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A Portal Special Presentation- Geometric Unity: A First Look (view source)

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==== Sector III: Toolkit for the Unified Field Content ====
==== Sector III: Toolkit for the Unified Field Content ====


Now in Sector 3, there are payoffs to the magic beans trade.
[[File:GU Presentation Powerpoint Sector III Intro Slide.png|thumb|right]]


[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.
''[https://youtu.be/Z7rd04KzLcg?t=9009 02:30:09]''<br>
Now, in sector III, there are payoffs to the magic beans trade. 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. 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 else.


[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.
[[File:GU Presentation Powerpoint Spinorial Levi-Civita Slide.png|thumb|right]]


[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[joint]-valued i forms.
''[https://youtu.be/Z7rd04KzLcg?t=9058 02:30:58]''<br>
So, what do we get? Well as promised, there is a '''tilted homomorphism''', 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. 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.


[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[joint]-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.
===== Shiab Operators =====


[02:32:38] Now the augmented torsion, the torsion is a very strange object. It's introduced sort of right at the beginning of learning a differential geometry, but it really doesn't get used very much. One of the reasons it doesn't get used, gauge theory is that it's not gauge invariant. It has a gauge and variant piece to it, but then a piece that spoils the gauge invariant.
[[File:GU Presentation Powerpoint Shiab Operators Slide.png|thumb|right]]


[02:32:56] But we, because we have two connections, one of the ideas was to introduce two diseases and then to take a difference, and as long as the disease is the same in both, the difference will not have the disease because both diseases are included, but with a minus sign between them. So the augmented torsion is relatively well behaved relative to this particular slanted or tilted embedding of the gauge group in its inhomogeneous extension.
''[https://youtu.be/Z7rd04KzLcg?t=9094 02:31:34]''<br>
We then get to shiab operators. Now, a '''shiab operator''' is a map from the group crossed the ad-valued i-forms. In this case, the particular shiab operator we're interested in is mapping i-forms to d-minus-three-plus-i-forms. So for example, you would map a two-form to d-minus-three-plus-i. So if d, for example, were 14, 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 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.


[02:33:28] Which this is very nice because now we actually have use for the torsion. We have an understanding of why it may never have figured, particularly into geometry, is that you need to have two connections rather than one to see the advantages of torsion at all. So here's an example of one Ship in a Bottle (Shiab) operator.
[[File:GU Presentation Powerpoint Augmented Torsion-1 Slide.png|thumb|right]]
[[File:GU Presentation Powerpoint Augmented Torsion-2 Slide.png|thumb|right]]


[02:33:47] I think this would be sort of analogous, if I'm not mistaken, to trying to take the [[Ricci curvature]] from the entire [[Riemann curvature]], but if you think about what Einstein did, Einstein had to go further and reduce the [[Ricci curvature]] to the scalar curvature. And then, sort of dial the components of the traceless Ricci and the scalar curvature to get the right proportions.
''[https://youtu.be/Z7rd04KzLcg?t=9158 02:32:38]''<br>
Now the '''augmented torsion''', the torsion is a very strange object. It's introduced sort of right at the beginning of learning differential geometry, but it really doesn't get used very much. One of the reasons it doesn't get used in gauge theory is that it's not gauge invariant. It has a gauge invariant piece to it, but then a piece that spoils the gauge invariance. But because we have two connections, one of the ideas was to introduce two diseases and then to take a difference, and as long as the disease is the same in both, the difference will not have the disease because both diseases are included but with a minus sign between them.


[02:34:11] So, there are many Shiab operators and you have to be very careful about which one you want. And once you know exactly what it is you're trying to hit, you can choose the Shiab operator to be bespoke and get the contraction that you need. Now I've, I've made you guys sit through a lot. So I wanted to give you a sort of humorously, a feeling of positivity for the exhaustion.
''[https://youtu.be/Z7rd04KzLcg?t=9193 02:33:13]''<br>
So the augmented torsion is relatively well behaved, relative to this particular slanted, or tilted, embedding of the gauge group in its inhomogeneous extension. This is very nice, because now we actually have a use for the torsion. We have an understanding of why it may never have figured, particularly into geometry, is that you need two connections rather than one to see the advantages of torsion at all.


<blockquote>
[[File:GU Presentation Powerpoint Shiab Example Slide.png|thumb|right]]
The years of anxious searching in the dark, with their intense longing, their alternations of confidence and exhaustion and the final emergence into the light—only those who have experienced it can understand it. -Albert Einstein
</blockquote>


[02:34:41] I've just always thought this was like the most sensitive and beautiful quote, and I wish it were one of his better known quotes, but I think it's so singular that it's hard to, it's hard to feel what it was that he was talking about because in fact, he sort of explains this in the last line.
''[https://youtu.be/Z7rd04KzLcg?t=9223 02:33:43]''<br>
So here's an example of one ship in a bottle (shiab) operator. I think this would be sort of analogous, if I'm not mistaken, to trying to take the Ricci curvature from the entire Riemann curvature. But if you think about what Einstein did, Einstein had to go further and reduce the Ricci curvature to the scalar curvature, and then sort of dial the components of the traceless Ricci and the scalar curvature to get the right proportions. So, there are many shiab operators, and you have to be very careful about which one you want. And once you know exactly what it is you're trying to hit, you can choose the shiab operator to be bespoke and get the contractions that you need.


[02:35:10] So, given that you've been on a long journey, here is something of what Geometric Unity equations might look like. So, in the first place, you have the swerved curvature, the Shiab applied to the curvature tensor. That, in general, does not work out to be exact, so you can't have it as the differential of a Lagrangian.
''[https://youtu.be/Z7rd04KzLcg?t=9269 02:34:29]''<br>
Now, I've made you guys sit through a lot, so I wanted to give you, sort of humorously, a feeling of positivity for the exhaustion:


[02:35:31] And, in fact that we talked about swirls, swerves twirls eddies. There has to be a quadratic Eddy tensor. That I occasionally forget when I pull this thing out of mothballs and the two of those together make up what I call the total swervature. And, on the other side of that equation, you have the displaced torsion, which I've called the displasion.
''[https://youtu.be/Z7rd04KzLcg?t=9282 02:34:42]''<br>
"The years of anxious searching in the dark, with their intense longing, their intense alternations of confidence and exhaustion and the final emergence into the light—only those who have experienced it can understand it."<br>-Albert Einstein


[02:35:54] And to get rid of the pesky sort of minus sign and Hodge star operator. This would be the replacement for the Einstein equation, not on $$X$$ where we would perceive it, but on $$Y$$ before being pulled back onto the manifold $$X$$. So a condensation of that would be very simple. In simplest terms, we would be saying that the swervature is equal to the displasion, and at least in this sector of the four main equations of theoretical physics.  
''[https://youtu.be/Z7rd04KzLcg?t=9281 02:34:41]''<br>
I've just always thought this was, like, the most sensitive and beautiful quote, and I wish it were one of his better known quotes, but I think it's so singular that it's hard to, it's hard to feel what it was that he was talking about because, in fact, he sort of explains this in the last line.


[02:36:25] This would be the replacement for the Einstein equations again on $$Y$$ before being pulled back to $$X$$. Next is the sketch of the fermionic field content. I'm not sure whether that should have been sector four, sector three, but it's going to be very brief. I showed some pictures during the lecture and I'm not going to go back through them, but I wanted to just give you an idea of where this mysterious third generation I think comes [from].
===== GU Equations: Swervature and Displasion =====
 
[[File:GU Presentation Powerpoint Gauge Invariant Einstein Replacement Slide.png|thumb|right]]
 
''[https://youtu.be/Z7rd04KzLcg?t=9310 02:35:10]''<br>
So, given that you've been on a long journey, here is something of what Geometric Unity equations might look like. So in the first place, you have the swerved curvature, the shiab applied to the curvature tensor. That, in general, does not work out to be exact, so you can't have it as the differential of a Lagrangian. And in fact, you know, we talked about swirls, swerves, twirls, eddies—there has to be a quadratic '''eddy tensor''', that I occasionally forget when I pull this thing out of mothballs, and the two of those together make up what I call the '''total swervature'''. And, on the other side of that equation, you have the displaced torsion, which I've called the '''displasion'''. And to get rid of the pesky, sort of, minus sign and Hodge star operator... This would be the replacement for the Einstein equation, not on \(X\) where we would perceive it, but on \(Y\) before being pulled back onto the manifold \(X\).
 
[[File:GU Presentation Powerpoint Condensation Slide.png|thumb|right]]
 
''[https://youtu.be/Z7rd04KzLcg?t=9372 02:36:12]''<br>
So, a condensation of that would be very simple. In simplest terms, we would be saying that the swervature is equal to the displasion, and at least in this sector of the four main equations of theoretical physics, this would be the replacement for the Einstein equations, again on \(Y\) before being pulled back to \(X\).
 
==== Sector IV: Fermionic Field Content ====
 
Next is the sketch of the fermionic field content. I'm not sure whether that should have been sector four, sector three, but it's going to be very brief. I showed some pictures during the lecture and I'm not going to go back through them, but I wanted to just give you an idea of where this mysterious third generation I think comes [from].


[02:36:55] So, if we review the three identities here, we see that if we have a space $$V$$, thought of like as a tangent bundle, and then you have spinors built on the tangent bundle. When you tensor product the tangent bundle with its own spinors, it breaks up into two pieces. One piece is the so-called Cartan product, which is sort of the some of the highest weights, and the other is a second copy of the spinors gotten through the Clifford contraction.
[02:36:55] So, if we review the three identities here, we see that if we have a space $$V$$, thought of like as a tangent bundle, and then you have spinors built on the tangent bundle. When you tensor product the tangent bundle with its own spinors, it breaks up into two pieces. One piece is the so-called Cartan product, which is sort of the some of the highest weights, and the other is a second copy of the spinors gotten through the Clifford contraction.
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