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

TL;DR

The work develops a framework for non-commutative, twist-deformed gauge theories that preserve relativistic scale invariance by using Drinfel'd twists built from the dilatation and Poincaré generators. A twisted differential calculus is constructed, including a generalized Hodge dual, deformed Levi-Civita symbol, and a unimodular-consistent integration, enabling gauge-invariant actions for Yang–Mills and matter fields under star products. A key result is that, at the planar level, deformations affect only external legs of Feynman diagrams, leaving amputated diagrams undeformed, and a planar equivalence theorem is established for twists including scale transformations. The authors also present an alternative, activity-based approach to twists, proving its equivalence to the differential-calculus formulation and discussing how twisted symmetries and their coproducts act on products of fields. The construction prepares twist-deformed versions of $ ext{N}=4$ super Yang–Mills, potentially holographically dual to homogeneous Yang–Baxter deformations of $AdS_5 imes S^5$, and outlines numerous open questions about quantization and holographic implications.

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 $AdS_5\times S^5$.