Sergei L. Lukyanov

Integrable structure of conformal field theory, quantum KdV theory and Thermodynamic Bethe Ansatz

AbstractWe construct the quantum versions of the monodromy matrices of KdV theory. The traces of these quantum monodromy matrices, which will be called as “T-operators,” act in highest weight Virasoro modules. TheT-operators depend on the spectral parameter λ and their expansion around λ=∞ generates an infinite set of commuting Hamiltonians of the quantum KdV system. TheT-operators can be viewed as the continuous field theory versions of the commuting transfermatrices of integrable lattice theory. In particular, we show that for the values

Integrable Structure of Conformal Field Theory III. The Yang–Baxter Relation

c = 1 - 3\frac{{3(2n + 1)^2 }}{{2n + 3}}

Exact expectation values of local fields in the quantum sine-Gordon model

,n=1,2,3 .... of the Virasoro central charge the eigenvalues of theT-operators satisfy a closed system of functional equations sufficient for determining the spectrum. For the ground-state eigenvalue these functional equations are equivalent to those of the massless Thermodynamic Bethe Ansatz for the minimal conformal field theoryM2,2n+3; in general they provide a way to generalize the technique of the Thermodynamic Bethe Ansatz to the excited states. We discuss a generalization of our approach to the cases of massive field theories obtained by perturbing these Conformal Field Theories with the operator Φ1,3. The relation of theseT-operators to the boundary states is also briefly described.

Free field representation for massive integrable models

Abstract: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

MULTI-POINT LOCAL HEIGHT PROBABILITIES IN THE INTEGRABLE RSOS MODEL

{\bf Q}_{\pm}(\lambda)

Spectral Determinants for Schrödinger Equation and Q-Operators of Conformal Field Theory

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 Baxters famous T- Q equation. We also show that under natural assumptions about analytic properties of the operators as the functions of λ the Baxters 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.

Form Factors of Exponential Fields in the Sine–Gordon Model

Abstract:In this paper we fill some gaps in the arguments of our previous papers [1,2]. In particular, we give a proof that the L operators of Conformal Field Theory indeed satisfy the defining relations of the Yang–Baxter algebra. Among other results we present a derivation of the functional relations satisfied by T and Q operators and a proof of the basic analyticity assumptions for these operators used in [1,2].

Expectation values of local fields in the Bullough-Dodd model and integrable perturbed conformal field theories

Abstract We develop a method of computing the excited state energies in Integrable Quantum Field Theories (IQFT) in finite geometry, with the spatial coordinate compactified on a circle of circumference R . The IQFT “commuting transfer matrices” introduced earlier [Commun. Math. Phys. 177 (1996) 381] for Conformal Field Theories (CFT) are generalized to non-conformal IQFT obtained by perturbing CFT with the operator Φ 1,3 . We study the models in which the fusion relations for these “transfer matrices” truncate and provide closed integral equations which generalize the equations of the thermodynamic Bethe ansatz to excited states. The explicit calculations are done for the first excited state in the “scaling Lee-Yang model”.