Virtues of principal/associated bundle formulation of covariant deriv.

[center o bass]

For someone who does not already know Lie group and bundle theory, the formulation of covariant derivatives through parallel transport in the principal, and associated vector bundles, might seem unnecessarily complicated.

In that light, I wondered what the virtues of the principal/associated bundle formulation is?
In particular what does one get from this formulation, that one does not get from the formulation of the covariant derivative as a map from smooth vector fields to smooth vector fields on a manifold M?

Does it give some new insights? Or is it somehow more general?

 

[homeomorphic]

It is more general and, I think, more insightful.

First of all, a principal bundle allows you to forget what the fiber is (provided you are given a suitable group action on the fiber). So, rather than having the fiber be the tangent space, it could be anything you want. To get the parallel transported whatever-it-is, you can just apply the parallel transported group element to it.

I don't really know the details of the Standard Model, but from what I understand, it depends on having a notion of parallel transport for group elements, not just tangent vectors.

The Lie-algebra-valued 1-form definition is essential to the ideas of gauge theory. One idea there is that you have a curvature map (covariant exterior derivative) going from those one forms to 2-forms (technically, ad-P-valued). Then, for example, flat connections would be the connections that have 0 curvature, and therefore they are the pre-image of 0 under this map. So, basically, what happens is you are studying a level surface--the pre-image of 0, the moduli space of flat connections. So, then, the trick is kind of like the implicit function theorem, except it's infinite-dimensional. So, lots of very complicated topology comes up here. This is essentially how Chern-Simons theory goes. Similar, but somewhat modified ideas, apply in the case of Donaldson theory or Sieberg-Witten theory. It's pretty gory when you get into the details, but out of this comes a lot of weird stuff about 4-manifolds, like ones than have no smooth structure or versions of the same manifold that have different smooth structures.

The idea of horizontal tangent planes is a nice way to picture the parallel transport, which just corresponds to covariant derivatives along the path being 0. And it's the geometric idea corresponding to the Lie-algebra-valued 1-forms.