Open Gromov-Witten Theory and the Crepant Resolution Conjecture

TL;DR

This paper advances open Gromov-Witten theory by computing invariants for specific geometries, formulating and verifying an open crepant resolution conjecture, and connecting open invariants to the Bryan-Graber closed conjecture.

Contribution

It introduces an open crepant resolution conjecture, verifies it for a specific pair, and links open invariants to the Bryan-Graber closed conjecture for certain orbifolds.

Findings

Computed open GW invariants for specific geometries.

Formulated and verified an open crepant resolution conjecture.

Connected open invariants to the Bryan-Graber closed 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 $[\mathcal{O}_{\mathbb{P}^1}(-1)\oplus\mathcal{O}_{\mathbb{P}^1}(-1)/\Z_2]$ and its crepant resolution $\mathcal{K}_{\mathbb{P}^1\times\mathbb{P}^1}$.