## Kyoto Journal of Mathematics

### 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 triples aimed at dealing with some type III geometric situations. In the first part of the article, 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 clarifies 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 triple, in both the invertible and the noninvertible cases. Combining these two parts we obtain an analogue of the Atiyah–Singer index formula for twisted spectral triples.

#### Article information

Source
Kyoto J. Math. Volume 56, Number 2 (2016), 347-399.

Dates
Received: 21 February 2015
Accepted: 24 March 2015
First available in Project Euclid: 10 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1462901082

Digital Object Identifier
doi:10.1215/21562261-3478907

Mathematical Reviews number (MathSciNet)
MR3500845

Zentralblatt MATH identifier
06591223

#### Citation

Ponge, Raphaël; Wang, Hang. Index map, $\sigma$ -connections, and Connes–Chern character in the setting of twisted spectral triples. Kyoto J. Math. 56 (2016), no. 2, 347--399. doi:10.1215/21562261-3478907. https://projecteuclid.org/euclid.kjm/1462901082

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