Linearized Poisson geometry and gauge fields

Abstract:
In the paper, we show how classical dynamics of particles in a gravitational and Yang-Mills field emerges naturally from the geometry of a general Poisson manifold as a second order approximation of a Hamiltonian system on this manifold. The Hamiltonian only has to have vanishing differential on some Lagrangian submanifold X of a locally minimal, polarized symplectic leaf and satisfy a non-degeneracy condition. Furthermore, Higgs fields are naturally present if the systems in coisotropically constraint. The most important feature of the work is the definition of an E-connection form associated to such a Hamiltonian on X, where E is a natural Lie algebroid over X.