Asymptotic symmetries in 3D gravity with torsion

Abstract

We study the nature of asymptotic symmetries in topological 3d gravity with torsion. After introducing the concept of asymptotically anti-de Sitter configuration, we find that the canonical realization of the asymptotic symmetry is characterized by the Virasoro algebra with classical central charge, the value of which is the same as in general relativity: c = 3ℓ/2G.

Three-dimensional gravity with torsion as a Chern-Simons gauge theory

“…For A = 2/ℓ, and consequently B = 0, this solution reduces to the teleparalell black hole [14]. General solution of the MB model possessing maximal number of global symmetries defines the Riemann-Cartan AdS solution, AdS 3 .…”

“…This equation can be considered as an equivalent of the second equation in (2.3). There are two interesting special cases of the general MB model: a) for α 3 = α 4 = 0, the Riemann-Cartan theory leads to Riemannian geometry (A = 0), in the context of which the Chern-Simons structure of gravity has been first discovered [3]; b) for α 3 = (α 4 ) 2 + aΛ = 0, the vacuum geometry becomes teleparallel (B = 0), but the related Chern-Simons structure remains the same as in Riemannian case [14,15].…”

“…In Riemann-Cartan geometry one can use a well known identity to express the curvature 2-form R ij in terms of its Riemannian pieceR ij and the contortion [14], whereupon the second field equation in (2.3) can be transformed into an equivalent form,…”

“…[14,15]: the Latin indices (i, j, k, ...) refer to the local Lorentz frame, the Greek indices (µ, ν, ρ, ...) refer to the coordinate frame, and both run over 0, 1, 2; η ij = (+, −, −) and g µν = b i µ b j ν η ij are metric components in the local Lorentz and coordinate frame, respectively; totally antisymmetric tensor ε ijk and the related tensor density ε µνρ are both normalized by ε 012 = +1.…”

“…The model generalizes general relativity with a cosmological constant (GR Λ ) to a topological gravity in Riemann-Cartan spacetime [13]. A specific version of the MB model, characterized by the teleparallel geometry of spacetime without matter, has been recently used to study the influence of torsion on the asymptotic structure of gravity [14,15]. The results show that both GR Λ and the teleparallel 3D gravity have identical asymptotic structures (Chern-Simons formulation, conformal symmetry and Liouville dynamics at the boundary), at least at the classical level.…”