Wall-crossings in toric Gromov–Witten theory I: crepant examples

Coates, Tom; Iritani, Hiroshi; Tseng, Hsian-Hua

doi:10.2140/gt.2009.13.2675

Cited by 81 publications

(207 citation statements)

References 57 publications

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“…In [1] this was used to calculate generating functions of orbifold GromovWitten invariants in the case of the C 3 /Z 3 orbifold, which corresponds to a phase in the moduli space of local P 2 , its crepant resolution. The predictions obtained in this way have been later verified mathematically in orbifold Gromov-Witten theory [11,17,20,25], and other examples have been recently calculated [18,26].…”

Section: Introductionmentioning

confidence: 55%

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Topological Open Strings on Orbifolds

Bouchard

Klemm

Mariño

et al. 2010

Commun. Math. Phys.

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We use the remodeling approach to the B-model topological string in terms of recursion relations to study open string amplitudes at orbifold points. To this end, we clarify modular properties of the open amplitudes and rewrite them in a form that makes their transformation properties under the modular group manifest. We exemplify this procedure for the C 3 /Z 3 orbifold point of local P 2 , where we present results for topological string amplitudes for genus zero and up to three holes, and for the one-holed torus. These amplitudes can be understood as generating functions for either open orbifold GromovWitten invariants of C 3 /Z 3 , or correlation functions in the orbifold CFT involving insertions of both bulk and boundary operators.

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Section: Introductionmentioning

confidence: 55%

“…. , k. Its mirror [37,42] -see also [24] for a clear explanation -is a B-twisted Landau-Ginzburg model on the family of algebraic tori π : V → M, with V = (C * ) 3+k , and M = (C * ) k , with projection map π : (y 0 , . .…”

Section: A Landau-ginzburg Vs Sigma Modelmentioning

confidence: 99%

Topological Open Strings on Orbifolds

Bouchard

Klemm

Mariño

et al. 2010

Commun. Math. Phys.

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“…Since ρ is crepant, we have h = (1/ n i=0 w i )( n i=0 b i + d j=1 e j ) (see Fulton [11]). Since H is an ample line bundle (see [11,Section 3.4]), Lemma 3.1.1 shows that the Mori cone M ρ (Z) is generated by the set (10) [ (Z; C)(q 1 , q 2 , q 3 , q 4 ) ∼ = H CR P (1, 3, 4, 4); C which is an isometry with respect to the Poincaré pairing on H ρ (Z; C)(q 1 , q 2 , q 3 , q 4 ) and with respect to the Chen-Ruan pairing on H CR (P (1, 3, 4, 4); C).…”

Section: The Cohomological Crepant Resolution Conjecturementioning

confidence: 99%

Computing certain Gromov-Witten invariants of the crepant resolution of ℙ(1, 3, 4, 4)

Boissière¹,

Mann²,

Perroni³

2011

Nagoya Math. J.

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Abstract. We prove a formula computing the Gromov-Witten invariants of genus zero with three marked points of the resolution of the transversal A3-singularity of the weighted projective space P(1, 3, 4, 4) using the theory of deformations of surfaces with An-singularities. We use this result to check Ruan's conjecture for the stack P(1, 3, 4, 4). §1. IntroductionThe main results of this paper concern the weighted projective space P(1, 3, 4, 4). The singular locus of its coarse moduli space |P(1, 3, 4, 4)| is the disjoint union of an isolated singularity of type (1/3)(1, 1, 1) (we use Reid's notation in [17]) and a transversal A 3 -singularity. Using toric methods, we construct a crepant resolution Z of |P(1, 3, 4, 4)|. In Theorem 3.3.1, we determine a formula for certain Gromov-Witten invariants of Z over the A 3 -singularity using the theory of deformations of surfaces with rational double points and the deformation invariance property of Gromov-Witten invariants. We then apply this result in Theorem 5.2.1 to construct a ring isomorphism-predicted in [18] by Ruan's cohomological crepant resolution conjecture -between the quantum corrected cohomology ring of Z and the Chen-Ruan orbifold cohomology of P(1, 3, 4, 4), after evaluating the quantum parameters related to the transversal A 3 -singularity to a fourth root of the unity and putting the last parameter to zero. This last evaluation is quite surprising (in [2], we show that this parameter can be evaluated to 1). To confirm this property, we show in Proposition 5.3.1 that, for all weighted projective spaces P(1, . . . , 1, n) with n weights equal to 1 which have only

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“…Ruan's conjecture is discussed further and revised in §8 and §11 below. Bryan and Graber [7] recently proposed a refinement of Ruan's conjecture, applicable whenever X satisfies a Hard Lefschetz condition on orbifold cohomology [14]. They suggest that in this case the big quantum cohomology algebras of X and Y coincide after analytic continuation and specialization of quantum parameters, via a linear isomorphism that also matches certain pairings on the algebras.…”

Section: Introductionmentioning

confidence: 99%

Quantum Cohomology and Crepant Resolutions: A Conjecture

Coates¹,

Ruan²

2013

Ann. inst. Fourier

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We give an expository account of a conjecture, developed by Coates-Corti-Iritani-Tseng and Ruan, which relates the quantum cohomology of a Gorenstein orbifold X to the quantum cohomology of a crepant resolution Y of X . We explore some consequences of this conjecture, showing that it implies versions of both the Cohomological Crepant Resolution Conjecture and of the Crepant Resolution Conjectures of Ruan and Bryan-Graber. We also give a 'quantized' version of the conjecture, which determines higher-genus Gromov-Witten invariants of X from those of Y .

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Wall-crossings in toric Gromov–Witten theory I: crepant examples

Cited by 81 publications

References 57 publications

Topological Open Strings on Orbifolds

Topological Open Strings on Orbifolds

Computing certain Gromov-Witten invariants of the crepant resolution of ℙ(1, 3, 4, 4)

Quantum Cohomology and Crepant Resolutions: A Conjecture

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