(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 a topological theory with no gravitational local degrees of freedom.

Edward Witten has argued this 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, which can be exactly solved, making this 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.