Geometric actions for three-dimensional gravity

TL;DR

This paper develops a group-theoretical approach to derive geometric actions for the solution spaces of 3D gravity, providing an alternative to the traditional Chern-Simons method and exploring potential broader applications.

Contribution

It introduces a novel group-theoretical method to construct geometric actions for coadjoint orbits in 3D gravity, offering new insights beyond existing boundary condition approaches.

Findings

Derived geometric actions from coadjoint orbits for 3D gravity

Connected these actions to the solution space of Einstein gravity in 3D

Outlined potential generalizations and applications beyond 3D gravity

Abstract

The solution space of three-dimensional asymptotically anti-de Sitter or flat Einstein gravity is given by the coadjoint representation of two copies of the Virasoro group in the former and the centrally extended BMS$_3$ group in the latter case. Dynamical actions that control these solution spaces are usually constructed by starting from the Chern-Simons formulation and imposing all boundary conditions. In this note, an alternative route is followed. We study in detail how to derive these actions from a group-theoretical viewpoint by constructing geometric actions for each of the coadjoint orbits, including the appropriate Hamiltonians. We briefly sketch relevant generalizations and potential applications beyond three-dimensional gravity.