We consider the ABF background underlying the η-deformed AdS 5 × S 5 sigma model. This background fails to satisfy the standard IIB supergravity equations which indicates that the corresponding sigma model is not Weyl invariant, i.e. does not define a critical string theory in the usual sense. We argue that the ABF background should still define a UV finite theory on a flat 2d world-sheet implying that the η-deformed model is scale invariant. This property follows from the formal relation via T-duality between the η-deformed model and the one defined by an exact type IIB supergravity solution that has 6 isometries albeit broken by a linear dilaton. We find that the ABF background satisfies candidate type IIB scale invariance conditions which for the R-R field strengths are of the second order in derivatives. Surprisingly, we also find that the ABF background obeys an interesting modification of the standard IIB supergravity equations that are first order in derivatives of R-R fields. These modified equations explicitly depend on Killing vectors of the ABF background and, although not universal, they imply the universal scale invariance conditions. Moreover, we show that it is precisely the non-isometric dilaton of the T-dual solution that leads, after T-duality, to modification of type II equations from their standard form. We conjecture that the modified equations should follow from κ-symmetry of the η-deformed model. All our observations apply also to η-deformations of AdS 3 × S 3 × T 4 and AdS 2 × S 2 × T 6 models.conformality. In particular, the TsT deformed background is a solution of type IIB supergravity.2 This additional term is certainly required to reproduce the standard 1-loop Weyl-invariance conditions for the G and B-field couplings or supergravity equations in NS-NS sector. This term should also be required to cancel the quantum anomaly of κ-symmetry.(i) the scale invariance conditions for the type II superstring sigma model (with equations on the R-R fields F being of 2nd order in derivatives) (ii) a set of equations that are structurally similar to those of type II supergravity (with 1storder equations for the R-R fields F) but involving, instead of derivatives of the dilaton, a certain co-vector Z m playing now the role of the dilaton one-form and a Killing vector I m responsible for the "modification" of the equations from their standard form. 4While the scale invariance conditions are universal, the second set of equations (which we shall refer to as "I-modified" type II equations) only apply to particular backgrounds with isometric G, B and F-fields, which are related by formal T-duality to a type II solution (Ĝ,B,F,φ) with the dilatonφ containing a term linear in the isometric coordinates. Such a dilaton background, breaking isometries by a linear term only, is special. As the type II supergravity equations written in terms of the F-fields only depend on the dilaton through its derivatives, they remain independent of the isometric directions. As a result, the standard type II supergravity equ...

We study the deformed AdS 5 × S 5 supercoset model of arXiv:1309.5850 which depends on one parameter κ and has classical quantum group symmetry. We confirm the conjecture that in the "maximal" deformation limit, κ → ∞, this model is T-dual to "flipped" double Wick rotation of the target space AdS 5 × S 5 , i.e. dS 5 × H 5 space supported by an imaginary 5-form flux. In the imaginary deformation limit, κ → i, the corresponding target space metric is of a pp-wave type and thus the resulting light-cone gauge S-matrix becomes relativistically invariant. Omitting non-unitary contributions of imaginary WZ terms, we find that this tree-level S-matrix is equivalent to that of the generalized sine-Gordon model representing the Pohlmeyer reduction of the undeformed AdS 5 × S 5 superstring model. We also study in some detail similar deformations of the AdS 3 × S 3 and AdS 2 × S 2 supercosets. The bosonic part of the deformed AdS 3 × S 3 model happens to be equivalent to the symmetric case of the sum of the Fateev integrable deformation of the SL(2) and SU(2) principal chiral models, while in the AdS 2 × S 2 case the role of the Fateev model is played by the 2d "sausage" model. The κ = i limits are again directly related to the Pohlmeyer reductions of the corresponding AdS n × S n supercosets: (2,2) super sine-Gordon model and its complex sine-Gordon analog. We also discuss possible deformations of AdS 3 × S 3 with more than one parameter.

We consider superstring theory on AdS 3 × S 3 × T 4 supported by a combination of RR and NSNS 3-form fluxes (with parameter of the NSNS 3-form q). This theory interpolates between the pure RR flux model (q = 0) whose spectrum is expected to be described by a (thermodynamic) Bethe ansatz and the pure NSNS flux model (q = 1) which is described by the supersymmetric extension of the SL(2, R) × SU (2) WZW model. As a first step towards the solution of this integrable theory for generic value of q we compute the corresponding tree-level S-matrix for massive BMN-type excitations. We find that this S-matrix has a surprisingly simple dependence on q: the diagonal amplitudes have exactly the same structure as in the q = 0 case but with the BMN dispersion relation e 2 = p 2 + 1 replaced by the one with shifted momentum and mass, e 2 = (p ± q) 2 + 1 − q 2 . The off-diagonal amplitudes are then determined from the classical Yang-Baxter equation. We also construct the Pohlmeyer reduced model corresponding to this superstring theory and find that it depends on q only through the rescaled mass parameter, µ → 1 − q 2 µ, implying that its relativistic S-matrix is q-independent.

We propose that the Yang-Baxter deformation of the symmetric space σ-model parame- This also includes the special case of deformations based on abelian r-matrices, which correspond to TsT transformations: they are equivalent to non-abelian duals of the original model with respect to a central extension of abelian subalgebras.

A set of four factorizable non-relativistic S-matrices for a multiplet of fundamental particles are defined based on the R-matrix of the quantum group deformation of the centrally extended superalgebra su(2|2). The S-matrices are a function of two independent couplings g and q = e iπ/k . The main result is to find the scalar factor, or dressing phase, which ensures that the unitarity and crossing equations are satisfied. For generic (g, k), the S-matrices are branched functions on a product of rapidity tori. In the limit k → ∞, one of them is identified with the S-matrix describing the magnon excitations on the string world sheet in AdS 5 × S 5 , while another is the mirror S-matrix that is needed for the TBA. In the g → ∞ limit, the rapidity torus degenerates, the branch points disappear and the S-matrices become meromorphic functions, as required by relativistic S-matrix theory. However, it is only the mirror S-matrix which satisfies the correct relativistic crossing equation. The mirror S-matrix in the relativistic limit is then closely related to that of the semi-symmetric space sine-Gordon theory obtained from the string theory by the Pohlmeyer reduction, but has anti-symmetric rather than symmetric bound states. The interpolating S-matrix realizes at the quantum level the fact that at the classical level the two theories correspond to different limits of a one-parameter family of symplectic structures of the same integrable system. arXiv:1112.4485v1 [hep-th]