Real structures on almost-commutative spectral triples

TL;DR

This paper refines the reconstruction theorem for real almost-commutative spectral triples, clarifies their definitions, and explores their geometric and KK-theoretic implications, including the encoding of base manifolds and families of spectral triples.

Contribution

It extends the reconstruction theorem to real spectral triples, clarifies definitions, and introduces smooth families and twisting, with implications for KK-theory and gauge theory.

Findings

Real spectral triples encode the base manifold algebraically.

Existence of real spectral triples of arbitrary KO-dimension on manifolds.

Introduction of smooth families and twisting of spectral triples with KK-theoretic implications.

Abstract

We refine the reconstruction theorem for almost-commutative spectral triples to a result for real almost-commutative spectral triples, clarifying, in the process, both concrete and abstract definitions of real commutative and almost-commutative spectral triples. In particular, we find that a real almost-commutative spectral triple algebraically encodes the commutative *-algebra of the base manifold in a canonical way, and that a compact oriented Riemannian manifold admits real (almost-)commutative spectral triples of arbitrary KO-dimension. Moreover, we define a notion of smooth family of real finite spectral triples and of the twisting of a concrete real commutative spectral triple by such a family, with interesting KK-theoretic and gauge-theoretic implications.