• massive gravity
• bimetric gravity
• multimetric gravity
• supersymmetry
• multimetric supergravity
• supergravity

In this thesis we take the first steps on the road that goes from studying nonlinear interactions between spin-2 fields to investigating those between spin-2 supermultiplets. After the recent discovery of the nonlinear theory of massive gravity by de Rham, Gabadadze and Tolley and its subsequent generalization to multimetric theories of gravity, in which we see nonlinear interactions between multiple massive spin-2 fields and a single massless one, much progress has been done in the understanding of these theories, which admit two formulations: one exploits metric tensors, while the other employs vielbein variables. The latter formulation in particular allows to avoid the square root tensor apparently unavoidable in the metric approach. Originally, the main motivation for the study of these theories was that they seem to naturally exhibit self-acceleration, and thus to have the potential to address the issue of dark energy. Recently it has been suggested that the additional spin-2 field may also be an interesting dark matter candidate.

The main result of this thesis is to provide the supersymmetric extension of multimetric gravities in the vielbein formulation for space-time dimensions up to D = 4. These classes of supergravity models were not explored so far.
Apart from their intrinsic interest, multimetric supergravities may be relevant for a number of reasons. For instance, they may shed light on the positive energy branch of the non supersymmetric theory, in the same way as the proof of the positive energy theorem for General Relativity was greatly simplified by Witten, using results coming from Supergravity. Further, taking into account the presence of massive supermultiplets, there may be room for the implementation of new types of supersymmetry breaking scenarios. In the present work we focused on the general construction, leaving to future investigation the analysis of possible applications.

Due to the particular form of the interaction potential, usual superfield techniques could not be straightforwardly applied to the case at hand. For this reason, in order to tackle the problem we resorted to recent developments on the theory of integration over supermanifolds, generalizations of the usual differentiable manifolds to the case in which anticommuting (fermionic) coordinates are considered. In particular, we made systematic use of the calculus of integral forms: if one considers ordinary differential forms as polynomials of the differentials (with multiplication given by the wedge product), integral forms will be instead distributions in the same variables with support in the origin. This approach has the advantage of allowing to describe integration on supermanifolds using forms in a very similar way as in the usual case, and thus to write manifestly supercovariant actions. We were able, using this tool, to write the actions which generalize the multimetric gravity models by displaying manifest local supersymmetry.

We observed that multimetric gravity theories incarnate particular instances of a more general mechanism that gives mass to gauge fields. We implemented this mechanism both in the Yang-Mills case and in the case of a supersymmetric spin-1 multiplet: for the latter, interestingly enough, we could see explicitly how the multiplets recombine in the right way giving full massive multiplets together with a single massless one. Also, the massless and massive combinations are the same in this case and in gravity, again showing the generality of the underlying mechanism. While our description holds at the classical level, it would be interesting to study the quantum theory of the vector case. Our work on these new supersymmetric theories, both in the spin-1 and in the spin-2 case, led to the article "Multimetric Supergravities" (arXiv:1605.06793 [hep-th]).

We also investigated some aspects of multimetric theories in the non-supersymmetric context. Indeed, one of conceptual conundrums of multimetric gravity is that the underlying geometry maybe is not yet fully transparent. In this thesis we tried to shed some additional light in this respect: we generalized a covariant constraint analysis made by Deser et al. for the case of dRGT massive gravity to that of bimetric gravity, in which we have one diffeomorphism and one local Lorentz invariance. This allowed us to give a clear direct interpretation in terms of gauge symmetries of some of the constraints arising from the equations of motion. With this information we were able to give a group manifold formulation of bimetric gravity: in this type of treatment, one sees the fibre bundle structure of the theory, which is usually taken as the starting point, emerging from its field equations. This allows to see clearly that the geometric structure underlying bimetric gravity is exactly the same as that of General Relativity: a “Poincaré bundle” in which the diffeomorphisms can be interpreted as a “gauging” of the translation sector of the Poincaré group. An interesting perspective would be to see if one can interpret the limitations to the possible interaction terms one can consider in the case of more than two vielbeins in light of these new results.

In addition, we also tried to develop an alternative view on the geometry of the vielbein formulation. In particular, we proposed a new set of variables providing a nonlinear extension of the linear massless mode of bimetric gravity. In our opinion this nonlinear extension looks more natural than those previously considered in the context of the metric formulation. Further, in these new variables, for a specific choice of the parameters, it appears that the action of bimetric gravity in the metric formulation may be rewritten without any square-root tensor.