Calabi-Yau four-folds for M- and F-theory compactifications

We investigate topological properties of Calabi-Yau four-folds and consider a wide class of explicit constructions in weighted projective spaces and, more generally, toric varieties. Divisors which lead to a non-perturbative superpotential in the effective theory have a very simple description in the toric construction. Relevant properties of them follow just by counting lattice points and can also be used to construct examples with negative Euler number. We study nets of transitions between cases with generically smooth elliptic fibres and cases with ADE gauge symmetries in the N = 1 theory due to degenerations of the fibre over codimension one loci in the base. Finally we investigate the quantum cohomology ring of this four-folds using Frobenius algebras. (C) 1998 Elsevier Science B.V.