Integrable deformations of coupled sigma-models

We construct integrability-preserving deformations of the integrable $\sigma$-model coupling together $N$ copies of the Principal Chiral Model. These deformed theories are obtained using the formalism of affine Gaudin models, by applying various combinations of Yang-Baxter and $\lambda$-deformations to the different copies of the undeformed model. We describe these models both in the Hamiltonian and Lagrangian formulation and give explicit expressions of their action and Lax pair. In particular, we recover through this construction various integrable $\lambda$-deformed models previously introduced in the literature. Finally, we discuss the relation of the present work with the semi-homolomorphic four-dimensional Chern-Simons theory.