Mirror symmetry for open strings

We discuss the generation of superpotentials in $d=4,$ $\mathcal{N}=1\mathrm{}$ supersymmetric field theories arising from type-IIA $D6\mathrm{}$-branes wrapped on supersymmetric three-cycles of a Calabi-Yau threefold. In general, nontrivial superpotentials arise from sums over disc instantons. We then find several examples of special Lagrangian three-cycles with nontrivial topology which are mirror to obstructed rational curves, conclusively demonstrating the existence of such instanton effects. In addition, we present explicit examples of disc instantons ending on the relevant three-cycles. Finally, we give a preliminary construction of a mirror map for the open string moduli, in a large-radius limit of the type IIA compactification.