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30
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 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 56 (20 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
LANDAUGINZBURG/CALABIYAU CORRESPONDENCE, GLOBAL MIRROR SYMMETRY AND ORLOV EQUIVALENCE
, 2013
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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
Localization in GromovWitten Theory and Orbifold GromovWitten Theory
 INTERNATIONAL PRESS AND HIGHER EDUCATION PRESS
, 2012
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The Crepant Resolution Conjecture for Type A Surface Singularities, Preprint
"... Abstract. Let X be an orbifold with crepant resolution Y. The Crepant Resolution Conjectures of Ruan and Bryan–Graber assert, roughly speaking, that the quantum cohomology of X becomes isomorphic to the quantum cohomology of Y after analytic continuation in certain parameters followed by the special ..."
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Cited by 11 (1 self)
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Abstract. Let X be an orbifold with crepant resolution Y. The Crepant Resolution Conjectures of Ruan and Bryan–Graber assert, roughly speaking, that the quantum cohomology of X becomes isomorphic to the quantum cohomology of Y after analytic continuation in certain parameters followed by the specialization of some of these parameters to roots of unity. We prove these conjectures in the case where X is a surface singularity of type A. The key ingredient is mirror symmetry for toric orbifolds. 1.
Gross fibrations, SYZ mirror symmetry, and open GromovWitten invariants for toric CalabiYau orbifolds
, 2013
"... Given a toric CalabiYau orbifold X whose underlying toric variety is semiprojective, we construct and study a nontoric Lagrangian torus fibration on X, which we call the Gross fibration. We apply the StromingerYauZaslow recipe to the Gross fibration of (a toric modification of) X to construct ..."
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Cited by 10 (6 self)
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Given a toric CalabiYau orbifold X whose underlying toric variety is semiprojective, we construct and study a nontoric Lagrangian torus fibration on X, which we call the Gross fibration. We apply the StromingerYauZaslow recipe to the Gross fibration of (a toric modification of) X to construct its instantoncorrected mirror, where the instanton corrections come from genus 0 open orbifold GromovWitten invariants, which are virtual counts of holomorphic orbidisks in X bounded by fibers of the Gross fibration. We explicitly evaluate all these invariants by first proving an open/closed equality and then employing the toric mirror theorem for suitable toric (parital) compactifications of X. Our calculations are then applied to (1) prove a conjecture of GrossSiebert on a relation between genus 0 open orbifold GromovWitten invariants and mirror maps of X – this is called the open mirror theorem, which leads to an enumerative meaning of mirror maps, and (2) demonstrate how open (orbifold) GromovWitten invariants for toric CalabiYau orbifolds change under toric crepant resolutions – this is an open analogue of Ruan’s crepant resolution conjecture.
GIVENTAL’S LAGRANGIAN CONE AND S 1EQUIVARIANT GROMOV–WITTEN THEORY
, 2007
"... Abstract. In the approach to Gromov–Witten theory developed by Givental, genuszero Gromov–Witten invariants of a manifold X are encoded by a Lagrangian cone in a certain infinitedimensional symplectic vector space. We give a construction of this cone, in the spirit of S 1equivariant Floer theory, ..."
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Cited by 5 (4 self)
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Abstract. In the approach to Gromov–Witten theory developed by Givental, genuszero Gromov–Witten invariants of a manifold X are encoded by a Lagrangian cone in a certain infinitedimensional symplectic vector space. We give a construction of this cone, in the spirit of S 1equivariant Floer theory, in terms of S 1equivariant Gromov–Witten theory of X × P 1. This gives a conceptual understanding of the “dilaton shift”: a changeofvariables which plays an essential role in Givental’s theory. 1.