Darboux Soft Hair in 3D Asymptotically Flat Spacetimes

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.