Generalized KdV and Quantum Inverse Scattering Description of Conformal Minimal Models

Abstract

We propose an alternative description of 2 dimensional Conformal Field Theory in terms of Quantum Inverse Scattering. It is based on the generalized KdV systems attached to $A_2^{(2)}$, yielding the classical limit of Virasoro as Poisson bracket structure. The corresponding T-system is shown to coincide with the one recently proposed by Kuniba and Suzuki. We classify the primary operators of the minimal models that commute with all the Integrals of Motion, and that are therefore candidates to perturb the model by keeping the conservation laws. For our $A_2^{(2)}$ structure these happen to be $\phi_{1,2},\phi_{2,1},\phi_{1,5}$, in contrast to the $A_1^{(1)}$ case, studied by Bazhanov, Lukyanov and Zamolodchikov~\cite{BLZ}, related to $\phi_{1,3}$.