3d gravity in Bondi-Weyl gauge: charges, corners, and integrability

TL;DR

This paper develops a new gauge for 3D gravity that incorporates Weyl charges, allowing for a broad class of asymptotic spacetimes, and analyzes the associated charges, symmetries, and corner terms.

Contribution

It introduces a novel Bondi-Weyl gauge for 3D gravity, revealing a rich algebra of charges and clarifying the role of corner terms and integrability.

Findings

Charges form a $ ext{Vir} imes ext{Vir} imes ext{Heisenberg}$ algebra with three central extensions.

Charges are finite, symplectic, integrable, but not conserved.

The study applies both metric and triad variables, highlighting the covariant origin of corner terms.

Abstract

We introduce a new gauge and solution space for three-dimensional gravity. As its name Bondi-Weyl suggests, it leads to non-trivial Weyl charges, and uses Bondi-like coordinates to allow for an arbitrary cosmological constant and therefore spacetimes which are asymptotically locally (A)dS or flat. We explain how integrability requires a choice of integrable slicing and also the introduction of a corner term. After discussing the holographic renormalization of the action and of the symplectic potential, we show that the charges are finite, symplectic and integrable, yet not conserved. We find four towers of charges forming an algebroid given by $\mathfrak{vir}\oplus\mathfrak{vir}\oplus\text{Heisenberg}$ with three central extensions, where the base space is parametrized by the retarded time. These four charges generate diffeomorphisms of the boundary cylinder, Weyl rescalings of the boundary metric, and radial translations. We perform this study both in metric and triad variables, and use the triad to explain the covariant origin of the corner terms needed for renormalization and integrability.