From c482286793cbc6083c25ee8f2ea9e472fd8025db Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Sun, 6 Feb 2022 11:08:13 +0100
Subject: Added introduction to hessian discussion.

---
 stokes.tex | 47 +++++++++++++++++++++++++++++++++++++++--------
 1 file changed, 39 insertions(+), 8 deletions(-)

diff --git a/stokes.tex b/stokes.tex
index 26117d2..752ba87 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -362,28 +362,59 @@ keeping track of the resulting weights, and analytic continuation is intractable
 
 \subsection{The structure of stationary points}
 
-\begin{eqnarray}
-  \operatorname{Hess}\operatorname{Re}\beta\mathcal S
-  &=\left[\matrix{
+The shape of each thimble in the vicinity of a stationary point can be
+described using an analysis of the hessian of the real part of the action at
+the stationary point. Here we'll review some general properties of this
+hessian, which because the action is holomorphic has rich structure.
+
+First, consider the problem as one of $2N$ real variables $x,y\in\mathbb R^N$
+with $z=x+iy$. The hessian of the real part of the action with respect to these
+real variables is
+\begin{equation} \label{eq:real.hessian}
+  \operatorname{Hess}_{x,y}\operatorname{Re}\beta\mathcal S
+  =\left[\matrix{
     \partial_x\partial_x\operatorname{Re}\beta\mathcal S &
     \partial_y\partial_x\operatorname{Re}\beta\mathcal S \cr
     \partial_x\partial_y\operatorname{Re}\beta\mathcal S &
     \partial_y\partial_y\operatorname{Re}\beta\mathcal S
-  }\right] \\
-  &=\left[\matrix{
+  }\right]
+\end{equation}
+This con be simplified using the fact that the action is holomorphic, which
+means that it obeys the Cauchy--Riemann equations
+\begin{equation}
+  \partial_x\operatorname{Re}\mathcal S=\partial_y\operatorname{Im}\mathcal S
+  \qquad
+  \partial_y\operatorname{Re}\mathcal S=-\partial_x\operatorname{Im}\mathcal S
+\end{equation}
+Using these relationships alongside the Wirtinger derivative
+$\partial\equiv\frac12(\partial_x-i\partial_y)$ allows the order of the
+derivatives and the real or imaginary parts to be commuted, with
+\begin{eqnarray}
+  \partial_x\operatorname{Re}\mathcal S=\operatorname{Re}\partial\mathcal S
+  \qquad
+  \partial_y\operatorname{Re}\mathcal S=-\operatorname{Im}\partial\mathcal S \\
+  \partial_x\operatorname{Im}\mathcal S=\operatorname{Im}\partial\mathcal S
+  \qquad
+  \partial_y\operatorname{Im}\mathcal S=\operatorname{Re}\partial\mathcal S
+\end{eqnarray}
+Using these relationships, the hessian \eref{eq:real.hessian} can be written in
+the more manifestly complex way
+\begin{equation}
+  \operatorname{Hess}_{x,y}\operatorname{Re}\beta\mathcal S
+  =\left[\matrix{
     \hphantom{-}\operatorname{Re}\beta\partial\partial\mathcal S &
     -\operatorname{Im}\beta\partial\partial\mathcal S \cr
     -\operatorname{Im}\beta\partial\partial\mathcal S &
     -\operatorname{Re}\beta\partial\partial\mathcal S
   }\right]
-\end{eqnarray}
+\end{equation}
 
 The eigenvalues and eigenvectors of the Hessian are important for evaluating
 thimble integrals, because those associated with upward directions provide a
 local basis for the surface of the thimble. Suppose that $v_x,v_y\in\mathbb
 R^N$ are such that
 \begin{equation}
-  (\operatorname{Hess}\operatorname{Re}\beta\mathcal S)\left[\matrix{v_x \cr v_y}\right]
+  (\operatorname{Hess}_{x,y}\operatorname{Re}\beta\mathcal S)\left[\matrix{v_x \cr v_y}\right]
   =\lambda\left[\matrix{v_x \cr v_y}\right]
 \end{equation}
 where the eigenvalue $\lambda$ must be real because the hessian is real symmetric. The problem can be put into a more obviously complex form by a change of basis. Writing $v=v_x+iv_y$, we find
@@ -391,7 +422,7 @@ where the eigenvalue $\lambda$ must be real because the hessian is real symmetri
   &\left[\matrix{0&-i(\beta\partial\partial\mathcal S)^*\cr i\beta\partial\partial\mathcal S&0}\right]
   \left[\matrix{v \cr iv^*}\right]\\
   &\qquad=\left[\matrix{1&i\cr i&1}\right]
-  (\operatorname{Hess}\operatorname{Re}\beta\mathcal S)
+  (\operatorname{Hess}_{x,y}\operatorname{Re}\beta\mathcal S)
   \left[\matrix{1&i\cr i&1}\right]^{-1}
   \left[\matrix{1&i\cr i&1}\right]
   \left[\matrix{v_x \cr v_y}\right] \\
-- 
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