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@@ -329,6 +329,30 @@ behavior can be seen in Fig.~\ref{fig:1d.stokes}. } \label{fig:1d.stokes} \end{figure} +\begin{figure} + \includegraphics{figs/thimble_orientation_1.pdf}\hfill + \includegraphics{figs/thimble_orientation_2.pdf}\hfill + \includegraphics{figs/thimble_orientation_3.pdf} + + \caption{ + The behavior of thimble contours near $\arg\beta=0$ for real actions. In all + pictures, green arrows depict a canonical orientation of the thimbles + relative to the real axis, while purple arrows show the direction of + integration implied by the orientation. \textbf{Left:} $\arg\beta=-0.1$. To + progress from left to right, one must follow the thimble from the minimum + $\blacklozenge$ in the direction implied by its orientation, and then + follow the thimble from the maximum $\blacktriangledown$ \emph{against} the + direction implied by its orientation, from top to bottom. Therefore, + $\mathcal C=\mathcal J_\blacklozenge-\mathcal J_\blacktriangledown$. + \textbf{Center:} $\arg\beta=0$. Here the thimble of the minimum covers + almost all of the real axis, reducing the problem to the real phase space + integral. This is also a Stokes point. \textbf{Right:} $\arg\beta=0.1$. Here, one follows the thimble of + the minimum from left to right again, but now follows that of the maximum + in the direction implied by its orientation, from bottom to top. Therefore, + $\mathcal C=\mathcal J_\blacklozenge+\mathcal J_\blacktriangledown$. + } \label{fig:thimble.orientation} +\end{figure} + The prevalence (or not) of Stokes points in a given continuation, and whether those that do appear affect the weights of critical points of interest, is a concern for the analytic continuation of theories. If they do not occur or |