On Orbifold Quantum Cohomology of Foldings of $ADE$ Resolutions

Abstract

We compute the untwisted part of the $\mathbb{C}^\times$-equivariant orbifold quantum cohomology of certain finite cyclic quotients of the minimal resolutions of Kleinian singularities, which we refer to as their foldings. We then formulate a conjecture for the full $\mathbb{C}^\times$-equivariant orbifold quantum cohomology, motivated by the Crepant Resolution Conjecture, and provide two pieces of supporting evidence. We also identify the resulting Frobenius structure with known Frobenius structures associated with the corresponding non-simply-laced root systems.