Reduction of 𝐿∞-Algebras of Observables on Multisymplectic Manifolds

Abstract

We develop a reduction scheme for the 𝐿∞-algebra of observables on a premultisymplectic manifold (𝑀, 𝜔) in the presence of a compatible Lie algebra action 𝖌 ↷ 𝑀 and a subset 𝑁 ⊂ 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.