Abstract. We present the complete form of the decoupling limit of ghost-free massive gravity with a Minkowski reference metric, including the full interactions of the helicity-1 and helicity-0 modes of the massive spin-2 field. While in the metric language the square root structure of the mass terms makes it difficult to find a simple way to write down the interactions, we show that using the vierbein formulation of massive gravity, including Stückelberg fields for both diffeomorphism and local Lorentz symmetries, we can find an explicitly resummed expression for the helicity-1 field interactions. We clarify the equations of motion for the Lorentz Stückelberg fields and how these generate the symmetric vierbein condition which guarantees equivalence between the vierbein and metric formulations of massive gravity.

In light of recent progress in ghost-free theories of massive gravity and multi-gravity, we reconsider the problem of constructing a ghost-free theory of an interacting spin-2 field charged under a U(1) gauge symmetry. Our starting point is the theory originally proposed by Federbush, which is essentially Fierz-Pauli generalized to include a minimal coupling to a U(1) gauge field. We show the Federbush theory with a dynamical U(1) field is in fact ghost-free and can be treated as a healthy effective field theory to describe a massive charged spin-2 particle. It can even potentially have healthy dynamics above its strong-coupling scale. We then construct candidate gravitational extensions to the Federbush theory both by using Dimensional Deconstruction, and by constructing a general non-linear completion. However, we find that the U(1) symmetry forces us to modify the form of the Einstein-Hilbert kinetic term. By performing a constraint analysis directly in the first-order form, we show that these modified kinetic terms inevitably reintroduce the Boulware-Deser ghost. As a by-product of our analysis, we present a new proof for ghost-freedom of bi-gravity in 2+1 dimensions (also known as Zwei-Dreibein gravity). We also give a complementary algebraic argument that the Einstein-Hilbert kinetic term is incompatible with a U(1) symmetry, for a finite number of gravitons.(1) ij 43 1 IntroductionIt has been more than seventy years since Wigner demonstrated that all consistent, relativistic, quantum particles can be classified by their mass m and their spin j [1, 2]. Experimentally, particle accelerators have established the existence of composite, charged massive higher spin particles [3]. Nevertheless, the theoretical understanding of higher spin fields is considerably less developed than their lower spin counterparts.The most obvious bosonic higher spin theory to consider is spin-2. There are arguments that the only consistent theory of a massless, self-interacting, Lorentz-invariant spin-2 field is General Relativity [4][5][6][7][8]. In fact, recent work has established that these assumptions may be weakened somewhat. Ghost-freedom alone is sufficient to derive the Einstein-Hilbert action as the kinetic term for Lorentz-invariant massive fields [9] or for massless gravity theories where time translation invariance is broken explicitly [10,11].However the massive case is less well understood. In the 1930's, Fierz and Pauli wrote down the linearized, non-interacting theory of a single massive spin-2 field [12,13]. There are several issues. The first is the vDVZ discontinuity of this model [14,15]. The vDVZ discontinuity is a curious feature of the Fierz-Pauli action, which is that in the limit m → 0, the Fierz-Pauli predictions do not become equivalent to that of the linearized Einstein-Hilbert Lagrangian. Vainshtein was the first to see that this discontinuity could be avoid by adding self-interactions for the massive spin-2 field, and associating the regime of validity of the linear approximation [16]. However, Boulwa...

Given the success of the deconstruction program in obtaining ghost-free massive gravity from 5-D Einstein gravity, we propose a modification of the deconstruction procedure that incorporates supersymmetry at the linear level. We discuss the relevant limits of a conjectured interacting theory of a massive spin 2 supermultiplet, and determine the linear theory to be the N = 1 Zinoviev theory, a supersymmetric extension of Fierz-Pauli theory. We develop a family of 1-site deconstruction procedures for fermionic fields (yielding Dirac and Majorana mass terms). The deconstruction procedure appropriate for giving fermions a Dirac mass is found to preserve half of the supersymmetry of the 5-D theory. We explicitly check this by deconstructing 5-D N = 2 super-Maxwell theory down to 4-D N = 1 super-Proca theory, and deconstructing linear 5-D N = 2 supergravity down to 4-D N = 1 Zinoviev theory, and derive the full 4-D supersymmetry algebras and Stückelberg symmetries from the 5-D superalgebras and gauge symmetries, respectively. We conjecture that this procedure should admit a generalization to fully non-linear theories.

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