From e95106056fe738564a4e57fcdfad81e4799e47de Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 2 Feb 2022 21:21:41 +0100
Subject: Some writing and editing.

---
 stokes.tex | 23 ++++++++++-------------
 1 file changed, 10 insertions(+), 13 deletions(-)

diff --git a/stokes.tex b/stokes.tex
index 2edf7cc..230ed01 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -288,10 +288,8 @@ holomorphic property of $\mathcal S$:
     \|\partial\mathcal S\|^2-(\|\partial\mathcal S\|^*)^2
   \right)=0.
 \end{eqnarray}
-As a result of this conservation law, surfaces of constant imaginary action
-will be important when evaluting the possible endpoints of trajectories. A
-consequence of this conservation is that the flow in the action takes a simple
-form:
+A consequence of this conservation is that the flow in the action takes a
+simple form:
 \begin{equation}
   \dot{\mathcal S}
   =\dot z\partial\mathcal S
@@ -301,22 +299,21 @@ form:
 In the complex-$\mathcal S$ plane, dynamics is occurs along straight lines in
 a direction set by the argument of $\beta$.
 
-Let us consider the generic case, where the critical points of $\mathcal S$ have
-distinct energies. What is the topology of the $C=\operatorname{Im}\mathcal S$ level
-set? We shall argue its form by construction. Consider initially the situation
-in the absence of any critical point. In this case the level set consists of a
-single simply connected surface, locally diffeomorphic to $\mathbb R^{2D-1}$. Now, `place' a generic
-(nondegenerate) critical point in the function at $z_0$. In the vicinity of the critical
-point, the flow is locally
+What does the topology of the space of thimbles look like?  Let us consider the
+generic case, where the critical points of $\beta\mathcal S$ have distinct
+energies. Consider initially the situation in the absence of any critical
+point, e.g., as for a constant or linear function. Now, `place' a generic
+(nondegenerate) critical point in the function at $z_0$. In the vicinity of the
+critical point, the flow is locally
 \begin{equation}
   \dot z
-  \simeq-\frac12(\partial\partial\mathcal S|_{z=z_0})^\dagger P(z-z_0)^*
+  \simeq-\frac{\beta^*}2(\partial\partial\mathcal S|_{z=z_0})^\dagger P(z-z_0)^*
 \end{equation}
 The matrix $(\partial\partial\mathcal S)^\dagger P$ has a spectrum identical to that of
 $(\partial\partial\mathcal S)^\dagger$ save marginal directions corresponding to the normals to
 manifold. Assuming we are working in a diagonal basis, this becomes
 \begin{equation}
-  \dot z_i=-\frac12\lambda_i\Delta z_i^*+O(\Delta z^2,(\Delta z^*)^2)
+  \dot z_i=-\frac12(\beta\lambda_i)^*\Delta z_i^*+O(\Delta z^2,(\Delta z^*)^2)
 \end{equation}
 Breaking into real and imaginary parts gives
 \begin{eqnarray}
-- 
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