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To better understand papers like this for example, which makes heavy use of fibre bundles and gauge connections to represent gauge fields, I am looking for a nice introduction to this topic.

The only thing I have read so far is the corresponding chapter 15 of Roger Penrose's "Road to Reality".

I do not want to read a whole book, I am rather thinking about an appropriate introductory paper, lecture notes, or a tutorial.

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I have noted that this question is related and looks similar, but the answers do not contain what I am looking for. – Dilaton Feb 6 '13 at 10:04
up vote 3 down vote accepted

You can find the definition of a fiber bundle and some examples on pp 376-379 of Hatcher's online book Algebraic Topology. You might also consult "Fiber Bundles," chapter 4 of Lecture Notes in Algebraic Topology, by Davis-Kirk. A fast introduction to connections and curvature can be found here. In the case of surfaces, chapter 3 of these lecture notes might be useful to you.

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for fiber bundles,you may look into novikov's modern geometry part 2. it gives nice explanation and a good place to do learn some "real geometry"

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Thanks for this hint, I hope you do not mind that I inserted a link to the book. – Dilaton Feb 6 '13 at 11:44

I do not know what are gauge connections. However for connections on bundles, a "lecture note" reference is J.-L. Koszul's Lectures on Fibre Bundles and Differential Geometry.

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The term "gauge connection" is not correct. Physicists use the term "gauge field" to denote a connection on a principal bundle. In fact, they usually denote by "gauge field" the local pull-back to the base of the connection one-form. A gauge transformation would be nothing but a right action of the Lie group of the bundle (assuming the transition functions act on the left). – Bilateral Mar 25 '15 at 15:33

For a 'physicsy' viewpoint, checkout "Geometry of Physics" by Frankel.

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Thanks, this looks nice too. – Dilaton Feb 6 '13 at 16:13

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