The Quantum McKay Correspondence for Singularities of type D

TL;DR

This paper proves the quantum McKay correspondence for type D singularities using induction, polynomiality techniques, and building on previous results for type A groups, advancing understanding of orbifold Gromov-Witten invariants.

Contribution

It establishes the quantum McKay correspondence for type D polyhedral groups, extending prior results and introducing new polynomiality methods.

Findings

Proof of the quantum McKay correspondence for type D groups.

Development of polynomiality techniques for these singularities.

Extension of known results from type A to type D groups.

Abstract

We prove the quantum McKay correspondence formulae conjectured by J. Bryan and A. Gholampour for the type D (binary) polyhedral groups in SU(2) and SO(3). We use the method of induction by the WDVV equation and from the normal subgroups by J. Bryan and A. Gholampour, and the polynomiality technique developed in this article. We are also based on the validity of the corresponding conjecture for type A groups, which is proved by T. Coates, A. Corti, H. Iritani, and H.-H. Tseng.