Results 11  20
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110
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.
Hodge integrals and invariants of the unknot
"... Abstract We prove the Gopakumar{Mariño{Vafa formula for special cubic Hodge integrals. The GMV formula arises from Chern{Simons/string duality applied to the unknot in the three sphere. The GMV formula is a q {analog of the ELSV formula for linear Hodge integrals. We nd a system of bilinear localiz ..."
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Cited by 31 (4 self)
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Abstract We prove the Gopakumar{Mariño{Vafa formula for special cubic Hodge integrals. The GMV formula arises from Chern{Simons/string duality applied to the unknot in the three sphere. The GMV formula is a q {analog of the ELSV formula for linear Hodge integrals. We nd a system of bilinear localization equations relating linear and special cubic Hodge integrals. The GMV formula then follows easily from the ELSV formula. An operator form of the GMV formula is presented in the last section of the paper.
All orders asymptotic expansion of large partitions
, 2008
"... The generating function which counts partitions with the Plancherel measure (and its qdeformed version), can be rewritten as a matrix integral, which allows to compute its asymptotic expansion to all orders. There are applications in statistical physics of growing/melting crystals, T.A.S.E.P., and ..."
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Cited by 28 (6 self)
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The generating function which counts partitions with the Plancherel measure (and its qdeformed version), can be rewritten as a matrix integral, which allows to compute its asymptotic expansion to all orders. There are applications in statistical physics of growing/melting crystals, T.A.S.E.P., and also in algebraic geometry. In particular we compute the GromovWitten invariants of the Xp = O(p − 2) ⊕ O(−p) → P1 CalabiYau 3fold, and we prove a conjecture of M. Mariño, that the generating functions Fg of Gromov–Witten invariants of Xp, come from a matrix model, and are the symplectic invariants of the mirror spectral curve.
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 extended Toda hierarchy
"... We present the Lax pair formalism for certain extension of the continuous limit of the classical Toda lattice hierarchy, provide a well defined notion of tau function for its solutions, and give an explicit formulation of the relationship between the CP 1 topological sigma model and the extended Tod ..."
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Cited by 27 (5 self)
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We present the Lax pair formalism for certain extension of the continuous limit of the classical Toda lattice hierarchy, provide a well defined notion of tau function for its solutions, and give an explicit formulation of the relationship between the CP 1 topological sigma model and the extended Toda hierarchy. We also establish an equivalence of the latter with certain extension of the nonlinear Schrödinger hierarchy. 1
Phase transitions, double–scaling limit, and topological strings
, 2007
"... Topological strings on Calabi–Yau manifolds are known to undergo phase transitions at small distances. We study this issue in the case of perturbative topological strings on local Calabi–Yau threefolds given by a bundle over a twosphere. This theory can be regarded as a q–deformation of Hurwitz the ..."
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Cited by 27 (8 self)
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Topological strings on Calabi–Yau manifolds are known to undergo phase transitions at small distances. We study this issue in the case of perturbative topological strings on local Calabi–Yau threefolds given by a bundle over a twosphere. This theory can be regarded as a q–deformation of Hurwitz theory, and it has a conjectural nonperturbative description in terms of q–deformed 2d Yang–Mills theory. We solve the planar model and find a phase transition at small radius in the universality class of 2d gravity. We give strong evidence that there is a double–scaled theory at the critical point whose all genus free energy is governed by the Painlevé I equation. We compare the critical behavior of the perturbative theory to the critical behavior of its nonperturbative description, which belongs to the universality class of 2d supergravity, and we comment on possible implications for nonperturbative 2d gravity. We also give evidence for a new open/closed duality relating these Calabi–Yau backgrounds to open strings with framing.
The moduli space of curves and GromovWitten theory
, 2006
"... The goal of this article is to motivate and describe how GromovWitten theory can and has provided tools to understand the moduli space of curves. For example, ideas and methods from GromovWitten theory have led to both conjectures and theorems showing that the tautological part of the cohomology r ..."
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Cited by 26 (4 self)
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The goal of this article is to motivate and describe how GromovWitten theory can and has provided tools to understand the moduli space of curves. For example, ideas and methods from GromovWitten theory have led to both conjectures and theorems showing that the tautological part of the cohomology ring has a remarkable and profound structure. As an illustration, we describe a new approach to Faber’s intersection number conjecture via branched covers of the projective line (work with I.P. Goulden and D.M. Jackson, based on work with T. Graber). En route we review the work of a large number of mathematicians.
Generating functions for intersection numbers on moduli spaces of curves
, 2000
"... Using the connection between intersection theory on the DeligneMumford spaces Mg,n and the edge scaling of the GUE matrix model (see [12, 14]), we express the npoint functions for the intersection numbers as ndimensional errorfunctiontype integrals and also give a derivation of Witten’s KdV equ ..."
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Cited by 23 (4 self)
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Using the connection between intersection theory on the DeligneMumford spaces Mg,n and the edge scaling of the GUE matrix model (see [12, 14]), we express the npoint functions for the intersection numbers as ndimensional errorfunctiontype integrals and also give a derivation of Witten’s KdV equations using the higher Fay identities of Adler, Shiota, and van Moerbeke.
BIRATIONAL COBORDISM INVARIANCE OF UNIRULED SYMPLECTIC MANIFOLDS
, 2006
"... 2. Birational cobordism 3 2.1. Coupling form and linear deformations 4 2.2. Blowup and blowdown 6 ..."
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Cited by 23 (1 self)
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2. Birational cobordism 3 2.1. Coupling form and linear deformations 4 2.2. Blowup and blowdown 6