We explain how to obtain new classical integrable field theories by assembling two affine Gaudin models into a single one. We show that the resulting affine Gaudin model depends on a parameter γ in such a way that the limit γ → 0 corresponds to the decoupling limit. Simple conditions ensuring Lorentz invariance are also presented. A first application of this method for σ-models leads to the action announced in [1] and which couples an arbitrary number N of principal chiral model fields on the same Lie group, each with a Wess-Zumino term. The affine Gaudin model descriptions of various integrable σ-models that can be used as elementary building blocks in the assembling construction are then given. This is in particular used in a second application of the method which consists in assembling N − 1 copies of the principal chiral model each with a Wess-Zumino term and one homogeneous Yang-Baxter deformation of the principal chiral model.are expressed in terms of dual bases of g. Here I a n := I a ⊗ t n ∈ g and I a,−n := I a ⊗ t −n ∈ g, while k is the central element of g and d is the derivation element corresponding to the homogeneous gradation of g. In this article we focus on the local realisation of affine Gaudin models, using the terminology of [4], whereby the first tensor factor of J (i) is realised in terms of g-valued connections on the circle and the central elements k (i) are realised as complex numbers i i , called the levels. Explicitly, under this realisation we haveis a g-valued field on the circle with J a(i) (x) := n∈Z I a(i) n e −inx . The Kostant-Kirillov bracket (1.1) for the infinite basis I a(i) translates to the statement that the J (i) (x) are pairwise Poisson commuting Kac-Moody currents with levels i . The Lax matrix of the local affine Gaudin model thus takes the form ϕ(z)∂ x + Γ(z, x) where 2.2 Lax matrix, Hamiltonian and integrability 2.2.1 Lax matrix and twist function Position of the sites. In addition to the Takiff datum defining its algebra of observables T , a local AGM depends on other parameters, including the positions of the sites. These are points z α ∈ R, for α ∈ Σ r and z α ∈ C, for α ∈ Σ c

In the approach recently proposed by K. Costello and M. Yamazaki, which is based on a four-dimensional variant of Chern-Simons theory, we derive a simple and unifying two-dimensional form for the action of many integrable σ-models which are known to admit descriptions as affine Gaudin models. This includes both the Yang-Baxter deformation and the λ-deformation of the principal chiral model. We also give an interpretation of Poisson-Lie T -duality in this setting and derive the action of the E-model. The four-dimensional actionLet G C be a complex semisimple Lie group with Lie algebra g C , on which we fix a choice of non-degenerate invariant symmetric bilinear form ·, · : g C × g C → C.Let CP 1 := C ∪ {∞} denote the Riemann sphere. We shall fix a choice of global holomorphic coordinate z on C ⊂ CP 1 .2.1. Bulk and boundary equations of motion. Consider the action (1.2) where ω is a meromorphic 1-form on CP 1 and the Chern-Simons 3-form for the 1-form A = A σ dσ + A τ dτ + Azdz is given by

A systematic procedure for constructing classical integrable field theories with arbitrarily many free parameters is outlined. It is based on the recent interpretation of integrable field theories as realisations of affine Gaudin models. In this language, one can associate integrable field theories with affine Gaudin models having arbitrarily many sites. We present the result of applying this general procedure to couple together an arbitrary number of principal chiral model fields on the same Lie group, each with a Wess-Zumino term.

We construct the actions of a very broad family of 2d integrable σ-models. Our starting point is a universal 2d action obtained in [arXiv:2008.01829] using the framework of Costello and Yamazaki based on 4d Chern-Simons theory. This 2d action depends on a pair of 2d fields h and L, with L depending rationally on an auxiliary complex parameter, which are tied together by a constraint. When the latter can be solved for L in terms of h this produces a 2d integrable field theory for the 2d field h whose Lax connection is given by L(h). We construct a general class of solutions to this constraint and show that the resulting 2d integrable field theories can all naturally be described as E-models.

By using the general framework of affine Gaudin models, we construct a new class of integrable sigma models. They are defined on a coset of the direct product of N copies of a Lie group over some diagonal subgroup and they depend on 3N − 2 free parameters. For N = 1 the corresponding model coincides with the well-known symmetric space sigma model. Starting from the Hamiltonian formulation, we derive the Lagrangian for the N = 2 case and show that it admits a remarkably simple form in terms of the classical ℛ-matrix underlying the integrability of these models. We conjecture that a similar form of the Lagrangian holds for arbitrary N. Specifying our general construction to the case of SU(2) and N = 2, and eliminating one of the parameters, we find a new three-parametric integrable model with the manifold T1,1 as its target space. We further comment on the connection of our results with those existing in the literature.

Integrable σ-models, such as the principal chiral model, Z T -coset models for T ∈ Z ≥2 and their various integrable deformations, are examples of non-ultralocal integrable field theories described by r/s-systems with twist function. In this general setting, and when the Lie algebra g underlying the r/s-system is of classical type, we construct an infinite algebra of local conserved charges in involution, extending the approach of Evans, Hassan, MacKay and Mountain developed for the principal chiral model and symmetric space σ-model. In the present context, the local charges are attached to certain 'regular' zeros of the twist function and have increasing degrees related to the exponents of the untwisted affine Kac-Moody algebra g associated with g. The Hamiltonian flows of these charges are shown to generate an infinite hierarchy of compatible integrable equations.

The construction of conformal blocks for the analysis of multipoint correlation functions with N > 4 local field insertions is an important open problem in higher dimensional conformal field theory. This is the first in a series of papers in which we address this challenge, following and extending our short announcement in [1]. According to Dolan and Osborn, conformal blocks can be determined from the set of differential eigenvalue equations that they satisfy. We construct a complete set of commuting differential operators that characterize multipoint conformal blocks for any number N of points in any dimension and for any choice of OPE channel through the relation with Gaudin integrable models we uncovered in [1]. For 5-point conformal blocks, there exist five such operators which are worked out smoothly in the dimension d.

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