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| -rw-r--r-- | stokes.bib | 15 | ||||
| -rw-r--r-- | stokes.tex | 116 |
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@@ -22,4 +22,19 @@ isbn = {9780821840559} } +@incollection{Witten_2011_Analytic, + author = {Witten, Edward}, + title = {Analytic continuation of Chern-Simons theory}, + publisher = {American Mathematical Society}, + year = {2011}, + month = {7}, + volume = {50}, + pages = {347--446}, + url = {https://doi.org/10.1090%2Famsip%2F050%2F19}, + doi = {10.1090/amsip/050/19}, + booktitle = {Chern-Simons Gauge Theory: 20 Years After}, + editor = {Andersen, Jørgen E. and Boden, Hans U. and Hahn, Atle and Himpel, Benjamin}, + series = {AMS/IP Studies in Advanced Mathematics} +} + @@ -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\\ |