliased” based merely upon the Fourier technique; they require [ **additional and independent information**]{}, as discussed ahead.
Consider a PAM signal in which the amplitude modulation is due to a pure tone of frequency $\nu$ \[cpp\]. The amplitudes occur at frequencies $\nu$ \[cpp\], $1 \pm \nu$ \[cpp\], $2 \pm \nu$ \[cpp\], etc; and there is no difference in the amplitude spectra of the original ($\nu < 0.5$) and the aliased ($\nu > 0.5$) signal. However the phase spectra differ. Therefore one can distinguish between the original and aliased PAM signals only by using additional information such as the phase of the modulating signal. In the PPM case, the amplitudes in the spectra occur at frequencies $\nu$ \[cpp\], $1 \pm
m \times \nu$ \[cpp\], $2 \pm m \times \nu$ \[cpp\], etc., where $m = 1, 2, 3, ...$ is the order of the harmonic (see eq. 6-2-13 on page 252 of SBS1966). Now, both amplitude and phase spectra differ for the original and aliased signals. Therefore once again, one can distinguish between the original and aliased PPM signals only by using additional information such as the exact shape of the two amplitude spectra. In practice it is impossible to predict this exactly in the current pulsar context.
![ Simulated amplitude spectra of a pulsar signal showing the drifting sub pulse phenomenon. Only a small range of frequency has been shown for better visual comparison. The time series consists of $4 \times 1024 \times 1024$ samples, each of duration 0.25 milli seconds (ms). The pulsar parameters are those of 0943, taken from DR2001: period $P = 1.0977$ (sec), Gaussian integrated profile of width $= 0.031 \times P$ (sec), maximum time departure of drifting sub pulses $= 0.021 \times P$ (sec). The sub pulses are assumed to be Gaussian in shape of very narrow intrinsic width, of $\approx$ one sampling interval; making the width a more realistic value merely suppresses the spectra at higher frequencies. The drifting sub pulse phenomenon is modeled as a PPM signal with a saw-tooth modulation in time. [**Top panel**]{}: Original modulating frequency $= 0.465$ \[cpp\], or $P3 = 2.1505376$ periods; [**Bottom panel**]{}: aliased modulating frequency $= 0.535$ \[cpp\], or $P3 = 1.8691589$ periods, and with opposite drift direction. []{data-label="fig1"}](vivek2-fig1.eps){width="12.5cm"}
Fig. \[fig1\] simulates the amplitude spectra of a pulsar signal showing the drifting sub pulse phenomenon, modeled as a PPM signal with a saw-tooth modulation in time, for both the original modulating frequency (top panel), and its alias but with the opposite drift direction (bottom panel). The pulsar parameters are taken from DR2001 for 0943; however the figure is insensitive to small variations in these parameters. The algorithm was checked by reproducing the spectrum in eq. 6-2-13 of SBS1966. The difference between the two amplitude spectra in fig. \[fig1\] is generally $\le$ 5% of the maximum amplitude of pulsar harmonics, upto a frequency of 100 \[cpp\], for the harmonics $m = 1$ to $3$. For example, at $59.93$ \[cpp\], the second harmonics differ by $\approx$ 12.5% in the two panels; but their average amplitude is $\le 33$% of the maximum amplitude. The third harmonics, aliased to $59.605$ \[cpp\], differ by $\approx$ 9%; but their average amplitude is only about 4.5% of the maximum amplitude. At $110.395$ \[cpp\], the harmonics differ by $\approx$ 20%; but their average amplitude is now about 37.5% of the peak amplitude of the pulsar harmonics; this is also insignificant.
Therefore at lower frequencies ($< 100$ \[cpp\]; the range analyzed by DR2001) the spectra of the original and the aliased signals should be predicted to an accuracy of $\le$ 5% to tell the difference. This is almost impossible, considering other factors ignored here: random variations in shape, size and intensity from pulse to pulse, lack of knowledge of the exact modulating function, the exact shape of the integrated profile, etc. Maybe there is some hope of making better prediction about the higher harmonics ($m > 3$) at higher frequencies ($> 100$ \[cpp\]), but DR2001 do not analyze these frequencies. The next section indicates why, if at all, the two spectra can probably be distinguished only at higher frequencies.
Time domain discussion
----------------------
Let $t_n$ be the time of occurrence of the $n^{\mathrm{th}}$ pulse in a PPM signal that is modulated by a pure tone, of phase $\phi_m$, and frequency $1 / P3$ \[cpp\], implying angular frequency $\omega_m = 2 \pi / (P3 P)$, $$t_n + \tau \sin \left ( \omega_m t_n + \phi_m \right ) = n P,$$ where $\tau$ is the maximum position departure (see eq. 6-2-2 of SBS1966). Let $\Delta
t_n$ be the difference between $t_n$ and $n P$, its time of occurrence in the un modulated case, $$\Delta t_n = - \tau \sin \left ( \Phi_n + \omega_m \Delta t_n \right ),$$ where $\Phi_n = 2 \pi n / P3 + \phi_m$. In 0943, $\tau \approx 0.021 P$, and the original $P3 \approx 2.15$, so the phase $\omega_m \tau = 2 \pi \tau / (P3 P) \approx
0.06$, which is a small quantity. Furthermore, $\Delta t_n \le \tau$; therefore, $$\Delta t_n \approx - \tau \sin \left ( \Phi_n \right ) \left [ 1 - \tau \left \{
0.5 \omega_c - \Delta \omega_m \right \} \cos \left ( \Phi_n \right ) \right ]$$ where $\omega_c = 2 \pi / P$ is the sampling frequency (angular) and $\Delta \omega_m
= 0.5 \omega_c - \omega_m$. In 0943, $\Delta \omega_m$ ($= 2.862 - 2.662 = 0.2$) is much smaller than either $0.5 \omega_c$ or $\omega_m$.
Now consider a PPM signal that is modulated by a pure tone, of phase $\phi^\prime_m =
-\phi_m$, and a frequency that is the alias of the earlier one; i.e., of angular frequency $\omega^\prime_m = \omega_c - \omega_m$. Let it have the same magnitude of drift but opposite in direction, i.e., $\tau^\prime = - \tau$; Then it can be shown that $$\Delta t^\prime_n \approx - \tau \sin \left ( \Phi_n \right ) \left [ 1 + \tau \left \{
0.5 \omega_c + \Delta \omega_m \right \} \cos \left ( \Phi_n \right ) \right ],$$ for integer $n$. For non-integer $n$, which implies that the pulses start at some arbitrary phase within the period, a constant phase adds to $\Phi_n$ in eq. (4).
Now, the observed time series corresponds to either eq. (3) or eq. (4). If one knows its absolute phase, then one directly compares the observation to those two equations, and thus determines if the observed frequency is original or aliased. However, this is rarely the case. So, one has to distinguish between the the time series $\pm 0.25
\omega_c \tau^2 \sin \left ( 2 \Phi_n \right )$ of eq. (3) and eq. (4). This is possible only if the sampling interval used for the observations is much smaller than the quantity $0.5 \omega_c \times \tau^2$, which is $\approx 1.52$ ms in 0943. However the sampling interval of DR2001 (1.006 ms) is barely smaller than this. Therefore it is probably impossible for DR1999 and DR2001 to distinguish between the original drift frequency and its alias with opposite drift direction. This argument also holds for saw-tooth modulation drifting pattern, which can be Fourier decomposed into harmonics of the fundamental modulating frequency $\omega_m$.
Now one can understand the result of the previous section - a time resolution $<<
1.52$ ms implies a minimum Nyquist frequency of $ >> 0.5 / 0.00152 \approx 329$ Hz or 361 \[cpp\]. Indeed, the spectra in fig. \[fig1\] differ most in the range 500 to 900 \[cpp\], mainly in the higher harmonics $m >> 3$. For example, the 15$^{\mathsf{th}}$ harmonics (aliased) at frequency $679.975$ \[cpp\] differ in amplitude by $\approx$ 107%; and their average amplitude is about 24% of the maximum amplitude of pulsar harmonics.
In summary, DR2001 can hope to distinguish between the original and aliased spectral feature in 0943 by predicting the two amplitude spectra at frequencies much larger than 100 \[cpp\], particularly for the harmonics $m \ge 3$ to $15$, and comparing those with observations. This they have not done.
Aliasing claim by DR2001
========================
This section discusses the three main arguments that DR2001 appear to offer (their sections 3 and 5) regarding the spectral feature in 0943.
Amplitude spectrum argument
---------------------------
In the longitude resolved spectrum in fig. 1 of DR2001, the narrow spectral feature falls at frequency 0.465 \[cpp\]. The spectrum of such a signal will naturally have a Nyquist sampling limit of 0.5 \[cpp\], by definition. To test whether this feature is aliased or not, one needs data with much faster sampling. DR2001 achieve this by obtaining the spectrum of the original time series (including zero padding) which is sampled at $1.006$ (ms). The corresponding spectrum certainly shows that the spectral feature is much stronger at 0.535 \[cpp\] than at 0.465 \[cpp\] (their fig. 4).
Therefore they argue (on the right side of page 442) that “... the asymmetry between the 0.535 and 0.465 \[cpp\] features in fig. 4 is so great that we can regard them as a signature of an almost pure phase modulation”. This is justified, as seen in fig. \[fig1\] above. Then, after a short paragraph offering no new argument or evidence, they state that “We can now be certain that the principal feature in fig. 1 is the alias of a fluctuation the actual frequency which is greater than 0.5 \[cpp\] ...”. This is unjustified.
![ [**Top panel**]{}: Amplitude spectrum of 1024 periods of radio flux data of 0031, obtained on 1996 March 31 using the ORT. The data was acquired in a gated fashion and later zero padded., as in DR2001. The sampling interval was 6.5 ms and the pulsar has period 0.9430 (sec). Only a small range of frequency has been shown for better visual comparison. [**Bottom panel**]{}: The same as in the top panel, except that the time sequence within each period is reversed, while the period sequence itself is retained. []{data-label="fig2"}](vivek2-fig2.eps){width="14.0cm"}
It is purely fortuitous that the aliased feature is stronger than the original in their fig. 4. It depends upon the specific combination of [**true**]{} $P3$, [**true**]{} drift direction and the [**exact**]{} shape of the IP of 0943. Fig. \[fig2\] verifies the above argument. It shows the amplitude spectra of 0031, another drifting pulsar, observed using Ooty Radio Telescope (ORT). The top panel corresponds to fig. 3 of DR2001. In the bottom panel of fig. \[fig2\] the data is altered: the period sequence is retained, but within each period the time sequence is reversed. This could have occurred either due to (1) 0031 having the aliased drift frequency but the same drift direction, or (2) it having the original drift frequency but opposite drift direction. The asymmetry has also got reversed in the bottom panel of fig. \[fig2\].
Phase spectrum argument
-----------------------
DR2001 do not use the phase information corresponding to their fig. 4, but use that available in the longitude resolved spectrum (their fig. 1). They compare the rate of change of phase, with longitude, of the 0.465 \[cpp\] spectral feature, with that of two other much weaker spectral features occurring at 0.071 \[cpp\] and 0.607 \[cpp\], which are supposed to be the aliased first and second harmonics of the fundamental. They claim on page 443 that “the harmonicity of these phase rates argues strongly that the 0.071 \[cpp\] and 0.607 \[cpp\] features are indeed the aliases of the second and third harmonics of the primary feature ...”, which is justified.
However, this will be true irrespective of whether the original frequency is 0.465 \[cpp\] or 0.535 \[cpp\]; this is obvious from Fourier theory. The only difference is the inverted phase relation between the fundamental and its (aliased) harmonics, about which one has no independent information anyway. This author is confused about what DR2001 intended in invoking this discussion. This author is most confused about a sentence in the last para of their section 3, the section that is supposed to justify their claim of aliasing. It goes “The ambiguity, however, between the remaining two possible combinations – namely, $P3 < 2 P$ (thus implying negative drift) and $P3 > 2 P$ (positive drift) can not be fully resolved through the above analysis, ...”. Presumably this is what [@SO1975] implied, with which DR2001 apparently disagreed earlier. It appears that either DR2001 are contradicting themselves, or this author has not understood what was it in 0943 that was “otherwise multiply folded” which DR2001 have managed to “unfold” (last para of their section 3).
“Modulation on modulation” argument
-----------------------------------
![ Same as in fig. \[fig1\], except (1) $1024 \times 1024$ samples used, (2) sampling interval of 1.0 ms, (3) modulating $P3 = 3$ periods. Only a small range of frequency has been shown for better visual comparison. [**Top panel**]{}: The PPM signal is additionally amplitude modulated by a tone of period 20 \[P\], and relative amplitude 0.5. [**Bottom panel**]{}: The PPM signal is additionally position modulated by a tone of period 20 \[P\], and maximum time departure of 0.010 \[P\]. []{data-label="fig3"}](vivek2-fig3.eps){width="14.0cm"}
DR1999 claim on their page 1010 that the symmetrical sidebands associated with the primary feature in the fluctuation spectrum represent “an amplitude modulation on the phase modulation”. These sidebands fall 0.027 \[cpp\] higher and lower than the primary feature. Since $0.535 / 0.027 \approx 20$, an integer value, DR2001 claim in their conclusion that “it is at this point that we have conclusive evidence that the aliasing question is resolved ...”.
Fig. \[fig3\] is similar to fig. \[fig1\] except that (1) 4 times fewer samples and 4 times larger sampling interval, and (2) the drifting $P3 = 3$ periods, for better visibility. However, in the top panel the pulsar signal is additionally amplitude modulated by a sinusoid of frequency 0.05 \[cpp\]; of relative magnitude of modulation of 50%, for the sidebands on the position modulating harmonics to show up significantly. The pulsar harmonics also have the symmetric sidebands, but of much larger amplitude. In the bottom panel of fig. \[fig3\] the above signal is instead additionally position modulated, by a sinusoid of the the same frequency, and a maximum time departure of 0.010 \[P\]. Now also the pulsar harmonics have the symmetric sidebands, but much smaller in amplitude.
From fig. \[fig3\] shows that both an additional amplitude modulation or an additional position modulation of the pulsar signal in fig. \[fig1\] gives rise to sidebands around the drifting harmonics. However, in the former case the side bands are much stronger around the pulsar harmonics (at 1 \[cpp\], 2 \[cpp\], etc.), and in the latter case they are much stronger around the drifting harmonics, which is what DR2001 observe. Therefore it is as likely, if not more likely, that DR2001 have noticed an additional position modulation over and above their drifting sub pulse pattern. DR2001 do not reason why they prefer the former over the latter.
The foundation of DR2001 vanishes if they are unable to justify the amplitude modulation of the drifting sub pulses.
Summary
=======
DR2001’s claim of aliasing in 0943 may rest purely upon an arbitrary assumption, that of a very steady “amplitude modulation on the phase modulation”. Recent claims [@ES2002] to have independently verified the results of DR2001 are, in the opinion of this author, not relevant here because (1) they are model dependent, and (2) they work with the phase resolved spectrum, which is inadequate by definition to discuss the issue of aliasing.
This research has made use of NASA’s Astrophysics Data System (ADS) Bibliographic Services.
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[^1]: vivek@ncra.tifr.res.in
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
The conjecture of Bollobás and Komlós, recently proved by Böttcher, Schacht, and Taraz \[Math. Ann. 343(1), 175–205, 2009\], implies that for any $\gamma>0$, every balanced bipartite graph on $2n$ vertices with bounded degree and sublinear bandwidth appears as a subgraph of any $2n$-vertex graph $G$ with minimum degree $(1+\gamma)n$, provided that $n$ is sufficiently large. We show that this threshold can be cut in half to an essentially best-possible minimum degree of $(\frac12+\gamma)n$ when we have the additional structural information of the host graph $G$ being balanced bipartite.
This complements results of Zhao \[to appear in SIAM J. Discrete Math.\], as well as Hladký and Schacht \[to appear in SIAM J. Discrete Math.\], who determined a corresponding minimum degree threshold for $K_{r,s}$-factors, with $r$ and $s$ fixed. Moreover, it implies that the set of Hamilton cycles of $G$ is a generating system for its cycle space.
[*Keywords:*]{} Graph theory (05Cxx), Extremal combinatorics (05Dxx), Graph embedding
address:
- 'Zentrum Mathematik, Technische Universität München, Boltzmannstraße 3, D-85747 Garching bei München, Germany'
- 'Zentrum Mathematik, Technische Universität München, Boltzmannstraße 3, D-85747 Garching bei München, Germany'
- 'Zentrum Mathematik, Technische Universität München, Boltzmannstraße 3, D-85747 Garching bei München, Germany'
author:
- Julia Böttcher
- Peter Heinig
- Anusch Taraz
bibliography:
- 'bipartite.bib'
title: Embedding into bipartite graphs
---
Introduction
============
The Bollobás–Komlós conjecture, recently proved in [@BST09], provides a sufficient and essentially best possible minimum degree condition for the containment of $r$-chromatic spanning graphs $H$ of bounded maximum degree and small bandwidth. Here, a graph is said to have bandwidth at most $b$, if there exists a labelling of the vertices by numbers $1,\dots,n$, such that for every edge $\{i,j\}$ of the graph we have $|i-j| \le b$.
\[thm:bk\] For all $r,\Delta\in\mathbb{N}$ and $\gamma>0$, there exist constants $\beta>0$ and $n_0\in\mathbb{N}$ such that for every $n\geq n_0$ the following holds. If $H$ is an $r$-chromatic graph on $n$ vertices with $\Delta(H) \leq \Delta$ and bandwidth at most $\beta n$ and if $G$ is a graph on $n$ vertices with minimum degree $\delta(G) \geq (\frac{r-1}{r}+\gamma)n$, then $G$ contains a copy of $H$.
This theorem in particular implies that for any $\gamma>0$, every [bipartite]{} graph $H$ on $2n$ vertices with bounded degree and sublinear bandwidth appears as a subgraph of any $2n$-vertex graph $G$ with minimum degree $(1+\gamma)n$, provided that $n$ is sufficiently large. This bound is essentially best possible for an almost trivial reason: there are graphs $G$ with minimum degree just slightly below $n$ that are not connected. Such $G$ clearly do not contain a connected $H$ as a subgraph. These graphs are simply too different in structure from $H$.
One may ask, however, whether it is possible to lower the minimum degree threshold in Theorem \[thm:bk\] for graphs $G$ and $H$ that are structurally more similar and, in particular, have the same chromatic number. In this paper we will pursue this question for the case of balanced bipartite graphs, i.e., bipartite graphs on $2n$ vertices with $n$ vertices in each colour class.
Dirac’s theorem [@Dir] implies that a $2n$-vertex graph $G$ with minimum degree at least $n$ contains a Hamilton cycle. If $G$ is balanced bipartite, it follows from a theorem of Moon and Moser [@MooMos] that this minimum degree threshold can be cut almost in half.
\[thm:moon\] Let $G$ be a balanced bipartite graph on $2n$ vertices. If $\delta(G)\geq
\frac{n}{2}+1$, then $G$ contains a Hamilton cycle.
We prove that slightly increasing this minimum degree bound suffices to obtain all balanced bipartite graphs with bounded maximum degree and sublinear bandwidth as subgraphs, and thereby establishing the following bipartite analogue of Theorem \[thm:bk\], halving the minimum degree threshold in that result.
\[thm:bipbw\] For all $\gamma$ and $\Delta$ there is a positive constant $\beta$ and an integer $n_0$ such that for all $n\ge n_0$ the following holds. Let $G$ and $H$ be balanced bipartite graphs on $2n$ vertices such that $G$ has minimum degree $\delta(G)\ge(\frac12+\gamma)n$ and $H$ has maximum degree $\Delta$ and bandwidth at most $\beta n$. Then $G$ contains a copy of $H$.
Results of a similar nature have recently been established by Zhao [@Zhao_bip], and by Hladký and Schacht [@HlaSch] who considered the special case of coverings of $G$ with disjoint copies of complete bipartite graphs. Moreover, as a first step towards Theorem \[thm:bipbw\], in [@heinigbsc] this result was proved for a special balanced bipartite connected graph (the so-called Möbius ladder).
We remark that the bandwidth condition in Theorem \[thm:bipbw\] cannot be omitted. Indeed, Abbasi [@Abbasi] proved that the assertion of Theorem \[thm:bk\] gets false if $\beta>4\gamma$. The graph $H$ he constructs for this purpose is a balanced bipartite graph and it is not difficult to see that Abbasi’s host graph contains a bipartite subgraph meeting our conditions but not containing $H$. However, the bound on $\beta$ coming from our proof is very small, having a tower-type dependence on $1/\gamma$.
The proof of Theorem \[thm:bipbw\] is given in Section \[sec:proof\]. It is based on Szemerédi’s regularity lemma which we introduce in the following section. In Sections \[sec:G\] and \[sec:H\] we provide the proofs of the remaining lemmas that are used in the proof of Theorem \[thm:bipbw\].
The regularity method {#sec:reg}
=====================
In this section we formulate a version of Szemerédi’s regularity lemma [@Szemeredi76] that is convenient for our application (Lemma \[lem:reg\]), introduce all necessary definitions, and formulate an embedding lemma for spanning subgraphs (Lemma \[lem:gel\]).
The regularity lemma relies on the concept of a regular pair. To define this, let $G=(V,E)$ be a graph and $0\le \eps,d\le 1$. For disjoint nonempty vertex sets $U,W\subset V$ the *density* $d(U,W)$ of the pair $(U,W)$ is the number of edges that run between $U$ and $W$ divided by $|U||W|$. A pair $(U,W)$ with density at least $d$ is *$(\eps,d)$-regular* if $|d(U',W')-d(U,W)|\le\eps$ for all $U'\subset U$ and $W'\subset W$ with $|U'|\ge\eps|U|$ and $|W'|\ge\eps|W|$. The following useful property of regular pairs follows immediately from the definition.
\[lem:typical\] Let $G=(A,B)$ be an $(\eps, d)$-regular pair. Let $B'$ be a subset of $B$ with $|B'|\ge\eps|B|$. Then there are at most $\eps|A|$ vertices in $A$ with less than $(d-\eps)|B'|$ neighbours in $B'$.
The regularity lemma asserts that each graph admits a partition into relatively few vertex classes of equal size such that most pairs of these classes form an $\eps$-regular pair. The following definition makes this precise. A partition $V_0\dcup V_1 \dcup\dotsm\dcup V_k$ of $V$ with $|V_0|\le\eps|V|$ is *$(\eps,d)$-regular on* a graph $R=([k],E_R)$ if $ij \in E_R$ implies that $(V_i,V_j)$ is an $(\eps,d)$-regular pair in $G$. If such a partition exists, we also say that $R$ is an *$(\eps,d)$-reduced graph* of $G$. Moreover, $R$ is the *maximal* $(\eps,d)$-reduced graph of the partition $V_0\dcup V_1 \dcup\dotsm\dcup V_k$ if there is no $ij\not\in E_R$ with $i,j\in[k]$ such that $(V_i,V_j)$ is $(\eps,d)$-regular. A partition $V_0\dcup V_1 \dcup\dotsm\dcup V_k$ of $V$ is an *equipartition* if $|V_i|=|V_j|$ for all $i,j\in[k]$. The partition classes $V_i$ with $i\in[k]$ are also called *clusters* of $G$ and $V_0$ is the *exceptional set*. When the exceptional set $V_0$ is empty (or when we want to ignore it as well as its size) then we may omit it and say that $V_1 \dcup\dots\dcup V_k$ is regular on $R$. An $(\eps,d)$-regular pair $(U,W)$ is *$(\eps,d)$-super-regular* if every vertex $u\in U$ has degree $\deg_W(u)\ge
d|W|$ and every $w\in W$ has $\deg_U(w)\ge d|U|$. For a graph $G=(V,E)$ a partition $V=V_0\dcup V_1\dcup\dotsm\dcup V_k$ is said to be *super-regular on a graph $R$* with vertex set $V_R$, $V_R\subset[k]$, if $(V_i,V_j)$ is super-regular whenever $ij$ is an edge of $R$.
In this paper we consider bipartite graphs and the regular partitions that appear in the proof of Theorem \[thm:bipbw\] refine some bipartition and their reduced graphs are bipartite. More precisely, for a bipartite graph $G=(A\dcup
B,E)$ we will obtain a partition $(A_0 \dcup B_0)\dcup A_1 \dcup
B_1 \dcup \dots \dcup A_k \dcup B_k$ that is $(\eps,d)$-regular (or super-regular) on some bipartite graph $R$ such that $A=A_0 \dcup\dots\dcup
A_k$ and $B=B_0\dcup\dots\dcup B_k$. In particular we have two different exceptional sets now, one in $A$ called $A_0$ and one in $B$ called $B_0$, each of size $\eps n$ at most. Such a partition is an equipartition if $|A_1|=|B_1|=|A_2|=\dots=|A_k|=|B_k|$. In addition, we consider only regular pairs running between the bipartition classes, i.e., pairs of the form $(A_i,B_j)$. Consequently, all reduced graphs (also the maximal reduced graph of a partition) are bipartite.
We now state the version of the regularity lemma that we will use. This is a corollary of the degree form of the regularity lemma (see, e.g., [@KS96 Theorem 1.10]) and is tailored for embedding applications in balanced bipartite graphs satisfying some minimum degree condition. We sketch its proof below.
\[lem:reg\] For every $\eps'>0$ and for every $\Delta,k_0\in {\ensuremath{\mathbb{N}}}$ there exists $K_0 = K_0(\eps',k_0) \in {\ensuremath{\mathbb{N}}}$ such that for every $0\leq d' \leq 1$, for $$\eps'':= \frac{2\Delta\eps'}{1-\eps'\Delta}
\quad \text{ and } \quad
d'':= d'-2\eps'\Delta\,,$$ and for every bipartite graph $G = (A\dcup B, E)$ with $|A|=|B|\geq K_0$ and $\delta(G)\ge \nu |G|$ for some $0<\nu<1$ there exists a graph $R$ and an integer $k$ with $k_0\le k \le K_0$ with the following properties:
1. \[lem:reg:a\] $R$ is an $(\eps',d')$-reduced graph of an equipartition of $G$ and $|V(R)|=2k$.
2. \[lem:reg:b\] $\delta(R) \ge (\nu-d'-\eps'')|R|$.
3. \[lem:reg:c\] For every subgraph $R^*\subseteq R$ with $\Delta(R^*)\le \Delta$ there is an equipartition $$A \dcup B = A''_0 \dcup B''_0 \dcup A''_1 \dcup B''_1 \dcup \dots \dcup
A''_k \dcup B''_k$$ with $A''_i \subseteq A$ and $B''_i \subseteq B$ for all $0\le i \le k$ and $(\eps'',d'')$-reduced graph $R$, which in addition is $(\eps'',d'')$-super-regular on $R^*$.
The proof of this lemma is a standard combination of three standard tools. As a first step we simulate the proof of the degree-form (see [@KSS98], Lemma 2.1, or the survey [@KS96]) of the regularity lemma starting with $A\dcup B$ as the initial partition (see also [@Diestel Chapter 7.4]). This yields a partition into clusters $A_0,\dots,B_k$ such that for all vertices $v\not\in A_0\cup B_0$ there are at most $(d'+\eps')n$ edges $e\in E$ with $v\in e$ such that $e$ is not in some $(\eps',d')$-regular pair $(A_i,B_j)$. Hence we get \[lem:reg:a\]. Let $R$ be the maximal (bipartite) $(\eps',d')$-reduced graph of this partition. Then it is easy to see that $R$ inherits the minimum degree condition of $G$ (except for a small loss), see [@KueOstTar Proposition 9]. This yields \[lem:reg:b\]. Finally, for all pairs $(A_i,B_j)$ with $i,j\in[k]$ that correspond to edges in $R^*$ we take those vertices in $A_i$ or $B_i$ that have too few edges in $(A_i,B_j)$ and move them to $A_0$ or $B_0$, respectively. See [@KueOstTar Proposition 8] for details. This yields \[lem:reg:c\].
Embedding into regular partitions
---------------------------------
For embedding *spanning subgraphs* $H$ into graphs $G$ with high minimum degree the blow-up lemma of Komlós, Sárközy and Szemerédi [@KSS_bl] has proved to be an extremely valuable tool. The blow-up lemma guarantees that bipartite spanning graphs of bounded degree can be embedded into sufficiently super-regular pairs. In fact, this lemma is more general and allows the embedding of graphs $H$ into partitions that are super-regular on some graph $R$ if there is a homomorphism from $H$ to $R$ that does not send too many vertices of $H$ to each cluster of $R$.
When embedding a spanning graph $H$ into a host graph $G$ a well-established strategy is to utilise the blow-up lemma on small super-regular “spots” in a regular partition of $G$ for embedding most of the vertices of $H$, and to use a greedy embedding method to embed the few other vertices first. This embedding method is summarised in the next lemma, the general embedding lemma. Before stating it we need to identify conditions under which it is possible to proceed in the way just described. This is addressed in the following definition that specifies when a partition of $H$ is “compatible” with a regular partition of $G$ with reduced graph $R$ and a subgraph $R'$ of $R$ such that edges of $R'$ correspond to dense super-regular pairs. In this definition we require that the partition of $H$ has smaller partition classes than the partition of $G$ (condition \[def:comp:0\]), and that edges of $H$ run only between partition classes that correspond to a dense regular pair in $G$ (condition \[def:comp:1\]). Further, in each partition class $W_i$ of $H$ we identify two subsets $S_i$ and $T_i$ that are both supposed to be small (condition \[def:comp:2\]). The set $S_i$ contains those vertices that send edges over pairs that do not belong to the super-regular pairs specified by $R'$ and $T_i$ contains neighbours of such vertices.
\[def:comp\] Let $H=(W,E_H)$ and $R=([k],E_R)$ be graphs and let $R'=([k],E_{R'})$ be a subgraph of $R$. We say that a vertex partition $W=(W_i)_{i\in[k]}$ of $H$ is *$\eps$-compatible* with an integer partition $(n_i)_{i\in[k]}$ of $n$ and with $R'\subset R$ if the following holds. For $i\in[k]$ let $S_i$ be the set of vertices in $W_i$ with neighbours in some $W_j$ with $ij\not\in E_{R'}$ and $i\neq j$, set $S:=\bigcup S_i$ and $T_i:=N_H(S)\cap (W_i\setminus S)$. Then for all $i,j\in[k]$ we have that
1. \[def:comp:0\] $|W_i|\le n_i$,
2. \[def:comp:1\] $xy\in E_H$ for $x\in W_i$ and $y\in W_j$ implies $ij\in E_R$,
3. \[def:comp:2\] $|S_i|\le\eps n_i$ and $|T_i|\le\eps\cdot\min\{n_j:\text{$i$ and $j$ are in the same
component of $R'$}\}$.
The partition $W=(W_i)_{i\in[k]}$ of $H$ is $\eps$-compatible with a partition $V=(V_i)_{i\in[k]}$ of a graph $G$ and with $R'\subset R$ if $W=(W_i)_{i\in[k]}$ is $\eps$-compatible with $(|V_i|)_{i\in[k]}$ and with $R'\subset R$.
The general embedding lemma asserts that a bounded-degree graph $H$ can be embedded into a graph $G$ if $H$ and $G$ have compatible partitions. A proof can be found in [@JuliasDiss Section 3.3.3].
\[lem:gel\] For all $d,\Delta,r>0$ there is a constant $\eps=\eps(d,\Delta,r)>0$ such that the following holds. Let $G=(V,E)$ be an $n$-vertex graph that has a partition $V=(V_i)_{i\in[k]}$ with $(\eps,d)$-reduced graph $R$ on $[k]$ which is $(\eps,d)$-super-regular on a graph $R'\subset R$ with connected components having at most $r$ vertices each. Further, let $H=(W,E_H)$ be an $n$-vertex graph with maximum degree $\Delta(H)\le\Delta$ that has a vertex partition $W=(W_i)_{i\in[k]}$ which is $\eps$-compatible with $V=(V_i)_{i\in[k]}$ and $R'\subset R$. Then $H\subset G$.
For applying the general embedding lemma to *spanning* graphs $H$ we need a partition of the graph $H$ whose partition classes match the sizes of a regular partition of $G$ *precisely*. However, usually we cannot guarantee that this is the case for a regular partition obtained from Lemma \[lem:reg\]. Hence it will become necessary to modify such a regular partition slightly by moving some vertices into different clusters. The following lemma asserts that the resulting partition is still regular with somewhat worse parameters. For a proof see [@BoeSchTar_JCTB Proposition 8].
\[prop:moving-vertices\] Let $(A,B)$ be an $(\eps, d)$-regular pair and let $\sh{A}$ and $\sh{B}$ be vertex sets with $|\sh{A}\triangle A|\le\alpha|\sh{A}|$ and $|\sh{B}\triangle B|\le\beta|\sh{B}|$. Then $(\sh{A},\sh{B})$ is an $(\sh{\eps},\sh{d})$-regular pair where $$\sh{\eps} := \eps + 3(\sqrt{\alpha} + \sqrt{\beta})
\qquad \text{and} \qquad \sh{d} := d - 2(\alpha + \beta).$$ If, moreover, $(A,B)$ is $(\eps, d)$-super-regular and each vertex $v$ in $\sh{A}$ has at least $d |\sh{B}|$ neighbours in $\sh{B}$ and each vertex $v$ in $\sh{B}$ has at least $d |\sh{A}|$ neighbours in $\sh{A}$, then $(\sh{A},\sh{B})$ is $(\sh{\eps},
\sh{d})$-super-regular with $\sh{\eps}$ and $\sh{d}$ as above.
The proof of the main theorem {#sec:proof}
=============================
In the proof of Theorem \[thm:bipbw\] we will use the general embedding lemma (Lemma \[lem:gel\]). For applying this lemma we need compatible partitions of the graphs $G$ and $H$ which are provided by the next two lemmas. We start with the lemma for $G$ which constructs a regular partition of $G$ whose reduced graph $R$ contains a perfect matching within a Hamilton cycle of $R$. The lemma guarantees, moreover, that the regular partition is super-regular on this perfect matching (see Figure \[fig:G\]) and that the cluster sizes in the partition can be slightly changed.
We remark that, throughout, $A\dcup B$ will denote the vertex set of the host graph $G$ while $X\dcup Y$ is the vertex set of the bipartite graph $H$ we would like to embed. The sets $A_i$ and $B_i$ with $i\in[k]$ for some integer $k$ will denote the clusters of a regular partition of $G$ as well as for the vertices of a corresponding reduced graph.
\[lem:G\] For every $\gamma > 0$ there exists $d_{\subsc{lg}}>0$ such that for every $\eps>0$ and every $k_0\in {\ensuremath{\mathbb{N}}}$ there exist $K_0\in {\ensuremath{\mathbb{N}}}$ and $\xi_{\subsc{lg}} > 0$ with the following properties: For every $n\geq K_0$ and for every balanced bipartite graph $G = (A \dcup B,
E)$ on $2n$ vertices with $\delta(G)\geq \bigl (1/2 + \gamma \bigr) n$ there exists $k_0\leq k\leq K_0$ and a partition $(n_i)_{i\in [k]}$ of $n$ with $n_i\ge n/(2k)$ such that for every partition ${(a_i)_{i\in[k]}}$ of $n$ and ${(b_i)_{i\in[k]}}$ of $n$ satisfying $a_i \le n_i + \xi_{\subsc{lg}} n$ and $b_i \le n_i +
\xi_{\subsc{lg}} n$, for all $i\in [k]$, there exist partitions $$A= A_1 \dcup \dotsm \dcup A_k
\quad \text{and} \quad
B= B_1 \dcup \dotsm \dcup B_k$$ such that
1. \[lem:G:1\] $|A_i|=a_i$ and $|B_i|=b_i$ for all $i\in [k]$,
2. \[lem:G:2\] $(A_i, B_i)$ is $(\eps, d_{\subsc{lg}})$-super-regular for every $i\in [k]$.
3. \[lem:G:3\] $(A_i, B_{i+1})$ is $(\eps, d_{\subsc{lg}})$-regular for every $i\in [k]$.
![The regular partition constructed by Lemma \[lem:G\] with super-regular pairs $(A_i, B_i)$ and regular pairs $(A_i, B_{i+1})$.[]{data-label="fig:G"}](BipAB)
The proof of this lemma is presented in Section \[sec:G\]. The following lemma, which we will prove in Section \[sec:H\], constructs the corresponding partition of $H$. It guarantees that the $2k$ partition classes of $H$ are roughly of the same sizes as the corresponding partition classes of $G$ (see \[lem:H:3\]), and that all edges of $H$ are mapped to edges of a cycle $C$ on $2k$ vertices and all edges except those incident to a very small set $S$ (see \[lem:H:1\]) are in fact mapped to the edges of a perfect matching in $C$ (see \[lem:H:2\]).
\[lem:H\] For every $k\in {\ensuremath{\mathbb{N}}}$ and every $\xi > 0$ there exists $\beta > 0$ and $n_0\in {\ensuremath{\mathbb{N}}}$ such that for every $n\geq n_0$ and for every balanced bipartite graph $H = (X\dcup Y, F)$ on $2n$ vertices having ${\operatorname{bw}}(H)\leq \beta n$ and for every integer partition $n=n_1+\dotsm + n_k$ with $n_i \leq n/8$ there exists a set $S\subseteq V(H)$ and a graph homomorphism $f\colon V(H) \rightarrow V(C)$, where $C$ is the cycle on the vertices $A_1,B_2,A_2,\dotsc, B_k, A_k, B_1, A_1$, such that
1. \[lem:H:1\] $|S|\leq \xi \cdot 2k \cdot n$,
2. \[lem:H:2\] for every $\{x,y\}\in F$ with $x\in X\backslash S$ and $y\in Y\backslash S$ there is $i\in [k]$ such that $f(x)\in A_i$ and $f(y)\in B_i$,
3. \[lem:H:3\] $|f^{-1}(A_i)| < n_i + \xi n$ and $|f^{-1}(B_i)| < n_i +
\xi n$ for every $i\in [k]$.
With Lemmas \[lem:gel\] (the general embedding lemma), Lemma \[lem:G\] (the lemma for $G$) and Lemma \[lem:H\] (the lemma for $H$) at our disposal, we are ready to give the proof of the main theorem.
Given $\gamma$ and $\Delta$, let $d$ be the constant provided by Lemma \[lem:G\] for input $\gamma$. Let $\eps$ be the constant Lemma \[lem:gel\] returns for input $d$, $\Delta$, and $r=2$. We continue the application of Lemma \[lem:G\] with input $\eps$ and $k_0:=2$ and get constants $K_0$ and $\xi_\subsc{lg}$ and set $\xi_\subsc{lh}:=\xi_\subsc{lg}\eps/(100\Delta K_0^2)$. Further let $\beta$ be the minimum of all the values $\beta_k$ and $n'_0$ be the maximum of all the values $n_0^{(k)}$ that Lemma \[lem:H\] returns for input $k$ and $\xi$ where $k$ runs from $k_0$ to $K_0$. Finally, we set $n_0:=\max\{n'_0,K_0\}$.
Let $G=(A\dcup B,E)$ and $H=(X\dcup Y,F)$ be balanced bipartite graphs on $2n$ vertices with $n\ge n_0$, $\delta(G)\ge(\frac12+\gamma)n$, $\Delta(H)\le\Delta$, and ${\operatorname{bw}}(H)\le\beta n$. We apply Lemma \[lem:G\] to the graph $G$ in order to obtain an integer $k$ and an integer partition $(n_i)_{i\in[k]}$ with $n_i\ge \frac12 n/k$ for all $i\in[k]$. Next, we apply Lemma \[lem:H\] to the graph $H$ and the integer partition $(n_i)_{i\in[k]}$ and get a vertex set $S\subset X\cup Y$ and a homomorphism $f$ from $H$ to the cycle $C$ on vertices $A_1,B_2,A_2,\dots B_k,A_k,B_1,A_1$ such that \[lem:H:1\]–\[lem:H:3\] are satisfied. With this we can define the integer partitions $(a_i)_{i\in[k]}$ and $(b_i)_{i\in[k]}$ required for the continuation of Lemma \[lem:G\]: set $a_i:=|f^{-1}(A_i)|$ and $b_i:=|f^{-1}(B_i)|$ for all $i\in[k]$. By \[lem:H:3\] we have $a_i\le
n_i+\xi_\subsc{lh}n\le n_i+\xi_\subsc{lg}n$ and $b_i\le n_i+\xi_\subsc{lg}n$ for all $i\in[k]$. It follows that Lemma \[lem:G\] now gives us vertex partitions $A=(A_i)_{i\in[k]}$ and $B=(B_i)_{i\in[k]}$ for $G$ such that \[lem:G:1\]–\[lem:G:3\] hold. We complement this with vertex partitions $X=(X_i)_{i\in[k]}$ and $Y=(Y_i)_{i\in[k]}$ for $H$ defined by $X_i:=f^{-1}(A_i)$ and $Y_i:=f^{-1}(B_i)$ and claim that we can use the general embedding lemma (Lemma \[lem:gel\]) for these vertex partitions of $G$ and $H$.
Indeed, first observe that \[lem:G:2\] and \[lem:G:3\] imply that the partition $V(G)=(A_i)_{i\in[k]}\dcup(B_i)_{i\in[k]}$ is $(\eps,d)$-regular on the graph $C$. Further, by \[lem:G:3\] this partition is $(\eps,d)$-super-regular on the graph $R'$ on the same vertices as $C$ and with edges $A_iB_i$ for all $i\in[k]$. Notice that the components of $R'$ have size $r=2$. It follows that we can apply Lemma \[lem:gel\] if the vertex partition $V(H)=(X_i)_{i\in[k]}\dcup(Y_i)_{i\in[k]}$ is $\eps$-compatible with the partition $V(G)=(A_i)_{i\in[k]}\dcup(B_i)_{i\in[k]}$ and with $R'\subset C$. To check this first note that by \[lem:G:1\] we have $|A_i|=a_i=|X_i|$ and $|B_i|=b_i=|Y_i|$ for all $i\in[k]$ and thus Property \[def:comp:0\] of an $\eps$-compatible partition is satisfied. Since $f$ is a homomorphism from $H$ to $C$ we also immediately get Property \[def:comp:1\] for $(X_i)_{i\in[k]}\dcup(Y_i)_{i\in[k]}$. In addition, since $|A_i|=a_i\le
n_i+\xi_\subsc{lh}n$ for all $i\in[k]$, we also have $|A_i|\ge
n_i-k\xi_\subsc{lh}n \ge\frac12n/k-k\xi_\subsc{lh}n\ge \Delta\xi_\subsc{lh}2k
n/\eps$ by the choice of $\xi_\subsc{lh}$. This together with \[lem:H:1\] implies that $|S\cap A_i|\le\xi_{\subsc{lh}}2k n\le\eps|A_i|$ and $|N_H(S)\cap
A_i|\le\Delta|S|\le\Delta\xi_{\subsc{lh}}2k n\le\eps|A_j|$ for all $i,j\in[k]$. Similarly we get $|S\cap B_i|\le\eps|B_i|$ and $|N_H(S)\cap B_i|\le\eps|B_j|$ for all $i,j\in[k]$. This clearly implies Property \[def:comp:2\] of an $\eps$-compatible partition.
Accordingly we can apply Lemma \[lem:gel\] to the graphs $G$ and $H$ with their partitions $V(G)=(A_i)_{i\in[k]}\dcup(B_i)_{i\in[k]}$ and $V(H)=(X_i)_{i\in[k]}\dcup(Y_i)_{i\in[k]}$, respectively, which implies that $H$ is a subgraph of $G$.
A regular partition of $G$ with a spanning cycle {#sec:G}
================================================
In this section we will prove the Lemma for $G$. This lemma is a consequence of the regularity lemma (Lemma \[lem:reg\]), Theorem \[thm:moon\], and the following lemma which states that, under certain circumstances, we can adjust a (super)-regular partition in order to meet a request for slightly differing cluster sizes.
\[lem:adjust\] Let $k\ge 1$ be an integer, $0 < \xi \leq 1/(20k^2)$ and let $G = (A\dcup B, E)$ be a balanced bipartite graph on $2n$ vertices with partitions $A=A'_1\dcup\dotsm\dcup A'_k$ and $B=B'_1\dcup\dotsm\dcup B'_k$ such that $|A'_i|, |B'_i|\geq n/(2k)$ and $(A'_i,B'_i)$ is $(\eps', d')$-super-regular and $(A'_i, B'_{i+1})$ is $(\eps',d')$-regular for all $i\in[k]$. Let ${(a'_i)_{i\in[k]}}$ and ${(b'_i)_{i\in[k]}}$ be integers such that $a'_i,b'_i\le\xi n$ for all $i\in[k]$ and $\sum_{i\in [k]}a'_i=\sum_{i\in [k]}b'_i = 0$. Then there are partitions $A=A_1\dcup\dotsm\dcup A_k$ and $B=B_1\dcup\dotsm\dcup B_k$ with $|A_i|=|A'_i|+a'_i$ and $|B_i|=|B'_i|+b'_i$ and such that $(A_i,B_i)$ is $(\eps,d)$-super-regular and $(A_i,B_{i+1})$ is $(\eps,d)$-regular for all $i\in[k]$ where $\eps:=\eps' + 100 k\sqrt{\xi}$ and $d:=d' - 100 k^2 \sqrt{\xi} - \eps'$.
The lemma will be proved by performing a simple redistribution algorithm that will iteratively adjust the cluster sizes. Throughout the process, we denote by $A_i$ and $B_i$ the changing clusters, beginning with $A_i:=A'_i$ and $B_i:=B'_i$. We call $A_i$ a *sink* when $|A_i|<|A'_i|+a'_i$, and a *source* when $|A_i|>|A'_i|+a'_i$, and analogously for $B'_i$. Each iteration of the algorithm will have the effect that the number of vertices in a single source decreases by one, the number of vertices in a single sink increases by one, and all other cluster cardinalities stay the same.
We start by describing one iteration of the algorithm. Obviously, as long as not every cluster in $A$ has exactly the desired size, there is at least one source. We choose an arbitrary source $A_i$, and, as will be further explained below, the regularity of the pair $(A_i,B_{i+1})$ implies that within $A_i$ there is a large set of vertices each of which can be added to the neighbouring cluster $A_{i+1}$ while preserving the super-regularity of the pair $(A_{i+1},B_{i+1})$. We do this with one arbitrary vertex from this set. Thereafter, within $A_{i+1}$ there is again a large set of vertices (the newly arrived vertex may or may not be one of them) suitable for being moved into $A_{i+2}$ while preserving the super-regularity of the pair $(A_{i+2},B_{i+2})$, and we again do this with one arbitrary vertex from this set. We then continue in this way until for the first time we move a vertex into a sink. (It may happen that it is not the vertex we initially took out of $A_i$ that arrives in the sink.) This is the end of the iteration.
We repeat such iterations as long as there are sources, i.e. we choose an arbitrary source and repeat what we have just described. Since each iteration ends with adding a vertex to a sink while not changing the cardinality of the clusters visited along the way, we do not increase the number of vertices in any source, let alone create a new source, and hence after a finite number of iterations (which we will estimate below) the algorithm ends with no sources remaining and therefore all clusters within $A$ having exactly the desired size.
We then repeat what we have just described for the clusters within $B$, the only difference being that vertices get moved from $B_i$ into $B_{i-1}$, not $B_{i+1}$, since only in this direction a regular pair can be used ($(A_{i-1},B_i)$ is regular, $(A_{i+1},B_i)$ need not be regular).
We now analyse the algorithm quantitatively. Clearly, the total number of iterations (we call it $t$) is at most the sum of all positive $a'_i$ and all positive $b'_i$. Obviously, both the sum of all positive $a'_i$ and the sum of all positive $b'_i$ is bounded from above by $\frac{1}{2} k \xi n$, hence $$\label{eq:totalNumberOfChanges}
t\leq \tfrac{1}{2} k \xi n+\tfrac{1}{2} k \xi n = k\xi n.$$
We will now use this bound together with Proposition \[prop:moving-vertices\] to estimate the effect of the redistribution on the regularity and density parameters. Since in each iteration each cluster receives at most one vertex and loses at most one vertex, for every $i\in [k]$ and after any step of the algorithm, we have $$|A_i\Delta A'_i| \leq 2 t \leq 2 k \xi n\,,$$ and analogously $|B_i\Delta B'_i|\leq 2 k \xi n$. We now invoke Proposition \[prop:moving-vertices\] on the pairs $(A_i,B_i)$ and $(A_i,B_{i+1})$, once with $\sh{A}:=A_i$, $\sh{B}:=B_i$ then with $\sh{A}:=A_i$, $\sh{B}:=B_{i+1}$ and we claim that we may use $\alpha := \beta := 16 k^2 \xi$. Indeed, we have $|A_i|\geq |A'_i| - t \geq n/(2k)-2k\xi n$ and because $\xi \leq 1/(20k^2)$ implies $2k\xi n \leq 5 k \xi n - 20k^3\xi^2 n$, hence $|A_i\Delta A'_i| \leq 2 k \xi n \leq (5k\xi - 20k^3\xi^2)n
= 10 k^2\xi (n/(2k) - 2k\xi n) \leq \alpha |A_i'|$, and analogously $|B_i\Delta B'_i|\leq \beta |B_i'|$. By Proposition \[prop:moving-vertices\], every pair $(A_i,B_i)$ and $(A_i,B_{i+1})$ is $\bigl( \sh{\eps}, \sh{d} \bigr )$-regular with $\sh{\eps}:=\eps' + 24k\sqrt{\xi}$ and $\sh{d}:=d'-64k^2\xi$, hence $\sh{\eps}\leq \eps$ and $\sh{d}\geq d$, proving the parameters claimed in the lemma, as far as mere regularity goes.
As for the claimed super-regularity of the vertical pairs, let $A_i$, $B_i$ and $B_{i+1}$ be clusters at an arbitrary step of the algorithm. Using Proposition \[lem:typical\] and we know that the pairs $(A_i,B_i)$ and $(A_i,B_{i+1})$ being $(\sh{\eps},\sh{d})$-regular implies that there are at least $(1 - \sh{\eps})|A_i|$ vertices in $A_i$ having at least $(\sh{d}-\sh{\eps})|B_{i+1}|-t\geq
(\sh{d}-\sh{\eps})|B_{i+1}| - 2k\xi n$ neighbours in $B_{i+1}$, and it remains to prove that $(\sh{d}-\sh{\eps})|B_{i+1}| - 2k\xi n \geq d|B_{i+1}|$ which is equivalent to $2k\xi n / |B_{i+1}| \leq 100 k^2\sqrt{\xi}-64k^2\xi-24k\xi$. Because of $2k\xi n / |B_{i+1}| \leq 2k\xi n/(|B'_{i+1}|-t)
\leq 2k\xi n/(n/2k - 2k\xi n) = 4k^2\xi/(1-4k^2\xi)$ it is therefore sufficient that $4k^2\xi/(1-4k^2\xi)\leq 100k^2\sqrt{\xi}-64k^2\xi-24k\sqrt{\xi}$ and it is easy to check that this is true by the hypothesis on $\xi$.
Now we will prove Lemma \[lem:G\]. To this end we will apply Lemma \[lem:reg\] to the input graph $G$. By \[lem:reg:a\] and \[lem:reg:b\] of Lemma \[lem:reg\] we obtain a regular partition with a bipartite reduced graph $R$ of high minimum degree. Theorem \[thm:moon\] then guarantees the existence of a Hamilton cycle in $R$ which will imply property \[lem:G:3\]. This Hamilton cycle serves as $R^*$ in Lemma \[lem:reg\]\[lem:reg:c\] which promises a regular partition of $G$ that is super-regular on $R^*$. For finishing the proof we will use a greedy strategy for distributing the vertices into the exceptional sets over the clusters of this partition (without destroying the super-regularity required for \[lem:G:2\]) and then apply Lemma \[lem:adjust\] to adjust the cluster sizes as needed for \[lem:G:1\].
Let $\gamma > 0$ given. We assume without loss of generality that $\gamma<1/20$ and set $d_{\subsc{lg}}:=\gamma^2/100$. Now let $\eps>0$ and $k_0\in{\ensuremath{\mathbb{N}}}$ be given. We assume that $\eps\le\gamma^2/1000$, since otherwise we can set $\eps:=\gamma^2/1000$, prove the lemma, and all statements will still hold for any larger $\eps$.
Our next task is to choose $\eps'$ and $d'$. For this, consider the following functions in $\eps'$ and $d'$: $$\label{lem:G:eps}
\begin{aligned}
\eps'' &:= \frac{\eps'}{1-2\eps'}\,, &\qquad
\sh{\eps} &:= \eps'' + 6\sqrt{\eps''/\gamma(1-\eps'')}\,,
\\ d'' &:= d'-4\eps'\,, &
\sh{d} &:= d'' - 4\eps''/\gamma(1-\eps'')\,.
\end{aligned}$$ Observe that $$\eps' \ll \eps'' \ll \sh{\eps}\, \qquad\text{and}\qquad
\sh{d} \ll d'' \ll d'\,,$$ by which we mean, for example, that $\eps'\le \eps''$ but that we can make $\eps''$ arbitrarily small by choosing $\eps'$ sufficiently small. Keeping in mind that $\gamma <1/20$, it is easy to check that when setting $\eps':=\eps^3\gamma^3$ and $d':=\eps+\gamma^2$, the following inequalities are all satisfied: $$\begin{gathered}
\label{eq:lg1}
\sh{\eps} \le\tfrac1{10}\eps\,, \qquad
\sh{d} - \eps \ge 2d_{\subsc{lg}}\,, \qquad
\gamma-d'-\eps'' > 0\, \\
\label{eq:lg2}
(\tfrac12+\gamma-\eps'')(1-d'')^{-1} \ge \tfrac12 + \tfrac23 \gamma\,,
\qquad {d''}(1-{d''})^{-1} \le \tfrac16 \gamma\,.\end{gathered}$$ Next, using , we can choose an integer $k'_0$ with $k_0\le k'_0$ such that for all integers $k$ with $k'_0\le k$ we have $$\label{eq:lg6}
(\gamma-d'-\eps'')k \ge 1\,.$$
Apply Lemma \[lem:reg\] with $\eps'$, $\Delta:=2$, and with $k_0$ replaced by $k'_0$, to obtain $K_0$. Choose $\xi_{\subsc{lg}}>0$ such that $$\label{eq:lg7}
100 K_0 \sqrt{\xi_{\subsc{lg}}} \le \tfrac1{10}{\eps}, \quad
100 (K_0)^2 \sqrt{\xi_{\subsc{lg}}} \le d_{\subsc{lg}}.$$ Now let $G$ be given. Feed $d'$ and $G$ into Lemma \[lem:reg\] and obtain $k\in{\ensuremath{\mathbb{N}}}$ with $k_0 \le k'_0\le k \le K_0$ together with an equipartition of $G$ into $2k+2$ classes and an $(\eps',d')$-reduced graph $R$ on $2k$ vertices by \[lem:reg:a\] of Lemma \[lem:reg\]. By assumption $\delta(G)\ge (\frac12 +\gamma)n$, so setting $\nu:= 1/2+\gamma$ and making use of part \[lem:reg:b\] of Lemma \[lem:reg\], we get $$\delta(R) \ge (\tfrac12 + \gamma - d'-\eps'')|V(R)|
= \tfrac12 |V(R)| + (\gamma-d'-\eps'')k
{ { {\overset{\mbox{\tiny{\eqref{eq:lg6}}}}{\ge}} } } \tfrac12 |V(R)| + 1.$$ We infer from Theorem \[thm:moon\] that $R$ contains a Hamilton cycle $R^*$. Now apply part \[lem:reg:c\] of Lemma \[lem:reg\] and obtain an equipartition of $G$ which is $(\eps'',d'')$-regular on $R$, $(\eps'',d'')$-super-regular on $R^*$, and has classes $$A= A''_0 \dcup \dots \dcup A''_k
\quad \text{and} \quad
B= B''_0 \dcup \dots \dcup B''_k.$$ Obviously, $R$ and thus $R^*$ are bipartite and so, without loss of generality (renumbering the clusters if necessary), we can assume that the Hamilton cycle $R^*$ consists of the vertices representing the classes $$A''_1,B''_2,A''_2,B''_3,\dots,B''_k,A''_k,B''_1,A''_1$$ with edges in this order. Therefore, we know that the pairs $(A''_i,B''_i)$ and $(A''_i,B''_{i+1})$ are $(\eps'',d'')$-super-regular for all $i\in [k]$. Let $L:=|A''_i|=|B''_i|$ and observe that $$(1-\eps'')\frac{n}{k} \le L \le \frac{n}{k}\,.$$
Our next aim is to get rid of the classes $A''_0$ and $B''_0$ by moving their vertices to other classes. We will do this, roughly speaking, as follows. When moving a vertex $x\in A''_0$ to some class $A''_i$, say, we will move an arbitrary vertex $y\in B''_0$ to the corresponding class $B''_i$ at the same time. We will also make sure that $x$ has at least $d'' |B''_i|$ neighbours in $B''_i$ and $y$ has at least $d'' |A''_i|$ neighbours in $A''_i$. Here are the details for this procedure. For an arbitrary pair $(x,y)\in A''_0\times B''_0$ we define $$I_{}(x,y) := \Big\{ i\in [ k ]\colon\quad |N_G (x)\cap B''_i| \geq d'' \;
|B''_i|\quad\text{and}\quad|N_G (y)\cap A''_i| \geq d'' \; |A''_i|\Big\}\,.$$ We claim that for every $(a,b)\in A''_0\times B''_0$ we have $
|I_{}(x,y)| \ge \gamma k
$. To prove this claim, first recall that $L=|A''_i|=|B''_i|$ for all $i\in [k]$. Define $$\begin{aligned}
I_{}(x) &:= \big\{ i\in [ k ]\colon |N_G (x)\cap B''_i| \geq d'' |B''_i|
\big\}\,, \\
I_{}(y) &:= \big\{ i\in [ k ]\colon |N_G (y)\cap A''_i| \geq d'' |A''_i|
\big\}\,.\end{aligned}$$ As $|A''_0|=|B''_0| \le\eps'' n$ we have $$\begin{aligned}
(\tfrac12 + \gamma)n
&\,\le\, \deg_G(x)
\,\le\, |I_{}(x)| L + (k-|I_{}(x)|) \,d'' L + \eps'' n \\
&\,=\, |I_{}(x)| (1-d'') L + k d'' L + \eps'' n\,. \end{aligned}$$ and hence $$\begin{aligned}
|I_{}(x)|
&\ge \frac{(\frac12+\gamma)n -kd'' L - \eps'' n}{(1-d'')L}
= \frac{(\frac12+\gamma-\eps'')}{1-d''} \frac{n}{L}
-\frac{d''}{1-d''} \,k \\
&{ { {\overset{\mbox{\tiny{\eqref{eq:lg2}}}}{\ge}} } } (\tfrac12 + \tfrac23\gamma)k -
\tfrac16\gamma k = (\tfrac12 + \tfrac12\gamma)k \,.\end{aligned}$$ Similarly, $|I_{}(y)| \ge (\frac12 + \frac12\gamma)k$. Since $I_{}(x)$ and $I_{}(y)$ are both subsets of $[k]$, this implies that $
|I(x,y)|= |I_{}(x) \cap I_{}(y)| \ge \gamma k
$, which proves the claim.
We group the vertices in $A''_0 \cup B''_0$ into (at most $\eps'' n$) pairs $(x,y)\in A''_0\times B''_0$ and choose an index $i\in I(x,y)$ which has the property that $(A''_i,B''_i)$ has so far received a minimal number of additional vertices. Then we move $x$ into $A''_i$ and $y$ into $B''_i$. Hence, at the end, every cluster $A''_i$, or $B''_i$ gains at most $\eps'' n / (\gamma k)$ additional vertices. Denote the final partition obtained in this way by $$A \dcup B = \sh{A}_1 \dcup \sh{B}_1 \dcup \dots \dcup \sh{A}_k \dcup \sh{B}_k\,.$$ Set $\alpha:=\beta:=\eps''/\gamma(1-\eps'')$ and observe that $$\frac{\eps'' n}{\gamma k} = \alpha (1-\eps'')\frac{n}{k} \le \alpha L\,.$$ So Proposition \[prop:moving-vertices\] tells us that for all $i\in [k]$ the pairs $(\sh{A}_i,\sh{B}_i)$ are still $(\sh{\eps},\sh{d})$-super-regular and the pairs $(\sh{A}_i,\sh{B}_{i+1})$ are still $(\sh{\eps},\sh{d})$-regular, because $$\begin{aligned}
\sh{\eps}
&{ { {\overset{\mbox{\tiny{\eqref{lem:G:eps}}}}{=}} } }\eps'' + 6\sqrt{\eps''/\gamma(1-\eps'')}
=\eps''+3(\sqrt{\alpha}+\sqrt{\beta})
\qquad\text{and} \\
\sh{d}
&{ { {\overset{\mbox{\tiny{\eqref{lem:G:eps}}}}{=}} } } d'' - 4\eps''/\gamma(1-\eps'')
= d'' -4\alpha
= d'' -2(\alpha+\beta) \,.\end{aligned}$$ Now we return to the statement of Lemma \[lem:G\]. We set $n_i:=|\sh{A_i}|=|\sh{B_i}|$ for all $i\in [k]$. Let ${(a_i)_{i\in[k]}}$ and ${(b_i)_{i\in[k]}}$ be given and set $a''_i:= a_i- n_i$ and $b''_i:= b_i- n_i$. Then $$a''_i \le \xi_\subsc{lg}n, \quad
b''_i \le \xi_\subsc{lg}n, \quad
\sum_{i\in[k]} a''_i = \sum_{i\in[k]} a_i - \sum_{i\in[k]} n_i = n-n =0 =
\sum_{i\in[k]} b''_i\,.$$ Therefore we can apply Lemma \[lem:adjust\] with parameter $\xi_\subsc{lg}$ to the graph $G$ with partitions $\sh{A}_1 \dcup \dots \dcup \sh{A}_k$ and $\sh{B}_1 \dcup \dots \dcup \sh{B}_k$. Since $$\begin{aligned}
\sh{\eps} + 100 k \sqrt{\xi_\subsc{lg}}
{ {\overset{\mbox{\tiny{\eqref{eq:lg1},\eqref{eq:lg7}}}}{\le}} }\tfrac{1}{10}\eps+\tfrac{1}{10}\eps
&\le \eps \qquad\text{and} \\
\sh{d} - 100 k^2 \sqrt{\xi_\subsc{lg}}-\eps
{ {\overset{\mbox{\tiny{\eqref{eq:lg1},\eqref{eq:lg7}}}}{\ge}} } 2d_\subsc{lg} - d_\subsc{lg}
&= d_\subsc{lg}\,,\end{aligned}$$ we obtain sets $A_i$ and $B_i$ for each $i\in [k]$ such that $|A_i| = |\sh{A}_i| + a''_i = n_i + a''_i = a_i$ and $|B_i|=b_i$, and with the property that $(A_i,B_i)$ is $(\eps,d)$-super-regular and $(A_i,B_{i+1})$ is $(\eps,d)$-regular. This completes the proof of Lemma \[lem:G\].
Distributing $H$ among the edges of a cycle {#sec:H}
===========================================
In this section we will provide the proof of the Lemma for $H$ (Lemma \[lem:H\]). The idea is to cut $H$ into small pieces along its bandwidth ordering, that is, an ordering of the vertices $H$ that respects the bandwidth bound. These pieces are then distributed to the edges $A_iB_i$ of the cycle $C$ in such a way that the following holds. Let $X_i$ be all the vertices from $X$, and $Y_i$ all the vertices from $Y$ that were assigned to the edge $A_iB_i$. Then we require that $X_i$ and $Y_i$ are roughly of size $n_i$. Observe that this goal would be easy to achieve if $H$ were *locally balanced*, i.e., if each of the small pieces had colour classes of equal size. While this need not be the case, we know, however, that $H$ itself is a *balanced* bipartite graph. Therefore we use a probabilistic argument to show that the pieces of $H$ can be grouped in such a way that the resulting packages form balanced bipartite subgraphs of $H$. The details of this argument are given in Section \[subsec:balance\].
After this distribution of the pieces to the edges $A_iB_i$ we will construct the desired homomorphism $f$ in the following way. We will map most vertices of $X_i$ to $A_i$ and most vertices of $Y_i$ to $B_i$.
Balancing $H$ locally {#subsec:balance}
---------------------
Our goal is to group small pieces $W_1,\dots,W_\ell$ of the balanced bipartite graph $H$ on $2n$ vertices into packages $P_1,\dots,P_k$ that form balanced bipartite subgraphs of $H$. This is equivalent to the following problem. Given the sizes $a_j$ and $b_j$ of the colour classes of each piece $W_j$ (i.e., $a_j$ counts the vertices of $W_j$ that are in $X$ and $b_j$ those that are in $Y$) we know that the $a_j$’s sum up to $n$ and the $b_j$’s sum up to $n$. Then we would like to have a mapping $\phi:[\ell]\to[k]$ such that for all $i\in[k]$ the $a_j$ with $j\in \phi^{-1}(i)$ sum up approximately to the same value as the $b_j$ with $j\in \phi^{-1}(i)$. The following lemma asserts that such a mapping $\phi$ exists. The package $P_i$ will then (in the proof of Lemma \[lem:H\]) consist of all pieces $W_j$ with $j\in \phi^{-1}(i)$.
\[lem:num\] For all $0 < \xi \leq 1/4$ and all positive integers $k$ there exists $\ell\in{\ensuremath{\mathbb{N}}}$ such that for all integers $n\ge\ell$ the following holds. Let $(n_i)_{i\in [k]}$, $(a_j)_{j\in [\ell]}$, and $(b_j)_{j\in [\ell]}$ be integer partitions of $n$ such that $n_i\leq
\frac{1}{8}n$ and $a_j+b_j \leq (1+\xi)\frac{2n}\ell$ for all $i\in [k]$, $j\in[\ell]$. Then there is a map $\phi:[\ell]\to[k]$ such that for all $i\in [k]$ and $\sbar a_i:=\sum_{j\in \phi^{-1}(i)} a_j$ and $\sbar b_i:=\sum_{j\in \phi^{-1}(i)} b_j$ we have $$\label{lem:num:1}
\sbar a_i < n_i + \xi n
\qquad\text{and}\qquad
\sbar b_i < n_i + \xi n\,.$$
In the proof of Lemma \[lem:num\] we will use a Chernoff bound and the following formulation of a concentration bound due to Hoeffding.
\[thm:hoeff\] Let $X_1,\dots,X_s$ be independent random variables with ${\mathbb{E}}X_i=0$ and $|X_i|\le 1$ for all $i\in[s]$ and let $X$ be their sum. Then ${\mathbb{P}}[|X|\ge a]\le2\exp(-a^2/(2s))$.
For the proof of this lemma we use a probabilistic argument and show that under a suitable probability distribution a random map satisfies the desired properties with positive probability. For this purpose set $\ell := \bigl \lceil 1000 k^5 / \xi^2 \bigr
\rceil $ and construct a random map $\phi\colon [\ell] \rightarrow [k]$ by choosing $\phi(j)=i$ with probability $n_i /n$ for $i\in [k]$, independently for each $j\in [\ell]$. To show that this map satisfies with positive probability we first estimate the sum of all $a_j$’s and $b_j$’s assigned to a fixed $i\in[k]$. To this end, let $\mathbbm 1_j$ be the indicator variable for the event $\phi(j)=i$ and define a random variable $S_i:=\sum_{j \in [\ell]}\mathbbm
1_j$. Clearly $S_i$ is binomially distributed, we have ${\mathbb{E}}S_i=\ell\frac{n_i}n$, and by the Chernoff bound ${\mathbb{P}}[|S_i|\ge{\mathbb{E}}S_i+t]\le2\exp(-2t^2/\ell)$ (cf. [@purpleBook Remark 2.5]) we get $${\mathbb{P}}\Big[\big|S_i-\ell\frac{n_i}n\big|\ge\tfrac12\xi\ell\Big]
\le2\exp(-\tfrac12\xi^2\ell).$$ Next, we examine the difference between the sum of the $a_j$’s assigned to $i$ and the sum of the $b_j$’s assigned to $i$. We define random variables $D_{i,j}:=\frac\ell{3n}(a_j - b_j)(\mathbbm 1_j-\frac{n_i}n)$ and set $D_i
:=\sum_{j\in[\ell]}D_{i,j}$. Then ${\mathbb{E}}D_{i,j}=0$ and as $a_j+b_j\le\frac{3n}{\ell}$ we have $|D_{i,j}|\le 1$. Thus Theorem \[thm:hoeff\] implies $${\mathbb{P}}\big[|D_i|\ge\tfrac16\xi\ell\big] \le2\exp(-\tfrac1{72}\xi^2\ell).$$ By the union bound, the probability that we have $$\label{eq:num:good}
|S_i-\ell\tfrac{n_i}n|<\tfrac12\xi\ell
\qquad\text{and}\qquad |D_i|<\tfrac16\xi\ell
\qquad\text{for all $i\in[k]$}$$ is therefore at least $1-k\cdot2\exp(-\tfrac12\xi^2\ell)-
k\cdot2\exp(-\tfrac1{72}\xi^2\ell)$ which is strictly greater than $0$ by our choice of $\ell$. Therefore there exists a map $\phi$ with . We claim that this map satisfies . To see this, observe first that $\frac{3n}\ell D_i=\sum_{j\in\phi^{-1}(i)}(a_j-b_j)=\sbar a_i-\sbar b_i$ which together with implies $\sbar a_i - \sbar b_i < \xi n$. Moreover, we have $S_i=|\phi^{-1}(i)|$ and $$\begin{split}
\sbar a_i &= \tfrac{1}{2}(\sbar a_i+\sbar b_i) + \tfrac12(\sbar a_i-\sbar
b_i) \le\tfrac{1}{2}(1+\xi)\tfrac{2n}\ell|\phi^{-1}(i)|
+\tfrac12\cdot\tfrac12\xi n \\
& { { {\overset{\mbox{\tiny{\eqref{eq:num:good}}}}{<}} } }
\frac{1}{2}(1+\xi)\frac{2n}\ell
\Big(\ell\frac{n_i}n+\frac12\xi\ell\Big)
+\frac14\xi n
\le n_i+\xi n
\end{split}$$ where the last inequality follows from $\xi \leq \frac{1}{4}$ and $n_i\leq
\frac{1}{8}n$. Since an entirely analogous calculation shows that $\sbar b_i
< n_i + \xi n$, this completes the proof of .
The proof of the Lemma for $H$ {#subsec:H}
------------------------------
For the proof of Lemma $H$ we will now use Lemma \[lem:num\] as outlined in the beginning of Section \[subsec:balance\]. In this way we obtain an assignment of pieces $W_1,\dots\,W_\ell$ of $H$ to edges $A_iB_i$ of $C$. This assignment, however, does not readily give a homomorphism from $H$ to $C$ as there might be edges between pieces $W_j$ and $W_{j+1}$ that end up on edges $A_iB_i$ and $A_{i'}B_{i'}$ which are not neighbouring in $C$. Nevertheless (owing to the small bandwidth of $H$) we will be able to transform it into a homomorphism by assigning some few vertices of $W_{j+1}$ to other vertices of $C$ along the path between $A_iB_i$ and $A_{i'}B_{i'}$ in $C$.
Let $k$ and $\xi$ be given. Give $\xi' := \xi/4$ and $k$ to Lemma \[lem:num\], get $\ell$, set $\beta := \xi'/(4\ell k)$ and $n_0 := \lceil \ell /(2\xi) \rceil$, and let $H$ and $(n_i)_{i\in [k]}$ be given as in the statement of the lemma for $H$.
We assume that the vertices of $H$ are given a bandwidth labelling, partition $V(H)$ along this labelling into $\ell$ sets $W_1,\dotsc, W_\ell$ of as equal sizes as possible and define $x_i := |W_i\cap X|$ and $y_i := |W_i\cap Y|$. Then $x_i + y_i = |W_i|\leq \lceil 2n/\ell \rceil
\leq 2n/\ell + 1 \leq (1+\xi) 2n/\ell$ and since $n_i\leq n/8$ by hypothesis we can give $(n_i)_{i\in [k]}$, $(x_i)_{i\in [\ell]}$ and $(y_i)_{i\in [\ell]}$ to Lemma \[lem:num\] and get a $\phi\colon [\ell]\rightarrow [k]$ with .
Let us discuss the main difficulty in our proof. Since the map $\phi$ is obtained via the probabilistic method, there is no control over how far apart in the Hamilton cycle $C$ two sets $W_{\varphi(i-1)}$ and $W_{\varphi(i)}$ will be assigned by $\varphi$. Hence these sets might end up in non-adjacent vertices of the cycle $C$. If there are edges between $W_{\varphi(i-1)}$ and $W_{\varphi(i)}$ we need to guarantee, however, that these edges are mapped to edges of $C$ in order to obtain the desired homomorphism $f$. Therefore, we resort to a greedy linking process which robs the pieces $W_i$ of a small number of vertices. These are then distributed over the clusters lying between the cluster pair $A_{\varphi(i-1)},B_{\varphi(i-1)}$ and the cluster pair $A_{\varphi(i)},B_{\varphi(i)}$ such that the corresponding edges of $H$ are placed on edges of $C$.
Let $w_i$ be the first vertex in $W_i$ and define sets of *linking vertices* by $$L_j^i := [w_i+(j-1)\beta n, w_i+j\beta n)\subset W_i$$ for every $j\in [2k]$ , and set $L^i := \bigcup_{j\in [2k]} L_j^i$. Then all $L_j^i$ have the common cardinality $\beta n$ and $|L^i|=2k\beta n$. Since $\beta \leq 1/(4k\ell)$ implies that $2k\beta n + \beta n \leq \lfloor 2n/\ell \rfloor \leq |W_i|$ for every $i\in [\ell]$, we have $L^i\subsetneq W_i$ for every $i\in [\ell]$ where $|W_i\backslash L^i| \geq \beta n$, i.e., at the end of every set $W_i$ there are at least $\beta n$ non-linking vertices (see the left hand side of Figure \[fig:link\]).
We now construct a map $f\colon V(H)\to\{A_1,\dotsc, A_k, B_1,\dotsc,B_k\}$ by defining, for every $i\in [\ell]$, $$f(x) :=
\begin{cases}
A_{\varphi(i-1) + \lfloor j/2 \rfloor } & \text{if $x\in L_j^i$ with $j\in \bigl [ 2\cdot \bigl( (\varphi(i)-\varphi(i-1)) \mod k \bigr)\bigr ]$,}\\
A_{\varphi(i)} & \text{else,}
\end{cases}$$ for every $x\in W_i\cap X$, and $$f(y) :=
\begin{cases}
B_{\varphi(i-1) + \lceil j/2 \rceil } & \text{if $y\in L_j^i$ with $j\in \bigl [ 2\cdot \bigl( (\varphi(i)-\varphi(i-1)) \mod k \bigr)\bigr ]$,}\\
B_{\varphi(i)} & \text{else,}
\end{cases}$$ for every $y\in W_i\cap Y$, and show that this is indeed a homomorphism (see also Figure \[fig:link\]). To do this, it is convenient to note that a set $\{A_{i}, B_{i'} \}$ is an edge of $C$ if and only if $0 \leq i' - i \leq 1$.
Let arbitrary vertices $x\in X$ and $y\in Y$ with $\{x,y\}\in F$ be given. Since the sets $W_i$ are defined along the bandwidth labelling, either $x$ and $y$ are both within the same $W_i$, or $x$ and $y$ lie in consecutive sets $W_i$ and $W_{i+1}$. We will now distinguish several cases. For brevity let $I_i := \bigl [ 2\cdot \bigl( (\varphi(i)-\varphi(i-1)) \mod k \bigr)\bigr ]$.
[*Case 1.*]{} Both $x$ and $y$ lie within the same set $W_i$.\
[*Case 1.1.*]{} There is $j\in I_i$ with $x\in L_j^i$, hence $f(x) = A_{\varphi(i-1)+\lfloor j/2 \rfloor}$. Due to the bandwidth condition together with $|L_j^i|=\beta n$, if $y\notin L_j^i$ and $j+1\in I_i$, then necessarily $y\in L_{j+1}^i$, which explains the following three sub-cases.\
[*Case 1.1.1.*]{} We have $y\in L_j^i$, hence $f(y) = B_{\varphi(i-1)+\lceil j/2 \rceil}$, hence the difference of the indices of $f(x)$ and $f(y)$ is $\lceil j/2 \rceil - \lfloor j/2 \rfloor$, which is either $0$ or $1$ according to whether $j$ is even or odd, hence $\{f(x),f(y)\}\in E(C)$.\
[*Case 1.1.2.*]{} We have $y\notin L_j^i$ and $j+1\in I_i$, hence $y\in L_{j+1}^i$, hence $f(y) = \varphi(i-1)+\lceil (j+1)/2 \rceil$, hence the difference of indices of $f(y)$ and $f(x)$ is $\lceil (j+1)/2 \rceil - \lfloor j/2 \rfloor$, and this is always $1$, whether $j$ is even or odd, so $\{f(x),f(y)\}\in E(C)$.\
[*Case 1.1.3.*]{} We have $y\notin L_j^i$ and $j+1\notin I_i$, hence $f(y) = B_{\varphi(i)}$. Here, $j+1\notin L_j^i$ implies that $j\geq 2\cdot \bigl( (\varphi(i)-\varphi(i-1)) \mod k \bigr)$ while being within Case 1.1 implies $j\in I_i$, hence $j\leq 2\cdot \bigl( (\varphi(i)-\varphi(i-1)) \mod k \bigr)$, so we have $j = 2\cdot \bigl( (\varphi(i)-\varphi(i-1)) \mod k \bigr)$, thus $f(x) = A_{\varphi(i-1)+\lfloor j/2 \rfloor} = A_{\varphi(i)}$, the index difference between $f(y)$ and $f(x)$ is $0$ and $\{f(x),f(y)\}\in E(C)$.\
[*Case 1.2.*]{} There is no $j\in I_i$ with $x\in L_j^i$, hence $f(x) = A_{\varphi(i)}$. Being within Case 1, i.e. $y\in W_i$, it follows that there are exactly two cases.\
[*Case 1.2.1.*]{} If $y$ precedes $x$ in the bandwidth labelling, then $y\in L_{2\cdot q}^i$ with $q=(\varphi(i)-\varphi(i-1)) \mod k$. Hence $f(y)=B_{\varphi(i)}$, so the index difference between $f(y)$ and $f(x)$ is $0$ and $\{f(x),f(y)\}\in E(C)$.\
[*Case 1.2.2.*]{} If $y$ succeeds $x$ in the bandwidth labelling, then, since $y\in W_i$ by being within Case 1, there is no $j\in I_i$ with $y\in I_i$, hence $f(y)=B_{\varphi(i)}$, so again the index difference between $f(y)$ and $f(x)$ is $0$ and $\{f(x),f(y)\}\in E(C)$.
[*Case 2.*]{} We have $x\in W_i$ and $y\in W_{i+1}$. Then, by the bandwidth condition and size of the sets of linking vertices, we must have $y\in L_1^{i+1}$, hence $f(y) = B_{\varphi((i+1)-1) + \lceil 1/2\rceil} = B_{\varphi(i)+1}$, and since there are at least $\beta n$ non-linking vertices to the right of $W_i$, the vertex $x$ cannot lie in a $L_j^i$, hence $f(x) = A_{\varphi(i)}$, so the index difference of $f(y)$ and $f(x)$ is $1$ and $\{f(x),f(y)\}\in E(C)$.
[*Case 3.*]{} We have $y\in W_i$ and $x\in W_{i+1}$. Then, by the bandwidth condition and size of the sets of linking vertices, we must have $x\in L_1^{i+1}$, hence $f(x) = A_{\varphi((i+1)-1) + \lfloor 1/2\rfloor} = A_{\varphi(i)}$, and since there are at least $\beta n$ non-linking vertices to the right of $W_i$, the vertex $y$ cannot lie in a $L_j^i$, hence $f(y)=B_{\varphi(i)}$, so the index difference of $f(y)$ and $f(x)$ is $0$ and $\{f(x),f(y)\}\in E(C)$. This completes the proof that $f$ is a homomorphism.
We now prove \[lem:H:1\] and \[lem:H:2\]. Define $S := \bigcup_{i\in [\ell]} L^i$. Then $|S|\leq \ell\cdot 2k \cdot \beta n \leq \ell\cdot 2k \cdot (\xi'/(2\ell k))\cdot n = \xi' n\leq \xi n$, which shows \[lem:H:1\], and \[lem:H:2\] is obvious from the definitions of $S$ and the map $f$ above.
We now prove \[lem:H:3\]. For this it suffices to note, rather crudely, that for every $j\in [k]$, no pre-image $f^{-1}(A_j)$ can become larger than the sum of the sizes of all sets $W_i$ assigned to $A_j$ by $\varphi$ (which by the definition of $f$ equals the sum of all $x_i = |X\cap W_i|$ with $\varphi(i)=j$) plus the total number of linking vertices, i.e. for every $j\in [k]$, using the choice of $\beta$ and using that $\varphi$ has the property promised by Lemma \[lem:num\], we have $|f^{-1}(A_j)|\leq \bigl ( \sum_{i\in \varphi^{-1}(j)} x_i \bigr ) + |\bigcup_{i\in [\ell]} L^i|
\leq n_j + \xi' n + \ell\cdot |L^i| = n_j + \xi' n + 2k\ell\beta n \leq n_j + 2\xi' n = n_j + \xi n$, completing the proof of \[lem:H:3\].
Concluding remarks
==================
[**Unbalanced $H$ and $G$.**]{} Essentially the same proof allows for an analogue of Theorem \[thm:bipbw\] for bipartite graphs $H$ and $G$ that are not balanced but whose colour classes have the same sizes. More precisely, let $H=(X\dcup Y,F)$ and $G=(A\dcup
B,E)$ be as in Theorem \[thm:bipbw\], except that $|X|=|A|=n_1$ and $|Y|=|B|=n_2$ (where $n_1+n_2=2n$) and the minimum degree condition on $G$ is replaced by the following condition. For all $v\in A$ we have $\deg_G(v)\ge(\frac12+\gamma)n_2$ and for all $w\in B$ we have $\deg_G(w)\ge(\frac12+\gamma)n_1$. Then $H$ is a subgraph of $G$.
[**Generating systems for the cycle space.**]{} As an application of Theorem \[thm:bipbw\], one can show the following result. For every $\gamma > 0$ there is $n_0 \in{\ensuremath{\mathbb{N}}}$ such that for every $n\geq n_0$ every balanced bipartite graph $G$ on $2n$ vertices with $\delta(G)\ge(\frac12+\gamma)n$ has the property that the edge-sets of all Hamilton cycles in $G$ form a generating system for the cycle space of $G$. A proof for this theorem will be given in a forthcoming paper [@heinighamspace]. It utilises the fact that a special balanced bipartite graph $H$ (the so-called Möbius ladder) of bounded maximum degree and bandwidth has this property and then shows that this gets translated to the graph $G$, using a result of Locke [@locke85].
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Manipulating the orbital occupation of valence electrons via epitaxial strain in an effort to induce new functional properties requires considerations of how changes in the local bonding environment affect the band structure at the Fermi level. Using synchrotron radiation to measure the x-ray linear dichroism of epitaxially strained films of the correlated oxide CaFeO~3~, we demonstrate that the orbital polarization of the Fe valence electrons is opposite from conventional understanding. Although the energetic ordering of the Fe $3d$ orbitals is confirmed by multiplet ligand field theory analysis to be consistent with previously reported strain-induced behavior, we find that the nominally higher energy orbital is more populated than the lower. We ascribe this inverted orbital polarization to an anisotropic bandwidth response to strain in a compound with nearly filled bands. These findings provide an important counterexample to the traditional understanding of strain-induced orbital polarization and reveal a new method to engineer otherwise unachievable orbital occupations in correlated oxides.'
author:
- 'Paul C. Rogge'
- 'Robert J. Green'
- Padraic Shafer
- Gilberto Fabbris
- 'Andi M. Barbour'
- 'Benjamin M. Lefler'
- Elke Arenholz
- 'Mark P. M. Dean'
- 'Steven J. May'
title: Inverted orbital polarization in strained correlated oxide films
---
The use of epitaxial strain to induce occupation of specific electron orbitals by removing orbital degeneracies has been pursued in transition metal oxides in an effort to engineer new electronic and magnetic properties [@Tokura_Manganites_XLD; @Aruta_LSMO_XLD; @Strain_OP_Csiszar; @Hansmann_Held_theory_nickelate_SL_OP; @Nickelate_holes_reduced_OP; @Chak_asymmetric_XLD; @Freeland_LNO_XLD; @Wu_nickelate_SL_XLD; @Wu_PrNiO3_orbital_polarization; @Bruno_nickelate_XLD; @Pesquera_LSMO_XLD]. Such strain-induced orbital polarization has been very successfully described by ligand field theory, which considers the overlap of electron orbitals between a central cation and its surrounding anions [@Cox_oxide_book; @Khomskii_book]. For transition metal perovskite oxides, the metal cation is octahedrally coordinated by six oxygen anions, or ligands. This $O_h$ symmetry splits the five degenerate $d$-levels into two groups: a lower, triply degenerate group ($t_{2g}$) and a doubly degenerate group ($e_g$) higher in energy by an amount $10Dq$. Whereas the lobes of the O $p$ orbitals point in between the $t_{2g}$ lobes, they directly overlap with the $e_g$ lobes, which comes at a coulombic energy cost that raises the $e_g$ orbitals in energy. Epitaxial strain alters the local crystal field and lifts the $t_{2g}$ and $e_g$ degeneracies. For example, tensile strain reduces the overlap between the $e_g$ orbital of $d_{x^2-y^2}$ symmetry and its ligands, thus lowering its energy relative to the other $e_g$ orbital, $d_{3z^2-r^2}$, by an amount $\Delta e_g$ \[see Fig. 1(a) inset\]. Unless the $e_g$ orbitals are fully filled, one subsequently expects $d_{x^2-y^2}$ to be preferentially occupied; the converse applies for compressive strain. This simple picture has been used to explain strain-induced orbital polarization in many systems, particularly *AB*O~3~ perovskite oxides [@Tokura_Manganites_XLD; @Aruta_LSMO_XLD; @Hansmann_Held_theory_nickelate_SL_OP; @Nickelate_holes_reduced_OP; @Chak_asymmetric_XLD; @Freeland_LNO_XLD; @Wu_nickelate_SL_XLD; @Wu_PrNiO3_orbital_polarization; @Bruno_nickelate_XLD; @Pesquera_LSMO_XLD]. In this Letter, we find that this model fails to explain orbital polarization in strained films of CaFeO~3~, which exhibit orbital polarization opposite to that described above.
To quantify the electron occupation of specific $e_g$ orbitals, we measure x-ray absorption across the Fe $L$- and O $K$-edge resonance energies using linearly polarized photons, which allows us to differentiate between $d_{x^2-y^2}$ and $d_{3z^2-r^2}$ occupations. Analyzing the x-ray linear dichroism using multiplet ligand field theory reveals that the effect of epitaxial strain on the energetic ordering of the $e_g$ orbitals is consistent with the aforementioned considerations–stretched bonds are lower in energy than unstretched. Given this energetic landscape, however, the expected orbital *occupations* do not follow: The out-of-plane ($d_{3z^2-r^2}$) orbitals are more populated under tensile strain (and *vice versa* for compressive strain). We propose that this inverted orbital polarization arises from strain-induced anisotropic changes in the Fe-O-Fe bond angles and the resulting anisotropic bandwidths in bands that are more-than-half-filled. Such conditions are not limited to ferrates but could arise in other strongly hybridized systems, such as the rare-earth nickelates [@Sawatzky_bond_disproportionation].
![Polarization-dependent x-ray absorption measured by total electron yield (TEY) across the Fe $L$-edge for CaFeO~3~ under (a) tensile strain and (b) compressive strain. Inset: Octahedral crystal field splitting of transition metal $d$ levels for a (001)-oriented film under no strain ($c=a$) and under biaxial tensile strain ($c<a$). \[Fig1\_Fe\_XLD\]](Fig1.eps)
CaFeO~3~ films of 40 pseudocubic unit cells (${\sim}$15 nm thick) were deposited by oxygen plasma-assisted molecular beam epitaxy. Epitaxial strain was achieved by deposition on single crystal, (001)-oriented substrates: YAlO$_3$ (YAO, -2.0% strain), SrLaAlO$_4$ (SLAO, -0.7%), LaAlO$_3$ (LAO, 0.2%), (La$_{0.18}$Sr$_{0.82}$)(Al$_{0.59}$Ta$_{0.41}$)O$_3$ (LSAT, 2.3%), and SrTiO$_3$ (STO, 3.3%). As previously reported, the films are coherently strained and exhibit bulk-like electrical transport, indicating high-quality, stoichiometric films [@Rogge_PRM]. Prior to all measurements, the films were reoxidized by heating to ${\sim}$600 $^\circ$C in oxygen plasma (200 Watts, 1x$10^{-5}$ Torr chamber pressure) and then slowly cooled to room temperature in oxygen plasma. X-ray absorption spectroscopy was performed at the Advanced Light Source, Beamline 4.0.2 and at the National Synchrotron Light Source-II, Beamline 23-ID-1. The spectra were recorded at 290 K, where CaFeO~3~ is paramagnetic with metallic conductivity [@Woodward_CFO]. The x-ray incident angle was $20^{\circ}$ from the film plane, and a geometric correction was applied to the absorption measured with photons polarized out of the film plane [@SI_XLD].
Although CaFeO~3~ has an unusually high formal oxidation state of Fe^4+^, its ground state exhibits a significant self-doped ligand hole density due to its negative charge transfer energy, $\Delta$ [@Kawasaki_CFO_first_transport; @Bocquet_SFO_ligand_holes; @Woodward_CFO; @Matsuno_CFO_dispro; @Takeda_CFO; @Rogge_PRM]. In this regime, the transition metal cation does not adopt its formal oxidation state but instead keeps an extra electron that results in a hole ($\underline{L}^1$) on the oxygen ligand [@ZSA; @Sawatzky_neg_charge_trans_1; @Matsuno_CFO_dispro]. So while CaFeO~3~ has a nominal Fe configuration of $d^4$ ($e_g^1$), its ground state has a strong $d^5\underline{L}^1$ contribution. Because of the half-filled $d$-shell, this $d^5\underline{L}^1$ ($e_g^2$) state has no significant orbital polarization and is expected to decrease the degree of orbital polarization achievable in the Fe states.
X-ray absorption across the Fe $L$-edge for a CaFeO~3~ film under tensile strain is shown in Fig. \[Fig1\_Fe\_XLD\](a). The $L_3$ peak exhibits primarily a single, broad peak (with a small shoulder) that is consistent with nominal Fe^4+^ [@Abbate_SFO_XAS; @Reduced_SFO_XAS] and significantly contrasts with the well-separated double peak structure seen in Fe^3+^ perovskites, such as LaFeO~3~ and EuFeO~3~ [@Reduced_SFO_XAS; @Amber_EuFeO3]. This spectral signature as well as the bulk-like electrical transport indicate that oxygen vacancies have been sufficiently suppressed. As seen in Fig. \[Fig1\_Fe\_XLD\](a), the x-ray absorption is polarization-dependent. The difference in absorption measured with photons polarized parallel to the film plane, $I_x$, and photons polarized out of the film plane, $I_z$, is termed x-ray linear dichroism (XLD = $I_x-I_z$). The XLD shows areas of both positive and negative intensity, and this lineshape is similar but nearly opposite in sign for the compressively strained film, CaFeO~3~/SLAO, shown in Fig. \[Fig1\_Fe\_XLD\](b). Because $I_x$ preferentially probes empty states in $d_{x^2-y^2}$ and $I_z$ probes $d_{3z^2-r^2}$, their difference in total integrated intensity is a measure of the orbital polarization [@Thole_dichroism_prediction; @van_der_Laan_XLD], and indeed the XLD integrals are non-zero.
Evaluating the sign of the integrated XLD, however, uncovers a surprising result: the $e_g$ electron occupation does not follow the conventional ligand field model. For tensile strain the positive XLD integral implies more empty $d_{x^2-y^2}$ states. Thus under tensile strain CaFeO~3~ has more electrons in $d_{3z^2-r^2}$, which is opposite of that predicted by ligand field theory. Under compressive strain, the integrated XLD sign implies that $d_{x^2-y^2}$ has more electrons. This behavior is consistent among the other films: The integrated XLD for tensile CaFeO~3~ on STO (+3.3% strain) is positive, compressed CaFeO~3~ on YAO (-2.0%) is negative, and the relatively unstrained CaFeO~3~ film on LAO (+0.2%) is approximately zero [@SI_XLD].
In order to verify these relative $e_g$ occupations, we repeated the XLD measurements at the O $K$-edge. This transition probes unoccupied states with O $2p$ character, which are strongly hybridized with Fe $3d$ states due to the negative charge transfer energy [@Abbate_SFO_XAS]. Because these ligand states have the same symmetry as the Fe $3d$ states that they hybridize with [@Sawatzky_bond_disproportionation], they are expected to mimic the Fe $e_g$ occupation. We particularly focus on the O $K$-edge prepeak feature between 526 and 529 eV because it directly probes the oxygen ligand hole states [@Abbate_SFO_XAS; @Chen_cuprate_O_prepeak; @Suntivich_O_Kedge_holes; @Pellegrin_holes_prepeak]. We note that oxygen in the substrates contributes only at energies above the prepeak. As seen in Fig. \[O\_XAS\], the oxygen prepeak exhibits linear dichroism, where the tensile strained film, CaFeO~3~/LSAT, has positive dichroism and the compressively strained film, CaFeO~3~/SLAO, has negative dichroism. A positive integrated XLD indicates more empty states in the $p_x$ and $p_y$ orbitals compared to $p_z$–that is, under tensile strain, more electrons have $p_z$ character than $p_x$ and $p_y$; the opposite situation exists for the film under compressive strain. This precisely mirrors the $e_g$ occupation measured for the Fe $3d$ states. The other strained films (CaFeO~3~/STO, CaFeO~3~/LAO, CaFeO~3~/YAO) exhibit an O prepeak XLD consistent with the two films highlighted here [@SI_XLD].
![Polarization-dependent x-ray absorption of the O $K$-edge prepeak (arrow) for CaFeO~3~ under (a) tensile and (b) compressive strain measured by total fluorescence yield. \[O\_XAS\]](Fig2.eps)
With the qualitative occupation of the Fe $e_g$ orbitals confirmed, we now quantitatively estimate the orbital polarization by computing the hole ratio, $\int I_x dE$/$\int I_z dE$ [@Thole_hole_ratio; @Haverkort_dissertation]. The hole ratio depends on the Fe valence filling, and for high-spin CaFeO~3~ there are three $t_{2g}$ electrons and, as will be shown below, we find that the total $e_g$ occupation is 1.85 electrons. For CaFeO~3~/LSAT (tensile), the hole ratio is 1.018, which gives 0.90 electrons in $d_{x^2-y^2}$ and 0.95 electrons in $d_{3z^2-r^2}$ [@SI_XLD], or an orbital polarization of ${\sim}$6%. Repeating for compressively strained CaFeO~3~/SLAO, we find 0.93 electrons in $d_{x^2-y^2}$ and 0.91 electrons in $d_{3z^2-r^2}$, or ${\sim}$2% polarized. This smaller orbital polarization is consistent with its lower strain state (-0.7%) compared to CaFeO~3~/LSAT (+2.3%).
To help interpret these findings, we analyze the x-ray absorption for CaFeO~3~ using multiplet ligand field theory of a FeO~6~ cluster [@Groot_XAS_multiplet_review]. We begin with the formal Fe$^{4+}$ ($3d^4$) occupation and full ligand orbitals while including a negative charge transfer energy [@Rogge_PRM] such that the configuration interaction ground state has primarily a $d^5\underline{L}^1$ character but still exhibits the same $S=2$ high spin symmetry of the $3d^4$ case [@Sawatzky_bond_disproportionation]. This $S=2$ configuration has a two-fold degeneracy due to the hybridized $e_g$ orbitals that is lifted by the imposed strain (corresponding to preferential occupation of $d_{x^2-y^2}$ and preferential occupation of $d_{3z^2-r^2}$, respectively). Further, each of these $S=2$ states has a five-fold spin degeneracy, which is lifted via the atomic spin-orbit interaction and non-tetragonal local crystal field distortions. We neglect the latter and hence label the spin-orbit split states by $J_z = 0,\pm 1,\pm 2$. Thus our XAS and XLD spectra are expected to be linear combinations of two sets of 5 spectra, one set corresponding to preferential $d_{x^2-y^2}$ occupation and one for preferential $d_{3z^2-r^2}$ occupation [@Wu_nickelate_SL_XLD]. The model parameters were optimized by comparing the calculated XLD to the experimental XLD [@SI_XLD].
The XLD from these two sets of five calculated spectra are shown in Fig. \[Fig\_Jz\](a) for moderate tensile strain ($\Delta e_g = +40$ meV). At finite temperature, the experimental spectrum is expected to be a combination of these $J_z$ spectra [@Haverkort_LaTiO3_Jz_boltzmann], depending on their relative energies due to the spin-orbit splitting and low symmetry crystal field distortions. Therefore, a least-squares fitting procedure was used to determine a coefficient value for each of the $J_z$ XLD spectra such that the resulting combination produces the best fit with experiment. This calculated XLD spectrum has a corresponding x-ray absorption spectrum, and the XLD fitting was constrained such that the resulting calculated x-ray absorption spectral weight ($I_x + I_z$) is within $\pm1$% of the experimental spectral weight.
As seen in Figs. \[Fig\_Jz\](b) and \[Fig\_Jz\](c), the experimental XLD is well-captured by the $J_z$ fit for both CaFeO~3~/LSAT and CaFeO~3~/SLAO. All major features of the $L_3$ and $L_2$ XLD peaks are replicated. The corresponding x-ray absorption spectrum of the optimized XLD fit for CaFeO~3~/LSAT, shown in Fig. \[Fig\_Jz\](d), also has excellent agreement with experiment. The coefficients for each $J_z$ spectrum are listed in the Supplemental Material [@SI_XLD]. Fig. \[Fig\_Jz\](e) highlights that the goodness of fit, $\chi^2$, is a strong function of $\Delta$, and the lowest $\chi^2$ values are obtained for $\Delta < 0$, further confirming that CaFeO~3~ is a negative charge transfer material. We find that $\Delta = -2.0$ eV provides the best fit to experiment, which is in good agreement with previously reported values for formal Fe^4+^ SrFeO~3~ [@Bocquet_SFO_ligand_holes], and is more negative than the rare-earth nickelates [@Wu_nickelate_SL_XLD; @Robert] but not so negative that the $t_{2g}$ and $e_g$ levels are inverted, as in some compounds [@Ushakov_crystal_field]. This value sets the number of self-doped ligand holes, and as seen in Fig. \[Fig\_Jz\](e), for $\Delta = -2.0$ eV the Fe $e_g$ occupation is 1.85 electrons. This large Fe $e_g$ occupation is consistent with the small measured Fe $3d$ orbital polarization.
The XLD fits also reproduce the measured Fe orbital polarization. Converting the preferential $x^2-y^2$ and the preferential $3z^2-r^2$ fit contributions to a $d_{x^2-y^2}$ and $d_{3z^2-r^2}$ occupation, we find that the orbital occupation for tensile CaFeO~3~/LSAT exhibits a small preference for $d_{3z^2-r^2}$, where $d_{x^2-y^2}$ has 0.91 electrons and $d_{3z^2-r^2}$ has 0.94 electrons [@SI_XLD]. This difference of 0.03 electrons agrees well with the difference of 0.05 electrons determined by the sum rule analysis of the XLD integrals. For the compressively strained film, CaFeO~3~/SLAO, the best $J_z$ fit is with equal occupation of 0.93 electrons in both $d_{x^2-y^2}$ and $d_{3z^2-r^2}$.
Of particular note is the sign of the strain-induced crystal field $e_g$ splitting, $\Delta e_g$, that produces the best agreement with experiment. As seen in Fig. \[Fig\_Jz\](f), for tensile strained CaFeO~3~/LSAT, the lowest $\chi^2$ value occurs for +40 meV; for compressively strained CaFeO~3~/SLAO, -30 meV produces the best fit. These magnitudes are of the same order as other similarly strained perovskite oxides [@Aruta_LSMO_XLD; @Wu_nickelate_SL_XLD; @Gilberto_Nickelate_XLD]. Importantly, the respective signs indicate an $e_g$ splitting consistent with the traditional ligand field model: $\Delta e_g > 0$ implies that $d_{x^2-y^2}$ is lower in energy than $d_{3z^2-r^2}$, which would be expected for tensile strain, and *vice versa* for compressive strain. This provides a critical insight: The energetic landscape of the Fe $3d$ orbitals follows the typical ligand field understanding, where, for example, tensile strain lowers $d_{x^2-y^2}$ in energy relative to $d_{3z^2-r^2}$. Despite this, the $d_{x^2-y^2}$ orbital has fewer electrons than $d_{3z^2-r^2}$ in the film under tensile strain, indicating an inversion in orbital polarization.
What, then, overrides the $\Delta e_g$ splitting and produces the inverted $e_g$ orbital occupation? Although oxygen vacancies can be equatorially or apically ordered under epitaxial strain [@Spaldin_O_vacancy_ordering], the resulting preferential orbital occupation would be opposite of the results here. Moreover, because our experimental findings are not replicated by previous density functional theory calculations [@Antonio_CFO_strain], we propose a new mechanism. It is well known that perovskites can accommodate epitaxial strain by changes in both bond lengths and rotations of the octahedral complexes surrounding the transition metal (TM) cation [@Miniotas_strain_rotations; @Xie_strain_rotations; @Steve_octahedral_rotations_strain; @Rondinelli_Spaldin_Adv_Mater]. Rotations alter the TM-O-TM bond angle, and angles less than $180^{\circ}$ have reduced orbital overlap and thus narrower bands. For a perovskite that exhibits rotations in its bulk form, such as CaFeO~3~, biaxial tensile strain increases the in-plane TM-O-TM bond angle towards $180^{\circ}$, whereas the out-of-plane angle decreases further and is typically more strongly affected than the in-plane angles [@Rondinelli_Spaldin_Adv_Mater; @Antonio_CFO_strain]. Thus in the simplest approximation where strain is accommodated predominantly by octahedral rotations, under tensile strain one would expect the in-plane (x, y) bandwidth to increase and the out-of-plane (z) bandwidth to decrease.
Such anisotropic bandwidth effects can lead to an inverted orbital polarization in compounds with greater-than-half-filled bands. As illustrated in Fig. \[Fig\_Jz\](g) for the case under tensile strain, $\Delta e_g > 0$ shifts the band center of masses, but a broadening of the $x^2-y^2$ band and a narrowing of the $3z^2-r^2$ band can result in $3z^2-r^2$ being more occupied than $x^2-y^2$. The precise orbital polarization is expected to depend on the specific bandstructure and Fermi level position. For bands with half-filling or less, the same anisotropic bandwidths result in the conventional orbital polarization and thus do not replicate our findings [@SI_XLD]. We further note that this effect does not require metallicity and indeed when repeating the Fe $L$-edge XLD measurements at lower temperatures (180 K) where CaFeO~3~ is insulating, the inverted orbital polarization is maintained [@SI_XLD].
In summary, we have shown that epitaxially strained films of CaFeO~3~ exhibit orbital polarization that responds to the strain state in a way that requires considerations beyond the commonly assumed ligand field model. By analyzing the x-ray linear dichroism with multiplet ligand field simulations, we find that under tensile strain the $e_g$ electronic population is weighted towards $d_{3z^2-r^2}$ orbitals, despite being ${\sim}$40 meV higher in energy than $d_{x^2-y^2}$. The opposite is observed under compressive strain. We propose an explanation for this behavior by considering anisotropic modifications of the bandwidth of the $e_g$ states, in which under tensile strain a broadened $d_{x^2-y^2}$ band and a narrowed $d_{3z^2-r^2}$ lead to this inverted orbital polarization configuration. This scenario is consistent with the orbital energetic landscape as determined by ligand field theory, as well as the measured film strain, under the assumption that strain is accommodated primarily by octahedral bond rotations. More generally, our results demonstrate that effects typically not considered in the conventional understanding of strain-induced orbital polarization can mitigate or even invert the orbital polarization. This highlights that the interpretation of orbital polarization in ultrathin films and short-period superlattices [@Chak_asymmetric_XLD; @Freeland_LNO_XLD; @Keimer_Nickelate_reflectometry; @Cao_XLD; @Wu_PrNiO3_orbital_polarization; @Wu_nickelate_SL_XLD; @Disa_Ahn_tricolor_SL], where non-bulk octahedral rotations can be induced, should include such considerations. Additionally, these results demonstrate that bandwidth control is a potentially new way to engineer orbital polarization in correlated oxides.
We thank G. Sawatzky and A. Fujimori for helpful discussions. PCR and SJM were supported by the Army Research Office, grant number W911NF-15-1-0133, and film synthesis at Drexel utilized deposition instrumentation acquired through an Army Research Office DURIP grant (W911NF-14-1-0493). RJG was supported by the Natural Sciences and Engineering Research Council of Canada. Work at Brookhaven National Laboratory was supported by the U.S. Department of Energy (DOE), Office of Basic Energy Sciences under Contract No. DE-SC0012704 and Early Career Award Program under Award No. 1047478. This work used resources at the Advanced Light Source, which is a DOE Office of Science User Facility under contract No. DE-AC02-05CH11231, and at Beamline 23-ID-1 of the National Synchrotron Light Source II, a DOE Office of Science User Facility operated for the DOE Office of Science by Brookhaven National Laboratory under Contract No. DE-SC0012704.
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**Supplemental Material: Inverted orbital polarization in strained correlated oxide films**
I. X-ray absorption experiment and additional data
--------------------------------------------------
![Fe $L$-edge x-ray absorption data of additional CaFeO~3~ films. Polarization-dependent x-ray absorption across the Fe $L$-edge for (a) tensile (+3.3%) CaFeO~3~ on SrTiO~3~, (b) relatively unstrained CaFeO~3~ on LaAlO~3~ (+0.2%), and (c) compressively strained (-2.0%) CaFeO~3~ on YAlO~3~. The resulting x-ray linear dichroism ($I_x - I_z$) and its integral are also shown. Measurements were made above CaFeO~3~’s metal-insulator transition temperature (*T*~MIT~). (d) The integrated XLD of all CaFeO~3~ films for $T>$*T*~MIT~ (metallic) show consistent behavior: Tensile strained films have a positive XLD integral and compressively strained films have a negative XLD integral. (e) This behavior is maintained in the insulating state ($T<$*T*~MIT~). Data for CaFeO~3~/STO sample was not obtained at 180 K. \[SI\_Fe\_XAS\]](SI_Fe_XAS.eps)
The x-ray incident angle was $20^{\circ}$ relative to the film plane and the polarization was controlled upstream. A geometric correction was applied to the absorption measured with photons polarized out of the film plane, $I_{\pi}$: $$I_z = \frac{(I_\pi - I_x\sin^2(\theta))}{\cos^2(\theta)},$$ where $I_x$ is the absorption intensity measured with photons polarized parallel to the film plane. At least 12 scans of each polarization were performed for the Fe $L$-edge measurements ($680-750$ eV), and at least four scans for the O $K$-edge ($513-555$ eV). The spectra were normalized by setting the pre-edge intensity to zero by subtracting a line fit to the pre-edge, followed by setting the post-$L_2$ intensity to unity at 750 eV for the Fe scans and setting the maximum intensity to unity for the O scans. The Fe x-ray absorption and XLD of the additional samples are shown in Fig. \[SI\_Fe\_XAS\], and the O $K$-edge prepeak XLD of the additional samples are shown in Fig. \[SI\_O\_XAS\].
![O $K$-edge prepeak x-ray linear dichroism for all CaFeO~3~ films measured by total fluorescence yield. The strain-dependent behavior is consistent with that seen for the Fe $3d$ $e_g$ occupation. \[SI\_O\_XAS\]](SI_O_XAS.eps)
II. Hole ratio derivation for CaFeO~3~
--------------------------------------
Following Refs. [@Thole_hole_ratio; @Haverkort_dissertation], the relative intensities for $x$, $y$, and $z$ polarized light are given by $$\begin{split}
I_x &= \frac{1}{n}(\frac{1}{2}n_{xy} + \frac{1}{2}n_{xz} + \frac{1}{6}n_{z^2} + \frac{1}{2}n_{x^2-y^2}) \\
I_y &= \frac{1}{n}(\frac{1}{2}n_{xy} + \frac{1}{2}n_{yz} + \frac{1}{6}n_{z^2} + \frac{1}{2}n_{x^2-y^2}) \\
I_z &= \frac{1}{n}(\frac{1}{2}n_{xz} + \frac{1}{2}n_{yz} + \frac{2}{3}n_{z^2} )
\end{split}$$ where $I_j$ is the normalized intensity along direction $j$, $n_i$ is the number of holes in orbital $i$, and $n$ is the total number of holes. For high-spin CaFeO~3~, the $t_{2g}$ shell is half-full, and under moderate strains we assume negligible polarization of the $t_{2g}$ orbital occupation, *i.e.*, $$\begin{split}
n_{xy} &= 1\\
n_{xz} &= 1\\
n_{yz} &= 1,\\
\end{split}$$ where we leave the $e_g$ occupation as unknown. This then gives $$\frac{I_x}{I_z} = \frac{\frac{1}{2} + \frac{1}{2} + \frac{1}{6}n_{z^2} + \frac{1}{2}n_{x^2-y^2}}{\frac{1}{2} + \frac{1}{2} + \frac{2}{3}n_{z^2}} = \frac{6 + n_{z^2}+3n_{x^2-y^2}}{6 + 4n_{z^2}},
\label{eq_hole_ratio}$$ where $n_{x^2-y^2}$ and $n_{z^2}$ are the number of holes in the $d_{x^2-y^2}$ and $d_{3z^2-r^2}$ orbitals, respectively. This equation can only be solved if the total $e_g$ occupation is known. As discussed in the main manuscript, we estimate that the total $e_g$ occupation for CaFeO~3~ is 1.85.
III. Optimization of the ligand field model parameters
------------------------------------------------------
We employed a standard multiplet ligand field theory model using the code *Quanty* [@Haverkort_Wannier_PRB_2010; @QuantyWeb], which computes the eigenstates and spectra using exact diagonalization. The model includes the Fe $3d$ shell, a ligand shell comprised of $d$-symmetry linear combinations of oxygen $2p$ orbitals, and the Fe core $2p$ shell (needed for the spectroscopy simulations). Parameters of the model include the local Fe Coulomb and exchange integrals ($F^k_{dd}$, $F^k_{pd}$, and $G^k_{pd}$), for which we use Hartree-Fock determined values [@Cowan_TASS_1981] that are subsequently rescaled to account for atomic as well as solid state corrections using inter- and intra-shell rescaling factors $\kappa_{dd}$ and $\kappa_{pd}$, respectively. We also include the Fe atomic $3d$ and $2p$ spin orbit interaction (66 meV and 8.199 eV, respectively). We further include the octahedral crystal field splitting $10Dq$, as well as the strain induced, tetragonal crystal field distortion $\Delta_{e_g}\left(\equiv 2\Delta_{t_{2g}}\right)$. Also included are the charge transfer energy $\Delta$ as well as the monopole parts of the valence-valence and core-valence Coulomb interactions ($U_{dd} = 6$ eV and $U_{pd}=8$ eV, respectively). Finally, hybridization between the Fe $3d$ and ligand orbitals is included via $O_h$ symmetry hopping integrals $V_{e_g}$ and $V_{t_{2g}}\left(\equiv 0.58V_{e_g}\right)$ [@Robert], and in the XAS final state the hopping integrals are rescaled by $V_f$ to account for orbital contraction due to the core hole.
The parameters used to calculate the CaFeO~3~ x-ray absorption spectra were optimized by comparing the calculated x-ray linear dichroism to the experimental data. A single model parameter was systematically varied and all 10 independent spectra (arising from orbital and spin degeneracies of the initial state as described in the main text) were calculated for each parameter setting. The resulting x-ray linear dichroism spectra from each data set were then used to obtain the best possible fit to the experimental spectrum, illustrated here using the CaFeO~3~/LSAT sample. The parameter value that gave the best fit to experiment (lowest $\chi^2$ value) was taken as the optimized value. After optimizing all six model parameters in this manner, this process was repeated until the optimized parameters converged. The final parameters are shown in Table \[table\_parameters\].
![Goodness of fit ($\chi^2$) for the calculated x-ray linear dichroism as a function of crystal field ($10D_q$), intra-shell Coulomb interaction rescaling ($\kappa_{dd}$), hopping integrals ($V_{e_g}$, with $V_{t_{2g}}=0.58V_{e_g}$ [@Robert]), final state hopping rescaling ($V_f$), and inter-shell Coulomb and exchange interaction rescaling ($\kappa_{pd}$). The relevant set of 10 independent spectra was calculated for each parameter value and then the weights of each spectrum were fit to the experimental CaFeO~3~/LSAT x-ray linear dichroism data to obtain the $\chi^2$ value. \[SI\_Jz\_fitting\]](SI_fig_Jz_fitting_process.eps)
----------- -------------- --------------- ------- --------------- --------------- ---------------
Parameter $10D_q$ (eV) $V_{eg}$ (eV) $V_f$ $\kappa_{dd}$ $\kappa_{pd}$ $\Delta$ (eV)
Value 0.5 2.80 0.80 0.65 0.80 -2.0
----------- -------------- --------------- ------- --------------- --------------- ---------------
: \[table\_parameters\] Optimized parameters for the ligand field simulations of the x-ray absorption and linear dichroism for CaFeO~3~: Crystal field ($10D_q$), hopping ($V_{eg}$), final state hopping rescaling factor ($V_f$), intra-shell Coulomb interaction rescaling factor ($\kappa_{dd}$), inter-shell Coulomb and exchange interaction rescaling factor ($\kappa_{pd}$), and charge transfer energy ($\Delta$).
IV. Breakdown of the $J_z$ fits
-------------------------------
Tables 2 and 3 show the breakdown of the fit of the calculated x-ray linear dichroism spectrum to the experimental CaFeO~3~/LSAT and CaFeO~3~/SLAO spectra, respectively. Under tensile strain, states having preferential $x^2-y^2$ occupation are lower in energy than those having preferential $3z^2-r^2$ occupation, and their respective spin multiplicity derived spectra (labelled by $J_z$) are split in energy by a few meV due to the Fe $3d$ spin-orbit coupling. The resulting coefficient value (fit weight) and its 95% confidence interval are shown for the optimized fit. Although the relative $J_z$ contributions do not follow a Boltzmann distribution [@Haverkort_LaTiO3_Jz_boltzmann], this is not unexpected given that a Boltzmann distribution would not reproduce the inverted orbital polarization. Moreover, small distortions of the local symmetry of the crystal field are expected to change the relative energy alignment of the $J_z$ spectra, particularly given the small separation in energy of roughly 5 meV.
To determine the $e_g$ orbital occupation, the weight of each $J_z$ spectrum is multiplied by its electron occupation in the $x^2-y^2$ and $3z^2-r^2$ orbitals. These reported occupations are the electron count in the Fe $3d$ orbitals only and do not include the ligand hole count. The resulting values are then summed for $d_{x^2-y^2}$ and $d_{3z^2-r^2}$ in order to obtain the $e_g$ electron occupation values.
$J_z$ spectrum Energy above ground state (meV) Fit weight $\pm$95% conf. int. Weight (%) 95% conf. int. (%) $x^2-y^2$ occup. $3z^2-r^2$ occup. Weight% \*occup $x^2-y^2$ Weight% \*occup $3z^2-r^2$
-------------------------------- --------------------------------- ------------ --------------------- ------------ -------------------- ------------------ ------------------- --------------------------- ----------------------------
pref $3z^2-r^2$, $J_z = 0$ 23 3.06 0.57 18.8 3.5 0.71 1.14 0.13 0.21
pref $3z^2-r^2$, $J_z = \pm 1$ 22 5.74 0.89 35.3 5.5 0.71 1.14 0.25 0.40
pref $3z^2-r^2$, $J_z = \pm 2$ 18 0.00 0.36 0.0 2.2 0.71 1.14 0.00 0.00
pref $x^2-y^2$, $J_z = \pm 2$ 5 4.26 0.46 26.2 2.8 1.14 0.71 0.30 0.19
pref $x^2-y^2$, $J_z = \pm 1$ 2 1.06 0.67 6.5 4.1 1.14 0.71 0.07 0.05
pref $x^2-y^2$, $J_z = 0$ 0 2.15 0.54 13.2 3.3 1.14 0.71 0.15 0.09
Sum 16.27 100.0 0.91 0.94
: \[\]$J_z$ fit breakdown for CaFeO~3~/LSAT, $\Delta e_g$ = +40 meV.
$J_z$ spectrum Energy above ground state (meV) Fit weight $\pm$95% conf. int. Weight (%) 95% conf. int. (%) $x^2-y^2$ occup. $3z^2-r^2$ occup. Weight% \*occup $x^2-y^2$ Weight% \*occup $3z^2-r^2$
------------------------------- --------------------------------- ------------ --------------------- ------------ -------------------- ------------------ ------------------- --------------------------- ----------------------------
pref $x^2-y^2$, $J_z = \pm$2 20 2.97 0.18 18.4 1.1 1.15 0.71 0.21 0.13
pref $x^2-y^2$, $J_z = \pm$1 17 3.63 0.3 22.5 1.9 1.14 0.72 0.26 0.16
pref $x^2-y^2$, $J_z = 0$ 16 1.44 0.16 8.9 1.0 1.13 0.72 0.10 0.06
pref $3z^2-r^2$, $J_z = 0$ 7 1.62 0.24 10.1 1.5 0.71 1.14 0.07 0.11
pref $3z^2-r^2$, $J_z = \pm1$ 4 1.96 0.26 12.2 1.6 0.71 1.14 0.09 0.14
pref $3z^2-r^2$, $J_z = \pm2$ 0 4.49 0.22 27.9 1.4 0.71 1.14 0.20 0.32
Sum 16.11 100.0 0.93 0.93
: \[\]$J_z$ fit breakdown for CaFeO~3~/SLAO, $\Delta e_g$ = -30 meV.
V. Anisotropic band broadening scenarios
----------------------------------------
Bandwidth effects are illustrated in Fig. \[Fig\_bandwidths\] for three scenarios: a less-than-half-filled band, a half-filled band, and a greater-than-half-filled band. In the simplest picture, the unstrained system has equal bandwidths in all three directions. Applying tensile strain ($\Delta e_g > 0$) lowers the $x^2-y^2$ band in energy relative to the $3z^2-r^2$ band and, as seen in Fig. \[Fig\_bandwidths\](a), the $x^2-y^2$ band is more occupied. For the less-than-half-filled band (Fig. \[Fig\_bandwidths\](a)) and the half-filled band (Fig. \[Fig\_bandwidths\](b)), adding in the tensile strain-induced anisotropic bandwidth effects, where the in-plane $x^2-y^2$ band broadens and the out-of-plane $3z^2-r^2$ band narrows, still results in $x^2-y^2$ being more occupied and thus does not replicate our findings.
In contrast, for a greater-than-half-filled band the band broadening can result in an inverted orbital polarization. As seen in Fig. \[Fig\_bandwidths\](c), band broadening results in the higher energy edge of the $x^2-y^2$ band surpassing the $3z^2-r^2$ band edge, and the $3z^2-r^2$ band becomes more occupied than $x^2-y^2$. From this simple illustration, an inverted orbital polarization can be expected when the net change in bandwidth is greater than the strain-induced $\Delta e_g$ and the bands are more-than-half-filled.
![Simplified schematic of the effect of changes in bandwidth on the resulting orbital polarization for a system under tensile strain ($\Delta e_g>0$), which shifts the band center of masses about the unstrained center. Broadening of the in-plane band and narrowing of the out-of-plane band for the (a) less-than-half-filled band and (b) half-filled band results in the conventional preferential $x^2-y^2$ orbital polarization. (c) For a band with greater than half-filling, the same broadening can result in an inverted orbital polarization with $3z^2-r^2$ preferentially occupied. \[Fig\_bandwidths\]](SI_fig_band_broadening.eps)
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'S. Théado'
- 'G. Alecian'
- 'F. LeBlanc'
- 'S. Vauclair'
date: 'Received September 15, 2011; accepted'
title: 'The new Toulouse-Geneva Stellar Evolution Code including radiative accelerations of heavy elements'
---
[Atomic diffusion has been recognized as an important process that has to be considered in any computations of stellar models. In solar-type and cooler stars, this process is dominated by gravitational settling, which is now included in most stellar evolution codes. In hotter stars, radiative accelerations compete with gravity and become the dominant ingredient in the diffusion flux for most heavy elements. Introducing radiative accelerations into the computations of stellar models modifies the internal element distribution and may have major consequences on the stellar structure. Coupling these processes with hydrodynamical stellar motions has important consequences that need to be investigated in detail.]{} [We aim to include the computations of radiative accelerations in a stellar evolution code (here the TGEC code) using a simplified method (SVP) so that it may be coupled with sophisticated macroscopic motions. We also compare the results with those of the Montreal code in specific cases for validation and study the consequences of these coupled processes on accurate models of A- and early-type stars.]{} [We implemented radiative accelerations computations into the Toulouse-Geneva stellar evolution code following the semi-analytical prescription proposed by Alecian and LeBlanc. This allows more rapid computations than the full description used in the Montreal code.]{} [We present results for A-type stellar models computed with this updated version of TGEC and compare them with similar published models obtained with the Montreal evolution code. We discuss the consequences for the coupling with macroscopic motions, including thermohaline convection.]{}
Introduction
============
Accurate stellar modeling has recently been given a new boost with the advent of asteroseismology. The observations of oscillating stars and the analysis of the stellar oscillation properties brought new and powerful constraints on stellar models and allowed major progress in our understanding of stellar internal structure. Furthermore, since the discovery of the first extrasolar planets [@Wolszczan92; @Mayor95], the spectacular development of the exoplanet research field has also sparked renewed interest in stellar physics, the accurate knowledge of the host star being a necessary condition for characterizing the surrounding planets.
One of the most important successes of astrophysics was the understanding of the basics of stellar internal structure and evolution. This progress was supported by the computations of numerical models. Growing computational resources contributed to refining the description of the physics of the stellar medium (equation of state, opacities, nuclear reactions, etc) and allowed building a simplified but efficient and widely used “standard model”. However, this standard model does not take the effects of rotation and magnetic fields or the occurrence of accretion or mass loss into account, and it considers convection as the only chemical transport process.
Observations of chemical abundance anomalies in stars and unexpected stellar seismic behaviors have proved the necessity of including “non standard processes” in stellar evolutionary computations. In this framework, the main challenges encountered today by stellar physicists are to better determine the effects of rotation and magnetic fields and understand the chemical transport processes better. This last point, and more specifically modeling of atomic diffusion including the radiative accelerations on individual elements, represents a key ingredient for accurate stellar modeling.
The importance of atomic diffusion inside stars is well established: not only can it modify the atmospheric abundances [@Michaud70], as observed in the so-called “chemically peculiar stars”, but it can also have strong implications for the stellar internal structure [e.g. @Richard01]. In main-sequence solar-type and cooler stars (below about 1.2 M$_{\odot}$), the radiative accelerations on the heavy elements are generally slower than gravity in absolute value, owing to the small radiation flux compared to hotter stars [@MichaudMiChVaetal1976]. The elements heavier than hydrogen sink, even if the efficiency of this sinking may be modulated by the radiative accelerations [@Turcotte98].
One of the biggest success of helioseismology was to prove the importance of atomic diffusion in the Sun. Its introduction in solar evolutionary models has significantly improved the agreement between the sound speed profile inside solar models and that deduced from helioseismic inversion techniques [e.g. @JCD96; @Richard96; @Brun98; @Turcotte98; @Schlattl02]. (This agreement has however been spoiled by the new abundances proposed by Asplund et al. 2005, 2009.) Gravitational settling is now introduced in most stellar evolution codes.
The effects of atomic diffusion on the seismic frequency of main-sequence, solar-type star models have been studied by [@Theado05]. They have shown that element segregation significantly alters the internal structure of the models and their oscillations frequencies. The frequency differences between models with and without diffusion reach several microHertz for stars with masses greater than 1.3M$_{\odot}$.
In hotter stars, the radiative accelerations may become significantly stronger than gravity for many metals, which are pushed up. The variations with depth of the radiative accelerations of specific elements combined with the selective effects of gravitational settling can lead to element accumulation or depletion in various stellar layers. In A and B-type stars, iron-peak element accumulation appear in the Z-opacity bump (located at $\simeq$200’000K). The induced opacity increase may lead to local convection [@Richer00; @Richard01].
The iron-peak element accumulation in the opacity bump region can help trigger stellar pulsations, therefore improving the agreement between seismic observations and theoretical frequency spectra in many stars: e.g. in $\gamma$ Doradus, Am, SPB, $\beta$ Cephei or sdB stars [see @Theado09 for a detailed discussion of this subject].
The effects of the radiative accelerations on the oscillation frequencies have been tested by @Escobar12. Very weak effects are observed for models with masses up to 1.28M$_{\odot}$, but significant effects are expected for more massive stars.
Progress in the understanding of the physics of early-type stars is limited by the radiative accelerations not being computed in most stellar evolutionary codes. Up to now, the Montreal stellar evolution code is the only one in which a complete, accurate, and consistent treatment of radiative accelerations has been introduced [@Richer00; @Richard01; @Turcotte00]. The price to pay for this accuracy is that the heavy and CPU-time consuming computations of atomic processes do not allow additional sophisticated treatments of macroscopic motions. In the Montreal code, turbulence is only treated as an extremely simplified process with a turbulent diffusion coefficient proportional to a parameterized power of the density.
In this context, we introduced into the Toulouse-Geneva Evolution Code (hereafter TGEC) new computations of the radiative accelerations on heavy elements, following the semi-analytical prescription proposed by @Alecian02 and @LeBlanc04. This prescription leads to fast but reasonably accurate computations, which represent a good compromise between accuracy and CPU-time consumption and allows coupling with macroscopic motions. Such a treatment of abundance variations inside early-type stars is necessary for a good understanding of these stars.
In the present paper, we do not introduce thermohaline convection as described in @Theado09 because we want to present a detailed comparison with the Montreal code in which this specific physical process is not included. Very precise tests of the results obtained by the two codes for iron accumulation inside nearly identical models with similar physics are still underway. In particular, the computations of iron fluxes seem to lead to differences in some cases, which still have to be understood. These tests are beyond the scope of the present paper and will be presented elsewhere in the near future. Here we present a first step in the comparisons, namely the detailed study of the radiative accelerations obtained with the two methods. Some abundance profiles are only shown as indicators of the results that the TGEC code is presently able to obtain.
In the following section, we explain in detail the major improvements implemented in the TGEC code. In Sect. 3, we present a comparison between the Montreal and TGEC computations for two similar stellar models. In Sect. 4, additional models are presented to illustrate the capabilities of the updated TGEC version and its application fields. Our conclusions are given in Sect. 5.
New opacities and atomic diffusion computations in TGEC {#implementation}
=======================================================
The Toulouse-Geneva stellar evolution code is described in detail in @Hui08; however, the code has recently undergone major improvements that are reported in the following sections. The code can follow the time-dependent abundance variations of 21 species (12 elements and their main isotopes: H, $^3$He, $^4$He, $^6$Li, $^7$Li, $^9$Be, $^{10}$B, $^{12}$C, $^{13}$C, $^{14}$N, $^{15}$N, $^{16}$O, $^{17}$O, $^{18}$O, $^{20}$Ne, $^{22}$Ne, $^{24}$Mg, $^{25}$Mg, $^{26}$Mg, $^{40}$Ca, and $^{56}$Fe) in detail. The remaining metals are collected into an average species Z.
Opacities {#opacity}
---------
In the TGEC code, the opacities are computed using the OPCD v3.3 codes and data [@Seaton05]. They allow computating self-consistent Rosseland opacities taking the detailed composition of the chemical mixture into account. The opacities are recalculated by considering the abundance variations at each time step and at each mesh point.
Atomic diffusion {#dif}
----------------
### Computational methods
The diffusion computations are based on the Boltzmann equation for dilute collision-dominated plasma. At equilibrium, the solution of the equation is the Maxwellian distribution function. Transport properties in stars are computed considering small deviations from the Maxwellian distribution. In this framework, two different formalism have been proposed to obtain approximate solutions to the Boltzmann equations.
The first method lies on the Chapman-Enskog theory [@Chapman70], which assumes that the total distribution function of a given species can be written as a convergent series, each term of the series representing successive approximations to the distribution function. The formalism is first applied to a test element diffusing in the stellar plasma, taking the diffusion of electrons and the induced electric field into account. The computations lead to a statistical “diffusion velocity" of the test-element with respect to the main component of the plasma. In stars, this basic component is generally hydrogen, so that the diffusion of every element is computed with respect to neutral hydrogen and/or to protons. Meanwhile, the hydrogen abundance is renormalized to recover the abundance consistency and local stellar equilibrium. For the case of helium, which has a non-negligible abundance, corrections on the diffusion velocity as proposed by [@Montmerle76] are added.
The second method has been developed by @Burgers69. It is based on the Grad 13 moment approximation and the use of a Fokker-Planck collision term in the Boltzmann equation. In this approach, separate flow and heat equations for each component of a multicomponent mixture are solved simultaneously. This method provides a more convenient way for handling multicomponent gases than the Chapman-Enskog one but is heavier to apply. In both methods, the diffusion coefficients can be written as functions of the collisions integrals that depend on the exact nature of the interaction between colliding particules.
In TGEC, the atomic diffusion is computed following the [@Chapman70] formalism. We use the gravitational and thermal diffusion coefficients as derived by @Paquette86, who performed a detailed computation of the collision integrals for ionized elements diffusing in an ionized medium. In the case of the diffusion of neutral atoms in ions, or reverse, the polarization of the neutrals is taken into account as proposed by @Michaud78. For neutrals atoms diffusing in a neutral medium, the rigid sphere approximation is used. The average diffusing charge $Z_i$ of the particules is computed by solving the Saha-Boltzmann equations. The partition functions are limited to the statistical weight of fundamental levels, and no correction for high density or degenerate electrons is introduced. The set of Saha-Boltzmann equations are solved by using a Newton-Raphson iterative scheme. When the computations of radiative accelerations are added to the diffusion computations, the Chapman-Enskog method is easier to handle than Burgers’. This situation helps treating cases with short time scales and rapid variations, as shown in Sect. 4.
### Radiative accelerations {#raddif}
The radiative accelerations on C, N, O, Ne, Mg, Ca, and Fe have been included in the TGEC code following the improved version of @LeBlanc04 of the semi-analytical prescription proposed by @Alecian02. This method allows very fast computation of radiative accelerations with a reasonable accuracy. Radiative accelerations due to bound-bound (@AlecianAl1985, @AlecianAlAr1990a) and bound-free [@AlecianAl1994] transitions are obtained using a parametric form of the radiative acceleration equation. The basic idea of this parametric method is to derive formula where the terms depending explicitly on atomic data (such as $gf$ values for instance) are separated from those depending on the stellar plasma and the abundances of the considered ion (with the aim of accounting for the saturation effects). In this framework, the radiative accelerations may be approximated by calculating a single value for each parameter found in the related equations. This is the so-called single-valued parameter (or SVP) approximation [@LeBlanc04].
The interface of the SVP method with the models consists of a set of six parameters per ion, which allows to estimate the radiative acceleration of each element through simple algebraic expressions [@LeBlanc04], for each time step of the run of the evolution code. These parameters are determined only at the beginning of the computation through interpolation as a function of the stellar mass, inside a pre-established grid. The computation of the total radiative acceleration for a given element with SVP also requires computing the ions’ relative populations. This is included in the set of the SVP numerical routines added to TGEC.
The SVP approximation may be implemented in existing codes. There are much less data to process than for detailed radiative acceleration calculations because complete and detailed monochromatic opacities for each element are not needed. A new grid of SVP-parameters, well fitted to the stellar mass range considered in this work, was computed following the procedure described in @LeBlanc04. An implementation of the SVP method was first used in @Theado09.
Comparison with the Montreal computations {#compar}
=========================================
In this section, we compare the TGEC results to those obtained with the Montreal code. For this purpose we compare the internal structures and the results of the diffusion computations for two models: one computed with the Montreal code, the other computed with TGEC. The Montreal model chosen for this comparison is the 5.3D50-3, 1.7M$_{\odot}$-model presented in @Richard01. A similar model was computed with TGEC, including input physics and initial parameters as close as possible to the Montreal model. As a first step, we present comparisons of seven elements (listed in Table 1), which are especially important for A type stars.
The detailed physics of the Montreal model can be found in @Richard01, @Richer00, and @Turcotte98. The key ingredients are reported in the following sections and compared to those introduced in the TGEC model.
Input physics
-------------
### Basic input physics {#classic}
Here is a list that compares the basic input physics used in the TGEC and Montreal models:
#### Equation of state:
the Montreal model uses the CEFF equation of state [@JCD92]. The TGEC model is computed using the OPAL2001 [@Rogers02] equation.
#### Opacities: {#opacities}
in both models, the opacities are computed at every point in the star and at each evolution time step. The Montreal computations take the local abundance of 21 chemical elements into account [see Table 1 of @Turcotte98] using the OPAL monochromatic data. In TGEC, the opacities are computed as described in Sect. \[opacity\] considering the detailed abundance of the elements listed in Table \[initabund\].
#### Nuclear reactions:
the Montreal code uses the nuclear energy generation routine of [@Bahcall92]. The nuclear reaction rates used in TGEC are from the analytical formulae of the NACRE compilation [@Angulo99].
#### Convection:
in both Montreal and TGEC models, the delimitation of the convective zones is based on the Schwarzschild criterion. The energy flux in the convection zones is computed following the Böhm-Vitense mixing length formalism. In the Montreal model, convective zone mixing is modeled as a diffusion process, described by a diffusion coefficient $D_{mix}$ (cf. Sect. \[transportmacro\]). A mixing length approximation is used for $D_{mix}$, which always leads to high $D_{mix}$ values and very homogeneous convective zones. In TGEC, convective zones are assumed to be instantaneously homogenized. The mixing length parameter $\alpha$ is taken to be equal to 1.68 in both the TGEC and Montreal models.
#### Initial chemical mixture:
the initial mixture used in the Montreal model is given in Table 1 of @Turcotte98. The initial composition introduced in the TGEC model is given in Table \[initabund\] of this paper: the chemical elements followed in both models are introduced with the same initial abundance.
[c c]{} H & 0.70\
($^3$He+$^4$He) & 0.28\
$^{12}$C & $3.466 \times 10^{-3}$\
$^{14}$N & $1.063 \times 10^{-3}$\
$^{16}$O & $9.645 \times 10^{-3}$\
$^{40}$Ca & $7.469 \times 10^{-5}$\
$^{56}$Fe & $1.436 \times 10^{-3}$\
### Atomic diffusion {#atomic-diffusion}
In the Montreal code, atomic diffusion is computed as described in @Richard01 and @Turcotte98. The Montreal calculations take the time-dependent variations of 28 species (including isotopes) into account: H, $^3$He, $^4$He,$^6$Li, $^7$Li, $^9$Be, $^{10}$B, $^{11}$B, $^{12}$C, $^{13}$C, N, O, Ne, Na, Mg, Al, Si, P, S, Cl, Ar, K, Ca, Ti, Cr, Mn, Fe, and Ni. Monochromatic OPAL data are used to calculate the Rosseland opacities and the radiative accelerations at each evolution time step. The abundances are updated at every iteration over the stellar structure. The diffusion coefficients and velocities are determined by solving the Burgers’s flow equations for ionized gases [@Burgers69] for all diffusing elements. The collisions integrals are from @Paquette86.
As a first step, we include, in the TGEC model, the atomic diffusion (including radiative accelerations) of seven elements (eight species including isotopes), especially important for A type stars: H, $^3$He, $^4$He, $^{12}$C, $^{14}$N, $^{16}$O, $^{40}$Ca, and $^{56}$Fe. The atomic diffusion is computed as described in Sect. \[dif\]. The diffusion velocities are computed following the @Chapman70 formalism in the test-atom approximation, and the diffusion coefficients are from @Paquette86.
### Macroscopic transport of chemical elements {#transportmacro}
*Macroscopic transport in the Montreal code.*\
In the Montreal code, macroscopic transport processes, including (semi)convection and turbulent mixing, are modeled as diffusion processes by adding a pure diffusion term to each element’s diffusion velocity $V_{pi}$: $$V_i(macro)=-(D_T+D_{mix}) \frac{\partial \ln X_i}{\partial r}
\label{eqvdif2}$$ where $D_T$ and $D_{mix}$ are turbulent diffusion coefficients with $D_{mix}$ representing the effects of convective and semiconvective motions and $D_T$ parametrizing turbulent transport. The variable $X_i$ represents the mass fraction of species $i$.
In radiative layers $D_{mix}=0$, while in convective zones it is computed using the mixing length theory (as stated in Sect. \[classic\]). In semiconvection zones, $\rm D_{mix}$ is assumed proportional to $(\nabla-\nabla_{ad})/(\nabla_L-\nabla)$, where $\nabla$ and $\nabla_{ad}$ stands for the local and adiabatic logarithmic temperature gradients and $\nabla_L$ is a function of $\nabla_{\mu} = d \ln \mu / d \ln P$. We refer the reader to Sect. 2.2 of @Richard01 for a more detailed description of the treatment of convection.
In the 5.3D50-3-model, the turbulent transport is chosen strong enough to guarantee, during the whole evolution period considered, a complete mixing throughout the region between the surface and the point where $\log T_0=5.3$ [see Fig. 2 of @Richard01]. This mixing mimics the effects of a Fe convection zone except that it is imposed throughout evolution. Below $\log T_0=5.3$, the diffusion coefficient $\rm D_T$ obeys the following algebraic dependence on density [Eq. 1. of @Richer00]: $$\label{eqdt}
D_T=\omega D(He_0) \biggl(\frac{\rho_0}{\rho} \biggr)^n$$ where $\rho_0=\rho(T_0)$, $\omega=50$, $n$=3 and $$D(He)=3.3 \times 10^{-15} T^{2.5} / [4 \rho \ln (1+1.125\times 10^{-16}T^3/\rho)].$$ *Macroscopic transport in the TGEC code.*\
In the TGEC code, the macroscopic motions of a given species $i$ are parametrized, as in the Montreal code, by including a turbulent diffusion term to the diffusion velocity of the considered species: $$V_i(turb)=-D_{T} \frac{\partial \ln X_i}{\partial r}.
\label{eqvdif3}$$ The TGEC code does not include a $D_{mix}$ term, since convection is treated as an instantaneous dilution. The diffusion coefficient $D_{T}$ is chosen to mimic the mixing $(D_T+D_{mix})$ introduced in the 5.3D50-3 Montreal model. Figure \[figdt\] represents this diffusion coefficient inside our model, it may be compared to Fig. 2 of [@Richard01]. From the surface down to $\log T=5.3$, $D_T$ is chosen to homogenize the stellar material, below this region $D_{turb}$ is computed using the same expression as introduced in the Montreal code (i.e. following Eq. \[eqdt\]).
Results
-------
The Montreal and TGEC codes compute stellar evolution from pre-main sequence up to the subgiant branch. They were assumed to be homogeneous on the pre-main sequence, atomic diffusion, and turbulent transport were introduced at the beginning of the main-sequence phase.
Figure \[diaghr2\] compares the evolutionary tracks of the two models (for clarity the pre-main-sequence evolution is not shown). In spite of different treatments of some physical ingredients (e.g. equation of state, opacities, nuclear reactions, diffusion), the two tracks appear relatively close to each other.
![Evolutionary tracks of two 1.7M$_{\odot}$ sequences computed with TGEC or with the Montreal code. The dashed curve represents the 5.3D50-3 model presented in @Richard01 (and computed with the Montreal code). The solid curve represents a model computed with TGEC and including similar input physic. The black dots and crosses respectively represent the 30 and 400Myr-models.[]{data-label="diaghr2"}](fig2.eps){width="50.00000%"}
We first give a detailed comparison between the Montreal and TGEC calculations for the two 30Myr models. Figure \[internalstruc\] compares the internal structure of these two models. The relative differences in pressure, temperature, and density never exceed 6% (and even less for P and $\rho$). The maximum relative differences are observed near $ \log (\Delta M/M) \simeq -6.3$, which corresponds to the base of the potential iron convective zone.
Similarly to Fig. 7 of @Richard01, Fig. \[grad\] displays the radiative accelerations and the local gravity of the two models. The discrepancy between g$_{MTR}$ and g$_{TGEC}$ remains smaller than around 0.3 dex except for O in the upper layers (in the convection zone). We notice that the average accuracy of the SVP method is estimated to $\pm$0.1 dex [@Alecian02] with respect to the detailed computations of [@Seaton97]. However, the SVP approximation is less accurate for light elements (lighter than O) than for heavy ones because this method uses parameters determined at the position of the maximum of each ion relative population. Because there are fewer ionization states for light elements, the gap in temperature between these maxima is larger for them. The comparison of Fig. \[grad\] is fairly satisfactory, since radiative accelerations are computed in completely different ways and use different atomic and opacity databases.
{width="32.00000%"}{width="32.00000%"}{width="32.00000%"} {width="32.00000%"}{width="32.00000%"}
Figure \[vdif\] presents the diffusion velocities of several elements for the same models. Despite the different atomic diffusion prescriptions used, the diffusion velocities are quite close in most of the star.
Figure \[abond\] shows the abundance profiles in the two 30 and 400Myr-models. At 30 Myr the profiles are quite close for all the species followed in TGEC. However, at 400 Myr discrepancies in the abundance profiles are observed. These differences are small for most elements (He, C, N, O, and Fe) except for Ca. As shown in Fig. \[diaghr2\], the positions of the two 400Myr models in the HR diagram are slightly different, and are in particular more distant from each other than the 30Myr models. As a consequence, their internal structure is also expected to be different. In this context, it seems difficult to disentangle the discrepancies in the models due to structure variations from those caused specifically by the difference in diffusion calculations.
{width="32.00000%"}{width="32.00000%"}{width="32.00000%"} {width="32.00000%"}{width="32.00000%"}{width="32.00000%"}
[cccccccc]{} & Age (Myr) & $X_c$ & $|\Delta X_c|/X_c$ & $L/L_{\odot}$ & $|\Delta L|/L$ & $T_{eff}$ (K) & $|\Delta T_{eff}|/T_{eff}$\
M$_{MTR}$ & 30.2 & 0.6938 & 0.03% & 8.289 & 1.25% & 7975.66 & 0.18%\
M$_{TGEC}$ & 30.2 & 0.6940 & & 8.393 & & 7989.90 &\
M$_{MTR}$ & 400 & 0.5778 & 0.81 % & 8.951 & 1.51% & 7659.49 & 0.70%\
M$_{TGEC}$ & 400 & 0.5731 & & 9.086 & & 7713.35 &\
\[tab\]
Application field of the TGEC code
==================================
In this section, we propose to demonstrate the abilities of the TGEC code and its application field. Models of A-F stars including atomic diffusion and minimal mixing (i.e. only a mild mixing below the convective zones to avoid steep and unrealistic composition gradients at the transition between radiative and convective regions) have already been presented in @Theado09. These models, which were evolved until the end of the main-sequence phase, have shown the ability of the code to compute complete evolutionary tracks including radiative levitation effects in the presence of minimal mixing.
In this paper, we propose another test of the abilities of the code to manage rapid variations in the chemical composition and the thermal structure. For this purpose we present a set of models computed with TGEC with masses ranging from 1.5 to 2.5M$_{\odot}$. Like the models presented in [@Theado09], they include atomic diffusion and a mild mixing below the convective zones. In the present models, the iron convective zone, when it appears, is assumed to be connected and completely mixed with the H/He convective envelope. These models were evolved from pre-main sequence up to hydrogen-core exhaustion. They were assumed to be chemically homogeneous on the pre-main sequence. Atomic diffusion was introduced at the beginning of the main-sequence phase. Because the present computations are done to test the TGEC code, we did not include here the thermohaline convection that must be added for comparison with real stars [@Theado09].
\[computations1\]
Input physics
-------------
As previously described, our models were computed using the OPAL2001 [@Rogers02] equation of state. The nuclear reactions were from the NACRE compilation [@Angulo99], and the opacities were computed as described in Sect. \[opacity\]. Atomic diffusion (including the radiative forces) was introduced as described in Sect. \[dif\], for the following elements: H, $^3$He, $^4$He, $^{12}$C, $^{14}$N,$^{16}$ O, $^{40}$Ca, and $^{56}$ Fe. The initial metal mixture was the solar mixture of [@Grevesse93]. The initial mass fractions for the diffusing species are given in Table \[table1\]. Convection was computed using the mixing length theory with a mixing length parameter $\alpha=1.8$. The HI and HeII convective zones were assumed to be connected by overshooting and mixed together. As stated previously, the iron convective region that may appear during main-sequence evolution was also assumed connected and completely mixed with the surface convective region.
[c c]{} H & 0.7112\
($^{3}$He+$^{4}$He) & 0.2714\
$^{12}$C & 0.2981447E-02\
$^{14}$N & 0.9218931E-03\
$^{16}$O & 0.8375229E-02\
$^{40}$Ca & 0.6283632E-04\
$^{56}$Fe & 0.1148674E-02\
To avoid the appearance of sharp and nonphysical abundance gradients at the transition between radiative and convective regions, we introduced mild mixing below the surface convective zone. This mixing was modeled as a diffusion process [@SchatzmanSc1969] as described in Eq. \[eqvdif3\] with
$$D_{T}=D_{czb} \exp \left( \ln 2 \frac{r-r_{czb}}{\Delta} \right),
\label{eq5}$$
where $D_{czb}$ and $r_{czb}$ are the value of $D _{T}$ and the value of the radius at the base of the convective zone, respectively and $\Delta$ is the half width of the mixing region, $D_{czb}$ and $\Delta$ are free parameters chosen to produce a mild mixing to a small extent. The value of $D _{czb}$ is taken as equal to $2 \times 10^5$cm$^2$.s$^{-1}$, and $\Delta$ is fixed to 0.5% of the radius below the surface convective region. Figure \[dtacho\] presents this diffusion coefficient below the surface convective zone of 1.7M$_{\odot}$ models at two evolutionary stages. At 170 Myr, the surface convective region includes the H and He convective zones (there is no Fe convective zone), and at 588 Myrs the external convective zone is composed of the H, He, and Fe convective zones assumed connected and mixed.
The physics introduced in these models is close to that of the r30-3M, 1.5M$_{\odot}$-model presented in @Richard01. Like our models, the r30-3M model includes mild mixing below the He convective zone. The turbulent diffusion coefficient is chosen as large as $\rm 10^4 cm.s^{-1}$ at the base of the convective zone. It then rapidly decreases with depth, varying like $\rho^{-3}$. Turbulent transport is chosen to be large enough to guarantee complete mixing between the Fe and He convective zones whenever the Fe convective zone appears. The initial metal mixture and the mixing length parameter are, however, different in the Montreal and TGEC models. For CPU-time consuming reasons, the evolution of the Montreal model was stopped at 89 Myrs, and the results for a 1.5M$_{\odot}$ model are the only ones presented. TGEC models with masses up to 2.5M$_{\odot}$ are computed along the whole main-sequence phase. Since the physics of our models and the r30-3M Montreal model are close but not similar, no detailed comparison between them can be carried out. However, qualitative similarities are underlined.
Results {#section:3.2}
-------
Figure \[diaghr\] displays the evolutionary tracks of our models. (For clarity pre-main-sequence evolution is not represented.) In the following, we describe, as a representative example, the results obtained for a 2.1M$_{\odot}$ model. We focus on iron-diffusion related features.
Figure \[masse21a\] presents the profiles of various quantities inside the 2.1 M$_{\odot}$ model along the main-sequence evolution. The left and middle columns illustrate early main-sequence evolution, the right hand column displays later evolutionary stages.
{width="32.00000%"} {width="32.00000%"} {width="32.00000%"}
{width="32.00000%"} {width="32.00000%"} {width="32.00000%"}
{width="32.00000%"} {width="32.00000%"} {width="32.00000%"}
On the main sequence, the competition between the gravitational settling and the radiative acceleration leads to iron accumulations in the outer regions of the model. We note iron enrichment in the surface H/He convective zone and in the Z-bump region (at around $\rm \log (\Delta M/M) =-7$ ($\rm T \simeq 200000K$). In this region, iron enrichment significantly increases the opacity, hence the radiative gradient, and when this radiative gradient exceeds the adiabatic gradient, the so-called iron convective zone appears. Here it is assumed to be connected and mixed up to the H/He surface convective zone through overshooting. As a result, the bottom of the surface convective zone sinks abruptly to $\rm \log (\Delta M/M) \simeq -7$. The dilution in this thick convective zone decreases the iron abundance in the opacity bump: the opacity and the radiative gradient consequently decrease, which leads to the disappearance of the iron convective zone. The surface convective zone then recedes. A new cycle starts as radiative forces proceed in accumulating iron in the opacity bump region. The middle column of Fig. \[masse21a\] shows a second iron accumulation/deep convection phase.
The alternation between convective and radiative episodes in the Z-bump proceeds from $\simeq$20 to $\simeq$115 Myr. A thick convective zone then stays for the subsequent main-sequence evolution. The right hand column of Fig. \[masse21a\] illustrates the internal structure variations of our model during the later evolutionary phases.
Figure \[zconvmzoom\] shows the position of the bottom of the surface convective zone of our model during the main sequence. At the beginning of the main-sequence phase, the convective zone includes the H and He ionization regions. After a few million years and because of the He gravitational settling, the HeII convective zone disappears leading to an abrupt recession of the convective region (whose bottom subsequently lies at $\rm \log (\Delta M/M) \simeq-11.25$). After this first receding episode, the convective region undergoes a 100Myr-period of rapid variations during which the convective depth oscillates between $\rm \log (\Delta M/M) \simeq -11.2$ and $\rm \log (\Delta M/M) \simeq -6.5$. After this period, a thick surface convective zone appears that slowly deepens inside the interior during the rest of the main-sequence phase.
{width="40.00000%"} {width="40.00000%"}
The lasting thick convective zone results from two features explained below.
- Because of structural variations forced by the star’s evolution, the iron accumulation needed to induce convection in the opacity bump varies along the main sequence. After 115 Myrs, in particular, smaller iron accumulations in the Z-bump are needed to initiate the iron convective zone.
- The succession of convective sinking and receding episodes, which occurs from 20 to $\simeq$115Myr, leads to rapid variations in the surface iron abundance. As an illustration of this phenomenom, Fig. \[fesurf50\] displays the iron abundance evolution over 30Myrs (including 5 convective sinking/receding episodes).
![Time dependent variations in the Fe surface abundance in the 2.1M$_{\odot}$ model presented in Fig. \[diaghr\]. The iron abundance is shown on a 30Myr period including 5 convective sinking/receding episodes. This graph is similar to Fig. 16 of @Richard01 (see text for more details).[]{data-label="fesurf50"}](fig11.eps){width="50.00000%"}
The global effect of these episodes is illustrated in Fig. \[fesurf\], which presents the time-dependent variations of the averaged iron surface abundance (averaged over 20 Myrs).
The diffusion-induced iron enrichments (in the Z-bump but also in the H/He convective zone) combined with the deep convective mixing episodes leads on average to an iron enrichment of the surface convective zone. For high values of this enrichment, the convective dilution in the presence of an iron convective zone may become unable to efficiently decrease the iron abundance in the Z-bump and to suppress the iron convective zone. A thick convective zone then persists during the subsequent evolution.
According to Fig. \[fesurf\], the iron abundance in the surface convective zone continues to increase after 115 Myrs (because of radiative accelerations), while the convective zone slowly deepens. After 400Myr, the bottom of the convective zone reaches iron-underabundant regions, which causes as a slow decrease in the iron surface abundance.
Figure \[zconvm\] presents the convective zone extension (vs time) in all computed models. For an easy comparison with other masses, the 2.1 M$_{\odot}$ model is shown in this figure. A succession of convective sinking and receding episodes is observed in all the computed models. This period of rapid oscillations of the convective depth is shorter for higher stellar masses. In the 1.5 M$_{\odot}$ model, it persists during most of the main-sequence phase, whereas in the 2.5 M$_{\odot}$ model, a thick convective zone appears in less than 80 Myrs.
{width="32.00000%"} {width="32.00000%"} {width="32.00000%"} {width="32.00000%"} {width="32.00000%"} {width="32.00000%"}
The patterns observed during the early main-sequence phase of our models are consistent with the results presented by [@Richard01] for their r30-3M, 1.5M$_{\odot}$ model. The alternation between radiative and convective episodes in the opacity bump that occurs in our models is similarly observed in the Montreal model as described in Sect. 3 (paragraph “Model with mixing episodes”) of [@Richard01]. The comparison between our Fig. \[fesurf50\] and Fig. 16 of [@Richard01] (middle panel) also shows similar behaviour to the surface iron abundances. But while the Montreal code is limited by time consuming issues (their r30-3M, 1.5M$_{\odot}$ model is evolved for less than 100 Myr), TGEC is able to manage the rapid structural and compositional variations due to the succession of convective sinking and receding episodes and allows computing complete main-sequence evolutionary tracks for masses up to at least 2.5M$_{\odot}$.
Conclusion
==========
The implementation of radiative diffusion effects in stellar modeling turns out as a necessary condition for any computations of accurate models of F, A, and B type stars. Different ways of introducing these effects are considered by theoreticians, but few reliable methods are presently known.
The most accurate method consists in computing the time-dependent variations of the chemical species present in the stellar mixture while using a precise description of the diffusion process. This includes an accurate determination of the radiative accelerations applied to each species and the resolution of the Burgers equations to obtain the diffusion velocities. Such a method has been successfully introduced in the Montreal code, which has been used for many years for the modeling of main-sequence F-A-B type stars. This accurate method allows precise determinations of abundance variations due to atomic diffusion, but it leads to highly time-consuming codes, which makes the computations of complete main-sequence evolutionary tracks difficult.
The development of asteroseismology during the past 30 years and, in particular, the discovery of a wide variety of pulsating F, A, and B-type stars have brought to light the necessity for developing stellar modeling tools able to provide accurate models adapted to seimic analysis for these stars. The seismic modeling of a target star is an iterative process that requires computating numerous models before getting the best one. In this context, using a code like the Montreal one appears particularly unadapted.
This encouraged us to develop an alternative code that includes a more flexible prescription of the radiative accelerations. In this context we included the radiative diffusion in the TGEC code using the semi-analytical prescription proposed by @Alecian02 and @LeBlanc04 (called the SVP method). This method, less accurate than the one introduced in the Montreal code, has been proven to give good results with reasonable precision. With the most recent version of TGEC, this method has been successfully introduced for the first time in a stellar evolution code. In TGEC, the diffusion velocities are then computed following the Chapman & Cowling formalism in the test-atom approximation. The time-dependent variations in the chemical species are computed consistently.
To validate our code, we did a comparison between a model computed with the Montreal code and a similar model computed with TGEC. (The two models include input physics that are as close as possible.) The comparison shows good agreement in the computations of the radiative acceleration for models with similar internal structures.
Nickel is not presently included in the TGEC computations, as its atomic data are not included in TOPBase [@Cunto93], the atomic data base of the Opacity Project [@Seaton92] from which the atomic data used in the SVP approximation are taken. This explains its absence in the SVP package. Its introduction in TOPBase[^1] as well as in the SVP package, is underway.
To demonstrate both the abilities of the TGEC code and its flexibility as compared to the Montreal one, we presented sets of models that include atomic diffusion with small additional mixing. In this context, models including “minimal mixing” were already presented in @Theado09; in Sect. \[computations1\] we chose to present a new set of models (with masses ranging from 1.5 to 2.5M$_{\odot}$) including physical ingredients close to the r30-3M, 1.5M$_{\odot}$ model of @Richard01. In these models, only mild mixing is introduced below the surface convective zone, and the iron convection region, when its appears, is assumed to be connected and mixed with the surface convective envelope. Under these assumptions, our models show similar behavior to the Montreal models, at least during the first 100Myr (where the Montreal computations stop). In particular, the introduced physical hypothesis leads to an alternation of convective and radiative episodes in the opacity bump region in both the Montreal and the TGEC models. We have also shown that the TGEC code allows computing complete evolutionary tracks (up to the subgiant branch) for A-F stars with masses up to 2.5M$_{\odot}$. It is able to manage rapid variations in the chemical composition, under cover of mild mixing at the transition between convective and radiative regions. The effects of quasi-pure atomic diffusion can then be evaluated in the context of stellar evolutionary models.
As discussed in the introduction, precise tests of the results obtained by the two codes for the case of iron accumulation inside stars, when thermohaline convection is not taken into account, still have to be performed, since differences in the maximum iron value appear in some cases. Such tests are underway and will be presented elsewhere in the near future.
The improvements brought to the atomic diffusion aspects of the TGEC code presented here and in @Theado09 makes it an excellent test bed for stellar evolution and asteroseismology. Its application to various types of stars will certainly help us understand them better.
We thank Olivier Richard for kindly providing its 5.3D50-3 1.70 M$_{\odot}$ model. We acknowledge the financial support of the Programme National de Physique Stellaire (PNPS) of CNRS/INSU, France and the Natural Sciences and Engineering Research Council of Canada.
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