M-theory on G2-manifolds


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The Kaluza-Klein reduction of 11-dimensional supergravity on G2 manifolds (notably Freund-Rubin compactifications and variants) yields an effective N=1N=1 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.

KK-compactifications of higher dimensional supergravity with minimal (N=1N=1) supersymmetry:

perspectiveKK-compactification with N=1N=1 supersymmetry
M-theoryM-theory on G2-manifolds
F-theoryF-theory on CY4-manifolds
heterotic string theoryheterotic 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.


Vacuum solutions

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 CC-field strength one finds “weak G 2G_2-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 (E a)(E^a) of the super-vielbein (E a,E α)(E^a, E^\alpha) implies (together with the Bianchi identities) the equations of motion of supergravity, hence here the vacuum Einstein equations in the 11d spacetime.

Complexified moduli space

For vanishing field strength of the supergravity C-field, the formal linear combination

τC 3+iϕ 3 \tau \coloneqq C_3 + i \phi_3

of the (flat) supergravity CC-field C 3C_3 and the the 3-form ϕ 3\phi_3 of the G 2G_2-structure is the natural holomorphic coordinate on the moduli space of the KK-compactification of a G 2G_2-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)).

Notice that restricted to associative submanifolds this combination becomes C 3+ivolC_3 + i vol, which also governs the membrane instanton-contributions (“complex volume”).

Nonabelian gauge groups and chiral fermions at orbifold singularities

The KK-compactification of vacuum 11-dimensional supergravity on a smooth G2-manifold YY results in a effective N=1 D=4 super Yang-Mills theory with abelian gauge group U(1) b 2(Y)U(1)^{b_2(Y)} and with b 3(Y)b_3(Y) complex scalar fields which are neutral (not charged) under this gauge group (with b (Y)b_\bullet(Y) the Betti numbers of YY) (e.g. Acharya 02, section 2.3). This is of course unsuitable for phenomenology.

But when YY is a G 2G_2-orbifold then:

  1. at an ADE singularity there is enhanced gauge symmetry in that the gauge group (which a priori is copies of the abelian group U(1)U(1) 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));

  2. at a (non-orbifold) conifold singularity chiral fermions appear spring (Witten 01, p. 3, Atiyah-Witten 01, Acharya-Witten 01).

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:

  1. 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).

  2. 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 2/Γ\mathbb{C}^2/\Gamma generically constitutes a point in the moduli space of vacua in the Higgs branch of a super Yang-Mills theory.

  3. ADE 2Cycle

    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.

  4. 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 SU(N)SU(N) gauge symmetry by the usual string theory argument about Chan-Paton gauge fields.

Also notice that at least the SU(N)SU(N)-enhancement of the effective field theory at k\mathbb{Z}_k-singularities matches the SU(N)SU(N)-enhancement of the worldvolume theory of NN-coincident M2-branes sitting at the orbifold singularity: this is the statement of the ABJM model.

Solutions with non-vanishing CC-field strength

In compactifications with weak G2 holonomy it is the defining 4-form ϕ 4\phi_4 (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 X 11=(X 4,g 4)×(X 7,g 7)X_{11} = (X_4,g_4) \times (X_7,g_7) and

  • F 4=fvol X 4F_4 = f vol_{X_4};

  • F 7=g˜e 7 *ϕ 4F_7 = \tilde g e_7^\ast \phi_4

where e 4e_4, e 7e_7 are given vielbein fields on X 4X_4 and X 7X_7 and ϕ 4\phi_4 is the Hodge dual of the associative 3-form. Then the Einstein equations of 11-dimensional supergravity give

R 4=13(f 2+72g˜ 2)g 4 R_4 = - \frac{1}{3}\left(f^2 + \frac{7}{2} \tilde g^2\right) g_4
R 7=16(f 2+5g˜ 2)g 7 R_7 = \frac{1}{6}\left(f^2 + 5 \tilde g^2\right) g_7

(where g 4g_4, g 7g_7 is the metric tensor) saying that both spaces are Einstein manifolds (BSS 02, (5.4)). The equations of motion for the supergravity C-field is

g˜(dϕfϕ) \tilde g\left( d \phi - f \star\phi \right)

for ϕ=e 7 *ϕ 3\phi = e_7^\ast \phi_3 the pullback of the associative 3-form (BSS 02, (5.5)), saying that ϕF 7\phi \propto \star F_7 exhibits weak G2-holonomy with weakness parameter given by the component of the C-field on X 4X_4.


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 G 2G_2 models and further pointers and see (Acharya 98) for general discussion of orbifolds with G 2G_2-structure.


The KK-compactification of 11d supergravity of fibers of special holonomy was originally considered in

Specifically string phenomenology for the case of compactification on G2-manifolds (or rather orbifolds ) goes back to

Detailed discussion is in

See also

More discussion of the non-abelian gauge group enhancement at orbifold singularities includes

Discussion of Freund-Rubin compactification on 4×X 7\mathbb{R}^4 \times X_7 “with flux”, hence non-vanishing supergravity C-field and how they preserve one supersymmetry if X 7X_7 is of weak G2 holonomy with λ\lambda = cosmological constant = C-field strength on 4\mathbb{R}^4 is in

  • Adel Bilal, J.-P. Derendinger, K. Sfetsos, (Weak) G 2G_2 Holonomy from Self-duality, Flux and Supersymmetry, Nucl.Phys. B628 (2002) 112-132 (arXiv:hep-th/0111274)

  • Thomas House, Andrei Micu, M-theory Compactifications on Manifolds with G 2G_2 Structure (arXiv:hep-th/0412006)

Further discussion of membrane instantons in this context includes

Survey and further discussion includes

The corresponding membrane instanton corrections to the superpotential? are discussed in

A survey of the corresponding string phenomenology is in

  • Bobby Acharya, G 2G_2-manifolds at the CERN Large Hadron collider and in the Galaxy, talk at G 2G_2-days (2012) (pdf)

the hierarchy problem is discussed in

the moduli space and moduli stabilization is discussed in

the strong CP problem is discussed in

and realization of GUTs in

Revised on November 25, 2015 15:32:22 by Urs Schreiber (