From dd8ff99eb6b9f7a5d4cad83940e212a367a00f3f Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 10 Jun 2021 14:17:03 +0200
Subject: Writing introducing Stokes lines.

---
 stokes.tex | 29 ++++++++++++++++++++++++++---
 1 file changed, 26 insertions(+), 3 deletions(-)

(limited to 'stokes.tex')

diff --git a/stokes.tex b/stokes.tex
index f417f07..447fb3b 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -104,12 +104,35 @@ $\operatorname{Re}(-\beta H)$ is bounded from above by its value at the
 critical point. Likewise, we shall see that the imaginary part of $\beta H$ is
 preserved under gradient descent on its real part.
 
-Morse theory provides the universal correspondence between contours and thimbles: one must produce an integer-weighted linear combination of thimbles such that the homology of the combination is equivalent to that of the contour. If $\Sigma$ is the set of critical points and for each $\sigma\in\Sigma$ $\mathcal J_\sigma$ is its Lefschetz thimble, then this gives
-\begin{equation}
+Morse theory provides the universal correspondence between contours and
+thimbles: one must produce an integer-weighted linear combination of thimbles
+such that the homology of the combination is equivalent to that of the contour.
+If $\Sigma$ is the set of critical points and for each $\sigma\in\Sigma$
+$\mathcal J_\sigma$ is its Lefschetz thimble, then this gives
+\begin{equation} \label{eq:thimble.integral}
   Z(\beta)=\sum_{\sigma\in\Sigma}n_\sigma\int_{\mathcal J_\sigma}du\,e^{-\beta H(u)}
 \end{equation}
 Each of these integrals is very well-behaved: convergent asymptotic series
-exist for their value about the critical point $\sigma$, for example. One must know the integer weights $n_\sigma$.
+exist for their value about the critical point $\sigma$, for example. One must
+know the integer weights $n_\sigma$.
+
+Under analytic continuation of, say, $\beta$, the form of
+\eqref{eq:thimble.integral} persists. When the homology of the thimbles is
+unchanged by the continuation, the integer weights are likewise unchanged, and
+one can therefore use the knowledge of these weights in one regime to compute
+the partition function in other. However, their homology can change, and when
+this happens the integer weights can be traded between critical points. These
+trades occur when two thimbles intersect, or alternatively when one critical
+point lies on the gradient descent of another. These places are called
+\emph{Stokes points}, and the gradient descent trajectories that join two
+critical points are called \emph{Stokes lines}.
+
+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
+occur order-one times, one could reasonably hope to perform such a procedure.
+If they occur exponentially often, there is little hope of keeping track of the
+resulting weights.
 
 \section{Gradient descent dynamics}
 
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