Integral and higher-genus quantum McKay correspondence in type D

Abstract

We establish Iritani's integral form of the $\mathbb{C}^\times$-equivariant Crepant Transformation Conjecture (CTC) for the minimal resolution of the surface singularity $\mathbb{C}^2/\mathbb{BD}_{4n}$, where $\mathbb{BD}_{4n}$ is the binary dihedral group of order $4n$. We then use Givental's quantisation formalism to prove the all-genus full-descendent CTC in that case. As an application, we deduce the Bryan-Cadman-Young degree-zero form of the orbifold Gromov-Witten/Donaldson-Thomas correspondence for the Calabi-Yau 3-orbifold $[\mathbb{C}^2/\mathbb{BD}_{4n}] \times \mathbb{C}$.