Quantum Cohomology and Crepant Resolutions: A Conjecture

Abstract

We give an expository account of a conjecture, developed by Coates--Corti--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.

34 pages. Version 2: added a paragraph (Remark 4.2) clarifying the history and origins of the conjecture. Version 3: corrected some typos in Section 8

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