An integrable deformation of the type IIB AdS5×S 5 superstring action is presented. The deformed field equations, Lax connection, and κ-symmetry transformations are given. The original psu(2, 2|4) symmetry is expected to become q-deformed.

A procedure is developed for constructing deformations of integrable σ-models which are themselves classically integrable. When applied to the principal chiral model on any compact Lie group F , one recovers the Yang-Baxter σ-model introduced a few years ago by C. Klimčík. In the case of the symmetric space σ-model on F/G we obtain a new one-parameter family of integrable σ-models. The actions of these models correspond to a deformation of the target space geometry and include a torsion term. An interesting feature of the construction is the q-deformation of the symmetry corresponding to left multiplication in the original models, which becomes replaced by a classical q-deformed Poisson-Hopf algebra. Another noteworthy aspect of the deformation in the coset σ-model case is that it interpolates between a compact and a non-compact symmetric space. This is exemplified in the case of the SU(2)/U(1) coset σ-model which interpolates all the way to the SU(1, 1)/U(1) coset σ-model.

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

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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