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

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)