Weyl Invariance and the Origins of Mass

TL;DR

The work shows that massless, partially massless, and massive fields in curved backgrounds can be described within a single tractor-based, Weyl-invariant framework, where masses arise from Weyl weights via the relation $m^2 = rac{2P}{d}ig[ig(rac{d-1}{2}ig)^2 - (w+rac{d-1}{2})^2ig]$. By employing Weyl compensators (scale fields) and the scale tractor, the authors keep Weyl invariance manifest and derive stability bounds of Breitenlohner–Freedman type for AdS spaces, linking these bounds to reality conditions on Weyl weights. They provide explicit tractor constructions for spin-1 and spin-2 systems that unify Maxwell/Proca/conformal spin-1 theories and massless/partially massless/massive spin-2 theories, and extend the formalism to arbitrary higher spins, yielding general mass–weight relations and depth-based partially massless gauge structures. The approach suggests a powerful, general method to generate new, Weyl-invariant theories while clarifying the geometric origin of mass and its relation to conformal structure and higher-spin dynamics.

Abstract

By a uniform and simple Weyl invariant coupling of scale and matter fields, we construct theories that unify massless, massive, and partially massless excitations. Masses are related to tractor Weyl weights, and Breitenlohner-Freedman stability bounds in anti de Sitter amount to reality of these weights. The method relies on tractor calculus -- mathematical machinery allowing Weyl invariance to be kept manifest at all stages. The equivalence between tractor and higher spin systems with arbitrary spins and masses is also considered.