2 + 1 dimensional gravity as an exactly soluble system

In the citing paper

“One can consider nontrivial spatial or spacetime topologies [277, 279]. Such “cosmological” solutions have moduli — a finite number of parameters that distinguish among geometrically inequivalent constant curvature manifolds — and these can become dynamical.”

In the citing paper

“Fortunately, the description is closely related to the more familiar first-order Chern-Simons formalism [108, 277, 279, 2], which, in turn, can connect us back to the standard metric formalism.”

In the citing paper

“As first noted by Achúcarro and Townsend [2] and further developed by Witten [277, 279], the first-order action (14) is equivalent to that of a Chern-Simons theory.”

In the citing paper

“The path integral for this action was first fully analyzed in two seminal papers by Witten [277, 279], who showed that it reduced to a ratio of determinants that has an elegant topological interpretation as the analytic or Ray-Singer torsion [230].”

In the citing paper

“this form is unique up to multiplication by a constant factor and defines a dual pairing between so (2 , 1) and R 2+1 .”

In the citing paper

“Witten's paper [2] contains the result that the Einstein equation R ij = 0 can be regarded as the Euler-Lagrange equations for the Chern-Simons form 〈 A,dA + 2 3 [ A,A ] 〉 where A is an ISO (2 , 1) connection and 〈 , 〉 is the invariant bilinear form on the Lie algebra iso (2 , 1) for which both the radical R 2+1 and any Levi factor so (2 , 1) are isotropic”

In the citing paper

“The degeneracy plays a crucial role in the calculation [27,28] of black hole entropy using a canonical partition function approach.”