Integrable structure of [formula omitted] 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 nonequilibrium boundary affine Toda field theory with zero bulk mass.