Principal noncommutative torus bundles

TL;DR

This paper classifies and analyzes continuous bundles of non-commutative C*-algebras resembling principal torus bundles, revealing their local triviality, RKK-theoretic properties, and conditions for classical T-duality.

Contribution

It provides a complete classification of principal non-commutative torus bundles up to equivariant Morita equivalence and characterizes when they are RKK-equivalent to commutative bundles.

Findings

All such bundles are locally trivial in RKK-theory.

Complete classification of these bundles up to Morita equivalence.

Conditions for non-commutative bundles to be RKK-equivalent to commutative ones.

Abstract

In this paper we study continuous bundles of C*-algebras which are non-commutative analogues of principal torus bundles. We show that all such bundles, although in general being very far away from being locally trivial bundles, are at least locally trivial with respect to a suitable bundle version of bivariant K-theory (denoted RKK-theory) due to Kasparov. Using earlier results of Echterhoff and Williams, we shall give a complete classification of principal non-commutative torus bundles up to equivariant Morita equivalence. We then study these bundles as topological fibrations (forgetting the group action) and give necessary and sufficient conditions for any non-commutative principal torus bundle being RKK-equivalent to a commutative one. As an application of our methods we shall also give a K-theoretic characterization of those principal torus-bundles with H-flux, as studied by Mathai and Rosenberg which possess "classical" T-duals.