Partial Masslessness and Conformal Gravity

TL;DR

The paper investigates the possibility of a self-consistent, non-linear partially massless spin-2 sector within four-dimensional conformal (Weyl) gravity. By employing the Bach tensor factorization $B_{\mu\nu} = {\bm M}_{\mu\nu}^{\rho\sigma}{\bm P}_{\rho\sigma}$ and the tractor/Yang–Mills detour framework, the authors demonstrate that PM propagation is restricted to Einstein backgrounds and cannot be consistently extended to generic Bach-flat geometries. While conformal gravity can generate cubic PM self-interaction vertices that respect linear PM gauge invariance, a full non-linear truncation of CG to a ghost-free PM theory is obstructed; higher-order (quartic and beyond) consistency fails due to unresolved divergence constraints. Consequently, CG does not realize a non-linear PM general relativity, although it yields valid cubic PM couplings and offers insights for higher-spin extensions and potential cosmological applications. The work clarifies fundamental obstacles in constructing interacting PM theories and highlights the delicate balance between gauge invariance, background geometry, and ghostly degrees of freedom.

Abstract

We use conformal, but ghostful, Weyl gravity to study its ghost-free, second derivative, partially massless (PM) spin 2 component in presence of Einstein gravity with positive cosmological constant. Specifically, we consider both gravitational- and self- interactions of PM via the fully non-linear factorization of conformal gravity's Bach tensor into Einstein times Schouten operators. We find that extending PM beyond linear order suffers from familiar higher-spin consistency obstructions: it propagates only in Einstein backgrounds, and the conformal gravity route generates only the usual safe, Noether, cubic order vertices.