Matrix integrals and Hurwitz numbers

TL;DR

This paper explores multi-matrix models involving tau functions of integrable hierarchies, demonstrating their ability to generate Hurwitz numbers related to both orientable and nonorientable surfaces, with results depending on matrix types and integrand composition.

Contribution

It introduces new multi-matrix models that produce Hurwitz numbers with specific topological characteristics, linking integrable hierarchies to enumerative geometry.

Findings

Models generate Hurwitz numbers with controlled Euler characteristic and branch points.

The type of matrices (complex or unitary) affects the topological properties of the generated surfaces.

Presence of BKP tau functions allows for nonorientable surface enumeration.

Abstract

We study multi-matrix models which may be viewed as integrals of products of tau functions which depend on the eigenvalues of products of random matrices. In the present paper we consider tau functions of the hierarchy the two-component KP (semiinfinite relativistic Toda lattice) and of hierarchy of the BKP introduced by Kac and van de Leur. Sometimes such integrals are tau functions themselves. We consider models which generate Hurwitz numbers $H^{e},f$, where $e$ is the Euler characteristic of the base surface and $f$ 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 $e\le 2$ and $f\le n+2$, both numbers $e$ and $f$ depend both on $n$ and on the order of the multipliers in the matrix product. The Euler characteristic $ e $ can be either an even or an odd number, that is, match both orientable and nonorientable (Klein) base surfaces, depending on the presence of the tau function of the BKP hierarchy in the integrand. We study two cases: the products of complex and the products of unitary matrices.