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@@ -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} |