nLab Freund-Rubin compactification

Contents

Context

Gravity

gravity, supergravity

Formalism

Definition

Einstein-Hilbert action, Einstein's equations, general relativity
- first-order formulation of gravity, D'Auria-Fre formulation of supergravity
- gravity as a BF-theory, Plebanski formulation of gravity, teleparallel gravity

Spacetime configurations

Properties

Spacetimes

black hole spacetimes	vanishing angular momentum	positive angular momentum
vanishing charge	Schwarzschild spacetime	Kerr spacetime
positive charge	Reissner-Nordstrom spacetime	Kerr-Newman spacetime

Quantum theory

quantum gravity
- graviton, gravitino
- string theory

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

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

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

a Lorentzian manifold $X^{(s)}$ , for $2 \leq s \leq D-2$ ,
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 the stress-energy tensor

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

\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) (stated there in slightly larger generality — 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.)

Related concepts

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)
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)