Integrable structure of conformal field theory II. Q-operator and DDV equation

Vladimir V. Bazhanov, Sergei L. Lukyanov, Alexander B. Zamolodchikov

Research output: Contribution to journalArticlepeer-review

306 Scopus citations


This paper is a direct continuation of [1] where we began the study of the integrable structures in Conformal Field Theory. We show here how to construct the operators Q±(λ) which act in the 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 [2], in particular they satisfy Baxter's famous T - Q equation. We also show that under natural assumptions about analytic properties of the operators Q(λ) as the functions of λ the Baxter's relation allows one to derive the nonlinear integral equations of Destri-de Vega (DDV) [3] for the eigenvalues of the Q-operators. We then use the DDV equation to obtain the asymptotic expansions of the Q - operators at large λ; it is remarkable that unlike the expansions of the T operators of [1]. the asymptotic series for 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 Q -operators and the stationary transport properties in the 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 the quantum Hall system.

Original languageEnglish (US)
Pages (from-to)247-278
Number of pages32
JournalCommunications In Mathematical Physics
Issue number2
StatePublished - 1997
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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