BMS-like algebras: Canonical realisations and BRST quantisation

We generalise Bondi–Metzner–Sachs (BMS) algebras in three dimensions by the introduction of an arbitrary real parameter λ, recovering the standard algebras (BMS, extended BMS and Weyl-BMS) for λ = −1. We exhibit a realisation of the (centreless) Weyl λ-BMS algebra in terms of the symplectic structure on the space of solutions of the massless Klein–Gordon equation in 2 + 1, using the eigenstates of the spacetime momentum operator. The quadratic Casimir of the Lorentz algebra plays an essential role in the construction. The Weyl λ-BMS algebra admits a three-parameter family of central extensions, resulting in the (centrally extended) Weyl-BMS algebra, which we reformulate in terms of operator product expansions. We construct the Becchi-Rouet-Stora-Tyutin (BRST) complex of a putative Weyl-BMS string and show that the BRST cohomology is isomorphic to the chiral ring of a topologically twisted N = 2 superconformal field theory. We also comment on the obstructions to obtaining a “conformal” BMS Lie algebra–that is, one that includes in addition the special conformal generators–and the need to consider a W-algebra. We then construct the quantum version of this W-algebra in terms of operator product expansions. We show that this W-algebra does not admit a BRST complex.