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PreprintCC BY 4.0arXiv

Poisson bracket and L∞ algebras

  • Vinícius Bernardes
  • Theodore Erler
  • Atakan Hilmi Fırat
  • Igor Khavkine
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Abstract

We describe the Poisson bracket of a Lagrangian field theory expressed in the framework of L∞ algebras. If the symplectic structure on phase space is defined following recent work, we show that the Poisson bracket can be computed with the Peierls formula, confirming the well-known result of the covariant phase space formalism. Of particular interest is the Poisson bracket in nonlocal theories. In p-adic string theory we find that a straightforward construction of the Poisson bracket is obstructed by higher derivative instabilities. We also discuss the inverse relation between the Poisson bracket and symplectic structure in the language of homological algebra, extending some ideas in the mathematical physics literature.

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Cite

BibTeXRIS
BibTeX
@misc{bernardes2026poisson,
  title = {{Poisson bracket and $L_\infty$ algebras}},
  author = {Vinícius Bernardes and Theodore Erler and Atakan Hilmi Fırat and Igor Khavkine},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2606.30630},
  eprint = {2606.30630},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.30630},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Poisson bracket and L_∞ algebras
AU  - Vinícius Bernardes
AU  - Theodore Erler
AU  - Atakan Hilmi Fırat
AU  - Igor Khavkine
PY  - 2026
DA  - 2026/11/06
PB  - arXiv
DO  - 10.48550/arxiv.2606.30630
UR  - https://arxiv.org/abs/2606.30630
AB  - We describe the Poisson bracket of a Lagrangian field theory expressed in the framework of L_∞ algebras. If the symplectic structure on phase space is defined following recent work, we show that the Poisson bracket can be computed with the Peierls formula, confirming the well-known result of the covariant phase space formalism. Of particular interest is the Poisson bracket in nonlocal theories. In p-adic string theory we find that a straightforward construction of the Poisson bracket is obstructed by higher derivative instabilities. We also discuss the inverse relation between the Poisson bracket and symplectic structure in the language of homological algebra, extending some ideas in the mathematical physics literature.
ER  -