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 W2,sm ⊗ W2,s′l. The construction is possible for s = s′ or s ≥ s′ + 2. We also obtain the BRST operator for W2,4m ⊗ W4l 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 W2,sm ⊗ W2l with s = 4,5,6, where W2 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 W2,4 algebra at c = −24 and the W2,6 algebra at c = −2 and −2863, at which values the algebras were previously believed not to exist.