TY - JOUR

T1 - Lifting a weak Poisson bracket to the algebra of forms

AU - Lyakhovich, S.

AU - Peddie, M.

AU - Sharapov, A.

N1 - Funding Information:
It is a pleasure to acknowledge Th. Voronov for many helpful discussions. M.P. appreciates the hospitality of Tomsk State University where he has begun this work, and thanks D. Kaparulin for all the help he provided during this visit. The authors are grateful to Jim Stasheff who provided comments on an earlier version of this text. The visit by M.P. to Tomsk State University was supported by the RFBR grant 14-31-50799. S.L.L. and A.A.Sh. acknowledge partial support from RFBR grant 16- 02-00284-A; Work of S.L.L. is supported by Ministry of Education and Science of the Russian Federation under the Project no 3.5204.2017.
Publisher Copyright:
© 2017 Elsevier B.V.

PY - 2017/6/1

Y1 - 2017/6/1

N2 - 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.

AB - 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.

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U2 - 10.1016/j.geomphys.2017.02.009

DO - 10.1016/j.geomphys.2017.02.009

M3 - Article

AN - SCOPUS:85014521096

VL - 116

SP - 330

EP - 344

JO - Journal of Geometry and Physics

JF - Journal of Geometry and Physics

SN - 0393-0440

ER -