# Matrix model generating function for quantum weighted Hurwitz numbers

## TL;DR

This paper develops a matrix model framework for quantum weighted Hurwitz numbers using KP tau-functions, expressing key functions as convergent series and Mellin-Barnes integrals, and deriving a matrix model representation.

## Contribution

It introduces a novel matrix model representation for the KP tau-function generating quantum weighted Hurwitz numbers, linking integrable systems and matrix models.

## Key findings

- Explicit formulas for Baker functions and basis elements as Laurent series
- Representation of these functions as Mellin-Barnes integrals with Gamma functions
- Derivation of a matrix model for the tau-function at trace invariants

## Abstract

The KP $\tau$-function of hypergeometric type serving as generating function for quantum weighted Hurwitz numbers is used to compute the Baker function and the corresponding adapted basis elements, expressed as absolutely convergent Laurent seriesin the spectral parameter. These are equivalently expressed as Mellin-Barnes integrals, analogously to Meijer $G$-functions, but with an infinite product of $\Gamma$-functions as integral kernel. A matrix model representation is derived for the $\tau$-function evaluated at trace invariants of an externally coupled matrix.