Differential geometry of gauge theory: an introduction
February 12, 2021
We propose an introduction to the differential geometry of connections on fiber bundles underlying the physics of (classical) gauge theories. Reminding first how Ehresmann connections are behind Yang-Mills-Utiyama theories, we will then be prepared to appreciate why Cartan connections provide a compelling geometric framework for gauge theories of gravity. A recipe of sort is then given, showing how these geometric data provide a kinematics that constrain and channel the construction of gauge theories, which specify the dynamics of the geometry. We end with a brief presentation of the “dressing field method”, a systematic tool to achieve gauge symmetry reduction that has a wide range of applications in gauge theory.
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