Torsion and Lorentz symmetry from Twisted Spectral Triples

TL;DR

This paper demonstrates how twisting spectral triples can generate torsion and Lorentz symmetry in geometric models, linking algebraic structures to physical concepts like energy-momentum and Lorentz transformations.

Contribution

It introduces a method to produce torsion from a torsionless Dirac operator via spectral triple twisting, connecting algebraic and geometric aspects with physical symmetries.

Findings

Torsion arises from twisted spectral triples as a co-exact three form.

The torsion term corresponds to a Lorentzian energy-momentum vector.

Lorentz group is a normal subgroup of twisted unitaries.

Abstract

By twisting the spectral triple of a riemannian spin manifold, we show how to generate an orthogonal and geodesic preserving torsion from a torsionless Dirac operator. We identify the group of twisted unitaries as the generator of torsion with co-exact three form. Through the fermionic action, the torsion term identifies with a Lorentzian energy-momentum 4-vector. The Lorentz group turns out to be a normal subgroup of the twisted unitaries. We also investigate the spectral action related to this model.