The electromagnetic field in Poisson gauge theory: the groupoidal approach
Fabio Di Cosmo, Vladislav G. Kupriyanov, Patrizia Vitale
TL;DR
This work develops a unified geometric framework for Poisson gauge theory by embedding electrodynamics on Poisson manifolds into the language of symplectic groupoids and Lie algebroids. It defines two principal field-strength notions, $F^s$ (covariant) and $F^t$ (invariant), as pullbacks of the groupoid symplectic form via the source and target maps, and relates them to a third tensor $\boldsymbol{\mathcal{F}}$ built from gauge-invariant momenta. The authors prove explicit, invertible relations among these field strengths, showing that their vanishing is equivalent and corresponds to bisections being Lagrangian submanifolds; they also connect these constructions to a Poisson-Chern-Simons action whose equations of motion are $F^s=0$. The framework is illustrated with canonical and Lie-algebra-type Poisson structures and extended to local symplectic groupoids, providing a robust semi-classical pathway toward gauge theories on nontrivial Poisson backgrounds and laying groundwork for Lagrangian formulations of Poisson gauge dynamics.
Abstract
We consider the problem of defining the field strength of abelian potentials when the spacetime is a Poisson manifold, within the groupoidal approach. The natural definition in terms of gauge invariant momenta is proved to be equivalent to covariant and invariant tensors of a local symplectic groupoid representing a symplectic realization of the Poisson manifold. A Poisson Chern-Simons model is then proposed and its equations of motion are shortly discussed.