Laplacians and gauged Laplacians on a quantum Hopf bundle

TL;DR

This paper explores connections, covariant derivatives, and Laplacians on a quantum Hopf bundle over the Podles sphere, utilizing Hodge duality to analyze geometric operators in a noncommutative setting.

Contribution

It introduces a framework for defining and studying Laplacians and gauged Laplacians on a quantum Hopf bundle with a novel application of Hodge duality.

Findings

Defined Laplacians and gauged Laplacians in the quantum setting

Established Hodge duality on total and base space exterior algebras

Analyzed properties of covariant derivatives on the quantum bundle

Abstract

This paper presents an analysis of the set of connections and covariant derivatives on a U(1) quantum Hopf bundle on the standard Podles sphere, whose total space quantum SU(2) is equipped with the 3d left covariant differential calculus by Woronowicz. The introduction of a Hodge duality on the exterior algebras on both total and base space of the bundle allows for the study of Laplacians and of gauged Laplacians.