#### Volume 16, issue 1 (2012)

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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
##### Abstract

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

##### Keywords
orbifold Gromov–Witten invariant, symmetric product, $\mathcal{A}_r$ resolution, Crepant Resolution Conjecture
Primary: 14N35
Secondary: 14H10