# The Laplace Transform of the Cut-and-Join Equation and the Bouchard–Mariño Conjecture on Hurwitz Numbers

### Bertrand Eynard

CEA Saclay, Gif-Sur-Yvette, France### Motohico Mulase

University of California at Davis, USA### Bradley Safnuk

Central Michigan University, Mount Pleasant, USA

## 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 Mariño using topological string theory.

## Cite this article

Bertrand Eynard, Motohico Mulase, Bradley Safnuk, The Laplace Transform of the Cut-and-Join Equation and the Bouchard–Mariño Conjecture on Hurwitz Numbers. Publ. Res. Inst. Math. Sci. 47 (2011), no. 2, pp. 629–670

DOI 10.2977/PRIMS/47