Constrained affine Gaudin models and diagonal Yang-Baxter deformations

We review and pursue further the study of constrained realisations of affine Gaudin models, which form a large class of two-dimensional integrable field theories with gauge symmetries. In particular, we develop a systematic gauging procedure which allows to reformulate the non-constrained realisations of affine Gaudin models considered recently in Delduc et al (2019 J. High Energy Phys. JHEP06(2019)017) as equivalent models with a gauge symmetry. This reformulation is then used to construct integrable deformations of these models breaking their diagonal symmetry. In the second part of the article, we apply these general methods to the integrable coupled σ-model introduced recently, whose target space is the N-fold Cartesian product G0N of a real semi-simple Lie group G0. We present its gauged formulation as a model on G0N+1 with a gauge symmetry acting as the right multiplication by the diagonal subgroup G0diag and construct its diagonal homogeneous Yang-Baxter deformation.