Unoriented Strings, Loop Equations, and N=1 Superpotentials from Matrix Models

TL;DR

This paper extends the Dijkgraaf-Vafa matrix model approach to analyze N=1 gauge theories with SO(N) and Sp(N) groups, deriving contributions to the superpotential and verifying relations between free energies.

Contribution

It applies matrix model techniques to SO(N) and Sp(N) gauge theories, deriving loop equations and refining the superpotential proposal for these groups.

Findings

Derived planar and non-planar free energy contributions.

Verified the relation between RP^2 and sphere free energies.

Achieved agreement with field theory predictions.

Abstract

We apply the proposal of Dijkgraaf and Vafa to analyze N=1 gauge theory with SO(N) and Sp(N) gauge groups with arbitrary tree-level superpotentials using matrix model techniques. We derive the planar and leading non-planar contributions to the large M SO(M) and Sp(M) matrix model free energy by applying the technology of higher-genus loop equations and by straightforward diagrammatics. The loop equations suggest that the RP^2 free energy is given as a derivative of the sphere contribution, a relation which we verify diagrammatically. With a refinement of the proposal of Dijkgraaf and Vafa for the effective superpotential, we find agreement with field theory expectations.