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

<p>[01:47:43] And , whereas 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, uh, 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, this object. And he was like this amazing guy in the resistance during world war II.

<p>[01:48:03] So I would like to name this, the somatic complex. after, I guess his name is I think so this, this complex, I'm going to choose to start filling in some operators, the exterior derivative coupled to a connection, but on the case of spinors, we're going to put a slash through it. Let's make this the identity.

<p>[01:51:31] there are going to be plus and minus signs, but it's a magic brag 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. Okay. And then there's going to be one extra term.

<p>[01:51:48] Where all these Ts have the Epsilon 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>[01:52:50] map to equations usually in the curvature. And when you linearize that if you're in low enough dimensions, you have Omega zero Omega one. Sometimes I make a zero again and then something that comes from Omega two and if you can get that sequence to terminate by looking at something like a half signature theorem or a bent back.

<p>[01:53:12] Durham 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.

<p>[01:53:30] 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 zero add Omega one add direct um Omega zero add

<p>[01:54:09] And this is now not just a great guess, it's actually the information for the deformation theory of the linearized replacement of the Einstein field equations. So this is computing the dirt Zariski tangent space. Just as if you were doing self duel theory or Chern-Simon's theory, you've got two somatic complexes, right?

<p>[01:55:03] Well, there are different, depending upon whether you take the degree zero piece together with the degree two piece, or you take the degree one piece, let's just take the degree one piece.

<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 one tensor spinors and a field nu.

<p>[01:55:40] Nu is omega zero tensor S . Okay.

<p>[01:58:06] I guess I didn't write them in. But you would have two differential operators over here, and you'd have this differential operator coming from this Maurer-Cartan form. So I apologize, I'm being a little loose here, but the idea is you have two of these terms are zeroth order. Three of these terms would be first order, and on this side, one term would be first order.

<p>[01:58:32] and that's not there. That's fine. That was a mistake. Oh, no, sorry. That was a mistake. Calling it a mistake. These are two separate equations, right? So you have two separate fields, Nu and Zeta, and you have a coupled set of differential equations that are playing the role of the Dirac theory. Coming from the Hodge theory of a complex who's obstruction to being cohomology theory would be the replacement to the Einstein field equations, which would be rendered.

<p>[01:59:05] Gauge invariant on a group relative to a tilted subgroup. Okay. What would, so now we've dealt with two of the three sectors. Is there any generalization of the yang mills equation? Well, if we were to take the Einstein field equation generalization and take the norm square of it, Oh, there's some point I should make here.

<p>[01:59:29] Just one second. Yeah. I've been treating this as if everything is first order, but what really happens here

<p>[02:00:01] And we're neglecting to draw the fact that there have to be equations here too. These equations are first order. So why do we get to call this a first order theory? If there are equations here, which are of second order, well, it's not a pure first order theory, but when I say a first order theory in this context, what I really mean.

<p>[02:00:22] I mean is that the second order equations are completely redundant on the first order equations. By virtue of the symmetry principle, that is any solution of the first order equation should automatically apply, imply his solution of the second order equation. So from that perspective, I can pretend that this isn't here because it is sufficient to solve the first order equations.

<p>[02:00:59] which we previously called alpha. I mean that alpha equal to Upsilon because I've actually been using Upsilon. The portion of that is just the first order equations and take the norm square of that. That gives me a new Lagrangian, and if I solve that new Lagrangian, it leads to equations of motion.

<p>[02:02:01] So in other words, this piece gives you some portion that looks like right from the swervature tensor there’s going to be some component that's playing the role of Einstein's field equations directly and the Ricci tensor, but generalized. And then you're going to have some differential operator here so that the replacement for the yang mills term instead of $$d_A^*$$ of FAA, you've got these two an F .

<p>[02:02:26] and an adjoint together in the center, generalizing the yang mills theory. You say, well, how come we don't just see the yang mills theory? Why don't we see general relativity as well? But in the full expansion there's also a term with zeroth order that's effectively acting like the identity.

<p>[02:02:45] Which hits this as well. So you have one piece that looks like the yang mills theory, and in these second order equations, you also have a piece that looks like the Einstein theory. And this is in the vacuum equations. So then the question is how do you see the Dirac theory coming out of this? And so what we're just trying to put together now before we come out with the manuscript for this is.

<p>[02:03:14] Putting these two elliptic complexes together, the Dirac terms go between the two complexes, right? So the idea is that the stress energy tensor,should be the up and back term and the Dirac equations should come out of the term that goes up and over versus the term that goes over and up, and you need some cancellations to make sure that everything is of zeroth order properly invariant, etc.

<p>[02:03:39] And that's taking a little time because frankly, I'm not good at keeping track of indices minus signs left, right? That it's a learning disabled nightmare.

<p>[02:04:18] We've got problems. We're not in four dimensions. We're in 14 we don't have great field content cause we've just got these unadorned spinors and we're doing gauge transformations effectively on the intrinsic geometric quantities, not on some safe auxiliary. data. That's tensor product did with with what?