Cut-and-join equation for monotone Hurwitz numbers revisited

TL;DR

This paper provides a new combinatorial proof of the cut-and-join equation for monotone Hurwitz numbers and demonstrates its connection to topological recursion, enhancing understanding of spectral curve methods.

Contribution

It offers a novel combinatorial proof of the cut-and-join equation and links it to the quadratic loop equation in topological recursion for monotone Hurwitz numbers.

Findings

New combinatorial proof of the cut-and-join equation

Reveals the relation to quadratic loop equations in spectral curve theory

Provides a new proof of topological recursion for monotone Hurwitz numbers

Abstract

We give a new proof of the cut-and-join equation for the monotone Hurwitz numbers, derived first by Goulden, Guay-Paquet, and Novak. Our proof in particular uses a combinatorial technique developed by Han. The main interest in this particular equation is its close relation to the quadratic loop equation in the theory of spectral curve topological recursion, and we recall this motivation giving a new proof of the topological recursion for monotone Hurwitz numbers, obtained first by Do, Dyer, and Mathews.