Line Bundle Hidden Sectors for Strongly Coupled Heterotic Standard Models.

The compactification from the eleven-dimensional Ho\v{r}ava-Witten orbifold to five-dimensional heterotic M-theory on a Schoen Calabi-Yau threefold is reviewed, as is the specific $SU(4)$ vector bundle leading to the "heterotic standard model" in the observable sector. A formalism for consistent hidden-sector line bundles, within the context of strongly coupled heterotic M-theory, is presented and a specific line bundle is introduced. Anomaly cancellation and the associated bulk space five-branes are discussed in this context. The further compactification to a four-dimensional effective theory on a linearized BPS double domain wall is then presented to order $\kappa_{11}^{4/3}$. Specifically, the generic constraints required for anomaly cancellation and by the linearized domain wall solution, as well as restrictions imposed by the necessity to have positive squared gauge couplings to order $\kappa_{11}^{4/3}$, are presented in detail. Four additional constraints are imposed, two guaranteeing that the orbifold interval is sufficiently large and that the effective strong coupling parameter has an acceptable value, and two enforcing that the hidden sector be compatible with both the mass scale and gauge coupling of the $SO(10)$ unification group in the observable sector. Finally, the expression for the Fayet-Iliopoulos term associated with an anomalous $U(1)$ symmetry is presented and its role in $N=1$ spontaneous supersymmetry breaking in the low-energy effective theory is discussed. It is shown that $N=1$ supersymmetry can be preserved, although at the cost of giving a mass to the $U(1)$ vector superfield in addition to its usual anomalous mass.