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

We study the large $N$ limit of the class of U(N) ${\CN}=1$ SUSY gauge theories with an adjoint scalar and a superpotential $W(¶)$. In each of the vacua of the quantum theory, the expectation values $\la$Tr$Φ^p$$\ra$ are determined by a master matrix $Φ_0$ with eigenvalue distribution $ρ_{GT}(ł)$. $ρ_{GT}(ł)$ is quite distinct from the eigenvalue distribution $ρ_{MM}(ł)$ 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 $ρ_{GT}(ł)$, knowing $ρ_{MM}(ł)$.

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