Metric-like Lagrangian Formulations for Higher-Spin Fields of Mixed Symmetry

TL;DR

This work surveys metric-like Lagrangian formulations for free higher-spin fields of mixed symmetry in flat space, detailing constrained (Labastida) and unconstrained approaches with compensators and Lagrange multipliers, including minimal and low-derivative variants. It shows how to construct gauge-invariant Lagrangians for Bose and Fermi fields using two-column projections of kinetic tensors and Bianchi identities, and explains how irreducible representations fit naturally into the framework. The analysis reveals Weyl-like symmetries that arise in low spacetime dimensions, leading to pathologies where actions vanish or gauge equivalence to Labastida form emerges only after accounting for these extra symmetries. The review connects these metric-like constructions to nonlocal and frame-like formulations, outlining pathways toward interacting theories and extensions to constant-curvature backgrounds, with implications for string theory and higher-spin geometry.

Abstract

We review the structure of local Lagrangians and field equations for free bosonic and fermionic gauge fields of mixed symmetry in flat space. These are first presented in a constrained setting extending the metric formulation of linearized gravity, and then the ($γ$-)trace constraints on fields and gauge parameters are eliminated via the introduction of auxiliary fields. We also display the emergence of Weyl-like symmetries in particular classes of models in low space-time dimensions.