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. During this period, Edward Witten argued that 2+1D topological gravity is equivalent to a

Chern–Simons theory The Chern–Simons theory is a 3-dimensional topological quantum field theory of Schwarz type developed by Edward Witten. It was discovered first by mathematical physicist Albert Schwarz. It is named after mathematicians Shiing-Shen Chern and Jam ...

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 Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics; it deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the vi ...

. The

Killing form In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. Cartan's criteria (criterion of solvability and criterion of semisimplicity) show ...

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 Functional may refer to: * Movements in architecture: ** Functionalism (architecture) ** Form follows function * Functional group, combination of atoms within molecules * Medical conditions without currently visible organic basis: ** Functional sy ...

is only over nonsingular

vielbein The tetrad formalism is an approach to general relativity that generalizes the choice of basis for the tangent bundle from a coordinate basis to the less restrictive choice of a local basis, i.e. a locally defined set of four linearly independe ...

s. He suggested the CFT dual is a monster conformal field theory, and computed the entropy of BTZ black holes.

References

Quantum gravity {{relativity-stub