  F-theory admits 7-branes with exceptional gauge symmetries, which can be
compactified to give phenomenological four-dimensional GUT models. Here we
study general supersymmetric compactifications of eight-dimensional Yang-Mills
theory. They are mathematically described by meromorphic Higgs bundles, and
therefore admit a spectral cover description. This allows us to give a rigorous
and intrinsic construction of local models in F-theory. We use our results to
prove a no-go theorem showing that local SU(5) models with three generations do
not exist for generic moduli. However we show that three-generation models do
exist on the Noether-Lefschetz locus. We explain how F-theory models can be
mapped to non-perturbative orientifold models using a scaling limit proposed by
Sen.
  Further we address the construction of global models that do not have
heterotic duals. We show how one may obtain a contractible worldvolume with a
two-cycle not inherited from the bulk, a necessary condition for implementing
GUT breaking using fluxes. We also show that the complex structure moduli in
global models can be arranged so that no dimension four or five proton decay
can be generated.
