From 0f2a0b73d1b76221a0e7c4f2df801b10869e0f56 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Thu, 10 Jun 2021 13:41:19 +0200 Subject: Changed section titles. --- stokes.tex | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/stokes.tex b/stokes.tex index fdc8e69..f417f07 100644 --- a/stokes.tex +++ b/stokes.tex @@ -74,7 +74,7 @@ landscapes, which are typically constructed from the limits of series or integrals of analytic functions which are not themselves analytic \cite{Cavagna_1999_Energy}. -\section{Dynamics} +\section{Integration by Lefschetz thimble} Consider an $N$-dimensional hermitian manifold $M$ and a Hamiltonian $H:M\to\mathbb C$. The partition function \begin{equation} @@ -111,6 +111,8 @@ 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 $n_\sigma$. +\section{Gradient descent 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 -- cgit v1.2.3-70-g09d2