Applications of relative multisymplectic geometry

Abstract

Relative multisymplectic geometry studies smooth maps $F\colon M\to N$ equipped with a closed, nondegenerate relative $(n+1)$-form $\varpi$ in the mapping-cone complex of $F$, together with the associated Lie $n$-algebras of relative observables and relative homotopy moment maps developed by the author in earlier work. The purpose of this article is to demonstrate the scope of this framework through a systematic collection of applications, each developed with complete statements and proofs. We construct a relative integration theory (relative cycles, a relative Stokes theorem, and periods) and use it to prove: (i) that closed relative forms are precisely the topological terms of action functionals -- in particular, Wess-Zumino terms are canonically relative objects, well defined modulo relative periods; (ii) a relative Weil-Kostant theorem characterizing prequantizability of relative $2$-forms by integrality of the relative class, which for quasi-Hamiltonian $G$-spaces recovers the level quantization of group-valued moment map theory via relative gerbes; (iii) a relative Noether theorem producing conserved charges supported on relative cycles, with bulk-boundary charge splitting; (iv) a rigidity theorem in the relative symplectic case: comoment maps, when they exist, are unique, automatically strict, and automatically equivariant -- the Kostant-Souriau cocycle vanishes; (v) canonical functoriality of the moment map theory of quasi-Hamiltonian $G$-spaces under fusion, with applications to moduli spaces of flat connections on surfaces with boundary. We conclude with an outlook on hydrodynamics, Dirac geometry, and reduction.