Lifting a Weak Poisson Bracket to the Algebra of Forms

TL;DR

This paper constructs a weak Poisson bracket on a submanifold and lifts it to a weak odd Poisson bracket on differential forms, with applications to gauge systems and BRST operators.

Contribution

It introduces a novel method to lift weak Poisson structures to differential forms via homotopy structures, linking geometric and physical gauge theories.

Findings

Constructed a weak Poisson bracket on submanifolds with foliation.

Lifted the weak Poisson bracket to a weak odd Poisson bracket on forms.

Connected the construction to gauge systems and BRST operators.

Abstract

We detail the construction of a weak Poisson bracket over a submanifold of a smooth manifold M with respect to a local foliation of this submanifold. Such a bracket satisfies a weak type Jacobi identity but may be viewed as a usual Poisson bracket on the space of leaves of the foliation. We then lift this weak Poisson bracket to a weak odd Poisson bracket on the odd tangent bundle, interpreted as a weak Koszul bracket on differential forms on M. This lift is achieved by encoding the weak Poisson structure into a homotopy Poisson structure on an extended manifold, and lifting the Hamiltonian function that generates this structure. Such a construction has direct physical interpretation. For a generic gauge system, the submanifold may be viewed as a stationary surface or a constraint surface, with the foliation given by the foliation of the gauge orbits. Through this interpretation, the lift of the weak Poisson structure is simply a lift of the action generating the corresponding BRST operator of the system.