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

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

<p>[01:48:03] So I would like to name this, the Somatic Complex, after, I guess his name is Piet Hein, I think. So 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>[01:52:26] Can you go farther? Well, look it up. 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're in low enough dimensions, you have $$\Omega^{0}$$, $$\Omega^{1}$$. Sometimes $$\Omega^{0}$$ again and then something that comes from $$\Omega^{3}$$

<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.

<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 $$\Omega^{0}(ad)$$, $$\Omega^{0}(ad)$$ direct sum $$\Omega^{0}(ad)$$

<p>[01:54:00] $$\Omega^{d-1}(ad)$$ and it's almost the same operators.

<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 [[[Zariski tangent space]] just as if you were doing self-dual theory or Chern-Simons theory. You've got two somatic complexes, right?

<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 $$\Omega^{0}$$ tensor S.  

<p>[01:55:47] What equation would they solve if we were doing Hodge theory relative to this complex? The equation would look something like this.

<p>[01:57:48] Now, if you have something like that, that would be a hell of a Dirac equation. Right. You've got differential operators over here. You've got differential operators. 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>[01:58:27] 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, whose obstruction to being cohomology theory, would be the replacement to the Einstein field equations, which would be rendered gauge invariant on a group relative to a tilted subgroup.

<p>[01:58:50] 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 is that you've got symmetries. You've got symmetric field content, you've got ordinary connections.

<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>[02:01:28] And it ends up defining an operator that looks something like this, $$d_A^*$$, the adjoint of the Shiab operator.