Soft matrix models and Chern-Simons partition functions

TL;DR

This paper explores soft matrix models with non-moment-determined weights, linking them to Chern-Simons theory, and demonstrates multiple matrix models can compute the same Chern-Simons partition function using q-deformed orthogonal polynomials.

Contribution

It introduces a class of soft matrix models characterized by non-moment-determined weights and connects them to Chern-Simons theory through a simple mapping.

Findings

Matrix models with soft confining potentials are characterized by non-moment-determined weights.

These models can be solved using q-deformed orthogonal polynomials, specifically Stieltjes-Wigert polynomials.

Multiple matrix models can compute the Chern-Simons partition function on S^3.

Abstract

We study the properties of matrix models with soft confining potentials. Their precise mathematical characterization is that their weight function is not determined by its moments. We mainly rely on simple considerations based on orthogonal polynomials and the moment problem. In addition, some of these models are equivalent, by a simple mapping, to matrix models that appear in Chern-Simons theory. The models can be solved with q deformed orthogonal polynomials (Stieltjes-Wigert polynomials), and the deformation parameter turns out to be the usual $q$ parameter in Chern-Simons theory. In this way, we give a matrix model computation of the Chern-Simons partition function on $S^{3}$ and show that there are infinitely many matrix models with this partition function.