I am essentially looking for a book that would hold my hand through basic concepts to more complicated ones. I am coming from physics. I am looking to make some connections with Classical mechanics and or quantum field theory. I have been told that QFT is really just geometry and have only seen a few manifest aspects of that. I must say I am not any kind of expert. I want to be one though. I am looking for a short book. I also want to be able to understand the whole complex cohomology and homology thing. I am guessing most of you know what I am talking about. Are there any books out there? Again, I am wanting to hold back on rigor and proofs to first get the concepts down well. Any video lecture links would be highly appreciated too. I am trying to study them for the next couple of months say three or four. Differential geometry and topology are things I would love to be invisible at. . . So please help me out. Recommend some books and video links. Feel free to throw any Yang- Mills material at me too. Thanks in advance.
closed as too broad by Andrey Rekalo, Benoît Kloeckner, David White, Karl Schwede, Misha Aug 24 '13 at 13:43
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I think the books "Topology, Geometry, and Gauge Fields: Foundations" and "Topology, Geometry, and Gauge Fields: Interactions" are exactly what you're looking for. They are not short, but you don't actually want a short book anyway because the only way to write a short differential geometry book is to give few terse examples and leave all the calculations to the reader. That's good once you have a foothold but frustrating at first. Mathematically, the books start with basically nothing and get through:
-Basic differential geometry: manifolds, Lie groups, connections on principal bundles, spin geometry -Basic algebraic topology: de Rham cohomology, Chern-Weil theory, basic homotopy theory, classification of principal bundles
Physically, it covers the rudiments of gauge theory up through the beginnings of Donaldson theory / Yang-Mills theory ("Foundations") and the beginnings of Seiberg-Witten theory ("Interactions"). It gives tons of detailed examples and exercises throughout, including some examples that you don't often see in textbooks (like the quaternionic Hopf fibration).
To decide if the books are for you, I would recommend just finding a copy of "Foundations" and reading Chapter 0 - it is a bird's eye view of the subject which sets the tone for the rest of the series.
Maybe the following books will be helpful: The Geometry of Physics: An Introduction, by Theodore Frankel. Geometry, Topology and Physics, by Mikio Nakahara.
For a quick introduction into the subject I would recommend (not physics oriented) Elementary Differential Geometry, by Andrew Pressley.
And finally, as the geometry of physics (both general relativity and gauge theory) is really a Cartan geometry, the following book (alas, also not physics oriented) is worth to mention: Cartan for Beginners: Differential Geometry Via Moving Frames and Exterior Differential Systems, by Thomas A. Ivey and J. M. Landsberg.