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| -rw-r--r-- | stokes.tex | 10 |
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@@ -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 |