Type III and spectral triples

TL;DR

This paper introduces a simple twisting of spectral triples to include type III examples, showing that the core index theory and cyclic cohomology structures remain valid without requiring cohomological twisting.

Contribution

It demonstrates that twisted spectral triples of type III can be analyzed using standard cyclic cohomology and index theory, without the need for cohomological twisting.

Findings

Chern character extends to twisted spectral triples in ordinary cyclic cohomology.

Index pairing remains valid with untwisted K-theory.

No cohomological twisting is necessary for the theory to hold.

Abstract

We explain how a simple twisting of the notion of spectral triple allows to incorporate type III examples, such as those arising from the transverse geometry of codimension one foliations. Since the twisting of the commutators turns the usual hypertrace constructed out of the Dixmier trace into a twisted trace on the coordinate algebra, one would be tempted to interpret that as a manifestation of twisting at the level of cyclic cohomology, akin to that introduced by the authors in the context of Hopf cyclic cohomology. The main point of this note, besides giving simple natural examples of the general notion and developing the first basic steps of the theory, is to show that contrary to the initial expectations no cohomological twisting is in fact required. The Chern character of finitely summable spectral triples extends to the twisted case, and lands in fact in ordinary (untwisted) cyclic cohomology. The same holds true for the local Hochschild character. The index pairing with ordinary (untwisted) K-theory continues to make sense and the index formula is still given by the pairing of the corresponding Chern characters. This opens the road to extending the local index formula, as well as the analogue of the hypoelliptic construction on the dual system together with the corresponding Thom isomorphism, to the context of twisted spectral triples of type III.