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Diffstat (limited to 'stokes.tex')
| -rw-r--r-- | stokes.tex | 34 |
1 files changed, 29 insertions, 5 deletions
@@ -10,11 +10,12 @@ filecolor=purple, linkcolor=purple ]{hyperref} % ref and cite links with pretty colors +\usepackage{bibentry} \usepackage{amsmath, graphicx, xcolor} % standard packages \begin{document} -\title{Analytic continuation of theories with complex landscapes} +\title{Analytic continuation over complex landscapes} \author{Jaron Kent-Dobias} \author{Jorge Kurchan} @@ -23,10 +24,33 @@ \date\today -\begin{abstract} -In this paper we follow up the study of `complex-complex landscapes' \cite{Kent-Dobias_2021_Complex}, rugged landscapes in spaces of many complex variables. -Contrary to the real case, the index of saddles is not itself relevant, as it is always half the total dimension. The relevant topological objects here are Stokes trajectories, gradient lines joining two saddles, where Lefschetz thimbles merge. The well-studied `threshold level', separating regions with minima from regions with saddles in the real case, here separates regions where Stokes lines are rare, from region where they proliferate. -Likewise, when a real landscape is prolonged to complex variables, the distinction between "one step replica-symmetry breaking" and "many step replica symmetry breaking" is that in the former case the saddles are at first free of Stokes lines, while in the latter these immediately proliferate. +\begin{abstract} \setcitestyle{authoryear,round} + In this paper we follow up the study of `complex complex landscapes' + [\cite{Kent-Dobias_2021_Complex}], rugged landscapes of many complex + variables. Unlike real landscapes, there is no useful classification of + saddles by index. Instead, the spectrum at critical points determines their + tendency to trade topological numbers under analytic continuation of the + theory. These trades, which occur at Stokes points, proliferate when the + spectrum includes marginal directions and are exponentially suppressed + otherwise. This gives a direct interpretation of the `threshold' energy---which + in the real case separates saddles from minima---where the spectrum of + typical critical points develops a gap. This leads to different consequences + for the analytic continuation of real landscapes with different structures: + the global minima of ``one step replica-symmetry broken'' landscapes lie + beyond a threshold and are locally protected from Stokes points, whereas + those of ``many step replica symmetry broken'' lie at the threshold and + Stokes points immediately proliferate. + +% Contrary to the real case, the index of saddles is not itself +% relevant, as it is always half the total dimension. The relevant topological +% objects here are Stokes trajectories, gradient lines joining two saddles, where +% Lefschetz thimbles merge. The well-studied `threshold level', separating +% regions with minima from regions with saddles in the real case, here separates +% regions where Stokes lines are rare, from region where they proliferate. +% Likewise, when a real landscape is prolonged to complex variables, the +% distinction between "one step replica-symmetry breaking" and "many step replica +% symmetry breaking" is that in the former case the saddles are at first free of +% Stokes lines, while in the latter these immediately proliferate. \end{abstract} \maketitle |