Geometry & Topology 16 (2012) 475–526

DOI: 10.2140/gt.2012.16.475

Abstract

Let A_r be the minimal resolution of the type A_r surface singularity. We study the equivariant orbifold Gromov–Witten theory of the n–fold symmetric product stack [Symⁿ(A_r)] of A_r. We calculate the divisor operators, which turn out to determine the entire theory under a nondegeneracy hypothesis. This, together with the results of Maulik and Oblomkov, shows that the Crepant Resolution Conjecture for Symⁿ(A_r) is valid. More strikingly, we complete a tetrahedron of equivalences relating the Gromov–Witten theories of [Symⁿ(A_r)] ∕ Hilbⁿ(A_r) and the relative Gromov–Witten/Donaldson–Thomas theories of A_r × P¹.

Keywords

orbifold Gromov–Witten invariant, symmetric product, A_r resolution, Crepant Resolution Conjecture

Mathematical Subject Classification

Primary: 14N35

Secondary: 14H10

References

Publication

Received: 24 October 2010
Revised: 3 September 2011
Accepted: 15 November 2011
Published: 29 March 2012
Proposed: Jim Bryan
Seconded: Richard Thomas, Simon Donaldson

Authors