Probing integrable perturbations of conformal theories using singular vectors

TL;DR

This paper explores how certain perturbations of extended conformal field theories preserve conserved currents that are also singular vectors, using algebraic conjectures to identify integrable cases.

Contribution

It extends the singular vector approach to W-algebra models, connecting conserved currents with integrable perturbations under a key algebraic conjecture.

Findings

Identifies conserved currents as singular vectors in extended models.

Recovers all known integrable perturbations using the singular-vector method.

Discovers that some conserved densities are subsingular vectors, not singular vectors.

Abstract

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.