arXiv:0907.5224v2 [math.AG] 20 Aug 2009 THE LAPLACE TRANSFORM OF THE CUT-AND-JOIN EQUATION AND THE BOUCHARD-MARI ˜ NO CONJECTURE ON HURWITZ NUMBERS BERTRAND EYNARD, MOTOHICO MULASE, AND BRAD SAFNUK 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 Mari˜ no using topological string theory. Contents 1. Introduction 1 2. The Laplace transform of the ELSV formula 6 3. The cut-and-join equation and its Laplace transform 9 4. The Bouchard-Mari˜ no recursion formula for Hurwitz numbers 14 5. Residue calculation 18 6. Analysis of Laplace transforms on the Lambert curve 21 7. Proof of the Bouchard-Mari˜ no topological recursion formula 28 Appendix. Examples of linear Hodge integrals and Hurwitz numbers 31 References 32 1. Introduction The purpose of this paper is to give a proof of the Bouchard-Mari˜ no conjecture [3] on Hurwitz numbers using the Laplace transform of the celebrated cut-and-join equation of Goulden, Jackson, and Vakil [15, 37]. The cut-and-join equation expresses the Hurwitz number of a given genus and profile (partition) in terms of those corresponding to profiles modified by either cutting a part into two pieces or joining two parts into one. This equation holds for an arbitrary partition µ. We calculate the Laplace transform of this equation with µ as the summation variable. The result is a polynomial equation [32]. A Hurwitz cover is a holomorphic mapping f : X → P 1 from a connected nonsingu- lar projective algebraic curve X of genus g to the projective line P 1 with only simple ramifications except for ∞∈ P 1 . Such a cover is further refined by specifying its pro- file, which is a partition µ =(µ 1 ≥ µ 2 ≥ ··· ≥ µ ℓ > 0) of the degree of the covering deg f = |µ| = µ 1 + ··· + µ ℓ . The length ℓ(µ)= ℓ of this partition is the number of points in the inverse image f −1 (∞)= {p 1 ,...,p ℓ } of ∞. Each part µ i gives a local description of the map f , which is given by z −→ z µ i in terms of a local coordinate z of X around p i . The number h g,µ of topological types of Hurwitz covers of given genus g and profile µ, counted with the weight factor 1/|Autf |, is the Hurwitz number we shall deal with in this 2000 Mathematics Subject Classification. 14H10, 14N10, 14N35; 05A15, 05A17; 81T45. Saclay preprint number: IPHT T09/101. 1