Strong Connections on Quantum Principal Bundles

TL;DR

This paper introduces a gauge invariant concept of strong connections on quantum principal bundles, characterizes their properties, and explores their implications for curvature, examples, and Yang-Mills theory in the quantum setting.

Contribution

It defines and characterizes strong connections on quantum bundles, linking them to curvature, examples, and Yang-Mills actions, and shows their moduli space independence from the deformation parameter q.

Findings

Strong connections are characterized and linked to curvature forms.

Examples include quantum deformations of classical fibrations like S^2 -> RP^2.

The Yang-Mills action on quantum bundles matches known formulations and is q-independent.

Abstract

A gauge invariant notion of a strong connection is presented and characterized. It is then used to justify the way in which a global curvature form is defined. Strong connections are interpreted as those that are induced from the base space of a quantum bundle. Examples of both strong and non-strong connections are provided. In particular, such connections are constructed on a quantum deformation of the fibration $S^2 -> RP^2$. A certain class of strong $U_q(2)$-connections on a trivial quantum principal bundle is shown to be equivalent to the class of connections on a free module that are compatible with the q-dependent hermitian metric. A particular form of the Yang-Mills action on a trivial $U\sb q(2)$-bundle is investigated. It is proved to coincide with the Yang-Mills action constructed by A.Connes and M.Rieffel. Furthermore, it is shown that the moduli space of critical points of this action functional is independent of q.