Difference between revisions of "A Portal Special Presentation- Geometric Unity: A First Look"
Jump to navigation
Jump to search
A Portal Special Presentation- Geometric Unity: A First Look (view source)
Revision as of 22:29, 15 April 2020
, 22:29, 15 April 2020→Part II: Unified Field Content
| Line 466: | Line 466: | ||
<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 $$\mathcal{G} \ | <p>[01:27:49] So, this magic being trade is going to start to enter more and more into our consciousness. If I take an element h and I map that in the obvious way into the first factor, but I map it onto the [[Maurer-Cartan]] form, I think that's when I wish I remembered more of this stuff, into the second factor. It turns out that this is actually a group homomorphism and so we have a nontrivial embedding, which is, in some sense, diagonal between the two factors. That subgroup we are going to refer to as the Tilted Gauge Group, and now our field content, at least in the Bosonic sector, is going to be a group manifold, an infinite dimensional function space [[Lie Group]], but a group nonetheless. And we can now look at $$\mathcal{G} \mod \mathcal{H_t}$$, and if we have any interesting representation of $$\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. | ||