The BRST quantisation of chiral BMS-like field theories

TL;DR

This work constructs the BRST quantisation of the one-parameter family of algebras $\hat{\mathfrak{g}}_\lambda$, of which BMS$_3$ at $\lambda=-1$ is a central case, using both semi-infinite cohomology and vertex-operator methods. It provides a free-field realization in terms of two bc-systems, shows how the BRST differential arises as the semi-infinite differential, and identifies a precise central-charge condition $c^{\text{mat}}=26+2(6\lambda^2-6\lambda+1)$ for square-zero BRST symmetry. The paper proves two BV-algebra-level embedding theorems: (i) Virasoro BRST cohomology at $c_L=26$ embeds into $\hat{\mathfrak{g}}_\lambda$ cohomology with a $\beta\gamma$-Koszul sector, and (ii) the BRST cohomology of $\hat{\mathfrak{g}}_\lambda$ is isomorphic to the chiral ring of a twisted $N=2$ SCFT. Special attention is given to the BMS$_3$ case ($\lambda=-1$): BRST nilpotency cannot hold with nonzero $c_M$, but physically relevant chiral BMS$_3$ realizations exist (e.g., ambitwistor and Nappi–Witten strings) with $c_M=0$; the work hence constrains the consistent BRST treatment of chiral BMS$_3$-theories and points to avenues for broader non-chiral or supersymmetric extensions.

Abstract

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

Figures (1)

• Figure 1: A diagram summarising the different explicit constructions of $\hat{g}_{\lambda\leq 0}$ field theories from weight $(1,0)$$\beta\gamma$-systems and field theories with $\widehat{\mathfrak{nw}}_{2n+2}$ symmetry.

Theorems & Definitions (55)

• Definition 2.1
• Definition 2.2
• Remark 2.3
• Definition 2.4
• Remark 2.5
• Definition 2.6
• Definition 3.1
• Definition 3.2
• Lemma 3.3
• Proposition 3.4