BRST Operators for Higher-spin Algebras

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.