Reduction of $L_\infty$-Algebras of Observables on Multisymplectic Manifolds

We develop a reduction scheme for the $L_\infty$-algebra of observables on a premultisymplectic manifold $(M,\omega)$ in the presence of a compatible Lie algebra action $\mathfrak{g}\curvearrowright M$ and subset $N\subset M$. This reproduces in the symplectic setting the Poisson algebra of observables on the Marsden–Weinstein–Meyer symplectic reduced space, whenever the reduced space exists, but is otherwise distinct from the Dirac, Śniatycki–Weinstein, and Arms–Cushman–Gotay observable reduction schemes. We examine various examples, including multicotangent bundles and multiphase spaces, and we conclude with a discussion of applications to classical field theories and quantization.