On $\widehat{sl}(3)$ reduction, quantum gauge transformations, and ${\cal W}-$ algebras singular vectors

Abstract

The problem of describing the singular vectors of $\cW_3$ and $\cW_3^{(2)}$ Verma modules is addressed, viewing these algebras as BRST quantized Drinfeld-Sokolov (DS) reductions of $A^{(1)}_2\,$. Singular vectors of an $A^{(1)}_2\,$ Verma module are mapped into $\W$ algebra singular vectors and are shown to differ from the latter by terms trivial in the BRST cohomology. These maps are realized by quantum versions of the highest weight DS gauge transformations.