Geometry & Topology

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

Wan Keng Cheong and Amin Gholampour

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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×1.

Article information

Geom. Topol., Volume 16, Number 1 (2012), 475-526.

Received: 24 October 2010
Revised: 3 September 2011
Accepted: 15 November 2011
First available in Project Euclid: 20 December 2017

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Zentralblatt MATH identifier

Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]
Secondary: 14H10: Families, moduli (algebraic)

orbifold Gromov–Witten invariant symmetric product $\mathcal{A}_r$ resolution Crepant Resolution Conjecture


Cheong, Wan Keng; Gholampour, Amin. Orbifold Gromov–Witten theory of the symmetric product of $\mathcal{A}_r$. Geom. Topol. 16 (2012), no. 1, 475--526. doi:10.2140/gt.2012.16.475.

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