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

We extend our program, of coupling theories to scale in order to make their Weyl invariance manifest, to include interacting theories, fermions and supersymmetric theories. The results produce mass terms coinciding with the standard ones for universes that are Einstein, but are novel in general backgrounds. They are generalizations of the gravitational couplings of a conformally improved scalar to fields with general scaling and tensor properties. The couplings we find are more general than just trivial ones following from the conformal compensating mechanisms. In particular, in the setting where a scale gauge field (or dilaton) is included, masses correspond to 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 a uniform framework. Also, bona fide Weyl invariant theories (invariant without coupling to scale) can be directly derived in this approach. The results are based on the tractor calculus approach to conformal geometry, in particular we show how to handle fermi fields, supersymmetry and Killing spinors using tractor techniques. Another useful consequence of the construction is that it automatically produces the (anti) de Sitter theories obtained by log-radial reduction of Minkowski theories in one higher dimension. Theories presented in detail include interacting scalars, spinors, Rarita--Schwinger fields, and the interacting Wess--Zumino model.