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@@ -58,61 +58,6 @@ acquires an imaginary component, various numeric and perturbative schemes for approximating its value can face immediate difficulties due to the emergence of a sign problem, resulting from rapid oscillations coinciding with saddles. -\section{Geometry} - -The surface $M\subset\mathbb C^N$ defined by $N=f(z)=z^2$ is an $N-1$ dimensional -\emph{Stein manifold}, a type of complex manifold defined by the level set of a -holomorphic function \cite{Forstneric_2017_Stein}. One can define a Hermitian -metric $h$ on $M$ by taking the restriction of the standard metric of $\mathbb -C^N$ to vectors tangent along $M$. For any smooth function $\phi:M\to\mathbb -R$, its gradient is a holomorphic vector field given by -\begin{equation} - \operatorname{grad}\phi=\bar\partial^\sharp\phi -\end{equation} -Suppose $u:M\to\mathbb C^{N-1}$ is a coordinate system. Then -\begin{equation} - \operatorname{grad}\phi=h^{\bar\beta\alpha}\bar\partial_{\bar\beta}\phi\frac{\partial}{\partial u^\alpha} -\end{equation} -Let $z=u^{-1}$. -\begin{equation} - \frac\partial{\partial u^\alpha}=\frac{\partial z^i}{\partial u^\alpha}\frac\partial{\partial z^i} -\end{equation} -\begin{equation} - \bar\partial_{\bar\beta}\phi=\frac{\partial\bar z^i}{\partial\bar u^{\bar\beta}}\frac{\partial\phi}{\partial\bar z^i} -\end{equation} -\begin{equation} - \operatorname{grad}\phi - =\frac{\partial\bar z^{\bar\jmath}}{\partial\bar u^{\bar\beta}}h^{\bar\beta\alpha}\frac{\partial z^i}{\partial u^\alpha}\frac{\partial\phi}{\partial\bar z^{\bar\jmath}}\frac\partial{\partial z^i} -\end{equation} -At any point $z_0\in M$, $z_0$ is parallel to the normal to $M$. Without loss of generality, take $z_0=e^N$. In a neighborhood of $z_0$, the map $u^\alpha=z^\alpha$ is a coordinate system. -its inverse is $z^i=u^i$ for $1\leq i\leq N-1$ and -\begin{equation} - z^N=\sqrt{N-u^2}. -\end{equation} -The Jacobian is -\begin{equation} - \frac{\partial z^i}{\partial u^\alpha}=\delta^i_\alpha-\delta_N^{i}\frac{u^\alpha}{\sqrt{N-u^2}} -\end{equation} -and therefore the Hermitian metric induced by the map is -\begin{equation} - h_{\alpha\bar\beta}=\frac{\partial\bar z^i}{\partial\bar u^\alpha}\frac{\partial z^{\bar\jmath}}{\partial u^{\bar\beta}}\delta_{i\bar\jmath} - =\delta_{\alpha\bar\beta}+\frac{\bar u^{\alpha}u^{\bar\beta}}{|N-u^2|} -\end{equation} -The metric can be inverted explicitly: -\begin{equation} - h^{\bar\beta\alpha} - =\delta^{\bar\beta\alpha}-\frac{\bar u^{\bar\beta}u^\alpha}{|N-u^2|+|u|^2}. -\end{equation} -Putting these pieces together, we find -\begin{equation} - \frac{\partial\bar z^{\bar\jmath}}{\partial\bar u^{\bar\beta}}h^{\bar\beta\alpha}\frac{\partial z^i}{\partial u^\alpha} - =\delta^{\bar\jmath i}-\frac{z^{\bar\jmath}\bar z^i}{|z|^2} -\end{equation} -\begin{equation} - \operatorname{grad}\phi - =\left(\delta^{\bar\jmath i}-\frac{z^{\bar\jmath}\bar z^i}{|z|^2}\right) - \frac{\partial\phi}{\partial\bar z^{\bar\jmath}}\frac\partial{\partial z^i} -\end{equation} \section{Dynamics} @@ -286,10 +231,70 @@ To confirm the presence of Stokes lines under certain processes in the $p$-spin, \end{equation} $\mathcal C$ has minimum of zero, which is reached only by functions $z(t)$ whose tangent is everywhere parallel to the direction of the dynamics. Therefore, finding functions which reach this minimum will find time-reparameterized Stokes lines. For the sake of numeric stability, we look for functions whose tangent has constant norm. + + \begin{acknowledgments} MIT mathematicians have been no help \end{acknowledgments} \bibliography{stokes} +\appendix + +\section{Geometry} + +The surface $M\subset\mathbb C^N$ defined by $N=f(z)=z^2$ is an $N-1$ dimensional +\emph{Stein manifold}, a type of complex manifold defined by the level set of a +holomorphic function \cite{Forstneric_2017_Stein}. One can define a Hermitian +metric $h$ on $M$ by taking the restriction of the standard metric of $\mathbb +C^N$ to vectors tangent along $M$. For any smooth function $\phi:M\to\mathbb +R$, its gradient is a holomorphic vector field given by +\begin{equation} + \operatorname{grad}\phi=\bar\partial^\sharp\phi +\end{equation} +Suppose $u:M\to\mathbb C^{N-1}$ is a coordinate system. Then +\begin{equation} + \operatorname{grad}\phi=h^{\bar\beta\alpha}\bar\partial_{\bar\beta}\phi\frac{\partial}{\partial u^\alpha} +\end{equation} +Let $z=u^{-1}$. +\begin{equation} + \frac\partial{\partial u^\alpha}=\frac{\partial z^i}{\partial u^\alpha}\frac\partial{\partial z^i} +\end{equation} +\begin{equation} + \bar\partial_{\bar\beta}\phi=\frac{\partial\bar z^i}{\partial\bar u^{\bar\beta}}\frac{\partial\phi}{\partial\bar z^i} +\end{equation} +\begin{equation} + \operatorname{grad}\phi + =\frac{\partial\bar z^{\bar\jmath}}{\partial\bar u^{\bar\beta}}h^{\bar\beta\alpha}\frac{\partial z^i}{\partial u^\alpha}\frac{\partial\phi}{\partial\bar z^{\bar\jmath}}\frac\partial{\partial z^i} +\end{equation} +At any point $z_0\in M$, $z_0$ is parallel to the normal to $M$. Without loss of generality, take $z_0=e^N$. In a neighborhood of $z_0$, the map $u^\alpha=z^\alpha$ is a coordinate system. +its inverse is $z^i=u^i$ for $1\leq i\leq N-1$ and +\begin{equation} + z^N=\sqrt{N-u^2}. +\end{equation} +The Jacobian is +\begin{equation} + \frac{\partial z^i}{\partial u^\alpha}=\delta^i_\alpha-\delta_N^{i}\frac{u^\alpha}{\sqrt{N-u^2}} +\end{equation} +and therefore the Hermitian metric induced by the map is +\begin{equation} + h_{\alpha\bar\beta}=\frac{\partial\bar z^i}{\partial\bar u^\alpha}\frac{\partial z^{\bar\jmath}}{\partial u^{\bar\beta}}\delta_{i\bar\jmath} + =\delta_{\alpha\bar\beta}+\frac{\bar u^{\alpha}u^{\bar\beta}}{|N-u^2|} +\end{equation} +The metric can be inverted explicitly: +\begin{equation} + h^{\bar\beta\alpha} + =\delta^{\bar\beta\alpha}-\frac{\bar u^{\bar\beta}u^\alpha}{|N-u^2|+|u|^2}. +\end{equation} +Putting these pieces together, we find +\begin{equation} + \frac{\partial\bar z^{\bar\jmath}}{\partial\bar u^{\bar\beta}}h^{\bar\beta\alpha}\frac{\partial z^i}{\partial u^\alpha} + =\delta^{\bar\jmath i}-\frac{z^{\bar\jmath}\bar z^i}{|z|^2} +\end{equation} +\begin{equation} + \operatorname{grad}\phi + =\left(\delta^{\bar\jmath i}-\frac{z^{\bar\jmath}\bar z^i}{|z|^2}\right) + \frac{\partial\phi}{\partial\bar z^{\bar\jmath}}\frac\partial{\partial z^i} +\end{equation} + \end{document} |