Weyl's gauge invariance: Conformal geometry, spinors, supersymmetry, and interactions

Abstract

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.

Weyl geometric gravity and electroweak symmetry “breaking”

“…Weyl geometry is ideally suited to the purpose of giving such questions a clear conceptual and mathematical foundation, at least on the classical level. In this respect it agrees with the intention of some recent studies in conformal field theory [34,58]. Sections 3.2 and 3.3 discuss what happens if the ordinary Higgs mechanism (in its non-quantized form) is transferred from special relativistic fields to Weyl geometrically "curved" spaces.…”

Boundary calculus for conformally compact manifolds

“…It should be emphasised that throughout our treatment is valid in any signature, and whereas the equation (∆ g o + s(n − s))u = 0 has mainly been studied with u a function, here we treat the I·D equation on sections of (weighted) tractor bundles, and on conformally compact structures of any signature. The operator I·D on tractor bundles determines natural equations on tensor and spinor fields, this is central to the treatments in [33,34,59]. This means our results can be used to deduce results for general tensor and spinor fields in the spirit of the Eastwood curved translation principle, see [27] for an early discussion of this.…”

The conformal transformation’s controversy: what are we missing?

“…Although, as long as we know, it has not been as strictly formulated as we have done above, the princi-ple of conformal equivalence has been assumed to play an important role in the understanding of the laws of physics in many influential papers before [2,[46][47][48]. In this regard we want to point out that whether the CEP is a fundamental principle of nature is not a subject of interest in the present paper.…”