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68
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 30 (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.
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 19 (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.
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 19 (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].
Orbifold cohomology of torus quotients
 Duke Math. J
"... ABSTRACT. We introduce the preorbifold cohomology ring PH ∗,⋄ T (Y) of a stably almost complex manifold carrying an action of a torus T. We show that in the case that Y has a locally free action by T, the preorbifold cohomology ring is isomorphic to the orbifold cohomology ring H ∗ orb(Y/T) (as defi ..."
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Cited by 8 (3 self)
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ABSTRACT. We introduce the preorbifold cohomology ring PH ∗,⋄ T (Y) of a stably almost complex manifold carrying an action of a torus T. We show that in the case that Y has a locally free action by T, the preorbifold cohomology ring is isomorphic to the orbifold cohomology ring H ∗ orb(Y/T) (as defined in [ChenRuan]) of the quotient orbifold Y/T. For Y a compact Hamiltonian Tspace, we extend to orbifold cohomology two techniques that are standard in ordinary cohomology. We show that PH ∗,⋄ T (Y) has a natural ring surjection onto H ∗ orb(Y//T), where Y//T is the symplectic reduction of Y by T at a regular value of the moment map. We extend to PH ∗,⋄ T (Y) the graphical GKM calculus (as detailed in e.g. [HaradaHenriquesHolm]), and the kernel computations of [TolmanWeitsman, Goldin]. We detail this technology in two examples: toric orbifolds and weight varieties, which are symplectic reductions of flag manifolds. Toric orbifolds were previously calculated over Q in [BorisovChenSmith]); we reproduce their results over Q for all symplectic toric orbifolds obtained by reduction by a connected torus (though with different computational methods), and extend them over Z in certain cases. 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 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 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.
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.