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Does anyone know of any good references for physical examples of gauge theory (as a mathematically precise theory of connection on principal bundles). Simple examples will do (e.g the $U(1)$ electromagnetic theory), for someone who has just started learning about principal bundles, and has little/no background in physics. An introduction to "Yang-Mills" theory would also be nice. What I really want is something to connect what mathematicians call "gauge theory" and what physicists call "gauge theory" (starting from the maths point of view).

I would also like references for physical examples of spin geometry. Again, in a mathematical style in the context of clifford and spin bundles (over spacetime manifold). Thanks.

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I think D. Bleecker - Gauge Theory and Variational Principles might be of great help. It's written with mathematical rigor and he always gives physical examples and shows how one can derive known physical equations such as the Klein-Gordon or Dirac Equation from the principal bundle formalism. Indeed, one complete chapter is concerned about the Dirac equation, where spin geometry comes into play too. However, I do not think it is an easy read, but very much worth your time! Therein are also more references for further reading.

Another reference is B. Felsager - Geometry, Particles, and Fields , which is not as rigorous as Bleeckers book, but covers more physics. It might aid you very well in understanding why Bleecker is doing things the way he does!

Lastly, I'd like to mention T. Frankel - The Geometry of Physics. It is a very well written all around book with several physical examples, including spin geometry and electromagnetism.

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