ODE/IM correspondence and Bethe ansatz for affine Toda field equations

TL;DR

This paper explores the connection between affine Toda field equations and quantum integrable models, establishing a correspondence with Bethe ansatz equations and T-Q relations through linear problems and connection coefficients.

Contribution

It demonstrates the ODE/IM correspondence for affine Toda equations related to the Langlands dual algebra, deriving Bethe ansatz equations and T-Q relations for these models.

Findings

Connection coefficients correspond to Q-functions in quantum integrable models.

Derived Bethe ansatz equations from the psi-system for affine Toda solutions.

Analyzed the massless limit of the A^{(2)}_{2r} affine Toda field equation.

Abstract

We study the linear problem associated with modified affine Toda field equation for the Langlands dual $\hat{\mathfrak{g}}^\vee$, where $\hat{\mathfrak{g}}$ is an untwisted affine Lie algebra. The connection coefficients for the asymptotic solutions of the linear problem are found to correspond to the $Q$-functions for $\mathfrak{g}$-type quantum integrable models. The $\psi$-system for the solutions associated with the fundamental representations of $\mathfrak{g}$ leads to Bethe ansatz equations associated with the affine Lie algebra $\hat{\mathfrak{g}}$. We also study the $A^{(2)}_{2r}$ affine Toda field equation in massless limit in detail and find its Bethe ansatz equations as well as T-Q relations.