Integrable structure of W_3 Conformal Field Theory, Quantum Boussinesq Theory and Boundary Affine Toda Theory

TL;DR

This paper explores the integrable structure of W_3 conformal field theory, constructing operators that reveal its conserved quantities and linking quantum and classical integrable models.

Contribution

It explicitly constructs T- and Q-operators for W_3 CFT, formulates conjectures on their properties, and connects quantum integrals of motion to classical Boussinesq Hamiltonians.

Findings

Asymptotic expansion of T-operators yields local integrals of motion.

Vacuum eigenvalues of Q-operators relate to boundary exponential fields.

Classical limit reproduces conserved Hamiltonians of Boussinesq equation.

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(\hat{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.