Chern-Simons matrix models and Stieltjes-Wigert polynomials

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.

25 pages, AMS-LaTex

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• Chern-Simons theory, matrix integrals, and perturbative three-manifold invariants
• Biorthogonal ensembles
• Chern-Simons Theory and Topological Strings
• Soft matrix models and Chern-Simons partition functions
• Characteristic polynomials of random matrices
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