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 S O ( 2 , 2 ) for a negative cosmological constant, and S O ( 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.