On Poisson Electrodynamics With Charged Fields

Poisson electrodynamics is the low-energy limit of a rank-one noncommutative gauge theory. It admits a closed formulation in terms of a Poisson structure on the space-time manifold and reproduces ordinary classical electrodynamics in the commutative limit. In this paper, we address and solve the problem of minimal coupling to charged matter fields with a proper commutative limit. Our construction essentially relies on the geometry of symplectic groupoids and works for all integrable Poisson manifolds. An additional advantage of our approach is that the corresponding Lagrangians can be defined on an arbitrary metric background.