Despite much recent progress in model building with D-branes, it has been problematic to find a completely convincing explanation of gauge coupling unification. We extend the class of models by considering F -theory compactifications, which may incorporate unification more naturally. We explain how to derive the charged chiral spectrum and Yukawa couplings in N = 1 compactifications of F -theory with G-flux. In a class of models which admit perturbative heterotic duals, we show that the F -theory and heterotic computations match.

The Coulomb branch of N = 2 supersymmetric gauge theories in four dimensions is described in general by an integrable Hamiltonian system in the holomorphic sense. A natural construction of such systems comes from two-dimensional gauge theory and spectral curves. Starting from this point of view, we propose an integrable system relevant to the N = 2 SU (n) gauge theory with a hypermultiplet in the adjoint representation, and offer much evidence that it is correct. The model has an SL(2, Z) S-duality group (with the central element −1 of SL(2, Z) acting as charge conjugation); SL(2, Z) permutes the Higgs, confining, and oblique confining phases in the expected fashion. We also study more exotic phases.

We consider the possibility of breaking the GUT group to the Standard Model gauge group in F -theory compactifications by turning on certain U(1) fluxes. We show that the requirement of massless hypercharge is equivalent to a topological constraint on the UV completion of the local model. The possibility of this mechanism is intrinsic to F -theory. We address some of the phenomenological signatures of this scenario. We show that our models predict monopoles as in conventional GUT models. We discuss in detail the leading threshold corrections to the gauge kinetic terms and their effect on unification. They turn out to be related to Ray-Singer torsion. We also discuss the issue of proton decay in Ftheory models and explain how to engineer models which satisfy current experimental bounds.

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Techniques are presented for computing the cohomology of stable, holomorphic vector bundles over elliptically fibered Calabi-Yau threefolds. These cohomology groups explicitly determine the spectrum of the low energy, four-dimensional theory. Generic points in vector bundle moduli space manifest an identical spectrum. However, it is shown that on subsets of moduli space of co-dimension one or higher, the spectrum can abruptly jump to many different values. Both analytic and numerical data illustrating this phenomenon are presented. This result opens the possibility of tunneling or phase transitions between different particle spectra in the same heterotic compactification.In the course of this discussion, a classification of SU (5) GUT theories within a specific context is presented. *