Index map, $\sigma$-connections, and Connes-Chern character in the setting of twisted spectral triples

Abstract

Twisted spectral triples are a twisting of the notion of spectral triple aiming at dealing with some type III geometric situations. In the first part of the paper, we give a geometric construction of the index map of a twisted spectral triple in terms of $\sigma$-connections on finitely generated projective modules. This makes it more transparent the analogy with the indices of Dirac operators with coefficients in vector bundles. In the second part, we give a direct construction of the Connes-Chern character of a twisted spectral, both in the invertible and non-invertible cases. Combining these two parts we obtain an analogue the Atiyah-Singer index formula for twisted spectral triples.

35 pages; v3: minor changes to the exposition on $\sigma$-connections

References in corpus (5)

• The Gauss-Bonnet Theorem for the noncommutative two torus
• A twisted spectral triple for quantum SU(2)
• Local index formula and twisted spectral triples
• On stochastic generation of ultrametrics in high-dimension Euclidean spaces
• Noncommutative Geometry and Conformal Geometry. III. Vafa-Witten Inequality and Poincaré Duality