On the completeness of the canonical reductions from Kac-Moody to W-algebras

We study the question as to whether the canonical reductions of Kac-Moody (KM) algebras to W-algebras are exhaustive. We first review and consolidate previous results. In particular we show that, apart from the two lowest grades, the canonical reductions are the only ones that respect the physically reasonable requirement that the W-algebra have no negative conformal weights. We then break new ground by formulating a condition that the W-algebra be differential polynomial. We apply the condition that the W-algebra be polynomial (in a particular gauge) and primary to the groups SL(N,R) with integral SL(2,R) embeddings. We find that, subject to some reasonable technical assumptions, the canonical reductions are exhaustive (except possibly at grade one). The derivation suggests that similar results hold for the other classes of simple groups. (orig.)