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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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 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

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

Subjects
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]

Keywords
noncommutative geometry twisted spectral triples index theory cyclic cohomology

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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