Spectral curves in gauge/string dualities: integrability, singular sectors and regularization

TL;DR

This paper explores the spectral curves associated with certain supersymmetric gauge theories, linking integrability, singularities, and regularization techniques, and introduces a new approach to analyze their moduli space and critical points.

Contribution

It provides a novel characterization of spectral curves via critical points of polynomial solutions and introduces a regularization method connecting to Painlevé equations.

Findings

Spectral curves are characterized by critical points of polynomial solutions.

Singular spectral curves correspond to degenerate critical points.

A regularization method leads to Painlevé I and its generalizations.

Abstract

We study the moduli space of the spectral curves $y^2=W'(z)^2+f(z)$ which characterize the vacua of $\mathcal{N}=1$ U(n) supersymmetric gauge theories with an adjoint Higgs field and a polynomial tree level potential $W(z)$. It is shown that there is a direct way to associate a spectral density and a prepotential functional to these spectral curves. The integrable structure of the Whitham equations is used to determine the spectral curves from their moduli. An alternative characterization of the spectral curves in terms of critical points of a family of polynomial solutions $\mathbb{W}$ to Euler-Poisson-Darboux equations is provided. The equations for these critical points are a generalization of the planar limit equations for one-cut random matrix models. Moreover, singular spectral curves with higher order branch points turn out to be described by degenerate critical points of $\mathbb{W}$. As a consequence we propose a multiple scaling limit method of regularization and show that, in the simplest cases, it leads to the Painlev\`{e}-I equation and its multi-component generalizations.