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

TL;DR

This paper explores the connection between integrable quantum field theories related to SU(3) and third-order differential equations, extending known correspondences and deriving new integral equations with numerical validation.

Contribution

It establishes a relationship between SU(3)-related integrable models and third-order differential equations, introducing a nonlinear integral equation for spectral problems and discussing duality and finite volume energies.

Findings

Derived a nonlinear integral equation for the spectral problem.

Numerical checks support the proposed correspondence.

Discussed duality properties and finite volume energy descriptions.

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.