Wall-Crossings in Toric Gromov-Witten Theory I: Crepant Examples

Abstract

Let X be a Gorenstein orbifold and let Y be a crepant resolution of X. We state a conjecture relating the genus-zero Gromov--Witten invariants of X to those of Y, which differs in general from the Crepant Resolution Conjectures of Ruan and Bryan--Graber, and prove our conjecture when X = P(1,1,2) and X = P(1,1,1,3). As a consequence, we see that the original form of the Bryan--Graber Conjecture holds for P(1,1,2) but is probably false for P(1,1,1,3). Our methods are based on mirror symmetry for toric orbifolds.