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

Abstract

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

6 pages, no figure, v2: accepted in PRD, clarifications and references added. The title slightly changed

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