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We derive some new relationships between matrix models of Chern-Simons gaugetheory and of two-dimensional Yang-Mills theory. We show that q-integration ofthe Stieltjes-Wigert matrix model is the discrete matrix model that describesq-deformed Yang-Mills theory on the 2-sphere. We demonstrate that thesemiclassical limit of the Chern-Simons matrix model is equivalent to theGross-Witten model in the weak coupling phase. We study the strong couplinglimit of the unitary Chern-Simons matrix model and show that it too induces theGross-Witten model, but as a first order deformation of Dyson's circularensemble. We show that the Sutherland model is intimately related toChern-Simons gauge theory on the 3-sphere, and hence to q-deformed Yang-Millstheory on the 2-sphere. In particular, the ground state wavefunction of theSutherland model in its classical equilibrium configuration describes theChern-Simons free energy. The correspondence is extended to Wilson lineobservables and to arbitrary simply-laced gauge groups.

Chern-Simons matrix models, two-dimensional Yang-Mills theory and the Sutherland model