The Universal Phase Space of AdS₃ Gravity

Abstract

We describe what can be called the “universal” phase space of AdS₃ gravity, in which the moduli spaces of globally hyperbolic AdS spacetimes with compact spatial sections, as well as the moduli spaces of multi-black-hole spacetimes are realized as submanifolds. The universal phase space is parametrized by two copies of the universal Teichmüller space ${\mathcal{T}(1)}$ and is obtained from the correspondence between maximal surfaces in AdS₃ and quasisymmetric homeomorphisms of the unit circle. We also relate our parametrization to the Chern-Simons formulation of 2+1 gravity and, infinitesimally, to the holographic (Fefferman-Graham) description. In particular, we obtain a relation between the generators of quasiconformal deformations in each ${\mathcal{T}(1)}$ sector and the chiral Brown-Henneaux vector fields. We also relate the charges arising in the holographic description (such as the mass and angular momentum of an AdS₃ spacetime) to the periods of the quadratic differentials arising via the Bers embedding of ${\mathcal{T}(1)\times\mathcal{T}(1)}$ . Our construction also yields a symplectic map ${T^ \ast \mathcal{T}(1) \rightarrow \mathcal{T}(1) \times \mathcal{T}(1)}$ generalizing the well-known Mess map in the compact spatial surface setting.