A Remark on Gelfand Duality for Spectral Triples

TL;DR

This paper establishes a duality between compact Riemannian spin manifolds with isometries and a category of spectral triples, extending Gelfand duality to the noncommutative geometric setting.

Contribution

It introduces a duality framework linking Riemannian spin manifolds with spectral triples and embeds a quotient category of spectral triples into this framework.

Findings

Duality between manifolds and spectral triples established

Embedding of spectral triple quotient category into the metric category

Discussion of duality with orientation and spin-preserving maps

Abstract

We present a duality between the category of compact Riemannian spin manifolds (equipped with a given spin bundle and charge conjugation) with isometries as morphisms and a suitable "metric" category of spectral triples over commutative pre-C*-algebras. We also construct an embedding of a "quotient" of the category of spectral triples introduced in arXiv:math/0502583v1 into the latter metric category. Finally we discuss a further related duality in the case of orientation and spin-preserving maps between manifolds of fixed dimension.