abstract: In this paper we construct a Poisson algebra bundle whose distributional sections are suitable to represent multilocal observables in classical field theory. To do this, we work with vector bundles over the unordered configuration space of a manifold $M$ and consider the structure of a $2$-monoidal category given by the usual (Hadamard) tensor product of bundles and a new (Cauchy) tensor product which provides a symmetrized version of the usual external tensor product of vector bundles on $M$. We use the symmetric algebras with respect to both products to obtain a Poisson 2-algebra bundle mimicking the construction of Peierls bracket from the causal propagator in field theory. The explicit description of observables from this Poisson algebra bundle will be carried out in a forthcoming paper.

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@misc{Frabetti-Poisson-bundles-over-2024,
 abstract = {In this paper we construct a Poisson algebra bundle whose distributional
sections are suitable to represent multilocal observables in classical field
theory. To do this, we work with vector bundles over the unordered
configuration space of a manifold $M$ and consider the structure of a
$2$-monoidal category given by the usual (Hadamard) tensor product of bundles
and a new (Cauchy) tensor product which provides a symmetrized version of the
usual external tensor product of vector bundles on $M$. We use the symmetric
algebras with respect to both products to obtain a Poisson 2-algebra bundle
mimicking the construction of Peierls bracket from the causal propagator in
field theory. The explicit description of observables from this Poisson algebra
bundle will be carried out in a forthcoming paper.},
 arxiv = {arXiv:2407.15287},
 author = {Frabetti, Alessandra and Kravchenko, Olga and Ryvkin, Leonid},
 eprint = {2407.15287},
 howpublished = {Preprint, {arXiv}:2407.15287 [math-ph] (2024)},
 keywords = {17B63,18N10,14D21,70S99},
 title = {Poisson bundles over unordered configurations},
 url = {https://arxiv.org/abs/2407.15287},
 year = {2024}
}