Probing integrable perturbations of conformal theories using singular vectors

It has been known for some time that the (1,3) perturbations of the (2 k + 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 (3 k ± 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.