Poincar\'e Duality and Supergravity

We study relative differential and integral forms on families of supermanifolds and their cohomology. We prove a relative Poincar\'e--Verdier duality and show that it relates the cohomology of differential and integral forms, admitting a concrete geometric realization via Berezin fiber integration. We further introduce the Poincar\'e--dual integral form associated to an embedded even family and prove that it satisfies the correct localization property. We then apply these results to supergravity, focusing on the $3d$ case. In this setting, we show that relative Poincar\'e duality provides the natural framework for encoding the data needed to relate a superspace formulation to the physical spacetime, thereby yielding a rigorous definition of picture changing operators used in the physics literature. Building on this, after a careful analysis of the space of fields and the relevant constraints, we prove that the component, superspace, and geometric formulation of the theory are all equivalent. Finally, under suitable hypotheses, we argue that our construction illustrates a general principle governing the mathematical formulation of classical field theories on supermanifolds.