Quantum Cohomology and Crepant Resolutions: A Conjecture

Coates, Tom; Ruan, Yongbin

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

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 ${\displaystyle Y}$ 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 ${\displaystyle Y}$.

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 $Y$.

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