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Tractors and twistors can be obtained via gauge reduction of the conformalCartan geometry, thanks to the dressing field method. Perhaps surprisingly, itis possible to reduce the gauge symmetry further still. One can indeed eraseWeyl symmetry thanks to the tractor field, in a way reminiscent of how theC^2-scalar field allows to erase SU(2) in the electroweak model. This suggestsan alternative to the Weyl or conformal spontaneous symmetry breaking (SSB)that some authors proposed as improvement for Standard Model physics orinflationary cosmology. Here, after gauge reduction via dressing, there remainsonly the Lorentz gauge symmetry and the twistor field becomes - for allpractical purposes - a Dirac spinor field. In a very simple and natural toymodel, the latter acquires a mass through Lorentz SSB due to the VEV of theWeyl-invariant tractor field.

Dilaton from Tractor and Matter Field from Twistor