Yang-Baxter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>σ</mml:mi></mml:math>-models, conformal twists, and noncommutative Yang-Mills theory

Abstract

The Yang-Baxter $\ensuremath{\sigma}$-model is a systematic way to generate integrable deformations of ${\mathrm{AdS}}_{5}\ifmmode\times\else\texttimes\fi{}{\mathrm{S}}^{5}$. We recast the deformations as seen by open strings, where the metric is undeformed ${\mathrm{AdS}}_{5}\ifmmode\times\else\texttimes\fi{}{\mathrm{S}}^{5}$ with constant string coupling, and all information about the deformation is encoded in the noncommutative (NC) parameter $\mathrm{\ensuremath{\Theta}}$. 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 $\mathrm{\ensuremath{\Theta}}$ being divergence-free. Integrability of the $\ensuremath{\sigma}$-model for unimodular $r$-matrices implies the existence and planar integrability of the dual NC gauge theory.