Disc Instantons in Linear Sigma Models

TL;DR

This paper develops a linear sigma model framework for open-string instantons on Calabi-Yau manifolds, providing a concrete approach to compute instanton contributions to the superpotential and confirming results with mirror symmetry techniques.

Contribution

It introduces a linear sigma model construction for open-string instantons on special Lagrangian cycles, linking moduli spaces to boundary toric varieties and deriving instanton effects via GKZ equations.

Findings

Constructed explicit models for open-string instanton moduli spaces.

Derived instanton contributions to the superpotential using GKZ equations.

Confirmed agreement with previous results on holomorphic disc counting.

Abstract

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.