The Third Way to Interacting $p$-form Theories

TL;DR

This paper introduces a new class of interacting p-form theories in various dimensions that are consistent on-shell despite not deriving from an action, expanding the landscape of gauge theories with novel interaction structures.

Contribution

It presents a novel 'third way' consistent construction of interacting (d-2)-form theories, including generalizations and higher derivative extensions, not obtainable from traditional action principles.

Findings

Constructed a new class of on-shell consistent p-form theories

Extended the method to include couplings with (d-3)-forms and 3-algebras

Recovered and generalized 3d Yang-Mills theory

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.