A Note concerning Subsingular Vectors and Embedding Diagrams of the N=2 Superconformal Algebras

Subsingular vectors of the N=2 superconformal algebras were discovered, and examples given, in 1996. Shortly afterwards Semikhatov and Tipunin claimed to have obtained a complete classification of the N=2 subsingular vectors in the paper ‘The Structure of Verma Modules over the N=2 Superconformal algebra’, hep-th/9704111, published in CMP 195 (1998) 129. Surprisingly, the only explicit examples of N=2 subsingular vectors known at that time did not fit into their classification. All the results presented in that paper, including the classification of subsingular vectors, were based on the following assumptions: i) The authors claimed that there are only two different types of submodules in N=2 Verma modules, overlooking from the very beginning indecomposable ‘no-label’ singular vectors, that had been discovered a few months before, and clearly do not fit into their two types of submodules, and ii) The authors claimed to have constructed ‘non-conventional’ singular vectors with the property of generating the two types of submodules maximally, i.e. with no subsingular vectors left outside. In this note we prove that both assumptions are incorrect. These facts also affect profoundly the results presented in several other publications, especially the papers: ‘On the Equivalence of Affine sl(2) and N=2 ....’, by Semikhatov, hep-th/9702074, ‘Embedding Diagrams of N=2 Verma Modules ....’, by Semikhatov and Sirota, hep-th/9712102, and ‘All Singular Vectors of the N=2 ....’, by Semikhatov and Tipunin, hep-th/9604176 (last revised version in September 98).