On the Crepant Resolution Conjecture in the Local Case

TL;DR

This paper verifies the Crepant Resolution Conjecture for several local Calabi-Yau 3-fold transformations using mirror symmetry, suggesting modifications to the original conjecture and its potential generalizations.

Contribution

It proves the Coates-Corti-Iritani-Tseng/Ruan form of the Crepant Resolution Conjecture for specific local Calabi-Yau cases and discusses necessary modifications to Ruan's original conjecture.

Findings

The conjecture holds for studied examples of crepant resolutions and flops.

Quantum corrections are necessary for Ruan's original conjecture.

No straightforward extension of the conjecture applies to crepant partial resolutions.

Abstract

In this paper we analyze four examples of birational transformations between local Calabi-Yau 3-folds: two crepant resolutions, a crepant partial resolution, and a flop. We study the effect of these transformations on genus-zero Gromov-Witten invariants, proving the Coates-Corti-Iritani-Tseng/Ruan form of the Crepant Resolution Conjecture in each case. Our results suggest that this form of the Crepant Resolution Conjecture may also hold for more general crepant birational transformations. They also suggest that Ruan's original Crepant Resolution Conjecture should be modified, by including appropriate "quantum corrections", and that there is no straightforward generalization of either Ruan's original Conjecture or the Cohomological Crepant Resolution Conjecture to the case of crepant partial resolutions. Our methods are based on mirror symmetry for toric orbifolds.