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We've come a long way from manifolds over Lie groups, Lie algebras in Beck, and we're now fully prepared to come to the at least technical climax of this course.

And that's our Chapter 5. That's Principle Fiber Bundles.

So what's a Principle Fiber Bundle, and why is it so important a notion?

Well, because this is going to require some technical developments, and some rather technical developments at some points, it's maybe good to know what this is all about.

So let's start by claiming or stating that very roughly speaking, and to make this precise is of course exactly the point of the technical developments,

but very roughly speaking, a Principle Fiber Bundle is a bundle whose fiber is a Lie group.

So this is why Lie groups are also so important because they provide the fibers of fiber

bundles, okay, so that is what it very roughly is

But why is this important?

This is so immensely important or principle fiber bundles are

so immensely important because they allow to understand

any fiber bundle

with a fiber F

That's a different guy any fiber bundle with a fiber F on which

The Lie algebra G acts in a particular way

Well, we could say acts well. It's always in a particular way, but it acts okay and

these

Fiber bundles on which G acts and that are associated with this principle bundle are called associate bundles

These are called

Associated bundles and in physics

Fiber bundles are really ubiquitous they appear everywhere

To just mention a few examples for instance in general relativity

They appear and the fiber is given by the

SO

1 3

Lorentz group or if you consider spinners

by SL 2 C which is the double cover of SO 1 3

Now also and

That is probably what mainly sparked this whole thing in physics is that in Yang-Mills theory

Yang-Mills theory and its generalizations. I mean not okay anyway

So this is a non-abelian gauge theories is where the vector potential the covectin

Non-abelian gauge theories is where the vector potential the covector potential of your in quotation mark

Electrodynamics all of a sudden is Lee algebra valued. Okay, that's non-abelian gauge theories

Well should probably write non-abelian gauge theories and but in the standard model of particle physics

That would be in for instance su2

Or it could also be su3. So this would be say roughly speaking the electroweak theory

And this is the the strong interaction

these guys

these theories can actually only be properly understood and the

subtleties in there by considering

principle fiber bundles that's in Yang-Mills theory and

But then not least well, that's not exactly physics. So let's say in mathematics

Well

Studying these principle fiber bundles is if you study the twists in that bundle how the

How the fibers twist, you know, there's need to be a trivial bundle

Studying that you learn a lot about the manifold and in particular there is a

field called K theory

In mathematics one hears a lot about and in K theory roughly speaking

the cohomology groups of some manifold and you heard we I told you about the cohomology groups, right the cohomology groups