On conformal higher spin wave operators

TL;DR

This work addresses conformal higher spin wave operators in arbitrary even dimensions, identifying when such operators factorize and when they fail due to curvature. It introduces an operator framework with a factorized Ansatz and gauge‑invariance constraints, showing that the Weyl tensor and its derivatives generically obstruct factorization, while yielding manifestly factorized forms on $(A)dS$ (for any spin) and on Einstein backgrounds for spin 2. The authors also compute the spin‑3 operator on Bach‑flat backgrounds to linear order in curvature, illustrating the complexity that arises beyond spin 2. Together, these results clarify the structure of conformal HS operators in curved spacetimes, provide practical factorized forms in key backgrounds, and lay groundwork for future explorations of interactions and HS algebras via tractor‑calculus methods.

Abstract

We analyze free conformal higher spin actions and the corresponding wave operators in arbitrary even dimensions and backgrounds. We show that the wave operators do not factorize in general, and identify the Weyl tensor and its derivatives as the obstruction to factorization. We give a manifestly factorized form for them on (A)dS backgrounds for arbitrary spin and on Einstein backgrounds for spin 2. We are also able to fix the conformal wave operator in d=4 for s=3 up to linear order in the Riemann tensor on generic Bach-flat backgrounds.