I'm looking for a textbook or set of lecture notes on gauge theory for mathematicians that assumes only minimal background in physics. I'd prefer a text that uses more sophisticated mathematical concepts like principal bundles and connections, and categorical language whenever convenient.
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$\begingroup$ Google is your friend... $\endgroup$– DanuOct 27, 2015 at 21:16
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2$\begingroup$ Check out "Gauge fields, knots and gravity" by Baez and Munian. I think it strikes a nice balance between mathematical formalism and physical intuition. $\endgroup$– SivaOct 27, 2015 at 21:16
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$\begingroup$ You might see "Gauge Fields Knots and Gravity" by John Baez and Javier Munian. $\endgroup$– Selene RoutleyOct 27, 2015 at 21:17
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I have been writing something in this direction in section 1 of the book Differential cohomology in a Cohesive topos (pdf). Have a look, just focus on section 1 and ignore the remaining sections on first reading.
The survey-part is presently also appearing as a series on PhysicsForums. See at Higher prequantum geometry I, II, III, IV, V and Examples of Prequantum Field Theories I -- Gauge fields, II -- Higher gauge fields.
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2$\begingroup$ Please don't edit away the link to the pdf (dl.dropboxusercontent.com/u/12630719/dcct.pdf). It's more comprehensive. I will update the arXiv version soon. $\endgroup$ Oct 27, 2015 at 21:29
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$\begingroup$ Thanks for the link! I just have a question: you say on page 4 that the curvature of a connection is a cocycle in nonabelian differential cohomology. I'm only familiar with the basics of the Chern-Weil homomorphism and characteristic classes, but it seems to me that the curvature is not in general a closed form. Is there some other sense in which the curvature form is a cocycle? $\endgroup$– ಠ_ಠOct 28, 2015 at 5:18
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$\begingroup$ Which page 4 do you mean? Abelian curvatures are closed, otherwise the evaluation of the curvature in a characteristic polynomial is closed. $\endgroup$ Oct 28, 2015 at 8:50
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One less well-known but great reference are the classical field theory notes by Deligne and Freed in the '99 IAS lectures. Some good things about them
- Very elegant treatment written for mathematicians
- Begins with a nice discussion of ordinary classical mechanics using principal bundles and connections
- Useful comments on supersymmetric gauge theories throughout.