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 $\hat{\mathfrak{g}}^\vee$, 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.Comment: 22 page

On Integrable Field Theories as Dihedral Affine Gaudin Models

“…Recall that in Corollary 4.5 we applied the tensor product of Lie algebra homomorphisms ⊗ ⊗ π to the Poisson brackets of formal fields from Proposition 4.4 to obtain the Poisson brackets of the g-valued classical fields (53). In this section we similarly apply the linear map ⊗ ⊗ π ⊗ id to the algebra of Lax matrices (63) from Proposition 4.7.…”

“…Specifically, they showed that the functional relations and analytic properties characterising the vacuum eigenvalues of the Q-operators of quantum sine/sinh-Gordon theory were the same as those satisfied by certain connection coefficients of the auxiliary linear problem of the classical modified sinh-Gordon equation for a suitably chosen classical solution. Subsequently, various higher rank generalisations of this massive ODE/IM correspondence for quantum affine g-Toda field theories were also conjectured, when g is of type A for rank 3 in [24] and for general rank n in [1], and more recently for a general untwisted affine Kac-Moody algebra g in [52,53] as well as examples of twisted type in [54]. Another important quantum integrable field theory for which a massive ODE/IM correspondence has been conjectured in [63], and further studied in [7,6], is the Fateev model [35].…”

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 the Argyres-Douglas theory

“…For a classical Lie algebra g the ODEs have been constructed by [19,27,22]. Let us consider the linear problem for the modified affine Toda field equation forĝ of a simply laced Lie algebra g of rank n. In the conformal limit [20,22,24], it takes the form:…”

“…The ordinary differential equations for the integrable models associated with the Bethe equations of a classical Lie algebra g have been proposed in [19]. For a general Lie algebra g, it is obtained from the conformal limit of the linear problem for the modified affine Toda field equations associated withĝ ∨ , while the full linear problem describes the massive version of the ODE/IM correspondence [20,21,22,23,24,25,26] (see also [27] for a classical Lie algebra case). Hereĝ denotes the untwisted affine Lie algebra of g andĝ ∨ the Langlands dual ofĝ.…”