We study aspects of Calabi-Yau four-folds as compactification manifold
s of F-theory, using mirror symmetry of toric hypersurfaces. Correlati
on functions of the topological field theory are determined directly i
n terms of a natural ring structure of divisors and the period integra
ls, and subsequently used to extract invariants of moduli spaces of ra
tional curves subject to certain conditions. We then turn to the discu
ssion of physical properties of the space-time theories, for a number
of examples which are dual to E-8 X E-8 heterotic N = 1 theories. Non-
critical strings of various kinds, with low tension for special values
of the moduli, lead to interesting physical effects. We give a comple
te classification of those divisors in toric manifolds that contribute
to the non-perturbative four-dimensional superpotential; the physical
singularities associated to it are related to the appearance of tensi
onless strings'. In some cases non-perturbative effects generate an ev
erywhere non-zero quantum tension leading to a combination of a conven
tional field theory with light strings hiding at a low energy scale re
lated to supersymmetry breaking. (C) 1997 Elsevier Science B.V.