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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.

References

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].

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Quantum gravity
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