(2 1)-dimensional Topological Gravity
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. During this period, Edward Witten 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 In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a ...
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