Metric formulation of the simple theory of 3d massive gravity

We have recently introduced a new and very simple action for three-dimensional massive gravity. This action is written in a first-order formulation where the triad and the connection play a manifestly symmetric role, but where internal Lorentz gauge symmetry is broken. The absence of Lorentz invariance, which in this model is the mechanism underlying the propagation of a massive graviton, does however prevent us from writing a purely metric nonlinear action for the theory. Nonetheless, in this paper, we explain how to disentangle, at the nonlinear level, the metric and nonmetric degrees of freedom in the equations of motion. Focusing on the metric part, we show that it satisfies modified Einstein equations with higher derivative terms. As a particular case, these equations reproduce a well-studied model known as minimal massive gravity. In the general case, we obtain new metric field equations for massive gravity in three dimensions starting from the simple first-order action. These field equations are consistent through a mechanism known as "third way consistency," which our theory therefore provides a new example of.