CALABI-YAU 4-FOLDS FOR M-THEORY AND F-THEORY COMPACTIFICATIONS

We investigate topological properties of Calabi-Yau four-folds and con sider a wide class of explicit constructions in weighted projective sp aces and, more generally, toric varieties. Divisors which lead to a no n-perturbative superpotential in the effective theory have a very simp le description in the toric construction. Relevant properties of them follow just by counting lattice points and can also be used to constru ct examples with negative Euler number. We study nets of transitions b etween cases with generically smooth elliptic fibres and cases with AD E gauge symmetries in the N = 1 theory due to degenerations of the fib re over codimension one loci in the base. Finally we investigate the q uantum cohomology ring of this four-folds using Frobenius algebras. (C ) 1998 Elsevier Science B.V.