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.[1]
During this period, Edward Witten[2] argued that 2+1D topological gravity is equivalent to a Chern–Simons theory with the gauge group for a negative cosmological constant, and 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.