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Quantum Cohomology and Crepant Resolutions: A Conjecture

Tom Coates, Yongbin Ruan (2013)

Annales de l’institut Fourier

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...

Quantum Singularity Theory for A ( r - 1 ) and r -Spin Theory

Huijun Fan, Tyler Jarvis, Yongbin Ruan (2011)

Annales de l’institut Fourier

We give a review of our construction of a cohomological field theory for quasi-homogeneous singularities and the r -spin theory of Jarvis-Kimura-Vaintrob. We further prove that for a singularity W of type A our construction of the stack of W -curves is canonically isomorphic to the stack of r -spin curves described by Abramovich and Jarvis. We further prove that our theory satisfies all the Jarvis-Kimura-Vaintrob axioms for an r -spin virtual class. Therefore, the Faber-Shadrin-Zvonkine proof of the...

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