The resolvent cocycle in twisted cyclic cohomology and a local index formula for the Podles sphere

TL;DR

This paper develops a resolvent cocycle in twisted cyclic cohomology, enabling a local index formula for the Podles sphere, thus advancing the understanding of twisted homology theories and index calculations in noncommutative geometry.

Contribution

It introduces a resolvent cocycle under weaker conditions and applies it to derive a local index formula for the Podles sphere in twisted cyclic cohomology.

Findings

Existence of the resolvent cocycle in modular spectral triples

Derivation of a local index formula for the Podles sphere

Twisted cyclic cocycle with non-vanishing Hochschild class

Abstract

We continue the investigation of twisted homology theories in the context of dimension drop phenomena. This work unifies previous equivariant index calculations in twisted cyclic cohomology. We do this by proving the existence of the resolvent cocycle, a finitely summable analogue of the JLO cocycle, under weaker smoothness hypotheses and in the more general setting of `modular' spectral triples. As an application we show that using our twisted resolvent cocycle, we can obtain a local index formula for the Podles sphere. The resulting twisted cyclic cocycle has non-vanishing Hochschild class which is in dimension 2.