Good Fiber Bundles reference for Physicists

Question

I'm a student of Physics and I have interest on the theory of Fiber Bundles because of the applications they have in Physics (gauge theory for example). What are good books to learn the theory of fiber bundles and connections that are rigorous but at the same time gives what we need to apply in Physics?

This post imported from StackExchange Physics at 2014-10-15 14:10 (UTC), posted by SE-user user1620696

Comments

Gauge Fields, Knots, and Gravity by Baez and Muniain, and Topology, Geometry and Gauge fields by Naber, both volumes.

This post imported from StackExchange Physics at 2014-10-15 14:10 (UTC), posted by SE-user Robin Ekman

Answer 1

The obvious reference for understanding fiber bundles is Mikio Nakahara's book entitled Geometry, Topology and Physics (http://www.amazon.com/Geometry-Topology-Physics-Edition-Graduate/dp/0750306068).

Furthermore, a book that I find really interesting is the book Topology, Geometry and Gauge Fields: Foundations, by Gregory Naber and note that it focuses more on understanding gauge theories through differential geometry (http://www.amazon.com/Topology-Geometry-Gauge-fields-Foundations/dp/1461426820/ref=sr_1_1?s=books&ie=UTF8&qid=1413385257&sr=1-1&keywords=topology+geometry+and+gauge+fields). It has another volume on interactions.

Finally a good reference are thew following lecture notes by Lionel Brits but they serve more as a reminder rather than to learn from zero (http://www.thetangentbundle.net/papers/gauge.pdf).

Comments

Here's a permanent link in the wayback machine for the lecture notes by Brits, as I've noticed that he's been taking down files from the tangent bundle recently (the precious wikis are now completely gone).

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Did you try to ask him why the useful and precious content of the wikis is getting removed?

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@Dilaton I tried once, but the email bounced - I think even his email isn't working any longer.

Answer 2

I think I good book for that may be C. J. Isham's Modern Differential Geometry for Physicists. I haven't gotten to the chapter of fiber bundles, but what I've read seems to be quite rigorous. And as it is written for physicists, I think it could please your needs.

This post imported from StackExchange Physics at 2014-10-15 14:10 (UTC), posted by SE-user guillefix

Answer 3

I like Naber's books too (I would have liked them even more, if he had used the "definition environment" and less boldface... the pages look a bit too dense and heavy). Other books on the topic (with similar style):

• Hamilton: Mathematical Gauge Theory https://www.springer.com/la/book/9783319684383
• Sontz: Principal bundles: the Classical Case https://www.springer.com/la/book/9783319147642#otherversion=9783319147659
• This is more oriented to General Relativity: Johannesen: Smooth Manifolds and Fibre Bundles with Applications to Theoretical Physics https://www.crcpress.com/Smooth-Manifolds-and-Fibre-Bundles-with-Applications-to-Theoretical-Physics/Johannesen/p/book/9781498796712
• I think this is quite old but useful for "general reference": Eguchi, Gilkey, Hanson Gravitation, gauge theories and differential geometry https://www.sciencedirect.com/science/article/pii/0370157380901301

Introductory books for mathematicians, but easy to read:

• Taubes: Differential geometry https://global.oup.com/academic/product/differential-geometry-9780199605873?cc=it&lang=en&
• Tu: Differential geometry https://www.springer.com/us/book/9783319550824

The book "Connections, Curvature, and Cohomology, Vol. 2" by Halperin, Greub, Vanstone is more advanced, useful for general reference; vector bundles, which are fibre bundles, are discussed in vol. 1.

Holomorphic vector bundles need some basic knowledge of sheaf theory and sheaf cohomology, they are interesting on their own but also useful for two-dimensional CFTs and Chern-Simon quantization (and geometric quantisation, in general). They are explained in Griffiths, Harris - Principles of Algebraic Geometry, Huybrechts - Complex geometry, Wells - Differential analysis on complex manifolds, for example.

The review by Viallet and Daniel https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.52.175 is a good place to start, if you're interested in gauge theories: it's really motivating.