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| -rw-r--r-- | stokes.tex | 30 |
1 files changed, 19 insertions, 11 deletions
@@ -121,13 +121,21 @@ 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 $u$, gradient descent follows the dynamics +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} - \dot u^\alpha - =-(\partial^*_{\beta}\operatorname{Re}H)h^{\beta\alpha} - =-\tfrac12(\partial_\beta H)^*h^{\beta\alpha} + \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} + \dot z=-\tfrac12(\partial H)^\dagger J^*[J^\dagger J]^{-1}J^T + =-\tfrac12(\partial H)^\dagger P +\end{equation} +which is nothing but the projection of $(\partial H)^*$ into the tangent space of the manifold, with $P=J^*[J^\dagger J]^{-1}J^T$. Note that $P$ is hermitian: $P^\dagger=(J^*[J^\dagger J]^{-1}J^T)^\dagger=J^*[J^\dagger J]^{-1}J^T=P$. + 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 @@ -135,12 +143,12 @@ $\operatorname{Im}H$ can be shown using \eqref{eq:flow} and the holomorphic prop \begin{equation} \begin{aligned} \frac d{dt}&\operatorname{Im}H - =\dot u^\alpha\partial_\alpha\operatorname{Im}H+(\dot u^\alpha)^*\partial_\alpha^*\operatorname{Im}H \\ + =\dot z\partial\operatorname{Im}H+\dot z^*\partial^*\operatorname{Im}H \\ &=\frac i4\left( - (\partial_\beta H)^*h^{\beta\alpha}\partial_\alpha H-\partial_\beta H(h^{\beta\alpha})^*\partial_\alpha^*H^* + (\partial H)^\dagger P\partial H-(\partial H)^TP^\dagger(\partial H)^* \right) \\ &=\frac i4\left( - (\partial_\beta H)^*h^{\beta\alpha}\partial_\alpha H-[(\partial_\beta H)^*h^{\beta\alpha}\partial_\alpha H]^* + (\partial H)^\dagger P\partial H-[(\partial H)^\dagger P\partial H]^* \right) \\ &=\frac i4\left( \|\partial H\|-\|\partial H\|^* @@ -160,13 +168,13 @@ single simply connected surface, locally diffeomorphic to $\mathbb R^{2N}$. Now, point, the flow is locally \begin{equation} \begin{aligned} - \dot u^\alpha - &\simeq-\frac12\operatorname{Hess}(H)^* + \dot z + &\simeq-\frac12(\partial\partial H)^\dagger P(z-z_0) \end{aligned} \end{equation} The matrix $(\partial\partial H)^\dagger P$ has a spectrum identical to that of -$\partial\partial H$ save a single marginal direction corresponding to $z_0$, -the normal to the constraint surface. Assuming we are working in a diagonal basis, we find +$\partial\partial H$ 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) \end{equation} |