Ruan’s conjecture on singular symplectic flops of mixed type

Chen, BoHui; Li, AnMin; Li, Xiaobin; Zhao, Guosong

doi:10.1007/s11425-014-4801-7

Cited by 4 publications

(4 citation statements)

References 31 publications

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“…In this case, the celebrated Crepant Transformation Conjecture of Y. Ruan predicts that the quantum (orbifold) cohomology algebras of X + and X − should be related by analytic continuation in the quantum parameters. This conjecture has stimulated a great deal of interest in the connections between quantum cohomology (or Gromov-Witten theory) and birational geometry: see, for example, [9,10,18,19,21,23,24,28,41,45,53,[56][57][58][59]62,68,71,75,76]. Ruan's original conjecture was subsequently refined, revised, and extended to higher genus Gromov-Witten invariants, first by Bryan-Graber [20] under some additional hypotheses, and then by Coates-Iritani-Tseng, Iritani, and Ruan in general [34,35,50].…”

Section: Introductionmentioning

confidence: 99%

The Crepant Transformation Conjecture for toric complete intersections

Coates

Iritani

Jiang

2018

Advances in Mathematics

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Let X and Y be K-equivalent toric Deligne-Mumford stacks related by a single toric wall-crossing. We prove the Crepant Transformation Conjecture in this case, fully-equivariantly and in genus zero. That is, we show that the equivariant quantum connections for X and Y become gauge-equivalent after analytic continuation in quantum parameters. Furthermore we identify the gauge transformation involved, which can be thought of as a linear symplectomorphism between the Givental spaces for X and Y , with a Fourier-Mukai transformation between the K-groups of X and Y , via an equivariant version of the Gamma-integral structure on quantum cohomology. We prove similar results for toric complete intersections. We impose only very weak geometric hypotheses on X and Y : they can be non-compact, for example, and need not be weak Fano or have Gorenstein coarse moduli space. Our main tools are the Mirror Theorems for toric Deligne-Mumford stacks and toric complete intersections, and the Mellin-Barnes method for analytic continuation of hypergeometric functions.

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Section: Introductionmentioning

confidence: 99%

The Crepant Transformation Conjecture for toric complete intersections

Coates

Iritani

Jiang

2018

Advances in Mathematics

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“…In [3,4], the authors initiated a program for studying quantum cohomology under birational transformation of orbifolds. In their papers, they considered the singularity W r = {(x, y, z, t) | xy − z 2r + t 2 }/µ r (a, −a, 1, 0) with r being a prime number.…”

Section: Introduction In [18]mentioning

confidence: 99%

“…They showed that Ruan cohomology is invariant under orbi-flops. Then, by using relative orbifold Gromov-Witten invariants and degeneration formulas, they proved that quantum cohomology is invariant under orbi-flops in [4].…”

Section: Introduction In [18]mentioning

confidence: 99%

Ruan cohomologies of the compactifications of resolved orbifold conifolds

Du¹,

Chen²,

Du³

et al. 2016

Rocky Mountain J. Math.

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In this paper, we study the Ruan cohomologies of X s and X sf , the natural compactifications of V s and V sf , where V s and V sf are the two small resolutions ofthe finite group quotient of the singular conifold. There is an additive isomorphism between the Chen-Ruan cohomologies ϕ : H * CR (X s ) → H * CR (X sf ). We study the three-point orbifold Gromov-Witten invariants of the exceptional curves Γ s on X s and Γ sf on X sf and show that the corresponding Ruan cohomology ring structures on the Chen-Ruan cohomologies of X s and X sf , defined by these three-point functions, are isomorphic to each other under the map ϕ and the 2010Such a function and the orbifold Poincaré pairing define the ring structure of Ruan cohomology RH * CR (X s ) over the Chen-Ruan cohomology group. Similarly, one can define F sf (β 1 , β 2 , β 3 ) on H * CR (X sf ) and the Ruan cohomology RH * CR (X sf ). Our main theorem isi.e., q ↔ q −1 . Hence, we get the isomorphism of rings: Compactifications of the resolved orbifold conifolds and their Chen-Ruan cohomologies. Resolved orbifold conifolds and their compactifications.The well-known (smooth) conifold singularity is the complex hyperplane given byIt has an isolated singular point at the origin. Given a prime number r, let µ r = ⟨ξ⟩, the cyclic group of rth roots of 1 with ξ = exp(2πi/r), act on C 4 : ξ · (x, y, z, w) = (ξx, ξ −1 y, z, w).

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“…The Crepant Transformation Conjecture has been verified to various extents of generalities and has become a guiding principle in the study of the relation between quantum cohomology and birational geometry. See for instance [3,4,10,11,13,[16][17][18][19][21][22][23]36,38,41,[45][46][47][48][49][50][51]57]. It has received particular success in the toric setting, where most required ingredients have explicit descriptions.…”

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confidence: 99%

The Open Crepant Transformation Conjecture for Toric Calabi-Yau 3-Orbifolds

2020

Preprint

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We prove an open version of Ruan's Crepant Transformation Conjecture for toric Calabi-Yau 3-orbifolds, which is an identification of disk invariants of K-equivalent semi-projective toric Calabi-Yau 3-orbifolds relative to corresponding Lagrangian suborbifolds of Aganagic-Vafa type. Our main tool is a mirror theorem of Fang-Liu-Tseng that relates these disk invariants to local coordinates on the B-model mirror curves. Treating toric crepant transformations as wall-crossings in the GKZ secondary fan, we establish the identification of disk invariants through constructing a global family of mirror curves over charts of the secondary variety and understanding analytic continuation on local coordinates. Our work generalizes previous results of Brini-Cavalieri-Ross on disk invariants of threefold type-A singularities and of Ke-Zhou on crepant resolutions with effective outer branes.

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Ruan’s conjecture on singular symplectic flops of mixed type

Abstract: We prove that the orbifold quantum ring is preserved under singular symplectic flops. Hence we verify Ruan's conjecture for this case. Contents

Cited by 4 publications

References 31 publications

The Crepant Transformation Conjecture for toric complete intersections

The Crepant Transformation Conjecture for toric complete intersections

Ruan cohomologies of the compactifications of resolved orbifold conifolds

The Open Crepant Transformation Conjecture for Toric Calabi-Yau 3-Orbifolds

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