Geometric actions for three-dimensional gravity

TL;DR

The paper addresses constructing dual two-dimensional actions for gauge-fixed solution spaces of three-dimensional gravity with AdS or flat asymptotics by geometric actions on coadjoint orbits of the asymptotic symmetry groups, namely two copies of the Virasoro group for AdS and the centrally extended BMS$_3$ group for flat spacetimes. It develops a group-theoretical route, extending to central extensions and semidirect products, and introduces Hamiltonians that preserve G-invariance, enabling explicit actions for each orbit. The results reproduce and unify boundary dynamics known from Chern–Simons/Holography approaches, connecting to chiral WZW models, chiral bosons, and Hill’s equation, and provide a versatile framework that accounts for holonomies and nontrivial boundary conditions. This group-theoretic construction offers potential applications to one-loop partition functions, Berry phases, and possible extensions to higher dimensions, including four-dimensional gravity and related symmetry structures.

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.