Conformal Structures in Noncommutative Geometry

TL;DR

This paper rigorously proves that in noncommutative geometry, the conformal class of a metric can be reconstructed from the spectral triple by using the sign of the Dirac operator, extending classical geometric results.

Contribution

It provides a precise formulation and proof that the conformal structure of a manifold can be recovered from the spectral triple via the sign of the Dirac operator.

Findings

The conformal class of a metric is reconstructible from the spectral triple.

The sign of the Dirac operator encodes conformal information.

The result formalizes a folklore belief in noncommutative geometry.

Abstract

It is well-known that a compact Riemannian spin manifold can be reconstructed from its canonical spectral triple which consists of the algebra of smooth functions, the Hilbert space of square integrable spinors and the Dirac operator. It seems to be a folklore fact that the metric can be reconstructed up to conformal equivalence if one replaces the Dirac operator D by sign(D). We give a precise formulation and proof of this fact.