Non-linear massive gravity as a gravitational σ-model

We show the direct analogy between the ghost-free non-linear formulation of massive gravity and the standard σ-models well understood in the literature. This issue explains why there are two non-trivial family of solutions for the spherically symmetric case inside the non-linear massive gravity formulations with two free parameters α and β. In general, the case $\beta <{\alpha }^2$ has a single physical vacuum state. On the other hand, the case $\beta ={\alpha }^2$ contains a natural vacuum degeneracy. This is in perfect analogy with the σ-model for scalar fields where depending on the values taken by the parameters of the theory the vacuum can be single or degenerate. (letter)