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| -rw-r--r-- | stokes.tex | 21 |
1 files changed, 14 insertions, 7 deletions
@@ -111,16 +111,24 @@ Morse theory provides the universal correspondence between contours and thimbles 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 $\operatorname{Re}H$. In hermitian geometry, the gradient is given by raising an index of the conjugate differential, or $\operatorname{grad}f=(\partial^*f)^\sharp$. This implies that, in terms of coordinates, gradient descent follows the dynamics -\begin{equation} \label{eq:flow} +For a holomorphic Hamiltonian $H$, dynamics are defined by gradient descent on +$\operatorname{Re}H$. In hermitian geometry, the gradient is given by raising +an index of the conjugate differential, or +$\operatorname{grad}f=(\partial^*f)^\sharp$. This implies that, in terms of +coordinates $u:M\to\mathbb C^N$, gradient descent follows the dynamics +\begin{equation} \label{eq:flow.raw} \dot z^i =-(\partial^*_{\beta}\operatorname{Re}H)h^{\beta\alpha}\partial_\alpha z^i =-\tfrac12(\partial_\beta H)^*h^{\beta\alpha}\partial_\alpha z^i \end{equation} where $h$ is the Hermitian metric and we have used the fact that, for holomorphic $H$, $\partial^*H=0$. -This can be simplied furthur by noting that $h^{\beta\alpha}=h^{-1}_{\beta\alpha}$ for $h_{\beta\alpha}=(\partial_\beta z^i)^*\partial_\alpha z^i=(J^\dagger J)_{\beta\alpha}$. -\begin{equation} +This can be simplied by noting that $h^{\beta\alpha}=h^{-1}_{\beta\alpha}$ for +$h_{\beta\alpha}=(\partial_\beta z^i)^*\partial_\alpha z^i=(J^\dagger +J)_{\beta\alpha}$ where $J$ is the Jacobian of the coordinate map. Writing +$\partial H=\partial H/\partial z$ and inserting Jacobians everywhere they +appear, \eqref{eq:flow.raw} becomes +\begin{equation} \label{eq:flow} \dot z=-\tfrac12(\partial H)^\dagger J^*[J^\dagger J]^{-1}J^T =-\tfrac12(\partial H)^\dagger P \end{equation} @@ -146,8 +154,7 @@ $\operatorname{Im}H$ can be shown using \eqref{eq:flow} and the holomorphic prop \end{aligned} \end{equation} As a result of this conservation law, surfaces of constant $\operatorname{Im}H$ -will be important when evaluting the possible endpoints of dynamic -trajectories. +will be important when evaluting the possible endpoints of 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 @@ -163,7 +170,7 @@ point, the flow is locally \end{aligned} \end{equation} The matrix $(\partial\partial H)^\dagger P$ has a spectrum identical to that of -$\partial\partial H$ save marginal directions corresponding to the normals to +$(\partial\partial H)^\dagger$ save marginal directions corresponding to the normals to manifold. Assuming we are working in a diagonal basis, we find \begin{equation} \dot z_i=-\frac12\lambda_i\Delta z_i^*+O(\Delta z^2) |