I was faced with a problem this week. Well, several problems. The start of it all (the meta-problem if you will) was te realisation that dawns on you constantly when you try to do anything in string theory, and any branch of theoretical physics for that matter; I don't know enough mathematics.
Fair enough, this is where we get some studying done. The plan for the coming weeks is to get a good grasp of differential geometry. It's pretty hard to find a good text that describes it at the level that I need. I don't need it to be too rigorous (my apologies to the mathematicians), but it needs to be rigorous enough. I think that my long search is over. Prof. Gary Gibbons from Cambridge was kind enough to post the following lecture notes on his website
Applications of Differential Geometry to Physics
The notes are from a Part III course in Cambridge, which seems to be exaclty what I need.
The main reason why I need this is that I want to understand more about vector bundles. Remember that I'm working on gravity in the string theory context? Well, gravity is described by General Relativity, who'se mathematical language is that of tensor calculus on manifolds. I'm trying to use methods found by my advisor to construct exact solutions that hopefully will shed some light on the high energy behaviour. These methods arised in the context of gauge theories, which are described mathematically by symmetry groups called Lie groups. Fibre bundles (at least to me) seem to be the natural bridge between the two. Apparently, gauge fields can be seen as connections on vector bundles. This is what I want to understand better.