Journal of Noncommutative Geometry

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Volume 8, Issue 1, 2014, pp. 179–215
DOI: 10.4171/JNCG/153

Published online: 2014-03-20

Projective Dirac operators, twisted K-theory, and local index formula

Dapeng Zhang[1]

(1) California Institute of Technology, Pasadena, USA

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 $\hat{A}$-genus of the manifold.

Keywords: Twisted K-theory, spectral triple, Chern character

Zhang Dapeng: Projective Dirac operators, twisted K-theory, and local index formula. J. Noncommut. Geom. 8 (2014), 179-215. doi: 10.4171/JNCG/153