  We show that certain superpotential and Kahler potential couplings of N=1
supersymmetric compactifications with branes or bundles can be computed from
Hodge theory and mirror symmetry. This applies to F-theory on a Calabi-Yau
four-fold and three-fold compactifications of type II and heterotic strings
with branes. The heterotic case includes a class of bundles on elliptic
manifolds constructed by Friedmann, Morgan and Witten. Mirror symmetry of the
four-fold computes non-perturbative corrections to mirror symmetry on the
three-folds, including D-instanton corrections. We also propose a physical
interpretation for the observation by Warner that relates the deformation
spaces of certain matrix factorizations and the periods of non-compact 4-folds
that are ALE fibrations.
