  The non-perturbative superpotential generated by a heterotic superstring
wrapped once around a genus-zero holomorphic curve is proportional to the
Pfaffian involving the determinant of a Dirac operator on this curve. We show
that the space of zero modes of this Dirac operator is the kernel of a linear
mapping that is dependent on the associated vector bundle moduli. By explicitly
computing the determinant of this map, one can deduce whether or not the
dimension of the space of zero modes vanishes. It is shown that this
information is sufficient to completely determine the Pfaffian and, hence, the
non-perturbative superpotential as explicit holomorphic functions of the vector
bundle moduli. This method is illustrated by a number of non-trivial examples.
