Projective Dirac Operators, Twisted K-Theory and Local Index Formula

TL;DR

This paper constructs a canonical noncommutative spectral triple for oriented closed Riemannian manifolds, linking twisted K-homology, local index formulas, and the A-hat genus in a novel way.

Contribution

It introduces the projective spectral triple representing the fundamental class in twisted K-homology and provides an explicit local formula for the twisted Chern character.

Findings

The projective spectral triple is Morita equivalent to the spin spectral triple for spin-c manifolds.

An explicit local formula for the twisted Chern character is derived.

The twisted Chern character matches the Poincaré dual of the A-hat genus.

Abstract

We construct a canonical noncommutative spectral triple for every oriented closed Riemannian manifold, which represents the fundamental class in the twisted K-homology of the manifold. This so-called "projective spectral triple" is Morita equivalent to the well-known commutative spin spectral triple provided that the manifold is spin-c. We give an explicit local formula for the twisted Chern character for K-theories twisted with torsion classes, and with this formula we show that the twisted Chern character of the projective spectral triple is identical to the Poincar\'e dual of the A-hat genus of the manifold.