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 ${\widehat{\mathfrak{g}}}_{\lambda }$, with BMS${}_3$ corresponding to the universal central extension of $\lambda =-1$. We construct the BRST complex for ${\widehat{\mathfrak{g}}}_{\lambda }$ 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 ${\widehat{\mathfrak{g}}}_{\lambda }$. The first is the construction of a quasi-isomorphic embedding of the chiral sector of any Virasoro string as a ${\widehat{\mathfrak{g}}}_{\lambda }$ string. The second is the isomorphism (as Batalin-Vilkovisky algebras) of any ${\widehat{\mathfrak{g}}}_{\lambda }$ BRST cohomology and the chiral ring of a topologically twisted $N=2$ superconformal field theory.