  We construct a linear sigma model for open-strings ending on special
Lagrangian cycles of a Calabi-Yau manifold. We illustrate the construction for
the cases considered by Aganagic and Vafa in hep-th/0012041. This leads
naturally to concrete models for the moduli space of open-string instantons.
These instanton moduli spaces can be seen to be intimately related to certain
auxiliary boundary toric varieties. By considering the relevant
Gelfand-Kapranov-Zelevinsky (GKZ) differential equations of the boundary toric
variety, we obtain the contributions to the worldvolume superpotential on the
A-branes from open-string instantons. By using an ansatz due to Aganagic, Klemm
and Vafa in hep-th/0105045, we obtain the relevant change of variables from the
linear sigma model to the non-linear sigma model variables - the open-string
mirror map. Using this mirror map, we obtain results in agreement with those of
AV and AKV for the counting of holomorphic disc instantons.
