Notes on the BMS group in three dimensions: II. Coadjoint representation

TL;DR

The paper classifies the coadjoint orbits of the centrally extended BMS3 group and demonstrates how these orbits encode the covariant phase space of three-dimensional asymptotically flat gravity. By analyzing the semi-direct product structure with Virasoro, it shows that generic orbits are bundles over Virasoro coadjoint orbits, with fibres given by little-group coadjoint orbits, and identifies when exceptional orbits arise. It connects this classical orbit structure to quantum representations via geometric quantization, yielding BMS3 particles as induced representations and clarifying how intrinsic angular momentum remains invariant under supertranslations. The work thus extends the AdS3 orbit method to flat space, linking gravitational solutions to group-theoretic data and setting the stage for a representation-theoretic understanding of 3D gravity in asymptotically flat spacetimes.

Abstract

The coadjoint representation of the BMS$_3$ group, which governs the covariant phase space of three-dimensional asymptotically flat gravity, is investigated. In particular, we classify coadjoint BMS$_3$ orbits and show that intrinsic angular momentum is free of supertranslation ambiguities. Finally, the link with induced representations upon geometric quantization is discussed.