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 ...

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. *

Yang-Baxter (YB) deformations of string sigma model provide deformed target spaces. We propose that homogeneous YB deformations always lead to a certain class of β-twisted backgrounds and represent the bosonic part of the supergravity fields in terms of the classical r-matrix associated with the YB deformation. We then show that various β-twisted backgrounds can be realized by considering generalized diffeomorphisms in the undeformed background. Our result extends the notable relation between the YB deformations and (non-commuting) TsT transformations. We also discuss more general deformations beyond the YB deformations.

Abstract:Recently, generalized equations of type IIB supergravity have been derived from the requirement of classical kappa-symmetry of type IIB superstring theory in the Green-Schwarz formulation. These equations are covariant under generalized T -duality transformations and hence one may expect a formulation similar to double field theory (DFT). In this paper, we consider a modification of the DFT equations of motion by relaxing a condition for the generalized covariant derivative with an extra generalized vector. In this modified double field theory (mDFT), we show that the flatness condition of the modified generalized Ricci tensor leads to the NS-NS part of the generalized equations of type IIB supergravity. In particular, the extra vector fields appearing in the generalized equations correspond to the extra generalized vector in mDFT. We also discuss duality symmetries and a modification of the string charge in mDFT.