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(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 (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.〔A. Achúcarro and P. Townsend, "A Chern-Simons Action for Three-Dimensional anti-De Sitter Supergravity Theories", ''Phys. Lett.'' B180 (1986) 89〕
During this period, Edward Witten〔url=http://srv2.fis.puc.cl/~mbanados/Cursos/TopicosRelatividadAvanzada/Witten2.pdf〕 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.
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.
== References ==
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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