A 2D Field Theory Equivalent to 3D Gravity with No Cosmological Constant

Abstract

In (2+1) space-time dimensions the Einstein theory of gravity has no local degrees of freedom. In fact, in the presence of a negative cosmological term, it is described by a (1+1) dimensional theory living on its boundary: Liouville theory. It is invariant under the action of the two-dimensional conformal group, which, in the gravitational context, corresponds to the asymptotic symmetries of asymptotically AdS geometries. In the flat case, when the cosmological term is turned off, a theory describing gravity at the boundary is absent. In this note we show that, in the Hamiltonian setup, such a theory may be constructed. The theory is BMS₃ invariant, as it should, corresponding to the asymptotic symmetry group of an asymptotically flat spacetime.