ODE/IM correspondence and modified affine Toda field equations

Abstract

We study the two-dimensional affine Toda field equations for affine Lie algebra $\hat{\mathfrak{g}}$ modified by a conformal transformation and the associated linear equations. In the conformal limit, the associated linear problem reduces to a (pseudo-)differential equation. For classical affine Lie algebra $\hat{\mathfrak{g}}$, we obtain a (pseudo-)differential equation corresponding to the Bethe equations for the Langlands dual of the Lie algebra $\mathfrak{g}$, which were found by Dorey et al. in study of the ODE/IM correspondence.Comment: 29 pages; added references and note, fixed typos, minor rewritin

Bethe ansatz equations for the classical $A^{(1)}_{n}$ affine Toda field theories

“…Note added -This paper is based on chapters 5 and 6 of [17], submitted by PA for the award of PhD in October 2013. As we were completing the article, we became aware of the preprint [21], which also reports results on the A (1) n massive ODE/IM correspondence and extends the analysis to all simply laced affine Lie algebras and A (2) 2n , D …”

Quantum transfer-matrices for the sausage model

“…These results were extended to massive Integrable Quantum Field Theories (IQFT) [27] (for recent developments, see also refs. [28][29][30][31][32][33][34][35]). The general relation of this type will be referred to in the paper as the ODE/IQFT correspondence.…”

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

“…This was generalized to a relation between the classical Tzitzéica-Bullough-Dodd equation (A (2) 2 algebra) and the quantum Izergin-Korepin model in [9], and was studied for type A (1) r affine Toda theories in [10,11]. In these works it was shown that connection coefficients for subdominant solutions to the linear problem associated with the affine Toda field equation correspond to the vacuum eigenvalues of Q-operators for g-type quantum integrable models.…”

“…In these works it was shown that connection coefficients for subdominant solutions to the linear problem associated with the affine Toda field equation correspond to the vacuum eigenvalues of Q-operators for g-type quantum integrable models. The work of [11] looked at ABCDG-type affine Lie algebras and found that the (pseudo-)ordinary differential equation associated with ĝ ∨ affine Toda field equation was the same as that of [6] for simple Lie algebra g after taking the conformal limit.…”

“…The case of A (2) 2r is unique in that it is non-simply laced yet its Langlands dual is equal to itself. Furthermore, the correspondence in [11] links massive theories associated to the Langlands dual affine algebra ĝ ∨ to conformal quantum theories associated with g in the massless limit, so it is interesting to understand A (2) 2r which does not fit into this scheme in more detail. To investigate the meaning in this case we also propose a new ψ -system for A (2) 2r and give evidence for it by studying the spectral determinant of the ordinary differential equation associated with the linear problem and find its T -Q relations and the Bethe ansatz equations satisfied by Q.…”