nLab
de Sitter spacetime

Contents

Contents

Definition

Full de Sitter spacetime is the Lorentzian manifold which is the coset space of Lorentz groups

dS dO(d,1)/O(d1,1) dS^d \simeq O(d,1)/O(d-1,1)

This models a cosmology FRW model for positive cosmological constant/dark energy.

Similarly anti de Sitter spacetime is

adS dO(d1,2)/O(d1,1) adS^d \simeq O(d-1,2)/O(d-1,1)

(e.g here)

This admits different spatial slicings of relevance:

graphics grabbed from this Physics.SE comment

geometric contextgauge groupstabilizer subgrouplocal model spacelocal geometryglobal geometrydifferential cohomologyfirst order formulation of gravity
differential geometryLie group/algebraic group GGsubgroup (monomorphism) HGH \hookrightarrow Gquotient (“coset space”) G/HG/HKlein geometryCartan geometryCartan connection
examplesEuclidean group Iso(d)Iso(d)rotation group O(d)O(d)Cartesian space d\mathbb{R}^dEuclidean geometryRiemannian geometryaffine connectionEuclidean gravity
Poincaré group Iso(d1,1)Iso(d-1,1)Lorentz group O(d1,1)O(d-1,1)Minkowski spacetime d1,1\mathbb{R}^{d-1,1}Lorentzian geometrypseudo-Riemannian geometryspin connectionEinstein gravity
anti de Sitter group O(d1,2)O(d-1,2)O(d1,1)O(d-1,1)anti de Sitter spacetime AdS dAdS^dAdS gravity
de Sitter group O(d,1)O(d,1)O(d1,1)O(d-1,1)de Sitter spacetime dS ddS^ddeSitter gravity
linear algebraic groupparabolic subgroup/Borel subgroupflag varietyparabolic geometry
conformal group O(d,t+1)O(d,t+1)conformal parabolic subgroupMöbius space S d,tS^{d,t}conformal geometryconformal connectionconformal gravity
supergeometrysuper Lie group GGsubgroup (monomorphism) HGH \hookrightarrow Gquotient (“coset space”) G/HG/Hsuper Klein geometrysuper Cartan geometryCartan superconnection
examplessuper Poincaré groupspin groupsuper Minkowski spacetime d1,1|N\mathbb{R}^{d-1,1\vert N}Lorentzian supergeometrysupergeometrysuperconnectionsupergravity
super anti de Sitter groupsuper anti de Sitter spacetime
higher differential geometrysmooth 2-group GG2-monomorphism HGH \to Ghomotopy quotient G//HG//HKlein 2-geometryCartan 2-geometry
cohesive ∞-group∞-monomorphism (i.e. any homomorphism) HGH \to Ghomotopy quotient G//HG//H of ∞-actionhigher Klein geometryhigher Cartan geometryhigher Cartan connection
examplesextended super Minkowski spacetimeextended supergeometryhigher supergravity: type II, heterotic, 11d

References

In classical gravity

See also

In perturbative quantum gravity

Discussion of de Sitter spacetime in perturbative quantum gravity and its infrared instability, includes the following:

In string/M-theory

A no-go theorem for KK-compactifications of supergravity down to de Sitter spacetime is claimed in

  • Juan Maldacena, Carlos Nunez, Supergravity description of field theories on curved manifolds and a no go theorem, Int.J.Mod.Phys. A16 (2001) 822-855 (arXiv:hep-th/0007018)

Discussion of de Sitter perturbative string theory vacua and their role in the landscape of string theory vacua or else in the swampland:

On de Sitter spacetime cosmology realized in brane world models in ambient \simAdS-bulk spacetime:

Discussion in the context of M-theory on G2-manifolds:

For analogous discussion of embedding of the ekpyrotic cosmology-scenario into string theory/type IIA supergravity:

Last revised on September 8, 2019 at 09:58:20. See the history of this page for a list of all contributions to it.