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

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 Poincaré dual of the A-hat genus of the manifold.

Provides complete proofs to the main theorems, and corrected errors in version 1. Removed the section on Lie Algebroid

References in corpus (4)

• Inner fluctuations of the spectral action
• Riemannian manifolds in noncommutative geometry
• Twisted K-theory and loop groups
• On a generalized Connes-Hochschild-Kostant-Rosenberg theorem

Cited by in corpus (3)

• Riemannian manifolds in noncommutative geometry
• On stochastic generation of ultrametrics in high-dimension Euclidean spaces
• Operator-valued pseudo-differential operators and the twisted index pairing