Local symmetries in gauge theories in a finite-dimensional setting

Abstract

It is shown that the correct mathematical implementation of symmetry in the geometric formulation of classical field theory leads naturally beyond the concept of Lie groups and their actions on manifolds, out into the realm of Lie group bundles and, more generally, of Lie groupoids and their actions on fiber bundles. This applies not only to local symmetries, which lie at the heart of gauge theories, but is already true even for global symmetries when one allows for fields that are sections of bundles with (possibly) non-trivial topology or, even when these are topologically trivial, in the absence of a preferred trivialization.

Lie groupoids in classical field theory I: Noether’s theorem

“…Of course, these actions extend the actions of the gauge group bundle P × G 0 G 0 on the principal bundle P itself and on the associated bundle P × G 0 Q, respectively, considered in Ref. [10].…”

“…As stated in the introduction, our main goal in this paper is to extend the results of Ref. [10] about invariance of the minimal coupling prescription and of the curvature map (Utiyama's theorem) from the context of Lie group bundles to that of Lie groupoids. To do so, let us begin by recalling the general definition of these two constructions.…”

“…Observe that the gauge group bundle associated with P employed in Ref. [10], also known as the adjoint bundle AdP = P × G 0 G 0 (where G 0 acts on itself by conjugation), is (up to a canonical isomorphism) just the isotropy subgroupoid of G, that is,…”

“…In Section 4, we then prove the main theorems concerning the invariance (or perhaps it might be better to say, the equivariance) of the minimal coupling prescription and the curvature map under the actions of the pertinent Lie groupoids introduced in the previous section, thus providing the desired extension of the results of Ref. [10] from the setting of Lie group bundles (internal symmetries) to that of Lie groupoids (space-time symmetries).…”

“…Thus our task in what follows will be to extend the results of Ref. [10] by applying the general formalism of Ref. [6] to this specific situation.…”