I am looking for the shortest way possible to study basic gauge theory. I am looking for some inspiring survey notes like this one: Christian Bär, Gauga Theory (rather than the great books by Kobayashi & Nomizu or Steenrod). Dealing, in particular, with:
- Problem of lifting 'basic' paths to horizontal paths.
- Why the reduction of the structural group is important.
- Links between: trivial bundle, flat bundle, $G$-local system
David Bleeckers Gauge Theory and Variational Principles is the quickest and dirtiest introduction to gauge theory I know. It’s pretty physics motivated but definitely doesn’t skimp on the math, and should deal with those first two points, though I’m not sure about the third explicitly.
A longer text is mathematical gauge theory by Hamilton, but if you’re already cool with Lie groups, basic representation theory, and quotient manifolds you could jump right into chapter 4 (fiber bundles and principal bundles) and read through to chapter 5 (connections and curvature).
Another longer text is Bruce Sontz’s Principal Bundles the classical case. I have less to say about this one (I mostly used to read through certain topics pertaining to connections in a different), but it’s a refreshing text, that employs some category theory if you’re interested in that at all. It also has a sequel that I have yet to look at, Principal Bundles the quantum case.
Another shorter text that’s completely physics motivated is Gauge Fields, Knots, Gravity by John Baez. I have found this one interesting but not great for a first introduction for the mathematics because it’s a little too focused on physics.