Non-commutative deformations of gauge theories via Drinfel'd twists of the scale symmetry

TL;DR

This paper develops a method to deform relativistic, scale-invariant gauge theories using Drinfel'd twists, resulting in non-commutative structures that preserve a twisted symmetry group and have implications for holography.

Contribution

It introduces a new approach to deform gauge theories via Drinfel'd twists involving scale symmetry, extending previous methods and enabling applications to super Yang-Mills theories.

Findings

Deformed gauge theories retain a twisted symmetry group.

Planar Feynman diagrams are deformed only on external legs.

Extension to $ abla$=4 super Yang-Mills suggests holographic duals.

Abstract

In this paper we consider gauge theories that are relativistic and scale-invariant, and we construct their deformed versions via suitable star products. In particular, the non-commutative structure is controlled by Drinfel'd twists that are built out of symmetry generators that include the scale transformation. To achieve this, we construct a twisted differential calculus that allows us to identify the proper gauge-covariant quantities. We also show that our construction is equivalent to twists where the symmetry generators are implemented as active transformations of fields. As a consequence of our construction, the deformed gauge theories possess a twisted version of the original symmetry group. Moreover, at the planar level, the deformation is encoded just on the external legs of Feynman diagrams, leaving then the amputated diagrams undeformed. This work extends previous constructions and allows us to define twist-deformations of $\mathcal N=4$ super Yang-Mills that are conjectured to be holographically dual to a class of homogeneous Yang-Baxter deformations of $AdS_5\times S^5$.