Root systems and the quantum cohomology of ADE resolutions

Bryan, Jenny; Gholampour, Amin

doi:10.2140/ant.2008.2.369

Cited by 25 publications

(52 citation statements)

References 12 publications

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“…For the case k = 1, we expect the result to only depend on the D p algebra data [79], similarly to what we discussed in subsection 5.2. Nevertheless, an analysis of the simplest cases…”

Section: Equivariant Quantum Cohomologysupporting

confidence: 58%

Quantum cohomology and quantum hydrodynamics from supersymmetric quiver gauge theories

Bonelli

Sciarappa

Tanzini

et al. 2016

Journal of Geometry and Physics

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We study the connection between N = 2 supersymmetric gauge theories, quantum cohomology and quantum integrable systems of hydrodynamic type. We consider gauge theories on ALE spaces of A and D-type and discuss how they describe the quantum cohomology of the corresponding Nakajima's quiver varieties. We also discuss how the exact evaluation of local BPS observables in the gauge theory can be used to calculate the spectrum of quantum Hamiltonians of spin Calogero integrable systems and spin Intermediate Long Wave hydrodynamics. This is explicitly obtained by a Bethe Ansatz Equation provided by the quiver gauge theory in terms of its adjacency matrix. *

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“…For the case k = 1, we expect the result to only depend on the D p algebra data [79], similarly to what we discussed in subsection 5.2. Nevertheless, an analysis of the simplest cases…”

Section: Equivariant Quantum Cohomologysupporting

confidence: 58%

Quantum cohomology and quantum hydrodynamics from supersymmetric quiver gauge theories

Bonelli

Sciarappa

Tanzini

et al. 2016

Journal of Geometry and Physics

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“…Davesh Maulik has computed the genus-zero Gromov-Witten potential of the type A surface singularity X = [C 2 /Z n ] for all n (as well as certain higher-genus Gromov-Witten invariants of X ) [48] and the reduced genus-zero Gromov-Witten potential of the crepant resolution Y [49]; Theorem 1 should follow from this. The quantum cohomology of the crepant resolutions of type ADE surface singularities has been computed by Bryan-Gholampour [11]. Skarke [55] and Hosono [36] have also studied the A n case, from a point of view very similar to ours, as part of their investigations of homological mirror symmetry.…”

Section: ) the Inverse To The Mirror Map (25) Ismentioning

confidence: 99%

“…When the topology on R is defined by ideals {I n } n≥0 , the topology on R Here the s i are parameters of the universal invertible multiplicative characteristic class c: see (11 Example. The correlator t with no insertion is the genus-zero descendant potential.…”

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confidence: 99%

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Computing genus-zero twisted Gromov-Witten invariants

et al. 2009

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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 genus-zero one-point invariants of complete intersections in X . We develop tools for computing genus-zero twisted Gromov-Witten invariants of orbifolds and apply them to several examples. We prove a "quantum Lefschetz theorem" which expresses genus-zero one-point Gromov-Witten invariants of a complete intersection in terms of those of the ambient orbifold X . We determine the genus-zero Gromov-Witten potential of the type A surface singularityˆC 2 /Zn˜. We also compute some genus-zero invariants ofˆC 3 /Z 3˜, verifying predictions of Aganagic-Bouchard-Klemm. In a self-contained 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.

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“…Quantum equivariant geometry. We compute the genus 0 GW invariants of Y via localization (extending the computations of [17] to a more general torus action):…”

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confidence: 99%

Crepant resolutions and open strings

Brini

Cavalieri

Ross

2017

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. We formulate a Crepant Resolution Correspondence for open Gromov-Witten invariants (OCRC) of toric Lagrangian branes inside Calabi-Yau 3-orbifolds by encoding the open theories into sections of Givental's symplectic vector space. The correspondence can be phrased as the identification of these sections via a linear morphism of Givental spaces. We deduce from this a Bryan-Graber-type statement for disk invariants, which we extend to arbitrary topologies in the Hard Lefschetz case. Motivated by ideas of Iritani, Coates-CortiIritani-Tseng and Ruan, we furthermore propose 1) a general form of the morphism entering the OCRC, which arises from a geometric correspondence between equivariant K-groups, and 2) an all-genus version of the OCRC for Hard Lefschetz targets. We provide a complete proof of both statements in the case of minimal resolutions of threefold An-singularities; as a necessary step of the proof we establish the all-genus closed Crepant Resolution Conjecture with descendents in its strongest form for this class of examples. Our methods rely on a new description of the quantum D-modules underlying the equivariant Gromov-Witten theory of this family of targets.

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Root systems and the quantum cohomology of ADE resolutions

Cited by 25 publications

References 12 publications

Quantum cohomology and quantum hydrodynamics from supersymmetric quiver gauge theories

Quantum cohomology and quantum hydrodynamics from supersymmetric quiver gauge theories

Computing genus-zero twisted Gromov-Witten invariants

Crepant resolutions and open strings

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