On Integrable Field Theories as Dihedral Affine Gaudin Models

We introduce the notion of a classical dihedral affine Gaudin model, associated with an untwisted affine Kac–Moody algebra $\widetilde{\mathfrak{g}}$ equipped with an action of the dihedral group $D_{2T}$, $T \geq 1$ through (anti-)linear automorphisms. We show that a very broad family of classical integrable field theories can be recast as examples of such classical dihedral affine Gaudin models. Among these are the principal chiral model on an arbitrary real Lie group $G_0$ and the $\mathbb{Z}_T$-graded coset $\sigma $-model on any coset of $G_0$ defined in terms of an order $T$ automorphism of its complexification. Most of the multi-parameter integrable deformations of these $\sigma $-models recently constructed in the literature provide further examples. The common feature shared by all these integrable field theories, which makes it possible to reformulate them as classical dihedral affine Gaudin models, is the fact that they are non-ultralocal. In particular, we also obtain affine Toda field theory in its lesser-known non-ultralocal formulation as another example of this construction. We propose that the interpretation of a given classical non-ultralocal integrable field theory as a classical dihedral affine Gaudin model provides a natural setting within which to address its quantisation. At the same time, it may also furnish a general framework for understanding the massive ordinary differential equations (ODE)/integrals of motion (IM) correspondence since the known examples of integrable field theories for which such a correspondence has been formulated can all be viewed as dihedral affine Gaudin models.