Kyoto Journal of Mathematics

Index map, σ-connections, and Connes–Chern character in the setting of twisted spectral triples

Raphaël Ponge and Hang Wang

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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 σ-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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46L87: Noncommutative differential geometry [See also 58B32, 58B34, 58J22] 58B34: Noncommutative geometry (à la Connes)
Secondary: 58J20: Index theory and related fixed point theorems [See also 19K56, 46L80] 19D55: $K$-theory and homology; cyclic homology and cohomology [See also 18G60]

noncommutative geometry twisted spectral triples index theory cyclic cohomology


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.

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