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Ponge, Raphaël; Wang, Hang

Index map, \(\sigma\)-connections, and Connes-Chern character in the setting of twisted spectral triples. (English) Zbl 1361.46055

Kyoto J. Math. 56, No. 2, 347-399 (2016).
Summary: 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.

Cited in 2 Documents

MSC:

46L87 Noncommutative differential geometry
58B34 Noncommutative geometry (à la Connes)
58J20 Index theory and related fixed-point theorems on manifolds
19D55 \(K\)-theory and homology; cyclic homology and cohomology

Keywords:

noncommutative geometry; twisted spectral triples; index theory; cyclic cohomology
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References:

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