Gauge transformations on quantum principal bundles

TL;DR

This paper develops a framework for quantum principal bundles using Hopf--Galois extensions, incorporating differential structures, gauge transformations, and curvature, with explicit examples like the noncommutative 2-torus.

Contribution

It introduces a compatible differential structure on quantum principal bundles and extends quantum gauge transformations to differential forms, providing a noncommutative geometric framework.

Findings

Established an exact noncommutative Atiyah sequence

Defined a graded-braided commutative structure on total space forms

Demonstrated gauge transformations act on connections and curvature

Abstract

We understand quantum principal bundle as faithfully flat Hopf--Galois extensions, with a structure Hopf algebra coacting on a total space algebra and with base algebra given by the coinvariant elements. To endow such bundles with a compatible differential structure, one requires the coaction to extend as a morphism of differential graded algebras. This leads to an exact noncommutative Atiyah sequence, a graded Hopf--Galois extension of differential forms and a canonical braiding on total space forms such that the latter are graded-braided commutative. We recall this approach to noncommutative differential geometry and further discuss the extension of quantum gauge transformations, in the sense of Brzezi\'nski, to differential forms. In this way we obtain an action of quantum gauge transformations on connections of the quantum principal bundle and their curvature. Explicit examples, such as the noncommutative 2-torus, the quantum Hopf fibration and smash product algebras are discussed.