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Diffstat (limited to 'stokes.tex')
| -rw-r--r-- | stokes.tex | 22 |
1 files changed, 13 insertions, 9 deletions
@@ -549,6 +549,19 @@ manifold: its tangent at any point $z$ is given by $\partial(z^Tz)=z$, and $Pz=z-z|z|^2/|z|^2=0$. For any vector $u$ perpendicular to $z$, i.e., $z^\dagger u=0$, $Pu=u$. +\begin{figure} + \includegraphics{figs/thimble_flow.pdf} + + \caption{Example of gradient descent flow on the action $\mathcal S$ featured + in Fig.~\ref{fig:example.action} in the complex-$\theta$ plane, with + $\arg\beta=0.4$. Symbols denote the stationary points, while thick blue and + red lines depict the thimbles and antithimbles, respectively. Streamlines + of the flow equations are plotted in a color set by their value of + $\operatorname{Im}\beta\mathcal S$; notice that the color is constant along + each streamline. + } \label{fig:flow.example} +\end{figure} + Gradient descent on $\operatorname{Re}\beta\mathcal S$ is equivalent to Hamiltonian dynamics with the Hamiltonian $\operatorname{Im}\beta\mathcal S$ and conjugate coordinates given by the real and imaginary parts of each complex @@ -616,15 +629,6 @@ $C\neq\operatorname{Im}\mathcal S(z_0)$ is split into two disconnected pieces, o lying in each of two quadrants corresponding with its value relative to that at the critical point. -\begin{figure} - \includegraphics{figs/local_flow.pdf} - \caption{ - Gradient descent in the vicinity of a critical point, in the $z$--$z^*$ - plane for an eigenvector $z$ of $(\partial\partial\mathcal S)^\dagger P$. The flow - lines are colored by the value of $\operatorname{Im}H$. - } \label{fig:local_flow} -\end{figure} - Continuing to `insert' critical points whose imaginary energy differs from $C$, one repeatedly partitions the space this way with each insertion. Therefore, for the generic case with $\mathcal N$ critical points, with $C$ differing in |