Dilaton from tractor and matter field from twistor
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Conformal 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. The Weyl symmetry can indeed be erased thanks to the tractor field, from which a dilaton is extracted. This suggests an alternative to the Weyl or conformal spontaneous symmetry breaking (SSB) that some authors proposed as improvement of the Standard Model or of inflationary cosmology, but also raises doubts as to the physical significance of such symmetries. Here, after gauge reduction via dressing, only the Lorentz gauge symmetry remains physically relevant, and the twistor field becomes — for all practical purposes — a Dirac spinor field. In a simple illustrative toy model, the latter acquires a mass through Lorentz SSB due to the VEV of the Weyl-invariant tractor field.
KeywordsConformal and W Symmetry Differential and Algebraic Geometry Gauge Symmetry Spontaneous Symmetry Breaking
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