Indefinite Kasparov modules and pseudo-Riemannian manifolds

TL;DR

This paper introduces indefinite Kasparov modules to model non-symmetric, non-elliptic operators like hyperbolic ones, establishing a reversible correspondence with traditional Kasparov modules and illustrating with examples including pseudo-Riemannian manifolds.

Contribution

It defines indefinite Kasparov modules and proves a reversible association with standard Kasparov modules, extending noncommutative geometry tools to indefinite metric spaces.

Findings

Established a correspondence between indefinite and standard Kasparov modules.

Applied the framework to pseudo-Riemannian spin manifolds and the harmonic oscillator.

Constructed indefinite spectral triples from families of spectral triples.

Abstract

We present a definition of indefinite Kasparov modules, a generalisation of unbounded Kasparov modules modelling non-symmetric and non-elliptic (e.g. hyperbolic) operators. Our main theorem shows that to each indefinite Kasparov module we can associate a pair of (genuine) Kasparov modules, and that this process is reversible. We present three examples of our framework: the Dirac operator on a pseudo-Riemannian spin manifold (i.e. a manifold with an indefinite metric), the harmonic oscillator, and the construction via the Kasparov product of an indefinite spectral triple from a family of spectral triples. This last construction corresponds to a foliation of a globally hyperbolic spacetime by spacelike hypersurfaces.