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59
The Quantum Orbifold Cohomology of Weighted Projective Spaces
, 2007
"... We calculate the small quantum orbifold cohomology of arbitrary weighted projective spaces. We generalize Givental’s heuristic argument, which relates small quantum cohomology to S 1equivariant Floer cohomology of loop space, to weighted projective spaces and use this to conjecture an explicit for ..."
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Cited by 29 (11 self)
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We calculate the small quantum orbifold cohomology of arbitrary weighted projective spaces. We generalize Givental’s heuristic argument, which relates small quantum cohomology to S 1equivariant Floer cohomology of loop space, to weighted projective spaces and use this to conjecture an explicit formula for the small Jfunction, a generating function for certain genuszero Gromov–Witten invariants. We prove this conjecture using a method due to Bertram. This provides the first nontrivial example of a family of orbifolds of arbitrary dimension for which the small quantum orbifold cohomology is known. In addition we obtain formulas for the small Jfunctions of weighted projective complete intersections satisfying a combinatorial condition; this condition
A de Rham model for ChenRuan cohomology ring of abelian orbifolds
, 408
"... Abstract. We present a deRham model for ChenRuan cohomology ring of abelian orbifolds. We introduce the notion of twist factors so that formally the stringy cohomology ring can be defined without going through pseudoholomorphic orbifold curves. Thus our model can be viewed as the classical descrip ..."
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Cited by 19 (2 self)
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Abstract. We present a deRham model for ChenRuan cohomology ring of abelian orbifolds. We introduce the notion of twist factors so that formally the stringy cohomology ring can be defined without going through pseudoholomorphic orbifold curves. Thus our model can be viewed as the classical description of ChenRuan cohomology for abelian orbifolds. The model simplifies computation of ChenRuan cohomology ring. Using our model, we give a version of wall crossing formula. 1.
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 18 (1 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].
INTEGRAL COHOMOLOGY AND MIRROR SYMMETRY FOR CalabiYau 3folds
, 2005
"... In this paper we compute the integral cohomology groups for all examples of CalabiYau 3folds obtained from hypersurfaces in 4dimensional Gorenstein toric Fano varieties. Among 473 800 776 families of CalabiYau 3folds X corresponding to 4dimensional reflexive polytopes there exist exactly 32 ..."
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Cited by 18 (7 self)
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In this paper we compute the integral cohomology groups for all examples of CalabiYau 3folds obtained from hypersurfaces in 4dimensional Gorenstein toric Fano varieties. Among 473 800 776 families of CalabiYau 3folds X corresponding to 4dimensional reflexive polytopes there exist exactly 32 families having nontrivial torsion in H ∗ (X, Z). We came to an interesting observation that the torsion subgroups in H 2 and H 3 are exchanged by the mirror symmetry involution.
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 8 (3 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.
On the C n /Zm fractional branes
, 2008
"... We construct several geometric representatives for the C n /Zm fractional branes on either a partially or the completely resolved orbifold. In the process we use large radius and conifoldtype monodromies, and provide a strong consistency check. In particular, for C 3 /Z5 we give three different set ..."
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Cited by 8 (4 self)
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We construct several geometric representatives for the C n /Zm fractional branes on either a partially or the completely resolved orbifold. In the process we use large radius and conifoldtype monodromies, and provide a strong consistency check. In particular, for C 3 /Z5 we give three different sets of geometric representatives. We also find the explicit Seibergduality which connects our fractional branes to the ones given by the McKay correspondence.
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 8 (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 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 8 (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 7 (3 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.