Type III and spectral triples

By Alain Connes, H. Moscovici. 2006

135 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6d88545a3de53268d20d644939ca3484cf0672c8

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Citations in the corpus

The Gauss-Bonnet Theorem for the noncommutative two torus (2009, background): “The twisted (or non-unimodular) spectral triples appeared very naturally [ 10 ] in the study of type III examples of foliation algebras.”
Twisted cyclic theory, equivariant KK-theory and KMS states (2008, result): “as an analytically manageable part of an equivariant cohomology. We make the observation that our constructions and resulting index theorems are not related to the approach of Connes and Moscovici in [15]. 2. The constructionof equivariantKKclassesfrom a circleaction 2.1. A Kasparov module from a ci”