Quantization of BMS$_3$ orbits: a perturbative approach

TL;DR

This work addresses the problem of obtaining representations for the infinite-dimensional $BMS_3$ group by perturbatively quantizing coadjoint orbits, motivated by the difficulties of full geometric quantization in infinite dimensions. The authors develop a perturbative framework around a given orbit, recasting the orbit's phase space into an infinite set of decoupled bosonic modes and realizing the Hilbert space as a tensor product $igotimes_{n\,>\,0} L^2(\mathbb{R}^2)_{(n)}$, enabling explicit character calculations. They compute the massive and vacuum $BMS_3$ characters, showing agreement with induced-representation results by Oblak and Barnich et al., and establish a concrete link between coadjoint-orbit quantization and induced representations for $BMS_3$. The findings suggest that perturbative methods capture essential representation-theoretic data for infinite-dimensional groups and hint at deeper index-theoretic underpinnings that could generalize geometric quantization via Lefschetz-type approaches.

Abstract

We compute characters of the BMS group in three dimensions. The approach is the same as that performed by Witten in the case of coadjoint orbits of the Virasoro group in the eighties, within the large central charge approximation. The procedure involves finding a Poisson bracket between classical variables and the corresponding commutator of observables in a Hilbert space, explaining why we call this a quantization. We provide first a pedagogical warm up by applying the method to both SL(2,R) and Poincaré3 groups. As for BMS3, our results coincide with the characters of induced representations recently studied in the literature. Moreover, we relate the 'coadjoint representations' to the induced representations.