From 83678dc44b7690e12cc9374e230938664b3afbae Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 10 Jun 2021 13:31:41 +0200
Subject: More writing.

---
 stokes.tex | 21 ++++++++++++++-------
 1 file changed, 14 insertions(+), 7 deletions(-)

(limited to 'stokes.tex')

diff --git a/stokes.tex b/stokes.tex
index 2db31bf..fdc8e69 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -111,16 +111,24 @@ 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 $\operatorname{Re}H$. 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, gradient descent follows the dynamics
-\begin{equation} \label{eq:flow}
+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 $u:M\to\mathbb C^N$, gradient descent follows the dynamics
+\begin{equation} \label{eq:flow.raw}
   \dot z^i
   =-(\partial^*_{\beta}\operatorname{Re}H)h^{\beta\alpha}\partial_\alpha z^i
   =-\tfrac12(\partial_\beta H)^*h^{\beta\alpha}\partial_\alpha z^i
 \end{equation}
 where $h$ is the Hermitian metric and we have used the fact that, for holomorphic $H$, $\partial^*H=0$.
-This can be simplied furthur by noting that $h^{\beta\alpha}=h^{-1}_{\beta\alpha}$ for $h_{\beta\alpha}=(\partial_\beta z^i)^*\partial_\alpha z^i=(J^\dagger J)_{\beta\alpha}$.
 
-\begin{equation}
+This can be simplied  by noting that $h^{\beta\alpha}=h^{-1}_{\beta\alpha}$ for
+$h_{\beta\alpha}=(\partial_\beta z^i)^*\partial_\alpha z^i=(J^\dagger
+J)_{\beta\alpha}$ where $J$ is the Jacobian of the coordinate map. Writing
+$\partial H=\partial H/\partial z$ and inserting Jacobians everywhere they
+appear, \eqref{eq:flow.raw} becomes
+\begin{equation} \label{eq:flow}
   \dot z=-\tfrac12(\partial H)^\dagger J^*[J^\dagger J]^{-1}J^T
   =-\tfrac12(\partial H)^\dagger P
 \end{equation}
@@ -146,8 +154,7 @@ $\operatorname{Im}H$ can be shown using \eqref{eq:flow} and the holomorphic prop
   \end{aligned}
 \end{equation}
 As a result of this conservation law, surfaces of constant $\operatorname{Im}H$
-will be important when evaluting the possible endpoints of dynamic
-trajectories.
+will be important when evaluting the possible endpoints of trajectories.
 
 Let us consider the generic case, where the critical points of $H$ have
 distinct energies. What is the topology of the $C=\operatorname{Im}H$ level
@@ -163,7 +170,7 @@ point, the flow is locally
   \end{aligned}
 \end{equation}
 The matrix $(\partial\partial H)^\dagger P$ has a spectrum identical to that of
-$\partial\partial H$ save marginal directions corresponding to the normals to
+$(\partial\partial H)^\dagger$ save marginal directions corresponding to the normals to
 manifold. Assuming we are working in a diagonal basis, we find
 \begin{equation}
   \dot z_i=-\frac12\lambda_i\Delta z_i^*+O(\Delta z^2)
-- 
cgit v1.2.3-70-g09d2

