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In this paper we explore contributions to non-perturbative superpotentialsarising from instantons wrapping effective divisors in smooth Calabi-Yaufour-folds. We concentrate on the case of manifolds constructed as completeintersections in products of projective spaces (CICYs) or generalizationsthereof (gCICYs). We systematically investigate the structure of the cone ofeffective (algebraic) divisors in the four-fold geometries and employ the sametools recently developed in arXiv:1507.03235 to construct more generalinstanton geometries than have previously been considered in the literature. Weprovide examples of instanton configurations on Calabi-Yau manifolds that areelliptically and $K3$-fibered and explore their consequences in the context ofstring dualities. The examples discussed include manifolds containing infinitefamilies of divisors with arithmetic genus, $\chi(D, \mathcal O_D)=1$ andsuperpotentials exhibiting modular symmetry.

On Instanton Superpotentials, Calabi-Yau Geometry, and Fibrations