Non-vanishing Superpotentials in Heterotic String Theory and Discrete Torsion

TL;DR

This paper investigates conditions under which non-perturbative superpotentials in heterotic string theory do not vanish, focusing on the role of discrete torsion and the limitations of the residue theorem in certain Calabi-Yau manifolds.

Contribution

It demonstrates that discrete torsion can lead to non-zero superpotentials by affecting curve homology classes, challenging previous assumptions based on the residue theorem.

Findings

Superpotential can be non-zero due to discrete torsion effects.

The residue theorem's applicability is limited when curves have different areas.

Explicit computation shows non-vanishing superpotential on specific manifolds.

Abstract

We study the non-perturbative superpotential in E_8 x 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 0 curves. According to the residue theorem of Beasley and Witten, the non-perturbative superpotential must vanish in a large class of heterotic vacua because the contributions from curves in the same homology class cancel each other. We point out, however, that in certain cases the curves treated in the residue theorem as lying in the same homology class, can actually have different area with respect to the physical Kahler form and can be in different homology classes. In these cases, the residue theorem is not directly applicable and the structure of the superpotential is more subtle. We show, in a specific example, that the superpotential is non-zero both on \tilde{X} and on X. On the non-simply connected manifold X, we explicitly compute the leading contribution to the superpotential from all holomorphic, isolated, genus 0 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, hence, do not cancel each other.