Added new figure discussing thimble orientation and the integer weights.

1 files changed, 24 insertions, 0 deletions

diff --git a/stokes.tex b/stokes.tex
index e0d9850..b10dce4 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -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