Cubic curves from matrix models and generalized Konishi anomalies

TL;DR

This paper explores the connection between matrix models and N=1 gauge theories with various matter representations, demonstrating how loop equations lead to cubic curves and establishing the equivalence of descriptions via anomaly equations and perturbative analysis.

Contribution

It explicitly derives cubic algebraic curves from matrix models and proves the equivalence with gauge theories using generalized Konishi anomalies and perturbative methods.

Findings

Loop equations produce cubic algebraic curves.

Matrix model and gauge theory descriptions are equivalent.

Gauge coupling matrix differs from the second derivative of the free energy.

Abstract

We study the matrix model/gauge theory connection for three different N=1 models: U(N) x U(N) with matter in bifundamental representations, U(N) with matter in the symmetric representation, and U(N) with matter in the antisymmetric representation. Using Ward identities, we explicitly show that the loop equations of the matrix models lead to cubic algebraic curves. We then establish the equivalence of the matrix model and gauge theory descriptions in two ways. First, we derive generalized Konishi anomaly equations in the gauge theories, showing that they are identical to the matrix-model equations. Second, we use a perturbative superspace analysis to establish the relation between the gauge theories and the matrix models. We find that the gauge coupling matrix for U(N) with matter in the symmetric or antisymmetric representations is_not_ given by the second derivative of the matrix-model free energy. However, the matrix-model prescription can be modified to give the gauge coupling matrix.