U_q(N) Gauge Theories

TL;DR

This paper constructs consistent gauge theories based on the quantum groups $U_q(N)$, extending classical gauge theory concepts to a quantum group framework with $q$-commuting fields and a generalized Yang-Mills Lagrangian.

Contribution

It introduces a novel formulation of $U_q(N)$ gauge theories that are compatible with ordinary spacetime and generalize classical gauge transformations and Lagrangians.

Findings

Gauge potentials and field strengths are $q$-commuting fields.

The $q$-Lagrangian has a Yang-Mills form with a quantum metric.

Theories are consistent with an ordinary spacetime.

Abstract

Improving on an earlier proposal, we construct the gauge theories of the quantum groups $U_q(N)$. We find that these theories are consistent also with an ordinary (commuting) spacetime. The bicovariance conditions of the quantum differential calculus are essential in our construction. The gauge potentials and the field strengths are $q$-commuting ``fields", and satisfy $q$-commutation relations with the gauge parameters. The transformation rules of the potentials are given explicitly, and generalize the ordinary infinitesimal gauge variations. The $q$-lagrangian invariant under the $U_q(N)$ variations has the Yang-Mills form $\Fmn^i \Fmn^j g_{ij}$, the ``quantum metric'' $g_{ij}$ being a generalization of the Killing metric.