From ba657f5720db080f1120e93cdf20098c844ccbe6 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 2 Feb 2022 21:21:20 +0100
Subject: Added a figure about the homology of stokes lines.

---
 stokes.tex | 21 +++++++++++++++++++++
 1 file changed, 21 insertions(+)

(limited to 'stokes.tex')

diff --git a/stokes.tex b/stokes.tex
index cfec1f2..2edf7cc 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -162,6 +162,27 @@ for our collection of thimbles to produce the correct contour, the composition
 of the thimbles must represent the same element of this relative homology
 group.
 
+\begin{figure}
+  \includegraphics{figs/thimble_homology.eps}
+  \hfill
+  \includegraphics{figs/antithimble_homology.eps}
+
+  \caption{
+    A demonstration of the rules of thimble homology. Both figures depict the
+    complex-$\theta$ plane of an $N=2$ spherical $3$-spin model. The black
+    symbols lie on the stationary points of the action, and the grey regions
+    depict the sets $\tilde\Omega_T$ of well-behaved regions at infinity (here
+    $T=5$). (Left) Lines show the thimbles of each stationary point. The
+    thimbles necessary to recreate the cyclic path from left to right are
+    darkly shaded, while those unnecessary for the task are lightly shaded.
+    Notice that all thimbles come and go from the well-behaved regions. (Right)
+    Lines show the antithimbles of each stationary point. Notice that those of
+    the stationary points involved in the contour (shaded darkly) all intersect
+    the desired contour (the real axis), while those not involved do not
+    intersect it.
+  } \label{fig:thimble.homology}
+\end{figure}
+
 Each thimble represents an element of the relative homology, since each thimble
 is a contour on which the real part of the action diverges in any direction.
 And, thankfully for us, Morse theory on our complex manifold $\tilde\Omega$
-- 
cgit v1.2.3-70-g09d2

