The Crepant Resolution Conjecture

TL;DR

This paper proposes a conjecture linking the Gromov-Witten theories of orbifolds with crepant resolutions and proves it for specific cases involving symmetric products and Hilbert schemes.

Contribution

It formulates a new conjecture relating orbifold and resolution Gromov-Witten theories and proves it for the equivariant theories of symmetric products and Hilbert schemes.

Findings

Conjectural equivalence between orbifold and resolution Gromov-Witten theories.

Proof of the conjecture for symmetric product of the plane and Hilbert scheme.

Verification under the hard Lefschetz condition.

Abstract

For orbifolds admitting a crepant resolution and satisfying a hard Lefschetz condition, we formulate a conjectural equivalence between the Gromov-Witten theories of the orbifold and the resolution. We prove the conjecture for the equivariant Gromov-Witten theories of the nth symmetric product of the complex plane and the Hilbert scheme of n points in the plane.