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 ${\mathbf{Q}}_{\pm }(\lambda )$ which act in highest weight Virasoro module and commute for different values of the parameter $\lambda$. These operators appear to be the CFT analogs of the $Q$ - matrix of Baxter\ \Baxn, in particular they satisfy famous Baxter's $\mathbf{T}-\mathbf{Q}$ equation. We also show that under natural assumptions about analytic properties of the operators $\mathbf{Q}(\lambda )$ as the functions of $\lambda$ the Baxter's relation allows one to derive the nonlinear integral equations of Destri-de Vega (DDV)\ \dVega\ for the eigenvalues of the $\mathbf{Q}$-operators. We then use the DDV equation to obtain the asymptotic expansions of the $\mathbf{Q}$ - operators at large $\lambda$; it is remarkable that unlike the expansions of the $\mathbf{T}$ operators of \ \BLZ, the asymptotic series for $\mathbf{Q}(\lambda )$ contains the ``dual'' nonlocal Integrals of Motion along with the local ones. We also discuss an intriguing relation between the vacuum eigenvalues of the $\mathbf{Q}$ - 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.

Revised version, 43 pages, harvmac.tex. Minor changes, references added

References in corpus (9)

• Quantum Inverse Scattering Method and Correlation Functions
• Integrable Structure of Conformal Field Theory III. The Yang-Baxter Relation
• Exact Conductance through Point Contacts in the $\nu =1/3$ Fractional Quantum Hall Effect
• Exact non-equilibrium transport through point contacts in quantum wires and fractional quantum Hall devices
• Unified Approach to Thermodynamic Bethe Ansatz and Finite Size Corrections for Lattice Models and Field Theories
• Resonant Tunneling Between Quantum Hall Edge States
• Exact solution of a massless scalar field with a relevant boundary interaction
• Exact perturbative solution of the Kondo problem
• The connection between statistical mechanics and quantum field theory

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