  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.
