Integrable structure of Conformal Field Theory, Quantum Boussinesq Theory and Boundary Affine Toda Theory

Abstract

In this paper we study the Yang-Baxter integrable structure of Conformal Field Theories with extended conformal symmetry generated by the W 3 algebra. We explicitly construct various T and Q-operators which act in the irreducible highest weight modules of the W 3 algebra. These operators can be viewed as continuous field theory analogues of the commuting transfer matrices and Q-matrices of the integrable lattice systems associated with the quantum algebra U q ( sl (3)). We formulate several conjectures detailing certain analytic characteristics of the Q-operators and propose exact asymptotic expansions of the T and Q-operators at large values of the spectral parameter. We show, in particular, that the asymptotic expansion of the T-operators generates an infinite set of local integrals of motion of the W 3 CFT which in the classical limit reproduces an infinite set of conserved Hamiltonians associated with the classical Boussinesq equation. We further study the vacuum eigenvalues of the Q-operators (corresponding to the highest weight vector of the W 3 module) and show that they are simply related to the expectation values of the boundary exponential fields in the non-equilibrium boundary affine Toda field theory with zero bulk mass.

An(1)affine Toda field theories with integrable boundary conditions revisited

“…[12,43,34]). In this case instead of assuming a c-number solution of the classical version of the q-twisted Yangian (2.11) we consider a generic -dynamical-representation of the algebra defined as [2]:…”

“…a q-oscillator. Such boundary conditions for the A ATFT have been analyzed in [12]. More precisely, in this case the entries of K are not cnumber anymore, but algebraic objects satisfying Poisson commutation relations dictated by the underlying classical algebra.…”

“…Generic affine Toda field theories with classical integrable boundary conditions were first analyzed more than a decade ago in [11]. A different point of view, although regarding the same class of boundary conditions 2 analyzed in [11], is presented in [12]. Specifically, in [12] the A (1) 2 ATFT with 'dynamical' boundary conditions -that is a quantum mechanical system is attached at the boundary-is investigated.…”

“…A different point of view, although regarding the same class of boundary conditions 2 analyzed in [11], is presented in [12]. Specifically, in [12] the A (1) 2 ATFT with 'dynamical' boundary conditions -that is a quantum mechanical system is attached at the boundary-is investigated. Further studies regarding the boundary ATFT at both classical and quantum level may be also found in various articles (see e.g.…”

Quantum supersymmetric Toda–mKdV hierarchies

“…[15], [16] and below). In this section we give the generalization of the constructions appeared in [4], [5]. First, let's quantize the scalar fields φ i :…”