Non-Perturbative Superpotentials and Discrete Torsion

Abstract: We discuss the non-perturbative superpotential in E₈×E₈ heterotic string theory on a non-simply connected Calabi–Yau manifold X, as well as on its simply connected covering space X. The superpotential is induced by the string wrapping holomorphic, isolated, genus zero curves. We show, in a specific example, that the superpotential is non-zero both on X and on X avoiding the no-go residue theorem of Beasley and Witten. On the non-simply connected manifold X, we explicitly compute the leading contribution to the superpotential from all holomorphic, isolated, genus zero curves with minimal area. The reason for the non-vanishing of the superpotental on X is that the second homology class contains a finite part called discrete torsion. As a result, the curves with the same area are distributed among different torsion classes and their contributions do not cancel each other.