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I would like to learn gauge theory, starting from the simplest case. I have heard that I should start with electromagnetism, which is just the $U(1)$-gauge theory. All the references I know are written for physics students.

Being familiar with bundle theory, it would be nice if I can start with a short exposition that explains electromagnetism using the language of bundles, characteristic classes, curvatures, etc.. . Any relevant pointers are appreciated.

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  • $\begingroup$ I like the lecture notes by José Figueroa-O'Farrill empg.maths.ed.ac.uk/Activities/GT $\endgroup$
    – Appliqué
    Commented Oct 9, 2019 at 13:28
  • $\begingroup$ It seems to be useful! I will grok the first 5 lectures soon. $\endgroup$
    – Student
    Commented Oct 9, 2019 at 14:54

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John Baez wrote a book called Gauge Fields, Knots, and Gravity. I think it's probably what you're looking for.

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  • $\begingroup$ His book is amazing indeed! But that book does not treat its readers as if they know bundle theory.. $\endgroup$
    – Student
    Commented Oct 9, 2019 at 14:24
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I think a comprehensive reference here would be Naber's Topology, Geometry and Gauge fields (Volume 1, Volume 2).

More specifically, Sections 0.2 of Volume 1 has a physical discussion of Maxwell's equations and Dirac monopoles, which is reformulated in differential-geometric language (i.e. principal bundles, connections, etc.) in Section 0.4.

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The book "The Geometry of Physics: An Introduction", by Theodore Frankel is, despite its name, a fairly complete treatment.

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