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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
KP hierarchy for Hodge integrals
, 2008
"... Starting from the ELSV formula, we derive a number of new equations on the generating functions for Hodge integrals over the moduli space of complex curves. This gives a new simple and uniform treatment of certain known results on Hodge integrals like Witten’s conjecture, Virasoro constrains, Faber’ ..."
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Cited by 33 (1 self)
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Starting from the ELSV formula, we derive a number of new equations on the generating functions for Hodge integrals over the moduli space of complex curves. This gives a new simple and uniform treatment of certain known results on Hodge integrals like Witten’s conjecture, Virasoro constrains, Faber’s λgconjecture etc. Among other results we show that a properly arranged generating function for Hodge integrals satisfies the equations of the KP hierarchy.
MIRZAKHANI’S RECURSION RELATIONS, VIRASORO CONSTRAINTS AND THE KDV HIERARCHY
"... Abstract. We present in this paper a differential version of Mirzakhani’s recursion relation for the WeilPetersson 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 ..."
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Cited by 28 (8 self)
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Abstract. We present in this paper a differential version of Mirzakhani’s recursion relation for the WeilPetersson 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 ψ and κ1 intersections on Mg,n is a 1parameter solution to the KdV hierarchy. It recovers the WittenKontsevich generating function when the parameter is set to be 0. 1.
The KP hierarchy, branched covers, and triangulations
, 2008
"... The KP hierarchy is a completely integrable system of quadratic, partial differential equations that generalizes the KdV hierarchy. A linear combination of Schur functions is a solution to the KP hierarchy if and only if its coefficients satisfy the Plücker relations from geometry. We give a soluti ..."
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Cited by 27 (1 self)
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The KP hierarchy is a completely integrable system of quadratic, partial differential equations that generalizes the KdV hierarchy. A linear combination of Schur functions is a solution to the KP hierarchy if and only if its coefficients satisfy the Plücker relations from geometry. We give a solution to the Plücker relations involving products of variables marking contents for a partition, and thus give a new proof of a content product solution to the KP hierarchy, previously given by Orlov and Shcherbin. In our main result, we specialize this content product solution to prove that the generating series for a general class of transitive ordered factorizations in the symmetric group satisfies the KP hierarchy. These factorizations appear in geometry as encodings of branched covers, and thus by specializing our transitive factorization result, we are able to prove that the generating series for two classes of branched covers satisfies the KP hierarchy. For the first of these, the double Hurwitz series, this result has been previously given by Okounkov. The second of these, that we call the mhypermap series, contains the double Hurwitz series polynomially, as the leading coefficient in m. The mhypermap series also specializes further, first to the series for hypermaps and then to the series for maps, both in an orientable surface. For the latter series, we apply one of the KP equations to obtain a new and remarkably simple recurrence for triangulations in a surface of given genus, with a given number of faces. This recurrence leads to explicit asymptotics for the number of triangulations with given genus and number of faces, in recent work by Bender, Gao and Richmond.
A simple proof of Witten conjecture through localization
"... Abstract. We obtain a system of relations between Hodge integrals with one λclass. As an application, we show that its first nontrivial relation implies the Witten’s Conjecture/Kontsevich Theorem [12, 6]. 1. ..."
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Cited by 25 (8 self)
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Abstract. We obtain a system of relations between Hodge integrals with one λclass. As an application, we show that its first nontrivial relation implies the Witten’s Conjecture/Kontsevich Theorem [12, 6]. 1.
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
Virasoro constraints for KontsevichHurwitz partition function
, 2008
"... In [1, 2] M.Kazarian and S.Lando found a 1parametric interpolation between Kontsevich and Hurwitz partition functions, which entirely lies within the space of KP τfunctions. In [3] V.Bouchard and M.Marino suggested that this interpolation satisfies some deformed Virasoro constraints. However, they ..."
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Cited by 17 (0 self)
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In [1, 2] M.Kazarian and S.Lando found a 1parametric interpolation between Kontsevich and Hurwitz partition functions, which entirely lies within the space of KP τfunctions. In [3] V.Bouchard and M.Marino suggested that this interpolation satisfies some deformed Virasoro constraints. However, they described the constraints in a somewhat sophisticated form of AMMEynard equations [4, 5, 6, 7] for the rather involved Lambert spectral curve. Here we present the relevant family of Virasoro constraints explicitly. They differ from the conventional continuous Virasoro constraints in the simplest possible way: by a constant shift u2 24 of the ˆ L−1 operator, where u is an interpolation parameter between Kontsevich and Hurwitz models. This trivial modification of the string equation gives rise to the entire deformation which is a conjugation of the Virasoro constraints ˆ Lm → Û ˆ LmÛ −1 and ”twists ” the partition function, ZKH { = ÛZK. The conjugation 2 u
The moduli space of curves, double Hurwitz numbers, and Faber’s intersection number conjecture
, 2006
"... We define the dimension 2g − 1 FaberHurwitz Chow/homology classes on the moduli space of curves, parametrizing curves expressible as branched covers of P 1 with given ramification over ∞ and sufficiently many fixed ramification points elsewhere. Degeneration of the target and judicious localizatio ..."
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Cited by 17 (2 self)
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We define the dimension 2g − 1 FaberHurwitz Chow/homology classes on the moduli space of curves, parametrizing curves expressible as branched covers of P 1 with given ramification over ∞ and sufficiently many fixed ramification points elsewhere. Degeneration of the target and judicious localization expresses such classes in terms localization trees weighted by “top intersections” of tautological classes and genus 0 double Hurwitz numbers. This identity of generating series can be inverted, yielding a “combinatorialization” of top intersections of ψclasses. As genus 0 double Hurwitz numbers with at most 3 parts over ∞ are well understood, we obtain Faber’s Intersection Number Conjecture for up to 3 parts, and an approach to the Conjecture in general (bypassing the Virasoro Conjecture). We also recover other geometric results in a unified manner, including Looijenga’s theorem, the socle theorem for curves with rational tails, and the hyperelliptic locus in terms of κg−2.
A short proof of the λgConjecture without GromovWitten theory: Hurwitz theory and the moduli of curves
"... Abstract. We give a short and direct proof of the λgConjecture. The approach is through the EkedahlLandoShapiroVainshtein theorem, which establishes the “polynomiality ” of Hurwitz numbers, from which we pick off the lowest degree terms. The proof is independent of GromovWitten theory. We brief ..."
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Cited by 16 (2 self)
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Abstract. We give a short and direct proof of the λgConjecture. The approach is through the EkedahlLandoShapiroVainshtein theorem, which establishes the “polynomiality ” of Hurwitz numbers, from which we pick off the lowest degree terms. The proof is independent of GromovWitten theory. We briefly describe the philosophy behind our general approach to intersection numbers and how it may be extended to other intersection number conjectures.
The npoint functions for intersection numbers on moduli spaces of curves
, 2009
"... Using the celebrated WittenKontsevich theorem, we prove a recursive formula of the npoint functions for intersection numbers on moduli spaces of curves. It has been used to prove the Faber intersection number conjecture and motivated us to find some conjectural vanishing identities for GromovWit ..."
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Cited by 11 (8 self)
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Using the celebrated WittenKontsevich theorem, we prove a recursive formula of the npoint functions for intersection numbers on moduli spaces of curves. It has been used to prove the Faber intersection number conjecture and motivated us to find some conjectural vanishing identities for GromovWitten invariants. The latter has been proved recently by X. Liu and R. Pandharipande. We also give a combinatorial interpretation of npoint functions in terms of summation over binary trees.