Cohomological techniques within the Batalin-Vilkovisky (BV) extension of the Becchi-Rouet-Stora-Tyutin (BRST) formalism have proved invaluable for classifying consistent deformations of gauge theories. In this work we investigate the application of this idea to massive field theories in the Stueckelberg formulation. Starting with a collection of free massive vectors, we show that the cohomological method reproduces the cubic and quartic vertices of massive Yang-Mills theory. In the same way, taking a Fierz-Pauli graviton on a maximally symmetric space as the starting point, we are able to recover the consistent cubic vertices of nonlinear massive gravity. The formalism further sheds light on the characterization of Stueckelberg gauge theories, by demonstrating for instance that the gauge algebra of such models is necessarily Abelian and that they admit a Born-Infeld-like formulation in which the action is simply a combination of the gauge-invariant structures of the free theory.

Starting from the dual Lagrangians recently obtained for (partially) massless spin-2 fields in the Stueckelberg formulation, we write the equations of motion for (partially) massless gravitons in (A)dS in the form of twisted-duality relations. In both cases, the latter admit a smooth flat limit. In the massless case, this limit reproduces the gravitational twistedduality relations previously known for Minkowski spacetime. In the partially-massless case, our twisted-duality relations preserve the number of degrees of freedom in the flat limit, in the sense that they split into a decoupled pair of dualities for spin-1 and spin-2 fields. Our results apply to spacetimes of any dimension greater than three. In four dimensions, the twisted-duality relations for partially massless fields that appeared in the literature are recovered by gauging away the Stueckelberg field.

We revisit the problem of building consistent interactions for a multiplet of partially massless spin-2 fields in (anti-)de Sitter space. After rederiving and strengthening the existing no-go result on the impossibility of Yang-Mills type non-abelian deformations of the partially massless gauge algebra, we prove the uniqueness of the cubic interaction vertex and field-dependent gauge transformation that generalize the structures known from single-field analyses. Unlike in the case of one partially massless field, however, we show that for two or more particle species the cubic deformations can be made consistent at the complete non-linear level, albeit at the expense of allowing for negative relative signs between kinetic terms, making our new theory akin to conformal gravity. Our construction thus provides the first example of an interacting theory containing only partially massless fields.

After proving the impossibility of consistent non-minimal coupling of a real Rarita-Schwinger gauge field to electromagnetism, we re-derive the necessity of introducing the graviton in order to couple a complex Rarita-Schwinger gauge field to electromagnetism, with or without a cosmological term, thereby obtaining N = 2 pure supergravity as the only possibility. These results are obtained with the BRST-BV deformation method around the flat and (A)dS backgrounds in 4 dimensions. The same method applied to n v vectors, N real spin-3/2 gauge fields and at most one real spinor field also requires gravity and yields N = 3 pure supergravity as well as N = 1 pure supergravity coupled to a vector supermultiplet, with or without cosmological terms. Independently of the matter content, we finally derive strong necessary quadratic constraints on the possible gaugings for an arbitrary number of spin-1 and spin-3/2 gauge fields, that are relevant for larger supergravities.

We perform a complete classification of the consistent two-derivative cubic couplings for a system containing an arbitrary number of massless spin-1, massless spin-2, and partially massless (PM) spin-2 fields in D-dimensional (anti-)de Sitter space. In addition to previously known results, we find a unique candidate mixing between spin-1 and PM spin-2 fields. We derive all the quadratic constraints on the structure constants of the theory, allowing for relative "wrong-sign" kinetic terms for any of the fields. In the particular case when the kinetic terms in each sector have no relative signs, we find that the unique consistent non-trivial theory is given by multiple independent copies of conformal gravity coupled to a Yang-Mills sector in D = 4. Our results strengthen the well-known no-go theorems on the absence of mutual interactions for massless and PM spin-2 fields.