Singular Vectors of ${\cal W}$ Algebras via DS Reduction of $A_2^(1)$

TL;DR

This paper systematically constructs singular vectors in ${ m W}_3$ and ${ m W}_3^{(2)}$ Verma modules using BRST quantisation of Drinfeld-Sokolov reduction for $A_2^{(1)}$, revealing their classification via affine Weyl groups.

Contribution

The paper introduces a unified BRST-based method for constructing and classifying singular vectors in ${ m W}$ algebra modules, extending previous results with a detailed algebraic framework.

Findings

Constructed singular vectors using BRST quantisation and DS gauge fixing.

Established the classification of singular vectors via affine Weyl groups.

Proved BRST equivalence of Malikov-Feigin-Fuks and ${ m W}$ algebra singular vectors.

Abstract

The BRST quantisation of the Drinfeld - Sokolov reduction applied to the case of $A^{(1)}_2\,$ is explored to construct in an unified and systematic way the general singular vectors in ${\cal W}_3$ and ${\cal W}_3^{(2)}$ Verma modules. The construction relies on the use of proper quantum analogues of the classical DS gauge fixing transformations. Furthermore the stability groups $\overline W^{(\eta)}\,$ of the highest weights of the ${\cal W}\,$ - Verma modules play an important role in the proof of the BRST equivalence of the Malikov-Feigin-Fuks singular vectors and the ${\cal W}$ algebra ones. The resulting singular vectors are essentially classified by the affine Weyl group $W\, $ modulo $\overline W^{(\eta)}\,$. This is a detailed presentation of the results announced in a recent paper of the authors (Phys. Lett. B318 (1993) 85).