The matrix model for hypergeometric Hurwitz numbers

TL;DR

This paper develops multi-matrix models as generating functions for hypergeometric Hurwitz numbers, linking them to integrable hierarchies and spectral curve analysis for advanced enumerative geometry.

Contribution

It introduces a novel chain of matrices model with nonstandard interactions for hypergeometric Hurwitz numbers, enabling spectral curve evaluation and topological recursion applications.

Findings

Models are tau functions of KP hierarchy.

Spectral curves are algebraic and suitable for topological recursion.

Provides a technique for spectral curve evaluation in complex matrix models.

Abstract

We present the multi-matrix models that are the generating functions for branched covers of the complex projective line ramified over $n$ fixed points $z_i$, $i=1,\dots,n$, (generalized Grotendieck's dessins d'enfants) of fixed genus, degree, and the ramification profiles at two points, $z_1$ and $z_n$. We take a sum over all possible ramifications at other $n-2$ points with the fixed length of the profile at $z_2$ and with the fixed total length of profiles at the remaining $n-3$ points. All these models belong to a class of hypergeometric Hurwitz models thus being tau functions of the Kadomtsev--Petviashvili (KP) hierarchy. In the case described above, we can present the obtained model as a chain of matrices with a (nonstandard) nearest-neighbor interaction of the type $\tr M_iM_{i+1}^{-1}$. We describe the technique for evaluating spectral curves of such models, which opens the possibility of applying the topological recursion for developing $1/N^2$-expansions of these model. These spectral curves turn out to be of an algebraic type.