From 482409f7570927fe6cf8cec4d4bf357a698291c3 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 3 Feb 2022 10:58:43 +0100
Subject: More caption writing, and fixed the function to match its shifted
reality.
---
stokes.tex | 10 ++++++----
1 file changed, 6 insertions(+), 4 deletions(-)
diff --git a/stokes.tex b/stokes.tex
index 095f6cb..e702bac 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -155,11 +155,13 @@ action potentially has more stationary points. We'll call $\Sigma$ the set of
An example of a simple action and its critical points. \textbf{Left:} An
action $\mathcal S$ for the $N=2$ spherical (or circular) $3$-spin model,
defined for $s_1,s_2\in\mathbb R$ on the circle $s_1^2+s_2^2=2$ by
- $\mathcal S(s_1,s_2)=1.051s_1^3+1.180s_1^2s_2+0.823s_1s_2^2+1.045s_2^3$. In
+ $\mathcal S(s_1,s_2)=-1.051s_1^3-1.180s_1^2s_2-0.823s_1s_2^2-1.045s_2^3$. In
the example figures in this section, we will mostly use the single angular
- variable $\theta=\arctan(s_1,s_2)$, which parameterizes the unit circle and
- its complex extension. \textbf{Right:} The stationary points of $\mathcal
- S$ in the complex-$\theta$ plane. In this example,
+ variable $\theta$ defined by $s_1=\sqrt2\cos\theta$,
+ $s_2=\sqrt2\sin\theta$, which parameterizes the unit circle and its complex
+ extension, as $\cos^2\theta+\sin^2\theta=1$ is true even for complex
+ $\theta$. \textbf{Right:} The stationary points of $\mathcal S$ in the
+ complex-$\theta$ plane. In this example,
$\Sigma=\{\blacklozenge,\bigstar,\blacktriangle,\blacktriangledown,\bullet,\blacksquare\}$
and $\Sigma_0=\{\blacklozenge,\blacktriangledown\}$. Symmetries exist
between the stationary points both as a result of the conjugation symmetry
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