From 5aafa92d1e4bc9f44b253b6601e00d2b17a1026a Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 19 Sep 2024 11:26:02 +0200
Subject: Fixed a small inconsistency in the derivation of the quenched
 characteristic.

---
 topology.tex | 9 +++++----
 1 file changed, 5 insertions(+), 4 deletions(-)

(limited to 'topology.tex')

diff --git a/topology.tex b/topology.tex
index 64a1215..6f8f9fa 100644
--- a/topology.tex
+++ b/topology.tex
@@ -1448,20 +1448,21 @@ and look for the place where such solutions become complex. Inserting these simp
 \begin{equation}
   \begin{aligned}
     \overline{\log\chi(\Omega)}
-    =\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\,d\hat\omega_0\,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)
+      -i\frac12n\omega_1^*\hat\omega_1r_df'(1)
       -\frac12\sum_{ab}^n
         \hat\omega_1^2f(C_{ab})
       +\frac12\log\det
-      \left(\frac{-i\hat\omega_1}{\omega_1r_d}C+I\right)
+      \left(\frac{-i\hat\omega_1}{\omega_1^*r_d}C+I\right)
     \Bigg\}
   \end{aligned}
 \end{equation}
-If we redefine $\hat\beta=-i\hat\omega_1$ and $\tilde r_d=\omega_1 r_d$, we find
+where $\omega_1^*$ is a constant set by satisfying the extremal equations for $D$.
+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)}
-- 
cgit v1.2.3-70-g09d2

