Minimal Local Lagrangians for Higher-Spin Geometry

TL;DR

The paper shows that removing Fronsdal's trace constraints allows a minimal, local Lagrangian description of unconstrained higher-spin fields. It introduces two additional fields for bosons, $α$ and $β$, and two for fermions, $ξ$ and $λ$, yielding gauge-invariant Lagrangians that reproduce the geometric (nonlocal) higher-spin equations in compensator form, e.g. $F=3\,∂^{3}α$ and $φ''=4\,∂·α+∂α'$. By explicit Noether construction, the authors demonstrate the equivalence between the local compensator framework and the nonlocal geometric approach, providing a streamlined off-shell description for free higher-spin fields and a foundation for future extensions to (A)dS and interactions. This work connects string-field insights with a minimal, Lagrangian formulation, offering a practical path toward unconstrained higher-spin dynamics.

Abstract

The Fronsdal Lagrangians for free totally symmetric rank-s tensors rest on suitable trace constraints for their gauge parameters and gauge fields. Only when these constraints are removed, however, the resulting equations reflect the expected free higher-spin geometry. We show that geometric equations, in both their local and non-local forms, can be simply recovered from local Lagrangians with only two additional fields, a rank-(s-3) compensator and a rank-(s-4) Lagrange multiplier. In a similar fashion, we show that geometric equations for unconstrained rank-n totally symmetric spinor-tensors can be simply recovered from local Lagrangians with only two additional spinor-tensors, a rank-(n-2) compensator and a rank-(n-3) Lagrange multiplier.