HOW COMPLETE IS THE CLASSIFICATION OF W-SYMMETRIES

In 2D conformal quantum field theory, we continue a systematic study of W-algebras with two and three generators and their highest weight representations focussing mainly on rational models. We review the known facts about rational models of W (2, delta)-algebras, in particular those models that are not related to some generically existing algebra. However, our new rational models of W-algebras with two generators all belong to one of the known series. The majority of W-algebras with three generators - including the new ones constructed in this letter - can be explained as subalgebras of Casimir algebras. Nonetheless, for one solution of W (2, 4, 6) we reveal some features that do not fit into the pattern of Casimir algebras or orbifolds thereof. This shows that even a classification of all Casimir algebras (or Toda field theories) will probably fail to classify all rational conformal field theories.