The Crepant Resolution Conjecture for Type A Surface Singularities

TL;DR

This paper proves the Crepant Resolution Conjecture for type A surface singularities by leveraging mirror symmetry for toric orbifolds, establishing an isomorphism between quantum cohomologies after specific analytic continuations.

Contribution

It provides a proof of the conjecture for type A surface singularities using mirror symmetry techniques, filling a gap in the understanding of quantum cohomology relations.

Findings

Confirmed the conjecture for type A surface singularities

Established isomorphism of quantum cohomologies after analytic continuation

Applied mirror symmetry for toric orbifolds in the proof

Abstract

Let X be an orbifold with crepant resolution Y. The Crepant Resolution Conjectures of Ruan and Bryan-Graber assert, roughly speaking, that the quantum cohomology of X becomes isomorphic to the quantum cohomology of Y after analytic continuation in certain parameters followed by the specialization of some of these parameters to roots of unity. We prove these conjectures in the case where X is a surface singularity of type A. The key ingredient is mirror symmetry for toric orbifolds.