On the Matter of the Dijkgraaf--Vafa Conjecture

TL;DR

This paper investigates the connection between gauge theories with matter and matrix models, extending the analysis to more general matter representations and identifying where the Dijkgraaf-Vafa conjecture holds or breaks down.

Contribution

It proves that for various two-index tensor matter in classical gauge groups, the perturbative superpotential reduces to matrix integrals and identifies the loop order where discrepancies occur.

Findings

Agreement with Dijkgraaf-Vafa up to N/2 loops

Disagreement at h=N/2+1 loops due to superfield relations

Provides explicit calculations for $Sp(N)$ with antisymmetric matter

Abstract

With the aim of extending the gauge theory -- matrix model connection to more general matter representations, we prove that for various two-index tensors of the classical gauge groups, the perturbative contributions to the glueball superpotential reduce to matrix integrals. Contributing diagrams consist of certain combinations of spheres, disks, and projective planes, which we evaluate to four and five loop order. In the case of $Sp(N)$ with antisymmetric matter, independent results are obtained by computing the nonperturbative superpotential for $N=4,6$ and 8. Comparison with the Dijkgraaf-Vafa approach reveals agreement up to $N/2$ loops in matrix model perturbation theory, with disagreement setting in at $h=N/2+1$ loops, $h$ being the dual Coxeter number. At this order, the glueball superfield $S$ begins to obey nontrivial relations due to its underlying structure as a product of fermionic superfields. We therefore find a relatively simple example of an ${\cal N}=1$ gauge theory admitting a large $N$ expansion, whose dynamically generated superpotential differs from the one obtained in the matrix model approach.