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

TL;DR

This paper explores the large N limit of ${ m U}(N)$ ${ m extbf{N}=1}$ supersymmetric gauge theories with an adjoint scalar, revealing a geometric approach to determine eigenvalue distributions that differ from traditional matrix models.

Contribution

It introduces a geometric master field framework for ${ m N}=1$ theories, linking eigenvalue distributions to the auxiliary Riemann surface geometry, distinct from standard matrix model predictions.

Findings

Eigenvalue distribution $ ho_{GT}()$ is different from the matrix model distribution $ ho_{MM}()$.

The geometry of the auxiliary Riemann surface prescribes a method to compute $ ho_{GT}()$.

A simple form of $ ho_{GT}()$ is obtained from the matrix model's geometry.

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 $\rho_{MM}(\l)$.