Gravity model from (A)dS Yang-Mills theory

We investigate the relationship between a one-parameter family of (anti-)de Sitter Yang-Mills models and a model of Einstein-Palatini gravity with matter, realized through In\"o\"nu-Wigner contraction of the (A)dS algebra. By setting the group parameter $\alpha$ to zero, the gauge transformation of the potential becomes consistent with the transformation properties of the tetrad form and spin connection. We show that a sector of the Yang-Mills dynamics exists in which the equations decouple. Moreover, a subset of the gauge transformations can be related to diffeomorphisms, leading to the identification of the tetrad field. Finally, the resulting dynamics is consistent with a gravitational dynamics in the first-order formalism.