# Untwisting twisted spectral triples

## TL;DR

This paper demonstrates that twisted spectral triples can be transformed into ordinary spectral triples through a functional calculus process, and provides examples with nontrivial index data where existing formulas vanish.

## Contribution

It introduces a method to 'untwist' Lipschitz regular twisted spectral triples and higher order spectral triples, simplifying their analysis.

## Key findings

- Lipschitz regular twisted spectral triples can be logarithmically dampened to become untwisted.
- Higher order spectral triples can be converted into spectral triples using the proposed method.
- Examples show nontrivial index data with vanishing twisted local index formula.

## Abstract

We examine the index data associated to twisted spectral triples and higher order spectral triples. In particular, we show that a Lipschitz regular twisted spectral triple can always be `logarithmically dampened' through functional calculus, to obtain an ordinary (i.e. untwisted) spectral triple. The same procedure turns higher order spectral triples into spectral triples. We provide examples of highly regular twisted spectral triples with nontrivial index data for which Moscovici's ansatz for a twisted local index formula is identically zero.