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

Abstract

The partition function of the Chern-Simons theory on the three-sphere with the unitary group U(N) provides a one-matrix model. The corresponding N-particle system can be mapped to the determinantal point process whose correlation kernel is expressed by using the Stieltjes-Wigert orthogonal polynomials. The matrix model and the point process are regarded as q-extensions of the random matrix model in the Gaussian unitary ensemble and its eigenvalue point process, respectively. We prove the convergence of the N-particle system to an infinite-dimensional determinantal point process in N → ∞, in which the correlation kernel is expressed by Jacobi's theta functions. We show that the matrix model obtained by this limit realizes the oscillatory matrix model in Chern-Simons theory discussed by de Haro and Tierz.