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Abstract

We construct a class of interacting $(d-2)$-form theories in $d$ dimensions
that are `third way' consistent. This refers to the fact that the interaction
terms in the $p$-form field equations of motion neither come from the variation
of an action nor are they off-shell conserved on their own. Nevertheless the
full equation is still on-shell consistent. Various generalizations, e.g.
coupling them to $(d-3)$-forms, where 3-algebras play a prominent role, are
also discussed. The method to construct these models also easily recovers the
modified 3$d$ Yang-Mills theory obtained earlier and straightforwardly allows
for higher derivative extensions.