# On integrable field theories as dihedral affine Gaudin models

## TL;DR

This paper introduces classical dihedral affine Gaudin models, unifying various non-ultralocal integrable field theories, including principal chiral models and affine Toda theories, within a common algebraic framework to facilitate quantization and deepen understanding of their structures.

## Contribution

It defines dihedral affine Gaudin models associated with affine Kac-Moody algebras and shows many known integrable field theories are special cases, providing a new algebraic perspective.

## Key findings

- Reformulation of principal chiral and coset sigma models as dihedral affine Gaudin models
- Identification of non-ultralocal integrable theories within this framework
- Potential for addressing quantization and the ODE/IM correspondence

## Abstract

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 ODE/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.