Differential equations and integrable models: the SU(3) case

Abstract

We exhibit a relationship between the massless \(a_2^{(2)}\) integrable quantum field theory and a certain third-order ordinary differential equation, thereby extending a recent result connecting the massless sine-Gordon model to the Schr\"odinger equation. This forms part of a more general correspondence involving \(A_2\)-related Bethe ansatz systems and third-order differential equations. A non-linear integral equation for the generalised spectral problem is derived, and some numerical checks are performed. Duality properties are discussed, and a simple variant of the nonlinear equation is suggested as a candidate to describe the finite volume ground state energies of minimal conformal field theories perturbed by the operators \(\phi_{12}\), \(\phi_{21}\) and \(\phi_{15}\). This is checked against previous results obtained using the thermodynamic Bethe ansatz.