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The Kaluza-Klein reduction of 11-dimensional supergravity on G2 manifolds (notably Freund-Rubin compactifications and variants) yields an effective $N=1$ 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 ($N=1$) supersymmetry:
perspective | KK-compactification with $N=1$ 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 $C$-field strength one finds “weak $G_2$-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 $(E^a)$ of the super-vielbein $(E^a, E^\alpha)$ implies (together with the Bianchi identities) the equations of motion of supergravity, hence here the vacuum Einstein equations in the 11d spacetime.
For vanishing field strength of the supergravity C-field, the formal linear combination
of the (flat) supergravity $C$-field $C_3$ and the the 3-form $\phi_3$ of the $G_2$-structure is the natural holomorphic coordinate on the moduli space of the KK-compactification of a $G_2$-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 $C_3 + i vol$, which also governs the membrane instanton-contributions (“complex volume”).
The KK-compactification of vacuum 11-dimensional supergravity on a smooth G2-manifold $Y$ results in a effective N=1 D=4 super Yang-Mills theory with abelian gauge group $U(1)^{b_2(Y)}$ and with $b_3(Y)$ complex scalar fields which are neutral (not charged) under this gauge group (with $b_\bullet(Y)$ the Betti numbers of $Y$) (e.g. Acharya 02, section 2.3). This is of course unsuitable for phenomenology.
But when $Y$ is a $G_2$-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 $U(1)$ 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));
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:
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 $\mathbb{C}^2/\Gamma$ 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 $SU(N)$ gauge symmetry by the usual string theory argument about Chan-Paton gauge fields.
Also notice that at least the $SU(N)$-enhancement of the effective field theory at $\mathbb{Z}_k$-singularities matches the $SU(N)$-enhancement of the worldvolume theory of $N$-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 $\phi_4$ (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 $X_{11} = (X_4,g_4) \times (X_7,g_7)$ and
$F_4 = f vol_{X_4}$;
$F_7 = \tilde g e_7^\ast \phi_4$
where $e_4$, $e_7$ are given vielbein fields on $X_4$ and $X_7$ and $\phi_4$ is the Hodge dual of the associative 3-form. Then the Einstein equations of 11-dimensional supergravity give
(where $g_4$, $g_7$ 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
for $\phi = e_7^\ast \phi_3$ the pullback of the associative 3-form (BSS 02, (5.5)), saying that $\phi \propto \star F_7$ exhibits weak G2-holonomy with weakness parameter given by the component of the C-field on $X_4$.
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 $G_2$ models and further pointers and see (Acharya 98) for general discussion of orbifolds with $G_2$-structure.
The KK-compactification of 11d supergravity of fibers of special holonomy was originally considered in
Leonardo Castellani, Riccardo D'Auria, Pietro Fré, chapter V.6 of Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991)
George Papadopoulos, Paul Townsend, Compactification of D=11 supergravity on spaces of exceptional holonomy, Phys.Lett.B357:300-306,1995 (arXiv:hep-th/9506150)
Specifically string phenomenology for the case of compactification on G2-manifolds (or rather orbifolds ) goes back to
Bobby Acharya, M theory, Joyce Orbifolds and Super Yang-Mills, Adv.Theor.Math.Phys. 3 (1999) 227-248 (arXiv:hep-th/9812205)
Bobby Acharya, On Realising $N=1$ Super Yang-Mills in M theory (arXiv:hep-th/0011089)
Michael Atiyah, Edward Witten $M$-Theory dynamics on a manifold of $G_2$-holonomy, Adv. Theor. Math. Phys. 6 (2001) (arXiv:hep-th/0107177)
Edward Witten, Anomaly Cancellation On Manifolds Of $G_2$ Holonomy (arXiv:hep-th/0108165)
Bobby Acharya, Edward Witten, Chiral Fermions from Manifolds of $G_2$ Holonomy (arXiv:hep-th/0109152)
Detailed discussion is in
See also
Mirjam Cvetic, Gary Gibbons, H. Lü and Christopher Pope, Supersymmetric M3-branes and G2 Manifolds (pdf)
Bobby Acharya, F. Denef, C. Hofman, Neil Lambert, Freund-Rubin Revisited (arXiv:hep-th/0308046)
More discussion of the non-abelian gauge group enhancement at orbifold singularities includes
Mirjam Cvetic, Gary Shiu, Angel Uranga, Chiral Four-Dimensional $N=1$ Supersymmetric Type IIA Orientifolds from Intersecting D6-Branes, Nucl.Phys.B615:3-32,2001 (arXiv:hep-th/0107166)
James Halverson, David Morrison, On Gauge Enhancement and Singular Limits in $G_2$ Compactifications of M-theory (arXiv:1507.05965)
Antonella Grassi, James Halverson, Julius L. Shaneson, Matter From Geometry Without Resolution, Journal of High Energy Physics October 2013, 2013:205 (arXiv:1306.1832)
Discussion of Freund-Rubin compactification on $\mathbb{R}^4 \times X_7$ “with flux”, hence non-vanishing supergravity C-field and how they preserve one supersymmetry if $X_7$ is of weak G2 holonomy with $\lambda$ = cosmological constant = C-field strength on $\mathbb{R}^4$ is in
Adel Bilal, J.-P. Derendinger, K. Sfetsos, (Weak) $G_2$ 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 $G_2$ Structure (arXiv:hep-th/0412006)
Further discussion of membrane instantons in this context includes
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, $G_2$-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 $G_2$ Holonomy (arXiv:hep-th/0612096)
James Halverson, David Morrison, The Landscape of M-theory Compactifications on Seven-Manifolds with $G_2$ Holonomy (arXiv:1412.4123)
The corresponding membrane instanton corrections to the superpotential? are discussed in
Jeffrey Harvey, Greg Moore, Superpotentials and Membrane Instantons (arXiv:hep-th/9907026)
Katrin Becker, Melanie Becker, John Schwarz, p. 333 of String Theory and M-Theory: A Modern Introduction, 2007
A survey of the corresponding string phenomenology is in
the hierarchy problem is discussed in
the moduli space and moduli stabilization is discussed in
Bobby Acharya, A Moduli Fixing Mechanism in M theory (arXiv:hep-th/0212294)
Bobby Acharya, Konstantin Bobkov, Gordon Kane, Piyush Kumar, Jing Shao, Explaining the Electroweak Scale and Stabilizing Moduli in M Theory, Phys.Rev.D76:126010,2007 (arXiv:hep-th/0701034)
Sergey Grigorian, Shing-Tung Yau, Local geometry of the $G_2$ moduli space, Commun.Math.Phys.287:459-488,2009 (arXiv:0802.0723)
Bobby Acharya, Konstantin Bobkov, Kähler Independence of the G2-MSSM, HEP 1009:001,2010 (arXiv:0810.3285)
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
Peter Svrcek, Edward Witten, section 6 of Axions In String Theory, JHEP 0606:051,2006 (arXiv:hep-th/0605206)
Bobby Acharya, Konstantin Bobkov, Piyush Kumar, An M Theory Solution to the Strong CP Problem and Constraints on the Axiverse, JHEP 1011:105,2010 (arXiv:1004.5138)
and realization of GUTs in
Edward Witten, Deconstruction, $G_2$ Holonomy, and Doublet-Triplet Splitting, (arXiv:hep-ph/0201018)
Bobby Acharya, Krzysztof Bozek, Miguel Crispim Romao, Stephen F. King, Chakrit Pongkitivanichkul, $SO(10)$ Grand Unification in M theory on a $G_2$ manifold (arXiv:1502.01727)