Effective actions for dual massive (super) p-forms

In $d$ dimensions, the model for a massless $p$-form in curved space is known to be reducible gauge theory $p>1$, and therefore its covariant quantisation cannot carried out using standard Faddeev-Popov scheme. However, adding mass term also introducing Stueckelberg reformulation of resulting model, one ends up with an irreducible which can quantised \`a la Faddeev Popov. We derive compact expression massive effective action, $\Gamma^{(m)}_p$, terms functional determinants Hodge-de Rham operators. then show that actions $\Gamma^{(m)}_p$ $\Gamma^{(m)}_{d-p-1}$ differ by topological invariant. This generalisation result case $\Gamma_p$ $\Gamma_{d-p-2}$ coincide modulo term. Finally, our analysis extended super $p$-forms coupled background ${\cal N}=1$ supergravity four dimensions. Specifically, we study quantum dynamics following $p$-forms: (i) vector multiplet; (ii) tensor (iii) three-form multiplet. It demonstrated multiplets coincide. The action shown sum those corresponding two scalar multiplets,