quantization of pure 3D gravity with Dirichlet boundary conditions on a finite boundary

3D Gravity in a Box

We apply covariant phase space methods to deduce the algebra of boundary observables of pure 3d gravity with dirichlet boundary conditions on a finite boundary.The result is a one-parameter nonlinear deformation of the usual virasoro algebra of asymptotically gravity.This algebra should be obeyed by the stresstensor in any 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 theories, although we only carry out the explicit comparison to in the expansion.