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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Orbifold Gromov–Witten theory of the symmetric product of $\mathcal{A}_r$

Wan Keng Cheong and Amin Gholampour

Geometry & Topology 16 (2012) 475–526
Abstract

Let Ar be the minimal resolution of the type Ar surface singularity. We study the equivariant orbifold Gromov–Witten theory of the n–fold symmetric product stack [Symn(Ar)] of Ar. 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(Ar) is valid. More strikingly, we complete a tetrahedron of equivalences relating the Gromov–Witten theories of [Symn(Ar)]∕Hilbn(Ar) and the relative Gromov–Witten/Donaldson–Thomas theories of Ar × ℙ1.

Keywords
orbifold Gromov–Witten invariant, symmetric product, $\mathcal{A}_r$ resolution, Crepant Resolution Conjecture
Mathematical Subject Classification 2010
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
Wan Keng Cheong
Department of Mathematics
National Cheng Kung University
Tainan City 701
Taiwan
http://www.math.ncku.edu.tw/~keng
Amin Gholampour
Department of Mathematics
University of Maryland
College Park
MD 20742
USA
http://www2.math.umd.edu/~amingh