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

<blockquote style="background: #f3f3ff; border-color: #ddd;">If one wants to summarize our knowledge of physics in the briefest possible terms, there are three really fundamental observations:  (i)  Space-time is a pseudo-Riemannian manifold \(M\), endowed with a metric tensor and governed by geometrical laws.  (ii)  Over \(M\) is a vector bundle \(X\) with a nonabelian gauge group \(G\).  (iii)  Fermions are sections of \((\hat{S}{+} \otimes V_{R}) \oplus (\hat{S}{-} \otimes V_{\tilde{R}})\).  \(R\) and \(\tilde{R}\) not isomorphic; their failure to be isomorphic explains why the light fermions are light and presumably has its origins in a representation difference \(\Delta\) in some underlying theory.  All of this must be supplemented with the understanding that the geometrical laws obeyed by the metric tensor, the gauge fields, and the fermions are to interpreted in quantum mechanical terms.<br>

So the first one 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. And of course, I've run into the margin. So it says that a piece of the Riemann curvature tensor, the Ricci tensor minus an even smaller piece, the scalar curvature multiplied by the metric, is equal—plus the cosmological constant—is equal to some amount of matter and energy, the stress-energy tensor. So it's intrinsically a curvature equation.

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} + \Lambda g_{\mu\nu} = 8 \pi T_{\mu\nu}$$</div>

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \text{SU}(3) \text{ (color)} \times \text{SU}(2) \text{ (weak isospin)} \times \text{U}(1) \text{ (weak hypercharge)}$$</div>  

Which breaks down to \(\text{SU}(3) \times \text{U}(1)\), where the broken \(\text{U}(1)\) is the electromagnetic symmetry. This equation is also a curvature equation—the corresponding equation—and it says that 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.

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$d^*_A F_A = J(\psi)$$</div>

The third point surrounds the matter in the system. And here we have a Dirac equation, again coupled to a connection. But one of the great insights is that the reason for the lightness of matter in the natural mass scale of physics has to do with the fact that this \(\psi\) really should have two components, and the differential operator should map to one component on the other side of the equation, but the mass operator should map to another. And so if one of the components is missing, if the equation is intrinsically lopsided—chiral, asymmetric—then the mass term and the differential term have difficulty interacting, which is sort of overcompensating for the mass scale of the universe, so you get to a point where you actually have to define a massless equation. But then, just like overshooting a putt, it's easier to knock it back by putting in a Higgs field in order to generate an as-if fundamental mass through the Yukawa couplings. So matter is asymmetric, and therefore light.

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \unicode{x2215}\kern-0.55em \partial_A \psi = m \psi $$</div>

Well, in the case of Einstein's general relativity, we can rewrite the Einstein theory by saying that there's a projection map due to Einstein of a curvature tensor, where I'm going to write that curvature tensor as I would in the Yang-Mills theory. That should be an LC for Levi-Civita. So, the Einstein projection of the curvature tensor of the Levi-Civita connection of the metric on this side, and on this side, I'm going to write down this differential operator, the adjoint of the exterior derivative coupled to a connection. And you begin to see that we're missing an opportunity, potentially. What if the \(F_A\)s were the same in both contexts? Then you're applying two separate operators, one zeroth-order and destructive, in the sense that it doesn't see the entire curvature tensor; the other inclusive, but of first-order. And so the question is, is there any opportunity to do anything that combines these two?

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ P_E(F_{\nabla^{LC}h}) \neq h^{-1}P_E(F_{\nabla^{LC}}) h $$</div>

What about the incompatibilities between the Einstein theory of general relativity and the Dirac theory of matter? I was very struck that, if we're going to try to quantize gravity and we associate gravity with the spin-2 field \(g_{\mu \nu}\), we actually have a pretty serious problem, which is if you think about spinors, electrons, quarks, as being waves in a medium, and you think about photons as being waves in a different medium, [the] photon's medium does not depend on the existence of a metric. One-forms are defined whether or not a metric is present yet spinors are not. So if we're going to take the spin-2 \(g_{\mu \nu}\) field to be quantum mechanical, if it blinks out and does whatever the quantum does between observations, in the case of the photon, it is saying that the waves may blink out, but the ocean need not blink out. In the case of the Dirac theory, it is the ocean, the medium, in which the waves live that becomes uncertain itself. So even if you're comfortable with the quantum, to me, this becomes a bridge too far. So the question is how do we liberate the definition? How do we get the metric out from its responsibilities? It's been assigned far too many responsibilities. It is responsible for a volume form, for differential operators, it's responsible for measurement, it's responsible for being a dynamical field, part of the field content of the system.

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$\delta^\omega(\mathscr{D}^\omega_2 \mathscr{D}^\omega_1) = 0$$</div>

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 of zeroth-order, and some sort of further differential operator saying that whatever those two operators are in composition is in some sense harmonic. Is such a program even possible? So if the universe is in fact capable of being the fire that lights itself, is it capable of managing its own flame as well and closing up? What I'd like to do is to set ourselves an almost impossible task, which is to begin with this little data in a sandbox, to use a computer science concept.

So, if here is physical reality, standard physics is over here, we're going to start with the sandbox, and all we're going to put in it is \(X^4\). And we're going to set ourselves a strait-jacketed task of seeing how close we can come to dragging out a model that looks like the natural world that follows this trajectory. While it may appear that that is not a particularly smart thing to do, I would like to think that we could agree that it is quite possible that if that were to be the case, we might say that this is what Einstein meant by a creator, which was his anthropomorphic concept for necessity and elegance and design, having no choice in the making of the world.

There's a completely exogenous flavor. And what I'm gonna do is I'm going to take the concept of observation and I'm going to break the world into two pieces: a place where we do our observation and a place where most of the activity takes place. And I'm going to try and do this without loss of generality. So in this case, we have \(X^4\) and it can map into some other space, and we are going to call this an '''observerse'''. The idea of an observerse is a bit like a stadium: you have a playing field and you have stands. They aren't distinct entities, they're coupled. And so fundamentally, we're going to replace one space with two. Exogenous model simply means that \(U\) is unrestricted, although larger than \(X^4\), so any manifold of four dimensions or higher that is capable of admitting \(X^4\) as an immersion.

The next model we have is the bundle theoretic, in which case, \(U\) sits over \(X\) as a fiber bundle.

The most exciting, which is the one we'll deal with today, is the endogenous model, where \(X^4\) actually grows the space \(U\) where the activity takes place. So we talk about extra dimensions, but these are, in some sense, not extra dimensions. They are implicit dimensions within \(X^4\).

And last, to proceed without loss of generality, we have the tautological model. In that case, \(X^4 = U\), and the immersion is the identity, and without loss of generality, we simply play our games on one space. Okay?

We take \(X^4\). We need metrics, we have none. We're not allowed to choose one, so we do the standard trick: we choose them all.

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$U^{14} = \text{met}(X^4)$$</div>

So we allow \(U^{14}\) to equal the space of metrics on \(X^4\) pointwise. Therefore, if we propagate on top of this—let me call this (\(\pi\)) the projection operator. If we propagate on \(U^{14}\), we are, in some sense, following a Feynman-like idea of propagating over the space of all metrics, but not at a field level, at a pointwise tensorial level.

Is there a metric on \(U^{14}\)? Well, we both want one and don't want one. If we had a metric from the space of all metrics, we could define fermions, but we would also lock out any ability to do dynamics. We want some choice over what this metric is, but we don't want full choice, because we want enough to be able to define the matter fields to begin with.

It turns out that if this is \(X^4\), and this is this particular endogenous choice of \(U^{14}\), we have a 10-dimensional metric along the fibers. So we have a \(g^{10}_{\mu \nu}\). Further, for every point in the fibers, we get a metric downstairs on the base space. So if we pull back the cotangent bundle, we get a metric \(g^4_{\mu \nu}\) on \(\pi*\) of the cotangent bundle of \(X\).

We now define the '''chimeric bundle''', right? And the chimeric bundle is this direct sum of the vertical tangent bundle along the fibers with the pullback, which we're going to call the horizontal bundle, from the base space. So the chimeric bundle is going to be the vertical tangent space of 10 dimensions to \(U\), direct sum the cotangent space, which we're going to call horizontal, to \(U\). And the great thing about the chimeric bundle is that it has an a priori metric. It's got a metric on the 4, a metric on the 10, and we can always decide that the two of them are naturally perpendicular to each other.

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ T_V^{10}(U) \oplus T_H^{4*}(U) $$</div>

Furthermore, it is almost canonically isomorphic to the tangent bundle or the cotangent bundle, because we either have 4 out of 14, or 10 out of 14 dimensions, on the nose. So the question is, what are we missing? And the answer is that we're missing exactly the data of a connection. So this bundle, chimeric \(C\)—we have \(C\) is equal to the tangent bundle of \(U\) up to a choice of a connection \(\theta\).

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ CU \overset{\theta}{=} TU $$</div>

And this is exactly what we wanted. We have a situation where we have some field on the manifold \(X\) in the form of a connection, which is amenable, more friendly to quantization, which is now determining a metric, turning around the Levi-Civita game. And the only problem is that we've had to buy ourselves into a different space than the one we thought we wanted to work on.

But now, as \(\theta\) changes, the fermions are defined on the chimeric bundle, and it's the isomorphism from the chimeric bundle to the tangent bundle of the space \(U\), which is variant—which means that the fermions no longer depend on the metric. They no longer depend on the \(\theta\) connection. They are there if things go quantum mechanical, and we've achieved our objective of putting the matter fields and the spin-1 fields on something of the same footing.

Okay. The next unit of GU is the unified field content. What does it mean for our fields to become unified? There are, in fact, only—at this moment—two fields that know about \(X\). \(\theta\), which is the connection that we've just talked about, and a section \(\sigma\), that takes us back so that we can communicate back and forth between \(U\) and \(X\). We now need field content that only knows about \(U\), which now has a metric depending on \(\theta\).

Okay. 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, you pick out one point, which then becomes the origin. That means that any connection \(A\) has a torsion tensor \(T_A\), which is equal to the connection minus the Levi-Civita connection. So we get a tensor that we don't usually have. Gauge potentials are not usually well-defined, they're are only defined up to a choice of gauge.

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ A \hookrightarrow T_A = A - \nabla^{LC} $$</div>

[The] first thing we need to do is we still have the right to choose intrinsic field content. We have an intrinsic field theory, so if you consider the structure bundle of the spinors—we've built the chimeric bundle, so we can define Dirac spinors on the chimeric bundle. If we're in Euclidean signature, a 14-dimensional manifold has Dirac spinors of dimension two to the dimension of the space divided by two. Right? So \(2^{14}\) over \(2^7\) is \(128\).

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ C ⇝ \unicode{x2215}\kern-0.55em S(C) $$</div>

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ P_{U(\unicode{x2215}\kern-0.4em S)} = P_{U(128)} $$</div>

So we have a map into a structure group of \(U(128)\), at least in Euclidean signature—we can get to mixed signatures later. From that, we can form the associated bundle.

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ P_{U(\unicode{x2215}\kern-0.4em S)} \times _{\text{Ad}}U(\unicode{x2215}\kern-0.55em S) $$</div>

And sections of this bundle are either, depending upon how you want to think about it, the gauge group \(\mathcal{H}\), or \(\Xi\), a space of \(\sigma\) fields—nonlinear.

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \Gamma^\infty(P_{U(\unicode{x2215}\kern-0.4em S)} \times _{\text{Ad}}U(\unicode{x2215}\kern-0.55em S)) = \begin{cases} \mathcal{H} \text{ gauge group}\\ \Xi \text{ sigma fields (non-linear)} \end{cases} $$</div>

Let's think about unified content. We know that we want a space of connections \(\mathscr{A}\) 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 1-forms as a vector space. The gauge group represents on ad-valued 1-forms.

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \mathscr{A} = \Omega^1(ad) $$</div>

So, if we also have the gauge group, but we think of that instead as a space of \(\sigma\) fields, what if we take the semi-direct product 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?''' '''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 1-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.

So we're going to call this the '''inhomogeneous gauge group''', or '''iggy'''. 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, \(\mathcal{H}\) includes into \(\mathcal{G}\) by just including onto the first factor, but in fact, there's a more interesting homomorphism brought to you by the Levi-Civita connection. So, this magic bean trade is going to start to enter more and more into our consciousness.

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \tau_{\mathcal{H}} : \mathcal{H} \hookrightarrow \mathcal{G} $$</div>

If I take an element \(h\), and I map that in the obvious way into the first factor, but I map it onto the Maurer-Cartan form—I think that's when I wish I remembered more of this stuff—into the second factor, it turns out that this is actually a group homomorphism.

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ h \rightarrow (h, h^{-1} d_{A_0} h) $$</div>

And so we have a nontrivial embedding, which is in some sense diagonal between the two factors. That subgroup we are going to refer to as the '''tilted gauge group''', and now our field content, at least in the bosonic sector, is going to be a group manifold, an infinite-dimensional function space Lie group, but a group nonetheless. And we can now look at \(\mathcal{G} / \mathcal{H}_\tau\), and if we have any interesting representation of \(\mathcal{H}\), we can form homogeneous vector bundles and work with induced representations. And that's what the fermions are going to be. So the fermions in our theory are going to be \(\mathcal{H}\) modules, and the idea is that we're going to work with vector bundles of the form inhomogeneous gauge group producted over the tilted gauge group.

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \mathcal{G} / \mathcal{H_{\tau}} $$</div>

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \mathcal{E} = \mathcal{G} \times_{H_{\tau}} \Upsilon $$</div>

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \Upsilon_H = \text{Fermion Cover} $$</div>

So, what we're talking about is something like a supersymmetric extension of the inhomogeneous gauge group analogous to supersymmetric extensions of the double cover of the inhomogeneous Lorentz or Poincaré group. Further, because this construction is at the level of groups, we've left a slot on the left-hand side on which to act. So, for example, if we want to take regular representations on the group, we can act by the group \(\mathcal{G}\) on the left-hand side, because we're allowing the tilted gauge group to act on the right-hand side. So it's perfectly built for representation theory, and if you think back to Wigner's classification, and the concept that a particle should correspond to an irreducible representation of the inhomogeneous Lorentz group, we may be able to play the same games here, up to the issue of infinite-dimensionality.

Now, what would it mean to be able to use a gauge group in an intrinsic theory like this? We would be talking about something like an action, let's say a first-order action. And it would take the group \(\mathcal{G}\), let's say to the real numbers, invariant, not under the full group, but under the tilted gauge subgroup.

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ S_1:\mathcal{G} \rightarrow \mathbb{R} $$</div>

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \text{Invariant } \mathcal{H}_\tau $$</div>

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 1-forms on the group.

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \alpha \in \Omega^1(\mathcal{G}) $$</div>

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ dS_1 = \alpha = 0 $$</div>

That means that it's graded by degrees. [The] chimeric bundle has dimension 14, so there's a zero-part, a one-part, a two-part, all the way up to 14. Plus, we have forms in the manifold. And so the question is, if I want to look at \(\Omega^i\) valued in the adjoint bundle, there's going to be some element \(\Phi_i\), which is pure trace.

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \Omega^i(ad) \ni \Phi_i $$</div>

Right? Because it's the same representations appearing where in the usually auxiliary directions, as well as the geometric directions. So we get an entire suite of invariance, together with trivially associated invariants that come from using the Hodge star operator on the forms. I'm just going to call them, for completeness, \(\tilde{\Phi}_i\). I'm not going to deal with them.

Now, this is a tremendous amount of freedom that we've just gained. Normally, we keep losing freedom, but this is the first time we actually begin to see that we have a lot of freedom, and we're going to actually retain some of this freedom to the end of the talk. But the idea being that I can now start to define operators which correspond to the "Ship in the Bottle" problem. I can take field content \(\varepsilon\) and \(\pi\), where these are elements of the inhomogeneous gauge group. In other words, \(\varepsilon\) is a gauge transformation, and \(\pi\) is a gauge potential.

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ (\varepsilon, \pi) \in \mathcal{G} $$</div>

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \bigcirc\kern-0.940em\circ\kern-0.58em·\text{}\kern0.3em_\varepsilon \eta = [\text{Ad}(\varepsilon^{-1}, \Phi), \eta] $$</div>

So in this case, if I have a \(\Phi\), which is one of these invariants, in the form piece I can either take a contraction or I can take a wedge product. In the Lie algebra piece, I can either take a Lie product, or because I'm looking at the unitary group there's a second possibility, which is I can multiply everything by \(i\) and go from skew-Hermitian to Hermitian and take a Jordan product using anti-commutators rather than commutators. So I actually have a fair amount of freedom, and I'm going to use a "magic bracket" notation, which in whatever situation I'm looking for, [the operator] knows what it wants to be. Does it want to do contraction? Does want to do wedge product, Lie product, Jordan product? But the point is, I now have a suite of ways of moving forms around.

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.

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \bigcirc\kern-0.940em\circ\kern-0.58em·\text{}\kern0.3em_\varepsilon \Omega^i(ad) \rightarrow \Omega^{d-3+i}(ad) $$</div>

So for this case, for example, it would take a 2-form to a \((d - 3 + 2)\) or a \((d - 1)\)-form. So curvature is an ad-valued 2-form. And, if I had such a shiab operator, it would take ad-valued 2-forms to ad-valued \((d - 1)\)-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 \(d - 3 + 1\)?

'''Eric Weinstein:''' \(3 + i\).

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ F_Ah = h^{-1}(F_A)h $$</div>

What about the torsion? Can we rescue the torsion? Here, again, we have good news. The torsion is problematic, but if I look at a different field—which I'm going to call the '''augmented torsion''', and I define it to be the regular torsion, which would be \(\Pi\) minus this expression, this turns out to be beautifully invariant again.

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ T_{\varepsilon, \pi} = \Pi - h^{-1}d_{A_0}h $$</div>

Okay. We're not doing physics yet. We're just building tools. We've built ourselves a little bit of freedom, we have some reprieves, we've still got some very big debts to pay back for this magic beans trade. We're in the wrong dimension. We don't have good field content. We're stuck on this one spinor. We've built ourselves some projection operators. We've picked up some symmetric, nonlinear \(\sigma\) field.

What can we write down in terms of equations of motion? Let's start with Einstein's concept. If we do shiab of the curvature tensor of a gauge potential, hit with an operator defined by the epsilon-sigma field, plus the star operator acting on the augmented torsion of the pair, this contains all of the information, when \(\pi\) is zero, in Einstein's tensor.

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \bigcirc\kern-0.940em\circ\kern-0.58em·\text{}\kern0.3em_\varepsilon F_\pi +[\bigcirc\kern-0.940em\circ\kern-0.58em·\text{}\kern0.3em_\varepsilon T_{\varepsilon,\eta}, T_{\varepsilon, \pi}] + *T_{\varepsilon, \pi} = 0 $$</div>

In other words, there is a spinorial analog of the Weyl curvature, a spinorial analog of the traceless Ricci curvature, and a spinorial analog of the scalar curvature. This operator should shear off the analog of the Weyl curvature just the way the Einstein's projection shears off the Weyl curvature when you're looking at the tangent bundle. And this term, which is now gauge invariant, may be considered as containing a piece that looks like \(\Lambda g_{\mu \nu}\), or a cosmological constant, and this piece here can be made to contain a piece that looks like Einstein's tensor, and so this looks very much like the vacuum field equations. But we have to add in something else. I'll be a little bit vague 'cause I'm still giving myself some freedom as we write this up.

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 1-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.

Well, we know that \(\unicode{x2215}\kern-0.5em d_A\) composed with itself is going to be the curvature, and we know that we want that to be hit by a shiab operator. And if shiab is a derivation, you can start to see that that's going to be curvature, so you want something like \(F_A\) followed by shiab over here to cancel. Then you think okay, how am I going to get at getting this augmented torsion? And, then you realize that the information in the inhomogeneous gauge group, you actually have information not for one connection, but for two connections.

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 1-form as well as having the same derivative coming from the Levi-Civita connection on \(U\).

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 \(\pi\) as well as the derivative, in the denominator, because I have no derivative here, I'm going to pick up this \(h^{-1} d_{A_0} h\).

I'm going to do that again on the other side. There are going to be plus and minus signs, but it's a magic bracket that knows whether or not it should be a plus sign or a minus sign and I apologize for that, but I'm not able to keep that straight. And then there's going to be one extra term, where all these \(T\)s have the \(\varepsilon\) and \(\pi\)s.

And when you linearize that, if you are in low enough dimension, you have \(\Omega^0\), \(\Omega^1\), sometimes \(\Omega^0\) again, and then something that comes from \(\Omega^2\), and if you can get that sequence to terminate by looking at something like a half-signature theorem or a bent back de Rham complex in the case of dimension three, you have an 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 and Singer and say we're in some infinite dimensional space, 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 \(\Omega^0(ad)\), \(\Omega^1(ad)\), direct sum \(\Omega^0(ad)\), \(\Omega^{d−1}(ad)\), and it's almost the same operators.

Well, they're 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. You get some kind of equation—so, I'm going to decide that I have a \(\zeta\) field, which is an \(\Omega^1(\unicode{x2215}\kern-0.55em S)\), and a field \(\nu\), which always strikes me as a Yiddish field. \(\nu\) is \(\Omega^0(\unicode{x2215}\kern-0.55em S)\).

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \zeta \in \Omega^1(\unicode{x2215}\kern-0.55em S) $$</div>

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \nu \in \Omega^0(\unicode{x2215}\kern-0.55em S) $$</div>

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ *d_A^* \zeta = *\nu $$</div>

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \bigcirc\kern-0.96em\circ\kern-0.60em·\text{}\kern0.3em_\varepsilon d_A \zeta + [\bigcirc\kern-0.96em\circ\kern-0.60em·\text{}\kern0.3em_\varepsilon\zeta, T] + [T, \bigcirc\kern-0.96em\circ\kern-0.60em·\text{}\kern0.3em_\varepsilon\zeta] + *\zeta = F_A \bigcirc\kern-0.96em\circ\kern-0.60em·\text{}\kern0.3em \nu + [[T, \nu], T] + [T, [T, \nu]] + [[T, \bigcirc\kern-0.96em\circ\kern-0.60em·\text{}\kern0.3em \nu], T] + *\nu $$</div>

''[https://youtu.be/Z7rd04KzLcg?t=7107 01:58:27]''<br>And that's not there, 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 [a] 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.

So I can now look—let's call that entire replacement, which we previously called \(\alpha\), I'm going to set \(\alpha\) equal to \(\Upsilon\), because I've actually been using \(\Upsilon\), the portion of it that is just the first-order equations, and take the norm squared of that, that gives me a new Lagrangian. And if I solve that new Lagrangian, it leads to equations of motion that look like exactly what we said before.

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \alpha = \Upsilon $$</div>

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ ||\Upsilon||^2 \rightarrow \delta^\omega(\mathscr{D}_2^\omega \circ \mathscr{D}_1^\omega) $$</div>

And it ends up defining an operator that looks something like this, \(d_A^∗\) the adjoint of the shiab operator.

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ *(d_A^* \bigcirc\kern-0.940em\circ\kern-0.58em·\text{}\kern0.3em^* + * + ...)(\bigcirc\kern-0.940em\circ\kern-0.58em·\text{}\kern0.3em_\varepsilon F_{A_\pi} + ...) $$</div>

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 \(d^∗_A F_A\), you've got these two shiab and an adjoint shiab together in the center, generalizing the Yang-Mills theory. Then 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 that's zeroth-order that's effectively acting like the identity 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?

We've got problems. We're not in four dimensions, we're in 14. We don't have great field content because 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 producted with what 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 \(U\). We've forgotten about \(X\). How does all of this look to \(X\)?

So \(X\) is sitting down here, and all the action is happening up here on \(U^{14}\). There's a projection operator—I've used \(\pi\) twice, it's not, here, the field content, it's just projection. And I've got a \(\sigma\), which is a section. What does \(\zeta\) pulled back or \(\nu\) pulled back look like on \(X^4\)?

We then choose to add some stuff that we can't see at all, that's dark. And this matter would be governed by forces that were dark too. There might be dark electromagnetism, and dark strong, and dark weak. It might be that things break in that sector completely differently, and it doesn't break down to an \(\text{SU}(3) \times \text{SU}(2) \times \text{U}(1)\) because these are different \(\text{SU}(3)\)s, \(\text{SU}(2)\)s, and \(\text{U}(1)\)s, and it may be that there would be like a high energy \(\text{SU}(5)\), or some Pati-Salam model. Imagine then that chirality was not fundamental, but it was emergent—that you had some complex, and as long as there were cross terms these two halves would talk to each other. But if the cross terms went away, the two terms would become decoupled.

And just the way we have a left hand and we have a right hand, and you ask me, right, imagine you have a neurological condition in an Oliver Sacks sort of an idiom, if somebody is only aware of one side of their body and they say, "Oh my God, I'm deformed, I'm asymmetric!" But we actually have a symmetry between the two things that can't see each other. Then we would still have a chiral world, but the chirality wouldn't be fundamental. There'd be something else keeping the fermions light, and that would be the absence of the cross terms. Now if you look at what happens in our replacement for the Einstein field equations, the term that would counterbalance the scalar curvature, if you put these equations on a sphere, they wouldn't be satisfied if the \(T\) term had a zero expectation value: because there would be non-trivial scalar curvature in the swervature terms, but there'd be nothing to counterbalance it. So it's fundamentally the scalar curvature that would coax the VEV (vacuum expectation value) on the augmented torsion out of the vacuum to have a non-zero level. And if you pumped up that sphere and it smeared out the curvature, which you can't get rid of because of topological considerations—let's say from Chern-Weil theory—you would have a very diffuse, very small term. And that term would be the term that was playing the role of the cosmological constant.

So in other words, just to recap, starting with nothing other than a four-manifold, we built a bundle \(U\). The bundle \(U\) had no metric, but it almost had a metric. It had a metric up to a connection. There was another bundle on top of that bundle called the chimeric bundle. The chimeric bundle had an intrinsic metric. We built our spinors on that. We restricted ourselves to those spinors. We moved most of our attention to the emergent metric on \(U^{14}\), which gave us a map between the chimeric bundle and the tangent bundle of \(U^{14}\). We built a toolkit, allowing us to choose symmetric field content, to define equations of motion on the cotangent space of that field content, to form a homogeneous vector bundle with the fermions, to come up with unifications of the Einstein field equations, Yang-Mills equations, and Dirac equations. We then broke those things apart under decomposition, pulling things back from \(U^{14}\), and we found a three-generation model where nothing has been put in by hand, and we have a 10-dimensional normal component, which looks like the \(\text{Spin}(10)\) theory.

Now there's a huge problem in the spinorial sector, which I don't know why more people don't worry about, which is that spinors aren't defined for representations of the double cover of \(\text{GL}(4, \mathbb{R})\), the general linear group's effective spin analog. And as such, if we imagine that we will one day quantize gravity, we will lose our definition, not of the electrons but, let's say, of the medium in which the electrons operate. That is, there will be no spinorial bundle until we have an observation of a metric. So, one thing we can do is to take a manifold \(X^d\) as the starting point and see if we can create an entire universe from no other data—not even with a 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 we will work over a bundle that is of a quite larger, quite a bit larger dimension.

So for example, if \(X\) was four-dimensional, therefore d equals 4, then \(Y\) in this case would be \(d^2\), which would be \(16 + 3d\), which would be 12, making 28, divided by two, which would be 14. So in other words, a four-dimensional proto-spacetime—not a spacetime, but a proto-spacetime with no metric—would give rise to a 14-dimensional observerse portion called \(Y\). Now I believe that in the lecture in 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 years.

So we have two separate spaces, and we have fields on the two spaces. Now what I'm going to do is I'm going to refer to fields on the \(X^d\) space by Hebrew letters. So instead of \(g_{\mu \nu}\) for a metric, I just wrote \(ℷ_{נ,מ}\) (''gimel mem nun''). And the idea being that I want to separate Latin and Greek fields on the \(Y\) space from the rather rarer fields that actually live directly on \(X\).

Most fields—and in this case we're going to call the collection of 2-tuples \(\omega\), so inside of \(\omega\) that will be, in the first tuple we'll have \(ϵ\) and \(ϖ\), written in sort of a nontraditional variation of how we write this symbol for \(\pi\); in the second tuple, we'll have the letters, \(\nu\) and \(\zeta\), and I would like them not to move because they honor particular people who are important. So most fields, in this case \(\omega\), are dancing on \(Y\), which was called \(U\) in the lecture, unfortunately. But, they are observed via pullback as if they lived on \(X\). In other words, if you're sitting in the stands, you might feel that you're actually literally on the pitch, even though that's not true. So what we've done is we've taken the U of Wheeler, we've put it on its back, and created a double-U structure.

And the double-U structure is meant to say that there's a bundle on top of a bundle. Again, Geometric Unity is more flexible than this, but I wanted to make the most concrete approach to this possible for at least this introduction. Sometimes, we don't need to state that \(Y\) would, in fact, be a bundle. It could be an immersion of \(X\) into any old manifold, but I'd like to go with the most ambitious version of GU first.

So, the two projection maps are \(\pi_2\) and \(\pi_1\), and what we're going to say up top is that we're going to have a symbol \(Z\) and an action of a group \(\rho\) on \(Z\), standing in for any bundle associated to the principal bundle, which is generated as the unitary bundle of the spinors on the '''chimeric tangent bundle''' to \(Y\). It's a bit of a mouthful, but the key issue was that on the manifold \(Y\), there happens to be a bundle which is isomorphic, non-canonically, to the tangent bundle of \(Y\), which has a definite and canonical metric. And in fact, that carries the spinors. So this is the way in which we get spinors without ever having to choose a metric, 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.

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 \(X\).

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 0-forms tensored with spinors direct sum 1-forms tensored with spinors all up on \(Y\) as the fermionic field content.

''* Note: Where Eric mistakes ''\(\alpha\)'' for ''\(\aleph\)'' it is written correctly in the transcript for the sake of clarity and parity with the diagram.''

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 two different origin stories, which is a little bit like Lilith and Genesis. We can't easily say we have a unified theory if spacetime and the \(\text{SU}(3) \times \text{SU}(2) \times \text{U}(1)\) group that lives on spacetime have different origins and cannot be related. In this situation, we tie our hands and we have no choice over the group content.

So just to fix bundle notation, we let \(H\) be the structure group of a bundle \(P_H\) over a base space \(B\). We use \(\pi\) for the projection map. We've reserved the variation in the \(\pi\) 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. We use \(H\) here, not \(G\), because we want to reserve \(G\) for the inhomogeneous extension of \(H\) once we move to function spaces.

So with function spaces, we can take the bundle of groups using the adjoint action of \(H\) on itself and form the associated bundle, and then move to \(C^\infty\) sections to get the so-called gauge group of automorphisms. We have a space of connections, typically denoted by \(\mathcal{A}\), and we're going to promote \(\Omega^1(B,\text{ ad}(P_H))\) to a notation of script \(\mathcal{N}\) as the affine group, which acts directly on the space of connections.

And then we have an action of \(\mathcal{G}\), that is the inhomogeneous gauge group, on the space of connections, because we have two different ways to act on connections. We can either act by gauge transformations or we can act by affine translations. So putting them together gives us something for the inhomogeneous gauge group to do.

We then get a bi-connection. In other words, because we have two different ways of pushing a connection around, if we have a choice of a base connection, we can push the base connection around in two different ways according to the portion of it that is coming from the gauge transformations of curly \(\mathcal{H}\) or the affine translations coming from curly \(\mathcal{N}\). 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 when we find an interesting embedding of the gauge group that is not the standard one in the inhomogeneous gauge group.

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 1-form.

Now, in sector III, there are payoffs to the magic beans trade. The big issue here 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 on top of that. We built spinors off of the chimeric tangent bundle, and we have not made any other choices. So we're dealing with, I think it's \(U^{128}\), \(U^{2^7}\). That is our structure group, and it's fixed by the choice of \(X^4\), not anything else.

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 \((d - 3 + i)\)-forms. So for example, you would map a 2-form to \((d - 3 + i)\). So if \(d\), for example, were 14, and i were equal to 2, then 14 minus 3 is equal to 11 plus 2 is equal to 13. So that would be an ad-valued \((14 - 1)\)-form, which is exactly the right place for something to form a current, that is, the differential of a Lagrangian on the space.

So, given that you've been on a long journey, here is something of what Geometric Unity equations might look like. So in the first place, you have the swerved curvature, the shiab applied to the curvature tensor. That, in general, does not work out to be exact, so you can't have it as the differential of a Lagrangian. And in fact, you know, we talked about swirls, swerves, twirls, eddies—there has to be a quadratic '''eddy tensor''', that I occasionally forget when I pull this thing out of mothballs, and the two of those together make up what I call the '''total swervature'''. And, on the other side of that equation, you have the displaced torsion, which I've called the '''displasion'''. And to get rid of the pesky, sort of, minus sign and Hodge star operator... This would be the replacement for the Einstein equation, not on \(X\) where we would perceive it, but on \(Y\) before being pulled back onto the manifold \(X\).

So, a condensation of that would be very simple. In simplest terms, we would be saying that the swervature is equal to the displasion, and at least in this sector of the four main equations of theoretical physics, this would be the replacement for the Einstein equations, again on \(Y\) before being pulled back to \(X\).

So, if we review the three identities here, we see that if we have a space \(V\), thought of like as a tangent bundle, and then you have spinors built on the tangent bundle, when you tensor product the tangent bundle with its own spinors, it breaks up into two pieces. One piece is the so-called Cartan product, which is sort of the sum of the highest weights, and the other is a second copy of the spinors gotten through the Clifford contraction. So, that's well known, but now what I think fewer people know—many people know that the spinors have a sort of an exponential property. That is, the spinors of a direct sum are the tensor product of the spinors of the two summands of the direct sum. So that's a very nice sort of version of an exponential—an exponential would take a sum and turn it into a product.

What happens when you're trying to think about a tangent space in \(Y\) as being broken up into a tangent space along an immersed \(X\), together with its normal bundle? So imagine that \(X\) and \(Y\) are the tangent space to \(X\) and a normal bundle. So the Rarita-Schwinger piece—that is, the spin-3/2 piece—has a funny kind of an almost exponential property. That is, the Rarita-Schwinger content of a direct sum of vector spaces is equal to the Rarita-Schwinger of the first, tensor producted with the ordinary spinors in the second, direct sum with the ordinary spinors in the first, tensor producted with the Rarita Schwinger content of the second summand. But then there's this extra interesting term, which is the spinors on the first summand tensor producted with the spinors on the second summand.

Now recalling that, when I started my career, we did not know that neutrinos were massive. And I figured that they probably had to be massive because I desperately wanted a 16-dimensional space of internal quantum numbers, not 15, because my ideas only work if the space of internal quantum numbers is of dimension \(2^n\). And, one of my favorite equations at the time was \(15 = 2^4\). Not literally true, but almost true. And thankfully in the late 1990s, the case for 16 particles in a generation was strengthened when neutrinos were found to have mass. But that remaining term in the southeast corner, the spinors on \(X\) tensor spinors on \(Y\) looks like the term above it in line 2.15. And that, in fact, is the third generation of matter, in my opinion. That is, it is not a true generation. It is broken off, and would unify very differently if we were able to heat the universe to the proper temperature.

This is the Einsteinian replacement, and it must be pulled back to \(X\). That's the first thing.

The metric does multiple duties. Here, it's the main field in this version of GU, with the sort of strongest assumptions, as field content that is originally on \(X\), whereas most of the rest of the field content is on \(Y\), but it also acts as the observer pulling back the full content of \(Y\) onto \(X\), to be interpreted as if it came from \(X\) all along, generating the sort of illusion of internal quantum numbers.

And I should say that the Pati-Salam theory, which is usually advertised as, I think as \(\text{SU}(4) \times \text{SU}(2) \times \text{SU}(2)\), is really much more naturally \(\text{Spin}(6) \times \text{Spin}(4)\) when the trace portion of the space of metrics is put in with the proper sign if you're trying to generate the sector that begins as \(X(1,3)\). Remember \(X^d\), where \(d = 4\), is the generic situation. But you have all these different sectors. I believe that these sectors probably exist if this model's correct, but we are trapped in the \((1,3)\) sector, so you have to figure out what the implications are for pushing that indefinite signature up into an indefinite signature on the \(Y\) manifold. And, there are signatures that make it look like the Pati-Salam rather than directly in the \(\text{Spin}(10)\), \(\text{SU}(5)\) line of thinking.