Noncommutative circle bundles and new Dirac operators

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.

25 pages, LaTeX, substantially rewritten

References in corpus (4)

• Strong Connections and Chern-Connes Pairing in the Hopf-Galois Theory
• Projective module description of the q-monopole
• On a Classification of Irreducible Almost Commutative Geometries
• A Note on First Order Differential Calculus on Quantum Principal Bundles

Cited by in corpus (8)

• Curved noncommutative torus and Gauss--Bonnet
• Gauge theory on noncommutative Riemannian principal bundles
• Wodzicki residue and minimal operators on a noncommutative 4-dimensional torus
• Part I, Free Actions of Compact Abelian Groups on C*-Algebras
• Dirac operator on noncommutative principal circle bundles
• The geometry of quantum lens spaces: real spectral triples and bundle structure
• Lifting spectral triples to noncommutative principal bundles
• Geometric foundations for classical $\mathrm{U}(1)$-gauge theory on noncommutative manifolds