Noncommutative circle bundles and new Dirac operators

TL;DR

This paper develops a framework for spectral triples over noncommutative principal U(1) bundles, introducing a new operatorial approach to connections and Dirac operators, with detailed analysis on noncommutative tori.

Contribution

It proposes an operatorial definition of connections compatible with Dirac operators in noncommutative bundles and constructs new Dirac operators from base-space data.

Findings

Classified all compatible connections on the noncommutative three-torus.

Constructed new Dirac operators from base-space Dirac operators and connections.

Analyzed the structure of noncommutative U(1) bundles and their spectral triples.

Abstract

We study spectral triples over noncommutative principal U(1) bundles. Basing on the classical situation and the abstract algebraic approach, we propose an operatorial definition for a connection and compatibility between the connection and the Dirac operator on the total space and on the base space of the bundle. We analyze in details the example of the noncommutative three-torus viewed as a U(1) bundle over the noncommutative two-torus and find all connections compatible with an admissible Dirac operator. Conversely, we find a family of new Dirac operators on the noncommutative tori, which arise from the base-space Dirac operator and a suitable connection.