nLab de Sitter gravity

Idea

What is sometimes called de Sitter gravity and anti de Sitter gravity is the analog of Einstein gravity with the local model geometry of Minkowski spacetime acted on by the Poincaré group replaced by anti de Sitter spacetime or de Sitter spacetime acted on by their isometry groups (anti de Sitter group and de Sitter group, respectively).

More in detail, in the first order formulation of gravity the field configurations of Einstein gravity (general relativity) are Cartan connections for the inclusion of the Spin group into the Poincaré group. Accordingly a field configuration of anti de Sitter (resp. de Sitter) gravity is a Cartan connection for a suitable stabilizer subgroup included into $O(d-1,2)$ (resp. $O(d,1)$).

Examples

• In the case of 3-dimensional spacetime both de Sitter and anti de Sitter gravity have been studie in much detail. See at 3d quantum gravity for more.

Related concepts

• AdS-CFT correspondence

• Randall-Sundrum model

• Chern-Simons gravity

geometric context	gauge group	stabilizer subgroup	local model space	local geometry	global geometry	differential cohomology	first order formulation of gravity
differential geometry	Lie group/algebraic group $G$	subgroup (monomorphism) $H \hookrightarrow G$	quotient (“coset space”) $G/H$	Klein geometry	Cartan geometry	Cartan connection
examples	Euclidean group $Iso(d)$	rotation group $O(d)$	Cartesian space $\mathbb{R}^d$	Euclidean geometry	Riemannian geometry	affine connection	Euclidean gravity
	Poincaré group $Iso(d-1,1)$	Lorentz group $O(d-1,1)$	Minkowski spacetime $\mathbb{R}^{d-1,1}$	Lorentzian geometry	pseudo-Riemannian geometry	spin connection	Einstein gravity
	anti de Sitter group $O(d-1,2)$	$O(d-1,1)$	anti de Sitter spacetime $AdS^d$				AdS gravity
	de Sitter group $O(d,1)$	$O(d-1,1)$	de Sitter spacetime $dS^d$				deSitter gravity
	linear algebraic group	parabolic subgroup/Borel subgroup	flag variety	parabolic geometry
	conformal group $O(d,t+1)$	conformal parabolic subgroup	Möbius space $S^{d,t}$		conformal geometry	conformal connection	conformal gravity
supergeometry	super Lie group $G$	subgroup (monomorphism) $H \hookrightarrow G$	quotient (“coset space”) $G/H$	super Klein geometry	super Cartan geometry	Cartan superconnection
examples	super Poincaré group	spin group	super Minkowski spacetime $\mathbb{R}^{d-1,1\vert N}$	Lorentzian supergeometry	supergeometry	superconnection	supergravity
	super anti de Sitter group		super anti de Sitter spacetime
higher differential geometry	smooth 2-group $G$	2-monomorphism $H \to G$	homotopy quotient $G//H$	Klein 2-geometry	Cartan 2-geometry
	cohesive ∞-group	∞-monomorphism (i.e. any homomorphism) $H \to G$	homotopy quotient $G//H$ of ∞-action	higher Klein geometry	higher Cartan geometry	higher Cartan connection
examples			extended super Minkowski spacetime		extended supergeometry		higher supergravity: type II, heterotic, 11d

References

• Paul Townsend, Small Scale Structure of Space-Time as the Origin of the Gravitational Constant, Phys.Rev. D15 (1977) 2795 (spire)

• Takeshi Fukuyama, Gauge theory of gravity, Zeitschrift für Physik C Particles and Fields 1981, Volume 10, Issue 1, pp 9-15 (publisher)

• A. Achúcarro and Paul Townsend, A Chern-Simons Action for Three-Dimensional anti-De Sitter Supergravity Theories , Phys. Lett. B180 (1986) 89.

• Edward Witten, (2+1)-Dimensional Gravity as an Exactly Soluble System Nucl. Phys. B311 (1988) 46. (web)

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, volume 1, chapter I.4.4 of Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991)

• Derek Wise, MacDowell-Mansouri gravity and Cartan geometry, Class.Quant.Grav.27:155010,2010 (arXiv:gr-qc/0611154)

• Theo Verwimp, Anti-de Sitter gauge theory for gravity (arXiv:1006.1614)

• Wikipedia, de Sitter invariant special relativity

• M. Gunaydin, Peter van Nieuwenhuizen, N.P. Warner, General Construction of the Unitary Representations of Anti-de Sitter Superalgebras and the Spectrum of the $S^7$ Compactification of Eleven-dimensional Supergravity, Nucl.Phys. B255 (1985) 63 (spire)