Boris Runov, Concordia University and CRM

Title: Matrix model for Quantum Weighted Hurwitz numbers.

Abstract: Pure Hurwirz numbers count inequivalent branched coverings of a Riemann surface with given ramification profile. Weighted Hurwitz numbers are certain weighted sums of pure Hurwitz numbers with weights determined by a generating function. In earlier works by J.Harnad et al weighted Hurwitz numbers were related to hypergeometric tau functions of KP or 2D Toda hierarchies. Such tau functions are defined by the same generating function. Recently M. Bertola and J. Harnad considered the case of rational generating functions and managed to show that hypergeometric tau function in this case coincides with the partition function of certain matrix model. Quantum weighted Hurwitz numbers are the particular case of weighted hurwitz numbers with generating function given by q-Pochammer symbol. In my talk I shall explain how the approach of Bertola and Harnad can be generalized to this case.