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| -rw-r--r-- | stokes.tex | 44 |
1 files changed, 31 insertions, 13 deletions
@@ -206,6 +206,27 @@ 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. +\begin{figure} + \includegraphics{figs/thimble_homology.eps} + \hfill + \includegraphics{figs/antithimble_homology.eps} + + \caption{ + A demonstration of the rules of thimble homology. Both figures depict the + complex-$\theta$ plane of an $N=2$ spherical $3$-spin model. The black + symbols lie on the stationary points of the action, and the grey regions + depict the sets $\tilde\Omega_T$ of well-behaved regions at infinity (here + $T=5$). (Left) Lines show the thimbles of each stationary point. The + thimbles necessary to recreate the cyclic path from left to right are + darkly shaded, while those unnecessary for the task are lightly shaded. + Notice that all thimbles come and go from the well-behaved regions. (Right) + Lines show the antithimbles of each stationary point. Notice that those of + the stationary points involved in the contour (shaded darkly) all intersect + the desired contour (the real axis), while those not involved do not + intersect it. + } \label{fig:thimble.homology} +\end{figure} + 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 $\tilde\Omega$ @@ -311,10 +332,8 @@ holomorphic property of $\mathcal S$: \|\partial\mathcal S\|^2-(\|\partial\mathcal S\|^*)^2 \right)=0. \end{eqnarray} -As a result of this conservation law, surfaces of constant imaginary action -will be important when evaluting the possible endpoints of trajectories. A -consequence of this conservation is that the flow in the action takes a simple -form: +A consequence of this conservation is that the flow in the action takes a +simple form: \begin{equation} \dot{\mathcal S} =\dot z\partial\mathcal S @@ -324,22 +343,21 @@ form: In the complex-$\mathcal S$ plane, dynamics is occurs along straight lines in a direction set by the argument of $\beta$. -Let us consider the generic case, where the critical points of $\mathcal S$ have -distinct energies. What is the topology of the $C=\operatorname{Im}\mathcal S$ 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 $\mathbb R^{2D-1}$. Now, `place' a generic -(nondegenerate) critical point in the function at $z_0$. In the vicinity of the critical -point, the flow is locally +What does the topology of the space of thimbles look like? Let us consider the +generic case, where the critical points of $\beta\mathcal S$ have distinct +energies. Consider initially the situation in the absence of any critical +point, e.g., as for a constant or linear function. 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} \dot z - \simeq-\frac12(\partial\partial\mathcal S|_{z=z_0})^\dagger P(z-z_0)^* + \simeq-\frac{\beta^*}2(\partial\partial\mathcal S|_{z=z_0})^\dagger P(z-z_0)^* \end{equation} The matrix $(\partial\partial\mathcal S)^\dagger P$ has a spectrum identical to that of $(\partial\partial\mathcal S)^\dagger$ save marginal directions corresponding to the normals to manifold. Assuming we are working in a diagonal basis, this becomes \begin{equation} - \dot z_i=-\frac12\lambda_i\Delta z_i^*+O(\Delta z^2,(\Delta z^*)^2) + \dot z_i=-\frac12(\beta\lambda_i)^*\Delta z_i^*+O(\Delta z^2,(\Delta z^*)^2) \end{equation} Breaking into real and imaginary parts gives \begin{eqnarray} |