Darboux Soft Hair in 3D Asymptotically Flat Spacetimes

TL;DR

This paper constructs a comprehensive solution space for 3D asymptotically flat Einstein gravity, revealing Darboux soft hair and rich algebraic structures, including multiple Kac-Moody, Virasoro, and BMS algebras, with implications for boundary dynamics.

Contribution

It introduces a novel Darboux coordinate framework for 3D flat gravity, uncovering new algebraic structures and clarifying the action principle without boundary Lagrangians.

Findings

Solution space parametrized by four functions representing soft hair.

Symplectic form adopts Darboux form, yielding Heisenberg algebra.

Construction of multiple Lie algebras including Kac-Moody, Virasoro, and BMS.

Abstract

In this paper, we construct a fully on-shell solution space for three-dimensional Einstein's gravity in asymptotically flat spacetimes using a finite coordinate transformation. This space is parametrized by four unconstrained codimension-one functions that parameterize geometrical deformations of the null infinity cylinder, known as Darboux soft hair. The symplectic form of the theory in terms of these functions adopts a Darboux form at the corner, thereby yielding two copies of the Heisenberg algebra. Utilizing a series of field redefinitions of boundary charges, we construct various Lie algebras, including four Kac-Moody algebras, four Virasoro algebras, and two centrally extended BMS$_3$ algebras. Intriguingly, we show that the bulk theory possesses a well-defined action principle without the need for any boundary Lagrangian in the Darboux frame. Conversely, in the hydrodynamics frame, a well-defined action principle with the Dirichlet boundary condition results in two Schwarzian actions at null infinity.