Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Tractors, Mass and Weyl Invariance

A. R. Gover, A. Shaukat, A. Waldron

TL;DR

This work constructs a unified, Weyl-invariant framework for scalars, vectors, spin-2, and higher-spin fields using tractor calculus and a scale (Weyl compensator). By packaging fields into $SO(d,2)$ multiplets and employing the scale tractor $I^M= rac{1}{d}D^M \sigma$, masses arise from Weyl weights through the relation $m^2= rac{2{ m P}}{d}ig[( rac{d-1}{2})^2-(w+ rac{d-1}{2})^2ig]$, with special weights yielding conformal, Maxwell, massless, or partially massless theories. The formalism naturally yields Breitenlohner–Freedman bounds and provides tractor-based actions and operator identities that unify massive, massless, and partially massless dynamics in constant curvature backgrounds. It also frames gravity with a Weyl compensator, linking Weyl invariance to the Einstein–Hilbert action and spontaneous symmetry breaking of the Weyl scale. Overall, the tractor approach offers a powerful, systematic route to Weyl-invariant ancestors of a broad spectrum of field theories and paves the way for exploring higher-spin interactions within a conformal-geometric setting.

Abstract

Deser and Nepomechie established a relationship between masslessness and rigid conformal invariance by coupling to a background metric and demanding local Weyl invariance, a method which applies neither to massive theories nor theories which rely upon gauge invariances for masslessness. We extend this method to describe massive and gauge invariant theories using Weyl invariance. The key idea is to introduce a new scalar field which is constant when evaluated at the scale corresponding to the metric of physical interest. This technique relies on being able to efficiently construct Weyl invariant theories. This is achieved using tractor calculus--a mathematical machinery designed for the study of conformal geometry. From a physics standpoint, this amounts to arranging fields in multiplets with respect to the conformal group but with novel Weyl transformation laws. Our approach gives a mechanism for generating masses from Weyl weights. Breitenlohner--Freedman stability bounds for Anti de Sitter theories arise naturally as do direct derivations of the novel Weyl invariant theories given by Deser and Nepomechie. In constant curvature spaces, partially massless theories--which rely on the interplay between mass and gauge invariance--are also generated by our method. Another simple consequence is conformal invariance of the maximal depth partially massless theories. Detailed examples for spins s<=2 are given including tractor and component actions, on-shell and off-shell approaches and gauge invariances. For all spins s>=2 we give tractor equations of motion unifying massive, massless, and partially massless theories.

Tractors, Mass and Weyl Invariance

TL;DR

This work constructs a unified, Weyl-invariant framework for scalars, vectors, spin-2, and higher-spin fields using tractor calculus and a scale (Weyl compensator). By packaging fields into multiplets and employing the scale tractor , masses arise from Weyl weights through the relation , with special weights yielding conformal, Maxwell, massless, or partially massless theories. The formalism naturally yields Breitenlohner–Freedman bounds and provides tractor-based actions and operator identities that unify massive, massless, and partially massless dynamics in constant curvature backgrounds. It also frames gravity with a Weyl compensator, linking Weyl invariance to the Einstein–Hilbert action and spontaneous symmetry breaking of the Weyl scale. Overall, the tractor approach offers a powerful, systematic route to Weyl-invariant ancestors of a broad spectrum of field theories and paves the way for exploring higher-spin interactions within a conformal-geometric setting.

Abstract

Deser and Nepomechie established a relationship between masslessness and rigid conformal invariance by coupling to a background metric and demanding local Weyl invariance, a method which applies neither to massive theories nor theories which rely upon gauge invariances for masslessness. We extend this method to describe massive and gauge invariant theories using Weyl invariance. The key idea is to introduce a new scalar field which is constant when evaluated at the scale corresponding to the metric of physical interest. This technique relies on being able to efficiently construct Weyl invariant theories. This is achieved using tractor calculus--a mathematical machinery designed for the study of conformal geometry. From a physics standpoint, this amounts to arranging fields in multiplets with respect to the conformal group but with novel Weyl transformation laws. Our approach gives a mechanism for generating masses from Weyl weights. Breitenlohner--Freedman stability bounds for Anti de Sitter theories arise naturally as do direct derivations of the novel Weyl invariant theories given by Deser and Nepomechie. In constant curvature spaces, partially massless theories--which rely on the interplay between mass and gauge invariance--are also generated by our method. Another simple consequence is conformal invariance of the maximal depth partially massless theories. Detailed examples for spins s<=2 are given including tractor and component actions, on-shell and off-shell approaches and gauge invariances. For all spins s>=2 we give tractor equations of motion unifying massive, massless, and partially massless theories.

Paper Structure

This paper contains 17 sections, 168 equations, 2 figures, 6 tables.

Table of Contents

  1. Introduction
  2. The Tractor Philosophy
  3. Scalars
  4. Vectors
  5. On-Shell Approach
  6. Spin Two
  7. On-Shell Approach
  8. Off-Shell Approach
  9. Arbitrary Spins
  10. On-Shell Approach
  11. Off-Shell Approach
  12. Gravity and Weyl Compensators
  13. Conclusions
  14. Compendium of Weyl Transformations
  15. Tractor Component Expressions
  16. ...and 2 more sections

Figures (2)

  • Figure 1: The Weyl weight $w$ can be reinterpreted as a scalar mass parameter. Generic values of $w$ (the thick line) give massive theories, while $w=\frac{1}{2} -\frac{d}{2}$ and $w=1-\frac{d}{2}$ describe a scalar saturating the Breitenlohner--Freedman bound (in Anti de Sitter space) and an improved scalar, respectively.
  • Figure 2: A plot of the theories described by the tractor Maxwell system as a function of Weyl weight and dimension. Theories saturating a vector Breitenlohner--Freedman bound appear at $w=\frac{1}{2}-\frac{d}{2}$.