Divisors on elliptic Calabi-Yau 4-folds and the superpotential in F-theory, I

Each smooth elliptic Calabi-Yau 4-fold determines both a three-dimensional physical theory (a compactification of "M-theory") and a four-dimensional p hysical theory (using the "F-theory" construction). A key issue in both the ories is the calculation of the "superpotential" of the theory, which by a result of Witten is determined by the divisors D on the 4-fold satisfying c hi(O-D) = 1. Fire propose a systematic approach to identify these divisors, and derive some criteria to determine whether a given divisor indeed contr ibutes. We then apply our techniques in explicit examples, in particular, w hen the base B of the elliptic fibration is a toric variety or a Fano 3-fol d. When B is Fano, we show how divisors contributing to the superpotential are always "exceptional" (in some sense) for the Calabi-Yau 4-fold X. This nat urally leads to certain transitions of X, i.e., birational transformations to a singular model (where the image of D no longer contributes) as well as certain smoothings of the singular model. The singularities which occur ar e "canonical", the same type of singularities of a (singular) Weierstrass m odel. We work out the transitions. If a smoothing exists, then the Hedge nu mbers change. We speculate that divisors contributing to the superpotential are always "e xceptional" (in some sense) for X, also in M-theory. In fact we show that t his is a consequence of the (log)-minimal model algorithm in dimension 4, w hich is still conjectural in its generality, but it has been worked out in various cases, among which are toric varieties. (C) 1998 Elsevier Science B .V.