More on Asymptotically Anti-de Sitter Spaces in Topologically Massive Gravity

TL;DR

This work analyzes asymptotically AdS spacetimes in three-dimensional topologically massive gravity by extending boundary conditions to include both pp-wave chiralities when $|\mu\ell|<1$. Using a Regge–Teitelboim canonical framework, it derives the surface charges and shows that integrability imposes a functional relation between the two chiral sectors, which generically destroys full conformal invariance at infinity except when one chirality is set to zero. Consequently, the asymptotic symmetry is typically Virasoro$\times$R, with central charges $c_{\pm}=(1\pm 1/(\mu\ell))\,c$ (where $c=3\ell/(2G)$) for the surviving chiral copy, while the fully conformal case is recovered only in the one-chirality limit. The paper also provides explicit formulas for the asymptotic charges for all values of the topological mass $\mu$, clarifying when the Brown–Henneaux conformal symmetry can be preserved and highlighting the role of the relaxed fall-off terms as massive graviton hair.

Abstract

Recently, the asymptotic behaviour of three-dimensional anti-de Sitter gravity with a topological mass term was investigated. Boundary conditions were given that were asymptotically invariant under the two-dimensional conformal group and that included a fall-off of the metric sufficiently slow to consistently allow pp-wave type of solutions. Now, pp-waves can have two different chiralities. Above the chiral point and at the chiral point, however, only one chirality can be considered, namely the chirality that has the milder behaviour at infinity. The other chirality blows up faster than AdS and does not define an asymptotically AdS spacetime. By contrast, both chiralities are subdominant with respect to the asymptotic behaviour of AdS spacetime below the chiral point. Nevertheless, the boundary conditions given in the earlier treatment only included one of the two chiralities (which could be either one) at a time. We investigate in this paper whether one can generalize these boundary conditions in order to consider simultaneously both chiralities below the chiral point. We show that this is not possible if one wants to keep the two-dimensional conformal group as asymptotic symmetry group. Hence, the boundary conditions given in the earlier treatment appear to be the best possible ones compatible with conformal symmetry. In the course of our investigations, we provide general formulas controlling the asymptotic charges for all values of the topological mass (not just below the chiral point).