Poincaré Duality for Supermanifolds, Higher Cartan Structures and Geometric Supergravity

Abstract: We study relative differential and integral forms on families of supermanifolds and investigate their cohomology. In particular, we establish a relative version of Poincar\'e-Verdier duality, relating the cohomology of differential and integral forms, and provide a concrete interpretation via Berezin fiber integration, which we introduce. To complement Poincar\'e duality, we prove compactly supported Poincar\'e lemmas for both differential and integral forms, filling a gap in the literature. We then apply our results to the mathematical foundations of supergravity. Specifically, we rigorously define picture-changing operators via relative Poincar\'e duality and use them to formulate a general action principle for geometric supergravity in a mathematically rigorous manner. As an example, we explicitly describe three-dimensional supergravity via higher Cartan structures, which are defined by certain classes of connections valued in $L_\infty$-superalgebras. Our construction provides a unified framework interpolating between two equivalent formulations of supergravity in the physics literature: the superspace approach and the group manifold approach.