From b953eb189459b1756f630cf8c528c09227df9106 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 13 Sep 2024 17:08:53 +0200
Subject: Some new writing.

---
 topology.tex | 43 +++++++++++++++++++++++++++++++++++++++++--
 1 file changed, 41 insertions(+), 2 deletions(-)

(limited to 'topology.tex')

diff --git a/topology.tex b/topology.tex
index 4b48e3f..277d8d3 100644
--- a/topology.tex
+++ b/topology.tex
@@ -287,7 +287,7 @@ where $g$ is a prefactor of $o(N^0)$, and $\mathcal S_\chi$ is an effective acti
 \begin{equation} \label{eq:euler.action}
   \begin{aligned}
     \mathcal S_\chi(R,D,m,\hat m\mid\alpha,V_0)
-    &=\hat m-\frac\alpha2\left[
+    &=-\hat m-\frac\alpha2\left[
       \log\left(1+\frac{f(1)D}{f'(1)R^2}\right)
       +\frac{V_0^2}{f(1)}\left(1+\frac{f'(1)R^2}{f(1)D}\right)^{-1}
     \right] \\
@@ -389,7 +389,46 @@ However, the action becomes complex in the region $m^2<m_\text{min}^2$ for
     2V_0^2f(1)(2-\alpha)-\alpha(V_0^4+f(1)^2)
   }
 \end{equation}
-When $m_*^2<m_\text{min}^2$, this solution is no longer valid. Likewise, when $m_\text{min}^2>0$, the solution at $m=0$ is also not valid. In fact, it is not clear what the average value of the Euler characteristic should be at all
+When $m_*^2<m_\text{min}^2$, this solution is no longer valid. Likewise, when
+$m_\text{min}^2>0$, the solution at $m=0$ is also not valid. In fact, it is not
+clear what the average value of the Euler characteristic should be at all when
+there is some range $-m_\text{min}<m<m_\text{min}$ where the effective action
+is complex. Such a result could arise from the breakdown of the large-deviation
+principle behind the calculation of the effective action, or it could be the
+result of a negative Euler characteristic.
+
+To address this problem, we compute also the average of the square of the Euler
+characteristic, $\overline{\chi(\Omega)^2}$. This has the benefit of always
+being positive, so that the saddle-point approach to the calculation at large
+$N$ does not produce complex values even when $\overline{\chi(\Omega)}$ is
+negative. We find three saddle points that could contribute to the value of
+$\overline{\chi(\Omega)^2}$: two at $\pm m^*$ where
+$\frac1N\log\overline{\chi(\Omega)^2}=\frac1N\log\overline{\chi(\Omega)}\simeq0$, and one at $m=0$ where
+\begin{equation}
+  \frac1N\log\overline{\chi(\Omega)^2}=2\operatorname{Re}\mathcal S_\chi(0)
+\end{equation}
+
+We therefore have four possible topological regimes:
+\begin{itemize}
+  \item \textbf{Simple connected:
+    \boldmath{$\frac1N\log\overline{\chi(\Omega)^2}=0$,
+    $\frac1N\log\overline{\chi(\Omega)}=0$}.} This regime occurs when $m_*^2>0$
+    and $\operatorname{Re}\mathcal S_\chi(0)<0$. Here,
+    $\overline{\chi(\Omega)}=2+o(1)$ for even $N-M-1$, strongly indicating a
+    topology homeomorphic to the $S^{N-M-1}$ sphere.
+  \item \textbf{Complex connected: \boldmath{$\frac1N\log\overline{\chi(\Omega)^2}>0$, $\overline{\chi(\Omega)}<0$}.}
+    This regime occurs when $m_\text{min}^2>0$ and $\operatorname{Re}\mathcal
+    S_\chi(0)>0$. Here the average Euler characteristic is large and negative.
+    While the topology of the manifold is not necessarily connected in this
+    regime, holes are more numerous than components.
+  \item \textbf{Disconnected: \boldmath{$\frac1N\log\overline{\chi(\Omega)^2}>0$, $\overline{\chi(\Omega)}>0$}.}
+    This regime occurs when $m_\text{min}^2<0$ and $\mathcal S(0)>0$. Here the
+    average Euler characteristic is large and positive. Large connected
+    components of the manifold may or may not exist, but disconnected
+    components outnumber holes.
+  \item \textbf{\textsc{Unsat}: \boldmath{$\frac1N\log\overline{\chi(\Omega)^2}<0$}.}
+    There is typically not a manifold at all, indicating that the equations cannot be satisfied.
+\end{itemize}
 
 However, when the magnitude of $V_0$ is sufficiently large, with
 \begin{equation}
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