nLab Freund-Rubin compactification




String theory



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 7X_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 4AdS_4.

If X 7X_7 has weak G2 holonomy with weakness parameter/cosmological constant λ\lambda the scale of the flux, then this yields N=1N = 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 P 3\mathbb{C}P^3 (Nilsson-Pope 84, ABJM 08)

If X 7=S 7/G ADEX_7 = S^7/G_{ADE} is an orbifold of the round 7-sphere by an finite group G ADESU(2)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

Early developments:

Identification as near horizon geometries of black M2-branes:

See also

A classification of symmetric solutions is discussed in

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.

Textbook account (in D'Auria-Fré formulation):

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

Discussion of the case that X 7X_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/ΓS^7/\Gamma for a finite group Γ\Gamma is in

Discussion of weak G2 holonomy on X 7X_7 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)

See also:

Last revised on February 19, 2024 at 17:33:08. See the history of this page for a list of all contributions to it.