From 4585d9eea2e2fefac7d3ba6f0dcd1db0703e94be Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Sat, 7 Sep 2024 21:49:44 +0200
Subject: More writing and tweaking.

---
 topology.tex | 93 +++++++++++++++++++++++++++++++++---------------------------
 1 file changed, 51 insertions(+), 42 deletions(-)

(limited to 'topology.tex')

diff --git a/topology.tex b/topology.tex
index 4354db3..baf0766 100644
--- a/topology.tex
+++ b/topology.tex
@@ -1,5 +1,9 @@
 \documentclass{SciPost}
 
+% Prevent all line breaks in inline equations.
+\binoppenalty=10000
+\relpenalty=10000
+
 \usepackage[utf8]{inputenc}
 \usepackage[T1]{fontenc}
 \usepackage{amsmath,latexsym,graphicx}
@@ -834,7 +838,7 @@ JK-D is supported by a \textsc{DynSysMath} Specific Initiative of the INFN.
 Our starting point is the expression \eqref{eq:kac-rice.lagrange} with the
 substitutions of the $\delta$-function and determinant \eqref{eq:delta.exp} and
 \eqref{eq:det.exp} made. To make the calculation compact, we introduce
-superspace coordinates. Introducing the Grassmann indices $\bar\theta_1$
+superspace coordinates \cite{DeWitt_1992_Supermanifolds}. Introducing the Grassmann indices $\bar\theta_1$
 and $\theta_1$, we define the supervectors
 \begin{align}
   \pmb\phi(1)=\mathbf x+\bar\theta_1\pmb\eta+\bar{\pmb\eta}\theta_1+\bar\theta_1\theta_1i\hat{\mathbf x}
@@ -859,7 +863,7 @@ The Euler characteristic can be expressed using these supervectors as
     \right]
   \right\} \notag
 \end{align}
-where $d1=d\bar\theta_1\,d\theta_1$ is the integral over the Grassmann
+where $d1=d\bar\theta_1\,d\theta_1$ is the integral over both Grassmann
 indices. Since this is an exponential integrand linear in the Gaussian
 functions $V_k$, we can take their average to find
 \begin{equation} \label{eq:χ.post-average}
@@ -892,7 +896,7 @@ Performing that integral yields
 \end{equation}
 The supervector $\pmb\phi$ enters this expression as a function only of the
 scalar product with itself and with the vector $\mathbf x_0$ inside the
-function $H(\mathbf x)=\mathbf x_0\cdot\mathbf x$. We therefore make a change
+height function $H(\mathbf x)=\mathbf x_0\cdot\mathbf x$. We therefore make a change
 of variables to the superoperator $\mathbb Q$ and the supervector $\mathbb M$
 defined by
 \begin{equation}
@@ -902,19 +906,21 @@ defined by
 \end{equation}
 These new variables can replace $\pmb\phi$ in the integral using a generalized Hubbard--Stratonovich transformation, which yields
 \begin{align}
-  \overline{\chi(\Omega)}
-    &=\int d\mathbb Q\,d\mathbb M\,d\sigma_0\,
-    \left[(\operatorname{sdet}\mathbb Q)^\frac12+O(N^{-1})\right]
+  &\overline{\chi(\Omega)}
+    =\frac12\int d\mathbb Q\,d\mathbb M\,d\sigma_0\,
+    \left([\operatorname{sdet}(\mathbb Q-\mathbb M\mathbb M^T)]^\frac12+O(N^{-1})\right)
     \,\exp\Bigg\{
-      N\int d1\left[
+    \frac N2\log\operatorname{sdet}(\mathbb Q-\mathbb M\mathbb M^T)
+  \label{eq:post.hubbard-strat}
+    \\
+  &+N\int d1\left[
         \mathbb M(1)
         +\frac12\sigma_0(1)\big(\mathbb Q(1,1)-1\big)
-      \right] \notag \\
-    &\hspace{3em}-\frac M2V_0^2\int d1\,d2\,f(\mathbb Q)^{-1}(1,2)
+      \right]
+    -\frac M2V_0^2\int d1\,d2\,f(\mathbb Q)^{-1}(1,2)
     -\frac M2\log\operatorname{sdet}f(\mathbb Q)
-    +\frac N2\log\operatorname{sdet}(\mathbb Q-\mathbb M\mathbb M^T)
   \Bigg\}
-  \label{eq:post.hubbard-strat}
+  \notag
 \end{align}
 where we show the asymptotic value of the prefactor in Appendix~\ref{sec:prefactor}.
 To move on from this expression,
@@ -1066,7 +1072,7 @@ We can perform the remaining Gaussian integral in $\tilde{\mathbb M}$ to find
 \end{equation}
 The integral over $\tilde{\mathbb Q}$ can be evaluated to leading order using the saddle point
 method. The integrand is stationary at $\tilde{\mathbb Q}=(\mathbb Q-\mathbb
-M\mathbb M^T)^{-1}$, whose substitution results in the term
+M\mathbb M^T)^{-1}$, and substituting this into the above expression results in the term
 $\frac12\log\det(\mathbb Q-\mathbb M\mathbb M^T)$ in the effective action from
 \eqref{eq:post.hubbard-strat}. The saddle point also yields a prefactor of the form
 \begin{equation}
@@ -1458,7 +1464,7 @@ by the Euler characteristic action $\mathcal S_\chi$, and not either of the
 values implied by the alternative solutions. This indicates that $\mathcal
 S_\mathcal N(m)=\mathcal S_\chi(m)$ in all regimes.
 
-\section{The quenched shattering energy in {\oldstylenums 1}\textsc{frsb} models}
+\section{The quenched shattering energy}
 \label{sec:1frsb}
 
 Here we share how the quenched shattering energy is calculated under a
@@ -1467,29 +1473,30 @@ the spherical spin glasses, we start with \eqref{eq:χ.post-average}. The
 formula in a quenched calculation is almost the same as that for the annealed,
 but the order parameters $C$, $R$, $D$, and $G$ must be understood as $n\times
 n$ matrices rather than scalars. In principle $m$, $\hat m$, $\omega_0$, $\hat\omega_0$, $\omega_1$, and $\hat\omega_1$ should be considered $n$-dimensional vectors, but since in our ansatz replica vectors are constant we can take them to be constant from the start. We have
-\begin{equation}
-  \begin{aligned}
-    \overline{\log\chi(\Omega)}
-    =\lim_{n\to0}\frac\partial{\partial n}\int dC\,dR\,dD\,dG\,dm\,d\hat m\,d\omega_0\,d\hat\omega_0\,d\omega_1\,d\hat\omega_1\,
-    \\
-    \exp N\Bigg\{
-      n\hat m+\frac i2\hat\omega_0\operatorname{Tr}(C-I)-\omega_0\operatorname{Tr}(G+R)
-      -in\hat\omega_1E_0
-      +\frac12\log\det\begin{bmatrix}
-        C-m^2 & i(R-m\hat m) \\ i(R-m\hat m) & D-\hat m^2
-      \end{bmatrix}
-      \\
-      -\frac12\log G^2-\frac12\sum_{ab}^n\left[
-        \hat\omega_1^2f(C_{ab})
-        +(2i\omega_1\hat\omega_1R_{ab}+\omega_1^2D_{ab})f'(C_{ab})
-        +\omega_1^2(G_{ab}^2-R_{ab}^2)f''(C_{ab})
-      \right]
-    \Bigg\}
-  \end{aligned}
-\end{equation}
+\begin{align}
+  &\overline{\log\chi(\Omega)}
+  =\lim_{n\to0}\frac\partial{\partial n}\int dC\,dR\,dD\,dG\,dm\,d\hat m\,d\omega_0\,d\hat\omega_0\,d\omega_1\,d\hat\omega_1\,
+  \exp N\Bigg\{
+    n\hat m
+    +\frac i2\hat\omega_0\operatorname{Tr}(C-I)
+    \notag \\
+  &\hspace{2em}-\omega_0\operatorname{Tr}(G+R)
+    -in\hat\omega_1E_0
+    +\frac12\log\det\begin{bmatrix}
+      C-m^2 & i(R-m\hat m) \\ i(R-m\hat m) & D-\hat m^2
+    \end{bmatrix}
+    -\frac12\log G^2
+    \notag \\
+    &\hspace{2em}-\frac12\sum_{ab}^n\left[
+      \hat\omega_1^2f(C_{ab})
+      +(2i\omega_1\hat\omega_1R_{ab}+\omega_1^2D_{ab})f'(C_{ab})
+      +\omega_1^2(G_{ab}^2-R_{ab}^2)f''(C_{ab})
+    \right]
+  \Bigg\}
+\end{align}
 which is completely general for the spherical spin glasses with $M=1$. We now
 make a series of simplifications. Ward identities associated with the BRST
-symmetry possessed by the original action indicate that
+symmetry possessed by the original action \cite{Annibale_2003_The, Annibale_2003_Supersymmetric, Annibale_2004_Coexistence} indicate that
 \begin{align}
   \omega_1D=-i\hat\omega_1R
   &&
@@ -1501,15 +1508,15 @@ Moreover, this problem with $m=0$ has a close resemblance to the complexity of
 the spherical spin glasses. In both, at the supersymmetric saddle point the
 matrix $R$ is diagonal with $R=r_dI$ \cite{Kent-Dobias_2023_How}.
 To investigate the shattering energy, we can restrict to solutions with $m=0$
-and look for the place where such solutions vanish. Inserting these simplifications, we have
+and look for the place where such solutions vanish. Inserting these simplifications, we have up to highest order in $N$
 \begin{equation}
   \begin{aligned}
     \overline{\log\chi(\Omega)}
-    \propto\lim_{n\to0}\frac\partial{\partial n}\int dC\,dR\,d\hat\omega_0\,d\omega_1\,d\hat\omega_1\,
+    =\lim_{n\to0}\frac\partial{\partial n}\int dC\,dR\,d\hat\omega_0\,d\omega_1\,d\hat\omega_1\,
     \exp N\Bigg\{
       \frac i2\hat\omega_0\operatorname{Tr}(C-I)
       -in\hat\omega_1E
-      \\
+      \qquad\\
       -i\frac12n\omega_1\hat\omega_1r_df'(1)
       -\frac12\sum_{ab}^n
         \hat\omega_1^2f(C_{ab})
@@ -1522,11 +1529,11 @@ If we redefine $\hat\beta=-i\hat\omega_1$ and $\tilde r_d=\omega_1 r_d$, we find
 \begin{equation}
   \begin{aligned}
     \overline{\log\chi(\Omega)}
-    \propto\lim_{n\to0}\frac\partial{\partial n}\int dC\,d\hat\beta\,d\tilde r_d\,\hat\omega_0\,
+    =\lim_{n\to0}\frac\partial{\partial n}\int dC\,d\hat\beta\,d\tilde r_d\,\hat\omega_0\,
     \exp N\Bigg\{
       \frac i2\hat\omega_0\operatorname{Tr}(C-I)
       +n\hat\beta E
-      \\
+      \qquad\\
       +n\frac12\hat\beta\tilde r_df'(1)
       +\frac12\sum_{ab}^n
         \hat\beta^2f(C_{ab})
@@ -1539,9 +1546,9 @@ which is exactly the effective action for the supersymmetric complexity in the
 spherical spin glasses when in the regime where minima dominate
 \cite{Kent-Dobias_2023_How}. As the effective action for the Euler characteristic, this expression is valid whether minima dominate or not. Following the same steps as in
 \cite{Kent-Dobias_2023_How}, we can write the continuum version of this action
-for arbitrary \textsc{rsb} structure as
+for arbitrary \textsc{rsb} structure in the matrix $C$ as
 \begin{equation} \label{eq:cont.action}
-  \overline{\log\chi(\Omega)}=\hat\beta E+\frac12\hat\beta\tilde r_df'(1)
+  \frac1N\overline{\log\chi(\Omega)}=\hat\beta E+\frac12\hat\beta\tilde r_df'(1)
   +\frac12\int_0^1dq\,\left[
     \hat\beta^2f''(q)\chi(q)+\frac1{\chi(q)+\tilde r_d\hat\beta^{-1}}
   \right]
@@ -1573,7 +1580,9 @@ between \textsc{frsb} and {\oldstylenums1}\textsc{frsb} in this setting is that
 in the former case the ground state has $q_0=1$, while in the latter the ground
 state has $q_0<1$.
 
-We use this action to find the shattering energy in the following way. First, setting $\overline{\log\chi(\Omega)}=0$ and solving for $E$ yields a formula for the ground state energy
+We use this action to find the shattering energy in the following way. First,
+we know that the ground state energy is the place where the manifold and therefore the average Euler characteristic vanishes. Therefore, setting $\overline{\log\chi(\Omega)}=0$ and solving for $E$ yields a formula
+for the ground state energy
 \begin{equation}
   E_\text{gs}=-\frac1{\hat\beta}\left\{
     \frac12\hat\beta\tilde r_df'(1)
-- 
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