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In two spatial and one time dimensions, general relativity turns out to have no propagating gravitational degrees of freedom. In fact, it can be shown that in a vacuum, spacetime will always be locally flat (or de Sitter or anti-de Sitter depending upon the cosmological constant). This makes (2+1)-dimensional topological gravity (2+1D topological gravity) a topological theory with no gravitational local degrees of freedom.

Physicists became interested in the relation between Chern–Simons theory and gravity during the 1980s.[1] During this period, Edward Witten[2] argued that 2+1D topological gravity is equivalent to a Chern–Simons theory with the gauge group SO(2,2) for a negative cosmological constant, and SO(3,1) for a positive one. This theory can be exactly solved, making it a toy model for quantum gravity. The Killing form involves the Hodge dual.

Witten later changed his mind,[3] and argued that nonperturbatively 2+1D topological gravity differs from Chern–Simons because the functional measure is only over nonsingular vielbeins. He suggested the CFT dual is a Monster conformal field theory, and computed the entropy of BTZ black holes.


Achúcarro, A.; Townsend, P. (1986). "A Chern-Simons Action for Three-Dimensional anti-De Sitter Supergravity Theories". Physics Letters B180 (1–2): 89. doi:10.1016/0370-2693(86)90140-1.
Witten, Edward (19 Dec 1988). "(2+1)-Dimensional Gravity as an Exactly Soluble System". Nuclear Physics B. 311 (1): 46–78. Bibcode:1988NuPhB.311...46W. doi:10.1016/0550-3213(88)90143-5. hdl:10338.dmlcz/143077.url=http://srv2.fis.puc.cl/~mbanados/Cursos/TopicosRelatividadAvanzada/Witten2.pdf

Witten, Edward (22 June 2007). "Three-Dimensional Gravity Revisited". arXiv:0706.3359 [hep-th].


Quantum gravity
Central concepts

AdS/CFT correspondence Ryu-Takayanagi Conjecture Causal patch Gravitational anomaly Graviton Holographic principle IR/UV mixing Planck scale Quantum foam Trans-Planckian problem Weinberg–Witten theorem Faddeev-Popov ghost

Toy models

2+1D topological gravity CGHS model Jackiw–Teitelboim gravity Liouville gravity RST model Topological quantum field theory

Quantum field theory in curved spacetime

Bunch–Davies vacuum Hawking radiation Semiclassical gravity Unruh effect

Black holes

Black hole complementarity Black hole information paradox Black-hole thermodynamics Bousso's holographic bound ER=EPR Firewall (physics) Gravitational singularity

String theory

Bosonic string theory M-theory Supergravity Superstring theory

Canonical quantum gravity

Loop quantum gravity Wheeler–DeWitt equation

Euclidean quantum gravity

Hartle–Hawking state


Causal dynamical triangulation Causal sets Noncommutative geometry Spin foam Group field theory Superfluid vacuum theory Twistor theory Dual graviton


Quantum cosmology
Eternal inflation Multiverse FRW/CFT duality


Theories of gravitation
Newtonian gravity (NG)

Newton's law of universal gravitation Gauss's law for gravity Poisson's equation for gravity History of gravitational theory

General relativity (GR)

Introduction History Mathematics Exact solutions Resources Tests Post-Newtonian formalism Linearized gravity ADM formalism Gibbons–Hawking–York boundary term

Alternatives to
general relativity

Classical theories of gravitation Quantum gravity Theory of everything


Einstein–Cartan Bimetric theories Gauge theory gravity Teleparallelism Composite gravity f(R) gravity Infinite derivative gravity Massive gravity Modified Newtonian dynamics, MOND
AQUAL Tensor–vector–scalar Nonsymmetric gravitation Scalar–tensor theories
Brans–Dicke Scalar–tensor–vector Conformal gravity Scalar theories
Nordström Whitehead Geometrodynamics Induced gravity Chameleon Pressuron Degenerate Higher-Order Scalar-Tensor theories



Kaluza–Klein theory
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Generalisations /
extensions of GR

Liouville gravity Lovelock theory (2+1)-dimensional topological gravity Gauss–Bonnet gravity Jackiw–Teitelboim gravity

theories and
toy models

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Related topics


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