Moved geometry of p-spin to an appendix.

1 files changed, 60 insertions, 55 deletions

diff --git a/stokes.tex b/stokes.tex
index 0968e1b..88c80fe 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -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}