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47
Relative virtual localization and vanishing of tautological classes on moduli spaces of curves
 Duke Math. J
"... ABSTRACT. We prove a localization formula for the moduli space of stable relative maps. As an application, we prove that all codimension i tautological classes on the moduli space of stable pointed curves vanish away from strata corresponding to curves with at least i − g + 1 genus 0 components. As ..."
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Cited by 68 (6 self)
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ABSTRACT. We prove a localization formula for the moduli space of stable relative maps. As an application, we prove that all codimension i tautological classes on the moduli space of stable pointed curves vanish away from strata corresponding to curves with at least i − g + 1 genus 0 components. As consequences, we prove and generalize various conjectures and theorems about various moduli spaces of curves (due to Getzler, Ionel, Faber, Looijenga, Pandharipande, Diaz, and others). This theorem appears to be the geometric content behind these results; the rest is straightforward graph combinatorics. The theorem also suggests the importance of the stratification of the moduli space by number of rational components. CONTENTS
GromovWitten/DonaldsonThomas correspondence for toric 3folds
, 2008
"... We prove the equivariant GromovWitten theory of a nonsingular toric 3fold X with primary insertions is equivalent to the equivariant DonaldsonThomas theory of X. As a corollary, the topological vertex calculations by Agangic, Klemm, Mariño, and Vafa of the GromovWitten theory of local CalabiYau ..."
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Cited by 59 (17 self)
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We prove the equivariant GromovWitten theory of a nonsingular toric 3fold X with primary insertions is equivalent to the equivariant DonaldsonThomas theory of X. As a corollary, the topological vertex calculations by Agangic, Klemm, Mariño, and Vafa of the GromovWitten theory of local CalabiYau toric 3folds are proven to be correct in the full 3leg setting.
A formula of twopartition Hodge integrals
"... Let Mg,n denote the DeligneMumford moduli stack of stable curves of genus g with n marked points. Let π: Mg,n+1 → Mg,n be the universal curve, and let ωπ be the relative dualizing sheaf. The Hodge bundle E = π∗ωπ ..."
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Cited by 36 (19 self)
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Let Mg,n denote the DeligneMumford moduli stack of stable curves of genus g with n marked points. Let π: Mg,n+1 → Mg,n be the universal curve, and let ωπ be the relative dualizing sheaf. The Hodge bundle E = π∗ωπ
A mathematical theory of the topological vertex
"... Abstract. We have developed a mathematical theory of the topological vertex— a theory that was originally proposed by M. Aganagic, A. Klemm, M. Mariño, and C. Vafa on effectively computing GromovWitten invariants of smooth toric CalabiYau threefolds derived from duality between open string theory ..."
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Cited by 36 (19 self)
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Abstract. We have developed a mathematical theory of the topological vertex— a theory that was originally proposed by M. Aganagic, A. Klemm, M. Mariño, and C. Vafa on effectively computing GromovWitten invariants of smooth toric CalabiYau threefolds derived from duality between open string theory of smooth CalabiYau threefolds and ChernSimons theory on three manifolds. 1.
Hodge integrals and invariants of the unknots
"... We prove the GopakumarMariñoVafa formula for special cubic Hodge integrals. The GMV formula arises from ChernSimons/string duality applied to the unknot in the three sphere. The GMV formula is a qanalog of the ELSV formula for linear Hodge integrals. We find a system of bilinear localization equ ..."
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Cited by 31 (4 self)
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We prove the GopakumarMariñoVafa formula for special cubic Hodge integrals. The GMV formula arises from ChernSimons/string duality applied to the unknot in the three sphere. The GMV formula is a qanalog of the ELSV formula for linear Hodge integrals. We find 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.
Computation of open GromovWitten invariants for toric CalabiYau 3folds by topological recursion, a proof of the BKMP conjecture
, 2013
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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.
Localization and gluing of topological amplitudes
 Commun. Math. Phys
, 2005
"... We develop a gluing algorithm for GromovWitten invariants of toric CalabiYau threefolds based on localization and gluing graphs. The main building block of this algorithm is a generating function of cubic Hodge integrals of special form. We conjecture a precise relation between this generating fun ..."
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Cited by 21 (1 self)
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We develop a gluing algorithm for GromovWitten invariants of toric CalabiYau threefolds based on localization and gluing graphs. The main building block of this algorithm is a generating function of cubic Hodge integrals of special form. We conjecture a precise relation between this generating function and the topological vertex at fractional framing.
A CONJECTURE ON HODGE INTEGRALS
, 2003
"... We propose a conjectural formula expressing the generating series of some Hodge integrals in terms of representation theory of KacMoody algebras. Such generating series appear in calculations of GromovWitten invariants by localization techniques. It generalizes a formula conjectured by Mariño an ..."
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Cited by 20 (13 self)
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We propose a conjectural formula expressing the generating series of some Hodge integrals in terms of representation theory of KacMoody algebras. Such generating series appear in calculations of GromovWitten invariants by localization techniques. It generalizes a formula conjectured by Mariño and Vafa, recently proved in joint work with ChiuChu Melissa Liu and Kefeng Liu. Some examples are presented. An integral of the form Mg,n