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 string theory on Calabi-Yau manifolds (see at string phenomenology).

Specifically for discussion of string phenomenology obtaining or approximating the standard model of particle physics by this procedure see at G2-MSSM.


Vacuum solutions and the supergravity torsion constraints

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) already implies (together with the Bianchi identities) the equations of motion of supergravity, hence here the vacuum Einstein equations in the 11d spacetime.

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 (Achary 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

See also

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)

Survey and further discussion includes

  • Michael Duff, M-theory on manifolds of G2 holonomy: the first twenty years (arXiv:hep-th/0201062)

  • Sergei Gukov, M-theory on manifolds with exceptional holonomy, Fortschr. Phys. 51 (2003), 719–731 (pdf)

  • Bobby Acharya, M Theory, G 2G_2-manifolds and Four Dimensional Physics, Classical and Quantum Gravity Volume 19 Number 22, 2002 (pdf)

  • 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 G 2G_2 Holonomy (arXiv:hep-th/0612096)

  • James Halverson, David Morrison, The Landscape of M-theory Compactifications on Seven-Manifolds with G 2G_2 Holonomy (arXiv:1412.4123)

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

The hierarchy problem in the context of G 2G_2-compactifications is 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)

Revised on January 29, 2015 00:47:15 by Urs Schreiber (