Principal Bundles and Gauge Theories

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Vákár, Matthijs
Universiteit Utrecht
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This set of lecture notes fills a hole that in the authors opinion currently exists in the available literature on (prequantum or first quantised) gauge theory: theoretical physics texts do lots of calculations in coordinates and fail to appreciate the underlying geometry of their equations; mathematics texts develop the geometric theory of fibre bundles in its full abtract glory, but, in their attempt to keep things abstract, completely miss out on describing fascinating physical interpretations of their mathematical objects. In these notes, the mathematical theory of fibre bundles is built up from the ground in great generality, slowly adding more structure and converging to the structures we see in modern physics. At this point, we are in the perfect position to fit some modern theories of physics in the framework. This unconventional exposition of the material can hopefully shed new light on the topic for both mathematicians and physicists, serving as a brief introduction to the other discipline's point of view.