Open Gromov-Witten Invariants and Superpotentials for Semi-Fano Toric Surfaces

In this paper, we compute the open Gromov-Witten invariants for every compact toric surface X which is semi-Fano (i.e., the anticanonical line bundle K-x(-1) is nef). Unlike the Fano case, this involves nontrivial obstructions in the corresponding moduli problem. As a consequence, an explicit formula for the Lagrangian Floer superpotential W is obtained, which in turn gives an explicit presentation of the small quantum cohomology ring of X. We also provide a computational verification of the conjectural ring isomorphism between the small quantum cohomology of X and the Jacobian ring of W.