Chern-Simons matrix models and Stieltjes-Wigert polynomials

TL;DR

This paper explores the use of Stieltjes-Wigert polynomials in exact computations for Chern-Simons matrix models on Seifert manifolds, introducing new biorthogonal polynomials for lens spaces and establishing connections with q-2D Yang-Mills and Rogers-Szego polynomials.

Contribution

It introduces a biorthogonal extension of Stieltjes-Wigert polynomials for lens space geometries and links Chern-Simons matrix models with other mathematical frameworks.

Findings

Established the equivalence between Stieltjes-Wigert and discrete q-2D Yang-Mills models.

Connected Stieltjes-Wigert polynomials with Rogers-Szego polynomials and unitary matrix models.

Proved a relation between quantum dimensions and Schur polynomial averages in the ensemble.

Abstract

Employing the random matrix formulation of Chern-Simons theory on Seifert manifolds, we show how the Stieltjes-Wigert orthogonal polynomials are useful in exact computations in Chern-Simons matrix models. We construct a biorthogonal extension of the Stieltjes-Wigert polynomials, not available in the literature, necessary to study Chern-Simons matrix models when the geometry is a lens space. We also discuss several other results based on the properties of the polynomials: the equivalence between the Stieltjes-Wigert matrix model and the discrete model that appears in q-2D Yang-Mills and the relationship with Rogers-Szego polynomials and the corresponding equivalence with an unitary matrix model. Finally, we also give a detailed proof of a result that relates quantum dimensions with averages of Schur polynomials in the Stieltjes-Wigert ensemble.