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

<p>[02:30:14] The big issue here is, is that we've forgone the privilege of being able to choose and dial in our own field content, and we've decided to remain restricted to anything we can generate only from $$X^{d}$$, in this case $$X^{4}$$. So we generated $$Y^{14}$$ from $$X^{4}$$. And then we generated chimeric tangent bundles. Uh, on top of that, we built spinors off of the chimeric tangent bundle, and we have not made any other choices.

<p>[02:30:42] So we're dealing with, I think it's $$U^{128}$$, $$U^{2^7}$$. That is our structure group and it's fixed by the choice of $$X^4$$ not anything. So what do we get? Well, as promised, there is a tilted homomomorphism which takes the gauge group into its inhomogeneous extension. It acts as inclusion on the first factor, but it uses the Levi-Civita connection to create a second sort of Maurer Cartan form.

<p>[02:31:17] I hope I remember the terminology right. It's been a long time. The map is injective because it's injective under the first factor, but it actually gives us a nontrivial embedding of the gauge group in its inhomogeneous extension, which makes the whole theory work. We then get to Shiab operators now, a Shiab operator a map from the group crossed the add-valued i forms.

<p>[02:31:43] In this case, the particular Shiab operators we’re interested in is mapping i form is to d minus three, plus i forms. So, for example, uh, you would map a two form to a d minus three plus i. So if d, for example, were, um, 14, then, uh. Um, and, and i were equal to two. Then 14 minus three is equal to 11 plus two is equal to 13. So that would be an add-valued 14 minus one form, which is exactly the right place for something to form a current. That is the differential of a Lagrangian on the space.