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

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

2010-08-04

arxiv.org/abs/1008.0707v1

Mathematics

Differential Geometry

27 pages, LaTeX

Scientific paper

We construct a canonical noncommutative spectral triple for every closed oriented Riemannian manifold, which represents of 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 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. In addition, certain relation between cohomology of Lie algebroids and twisted de Rham cohomology is discovered.

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