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<p>[01:47:32] So I'll make Omega D minus one just by duality. So imagine that there's a [https://en.wikipedia.org/wiki/Hodge_star_operator Hodge star] operator.
<p>[01:47:32] So I'll make Omega D minus one just by duality. So imagine that there's a [https://en.wikipedia.org/wiki/Hodge_star_operator Hodge star] operator.


<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>[01:47:43] And , whereas a little kid, I had the [https://en.wikipedia.org/wiki/Soma_cube 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 (Piet Hein)[https://en.wikipedia.org/wiki/Piet_Hein_(scientist) 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>[01:48:03] 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>[01:48:26] We'd now like to come up with a second operator here. Here in this second operator here should have the property that the complex should be exact and the obstruction to it being a true complex to nillpotency should be exactly the generalization of the Einstein equations. Thus unifying the spinnorial matter with the intrinsic replacement for the curvature equations.
<p>[01:48:26] We'd now like to come up with a second operator here. Here in this second operator here should have the property that the complex should be exact and the obstruction to it being a true complex to nillpotency should be exactly the generalization of the Einstein equations. Thus unifying the spinnorial matter with the intrinsic replacement for the curvature equations.
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<p>[01:51:17] I'm going to do that again. On the other side,
<p>[01:51:17] I'm going to do that again. On the other side,


<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>[01:51:31] There are going to be plus and minus signs, but it's a magic bean 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>[01:51:48] 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>[01:52:21] Well, that's pretty good. If true,
<p>[01:52:21] Well, that's pretty good. If true,
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<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
<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


<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>[01:52:50] 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>[01:53:12] 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>[01:53:30] 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>[01:54:00] Omega d minus one add, and it's almost the same operator.
<p>[01:54:00] Omega d minus one add, and it's almost the same operator.


<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>[01:54:09] 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 [https://en.wikipedia.org/wiki/Zariski_tangent_space 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>[01:54:33] 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>[01:55:03] 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>[01:55:03] 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>[01:55:16] 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>[01:55:32] which always strikes me as a Yiddish field.
<p>[01:55:32] which always strikes me as a Yiddish field.


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


<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>[01:55:47] What equation would they solve if we were doing Hodge theory relative to this complex, the equation would look something like this.
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<p>[01:57:48] Now, if you have something like that, that would be a hell of a Dirac equation. Alright, you've got differential operators over here. you've got differential operators. Um,
<p>[01:57:48] Now, if you have something like that, that would be a hell of a Dirac equation. Alright, you've got differential operators over here. you've got differential operators. Um,


<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>[01:58:06] 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 [https://en.wikipedia.org/wiki/Maurer%E2%80%93Cartan_form 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] Um,
<p>[01:58:27] Um,


<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>[01:58:32] 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 whose 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>[01:59:05] 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>[01:59:29] Just one second. Yeah. I've been treating this as if everything is first-order, but what really happens here


<p>[01:59:41] is that you've got symmetries. You've got symmetric field content,
<p>[01:59:41] is that you've got symmetries. You've got symmetric field content,
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<p>[01:59:52] you've got ordinary connections.
<p>[01:59:52] you've got ordinary connections.


<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>[02:00:01] 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>[02:00:22] 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 imply the 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>[02:00:49] So I can now look. Let's call that entire replacement,
<p>[02:00:49] So I can now look. Let's call that entire replacement,


<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>[02:00:59] 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>[02:01:17] That look like exactly what we said before.
<p>[02:01:17] That look like exactly what we said before.
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<p>[02:01:28] And it ends up defining an operator that looks something like this, $$d_A^*$$, the adjoint of the operator.
<p>[02:01:28] And it ends up defining an operator that looks something like this, $$d_A^*$$, the adjoint of the operator.


<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>[02:02:01] 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>[02:02:26] 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>[02:02:45] 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>[02:03:14] 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>[02:03:39] 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>[02:03:53] So
<p>[02:03:53] So
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<p>[02:03:59] we've got one more unit to go. I mean, there's a fifth unit that has to do with mathematical applications, but this is sort of a physics talk for today. Is there any questions before we go into the last unit and then really handle questions for real? All right, let me show you the next little bit.
<p>[02:03:59] we've got one more unit to go. I mean, there's a fifth unit that has to do with mathematical applications, but this is sort of a physics talk for today. Is there any questions before we go into the last unit and then really handle questions for real? All right, let me show you the next little bit.


<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?
<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?


<p>[02:04:38] Our, our, uh, our spinors are. How is it that we're going to find anything realistic? And then we have to remember everything we've been doing recently has been done on U.
<p>[02:04:38] Our, our, uh, our spinors are. How is it that we're going to find anything realistic? And then we have to remember everything we've been doing recently has been done on U.
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