All consistent interactions for exterior form gauge fields

We give the complete list of all first-order consistent interaction vertices for a set of exterior form gauge fields of form degree $>1\mathrm{}$, described in the free limit by the standard Maxwell-like action. Special attention is paid to the interactions that deform the gauge transformations. These are shown to be necessarily of the Noether form “conserved antisymmetric tensor” times “$p$-form potential” and exist only in particular spacetime dimensions. Conditions for consistency to all orders in the coupling constant are given. For illustrative purposes, the analysis is carried out explicitly for a system of forms with two different degrees $p$ and $q$ $\mathrm{}(1<p<q<n)\mathrm{}$.