nLab Freund-Rubin compactification

Idea

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 DeWolfe et al. 2001 (it is considered 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 $X_7$ 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 $AdS_4$ (see also at super AdS spacetime).

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.

Details

Preliminaries

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

Notice that this means that a cosmological constant $\Lambda$ 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

$$R^{a_1 a_2} \;=\; \tfrac { 2\, \mathrm{R} } { D(D-1) } \, e^{a_1}\, e^{a_2} \,, \;\;\;\;\;\; \text{hence} \;\;\;\;\;\; R^{a_1 a_2}{}_{b_1 b_2} \;=\; \tfrac { 2 \, \mathrm{R} } { D (D-1) } \delta^{a_1 a_2}_{b_1 b_2} \,,$$

with Ricci curvature

$$\mathrm{Ric}_{a b} \;\equiv\; \eta_{a a'} R^{a' c}{}_{b c} \;=\; \tfrac { 2 \, \mathrm{R} } { D (D-1) } \, \eta_{a a'} \underset{ \tfrac{D-1}{2} \delta^{a'}_b }{ \underbrace{ \delta^{a' \, c}_{b \, c} } } \;=\; \tfrac { \mathrm{R} } { D } \, \eta_{a b}$$

and hence with scalar curvature

$$\mathrm{R} \;=\; \eta^{a b} \mathrm{Ric}_{a b} \,,$$

then its Einstein tensor is

$$\mathrm{G}_{a b} \;\equiv\; \mathrm{Ric}_{a b} - \tfrac{1}{2} \mathrm{R} \, \eta_{a b} \;=\; \big( \tfrac {1} {D} - \tfrac{1}{2} \big) \, \mathrm{R} \, \eta_{a b} \;=\; - \tfrac { D - 2 } { 2 D } \, \mathrm{R} \; \eta_{a b}$$

so that Einstein's equations equivalently say

$$\mathrm{G}_{a b} \;=\; \Lambda \, \eta_{a b} \;\;\;\;\;\;\;\; \Leftrightarrow \;\;\;\;\;\;\;\; \mathrm{R} \;=\; - \tfrac{2 D}{D-2} \, \Lambda \,,$$

implying

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

The general Freund-Rubin solution

We spell out the main argument due to Freund & Rubin 1980.

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

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

of

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

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

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

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

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

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

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

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

$$\begin{array}{l} G_{a b} \;=\; \Big( \big(\tfrac{1}{s} - \tfrac{1}{2}\big) \mathrm{R}_s - \tfrac{1}{2} \mathrm{R}_{D-2} \Big) \, \eta_{a b} \\ G_{i j} \;=\; \Big( -\tfrac{1}{2} \mathrm{R}_s + \big( \tfrac{1}{D-s} - \tfrac{1}{2} \big) \mathrm{R}_{D-2} \Big) \, \delta_{i j} \,. \end{array}$$

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

$$F_{a_1 \cdots a_s} \;\equiv\; f\, \epsilon_{a_1 \cdots a_s}$$

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

Then its energy-momentum tensor

(2)$$T_{\mu \nu} \;=\; \big( \tfrac{1}{2s} F_{\mu_1 \cdots \mu_s} F^{\mu_1 \cdots \mu_s} \, g_{\mu \nu} - F_{\mu \, \mu_1 \cdots \mu_{s-1}} F_{\nu}{}^{ \mu_1 \cdots \mu_{s-1} } \big)$$

has non-vanishing components

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

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

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

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

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

This is the result originally reported in Freund & Rubin 1980 (7) (the case where their $g_s$ is Lorentzian, so that $det(g_s)$ is negative — except that they seem to drop the joint factor of $(s-1)!$; but for $s=2$ 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. $X^{(s)}$ is an anti de Sitter spacetime $AdS_s$

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

Examples

The case of D=11 supergravity is

• $D = 11$

with the supergravity C-field flux density of degree

• $s = 4$.

In this case the above Freund-Rubin solution is

$$AdS_4 \times S^7 \,,$$

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

For more see also at AdS4/CFT3-duality.

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 2 (1980) 233-235 [doi:10.1016/0370-2693(80)90590-0]

Early developments:

• Francois Englert, Spontaneous Compactification of Eleven-Dimensional Supergravity, Phys. Lett. B 119 4-6 (1982) 339-342 [spire:180130, doi:10.1016/0370-2693(82)90684-0]

• Riccardo D'Auria, Pietro Fré: Spontaneous generation of symmetry in the spontaneous compactification of $D=11$ supergravity, Physics Letters B 121 2–3 (1983) 141-146 [doi:10.1016/0370-2693(83)90903-6]

• 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)

• Dmitri P. Sorokin, Vladimir I. Tkach, Dmitrij V. Volkov, On the relationship between compactified vacua of $D=11$ and $D=10$ supergravities, Phys. Lett. B 161 (1985) 301-306 [doi:10.1016/0370-2693(85)90766-X]

• Mike Duff, Bengt Nilsson, Christopher Pope, Kaluza-Klein supergravity, Physics Reports 130 1–2 (1986) 1-142 [spire:229417, doi:10.1016/0370-1573(86)90163-8]

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 popular toy example for flux compactifications and moduli stabilization in string theory.

On stability of Freund-Rubin compactifications:

• Oliver DeWolfe, Daniel Z. Freedman, Steven Gubser, Gary Horowitz, Indrajit Mitra: Stability of $AdS_p \times M^q$ Compactifications Without Supersymmetry, Phys. Rev. D 65 (2002) 064033 [doi:10.1103/PhysRevD.65.064033, arXiv:hep-th/0105047]

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

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, volume 2, chapters V.4-V.8 of Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991) [doi:10.1142/0224, ch V.4: pdf]

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, Frederik Denef, C. Hofman, Neil Lambert, Freund-Rubin Revisited [arXiv:hep-th/0308046, spire:625121]

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)