${\cal N}=1$ Theories and a Geometric Master Field

Abstract

We study the large $N$ limit of the class of U(N) ${\CN}=1$ SUSY gauge theories with an adjoint scalar and a superpotential $W(\P)$. In each of the vacua of the quantum theory, the expectation values $\la$Tr$\Phi^p$$\ra$ are determined by a master matrix $\Phi_0$ with eigenvalue distribution $\rho_{GT}(\l)$. $\rho_{GT}(\l)$ is quite distinct from the eigenvalue distribution $\rho_{MM}(\l)$ of the corresponding large $N$ matrix model proposed by Dijkgraaf and Vafa. Nevertheless, it has a simple form on the auxiliary Riemann surface of the matrix model. Thus the underlying geometry of the matrix model leads to a definite prescription for computing $\rho_{GT}(\l)$, knowing $\rho_{MM}(\l)$.