Open Gromov-Witten Theory and the Crepant Resolution Conjecture

Abstract

We compute open GW invariants for ${\mathcal{K}}_{{\mathbb{P}}^1}\oplus {\mathcal{O}}_{{\mathbb{P}}^1}$, open orbifold GW invariants for $[\C^3/\Z_2]$, formulate an open crepant resolution conjecture and verify it for this pair. We show that open invariants can be glued together to deduce the Bryan-Graber closed crepant resolution conjecture for the orbifold $[{\mathcal{O}}_{{\mathbb{P}}^1}(-1)\oplus {\mathcal{O}}_{{\mathbb{P}}^1}(-1)/{\mathbb{Z}}_2]$ and its crepant resolution ${\mathcal{K}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}$.