Exact Superpotentials from Matrix Models

Dijkgraaf and Vafa (DV) have conjectured that the exact superpotential for alarge class of N=1 SUSY gauge theories can be extracted from the planar limitof a certain holomorphic matrix integral. We test their proposal againstexisting knowledge for a family of deformations of N=4 SUSY Yang-Mills theoryinvolving an arbitrary polynomial superpotential for one of the three adjointchiral superfields. Specifically, we compare the DV prediction for these modelswith earlier results based on the connection between SUSY gauge theories andintegrable systems. We find complete agreement between the two approaches. Inparticular we show how the DV proposal allows the extraction of the exacteigenvalues of the adjoint scalar in the confining vacuum and hence computesall related condensates of the finite-N gauge theory. We extend these resultsto include Leigh-Strassler deformations of the N=4 theory.