Geometrical structure of Weyl invariants for spin three gauge field in general gravitational background in $d=4$

Understand

We construct all possible Weyl invariant actions in $d=4$ for linearized spin three field in a general gravitational background.

• The first action is obtained as the square of the generalized Weyl tensor for a spin three gauge field in nonlinear gravitational background.
• It is, however, not invariant under spin three gauge transformations.
• We then construct two other nontrivial Weyl but not gauge invariant actions which are linear in the Weyl tensor of the background geometry.