On the Uniqueness of Ghost-Free Multi-Gravity -- II: Constraining antisymmetrised multi spin-2 interactions

TL;DR

This paper proves the uniqueness of ghost-free multi-spin-2 field interactions, showing that only the known multivielbein theory with a tree-structured interaction graph is consistent without ghost instabilities.

Contribution

It establishes the most general form of ghost-free multi-spin-2 interactions, confirming the uniqueness of the known multivielbein theory for multiple fields.

Findings

For two vielbeins, parameters remain unrestricted, reproducing ghost-free bimetric theory.

For more than two vielbeins, couplings are restricted to the known ghost-free multivielbein form.

Interactions with a tree-structured graph satisfy ghost-free conditions.

Abstract

So far, only a single theory of multiple spin-2 fields is known that features genuine multi-field interactions while remaining free of Boulware-Deser-type ghost instabilities. In this paper we show that this is the most general ghost-free multi spin-2 interaction type possible. We start with the general class of multivielbein interactions containing antisymmetrised products of vielbeins, considered earlier by Hinterbichler and Rosen. We formulate a necessary condition for these theories to be ghost-free. For two vielbeins the theory parameters remain unrestricted, reproducing the ghost-free bimetric theory. But for more than two vielbeins with genuine multi-field interactions, we show that the couplings are restricted precisely to yield the known ghost-free multivielbein theory, thus establishing its uniqueness. We also show that more general interactions, constructed using the ghost-free bimetric and multivielbein potentials as building blocks, satisfy the necessary ghost-free conditions provided the associated interaction graphs have a tree structure.