  It has recently been realised that polystable, holomorphic sums of line
bundles over smooth Calabi-Yau three-folds provide a fertile ground for
heterotic model building. Large numbers of phenomenologically promising such
models have been constructed for various classes of Calabi-Yau manifolds. In
this paper we focus on a case study for the tetra-quadric - a Calabi-Yau
hypersurface embedded in a product of four CP1 spaces. We address the question
of finiteness of the class of consistent and physically viable line bundle
models constructed on this manifold. Further, for a specific semi-realistic
example, we explore the embedding of the line bundle sum into the larger moduli
space of non-Abelian bundles, both by means of constructing specific polystable
non-Abelian bundles and by turning on VEVs in the associated low-energy theory.
In this context, we explore the fate of the Higgs doublets as we move in bundle
moduli space. The non-Abelian compactifications thus constructed lead to SU(5)
GUT models with an additional global B-L symmetry. The non-Abelian
compactifications inherit many of the appealing phenomenological features of
the Abelian model, such as the absence of dimension four and dimension five
operators triggering fast proton decay.
