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<p>[01:27:23] And this is going to be a really interesting space because it has a couple of properties. One is it has a very interesting subgroup. Now, of course, $$\mathcal{H}$$ includes into $$\mathcal{G}$$ by just including onto the first factor, but in fact, there's a more interesting homomorphism brought to you by the Levi-Civita connection.
<p>[01:27:23] And this is going to be a really interesting space because it has a couple of properties. One is it has a very interesting subgroup. Now, of course, $$\mathcal{H}$$ includes into $$\mathcal{G}$$ by just including onto the first factor, but in fact, there's a more interesting homomorphism brought to you by the Levi-Civita connection.


<p>[01:27:49] So, this magic being trade is going to start to enter more and more into our consciousness. If I take an element h and I map that in the obvious way into the first factor, but I map it onto the [[Maurer-Cartan]] form, I think that's when I wish I remembered more of this stuff, into the second factor. It turns out that this is actually a group homomorphism and so we have a nontrivial embedding, which is, in some sense, diagonal between the two factors. That subgroup we are going to refer to as the Tilted Gauge Group, and now our field content, at least in the Bosonic sector, is going to be a group manifold, an infinite dimensional function space [[Lie Group]], but a group nonetheless. And we can now look at $$\mathcal{G} \bmod \mathcal{H_{\tau}}$$, and if we have any interesting representation of $$\mathcal{H}$$, we can form homogeneous vector bundles and work with induced representation. And that's what the fermions are going to be. So the fermions in our theory are going to be $$\mathcal{H}$$ modules.
<p>[01:27:49] So, this magic bean trade is going to start to enter more and more into our consciousness. If I take an element h and I map that in the obvious way into the first factor, but I map it onto the [[Maurer-Cartan Form]], I think that's when I wish I remembered more of this stuff, into the second factor. It turns out that this is actually a group homomorphism and so we have a nontrivial embedding, which is, in some sense, diagonal between the two factors. That subgroup we are going to refer to as the Tilted Gauge Group, and now our field content, at least in the Bosonic sector, is going to be a group manifold, an infinite dimensional function space [[Lie Group]], but a group nonetheless. And we can now look at $$\mathcal{G} \bmod \mathcal{H_{\tau}}$$, and if we have any interesting representation of $$\mathcal{H}$$, we can form homogeneous vector bundles and work with induced representation. And that's what the fermions are going to be. So the fermions in our theory are going to be $$\mathcal{H}$$ modules.


<p>[01:29:14] And the idea is that we're going to work with vector bundles, $$\mathcal{E}$$, of the form inhomogeneous gauge group producted over the tilted gauge group.  
<p>[01:29:14] And the idea is that we're going to work with vector bundles, $$\mathcal{E}$$, of the form inhomogeneous gauge group producted over the tilted gauge group.  
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<p>[01:56:56] Yeah, look all these boards and I still feel like I'm managing to run out of room.
<p>[01:56:56] Yeah, look all these boards and I still feel like I'm managing to run out of room.


<p>[01:57:48] Now, if you have something like that, that would be a hell of a Dirac equation. Right. You've got differential operators over here. You've got differential operators. I guess I didn't write them in. But you would have two differential operators over here, and you'd have this differential operator coming from this [[Maurer-Cartan form]]. So I apologize, I'm being a little loose here, but the idea is you have two of these terms are zeroth-order. Three of these terms would be first order, and on this side, one term would be first-order.
<p>[01:57:48] Now, if you have something like that, that would be a hell of a Dirac equation. Right. You've got differential operators over here. You've got differential operators. I guess I didn't write them in. But you would have two differential operators over here, and you'd have this differential operator coming from this Maurer-Cartan form. So I apologize, I'm being a little loose here, but the idea is you have two of these terms are zeroth-order. Three of these terms would be first order, and on this side, one term would be first-order.


<p>[01:58:27] And that's not there. That's fine. That was a mistake. Oh, no, sorry. That was a mistake, calling it a mistake. These are two separate equations, right? So you have two separate fields, $$\nu$$ and $$\zeta$$, and you have a coupled set of differential equations that are playing the role of the Dirac theory. Coming from the Hodge theory of a complex, whose obstruction to being cohomology theory, would be the replacement to the Einstein field equations, which would be rendered gauge invariant on a group relative to a tilted subgroup.  
<p>[01:58:27] And that's not there. That's fine. That was a mistake. Oh, no, sorry. That was a mistake, calling it a mistake. These are two separate equations, right? So you have two separate fields, $$\nu$$ and $$\zeta$$, and you have a coupled set of differential equations that are playing the role of the Dirac theory. Coming from the Hodge theory of a complex, whose obstruction to being cohomology theory, would be the replacement to the Einstein field equations, which would be rendered gauge invariant on a group relative to a tilted subgroup.  
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