Quantum Cohomology and Crepant Resolutions: A Conjecture

TL;DR

This paper discusses a conjecture linking the quantum cohomology of Gorenstein orbifolds to their crepant resolutions, with implications for higher-genus Gromov--Witten invariants and related conjectures.

Contribution

It provides an expository overview of a conjecture connecting orbifold and resolution quantum cohomology, including a quantized version for higher-genus invariants.

Findings

The conjecture implies versions of the Cohomological Crepant Resolution Conjecture.

It suggests a method to determine higher-genus invariants of orbifolds from their resolutions.

The paper explores consequences and related conjectures in quantum cohomology.

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.