Dirac operator on noncommutative principal circle bundles

TL;DR

This paper investigates spectral triples on noncommutative principal U(1)-bundles, establishing compatibility conditions between connections and Dirac operators, with detailed examples on noncommutative tori and theta-deformed spheres.

Contribution

It introduces a compatibility framework for connections and Dirac operators on noncommutative principal bundles, and constructs new Dirac operators from base-space data.

Findings

Identified compatible connections on noncommutative tori.

Constructed new Dirac operators from base Dirac operators and connections.

Extended examples to theta-deformed spheres.

Abstract

We study spectral triples over noncommutative principal U(1)-bundles of arbitrary dimension and formulate a compatibility condition between the connection and the Dirac operator on the total space and on the base space of the bundle. Examples of low dimensional noncommutative tori are analyzed in more detail and all connections found that are compatible with an admissible Dirac operator. Conversely, a family of new Dirac operators on the noncommutative tori, which arise from the base-space Dirac operator and a suitable connection is exhibited. These examples are extended to the theta-deformed principal U(1)-bundle S^3_\theta -> S^2.