Reduction of \(L_\infty\)-algebras of observables on multisymplectic manifolds

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.

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@article{Blacker-Reduction-of-L_inftyalgebras-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, \'Sniatycki-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.},
 author = {Blacker, Casey and Miti, Antonio Michele and Ryvkin, Leonid},
 doi = {10.3842/SIGMA.2024.061},
 eprint = {2206.03137},
 fjournal = {SIGMA. Symmetry, Integrability and Geometry: Methods and Applications},
 issn = {1815-0659},
 journal = {SIGMA, Symmetry Integrability Geom. Methods Appl.},
 keywords = {53D35,53D05,53D20},
 language = {English},
 pages = {paper 061, 47},
 title = {Reduction of {{\(L_\infty\)}}-algebras of observables on multisymplectic manifolds},
 volume = {20},
 year = {2024},
 zbl = {1552.53074},
 zbmath = {7897512}
}