The open string McKay correspondence for type A singularities

TL;DR

This paper develops a new framework for relating open Gromov-Witten invariants of toric Calabi-Yau orbifolds through a symplectic vector space correspondence, extending known results to arbitrary genus zero topologies.

Contribution

It formulates a Crepant Resolution Correspondence for open invariants, extending Bryan-Graber-type results and providing a complete proof for threefold A_n singularities.

Findings

Establishes a linear map between Givental spaces for open invariants.

Extends Bryan-Graber-type disk invariants to genus zero topologies.

Provides a new description of equivariant quantum D-modules for these targets.

Abstract

We formulate a Crepant Resolution Correspondence for open Gromov-Witten invariants (OCRC) of toric Calabi-Yau orbifolds by viewing the open theories as sections of Givental's symplectic vector space and the correspondence as a linear map of Givental spaces which identifies them. We deduce a Bryan-Graber-type statement for disk invariants and extend it to arbitrary genus zero topologies in the Hard Lefschetz case. Upon leveraging Iritani's theory of integral structures to equivariant quantum cohomology, we conjecture a general form of the symplectomorphism entering the OCRC which arises from a geometric correspondence at the equivariant K-theory level. We give a complete proof of this in the case of minimal resolutions of threefold A_n singularities. Our methods rely on a new description of the equivariant quantum D-modules underlying the Gromov-Witten theory of this class of targets.