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==== Sector II: Unified Field Content ====
==== Sector II: Unified Field Content ====
[[File:GU Presentation Powerpoint Sector II Intro Slide.png|thumb|right]]
[[File:GU Presentation Powerpoint Mark of Zorro Slide.png|thumb|right]]


[02:23:30] This leads to the [[Mark of Zorro]]. That is, we know that whenever we have a metric by the fundamental theorem of Riemannian geometry, we always get a connection. It happens, however, that what is missing to turn the canonical chimeric bundle on $$Y$$ into the tangent bundle of $$Y$$ is, in fact, a connection on the space $$X$$.
''[https://youtu.be/Z7rd04KzLcg?t=8610 02:23:30]''<br>
This leads to the Mark of Zorro. That is, we know that whenever we have a metric, by the fundamental theorem of Riemannian geometry we always get a connection. It happens, however, that what is missing to turn the canonical chimeric bundle on \(Y\) into the tangent bundle of \(Y\) is, in fact, a connection on the space \(X\).


[02:23:52] So there is one way in which we've reversed the fundamental theorem of Riemannian in geometry where a connection on $$X$$ leads to a metric on $$Y$$. So if we do the full transmission mechanism out, a Gimmel on X leads to alpha sub Gimmel for the Levi-Civita connection on X. Alpha sub Gimmel live leads to a G sub alpha, which is, or sorry, the G sub Alef.
''[https://youtu.be/Z7rd04KzLcg?t=8632 02:23:52]''<br>
So, there is one way in which we've reversed the fundamental theorem of Riemannian geometry, where a connection on \(X\) leads to a metric on \(Y\). So if we do the full transmission mechanism out, \(ℷ\) on \(X\) leads to \(\aleph_ℷ\) for the Levi-Civita connection on \(X\). \(\aleph_{ℷ}\) leads to \(g_{\aleph}\), which is—sorry, \(g_\aleph\). I'm not used to using Hebrew in math.* So \(g_{\aleph}\), then, is a metric on \(Y\), and that creates a Levi-Civita connection of the metric on the space \(Y\) as well, which then induces one on the spinorial bundles. In sector II, the inhomogeneous gauge group on \(Y\) replaces the Poincaré group and the internal symmetries that are found on \(X\). And in fact, you use a fermionic extension of the inhomogeneous gauge group to replace the supersymmetric Poincaré group, and that would be with the field content zero forms tensored with spinors direct sum one-forms tensored with spinors all up on \(Y\) as the fermionic field content.


[02:24:17] I'm not used to using Hebrew in math. So G_aleph is a metric on $$Y$$ and that creates a Levi-Civita connection of the metric on the space $$Y$$ as well, which then induces one on the spinorial levels. In sector two, the inhomogenous gauge group on $$Y$$ replaces the Poincaré group and the internal symmetries that are found on $$X$$.
''* Note: Where Eric mistakes ''\(\alpha\)'' for ''\(\aleph\)'' it is written correctly in the transcript for the sake of clarity and parity with the diagram.''


[02:24:50] And in fact, you use a fermionic extension of the inhomogeneous gauge group (IGG) to replace the supersymmetric Poincaré group, and that would be with the field content, zero forms, tensors and spinors, tensors with spinors, a direct sum one-forms tensor to spinors all up on $$Y$$ as the fermionic field content.
''[https://youtu.be/Z7rd04KzLcg?t=8709 02:25:09]''<br>
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.


[02:25:09] 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 to have two different origin stories, which is a little bit like Lilith and Genesis. We can't easily say we have a unified theory.
[[File:GU Presentation Powerpoint Bundle Notation Slide.png|thumb|right]]


[02:25:32] If spacetime and the SU(3)xSU(2)xU(1) group that lives on space-time have different origins and cannot be related. In this situation, we tie our hands and we have no choice over the group content. So, just to fix bundle notation, we let $$H$$ be the structure group of a bundle piece of $$H$$ over a base space, $$B$$.
''[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.


[02:25:56] 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 moved to function spaces.
[[File:GU Presentation Powerpoint Function Spaces Slide.png|thumb|right]]


[02:26:23] 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 infinity 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 a $$\Omega_1$$, a tensor in the adjoint bundle to a notation of.
''[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.


[02:26:54] Script N a as the affine group, which acts directly on the space of connections. 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.
[[File:GU Presentation Powerpoint Inhomogeneous Gauge Group Slide.png|thumb|right]]


[02:27:26] I hope I got that one right. 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 trans transformations, or we can act by affine translations. So putting them together gives us something for the inhomogeneous gauge group too.
''[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.


[02:27:46] 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 curly H or the affine translations coming from curly N.
[[File:GU Presentation Powerpoint Action of G Slide.png|thumb|right]]


[02:28:10] Yeah. 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. 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=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.


[02:28:42] So our summary diagram looks something like this. Take a look at the Taus of $$A_0$$. We will find a homeomorphism of the gauge group into its inhomogeneous extension that isn't simply inclusion under the first factor. And I'm realizing that I have the wrong pi production. That should just be as simple $$\pi$$, projecting down, we have a map from the inhomogeneous gauge group, via the bi-connection to A cross A connections cross connections, and that that behaves well according to the difference operator $$\delta$$ that takes the difference of two connections and gives an honest ad[joint]-valued one-form.
[[File:GU Presentation Powerpoint Bi-Connection-1 Slide.png|thumb|right]]
[[File:File:GU Presentation Powerpoint Bi-Connection-2 Slide.png|thumb|right]]


[02:29:23] The infinitesimal action of the gauge transformation of a gauge transformation, or at least an infinite testable one on a point inside of the group is given by a somewhat, almost familiar expression, which should remind us of how the first term in the gauge deformation complex for Self-Dual Yang-Mills actually gets started.
''[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.


[02:29:47] And so by acting via this interesting embedding of the embedding of the gauge group inside its inhomogeneous extension, but the non-trivial one, we get something very close to the original first step of the two-step Deformation complex. Now in Sector 3, there are payoffs to the magic beans trade.
[[File:GU Presentation Powerpoint Summary Diagram Slide.png|thumb|right]]
 
''[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.
 
[[File:GU Presentation Powerpoint Infinitesimal Action Slide.png|thumb|right]]
 
''[https://youtu.be/Z7rd04KzLcg?t=8963 02:29:23]''<br>
The infinitesimal action of a gauge transformation, or at least an infinitesimal one, on a point inside of the group is given by a somewhat, almost familiar expression, which should remind us of how the first term in the gauge deformation complex for self-dual Yang-Mills actually gets started. And so, by acting via this interesting embedding of the gauge group inside its inhomogeneous extension, but the non-trivial one, we get something very close to the original first step of the two-step deformation complex.
 
 
==== Sector III: Toolkit for the Unified Field Content ====
 
Now in Sector 3, there are payoffs to the magic beans trade.


[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.
[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.
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