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

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.

34 pages, 2 figures, 2 tables

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• Weil-Petersson volume of moduli spaces, Mirzakhani's recursion and matrix models
• Mirzakhani's recursion relations, Virasoro constraints and the KdV hierarchy
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Cited by in corpus (15)

• Local Mirror Symmetry for One-Legged Topological Vertex
• A generalized topological recursion for arbitrary ramification
• The ABCD of topological recursion
• Bouchard-Klemm-Marino-Pasquetti Conjecture for ${\mathbb{C}}^3$
• Equivalence of ELSV and Bouchard-Mariño conjectures for $r$-spin Hurwitz numbers
• Local Mirror Symmetry for the Topological Vertex
• Monotone Hurwitz numbers and the HCIZ integral II
• Generalized string equations for double Hurwitz numbers
• Quantum Mirror Curves for ${\mathbb{C}}^3$ and the Resolved Confiold
• Geometrical interpretation of the topological recursion, and integrable string theories
• A matrix model for the topological string II: The spectral curve and mirror geometry
• Polynomial recursion formula for linear Hodge integrals
• Pruned Hurwitz numbers
• Hurwitz-Hodge integral identities from the cut-and-join equation
• Quenched free energy from spacetime D-branes