Types of quantum field thories
The Kaluza-Klein reduction of 11-dimensional supergravity on G2 manifolds (notably Freund-Rubin compactifications and variants) yields an effective 4-dimensional supergravity. This construction is the lift to M-theory of the KK-compactification of string theory on Calabi-Yau manifolds (see at string phenomenology).
In order for this to yield phenomenologically interesting effective physics the compactification space must be an orbifold (hence an orbifold of special holonomy), its stabilizer groups will encode the nonabelian gauge group of the effective theory by “geometric engineering of quantum field theory” (Acharya 98, Atiyah-Witten 01, section 6). Specifically for discussion of string phenomenology obtaining or approximating the standard model of particle physics by this procedure see at G2-MSSM.
Genuine G2-manifold/orbifold fibers, these having vanishing Ricci curvature, correspond to vacuum solutions of the Einstein equations of 11d supergravity, i.e. with vanishing field strength of the gravitino and the supergravity C-field (see e.g. Acharya 02, p. 9). (If one includes non-vanishing -field strength one finds “weak -holonomy” instead, see below).
Notice that vanishing gravitino field strength (i.e. covariant derivative) means that the torsion of the super-vielbein is in each tangent space the canonical torsion of the super Minkowski spacetime. This torsion constraint already just for the bosonic part of the super-vielbein already implies (together with the Bianchi identities) the equations of motion of supergravity, hence here the vacuum Einstein equations in the 11d spacetime.
In compactifications with weak G2 holonomy it is the defining 4-form (the one which for strict G2 manifolds is the Hodge dual of the associative 3-form) which is the flux/field strength of the supergravity C-field. See for instance (Bilal-Serendinger-Sfetos 02, section 6):
Consider a KK-compactification-Ansatz and
For realistic field content after Kaluza-Klein compactification one needs to consider not smooth (weak) G2-manifolds but conical singularities and orbifolds of these. see the first page of (Acharya-Denef-Hofman-Lambert) for discussion of phenomenology for such orbifold models and further pointers and see (Achary 98) for general discussion of orbifolds with -structure.
Leonardo Castellani, Riccardo D'Auria, Pietro Fré, chapter V.6 of Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991)
Discussion of Freund-Rubin compactification on “with flux”, hence non-vanishing supergravity C-field and how they preserve one supersymmetry if is of weak G2 holonomy with = cosmological constant = C-field strength on is in
Thomas House, Andrei Micu, M-theory Compactifications on Manifolds with Structure (arXiv:hep-th/0412006)
Survey and further discussion includes
Adil Belhaj, M-theory on G2 manifolds and the method of (p, q) brane webs (2004) (web)
Adam B. Barrett, M-Theory on Manifolds with Holonomy (arXiv:hep-th/0612096)
The hierarchy problem in the context of -compactifications is discussed in
A survey of the corresponding string phenomenology is in