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

TL;DR

This paper investigates the structure of singular vectors in $\\mathcal{W}_3$ and $\\mathcal{W}_3^{(2)}$ algebras through BRST quantization of Drinfeld-Sokolov reductions, revealing their relation to $A^{(1)}_2$ Verma modules.

Contribution

It introduces quantum gauge transformations that relate $A^{(1)}_2$ Verma module singular vectors to $\\mathcal{W}$ algebra singular vectors, clarifying their differences in BRST cohomology.

Findings

Singular vectors of $A^{(1)}_2$ Verma modules map to $\\mathcal{W}$ algebra singular vectors.

Differences between these vectors are trivial in BRST cohomology.

Quantum gauge transformations realize the mapping between these 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.