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The compactification from the eleven-dimensional Hov{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. Within the context of strongly coupled heterotic M-theory, a formalism for consistent hidden-sector bundles associated with a single line bundle is presented, and a specific line bundle is introduced as a concrete example. Anomaly cancellation and the associated bulk space five-branes are discussed in this context, as is the constraint that the hidden sector bundle be compatible with the slope-stability requirements of the observable sector $SU(4)$ gauge bundle. 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 the restrictions imposed by positive squared gauge couplings to order $kappa_{11}^{4/3}$ are presented in detail. Three additional constraints are imposed, one guaranteeing that the $S^{1}/{mathbb{Z}}_{2}$ orbifold length is sufficiently larger than the average Calabi-Yau radius, and two enforcing that the hidden sector be compatible with both the unification mass scale and unified gauge coupling of the $SO(10)$ 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 $N=1$ supersymmetry in the low-energy effective theory is discussed. It is shown that $N=1$ supersymmetry can be preserved by cancelling the tree-level and genus-one contributions against each another.
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Within the class of heterotic line bundle models, we argue that N=1 vacua which lead to a small number of low-energy chiral families are preferred. By imposing an upper limit on the volume of the internal manifold, as required in order to obtain finite values of the four-dimensional gauge couplings, and validity of the supergravity approximation we show that, for a given manifold, only a finite number of line bundle sums are consistent with supersymmetry. By explicitly scanning over this finite set of line bundle models on certain manifolds we show that, for a sufficiently small volume of the internal manifold, the family number distribution peaks at small values, consistent with three chiral families. The relation between the maximal number of families and the gauge coupling is discussed, which hints towards a possible explanation of the family problem.
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 $B-L$ 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 $E_{6} times SU(3)$ maximal subgroup of the hidden sector $E_{8}$ 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.
The compactification from the 11-dimensional Horava-Witten orbifold to 5-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 generic formalism for a consistent hidden sector gauge bundle, within the context of strongly coupled heterotic M-theory, is presented. Anomaly cancellation and the associated bulk space 5-branes are discussed in this context. The further compactification to a 4-dimensional effective field 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, the constraints imposed by the necessity for positive, perturbative squared gauge couplings to this order and the restrictions on the $D$-terms for preserving or spontaneously breaking $N=1$ supersymmetry are presented.
We present a novel string-derived $U(1)$ combination that satisfies necessary properties to survive to low scales. We discuss previous attempts at acquiring such an abelian gauge symmetry from two different string embeddings and the pitfalls associated with them. Finally, we give an example of how a satisfactory model may be constructed within our framework.
Grand unification groups (GUTs) are constructed from SO(32) heterotic string via $Z_{12-I}$ orbifold compactification. So far, most phenomenological studies from string compactification relied on $EE8$ heterotic string, and this invites the SO(32) heterotic string very useful for future phenomenological studies. Here, spontaneous symmetry breaking is achieved by Higgsing of the anti-symmetric tensor representations of SU($N$). The anti-SU($N$) presented in this paper is a completely different class from the flipped-SU($N$)s from the spinor representations of SO($2N$). Here, we realize chiral representations: $tsixoplus 5cdot ineb $ for a SU(9) GUT and $3{{ten}_Loplus {fiveb}_L}$ for a SU(5)$$ GUT. In particular, we confirm that the non-Abelian anomalies of SU(9) gauge group vanish and hence our compactification scheme achieves the key requirement. We also present the Yukawa couplings, in particular for the heaviest fermion, $t$, and lightest fermions, neutrinos. In the supersymmetric version, we present a scenario how supersymmetry can be broken dynamically via the confining gauge group SU(9). Three families in the visible sector are interpreted as the chiral spectra of SU(5)$$ GUT.
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