Orbifold Gromov–Witten theory of the symmetric product of 𝒜r

Cheong, Wan Keng; Gholampour, Amin

doi:10.2140/gt.2012.16.475

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Orbifold Gromov–Witten theory of the symmetric product of $\mathcal{A}_r$

Wan Keng Cheong and Amin Gholampour

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 $[{Sym}^{n} (A_{r})]$ 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 ${Sym}^{n} (A_{r})$ is valid. More strikingly, we complete a tetrahedron of equivalences relating the Gromov–Witten theories of $[{Sym}^{n} (A_{r})] ∕ {Hilb}^{n} (A_{r})$ and the relative Gromov–Witten/Donaldson–Thomas theories of $A_{r} \times ℙ^{1}$ .

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

Volume 16, issue 1 (2012)