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

P. Furlan·A. Ch. Ganchev·V. B. Petkova
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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 $\overline W^{(η)}\,$ 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 $\overline W^{(η)}\,$. This is a detailed presentation of the results announced in a recent paper of the authors (Phys. Lett. B318 (1993) 85).

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P. Furlan, A. Ch. Ganchev, V. B. Petkova. 1994-09-25. Singular Vectors of ${\cal W}$ Algebras via DS Reduction of $A_2^(1)$. https://doi.org/10.1016/0550-3213(94)90217-8

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