Operator Counting and Eigenvalue Distributions for 3D Supersymmetric Gauge Theories

TL;DR

This paper supports a conjecture linking eigenvalue distributions in a matrix model to operator counts in 3D supersymmetric gauge theories, extending it to non-critical R-charges and theories with different supersymmetry.

Contribution

It demonstrates the conjecture's validity for various supersymmetry levels and R-charges, and relates free energy to Sasaki manifold volume, broadening understanding of gauge theory matrix models.

Findings

Conjecture holds for non-critical R-charges.

Relation between free energy and Sasaki volume established.

Implications for chiral theories explored.

Abstract

We give further support for our conjecture relating eigenvalue distributions of the Kapustin-Willett-Yaakov matrix model in the large N limit to numbers of operators in the chiral ring of the corresponding supersymmetric three-dimensional gauge theory. We show that the relation holds for non-critical R-charges and for examples with {\mathcal N}=2 instead of {\mathcal N}=3 supersymmetry where the bifundamental matter fields are nonchiral. We prove that, for non-critical R-charges, the conjecture is equivalent to a relation between the free energy of the gauge theory on a three sphere and the volume of a Sasaki manifold that is part of the moduli space of the gauge theory. We also investigate the consequences of our conjecture for chiral theories where the matrix model is not well understood.