From 4387c1083d535a8957a6d95903481da937c27e7f Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 4 Jun 2021 15:52:28 +0200
Subject: Some writing introducing the thimbles.

---
 stokes.tex | 67 ++++++++++++++++++++++++++++++++++++++++++++++++++------------
 1 file changed, 54 insertions(+), 13 deletions(-)

(limited to 'stokes.tex')

diff --git a/stokes.tex b/stokes.tex
index a2a97f2..c92f14f 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -54,7 +54,6 @@
 
 \maketitle
 
-\cite{Witten_2011_Analytic}
 
 Consider a thermodynamic calculation involving the (real) $p$-spin model for a
 particular instantiation of the coupling tensor $J$
@@ -85,16 +84,54 @@ Unfortunately the study is not so relevant for low-dimensional `rugged'
 landscapes, which are typically series or integrals of analytic functions whose
 limit are not themselves analytic \cite{Cavagna_1999_Energy}.
 
-
 \section{Dynamics}
 
+Consider an $N$-dimensional hermitian manifold $M$ and a Hamiltonian $H:M\to\mathbb C$. The partition function
+\begin{equation}
+  Z(\beta)=\int_S du\,e^{-\beta H(u)}
+\end{equation}
+for $S$ a submanifold (not necessarily complex) of $M$. For instance, the
+$p$-spin spherical model can be defined on the complex space $M=\{z\mid
+z^2=N\}$, but typically one is interested in the subspace $S=\{z\mid
+z^2=N,z\in\mathbb R\}$.
+
+If $S$ is orientable, then the integral can be converted to one over a contour
+corresponding to $S$. In this case, the contour can be freely deformed without
+affecting the value of the integral. Two properties of this deformed contour
+would be ideal. First, that as $|u|\to\infty$ the real part of $-\beta H(u)$
+goes to $-\infty$. This ensures that the integral is well defined. Second, that
+the contours piecewise correspond to surfaces of constant phase of $-\beta H$,
+so as to ameliorate sign problems.
+
+Remarkably, there is a recipe for accomplishing both these criteria at once,
+courtesy of Morse theory. For a more thorough review, see
+\citet{Witten_2011_Analytic}. Consider a critical point of $H$. The union of
+all gradient descent trajectories on the real part of $-\beta H$ that terminate
+at the critical point as $t\to-\infty$ is known as the \emph{Lefschetz thimble}
+corresponding with that critical point. Since each point on the Lefschetz
+thimble is a descent from a critical point, the value of
+$\operatorname{Re}(-\beta H)$ is bounded from above by its value at the
+critical point. Likewise, we shall see that the imaginary part of $\beta H$ is
+preserved under gradient descent on its real part.
+
+Morse theory provides the universal correspondence between contours and thimbles: one must produce an integer-weighted linear combination of thimbles such that the homology of the combination is equivalent to that of the contour. If $\Sigma$ is the set of critical points and for each $\sigma\in\Sigma$ $\mathcal J_\sigma$ is its Lefschetz thimble, then this gives
+\begin{equation}
+  Z(\beta)=\sum_{\sigma\in\Sigma}n_\sigma\int_{\mathcal J_\sigma}du\,e^{-\beta H(u)}
+\end{equation}
+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
 \begin{equation} \label{eq:flow}
-  \dot z=-\operatorname{grad}\operatorname{Re}H
-  =-\frac12(\partial H)^\dagger\left(I-\frac{zz^\dagger}{|z|^2}\right)
-  =-\tfrac12(\partial H)^\dagger P,
+  \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^*,
 \end{equation}
-where $P=I-\hat z\hat z^\dagger$ is the projection matrix on to the tangent space of $M$.
 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
@@ -102,21 +139,25 @@ $\operatorname{Im}H$ can be seen from the Cauchy--Riemann equations and
 \eqref{eq:flow}:
 \begin{equation}
   \begin{aligned}
-    \frac d{dt}\operatorname{Im}H
-    &=\dot z_i\partial_i\operatorname{Im}H+\dot{\bar z}_i\bar\partial_i\operatorname{Im}H \\
+    \frac d{dt}&\operatorname{Im}H
+    =\dot z_i\partial_i\operatorname{Im}H+\dot z^*_i\partial_i^*\operatorname{Im}H \\
     &=\frac i4\left(
-      \bar\partial_j\bar HP_{ji}\partial_i H-\partial_j H\bar P_{ji}\bar\partial_i\bar H
-    \right)
-    =0
+      \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^*
+    \right) \\
+    &=\frac i4\left(
+      \|\partial H\|-\|\partial H\|^*
+    \right)=0.
   \end{aligned}
 \end{equation}
-since $P$ is a Hermitian operator. This conservation law indicates that surfaces of constant $\operatorname{Im}H$ will be important when evaluting the possible endpoints of dynamic trajectories.
+As a result of this conservation law, surfaces of constant $\operatorname{Im}H$
+will be important when evaluting the possible endpoints of dynamic
+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
 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 $\mathbb R^{2(N-1)-1}$. Now, `place' a generic
+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
 point, the flow is locally
 \begin{equation}
-- 
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