I'm extremely interested in the field of gauge theory because of its broad applications. Now that I'm going into my final BSc semester, I was wondering:
What are the foundations of gauge theory? What are key concepts necessary to "build up" the topic?
The context of the question: I want to do my bachelor thesis in differential geometry, and would like to ask for a topic (they are very flexible) that allows me to find out if gauge theory is something I want to / have the ability to actually pursue in my further education.
Gauge theory builds up on the theory of principle fibre bundles. For this in turn you need to be familiar with Lie groups and Lie algebras, as well as with standard differential geometry. A very good (but not easy to digest) book connecting the mathematics of principle fibre bundles and the physics of gauge theory is: David Bleecker, Gauge Theory and Variational Principles, Dover Publications