  The F-theory vacuum constructed from an elliptic Calabi-Yau threefold with
section yields an effective six-dimensional theory. The Lie algebra of the
gauge sector of this theory and its representation on the space of massless
hypermultiplets are shown to be determined by the intersection theory of the
homology of the Calabi-Yau threefold. (Similar statements hold for M-theory and
the type IIA string compactified on the threefold, where there is also a
dependence on the expectation values of the Ramond-Ramond fields.) We describe
general rules for computing the hypermultiplet spectrum of any F-theory vacuum,
including vacua with non-simply-laced gauge groups. The case of monodromy
acting on a curve of A_even singularities is shown to be particularly
interesting and leads to some unexpected rules for how 2-branes are allowed to
wrap certain 2-cycles. We also review the peculiar numerical predictions for
the geometry of elliptic Calabi-Yau threefolds with section which arise from
anomaly cancellation in six dimensions.
