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

@article{Eynard2009TheLT,
  title={The Laplace transform of the cut-and-join equation and the Bouchard-Marino conjecture on Hurwitz numbers},
  author={Bertrand Eynard and Motohico Mulase and Brad Safnuk},
  journal={Publications of The Research Institute for Mathematical Sciences},
  year={2009},
  volume={47},
  pages={629-670}
}
Author(s): Eynard, Bertrand; Mulase, Motohico; Safnuk, Brad | 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… 
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