the non-Abelian gauge groups of the low-energy theory. Scalars transforming as triplets of $SU(2)$ simply do not exist for such affine level one constructions [@Gross:1985fr; @Font:1990uw]. Indeed, scalars transforming in the adjoint representation appear only at affine level two, while representations such as the $\mathbf{120}$ and $\mathbf{210}$ of $SO(10)$ appear only at affine level four - and the $\mathbf{126}$ of $SO(10)$, which contains triplets of the Standard Model $SU(2)_L$, has been shown to never appear in free-field heterotic string constructions [@Dienes:1996yh].

Directly constructing four-dimensional string compactifications yielding higher affine-level gauge groups has proven to be a difficult task. But a group factor $\mathcal{G}$ can be effectively realized at affine level $k = n$ by simply requiring it to be the result of a breaking of $\mathcal{G}_1 \times
\mathcal{G}_2 \times \cdots \times \mathcal{G}_n$ to the overall diagonal subgroup. In fact, these two ways of understanding higher affine levels – picking a particular set of GSO projections in the underlying string construction and the low energy field theory picture of breaking to a diagonal subgroup – are equivalent pictures [@Dienes:1996yh]. With this as motivation, let us consider an appropriate variation on the superpotential of (\[Wnu\]). The breaking of the gauge group $SU(2)_A 