  As a first step towards studying vector bundle moduli in realistic heterotic
compactifications, we identify all holomorphic rational curves in a Calabi-Yau
threefold X with Z_3 x Z_3 Wilson lines. Computing the homology, we find that
H_2(X,Z)=Z^3+Z_3+Z_3. The torsion curves complicate our analysis, and we
develop techniques to distinguish the torsion part of curve classes and to deal
with the non-toric threefold X. In this paper, we use direct A-model
computations to find the instanton numbers in each integral homology class,
including torsion. One interesting result is that there are homology classes
that contain only a single instanton, ensuring that there cannot be any
unwanted cancellation in the non-perturbative superpotential.
