Crepant resolutions and open strings II

TL;DR

This paper advances the understanding of open Gromov-Witten invariants and crepant resolution conjectures by verifying them for new classes of singularities and resolutions, with implications for closed-string theories.

Contribution

It extends the verification of open CRCs to non-hard Lefschetz targets and completes proofs for all-genus cases in specific toric Calabi-Yau threefolds.

Findings

Verified disk CRC for local weighted projective planes.

Completed proof of all-genus open CRC for G-Hilb resolution of [C^3/G].

Implications for closed-string CRCs in related theories.

Abstract

We recently formulated a number of Crepant Resolution Conjectures (CRC) for open Gromov-Witten invariants of Aganagic-Vafa Lagrangian branes and verified them for the family of threefold type A-singularities. In this paper we enlarge the body of evidence in favor of our open CRCs, along two different strands. In one direction, we consider non-hard Lefschetz targets and verify the disk CRC for local weighted projective planes. In the other, we complete the proof of the quantized (all-genus) open CRC for hard Lefschetz toric Calabi-Yau three dimensional representations by a detailed study of the G-Hilb resolution of $[C^3/G]$ for $G=\mathbb{Z}_2 \times \mathbb{Z}_2$. Our results have implications for closed-string CRCs of Coates-Iritani-Tseng, Iritani, and Ruan for this class of examples.