Dual dynamics of three dimensional asymptotically flat Einstein gravity at null infinity

TL;DR

The paper develops a two-stage holographic description of three-dimensional asymptotically flat gravity at null infinity by starting from a Chern–Simons formulation to obtain a chiral WZW model based on $ISO(2,1)$, followed by a Hamiltonian reduction to a $BMS_3$-invariant Liouville theory. It carefully treats both AdS and flat sectors, deriving on-shell group elements, improved actions, and the resulting current algebras, and shows how the flat-space theory emerges as a limit of the AdS construction. The flat case yields a centrally extended $BMS_3$ structure with $c_2=12k=3/G$ and $c_1=0$, realized through a first-order Liouville form and a modified Sugawara map from chiral WZW currents. A key subtlety is the zero-mode issue, which implies that AdS/flat gravity corresponds to the chiral WZW theories rather than Liouville theory in the zero-mode sector, highlighting the nuances of flat-space holography for cosmological spacetimes.

Abstract

Starting from the Chern-Simons formulation, the two-dimensional dual theory for three-dimensional asymptotically flat Einstein gravity at null infinity is constructed. Solving the constraints together with suitable gauge fixing conditions gives in a first stage a chiral Wess-Zumino-Witten like model based on the Poincaré algebra in three dimensions. The next stage involves a Hamiltonian reduction to a BMS3 invariant Liouville theory. These results are connected to those originally derived in the anti-de Sitter case by rephrasing the latter in a suitable gauge before taking their flat-space limit.