Chern-Simons matrix models and Stieltjes-Wigert polynomials

Abstract

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 StieltjesWigert matrix model and the discrete model that appears in q-2D Yang-Mills and the relationship with Rogers-Szegö 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.

Torus Knots and Mirror Symmetry

“…We show that the spectral curve of this matrix model agrees with our natural proposal for the Bmodel geometry. Since this curve is a symplectic transformation of the resolved conifold geometry, and since symplectic transformations do not change the 1/N expansion of the partition function [19,20], our result explains the empirical observation of [16,6] that the partition functions of the matrix models for different torus knots are all equal to the partition function of Chern-Simons theory on S 3 (up to an unimportant framing factor).…”

“…This can be also deduced from the calculation in [16]. We also note that there is an obvious generalization of the matrix model representation (4.1) to the torus link (Q, P ), given by…”

“…This is because the colored U (N ) invariants of torus knots admit a matrix integral representation, as first pointed out in the SU (2) case in [34]. The calculation of [34] was generalized to U (N ) in the unpublished work [38] (see also [16]), and the matrix integral representation was rederived recently in [6,28] by a direct localization of the path integral. We show that the spectral curve of this matrix model agrees with our natural proposal for the Bmodel geometry.…”

“…Such a representation was first proposed for SU (2) in [34], and then extended to simply-laced groups in [38] (see also [16]). More recently, the matrix integral for torus knots was derived by localization of the Chern-Simons path integral [6] (another localization procedure which leads to the same result has been recently proposed in [28]).…”

Oscillatory matrix model in Chern-Simons theory and Jacobi-theta determinantal point process

“…These finite point processes (Ξ, P N ) and ( Ξ, P N ) are fully studied in [28,4,5,26]. The purpose of the present paper is to consider an N → ∞ limit of the systems (Ξ, P N ) and ( Ξ, P N ).…”