Multi-particle systems in $\kappa$-Poincar\'e inspired by 2+1D gravity

Inspired by a Chern-Simons description of 2+1D gravity coupled to point particles we propose a new Lagrangian of a multiparticle system living in $\kappa$-Minkowski/$\kappa$-Poincar\'e spacetime. We derive the dynamics of interacting particles with $\kappa$-momentum space, alternative to the one proposed in the "principle of relative locality" literature. The model that we obtain takes into account of the nonlocal topological interactions between the particles, so that the effective multi-particle action is not a sum of their free actions. In this construction the locality of particle processes is naturally implemented, even for distant observers. In particular a particle process is characterized by a local deformed energy-momentum conservation law. The spacetime transformations are generated by total charges/generators for the composite particle system, and leave unaffected the locality of individual particle processes.