We apply the proposal of Dijkgraaf and Vafa to analyze $\mathcal{N}=1\mathrm{}$ gauge theory with $\mathrm{SO}\left(N\right)\mathrm{}$ and $\mathrm{Sp}\left(N\right)\mathrm{}$ gauge groups with arbitrary tree-level superpotentials using matrix model techniques. We derive the planar and leading nonplanar contributions to the large M $\mathrm{SO}\left(M\right)\mathrm{}$ and $\mathrm{Sp}\left(M\right)\mathrm{}$ matrix model free energy by applying the technology of higher-genus loop equations and by straightforward diagrammatics. The loop equations suggest that the $R{P}^2$ 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.

• Received 16 December 2002

DOI:https://doi.org/10.1103/PhysRevD.67.086004