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

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.

💡 Research Summary

The paper constructs a canonical non‑commutative spectral triple for any oriented closed Riemannian manifold (M) equipped with a twisting class (