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Last time we studied connections on a principal bundle and we saw that a connection, which was just a distribution of horizontal subspaces,

which was a choice, there is no canonical such distribution, that this could be expressed quite conveniently as a Lie algebra valued one form on the principal bundle.

And today we will study how this Lie algebra valued one form on the principal bundle can be expressed locally as a Lie algebra valued one form on the base manifold.

And that is how you usually meet it in physics and depending on context it's then called the local connection coefficients, gamma, in say general relativity, you may have met that,

or the Young-Mills field if you do elementary particle physics.

So today's chapter is 5.5, local representation of a connection

and we will use invariantly the terminology Young-Mills fields.

And we will put that in quotation marks because that may apply to situations that a physicist would not call a Young-Mills field in the first place.

So what was the situation? So let's recall that in the last lecture we considered a principal G bundle.

So we had a principal bundle P with total bundle space P, a right action by some Lie group G, a projection down to some base space.

And on there we considered a connection in terms of a Lie algebra valued one form omega.

So the one form aspect of it is that it takes you from a section of the tangent bundle linearly to,

and that's now the Lie algebra valued aspect of it, to the Lie algebra of the underlying group and it had to satisfy two conditions.

First, whenever you plug in the vector field on P, XA that is generated by a Lie algebra element A,

the one form sends you back to this A. And B, if you consider the pullback under the right action of an element G on the group G,

then on this form, once you plug in an X, this equals add G inverse, push forward.

I'm sorry, these brackets are wrong. So you pull back the form and then you apply it to an X.

And this is the add G inverse push forward of omega. You first apply the X, so the result is a Lie algebra element.

And on that, of course, the push forward of the adjoint map add G inverse can act.

So this was a connection one form.

Now somebody asked the question, what precisely is this mathematically a Lie algebra valued one form?

Well, the simple answer is exactly this. It's a map from the section of the tangent bundle into, not into C infinity of P,

not into the real numbers, some number field, but into this Lie algebra.

And I can, of course, call this a Lie algebra valued one form. And you may ask, to what extent is this still a one form?

It's a nice name for it. It's maybe a suggestive name for it. But in what sense can I still treat this as a one form?

In what sense can I not treat this anymore as a one form?

And one particular question one could ask, so this is some further explanation on how this is to be read.

One particular question is, assume we have a map U that takes us again to this P.

And you know if we make this commute here, so we have the U here again,

you know if this is a, this U together with the F is a bundle morphism, a principal bundle morphism,

then it's already a principal bundle, principal bundle isomorphism.

And because this is the same P, this is already a principal bundle automorphism, principal bundle automorphism,

which is just an isomorphism that goes back to itself, therefore auto.

If you have such a principal bundle automorphism, you know that there are these two compatibility conditions

that the diagram commutes here and commutes here. We wrote this down several times.

But now the question is, if on here lives a Lie algebra valued one form omega,

let's say on here lives an omega of this type, if it's a one form, we should be able to pull it back via this U.

So we should be able to consider U pullback of this omega.

Okay, but now the point is we defined the pullback of one form if the result was in C infinity P, right,

or in the number field if you work at the point.

We didn't define this pullback if it's Lie algebra valued.

So a worry could be, do all the things we defined for one forms and we are heavily using like pullbacks,

do they generalize to this case? And in fact, even more so, they don't only generalize,

the pullback doesn't affect, isn't affected at all by what the destination is.

I'll show this here. So you know the pullback, you apply this to a vector in on here, okay,

and you know that the pullback is defined as the omega of the push forward of this vector.

Because of this, this object here, the pullback of omega is as much Lie algebra valued as this guy is,

because the pullback defined in terms of the push forward, this is just a normal vector,

the pullback doesn't care about where you land. That's the thing, okay.