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It has recently been realised that polystable, holomorphic sums of linebundles over smooth Calabi-Yau three-folds provide a fertile ground forheterotic model building. Large numbers of phenomenologically promising suchmodels have been constructed for various classes of Calabi-Yau manifolds. Inthis paper we focus on a case study for the tetra-quadric - a Calabi-Yauhypersurface embedded in a product of four CP1 spaces. We address the questionof finiteness of the class of consistent and physically viable line bundlemodels constructed on this manifold. Further, for a specific semi-realisticexample, we explore the embedding of the line bundle sum into the larger modulispace of non-Abelian bundles, both by means of constructing specific polystablenon-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 bundlemoduli space. The non-Abelian compactifications thus constructed lead to SU(5)GUT models with an additional global B-L symmetry. The non-Abeliancompactifications inherit many of the appealing phenomenological features ofthe Abelian model, such as the absence of dimension four and dimension fiveoperators triggering fast proton decay.

The Moduli Space of Heterotic Line Bundle Models: a Case Study for the Tetra-Quadric