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@@ -463,54 +463,88 @@ described using an analysis of the hessian of the real part of the action at the stationary point. Here we'll review some general properties of this hessian, which because the action is holomorphic has rich structure. -For the moment we will work in flat space, where the notation is nicer. -Everything follows for a curved manifold embedded in a higher-dimensional flat -space by projecting out the directions normal to the manifold, e.g., -$P\partial\partial\mathcal S P^T$ for the projection operator $P$ from the -previous section. - -First, consider -the problem as one of $2N$ real variables $x,y\in\mathbb R^N$ with $z=x+iy$. -The hessian of the real part of the action with respect to these real variables -is +Writing down the hessian using the complex geometry of the previous section +would be quite arduous. Luckily, we are only interested in the hessian at +stationary points, and our manifolds of interest are all constraint surfaces. +These two facts allow us to find the hessian at stationary points using a +simpler technique, that of Lagrange multipliers. + +Suppose that our complex manifold $\tilde\Omega$ is defined by all points +$z\in\mathbb C^N$ such that $g(z)=0$ for some holomorphic function $z$. In the +case of the spherical models, $g(z)=\frac12(z^Tz-N)$. Introducing the Lagrange +multiplier $\mu$, we define the constrained action +\begin{equation} + \tilde\mathcal S(z)=\mathcal S(z)-\mu g(z) +\end{equation} +The condition for a stationary point is that $\partial\tilde\mathcal S=0$. This implies that, at any stationary point, +$\partial\mathcal S=\mu\partial g$. In particular, if $\partial g^T\partial g\neq0$, we find the value for $\mu$ as +\begin{equation} + \mu=\frac{\partial g^T\partial\mathcal S}{\partial g^T\partial g} +\end{equation} +As a condition for a stationary point, this can be intuited as projecting out +the normal to the constraint surface $\partial g$ from the gradient of the +unconstrained action. It implies that the hessian with respect to the complex +embedding coordinates $z$ at any stationary point is +\begin{equation} \label{eq:complex.hessian} + \operatorname{Hess}\mathcal S + =\partial\partial\tilde\mathcal S + =\partial\partial\mathcal S-\frac{\partial g^T\partial\mathcal S}{\partial g^T\partial g}\partial\partial g +\end{equation} +In practice one must neglect the directions normal to the constraint surface by +projecting them out using $P$ from the previous section, i.e., +$P\operatorname{Hess}\mathcal SP^T$. For notational simplicity we will not +include this here. + +In order to describe the structure of thimbles, one must study the Hessian of +$\operatorname{Re}\beta\mathcal S$. We first pose the problem problem as one +of $2N$ real variables $x,y\in\mathbb R^N$ with $z=x+iy$, the hessian of the +real part of the action with respect to these real variables is \begin{equation} \label{eq:real.hessian} \operatorname{Hess}_{x,y}\operatorname{Re}\beta\mathcal S =\left[\matrix{ - \partial_x\partial_x\operatorname{Re}\beta\mathcal S & - \partial_y\partial_x\operatorname{Re}\beta\mathcal S \cr - \partial_x\partial_y\operatorname{Re}\beta\mathcal S & - \partial_y\partial_y\operatorname{Re}\beta\mathcal S + \partial_x\partial_x\operatorname{Re}\beta\tilde\mathcal S & + \partial_y\partial_x\operatorname{Re}\beta\tilde\mathcal S \cr + \partial_x\partial_y\operatorname{Re}\beta\tilde\mathcal S & + \partial_y\partial_y\operatorname{Re}\beta\tilde\mathcal S }\right] \end{equation} -This con be simplified using the fact that the action is holomorphic, which +This can be simplified using the fact that the action is holomorphic, which means that it obeys the Cauchy--Riemann equations \begin{equation} - \partial_x\operatorname{Re}\mathcal S=\partial_y\operatorname{Im}\mathcal S + \partial_x\operatorname{Re}\tilde\mathcal S=\partial_y\operatorname{Im}\tilde\mathcal S \qquad - \partial_y\operatorname{Re}\mathcal S=-\partial_x\operatorname{Im}\mathcal S + \partial_y\operatorname{Re}\tilde\mathcal S=-\partial_x\operatorname{Im}\tilde\mathcal S \end{equation} Using these relationships alongside the Wirtinger derivative $\partial\equiv\frac12(\partial_x-i\partial_y)$ allows the order of the derivatives and the real or imaginary parts to be commuted, with \begin{eqnarray} - \partial_x\operatorname{Re}\mathcal S=\operatorname{Re}\partial\mathcal S + \partial_x\operatorname{Re}\tilde\mathcal S=\operatorname{Re}\partial\tilde\mathcal S \qquad - \partial_y\operatorname{Re}\mathcal S=-\operatorname{Im}\partial\mathcal S \\ - \partial_x\operatorname{Im}\mathcal S=\operatorname{Im}\partial\mathcal S + \partial_y\operatorname{Re}\tilde\mathcal S=-\operatorname{Im}\partial\tilde\mathcal S \\ + \partial_x\operatorname{Im}\tilde\mathcal S=\operatorname{Im}\partial\tilde\mathcal S \qquad - \partial_y\operatorname{Im}\mathcal S=\operatorname{Re}\partial\mathcal S + \partial_y\operatorname{Im}\tilde\mathcal S=\operatorname{Re}\partial\tilde\mathcal S \end{eqnarray} Using these relationships, the hessian \eref{eq:real.hessian} can be written in the more manifestly complex way -\begin{equation} +\begin{eqnarray} \operatorname{Hess}_{x,y}\operatorname{Re}\beta\mathcal S - =\left[\matrix{ - \hphantom{-}\operatorname{Re}\beta\partial\partial\mathcal S & - -\operatorname{Im}\beta\partial\partial\mathcal S \cr - -\operatorname{Im}\beta\partial\partial\mathcal S & - -\operatorname{Re}\beta\partial\partial\mathcal S + &=\left[\matrix{ + \hphantom{-}\operatorname{Re}\beta\partial\partial\tilde\mathcal S & + -\operatorname{Im}\beta\partial\partial\tilde\mathcal S \cr + -\operatorname{Im}\beta\partial\partial\tilde\mathcal S & + -\operatorname{Re}\beta\partial\partial\tilde\mathcal S + }\right] \\ + &=\left[\matrix{ + \hphantom{-}\operatorname{Re}\beta\operatorname{Hess}\mathcal S & + -\operatorname{Im}\beta\operatorname{Hess}\mathcal S \cr + -\operatorname{Im}\beta\operatorname{Hess}\mathcal S & + -\operatorname{Re}\beta\operatorname{Hess}\mathcal S }\right] -\end{equation} +\end{eqnarray} +where $\operatorname{Hess}\mathcal S$ is the hessian with respect to $z$ given +in \eqref{eq:complex.hessian}. The eigenvalues and eigenvectors of the Hessian are important for evaluating thimble integrals, because those associated with upward directions provide a @@ -522,7 +556,7 @@ R^N$ are such that \end{equation} where the eigenvalue $\lambda$ must be real because the hessian is real symmetric. The problem can be put into a more obviously complex form by a change of basis. Writing $v=v_x+iv_y$, we find \begin{eqnarray} - &\left[\matrix{0&-i(\beta\partial\partial\mathcal S)^*\cr i\beta\partial\partial\mathcal S&0}\right] + &\left[\matrix{0&(i\beta\operatorname{Hess}\mathcal S)^*\cr i\beta\operatorname{Hess}\mathcal S&0}\right] \left[\matrix{v \cr iv^*}\right]\\ &\qquad=\left[\matrix{1&i\cr i&1}\right] (\operatorname{Hess}_{x,y}\operatorname{Re}\beta\mathcal S) @@ -534,17 +568,18 @@ where the eigenvalue $\lambda$ must be real because the hessian is real symmetri \end{eqnarray} It therefore follows that the eigenvalues and vectors of the real hessian satisfy the equation \begin{equation} \label{eq:generalized.eigenproblem} - \beta\partial\partial\mathcal S v=\lambda v^* + \beta\operatorname{Hess}\mathcal S v=\lambda v^* \end{equation} a sort of generalized eigenvalue problem. If we did not know the eigenvalues were real, we could -still see it from the second implied equation, $(\beta\partial\partial\mathcal -S)^*v^*=\lambda v$, which is the conjugate of the first if $\lambda^*=\lambda$. +still see it from the second implied equation, +$(\beta\operatorname{Hess}\mathcal S)^*v^*=\lambda v$, which is the conjugate +of the first if $\lambda^*=\lambda$. Something somewhat hidden in the structure of the real hessian but more clear in its complex form is that each eigenvalue comes in a pair, since \begin{equation} - \beta\partial\partial\mathcal S(iv)=i\lambda v^*=-\lambda(iv) + \beta\operatorname{Hess}\mathcal S(iv)=i\lambda v^*=-\lambda(iv) \end{equation} Therefore, if $\lambda$ is an eigenvalue of the hessian with eigenvector $v$, than so is $-\lambda$, with associated eigenvector $iv$, rotated in the complex @@ -561,12 +596,12 @@ The effect of changing the phase of $\beta$ is revealed by \eqref{eq:generalized.eigenproblem}. Writing $\beta=|\beta|e^{i\phi}$ and dividing both sides by $|\beta|e^{i\phi/2}$, one finds \begin{equation} - \partial\partial\mathcal S(e^{i\phi/2}v) + \operatorname{Hess}\mathcal S(e^{i\phi/2}v) =\frac{\lambda}{|\beta|}e^{-i\phi/2}v^* =\frac{\lambda}{|\beta|}(e^{i\phi/2}v)^* \end{equation} Therefore, one only needs to consider solutions to the eigenproblem for the -action alone, $\partial\partial\mathcal Sv_0=\lambda_0 v_0^*$, and then rotate the +action alone, $\operatorname{Hess}\mathcal Sv_0=\lambda_0 v_0^*$, and then rotate the resulting vectors by a constant phase corresponding to half the phase of $\beta$ or $v(\phi)=v_0e^{-i\phi/2}$. One can see this in the examples of Figs. \ref{fig:1d.stokes} and \ref{fig:thimble.orientation}, where increasing the argument of $\beta$ from @@ -574,30 +609,30 @@ left to right produces a clockwise rotation in the thimbles in the complex-$\theta$ plane. These eigenvalues and vectors can be further related to properties of the -complex symmetric matrix $\beta\partial\partial\mathcal S$. Suppose that +complex symmetric matrix $\beta\operatorname{Hess}\mathcal S$. Suppose that $u\in\mathbb R^N$ satisfies the eigenvalue equation \begin{equation} - (\beta\partial\partial S)^\dagger(\beta\partial\partial S)u + (\beta\operatorname{Hess} S)^\dagger(\beta\operatorname{Hess} S)u =\sigma u \end{equation} -for some positive real $\sigma$ (real because $(\beta\partial\partial -S)^\dagger(\beta\partial\partial S)$ is self-adjoint). The square root of these +for some positive real $\sigma$ (real because $(\beta\operatorname{Hess} +S)^\dagger(\beta\operatorname{Hess} S)$ is self-adjoint). The square root of these numbers, $\sqrt{\sigma}$, are the definition of the \emph{singular values} of -$\beta\partial\partial\mathcal S$. A direct relationship between these singular +$\beta\operatorname{Hess}\mathcal S$. A direct relationship between these singular values and the eigenvalues of the hessian immediately follows by taking an eigenvector $v\in\mathbb C$ that satisfies \eref{eq:generalized.eigenproblem}, and writing \begin{eqnarray} \sigma v^\dagger u - &=v^\dagger(\beta\partial\partial S)^\dagger(\beta\partial\partial S)u - =(\beta\partial\partial Sv)^\dagger(\beta\partial\partial S)u\\ - &=(\lambda v^*)^\dagger(\beta\partial\partial S)u - =\lambda v^T(\beta\partial\partial S)u + &=v^\dagger(\beta\operatorname{Hess} S)^\dagger(\beta\operatorname{Hess} S)u + =(\beta\operatorname{Hess} Sv)^\dagger(\beta\operatorname{Hess} S)u\\ + &=(\lambda v^*)^\dagger(\beta\operatorname{Hess} S)u + =\lambda v^T(\beta\operatorname{Hess} S)u =\lambda^2 v^\dagger u \end{eqnarray} Thus if $v^\dagger u\neq0$, $\lambda^2=\sigma$. It follows that the eigenvalues of the real hessian are the singular values of the complex matrix -$\beta\partial\partial\mathcal S$, and their eigenvectors coincide up to a +$\beta\operatorname{Hess}\mathcal S$, and their eigenvectors coincide up to a constant complex factor. \subsection{The conditions for Stokes points} @@ -612,10 +647,10 @@ point, e.g., as for a constant or linear function. Now, `place' a generic critical point, the flow is locally \begin{equation} \dot z - \simeq-\frac{\beta^*}2(\partial\partial\mathcal S|_{z=z_0})^\dagger P(z-z_0)^* + \simeq-\frac{\beta^*}2(\operatorname{Hess}\mathcal S|_{z=z_0})^\dagger P(z-z_0)^* \end{equation} -The matrix $(\partial\partial\mathcal S)^\dagger P$ has a spectrum identical to that of -$(\partial\partial\mathcal S)^\dagger$ save marginal directions corresponding to the normals to +The matrix $(\operatorname{Hess}\mathcal S)^\dagger P$ has a spectrum identical to that of +$(\operatorname{Hess}\mathcal S)^\dagger$ save marginal directions corresponding to the normals to manifold. Assuming we are working in a diagonal basis, this becomes \begin{equation} \dot z_i=-\frac12(\beta\lambda_i)^*\Delta z_i^*+O(\Delta z^2,(\Delta z^*)^2) @@ -630,7 +665,7 @@ Breaking into real and imaginary parts gives \right) \end{eqnarray} Therefore, in the complex plane defined by each eigenvector of -$(\partial\partial\mathcal S)^\dagger P$ there is a separatrix flow. +$(\operatorname{Hess}\mathcal S)^\dagger P$ there is a separatrix flow. Continuing to `insert' critical points whose imaginary energy differs from $C$, one repeatedly partitions the space this way with each insertion. Therefore, @@ -708,7 +743,7 @@ We are left with evaluating the determinant of the coordinate transformation. Th \begin{eqnarray} \beta\mathcal S(s(u)) &=\beta\mathcal S(s_\sigma) - +\frac12(s(u)-s_\sigma)^T(\beta\partial\partial\mathcal S)(s(u)-s_\sigma)+O((s(u)-s_\sigma)^3) \\ + +\frac12(s(u)-s_\sigma)^T(\beta\operatorname{Hess}\mathcal S)(s(u)-s_\sigma)+O((s(u)-s_\sigma)^3) \\ &=\beta\mathcal S(s_\sigma) +\frac{|\beta|}2\sum_i\sum_j\frac{v^{(i)}_k}{\sqrt{\lambda^{(i)}}}(\beta\partial_k\partial_\ell\mathcal S)\frac{v^{(j)}_\ell}{\sqrt{\lambda^{(j)}}}u_iu_j+\cdots \\ &=\beta\mathcal S(s_\sigma) @@ -744,6 +779,13 @@ $\det U=i^k$. As the argument of $\beta$ is changed, we know how the eigenvector Z_\sigma(\beta)\simeq\left(\frac{2\pi}\beta\right)^{D/2}i^{k_\sigma}\prod_{\lambda_0>0}\lambda_0^{-\frac12}e^{-\beta\mathcal S(s_\sigma)} \end{equation} +\begin{eqnarray} + Z(\beta)^* + =\sum_{\sigma\in\Sigma_0}n_\sigma Z_\sigma(\beta)^* + =\sum_{\sigma\in\Sigma_0}n_\sigma(-1)^{k_\sigma}Z_\sigma(\beta^*) + =Z(\beta^*) +\end{eqnarray} + \section{The \textit{p}-spin spherical models} The $p$-spin spherical models are statistical mechanics models defined by the |