From 49b2d884ed027103b019a5b22a2c6d851d1c34fc Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 4 Jun 2021 16:49:13 +0200
Subject: More work.

---
 stokes.tex | 38 ++++++++++++++------------------------
 1 file changed, 14 insertions(+), 24 deletions(-)

diff --git a/stokes.tex b/stokes.tex
index c92f14f..d833866 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -121,28 +121,26 @@ Morse theory provides the universal correspondence between contours and thimbles
 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 $n_\sigma$.
 
-For a holomorphic Hamiltonian $H$, dynamics are defined by gradient descent on $\operatorname{Re}H$, or
+For a holomorphic Hamiltonian $H$, dynamics are defined by gradient descent on $\operatorname{Re}H$. In hermitian geometry, the gradient is given by raising an index of the conjugate differential, or $\operatorname{grad}f=(\partial^*f)^\sharp$. This implies that, in terms of coordinates $u$, gradient descent follows the dynamics
 \begin{equation} \label{eq:flow}
-  \dot z_i
-  =-\operatorname{grad}_i\operatorname{Re}H
-  =-h^{\beta\alpha}\frac{\partial z_i}{\partial u^\alpha}\partial^*_{\beta}\operatorname{Re}H,
-\end{equation}
-where $h$ is the Hermitian metric. For holomorphic $H$, $\partial^*H=0$ and we have
-\begin{equation} \label{eq:flow.2}
-  \dot z_i
-  =-\frac12h^{\beta\alpha}\frac{\partial z_i}{\partial u^\alpha}\partial^*_{\beta}H^*,
+  \dot u^\alpha
+  =-(\partial^*_{\beta}\operatorname{Re}H)h^{\beta\alpha}
+  =-\tfrac12(\partial_\beta H)^*h^{\beta\alpha}
 \end{equation}
+where $h$ is the Hermitian metric and we have used the fact that, for holomorphic $H$, $\partial^*H=0$.
 Gradient descent on $\operatorname{Re}H$ is equivalent to Hamiltonian dynamics
 with the Hamiltonian $\operatorname{Im}H$. This is because $(M, h)$ is Kähler
 and therefore admits a symplectic structure, but that the flow conserves
-$\operatorname{Im}H$ can be seen from the Cauchy--Riemann equations and
-\eqref{eq:flow}:
+$\operatorname{Im}H$ can be shown using \eqref{eq:flow} and the holomorphic property of $H$:
 \begin{equation}
   \begin{aligned}
     \frac d{dt}&\operatorname{Im}H
-    =\dot z_i\partial_i\operatorname{Im}H+\dot z^*_i\partial_i^*\operatorname{Im}H \\
+    =\dot u^\alpha\partial_\alpha\operatorname{Im}H+(\dot u^\alpha)^*\partial_\alpha^*\operatorname{Im}H \\
+    &=\frac i4\left(
+      (\partial_\beta H)^*h^{\beta\alpha}\partial_\alpha H-\partial_\beta H(h^{\beta\alpha})^*\partial_\alpha^*H^*
+    \right) \\
     &=\frac i4\left(
-      \partial^*_\beta H^*h^{\beta\gamma}J_{i\gamma}J^{-1}_{i\alpha}\partial_\alpha H-\partial_\beta Hh^{\gamma\beta}J_{i\gamma}^*J^{*-1}_{i\alpha}\partial_\alpha^*H^*
+      (\partial_\beta H)^*h^{\beta\alpha}\partial_\alpha H-[(\partial_\beta H)^*h^{\beta\alpha}\partial_\alpha H]^*
     \right) \\
     &=\frac i4\left(
       \|\partial H\|-\|\partial H\|^*
@@ -158,20 +156,12 @@ distinct energies. What is the topology of the $C=\operatorname{Im}H$ 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 $\mathbb R^{2N}$. Now, `place' a generic
-(nondegenerate) critical point in the function at $z_0$. In the vicinity of the critical
+(nondegenerate) critical point in the function at $u_0$. In the vicinity of the critical
 point, the flow is locally
 \begin{equation}
   \begin{aligned}
-    \dot z_i
-    &\simeq-\frac12\left[
-      \partial_j\left(
-        H(z_0)+\frac12\partial_k\partial_\ell H(z_0)\Delta z_k\Delta z_\ell
-      \right)
-    \right]^* P_{ji} \\
-    &=-\frac12\left(
-      \partial_j\partial_kH(z_0)\Delta z_k
-    \right)^* P_{ji} \\
-    &=-\frac12\Delta z_k^*(\partial_k\partial_jH(z_0))^*P_{ji}
+    \dot u^\alpha
+    &\simeq-\frac12\operatorname{Hess}(H)^*
   \end{aligned}
 \end{equation}
 The matrix $(\partial\partial H)^\dagger P$ has a spectrum identical to that of
-- 
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