Branched matrix models and the scales of supersymmetric gauge theories

In the framework of the matrix model/gauge theory correspondence, we consider supersymmetric U(N) gauge theory with U(1)^N symmetry breaking pattern. Due to the presence of the Veneziano-Yankielowicz effective superpotential, in order to satisfy the F-term condition sum_iS_i = 0, we are forced to introduce additional terms in the free energy of the corresponding matrix model with respect to the usual formulation. This leads to a matrix model formulation with a cubic potential which is free of parameters and displays a branched structure. In this way we naturally solve the usual problem of the identification between dimensionful and dimensionless quantities. Furthermore, we need not introduce the Script N = 1 scale by hand in the matrix model. These facts are related to remarkable coincidences which arise at the critical point and lead to a branched bare coupling constant. The latter plays the role of the Script N = 1 and Script N = 2 scales tuning parameter. We then show that a suitable rescaling leads to the correct identification of the Script N = 2 variables. Finally, by means of the mentioned coincidences, we provide a direct expression for the Script N = 2 prepotential, including the gravitational corrections, in terms of the free energy. This suggests that the matrix model provides a triangulation of the istanton moduli space.