Abstract
We clarify the notion of the DS — generalized Drinfeld-Sokolov — reduction approach to classicalW-algebras. We first strengthen an earlier theorem which showed that ansl(2) embeddingL ⊏G can be associated to every DS reduction. We then use the fact that aW-algebra must have a quasi-primary basis to derive severe restrictions on the possible reductions corresponding to a givesl(2) embedding. In the known DS reductions found to data, for which theW-algebras are denoted byW G L -algebras and are called canonical, the quasi-primary basis corresponds to the highest weights of thesl(2). Here we find some examples of noncanonical DS reductions leading toW-algebras which are direct products ofW G L -algebras and “free field” algebras with conformal weights Δ∈{0, 1/2, 1}. We also show that if the conformal weights of the generators of aW-algebra obtained from DS reduction are nonnegative Δ≥0 (which is the case for all DS reductions known to date), then the Δ≥3/2 subsectors of the weights are necessarily the same as in the correspondingW G L -algebra. These results are consistent with an earlier result by Bowcock and Watts on the spectra ofW-algebras derived by different means. We are led to the conjecture that, up to free fields, the set ofW-algebras with nonnegative spectra Δ>-0 that may be obtained from DS reduction is exhausted by the canonical ones.
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Communicated by R. H. Dijkgraaf
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Fehér, L., O'Raifeartaigh, L., Ruelle, P. et al. On the completeness of the set of classicalW-algebras obtained from DS reductions-algebras obtained from DS reductions. Commun.Math. Phys. 162, 399–431 (1994). https://doi.org/10.1007/BF02102024
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DOI: https://doi.org/10.1007/BF02102024