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

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

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→‎The Levi-Civita Connection

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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>[01:20:35] 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 ($$\nabula_{LC}$$. 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.