Hurwitz numbers from matrix integrals over Gaussian measure

Abstract

We explain how Gaussian integrals over ensemble of complex matrices with source matrices generate Hurwitz numbers of the most general type, namely, Hurwitz numbers with an arbitrary orientable or non-orientable base surface and with arbitrary profiles at branch points. Our approach makes use of Feynman diagrams. We make connections with topological theories and also with certain classical and quantum integrable theories; in particular with Witten’s description of two-dimensional gauge theory. We generalize a model of quantum Hopf equation considered by Dubrovin.