JournalsjncgVol. 8, No. 1pp. 179–215

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

  • Dapeng Zhang

    California Institute of Technology, Pasadena, USA
Projective Dirac operators, twisted K-theory, and local index formula cover

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{A}-genus of the manifold.

Cite this article

Dapeng Zhang, Projective Dirac operators, twisted K-theory, and local index formula. J. Noncommut. Geom. 8 (2014), no. 1, pp. 179–215

DOI 10.4171/JNCG/153