Local symmetries in gauge theories in a finite-dimensional setting

TL;DR

This paper explores the mathematical structure of symmetries in classical field theories, extending from Lie groups to Lie groupoids and bundles to better capture local and global gauge symmetries.

Contribution

It introduces a geometric framework using Lie groupoids and bundles to accurately model local symmetries in gauge theories, surpassing traditional Lie group approaches.

Findings

Symmetries are naturally described by Lie group bundles and groupoids.

Global symmetries require fiber bundle sections with non-trivial topology.

The framework applies to both trivial and non-trivial topological field configurations.

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.