Free geometric equations for higher spins

We show how allowing non-local terms in the field equations of symmetric tensors uncovers a neat geometry that naturally generalizes the Maxwell and Einstein cases. The end results can be related to multiple traces of the generalized Riemann curvatures Rα1⋯αs;β1⋯βs introduced by de Wit and Freedman, divided by suitable powers of the DʹAlembertian operator □. The conventional local equations can be recovered by a partial gauge fixing involving the trace of the gauge parameters Λα1⋯αs−1, absent in the Fronsdal formulation. The same geometry underlies the fermionic equations, that, for all spins s+1/2, can be linked via the operator ∂̷□ to those of the spin-s bosons.