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129
AN INTEGRAL STRUCTURE IN QUANTUM COHOMOLOGY AND MIRROR SYMMETRY FOR TORIC ORBIFOLDS
, 2009
"... We introduce an integral structure in orbifold quantum cohomology associated to the Kgroup and the b Γclass. In the case of compact toric orbifolds, we show that this integral structure matches with the natural integral structure for the LandauGinzburg model under mirror symmetry. By assuming the ..."
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Cited by 72 (5 self)
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We introduce an integral structure in orbifold quantum cohomology associated to the Kgroup and the b Γclass. In the case of compact toric orbifolds, we show that this integral structure matches with the natural integral structure for the LandauGinzburg model under mirror symmetry. By assuming the existence of an integral structure, we give a natural explanation for the specialization to a root of unity in Y. Ruan’s crepant resolution conjecture [66].
The crepant resolution conjecture
, 2006
"... Abstract. For orbifolds admitting a crepant resolution and satisfying a hard Lefschetz condition, we formulate a conjectural equivalence between the GromovWitten theories of the orbifold and the resolution. We prove the conjecture for the equivariant GromovWitten theories of Sym n C 2 and Hilb n C ..."
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Cited by 42 (8 self)
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Abstract. For orbifolds admitting a crepant resolution and satisfying a hard Lefschetz condition, we formulate a conjectural equivalence between the GromovWitten theories of the orbifold and the resolution. We prove the conjecture for the equivariant GromovWitten theories of Sym n C 2 and Hilb n C 2. 1.
The tropical vertex
"... Abstract. Elements of the tropical vertex group are formal families of symplectomorphisms of the 2dimensional algebraic torus. We prove ordered product factorizations in the tropical vertex group are equivalent to calculations of certain genus 0 relative GromovWitten invariants of toric surfaces. ..."
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Cited by 33 (10 self)
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Abstract. Elements of the tropical vertex group are formal families of symplectomorphisms of the 2dimensional algebraic torus. We prove ordered product factorizations in the tropical vertex group are equivalent to calculations of certain genus 0 relative GromovWitten invariants of toric surfaces. The relative invariants which arise have full tangency to a toric divisor at a single unspecified point. The method uses scattering diagrams, tropical curve counts, degeneration formulas, and exact multiple cover calculations in orbifold GromovWitten theory. Contents
ORBIFOLD QUANTUM RIEMANNROCH, LEFSCHETZ AND SERRE
, 2009
"... Given a vector bundle F on a smooth DeligneMumford stack X and an invertible multiplicative characteristic class c, we define orbifold GromovWitten invariants of X twisted by F and c. We prove a “quantum RiemannRoch theorem” (Theorem 4.2.1) which expresses the generating function of the twisted i ..."
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Cited by 30 (9 self)
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Given a vector bundle F on a smooth DeligneMumford stack X and an invertible multiplicative characteristic class c, we define orbifold GromovWitten invariants of X twisted by F and c. We prove a “quantum RiemannRoch theorem” (Theorem 4.2.1) which expresses the generating function of the twisted invariants in terms of the generating function of the untwisted invariants. A quantum Lefschetz hyperplane theorem is derived from this by specializing to genus zero. As an application, we determine the relationship between genus0 orbifold GromovWitten invariants of X and that of a complete intersection, under additional assumptions. This provides a way to verify mirror symmetry predictions for some complete intersection orbifolds.
SMOOTH TORIC DELIGNEMUMFORD STACKS
, 2009
"... We give a geometric definition of smooth toric DeligneMumford stacks using the action of a “torus”. We show that our definition is equivalent to the one of Borisov, Chen and Smith in terms of stacky fans. In particular, we give a geometric interpretation of the combinatorial data contained in a sta ..."
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Cited by 27 (0 self)
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We give a geometric definition of smooth toric DeligneMumford stacks using the action of a “torus”. We show that our definition is equivalent to the one of Borisov, Chen and Smith in terms of stacky fans. In particular, we give a geometric interpretation of the combinatorial data contained in a stacky fan. We also give a bottom up classification in terms of simplicial toric varieties and fiber products of root stacks.
LANDAUGINZBURG/CALABIYAU CORRESPONDENCE, GLOBAL MIRROR SYMMETRY AND ORLOV EQUIVALENCE
, 2013
"... ..."
Sums over topological sectors and quantization of FayetIliopoulos parameters,” arXiv:1012.5999 [hepth
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Orbifold quantum cohomology of weighted projective spaces
 J. Algenraic Geom
"... Abstract. In this article, we prove the following results. • We show a mirror theorem: the Frobenius manifold associated to the orbifold quantum cohomology of weighted projective space is isomorphic to the one attached to a specific Laurent polynomial, • We show a reconstruction theorem, that is, we ..."
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Cited by 22 (3 self)
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Abstract. In this article, we prove the following results. • We show a mirror theorem: the Frobenius manifold associated to the orbifold quantum cohomology of weighted projective space is isomorphic to the one attached to a specific Laurent polynomial, • We show a reconstruction theorem, that is, we can reconstruct in an algorithmic way the full genus 0 GromovWitten potential from the 3point invariants. 1.
The Crepant Resolution Conjecture in all genera for type A singularities
, 2008
"... We prove an all genera version of the Crepant Resolution Conjecture of Ruan and BryanGraber for type A surface singularities. We are based on a method that explicitly computes HurwitzHodge integrals described in an earlier paper and some recent results by LiuXu for some intersection numbers on t ..."
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Cited by 14 (2 self)
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We prove an all genera version of the Crepant Resolution Conjecture of Ruan and BryanGraber for type A surface singularities. We are based on a method that explicitly computes HurwitzHodge integrals described in an earlier paper and some recent results by LiuXu for some intersection numbers on the DeligneMumford moduli spaces. We also generalize our results to some threedimensional orbifolds.