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… 

Figures and Tables from this paper

The spectral curve of the Eynard-Orantin recursion via the Laplace transform
The Eynard-Orantin recursion formula provides an effective tool for certain enumeration problems in geometry. The formula requires a spectral curve and the recursion kernel. We present a uniform
The spectral curve of the Eynard-Orantin recursion via the Laplace transform
The Eynard-Orantin recursion formula provides an effective tool for certain enumeration problems in geometry. The formula requires a spectral curve and the recursion kernel. We present a uniform
KP integrability of triple Hodge integrals. III. Cut-and-join description, KdV reduction, and topological recursions
In this paper, we continue our investigation of the triple Hodge integrals satisfying the Calabi–Yau condition. For the tau-functions, which generate these integrals, we derive the complete families
Topological recursion on the Bessel curve
The Witten-Kontsevich theorem states that a certain generating function for intersection numbers on the moduli space of stable curves is a tau-function for the KdV integrable hierarchy. This
Chiodo formulas for the r-th roots and topological recursion
We analyze Chiodo’s formulas for the Chern classes related to the r-th roots of the suitably twisted integer powers of the canonical class on the moduli space of curves. The intersection numbers of
Hurwitz-Hodge Integral Identities from the Cut-and-Join Equation
In this paper, we present some Hurwitz-Hodge integral identities which are derived from the Laplace transformation of the cut-and-join equation for the orbifold Hurwitz numbers. As an application, we
The Laplace transform, mirror symmetry, and the topological recursion of Eynard-Orantin
This paper is based on the author’s talk at the 2012 Workshop on Geometric Methods in Physics held in Bialowieza, Poland. The aim of the talk is to introduce the audience to the Eynard–Orantin
Mirror curve of orbifold Hurwitz numbers
Edge-contraction operations form an effective tool in various graph enumeration problems, such as counting Grothendieck's dessins d'enfants and simple and double Hurwitz numbers. These counting
Topological recursion and mirror curves
We study the constant contributions to the free energies obtained through the topological recursion applied to the complex curves mirror to toric Calabi-Yau threefolds. We show that the recursion
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 66 REFERENCES
A short proof of the λg -conjecture without Gromov-Witten theory: Hurwitz theory and the moduli of curves
Abstract We give a short and direct proof of Getzler and Pandharipande's λg -conjecture. The approach is through the Ekedahl-Lando-Shapiro-Vainshtein theorem, which establishes the “polynomiality” of
Mirzakhani's recursion relations, Virasoro constraints and the KdV hierarchy
We present in this paper a differential version of Mirzakhani's recursion relation for the Weil-Petersson volumes of the moduli spaces of bordered Riemann surfaces. We discover that the
A short proof of the lambda_g-conjecture without Gromov-Witten theory: Hurwitz theory and the moduli of curves
We give a short and direct proof of the $\lambda_g$-Conjecture. The approach is through the Ekedahl-Lando-Shapiro-Vainshtein theorem, which establishes the ``polynomiality'' of Hurwitz numbers, from
Relative virtual localization and vanishing of tautological classes on moduli spaces of curves
We prove a localization formula for the moduli space of stable relative maps. As an application, we prove that all codimension i tautological classes on the moduli space of stable pointed curves
Weil-Petersson volume of moduli spaces, Mirzakhani's recursion and matrix models
We show that Mirzakhani's recursions for the volumes of moduli space of Riemann surfaces are a special case of random matrix loop equations, and therefore we confirm again that Kontsevitch's integral
Hodge Integrals and Hurwitz Numbers via Virtual Localization
We give another proof of Ekedahl, Lando, Shapiro, and Vainshtein's remarkable formula expressing Hurwitz numbers (counting covers of P1 with specified simple branch points, and specified branching
All order asymptotic expansion of large partitions
The generating function which counts partitions with the Plancherel measure (and its q-deformed version) can be rewritten as a matrix integral, which allows one to compute its asymptotic expansion to
Hurwitz numbers, matrix models and enumerative geometry
We propose a new, conjectural recursion solution for Hurwitz numbers at all genera. This conjecture is based on recent progress in solving type B topological string theory on the mirrors of toric
The Gromov–Witten Potential of A Point, Hurwitz Numbers, and Hodge Integrals
Hurwitz numbers, which count certain covers of the projective line (or, equivalently, factorizations of permutations into transpositions), have been extensively studied for over a century. The
...
1
2
3
4
5
...