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38
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.
Witten’s conjecture, Virasoro conjecture, and semisimple Frobenius manifolds
, 2002
"... Abstract. The main goal of this paper is to prove the following two conjectures for genus up to two: (1) Witten’s conjecture on the relations between higher spin curves and Gelfand–Dickey hierarchy. (2) Virasoro conjecture for target manifolds with conformal semisimple Frobenius manifolds. The main ..."
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Cited by 21 (7 self)
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Abstract. The main goal of this paper is to prove the following two conjectures for genus up to two: (1) Witten’s conjecture on the relations between higher spin curves and Gelfand–Dickey hierarchy. (2) Virasoro conjecture for target manifolds with conformal semisimple Frobenius manifolds. The main technique used in the proof is the invariance of tautological equations under loop group action. 1.
GromovWitten invariants of varieties with holomorphic 2forms
"... Abstract. We show that a holomorphic twoform θ on a smooth algebraic variety X localizes the virtual fundamental class of the moduli of stable maps Mg,n(X, β) to the locus where θ degenerates; it then enables us to define the localized GWinvariant, an algebrogeometric analogue of the local invari ..."
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Cited by 21 (6 self)
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Abstract. We show that a holomorphic twoform θ on a smooth algebraic variety X localizes the virtual fundamental class of the moduli of stable maps Mg,n(X, β) to the locus where θ degenerates; it then enables us to define the localized GWinvariant, an algebrogeometric analogue of the local invariant of Lee and Parker in symplectic geometry [15], which coincides with the ordinary GWinvariant when X is proper. It is deformation invariant. Using this, we prove formulas for low degree GWinvariants of minimal general type surfaces with pg> 0 conjectured by Maulik and Pandharipande. 1.
GromovWitten/Pairs correspondence for the quintic 3fold
, 2012
"... We use the GromovWitten/Pairs descendent correspondence for toric 3folds and degeneration arguments to establish the GW/P correspondence for several compact CalabiYau 3folds (including all CY complete intersections in products of projective spaces). A crucial aspect of the proof is the study of ..."
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Cited by 18 (6 self)
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We use the GromovWitten/Pairs descendent correspondence for toric 3folds and degeneration arguments to establish the GW/P correspondence for several compact CalabiYau 3folds (including all CY complete intersections in products of projective spaces). A crucial aspect of the proof is the study of the GW/P correspondence for descendents in relative geometries. Projective bundles over surfaces relative to a section play a special role. The GW/P correspondence for CalabiYau complete intersections provides a structure result for the GromovWitten invariants in a fixed curve class. After change of variables, the GromovWitten series is a rational function in the variable −q = e iu invariant under q ↔ q −1.
MariñoVafa formula and Hodge integral identities
 J. Algebraic Geom
"... Abstract. We derive some Hodge integral identities by taking various limits of the MariñoVafa formula using the cutandjoin equation. These identities include the formula of general λgintegrals, the formula of λg−1integrals on Mg,1, the formula of cubic λ integrals on Mg, and the ELSV formula re ..."
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Cited by 17 (6 self)
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Abstract. We derive some Hodge integral identities by taking various limits of the MariñoVafa formula using the cutandjoin equation. These identities include the formula of general λgintegrals, the formula of λg−1integrals on Mg,1, the formula of cubic λ integrals on Mg, and the ELSV formula relating Hurwitz numbers and Hodge integrals. In particular, our proof of the MV formula by the cutandjoin equation leads to a new and simple proof of the λg conjecture. We also present a proof of the ELSV formula completely parallel to our proof of the MariñoVafa formula. 1.
BCOV theory on CalabiYau manifolds and the higher genus Bmodel, preprint
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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 local DonaldsonThomas theory of curves
, 2005
"... Dedicated to the memory of Raoul Bott The local DonaldsonThomas theory of curves is solved by localization and degeneration methods. The results complete a triangle of equivalences relating GromovWitten theory, DonaldsonThomas theory, and the quantum cohomology of the Hilbert scheme of points of ..."
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Cited by 13 (5 self)
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Dedicated to the memory of Raoul Bott The local DonaldsonThomas theory of curves is solved by localization and degeneration methods. The results complete a triangle of equivalences relating GromovWitten theory, DonaldsonThomas theory, and the quantum cohomology of the Hilbert scheme of points of
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