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.



Regarding understanding differential geometry
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.


