The motion of a classical particle in a gravitational and a Yang-Mills field was described by S. Sternberg and A. Weinstein by a particular Hamiltonian system on a Poisson manifold known under the name of Sternberg-Weinstein phase space. This system leads to the generalization of the Lorentz equation of motion first discovered by Wong. The aim of this work is to show that inversely, a Hamiltonian H on a general Poisson manifold, with the property that its differential vanishes on a Lagrangian submanifold X of a symplectic leaf and is generic in any other direction, naturally defines a metric on X, as well as a principal connection form on a canonical principal fiber bundle on X. These fields, which are credited to model a gravitational and a Yang-Mills field on X, respectively, define a linearized Hamiltonian system of Wong type on a canonical linearized Poisson manifold at X locally isomorphic to a Sternberg-Weinstein phase space. In addition, H is shown to define scalar fields which first appeared in a theory of Einstein and Mayer. In the presence of a coisotropic constraint, the reduced system can be regarded as the phase space of particles in gravitational, Yang-Mills and Higgs fields. We further show that all our constructions are locally related to usual gauge and Kaluza-Klein theory via symplectic realization.
Quantization commutes with reduction for coisotropic A-branes
The Institute of Mathematical Sciences and Department of Mathematics, The Chinese University of Hong Kong（香港中文大学数学科学研究所）；School of Mathematical Sciences, Shenzhen University（深圳大学数学科学学院）；Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo Institutes for Advanced Study, The University of Tokyo（东京大学宇宙物质起源研究机构（WPI） Kavli宇宙物理数学研究所）
