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In two spatial and one time dimensions, general relativity has no propagating gravitational degrees of freedom. In fact, 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. During this period, Edward Witten argued that 2+1D topological gravity is equivalent to a Chern–Simons theory with the gauge group ${\displaystyle SO(2,2)}$ for a negative cosmological constant, and ${\displaystyle SO(3,1)}$ 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, 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.

References

1. Achúcarro, A.; Townsend, P. (1986). "A Chern-Simons Action for Three-Dimensional anti-De Sitter Supergravity Theories". Phys. Lett. B180 (1–2): 89. Bibcode:1986PhLB..180...89A. doi:10.1016/0370-2693(86)90140-1.
2. 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
3. Witten, Edward (22 June 2007). "Three-Dimensional Gravity Revisited". arXiv:0706.3359 .

v t e Quantum gravity
Central concepts	AdS/CFT correspondence Batalin–Vilkovisky formalism CA-duality Causal patch Faddeev–Popov ghost Gravitational anomaly Graviton Holographic principle IR/UV mixing Planck units Quantum foam Ryu–Takayanagi conjecture Trans-Planckian problem Weinberg–Witten theorem
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 Bekenstein bound Bousso's holographic bound Cosmic censorship hypothesis ER = EPR Firewall (physics) Gravitational singularity
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Applications	Quantum cosmology Eternal inflation FRW/CFT duality Multiverse
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