Asymptotic symmetries of three dimensional gravity and the membrane paradigm

TL;DR

This paper presents a novel method for deriving the asymptotic charge algebra of three-dimensional (A)dS gravity by formulating boundary dynamics as a Hamiltonian system on the dual Lie algebra, simplifying the derivation process.

Contribution

It introduces a new approach to derive the asymptotic charge algebra using Lie-Poisson brackets, providing clearer insight and simplifying calculations compared to traditional methods.

Findings

Derived the asymptotic charge algebra via boundary Hamiltonian dynamics.

Clarified the analogy between gravity symmetries and fluid dynamics.

Streamlined the derivation process using Lie-Poisson brackets.

Abstract

The asymptotic symmetry group of three-dimensional (anti) de Sitter space is the two dimensional conformal group with central charge $c=3\ell/2G$. Usually the asymptotic charge algebra is derived using the symplectic structure of the bulk Einstein equations. Here, we derive the asymptotic charge algebra by a different route. First, we formulate the dynamics of the boundary as a 1+1-dimensional dynamical system. Then we realize the boundary equations of motion as a Hamiltonian system on the dual Lie algebra, $\mathfrak{g}^*$, of the two-dimensional conformal group. Finally, we use the Lie-Poisson bracket on $\mathfrak{g}^*$ to compute the asymptotic charge algebra. This streamlines the derivation of the asymptotic charge algebra because the Lie-Poisson bracket on the boundary is significantly simpler than the symplectic structure derived from the bulk Einstein equations. It also clarifies the analogy between the infinite dimensional symmetries of gravity and fluid dynamics.