Integrable structure of W₃ Conformal Field Theory, quantum boussinesq theory and boundary affine Toda theory

In this paper we study the Yang-Baxter integrable structure of Conformal Field Theories with extended conformal symmetry generated by the W₃ algebra. We explicitly construct various T and Q-operators which act in the irreducible highest weight modules of the W₃ 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 W₃ 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 W₃ 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.