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

Abstract

The BRST quantisation of the Drinfeld - Sokolov reduction applied to the case of $A_2^{(1)}$ is explored to construct in an unified and systematic way the general singular vectors in ${\mathcal{W}}_3$ and ${\mathcal{W}}_3^{(2)}$ 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 $\mathcal{W}$ - Verma modules play an important role in the proof of the BRST equivalence of the Malikov-Feigin-Fuks singular vectors and the $\mathcal{W}$ 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).

plain TeX, 39 pages, INFN/AE-94/10. (revised version, to appear in Nuclear Physics B; some extensions in section 6 and Note Added.)