On the index theory of twisted spectral triples on traceless C*-algebras

Bram Mesland (Max Planck Institute for Mathematics, Bonn)

Thursday 7th February 16:00-17:00 Maths 311B

Abstract

Noncommutative geometry (NCG) is built on Connes’ observation that the Dirac operator on a spin manifold encodes both the Riemannian metric as well as the fundamental class in K-homology. This observation leads to the proposal of spectral triples as the central objects of study in NCG. The landmark result of Connes-Moscovici is the local index formula for a finitely summable smooth spectral triple. It computes the pairing of the spectral triple with K-theory.

By a well known result of Connes, traceless C*-algebras do not admit finitely summable spectral triples. In this context it is natural to study several variations of spectral triples, such as twisted and higher order versions.

In this talk I discuss how functional calculus involving the logarithm can be employed to turn such "exotic" K-cycles into ordinary spectral triples. Subsequently, I discuss the existence of smooth finitely summable twisted spectral triples on the boundary crossed product of a free group. These spectral triples pair non-trivially with K-theory, but Moscovici's Ansatz for a twisted local index formula can be shown to fail.

This is joint work with Magnus Goffeng and Adam Rennie

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