Wall-crossings in Toric Gromov–witten Theory I: Crepant Examples

The Crepant Resolution Conjecture of Ruan and Bryan–Graber asserts that certain generating functions for genus-zero Gromov–Witten invariants of an orbifold X can be obtained from their counterparts for a crepant resolution of X by analytic continuation followed by specialization of parameters. In this paper we use mirror symmetry to determine the relationship between the genus-zero Gromov–Witten invariants of the weighted projective spaces P(1, 1, 2), P(1, 1, 1, 3) and those of their crepant resolutions. Our methods are applicable to other toric birational transformations. Our results verify the Crepant Resolution Conjecture when X = P(1, 1, 2) and suggest that it needs modification when X = P(1, 1, 1, 3).