A Calculus for Higher Spin Interactions

TL;DR

This work develops a tractor calculus framework to derive cubic interactions for totally symmetric higher-spin fields, unifying massless, massive, and partially massless regimes by coupling to scale through Weyl weights $w$ and the scale tractor $I_M$. It constructs conformally invariant vertex functionals with invariants $\mathcal{Y}_i$ and $\mathcal{Z}_i$, and imposes gauge consistency via a differential equation for the cubic vertex function $C(\mathcal{Y}_i,\mathcal{Z}_i)$, matching results from $(d+1)$-dimensional projective approaches. The approach uses a Noether procedure and an explicit vertex ansatz to generate a generating function for three-point interactions, revealing a direct dictionary between ambient/projective and tractor formulations and highlighting bulk/boundary conformal structures. This tractor-based calculus clarifies the role of conformal geometry in higher-spin interactions and provides a path toward higher-point functions and deeper links to AdS/CFT and conformal bootstrap analyses, while noting that quartic locality and second-class constraints for massive/PM fields require further study.

Abstract

Higher spin theories can be efficiently described in terms of auxiliary Stückelberg or projective space field multiplets. By considering how higher spin models couple to scale, these approaches can be unified in a conformal geometry/tractor calculus framework. We review these methods and apply them to higher spin vertices to obtain a generating function for massless, massive and partially massless three-point interactions.

Figures (3)

• Figure 1: Conformally related metrics are obtained by slicing the conformal cone. These are conformally Einstein when there exist slices admitting a parallel scale tractor $I$. The second picture depicts the slice inducing an Einstein metric.
• Figure 2: Coupling to scale through the scale tractor $I$ determines the evolution of physical fields with masses labeled by conformal weights. The scale tractor also determines how data is moved from the boundary (the zero scale slice) to the bulk (standardly described by a constant scale slice).
• Figure 3: Index-free tractor field equations for totally symmetric higher spins of any mass type. These also apply to the ambient description in terms of fields $\Phi(X,U)$ extended off the cone and subject to $\Phi\sim \Phi+X^2 S\, .$ In that case the weight condition is rewritten as the homogeneity one \ref{['hom']}.