# Vierbein interactions with antisymmetric components

## TL;DR

This paper introduces a new gravitational theory with two interacting vierbein fields, incorporating antisymmetric components as dynamical fields, and analyzes its degrees of freedom and stability.

## Contribution

It extends previous vierbein formulations by promoting antisymmetric components to dynamical fields with a kinetic term, inspired by ghost-free massive gravity and bimetric theories.

## Key findings

- The theory describes interactions of massless and massive spin-2 fields with an antisymmetric tensor.
- The model is free of extra degrees of freedom and potential pathologies.
- The antisymmetric tensor exhibits a tachyonic mass pole around symmetric backgrounds.

## Abstract

In this work we propose a new gravitational setup formulated in terms of two interacting vierbein fields. The theory is the fully diffeomorphism and local Lorentz invariant extension of a previous construction which involved a fixed reference vierbein. Certain vierbein components can be shifted by local Lorentz transformations and do not enter the associated metric tensors. We parameterize these components by an antisymmetric tensor field and give them a kinetic term in the action, thereby promoting them to dynamical variables. In addition, the action contains two Einstein-Hilbert terms and an interaction potential whose form is inspired by ghost-free massive gravity and bimetric theory. The resulting theory describes the interactions of a massless spin-2, a massive spin-2 and an antisymmetric tensor field. It can be generalized to the case of multiple massive spin-2 fields and multiple antisymmetric tensor fields. The absence of additional and potentially pathological degrees of freedom is verified in an ADM analysis. However, the antisymmetric tensor fluctuation around the maximally symmetric background solution has a tachyonic mass pole.