nLab Freund-Rubin compactification




String theory



In general, a Freund-Rubin compactification [Freund & Rubin 1980] is a Kaluza-Klein compactification of a theory of gravity coupled to (higher) gauge fields with flux (field strength) on the compact fiber spaces such that the result is stable (a basic 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 (see also at super AdS spacetime).

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.



We work in the convention where a round n n -sphere has negative scalar curvature (following Freund & Rubin 1980, below (4b), cf. this Example).

Notice that this means that a cosmological constant Λ\Lambda appears with a positive sign on the right hand side of Einstein's equations:

For if we have an Einstein-spacetime, hence with Riemann curvature of the form

R a 1a 2=2RD(D1)e a 1e a 2,henceR a 1a 2 b 1b 2=2RD(D1)δ b 1b 2 a 1a 2, R^{a_1 a_2} \;=\; \tfrac { 2\, \mathrm{R} } { D(D-1) } \, e^{a_1}\, e^{a_2} \,, \;\;\;\;\;\; \text{hence} \;\;\;\;\;\; R^{a_1 a_2}{}_{b_1 b_2} \;=\; \tfrac { 2 \, \mathrm{R} } { D (D-1) } \delta^{a_1 a_2}_{b_1 b_2} \,,

with Ricci curvature

Ric abη aaR ac bc=2RD(D1)η aaδ bc acD12δ b a=RDη ab \mathrm{Ric}_{a b} \;\equiv\; \eta_{a a'} R^{a' c}{}_{b c} \;=\; \tfrac { 2 \, \mathrm{R} } { D (D-1) } \, \eta_{a a'} \underset{ \tfrac{D-1}{2} \delta^{a'}_b }{ \underbrace{ \delta^{a' \, c}_{b \, c} } } \;=\; \tfrac { \mathrm{R} } { D } \, \eta_{a b}

and hence with scalar curvature

R=η abRic ab, \mathrm{R} \;=\; \eta^{a b} \mathrm{Ric}_{a b} \,,

then its Einstein tensor is

G abRic ab12Rη ab=(1D12)Rη ab=D22DRη ab \mathrm{G}_{a b} \;\equiv\; \mathrm{Ric}_{a b} - \tfrac{1}{2} \mathrm{R} \, \eta_{a b} \;=\; \big( \tfrac {1} {D} - \tfrac{1}{2} \big) \, \mathrm{R} \, \eta_{a b} \;=\; - \tfrac { D - 2 } { 2 D } \, \mathrm{R} \; \eta_{a b}

so that Einstein's equations equivalently say

G ab=Λη abR=2DD2Λ, \mathrm{G}_{a b} \;=\; \Lambda \, \eta_{a b} \;\;\;\;\;\;\;\; \Leftrightarrow \;\;\;\;\;\;\;\; \mathrm{R} \;=\; - \tfrac{2 D}{D-2} \, \Lambda \,,


(1)Λ>0 R<0 de Sitter spacetime Λ<0 R>0 anti de Sitter spacetime \begin{array}{ccccr} \Lambda \gt 0 &\Leftrightarrow& \mathrm{R} \lt 0 &\Leftrightarrow& \text{de Sitter spacetime} \\ \Lambda \lt 0 &\Leftrightarrow& \mathrm{R} \gt 0 &\Leftrightarrow& \text{anti de Sitter spacetime} \end{array}

The general Freund-Rubin solution

Consider a D3D \geq 3-dimensional spacetime which is the product

X (D)=X (s)×Y (Ds) X^{(D)} \;=\; X^{(s)} \times Y^{(D-s)}


  1. a Lorentzian manifold X (s)X^{(s)}, for 2sD22 \leq s \leq D-2,

  2. a Riemannian manifold Y (Ds)Y^{(D-s)},

and assume that both factors are Einstein manifolds by themselves, in that their Ricci tensors are of the form

Ric ab=R ssη ab Ric ij=R DsDsδ ij. \begin{array}{l} Ric_{a b} \;=\; \tfrac {\mathrm{R}_s} {s} \, \eta_{a b} \\ Ric_{i j} \;=\; \tfrac {\mathrm{R}_{D-s}} {D-s} \, \delta_{i j} \,. \end{array}

for R s,R Ds\mathrm{R}_s, \mathrm{R}_{D-s} \,\in\, \mathbb{R}, to be determined.

Then the total scalar curvature Rg μνRic μν\mathrm{R} \equiv g^{\mu \nu} Ric_{\mu \nu} is

R=R s+R Ds, \mathrm{R} \;=\; \mathrm{R}_s + \mathrm{R}_{D-s} \,,

and the non-vanishing components of the Einstein tensor GRic12RgG \equiv Ric - \tfrac{1}{2}\mathrm{R}g are

G ab=((1s12)R s12R D2)η ab G ij=(12R s+(1Ds12)R D2)δ ij. \begin{array}{l} G_{a b} \;=\; \Big( \big(\tfrac{1}{s} - \tfrac{1}{2}\big) \mathrm{R}_s - \tfrac{1}{2} \mathrm{R}_{D-2} \Big) \, \eta_{a b} \\ G_{i j} \;=\; \Big( -\tfrac{1}{2} \mathrm{R}_s + \big( \tfrac{1}{D-s} - \tfrac{1}{2} \big) \mathrm{R}_{D-2} \Big) \, \delta_{i j} \,. \end{array}

Next assume that the “matter” content is that of a higher gauge field with degree-ss flux density homogeneously extended over X (s)X^{(s)}:

F a 1a sfϵ a 1a s F_{a_1 \cdots a_s} \;\equiv\; f\, \epsilon_{a_1 \cdots a_s}

for some ff \in \mathbb{R}, and all other components vanishing.

Then its energy-momentum tensor

T μν=(12sF μ 1μ sF μ 1μ sg μνF μμ 1μ s1F ν μ 1μ s1) T_{\mu \nu} \;=\; \big( \tfrac{1}{2s} F_{\mu_1 \cdots \mu_s} F^{\mu_1 \cdots \mu_s} \, g_{\mu \nu} - F_{\mu \, \mu_1 \cdots \mu_{s-1}} F_{\nu}{}^{ \mu_1 \cdots \mu_{s-1} } \big)

has non-vanishing components

T ab=f 2((s1)!s!2s)+(s1)!2η ab T ij=f 2(s!2s)(s1)!2η ij. \begin{array}{l} T_{a b} \;=\; f^2 \underset{ +\,\tfrac{(s-1)!}{2} }{ \underbrace{ \big( (s-1)! - \tfrac{s!}{2s} \big) } } \, \eta_{a b} \\ T_{i j} \;=\; f^2 \underset{ -\,\tfrac{(s-1)!}{2} }{ \big( \underbrace{ - \tfrac{s!}{2s} } \big) } \, \eta_{i j} \mathrlap{\,.} \end{array}

Therefore the Einstein equation G=TG \;=\; T says in this case that

(1s12)R s12R Ds = +(s1)!2f 2 12R s+(1Ds12)R Ds = (s1)!2f 2. \begin{array}{rcl} \big( \tfrac{1}{s} - \tfrac{1}{2} \big) \mathrm{R}_s - \tfrac{1}{2}\mathrm{R}_{D-s} &=& + \tfrac{(s-1)!}{2}\, f^2 \\ -\tfrac{1}{2} \mathrm{R}_s + \big( \tfrac{1}{D-s} - \tfrac{1}{2} \big) \mathrm{R}_{D-s} &=& - \tfrac{(s-1)!}{2} \, f^2 \mathrlap{\,.} \end{array}

The unique solution to this system of linear equations for R s\mathrm{R}_s, R Ds\mathrm{R}_{D-s} is (see WolframAlpha here)

R s = +s(Ds1)D2(s1)!f 2 R Ds = (s1)(Ds)D2(s1)!f 2. \begin{array}{rcl} \mathrm{R}_s &=& + \, \frac { s (D - s - 1) } { D - 2 } \, (s-1)! \, f^2 \\ \mathrm{R}_{D-s} &=& - \, \frac { (s-1)(D-s) } { D-2 } \, (s-1)! \, f^2 \mathrlap{\,.} \end{array}

This is the result originally reported in Freund & Rubin 1980 (7) (the case where their g sg_s is Lorentzian, so that det(g s)det(g_s) is negative — except that they seem to drop the joint factor of (s1)!(s-1)!; but for s=2s=2 the factor disappears and we get their equation (4) on the nose.)

By comparison with (1) this means that for the maximally extended solution

  1. X (s)X^{(s)} is an anti de Sitter spacetime AdS sAdS_s

  2. X (Ds)X^{(D-s)} is a sphere S DsS^{D-s}.


The case of D=11 supergravity is

  • D=11D = 11

with the supergravity C-field flux density of degree

  • s=4s = 4.

In this case the above Freund-Rubin solution is

AdS 4×S 7, AdS_4 \times S^7 \,,

which may be understood as the near horizon geometry of black M2-branes, see there.

For more see also at AdS4/CFT3-duality.


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 May 17, 2024 at 13:01:03. See the history of this page for a list of all contributions to it.