Observables of Relative Structures and Lie 2-algebras associated with Quasi-Hamiltonian $G$-spaces

Abstract

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 \emph{relative $n$-plectic structures}, where the classical condition is replaced by a closed, nondegenerate \emph{relative} $(n+1)$-form defined with respect to a smooth map. Analogous to how $n$-plectic manifolds give rise to $L_\infty$-algebras of observables, we show that relative $n$-plectic structures naturally induce corresponding $L_\infty$-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 $L_\infty$-morphisms from the Lie algebra $\mathfrak{g}$ into the Lie $2$-algebra of relative observables, extending the moment map formalism to the higher and relative geometric setting.