Exact solution of Chern-Simons-matter matrix models with characteristic/orthogonal polynomials

TL;DR

This paper provides an exact finite N solution to a supersymmetric Chern-Simons matrix model with fundamental matter, using characteristic polynomial averages and special functions, connecting to known integrals and semiclassical limits.

Contribution

It introduces a novel exact solution for the matrix model with fundamental matter by linking it to inverse characteristic polynomial averages and special functions.

Findings

Exact finite N solution for the matrix model.

Connection with Mordell integrals and special functions.

Semiclassical limit expressed via Hermite polynomials.

Abstract

We solve for finite $N$ the matrix model of supersymmetric $U(N)$ Chern-Simons theory coupled to $N_{f}$ fundamental and $N_{f}$ anti-fundamental chiral multiplets of $R$-charge $1/2$ and of mass $m$, by identifying it with an average of inverse characteristic polynomials in a Stieltjes-Wigert ensemble. This requires the computation of the Cauchy transform of the Stieltjes-Wigert polynomials, which we carry out, finding a relationship with Mordell integrals, and hence with previous analytical results on the matrix model. The semiclassical limit of the model is expressed, for arbitrary $N_{f},$ in terms of a single Hermite polynomial. This result also holds for more general matter content, involving matrix models with double-sine functions.