Probing integrable perturbations of conformal theories using singular vectors

It has been known for some time that the (1, 3) perturbations of the (2k + 1, 2) Virasoro minimal models have conserved currents which are also singular vectors of the Virasoro algebra. This also turns out to hold for the (1, 2) perturbation of the (3k± 1, 3) models. In this paper we investigate the requirement that a perturbation of an extended conformal field theory has conserved currents which are also singular vectors. We consider conformal field theories with W3 and (bosonic) WBC2 = W (2, 4) extended symmetries. Our analysis relies heavily on the general conjecture of de Vos and van Driel relating the multiplicities of W -algebra irreducible modules to the Kazhdan-Lusztig polynomials of a certain double coset. Granting this conjecture, the singular-vector argument provides a direct way of recovering all known integrable perturbations. However, W models bring a slight complication in that the conserved densities of some (1, 2)-type perturbations are actually subsingular vectors, that is, they become singular vectors only in a quotient module. March 13, 1996 Address from March 1, 1996: Department of Mathematics, King’s College London, The Strand, London, WC2R 2LS