On the unexpected geometrical origin of the algebra of symmetries

TL;DR

This paper reveals a geometric origin of the symmetry algebra in gravity and gauge theories, linking it to vector fields on fiber bundles and showing its independence from the connection, with implications for understanding symmetries in spacetime.

Contribution

It establishes a geometric interpretation of the symmetry algebra as the Lie bracket of vector fields on fiber bundles, independent of the connection, and analyzes how specific connections simplify the algebra.

Findings

Symmetry algebra corresponds to the Lie bracket of vector fields on fiber bundles.

The algebra is independent of the choice of connection.

A specific connection simplifies the algebra in the presence of Killing vectors.

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.