Non-Perturbative Superpotentials and Discrete Torsion

Abstract

We discuss the non-perturbative superpotential in $${{E}_{8}} \times {{E}_{8}}$$ heterotic string theory on a non-simply connected Calabi–Yau manifold X, as well as on its simply connected covering space $$\tilde {X}.$$ 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 $$\tilde {X}$$ 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.