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 implies (together with the Bianchi identities) the equations of motion of supergravity, hence here the vacuum Einstein equations in the 11d spacetime.
of the (flat) supergravity -field and the the 3-form of the -structure is the natural holomorphic coordinate on the moduli space of the KK-compactification of a -manifold, in M-theoretic higher analogy of the complexified Kähler classes of CY compactifications of 10d string theory (Harvey-Moore 99, (2.7), Acharya 02, (32) (59) (74), Grigorian-Yau 08, (4.57), Acharya-Bobkov 08, (4))
The KK-compactification of vaccuum 11-dimensional supergravity on a smooth G2-manifold results in a effective N=1 D=4 super Yang-Mills theory with abelian gauge group and with complex scalar fields which are neutral (not charges) under this gauge group (with the Betti numbers of ) (e.g. Acharya 02, section 2.3). This is of course unsuitable for phenomenology.
But when is a -orbifold then:
The first point is argued from the duality between M-theory compactified on K3 and heterotic string theory on a 3-torus. Here it is fairly well understood that at the degenertion points of the K3-moduli space enhanced nonabelian gauge symmetry appears. The degeneration points correspond to vanishing 2-cycles and the ideas is that therefore the M2-brane BPS charges corresponding to these cycles become massless and hence contribute further states of the 4d effective field theory. See at enhanced gauge symmetry.
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 (Acharya 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)
A survey of the corresponding string phenomenology is in
the hierarchy problem is discussed in
and realization of GUTs in