Heterotic Instanton Superpotentials from Complete Intersection Calabi-Yau Manifolds

We study Pfaffians that appear in non-perturbative superpotential terms arising from worldsheet instantons in heterotic theories. A result by Beasley and Witten shows that these instanton contributions cancel among curves within a given homology class for Calabi-Yau manifolds that can be described as hypersurfaces or complete intersections in projective or toric ambient spaces. We provide a prescription that identifies all $\mathbb{P}^1$ curves in certain homology classes of complete intersection Calabi-Yau manifolds in products of projective spaces (CICYs) and cross-check our results by a comparison with the genus zero Gromov-Witten invariants. We then use this construction to study instanton superpotentials on those manifolds and their quotients. We identify a non-toric quotient of a non-favorable CICY with a single genus zero curve in a certain homology class, so that a cancellation \`a la Beasley-Witten is not possible. In another example, we study a non-toric quotient of a favorable CICY and check that the superpotential still vanishes. From this and related examples, we conjecture that the Beasley-Witten cancellation result can be extended to toric and non-toric quotients of CICYs, but can be avoided if the CICY is non-favorable.