Self-dual gravity and self-dual Yang-Mills theory in the context of the Macdowell-Mansouri formalism

In this work we propose an action which unifies self-dual gravity and self-dual Yang-Mills theory in the context of the Macdowell-Mansouri formalism. We claim that such an action may be used to find the $S$-dual action for both self-dual gravity and self-dual Yang-Mills theory.