Difference between revisions of "A Portal Special Presentation- Geometric Unity: A First Look"
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A Portal Special Presentation- Geometric Unity: A First Look (view source)
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<p>[00:12:14] And those supposedly physicists spent their time submitting papers to what's called, um, the high energy section of the so-called preprint [[arXiv]]. But in fact, most of these papers have nothing to do with high energy physics whatsoever. And if you're looking for the, the designation, it's hep-th high energy physics dash theory. | <p>[00:12:14] And those supposedly physicists spent their time submitting papers to what's called, um, the high energy section of the so-called preprint [[arXiv]]. But in fact, most of these papers have nothing to do with high energy physics whatsoever. And if you're looking for the, the designation, it's hep-th high energy physics dash theory. | ||
<p>[00:12:36] Now, if you look through those papers, they don't seem to have much to do with particles. They don't seem to have to do with forces and | <p>[00:12:36] Now, if you look through those papers, they don't seem to have much to do with particles. They don't seem to have to do with forces and spacetime. They seem to have to do with very strange and obscure mathematical issues. And in the years since the, uh, string theory program got particularly reanimated, uh, I guess that would be around 1984 with the [[anomaly cancellation]] of [[Green]] and [[Schwarz]]. | ||
<p>[00:12:58] What you'll find is, is that physics became very active and simultaneously ground to a halt. It failed to remain a physical subject. It became something like a medieval quest for the number of angels to dance on the head of a pin. Now, in this circumstance, I think it's very important to realize that this is not a paper and we are not submitting to the arXiv. | <p>[00:12:58] What you'll find is, is that physics became very active and simultaneously ground to a halt. It failed to remain a physical subject. It became something like a medieval quest for the number of angels to dance on the head of a pin. Now, in this circumstance, I think it's very important to realize that this is not a paper and we are not submitting to the arXiv. | ||
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<p>[00:21:50] The [[Schwarzschild singularities]], which give us black holes, and the initial singularities of the [[Robertson]] [[Walker]] [[Friedmann]] universe, which are associated with the Big Bang, are some clue that there is some subtle flaw in Einstein's theory. So how to go beyond Einstein. I mean, what Einstein did to [[Newton]] was to recover Newton as a special case of a more general theory that is more flexible. | <p>[00:21:50] The [[Schwarzschild singularities]], which give us black holes, and the initial singularities of the [[Robertson]] [[Walker]] [[Friedmann]] universe, which are associated with the Big Bang, are some clue that there is some subtle flaw in Einstein's theory. So how to go beyond Einstein. I mean, what Einstein did to [[Newton]] was to recover Newton as a special case of a more general theory that is more flexible. | ||
<p>[00:22:17] And in fact, this is the same problem that we have because Albert Einstein's theory is so fundamental. We effectively begin every theoretical physics seminar with a statement about | <p>[00:22:17] And in fact, this is the same problem that we have because Albert Einstein's theory is so fundamental. We effectively begin every theoretical physics seminar with a statement about spacetime. In other words, Albert Einstein is locked in at the ground floor. So if we can't get below the ground floor to the foundations, it's very difficult to make progress. | ||
<p>[00:22:37] This is one of the things that is making it almost impossible for us to go beyond the initial revolutions of the 20th century. Do I know that this new theory, if it works, will allow us to escape? I do not. And there's no one who can uh can say that and I don't think I have the skills to develop the physical consequences of the theory, even if the theory turns out to be mostly right. | <p>[00:22:37] This is one of the things that is making it almost impossible for us to go beyond the initial revolutions of the 20th century. Do I know that this new theory, if it works, will allow us to escape? I do not. And there's no one who can uh can say that and I don't think I have the skills to develop the physical consequences of the theory, even if the theory turns out to be mostly right. | ||
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<p>[00:23:24] Perhaps there are only two. Even though physicists tell us that there are at least three or perhaps more. I believe that physics tells us that the universe is [[chiral]] that is left, right asymmetric, but the theory is itself not chiral. Instead, it chooses to present a different idea, which is that perhaps chirality is emergent much the way our hands are individually left, right asymmetric as our pinky is not a reflection of our thumb, but the thumb on each hand is a pairing to the other one. As is the pinky. Now, what does that mean? It means that if perhaps there is matter and there is force that is decoupled from our ordinary world, that that matter might restore the parity or the chirality. | <p>[00:23:24] Perhaps there are only two. Even though physicists tell us that there are at least three or perhaps more. I believe that physics tells us that the universe is [[chiral]] that is left, right asymmetric, but the theory is itself not chiral. Instead, it chooses to present a different idea, which is that perhaps chirality is emergent much the way our hands are individually left, right asymmetric as our pinky is not a reflection of our thumb, but the thumb on each hand is a pairing to the other one. As is the pinky. Now, what does that mean? It means that if perhaps there is matter and there is force that is decoupled from our ordinary world, that that matter might restore the parity or the chirality. | ||
<p>[00:24:06] Um, rather it would break the chirality and restore parity between these two halves the matter we see and the matter that is missing. There are a good number of other things that happened in the theory. It replaces | <p>[00:24:06] Um, rather it would break the chirality and restore parity between these two halves the matter we see and the matter that is missing. There are a good number of other things that happened in the theory. It replaces spacetime with what I've termed an [[observerse]]. Now an observerse is an unusual gadget in that it's thought of as two separate places where physics takes place connected by a map. | ||
<p>[00:24:32] That means effectively that we are in something like a stadium where there is, there are stands and there is a pitch and the playing field that we think we see may not in fact be where most of the action is taking place. In fact, not all of the fields live on the same space. So when we see waves and particles dancing around, they may have separate origins in each of the two components of the observerse. | <p>[00:24:32] That means effectively that we are in something like a stadium where there is, there are stands and there is a pitch and the playing field that we think we see may not in fact be where most of the action is taking place. In fact, not all of the fields live on the same space. So when we see waves and particles dancing around, they may have separate origins in each of the two components of the observerse. | ||
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<p>[00:41:47] What is physics to physicists today? How do they see it different from the way in which we might imagine the lay person sees physics? [[Ed Witten]] was asked this question in a talk he gave on physics and geometry many years ago, and he pointed us to three fundamental insights, which were his big three insights in physics. | <p>[00:41:47] What is physics to physicists today? How do they see it different from the way in which we might imagine the lay person sees physics? [[Ed Witten]] was asked this question in a talk he gave on physics and geometry many years ago, and he pointed us to three fundamental insights, which were his big three insights in physics. | ||
<p>[00:42:13] And they correspond to the three great equations. So the first one is, is that somehow physics takes place in an arena and that arena is a [[manifold]] X together with some kind of [[semi-Riemannian]] [[metric structure]], something that allows us to take length and angle. So that we can perform measurements at every point in this | <p>[00:42:13] And they correspond to the three great equations. So the first one is, is that somehow physics takes place in an arena and that arena is a [[manifold]] X together with some kind of [[semi-Riemannian]] [[metric structure]], something that allows us to take length and angle. So that we can perform measurements at every point in this spacetime or higher dimensional structure, leaving us a little bit of head room. The equation most associated with this is the [[Einstein field equation]]. | ||
<p>[00:43:12] And of course I'm running into the margin. Okay. | <p>[00:43:12] And of course I'm running into the margin. Okay. | ||
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<p>[00:43:47] The second fundamental insight, I'm going to begin to start drawing pictures here as well. | <p>[00:43:47] The second fundamental insight, I'm going to begin to start drawing pictures here as well. | ||
<p>[00:43:55] So if this is the | <p>[00:43:55] So if this is the spacetime manifold, the arena, the second one concerns symmetry groups. Which cannot necessarily be deduced from any structure inside of the arena. They are additional data that come to us out of the blue without explanation and these symmetries for a non-Abelian group, which is currently SU(3) color cross SU(2) weak isospin cross U(1) week hyper charge, which breaks down to SU(3) cross U(1) where the broken U(1)is the electromagnetic symmetry. | ||
<p>[00:44:54] This equation is also a curvature equation, the corresponding equation, and it says. But this time, the curvature of an auxiliary structure known as a gauge potential when differentiated in a particular way is equal, again, to the amount of stuff in the system that is not directly involved in the left hand side of the equation. So it has many similarities to the above equation. Both involve curvature. One involves a projection or a series of projections. The other involves a differential operator. | <p>[00:44:54] This equation is also a curvature equation, the corresponding equation, and it says. But this time, the curvature of an auxiliary structure known as a gauge potential when differentiated in a particular way is equal, again, to the amount of stuff in the system that is not directly involved in the left hand side of the equation. So it has many similarities to the above equation. Both involve curvature. One involves a projection or a series of projections. The other involves a differential operator. | ||
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<p>[01:02:57] Let us imagine that we cannot figure out the puzzle of why is there something rather than nothing. But if we do have a something that that's something has his little structure as possible, but it still invites us to work mathematically. So to me, the two great theories that we have mathematically and in physics are calculus and linear algebra. | <p>[01:02:57] Let us imagine that we cannot figure out the puzzle of why is there something rather than nothing. But if we do have a something that that's something has his little structure as possible, but it still invites us to work mathematically. So to me, the two great theories that we have mathematically and in physics are calculus and linear algebra. | ||
<p>[01:03:17] If we have calculus and linear algebra, I mean, imagine that we have some manifold, at least one of dimension four, but it's not a | <p>[01:03:17] If we have calculus and linear algebra, I mean, imagine that we have some manifold, at least one of dimension four, but it's not a spacetime. It doesn't have a metric. It's not broken down into two different kinds of coordinates, which then have some bleed into each other, but still maintains a distinction. | ||
<p>[01:03:36] It's just some sort of flabby proto | <p>[01:03:36] It's just some sort of flabby proto-spacetime, and in the end it has got to fill up with stuff and give us some kind of an equation. So let me write an equation. | ||
<p>[01:04:13] So I have in mind differential operators parameterized by some fields, omega, which when composed are not of second order, if these are first order operators, but as zeroth order in some sort of further differential operator saying that whatever those two operators are in composition is in some sense harmonic. | <p>[01:04:13] So I have in mind differential operators parameterized by some fields, omega, which when composed are not of second order, if these are first order operators, but as zeroth order in some sort of further differential operator saying that whatever those two operators are in composition is in some sense harmonic. | ||
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<p>[01:20:01] Okay. | <p>[01:20:01] Okay. | ||
<p>[01:20:08] What is it that we get for the Levi-Civita connection? Well, not- much. One thing we get is that normally the space of connections is an affine space. Not a vector space, but an affine space, almost a vector space, a vector space up to a choice of origin. But with the Levi Civita connection, rather than having an infinite plane with an ability to take differences, but no real ability to have a group structure. | <p>[01:20:08] What is it that we get for the Levi-Civita connection? Well, not- much. One thing we get is that normally the space of connections is an affine space. Not a vector space, but an affine space, almost a vector space, a vector space up to a choice of origin. But with the Levi-Civita connection, rather than having an infinite plane with an ability to take differences, but no real ability to have a group structure. | ||
<p>[01:20:35] You pick out one point, which then becomes the origin. That means that any connection A has the torsion tensor A which is equal to the connection minus the Levi Civita connection. Okay, so we get a tensor that we don't usually have. Gauge potentials are not usually well-defined. They're only defined up to a choice of gauge. | <p>[01:20:35] You pick out one point, which then becomes the origin. That means that any connection A has the torsion tensor A which is equal to the connection minus the Levi-Civita connection. Okay, so we get a tensor that we don't usually have. Gauge potentials are not usually well-defined. They're only defined up to a choice of gauge. | ||
<p>[01:21:00] So that's one of the things we get for our Levi Civita connection, but because the gauge group is going to go missing, this has terrible properties from with respect to the gauge group, it almost looks like a representation, but in fact, if we let the gauge group act, there's going to be an affine shift. | <p>[01:21:00] So that's one of the things we get for our Levi-Civita connection, but because the gauge group is going to go missing, this has terrible properties from with respect to the gauge group, it almost looks like a representation, but in fact, if we let the gauge group act, there's going to be an affine shift. | ||
<p>[01:21:21] Furthermore, as we've said before. The ability to use projection operators together with the gauge group is frustrated by virtue of the fact that these two things do not commute with each other. So now the question is, how are we going to prove that we're actually making a good trade | <p>[01:21:21] Furthermore, as we've said before. The ability to use projection operators together with the gauge group is frustrated by virtue of the fact that these two things do not commute with each other. So now the question is, how are we going to prove that we're actually making a good trade | ||
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<p>[01:25:18] This is exactly what we're going to hope is going to save us in this bad trade that we've made because we're going to add fields, content that has the ability to lower the mass and bring the mass back up. We're going to hope to have a theory which is going to create a communitive situation. But then once we've had this idea, we start to get a little bolder. | <p>[01:25:18] This is exactly what we're going to hope is going to save us in this bad trade that we've made because we're going to add fields, content that has the ability to lower the mass and bring the mass back up. We're going to hope to have a theory which is going to create a communitive situation. But then once we've had this idea, we start to get a little bolder. | ||
<p>[01:25:43] Let's think about unified content. We know that we want a space of connections, A for our field theory, but we know because we have a Levi Civita connection, that this is going to be equal on the nose to add value one forms as a vector space. The gauge group represents an add valued one forms. So if we also have the gauge group, what we think of that instead as a space of Sigma fields. | <p>[01:25:43] Let's think about unified content. We know that we want a space of connections, A for our field theory, but we know because we have a Levi-Civita connection, that this is going to be equal on the nose to add value one forms as a vector space. The gauge group represents an add valued one forms. So if we also have the gauge group, what we think of that instead as a space of Sigma fields. | ||
<p>[01:26:16] What if we take the semi direct product at a group theoretic level between the two and call this our group of interests. Well by analogy, we've always had a problem with the Poincare group being too intrinsically tied to rigid flar Mankowski space. What if we wanted to do quantum field theory in some situation, which was more amenable to a curved space situation? | <p>[01:26:16] What if we take the semi direct product at a group theoretic level between the two and call this our group of interests. Well by analogy, we've always had a problem with the Poincare group being too intrinsically tied to rigid flar Mankowski space. What if we wanted to do quantum field theory in some situation, which was more amenable to a curved space situation? | ||
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<p>[01:27:12] So we're going to call this the inhomogeneous gauge group or IGG. | <p>[01:27:12] So we're going to call this the inhomogeneous gauge group or IGG. | ||
<p>[01:27:23] And this is going to be a really interesting space because it has a couple of properties. One is it has a very interesting subgroup. Now, of course, H includes into G by just including onto the first factor, but in fact, there's a more interesting homomomorphism brought to you by the Levi Civita connection. | <p>[01:27:23] And this is going to be a really interesting space because it has a couple of properties. One is it has a very interesting subgroup. Now, of course, H includes into G by just including onto the first factor, but in fact, there's a more interesting homomomorphism brought to you by the Levi-Civita connection. | ||
<p>[01:27:49] So this magic being trade is going to start to enter more and more into our consciousness. If I take an element H and I map that in the obvious way into the first factor, but I map it | <p>[01:27:49] So this magic being trade is going to start to enter more and more into our consciousness. If I take an element H and I map that in the obvious way into the first factor, but I map it | ||
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<p>[01:49:59] But I'm 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 inverse, $$d_{A_0}$$ not H, which defines a connection one form as well as having the same. | <p>[01:49:59] But I'm 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 inverse, $$d_{A_0}$$ not H, which defines a connection one form as well as having the same. | ||
<p>[01:50:21] Derivative, uh, coming from the Levi Civita connection on U. So in other words, I have two derivative operators here. I have two add value one forms. The difference between them has been to be a zero with order, and it's going to be precisely the augmented torsion. And that's the same game I'm going to repeat here. | <p>[01:50:21] Derivative, uh, coming from the Levi-Civita connection on U. So in other words, I have two derivative operators here. I have two add value one forms. The difference between them has been to be a zero with 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] So I'm going to do the same thing here. I'm going to define a bunch of terms where in the numerator, I'm going to pick up the pie as well as the derivative in the denominator, because I have no derivative here. I'm going to pick up this H inverse $$d_{A_0}$$, not H. | <p>[01:50:52] So I'm going to do the same thing here. I'm going to define a bunch of terms where in the numerator, I'm going to pick up the pie as well as the derivative in the denominator, because I have no derivative here. I'm going to pick up this H inverse $$d_{A_0}$$, not H. | ||
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<p>[02:17:03] that which generates, um, all of the content and all of the action, you're, you're left with a puzzle as to how would you move forward from this? It might be easier in a mathematical sense, uh, to temporarily. put the U on its back, to put it more in line with a standard picture that many mathematicians and physicists will be familiar with. | <p>[02:17:03] that which generates, um, all of the content and all of the action, you're, you're left with a puzzle as to how would you move forward from this? It might be easier in a mathematical sense, uh, to temporarily. put the U on its back, to put it more in line with a standard picture that many mathematicians and physicists will be familiar with. | ||
<p>[02:17:28] In sector one of the geometric, uh, unity theory | <p>[02:17:28] In sector one of the geometric, uh, unity theory spacetime is replaced and recovered by the observerse contemplating itself. And so there are several sectors of GU and I wanted to go through a, at least four of them. In Einstein spacetime. We have not only four degrees of freedom, but also a spacetime metric representing rulers and protractors. | ||
<p>[02:17:56] If we're going to replace that. It's very tricky because it's almost impossible to think about what would be underneath Einstein’s theory. Now, there's a huge problem in the spinorial sector, which I don't. why more people don't worry about which is that spinors aren't defined for representations of the double cover of GL four R the general linear group’s. | <p>[02:17:56] If we're going to replace that. It's very tricky because it's almost impossible to think about what would be underneath Einstein’s theory. Now, there's a huge problem in the spinorial sector, which I don't. why more people don't worry about which is that spinors aren't defined for representations of the double cover of GL four R the general linear group’s. | ||
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<p>[02:18:39] Is to take a manifold $$X^d$$ as the starting point and see if we can create an entire universe from no other data and not even with the metric. So since we don't choose a metric, what we instead do is to work over the space of all possible point-wise metrics. So not quite in the Feynman sense, but in the sense that, um, we will work over a bundle that is of. | <p>[02:18:39] Is to take a manifold $$X^d$$ as the starting point and see if we can create an entire universe from no other data and not even with the metric. So since we don't choose a metric, what we instead do is to work over the space of all possible point-wise metrics. So not quite in the Feynman sense, but in the sense that, um, we will work over a bundle that is of. | ||
<p>[02:19:06] A quite larger, quite a bit larger dimension. So for example, if X, uh, was a four dimensional, therefore d equals four, then Y in this case would be d squared, which would be 16 plus three d, which would be 12 making 28 divided by two, which would be 14. So in other words, a four dimensional universe, or sorry, a four dimensional proto | <p>[02:19:06] A quite larger, quite a bit larger dimension. So for example, if X, uh, was a four dimensional, therefore d equals four, then Y in this case would be d squared, which would be 16 plus three d, which would be 12 making 28 divided by two, which would be 14. So in other words, a four dimensional universe, or sorry, a four dimensional proto-spacetime, not a space-time, but a produce based on, with no metric would give rise to a 14 dimensional. | ||
<p>[02:19:36] Um, observerse portion called Y. Now, I believe that in the lecture in, um, Oxford, I called that U, so I'm sorry for the confusion, but of course, this shifts around every time I take it out of the garage. And that's one of the problems with working on a theory in solitude for many . So we have two separate spaces and we have fields on the two spaces. | <p>[02:19:36] Um, observerse portion called Y. Now, I believe that in the lecture in, um, Oxford, I called that U, so I'm sorry for the confusion, but of course, this shifts around every time I take it out of the garage. And that's one of the problems with working on a theory in solitude for many . So we have two separate spaces and we have fields on the two spaces. | ||
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<p>[02:23:17] But we pick up some technical debt to use the computer science concept by actually having to now work on two different spaces, X and Y. And we're not merely working on X anymore. | <p>[02:23:17] But we pick up some technical debt to use the computer science concept by actually having to now work on two different spaces, X and Y. And we're not merely working on X anymore. | ||
<p>[02:23:30] This leads to the Mark of Zorro. That is, we know that whenever we have a metric by the fundamental theorem of | <p>[02:23:30] This leads to the Mark of Zorro. That is, we know that whenever we have a metric by the fundamental theorem of Riemannian geometry, we always get a connection. It happens, however, that what is missing to turn the canonical. chimeric bundle on Y into the tangent bundle of Y is in fact a connection on the space X. | ||
<p>[02:23:52] So there is one way in which we've reversed the fundamental theorem of | <p>[02:23:52] So there is one way in which we've reversed the fundamental theorem of Riemannian in geometry where a connection on X leads to a metric on Y. So if we do the full transmission mechanism out, a Gimmel on X leads to alpha sub Gimmel for the Levi-Civita connection on X. Alpha sub Gimmel live leads to a G sub alpha, which is, or sorry, the G sub Alef. | ||
<p>[02:24:17] Um, I'm not used to using Hebrew, uh, in, in math. Um, so G sub Alef, uh, then is a metric on Y and that creates a Levi Civita connection of the metric on the space Y as well, which then induces one on the spinorial levels. In sector two the inhomogenous gauge group on Y replaces the Poincare. Group and the internal symmetries that are found on X. | <p>[02:24:17] Um, I'm not used to using Hebrew, uh, in, in math. Um, so G sub Alef, uh, then is a metric on Y and that creates a Levi-Civita connection of the metric on the space Y as well, which then induces one on the spinorial levels. In sector two the inhomogenous gauge group on Y replaces the Poincare. Group and the internal symmetries that are found on X. | ||
<p>[02:24:50] And in fact, you use a Fermionic extension of the inhomogeneous gauge group to replace the supersymmetric Poincare group, and that would be with the field content, zero forms, tensors and spinner, tensors with spinors, a direct, sum one forms tensor to spinors all up on Y as the Fermionic field content. | <p>[02:24:50] And in fact, you use a Fermionic extension of the inhomogeneous gauge group to replace the supersymmetric Poincare group, and that would be with the field content, zero forms, tensors and spinner, tensors with spinors, a direct, sum one forms tensor to spinors all up on Y as the Fermionic field content. | ||
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<p>[02:25:09] So that gets rid of the biggest problem because the internal symmetry group is what causes the failure, I think, of supersymmetric particles to be seen in nature, which is we have to have two different origin stories, which is a little bit like Lilith and, uh, and Genesis. Um. We can't easily say we have a unified theory. | <p>[02:25:09] So that gets rid of the biggest problem because the internal symmetry group is what causes the failure, I think, of supersymmetric particles to be seen in nature, which is we have to have two different origin stories, which is a little bit like Lilith and, uh, and Genesis. Um. We can't easily say we have a unified theory. | ||
<p>[02:25:32] If | <p>[02:25:32] If spacetime and the SU(3) cross SU(2) cross U(1) group that lives on space-time have different origins and cannot be related. In this situation, we tie our hands and we have no choice over the, the uh, group content. So just to fix bundle notation, we let H, um, be the structure group of a bundle piece of H over a base space B. | ||
<p>[02:25:56] We use pi, um, for the projection map. We've reserved the variation in the pi. Um, orthography. For the field content and we try to use right principal actions. I'm terrible with left and right, but we do our best. Uh, we use H here, not G because we want to reserve G for the inhomogeneous extension of H once we moved to function spaces. | <p>[02:25:56] We use pi, um, for the projection map. We've reserved the variation in the pi. Um, orthography. For the field content and we try to use right principal actions. I'm terrible with left and right, but we do our best. Uh, we use H here, not G because we want to reserve G for the inhomogeneous extension of H once we moved to function spaces. | ||
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<p>[02:30:14] The big issue here is, is that we've forgone the privilege of being able to choose and dial in our own field content, and we've decided to remain restricted to anything we can generate only from $$X^d$$, in this case $$X ^4$$. So we generated $$Y^14$$ from $$X^4$$. And then we generated chimeric tangent bundles. Uh, on top of that, we built spinors off of the chimeric tangent bundle, and we have not made any other choices. | <p>[02:30:14] The big issue here is, is that we've forgone the privilege of being able to choose and dial in our own field content, and we've decided to remain restricted to anything we can generate only from $$X^d$$, in this case $$X ^4$$. So we generated $$Y^14$$ from $$X^4$$. And then we generated chimeric tangent bundles. Uh, on top of that, we built spinors off of the chimeric tangent bundle, and we have not made any other choices. | ||
<p>[02:30:42] So we're dealing with, I think it's $$U^128$$, um, w you, $$U^{2^7}$$. That is our, uh, structure group and we, it's fixed by the choice of $$X^4$$ not anything. So what do we get? Well, as promised, there is a tilted homomomorphism um, which, uh, takes the gauge group into its inhomogeneous extension. It acts as inclusion on the first factor, but it uses the Levi Civita connection to create a second sort of Maurer Cartan form. | <p>[02:30:42] So we're dealing with, I think it's $$U^128$$, um, w you, $$U^{2^7}$$. That is our, uh, structure group and we, it's fixed by the choice of $$X^4$$ not anything. So what do we get? Well, as promised, there is a tilted homomomorphism um, which, uh, takes the gauge group into its inhomogeneous extension. It acts as inclusion on the first factor, but it uses the Levi-Civita connection to create a second sort of Maurer Cartan form. | ||
<p>[02:31:17] I hope I remember the terminology right. It's been a long time. Um. The map is injective because it's injective under the first factor, but it actually gives us a nontrivial embedding of the, uh, gauge group in its, inhomogeneous extension, which makes the whole theory work. We then get to Shiab operators now, a shiab operator, um, is a map from the group crossed the add valued i forms. | <p>[02:31:17] I hope I remember the terminology right. It's been a long time. Um. The map is injective because it's injective under the first factor, but it actually gives us a nontrivial embedding of the, uh, gauge group in its, inhomogeneous extension, which makes the whole theory work. We then get to Shiab operators now, a shiab operator, um, is a map from the group crossed the add valued i forms. | ||