On the Crepant Resolution Conjecture for Gromov-Witten Gravitational Ancestors in All Genera for Surface Singularities

TL;DR

This paper formulates a version of the crepant resolution conjecture for Gromov-Witten invariants of surface singularities, proves it for type A singularities, and proposes a method for general cases using Teleman's theorem.

Contribution

It introduces a new formulation of the crepant resolution conjecture for surface singularities and proves it for type A, linking it to the quantum McKay correspondence.

Findings

Conjecture reduced to quantum McKay correspondence and vanishing Hurwitz-Hodge integrals

Proved for surface singularities of type A

Proposes approach for general cases using Teleman's reconstruction theorem

Abstract

We state a version of the crepant resolution conjecture for total ancestor potentials for surface singularities, and reduce the conjecture to the quantum McKay correspondence conjecture of J.Bryan and A.Gholampour and a vanishing conjecture for Hurwitz-Hodge integrals. In particular, for singularities of type A, we prove the conjecture. We also suggest an approach towards a proof for the general cases by Teleman's reconstruction theorem for semisimple cohomological field theories.