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 $\mathcal{N}=1$ 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. In particular we show how prepotentials and superpotentials can be associated to spectral curves without relying on any holomorphic matrix model. 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.