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28
The Laplace transform of the cutandjoin equation and the BouchardMarino conjecture on Hurwitz numbers
"... Abstract. We calculate the Laplace transform of the cutandjoin 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 WeilPetersson volume of the moduli space of bordered hyperbolic surfa ..."
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Cited by 44 (16 self)
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Abstract. We calculate the Laplace transform of the cutandjoin 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 WeilPetersson 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 Wfunction is the topological recursion formula for Hurwitz numbers conjectured by Bouchard and Mariño using topological string theory. Contents
Recursion between Mumford volumes of moduli spaces
, 2007
"... We propose a new proof, as well as a generalization of Mirzakhani’s recursion for volumes of moduli spaces. We interpret those recursion relations in terms of expectation values in Kontsevich’s integral, i.e. we relate them to a Ribbon graph decomposition of Riemann surfaces. We find a generalizatio ..."
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Cited by 30 (11 self)
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We propose a new proof, as well as a generalization of Mirzakhani’s recursion for volumes of moduli spaces. We interpret those recursion relations in terms of expectation values in Kontsevich’s integral, i.e. we relate them to a Ribbon graph decomposition of Riemann surfaces. We find a generalization of Mirzakhani’s recursions to measures containing all higher Mumford’s κ classes, and not only κ1 as in the WeilPetersson case.
Computation of open GromovWitten invariants for toric CalabiYau 3folds by topological recursion, a proof of the BKMP conjecture
, 2013
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POLYNOMIAL RECURSION FORMULA FOR LINEAR HODGE INTEGRALS
"... Abstract. We establish a polynomial recursion formula for linear Hodge integrals. It is obtained as the cutandjoin equation for the Laplace transform of the Hurwitz numbers. We show that the recursion recovers the WittenKontsevich theorem when restricted to the top degree terms, and also the comb ..."
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Cited by 19 (9 self)
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Abstract. We establish a polynomial recursion formula for linear Hodge integrals. It is obtained as the cutandjoin equation for the Laplace transform of the Hurwitz numbers. We show that the recursion recovers the WittenKontsevich theorem when restricted to the top degree terms, and also the combinatorial factor of the λg formula as the lowest degree terms. Dedicated to Herbert Kurke on the occasion of his 70th birthday Contents
Recursion formulae of higher WeilPetersson volumes
 Inter. Math. Res. Notices
"... Abstract. In this paper we study effective recursion formulae for computing intersection numbers of mixed ψ and κ classes on moduli spaces of curves. By using the celebrated WittenKontsevich theorem, we generalize MulaseSafnuk form of Mirzakhani’s recursion and prove a recursion formula of higher ..."
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Cited by 13 (4 self)
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Abstract. In this paper we study effective recursion formulae for computing intersection numbers of mixed ψ and κ classes on moduli spaces of curves. By using the celebrated WittenKontsevich theorem, we generalize MulaseSafnuk form of Mirzakhani’s recursion and prove a recursion formula of higher WeilPetersson volumes. We also present recursion formulae to compute intersection pairings in the tautological rings of moduli spaces of curves. 1.
Invariants of spectral curves and intersection theory of moduli spaces of complex curves
, 2011
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Topological recursion for the Poincare polynomial of the combinatorial moduli space of curves
, 2010
"... We show that the Poincare polynomial associated with the orbifold cell decomposition of the moduli space of smooth algebraic curves with distinct marked points satisfies a topological recursion formula of the EynardOrantin type. The recursion uniquely determines the Poincaré polynomials from the ..."
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Cited by 10 (4 self)
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We show that the Poincare polynomial associated with the orbifold cell decomposition of the moduli space of smooth algebraic curves with distinct marked points satisfies a topological recursion formula of the EynardOrantin type. The recursion uniquely determines the Poincaré polynomials from the initial data. Our key discovery is that the Poincare ́ polynomial is the Laplace transform of the number of Grothendieck’s dessins d’enfants.
Cusps and the family hyperbolic metric
 Duke Math. J
, 2007
"... The hyperbolic metric for the punctured unit disc in the Euclidean plane is singular at the origin. A renormalization of the metric at the origin is provided by the Euclidean metric. For Riemann surfaces there is a unique germ for the isometry class of a complete hyperbolic metric at a cusp. The ren ..."
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Cited by 9 (5 self)
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The hyperbolic metric for the punctured unit disc in the Euclidean plane is singular at the origin. A renormalization of the metric at the origin is provided by the Euclidean metric. For Riemann surfaces there is a unique germ for the isometry class of a complete hyperbolic metric at a cusp. The renormalization for the punctured unit disc provides a renormalization for a hyperbolic metric at a cusp. For a holomorphic family of punctured Riemann surfaces the family of (co)tangent spaces along a puncture defines a tautological holomorphic line bundle over the base of the family. The Hermitian connection and Chern form for the renormalized metric are determined. Connections to the work of M. Mirzakhani, L. Takhtajan and P. Zograf, and intersection numbers for the moduli space of punctured Riemann surfaces studied by E. Witten are presented. 1 Comparing cusps The renormalization of a hyperbolic metric at a cusp is introduced. The setting is used to present an intrinsic norm for the germ of a holomorphic map at a cusp. A compact Riemann surface R having punctures and negative Euler characteristic has a complete hyperbolic metric, [Ahl73]. The geometry of a cusp of a hyperbolic metric is standard. From the uniformization theorem for a puncture p there is a distinguished local conformal coordinate with z(p) = 0 and the metric locally given by the germ of ds 2 =
THE SPECTRAL CURVE OF THE EYNARDORANTIN RECURSION VIA THE LAPLACE TRANSFORM
"... Abstract. The EynardOrantin 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 construction of the spectral curve and the recursion kernel from the unstable geometries of th ..."
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Cited by 9 (2 self)
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Abstract. The EynardOrantin 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 construction of the spectral curve and the recursion kernel from the unstable geometries of the original counting problem. We examine this construction using four concrete examples: Grothendieck’s dessins d’enfants (or highergenus analogue of the Catalan numbers), the intersection numbers of tautological cotangent classes on the moduli stack of stable pointed curves, single Hurwitz numbers, and the stationary GromovWitten invariants of the complex projective line. Contents