The Poisson bracket in L_infty formulation of field theory is computed via the Peierls formula from the symplectic structure, illustrated in p-adic string theory with a homological algebra interpretation of the inverse relation.
Covariant Poisson Brackets in Geometric Field Theory
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abstract
We establish a link between the multisymplectic and the covariant phase space approach to geometric field theory by showing how to derive the symplectic form on the latter, as introduced by Crnkovic-Witten and Zuckerman, from the multisymplectic form. The main result is that the Poisson bracket associated with this symplectic structure, according to the standard rules, is precisely the covariant bracket due to Peierls and DeWitt.
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Authors apply multisymplectic and polysymplectic formalisms to the known Palatini-Cartan model, recovering torsion-free and Einstein equations, constructing momentum maps and Noether currents, and performing a space-time decomposition into instantaneous Hamiltonian form.
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Poisson bracket and $L_\infty$ algebras
The Poisson bracket in L_infty formulation of field theory is computed via the Peierls formula from the symplectic structure, illustrated in p-adic string theory with a homological algebra interpretation of the inverse relation.