Inconsistency of interacting, multi-graviton theories Original Research Article

We investigate, in any space–time dimension ⩾3, the problem of consistent couplings for a (finite or infinite) collection of massless, spin-2 fields described, in the free limit, by a sum of Pauli–Fierz actions. We show that there is no consistent (ghost-free) coupling, with at most two derivatives of the fields, that can mix the various “gravitons”. In other words, there are no Yang–Mills-like spin-2 theories. The only possible deformations are given by a sum (or integral) of individual Einstein–Hilbert actions. The impossibility of cross-couplings subsists in the presence of scalar matter. Our approach is based on the BRST-based deformation point of view and uses results on the so-called “characteristic cohomology” for massless spin-2 fields which are explained in detail.