Multi-Trace Superpotentials vs. Matrix Models

TL;DR

This paper explores the computation of effective superpotentials in N=1 supersymmetric U(N) gauge theories with multi-trace superpotentials, revealing unique contributions from non-planar diagrams and connections to matrix models and Seiberg-Witten theory.

Contribution

It introduces a novel approach to compute the effective action for multi-trace superpotentials, highlighting differences from single-trace cases and establishing a link to multi-trace matrix models.

Findings

Only a subset of planar diagrams contribute to the exact result.

The effective action differs from the derivative of a multi-trace matrix model free energy.

The superpotential can be derived from the N=2 Seiberg-Witten solution.

Abstract

We consider N = 1 supersymmetric U(N) field theories in four dimensions with adjoint chiral matter and a multi-trace tree-level superpotential. We show that the computation of the effective action as a function of the glueball superfield localizes to computing matrix integrals. Unlike the single-trace case, holomorphy and symmetries do not forbid non-planar contributions. Nevertheless, only a special subset of the planar diagrams contributes to the exact result. Some of the data of this subset can be computed from the large-N limit of an associated multi-trace Matrix model. However, the prescription differs in important respects from that of Dijkgraaf and Vafa for single-trace superpotentials in that the field theory effective action is not the derivative of a multi-trace matrix model free energy. The basic subtlety involves the correct identification of the field theory glueball as a variable in the Matrix model, as we show via an auxiliary construction involving a single-trace matrix model with additional singlet fields which are integrated out to compute the multi-trace results. Along the way we also describe a general technique for computing the large-N limits of multi-trace Matrix models and raise the challenge of finding the field theories whose effective actions they may compute. Since our models can be treated as N = 1 deformations of pure N =2 gauge theory, we show that the effective superpotential that we compute also follows from the N = 2 Seiberg-Witten solution. Finally, we observe an interesting connection between multi-trace local theories and non-local field theory.