Lifting a weak Poisson bracket to the algebra of forms

Research output: Contribution to journalArticle

  • Authors:
  • S. Lyakhovich
  • M. Peddie
  • A. Sharapov


We detail the construction of a weak Poisson bracket over a submanifold Σ 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 ΠTM, 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 Σ 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.

Bibliographical metadata

Original languageEnglish
Pages (from-to)330-344
Number of pages15
JournalJournal of Geometry and Physics
Early online date22 Feb 2017
StatePublished - 1 Jun 2017