Types of quantum field thories
The Kaluza-Klein reduction of 11-dimensional supergravity on G2 manifolds 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.
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 towards the end of (Bilal-Serendinger-Sfetos) for a derivation.
For realistic field content after Kaluza-Klein compactification one needs to consider not smooth (weak) G2-manifolds but 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.
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)
Adil Belhaj, M-theory on G2 manifolds and the method of (p, q) brane webs (2004) (web)
The hierarchy problem in the context of -compactifications is discussed in
A survey of the corresponding string phenomenology is in