nLab de Sitter spacetime

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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. Blau §39.1)

This admits different spatial slicings of relevance:

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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:

With focus on methods of conformal geometry:

Further discussion of the de Sitter group and its representation theory:

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:

In view of alpha' higher curvature corrections:

  • Ivano Basile, Alessia Platania, String Tension between de Sitter vacua and Curvature Corrections (arXiv:2103.06276)

    “We show that the resulting higher-derivative α\alpha'-corrections forbid the existence of de Sitter vacua, at least in regimes where string-loop corrections can be neglected.”

  • Liam McAllister, Jakob Moritz, Richard Nally, Andreas Schachner, Candidate de Sitter Vacua [arXiv:2406.13751]

    “Finding vacua that demonstrably survive subleading corrections, and in which the quantization conditions are completely understood, is an important open problem for which this work has prepared the foundations.”

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:

Discussion of asymptotic de Sitter spacetimes from time-dependent KK-compactification of type II supergravity:

reviewed in:

and further discussed in:

Last revised on July 18, 2024 at 11:42:14. See the history of this page for a list of all contributions to it.