^{1}

^{2}

^{3}.

Thus far, all known ghost-free interactions of multiple spin-2 fields have involved at most pairwise couplings of the fields, which are direct generalizations of bimetric interactions. We present a class of spin-2 theories with genuine multifield interactions and explicitly demonstrate the absence of ghost instabilities. The construction involves integrating out a nondynamical field in a theory of spin-2 fields with only pairwise ghost-free interactions. The new multivierbein interactions generated are not always expressible in terms of the associated metrics.

Interacting theories for multiple fields with spin 0,

In a covariant setup, spin-2 fields have more components than physically needed, and generic theories do not have enough symmetries and constraints to remove the unphysical components. Some of these, if not eliminated, give rise to ghost instabilities, with an example being the Boulware-Deser ghost

Theories for more than two spin-2 fields are also strongly restricted by the absence of ghosts. From the analysis of Ref.

An important class of multiple spin-2 theories was constructed in Ref.

The question is whether there exist interactions between multiple spin-2 fields beyond the pairwise ones, and thus beyond the class conjectured in Ref.

Summary of results: In this work we derive a class of ghost-free interactions for multiple spin-2 fields by integrating out a nondynamical field in a theory with ghost-free bimetric interactions. The result is an interaction term for

The starting point is a theory for

Lorentz constraints: Each vierbein contains six Lorentz parameters that drop out of the corresponding metric and hence appear in the action

Multiple spin-2 action: Let us set the coupling

The

In the following we will use

We decompose a vierbein

In the action, we write

It is easy to show that the potential

The dynamical fields

First,

Finally, the

Of the remaining constraints, one combination of the

The degree of freedom count is now easy. The fields

Ghost-free multivierbein theories with only pairwise interactions can be expressed in terms of the corresponding metrics, although some extra information contained in the vierbeins is retained (corresponding to the choices of square-root matrices that appear in the theory). Here we explore the existence of a similar metric formulation for the nonpairwise interaction in Eq.

By extracting a factor of

Metric formulation for three fields: We now consider metric representations for the case

The analysis partially generalizes to

Finally, it is important to note that, while the vierbeins were restricted by hand to obtain a metric formulation, such restrictions are inbuilt in the final multimetric theory. Hence, the resulting multimetric theories can be considered in their own right, independent of the starting vierbein formulations.

To summarize, we have constructed nontrivial interactions of multiple spin-2 fields, beyond the known pairwise potentials, and have demonstrated the existence of constraints that eliminate the extra ghost modes. The interactions are given in terms of the

The class of theories obtained here can be further generalized. First, note that the interactions

It is straightforward to introduce a standard coupling to matter via any of the dynamical vierbeins

The interactions for

Our couplings are the first instance of ghost-free spin-2 interactions where the vierbein formulation admits more general configurations than the associated, more restrictive metric formulation. Hence, it is interesting to directly investigate the associated multimetric theories, without recourse to the vierbein formulation. One expects that the extra restrictions on the metrics, already encoded in the multimetric interactions

The work of A. S. M. is supported by a grant from the Max Planck Society. S. F. H. acknowledges support from the Swedish Research Council.