Asymptotically anti-de Sitter spacetimes in topologically massive gravity

TL;DR

This work establishes consistent boundary conditions for asymptotically AdS spacetimes in three-dimensional topologically massive gravity across all nonzero mass parameters $μ$. For $0<|μl|<1$ it identifies two boundary-condition branches that preserve the conformal asymptotic symmetry and yield Virasoro algebras with central charges $c_{±}=(1±1/(μl))c$, while slower fall-off modes act as hair and do not alter charges. At the chiral point $μl=1$ the boundary conditions require logarithmic terms and both Virasoro sectors remain nonzero, with the left central charge vanishing ($c_{-}=0$) despite a nonzero left-moving charge, suggesting a link to logarithmic CFT structure. In the regime $|μl|>1$ the analysis enforces $h_{++}=0$ to maintain AdS asymptotics, with the same charge structure, and thus the results unify the treatment of TMG boundary conditions with and without slower fall-off while clarifying the chiral point behavior.

Abstract

We consider asymptotically anti-de Sitter spacetimes in three-dimensional topologically massive gravity with a negative cosmological constant, for all values of the mass parameter $μ$ ($μ\neq0$). We provide consistent boundary conditions that accommodate the recent solutions considered in the literature, which may have a slower fall-off than the one relevant for General Relativity. These conditions are such that the asymptotic symmetry is in all cases the conformal group, in the sense that they are invariant under asymptotic conformal transformations and that the corresponding Virasoro generators are finite. It is found in particular that at the chiral point $|μl|=1$ (where $l$ is the anti-de Sitter radius), one must allow for logarithmic terms (absent for General Relativity) in the asymptotic behaviour of the metric in order to accommodate the new solutions present in topologically massive gravity, and that these logarithmic terms make both sets of Virasoro generators non-zero even though one of the central charges vanishes.