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| -rw-r--r-- | stokes.tex | 47 |
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@@ -362,28 +362,59 @@ keeping track of the resulting weights, and analytic continuation is intractable \subsection{The structure of stationary points} -\begin{eqnarray} - \operatorname{Hess}\operatorname{Re}\beta\mathcal S - &=\left[\matrix{ +The shape of each thimble in the vicinity of a stationary point can be +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. + +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 +\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 - }\right] \\ - &=\left[\matrix{ + }\right] +\end{equation} +This con 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 + \qquad + \partial_y\operatorname{Re}\mathcal S=-\partial_x\operatorname{Im}\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 + \qquad + \partial_y\operatorname{Re}\mathcal S=-\operatorname{Im}\partial\mathcal S \\ + \partial_x\operatorname{Im}\mathcal S=\operatorname{Im}\partial\mathcal S + \qquad + \partial_y\operatorname{Im}\mathcal S=\operatorname{Re}\partial\mathcal S +\end{eqnarray} +Using these relationships, the hessian \eref{eq:real.hessian} can be written in +the more manifestly complex way +\begin{equation} + \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 }\right] -\end{eqnarray} +\end{equation} The eigenvalues and eigenvectors of the Hessian are important for evaluating thimble integrals, because those associated with upward directions provide a local basis for the surface of the thimble. Suppose that $v_x,v_y\in\mathbb R^N$ are such that \begin{equation} - (\operatorname{Hess}\operatorname{Re}\beta\mathcal S)\left[\matrix{v_x \cr v_y}\right] + (\operatorname{Hess}_{x,y}\operatorname{Re}\beta\mathcal S)\left[\matrix{v_x \cr v_y}\right] =\lambda\left[\matrix{v_x \cr v_y}\right] \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 @@ -391,7 +422,7 @@ where the eigenvalue $\lambda$ must be real because the hessian is real symmetri &\left[\matrix{0&-i(\beta\partial\partial\mathcal S)^*\cr i\beta\partial\partial\mathcal S&0}\right] \left[\matrix{v \cr iv^*}\right]\\ &\qquad=\left[\matrix{1&i\cr i&1}\right] - (\operatorname{Hess}\operatorname{Re}\beta\mathcal S) + (\operatorname{Hess}_{x,y}\operatorname{Re}\beta\mathcal S) \left[\matrix{1&i\cr i&1}\right]^{-1} \left[\matrix{1&i\cr i&1}\right] \left[\matrix{v_x \cr v_y}\right] \\ |