Abstract
We discuss an enhancement of the Brown-Henneaux boundary conditions in three-dimensional AdS General Relativity to encompass Weyl transformations of the boundary metric. The resulting asymptotic symmetry algebra, after a field-dependent redefinition of the generators, is a direct sum of two copies of the Witt algebra and the Weyl abelian sector. The charges associated to Weyl transformations are non-vanishing, integrable but not conserved due to a flux driven by the Weyl anomaly coefficient. The charge algebra admits an additional non-trivial central extension in the Weyl sector, related to the well-known Weyl anomaly. We then construct the holographic Weyl current and show that it satisfies an anomalous Ward-Takahashi identity of the boundary theory.
Highlights
Three-dimensional general relativity is one of the simplest gravitational systems [1,2] and, in particular, solutions with negative cosmological constant (AdS3) have received special attention, due to their holographic nature [3,4]
We compute the asymptotic Killing vectors preserving these choices and their algebra. We show that the latter comprises, besides the usual left and right Witt sectors, a new Abelian sector corresponding to Weyl rescalings of the boundary metric
The boundary Weyl symmetry is broken, for the bulk counterpart Weyl charges are not conserved. This process is driven by the anomaly coefficient: for flat boundary metrics the current is conserved [30], as we thoroughly review in Appendix A
Summary
Three-dimensional general relativity is one of the simplest gravitational systems [1,2] and, in particular, solutions with negative cosmological constant (AdS3) have received special attention, due to their holographic nature [3,4]. AdS3, under certain boundary conditions encompassing Bañados, Teitelboim, and Zanelli (BTZ) black holes [39,40,41], consists in two commuting copies of the Virasoro algebra with central extensions cÆ 1⁄4 23Gl, l being the AdS3 radius and G the Newton constant This result is considered as a precursor of the AdS=CFT correspondence [4,42,43,44,45], which, applied to three-dimensional general relativity, conjectures the existence of a dual confomal field theory (CFT) living on the two-dimensional boundary. In BH, a particular representative of the equivalence class is picked up, namely the flat Minkowski metric η, and kept fixed under the action of the asymptotic symmetry algebra This defines asymptotically (globally) AdS3 spacetimes (AAdS3). Appendix A contains a brief comparison of this work with [30], while in Appendix B we translate our results to the Chern-Simons formulation of the theory
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