Local index formula and twisted spectral triples

TL;DR

This paper establishes a local index formula for a class of twisted spectral triples related to conformal foliations, addressing challenges posed by the twist and demonstrating how certain limits restore cocycle properties.

Contribution

It introduces a new local index formula for twisted spectral triples of type III, extending previous untwisted results and handling the non-cocycle nature of associated cocycles.

Findings

Proves a local index formula for twisted spectral triples of type III.

Shows that infinite temperature limits restore cocycle properties.

Extends index theory to twisted geometric settings.

Abstract

We prove a local index formula for a class of twisted spectral triples of type III modeled on the transverse geometry of conformal foliations with locally constant transverse conformal factor. Compared with the earlier proof of the untwisted case, the novel aspect resides in the fact that the twisted analogues of the JLO entire cocycle and of its retraction are no longer cocycles in their respective Connes bicomplexes. We show however that the passage to the infinite temperature limit, respectively the integration along the full temperature range against the Haar measure of the positive half-line, has the remarkable effect of curing in both cases the deviations from the cocycle identity.