A new perspective on non-commutative deformations of field and gauge theories

Abstract

We construct non-commutative deformations of field and gauge theories based on star-products implemented by Drinfel'd twists. We are able to encompass a large family of twists, including those built out of conformal symmetries and supersymmetries. The main idea behind our construction is to work with twists constructed from symmetries of the undeformed theory, that are realised as active symmetry transformations. We argue that our construction amounts to a reformulation of known deformations of gauge theories, and that it significantly extends the range of applicable examples. To ensure consistency with gauge invariance, we also identify a unimodularity condition that is weaker than the one that is normally employed in the literature, so that we can apply twists that would otherwise be left out. Finally, we also prove a planar equivalence theorem stating that the Feynman diagrams of the deformed theories retain an undeformed internal structure, with the twist acting only on their external legs. All these results are important to identify and work with deformations of $\mathcal N=4$ super Yang-Mills that are proposed to be dual to homogeneous Yang-Baxter deformations of the $AdS_5\times S^5$ superstring, but the applicability of our construction and results goes beyond that.