The BRST quantisation of chiral BMS-like field theories

The BMS$_3$ Lie algebra belongs to a one-parameter family of Lie algebras obtained by centrally extending abelian extensions of the Witt algebra by a tensor density representation. In this paper we call such Lie algebras $\hat{\mathfrak{g}}_\lambda$, with BMS$_3$ corresponding to the universal central extension of $\lambda = -1$. We construct the BRST complex for $\hat{\mathfrak{g}}_\lambda$ in two different ways: one in the language of semi-infinite cohomology and the other using the formalism of vertex operator algebras. We pay particular attention to the case of BMS$_3$ and discuss some natural field-theoretical realisations. We prove two theorems about the BRST cohomology of $\hat{\mathfrak{g}}_\lambda$. The first is the construction of a quasi-isomorphic embedding of the chiral sector of any Virasoro string as a $\hat{\mathfrak{g}}_\lambda$ string. The second is the isomorphism (as Batalin-Vilkovisky algebras) of any $\hat{\mathfrak{g}}_\lambda$ BRST cohomology and the chiral ring of a topologically twisted $N{=}2$ superconformal field theory.