More caption writing, and fixed the function to match its shifted

reality.

1 files 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