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@@ -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 $\tilde\Omega$ |