From e7f379207b9e60591b07ae02fc304872e482ee33 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 5 Sep 2024 17:49:59 +0200
Subject: Some changes.

---
 topology.tex | 101 +++++++++++++++++++++++++++++++++++++----------------------
 1 file changed, 63 insertions(+), 38 deletions(-)

(limited to 'topology.tex')

diff --git a/topology.tex b/topology.tex
index daacab2..6570ca4 100644
--- a/topology.tex
+++ b/topology.tex
@@ -6,6 +6,7 @@
 \usepackage[bitstream-charter]{mathdesign}
 \usepackage[dvipsnames]{xcolor}
 \usepackage{anyfontsize,authblk}
+\usepackage{xfrac} % sfrac
 
 \urlstyle{sf}
 
@@ -448,10 +449,10 @@ make use of the identity
     |\det A|
   &=\lim_{\epsilon\to0}\frac{(\det A)^2}{\sqrt{\det(A+i\epsilon I)}\sqrt{\det(A-i\epsilon I)}}
   \\
-  &=\frac1{(2\pi)^N}\int d\bar{\pmb\eta}_1\,d\pmb\eta_1\,d\bar{\pmb\eta}_2\,d\pmb\eta_2\,d\mathbf a_+\,d\mathbf a_-\,
+  &=\frac1{(2\pi)^N}\int d\bar{\pmb\eta}_1\,d\pmb\eta_1\,d\bar{\pmb\eta}_2\,d\pmb\eta_2\,d\mathbf a_+\,d\mathbf a_2\,
   e^{
     -\bar{\pmb\eta}_1^TA\pmb\eta_1-\bar{\pmb\eta}_2^TA\pmb\eta_2
-    -\frac12\mathbf a_+^T(A+i\epsilon I)\mathbf a_+-\frac12\mathbf a_-^T(A-i\epsilon I)\mathbf a_-
+    -\frac12\mathbf a_+^T(A+i\epsilon I)\mathbf a_+-\frac12\mathbf a_2^T(A-i\epsilon I)\mathbf a_2
   }
 \end{aligned}
 \end{equation}
@@ -1082,18 +1083,18 @@ Starting from \eqref{eq:abs.kac-rice}, we make the substitution
   &\big|\det\partial\partial L(\mathbf x,\pmb\omega)\big|
   =\lim_{\epsilon\to0}\frac1{(2\pi)^{N+M+1}}\int d\bar{\pmb\eta}_1\,d\pmb\eta_1\,d\bar{\pmb\eta}_2\,d\pmb\eta_2\,
   d\bar{\pmb\gamma}_1\,d\pmb\gamma_1\,d\bar{\pmb\gamma}_2\,d\pmb\gamma_2\,
-  d\mathbf a_+\,d\mathbf a_-\,d\mathbf b_+\,d\mathbf b_-\, \notag \\
+  d\mathbf a_1\,d\mathbf a_2\,d\mathbf b_1\,d\mathbf b_2\, \notag \\
   &\qquad\times\exp\Bigg\{
-    -\frac12\begin{bmatrix}\mathbf a_+^T&\mathbf b_+^T\end{bmatrix}\big(\partial\partial L(\mathbf x,\pmb\omega)+i\epsilon I\big)\begin{bmatrix}\mathbf a_+\\\mathbf b_+\end{bmatrix}
-    -\frac12\begin{bmatrix}\mathbf a_-^T&\mathbf b_-^T\end{bmatrix}\big(\partial\partial L(\mathbf x,\pmb\omega)-i\epsilon I\big)\begin{bmatrix}\mathbf a_-\\\mathbf b_-\end{bmatrix}
+    -\frac12\begin{bmatrix}\mathbf a_1^T&\mathbf b_1^T\end{bmatrix}\big(\partial\partial L(\mathbf x,\pmb\omega)+i\epsilon I\big)\begin{bmatrix}\mathbf a_1\\\mathbf b_1\end{bmatrix}
+    -\frac12\begin{bmatrix}\mathbf a_2^T&\mathbf b_2^T\end{bmatrix}\big(\partial\partial L(\mathbf x,\pmb\omega)-i\epsilon I\big)\begin{bmatrix}\mathbf a_2\\\mathbf b_2\end{bmatrix}
     \notag \\
   &\hspace{6em}-\begin{bmatrix}\bar{\pmb\eta}_1^T&\bar{\pmb\gamma}_1^T\end{bmatrix}\partial\partial L(\mathbf x,\pmb\omega)\begin{bmatrix}\pmb\eta_1\\\pmb\gamma_1\end{bmatrix}
     -\begin{bmatrix}\bar{\pmb\eta}_2^T&\bar{\pmb\gamma}_2^T\end{bmatrix}\partial\partial L(\mathbf x,\pmb\omega)\begin{bmatrix}\pmb\eta_2\\\pmb\gamma_2\end{bmatrix}
   \Bigg\}
   \label{eq:abs.det.exp}
 \end{align}
-where the $\mathbf a_\pm$ are in $\mathbb R^N$, the $\mathbf b_\pm$ are in
-$\mathbb R^M$, and the $\pmb\eta$ and $\pmb\gamma$ are Grassmann vectors just as in
+where the $\mathbf a_{\sfrac12}$ are in $\mathbb R^N$, the $\mathbf b_{\sfrac12}$ are in
+$\mathbb R^M$, and the $\pmb\eta_{\sfrac12}$ and $\pmb\gamma_{\sfrac12}$ are Grassmann vectors just as in
 \eqref{eq:det.exp}, except with an extra copy of each. This zoo of vectors
 quickly becomes tiring. Thankfully, there is a way to compactly represent this
 calculation again using superspace vectors.
@@ -1104,16 +1105,16 @@ Consider the vectors
   &=\mathbf x
   +\bar\theta_1\pmb\eta_1+\bar{\pmb\eta}_1\theta_1\bar\theta_2\theta_2
   +\bar\theta_2\pmb\eta_2+\bar{\pmb\eta}_2\theta_2\bar\theta_1\theta_1 \\
-  &\qquad+\frac1{\sqrt2}(\bar\theta_1\theta_2+\bar\theta_2\theta_1)\mathbf a_+
-  +\frac i{\sqrt2}(\bar\theta_1\theta_1+\bar\theta_2\theta_2)\mathbf a_-
+  &\qquad+\frac1{\sqrt2}(\bar\theta_1\theta_2+\bar\theta_2\theta_1)\mathbf a_1
+  +\frac i{\sqrt2}(\bar\theta_1\theta_1+\bar\theta_2\theta_2)\mathbf a_2
   +\bar\theta_1\theta_1\bar\theta_2\theta_2i\hat{\mathbf x}
   \notag \\
   \sigma_k(1,2)
   &=\omega_k
   +\bar\theta_1\gamma_{1k}+\bar{\gamma}_{1k}\theta_1\bar\theta_2\theta_2
   +\bar\theta_2\gamma_{2k}+\bar{\gamma}_{2k}\theta_2\bar\theta_1\theta_1 \\
-  &\qquad+\frac1{\sqrt2}(\bar\theta_1\theta_2+\bar\theta_2\theta_1)b_{-k}
-  +\frac i{\sqrt2}(\bar\theta_1\theta_1+\bar\theta_2\theta_2)b_{+k}
+  &\qquad+\frac1{\sqrt2}(\bar\theta_1\theta_2+\bar\theta_2\theta_1)b_{1k}
+  +\frac i{\sqrt2}(\bar\theta_1\theta_1+\bar\theta_2\theta_2)b_{2k}
   +\bar\theta_1\theta_1\bar\theta_2\theta_2\hat\omega_k
   \notag
 \end{align}
@@ -1125,7 +1126,7 @@ form
   =\lim_{\epsilon\to0}\int d\pmb\phi\,d\pmb\sigma\,e^{
    \int d1\,d2\,L(\pmb\phi(1,2),\pmb\sigma(1,2))
    -\frac{i\epsilon}2
-   (\|\mathbf a_+\|^2-\|\mathbf a_-\|^2+\|\mathbf b_+\|^2-\|\mathbf b_-\|^2)
+   (\|\mathbf a_1\|^2-\|\mathbf a_2\|^2+\|\mathbf b_1\|^2-\|\mathbf b_2\|^2)
  }
 \end{equation}
 Note that unlike in the derivation of the average Euler characteristic,
@@ -1151,49 +1152,42 @@ Following similar steps as for the Euler characteristic, we first take the avera
 We once again have a Gaussian integral in the parameters inside the Lagrange
 multipliers $\sigma_k$ for $k=1,\ldots,M$. However, here we must expand the
 superfields at this stage. Before that, we introduce order parameters analogous
-to before. Alongside those in \eqref{eq:ops.bos} and \eqref{eq:ops.ferm}, we also define
+to before. Alongside those in \eqref{eq:ops.bos}, we have
 \begin{align}
-  A_{++}=\frac{\mathbf a_+\cdot\mathbf a_+}N
+  A_{ij}=\frac{\mathbf a_i\cdot\mathbf a_j}N
   &&
-  A_{--}=\frac{\mathbf a_-\cdot\mathbf a_-}N
+  X_i=\frac{\mathbf x\cdot\mathbf a_i}N
   &&
-  A_{+-}=\frac{\mathbf a_+\cdot\mathbf a_-}N
-  &&
-  X_+=\frac{\mathbf x\cdot\mathbf a_+}N
-  &&
-  X_-=\frac{\mathbf x\cdot\mathbf a_-}N
-  \\
-  \hat X_+=-i\frac{\hat{\mathbf x}\cdot\mathbf a_+}N
-  &&
-  \hat X_-=-i\frac{\hat{\mathbf x}\cdot\mathbf a_-}N
+  \hat X_i=-i\frac{\hat{\mathbf x}\cdot\mathbf a_i}N
   &&
   H_{ij}=\frac{\bar{\pmb\eta}_i\cdot\pmb\eta_j}N
   &&
   J=\frac{\pmb\eta_1\cdot\pmb\eta_2}N
   &&
-  \bar J=\frac{\bar{\pmb\eta}_1\cdot\bar{\pmb\eta}_2}N
+  m_i = \frac{\mathbf x_0\cdot\mathbf a_i}N
 \end{align}
+where the $i$ and $j$ run over $1$ and $2$.
 These come with a volume term in the action
 \begin{equation}
   \begin{aligned}
     \mathcal V=\frac12\log\det\left(
       \begin{bmatrix}
-        C   & iR        & X_+       & X_-       \\
-        iR  & D         & i\hat X_+ & i\hat X_- \\
-        X_+ & i\hat X_+ & A_{++}    & A_{+-}    \\
-        X_- & i\hat X_- & A_{+-}    & A_{--}
+        C   & iR        & X_1       & X_2       \\
+        iR  & D         & i\hat X_1 & i\hat X_2 \\
+        X_1 & i\hat X_1 & A_{11}    & A_{12}    \\
+        X_2 & i\hat X_2 & A_{12}    & A_{22}
       \end{bmatrix}
       -\begin{bmatrix}
         m \\
         i\hat m \\
-        m_+ \\
-        m_-
+        m_1 \\
+        m_2
       \end{bmatrix}
       \begin{bmatrix}
         m &
         i\hat m &
-        m_+ &
-        m_-
+        m_1 &
+        m_2
       \end{bmatrix}
     \right) \qquad \\
     -\frac12\log\det\begin{bmatrix}
@@ -1207,9 +1201,9 @@ These come with a volume term in the action
 As before, we can immediately integrate out $\sigma_0$, which fixes certain of the order parameters. In particular, we find
 \begin{align}
   C=1 &&
-  X_+=0 &&
-  X_-=0 &&
-  G_+=-R-\frac12(A_{++}+A_{--})
+  X_1=0 &&
+  X_2=0 &&
+  G_+=-R-\frac12A_+
 \end{align}
 where we have defined the symmetric and antisymmetric combinations of $G_{11}$ and $G_{22}$
 \begin{align}
@@ -1249,11 +1243,42 @@ and likewise a Gaussian integral in the Grassmann variables with kernel
 The effective action then has the form
 \begin{equation}
   \mathcal S
-  =-\frac\alpha2\left(
+  =\hat m+\frac12i\epsilon A_--\frac\alpha2\left(
     \log\det K-\log\det K'+V_0^2(K^{-1})_{22}
   \right)+\mathcal V
 \end{equation}
-
+It is not helpful to write out this entire expression, which is quite large.
+However, we only find saddle points of this action with $\bar
+J=J=G_{12}=G_{21}=G_-=\hat X_1=\hat X_2=m_1=m_2=A_{12}=0$. Setting these variables to zero, we find
+\begin{align}
+  \notag
+  \mathcal S
+  =\hat m
+  +\frac12i\epsilon A_-
+  -\alpha\frac12\log\frac{
+    f'(1)(Df(1)+R^2f'(1))-\frac12((R-\frac12A_+)^2+\frac12 A_-^2)f(1)f''(1)
+  }{
+    \frac14(R+\frac12A_+)^2f'(1)^2
+  }
+  \\
+  \notag
+  -\alpha\frac12\log\frac{
+    \epsilon^2+i\epsilon f'(1)A_-+\frac14(A_+^2-A_-^2)f'(1)^2
+  }{
+    \frac14(R+\frac12A_+)^2f'(1)^2
+  }
+  +\frac12\log\frac{D(1-m^2)+R^2-2Rm\hat m+\hat m^2}{\frac14(R+\frac12A_+)^2}
+  \\
+  +\frac12\log\frac{A_+^2-A_-^2}{(R+\frac12A_+)^2}
+  -\frac12\alpha V_0^2\left(
+    f(1)+\frac{R^2f'(1)^2}{Df'(1)-\frac12((R-\frac12A_+)^2+\frac12A_-^2)f''(1)}
+  \right)^{-1}
+\end{align}
+One solution to these equations at $\epsilon=0$ is $A_-=0$ and $A_+=2R^*$ with
+$D=\hat m=0$ and $R=R^*$, exactly as for the Euler characteristic. The resulting effective
+action as a function of $m$ is also exactly the same. We find two other
+solutions that differ from this one, but with formulae too complex to share in
+the text. One alternate solution has $A_-=0$ and one has $A_-\neq0$.
 
 
 \section{The quenched shattering energy in {\oldstylenums 1}\textsc{frsb} models}
-- 
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