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 heterotic string theory on Calabi-Yau manifolds (see at string phenomenology), and of F-theory on CY4-manifolds.
|perspective||KK-compactification with supersymmetry|
|M-theory||M-theory on G2-manifolds|
|F-theory||F-theory on CY4-manifolds|
|heterotic string theory||heterotic string theory on CY3-manifolds|
In order for this to yield phenomenologically interesting effective physics the compactification space must be a G2-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), see below. 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 vacuum 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 charged) 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:
at an ADE singularity there is enhanced gauge symmetry in that the gauge group (which a priori is copies of the abelian group of the supergravity C-field) becomes nonabelian (Acharya 98, Acharya 00, review includes Acharya 02, section 3, BBS 07, p. 422, 436, Ibáñez-Uranga 12, section 6.3.3, Wijnholt 14, part III (from which the graphics below is grabbed));
The conifold singularities are supposed/assumed to be is isolated (Witten 01, section 2), while the ADE singularities are supposed/assumed to be of codimension-4 in the 7-dimensional fibers (Witten 01, section 3, Barrett 06).
In the absence of a proper microscopic definition of M-theory, the first point is argued for indirectly in at least these ways:
The fact that under KK-compactification to type IIA string theory the singularity becomes special points of intersecting D6-branes for which the gauge enhancement is known (Witten 01, p. 1, based on Cvetic-Shiu-Uranga 01).
The duality between M-theory compactified on K3 and heterotic string theory on a 3-torus (Acharya-Witten 01). Here it is fairly well understood that at the degenertion points of the K3-moduli space enhanced nonabelian gauge symmetry appears. This comes down (Intriligator-Seiberg 96) to the fact that an ADE singularity generically constitutes a point in the moduli space of vacua in the Higgs branch of a super Yang-Mills theory.
The blow-up of an ADE-singularity happens to be a union of 2-spheres touching pairwise in one point, such as to form the Dynkin diagram of the simple Lie group which under the ADE classification corresponds to the given orbifold isotropy group. M2-branes may wrap these 2-cycles and since before blow-up they are of vanishing size, this corresponds to double dimensional reduction under which the M2-branes become strings stretching between coincident D-branes. These are well-understood to be the quanta of nonabelian gauge Chan-Paton gauge fields located on the D-branes, and hence these same nonabelian degrees of freedom have had to be present already at the level of the M2-branes.
In the F-theory description the ADE singularity maps to the locus where the F-theory elliptic fibration degenerates with 2-cycles in the elliptic fibers shrinking to 0. Via double dimensional reduction this manifestly takes the M2-brane wrapping these elliptic fibers to an open string stretching between D7-branes. This yields at least gauge symmetry by the usual string theory argument about Chan-Paton gauge fields.
Also notice that at least the -enhancement of the effective field theory at -singularities matches the -enhancement of the worldvolume theory of -coincident M2-branes sitting at the orbifold singularity: this is the statement of the ABJM model.
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)
Detailed discussion is in
More discussion of the non-abelian gauge group enhancement at orbifold singularities includes
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)
Further discussion of membrane instantons in this context includes
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
Spiro Karigiannis, Naichung Conan Leung_, Hodge Theory for G2-manifolds: Intermediate Jacobians and Abel-Jacobi maps, Proceedings of the London Mathematical Society (3) 99, 297-325 (2009) (arXiv:0709.2987 talk slides pdf
the strong CP problem is discussed in
and realization of GUTs in