Restrictions for n-point vertices in higher-spin theories

We give a simple classification of the independent n-point interaction vertices for bosonic higher-spin gauge fields in d-dimensional Minkowski spacetimes. We first give a characterisation of such vertices for large dimensions, d ≥ 2n − 1, where one does not have to consider Schouten identities due to over-antisymmetrisation of spacetime indices. When the dimension is lowered, such identities have to be considered, but their appearance only leads to equivalences of large-d vertices and does not lead to new types of vertices. We consider the case of low dimensions (d < n) in detail, where a large number of Schouten identities leads to strong restrictions on independent vertices. We also comment on the generalisation of our results to the intermediate region n ≤ d ≤ 2n − 2. In all cases, the independent vertices are expressed in terms of elementary manifestly gauge-invariant quantities, suggesting that no deformations of the gauge transformations are induced.