# Covariant momentum map for non-Abelian topological BF field theory

## TL;DR

This paper develops a covariant framework for analyzing symmetries and gauge generators in non-Abelian topological BF theory, connecting multisymplectic geometry with Hamiltonian and gauge structures.

## Contribution

It introduces covariant momentum maps in the multisymplectic formalism for BF theory, linking symmetries to gauge generators and extending the understanding of their geometric origin.

## Key findings

- Constructed covariant momentum maps for BF theory.
- Connected multisymplectic symmetries to Hamiltonian gauge generators.
- Provided the first multisymplectic analysis of symmetries in a model related to General Relativity.

## Abstract

We analyze the inherent symmetries associated to the non-Abelian topological BF theory from the geometric and covariant perspectives of the Lagrangian and the multisymplectic formalisms. At the Lagrangian level, we classify the symmetries of the theory as natural and Noether symmetries and construct the associated Noether currents, while at the multisymplectic level the symmetries of the theory arise as covariant canonical transformations. These transformations allowed us to build within the multisymplectic approach, in a complete covariant way, the momentum maps which are analogous to the conserved Noether currents. The covariant momentum maps are fundamental to recover, after the space plus time decomposition of the background manifold, not only the extended Hamiltonian of the BF theory but also the generators of the gauge transformations which arise in the instantaneous Dirac-Hamiltonian analysis of the first-class constraint structure that characterizes the BF model under study. To the best of our knowledge, this is the first non-trivial physical model associated to General Relativity for which both natural and Noether symmetries have been analyzed at the multisymplectic level. Our study shed some light on the understanding of the manner in which the generators of gauge transformations may be recovered from the multisymplectic formalism for field theory.