We give a formalism for constructing hidden sector bundles as extensions of
sums of line bundles in heterotic M-theory. Although this construction is
generic, we present it within the context of the specific Schoen threefold that
leads to the physically realistic BβL MSSM model. We discuss the embedding of
the line bundles, the existence of the extension bundle, and a number of
necessary conditions for the resulting bundle to be slope-stable and thus N=1
supersymmetric. An explicit example is presented, where two line bundles are
embedded into the SU(3) factor of the E6βΓSU(3) maximal subgroup
of the hidden sector E8β gauge group, and then enhanced to a non-Abelian
SU(3) bundle by extension. For this example, there are in fact six
inequivalent extension branches, significantly generalizing that space of
solutions compared with hidden sectors constructed from a single line bundle.Comment: 51 pages, 5 figure