Quantum Cohomology and Crepant Resolutions: A Conjecture

Coates, Tom; Ruan, Yongbin

doi:10.5802/aif.2766

Quantum Cohomology and Crepant Resolutions: A Conjecture
[Cohomologie Quantique et Résolutions Crépantes : Une Conjecture]

Coates, Tom ¹ ; Ruan, Yongbin ²

¹ Imperial College London Department of Mathematics London SW7 2AZ United Kingdom
² Department of Mathematics University of Michigan Ann Arbor MI 48105 USA

Annales de l'Institut Fourier, Tome 63 (2013) no. 2, pp. 431-478.

Résumé
Abstract

Nous présentons une conjecture développée par Coates-Iritani-Tseng et Ruan, qui relie la cohomologie quantique d’un orbifold de Gorenstein $𝒳$ à la cohomologie quantique d’une résolution crépante $Y$ de $𝒳$ . Nous explorons quelque conséquences de cette conjecture et montrons qu’elle implique des versions de la Conjecture de la Résolution Crépante Cohmologique et des Conjectures de la Résolution Crépante de Ruan et Bryan-Graber. Nous donnons aussi une version « quantisée » de la conjecture, qui détermine les invariants de Gromov-Witten de genre supérieur de $𝒳$ à partir de ceux de $Y$ .

We give an expository account of a conjecture, developed by Coates–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 $Y$ .

MR Zbl

DOI : 10.5802/aif.2766

Classification : 53D45, 14N35, 83E30
Keywords: Quantum cohomology, orbifold, crepant resolution, Gromov–Witten invariants.
Mot clés : Cohomologie quantique, orbifold, résolution crépante, les invariants de Gromov-Witten.

Affiliations des auteurs :

Coates, Tom ¹ ; Ruan, Yongbin ²

¹ Imperial College London Department of Mathematics London SW7 2AZ United Kingdom
² Department of Mathematics University of Michigan Ann Arbor MI 48105 USA

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     author = {Coates, Tom and Ruan, Yongbin},
     title = {Quantum {Cohomology} and {Crepant} {Resolutions:} {A} {Conjecture}},
     journal = {Annales de l'Institut Fourier},
     pages = {431--478},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {63},
     number = {2},
     year = {2013},
     doi = {10.5802/aif.2766},
     zbl = {1275.53083},
     mrnumber = {3112518},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2766/}
}

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Coates, Tom; Ruan, Yongbin. Quantum Cohomology and Crepant Resolutions: A Conjecture. Annales de l'Institut Fourier, Tome 63 (2013) no. 2, pp. 431-478. doi : 10.5802/aif.2766. http://www.numdam.org/articles/10.5802/aif.2766/

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