More on DBI action in 4D $\mathcal{N}=1$ supergravity

We construct a Dirac-Born-Infeld (DBI) action coupled to a two-form field in four dimensional $\mathcal{N}=1$ supergravity. Our superconformal formulation of the action shows a universal way to construct it in various Poincar\'e supergravity formulations. We generalize the DBI action to that coupled to matter sector. We also discuss duality transformations of the DBI action, which are useful for phenomenological and cosmological applications.