Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 061, 47 pages      arXiv:2206.03137      https://doi.org/10.3842/SIGMA.2024.061

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

Casey Blacker a, Antonio Michele Miti b and Leonid Ryvkin c
a) Department of Mathematical Sciences, George Mason University, 4400 University Dr, Fairfax, VA 22030, USA
b) Dipartimento di Matematica, Sapienza Università di Roma, Piazzale Aldo Moro 5, 00185 Roma, Italy
c) Institut Camille Jordan, Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, 69622 Villeurbann, France

Received October 24, 2023, in final form June 24, 2024; Published online July 03, 2024

Abstract
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.

Key words: $L_\infty$-algebras; multisymplectic manifolds; moment maps.

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