Algebraic backgrounds for noncommutative Kaluza-Klein theory. I. Motivations and general framework

Abstract

We investigate the representation of diffeomorphisms in Connes’ spectral triple formalism. By encoding the metric and spin structure in a moving frame, it is shown on the paradigmatic example of spin semi-Riemannian manifolds that the bimodule of noncommutative 1-forms Ω1 is an invariant structure in addition to the chirality, real structure, and Krein product. Adding Ω1 and removing the Dirac operator from an indefinite spectral triple, we obtain a structure which we call an algebraic background. All the Dirac operators compatible with this structure then form the configuration space of a noncommutative Kaluza-Klein theory. We explore the algebraic background canonically attached to a spin manifold, showing that its automorphism group is generated by diffeomorphisms and spin structure equivalences, and that its configuration space contains the Dirac operators associated with metrics and compatible spin structures, as well as additional centralizing fields. We explain how the latter can be removed without breaking the symmetries.