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We extend our program, of coupling theories to scale in order to make theirWeyl invariance manifest, to include interacting theories, fermions andsupersymmetric theories. The results produce mass terms coinciding with thestandard ones for universes that are Einstein, but are novel in generalbackgrounds. They are generalizations of the gravitational couplings of aconformally improved scalar to fields with general scaling and tensorproperties. The couplings we find are more general than just trivial onesfollowing from the conformal compensating mechanisms. In particular, in thesetting where a scale gauge field (or dilaton) is included, masses correspondto Weyl weights of fields organized in ``tractor'' multiplets.Breitenlohner--Freedman bounds follow directly from reality of these weights.Moreover, massive, massless and partially massless theories are handled in auniform framework. Also, bona fide Weyl invariant theories (invariant withoutcoupling to scale) can be directly derived in this approach. The results arebased on the tractor calculus approach to conformal geometry, in particular weshow how to handle fermi fields, supersymmetry and Killing spinors usingtractor techniques. Another useful consequence of the construction is that itautomatically produces the (anti) de Sitter theories obtained by log-radialreduction of Minkowski theories in one higher dimension. Theories presented indetail include interacting scalars, spinors, Rarita--Schwinger fields, and theinteracting Wess--Zumino model.

Weyl's Gauge Invariance: Conformal Geometry, Spinors, Supersymmetry, and Interactions