Dilaton from Tractor and Matter Field from Twistor

TL;DR

This paper explores a novel gauge reduction method in conformal geometry that eliminates Weyl symmetry, transforming tractor and twistor fields into matter-like fields with potential implications for Standard Model physics and cosmology.

Contribution

It introduces a new gauge reduction approach that erases Weyl symmetry, leading to matter fields from tractors and twistors, offering an alternative to traditional symmetry breaking mechanisms.

Findings

Weyl symmetry can be erased using the tractor field.

Twistor fields can effectively become Dirac spinors.

A toy model shows mass generation via Lorentz symmetry breaking.

Abstract

Tractors and twistors can be obtained via gauge reduction of the conformal Cartan geometry, thanks to the dressing field method. Perhaps surprisingly, it is possible to reduce the gauge symmetry further still. One can indeed erase Weyl symmetry thanks to the tractor field, in a way reminiscent of how the C^2-scalar field allows to erase SU(2) in the electroweak model. This suggests an alternative to the Weyl or conformal spontaneous symmetry breaking (SSB) that some authors proposed as improvement for Standard Model physics or inflationary cosmology. Here, after gauge reduction via dressing, there remains only the Lorentz gauge symmetry and the twistor field becomes - for all practical purposes - a Dirac spinor field. In a very simple and natural toy model, the latter acquires a mass through Lorentz SSB due to the VEV of the Weyl-invariant tractor field.