Integrable Structure of Conformal Field Theory II. Q-operator and DDV equation

Abstract

This paper is a direct continuation of\ \BLZ\ where we begun the study of the integrable structures in Conformal Field Theory. We show here how to construct the operators ${\bf Q}_{\pm}(λ)$ which act in highest weight Virasoro module and commute for different values of the parameter $λ$. These operators appear to be the CFT analogs of the $Q$ - matrix of Baxter\ \Baxn, in particular they satisfy famous Baxter's ${\bf T}-{\bf Q}$ equation. We also show that under natural assumptions about analytic properties of the operators ${\bf Q}(λ)$ as the functions of $λ$ the Baxter's relation allows one to derive the nonlinear integral equations of Destri-de Vega (DDV)\ \dVega\ for the eigenvalues of the ${\bf Q}$-operators. We then use the DDV equation to obtain the asymptotic expansions of the ${\bf Q}$ - operators at large $λ$; it is remarkable that unlike the expansions of the ${\bf T}$ operators of \ \BLZ, the asymptotic series for ${\bf Q}(λ)$ contains the ``dual'' nonlocal Integrals of Motion along with the local ones. We also discuss an intriguing relation between the vacuum eigenvalues of the ${\bf Q}$ - operators and the stationary transport properties in boundary sine-Gordon model. On this basis we propose a number of new exact results about finite voltage charge transport through the point contact in quantum Hall system.