Massive ODE/IM Correspondence and Non-linear Integral Equations for $A_r^{(1)}$-type modified Affine Toda Field Equations

TL;DR

This paper explores the connection between linear problems and integrable models in affine Toda field equations, deriving Bethe ansatz and non-linear integral equations, and analyzing the UV limit for $A_r^{(1)}$-type models.

Contribution

It derives non-linear integral equations for the $A_r^{(1)}$-type modified affine Toda equations from the Bethe ansatz, advancing understanding of the massive ODE/IM correspondence.

Findings

Derived Bethe ansatz equations from the linear problem.

Obtained non-linear integral equations for the Q-functions.

Computed the effective central charge in the UV limit.

Abstract

The massive ODE/IM correspondence is a relation between the linear problem associated with modified affine Toda field equations and two-dimensional massive integrable models. We study the massive ODE/IM correspondence for the $A_r^{(1)}$-type modified affine Toda field equations. Based on the $\psi$-system satisfied by the solutions of the linear problem, we derive the Bethe ansatz equations and determine the asymptotic behavior of the Q-functions for large value of the spectral parameter. We derive the non-linear integral equations for the Q-functions from the Bethe ansatz equations. We compute the effective central charge in the UV limit, which is identified with the one of the non-unitary $WA_r$ minimal models when the solution has trivial monodromy around the origin of the complex plane.