Quantum Group Gauge Theory on Classical Spaces

TL;DR

This paper investigates the limit of quantum group gauge theory on classical spaces, showing it reduces to standard gauge theory with some differences in matter coupling, and clarifies issues in prior research.

Contribution

It demonstrates that quantum group gauge theory on classical spaces is isomorphic to traditional gauge theory, correcting previous misconceptions about the role of quantum traces.

Findings

Quantum group gauge theory reduces to classical gauge theory on classical spaces.

Differences in matter coupling are identified compared to standard gauge theory.

The use of quantum trace in defining Yang-Mills action is shown to be inappropriate.

Abstract

We study the quantum group gauge theory developed elsewhere in the limit when the base space (spacetime) is a classical space rather than a general quantum space. We show that this limit of the theory for gauge quantum group $U_q(g)$ is isomorphic to usual gauge theory with Lie algebra $g$. Thus a new kind of gauge theory is not obtained in this way, although we do find some differences in the coupling to matter. Our analysis also illuminates certain inconsistencies in previous work on this topic where a different conclusion had been reached. In particular, we show that the use of the quantum trace in defining a Yang-Mills action in this setting is not appropriate.