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

## TL;DR

This paper establishes the most general flat space boundary conditions in three-dimensional Einstein gravity, revealing an isl(2)_k current algebra and connecting various known boundary conditions through contractions, also introducing Carroll gravity boundary conditions.

## Contribution

It introduces the most general asymptotically flat boundary conditions in 3D Einstein gravity, expanding the understanding of boundary symmetries and their contractions.

## Key findings

- Identified an isl(2)_k current algebra as the asymptotic symmetry.
- Recovered known boundary conditions via contractions from the general case.
- Proposed new boundary conditions for Carroll gravity.

## 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.