Enhanced Asymptotic Symmetry Algebra of 2+1 Dimensional Flat Space

TL;DR

This work introduces a new set of boundary conditions for 2+1 dimensional flat-space gravity that extend Barnich-Compère by yielding a $\mathfrak{bms}_3$ semidirect product with two affine $\hat{\mathfrak{u}}(1)$ currents. The authors develop the Chern-Simons and metric formulations, compute the canonical charges, and derive the full asymptotic symmetry algebra, including central extensions that deform in Topologically Massive Gravity (TMG). They demonstrate that the thermal entropy for solutions under these BCs can be obtained from horizon area and holonomy/Wilson-line methods, and they extend the framework to TMG where CS terms contribute to the entropy and modify the central charges. A limiting relation to Troessaert BCs is established, showing the flat-space BCs arise from a vanishing cosmological constant limit. The results provide a robust platform for exploring flat-space holography, entropy, and possible generalizations to higher-spin or higher-derivative theories.

Abstract

In this paper we present a new set of asymptotic boundary conditions for Einstein gravity in 2+1 dimensions with vanishing cosmological constant that are a generalization of the Barnich-Comp{è}re boundary conditions gr-qc/0610130. These new boundary conditions lead to an asymptotic symmetry algebra that is generated by a $\mathfrak{bms}_3$ algebra and two affine $\hat{\mathfrak{u}}(1)$ current algebras. We then apply these boundary conditions to Topologically Massive Gravity (TMG) and determine how the presence of the gravitational Chern-Simons term affects the central extensions of the asymptotic symmetry algebra. We furthermore determine the thermal entropy of solutions obeying our new boundary conditions for both Einstein gravity and TMG.