(2+1)-dimensional topological gravity

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The relation between Chern–Simons theory and gravity, had not been interested so much the 1970s in physics, but It began to be very interested in the 1980s.[1] Edward Witten[2] 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.