The Fronsdal Lagrangians for free totally symmetric rank-s tensors φμ1...μsrest 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 αμ1...μs-3and a rank-(s - 4) Lagrange multiplier βμ1...μs-4. In a similar fashion, we show that geometric equations for unconstrained rank-n totally symmetric spinor-tensors ψμ1...μncan be simply recovered from local Lagrangians with only two additional spinor-tensors, a rank-(n - 2) compensator ξμ1...μn-2and a rank-(n - 3) Lagrange multiplier λμ1...αn-3. © 2005 Elsevier B.V. All rights reserved.
Francia, D., & Sagnotti, A. (2005). Minimal local Lagrangians for higher-spin geometry. Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics, 624(1–2), 93–104. https://doi.org/10.1016/j.physletb.2005.08.002