(2+1)-dimensional topological gravity explained

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.

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.

Notes and References

Achúcarro . A. . Townsend . P. . 1986 . A Chern-Simons Action for Three-Dimensional anti-De Sitter Supergravity Theories . Phys. Lett. . B180 . 1–2. 89 . 10.1016/0370-2693(86)90140-1 . 1986PhLB..180...89A .
Witten . Edward . Edward Witten . 19 Dec 1988 . (2+1)-Dimensional Gravity as an Exactly Soluble System . . 311 . 1 . 46 - 78 . 10.1016/0550-3213(88)90143-5. 1988NuPhB.311...46W. 10338.dmlcz/143077 . free . url=http://srv2.fis.puc.cl/~mbanados/Cursos/TopicosRelatividadAvanzada/Witten2.pdf
Witten . Edward . Edward Witten . 22 June 2007 . Three-Dimensional Gravity Revisited . 0706.3359 . hep-th .