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

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

2009-07-29

arxiv.org/abs/0907.5224v3

Mathematics

Algebraic Geometry

34 pages, 2 figures, 2 tables

Scientific paper

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.

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