From aec7b1fb31e146d4e192081cc600caa121b692c8 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 8 Mar 2021 17:16:30 +0100
Subject: More work.

---
 stokes.tex | 116 ++++++++++++++++++++++++++++++++++++++-----------------------
 1 file changed, 73 insertions(+), 43 deletions(-)

(limited to 'stokes.tex')

diff --git a/stokes.tex b/stokes.tex
index 13138f7..1ac2298 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -28,6 +28,8 @@
 
 \maketitle
 
+\cite{Witten_2011_Analytic}
+
 Consider a thermodynamic calculation involving the (real) $p$-spin model for a
 particular instantiation of the coupling tensor $J$
 \begin{equation} \label{eq:partition.function}
@@ -55,68 +57,75 @@ 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.
 
-The surface $M\subset\mathbb C^N$ defined by $z^2=N$ is an $N-1$ dimensional
+\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}. Suppose that $z:\mathbb
-C^{N-1}\to M$ is a holomorphic map. The Jacobian $J$ of the map is
+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 $\nabla\phi$ is a holomorphic vector field given by
 \begin{equation}
-  J_{i\alpha}=\frac{\partial z_i}{\partial u_\alpha}=\partial_\alpha z_i
+  \nabla^\alpha\phi=h^{\bar\beta\alpha}\bar\partial_{\bar\beta}\phi
 \end{equation}
-where Greek coefficients run from $1$ to $N-1$ and Latin coefficients from $1$
-to $N$. The hermitian metric is $g=J^\dagger J$. For any smooth function
-$\phi:M\to\mathbb R$, its gradient $\nabla\phi$ is a holomorphic vector field
-given by
-\begin{equation}
-  \nabla\phi=(\partial^*\phi)^\sharp=(\partial^*\phi)g^{-1}
+Dynamics consists of gradient descent on $\operatorname{Re}H$, or
+\begin{equation} \label{eq:flow}
+  \dot u^\alpha=-\nabla^\alpha\operatorname{Re}H=-\tfrac12h^{\bar\beta\alpha}\bar\partial_{\bar\beta}\bar H
 \end{equation}
-
-For
-coordinates $u\in\mathbb C^{N-1}$, dynamics consists of gradient descent on
-$\operatorname{Re}H$, or
+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}:
 \begin{equation}
-  \dot u=-\nabla\operatorname{Re}H=-\tfrac12(\partial H)^\dagger g^{-1}
+  \begin{aligned}
+    \frac d{dt}\operatorname{Im}H
+    &=\dot u^\alpha\partial_\alpha\operatorname{Im}H+\dot{\bar u}^{\bar\alpha}\bar\partial_{\bar\alpha}\operatorname{Im}H \\
+    &=\tfrac i4\left(\bar\partial_{\bar\beta}\bar Hh^{\bar\beta\alpha}\partial_\alpha H-\partial_\beta H\bar h^{\beta\bar\alpha}\bar\partial_{\bar\alpha}\bar H  \right)\\
+    &=0
+  \end{aligned}
 \end{equation}
-These dynamics preserve $\operatorname{Im}H$ and in fact correspond to
-Hamiltonian dynamics, with the real and imaginary parts of the coordinates
-taking the role of conjugate variables. \cite{Morrow_2006_Complex}
+since $h$ is a Hermitian operator with $\bar h=h^T$.
 
-Working with a particular map is inconvenient, and we would like to develop a map-independent dynamics. Using the chain rule, one finds
+Working with a particular map is inconvenient, and we would like to develop a
+map-independent dynamics. Suppose that $z:\mathbb C^{N-1}\to M$ is a map. Using
+the chain rule, one finds
 \begin{equation}
   \begin{aligned}
-    \dot z_i
-    &=\dot u_\alpha\partial_\alpha z_i
-    =-\tfrac12(\partial_\beta H)^*g^{-1}_{\beta\alpha}\partial_\alpha z_i\\
-    &=-\tfrac12(\partial_j H)^*(\partial_\beta z_j)^*g^{-1}_{\beta\alpha}\partial_\alpha z_i
-    =-\tfrac12(\partial H)^\dagger(J^\dagger g^{-1}J)\\
+    \dot z^i
+    &=\dot u^\alpha\partial_\alpha z^i
+    =-\tfrac12\bar\partial_{\bar\beta}\bar Hh^{\bar\beta\alpha}\partial_\alpha z^i
+    =-\tfrac12\bar\partial_j\bar H\partial_{\bar\beta}\bar z^{\bar\jmath}h^{\bar\beta\alpha}\partial_\alpha z^i \\
+    &=-\tfrac12(J^\dagger h^{-1}J)^{\bar\jmath i}\bar\partial_{\bar\jmath}\bar H\\
   \end{aligned}
 \end{equation}
-where $J$ is the Jacobian of the coordinate map and $g$ is the metric. In stereographic coordinates this can be worked out directly.
-Consider the coordinates $z_i=u_i$ for $1\leq i\leq N-1$ and
+where $J$ is the Jacobian of the coordinate map and $h$ is the Hermitian
+metric. In stereographic coordinates this can be worked out directly.
+Consider the coordinates $z^i=u^i$ for $1\leq i\leq N-1$ and
 \begin{equation}
-  z_N=\sqrt{N-u_\alpha u_\alpha}
+  z^N=\sqrt{N-u^2}
 \end{equation}
 The Jacobian is
 \begin{equation}
-  J_{\alpha i}=\partial_\alpha z_i=\delta_{\alpha i}-\delta_{Ni}\frac{u_\alpha}{\sqrt{N-u_\beta u_\beta}}
+  J_\alpha^{\hphantom\alpha i}=\partial_\alpha z^i=\delta_\alpha^{\hphantom\alpha i}-\delta_N^{\hphantom Ni}\frac{u_\beta}{\sqrt{N-u^2}}
 \end{equation}
-and the corresponding hermitian metric is
+and therefore the Hermitian metric induced by the map is
 \begin{equation}
-  g_{\alpha\beta}=J_{i\alpha}^*J_{i\beta}
-  =\delta_{\alpha\beta}+\frac{u_\alpha^*u_\beta}{|N-u_\gamma u_\gamma|}
+  h_{\alpha\bar\beta}=\bar J_{\alpha}^{\hphantom\alpha i}J_{\bar\beta}^{\hphantom\beta\bar\jmath}\delta_{i\bar\jmath}
+  =\delta_{\bar\alpha\beta}+\frac{\bar u^{\alpha}u^{\bar\beta}}{|N-u^2|}
 \end{equation}
 The metric can be inverted explicitly:
 \begin{equation}
-  g^{-1}_{\alpha\beta}
-  =\delta_{\alpha\beta}-\frac{u_\alpha^*u_\beta}{|N-u_\gamma u_\gamma|+|u|^2}.
+  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}
-  (J^\dagger g^{-1}J)_{ij}
-  =\delta_{ij}-\frac{z_iz_j^*}{|z|^2}
+  (J^\dagger h^{-1}J)^{\bar\jmath i}
+  =\delta^{\bar\jmath i}-\frac{z^{\bar\jmath}\bar z^i}{|z|^2}
 \end{equation}
-which is just the projector onto the constraint manifold.
-
-Therefore, a map-independent dynamics is given by
+which is just the projector onto the constraint manifold \cite{Morrow_2006_Complex}.
+Therefore, a map-independent dynamics for $z\in M$ is given by
 \begin{equation}
   \dot z
   =-\frac12(\partial H)^\dagger\left(I-\frac{zz^\dagger}{|z|^2}\right)
@@ -132,15 +141,35 @@ The level sets defined by $\operatorname{Im}H=c$ for $c\in\mathbb R$ are surface
 $2(N-1)-1$ real dimensions. They must be simply connected, since gradient
 descent in $\operatorname{Re}H$ cannot pass the same point twice.
 
+
+
 \section{2-spin}
 
+The Hamiltonian of the $2$-spin model is defined for $z\in\mathbb C^N$ by
 \begin{equation}
-  H_0=\frac12z^TJz
+  H_0=\frac12z^TJz.
 \end{equation}
 $J$ is generically diagonalizable by a complex orthogonal matrix $P$. With
-$z\mapsto Pz$, $J\mapsto D$ for diagonal $D$. Then $\partial_i H=d_iz_i$.
+$z\mapsto Pz$, $J\mapsto D$ for diagonal $D$. Then $\partial_i H=d_iz_i$. We will henceforth assume to be working in this basis. To find the critical points, we use the method of Lagrange multipliers. Introducing $\epsilon$, the constrained Hamiltonian is
+\begin{equation}
+  H=H_0+\epsilon(N-z^2)
+\end{equation}
+As usual, $\epsilon$ is equivalent to the energy per spin at any critical point.
+Critical points must satisfy
+\begin{equation}
+  0=\partial_iH=(d_i-2\epsilon)z_i
+\end{equation}
+which is only possible for $z_i=0$ or $\epsilon=\frac12 d_i$. Generically the $d_i$ will all differ, so this can only be satisfied for one $d_i$ at a time, and to be a critical point all other $z_j$ must be zero. In the direction in question,
+\begin{equation}
+  \epsilon=\frac1N\frac12d_iz_i^2=\frac12 d_i,
+\end{equation}
+whence $z_i=\pm\sqrt{N}$. Thus there are $2N$ critical points, each corresponding to $\pm$ the cardinal directions in the diagonalized basis.
+
 Suppose that two critical points have the same imaginary energy; without loss
-of generality, assume these are the first and second components. Since the gradient is proportional to $z$, any components that are zero at some time will be zero at all times. The dynamics for the components of interest assuming all others are zero are
+of generality, assume these are associated with the first and second
+cardinal directions. Since the gradient is proportional to $z$, any components that are
+zero at some time will be zero at all times. The dynamics for the components of
+interest assuming all others are zero are
 \begin{equation}
   \begin{aligned}
     \dot z_1
@@ -148,8 +177,9 @@ of generality, assume these are the first and second components. Since the gradi
     &=-(d_1-d_2)^*z_1^*\frac{|z_2|^2}{|z_1|^2+|z_2|^2}
   \end{aligned}
 \end{equation}
-and the same for $z_2$ with all indices swapped.  Since $\Delta=d_1-d_N$ is
-real, if $z_1$ begins real it remains real, with the same for $z_2$. Since the critical points are at real $z$, we make this restriction, and find
+and the same for $z_2$ with all indices swapped.  Since $\Delta=d_1-d_2$ is
+real, if $z_1$ begins real it remains real, with the same for $z_2$. Since the
+critical points are at real $z$, we make this restriction, and find
 \begin{equation}
   \begin{aligned}
     \frac d{dt}(z_1^2+z_2^2)=0\\
-- 
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