BMS-like algebras: canonical realisations and BRST quantisation

We generalise BMS algebras in three dimensions by the introduction of an arbitrary real parameter $\lambda$, recovering the standard algebras (BMS, extended BMS and Weyl-BMS) for $\lambda=-1$. We exhibit a realisation of the (centreless) Weyl $\lambda$-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 r\^ole in the construction. The Weyl $\lambda$-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 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.