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@@ -104,12 +104,35 @@ $\operatorname{Re}(-\beta H)$ 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} +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} \label{eq:thimble.integral} 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 $n_\sigma$. +exist for their value about the critical point $\sigma$, for example. One must +know the integer weights $n_\sigma$. + +Under analytic continuation of, say, $\beta$, the form of +\eqref{eq:thimble.integral} persists. When the homology of the thimbles is +unchanged by the continuation, the integer weights are likewise unchanged, and +one can therefore use the knowledge of these weights in one regime to compute +the partition function in other. However, their homology can change, and when +this happens the integer weights can be traded between critical points. These +trades occur when two thimbles intersect, or alternatively when one critical +point lies on the gradient descent of another. These places are called +\emph{Stokes points}, and the gradient descent trajectories that join two +critical points are called \emph{Stokes lines}. + +The prevalence (or not) of Stokes points in a given continuation, and whether +those that do appear affect the weights of critical points of interest, is a +concern for the analytic continuation of theories. If they do not occur or +occur order-one times, one could reasonably hope to perform such a procedure. +If they occur exponentially often, there is little hope of keeping track of the +resulting weights. \section{Gradient descent dynamics} |