Worldsheet Instantons and Torsion Curves, Part A: Direct Computation

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.