(2+1)-dimensional topological gravity

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

1. A. Achúcarro and P. Townsend, "A Chern-Simons Action for Three-Dimensional anti-De Sitter Supergravity Theories", Phys. Lett. B180 (1986) 89
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.url=http://srv2.fis.puc.cl/~mbanados/Cursos/TopicosRelatividadAvanzada/Witten2.pdf
3. Witten, Edward (22 June 2007). "Three-Dimensional Gravity Revisited". arXiv:0706.3359.