F-theory on Calabi-Yau fourfolds

We discuss some aspects of F-theory in four dimensions on elliptically fibered Calabi-Yau fourfolds which are Calabi-Yau threefold fibrations, A particularly simple class of such manifolds emerges for fourfolds in which the generic Calabi-Yau threefold fiber is itself an elliptic fibration and is K3 fibered. Duality between F-theory compactified on Calabi-Yau fourfolds and heterotic strings on Calabi-Yau threefolds puts constraints on the cohomology of the fourfold. By computing the Hedge diamond of Calabi-Yau fourfolds we provide first numerical evidence for F-theory dualities in four dimensions.