Zeta Functions, Excision in Cyclic Cohomology and Index Problems

TL;DR

This paper combines zeta functions and cyclic cohomology excision to derive index theorems, including a local formula for spectral triples and an application to hypoelliptic Heisenberg operators on foliations.

Contribution

It introduces a local index formula for spectral triples with multiple poles and applies it to hypoelliptic operators on foliations, extending index theory to non-elliptic cases.

Findings

Derived a local index formula for spectral triples with multiple poles

Applied the formula to hypoelliptic Heisenberg operators on foliations

Discussed spectral triple regularity on manifolds with conic singularities

Abstract

The aim of this paper is to show how zeta functions and excision in cyclic cohomology may be combined to obtain index theorems. In the first part, we obtain a local index formula for "abstract elliptic pseudodifferential operators" associated to spectral triples. This formula is notably well adapted when the zeta function has multiple poles. The second part is devoted to give a concrete realization of this formula by deriving an index theorem on the simple, but significant example of Heisenberg elliptic operators on a trivial foliation, which are in general non-elliptic but hypoelliptic. The last part contains a discussion on manifolds with conic singularity, more precisely about the regularity of spectral triples in this context.