Indexmap, Σ-connections, and Connes-Chern character in the setting of twisted spectral triples

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 indexmap 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.