Metric Formulation of Ghost-Free Multivielbein Theory

TL;DR

This paper provides a ghost-free, gauge-invariant reformulation of the ghost-free multivielbein theory for an arbitrary number ${\cal N}$ of spin-2 fields by reintroducing the ${\cal N}-1$ broken local Lorentz invariances and trading the vielbein description for a fully metric (multimetric) formulation. It introduces Stückelberg Lorentz fields to preserve all ${\cal N}$ local Lorentz invariances, derives a metric-based mass potential that depends on auxiliary matrices $L(I)$ and the square-root matrices $\sqrt{g^{-1}(1)g(I)}$, and shows that the $L(I)$ are auxiliary fields constrained by equations equivalent to the vielbein antisymmetric constraints. The authors demonstrate the no-ghost property persists in this gauge-invariant setting and establish on-shell equivalence between the metric and vielbein formulations, including explicit results for ${\cal N}=2$ (bimetric) and ${\cal N}=3$ (tri-metric) cases. This metric formulation generalizes the deformed-determinant structure of prior bimetric theories to the multimetric context, facilitating covariant spin-2 interactions and potential phenomenological applications while maintaining the correct number of propagating degrees of freedom.

Abstract

We formulate the recently proposed ghost-free theory of multiple interacting vielbeins in terms of their corresponding metrics. This is achieved by reintroducing all local Lorentz invariances broken by the multivielbein interaction potential which, in turn, allows us to explicitly separate the gauge degrees of freedom in the vielbeins from the components of the metrics by an appropriate gauge choice. We argue that the gauge choice does not spoil the no-ghost proof of the multivielbein theory, hence the multimetric theory is ghost-free. We further show the on-shell equivalence of the metric and vielbein descriptions, first in general and thereafter in two illustrative examples.