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| -rw-r--r-- | stokes.tex | 4 |
1 files changed, 3 insertions, 1 deletions
@@ -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 |