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

[01:31:38] Now, what would it mean to be able to use a gauge group in an intrinsic theory like this? We would be talking about something like an action. Let's say a first-order action. And it would take the group $$\mathcal{G}$$ let's say to the $$\mathbb{R}$$. Invariant, not under the full group, but under the tilted gauge subgroup, $$\mathcal{H_{\tau}}$$.  

[01:32:38] Just humor me for this talk, and let me call it the Einstein-Grossmann Lagrangian. Hilbert's certainly done fantastic things and has a lot of credit elsewhere, and he did do it first. But here, what we had was that Einstein thought in terms of the differential of the action, not the action itself. So, what we're looking for is equations of motion or some field, $$\alpha$$ where $$\alpha\in\Omega^{1}(\mathcal{G})$$.

[01:33:43] So we've, we've restricted ourselves to one gauge group, this big unitary group on the spinors using whatever sort of inner product naturally exists on the spinors. And not spinors valued in an auxiliary structure, but intrinsic spinors.

[01:34:28] Which is just looking like the exterior algebra, $$\wedge^{*}(C)$$ on the chimeric bundle; that means that it is graded by degrees. [The] chimeric bundle has dimension-14, so there's a zero-part, a one-part, a two-part, all the way up to 14. Plus, we have forms in the manifold. And so the question is if I want to look at $$\Omega^i$$ valued in the adjoint bundle, there's going to be some element $$\Phi_{i}$$, which is pure trace.

[01:35:41] Now, this is a tremendous amount of freedom that we've just gained. Normally we keep losing freedom, but this is the first time we actually begin to see that we have a lot of freedom and we're going to actually retain some of this freedom to the end of the talk. But the idea being that I can now start to define operators which correspond to the "Ship in the Bottle" problem.

[01:36:04] I can take field content: $$\epsilon$$ and $$\pi$$ where these are elements of the inhomogeneous gauge group. In other words, where $$\epsilon$$, is a gauge transformation and $$\pi$$ is a gauge potential.

[01:36:44] So in this case, if I have a $$\Phi$$, which is one of these invariants in the form piece, I can either take a contraction or I can take a wedge product. In the Lie algebra piece, I can either take a Lie product or because I'm looking at the unitary group, there's a second possibility, which is I can multiply everything by $$i$$ and go from [[Skew-Hermitian]] to [[Hermitian]] and take a [[Jordan product]] using anti-commutators rather than commutators.

[01:37:34] So, for example, I can define a Shiab ("Ship in a bottle") operator that takes [$$\Omega^{i}$$] $$i$$-forms valued in the adjoint bundle to much higher-degree forms valued in the adjoint bundle.  

[01:38:38] This is exactly what Einstein was doing. He took the curvature, which was large, and he bent it back and he sheared off the [[Weyl curvature]] and he took that part and he pushed it back along the space of metrics to give us something which we nowadays call [[Ricci Flow]] and ability for the curvature to direct us to the next structure.

[01:39:28] Can you just clarify the index on the Shiab is? So you take i-forms to d-minus-three-plus-1? [No,] d-minus-three-plus-i.

[01:39:43] So, in this case, the idea is, is that we've actually got something for our magic beans. We have an ability now to get equations of motion, which go along the group. In some sense, it's as if it was a gradient vector field, except we're using forms rather than vectors. But now what are the transformation properties?

[01:40:08] Well, because the curvature... so, you have Shiab. Because the curvature of a connection hit by a gauge transformation is equal on-the-nose to the adjoint action on the Lie algebra of the curvature. We know that if we have two possible actions of conjugation under a bracket and the bracket respects the action of the gauge group, we know that this is going to be well-preserved. In other words, we're going to get a form that is gauged invariant relative to the tilted gauge group.

[01:41:26] What about the torsion, $$T_{\epsilon, \pi}$$? Can we rescue the torsion? Here again, we have good news. The torsion is problematic. But if I look at a different field which I'm going to call the augmented torsion and I define it to be the regular torsion, which would be $$\Pi$$ minus this expression. Okay? This turns out to be beautifully invariant again.

[01:42:16] This is true in [[Renormalization theory]]. It's true in [[Black-Scholes]] theory. It's true all over the place. So, what we have here is we've gotten more diseases into the theory, but an even number of diseases allow us to have no disease at all. So, now we have two great tensors. We've got one tensor coming from the curvature and Shiab operator.

[01:42:54] We're doing okay.