2025/09/09 by Dinamo Djounvouna, Djounvouna, Dinamo · 1 citation
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.2509.08153
openalex publication_date 2025/09/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A manifold is said to be n-plectic if it is equipped with a closed, nondegenerate (n+1)-form. This thesis develops the theory of relative n-plectic structures, where the classical condition is replaced by a closed, nondegenerate relative (n+1)-form defined with respect to a smooth map. Analogous to how n-plectic manifolds give rise to L_∞-algebras of observables, we show that relative n-plectic structures naturally induce corresponding L_∞-algebras. These structures provide a conceptual bridge between the frameworks of quasi-Hamiltonian G-spaces and 2-plectic geometry. As an application, we examine the relative 2-plectic structure canonically associated to quasi-Hamiltonian G-spaces. We show that every quasi-Hamiltonian G-space defines a closed, nondegenerate relative 3-form, and that the group action induces a Hamiltonian infinitesimal action compatible with this structure. We then construct explicit homotopy moment maps as L_∞-morphisms from the Lie algebra \mathfrakg into the Lie 2-algebra of relative observables, extending the moment map formalism to the higher and relative geometric setting.