Inconsistency of interacting, multi-graviton theories

TL;DR

This paper proves that in any spacetime dimension >=3, multiple massless spin-2 fields cannot be consistently coupled together in a ghost-free manner, implying no Yang-Mills-like theories for gravitons.

Contribution

It demonstrates the fundamental inconsistency of cross-couplings among multiple massless gravitons using BRST cohomology, extending previous no-go results.

Findings

No ghost-free, two-derivative cross-couplings among multiple gravitons.

The only consistent theories are sums of individual Einstein-Hilbert actions.

Cross-couplings remain impossible even with scalar matter present.

Abstract

We investigate, in any spacetime dimension >=3, the problem of consistent couplings for a finite 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 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.