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

Abstract

We study the moduli space of the spectral curves y2 = W′(z)2 + f(z) which characterize the vacua of U(n) supersymmetric gauge theories with an adjoint Higgs field and a polynomial tree level potential W(z). 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 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 . As a consequence we propose a multiple scaling limit method of regularization and show that, in the simplest cases, it leads to the Painlevè-I equation and its multi-component generalizations.