Beyond the Standard Embedding in $M$-Theory on $S^{1}/Z_{2}$

In this paper we discuss compactifications of M-theory to four dimensions on X \times S^1/Z_2, in which nonstandard embeddings in the E_8 \times E_8 vacuum gauge bundle are considered. At the level of the effective field theory description of Horava and Witten, this provides a natural extension of well known results at weak coupling, to strongly coupled E_8 \times E_8 heterotic strings. As an application of our results, we discuss models which exhibit an anomalous U(1)_A symmetry in four dimensions, and show how this emerges from the reduction of the d = 11 toplogical term C \wedge G \wedge G, and how it is consistent with d = 4 anomaly cancellation in M-theory. As a further application of nonstandard embeddings, we show how it is possible to obtain an inverse hierarchy of gauge couplings, where the observable sector is more strongly coupled than the hidden one. The basic construction and phenomenological viability of these scenarios is demonstrated.