# Geometry of gauge fields

1. Jun 3, 2009

### jdstokes

Recently I've been trying to understand the geometric formulation of gauge field theory.

The mantra I've been hearing is that a gauge field is a connection for a principal bundle where the structure group corresponds to the gauge group. Fields which are charged under the gauge group form sections of associated vector bundles.

John Baez's book has a slight different take, however, in which he essentially defines the gauge field as the connection for a G-bundle, which is vector bundle obtained by gluing together trivial bundles subject to G-identifications on the overlaps, what need then for the principal bundle?

I'm also grappling with the idea that a choice of gauge corresponds to a local trivialization. The idea seems to be that sufficiently small regions of the bundle are diffeomorphic to the product $U \times F$ where F is the typical fiber. I don't quite see how the different gauge possibilities arise out of this, however.

Thanks.

2. Jun 3, 2009

### tom.stoer

Regarding you last remark "I don't quite see how the different gauge possibilities arise out of this". I think one can't see that. One has to chose a global gauge function and check if it induces the correct local gauge sections. The other way round, namely to start with a local condition and try to "globalize" it may work if you have a trivial bundle,but in general it will not work.

Regarding Baez's book: I am not familiar with these ideas.

3. Jun 3, 2009

4. Jun 3, 2009

### jdstokes

Well, in my opinion it would make more sense to define a gauge choice on some spacetime patch as a section of a local trivialization of the principal fiber bundle.

Is this the correct way of thinking about it?

5. Jun 3, 2009

### jdstokes

I think I can see where this is going.

It makes sense to me that changing the local trivialization corresponds to a gauge transformation since it is assumed from the outset that the fibers are sewn together using smooth (group-valued) transition maps.

If I want to do a gauge tranformation on some specified patch, then I need to figure out another way of assigning a local trivialization to that patch. But since I am working with a vector bundle with structure group G, the only way I know how to do this is to do a continuous G-tranformation of the fibers.

Suppose that I want to do a gauge transformation on