On the geometry of higher-spin gauge fields

TL;DR

This work surveys a geometric reformulation of free higher-spin equations for symmetric bosons and tensor-spinors, emphasizing unconstrained gauge invariance and non-local operators that generalize the Fronsdal framework. It connects this geometry to the de Wit–Freedman curvatures, shows how local Fronsdal equations emerge after partial gauge fixing or via compensator fields, and links the formalism to free String Field Theory through triplet systems. The discussion spans flat and (A)dS backgrounds, including local compensator forms and their BRST-inspired origins, and highlights potential ties to Vasiliev-type theories and Segal's conformal higher-spin program. Collectively, the results illuminate how higher-spin dynamics can be structured by underlying geometric objects and BRST-based constructions, with implications for string theory and beyond.

Abstract

We review a recent construction of the free field equations for totally symmetric tensors and tensor-spinors that exhibits the corresponding linearized geometry. These equations are not local for all spins >2, involve unconstrained fields and gauge parameters, rest on the curvatures introduced long ago by de Wit and Freedman, and reduce to the local (Fang-)Fronsdal form upon partial gauge fixing. We also describe how the higher-spin geometry is realized in free String Field Theory, and how the gauge fixing to the light cone can be effected. Finally, we review the essential features of local compensator forms for the higher-spin bosonic and fermionic equations with the same unconstrained gauge symmetry.