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

Boris Konopelchenko, Luis Martínez Alonso, Elena Medina
2013 Journal of Physics A: Mathematical and Theoretical  
We study the moduli space of the spectral curves y^2=W'(z)^2+f(z) which characterize the vacua of 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
more » ... the spectral curves in terms of critical points of a family of polynomial solutions 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 W. 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.
doi:10.1088/1751-8113/46/22/225203 fatcat:n72kdakdvzhehp4p6qtwhvhchy