Most general flat space boundary conditions in three-dimensional Einstein gravity

TL;DR

This work identifies the most general asymptotically flat boundary conditions for three‑dimensional Einstein gravity in the Chern–Simons formulation, yielding an isl(2)_k current algebra as the asymptotic symmetry algebra with six charges and six chemical potentials. It shows that all previously studied flat-space boundary conditions (including BMS_3, Heisenberg, and Detournay–Riegler) arise as contractions from AdS_3 constructions, and it further explores new Carroll gravity boundary conditions via flat/AdS contractions. The paper also translates these results into the metric language, analyzes canonical charges and their conservation, and discusses holographic interpretations, including prospects for an isl(2)_k WZW description and Carroll holography. Overall, it provides a unifying, maximally general framework for flat-space holography in 3D gravity and opens avenues for Carrollian boundary dynamics and further deformations.

Abstract

We consider the most general asymptotically flat boundary conditions in three-dimensional Einstein gravity in the sense that we allow for the maximal number of independent free functions in the metric, leading to six towers of boundary charges and six associated chemical potentials. We find as associated asymptotic symmetry algebra an isl(2)_k current algebra. Restricting the charges and chemical potentials in various ways recovers previous cases, such as BMS_3, Heisenberg or Detournay-Riegler, all of which can be obtained as contractions of corresponding AdS_3 constructions. Finally, we show that a flat space contraction can induce an additional Carrollian contraction. As examples we provide two novel sets of boundary conditions for Carroll gravity.