Orbifold Gromov–Witten theory of the symmetric product of 𝒜r

Cheong, Wan Keng¹ ; Gholampour, Amin²

¹ Department of Mathematics, National Cheng Kung University, Tainan City 701, Taiwan
² Department of Mathematics, University of Maryland, College Park, MD 20742, USA

Geometry & topology, Tome 16 (2012) no. 1, pp. 475-526.

Résumé

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

Détail

DOI : 10.2140/gt.2012.16.475

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

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