From 9688a70b3b0964bbab687e086017e8c089069afe Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 30 Aug 2023 22:47:38 +0200
Subject: Some tenative work.

---
 marginal.tex | 31 +++++++++++++++++++++++++++++--
 1 file changed, 29 insertions(+), 2 deletions(-)

diff --git a/marginal.tex b/marginal.tex
index fb212db..59c9bc8 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -71,18 +71,45 @@ associated with the minimal eigenvalue $\lambda_\mathrm{min}$.
   \end{aligned}
 \end{equation}
 
+\begin{equation}
+  \beta^2f''(1)\sum A_{ab}^2+\hat x^2f''(1)A_{11}^2+\beta\hat xf''(1)\sum_a A_{1a}+\hat\beta^2f(C_{ab})+(2\hat\beta(R_{ab}-F_{ab})-D_{ab})f'(C_{a-b})+(R_{ab}^2-F_{ab}^2)f''(C_{ab})
+  +\log\det\begin{bmatrix}C&iR\\iR&D\end{bmatrix}-\log\det F
+\end{equation}
+
 \section{Superfield formalism}
 
 \begin{equation}
-  \pmb\phi=\pmb\sigma+\bar\theta\pmb\eta+\bar{\pmb\eta}\theta+\hat{\pmb\sigma}\bar\theta\theta+\mathbf x\bar\vartheta\theta+\mathbf x\bar\theta\vartheta
+  \pmb\phi_a(1)=\pmb\sigma_a+\bar\theta(1)\pmb\eta_a+\bar{\pmb\eta}_a\theta(1)+\hat{\pmb\sigma}_a\bar\theta(1)\theta(1)
 \end{equation}
 \begin{equation}
-  \int d\theta\,d\bar\theta\,d\vartheta\,d\bar\vartheta\,(\bar\vartheta\vartheta+\beta+\hat\beta\bar\vartheta\vartheta\bar\theta\theta)H(\pmb\phi)
+  \pmb\xi_b(1)=\pmb\sigma_1+\mathbf x_b\bar\vartheta(1)\theta(1)+\mathbf x_b\bar\theta(1)\vartheta(1)
+\end{equation}
+\begin{equation}
+  \int d\theta\,d\bar\theta\,\left[
+    (1+\hat\beta\bar\theta\theta)H(\pmb\phi)
+    +\int d\vartheta\,d\bar\vartheta\,H(\pmb\xi)
+  \right]
   =\hat{\pmb\sigma}^T\partial H(\pmb\sigma)
   +\pmb\eta^T\partial\partial H(\pmb\sigma)\pmb\eta
   +\beta\mathbf x^T\partial\partial H(\pmb\sigma)\mathbf x
   +\hat\beta H(\pmb\sigma)
 \end{equation}
+\begin{equation}
+  \int d1\,d2\,(1+\hat\beta\bar\theta(1)\theta(1))(1+\hat\beta\bar\theta(2)\theta(2))
+  f\left(\frac{\pmb\phi_a(1)\cdot\pmb\phi_b(2)}N\right)
+  +\int d1\,(1+\hat\beta\bar\theta(1)\theta(1))f\left(\frac{\pmb\phi_a(1)\cdot\pmb\xi_b(2)}N\right)
+\end{equation}
+
+\section{Twin spherical model}
+
+$\Omega=S^{N-1}\times S^{N-1}$
+\begin{equation}
+  H(\pmb\sigma)=H_1(\pmb\sigma^{(1)})+H_2(\pmb\sigma^{(2)})+\epsilon\pmb\sigma^{(1)}\cdot\pmb\sigma^{(2)}
+\end{equation}
+\begin{equation}
+  \overline{H_s(\pmb\sigma_1)H_s(\pmb\sigma_2)}
+  =Nf_s\left(\frac{\pmb\sigma_1\cdot\pmb\sigma_2}N\right)
+\end{equation}
 
 \section{Multi-species spherical model}
 
-- 
cgit v1.2.3-70-g09d2