Effective actions for dual massive (super) p-forms

TL;DR

This paper derives a compact expression for the effective action of massive p-forms in curved space, explores their topological relations, and extends the analysis to massive super p-forms coupled to supergravity.

Contribution

It provides a new formula for massive p-form effective actions, reveals their topological differences, and extends the framework to supergravity coupled massive super p-forms.

Findings

Effective actions expressed via Hodge-de Rham determinants.

Effective actions of dual massive p-forms differ by a topological invariant.

Massive vector and tensor multiplet actions coincide; three-form action relates to scalar multiplets.

Abstract

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