Yang–Baxter deformation as an $\boldsymbol {O(d,d)}$ transformation

Abstract

We show that the homogeneous Yang-Baxter (YB) deformation of Green-Schwarz sigma models manifests itself as the action of a coordinate dependent O(d, d) matrix on the target space fields both in the NS-NS and the RR sectors. When the R-matrix that determines the YB deformation is Abelian, the O(d, d) matrix reduces to the constant matrix that produces Lunin-Maldacena deformations (TsT deformations), in agreement with the well established fact that homogeneous YB deformations are a generalization of LM deformations. Our approach gives a natural embedding of the homogeneous YB model in Double Field Theory (DFT), a framework which provides an O(d, d) covariant formulation for effective string actions. We show that the YB deformed fields can be regarded as duality twisted fields in the context of Gauged Double Field Theory (GDFT). We compute the fluxes associated with the twist and show that the conditions on the R-matrix determining the YB deformation give rise to conditions on the fluxes on the (G)DFT side. We find that the R-flux vanishes if and only if the R-matrix satisfies the classical Yang-Baxter equation, and the unimodularity of the R-matrix implies that the Q-flux is traceless. Non-unimodularity of the R-matrix forces the generalized dilaton field to pick up a linear dependence on the winding type coordinates of DFT, implying that the corresponding supergravity fields should satisfy generalized supergravity equations. ♯ ozerayb@itu.edu.tr ♭ tunali16@itu.edu.tr IntroductionConstructing integrable deformations of string backgrounds relevant for AdS/CFT correspondence is an active line of research. An important milestone in this direction was the work of Klimcik [1], where he introduced a particular type of deformation for Principal Chiral Models (PCM) with simple compact Lie group as the target manifold. The resulting deformed sigma model was dubbed Yang-Baxter sigma model, as the deformation is based on solutions of modified classical Yang-Baxter equation (mCYBE). The integrability of the YB sigma model was proved later in [2]. The applicability of YB deformations was extended to symmetric coset spaces in [3], which in turn was applied to AdS 5 × S 5 in [4]. The NS-NS sector of the corresponding deformed supergravity background was found in [5]. Similar deformations for the NS-NS sector of AdS n × S n supercosets was studied in [6]. The RR sectors of these deformed backgrounds was worked out in [7], and was successfully established for the n = 2 and n = 3 cases. The most interesting n = 5 case could not be understood fully in that work and the full Lagrangian for the deformed AdS 5 × S 5 was found later in [8]. We should note that such deformations of the supergravity backgrounds are usually called η-deformations.It is also possible to consider similar deformations based on the classical Yang-Baxter equation (CYBE), as opposed to mCYBE. Following the literature, we call the resulting models homogeneous Yang-Baxter models. Such deformations of the AdS 5 × S 5 string was first studied in [9]. These deformatio...

Supergravity solution-generating techniques and canonical transformations of σ-models from O(D, D)

“…which coincides with the known parametrisation commonly used for the generalised metric of DFT. 4 Canonical Poisson brackets for x m and p m translate into Poisson 2 Invariance of the generalised fluxes under NATD and PLTD was observed and used already in [13,14,[21][22][23][24], and in the case of the Yang-Baxter deformations that we discuss later in [25,26]. Ref.…”

“…In the non-compact case there is no general theorem relating the Lie algebra and de Rham cohomologies. 25 Putting global issues aside, the equation H ijk = 0 may be solved for example by b ij =b ij +b ij withb ij = ωT i , T j andb ij = ω g T i , T j , whereω g = W −1 •ωḡ •W , ωḡ = Ad −1 g •ω • Adḡ and bothω andω are constant 2-cocycles that are not 2-coboundaries. The equation for β ij coming from Q i jk = 0 is…”

Nonlocal charges from marginal deformations of 2D CFTs: Holographic TT¯ & TJ¯<

“…As one important example of the power of twodimensional theories for our present work, there is a theorem which determines the necessary and sufficient conditions for exactly marginality of 2D CFT deformations given by a product of holomorphic and antiholomorphic currents, namely O ð1;1Þ ¼ JðzÞJðzÞ [43,44]; see also [45]. Additionally, it has been argued that this class of marginal deformations can be written as Oðd; dÞ transformations [41,42,[45][46][47][48][49][50][51]. In this respect, the authors [42] extended important results on integrable exactly marginal deformations of two-dimensional CFTs using Oðd; dÞ deformations, and they have also shown how to build the nonlocal Lax pairs of the deformed models which guarantee their classical integrability.…”

E6(6) exceptional Drinfel’d algebras

“…In the context of DFT, Yang-Baxter deformations can be understood as the O (d, d) transformation, generated by a bivector, acting on a Drinfel'd double with vanishing dual structure constants [49][50][51][52][53][54]. The bivector that generates the transformation is then related to the classical r-matrix, the dual structure constants are coboundaries and the requirement that the O(d, d) transformed algebra is a Drinfel'd double is precisely the classical Yang-Baxter equation.…”