Introduction to Model Building in Heterotic M-Theory

These lectures are an introduction to building semi-realistic models in heterotic M-theory and a continuation of those given by Burt Ovrut. First the phenomenological constraints on the compactification of supersymmetry, low-energy GUT groups and three families of charged matter are summarized. This involves a discussion of Calabi-Yau manifolds and, in particular, supersymmetric gauge field backgrounds as a special class of holomorphic bundles. We review the construction of supersymmetric bundles in the very simple example of a two-torus and show how this can be naturally understood in terms of the "Fourier-Mukai transform", which is really just the action of T-duality. We then review the structure of elliptically fibred Calabi-Yau manifolds X. Supersymmetric bundles can be constructed on X by extending the transform fibre by fibre. This is the spectral cover construction. Finally, we give an explicit example with GUT group SU(5) and three families of matter. Although discussed in the context of M-theory, these constructions work equally well for the heterotic string and have applications to D-brane physics.