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The Yang-Baxter $\sigma$-model is a systematic way to generate integrabledeformations of AdS$_5\times$S$^5$. We recast the deformations as seen by openstrings, where the metric is undeformed AdS$_5\times$S$^5$ with constant stringcoupling, and all information about the deformation is encoded in thenoncommutative (NC) parameter $\Theta$. We identify the deformations of AdS$_5$as twists of the conformal algebra, thus explaining the noncommutativity. Weshow that the unimodularity conditon on $r$-matrices for supergravity solutionstranslates into $\Theta$ being divergence-free. Integrability of the$\sigma$-model for unimodular $r$-matrices implies the existence and planarintegrability of the dual NC gauge theory.

Yang-Baxter $\sigma$-models, conformal twists, and noncommutative Yang-Mills theory