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

Abstract

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 ${\mathcal{A}}_r\times {\mathbb{P}}^1$.