Superpotentials, quantum parameter space and phase transitions in N=1 supersymmetric gauge theories

We study the superpotentials, quantum parameter space and phase transitions that arise in the study of large N dualities between N = 1 SUSY U(N) gauge theories and string models on local Calabi-Yau manifolds. The main tool of our analysis is a notion of spectral curve characterized by a set of complex partial ’t Hooft parameters and cuts given by projections on the spectral curve of minimal supersymmetric cycles of the underlying Calabi-Yau manifold. We introduce a prepotential functional via a variational problem which determines the complex density as an extremal constrained by the period conditions. This prepotential is shown to satisfy the special geometry relations of the spectral curve. We give a system of equations for the branch points of the spectral curves and determine the appropriate branch cuts as Stokes lines of a suitable set of polynomials. As an application, we use a combination of analytical and numerical methods to study the cubic model, determine the analytic condition satisfied by critical one-cut spectral curves, and characterize the transition curves between the one-cut and two-cut phases both in the space of spectral curves and in the quantum parameter space