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56
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].
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.
LANDAUGINZBURG/CALABIYAU CORRESPONDENCE, GLOBAL MIRROR SYMMETRY AND ORLOV EQUIVALENCE
, 2013
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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.
COMPUTING GENUSZERO TWISTED GROMOV–WITTEN INVARIANTS
, 2008
"... Abstract. Twisted Gromov–Witten invariants are intersection numbers in moduli spaces of stable maps to a manifold or orbifold X which depend in addition on a vector bundle over X and an invertible multiplicative characteristic class. Special cases are closely related to local Gromov–Witten invariant ..."
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Cited by 22 (7 self)
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Abstract. Twisted Gromov–Witten invariants are intersection numbers in moduli spaces of stable maps to a manifold or orbifold X which depend in addition on a vector bundle over X and an invertible multiplicative characteristic class. Special cases are closely related to local Gromov–Witten invariants of the bundle, and to genuszero onepoint invariants of complete intersections in X. We develop tools for computing genuszero twisted Gromov–Witten invariants of orbifolds and apply them to several examples. We prove a “quantum Lefschetz theorem ” which expresses genuszero onepoint Gromov–Witten invariants of a complete intersection in terms of those of the ambient orbifold X. We determine the genuszero Gromov–Witten potential of the type A surface singularity ˆ C2 ˜ ˆ ˜ /Zn. We also compute some genuszero invariants of C3 /Z3, verifying predictions of Aganagic–Bouchard–Klemm. In a selfcontained Appendix, we determine the relationship between the quantum cohomology of the An surface singularity and that of its crepant resolution, thereby proving the Crepant Resolution Conjectures of Ruan and Bryan–Graber in this case.
WALLCROSSINGS IN TORIC GROMOV–WITTEN THEORY I: CREPANT EXAMPLES
, 2006
"... Graber asserts that certain generating functions for genuszero Gromov–Witten invariants of an orbifold X can be obtained from their counterparts for a crepant resolution of X by analytic continuation followed by specialization of parameters. In this paper we use mirror symmetry to determine the rel ..."
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Cited by 18 (4 self)
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Graber asserts that certain generating functions for genuszero Gromov–Witten invariants of an orbifold X can be obtained from their counterparts for a crepant resolution of X by analytic continuation followed by specialization of parameters. In this paper we use mirror symmetry to determine the relationship between the genuszero Gromov–Witten invariants of the weighted projective spaces P(1, 1, 2), P(1, 1, 1, 3) and those of their crepant resolutions. Our methods are applicable to other toric birational transformations. Our results verify the Crepant Resolution Conjecture when X = P(1, 1, 2) and suggest that it needs modification when
On the crepant resolution conjecture in the local case
"... Abstract. In this paper we analyze four examples of birational transformations between local Calabi– Yau 3folds: two crepant resolutions, a crepant partial resolution, and a flop. We study the effect of these transformations on genuszero Gromov–Witten invariants, proving the Coates–Corti–Iritani– ..."
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Cited by 15 (2 self)
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Abstract. In this paper we analyze four examples of birational transformations between local Calabi– Yau 3folds: two crepant resolutions, a crepant partial resolution, and a flop. We study the effect of these transformations on genuszero Gromov–Witten invariants, proving the Coates–Corti–Iritani– Tseng/Ruan form of the Crepant Resolution Conjecture in each case. Our results suggest that this form of the Crepant Resolution Conjecture may also hold for more general crepant birational transformations. They also suggest that Ruan’s original Crepant Resolution Conjecture should be modified, by including appropriate “quantum corrections”, and that there is no straightforward generalization of either Ruan’s original Conjecture or the Cohomological Crepant Resolution Conjecture to the case of crepant partial resolutions. Our methods are based on mirror symmetry for toric orbifolds. 1.
ON THE COHOMOLOGICAL CREPANT RESOLUTION CONJECTURE FOR WEIGHTED PROJECTIVE SPACES
, 2006
"... Abstract. We investigate the Cohomological Crepant Resolution Conjecture for reduced Gorenstein weighted projective spaces. Using toric methods, we prove this conjecture in some new cases. As an intermediate step, we show that weighted projective spaces are toric DeligneMumford stacks. We also desc ..."
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Cited by 13 (2 self)
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Abstract. We investigate the Cohomological Crepant Resolution Conjecture for reduced Gorenstein weighted projective spaces. Using toric methods, we prove this conjecture in some new cases. As an intermediate step, we show that weighted projective spaces are toric DeligneMumford stacks. We also describe a combinatorial model for the orbifold cohomology of weighted projective spaces. 1.
Orbifold quantum Dmodules associated to weighted projective spaces
, 2008
"... We construct in an abstract fashion the orbifold quantum cohomology (quantum orbifold cohomology) of weighted projective space, starting from the orbifold quantum differential operator. We obtain the product, grading, and intersection form by making use of the associated selfadjoint Dmodule and t ..."
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Cited by 12 (4 self)
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We construct in an abstract fashion the orbifold quantum cohomology (quantum orbifold cohomology) of weighted projective space, starting from the orbifold quantum differential operator. We obtain the product, grading, and intersection form by making use of the associated selfadjoint Dmodule and the Birkhoff factorization procedure. The method extends to the more difficult case of Fano hypersurfaces in weighted projective space. However, in contrast to the case of weighted projective space itself or a Fano hypersurface in projective space, a “small Birkhoff cell” can appear in the construction; we give an example of this phenomenon.