A Jordanian deformation of AdS space in type IIB supergravity

We consider a Jordanian deformation of the AdS_5xS^5 superstring action by taking a simple R-operator which satisfies the classical Yang-Baxter equation. The metric and NS-NS two-form are explicitly derived with a coordinate system. Only the AdS part is deformed and the resulting geometry contains the 3D Schrodinger spacetime as a subspace. Then we present the full solution in type IIB supergravity by determining the other field components. In particular, the dilaton is constant and a R-R three-form field strength is turned on. The symmetry of the solution is [SL(2,R)xU(1)^2] x [SU(3)xU(1)] and contains an anisotropic scale symmetry.Comment: 29 pages, no figure, LaTeX, typos corrected, references added, further clarification adde

Jordanian deformations of the AdS_5xS^5 superstring

We consider Jordanian deformations of the AdS_5xS^5 superstring action. The deformations correspond to non-standard q-deformation. In particular, it is possible to perform partial deformations, for example, only for the S^5 part. Then the classical action and the Lax pair are constructed with a linear, twisted and extended R operator. It is shown that the action preserves the kappa-symmetry.Comment: 22 pages, no figure, LaTeX, typos corrected and further clarification adde

Hybrid classical integrable structure of squashed sigma models -- a short summary

We give a short summary of our recent works on the classical integrable structure of two-dimensional non-linear sigma models defined on squashed three-dimensional spheres. There are two descriptions to describe the classical dynamics, 1) the rational description and 2) the trigonometric description. It is possible to construct two different types of Lax pairs depending on the descriptions, and the classical integrability is shown by computing classical r/s-matrices satisfying the extended Yang-Baxter equation in both descriptions. In the former the system is described as an integrable system of rational type. On the other hand, in the latter it is described as trigonometric type. There exists a non-local map between the two descriptions and those are equivalent. This is a non-local generalization of the left-right duality in principal chiral models.Comment: 10 pages, Proceedings of QTS7, Prague, Czech Republic, 201

On the classical equivalence of monodromy matrices in squashed sigma model

We proceed to study the hybrid integrable structure in two-dimensional non-linear sigma models with target space three-dimensional squashed spheres. A quantum affine algebra and a pair of Yangian algebras are realized in the sigma models and, according to them, there are two descriptions to describe the classical dynamics 1) the trigonometric description and 2) the rational description, respectively. For every description, a Lax pair is constructed and the associated monodromy matrix is also constructed. In this paper we show the gauge-equivalence of the monodromy matrices in the trigonometric and rational description under a certain relation between spectral parameters and the rescalings of sl(2) generators.Comment: 32pages, 3figures, references added, introduction and discussion sections revise

Classical integrability of Schrodinger sigma models and q-deformed Poincare symmetry

We discuss classical integrable structure of two-dimensional sigma models which have three-dimensional Schrodinger spacetimes as target spaces. The Schrodinger spacetimes are regarded as null-like deformations of AdS_3. The original AdS_3 isometry SL(2,R)_L x SL(2,R)_R is broken to SL(2,R)_L x U(1)_R due to the deformation. According to this symmetry, there are two descriptions to describe the classical dynamics of the system, 1) the SL(2,R)_L description and 2) the enhanced U(1)_R description. In the former 1), we show that the Yangian symmetry is realized by improving the SL(2,R)_L Noether current. Then a Lax pair is constructed with the improved current and the classical integrability is shown by deriving the r/s-matrix algebra. In the latter 2), we find a non-local current by using a scaling limit of warped AdS_3 and that it enhances U(1)_R to a q-deformed Poincare algebra. Then another Lax pair is presented and the corresponding r/s-matrices are also computed. The two descriptions are equivalent via a non-local map.Comment: 20 pages, no figure, further clarification and references adde

The classical origin of quantum affine algebra in squashed sigma models

We consider a quantum affine algebra realized in two-dimensional non-linear sigma models with target space three-dimensional squashed sphere. Its affine generators are explicitly constructed and the Poisson brackets are computed. The defining relations of quantum affine algebra in the sense of the Drinfeld first realization are satisfied at classical level. The relation to the Drinfeld second realization is also discussed including higher conserved charges. Finally we comment on a semiclassical limit of quantum affine algebra at quantum level.Comment: 25 pages, 2 figure

Hybrid classical integrability in squashed sigma models

We show that SU(2)_L Yangian and q-deformed SU(2)_R symmetries are realized in a two-dimensional sigma model defined on a three-dimensional squashed sphere. These symmetries enable us to develop the two descriptions to describe its classical dynamics, 1) rational and 2) trigonometric descriptions. The former 1) is based on the SU(2)_L symmetry and the latter 2) comes from the broken SU(2)_R symmetry. Each of the Lax pairs constructed in both ways leads to the same equations of motion. The two descriptions are related one another through a non-local map.Comment: 12 pages, LaTeX, typos corrected and references added, further clarification adde

Yangian symmetry in deformed WZNW models on squashed spheres

We introduce a deformation of the Wess-Zumino-Novikov-Witten model with three-dimensional squashed sphere target space. We show how with an appropriate choice of Wess--Zumino and boundary terms it is possible to construct an infinite family of conserved charges realizing an SU(2) Yangian. Finally we discuss the running of the squashing parameter under renormalization group flow.Comment: 12 pages, 1 figure, references adde

Hidden Yangian symmetry in sigma model on squashed sphere

We discuss a hidden symmetry of a two-dimensional sigma model on a squashed S^3. The SU(2) current can be improved so that it can be regarded as a flat connection. Then we can obtain an infinite number of conserved non-local charges and show the Yangian algebra by directly checking the Serre relation. This symmetry is also deduced from the coset structure of the squashed sphere. The same argument is applicable to the warped AdS_3 spaces via double Wick rotations.Comment: 11 pages, 1 figure, typos corrected, references adde