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.

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.

Abstract. We study N = 2 supersymmetric U (N ) Chern-Simons with N f fundamental and N f antifundamental chiral multiplets of mass m in the parameter space spanned by (g, m, N, N f ), where g denotes the coupling constant. In particular, we analyze the matrix model description of its partition function, both at finite N using the method of orthogonal polynomials together with Mordell integrals and, at large N with fixed g, using the theory of Toeplitz determinants. We show for the massless case that there is an explicit realization of the Giveon-Kutasov duality. For finite N , with N > N f , three regimes that exactly correspond to the known three large N phases of theory are identified and characterized.

Abstract. We derive discrete and oscillatory Chern-Simons matrix models. The method is based on fundamental properties of the associated orthogonal polynomials. As an application, we show that the discrete model allows to prove and extend the recently found relationship between Chern-Simons theory and q-deformed 2dYM. In addition, the equivalence of the Chern-Simons matrix models gives a complementary view on the equivalence of effective superpotentials in N = 1 gauge theories.