The open string McKay correspondence for type A singularities

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.