Abstract
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 T\overline{T}TT¯-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_33 gravity. This algebra should be obeyed by the stress tensor in any T\overline{T}TT¯-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 T\overline{T}TT¯-deformed theories, although we only carry out the explicit comparison to \mathcal{O}(1/\sqrt{c})𝒪(1/c) in the 1/c1/c expansion.
Highlights
We consider the space of metrics with fixed topology while demanding that the metric on the boundary is ds2
We first compute the Poisson bracket algebra of observables by putting the classical theory in canonical form. This involves characterizing the general classical solution and writing down a candidate symplectic form on this space. To render this form non-degenerate, we need to identifying appropriate gauge orbits, each of which defines a point in covariant phase space
We discuss the gravitational symplectic form and find that large diffeomorphisms are generated by charges that can be written in terms of the stress tensor on the spatial boundary
Summary
Within the framework of our current understanding of quantum gravity, the only observables with a mathematically precise definition involve asymptotic quantities, such as the S-matrix in Minkowski space or boundary correlators in (A)dS. In the asymptotically AdS case Brown and Henneaux demonstrated the emergence of the Virasoro algebras present in the CFT; how are these algebras deformed if we impose Dirichlet boundary conditions at finite ρc?4. Another goal is to gain a better understanding of observables that probe the UV structure of T T -deformed theories; there are reasons to expect At the level of classical pure gravity these two descriptions are equivalent, as adding the T T deformation can be shown to coincide with subtracting the bulk action associated with the spacetime region between the cutoff and the asymptotic boundary [53]. In the present work we focus on the Dirichlet cutoff picture
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