Minimal local Lagrangians for higher-spin geometry

The Fronsdal Lagrangians for free totally symmetric rank-s tensors φμ1…μs 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 αμ1…μs−3 and 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…μn can be simply recovered from local Lagrangians with only two additional spinor-tensors, a rank-(n−2) compensator ξμ1…μn−2 and a rank-(n−3) Lagrange multiplier λμ1…μn−3.