An Introduction to Free Higher-Spin Fields

TL;DR

The paper surveys the development of free higher-spin field theories, tracing the progression from Fierz–Pauli and Singh–Hagen formalisms to Fronsdal's constrained massless equations for integer and half-integer spins. It then presents non-local geometric formulations that remove trace constraints via non-local Fronsdal-like operators and higher-spin curvatures, and introduces local compensator formulations that realize unconstrained gauge symmetry. The triplet framework, arising naturally from open string field theory in the tensionless limit, provides a BRST-based description and its AdS extensions for bosons and fermions, linking to discussions of Aragone–Deser obstructions and Vasiliev-type approaches. Together these formalisms show how unconstrained, local or non-local descriptions reproduce the same physical DOF and recover Fronsdal equations after partial gauge fixing, while extending to (A)dS backgrounds. The work thus connects free higher-spin dynamics with string theory and larger higher-spin gravity programs, offering robust tools for exploring consistent gauge theories of arbitrarily high spin.

Abstract

In this article we begin by reviewing the (Fang-)Fronsdal construction and the non-local geometric equations with unconstrained gauge fields and parameters built by Francia and the senior author from the higher-spin curvatures of de Wit and Freedman. We then turn to the triplet structure of totally symmetric tensors that emerges from free String Field Theory in the $α' \to 0$ limit and to its generalization to (A)dS backgrounds, and conclude with a discussion of a simple local compensator form of the field equations that displays the unconstrained gauge symmetry of the non-local equations. Based on the lectures presented by A. Sagnotti at the First Solvay Workshop on Higher-Spin Gauge Theories held in Brussels on May 12-14, 2004