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

But the problem is that the hallmark of the Yang-Mills theory is the freedom to choose the data, the internal quantum numbers that give all the particles their personalities beyond the mass and the spin. 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 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 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. 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, and the claim that Einstein's theory is a gauge theory relies more on analogy than on an exact mapping between the two theories.

So when I was thinking about this, I used to be amazed by ships in bottles, and 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. 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.

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 as a vector space. The gauge group represents on ad-valued one-forms.

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?''' '''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 Minkowski space, and 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.

And now the question is, do we have any such actions that are particularly nice, and could we recognize them the way Einstein did, by trying to write down not the action—and Hilbert was the first one to write that down but I, you know, I always feel defensive, because I think Einstein and Grossmann did so much more to begin the theory, and that the Lagrangian that got written down was really just an inevitability. So, just humor me for this talk, and let me call it the Einstein-Grossmann Lagrangian. Hilbert's certainly done fantastic things and has a lot of credit elsewhere, and he did do it first. But here, what we had was that Einstein thought in terms of the differential of the action, not the action itself. So, what we're looking for is equations of motion or some field \(\alpha\), where \(\alpha\) belongs to the one-forms on the group.

So for example, I can define a '''shiab operator '''(ship in a bottle operator) that takes i-forms valued in the adjoint bundle to much higher-degree forms valued in the adjoint bundle.

So for this case, for example, 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.

'''Audience Member:''' Can you just clarify what the index on the shiab is? So you take i-forms to d-minus-three-plus-1?

'''Eric Weinstein:''' Three-plus-i.

Let's define matter content in the form of Omega-0 tensored in the spinors, which is a fancy way of saying spinors, together with a copy of the one-forms tensored in the spinors. And let me come up with two other copies of the same data—so I'll make \(\Omega^{d-1}\) just by duality, so imagine that there's a Hodge star operator.

So in one case, I can do plus star to pick up the \(A_{\pi}\). 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, which defines a connection one-form as well as having the same derivative coming from the Levi-Civita connection on \(U\).

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 zeroth-order, and it's going to be precisely the augmented torsion. And that's the same game I'm going to repeat here.

So, there is one way in which we've reversed the fundamental theorem of Riemannian geometry, where a connection on \(X\) leads to a metric on \(Y\). So if we do the full transmission mechanism out, \(ℷ\) on \(X\) leads to \(\aleph_ℷ\) for the Levi-Civita connection on \(X\). \(\aleph_{ℷ}\) leads to \(g_{\aleph}\), which is—sorry, \(g_\aleph\). I'm not used to using Hebrew in math.* So \(g_{\aleph}\), 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 bundles. In sector II, the inhomogeneous gauge group on \(Y\) replaces the Poincaré group and the internal symmetries that are found on \(X\). And in fact, you use a fermionic extension of the inhomogeneous gauge group to replace the supersymmetric Poincaré group, and that would be with the field content zero forms tensored with spinors direct sum one-forms tensored with spinors all up on \(Y\) as the fermionic field content.

So our summary diagram looks something like this. Take a look at the \(\tau_{A_0}\). We will find a homomorphism of the gauge group into its inhomogeneous extension that isn't simply inclusion under the first factor. We—and I'm realizing that I have the wrong pi projection, that should just be a simple \(\pi\) projecting down. We have a map from the inhomogeneous gauge group via the bi-connection to \(A \times A\), connections cross connections, and that behaves well according to the difference operator \(\delta\) that takes the difference of two connections and gives an honest ad-valued one-form.

We then get to shiab operators. Now, a '''shiab operator''' is a map from the group crossed the ad-valued i-forms. In this case, the particular shiab operator we're interested in is mapping i-forms to d-minus-three-plus-i-forms. So for example, you would map a two-form to d-minus-three-plus-i. So if d, for example, were 14, and i were 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.