Difference between revisions of "A Portal Special Presentation- Geometric Unity: A First Look"

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<p>[01:20:01] Okay. What is it that we get for the Levi-Civita connection? Well, not much. One thing we get is that normally the space of connections is an affine space ($$\mathcal(A)$$). Not a vector space, but an affine space. Almost a vector space, a vector space up to a choice of origin. But with the Levi-Civita connection, rather than having an infinite plane with an ability to take differences, but no real ability to have a group structure.
<p>[01:20:01] Okay. What is it that we get for the Levi-Civita connection? Well, not much. One thing we get is that normally the space of connections is an affine space ($$\mathcal(A)$$). Not a vector space, but an affine space. Almost a vector space, a vector space up to a choice of origin. But with the Levi-Civita connection, rather than having an infinite plane with an ability to take differences, but no real ability to have a group structure.


<p>[01:20:35] You pick out one point, which then becomes the origin. That means that any connection $$A$$ has a torsion tensor, $$T_A$$, which is equal to the connection, $$A$$ minus the Levi-Civita connection ($$\nabla^{LC}$$. So we get a tensor that we don't usually have. Gauge potentials are not usually well-defined. They're are only defined up to a choice of gauge.
<p>[01:20:35] You pick out one point, which then becomes the origin. That means that any connection $$A$$ has a torsion tensor, $$T_A$$, which is equal to the connection, $$A$$ minus the Levi-Civita connection ($$\nabla^{LC}$$). So we get a tensor that we don't usually have. Gauge potentials are not usually well-defined. They're are only defined up to a choice of gauge.


<p>[01:21:00] So that's one of the things we get for our Levi-Civita connection, but because the gauge group is going to go missing, this has terrible properties from with respect to the gauge group. It almost looks like a representation. But, in fact, if we let the gauge group act, there's going to be an affine shift.
<p>[01:21:00] So that's one of the things we get for our Levi-Civita connection, but because the gauge group is going to go missing, this has terrible properties from with respect to the gauge group. It almost looks like a representation. But, in fact, if we let the gauge group act, there's going to be an affine shift.
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<p>[01:21:48] [The] first thing we need to do is we still have the right to choose intrinsic field content. [We] have an intrinsic field theory. So, if you consider the structure bundle of the spinors; we built the chimeric bundle, so we can define Dirac spinors on the chimeric bundle if we're in Euclidean signature. A 14-dimensional manifold has Dirac spinors of dimension-two to the dimension of the space divided by two.
<p>[01:21:48] [The] first thing we need to do is we still have the right to choose intrinsic field content. [We] have an intrinsic field theory. So, if you consider the structure bundle of the spinors; we built the chimeric bundle, so we can define Dirac spinors on the chimeric bundle if we're in Euclidean signature. A 14-dimensional manifold has Dirac spinors of dimension-two to the dimension of the space divided by two.


<p>[01:22:20] Right? So 2^14 over 2^7 is 128, so we have a map into a structured group of U(128)
<p>[01:22:20] Right? So $$2^14$$ over $$2^7$$ is $$128$$, so we have a map into a structured group of $$U(128)$$


<p>[01:22:36] At least in Euclidean signature. We can get to mixed signatures later. From that, we can form the associated bundle.
<p>[01:22:36] At least in Euclidean signature. We can get to mixed signatures later. From that, we can form the associated bundle.
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