# Rationally weighted Hurwitz numbers, Meijer $G$-functions and matrix integrals

## TL;DR

This paper links rationally weighted Hurwitz numbers to Meijer G-functions and matrix integrals, providing explicit solutions to quantum spectral curve equations via hypergeometric series.

## Contribution

It demonstrates that basis elements are Meijer G-functions and expresses the tau-function as a matrix integral using Mellin representations.

## Key findings

- Basis elements are Meijer G-functions or their asymptotic series.
- Tau-function can be expressed as a matrix integral.
- Provides explicit solutions to quantum spectral curve equations.

## Abstract

The quantum spectral curve equation associated to KP $\tau$-functions of hypergeometric type serving as generating functions for rationally weighted Hurwitz numbers is solved by generalized hypergeometric series. The basis elements spanning the corresponding Sato Grassmannian element are shown to be Meijer $G$-functions, or their asymptotic series. Using their Mellin integral representation the $\tau$-function, evaluated at the trace invariants of an externally coupled matrix, is expressed as a matrix integral.