Gauge theories of quantum groups

TL;DR

This paper introduces two q-generalizations of Yang-Mills theories that maintain gauge invariance under quantum group symmetries, with explicit formulations for a q-deformed SU(2)×U(1) and novel properties of gauge potentials and field strengths.

Contribution

It presents two new q-deformed Yang-Mills frameworks with explicit lagrangians and transformation rules, expanding the understanding of gauge theories within quantum group geometry.

Findings

One scheme features non-commuting gauge potentials and field strengths.

The other scheme has commuting field strengths despite q-deformed gauge potentials.

Both schemes preserve gauge invariance under q-analogues of gauge transformations.

Abstract

We find two different q-generalizations of Yang-Mills theories. The corresponding lagrangians are invariant under the q-analogue of infinitesimal gauge transformations. We explicitly give the lagrangian and the transformation rules for the bicovariant q-deformation of $SU(2) \times U(1)$. The gauge potentials satisfy q-commutations, as one expects from the differential geometry of quantum groups. However, in one of the two schemes we present, the field strengths do commute.