(2+1)-dimensional topological gravity

In two spatial and one time dimensions, <a href="/facts/General_relativity/jVsoIg8X">general relativity</a> has no propagating gravitational degrees of freedom. In fact, in a vacuum, spacetime will always be locally flat (or <a href="/facts/De_Sitter_space/gwIURvVf">de Sitter</a> or <a href="/facts/Anti-de_Sitter_space/xoDubsIe">anti-de Sitter</a> depending upon the <a href="/facts/Cosmological_constant/bjMksSA2">cosmological constant</a>). This makes (2+1)-dimensional topological gravity (2+1D topological gravity) a <a href="/facts/Topological_quantum_field_theory/zHjrp3b8">topological theory</a> 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, <a href="/facts/Edward_Witten/lPUylPmj">Edward Witten</a> argued that 2+1D topological gravity is equivalent to a <a href="/facts/Chern%25E2%2580%2593Simons_theory/AQ3Kayn3">Chern–Simons theory</a> with the <a href="/facts/Gauge_group/fSW3gggv">gauge group</a> 
 
 
 
 S
 O
 (
 2
 ,
 2
 )
 
 
 {\displaystyle SO(2,2)}
 
 for a negative cosmological constant, and 
 
 
 
 S
 O
 (
 3
 ,
 1
 )
 
 
 {\displaystyle SO(3,1)}
 
 for a positive one. This theory can be exactly <a href="/facts/Solvable_group/Na7lKIns">solved</a>, making it a <a href="/facts/Toy_model/xbSCc9qA">toy model</a> for <a href="/facts/Quantum_gravity/t7whBMNE">quantum gravity</a>. The <a href="/facts/Killing_form/u4HxRh5o">Killing form</a> involves the <a href="/facts/Hodge_dual/AH6tmHhU">Hodge dual</a>.
Witten later changed his mind, and argued that nonperturbatively 2+1D topological gravity differs from Chern–Simons because the <a href="/facts/Functional_integration/VBGTnDzq">functional measure</a> is only over nonsingular <a href="/facts/Vielbein/jyEhSaHM">vielbeins</a>. He suggested the <a href="/facts/AdS%2fCFT_correspondence/8Ide9C9m">CFT dual</a> is a <a href="/facts/Monster_group/MlwslQ7H">monster conformal field theory</a>, and computed the entropy of <a href="/facts/BTZ_black_hole/9u2gYja8">BTZ black holes</a>.

(2+1)-dimensional topological gravity open-in-new

(2+1)-dimensional topological gravity