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Expanding upon earlier results [arXiv:1702.02861], we present a compendium of$\sigma$-models associated with integrable deformations of AdS$_5$ generated bysolutions to homogenous classical Yang-Baxter equation. Each example we studyfrom four viewpoints: conformal (Drinfeld) twists, closed string gravitybackgrounds, open string parameters and proposed dual noncommutative (NC) gaugetheory. Irrespective of whether the deformed background is a solution tosupergravity or generalized supergravity, we show that the open string metricassociated with each gravity background is undeformed AdS$_5$ with constantopen string coupling and the NC structure $\Theta$ is directly related to theconformal twist. One novel feature is that $\Theta$ exhibits "holographicnoncommutativity": while it may exhibit non-trivial dependence on theholographic direction, its value everywhere in the bulk is uniquely determinedby its value at the boundary, thus facilitating introduction of a dual NC gaugetheory. We show that the divergence of the NC structure $\Theta$ is directlyrelated to the unimodularity of the twist. We discuss the implementation of anouter automorphism of the conformal algebra as a coordinate transformation inthe AdS bulk and discuss its implications for Yang-Baxter $\sigma$-models andself-T-duality based on fermionic T-duality. Finally, we comment onimplications of our results for the integrability of associated open stringsand planar integrability of dual NC gauge theories.

Conformal Twists, Yang-Baxter $\sigma$-models & Holographic Noncommutativity