paper

Crepant resolutions and open strings

arXiv:1309.4438 · doi:10.1515/crelle-2017-0011

Abstract

We formulate a Crepant Resolution Correspondence for open Gromov-Witten invariants (OCRC) of toric Lagrangian branes inside Calabi-Yau 3-orbifolds by encoding the open theories into sections of Givental's symplectic vector space. The correspondence can be phrased as the identification of these sections via a linear morphism of Givental spaces. We deduce from this a Bryan-Graber-type statement for disk invariants, which we extend to arbitrary topologies in the Hard Lefschetz case. Motivated by ideas of Iritani, Coates-Corti-Iritani-Tseng and Ruan, we furthermore propose 1) a general form of the morphism entering the OCRC, which arises from a geometric correspondence between equivariant K-groups, and 2) an all-genus version of the OCRC for Hard Lefschetz targets. We provide a complete proof of both statements in the case of minimal resolutions of threefold An singularities; as a necessary step of the proof we establish the all-genus closed Crepant Resolution Conjecture with descendents in its strongest form for this class of examples. Our methods rely on a new description of the quantum D-modules underlying the equivariant Gromov-Witten theory of this family of targets.

This paper supersedes arXiv:1303.0723 by the same authors, which will be withdrawn. v2: minor changes, references added. v3: arguments strengthened in Section 6.1 with reference to Teleman's theorem, statements about analytic continuation of flat sections of the Dubrovin connection have been clarified in Section 5.3 (Lemma 5.8); version accepted for publication in Crelle. 48 pages, 6 figures

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