# Lorentzian fermionic action by twisting euclidean spectral triples

## TL;DR

This paper demonstrates how twisting spectral triples induces a transition from euclidean to lorentzian noncommutative geometry, deriving lorentzian fermionic actions and equations from euclidean models, with invariance under Lorentz transformations.

## Contribution

It introduces a method to obtain lorentzian fermionic actions from euclidean spectral triples through twisting, connecting noncommutative geometry with Lorentzian physics.

## Key findings

- Derived lorentzian Weyl and Dirac equations from euclidean models.
- Showed invariance of twisted fermionic action under Lorentz group.
- Interpreted the 1-form field as dual to energy-momentum 4-vector.

## Abstract

We show how the twisting of spectral triples induces a transition from an euclidean to a lorentzian noncommutative geometry, at the level of the fermionic action. More specifically, we compute the fermionic action for the twisting of a closed euclidean manifold, then that of a two-sheet euclidean manifold, and finally the twisting of the spectral triple of electrodynamics in euclidean signature. We obtain the Weyl and the Dirac equations in lorentzian signature (and in the temporal gauge). The twisted fermionic action is then shown to be invariant under an action of the Lorentz group. This permits to interprete the field of 1-form that parametrizes the twisted fluctuation of a manifold as the (dual) of the energy momentum 4-vector.