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

The Yang-Baxter $\sigma$-model is a systematic way to generate integrable deformations of ${\mathrm{AdS}}_5\times {\mathrm{S}}^5$. We recast the deformations as seen by open strings, where the metric is undeformed ${\mathrm{AdS}}_5\times {\mathrm{S}}^5$ with constant string coupling, and all information about the deformation is encoded in the noncommutative (NC) parameter $\mathrm{\Theta }$. We identify the deformations of ${\mathrm{AdS}}_5$ as twists of the conformal algebra, thus explaining the noncommutativity. We show that the unimodularity condition on $r$-matrices for supergravity solutions translates into $\mathrm{\Theta }$ being divergence-free. Integrability of the $\sigma$-model for unimodular $r$-matrices implies the existence and planar integrability of the dual NC gauge theory.