From 3ad65179643c7a9df2820acd608897f0debc8421 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 7 Jun 2021 14:41:04 +0200
Subject: Found better way to generically introduce dynamics.

---
 stokes.tex | 30 +++++++++++++++++++-----------
 1 file changed, 19 insertions(+), 11 deletions(-)

diff --git a/stokes.tex b/stokes.tex
index d833866..c64f934 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -121,13 +121,21 @@ 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 $u$, gradient descent follows the dynamics
+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}
-  \dot u^\alpha
-  =-(\partial^*_{\beta}\operatorname{Re}H)h^{\beta\alpha}
-  =-\tfrac12(\partial_\beta H)^*h^{\beta\alpha}
+  \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}
+  \dot z=-\tfrac12(\partial H)^\dagger J^*[J^\dagger J]^{-1}J^T
+  =-\tfrac12(\partial H)^\dagger P
+\end{equation}
+which is nothing but the projection of $(\partial H)^*$ into the tangent space of the manifold, with $P=J^*[J^\dagger J]^{-1}J^T$. Note that $P$ is hermitian: $P^\dagger=(J^*[J^\dagger J]^{-1}J^T)^\dagger=J^*[J^\dagger J]^{-1}J^T=P$.
+
 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
@@ -135,12 +143,12 @@ $\operatorname{Im}H$ can be shown using \eqref{eq:flow} and the holomorphic prop
 \begin{equation}
   \begin{aligned}
     \frac d{dt}&\operatorname{Im}H
-    =\dot u^\alpha\partial_\alpha\operatorname{Im}H+(\dot u^\alpha)^*\partial_\alpha^*\operatorname{Im}H \\
+    =\dot z\partial\operatorname{Im}H+\dot z^*\partial^*\operatorname{Im}H \\
     &=\frac i4\left(
-      (\partial_\beta H)^*h^{\beta\alpha}\partial_\alpha H-\partial_\beta H(h^{\beta\alpha})^*\partial_\alpha^*H^*
+      (\partial H)^\dagger P\partial H-(\partial H)^TP^\dagger(\partial H)^*
     \right) \\
     &=\frac i4\left(
-      (\partial_\beta H)^*h^{\beta\alpha}\partial_\alpha H-[(\partial_\beta H)^*h^{\beta\alpha}\partial_\alpha H]^*
+      (\partial H)^\dagger P\partial H-[(\partial H)^\dagger P\partial H]^*
     \right) \\
     &=\frac i4\left(
       \|\partial H\|-\|\partial H\|^*
@@ -160,13 +168,13 @@ single simply connected surface, locally diffeomorphic to $\mathbb R^{2N}$. Now,
 point, the flow is locally
 \begin{equation}
   \begin{aligned}
-    \dot u^\alpha
-    &\simeq-\frac12\operatorname{Hess}(H)^*
+    \dot z
+    &\simeq-\frac12(\partial\partial H)^\dagger P(z-z_0)
   \end{aligned}
 \end{equation}
 The matrix $(\partial\partial H)^\dagger P$ has a spectrum identical to that of
-$\partial\partial H$ save a single marginal direction corresponding to $z_0$,
-the normal to the constraint surface. Assuming we are working in a diagonal basis, we find
+$\partial\partial H$ 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)
 \end{equation}
-- 
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