Signature change by a morphism of spectral triples

Abstract

We present a connection between twisted spectral triples and pseudo-Riemannian spectral triples, rooted in the fundamental interplay between twists and Krein products. A concept of morphism of spectral triples is introduced, transforming one spectral triple into its dual. In the case of even-dimensional manifolds, we demonstrate how this construction implements a local signature change via the parity operator induced by the twist. Consequently, the signature change transformation is governed solely by the unitary operator that implements the twist. This unitary is a central element of our approach, as it is directly linked to the Krein product, the twist, and the parity operator that implements the signature change.