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<p>[00:51:34] In addition to all of that freedom is some means of taking away some of the redundancy that comes with that freedom, which is the action of the gauge group. Now we can allow the gauge group of symmetries to act on both sides of the equation, but the key problem is that.
<p>[00:51:34] In addition to all of that freedom is some means of taking away some of the redundancy that comes with that freedom, which is the action of the gauge group. Now we can allow the gauge group of symmetries to act on both sides of the equation, but the key problem is that.


<p>[00:52:03] If I act on connections on the right and then take the Einstein projection, this is not equal to first taking the projection and then conjugating with the gauge action. So the problem is, is that the projection is based on the fact that you have a relationship between the intrinsic geometry. If this is an ad-valued two-form, the two-form portion of this and the adjoint portion of this are both associated to the structure group of the tangent bundle.
<p>[00:52:03] If I act on connections on the right and then take the Einstein projection, this is not equal to first taking the projection and then conjugating with the gauge action. So the problem is, is that the projection is based on the fact that you have a relationship between the intrinsic geometry. If this is an ad[joint]-valued two-form, the two-form portion of this and the adjoint portion of this are both associated to the structure group of the tangent bundle.


<p>[00:52:52] But the gauge rotation is only acting on one of the two factors. Yet the projection is making use of both of them. So there is a fundamental incompatibility in the claim that Einstein's theory is a gauge theory relies more on analogy than an exact mapping between the two theories. What about the incompatibilities between the Einstein theory of general relativity and the Dirac theory of matter?
<p>[00:52:52] But the gauge rotation is only acting on one of the two factors. Yet the projection is making use of both of them. So there is a fundamental incompatibility in the claim that Einstein's theory is a gauge theory relies more on analogy than an exact mapping between the two theories. What about the incompatibilities between the Einstein theory of general relativity and the Dirac theory of matter?
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<p>[01:24:06] So when I was thinking about this, I used to be amazed by ships in bottles. I must confess that I never figured out what the trick was for ships in bottles, but once I saw it, I remembered thinking, that's really clever. So if you've never seen it, you have a ship, which is like a curvature tensor. And imagine that the mast is the Ricci curvature.
<p>[01:24:06] So when I was thinking about this, I used to be amazed by ships in bottles. I must confess that I never figured out what the trick was for ships in bottles, but once I saw it, I remembered thinking, that's really clever. So if you've never seen it, you have a ship, which is like a curvature tensor. And imagine that the mast is the Ricci curvature.


<p>[01:24:30] If you just try to shove it into the bottle, you're undoubtedly going to snap the mast. So, you imagine that you've transformed your gauge fields, you've kept track of where the Ricci curvature was, you try to push it from one space, like ad-valued two-forms into another space like ad-valued one-forms, where connections live.
<p>[01:24:30] If you just try to shove it into the bottle, you're undoubtedly going to snap the mast. So, you imagine that you've transformed your gauge fields, you've kept track of where the Ricci curvature was, you try to push it from one space, like ad[joint]-valued two-forms into another space like ad[joint]-valued one-forms, where connections live.


<p>[01:24:54] That's not a good idea. Instead, what we do is the following: imagine that you're carrying around group theoretic information and what you do is you do a transformation based on the group theory. So you lower the mast. You push it through the neck, having some string attached to the mast, and then you undo the transformation on the other side.
<p>[01:24:54] That's not a good idea. Instead, what we do is the following: imagine that you're carrying around group theoretic information and what you do is you do a transformation based on the group theory. So you lower the mast. You push it through the neck, having some string attached to the mast, and then you undo the transformation on the other side.
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====== Unified Content ======
====== Unified Content ======


<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 ad-valued one-forms $$(\Omega^{1}(Ad))$$ as a vector space. The gauge group represents an ad-valued one-forms. So, if we also have the gauge group ($$\mathcal{H}$$), but 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 ad[joint]-valued one-forms $$(\Omega^{1}(Ad))$$ as a vector space. The gauge group represents an ad[joint]-valued one-forms. So, if we also have the gauge group ($$\mathcal{H}$$), but we think of that instead as a space of sigma fields.


<p>[01:26:16] What if we take the semi-direct product ($$\ltimes$$) at a group theoretic level between the two and call this our group of interest. Well, by analogy, we've always had a problem with the Poincaré group being too intrinsically tied to rigid, flat Minkowski 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 ($$\ltimes$$) at a group theoretic level between the two and call this our group of interest. Well, by analogy, we've always had a problem with the Poincaré group being too intrinsically tied to rigid, flat Minkowski space. What if we wanted to do quantum field theory in some situation which was more amenable to a curved space situation?


<p>[01:26:44] It's possible that we should be basing it around something more akin to the gauge group. And in this case, we're mimicking the construction where $$\Xi$$ here would be analogous to the Lorentz group fixing a point in Mankowski space. And the ad-valued one-forms would be analogous to the four momentums we take in the semi-direct product to create the inhomogeneous Lorentz group, otherwise known as the Poincaré group, or rather its double cover to allow spin.
<p>[01:26:44] It's possible that we should be basing it around something more akin to the gauge group. And in this case, we're mimicking the construction where $$\Xi$$ here would be analogous to the Lorentz group fixing a point in Mankowski space. And the ad[joint]-valued one-forms would be analogous to the four momentums we take in the semi-direct product to create the inhomogeneous Lorentz group, otherwise known as the Poincaré group, or rather its double cover to allow spin.


<p>[01:27:12] So we're going to call this the inhomogeneous gauge group, or iggy.
<p>[01:27:12] So we're going to call this the inhomogeneous gauge group, or iggy.
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<p>[01:37:34] So, for example, I can define a Shiab ("Ship in a bottle") operator that takes [$$\Omega^{i}$$] $$i$$-forms valued in the adjoint bundle to much higher-degree forms valued in the adjoint bundle.  
<p>[01:37:34] So, for example, I can define a Shiab ("Ship in a bottle") operator that takes [$$\Omega^{i}$$] $$i$$-forms valued in the adjoint bundle to much higher-degree forms valued in the adjoint bundle.  


<p>[01:38:00] So, for in this case, for example [where i = 2], it would take a two-form to a d-minus-three-plus-two or a d-minus-one-form. So, curvature is an ad-valued two-form. And, if I had such a Shiab operator, it would take ad-valued two-forms to ad-valued d-minus-one-forms, which is exactly the right space to be an $$\alpha$$ coming from the derivative of an action.
<p>[01:38:00] So, for in this case, for example [where i = 2], it would take a two-form to a d-minus-three-plus-two or a d-minus-one-form. So, curvature is an ad[joint]-valued two-form. And, if I had such a Shiab operator, it would take ad[joint]-valued two-forms to ad[joint]-valued d-minus-one-forms, which is exactly the right space to be an $$\alpha$$ coming from the derivative of an action.


<p>[01:38:38] This is exactly what Einstein was doing. He took the curvature, which was large, and he bent it back and he sheared off the [[Weyl curvature]] and he took that part and he pushed it back along the space of metrics to give us something which we nowadays call [[Ricci Flow]] and ability for the curvature to direct us to the next structure.
<p>[01:38:38] This is exactly what Einstein was doing. He took the curvature, which was large, and he bent it back and he sheared off the [[Weyl curvature]] and he took that part and he pushed it back along the space of metrics to give us something which we nowadays call [[Ricci Flow]] and ability for the curvature to direct us to the next structure.
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<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>[01:49:59] 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>[01:50:15] So in other words, I have two derivative operators here. I have two ad[joint]-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>[01:50:52] 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 $$H^{-1}$$ $$d_{A_0}$$ $$H$$.
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<p>[02:28:10] Yeah. We can call this map the bi-connection, which gives us two separate connections for any point in the inhomogeneous gauge group. And we can notice that it can be viewed as a section of a bundle over the base space to come. We find an interesting embedding of the gauge group that is not the standard one in the inhomogeneous gauge group.
<p>[02:28:10] Yeah. We can call this map the bi-connection, which gives us two separate connections for any point in the inhomogeneous gauge group. And we can notice that it can be viewed as a section of a bundle over the base space to come. We find an interesting embedding of the gauge group that is not the standard one in the inhomogeneous gauge group.


<p>[02:28:42] So our summary diagram looks something like this. Take a look at the Taus of $$A_0$$. We will find a homeomorphism of the gauge group into its inhomogeneous extension that isn't simply inclusion under the first factor. And I'm realizing that I have the wrong pi production. That should just be as simple $$\pi$$, projecting down, we have a map from the inhomogeneous gauge group, via the bi-connection to A cross A connections cross connections, and that that behaves well according to the difference operator $$\delta$$ that takes the difference of two connections and gives an honest ad-valued one-form.
<p>[02:28:42] So our summary diagram looks something like this. Take a look at the Taus of $$A_0$$. We will find a homeomorphism of the gauge group into its inhomogeneous extension that isn't simply inclusion under the first factor. And I'm realizing that I have the wrong pi production. That should just be as simple $$\pi$$, projecting down, we have a map from the inhomogeneous gauge group, via the bi-connection to A cross A connections cross connections, and that that behaves well according to the difference operator $$\delta$$ that takes the difference of two connections and gives an honest ad[joint]-valued one-form.


<p>[02:29:23] The infinitesimal action of the gauge transformation of a gauge transformation, or at least an infinite testable one on a point inside of the group is given by a somewhat, almost familiar expression, which should remind us of how the first term in the gauge deformation complex for Self-Dual Yang-Mills actually gets started.
<p>[02:29:23] The infinitesimal action of the gauge transformation of a gauge transformation, or at least an infinite testable one on a point inside of the group is given by a somewhat, almost familiar expression, which should remind us of how the first term in the gauge deformation complex for Self-Dual Yang-Mills actually gets started.
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<p>[02:30:42] So we're dealing with, I think it's $$U^{128}$$, $$U^{2^7}$$. That is our structure group and it is fixed by the choice of $$X^4$$ not anything else. So what do we get? Well, as promised, there is a tilted homomomorphism which 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 $$U^{128}$$, $$U^{2^7}$$. That is our structure group and it is fixed by the choice of $$X^4$$ not anything else. So what do we get? Well, as promised, there is a tilted homomomorphism which 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. The map is injective because it's injective under the first factor, but it actually gives us a nontrivial embedding of the gauge group in its inhomogeneous extension, which makes the whole theory work. We then get to Shiab operators. Now, a Shiab operator a map from the group crossed the ad-valued i forms.
<p>[02:31:17] I hope I remember the terminology right. It's been a long time. The map is injective because it's injective under the first factor, but it actually gives us a nontrivial embedding of the gauge group in its inhomogeneous extension, which makes the whole theory work. We then get to Shiab operators. Now, a Shiab operator a map from the group crossed the ad[joint]-valued i forms.


<p>[02:31:43] In this case, the particular Shiab operators we’re interested in is mapping i form is to d minus three, plus i forms. So, for example, you would map a two form to a d minus three plus i. So if d, for example, were 14, ..., and i was equal to two. Then 14 minus three is equal to 11 plus two is equal to 13. So that would be an ad-valued 14 minus one form, which is exactly the right place for something to form a current. That is the differential of a Lagrangian on the space.
<p>[02:31:43] In this case, the particular Shiab operators we’re interested in is mapping i form is to d minus three, plus i forms. So, for example, you would map a two form to a d minus three plus i. So if d, for example, were 14, ..., and i was equal to two. Then 14 minus three is equal to 11 plus two is equal to 13. So that would be an ad[joint]-valued 14 minus one form, which is exactly the right place for something to form a current. That is the differential of a Lagrangian on the space.


<p>[02:32:38] Now the augmented torsion, the torsion is a very strange object. It's introduced sort of right at the beginning of learning a differential geometry, but it really doesn't get used very much. One of the reasons it doesn't get used, gauge theory is that it's not gauge invariant. It has a gauge and variant piece to it, but then a piece that spoils the gauge invariant.
<p>[02:32:38] Now the augmented torsion, the torsion is a very strange object. It's introduced sort of right at the beginning of learning a differential geometry, but it really doesn't get used very much. One of the reasons it doesn't get used, gauge theory is that it's not gauge invariant. It has a gauge and variant piece to it, but then a piece that spoils the gauge invariant.
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