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 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 N=1 and N=2 scale tuning parameter. We then show that a suitable rescaling leads to the correct identification of the N=2 variables. Finally, by means of the the mentioned coincidences, we provide a direct expression for the 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.