Conformal twists, Yang–Baxter σ-models & holographic noncommutativity

Abstract

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.

T -folds from Yang-Baxter deformations

“…Various solutions of GSE have been obtained from YB deformations with non-unimodular classical r-matrices [52,54]. As it has been shown in [23][24][25][26], classical r-matrices, which characterize homogeneous YB deformations of AdS 5 geometry, are closely related to non-commutative parameters in the dual open-string description and, as pointed out in [33], they are nothing but β-fields [66]. In terms of the…”

“…In fact, an application of this scheme to type IIB superstring on AdS 5 ×S 5 [10][11][12] opened up a lot of new perspectives and directions including intriguing relations among YB deformations and non-commutative gauge theories, non-Abelian T -duality [13][14][15][16][17][18][19][20][21][22], and manifestly T -/U -duality covariant formulations, which have been discovered in [23][24][25][26][27][28][29], and [30][31][32][33], respectively. Our concern here is to delve deeper into the relation to a manifestly T -duality covariant formulation called Double Field Theory (DFT) [34][35][36][37][38] with particular emphasis on non-geometric aspects (i.e., the global nature) of YB-deformed backgrounds.…”

Classical Yang-Baxter equation from supergravity

“…The NS sector of generalized supergravity is hence characterized by ðg; B; Φ; IÞ. We note that the B-field is specified up to the Λ-gauge transformation, B → B þ dΛ and (1) and (2) are written in a particular Λ-gauge, while X, which appears in the EOMs of generalized supergravity, is Λ-gauge invariant [20,30]. However, this leaves the residual symmetry of shifting Φ by a constant, Φ → Φ þ Φ 0 , without changing B.…”

“…For TsT transformations, Θ is a constant [30], so that I ¼ 0 and the final solution is a bona fide supergravity solution. This completes our description of the NS sector transformation.…”

“…Its connection to T-duality has been exploited in actions that make nongeometric fluxes manifest [23,24] and string theory explanations [25,26] of the Ω-deformation [27,28]. More recently, it was noted [29,30] that the closed-open string map undoes integrable deformations of σ-models [14][15][16][17].…”