Results 1  10
of
38
equations for Hurwitz numbers
"... We consider ramified coverings of P 1 with arbitrary ramification type over 0, ∞ ∈ P 1 and simple ramifications elsewhere and prove that the generating function for the numbers of such coverings is a τfunction for the Toda lattice hierarchy of Ueno and Takasaki. ..."
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Cited by 66 (3 self)
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We consider ramified coverings of P 1 with arbitrary ramification type over 0, ∞ ∈ P 1 and simple ramifications elsewhere and prove that the generating function for the numbers of such coverings is a τfunction for the Toda lattice hierarchy of Ueno and Takasaki.
An algebrogeometric proof of Witten’s conjecture
 J. Amer. Math. Soc
"... Abstract We present a new proof of Witten's conjecture. The proof is based on the analysis of the relationship between intersection indices on moduli spaces of complex curves and Hurwitz numbers enumerating ramified coverings of the 2sphere. ..."
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Cited by 57 (2 self)
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Abstract We present a new proof of Witten's conjecture. The proof is based on the analysis of the relationship between intersection indices on moduli spaces of complex curves and Hurwitz numbers enumerating ramified coverings of the 2sphere.
Towards the geometry of double Hurwitz numbers
 Advances Math
"... ABSTRACT. Double Hurwitz numbers count branched covers of CP 1 with fixed branch points, with simple branching required over all but two points 0 and ∞, and the branching over 0 and ∞ points specified by partitions of the degree (with m and n parts respectively). Single Hurwitz numbers (or more usua ..."
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Cited by 45 (6 self)
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ABSTRACT. Double Hurwitz numbers count branched covers of CP 1 with fixed branch points, with simple branching required over all but two points 0 and ∞, and the branching over 0 and ∞ points specified by partitions of the degree (with m and n parts respectively). Single Hurwitz numbers (or more usually, Hurwitz numbers) have a rich structure, explored by many authors in fields as diverse as algebraic geometry, symplectic geometry, combinatorics, representation theory, and mathematical physics. A remarkable formula of Ekedahl, Lando, M. Shapiro, and Vainshtein (the ELSV formula) relates single Hurwitz numbers to intersection theory on the moduli space of curves. This connection has led to many consequences, including Okounkov and Pandharipande’s proof of Witten’s conjecture (Kontsevich’s theorem) connecting intersection theory on the moduli space of curves to integrable systems. In this paper, we determine the structure of double Hurwitz numbers using techniques from geometry, algebra, and representation theory. Our motivation is geometric: we give strong evidence that double Hurwitz numbers are top intersections on a moduli space of curves with a line bundle (a universal Picard variety). In particular, we prove a piecewisepolynomiality
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
The Toda equations and the GromovWitten theory of the Riemann sphere
 Lett. Math. Phys
"... 0.1. Toda equations. The GromovWitten theory of P1 has been intensively studied in a sequence of remarkable papers by Eguchi, Hori, Xiong, Yamada, and Yang [EHY], [EY], [EYY], [EHX]. A major step in this analysis was the discovery of a (conjectural) matrix model for ..."
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Cited by 40 (2 self)
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0.1. Toda equations. The GromovWitten theory of P1 has been intensively studied in a sequence of remarkable papers by Eguchi, Hori, Xiong, Yamada, and Yang [EHY], [EY], [EYY], [EHX]. A major step in this analysis was the discovery of a (conjectural) matrix model for
HODGE INTEGRALS AND HURWITZ NUMBERS VIA VIRTUAL LOCALIZATION
, 2000
"... Abstract. Ekedahl, Lando, Shapiro, and Vainshtein announced a remarkable formula ([ELSV]) expressing Hurwitz numbers (counting covers of P 1 with specified simple branch points, and specified branching over one other point) in terms of Hodge integrals. We give a proof of this formula using virtual l ..."
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Cited by 39 (5 self)
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Abstract. Ekedahl, Lando, Shapiro, and Vainshtein announced a remarkable formula ([ELSV]) expressing Hurwitz numbers (counting covers of P 1 with specified simple branch points, and specified branching over one other point) in terms of Hodge integrals. We give a proof of this formula using virtual localization on the moduli space of stable maps, and describe how the proof could be simplified by the proper algebrogeometric definition of a “relative
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.
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 [4]. As an application, we prove the conjecture proposed by Bouchard and Mariño [2], relating Hurw ..."
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Cited by 30 (5 self)
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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 [4]. As an application, we prove the conjecture proposed by Bouchard and Mariño [2], relating Hurwitz numbers to the spectral invariants of the Lambert curve ex = ye−y.