diff options
| -rw-r--r-- | stokes.tex | 23 |
1 files changed, 10 insertions, 13 deletions
@@ -288,10 +288,8 @@ 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 @@ -301,22 +299,21 @@ 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} |