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| -rw-r--r-- | stokes.tex | 38 |
1 files changed, 14 insertions, 24 deletions
@@ -121,28 +121,26 @@ 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 $\operatorname{Re}H$, or +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 $u$, gradient descent follows the dynamics \begin{equation} \label{eq:flow} - \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^*, + \dot u^\alpha + =-(\partial^*_{\beta}\operatorname{Re}H)h^{\beta\alpha} + =-\tfrac12(\partial_\beta H)^*h^{\beta\alpha} \end{equation} +where $h$ is the Hermitian metric and we have used the fact that, for holomorphic $H$, $\partial^*H=0$. 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 -$\operatorname{Im}H$ can be seen from the Cauchy--Riemann equations and -\eqref{eq:flow}: +$\operatorname{Im}H$ can be shown using \eqref{eq:flow} and the holomorphic property of $H$: \begin{equation} \begin{aligned} \frac d{dt}&\operatorname{Im}H - =\dot z_i\partial_i\operatorname{Im}H+\dot z^*_i\partial_i^*\operatorname{Im}H \\ + =\dot u^\alpha\partial_\alpha\operatorname{Im}H+(\dot u^\alpha)^*\partial_\alpha^*\operatorname{Im}H \\ + &=\frac i4\left( + (\partial_\beta H)^*h^{\beta\alpha}\partial_\alpha H-\partial_\beta H(h^{\beta\alpha})^*\partial_\alpha^*H^* + \right) \\ &=\frac i4\left( - \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^* + (\partial_\beta H)^*h^{\beta\alpha}\partial_\alpha H-[(\partial_\beta H)^*h^{\beta\alpha}\partial_\alpha H]^* \right) \\ &=\frac i4\left( \|\partial H\|-\|\partial H\|^* @@ -158,20 +156,12 @@ 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 $\mathbb R^{2N}$. Now, `place' a generic -(nondegenerate) critical point in the function at $z_0$. In the vicinity of the critical +(nondegenerate) critical point in the function at $u_0$. In the vicinity of the critical point, the flow is locally \begin{equation} \begin{aligned} - \dot z_i - &\simeq-\frac12\left[ - \partial_j\left( - H(z_0)+\frac12\partial_k\partial_\ell H(z_0)\Delta z_k\Delta z_\ell - \right) - \right]^* P_{ji} \\ - &=-\frac12\left( - \partial_j\partial_kH(z_0)\Delta z_k - \right)^* P_{ji} \\ - &=-\frac12\Delta z_k^*(\partial_k\partial_jH(z_0))^*P_{ji} + \dot u^\alpha + &\simeq-\frac12\operatorname{Hess}(H)^* \end{aligned} \end{equation} The matrix $(\partial\partial H)^\dagger P$ has a spectrum identical to that of |