From d9609a46855a879d83dc6c9c705dd3f1cc199b6a Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 28 Jan 2022 15:59:18 +0100
Subject: Big pedagogical rewrite of the first section.

---
 stokes.tex | 193 +++++++++++++++++++++++++++++++++++++++----------------------
 1 file changed, 123 insertions(+), 70 deletions(-)

(limited to 'stokes.tex')

diff --git a/stokes.tex b/stokes.tex
index 82270bd..6a8c35d 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -19,7 +19,6 @@
 \author{Jaron Kent-Dobias and Jorge Kurchan}
 
 \address{Laboratoire de Physique de l'Ecole Normale Supérieure, Paris, France}
-\ead{jaron.dobias@phys.ens.fr}
 
 \begin{abstract}
   In this paper we follow up the study of `complex complex landscapes'
@@ -41,6 +40,8 @@
 
 \maketitle
 
+\section{Introduction}
+
 Analytic continuation of physical theories is sometimes useful. Some theories
 have a well-motivated hamiltonian or action that nevertheless results in a
 divergent partition function, and can only be properly defined by continuation
@@ -49,81 +50,133 @@ in oscillatory phase space measures that spoil the use of Monte Carlo or saddle
 point techniques, but can be treated in a regime where the measure does not
 oscillated and the results continued to the desired model \cite{}.
 
-Consider an action $\mathcal S_\lambda$ defined on the phase space $\Omega$ and
-depending on parameters $\lambda$. In the context of statistical mechanics,
-$\mathcal S_{\beta,J}=-\beta H_J$ for some hamiltonian $H_J$ with quenched
-parameters $J$ at inverse temperature $\beta$. A typical calculation stems from
-the partition function
+In any case, the nicest modern technique (which we will describe in some detail
+later) consists of deforming the phase space integral into a complex phase
+space and then breaking it into pieces associated with stationary points of the
+action. Each of these pieces, known as \emph{thimbles}, has wonderful
+properties that guarantee convergence and prevent oscillations. Once such a
+decomposition is made, analytic continuation is mostly easy, save for instances
+where the thimbles interact, which must be accounted for.
+
+When your action has a manageable set of stationary points, this process is
+usually tractable. However, many actions of interest are complex, having many
+stationary points with no simple symmetry relating them, far too many to
+individually monitor. Besides appearing in classical descriptions of structural
+and spin glasses, complex landscapes are recently become important objects of
+study in the computer science of machine learning, the condensed matter theory
+of strange metals, and the high energy physics of black holes. What becomes of
+analytic continuation under these conditions?
+
+\section{Thimble integration and analytic continuation}
+
+Consider an action $\mathcal S$ defined on the (real) phase space $\Omega$. A
+typical calculation stems from the partition function
 \begin{equation} \label{eq:partition.function}
-  Z(\lambda)=\int_\Omega ds\,e^{\mathcal S_\lambda(s)}.
-\end{equation}
-This integral is often dominated by its behavior near stationary points of the
-action, and understanding these points is usually important to evaluate the
-partition function.
-
-Recent developments have found that stationary points of the action are
-important for understanding another aspect of the partition function: its
-analytic continuation. The integral \eref{eq:partition.function} is first
-interpreted as a contour in a larger complex phase space, then deformed into a
-linear combination of specially constructed contours each enumerated by a
-stationary point. Analytic continuation of parameters preserves this
-decomposition except at nongeneric points where contours intersect.
-
-We investigate the plausibility of analytic continuation in complex models
-where the action has a macroscopic number of stationary points. Such actions
-are common in studies of glasses, spin glass, machine learning, black holes,
-\dots We find that the geometry of the landscape, and in particular the
-relative position and spectrum of stationary points, is key.
-
-Analytic continuation of partition functions is useful for many reasons. Some
-physically motivated theories have actions whose partition function is formally
-infinite, but can be defined by continuing from a set of parameters where it
-converges. Other theories have oscillatory actions that lead to a severe sign
-problem in estimating the partition function, which can be addressed by taking
-advantage of a deformed phase space where the phase of the action varies slowly.
-
-Unfortunately the study is not so relevant for low-dimensional `rugged'
-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{Integration by Lefschetz thimble}
-
-We return to the partition function \eref{eq:partition.function}.  If
-the action can be continued to a holomorphic function on the Kähler
-manifold $\tilde\Omega\supset\Omega$ and $\Omega$ is orientable in $\tilde\Omega$,
-then \eref{eq:partition.function} can be considered a contour integral. 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 $|s|\to\infty$ the real part of the action goes to $-\infty$, to ensure
-the integral converges. Second, that the contours piecewise correspond to
-surfaces of slowing vary phase of the action, so as to ameliorate sign
-problems.
+  Z(\beta)=\int_\Omega ds\,e^{-\beta\mathcal S(s)}
+\end{equation}
+We've defined $Z$ in a way that strongly suggests application in statistical
+mechanics, but everything here is general: the action can be complex- or even
+imaginary-valued, and $\Omega$ could be infinite-dimensional. In typical
+contexts, $\Omega$ will be the euclidean real space $\mathbb R^N$ or some
+subspace of this like the sphere $S^{N-1}$ (as in the $p$-spin spherical models
+we will study later). We will consider only the analytic continuation of the
+parameter $\beta$, but any other would work equally well, e.g., of some
+parameter inside the action. The action will have some stationary points, e.g.,
+minima, maxima, saddles, and the set of those points in $\Omega$ we will call
+$\Sigma_0$, the set of real stationary points.
+
+In order to analytically continue \eref{eq:partition.function} by the method we
+will describe, $\mathcal S$ must have an extension to a holomorphic function on
+a larger complex phase space $\tilde\Omega$ containing $\Omega$. In many cases
+this is accomplished by simply noticing that the action is some sum or product
+of holomorphic functions, e.g., polynomials. For $\mathbb R^N$ the complex
+phase space $\tilde\Omega$ will be $\mathbb C^N$, while for the sphere
+$S^{N-1}$ it takes a little more effort. $S^{N-1}$ can be defined by all points
+$x\in\mathbb R^N$ such that $x^Tx=1$. A complex extension of the sphere is made
+by extending this constraint: all points $z\in \mathbb C^N$ such that $z^Tz=1$.
+Both cases are complex manifolds and moreover K\"ahler manifolds, since they
+are defined by holomorphic constraints, and therefore admit a hermitian
+metric and a symplectic structure. In the extended complex phase space, the
+action potentially has more stationary points. We'll call $\Sigma$ the set of
+\emph{all} stationary points of the action, which naturally contains the set of
+\emph{real} stationary points $\Sigma_0$.
+
+Assuming $\mathcal S$ is holomorphic (and that the phase space $\Omega$ is
+orientable, which is usually true) the integral in \eref{eq:partition.function}
+can be considered an integral over a contour in the complex phase space $\tilde\Omega$,
+or
+\begin{equation} \label{eq:contour.partition.function}
+  Z(\beta)=\oint_\Omega dz\,e^{-\beta\mathcal S(z)}
+\end{equation}
+For the moment this translation has only changed some of our symbols from
+\eref{eq:partition.function}, but conceptually it is very important: contour
+integrals can have their contour freely deformed (under some constraints)
+without changing their value. This means that we are free to choose a nicer
+contour than our initial phase space $\Omega$.
+
+What contour properties are desirable? Consider the two main motivations cited
+in the introduction for performing analytic continuation in the first place: we
+want our partition function to be well-defined, e.g., for the phase space
+integral to converge, and we want to avoid oscillations in the phase of the
+integrand. The first condition, convergence, necessitates that the real part of
+the action $\operatorname{Re}\mathcal S$ be bounded from below, and that it
+approach infinity in any limiting direction along the contour. The second,
+constant phase, necessitates that the imaginary part of the action
+$\operatorname{Im}\mathcal S$ be constant.
 
 Remarkably, there is an elegant recipe for accomplishing both these criteria at
 once, courtesy of Morse theory. For a more thorough review, see
-\cite{Witten_2011_Analytic}. Consider a stationary point of the action. The
-union of all gradient descent trajectories on the real part of the action that
-begin at the stationary point is known as a \emph{Lefschetz thimble}. Since
-each point on the Lefschetz thimble is found through descent from the
-stationary point, the real part of the action is bounded from above by its
-value at the stationary point. Likewise, we shall see that the imaginary part
-of the action is constant on a thimble.
-
-Morse theory provides a universal correspondence between contours and thimbles.
-For any contour $\Omega$, there exists a linear combination of thimbles such
-that the relative homology of the combination with respect to decent int he
-action is equivalent to that of the contour.  If $\Sigma$ is the set of
-stationary points of the action and for each $\sigma\in\Sigma$ the set
-$\mathcal J_\sigma\subset\tilde\Omega$ is its thimble, then this gives
+\cite{Witten_2011_Analytic}. We are going to construct our deformed contour out
+of a collection of pieces called \emph{Lefschetz thimbles}, or just thimbles.
+There is one thimble $\mathcal J_\sigma$ associated with each of the stationary
+points $\sigma\in\Sigma$ of the action, and it is defined by all points that
+approach the stationary point $z_\sigma$ under gradient descent on
+$\operatorname{Re}\mathcal S$.
+
+Thimbles guarantee convergent integrals by construction: the value of
+$\operatorname{Re}\mathcal S$ is bounded from below on the thimble $\mathcal J_\sigma$ by its value
+$\operatorname{Re}\mathcal S(z_\sigma)$ at the stationary point,
+since all other points on the thimble must descend to reach it. And, as we will
+see in a moment, thimbles guarantee constant phase for the integrand as well, a
+result of the underlying complex geometry of the problem.
+
+What thimbles are necessary to reproduce our original contour, $\Omega$? The
+answer is, we need the minimal set which produces a contour between the same
+places. Simply stated, if $\Omega=\mathbb R$ produced a phase space integral
+running along the real line from left to right, then our contour must likewise
+take one continuously from left to right, perhaps with detours to well-behaved
+places at infinity (see Fig.~\ref{fig:1d.thimble}). The less simply stated versions follows.
+
+Let $\tilde\Omega_T$ be the set of all points $z\in\tilde\Omega$ such that
+$\operatorname{Re}\mathcal S(z)\geq T$, where we will take $T$ to be a very,
+very large number. $\tilde\Omega_T$ is then the parts of the manifold where it
+is safe for any contour to end up if it wants its integral to converge, since
+these are the places where the real part of the action is very large and the
+integrand vanishes exponentially. The relative homology group
+$H_N(\tilde\Omega,\tilde\Omega_T)$ describes the homology of cycles which begin
+and end in $\Omega_T$, i.e., are well-behaved. Therefore, any well-behaved
+cycle must represent an element of $H_N(\tilde\Omega,\tilde\Omega_T)$. In order
+for our collection of thimbles to produce the correct contour, the composition
+of the thimbles must represent the same element of this relative homology
+group.
+
+Each thimble represents an element of the relative homology, since each thimble
+is a contour on which the real part of the action diverges in any direction.
+And, thankfully for us, Morse theory on our complex manifold $\tilde\Omega$
+implies that the set of all thimbles produces a basis for this relative
+homology group, and therefore any contour can be represented by some
+composition of thimbles! There is even a systematic way to determine the
+contribution from each thimble: for the critical point $\sigma\in\Sigma$, let
+$\mathcal K_\sigma$ be its \emph{antithimble}, defined by all points brought to
+$z_\sigma$ by gradient \emph{ascent} (and representing an element of the
+relative homology group $H_N(\tilde\Omega,\tilde\Omega_{-T})$). Then each
+thimble $\mathcal J_\sigma$ contributes to the contour with a weight given by
+its intersection pairing $n_\sigma=\langle\mathcal C,\mathcal K_\sigma\rangle$.
+
+With these tools in hands, we can finally write the partition function as a sum over contributions over thimbles, or
 \begin{equation} \label{eq:thimble.integral}
-  Z(\lambda)=\sum_{\sigma\in\Sigma}n_\sigma\int_{\mathcal J_\sigma}ds\,e^{\mathcal S_\lambda(s)}.
+  Z(\beta)=\sum_{\sigma\in\Sigma}n_\sigma\oint_{\mathcal J_\sigma}dz\,e^{-\beta\mathcal S(z)}.
 \end{equation}
-Each of these integrals is very well-behaved: convergent asymptotic series
-exist for their value about each critical point. The integer weights $n_\sigma$
-are fixed by comparison with the initial contour. For a real action, all maxima
-in $\Omega$ contribute in equal magnitude.
-
 Under analytic continuation, the form of \eref{eq:thimble.integral}
 generically persists. When the relative homology of the thimbles is unchanged
 by the continuation, the integer weights are likewise unchanged, and one can
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