Matrix integrals and Hurwitz numbers

A. Yu. Orlov

Published 2017-01-09Version 1

We consider multi-matrix models which may be viewed as integrals of products of tau functions of matrix argument. Sometimes such integrals are tau functions themselves. We consider models which generate Hurwitz numbers $H^{\textsc{e},\textsc{f}}$, where $\textsc{e}$ is the Euler characteristic of the base surface and $\textsc{f}$ is the number of branch points. We show that in case the integrands contains the product of $n > 2$ matrices the integral generates Hurwitz numbers with Euler characteristic $\textsc{e}\le 2$ and the number of branch points $\textsc{f}\le n+2$, both numbers $\textsc{e}$ and $\textsc{f}$ depend on $n$ and the order of the multipliers in the matrix product. The number $\textsc{e}$ may be even or odd (respectively describes Riemann (and cerian Klein) or only Klein (non-orientable) base surface) depending on the presence of the BKP tau functions in the integrand.