black hole spacetimes | vanishing angular momentum | positive angular momentum |
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vanishing charge | Schwarzschild spacetime | Kerr spacetime |
positive charge | Reissner-Nordstrom spacetime | Kerr-Newman spacetime |
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.
The original article is
and early developments include:
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)
Francois Englert, Spontaneous Compactification of Eleven-Dimensional Supergravity Phys.Lett. B119 (1982) 339 (inSPIRE)
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)
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
now a popoular toy example for flux compactifications and moduli stabilization in string theory.
A detailed textbook account is in
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
Specifically, discussion of an ADE classification of 1/2 BPS-compactifications on $S^7/\Gamma$ for a finite group $\Gamma$ is in
Discussion of weak G2 holonomy on $X_7$ is in
Last revised on December 6, 2019 at 10:08:40. See the history of this page for a list of all contributions to it.