Massive Higher Spins from BRST and Tractors

TL;DR

This work proves that the higher-spin tractor equations of motion proposed by Gover and collaborators arise from a BRST construction and correctly describe massive, partially massless, and massless higher spins in conformally flat backgrounds. By developing both ambient/radial reductions and a comprehensive BRST (including a parent BRST) formulation, the authors unify tractor, ambient, and AdS descriptions and show how masses emerge as geometric weights $w$ within conformal geometry. They demonstrate that the tractor equations reproduce on-shell massive dynamics, provide gauge-structure insights through cohomology, and connect bulk conformal geometry to boundary data, thereby enhancing control over bulk–boundary correspondences. The results establish a robust, scale-invariant framework for higher-spin fields that naturally encodes gauge symmetry, mass, and holographic relations in a conformal-geometric setting.

Abstract

We obtain the higher spin tractor equations of motion conjectured by Gover et al. from a BRST approach and use those methods to prove that they describe massive, partially massless and massless higher spins in conformally flat backgrounds. The tractor description makes invariance under local choices of unit system manifest. In this approach, physical systems are described by conformal, rather than (pseudo-)Riemannian geometry. In particular masses become geometric quantities, namely the weights of tractor fields. Massive systems can therefore be handled in a unified and simple manner mimicking the gauge principle usually employed for massless models. From a holographic viewpoint, these models describe both the bulk and boundary theories in terms of conformal geometry. This is an important advance, because tying the boundary conformal structure to that of the bulk theory gives greater control over a bulk--boundary correspondence.

Figures (5)

• Figure 1: Tractor equations of motion unifying massive, massless and partially massless higher spins in standard and index-free notations.
• Figure 2: The flat conformal space $\mathscr{M}$ in Riemannian signature realized as null rays.
• Figure 3: Tractor constraint algebra on conformally flat spaces. The Lie algebra cohomology $H^1(\frak{g},{\cal V})$ describes massive higher spins.
• Figure 4: The ambient space description of a conformal manifold.
• Figure 5: Projective model for conformal geometry depicted in Euclidean signature.