Root systems and the quantum cohomology of ADE resolutions

Abstract

We compute the ${ℂ}^{\ast }$-equivariant quantum cohomology ring of $Y$, the minimal resolution of the DuVal singularity ${ℂ}^2∕G$ where $G$ is a finite subgroup of $SU\left(2\right)$. The quantum product is expressed in terms of an ADE root system canonically associated to $G$. We generalize the resulting Frobenius manifold to nonsimply laced root systems to obtain an $n$ parameter family of algebra structures on the affine root lattice of any root system. Using the Crepant Resolution Conjecture, we obtain a prediction for the orbifold Gromov–Witten potential of $\left[{ℂ}^2∕G\right]$.