Connections on fibre bundles

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Whether we recognise them as such or not, connections on fibre bundles appear quite frequently in modern theoretical physics. This treatise is intended to be the document I wish I had as a theoretical physics student - and thus takes a unique route through the subject. Rather than present Riemannian geometry, Yang-Mills theory and spin connections as philosophically similar but seemingly distinct subjects, I have endeavoured to present them all under the banner of a single unifying theory. As usual, abstraction is the price of generality, but I think in this case it is well worth paying. Unfortunately, unless I eventually expand these musings into a complete book, the reader may require a somewhat eclectic base of pre-requisite differential geometry knowledge to comprehend what follows. As such this presentation is not very useful without a prior “first-pass” exposure to some of the topics I discuss. Nothing I present here is original - my only contribution is to collate results spread across various sources and various sections within each source. I have relied heavily on Jack Smith’s Cambridge Part III course on differential geometry and Mikio Nakahara’s “Geometry, topology and physics.”

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