Homogeneous Yang-Baxter (YB) deformation of AdS 5 × S 5 superstring is revisited.We calculate the YB sigma model action up to quadratic order in fermions and show that homogeneous YB deformations are equivalent to β-deformations of the AdS 5 × S 5 background when the classical r-matrices consist of bosonic generators. In order to make our discussion clearer, we discuss YB deformations in terms of the double-vielbein formalism of double field theory. We further provide an O(10, 10)-invariant string action that reproduces the Green-Schwarz type II superstring action up to quadratic order in fermions. When an AdS background contains a non-vanishing H-flux, it is not straightforward to perform homogeneous YB deformations. In order to get any hint for such YB deformations, we study β-deformations of H-fluxed AdS backgrounds and obtain various solutions of (generalized) type II Yang-Baxter (YB) sigma model was originally introduced by Klimčík [1] as a class of Poisson-Lie symmetric sigma models. It is characterized by a classical r-matrix that satisfies the modified classical YB equation (mCYBE). It was later shown to be integrable by constructing the Lax pair [2]. The original YB sigma model can be applied only to sigma models on group manifolds, but it was later generalized to coset sigma models in [3] and to the case of the homogeneous classical YB equation (CYBE) in [4].An interesting application of YB deformations is an integrable deformation of type IIB superstring theory on the AdS 5 × S 5 background [5][6][7], that has been studied in the context of the AdS/CFT correspondence. Through various examples [8][9][10][11][12][13], it turned out that, when we employ an Abelian classical r-matrix, the YB-deformed AdS 5 ×S 5 superstring can be described as type IIB superstring on a TsT-transformed 1 AdS 5 × S 5 background [14-20] (see [21] for a clear explanation and generalizations). Namely, Abelian YB deformation was found to be equivalent to a TsT-transformation. For non-Abelian classical r-matrices, the deformations of the AdS 5 × S 5 background have not been understood clearly; some deformed backgrounds were obtained through non-commuting TsT-transformations (see for example [22]) and some were obtained through a combination of diffeomorphisms and T -dualities [23], but it is not clear whether an arbitrary YB deformation can be realized as a combination of Abelian Tdualities and gauge symmetries of the supergravity (it was recently shown in [24][25][26][27][28] that YB deformations can be also reproduced from non-Abelian T -dualities [29][30][31][32][33][34][35][36][37][38]). As shown in a seminal paper [22], at least when an r-matrix satisfies a certain criterion called unimodularity, the deformed AdS 5 × S 5 background are solutions of type IIB supergravity. Moreover, for a non-unimodular r-matrix, the deformed AdS 5 × S 5 background was shown to satisfy the generalized supergravity equations of motion (GSE) [39,40], and a Killing vector I m appearing in the GSE was determined for a general r-matrix. In a recent ...

We study Yang-Baxter deformations of the AdS 5 ×S 5 superstring with non-Abelian classical r-matrices which satisfy the homogeneous classical Yang-Baxter equation (CYBE). By performing a supercoset construction, we can get deformed AdS 5 × S 5 backgrounds. While this is a new area of research, the current understanding is that Abelian classical r-matrices give rise to solutions of type IIB supergravity, while nonAbelian classical r-matrices lead to solutions of the generalized supergravity equations.We examine here some examples of non-Abelian classical r-matrices and derive the associated backgrounds explicitly. All of the resulting backgrounds satisfy the generalized equations. For some of them, we derive "T-dualized" backgrounds by adding a linear coordinate dependence to the dilaton and show that these satisfy the usual type IIB supergravity equations. Remarkably, some of the "T-dualized" backgrounds are locally identical to undeformed AdS 5 × S 5 after an appropriate coordinate transformation, but this seems not to be generally the case.arXiv:1607.00795v2 [hep-th]

The Yang-Baxter σ-model is a systematic way to generate integrable deformations of AdS5×S 5 . We recast the deformations as seen by open strings, where the metric is undeformed AdS5×S 5 with constant string coupling, and all information about the deformation is encoded in the noncommutative (NC) parameter Θ. We identify the deformations of AdS5 as twists of the conformal algebra, thus explaining the noncommutativity. We show that the unimodularity condition on r-matrices for supergravity solutions translates into Θ being divergence-free. Integrability of the σ-model for unimodular r-matrices implies the existence and planar integrability of the dual NC gauge theory.

It has recently been shown that a set of the generalized type IIB supergravity equations follows from the requirement of kappa symmetry of the type IIB Green-Schwarz superstring theory defined on an arbitrary background. In this paper, we show that the whole bosonic part of the generalized type II supergravity equations can be reproduced from the T -duality covariant equations of motion of the double field theory by choosing a non-standard solution of the strong constraint. Then, by using the doubled formalism, we show the Weyl invariance of the bosonic string sigma model on a generalized gravity background. According to the dual-coordinate dependence of the dilaton, the Fradkin-Tseytlin term nicely removes the Weyl anomaly. This result seems likely to support that string theories can be consistently defined on arbitrary generalized supergravity backgrounds. *

Expanding upon earlier results (Araujo et al 2017 Phys. Rev. D 95 105006), we present a compendium of σ-models associated with integrable deformations of AdS5 generated by solutions to homogenous classical Yang–Baxter equation. Each example we study from four viewpoints: conformal (Drinfeld) twists, closed string gravity backgrounds, open string parameters and proposed dual noncommutative (NC) gauge theory. Irrespective of whether the deformed background is a solution to supergravity or generalized supergravity, we show that the open string metric associated with each gravity background is undeformed AdS5 with constant open string coupling and the NC structure Θ is directly related to the conformal twist. One novel feature is that Θ exhibits ‘holographic noncommutativity’: while it may exhibit non-trivial dependence on the holographic direction, its value everywhere in the bulk is uniquely determined by its value at the boundary, thus facilitating introduction of a dual NC gauge theory. We show that the divergence of the NC structure Θ is directly related to the unimodularity of the twist. We discuss the implementation of an outer automorphism of the conformal algebra as a coordinate transformation in the AdS bulk and discuss its implications for Yang–Baxter σ-models and self-T-duality based on fermionic T-duality. Finally, we comment on implications of our results for the integrability of associated open strings and planar integrability of dual NC gauge theories.