The Laplace transform of the cut-and-join equation and the Bouchard-Marino conjecture on Hurwitz numbers

TL;DR

This paper derives a polynomial equation from the Laplace transform of the cut-and-join equation, revealing a topological recursion structure for Hurwitz numbers that aligns with conjectures from topological string theory.

Contribution

It connects the Laplace transform of the cut-and-join equation to the Bouchard-Marino conjecture, establishing a new link between Hurwitz numbers and topological recursion.

Findings

Laplace transform yields a polynomial equation with Mirzakhani-like structure

Inverse Lambert W-function maps the equation to topological recursion for Hurwitz numbers

Supports the Bouchard-Marino conjecture using new analytical methods

Abstract

We calculate the Laplace transform of the cut-and-join equation of Goulden, Jackson and Vakil. The result is a polynomial equation that has the topological structure identical to the Mirzakhani recursion formula for the Weil-Petersson volume of the moduli space of bordered hyperbolic surfaces. We find that the direct image of this Laplace transformed equation via the inverse of the Lambert W-function is the topological recursion formula for Hurwitz numbers conjectured by Bouchard and Marino using topological string theory.