A Portal Special Presentation- Geometric Unity: A First Look: Difference between revisions
A Portal Special Presentation- Geometric Unity: A First Look (edit)
Revision as of 19:20, 25 April 2020
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=== Supplementary Explainer Presentation === | === Supplementary Explainer Presentation === | ||
<p>[02:13:25] So, thanks for watching that video. What I thought I would do since that was the first time I'd really presented the theory at all in public and I had gotten somewhat turned around on my trip to England and trying, probably stupidly, to do last minute corrections got me a bit confused in a few places, and I wrote some things on the board I probably shouldn't have. | <p>[02:13:25] So, thanks for watching that video. What I thought I would do since that was the first time I'd really presented the theory at all in public and I had gotten somewhat turned around on my trip to England and trying, probably stupidly, to do last-minute corrections got me a bit confused in a few places, and I wrote some things on the board I probably shouldn't have. | ||
<p>[02:13:48] I thought I would try a partial explainer for technically-oriented people so that they're not mystified by the video. And any errors here or my own and I'm known to make many. So, hopefully they won't be too serious, but we'll find out. So this is a supplementary explainer for the Geometric Unity talk at Oxford that you just saw. | <p>[02:13:48] I thought I would try a partial explainer for technically-oriented people so that they're not mystified by the video. And any errors here or my own and I'm known to make many. So, hopefully they won't be too serious, but we'll find out. So this is a supplementary explainer for the Geometric Unity talk at Oxford that you just saw. | ||
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<p>[02:30:14] The big issue here 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. On top of that, we built spinors off of the chimeric tangent bundle, and we have not made any other choices. | <p>[02:30:14] The big issue here 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. 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 | <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 is fixed by the choice of $$X^4$$ not anything else. 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 | <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 ad-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, you would map a two form to a d minus three plus i. So if d, for example, were 14, ..., and i was equal to two. Then 14 minus three is equal to 11 plus two is equal to 13. So that would be an ad-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. | <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, you would map a two form to a d minus three plus i. So if d, for example, were 14, ..., and i was equal to two. Then 14 minus three is equal to 11 plus two is equal to 13. So that would be an ad-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. | ||