Branched Matrix Models and the Scales of Supersymmetric Gauge Theories

Abstract

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 ${\sum }_i{S}_i=0$, 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 $\mathbb{N}=1$ 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 $\mathbb{N}=1$ and $\mathbb{N}=2$ scale tuning parameter. We then show that a suitable rescaling leads to the correct identification of the $\mathbb{N}=2$ variables. Finally, by means of the the mentioned coincidences, we provide a direct expression for the $\mathbb{N}=2$ 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.

Comments: 1+18 pages, harvmac. Added discussion on the CSW relative shifts of theta vacua and the odd phases at the critical point. References added and typos corrected