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THE LOCAL GROMOVWITTEN INVARIANTS OF CONFIGURATIONS OF RATIONAL CURVES
, 2005
"... ABSTRACT. We compute the local GromovWitten invariants of certain configurations of rational curves in a CalabiYau threefold. These configurations are connected subcurves of the “minimal trivalent configuration”, which is a particular tree of P 1 ’s with specified formal neighborhood. We show that ..."
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ABSTRACT. We compute the local GromovWitten invariants of certain configurations of rational curves in a CalabiYau threefold. These configurations are connected subcurves of the “minimal trivalent configuration”, which is a particular tree of P 1 ’s with specified formal neighborhood. We show that these local invariants are equal to certain global or ordinary GromovWitten invariants of a blowup of P 3 at points, and we compute these ordinary invariants using the geometry of the Cremona transform. We also realize the configurations in question as formal toric schemes and compute their formal GromovWitten invariants using the mathematical and physical theories of the topological vertex. In particular, we provide further evidence equating the vertex amplitudes derived from physical and mathematical theories of the topological vertex. 1.
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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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 DonaldsonThomas invariants under blowups and flops
"... Abstract. Using the degeneration formula for DoanldsonThomas invariants, we proved formulae for blowing up a point and simple flops. 1. ..."
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Cited by 11 (1 self)
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Abstract. Using the degeneration formula for DoanldsonThomas invariants, we proved formulae for blowing up a point and simple flops. 1.
Formulae of onepartition and twopartition Hodge integrals
, 2006
"... Prompted by the duality between open string theory on noncompact Calabi–Yau threefolds and Chern–Simons theory on threemanifolds, M Mariño and C Vafa conjectured a formula of onepartition Hodge integrals in term of invariants of the unknot. Many Hodge integral identities, including the λg conjectu ..."
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Cited by 9 (1 self)
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Prompted by the duality between open string theory on noncompact Calabi–Yau threefolds and Chern–Simons theory on threemanifolds, M Mariño and C Vafa conjectured a formula of onepartition Hodge integrals in term of invariants of the unknot. Many Hodge integral identities, including the λg conjecture and the ELSV formula, can be obtained by taking limits of the Mariño–Vafa formula. Motivated by the Mariño–Vafa formula and formula of Gromov–Witten invariants of local toric Calabi–Yau threefolds predicted by physicists, J Zhou conjectured a formula of twopartition Hodge integrals in terms of invariants of the Hopf link and used it to justify the physicists ’ predictions. In this expository article, we describe proofs and applications of these two formulae of Hodge integrals based on joint works of K Liu, J Zhou and the author. This is an expansion of the author’s talk of the same title at the BIRS workshop The
A compactification of the space of maps from curves
 Trans. Amer. Math. Soc
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Curves in CalabiYau threefolds and Topological Quantum Field Theory
, 2008
"... We continue our study of the local GromovWitten invariants of curves in CalabiYau threefolds. We define relative invariants for the local theory which give rise to a 1+1dimensional TQFT taking values in the ring Q[[t]]. The associated Frobenius algebra over Q[[t]] is semisimple. Consequently, we ..."
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Cited by 8 (3 self)
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We continue our study of the local GromovWitten invariants of curves in CalabiYau threefolds. We define relative invariants for the local theory which give rise to a 1+1dimensional TQFT taking values in the ring Q[[t]]. The associated Frobenius algebra over Q[[t]] is semisimple. Consequently, we obtain a structure result for the local invariants. As an easy consequence of our structure formula, we recover the closed formulas for the local invariants in case either the target genus or the degree equals 1.
Invariance of Gromov–Witten theory under simple flops
 J. Reine Angew. Math
"... ABSTRACT. We show that the generating functions of Gromov–Witten invariants with ancestors are invariant under a simple flop, for all genera, after an analytic continuation in the extended Kähler moduli space. This is a sequel to [14]. 0.1. Statement of the main results. Let X be a smooth complex p ..."
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Cited by 7 (5 self)
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ABSTRACT. We show that the generating functions of Gromov–Witten invariants with ancestors are invariant under a simple flop, for all genera, after an analytic continuation in the extended Kähler moduli space. This is a sequel to [14]. 0.1. Statement of the main results. Let X be a smooth complex projective manifold and ψ: X → X ̄ a flopping contraction in the sense of minimal model theory, with ψ ̄ : Z ∼ = Pr → pt the restriction map to the extremal contraction. Assume that NZ/X ∼ = OPr(−1)⊕(r+1). It was shown