On the unexpected geometrical origin of the algebra of symmetries

Abstract

The fundamental symmetries in gravity and gauge theories, formulated using differential forms, are gauge transformations and diffeomorphisms. These symmetries act in distinct ways on different dynamical fields. Yet, the commutator of these symmetries forms a closed, field-independent algebra. This work uncovers a natural correspondence between this algebra and the Lie bracket of some vector fields on the principal fiber bundle associated with the physical theory, providing a geometric interpretation of the symmetry algebra. Furthermore, we demonstrate that the symmetry algebra is independent of the connection. Finally, we analyze an example illustrating how a specific connection, associated with Lorentz-Lie transformations, simplifies the symmetry algebra in the presence of spacetime Killing vector fields.