Hurwitz numbers from matrix integrals over Gaussian measure

TL;DR

This paper demonstrates how Gaussian matrix integrals can generate general Hurwitz numbers, linking matrix models, topological theories, and integrable systems through Feynman diagrams.

Contribution

It introduces a novel matrix integral approach to compute Hurwitz numbers for various base surfaces and branch profiles, connecting to topological and integrable theories.

Findings

Matrix integrals produce Hurwitz numbers with arbitrary surface types.

Feynman diagrams elucidate the connection between matrix models and topological invariants.

Links established between matrix integrals, topological theories, and integrable systems.

Abstract

We explain how Gaussian integrals over ensemble of complex matrices with source matrices generate Hurwitz numbers of the most general type, namely, Hurwitz numbers with arbitrary orientable or non-orientable base surface and arbitrary profiles at branch points. We use the Feynman diagram approach. The connections with topological theories and also with certain classical and quantum integrable theories in particular with Witten's description of two-dimensional gauge theory are shown.