Local β-deformations and Yang-Baxter sigma model

Abstract

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 ...

Tri-vector deformations in d = 11 supergravity

“…Yang-Baxter deformations [1][2][3], η-deformed [4,5] and λ-deformed [6,7] σ-models may all be represented by combinations of T-dualities [8,9], as well as their non-abelian [10][11][12] and Poisson-Lie [13] extensions. An element of the T-duality group O(d, d), acting on a supergravity background, can be conveniently represented by the so-called β-shift, parametrised by a bivector β [14,15]. Basic building blocks of integrable deformations in the supergravity language, the Lunin-Maldacena (TsT) [16,17] transformations, correspond to constant β [18].…”

“…Supergravity formulations that are natural to look at in this context are Double [26] and Exceptional [27] Field Theories (DFT and ExFT, respectively). Specifically designed to render supergravities in various dimensions covariant under T-and U-duality groups at the expense of extending the spacetime dimension, they are useful in describing Yang-Baxter deformations [15,[28][29][30][31] and Poisson-Lie T-duality [32][33][34][35][36][37]. The proof of [23] that (1.1), (1.2) is a supergravity symmetry relied upon DFT techniques, in particular the β-supergravity formalism [38][39][40].…”

“…To arrive at the same conclusions one may follow arguments based on generalised Killing vectors of the initial undeformed backgrounds in the spirit of[15].…”

Two-loop conformal invariance for Yang-Baxter deformed strings

“…3 Note however that the form of the α ′ -corrections are only know in special cases and to low loop order, e.g. [24].4 Homogeneous YB deformations also have an O(d, d) interpretation as so called β-shifts [26,27].…”

T-dual superstring Lagrangian with double zweibeins

“…Although we began by only the selfdual currents [10,20], the dimensional reduction constraints involve the antiselfdual currents leading to the Weyl invariant and Lorentz covariant worldsheet Lagrangian. As earlier studies many aspects of superstring Lagrangians with T-duality are examined such as the NS/NS superstring [11], the doubled-yet-gauged spacetime formulation [12][13][14] and the pure spinor [15]. The string background is described by the gravity with the T-duality symmetry.…”