A Portal Special Presentation- Geometric Unity: A First Look: Difference between revisions

So that gets rid of the biggest problem, because the internal symmetry group is what causes the failure, I think, of supersymmetric particles to be seen in nature, which is we have two different origin stories, which is a little bit like Lilith and Genesis. We can't easily say we have a unified theory if spacetime and the \(\text{SU}(3) \times \text{SU}(2) \times \text{U}(1)\) group that lives on spacetime have different origins and cannot be related. In this situation, we tie our hands and we have no choice over the group content.

''[https://youtu.be/Z7rd04KzLcg?t=8746 02:25:46]''<br>

So just to fix bundle notation, we let \(H\) be the structure group of a bundle \(P_H\) over a base space \(B\). We use \(\pi\) for the projection map. We've reserved the variation in the \(\pi\) orthography for the field content, and we try to use right principal actions. I'm terrible with left and right, but we do our best. We use \(H\) here, not \(G\), because we want to reserve \(G\) for the inhomogeneous extension of \(H\) once we move to function spaces.

''[https://youtu.be/Z7rd04KzLcg?t=8783 02:26:23]''<br>

So with function spaces, we can take the bundle of groups using the adjoint action of \(H\) on itself and form the associated bundle, and then move to \(C^\infty\) sections to get the so-called gauge group of automorphisms. We have a space of connections, typically denoted by \(A\), and we're going to promote \(\Omega^1(B,\text{ ad}(P_H))\) to a notation of script \(\mathcal{N}\) as the affine group, which acts directly on the space of connections.

''[https://youtu.be/Z7rd04KzLcg?t=8827 02:27:07]''<br>

Now, the '''inhomogeneous gauge group''' is formed as the semi-direct product of the gauge group of automorphisms together with the affine group of translations of the space of connections. And you can see here is a version of an explicit group multiplication rule, I hope I got that one right.

''[https://youtu.be/Z7rd04KzLcg?t=8847 02:27:27]''<br>

And then we have an action of \(G\), that is the inhomogeneous gauge group, on the space of connections, because we have two different ways to act on connections. We can either act by gauge transformations or we can act by affine translations. So putting them together gives us something for the inhomogeneous gauge group to do.

''[https://youtu.be/Z7rd04KzLcg?t=8866 02:27:46]''<br>

We then get a bi-connection. In other words, because we have two different ways of pushing a connection around, if we have a choice of a base connection, we can push the base connection around in two different ways according to the portion of it that is coming from the gauge transformations of curly \(\mathcal{H}\) or the affine translations coming from curly \(\mathcal{N}\). We can call this map the '''bi-connection''', which gives us two separate connections for any point in the inhomogeneous gauge group. And we can notice that it can be viewed as a section of a bundle over the base space to come when we find an interesting embedding of the gauge group that is not the standard one in the inhomogeneous gauge group.

''[https://youtu.be/Z7rd04KzLcg?t=8922 02:28:42]''<br>

So our summary diagram looks something like this. Take a look at the \(\tau_{A_0}\). We will find a homomorphism of the gauge group into its inhomogeneous extension that isn't simply inclusion under the first factor. We—and I'm realizing that I have the wrong pi projection, that should just be a simple \(\pi\) projecting down. We have a map from the inhomogeneous gauge group via the bi-connection to \(A \times A\), connections cross connections, and that behaves well according to the difference operator \(\delta\) that takes the difference of two connections and gives an honest ad-valued one-form.

''[https://youtu.be/Z7rd04KzLcg?t=8963 02:29:23]''<br>