nLab Freund-Rubin compactification

Idea

In general a Freund-Rubin compactification (Freund-Rubin 80) is a Kaluza-Klein compactification of a theory of gravity coupled to gauge fields or more generally higher gauge fields with flux (field strength) on the compact fiber spaces such that the result is stable (an example of moduli stabilization via flux compactification).

One example are Kaluza-Klein compactifications of 6d Einstein-Maxwell theory with magnetic flux on a 2-dimensional fiber space (sphere or torus) (RDSS 83). This serves these days as a toy example for flux compactifications and moduli stabilization in string theory.

In the string theory literature often the Freund-Rubin compactification refers by default to a Kaluza-Klein compactification of 11-dimensional supergravity on a manifold $X_7$ of dimension 7 (in the original model a round 7-sphere) with non-vanishing constant 4-form field strength (“flux”) of the supergravity C-field in the remaining four dimensional anti-de Sitter spacetimes $AdS_4$.

If $X_7$ has weak G2 holonomy with weakness parameter/cosmological constant $\lambda$ the scale of the flux, then this yields $N = 1$ supersymmetry in the effective QFT in four dimensions, discussed at M-theory on G2-manifolds. The KK-reduction on the circle fiber of these solutions to type IIA supergravity yields type IIA sugra on complex projective space $\mathbb{C}P^3$ (Nilsson-Pope 84, ABJM 08)

If $X_7 = S^7/G_{ADE}$ is an orbifold of the round 7-sphere by an finite group $G_{ADE} \subset SU(2)$ in the ADE-classification, then Freund-Rubin describes the near horizon geometry of coincident black M2-branes at an ADE-singularity, see at M2-brane – As a black brane.

Related concepts

• Kaluza-Klein mechanism

• M-theory on G2-manifolds, G2-MSSM

• AdS-CFT

References

The original article is

• Peter Freund, M. A. Rubin, Dynamics Of Dimensional Reduction, Phys. Lett. B 97, 233 (1980). (doi:10.1016/0370-2693(80)90590-0)

Early developments:

• Francois Englert, Spontaneous Compactification of Eleven-Dimensional Supergravity, Phys. Lett. B119 (1982) 339 (spire:180130)

• Riccardo D'Auria, Pietro Fre, Peter van Nieuwenhuizen, Symmetry Breaking in $d=11$ Supergravity on the Parallelized Seven Sphere, Phys. Lett. B 122 (1983) 225 (spire:181634, doi:10.1016/0370-2693(83)90689-5)

Identification as near horizon geometries of black M2-branes:

• Don Page, Classical stability of round and squashed seven-spheres in eleven-dimensional supergravity, Phys. Rev. D 28, 2976 (1983) (spire:14480 doi:10.1103/PhysRevD.28.2976)

• Mike Duff, Kellogg Stelle, Multi-membrane solutions of $D = 11$ supergravity, Phys. Lett. B 253, 113 (1991) (spire:299386, doi:10.1016/0370-2693(91)91371-2)

See also

• Hermann Nicolai, Krzysztof Pilch, Consistent truncation of $d = 11$ supergravity on $AdS_4 \times S^7$, JHEP 03 (2012) 099 (arXiv:1112.6131)

A classification of symmetric solutions is discussed in

• José Figueroa-O'Farrill, Symmetric M-Theory Backgrounds (arXiv:1112.4967)

• Linus Wulff, All symmetric space solutions of eleven-dimensional supergravity (arXiv:1611.06139)

The class of Freund-Rubin compactifications of 6d Einstein-Maxwell theory down to 4d is due to

• S. Randjbar-Daemi, Abdus Salam and J. A. Strathdee, Spontaneous Compactification In Six-Dimensional Einstein-Maxwell Theory, Nucl. Phys. B 214, 491 (1983) (spire)

now a popoular toy example for flux compactifications and moduli stabilization in string theory.

A detailed textbook account is in

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, volume 2, chapters V.4-V.8 of Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991)

Discussion of compactification along the fibration $S^1 \to S^7 \to \mathbb{C}P^3$ is in

• Bengt Nilsson, Christopher Pope, Hopf Fibration of Eleven-dimensional Supergravity, Class.Quant.Grav. 1 (1984) 499 (Spire)

• Ofer Aharony, Oren Bergman, Daniel Louis Jafferis, Juan Maldacena, $N=6$ superconformal Chern-Simons-matter theories, M2-branes and their gravity duals (arXiv:0806.1218, ABJM model)

Discussion of the case that $X_7$ is an orbifold or has other singularities (the case of interest for realistic phenomenology in M-theory on G2-manifolds) includes

• Bobby Acharya, F. Denef, C. Hofman, Neil Lambert, Freund-Rubin Revisited (arXiv:hep-th/0308046)

Specifically, discussion of an ADE classification of 1/2 BPS-compactifications on $S^7/\Gamma$ for a finite group $\Gamma$ is in

• Paul de Medeiros, José Figueroa-O'Farrill, Sunil Gadhia, Elena Méndez-Escobar, Half-BPS quotients in M-theory: ADE with a twist, JHEP 0910:038,2009 (arXiv:0909.0163, pdf slides)

Discussion of weak G2 holonomy on $X_7$ 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)

See also:

• Ergin Sezgin, 11D Supergravity on $AdS_4 \times S^7$ versus $AdS_7 \times S^4$ (arXiv:2003.01135)