3D gravity in a box

The quantization of pure 3D gravity with Dirichlet boundary conditions on a finite boundary is of interest both as a model of quantum gravity in which one can compute quantities which are ``more local" than S-matrices or asymptotic boundary correlators, and for its proposed holographic duality to TT̄ T T ¯ -deformed CFTs. In this work we apply covariant phase space methods to deduce the Poisson bracket algebra of boundary observables. The result is a one-parameter nonlinear deformation of the usual Virasoro algebra of asymptotically AdS _3 3 gravity. This algebra should be obeyed by the stress tensor in any TT̄ T T ¯ -deformed holographic CFT. We next initiate quantization of this system within the general framework of coadjoint orbits, obtaining — in perturbation theory — a deformed version of the Alekseev-Shatashvili symplectic form and its associated geometric action. The resulting energy spectrum is consistent with the expected spectrum of TT̄ T T ¯ -deformed theories, although we only carry out the explicit comparison to O(1/√c) 𝒪 ( 1 / c ) in the 1/c 1 / c expansion.