Geometric Lagrangians for massive higher-spin fields

TL;DR

The paper develops a geometric, unconstrained formulation for massive higher-spin bosons and fermions by deforming massless, curvature-based Lagrangians with a generalised Fierz-Pauli mass term that includes all traces of the field. It constructs bosonic and fermionic sectors using de Wit–Freedman curvatures, defines divergenceless Einstein tensors, and shows how ordinary-derivative local Lagrangians can reproduce the same dynamics with an extended field content. Spin-by-spin analyses (spin 3, 4, and general spin $s$ for bosons; spin $5/2$, $7/2$, and general spin for fermions) demonstrate how the mass terms enforce the Fierz–Pauli constraints on-shell, yielding correct massive propagation without auxiliary fields. The work discusses the non-uniqueness of geometric Lagrangians, the role of external currents in selecting a preferred theory, and the connection between non-local geometric descriptions and local unconstrained formulations, providing a comprehensive framework for massive higher-spin dynamics with potential links to string field theory. The results advance the understanding of how geometric, curvature-based approaches can describe massive higher-spin representations in a consistent, gauge-invariant way.

Abstract

Lagrangians for massive, unconstrained, higher-spin bosons and fermions are proposed. The idea is to modify the geometric, gauge invariant Lagrangians describing the corresponding massless theories by the addition of suitable quadratic polynomials. These polynomials provide generalisations of the Fierz-Pauli mass term containing all possible traces of the basic field. No auxiliary fields are needed.