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 -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.
"Index map, -connections, and Connes–Chern character in the setting of twisted spectral triples." Kyoto J. Math. 56 (2) 347 - 399, June 2016. https://doi.org/10.1215/21562261-3478907