13 research outputs found
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 differential relation, which is
equivalent to the original integral formula of Mirzakhani, is a Virasoro
constraint condition on a generating function for these volumes. We also show
that the generating function for psi and kappa_1 intersections on the moduli
space of stable algebraic curves is a 1-parameter solution to the KdV
hierarchy. It recovers the Witten-Kontsevich generating function when the
parameter is set to be 0.Comment: 21 pages, 3 figures; v3. new introduction, minor revision
The Laplace transform of the cut-and-join equation and the Bouchard-Marino conjecture on Hurwitz numbers
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.Comment: 34 pages, 2 figures, 2 table
The Kontsevich constants for the volume of the moduli of curves and topological recursion
We give an Eynard-Orantin type topological recursion formula for the
canonical Euclidean volume of the combinatorial moduli space of pointed smooth
algebraic curves. The recursion comes from the edge removal operation on the
space of ribbon graphs. As an application we obtain a new proof of the
Kontsevich constants for the ratio of the Euclidean and the symplectic volumes
of the moduli space of curves.Comment: 37 pages with 20 figure
A matrix model for simple Hurwitz numbers, and topological recursion
We introduce a new matrix model representation for the generating function of
simple Hurwitz numbers. We calculate the spectral curve of the model and the
associated symplectic invariants developed in [Eynard-Orantin]. As an
application, we prove the conjecture proposed by Bouchard and Marino, relating
Hurwitz numbers to the spectral invariants of the Lambert curve exp(x)=y
exp(-y).Comment: 24 pages, 3 figure
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The Laplace transform of the cut-and-join equation and the Bouchard-Marino conjecture on Hurwitz numbers
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