From 308ce96fef4d14a1be1330e03d4713db753af792 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 7 Mar 2025 15:07:44 -0300
Subject: Change addressing report #3, weakness 5.1.

Added a footnote explaining the use of subscript notation for the
superdeterminant of an operator with more than two indicies.
---
 topology.tex | 6 ++++--
 1 file changed, 4 insertions(+), 2 deletions(-)

(limited to 'topology.tex')

diff --git a/topology.tex b/topology.tex
index 9000b12..2a8eb52 100644
--- a/topology.tex
+++ b/topology.tex
@@ -1069,12 +1069,14 @@ method. The integrand is stationary at $\tilde{\mathbb Q}=(\mathbb Q-\mathbb
 M\mathbb M^T)^{-1}$, 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}
+\begin{equation} \label{eq:supermatrix.saddle}
   \left(\operatorname{sdet}_{\{1,2\},\{3,4\}}\frac{\partial^2\frac12\log\operatorname{sdet}\tilde{\mathbb Q}}{\partial\tilde{\mathbb Q}(1,2)\partial\tilde{\mathbb Q}(3,4)}\right)^{-\frac12}
   =\left(\operatorname{sdet}_{\{1,2\},\{3,4\}}\tilde{\mathbb Q}^{-1}(3,1)\tilde{\mathbb Q}^{-1}(2,4)\right)^{-\frac12}
   =1
 \end{equation}
-where the final superdeterminant is identically 1 for any superoperator $\tilde{\mathbb Q}$, not just its saddle-point value.
+where the final superdeterminant is identically 1 for any superoperator $\tilde{\mathbb Q}$, not just its saddle-point value.\footnote{
+  The subscript notation in \eqref{eq:supermatrix.saddle} indicates which superindices of the four-index superoperator associated with the Hessian belong to the domain and codomain, analogous to writing $\det A=\det_{ij}A_{ij}$ for a two-index complex-valued matrix. In this case, the domain is indexed by $\{3,4\}$ and the codomain is indexed by $\{1,2\}$.
+}
 The Hubbard--Stratonovich transformation therefore contributes a factor of
 \begin{equation}
   \frac12\operatorname{sdet}(\mathbb Q-\mathbb M\mathbb M^T)^{\frac12}
-- 
cgit v1.2.3-70-g09d2

