Toric Calabi - Yau Fourfolds Duality Between N = 1 Theories and Divisors that Contribute to the Superpotential *

We study issues related to F -theory on Calabi-Yau fourfolds and its duality to heterotic theory for Calabi–Yau threefolds. We discuss principally fourfolds that are described by reflexive polyhedra and show how to read off some of the data for the heterotic theory from the polyhedron. We give a procedure for constructing examples with given gauge groups and describe some of these examples in detail. Interesting features arise when the local pieces are fitted into a global manifold. An important issue is how to compute the superpotential explicitly. Witten has shown that the condition for a divisor to contribute to the superpotential is that it have arithmetic genus 1. Divisors associated with the short roots of non-simply laced gauge groups do not always satisfy this condition while the divisors associated to all other roots do. For such a ‘dissident’ divisor we distinguish cases for which χ(OD) > 1 corresponding to an X that is not general in moduli (in the toric case this corresponds to the existence of non-toric parameters). In these cases the ‘dissident’ divisor D does not remain an effective divisor for general complex structure. If however χ(OD) ≤ 0, then the divisor is general in moduli and there is a genuine instability. * Supported in part by the Robert A. Welch Foundation, N.S.F. grants DMS9706707 and PHY-9511632.