Singular Vectors of the Virasoro Algebra

We give expressions for the singular vectors in the highest weight representations of the Virasoro algebra. We verify that the expressions — which take the form of a product of operators applied to the highest weight vector — do indeed define singular vectors. These results explain the patterns of embeddings amongst Virasoro algebra highest weight representations. Conformal field theory relies on a description of the Virasoro algebra’s highest weight representations, and in particular on the classifications of the levels at which representations have singular vectors and the embedding relations amongst these vectors. These embedding relations are encoded in the irreducible Virasoro characters, from which the partition functions of conformal field theories are built. While this information is enough for most conformal field theoretic purposes, there are several applications in which explicit expressions for the singular vectors are needed. The first relevant work was the beautiful paper of Malikov, Feigin and Fuchs, which gives expressions for the general singular vectors in finite dimensional Lie algebra Verma modules. Feigin and Fuchs have also presented some partial results describing projections, and asymptotic properties, of the Virasoro algebra singular vectors. Benoit and Saint-Aubin (BSA) found remarkable explicit expressions for the sub-class of the singular vectors vp,q in which either p or q is 1. Recently, Bauer et al. have rewritten the BSA expressions in a compact form in which their singularity is manifest, and in which very interesting connections to integrable systems and to W-algebra theory appear. Bauer et al. have also given a new algorithm by which any vector vp,q can in principle be calculated: we shall discuss this later. In this letter, we give expressions for all the singular vectors vp,q, show how these expressions explain the embeddings of the Virasoro algebra’s highest weight representations, and sketch proofs of these results. First let us recall some basic facts and describe the results of Benoit and Saint-Aubin. The Virasoro algebra has commutation relations [Lm, Ln] = (m− n)Lm+n + m3−m 12 δm,−nC , [Lm, C] = 0 . (1) The Verma module V (h, c) is the representation which contains a vector |h〉 such