  We consider non-perturbative superpotentials from world-sheet instantons
wrapped on holomorphic genus zero curves in heterotic string theory. These
superpotential contributions feature prominently in moduli stabilization and
large field axion inflation, which makes their presence or absence, as well as
their functional dependence on moduli, an important issue. We develop geometric
methods to compute the instanton superpotentials for heterotic string theory
with monad and extension bundles. Using our methods, we find a variety of
examples with a non-vanishing superpotential. In view of standard vanishing
theorems, we speculate that these results are likely to be attributed to the
non-compactness of the instanton moduli space. We test this proposal, for the
case of monad bundles, by considering gauged linear sigma models where
compactness of the instanton moduli space can be explicitly checked. In all
such cases, we find that the geometric results are consistent with the
vanishing theorems. Surprisingly, linearly dependent Pfaffians even arise for
cases with a non-compact instanton moduli space. This suggests some gauged
linear sigma models with a non-compact instanton moduli space may still have a
vanishing instanton superpotential.
