Disc Instantons in Linear Sigma Models

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(AV). 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 GelfandKapranov-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 (AKV), 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. Email: [email protected] Email: [email protected] Email: [email protected]