Classifying Galileon p-form theories

We provide a complete classification of all abelian gauge invariant p-form theories with equations of motion depending only on the second derivative of the field-the p-form analogues of the Galileon scalar field theory. We construct explicitly the nontrivial actions that exist for spacetime dimension D <= 11, but our methods are general enough and can be extended to arbitrary D. We uncover in particular a new 4-form Galileon cubic theory in D >= 8 dimensions. As a by-product we give a simple proof of the fact that the equations of motion depend on the p-form gauge fields only through their field strengths and show this explicitly for the recently discovered 3-form Galileon quartic theory.