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

<p>[01:43:33] What can we write down in terms of equations of motion. Let's start with Einstein's concept. If we do Shiab of the curvature tensor of the gauge potential hit with an operator defined by the $$\epsilon$$-sigma field plus the star operator acting on the augmented torsion of the pair. This contains all of the information when $$\pi$$ is zero in Einstein's tensor.

<p>[01:44:16] In other words, there is a spinorial analog of the Weyl curvature, a spinorial analog of the traceless Ricci curvature, and a spinorial analog of the scalar curvature. This operator should shear off the analog of the Weyl curvature -- just the way the Einstein's projection shears off the Weyl curvature when you're looking at the tangent bundle. And this term, which is now gauge invariant may be considered as containing a piece that looks like $$\Lambda$$ times $$G_{\mu \nu}$$ or a cosmological constant.

<p>[01:44:51] And this piece here can be made to contain a piece that looks like Einstein's tensor. And so this looks very much like the vacuum field equations. Okay, well we have to add in something else. I'll be a little bit vague cause I'm still giving myself some freedom as we write this up.

<p>[01:45:25] But we are going to define whatever tensor we need. This is gauge invariant. This is gauge invariant. And this is gauge invariant with respect to the tilted gauge group. These two tensors together should be exact. And this tensor on its own should be exact.

<p>[01:45:53] We're going to call the exact tensor, the ***swervature***.

<p>[01:45:59] The particular Shiab operator we call the ***swerve***. So that's 'swerve-curvature' plus the adjustment needed for exactness and another gauge invariant term which is not usually gauge invariant.

<p>[01:46:17] That is pretty cool. If that works, we've now taken the Einstein equation and we've put it not on the space of metrics, but we've put a generalization and an analog on the space of gauge potentials, much more amenable to quantization with much more algebraic structure and symmetry in the form of the inhomogeneous gauge group and its homogeneous vector bundle, some of which may be supersymmetric.

<p>[01:46:44] Now, the question is: "We've integrated so tightly with the matter field" -- we have to ask ourselves the question -- "can we see unification here?"

<p>[01:47:01] Let's define matter content in the form of $$\Omega^{0}($)$$, which is a fancy way of saying spinors, together with a copy of the $$\Omega^{1}($)$$. And, let me come up with two other copies of the same data, so I'll make $$\Omega^{d-1}$$ just by duality so imagine that there's a Hodge star operator.

<p>[01:47:43] And, when I was a little kid, I had the Soma cube. I don't know if you've ever played with one of these things. They're fantastic. And I later found out that this guy who invented the Soma cube which you had to put together as pieces. There was one piece that looked like this object. And, he was like this amazing guy in the Resistance during World War II.

So I would like to name this, the Somatic Complex after -- I guess his name -- is Piet Hein.

<p>[01:48:03] So this complex -- I'm going to choose to start filling in some operators: the exterior derivative coupled to a connection, but in the case of spinors we're going to put a slash through it. Let's make this the identity.

<p>[01:48:26] We'd now like to come up with a second operator here. This second operator here should have the property that the complex should be exact and the obstruction to it being a true complex -- to nilpotency -- should be exactly the generalization of the Einstein equations. Thus, unifying the spinorial matter with the intrinsic replacement for the curvature equations.

<p>[01:48:59] Well, we know that $d_A$ composed with itself is going to be the curvature. And we know that we want that to be hit by a Shiab operator. And Shiab is a derivation, you can start to see that that's going to be curvature, so you want something like $$F_A$$ followed by Shiab over here to cancel. Then you think, okay, how am I going to get at getting this augmented torsion?

<p>[01:49:32] And, then you realize that the information in the inhomogeneous gauge group, you actually have information not for one connection, but for two connections.

<p>[01:49:44] In one case, I can do $$+ *$$ to pick up the $$A_\{pi}$$.

<p>[01:49:59] But I am also going to have a derivative operator if I just do a star operation. So, I need another derivative operator to kill it off here. So I'm going to take minus the derivative with respect to the connection, $$H^{-1}$$ $$d_{A_0}$$ $$H$$ which defines a connection one-form as well as having the same derivative coming from the Levi-Civita connection on $$U$$.

<p>[01:50:15] So in other words, I have two derivative operators here. I have two ad-value one-forms. The difference between them has been to be a zero-th order, and it's going to be precisely the augmented torsion. And that's the same game I'm going to repeat here.

<p>[01:50:52] I'm going to do the same thing here. I'm going to define a bunch of terms. In the numerator I'm going to pick up a $$\pi$$ as well as the derivative in the denominator -- because I have no derivative here -- I'm going to pick up this $$H^{-1}$$ $$d_{A_0}$$ $$H$$.

<p>[01:51:17] I'm going to do that again on the other side. There are going to be plus and minus signs, but it's a magic bracket that knows whether or not should be a plus sign or a minus sign, and I apologize for that, but I'm not able to keep that straight. And then there's going to be one extra term.

<p>[01:51:48] Where all these Ts have the epsilons and pis. Okay. So some crazy series of differential operators on the northern route. So if you take the high road or you take the low road, when you take the composition of the two, the differential operators fall out and you're left with an obstruction term that looks like the Einstein field equation.

<p>[01:52:26] Can you go farther? Well, look it how close to this field content is to the picture from [[Deformation theory]] that we learned about in low dimensions. The low-dimensional world works by saying that symmetries map to field content map to equations usually in the curvature. And when you linearize that if you are in low enough dimensions, you have $$\Omega^{0}$$, $$\Omega^{1}$$, sometimes $$\Omega^{0}$$ again, and then something that comes from $$\Omega^{2}$$

<p>[01:53:12] and if you can get that sequence to terminate by looking at something like a half-signature theorem or a bent-back De Rahm complex in the case of dimension three, you have [[Atiyah-Singer]] theory, and remember we need some way to get out of infinite dimensional trouble, right? You have to have someone to call when things go wrong overseas and you have to be able to get your way home. And in some sense, we call on [[Atiyah-Singer]] and say, we're in some infinite dimensional space can't you cut out some finite dimensional problem that we can solve even though we start getting ourselves into serious trouble? And so we're going to do the same thing down here. We're going to have $$\Omega^{0}(ad)$$, $$\Omega^{1}(ad)$$ direct sum $$\Omega^{0}(ad)$$, $$\Omega^{d-1}(ad)$$ and it's almost the same operators.

<p>[01:54:33] One of them is [[Bose]]. One of them is [[Fermi]]. The obstruction to both of them is a common generalization of the Einstein field equations. What's more is if you, if you start to think about this. This is some version of [[Hodge theory]] with funky operators, so you can ask yourself: "Well, what are the harmonic forms in a fractional spin context?"

<p>[01:55:16] You get some kind of equation. So, I'm going to decide that I have a $$\zeta$$ field, which is an $$\Omega^{1}$$ tensor spinors and a field $$\nu$$, which always strikes me as a Yiddish field. $$\nu$$ is $$\Omega^{0}($)$$.