A model for massless higher spin field interacting with a geometrical background

We study a very general four dimensional Field Theory model describing the dynamics of a massless higher spin $N$ symmetric tensor field particle interacting with a geometrical background.This model is invariant under the action of an extended linear diffeomorphism. We investigate the consistency of the equations of motion, and the highest spin degrees of freedom are extracted by means of a set of covariant constraints. Moreover the the highest spin equations of motions (and in general all the highest spin field 1-PI irreducible Green functions) are invariant under a chain of transformations induced by a set of $N-2$ Ward operators, while the auxiliary fields equations of motion spoil this symmetry. The first steps to a quantum extension of the model are discussed on the basis of the Algebraic Field Theory.Technical aspects are reported in Appendices; in particular one of them is devoted to illustrate the spin-$2$ case.