Branched Matrix Models and the Scales of Supersymmetric Gauge Theories

TL;DR

This paper develops a branched matrix model with a cubic potential to analyze supersymmetric gauge theories, naturally resolving scale identification issues and providing a direct expression for the N=2 prepotential including gravitational corrections.

Contribution

It introduces a branched matrix model formulation with a cubic potential that naturally incorporates scale identification and relates to the N=2 prepotential with gravitational corrections.

Findings

The matrix model has a branched structure with a cubic potential.

The model naturally resolves the scale identification problem.

Provides a direct expression for the N=2 prepotential including gravitational corrections.

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 $\sum_iS_i=0$, 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.