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

Abstract

We study the linear problem associated with modified affine Toda field equation for the Langlands dual g^∨\hat{\mathfrak{g}}^\vee, where g^\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 QQ-functions for g\mathfrak{g}-type quantum integrable models. The ψ\psi-system for the solutions associated with the fundamental representations of g\mathfrak{g} leads to Bethe ansatz equations associated with the affine Lie algebra g^\hat{\mathfrak{g}}. We also study the A2r(2)A^{(2)}_{2r} affine Toda field equation in massless limit in detail and find its Bethe ansatz equations as well as T-Q relations.Comment: 22 page

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