2023/12/08 by Konstantin Eder, Eder, Konstantin, John Huerta +3 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #High Energy Physics - Theory (hep-th) #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2312.05224
openalex publication_date 2023/12/08 · openalex created_date 2023/12/12 · openalex updated_date 2026/07/31
We study relative differential and integral forms on families of supermanifolds and their cohomology. We prove a relative Poincaré--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 Poincaré--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 Poincaré 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.