  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.
