BRST Operators for Higher-spin Algebras

TL;DR

This paper constructs and analyzes BRST operators for higher-spin W-algebras in matter and Liouville systems, revealing new algebraic structures and correcting previous beliefs about their existence at specific central charges.

Contribution

It introduces new classical and quantum BRST operators for various W-algebras, including cases previously thought impossible, using free scalar realizations and algebraic methods.

Findings

Constructed classical BRST operators for W^{M}_{2,s} mp; W^{L}_{2,s'}.

Derived quantum BRST operators for W^{M}_{2,s} mp; W^{L}_2 with s=4,5,6.

Discovered W_{2,4} and W_{2,6} algebras at specific central charges where they were previously believed not to exist.

Abstract

In this paper, we construct non-critical BRST operators for matter and Liouville systems whose currents generate two different $W$ algebras. At the classical level, we construct the BRST operators for $W^{\rm M}_{2,s}\otimes W^{\rm L}_{2,s'}$. The construction is possible for $s=s'$ or $s\ge s'+2$. We also obtain the BRST operator for $W^{\rm M}_{2,4}\otimes W^{\rm L}_4$ at the classical level. We use free scalar realisations for the matter currents in the above constructions. At the full quantum level, we obtain the BRST operators for $W^{\rm M}_{2,s}\otimes W^{\rm L}_2$ with $s=4,5, 6$, where $W_2$ denotes the Virasoro algebra. For the first and last cases, the BRST operators are expressed in terms of abstract matter and Liouville currents. As a by-product, we obtain the $W_{2,4}$ algebra at $c=-24$ and the $W_{2,6}$ algebra at $c=-2$ and $-\ft{286}3$, at which values the algebras were previously believed not to exist.