The Geometric Unity Iceberg... Oh Boy.

Welcome to the Iceberg of Geometric Unity, a comprehensive and technical edition. This iceberg format is one that will guide you through the intricacies of this theory of everything, beginning with foundational concepts and then advancing into the more sophisticated hinterlands. In this special episode, we rigorously explore Eric Weinstein's geometric unity, moving beyond metaphorical explanations to engage directly with the mathematical underpinnings of the theory. If you skip the rigor and opt for explanations aimed at a 5-year-old, well, I'm not sure how many 5-year-olds you've spoken to, but sure, it's cute, you can't explain what a Dirac operator

is to them outside of making a TikTok video that gives the impression of knowing without actually understanding. My name's Curt Jaimungal, and on Theories of Everything, I use my background in mathematical physics from the University of Toronto to explore the unification of gravity with the Standard Model and have also become interested in fundamental laws in general as they relate to explanations of some of the largest philosophical questions we have, such as what is consciousness and how does it arise? In other words, it's a peregrination into the all-encompassing nature of the universe. Today we'll cover the

abstruse math of bundle theory, of index theory, of course the Standard Model with general relativity. Just so you know, this episode took a combined 250 hours across three different editors and several rewrites on my part. It's on par with the most labor that's gone into any single Theories of Everything video, comparable to the iceberg of string theory, and that's saying something. If you're confused at any point by the exposition, don't worry, GU may seem like a formidable subject. That's what I thought before I started reading what Eric's write-ups were. And then I realized that it

only uses standard notions in differential geometry, the primary challenge of which lies in the novel constructions and the terminology introduced by Eric, yet these are accessible to those with a graduate-level understanding of mathematical physics. Even if you're not at that level, don't worry because I'll explain and I'll re-explain several points. First, I'll provide a quick overview of geometric unity, followed by an overview of modern physics. Then, I'll give a more detailed explanation of GU to thoroughly explain the derivations. Finally, I'll relate it back to modern physics. There are timestamps in the description to help navigate

around. Don't worry if you get lost, this video is meant to be watched and re-watched, where each time you'll glean something new. So let's begin with the first layer of the iceberg. Layer 1. Firstly, let's ask, what is a theory of everything? Most of the lay public thinks that it has something to do with quantum gravity. However, that's just a single approach to reconciling general relativity with the quantum world. Furthermore, quantum gravity isn't a TOE, it's not a theory of everything. A theory of everything in the physics sense is a framework that encompasses both the

standard model of particle physics as well as general relativity. In other words, it's not just about something being quantum. You can then ask the question, okay, well what's the minimal input that such a model has in order to recover the particles that we see, the gauge groups, the Lorentz group, the Yang-Mills action, and other ingredients of modern physics? There's always the temptation to make your theory more tortuous in terms of what's added to it as elements to the stew. But the goal of a TOE has always been an elegant one. This means that you start

with a tiny set of assumptions, and you recover a plethora. Now this iceberg isn't going to be hand-wavy or vague. It will give you analogies, yes, to help you if you don't understand the math. But if you do know these topics on screen, then that's enough to understand all of the conclusions, the derivations, and the claims of geometric unity. By the end of this iceberg, you'll not only be familiar with geometric unity, but also with the current state of fundamental physics as a whole. I'll explain GU in 4 words, in 30 words, and then in

20,000 words. In 4 words, Einstein knows Pati-Salam. Not terribly informative to people without a physics background. However, you can see my notes here on this Substack on eschewing simplistic explanations, as you only get to choose two of these three, simplicity, accuracy, and succinctness. Now slightly more accurate is the 30 word explanation, General Relativity grand unifies the standard model's first generation by pulling back Weyl spinor from the space of metrics after trace-reversing the Frobenius metric on the fibers. Again, that's a handful, that's actually only 28 words, and that will make sense to someone who has a

differential geometric background, but maybe not to you yet. Now the 20,000 word explanation is the rest of this iceberg. One problem is that there are three legs to this mathematical physics stool, geometry, algebra, and analysis, or in other words, calculus. The issue with quantum field theory is that it developed in an unbalanced manner. It's predominantly analysis. And we discovered super late that we'd been neglecting certain topological, geometric, and algebraic aspects. Starting in the 1970s with Jim Simons, for the last 50 years, people like Witten, Segal, Quillen, Singer, Atiyah, Hitchin, Donaldson, Dan Fried, C.N. Yang, and

Alvarez-Gaume have been making quantum field theory more geometric. When you're taught quantum field theory in graduate school, it's generally from the point of view of effectively a generalization of multivariate calculus. But this tool of analysis is too crude of a tool to bear the responsibility of advancing fundamental physics. Now I'll give some simple examples. How do you avoid issues with pseudotensors by ensuring that physical quantities are tensorial and coordinate-independent? Or number two, how do you ensure that the time evolution of a quantum state preserves the non-negativity of the probability density during propagation? Number three, how

can you tell if you have an anomaly in quantum field theory? Number four, how do non-local spectral contributions arise in ostensibly local theories? And number five, of course, how do you formulate quantum theory on curved spaces? None of these considerations that I've just mentioned are natural in an analytic framework in analysis, but geometric principles ensure that all of these conditions are met. I won't say the answers now, but I will address them later in the iceberg. Needless to say, it's exactly the same issue where phenomenon that are difficult to prove regarding convergence and analyticity in

the real case become completely obvious or even trivial when you extend to the complex case. For now, let's look at the current state of fundamental physics. Hi everyone, hope you're enjoying today's episode. If you're hungry for deeper dives into physics, AI, consciousness, philosophy, along with my personal reflections, you'll find it all on my Substack. Subscribers get first access to new episodes, new posts as well, behind the scenes insights, and the chance to be a part of a thriving community of like-minded pilgrimers. By joining, you'll directly be supporting my work and helping keep these conversations at

the cutting edge. So click the link on screen here, hit subscribe, and let's keep pushing the boundaries of knowledge together. Thank you and enjoy the show. Just so you know, if you're listening, it's C-U-R-T-J-A-I-M-U-N-G-A-L dot org. CURTJAIMUNGAL.org. What's on screen are terms which are the ingredients of modern physics. In the same way that you have flour, sugar, eggs, butter, and milk to form a cake, the following are ingredients of the universe as we know it. The universe combines these all, but we don't know how. And further, we don't know of a source that could give

rise to all of these. Is there something like the four simple base pairs of DNA that give rise to the protein bonanza that we call life? Let me go through these one by one. Firstly, we have Einstein. Now specifically, this term represents the Ricci curvature tensor, defined as the contraction of the Riemann tensor. Most people don't realize that this is quite odd, as the Riemann tensor has its antisymmetry in the last two factors, coming from it being a two-form, whereas the first two factors have antisymmetry coming from the Lie algebra, which is actually after you

pull down the I with the metric. More on this later, but intuitively, the Ricci scalar measures how volume changes in a curved spacetime, capturing the gravitational effects of matter. Next is Yang-Mills-Maxwell, and this term is the Yang-Mills action for gauge fields, where F-A is the field strength, also known as the curvature, associated with the gauge connection A, given by this formula here. And this inner product is something called the killing form on the Lie algebra of the gauge group. Intuitively, this term represents the energy stored in the gauge fields. The next term describes the action

for fermionic fields, psi and psi-bar, where psi-bar is the Dirac adjoint of psi, and this D with a slash is the Dirac operator coupled to the gauge connection. The gamma matrices here satisfy the Clifford algebra, and this guy describes how particles like electrons and quarks move and interact with gauge fields within spacetime. Confusingly, this psi here isn't the same psi as a wave function. This is just one of the many ambiguities in physics. You're just going to have to get used to it. The next is Higgs, and this governs the dynamics of the Higgs field,

and through the potential here, it gives mass to gauge bosons like the W and Z ones via spontaneous symmetry breaking when this Higgs field acquires a vacuum expectation value. And this next factor here is when the Higgs field acquires a VEV of vacuum expectation value, this time giving rise to mass for fermions, which is in proportion to the strength of the Yukawa coupling. At this point, don't worry if you're confused. Again, this video is meant to be watched and re-watched, and furthermore, this is standard physics, so nothing here is specific to geometric unity. All of

this will make sense, and I'll explain and re-explain, just keep watching. Now this next term, spin-1-3, I could have also said SL2C technically, which is the double cover of the proper orthochronous Lorentz group SO plus 1 comma 3, which is actually necessary for representing spinor fields, and anyhow, this is what allows us to correctly describe particles with half integer spins. Then there's this, which you hear of plenty, SU(3) cross SU(2) cross U(1). This product group is the gauge symmetry group of the standard model, where roughly speaking, SU(3) corresponds to the strong force, also known as

quantum chromodynamics, and SU(2) corresponds to the weak force, technically isospin, and U(1) corresponds to electromagnetism, though technically hypercharged. Next we have the family quantum numbers. These will be outlined later in this talk, but intuitively speaking, this space encodes all the intrinsic properties that distinguish different types of particles within a particular generation. Now next, speaking of these generations, this denotes that there are three generations of matter. Now some people call them three families. I've also heard it called three flavors of matter, confusingly again. This field here, psi, can be decomposed as follows. That's what this symbol

here means, which represents these three generations, or families, or flavors, or whatever you call it. This accounts for the existence of particles like the electron, the muon, and the tau, which are all similar in behavior, but they differ in mass. And lastly, please don't make me pronounce these, the CKM matrix explains why quarks can change types or flavors during weak decays, leading to phenomenon like the decay of a strange quark into an up quark. And the PMNS does something similar, but for neutrino mixing, or oscillations as they're sometimes called. There are two key equations which

can be derived from these terms, but I'll include them on screen anyhow for completeness. So one is the Einstein field equation, and the other is the Yang-Mills equation. Since the Einstein tensor on the left hand side can be thought of as an operator that acts on the Riemann tensor, you can rewrite it into this form. Now I'm going to write both of these equations suggestively, as Weinstein suggests, in a suggestive manner. If you squint, you'll see an analogy here between Yang-Mills and Einstein. Both theories involve operations acting on curvature tensors to map them back into

field variables and sources. However, one huge difference is how these operators interact with symmetries. In Yang-Mills theory, gauge covariance is preserved under gauge transformations due to the structure of the covariant derivative and field strength. In contrast, the linear contraction operator, which I've denoted P, which is used to form the Einstein tensor, denoted G, in general relativity, doesn't commute with gauge transformations. I'll leave my proof of this on screen. For those who don't believe me, you can just pause, but also there are show notes in the description with a full breakdown of everything in this video.

This, by the way, is partly what Eric means when he talks about the twin origin problems, but more on that later. In Yang-Mills theory, this involves Lie algebra-valued one-forms, which are also known as the gauge potentials or connections, while in general relativity it involves symmetric rank two tensors, or the metric tensor as you may know it. What other analogies exist, and how can we make these analogies not only precise, but derived? Furthermore, how can this be done with the smallest set of simple assumptions possible? That's what geometric unity aims to achieve. Layer two. Geometric unity

begins with the four-dimensional manifold x4, and then it constructs a bundle of metrics called y14. This 14-dimensional space comes about naturally. How? Well, at each point in the original manifold, you can assign a symmetric bilinear form, which is a metric of ten independent components plus, of course, the four dimensions of the base space to account for each point in the manifold. Rather than choosing a metric, which is what's ordinarily done, GU instead works with the space of all possible metrics simultaneously point-wise. This is extremely important because traditionally, spinors require a metric for their definition, and

this creates a chicken-and-the-egg problem with quantum gravity. How is it that matter can exist for fermionic matter when a metric isn't well-defined? We'll speak more about this later. The key innovation is what Eric calls the chimeric bundle over the space, constructed as follows, where v is the vertical bundle, so changes in the metric at a point, and h star is the dual of the horizontal bundle, so movement in the base space. The vertical space inherits a natural metric via something called the Frobenius inner product, which is a fancy word for this formula on screen here.

Again, all of this will be explained with examples. Now, what about that horizontal space? Well, it gets its structure from a connection choice. This allows us to define spinors without first choosing a metric on x4. Now, the structure group of GU is precipitated from spin 7,7 or spin 5,9, and this signature, by the way, is because we have this decomposition here between the vertical and horizontal components. This leads to a complex spinor representation. This dimension 64, by the way, comes about because the spinor representation of spin 7,7 has dimension 2 to the 7, which is

128, and that splits into two 64-dimensional pieces. Note, these Weyl spinor are not of definite signature. Importantly, they are two Weyl spinor of split signature, 32,32, and then of course another 32,32. Eric then introduces something called the inhomogeneous gauge group, and that combines gauge transformations, which are H, with so-called translations in the space of connections. That's just an analogy, and we call that N. This structure parallels how the Poincaré group combines Lorentz transformations with spacetime ones, but this time we're in the context of gauge theory. Later on, Eric then introduces something called the augmented torsion

tensor. Again, this is plenty of jargon and new terminology. It does need to be introduced because when you name something, you can then use that concept on its own rather than having to construct it from scratch every single time. Now, this augmented torsion tensor, which again is on screen but will be explained more later, is what combines aspects of gravity and maintains that gauge covariance that we talked about before. This is what resolves Einstein's quote-unquote twin origins problem, or more specifically, Eric's coining of the twin origins problem. Now, where does the standard model's gauge group,

SU(3) x SU(2) x U(1), come about? Well, what Eric does is instead of using a compact group, he uses a larger non-compact one, which has as a maximal compact subgroup the standard model. In other words, the gauge group of the standard model comes about not by an arbitrary choice, certainly not three arbitrary choices, but rather as a necessary consequence of the geometry itself. The three generations of matter come about from decomposing spinor-valued forms on this larger space Y14, with something special happening for the two generations, and then the third is more like a remnant. So

the first two come about from what's on screen here, a function, a spinor-valued field, so spinors, and then a spinor-valued one form. However, the third generation is actually a Rarita-Schwinger field in the decomposition of Zeta. And all of this comes about from minimal assumptions on the original manifold X4. Again, Eric's theory, geometric unity, makes unification not by adding new structures, but via recognizing that the ingredients of modern physics, remember that Einstein equation, Yang-Mills theory, fermions, Higgs, etc. They all come about from geometry, and moreover, from X4 itself. Don't be like the economy. Instead, read The

Economist. I thought all The Economist was was something that CEOs read to stay up to date on world trends, and that's true, but that's not only true. What I found more than useful for myself, personally, is their coverage of math, physics, philosophy, and AI, especially how something is perceived by other countries and how it may impact markets. For instance, The Economist had an interview with some of the people behind DeepSeek the week DeepSeek was launched. No one else had that. Another example is The Economist has this fantastic article on the recent dark energy data, which

surpasses even Scientific American's coverage, in my opinion. They also have the chart of everything. It's like the chart version of this channel. It's something which is a pleasure to scroll through and learn from. Links to all of these will be in the description, of course. Now, The Economist's commitment to rigorous journalism means that you get a clear picture of the world's most significant developments. I am personally interested in the more scientific ones, like this one on extending life via mitochondrial transplants, which creates actually a new field of medicine, something that would make Michael Levin proud.

The Economist also covers culture, finance and economics, business, international affairs, Britain, Europe, the Middle East, Africa, China, Asia, the Americas, and, of course, the USA. Whether it's the latest in scientific innovation or the shifting landscape of global politics, The Economist provides comprehensive coverage, and it goes far beyond just headlines. Look, if you're passionate about expanding your knowledge and gaining a new understanding, a deeper one, of the forces that shape our world, then I highly recommend subscribing to The Economist. I subscribe to them, and it's an investment into my, into your, intellectual growth. It's one that

you won't regret. As a listener of this podcast, you'll get a special 20% off discount. Now you can enjoy The Economist and all it has to offer, for less. Head over to their website, www.economist.com slash TOE, T-O-E, to get started. Thanks for tuning in, and now let's get back to the exploration of the mysteries of our universe. Again, that's economist.com slash TOE. Step 1, the Observer's Construction. Let's begin with what's familiar, that is a four-dimensional manifold. This is that tiny input. It's the priming of the pump necessary to construct the rest of geometric unity. There's

no metric needed. There's no connection. There's no additional structure yet, other than being spin, or orientable, or connected, etc. But there are generalizations of GU without these facets. I personally find it easier to assume these. However, instead of working with X4 directly, Eric constructs what's called the Observer's. Now this Observer's is actually a triple, even though before I denoted it simply as Y14, it's technically the base space X4, the total space Y14, and then the map pi that projects down between them. So recall, at every point in the manifold, there's a fiber, and this consists

of all possible metrics that one could have had at that point. Now, let's be precise. What the heck is meant when I say all possible metric tensors at this point? Well, okay. At any point, X in X4, a metric tensor is a symmetric, non-degenerate, bilinear form, G sub X, say, which takes in two vectors at that same point, and then outputs, linearly, something from the real numbers. The space of such metrics forms a 10-dimensional manifold, because a symmetric 4x4 matrix has 10 independent components, and then you get the extra 4 because of the base space.

So what's different about this construction? Rather than fixing a metric, which is what we ordinarily do in general relativity, we consider all possible metrics simultaneously, and then this allows one to study, how would physics look if you didn't make any specific choice of a metric? Eric's maneuver, which I haven't seen done before, is instead of quantizing gravity directly on the base space X4, which, as I'm sure you know, has problems with renormalization. Instead, Eric works on this larger space, Y14, and then he pulls back information to X4 via what he calls observation maps. So I

should spell out what a pullback is and what an observation map is. Eric, confusingly to me initially, calls the local sections of this bundle observations. So that's all it is. It's nothing more than you look at a local patch of the manifold and you think, Okay, what am I smoothly going to assign as a metric from the larger space Y14? Okay, let's clarify what's meant by pullback. Anytime you have a map, let's say something that goes from A to B, and then you have some structure on B in that target space, whether it's a differential

form, a tensor field, or a function, a pullback is the corresponding structure on A, which you can actually define because you already have a map which goes from A to B. And the definition is on screen here. Step two, the frame bundle and its double cover. Okay, let's construct one of the other essentials, the frame bundle over our base space. At each point in our base space for this frame bundle, the fibers consist of all possible frames, or bases, for the tangent space. The group structure here is GL4R. Nothing here is unique to GU, this

is standard in differential geometry. But what exactly is a frame? Well, again, at each point, a frame is an ordered basis for the tangent space. And I'll give you some examples. Here's the standard basis that you know probably by X, Y, Z, and T in some coordinate system. And then another frame could look like so. In differential geometry, by the way, your bases are vectors, which are differential operators. These frames are related to one another via an element of GL4R. In other words, if you had a base and another base, there exists a transformation between

them, which is a member of GL4, in order to get you from one to the other. But here we hit our first snag. GL4 does not have a finite dimensional spinor representation. Why? Well, if you take an element of it, say A, and you square it, and you get negative I, any finite dimensional representation A would need to satisfy that A also squares to negative I, which is impossible over the reals. This is an extremely important point, so it deserves some elaboration. Let's consider this matrix, which satisfies that condition of A squared equals minus I.

The eigenvalues of this matrix would also need to be square roots of minus one, which don't exist in R. So to fix this, we need to do something called passing to the double cover. The map is on screen here. And then we do something called the lifted frame bundle, which has the structure group of this double cover of GL4R, also known as the meta-linear group. Step three, we construct observation maps. Now what exactly do we mean by observation map? I understand this is plenty of new terminology, as we have an observer, so now we have

observation maps, and you may have heard of the Shiab operator, or the chimeric bundle. Eric is evidently an enormous fan of these neologisms, and this is just something we'll have to get used to. There is a method to the madness, though, because when we give a name to something, it means that we can then pick it out again and reference it without having to construct it anew each time. So back to the question, what are observation maps? These are local sections, so some map from a neighborhood of a point X going into the larger space

Y, but why do we call these observations? More on this later, but for now, you can think of it as how one measures geometry. Each of these iotas, each of these maps, picks out a specific metric at each point in its domain. For any such of these maps, we get a pullback map that brings geometric data from Y back down to X4. Specifically speaking, if you have any tensor, say omega, on Y, then you can pull back the corresponding tensor on X4. Think of this like a probe into the space of all possible metrics. The

spin 1,3 bundle comes about as follows. At each point little y in the larger space big Y, we have a metric g little y on X4. This metric determines a principle SO 1,3 bundle of orthonormal frames. The insight from Eric is that this bundle naturally admits a canonical double cover by a spin 1,3 bundle by this construction on screen here. The other insight from Eric is that this bundle has a 14 dimensional matrix representation broken up into the 10 vertical and then 4 horizontal, actually asterisk on that, dual horizontal. And that is the bundle that

we're calling C, the chimeric bundle. Now this C has a metric on it, and part of it is actually canonically isomorphic to the original 14 dimensional tangent bundle. The other part requires some extra structure. By the way, when someone says a bundle admits something, they use this word admit. What's meant is that there then exists something. In this case, when we say that a bundle admits a double cover, we mean there exists a bundle map that's locally two to one. More precisely, for any point in here, there are exactly two points in here that map

from here to here. And this mapping here preserves the bundle structure. This is analogous to how the complex function, say z, which maps to z squared, gives a two to one map from the circle to itself. Why do we need this double cover? Because SO1,3 doesn't admit spinor representations. More precisely, there's no finite dimensional representations of SO1,3 that, when restricted to rotations, gives the spin half representations of SO3. However, spin 1,3 does admit such representations. And this is how Eric will eventually describe fermions. Step four, we construct the tangent bundle and its dual. At each

point in the larger observer's capital Y, we have a tangent space, which has the same amount of dimensions, namely 14. And again, 10 of them come from the symmetric matrices at each point, which represent all the possible metrics. And we have the other four dimensions coming from the base space below. The dual space consists of linear functionals from the tangent space to whichever is your underlying field, namely the reals in this context. To be specific, at a particular point in the observer's Y, it can be decomposed as the base space, x, and then the metric,

g. Specifically speaking, you can see this isomorphism on screen here, where this guy represents the symmetric 0,2 tensors. And recall that in four dimensions, symmetric matrices have 10 independent components. Now, you may ask, why do we need both the tangent space and its dual tangent space? The answer lies in how Eric defines field content later. Some fields will naturally live in one bundle and others in its dual. In fact, we'll see that without a metric on Y, these bundles are not naturally isomorphic. And this is what leads Eric to later construct the chimeric bundle. I'll

be using this phrase field content, so I should define it. In physics, when people talk about field content, we just mean the collection of fields that appear in the theory. So these could be a scalar field, like the Higgs, for instance. It could also be a vector field, like the electromagnetic potential. It could also be tensor fields, like the metric. All the previous examples were tensor fields as well. Or spinor fields, like electrons. The word content just means collection or set. It's like the complete list of ingredients in our theory. Step five, the vertical and

horizontal bundles over Y. At each point in the observer's Y, we have a splitting of the tangent space, decomposed as follows. The V is the vertical space, which is tangent to the fiber. And the H is the horizontal space, which is non-canonical, meaning one needs to make a choice. Precisely speaking, the vertical bundle is defined as follows. To visualize this, you can think of the vertical subspace at Y as the space of all possible velocities for the changing metric at X, while staying in the same fiber over that X. In other words, if H of

T is a path in this metric space where H of 0 is G, your initial G, then taking the derivative with respect to time and setting it to 0 is an element of the vertical tangent space. This is all standard in differential geometry. Step six, the Zorro construction is named for its zigzag pattern. And this is what provides a canonical manner of defining the horizontal distribution. Now when I say canonical here, what I mean, or what is generally meant by canonical, is that there's a natural choice or a natural option that doesn't depend on arbitrary

decisions. It's similar to how when you have a vector space, there is no canonical basis. However, you do have a canonical dual space. Here, the Zorro construction gives us a natural method to split TY. TY being the tangent space of Y, or the observer, or the metric bundle, in other words, without making loathsome arbitrary choices. Here's how it works. You'll see this construction here, which looks like the backward Z of Zorro, which is why Eric denoted it the Zorro construction. And this funny symbol here is a gimel, which is a Hebrew letter. And then this

symbol here, which looks like an N, is an aleph. Note, I'm unfamiliar with using Hebrew letters in math unless it's aleph for cardinality. Now these two symbols on screen here look almost identical to me, which are gimel and beth, respectively. You may see me confuse these symbols throughout, but don't worry, because anytime they're referenced, they mean the same thing, namely, a section of the metric bundle. Note that the augmented torsion tensor is now called the displacement torsion tensor by Eric. I've also heard him call it distortion, but I'm going to continue to call it the

augmented torsion tensor for the remainder of this iceberg. To me, I wouldn't use these Hebrew letters, and in fact, I changed this gimel to the iota from before. The only difference is that the gimel is a global section, whereas the iota from before is a local one. Now the aleph represents the Levi-Cevita connection. G-sub-aleph is the induced metric on Y, and A-sub-aleph is the resulting connection on Y. However, I should point out that Eric uses this gimel symbol and the lowercase g to prevent confusion that was coming about from calling two different metrics on two

different spaces, both by the traditional G. Thus, I understand that the gimel and the aleph aren't there arbitrarily, as the way Eric sees it, all of the drama takes place on this larger observer. The reason Eric has this induction from the Zorro construction is that he wants the freedom to later not have a metric on the base space when not observing the system, for instance. And this is Eric's move to make the quantum metric make sense later. Let's break this down step by step. We start with a metric, which is a choice of a global

section on X4, akin to iota from before. This determines a unique Levi-Cevita connection, which Eric denotes as aleph on X4, and this comes about by the fundamental theorem of Riemannian geometry, nothing not standard here. Next, this aleph then introduces a metric on the larger space Y through the Frobenius inner product on symmetric matrices. And finally, the G of aleph determines its own Levi-Cevita connection, a sub-aleph on Y. This process gives one a canonical manner of splitting TY, so the tangent space, into the vertical and horizontal parts, without making arbitrary choices. Recall that the horizontal subspace

at Y is precisely the space of vectors that are deemed to be parallel, quote-unquote parallel, to X4, according to a sub-aleph. And a smooth choice of a horizontal subspace is the same as a connection, so to answer the question of why do we need this Zorro construction, it's because it gives us a canonical method to lift, quote-unquote lift, vectors from X4 to Y, the larger space, without making choices. The horizontal subspace is precisely the space of such lifted vectors. When I say lift, by the way, what I mean is we take something that's defined on

a lower space, and we find a corresponding object to it in a higher space, such that it projects downward to give us what we started with. More precisely, if this here is a fiber bundle, and we have a vector that's a tangent vector at, say, B, at the base space, then a lift of the vector from the base space is another vector in this larger space, such that when you project down, you get the same vector. Now the horizontal distribution, or the choice of horizontal subspace, gives us an approach to choose such lifts. Step seven,

the chimeric bundle. You'll notice we're introducing plenty of terminology. There's chimeric bundle, there's the observers. This isn't just jargon for jargon's sake. Instead, we're actually enhancing the clarity, because we're avoiding repetitive exposition, having to define these over and over. These terms will become familiar as we proceed, and recall this entire iceberg is meant to be watched and re-watched where you learn something new every single time. All right, let's define this beast. First of all, notice that if we take Y14 as its own bundle, the observers as its own entire bundle, and we take the tangent

space at a particular point in it, it can always be decomposed as follows, a vertical component and a horizontal one. You know from differential geometry, a choice of a horizontal subspace is a choice. That is the same as a connection, which then becomes something like curvature. However, in GU, you're always trying to minimize the amount of choices you make. You can even think of summing up GU as, if you cannot have one, then you must have them all. Anyhow, let's take a look at that Frobenius inner product that we referenced before, and let me just

give you an example of two matrices. So, this isn't actually from the bundle, it's just what I could write on screen, but let's imagine you have 1, 2, 2, 3, 0, 1, 1, minus 1. Well, you can just do the math and compute its Frobenius inner product, and it works out to minus 1. Now, this chimeric bundle differs in the second component, its H dual. It's not the horizontal bundle, but the dual of it. This asymmetry will turn out to be required for the theory's ability to unify gravity with gauge theory. Hi, Curt here. If

you're enjoying this conversation, please take a second to like and to share this video with someone who may appreciate it. It actually makes a difference in getting these ideas out there. Subscribe, of course. Thank you. Step 8. The Frobenius inner product. At each point in the large space Y, we need to define an inner product on the chimeric bundle. So, how do we do this without already having a metric on Y? The way Eric goes about doing this is by noticing that V inherits a natural metric via the Frobenius inner product. Again, for symmetric matrices

A and B, the Frobenius inner product is defined as follows on screen. Also notice that you can decompose the trace and the traceless parts of a symmetric 2 tensor. The trace part has dimension 1, and the traceless part has all the other dimensions. To see why this decomposition is reasonable and valid, consider that for any symmetric matrix A, you can always write it as follows, where you have a trace part and a traceless part. The traceless part has a signature, in this case, 3,6, whereas the trace itself contributes either a 1,0 or a 0,1, depending

on a choice. Here is where you make a specific choice. So we can either choose 4,6 or 3,7. For geometric unity, Eric chooses 4,6, for reasons that will become clear later, though there are generalizations of geometric unity with other choices. Step 9. Choosing a signature. Why is it that we have to be so careful about this signature? The answer is representation theory. The signature determines which spinor representations are possible, with our choice of 4,6 for the vertical space and 1,3 for the horizontal space from spacetime, we get a total signature of 7,7. Note, GU could

have had a signature of 5,9, and I believe Eric isn't sure which of these is the sector of our universe, in his theory, but for the remainder of this iceberg, we're going to select spin 7,7. But why these particular signatures, Curt, you may ask? Now the magic lies in representation theory, the representation theory of spin 7,7. When we have a metric of a signature, an arbitrary one, p,6, the real spinor representation has dimensions 2 raised to the floor of p plus q over 2. Now for 7,7, it gives us 2 to the power of 7.

This signature is essential because if you take the dimension of a spinor bundle, it equals this formula on screen here, where you get 2 raised to some floor function, in this case, it becomes 2 raised to 7, which equals 128, and this 128 splits not into c64 plus c64, but into the equally split signature that we talked about before, c32,32 and then another c32,32. Remember, these are Weyl spinor of split signature. Again, the same is also true for a spin 5,9 bundle, matching what's required for the standard model. Again, more later, this is somewhat of

a flyby overview. Step 10, defining spinors without a metric. Here's where everything so far comes together. The spinor bundle on the chimeric bundle decomposes as follows. Now what's so special about this decomposition? Well, it's the exponential property of spinors at work. For any direct sum of vectors, v, say, direct sum with w, as long as they have metrics, we have the following. Think of it like if you have a particle that can move in two independent directions, its quantum states multiply, rather than adding. This is the quote-unquote exponential property of spinors that Eric mentions. When

you pull this back via an observation map, iota, you get the following decomposition into tensor products. This decomposition eventuates in both spacetime spinors and internal quantum numbers exactly what's required for the standard model fermions. Now this alchemy happens because the vertical part v contributes internal symmetries, whereas the horizontal part, or more specifically the dual to the horizontal part, gives us spacetime properties. When one pulls this back to x4, the spinors decompose perfectly to give both the spacetime transformations of the particles and their internal quantum numbers like color and isospin. The implication? Eric has now constructed

spinors without choosing a metric on spacetime. Instead, they're fermented from the geometry of the observers. This resolves a long-standing chicken and egg problem in quantum gravity, which is how can matter exist between measurements if a metric is required for that matter to be defined? The answer, according to Eric, is that matter lives in the observers, namely that larger y-space where spinors exist prior to any choice of a metric on spacetime. Step 11. The structure group. The structure group of our theory originates from the spinor representations of spin 7,7. Why these dimensions? Recall, 7,7 comes about

from combining 4,6 with 1,3. And you can see that one is vertical and then the latter is horizontal. Let's pause and ask, why do these signatures even matter? Again, the 4,6 signature comes from the Frobenius inner product on symmetric matrices, while the 1,3 comes from spacetime. It's actually here that we make a choice. Because we could have had anything that summed to 4 and we're just choosing 1,3, this is one of the only places in this entire theory that I can see a choice being made. Now you may be wondering, why can't we just use

any group here? And the answer lies in representation theory. We would like a group whose representations can accommodate both gauge fields and fermions. The spinor representations of spin 7,7 do exactly that. Why do we need a structure group at all? When we started, we had a base space, an x4 manifold, and then we immediately got the frame bundle with the structure group gl4. But now we're working on y14, and we require a group that preserves the structure that we've built. The signature 7,7 is not arbitrary. It comes from the natural metric structure on the chimeric

bundle. Now you may be wondering, why can't we just use any old group here? We need a group whose representations can accommodate both gauge fields and fermions. The spinor representations of spin 7,7 do precisely that. So why 7,7 and not, say, 14? The key is that we would like to preserve the signature that comes about naturally from the vertical and horizontal decomposition. But here's the rub. The Frobenius inner product, which by the way, Eric sometimes calls the Frobenius metric, but I'm going to continue to call it the inner product, is given by this formula on

screen here in components. When you have 4x4 symmetric matrices, this naturally gives a 4,6 signature because the space of symmetric matrices compose into what's called a trace and a traceless part. Again, let me just be extremely specific. A 4x4 matrix actually gives different signatures depending on its signatures. Now in the 1,3 case, it naturally gives a 3,7. And this isn't often remarked on. Seeing this 4,6 signature here is extremely subtle because it requires remembering that you can do something called flipping the trace, which is what Eric says, and technically that's a trace reversal. And that's,

by the way, what Einstein did when he realized that his equation couldn't just be the Ricci tensor equal to the stress-energy tensor. Instead, you require the minus r over 2 correction. So let's take a look here. Recall the trace is a single number, which is why you can represent it by something of dimension 1, which is just the real numbers here, and then the rest becomes the traceless component. And then you wonder, well, what's a representation space of 7,7? And it's U64,64. And that 64, again, comes from half of 2 to the power of 7.

Thus, we preserve the Z2 grading on the spinors. Step 12, the principal bundle construction. Let's pause again. We've constructed this elaborate chimeric bundle with its spinor representations, but how does one actually implement gauge theory here? You may think, well, let's just use this spin 7,7 directly. However, there is a subtler approach that Eric takes. His idea is that spin 7,7 acts on that 128-dimensional complex vector space via its spinor representation. This space splits into two 64-dimensional pieces, as we've said before, and we wonder why is it we're using this unitary 64,64 rather than just U

of 128? It's because the spinor representation preserves a metric of signature 64,64. This brings us to our principal bundle on screen here, where we finally have a gauge group or a structure group, namely U64,64, and then we have what's called an associated bundle. This is from taking the frame bundle of the chimeric bundle and then doing what's called a lift to its double cover. We then use that row representation to convert the spin 7,7 transformations into U64,64 transformations. Step 13, the inhomogeneous gauge group. Okay, let's carefully build up our gauge structure. First, what do we

mean by gauge group? In physics, gauge groups represent redundancies in the descriptions of nature. In other words, they're different mathematical descriptions of the same underlying physical reality, which is unobservable. So for instance, you can measure your height in inches, you can measure it in centimeters, you can measure it in meters. It doesn't change your height. Those are just different representations of your height. Now, let's think about electromagnetism. You can add a gradient to the vector potential without changing the physics. That's a gauge transformation, but there's something deeper going on here that we're going to explore.

So let's clarify an important distinction in gauge theory. There are two related but distinct concepts that are often confused and so I'd like to spell it out. There's a choice of connection and then there's gauge transformations. The choice of connection, let's call it a one-form A on a principal G bundle, which has a total space of P going down to M, is a Lie algebra-valued one-form satisfying certain properties. The space of all of these connections, calligraphic A, is an affine space. Here's what we mean by affine space, by the way. A vector space has an

origin. Some people like to say a vector space has a preferred origin. Actually, anything that has an origin is not an affine space. So you don't even need to put the word preferred there. Okay, now what about gauge transformations? Gauge transformations are bundle automorphisms that preserve the fiber structure. Again, they're not just bundle automorphisms, which are often said. They have to preserve the fiber structure. They form a group, calligraphic H, acting on connections via what we see here. Now let's unpack this. The first term here, you can think of as a rotation of a sort

of the connection. And the second term is a correction to the connection that you need in order to preserve the transformational properties. In physics language, this ensures that the quote-unquote covariant derivative transforms properly. For a concrete example, let's just take the non-Abelian gauge theory case of QCD. If this A mu here is a gluon field and this G of X here is some varying space-time dependent element of SU(3), and that SU(3) comes from the SU(3) cross SU(2) cross U(1), then you have this situation over here being satisfied. And importantly, this describes the same physics, which

is why people call it a redundancy. So the choice of connection is not redundant, but the gauge transformation is. And that's something that's quite confusing when you first learn about it. Now here, this calligraphic H, by the way, is the smooth sections of the associated bundle P sub H with the adjoint action of H, where H, again, not calligraphic, is U of 64 comma 64. In fact, I'm going to stop just calling it H because that's confusing. I'm going to say U 64 comma 64 from now on. In simpler terms, we have a principal U

64 comma 64 bundle, and these are our gauge transformations of it. Again, I can be even more precise. Let's say we have a Lie group G, and we have its adjoint representation, where G goes into the automorphisms on the Lie algebra, which acts on the Lie algebra by conjugation. Then the adjoint bundle with the lowercase a is the vector bundle associated to the principal bundle P subscript U 64, 64 via this representation. The inhomogeneous gauge group is then this calligraphic G, which has the calligraphic H semi-direct producted with this calligraphic scripted N. Now this is

like translations in the space of connections, but not in a trivial manner. There's a multiplication rule here, and this is what shows how gauge transformations act on the translation part. It seems like Eric is creating this structure to parallel Poincaré's group's combination of Lorentz transformations with spacetime translations, but this time in the context of gauge theory rather than spacetime symmetry. It should be noted here that this is quite a large move. We're neither working on a space of metrics like Einstein, nor on a space of connections like Yang-Mills or Yang-Mills-Maxwell, as Eric says. Instead, what

I'll do is I'll overlay a dictionary here that you can take a screenshot of, but recall there are further notes in the Substack at c-u-r-t-j-a-i-m-u-n-g-a-l-dot-org-curtjaimungal.org, myname.org, so you can sign up there if you'd like the PDF. This is, as far as I can tell, Eric's interpretation of Einstein's unified field. Unified means algebraic in the eyes of GU. Step 14, defining a right action on connections. Now we need to understand how our gauge group acts on connections. For any connection a in this calligraphic a, and any element lowercase g in this calligraphic g, we define the

following. And you may ask, Curt, are you going to read that? No, it's tedious, just look, take a screenshot. Why this complicated formula? Well, let's be like Curtis Blow and break it down. This term here is the familiar gauge transformation. This next term represents translations in the space of connections. And this term here is a correction term to ensure the consistency. This right action was constructed to satisfy this specific property, making calligraphic a into a right calligraphic g space. Step 15, the augmented torsion tensor. So what happens when we try to combine gauge theory with

gravity? Well, there's several problems, but an immediate one is that gauge transformations don't play well with Einstein's way of contracting indices. However, what if we could find a quantity that transforms correctly under both? That's what Eric has found with the augmented torsion tensor, defined on screen as follows. And now this variational pi, at first I thought this was an omega, it's technically a variational pi, is a member of the adjoint valued one forms on the larger space y. And that's going to be the gauge potential. Whereas this variation on epsilon here, belonging to calligraphic h,

is a gauge transformation. Now the key property is that under gauge transformations, we have this formula on screen here, so let's break this down piece by piece. What is, firstly, this variational pi? Well, it's an adjoint valued one form, as I said before, which means it takes in vectors from ty, and it returns elements of the Lie algebra h of the gauge group h, which again is u64,64. And now this other term here looks complicated, however, it's just the gauge transformation of the base connection a0. By the way, when I use the word variational pi,

it's not in reference to anything about variational calculus. It's instead because Eric's notation is to use this symbol right here, and in LaTeX it's written with a slash and then var pi. It's a variation of pi in the same way that there's a variational phi, so var phi, or var phi as some people call it. Just letting you know, because you'll be hearing me use this term variational pi, and upon re-watching this iceberg for maybe the fourth, fifth time, too many times to count, I realize that this can be confusing. Now let me make a

concrete example. Let's say we have a u1 gauge theory, and our variational epsilon is e to the i theta. Then what we have is this on screen here. This is exactly the gauge-covariant combination that appears in electromagnetism. Now this is quite interesting, because it combines aspects of both gravity, so torsion, and gauge theory, so covariant derivatives, while maintaining gauge covariance. The way that I see it is it's like finding a method to make Einstein's gravitational theory speak the language of Yang-Mills theory. Step 16. The Shiab operator. How do we generalize Einstein's contraction of the Riemann

tensor in a gauge-covariant manner? The answer lies in what Eric calls the Shiab operator. That's spelled S-H-I-A-B. Now for a gauge-covariant 2-form, C, which some people call Casi, but I'm just going to say C, and it's not the letter C, it's this symbol on screen. You have this formula here, where the Shiab operator acts on this 2-form and gives you a Ricci-like term, which we'll explain more later, and then a scalar-curvature-like term. Again, more will be explained later. Note, the circle with a dot in the center is my notation for the Shiab operator. Eric actually

writes two concentric circles with a dot, but I wasn't able to get this to consistently render in my workflow, so just note that whenever you see this Shiab online outside of this video, it will likely have two circles and a dot. Let's further break it down, piece by piece. First, what is this operator doing? It's taking a gauge-covariant 2-form, and then it's returning another differential form that transforms, this time properly, under gauge transformations. So why do we need such an operator? Think about Einstein's theory. When you contract the Riemann tensor to get the Ricci tensor,

you're using the metric to raise and to lower indices. However, this operation doesn't respect gauge symmetries. It treats all copies of 2-forms in the same way. The Shiab operator fixes this by incorporating the gauge transformation epsilon explicitly. Here's how it works. These forms, phi, phi1, phi2, for instance, are invariant under the action of spin 7,7. When one conjugates them by an epsilon, we get objects that transform covariantly under gauge transformations. Now you also may ask, hey Curt, why these particular combinations of wedge products and Hodge stars? And again, the answer lies in representation theory. Just

as the Einstein tensor splits into trace and traceless parts, the Shiab operator respects a similar decomposition. However, this time it's one that's compatible with gauge structure. For a concrete example, consider the case of U(1) gauge theory. Here, the C would be the electromagnetic field strength, and the Shiab operator would give us something akin to the following. This is gauge invariant because F itself is gauge invariant in abelian theory. The general non-abelian case is more subtle, but the principle is analogous. Eric is building an operator that combines the metric and gauge structures consistently. The action principle

in GU takes a form reminiscent of both Einstein-Hilbert and Chern-Simons. Now you may look at this and say, this looks nothing like Einstein. It looks nothing like Chern-Simons. What's remarkable about this action? First, notice that it's first order in its derivatives, like Chern-Simons theory, but unlike the second order of Einstein-Hilbert action. Also notice that the field variables are omega, which actually comprise this epsilon and then this variational pi here, where epsilon is a gauge transformation and pi is a gauge potential. The first term combines the augmented torsion tensor with the curvature through the Shiab operator.

This generalizes both the Einstein-Hilbert term and the Yang-Mills term. This second term is like a mass term for the torsion with coupling constant kappa. Remember, in vanilla gauge theory, one can't put the gauge potential directly in the action because it's not gauge covariant. So in general relativity, you can't put the connection directly in the action because it's not diffeomorphism invariant. But here, the augmented torsion gives us a covariant object we can use directly. Note, you also have to recall that every element omega that comes from the inhomogeneous gauge group actually produces two connections. One is

A and another is B. The difference between these two is called T. Also, you should note that at this point, the theory is purely bosonic. The fermions haven't come about yet. This reminds me of how string theory was initially bosonic prior to being fermionic or having both. Step 18, field equations. From our action principle, we derive the field equations through a variational principle. The result is deceptively simple. It's on screen here. So what is going on? The Shiab operator acts on the curvature here, much like Einstein's contraction acts on the Riemann tensor. The primary difference

is that this operation preserves gauge covariance. The augmented torsion tensor term, T omega, enters with a coupling constant kappa. This E term here is essentially something that helps the theory's consistency as it makes the equation properly reflect the variational principle from which it's derived. I think of it like an error term. Eric demands this because it's necessary for recovering both Einstein's equations and Yang-Mills theory in the appropriate limits. Step 19, fermions and supersymmetry. Where do fermions enter? Recall that fermions act as quote-unquote square roots of gauge potentials. For spinor-valued forms, which we see here as

a zero-valued spinor form and a one-valued spinor form, we get a Dirac-like operator. Now this is fascinating because traditional supersymmetry relates bosons and fermions through spacetime translations. Here, we're seeing a different sort of supersymmetry based on the affine space of connections. The operator here combines aspects of the Dirac operator with our gauge structure. The upper left block involves the Shiab operator acting on the derivative of a spinor-valued one form. This is like the square root, quote-unquote square root, of the Yang-Mills operator. The off-diagonal blocks couple scalar spinors to vector spinors, similar to how supersymmetry transforms

fermions into bosons and vice versa. When one decomposes the spinor representations under spin7,7, one finds the following formula. Now this is how the three generations of fermions come about. Notice that upper index of 3 there. The first two generations come from a spinor-valued zero form and a spinor-valued one form directly, whereas the third generation comes about from Rarita-Schwinger fields in the decomposition of zeta. Step 20, the deformation complex. How do we study small perturbations around solutions? We require a complex. Now this complex is what's necessary for understanding the physical content or the field content of

the theory. The first map here encodes infinitesimal gauge transformations. It tells us how the fields change under small symmetry transformations. The second map gives us the linearized field equations. It tells us how the field propagates. The cohomology of this complex, which is the kernel module of the images, describes the physical degrees of freedom. At the first level, we have the first homology, and it gives us the gauge-inequivalent perturbations. This is analogous to how in electromagnetism, two gauge potentials differing by a gradient describe the same physics. Explicitly, these operations take the form on screen here. And

we also have the d squared equals zero property that makes this a complex, ensuring the gauge invariance of the linearized theory. Note that the second term is actually slightly more complicated, but I will cover that later and or in the PDF notes on my Substack and or with the upcoming podcast with Eric Weinstein himself. Step 21, the seesaw mechanism. Now here's where it gets fascinating if it wasn't already. Our Dirac-Rarita-Schwinger complex leads us to an operator of the form, which is on screen here that we've talked about before. Now, why is this interesting at all?

It's because this structure mirrors the neutrino seesaw mechanism. So the seesaw mechanism explains neutrino masses through the mixing between light and heavy states. Here, we're mixing between different spinorial sectors. The zero block in the lower right corner is actually essential because this is what allows for the hierarchy between different types of fermions. And this potentially explains why we see three generations of matter with such different masses. Step 22, analyzing the structure group reduction. I debated whether this section should go earlier in the script or later just because it's representation theory, and there's nothing specific to

GU here. However, I placed it here due to its length. The reduction of spin 7,7 to the standard model gauge group follows a path through intermediate subgroups. So let's go over how does this reduction work. Let's peel back the layers. Firstly, you have a spin 7,7 acting on a 128 dimensional space of spinors that we've talked about ad nauseum, and it splits into positive and negative chirality parts. Now here's where we get something different. You can reduce this to a maximal compact subgroup, spin 7 cross spin 7. Why this particular reduction of all the reductions

we could make? Well, experimentalists haven't ever observed non-compact internal symmetries in particle physics. Indeed, there are compelling theoretical reasons why physicists don't consider non-compact groups. So for instance, the famous or infamous Coleman-Mandula theorem, which essentially states that the symmetries of the S matrix, which describes particle interactions, must be a direct product of the Poincaré group and an internal symmetry group, though there are some assumptions here. This internal group must be compact for unitary representations, which means you need this for a consistent quantum theory. But why must it be compact? Well, it boils down to these

two reasons, unitarity that we mentioned already, and then positive energy. Non-compact groups often lead to these theories with negative energy states, which are physically problematic. We'll discuss the reasons why Eric's model circumvents these objections later. For now, let's break this down into the standard model's gauge group step by step. Again, the complete structure group reduction path begins with U64,64. In low gravity, and in Eric's model, this decouples into two vial halves, bringing us to spin 7,7. This contains a spin 1,3 cross spin 6,4, where the first term, the spin 1,3 represents spacetime, or specifically the

spacetime symmetries. Then you have to notice that the spin 6,4 part has spin 6 cross spin 4 as its unique maximal compact subgroup. And then this gives us SU4 cross SU(2) cross SU(2) via an isomorphism, which is precisely the Patis-Salam model. This answers a constitutional question, what is the maximal compact subgroup of the fiber structure group of our observable universe? The final reduction down to SU(3) cross SU(2) cross SU1, our standard model, comes about when the metrics carry an additional special unitary structure. Keep in mind that much of the above is somewhat standard in GUT

circles, so Grand Unified Theory circles. Eric, though, follows a different path, by using the non-compact group SU(3),2, which is a real form of SL5,C. The standard model gauge group comes about now as the maximal compact subgroup of SU(3),2. This specific real form corresponds to the A4 Dynkin diagram, and this distinction is what allows Eric to resolve the proton decay controversy and other issues that plague 1970s Grand Unified Theory schemes and approaches or what have you, because they used real forms that in Eric's eyes were incorrect. So how do we get this SU(3),2? Well, spin 7,7

has a spacetime split given on screen here. One of those, the 6,4, has SU(3),2 as a complex structure inside. We can then take the maximal compact subgroup of that, which is taking the special part of U3 cross U2. And we can further make an isomorphism of that to the standard model. Keep in mind that each step involves symmetry breaking, which in physics corresponds to the vacuum state not respecting the full symmetry that used to be there in the Lagrangian. This breaking can be spontaneous, dynamical, or explicit. You put it into the Lagrangian by hand. Now,

in this section, there are a plethora of subgroups being taken. So one question naturally comes up, which is, can we just take any subgroup willy-nilly in physics? And the short answer is no. Each time you take a subgroup, you're saying that the symmetry is broken, and thus you're introducing new physics. So for instance, the breaking from SU4 to SU(3) cross U(1) in step 4 corresponds to the separation of leptons and quarks. This is both a boon and a curse. Since there's new physics, it means you're deviating from what's standard. However, it also means that there

are predictions which can be falsified. Step 23. Three generations from the complex. Again, we're going to make this painfully clear. The Dirac-Rarita-Schwinger complex on Y14 gives us this generalization of the Diram complex, which is what we need to deal with the spinor-valued forms. On screen here, you take a one-form with the Shiab operator, technically the Hodge star of the Shiab operator and then some other differential, which takes you from a one-form to one dimension less than the manifold. So 13 in this case. Note, you may be wondering why you haven't seen this complex before, and

that's because as far as I can tell, it's a novel Dirac-Rarita-Schwinger-like complex introduced by Eric. This is brand new in the physics literature. This complex yields three distinct sets of fermions. How? Well, the scalar spinor here, new, gives the first generation. The zeta vector spinor here splits into two parts. A gamma traceless part, which gives the second generation, and a gamma trace part, and that's what gives the third generation. So the first generation, again, this space is the scalar-valued spinors on Y14, and when you pull it back to X4, these correspond to the familiar first

generation fermions like the electron, the electron neutrino, and the up and the down quark. The second and third come from zeta, which is a spinor-valued one-form. The decomposition of this is where it gets interesting. The first term here, on the left side of the Dirac sum, gives what Eric considers to be the second generation. And the last term, which is on the right-hand side here of the Dirac sum, is where the third generation comes from. These decompose to give us distinct generations because of how these spaces transform under the Lorentz group and internal symmetries when

pulled back to X4. This is Eric's explanation for the three generations of matter. Step 24. Higgs from Yang-Mills. So where's the Higgs field at? Well, it's lurking in the gauge potential variational pie here. Here's how. Firstly, we decompose this form as follows. The second term here, on the right-hand side, contains the Higgs field. Why? Because symmetric two-tensors decompose, remember, into a trace and a traceless part that we talked about ad nauseum again. Let's unpack this further. The gauge potential is a one-form on the Y14 space, which is valued in the adjoint bundle of the principal

bundle P sub g. When we decompose the one-forms on this space, we're essentially splitting it into parts that live on X and parts that live in the vertical direction in the fiber. The key term is this one. This is the space of symmetric two-tensors on X4, and it contains scalar fields from the perspective of X4. Among these scalar fields is our Higgs field, according to Eric. But what's the justification for even calling it the Higgs field? Well, it behaves like a Higgs field because of how it transforms under gauge transformations and diffeomorphisms of X4. The

trace component R in the decomposition transforms as a scalar under diffeomorphisms just like the Higgs field should. Moreover, under gauge transformations of the structure group G, this component transforms in the adjoint representation exactly how one would expect the Higgs to transform in gauge theories. Step 25. Trace and traceless contributions. The decomposition of the symmetric tensors into trace and traceless parts, it sounds like some mathematical pedantry, but it's not. Why? Consider the symmetric tensor aij. Which you can write as follows, where you decompose it into trace and traceless parts. We've done this many times. The traceless

part gives spin 2 contributions, and the trace part gives spin 0 contributions. This mirrors exactly what happens with gravitons and the Higgs. Now, we've discussed the trace and traceless decompositions before in the context of the Frobenius inner product, but let's go further. So we have this gravitational sector here, where the traceless part is on screen and it corresponds to the spin 2 field. In general relativity, this represents gravitational waves or gravitons in quantum theory. So why is it spin 2? Because it's a symmetric traceless rank 2 tensor, which transforms under the spin 2 representation of

the Lorentz group. Now, the scalar sector here is what engenders the Higgs field, in geometric unity at least. This decomposition has these as a consequence. It suggests that the graviton, which is spin 2, and the Higgs field, which is spin 0, are intimately related. Interestingly, it's called geometric unity for a reason, because these two are different aspects of the same geometric object on y14. Step 26. Natural quartic potential. Why does the Higgs field need that particular Mexican hat potential? Is it possible to get it to emerge in something that resembles something natural? Well, the Yang-Mills

action contains terms like the following, where you take the normed squared and you get quartic terms. This matches the structure of the Higgs potential. So, is it possible that the emergence of the Higgs potential comes from the Yang-Mills structure? Let's continue to explore this. In standard Yang-Mills theory, we have this term here, and that represents the self-interaction of gauge fields. However, in GU, remember that A contains components that we identify as the Higgs field. Let's write this out explicitly. Here, phi represents Higgs-like components. Now then, we expand the A wedge A squared term to get

terms like the following. Notice this last term here. This looks precisely like a quartic term in phi. Is this the origin of the famous Mexican hat potential? Eric says, of course, bro. You may wonder, where's the negative mass that gives the potential its characteristic shape? Now, this comes from the coupling between phi and the other components of A, specifically from terms like ADX wedge phi comma ADX wedge phi. This geometric unity-specific derivation is, to me, phenomenal because it shows that the Higgs potential, far from being an ad-hoc addition to the standard model, emanates, inevitably, from

the geometry of gauge fields. Step 27. Yukawa as minimal coupling. The traditional Yukawa coupling also looks ad-hoc, at least to me and to most other physicists and mathematicians. But in GU, it comes about inevitably. This time, it stems from viewing the Yukawa coupling as minimal coupling. Minimal coupling, by the way, to a mathematician just means a gauge covariant derivative. This A mu term, when it contains the Higgs component, gives the Yukawa interaction. This is the reinterpretation of the Yukawa coupling, and it's a prime example of how geometric unity unifies seemingly disparate aspects of particle physics.

To be specific, in the standard model, the Yukawa coupling is introduced by hand to give fermions mass. In contrast to geometric unity, this Yukawa coupling comes about from the geometry. Here's how. Recall that in GU, the Higgs field, phi, is part of the gauge potential A. This Dirac operator coupled to A is D slash A, here, on screen. Expanding this, you get this plus phi mu, where this new A with a little tilde on top are the usual gauge fields, and this Higgs is the component, and we get this full formula on screen here. This

last term is precisely the Yukawa coupling. Step 28. The correspondence between the Higgs and the Yang-Mills sectors. First, let's write out the Yang-Mills-Dirac action in the language of geometric unity. Here, this alpha is our gauge potential, and F sub A is the field strength as usual, and we also have some left and right-handed fermions. The covariant derivative acts as follows. Now compare this to the Higgs-Yukawa action. Here, this capital phi is traditionally viewed as a scalar field, which is valued in a representation of the gauge group, but in geometric unity, it comes about as a

component of this variational pi under the decomposition of one forms when pulled back to x4. This correspondence is surprising, because we have kinetic terms like D sub A alpha squared and D sub A phi squared, and they match, and we also have this quartic term, which correspond to one another. We also have this quadratic coupling, which parallels this. We also have the fermion couplings, and it takes on analogous forms. But how can a gauge field component act like a scalar field? Just remember, there either is right now or is going to be an accompanying PDF

to this geometric unity iceberg, so if you'd like more notes, such as expansions on these topics and proofs that I wasn't able to get to in this iceberg, then subscribe to my Substack, as that's where I'll publish it. It's CURTJAIMUNGAL.org, or c-u-r-t-j-a-i-m-u-n-g-a-l.org. Step 29. The Missing Quadratic Term The Einstein field equations have traditionally been written on screen here, with g mu nu plus the cosmological constant, and we set that all as something proportional to the stress-energy tensor. However, in GU, one needs to include a quadratic term in the field equations, so you get a modified

equation, which takes the form on screen here. This term here, with the T omega comma T omega, is the self-interaction of the augmented torsion, similar to how the Yang-Mills field strength contains A comma A, and it means that the theory maintains gauge covariance, while it preserves Einstein's intuition about geometric contraction. Think about it. In Yang-Mills theory, the field strength is F sub A, which equals dA plus A comma A, and it needs a quadratic term to be covariant, namely that last part. Similarly, our augmented torsion needs its quadratic interactions to maintain gauge covariance, while allowing

for some Einstein-like contraction. Step 30. The Emergence of the Cosmological Constant and CKM Matrix Lastly, both the cosmological constant and the CKM matrix come about from components of the gauge potential, variational pi here. How does that work? The gauge potential decomposes as follows. You see that it splits into these components based on how they transform under the structure group, when pulled back via an observation, which, recall, is iota. So the first part is what gives the standard model gauge fields. The second is what gives the Higgs field, as we've discussed earlier. The third contributes a

constant term into the Einstein equations, and the fourth determines mixing between generations. When one pulls this back to X4 via an observation, the component variational pi sub lambda gives the cosmological constant term in Einstein's equations. And then this variational pi sub CKM is what gives the mixing angles between the quark generations. Seemingly disparate physical phenomenon like dark energy, quark mixing, all derived from the geometry of the observers. Ordinarily, we think of these as independent parameters that we need to add in by hand. However, in Geometric Unity, they're intrinsic parts of the geometric structure. Just so

you know, if you have any questions, which you likely do, that's alright, I'm going to have a large solo podcast with Eric just on Geometric Unity. It will be unlike any other podcast because we'll delve into the particularities of it, especially now that they've been explored in a fair amount of detail. Therefore, feel free to subscribe to be notified of this upcoming podcast with Eric Weinstein. Layer 3. Welcome to Layer 3 of the Geometric Unity iceberg. Let's recap what's been done so far. In Layer 1, we gave a brief overview of Geometric Unity, as well

as the universe as we know it. In Layer 2, just now, we went over Geometric Unity in 4 minutes, and then I gave the longer 1 hour or so version of it as well. It's been quite a journey, and now in Layer 3, here's where it all starts to come together. Again, if you haven't understood much of this so far, then that's entirely fine, because this iceberg is designed to be watched and rewatched, where you glean something new every single time, not only from Geometric Unity, but perhaps from physics and math as well. I've worked

on this like I would work on a documentary with hundreds of hours put in. Also note that I will be recapitulating often so that your hand is held, metaphorically so, unless you're into that. Now as I mentioned, in Layer 1, we talked about the universe as it's currently known, and we went over symbols and equations that are at the heart of our models of the physical universe. They were the Ricci scalar, which is at the heart of the Einstein-Hilbert action, as well as for other actions, there's the Yang-Mills-Maxwell one, there's the Dirac one, there's the

Higgs one, there's also the Yukawa coupling, there's also the Lorentz group, or the double cover of it, namely spin 1,3, there's the internal gauge group of the standard model, so SU(3) cross SU(2) cross U(1), there's also the family of quantum numbers, there are also the three generations of matter, or families of matter, or flavors of matter, I've heard some people call it, there's also the CKM matrix, and the PMNS matrix, for quark and neutrino mixing respectively. Now there are also these which are alternative writings of what I've just mentioned. So there's the Einstein field equations,

there's the Dirac equation, the Klein-Gordon equation, the Yang-Mills equation, and the Higgs field equation. While I've reviewed the different steps of geometric unity as I see it, the question is, well, how do we integrate these concepts that I've just mentioned within geometric unity exactly? Now GU's explanation for these equations and concepts encompass essentially everything in modern physics, and the derivation of which is what this layer is about. Einstein-Hilbert action in GU In standard GR, one writes the Einstein-Hilbert action as follows, where the Ricci tensor is computed by contracting the Riemann curvature tensor with the inverse

metric, and then you further do a trace. In geometric unity, however, there's no single metric that's privileged, as you know, and instead, Eric begins with the curvature of a distinguished connection, A0, which is defined on the frame bundle over the manifold lifted to the double cover, so that spinors can exist. Now this A0 is obtained via the Zorro construction that we talked about earlier, and more precisely, one starts with the frame bundle of X4, lifts it to its double cover, so F tilde, so that spinors can be defined. Then, Eric uses the unique Levi-Cevita connection

associated with any metric, which itself, again, is not fixed on X4, but instead it varies over the metric bundle, over the observers, and uses that to define A0. Notice I use the term lifted here, and I do so in the sense of a standard bundle-theoretic lift. That is to say, given a projection from F tilde to F, we can look at a connection, which is A0 on the regular F, and that can be lifted, quote-unquote, to F tilde by composing with the covering map. Precisely, this means what's on screen here, which is a two-to-one covering,

and A0 is a connection on the base F, which is then lifted to an A tilde on F tilde, and it satisfies this equation here. In other words, for any vector in the tangent bundle of F tilde, you can define a connection by just pushing that vector forward. This allows for the proper transformation properties that Eric wants for spinors. As I'm reviewing this iceberg several weeks later, recall this is many months, many, many months in the making, I'm realizing that I sometimes use A0, and I sometimes use A aleph. You may be wondering, what is

the relationship between these? These are the same. My mistake is that in ordinary math notation, one would use A0, but in GU, because Eric is referencing something specific with the aleph, then the A sub aleph is used, and it refers to the choice made in the Zorro construction. Now, geometric unity reinterprets this contraction process as coming from an algebraic operator, which compresses the full curvature 2 form on Y, the metric bundle, the observers, into a symmetric 2 tensor. You'll notice that the domain of PE consists of a tensor product with two fundamentally different factors. So

this first component is a differential form, which is intrinsically tied to the manifold Y itself. The second factor here has a Lie algebraic character. In Einstein, it's SO1,3, the Lie algebra. So what's interesting is that this PE operator contracts mathematically distinct structures using a metric on Y. So the operator P sub E, it contracts these two using the metric on Y, much in the same spirit as how Einstein in his original approach obtained the Ricci tensor by contracting the indices of the Riemann tensor like we talked about just a minute ago. Now, where did these

two 2 forms come from? Well, one comes directly from the curvature of the distinguished connection, namely what's on screen here, while the other is an internal 2 form from the contraction mechanism itself induced by the metric on Y. Thus, both copies of the 2 form are essentially the same curvature data. However, Erik regards them as having different transformation behaviors under the gauge group. This is reminiscent, again, to how Einstein in his approach also had two appearances of two forms in the Riemann tensor treated identically by the metric contraction. In fact, one way that I think

of this algebraic operator is that it encapsulates the same idea as forming inner products via the Frobenius metric. If you take two symmetric matrices and multiply them together using the natural inner product on screen here, you obtain a contraction that produces a scalar. Here, however, this algebraic operator acts extended bilinearly to the tensor product to yield an element of the symmetric 0,2 tensors, thus compressing, quote-unquote, the curvature information into a form that mimics the Einstein tensor. What you're doing is you're allowing an algebraic operator, which is defined to act on a curvature tensor, and then

it's supposed to return something from the parameter space of field content. In Yang-Mills, this involves a Lie algebra-valued 1-form, while in general relativity, it involves symmetric 2-tensors. After this algebraic operator on y, we then pull back the result along the section, which was that observation map, so that the effective Einstein tensor on x4 is given by this formula on screen here. So you begin, again, with fA0, which is a 2-form. The contraction operator is defined locally, such that we have this formula on screen here, so that we can then pull back with an observation, and

we get the formula on screen here. And that's where the r comes from in geometric unity, re-derived from a gauge covariant contraction on y. The Yang-Mills-Maxwell term. Next, we will derive the Yang-Mills-Maxwell term within geometric unity. In conventional settings, on a fixed spacetime x4, the Yang-Mills action is written as follows, where the fA here is the curvature, as usual, of the connection A on a principal G bundle, and the inner product denotes the invariant bilinear form, also known as the killing form, on the Lie algebra G. Editor's note, in a local trivialization, that is, on

a patch of the base space where the bundle is trivial, each connection is represented by a Lie algebra-valued 1-form A. Its curvature is the following, but when you're comparing two different connections, say A and B, then you often write it as follows. The connection B here can be viewed as a reference or a base connection. Now when you set B to equal zero, that is the trivial connection, you then recover the usual single connection formula. This will become important later in the Higgs sector portion of this geometric unity iceberg. Eric considers the principal bundle here,

with the structure group of U64,64, coming from that spinor representation of spin 7,7. The gauge field is defined on Y as a curvature 2-form, with the transformation property as follows. Now recall that the key step is that this inner product here, which we use to contract fA, is actually induced from the metric on the larger space Y, which is in turn built from the vertical Frobenius metric on the symmetric two tensors from the base manifold, and the horizontal lift via the Zorro construction. Explicitly for local coordinates on Y, we can define the following slightly hairy

formula, at least to look at on screen. This ensures that the contraction is performed in a manner that's compatible with the dynamical nature of Y. Now in our current context, A represents the full gauge field on Y, and A0 denotes that distinguished background connection which is chosen by the Zorro construction. Therefore, you should actually understand that this connection is supposed to be expressed as A equals A0 plus alpha, where alpha is like a correction or a fluctuation. In contrast, this fA0, that appears in some derivations, is simply the curvature of the background connection alone. Our

full curvature, fA, is thus containing extra pieces, and then the contraction via the Shiab operator is engineered to handle these extra terms. Moreover, the domain of this operator is the space of Lie algebra value 2-forms on Y, and its target space is typically the space of scalar fields or symmetric 2-tensors, for instance, because it compresses the 2-forms into a lower degree object using the metric structure on Y. This guarantees that under the gauge transformations H, for instance, we have the following, which preserves gauge covariance. Technically, it's the below, but the above is much more succinct.

Now you may be confused. Look, is PE an example of the Shiab operator, or vice versa, or nothing? Now the answer is that the PE in Einstein's theory is an algebraic contraction that maps two of something, this is a 2-form, and then some Lie algebra data. Their domains and target are different. Also you'll see that this lambda with the bullet is what represents, in Eric's theory, the U64,64 Lie algebra structure, while the Shiab operator maps these 2-forms, which locally involve both a 2-form part and a Lie algebra data part, to a 1-form, or another suitable

counterpart. The idea here is that the Shiab operator is a generalization of Einstein's contraction operator, except Eric constructs it to be compatible with both the base manifold and the fiber gauge structure. At this point, I should point out that the function space for the bosonic theory is the inhomogeneous gauge group. It has two components. One looks like the gauge potential and functions like a connection. The other piece does the symmetry work. It allows the Shiab to be defined without destroying equivariance in contraction. Why don't we just do a step-by-step derivation? So, step A, let's say,

you get that distinguished connection from the Zorro construction. Step B, you define the gauge potential correction. Step C, you compute the curvature. Step D, you then apply the Shiab operator. Now, you'll notice that this form is analogous to the standard Yang-Mills action, except contracted by the Shiab operator here and replacing the usual inner product contraction on FA. Now, one has to show that in a suitable limit, after pulling back to X4, that this actually reduces to the familiar expression. For an explicit demonstration, you would take a local section from X4 to Y. Then, you would

note that the induced metric on X4 satisfies that the metric is a pulled-back metric and that the Shiab operator on the curvature when pulled back is the same as the inner product of the curvature with itself. And that completes our derivation of the Yang-Mills-Maxwell term in GU. The Dirac action. Eric derives the Dirac action in the framework of geometric unity, solving the previously unsolved problem of defining spinors without a prior metric. Now, in conventional formulations, the Dirac action is given as follows, with the Dirac operator also written on screen here. And the gamma matrices, of

course, satisfy that Clifford algebra so equals twice the metric. Even though many people think Clifford algebras don't rely on a metric, there is an implicit pre-assigned metric. And that's the dilemma. The very existence of spinors demand a metric. However, one would like to work in a framework where the metric is allowed to be dynamic, allowed to vary. Now, geometric unity sidesteps this by constructing the chimeric bundle, which recall is the vertical component Dirac summed with the dual of the horizontal. Eric then uses the exponential property of spinors, which you can recall if you have different

vector spaces V and W Dirac summed together, that their spinor bundle is isomorphic to the tensor of the individual components. Now, in our case, because our components are V and the dual of H, we have the following. The prowess of this construction is that it's like liberating. It's like liberating the definition of fermions from an a priori choice of a metric. Instead, the metrics are actually determined by the point that they are in Y. This means that the Dirac operator can be defined on Y, the larger space, rather than X4, the base space. Concretely speaking,

you construct a Dirac operator acting on the sections of the spinor bundle by creating something familiar here, where the nabla is, of course, the covariant derivative associated with the distinguished connection A0, which is from the Zorro construction. The gammas are the gamma matrices, or more specifically, the corresponding gamma matrices to the 7,7 signature, determined by the vertical Frobenius metric on V and the induced metric on the horizontal space, or the dual of the horizontal space, H star. The I, of course, runs over the entire indices of 14 dimensions, and the alpha represents additional potential coming

from the inhomogeneous gauge group structure. We're going to talk more about that. Now one uses these observation maps to pull back the spinor fields from the larger space Y to X4, and this is done using the standard pullback of vector bundles on screen here. You'll notice that this pullback yields a decomposition, and what happens is that the V part becomes the internal quantum numbers, whereas the H part becomes the spacetime Lorentz group spin 1,3. Now to spell it out in detail, if you look at the chimeric bundle, and let's just pick a single point Y,

then we have the following, and we then construct the spinor bundle as follows, again at a single point, and we take the pullback, which here I'll just show on screen every element. By this construction, this last factor here is equivalent to the conventional spinor bundle on X4, since H star, evaluated at the pullback, corresponds via the Zorro construction to the cotangent space on the base manifold, which we denote by that S and then the slash and spacetime. Meanwhile, the other factor here with the V, that's where the internal symmetry space comes from, so that's why

I placed a S with a slash through it and said internal as an indice, whose dimension encodes the family quantum numbers, thus that decomposition that you see above. Finally, in GU, the action, or the Dirac action to be specific, is defined as follows, where that field there is a section of the chimeric spinor bundle, and mu is the natural measure on Y. What I mean by natural is that the measure actually comes from the induced metric on Y itself, specifically the square root of the absolute value of the determinant of G with the 14 dimensions

dy. This measure has the required properties, that is, it's locally given by the Lebesgue measure in suitable coordinates, and it transforms appropriately under coordinate changes, so that the integrals such as the Dirac action are well defined and gauge invariant. The Higgs sector, the derivation of the Higgs kinetic and potential terms. In geometric unity, the origin of the Higgs sector is the gauge potential, written on screen here with this variational pi, and the origin of the Higgs sector is when we decompose this according to the splitting of the observers. Now here, let me clarify what this

gauge potential means. In GU, this gauge potential, variational pi here, is a one-form, defined on the larger observers, whose domain is the tangent space of Y, and whose target is the Lie algebra of the gauge group, specifically lowercase u, 64 comma 64. You can think of this as a rule that goes from here to here, that tells you how to differentiate sections of the associated bundle. There's nothing miraculous about this specific one-form. It's the canonical gauge potential, coming from the inhomogeneous gauge group structure on Y. In other words, any connection on the principal bundle Y

provides an adjoint valued one-form, but this variational pi is our chosen representative that encodes fluctuations over the distinguished background connection. Now, the gauge potential will decompose when pulled back to X4, via a local section. See it decomposing as follows with alpha and phi, where the alpha represents the Lie algebra valued one-form component corresponding to the Yang-Mills fields, and the phi is a scalar field, or at least appears as one, coming from additional degrees of freedom in the vertical direction. When one pulls back a one-form, variational pi from Y, via the section, we then decompose, or

Eric then decomposes the tangent vector into horizontal and vertical components. The horizontal part gives a one-form on the base manifold, since it involves directions along the base, and we denote that by alpha. In contrast, the vertical part is associated with the variations along the fiber, so that is the variations of the metric at a fixed point, which when pulled back, effectively lose their quote-unquote direction along X4, becoming functions. That is to say, they become zero-forms, valued in the symmetric two tensors, which we denote by phi. Editor's note, in a more fine-grained analysis, and this is

as Eric points out, you have to decompose both the one-form part, the horizontal versus the vertical, and the adjoint part, which is the pure horizontal, pure vertical, and mixed. And concretely, the pulled back one-form is written as a direct sum of six different pieces. Each of the sumands here are different sectors in 4D, so some give a spin-1 field, like the usual gauge bosons, and others give a spin-0 field, like the Higgs scalar, and it just depends on how it transforms. This is a refinement about the splitting of what I've just showed of alpha plus

phi, but my initial way of writing it was much more compact. As you can see, it's quite tortuous otherwise, to show every pairing of horizontal versus vertical in both the one-form and the adjoint representation. Note that the vertical component corresponds to the infinitesimal changes in the metric at a fixed point. Given that they're symmetric matrices, they contain both a trace and a traceless part. In our construction, or more specifically in Eric's construction, we use a contraction, so the trace, to isolate the scalar degree of freedom, which is then identified with the Higgs field. To derive

the Higgs Lagrangian, Eric starts with the Yang-Mills action on Y as follows, where the curvature is here. When we decompose A as A0 plus variational pi, with A0 being that distinguished connection induced by the Levi-Cevita connection on the base manifold, the field strength splits into parts involving our gauge connection. In particular, the terms that are quadratic in the gauge connection lead, upon pullback, to terms like d phi squared and phi wedge phi squared. Now remember, A0 is that classical background connection that encodes the standard geometric data, while the variational pi, the gauge, the full gauge

potential, represents the fluctuations. Now by writing A equals A0 plus variational pi, you separate this unchanging classical piece from the quantum corrections, which you then pull back variational pi, its vertical part, which then loses its vectorial character, and becomes a scalar field phi, a candidate for the Higgs field. To explain this clearly, we begin with a gauge connection that gets decomposed as follows, then you insert that into the curvature, and that leads to this formula on screen here. When you then pull back this variational pi via the local observations, its vertical component yields phi. The

derivative then contains a term that projects onto the derivative of phi, giving a kinetic term here. Meanwhile, the gauge potential wedged with itself, when restricted to the vertical directions, and then contracted via the Frobenius inner product, provides a quartic self-interaction term of the form on screen here. In GU's language, after the appropriate contractions on the metric on X4, these pieces match the standard Higgs kinetic term and the Mexican hat potential, typically written as follows. What I find remarkable, personally, is that the gauge potential, when viewed through the lens of the observers and decomposed via the

pullback, bifurcates into these two sectors. One is that alpha, which reproduces conventional Yang-Mills, as we'll see, and the other is the phi, which embodies the scalar fluctuations of the metric, which then become the Higgs field. In this way, GU unifies the disparate sectors of gauge theory and spontaneous symmetry breaking within a geometric framework, all following from the metric. Every step as follows, let me just repeat that, is you start with the full gauge potential, you then split it into horizontal and vertical components, using iota to pull it back, so that the vertical part, which carries

the metric fluctuation degrees of freedom, provide a natural candidate for the Higgs field. The kinetic term is then given by the norm squared here, while the quartic self-interaction is from the contraction, Both are computed via the Frobenius inner product on the space of symmetric 0-2 tensors. Yukawa coupling, the derivation via minimal coupling. The Yukawa coupling term that gives mass to fermions comes out as part of the minimal coupling of the Dirac operator on the spinor bundle to the gauge potential. Start with the Dirac action on the observers. So we have this action here, and of

course this D slash is as follows, which is the Dirac operator, coupled to the full gauge connection A, and psi is a section of the chimeric spinor bundle, S of C, or S slash of C actually. Now, the minimal coupling is another way of saying that the derivative in the Dirac operator is replaced by a covariant derivative. Inserting the decomposition of A we saw just a minute ago, we obtain the following. In standard physics, the term here, phi psi, isn't actually present in the definition of a derivative because the Higgs field is taken to be

a separate independent scalar. However, Eric's identification is that the Higgs field originates from a particular component of that variational pi. Now in practice, one arranges the theory so that the coupling of phi to the fermions takes the form on screen here, where y is the Yukawa coupling constant. Let's clarify the connection between this guy and this guy. Now in GU, the gauge connection again decomposes as follows, so that the minimal coupling replaces the ordinary derivative as follows, and next, one decomposes the chimeric spinor into its left and right components. This is so that the Dirac

operator then splits into off-diagonal blocks as follows. Here, the splitting into the left and right-handed parts is fermented from the Z2 grating of the chimeric spinor bundle S of C as constructed via that exponential property we talked about before ad nauseum. In more detail, once a spinor bundle is defined on a space with a metric of indefinite signature, that is, 7,7, the associated Clifford algebra admits a decomposition into even and odd parts. This decomposition allows one to identify chiral or vial sub-bundles, usually denoted by S-slash-L and S-slash-R. Some people put a plus and a minus

instead. Thus, psi is a section of the chimeric spin bundle which decomposes into its chiral components. Here, the operator D-minus-A and D-plus-A incorporate the corresponding parts of the connection. In particular, the Higgs field contribution appears precisely in the following blocks, where the first part, the A0 parts, denote the chiral pieces of the Dirac operator associated with the background connection A0, and phi, after some projection or contraction, behaves like a scalar. Inserting these into the Dirac action yield the following. Now, these two terms here combine after appropriate normalization, absorbing constants into the Yukawa coupling, to yield

the Yukawa action as follows. Thus, geometric unity derives the Yukawa action from the original Dirac action by expanding the covariant derivative to include the gauge potential's extra component phi, which is associated with the Higgs field. Now, this derivation is entirely quote-unquote minimal once one posits that the gauge potential contains both the conventional gauge fields and an extra component, phi. The Lorentz group and the standard model gauge group. The chimeric bundle is defined as the vertical component plus the dual of the horizontal component, with the overall signature of 7 comma 7. Now, the relevant real spin

group is spin 7 comma 7. However, Eric works with complex Dirac spinors, thus the natural symmetry group is U64 comma 64. The real spinor representation of spin 7 comma 7 has, of real dimension, 128. However, once you complexify and then you split into chiral components, it decomposes into two 64 dimensional complex vial representations. In this complex setting, the indefinite unitary group, U64 comma 64, is what preserves the Hermitian form, and this is what acts as the full symmetry group of the spinor bundle, even though the spin 7 comma 7 governs the underlying real Clifford structure

at signature 7 comma 7. All of this is as far as I understand it, and I could be incorrect, so I'm just telling you what I've understood. Now, to recover spacetime versus internal degrees of freedom, one uses the observation map, written on screen here, and the differential of this, or the push forward of this, maps from the tangent space on the base space to the upper space, so that the image is isomorphic to a four dimensional subspace, the dual of H. While the complement, V of dimension 10, corresponds to the vertical variations, now this splitting

is what is allowing this reduction of the structure group. So we start with spin 7 comma 7, that then gets reduced to spin 1 comma 3 cross spin 6 comma 4. Now this spin 1 comma 3 acts as the spacetime horizontal part, and it's isomorphic to the double cover SL2C of the Lorentz group. The remaining factor, 6 comma 4, acts on the vertical 10 dimensional space. Now since 10 equals 2 mod 4, this is just a result, that means that the vertical space admits a complex structure, and you can reduce 6 comma 4 in at

least two ways. So one approach is to reduce it to a non-compact real form, SU(3) comma 2. Alternatively, you can reduce spin 6 comma 4 to its maximal compact subgroup, and then you'll find spin 6 cross spin 4. Now there are some isomorphic coincidences, where spin 6 is SU4, and spin 4 is two copies of SU(2), and then that is precisely the Pati-Salam group. If you know anything about grand unified theories, this is quite remarkable and should be surprising. At least it was to me. Now to obtain the standard model gauge group, we then do

a further reduction, and from here this is standard in GUT folklore. So provided you choose a correct embedding of the residual U(1) factor as the difference between the two SU(2) factors, or equivalently by requiring certain trace conditions on the symmetric tensors, then the physical gauge group comes about. So you take U3 cross U2 and demand that the determinant equals 1, and then you get something that's isomorphic to the standard model gauge group SU(3) cross SU(2) cross U(1) up to discrete identifications. Note that the left hand side is the maximal compact subgroup of SU(3) comma 2.

Now this identification is enforced by the requirement that the internal quantum numbers, which ultimately turn out to be 16-dimensional for a single family, and we'll see more about that soon, they match the observed hypercharge and color assignments of elementary particles. So let me just sum up. First we start with the full symmetry group spin 7 comma 7 on the chimeric bundle. Then by application of the observation map, we split the geometry into a four-dimensional spacetime with the spin 1 comma 3 symmetry, and then a 10-dimensional vertical part which has a different structure group spin 6

comma 4, which then can get reduced to SU4 cross SU(2) SU(2), and that further gets reduced to the standard model. Now GU is baroque with many alternate paths, like a labyrinthian roguelike video game as far as I could see. So one path that I mentioned is that Pati-Salam first, which is spin 6 comma 4 reducing to Pati-Salam, providing an immediate quark-lepton unification picture. Now the other path is the one that goes through what I mentioned before, where you take U3 cross U2 and you take the special part or you enforce speciality, which means the determinant

equals 1, and then you recover up to discrete factors, the standard model gauge group. Neither of these as far as I can tell is more fundamental. They're just complementary ways to see how the indefinite group spin 7 comma 7 breaks into the spin 1 comma 3 spacetime group and the standard model gauge group. The family of quantum numbers. In conventional SO10 grand unification, the fundamental spinor representation is 16-dimensional. Here, however, in GU, the 16 is reinterpreted as coming from the structure of the metric bundle. So let's recall that the chimeric bundle has that 7 comma

7 signature. In Clifford algebra theory, for a space of signature PQ, I'm being general now, the real spinor dimension is given by 2 to the P plus Q over 2, and you have to take the floor function of that. Now for 7 comma 7, that works out to 2 to the power of 7, which is 128. This 128-dimensional complex representation on Dirac spinors naturally splits by chirality into two 64-dimensional Weyl spinors. By the way, you may wonder why is this with the W pronounced Weyl? It's just it's German. As I mentioned a couple minutes ago,

the properties of the Clifford algebra are such that there's a Z2 grading into positive and negative chirality parts, each of that 64 dimension. In technical terms, consider that for Clifford algebra CLPQ, the real spinor representation, if it exists, has this dimension here, and it decomposes as follows. Notice that this time I'm calling it S plus and then S minus. This operator is unique up to scaling so that the splitting is canonical. Now this action is defined by the standard representation induced by the Clifford algebra multiplication on this space. In our context, the full chimeric bundle's

spinor representation later will be pulled back and partially reduced through projection onto the internal degrees of freedom to yield a 16-dimensional space, corresponding to exactly the family of quantum numbers. Let me explain this reduction in more detail. Here, think of the spin bundle on the vertical part as encoding the internal degrees of freedom associated with the variations of the metric, a kind of quote-unquote gauge content, and then the spin bundle associated with the dual horizontal, as encoding spacetime properties. When one performs an observation and one pulls back via this observation back onto your base manifold,

then you get the decomposition on screen as follows. The careful matching of these components with the quantum numbers like weak hypercharge and isospin forces the internal part to be 16-dimensional. Now this matching process exploits the fact that for any given element, say k of u1, one can assign it a weight. And the sum of these weights in a chiral multiplet typically adds up to produce the correct quantization conditions. Now for instance, representing all of these components as little quote-unquote bit strings, so not in the string theory sense but in the computer science sense, as in

conventional SU5 language, you'll find that the total number of independent states is exactly 64. You can actually prove this to yourself by looking at the exterior algebra of C5, which has dimension 2 to the power 5, which is 32, and if particles split into particles and then antiparticles, then you get 16, one for each chirality. However, GU doesn't just postulate SU5. Instead it recovers the same number from its more fundamental chimeric construction. Explaining the three generations of matter. Some of you may know that in Hodge theory you obtain cohomological information from harmonic forms, but this

requires a metric. However, there's also the Dirac theory. Now in Diram theory there's the metric independence. So where is the Diram version of the Dirac operator? What I mean to say is that Nigel Hitchin showed that the dimension of the space of harmonic forms changes with respect to variations in the metric. It's only the difference between these dimensions that's topologically determined by the A-roof genus. Now unlike harmonic forms, the individual kernels don't behave consistently. In Hodge theory, regardless of the metric chosen, the dimension of the kernel of the Laplacian remains constant and it's topologically determined.

So this then raises the question, can you make Dirac theory resemble Diram theory? The way that I see it is that in answering this question, Eric derives the three generations of matter. To explain the three generations, we study the Dirac-Rarita-Schwinger complex on the full metric bundle Y14, also known as the observer's. I know I keep saying this over and over. By the way, the way I'm calling it a complex may give you the false impression that this complex exists in the literature, but as far as I can tell, this is a novel construction by Eric

himself. Instead of merely considering spinor-valued zero forms, which are just regular spinor fields, Eric assembles the combined complex as follows. And then he defines a Dirac operator, or a Dirac-like operator, because it's not exactly Dirac, where here the D with the subscript A, with the further subscript omega, is a gauge-covariant exterior derivative involving the full connection of A, and then the D with the star is the adjoint constructed with the Hodge star, and then this little dot-circle guy is our Shiab operator that everyone knows and loves. It ensures that the quote-unquote projection is gauge-invariant. I

call it a contraction. I've heard Eric repeatedly call it a projection. Also this operator itself is another novel and central contribution of geometric unity. You can tell that by the presence of the Shiab operator, but I'm just hammering the point home anyhow. What does it mean for this operator to quote-unquote mix different spinor-valued forms? Okay, let's take a look at the block matrix form. The off-diagonal operators couple zero forms and one forms. Consequently, any solution, like let's just say there's a psi from the kernel of this operator, it will take the form of these two

guys here, with one guy coming from a spinor-valued zero form, and the other guy coming from a spinor-valued one form. Now a detailed index-theoretic analysis, which uses the Atiyah-Singer Index Theorem, adapted to this context, shows that the solution space, so the kernel of this, splits into three gauge-inequivalent sectors. Note that I do need to be careful, as the Atiyah-Singer Index Theorem only applies to Euclidean signature, thus more work needs to be done here, which I've glossed over. In simple terms, the first generation comes directly from the zero-form spinors, and I'll just rewrite that here with

psi 1. Now the second part is where it's interesting, because the second part is actually a one-form, and this decomposes under the action of the Clifford Algebra into two components. So you'll see here there's one that's traceless or gamma-traceless, then there's another gamma-trace part. The gamma-traceless piece is gotten to by contracting with the gamma matrices, and requiring that this contraction vanish, yielding the second generation, according to Eric. And the complement is the gamma-trace part, which produces the third generation, also known as the imposter generation by Eric. Those are Eric's words. What you may not know

is why he calls it an imposter. So he calls it an imposter in GU, because unlike the first two generations, in Eric's framework, the third one has different unification properties as one goes up toward U64,64. Now I should emphasize that each of these components is isomorphic to the 16-dimensional representation predicted by grand unification schemes. That is, isomorphic at the level of reduction to spin-6 cross spin-4 representations. But how do we know these pieces are exactly 16-dimensional? Well, recall that the internal part of the chimeric spinor bundle, here, comes from the vertical bundle of metrics, and

by representation theory of the Clifford algebra for signature 7,7, you get that the full spinor representation's dimension is 128, and then you just divide by 2 to get the 264-dimensional vial representations. Now, you have to then pull this back, and when you do, the horizontal, so spacetime, component factors out as the standard 4-dimensional spinor representation of spin-1,3, leaving out a factor of 4, which you then just divide 64 by to get 16. Thus, you get the full matter field on spacetime, as follows. Now, not only does this yield the correct field content for a single

generation in conventional SL10, but you get the other 3, you get this splitting of the 3 generations. Again, this is as far as my reading of Eric's theory goes. Now, why does this Rorida-Dirac-Schwinger complex give 3 dimensions instead of, let's say, 2 or 4? Well, let's just take a precise look at the structure of the differential complex, and its interaction with the Clifford algebra. The zero-form part here contributes one block. The one-form part, through the action of the gamma matrices, splits into a part that vanishes upon contraction, so the gamma-traceless part, and a complementary gamma-trace

part. An index theory argument, which I'll leave as a PDF, shows that the net index forces the total kernel to split into 3 parts, and then you use spectral theory computation to confirm that this has 16 dimensions. You can show step-by-step that this Dirac-like operator, when acting on the zero-form plus the one-form part, is Fredholm, and that its index, which is the difference between the dimensions of its kernel and co-kernel, is topologically determined by the A-roof genus of the observers. Now, I should repeat that this Dirac-like operator is novel to GU, so perhaps we should

call it the GU fermionic operator. Now, let me just recapitulate this all. The operator structure on the complex here implies that its kernel splits as follows, with each of these guys here individually being 16-dimensional. By the way, to clarify the distinction between operators and complexes in the context of Dirac, Roweta, Schwinger, etc. and these quote-unquote Dirac-Roweta-Schwinger types, formalisms, and so on, let me just go through this one by one. First, let's take a look at the Dirac operator on an n-dimensional spin manifold. Now, this is standard. Nothing here I'm saying is new. We have a

spinor bundle S with sections as follows, and the Dirac operator is an endomorphism on this space. And the gammas here are the regular gamma matrices, and that's the spin connection. In even dimensions, S splits into its chirality parts, its chiral parts, and you have this as follows, where it's not exactly an endomorphism. You map instead the left to the right and the right to the left. Now, D alone is just an operator, a single first-order map actually, and one often embeds it into a Dirac complex. So you'll see this on screen here. Complexes just mean

that you have many vector spaces, and you tie them together by differential maps that eventually terminate to zero. And then there's some other conditions such that you want subsequent compositions to equal zero. And this is what allows people to study the cohomological properties and the index. Now, if none of those words meant anything, it doesn't actually matter. Next is the Rarita-Schwinger operator, R, which deals with spin 3-half fields. And these are usually realized as vector spinors. Concretely, you can write it as follows, plus some other possible constraints. These are rarely encountered in standard physics, because

there isn't any known Rarita-Schwinger fundamental particle. But this too, anyhow, can be placed into a complex, as follows here, where you see this sequence here. And again, you have these consecutive maps that go to zero. Actually, if you pause here, this should confuse you, because the one-form spinor part is not spin 3-halves. The reason is that the one-form spinor part, it's that spinor trace sector that's the ordinary spinors. And only the spinor traceless part has the spin 3-halves. And it's that part that gets mapped to the two forms of the spinor bundle. This two-form of

the spinor bundle contains a part that looks like pure spinors, so the zero forms valued in spinors. And it has a piece that looks like a Rarita-Schwinger spin 3-half part. However, it has another piece that we haven't seen yet. And that part looks like pure two-forms valued in spinors. And these will vanish under any contraction in Eric's framework. So this becomes analogous to scalar, traceless Ricci, and vial sectors of the curvature. See now, in geometric unity, you actually merge the spin-half and the spin-3-half fields into a single structure. And it's that that gives this Dirac-Rarita-Schwinger

type operator here. Where the zeta is what contains the Rarita-Schwinger part, it's just that it also has a trace piece that has ordinary spinors as well. And the new is the Dirac part. This block matrix operator generalizes both Dirac and Rarita-Schwinger simultaneously. Why? Because it mixes the zero-form and the one-form spinor fields. It's neither purely Dirac, nor is it purely Rorida. Hence, I've heard Eric call it a Dirac-Rarita-Schwinger type system. The CKM and PMNS Mixing Matrices In the standard model, the CKM matrix comes about when you consider flavor mixing of quarks, also known as generation

mixing, which happens under the weak interaction. Now, in conventional math terms, the CKM matrix is a unitary 3 by 3 matrix, and it comes about because the weak interaction eigenstates, so that is, the states that participate in the charged weak currents, don't align with the mass eigenstates, that is, the states with definite mass. So I actually talked about this here in this podcast with Lawrence Krauss. If you're interested, link is on screen and in the description. Let's say U, C, and T are the weak eigenstates, and I'm just putting a subscript of L here because

we're dealing with left chirality. Now, these are weak eigenstates of quote-unquote up type, and now we have the DSB for the down type quarks. Then the physical mass eigenstate down quarks are in a linear combination of these via the CKM matrix. Mathematically, we express it as follows, where the primed guys here are the mass eigenstates, and the B unprimed are the weak eigenstates. Now the question is whether this has an interpretation or an explanation or a derivation in geometric unity. Let's proceed by recalling again that GU constructs this unified field associated with the gauge interactions

from an inhomogeneous gauge group, and it's on screen here with the variational pi. This then gets pulled back via the observation, and after we perform this pullback, the spinor representation gets decomposed into those three generations that we talked about a few minutes ago. Concretely, we can write it as follows. Now, what about this indicates mixing? Well, because as you move in the internal space determined by the 14-dimensional construction, the observer's construction, the representation of the unified spinor field isn't fixed in a way that preserves the labels of the individual generations. So, in other words, that

internal bundle that we talked about before, which is the spin bundle of the vertical part, has extra substructure. If you change trivializations or perform gauge transformations that act non-trivially on these three families, then the mass term acquires off-diagonal contributions, generating a non-diagonal mass matrix MAB. In standard physics, you usually allow MAB to be non-diagonal by fiat, meaning that you just impose it. However, from the GU standpoint, this non-diagonally is viewed as a consequence of leftover gauge freedom, the residual transformations that don't block-diagonalize the three different families. So, let's say if you write the following, then

you have a Yukawa coupling here, which is the capital phi, and the diagonalization of MAB is gotten to by finding two unitary matrices that separate the uptype and the downtype quark sectors. The physical misalignment between these two guys here, these two block diagonalizations, that is what gives a unitary matrix the CKM matrix. So, geometrically, you can see the gauge freedom in the internal 16-dimensional representation spaces permitting these off-diagonal transformations among, say, psi 1, psi 2, psi 3. The way that I understand it is that suppose you write a gauge transformation as epsilon, like a 3x3

array of blocks, let's say epsilon AB. Each block acts between the spaces corresponding to the different generations. In the absence of any of these off-diagonal blocks, the three families are unmixed, they do not mix. However, if some of these guys with A not equal to B, so the off-diagonal parts are non-zero, in which the mass matrix appears block-diagonal, can be altered, producing mixing. So, we have under a gauge transformation epsilon, the following here. And similarly for, say, psi 2 and psi 3. Because weak interaction eigenstates differ from the mass eigenstates by these transformations, the resulting

mixing is codified in the CKM matrix. You can view the CKM matrix as the relative rotation required to diagonalize the up- and the down-type quark mass matrices. So that we have this equation here as the overall residual rotation. So why would this be unitary? The way that I understand this is that because the gauge transformations in geometric unity, as in typical quantum gauge theories, are unitary at the relevant stage, the resulting mixing matrix must itself be unitary. Preserving probability. That's the advantage, or the main want, of unitarity. In effect, you no longer impose by hand

that psi phi psi can be off-diagonal in generation space. It's forced by the leftover gauge transformations that don't preserve the decomposition here. The GU derivation of neutrino mixing is essentially a copy-and-paste of the quark sector derivation, just applied to leptons, particularly the neutrino mass matrix. The only difference is that quarks and leptons reside in different parts of the 16-dimensional multiplets. Now because of this, the unitary matrix that diagonalizes the quark mass matrix is called the CKM matrix, while the one that diagonalizes the neutrino mass matrix is called the PMNS matrix. The Dirac equation in geometric

unity. Let's begin by recalling the conventional Dirac equation, written as follows, where psi is a spinor field defined on a 4-dimensional spacetime. Here, the gamma matrices satisfy the Clifford relation, and that's what ensures compatibility with the spacetime metric. See, there's always an implicit metric used to define spinors. Next, any other connection A on this bundle can be expressed in the affine space of connections as A0 plus an alpha. Again, this alpha is an adjoint value to one form, representing fluctuations. In conventional Dirac theory, you usually couple a spinor, psi, to a connection A with a

derivative, so the Dirac operator here. However, in GU, the Dirac operator must be defined on the chimeric spinor bundle. And it takes the form of this calligraphic D, where the omega, the subscript, has two components, an epsilon and the variational pi. And that represents the overall gauge data, where we have a gauge transformation, which is epsilon, and a gauge potential, which is that variational pi. And precisely, we define, or Eric defines, this calligraphic D operator as follows. You can clearly see that this acts on a two-component object. Which one? Well, let's just write it as

follows, zeta and nu. The top one, zeta, is a one form, and the bottom one is a zero form, so just regular spinors. Here, the operator is the Shiab, which is designed, or engineered, to perform that generalization of a contraction that Einstein used when forming the Einstein tensor. But at least this time, it's gauge covariant. Now notice that D, the exterior covariant derivative, defined relative to the connection A omega, itself is built from that distinguished A zero, and a fluctuation, so the variational pi. Now see this decomposition into zeta and nu, it seems like it's

different than the original decomposition into this tensor, the spinor bundle of H dual. It actually is consistent, because the decomposition into zero and one forms come about because of the exterior algebra structure on Y. And it later plays a role in distinguishing the different generations. Recall just a couple minutes ago, maybe 10 or 20 minutes ago at most, I talked about the calligraphic D's kernel decomposition into subspaces of the first and the second and third generation. Now when someone says that fermions are quote-unquote square roots of gauge potential, someone like Eric for instance, this is

because the Dirac operator is a linear first order differential operator whose square yields the Laplacian, up to curvature terms, and it's exactly like taking the square root of the Klein-Gordon operator, which leads to the Dirac operator. This isn't language that I use personally. In GU, the Dirac operator, so the calligraphic D, plays a dual role. It not only governs the dynamics of the matter fields, for a given connection, but it also encodes via the square, the meshing of gravitational and gauge degrees of freedom. Let's now derive the gauge covariant form of the Dirac operator in

GU, step by step. So firstly, we start with the chimeric spinor, which is the decomposition of zeta and nu, the 1-form and the 0-form. Respectively, you can think of them as a vector-valued spinor versus just a regular spinor. Step 2, you introduce that distinguished connection A0 on the principal bundle, and you can get that however you like with an observation or what have you. Step 3, you define the covariant differential, coupled to a connection A omega, acting on differential forms valued in the spinor bundle, and the operator satisfies what you see here, which is just

you move it up a form. This obeys the Leibniz rule, and when I say you move it up a form, I mean if you were a p-form, you become a p plus 1 form. Step 4 is you construct the adjoint with the Hodge star operator, which is actually defined because you do have a metric now, the Hodge star requires a metric, or at least it requires a volume form, which usually comes from a metric, which we do have here on y. Step 5, you introduce the Shiab operator, which is a contraction map, and it's constructed

to mimic the contraction of the Riemann curvature tensor into the Einstein tensor while maintaining gauge covariance. And when I say the Shiab, I mean Eric's Shiab, because this is something Eric is introducing, and it's not a standard term in the physics or math literature. Neither is the following. Step 6, you assemble the Dirac-Rarita-Schwinger operator as follows. Also note that this could be called the GU fermionic operator. Step 7, you verify that the composition of the lower and upper blocks satisfy that the square is approximately the following, analogous to the standard property that we have with

the regular Dirac operator, modulo curvature corrections. Step 8, you notice that under gauge transformations of the entire operator, that it transforms covariantly. In other words, if you take a capital psi transformed by an element of the inhomogeneous gauge group, calligraphic G, then the calligraphic D, capital psi, transforms in the same way, preserving the physical content. Step 9, you conclude that the Dirac equation in GU may be written as follows, which covers both the propagation of the fermionic fields and their coupling to the gauge and gravitational degrees of freedom. The Klein-Gordon equation in geometric unity. Many

of the ideas I'm about to lay out are inspired by geometric unity. However, I wasn't able to find specific source notes on it, so this is my best attempt to piece things together. It may not align with how Eric sees it. Let's now turn our attention to the Klein-Gordon equation, which is the conventional way that we talk about spin-zero fields. In standard quantum field theory, the Klein-Gordon equation is written as follows, where the square is that double Laplacian, also known as the d'Alembertian operator, and phi is the scalar field. This equation is inherently second order

in its derivatives, and it comes about from the Euler-Lagrange equation, which is on screen here, and of course we're up to a potential term. So how does this come about in geometric unity? Well first, remember that we decompose the symmetric zero-two tensors as follows with a trace component and a traceless one, and the one-dimensional R subspace, so the trace subspace, is going to be the Higgs-like scalar field. Now in GU, the gauge potential, A with a subscript omega, includes both the usual Yang-Mills vector components and the scalar components coming from the vertical decomposition. When we

form the kinetic term, which is the square of taking the derivative of phi, it comes about from the Yang-Mills-Lagrangean term on screen here, after decomposing the curvature into the parts that are quote-unquote purely horizontal and those that are mixed, or vertical. More concretely, begin with the gauge curvature here. This is standard. Now note that A decomposes into the following. Now, the first part represents the conventional gauge field, like I mentioned Yang-Mills, and the second part is going to be something like the Higgs component. So, then you expand this expression, and you obtain cross-terms. Cross-terms that

are linear with respect to the Higgs, so the variational pi subscript h, and quadratic in A. You'll get terms like the following. That's actually quartic in the variational pi. Now this quartic term then plays the role of the Mexican hat potential in the Higgs sector, typically written as follows. Keep in mind that in Eric's framework, the structure and normalization of this potential are not arbitrary. They instead come from these contraction rules, when one projects the curvature, so F A, onto the vertical subspace of the chimeric bundle. To be precise, we can define an algebraic operator

as follows, which compresses the curvature tensor in a gauge covariant fashion. When applied to the curvature, the operator, P E in this case, yields a symmetric 2 tensor, whose trace reproduces the scalar curvature relevant to gravitational dynamics. And then there's some deviation, and that deviation from the idealized form is what encodes the self-interaction of the Higgs. You can vary the action with respect to phi, and it leads to the following equation, which one recognizes as the Klein-Gordon equation, except with a nonlinear potential. To say this differently, the same contraction mechanism that transforms the full gauge

curvature into a form suitable for gravity, also, when applied to the vertical fluctuations, yields the kinetic and the potential via the quartic self-interaction. And these terms are characteristic of the Klein-Gordon equation for a scalar field. The Einstein field equation in geometric unity. Again, much of this is based on my own reading of Eric's notes and discussions about this, but this is my current understanding, so I'll share it with you. This is about Einstein's field equations. So, in standard form, they read as follows, G plus the lambda with the lowercase g equals the kappa times t.

It's quite familiar to most of you. Now, in geometric unity, you still want to write some Einstein-like tensor, capital G, but defined on the 14-dimensional observers, or the metric bundle. Instead of simply writing the following, Eric introduces a projection operator, which I'm calling P subscript E, which acts on the spin-connection curvature, F of that distinguished connection. Concretely, we have the following. Now, because naive contraction would break gauge covariance, Eric adjusts the vertical Frobenius metric via trace reversal. This ensures that the curvature's quote-unquote traceful part is treated so as to preserve the covariance. Now, the result

is a modified Einstein tensor G, satisfying G plus lambda lowercase g equals kappa t, where the t comes about from the matter fields that are on Y, except when pulled back via an observation. Now, something I was confused about is what is the relationship between this P and the Shiab operator. Well, the Shiab operator is a more general mechanism for contracting curvature in a gauge covariant manner. In GU, one usually denotes the Shiab operator that mixes the torsion-type terms with the curvature to project out a Ricci-like combination. Thus, this P operator can be seen as

an Einstein-specific restriction of the broader Shiab framework, focusing on producing the usual Ricci tensor minus half the Ricci scalar G structure, except while remaining compatible with the inhomogeneous gauge group. Consequently, the final Einstein equation on Y, the larger space, is not a, like I mentioned, a naive contraction. Instead, it's from the Shiab operator or a specific instance of it to get this Einstein-type decomposition. This modified scheme is what recovers a gauge covariant version of G, the Einstein tensor, plus a cosmological term, equated to a derived stress-energy tensor. The Yang-Mills equation in geometric unity Finally, let's

examine the Yang-Mills equation in the context of geometric unity, beginning with the classical expression of DF equals J, where F is the curvature of the gauge connection A and we are on a principal G bundle over spacetime in the standard formulation. Now, again in standard Yang-Mills theory, we're in four dimensions, and the connection A is a Lie algebra G, so a lowercase g algebra valued one form, and its curvature is given as we've seen here ad nauseum, where the wedge product incorporates both the exterior derivative and the Lie algebraic structure. The gauge covariant derivative acts

on sections of the adjoint bundle as follows. Now, under a gauge transformation, the connection and curvature transform like so, which guarantees that the DF transforms homogeneously and makes it already gauge invariant. In geometric unity, however, Eric sees the Yang-Mills as recast, except on the observers, of course, where the gauge fields aren't extensions of some external field, but a component of the chimeric bundle associated with the space of metrics. More precisely, the connection on the principal bundle over Y is written as follows, where we have, like I mentioned, the distinguished Levi-Cevita connection coming about from the

Zorro construction, and the variational pi, which is just a one form that is Lie algebra valued, representing gauge potential fluctuations. The Yang-Mills curvature in Eric's framework is then given as follows, which is a Lie algebra value two form, so there's nothing new here, except that in GU, Eric's trying to solve the problem that the usual contraction of the curvature tensor that appears in Einstein's equations don't straightforwardly commute with gauge transformations as we discussed earlier, and something similar occurs in Yang-Mills theory if one tries to insert the gauge potential directly into the action. To remedy this,

Eric treats the gauge potential as living in an affine space, which is modeled on the one forms, and defines its action indirectly via the curvature, which is manifestly gauge covariant. Now, the Yang-Mills action in GU is defined as follows with the inner product defined by the killing form on the Lie algebra, and the volume being the volume form on Y, which is induced by the chimeric metric. This affine space that I mentioned, the structure guarantees that every term, especially the curvature, transforms correctly under the inhomogeneous gauge group, and I'll just write it here again for

reminder. In other words, while the final expression mirrors the familiar Yang-Mills action, its derivation is fully rooted in GU's framework, so Eric ensures gauge covariance and a natural incorporation of the horizontal and vertical degrees of freedom. When you try and vary that action with respect to A sub omega, the Euler-Lagrange equation gives a familiar Yang-Mills equation form, where the D with the star is the adjoint of the covariant derivative, and J will represent the current from derived matter fields. Now, there's a subtlety here in that GU with the curvature has to further be processed by

the Shiab operator. For clarity, the Shiab operator is defined as an algebraic contraction map designed to replace what I've called the naïve metric contraction present in Einstein's quote-unquote projection, although I see it as a contraction. Concretely, let's say we have a two-form Kazai. The Shiab operator combines other forms, phi1 and phi2, which are built from the geometric data of Y via the construction as follows, where the star is the Hodge star operator associated with the induced metric on Y, and the conjugation by epsilon is what ensures gauge covariance. In effect, this operator is what compresses

the two-form curvature into the lower-degree structure, analogous to a symmetric two-tensor, in a way that commutes with the residual gauge transformations. This construction is baroque and carefully done. It's necessary because Einstein's usual contraction doesn't commute with gauge transformations, leading to inconsistencies when unifying gravity with gauge theory. It's another reason why Eric calls it the ship-in-the-bottle when I was telling him, hey, isn't this just conjugation, which is just something viewed from another perspective? And it's from here that you can see why the ship-in-the-bottle reference is more appropriate, because of, like I mentioned, it's quite baroque and

carefully constructed. My current understanding is that the effective Yang-Mills equation in GU takes the form as follows. I'm writing it like this, but also know that the following equations are already gauge-invariant by construction, and that's why they don't require the same specialized operator. There will be an accompanying PDF to this, so if you would like more notes, such as expansions on these topics and proofs that I wasn't able to get to in this iceberg, then subscribe to this sub-stag, as that's where I will publish it. And that's it. That's all of physics explained in Geometric

Unity. You can breathe now. The rest of this iceberg will just be reprises of the derivations and the concepts that you've already encountered. So congratulations. Before we move on to the next and final layer, layer 4, which will be summaries and open questions, I want to cover some notes on the simplicity of the equations that we've just outlined, that underlie theoretical physics. The Dirac equation, for instance, it's the quote-unquote simplest of its kind. Why? Well, in part, because it generates K-theory. So what is K-theory? K-theory is about the differences between vector bundles. So, sure, we

can add vector bundles, but then the question is, what does it mean for one vector bundle to be subtracted by another? In order to make vector bundles into a complete abelian group, you require a way of talking about their quote-unquote formal differences. This is done by Grothendieck, called the Grothendieck completion of the monoid of isomorphism classes of vector bundles under direct sum. It's quite a hairy term, but the point is that you have something called the index, which generates the K-theory classification of vector bundles through something called the Atiyah-Singer index theorem, which we've talked about,

and the Dirac operator is the simplest operator in K-theory. Now how about the Yang-Mills equation? Well, it's the simplest geometric equation involving the curvature of a connection. What about the Einstein field equation? Well, it's simple in the sense that it's derived from the simplest Lagrangian possible. It's built not even from the curvature tensor, but just the scalar curvature. And also the Klein-Gordon is seen as one of the simplest equations, if not the simplest, of its class. Paradoxically though, the Klein-Gordon equation, it appears to be the most geometric one with its metric hat potential, is actually

the least intrinsically geometric of the four equations that I've just mentioned. Differential geometers often overlook the Higgs sector. Note, what I mean is that the Mexican hat is easily seen to be visually geometric, but not easily seen to be differential geometric. However, to Eric, the Higgs sector comes about as a vertical component of a connection form. So in GU, when decomposing the gauge potential, the Higgs field appears naturally. And the Yukawa coupling is just minimal coupling. Now conventional field theories introduce various vector bundles that are seemingly disconnected structures from the derived spacetime geometry. In general

relativity, symmetric 0-2 tensors decompose, like we've mentioned, into the trace and the traceless component, with the trace component carrying in Eric's formulation the spin 0. So Eric's innovation is that he lifts the frame bundle to the double cover, with the GL double cover, also known as the meta-linear group. And this structure may be new to most of you, even though you're familiar with physics, and it's because mathematicians tend to avoid the double cover, as there's no finite dimensional representation that can do the job of representing spinors. A fundamental group's obstruction exists because the GL4,R is

real. Now GU overcomes this by quickly passing to the spin subgroup, and moving to the observer's rather than working on the base X4 directly. This then is topological rather than metric dependent. Now the connection between these spaces allow the vertical components of the connection to appear as the Higgs field when pulled back to spacetime. Eric constructs these chimeric versions of spinor representations in indefinite signature space, with the spin 7,7 that we mentioned before. And then you reduce it through its maximal compact reduction in one of the paths that I mentioned. And that's how he unifies

previously disconnected geometric structures. Just so you know, if you have any questions, which you likely do, that's alright, I'm going to have a large solo podcast with Eric just on geometric unity. It will be unlike any other podcast because we'll delve into the particularities of it. Especially now that they've been explored in a fair amount of detail. Therefore, feel free to subscribe to be notified of this upcoming podcast with Eric Weinstein. Layer 4. Finally, we're in the deepest layer, which is actually the most accessible now that you've had the previous 2 or 3 hours of

background. Man, you don't see this, but this iceberg took hundreds of hours of work across 10 months on and off. I hope you enjoy it. The feeling that I get from geometric unity, I figured out how to make it into an analogy, is that firstly, if you take a look at the standard model, it resembles a jigsaw puzzle. It's baroque, the edges aren't clean, there are little protrusions. It's unclear where these pieces came from or what the source of the jagged edges, the spikiness, the messiness is. What is clear is that it fits together and

works somehow, except there's this other piece beside it, which is a pristine disk, also known as gravity. And it's unclear how to combine these, they look like they're different elements. One is like that colorful, baroque object that I talked about, the jigsaw, and the other is the gravity, the nice porcelain disk. What geometric unity looks or feels like to me, is if you start with this disk, generalize it, you get, say, a sphere, a perfect pearl. Then, if you allow this pearl to drop on the floor, it will shatter into different pieces, but what's remarkable

is that some of these pieces exactly outline the aforementioned messy jigsaw puzzle. Then you wonder, what are the odds? Look, I was trying to combine these two different pieces, the jigsaw puzzle and this disk, and I couldn't make it work. But actually, they're not just meant to naively be combined, instead they all fall out of the same structure. And, you don't need to do any work to get it to do so. You just let it drop on the floor and examine the pieces. Well, that's the feeling of geometric unity, at least to me. Now, enough

about feelings, let's get to this layer. Right now, I'd like to give you an explanation at four different levels. So, one is the explain like I'm five level, even though I have my issues with that, and you can read this Substack here, where I go into detail about how misleading and foolish this enterprise of, hey, if you can't explain it to a five-year-old, you don't understand it, is. After the ELI five, you'll get the ELI U, so that is explain it like I'm an undergrad, and then after it's explained at the undergrad level, I'll explain

it at the graduate level, and then I'll explain it at the PhD level. Then we're going to get to open questions, and then there's one more treat. Hi everyone, hope you're enjoying today's episode. If you're hungry for deeper dives into physics, AI, consciousness, philosophy, along with my personal reflections, you'll find it all on my Substack. Subscribers get first access to new episodes, new posts as well, behind-the-scenes insights, and the chance to be a part of a thriving community of like-minded pilgrimers. By joining, you'll directly be supporting my work and helping keep these conversations at the

cutting edge. So click the link on screen here, hit subscribe, and let's keep pushing the boundaries of knowledge together. Thank you and enjoy the show. Just so you know, if you're listening, it's c-u-r-t-j-a-i-m-u-n-g-a-l dot org, CURTJAIMUNGAL.org. Explaining Geometric Unity to a Five-Year-Old. In physics today, we have these two primary theories that don't get along well. One is about gravity, so general relativity, and one is also about particles, so it's the standard model. The universe seems governed by particles, but it also seems to be governed by gravity. Since you're made up of both particles, and you

stick to the ground, and you orbit the sun, etc. However, both of these theories, even though they describe the universe, they don't combine well together, and the prefix of universe is uni, which means one. So is there a unification that combines these? That's what Geometric Unity attempts. The key insight from Eric Weinstein is to take a look at this 4D spacetime, and instead of putting a metric on it, so it's actually not even a spacetime, it's just a 4D space, you think of, what are all the possible Einstein theories that can be placed on this?

Mathematicians sometimes call this a modulized space, but technically, in this case, it's a 14-dimensional manifold, what Eric calls the observers. Now in this higher dimensional space, forces, and even the three generations of matter that we observe in the standard model, they aren't added in by hand. Instead, they're engendered from the geometry of this 14-dimensional space itself. Explain Geometric Unity like I'm an undergrad, in math or physics. Start with a 4-manifold. You then think about, how do I make it geometric, since currently it's topological. Now geometric, in this case, means metrical, so adding a metric. However,

you want to be general. You want to think about all metrics. One way of thinking about this is by attaching all possible metrics at a single point, and that is technically called the metric bundle. You then think about, if this metric bundle itself carries a metric, that would be meta, and it turns out it doesn't, but there's a way that you can metricize part of it, namely the vertical parts of the tangent space. You do so with a certain type of metric called a Frobenius metric. The name doesn't matter. It's actually a somewhat natural metric,

and there are two choices here. There's a regular Frobenius metric, and then there's something called a trace-reversed one, or just the reversed one. It doesn't matter, they're both choices. The reversed one is preferred for various reasons. From there, you can then think about what is the signature, because you have a metric now. Now that you have the signature, you can think about the spin group of this signature, so spin 7,7. Now that you have a spin group, you can think about what does this spin group act on, and it turns out one of the spaces

it can act on is a real dimensional space except of 128 dimensions, so r to the 128. Since chiral fermions are complex chiral, and since real vector spaces can always be complexified, Eric complexifies here, and that also splits into two copies of C32,32. Note, I'm not saying anything special, like a Newlander-Nirenberg integrability condition, or that there's the vanishing of a certain Nijenhuis tensor. We're not dealing with complex manifolds. I'm just making the observation that anytime you have copies of r, you can complexify it. That ability is there in the same way that when you have

a manifold, you get with it a tangent space. You don't have to provide anything special to the manifold, it just comes with a tangent space. We also know that a manifold has a cotangent space, just like I said it has a tangent space. However, there is not an isomorphism between the tangent and the cotangent without a metric, or without additional structure like a symplectic form. A metric in this case for the base manifold would be a choice of a section of the metric bundle. Now you can think about why that is, and you'll be able

to convince yourself that it's an equivalent notion. So let's assume that you make that selection of a section. Now although we won't fix what type of section, we'll just say that you choose some section. At that point, you can make an isomorphism between the tangent and the cotangent, and from there, given that your tangent space of the full metric bundle will split into vertical and horizontal parts, you can think of the dual of the horizontal now. I should say at this point that there's plenty of twists and turns, like there's so much to remember, but

the point is that you start from something simple, and you look at what structure is contained within. In many ways, you can think of the claim of geometric unity as the claim of what we think of as simple actually has extreme complexity inside it, and furthermore, a subset of that complexity matches the standard model almost verbatim. Okay, so now we keep moving around in our little space of complexity, and we use a result from differential geometry about spinor bundles, which is that if you have A direct sum B as bundles, and you take the spinor

bundle of that, it's the same as the spinor bundle of A tensor the spinor bundle of B. You can use this along with the isomorphism provided before to form spinors. And now you wonder, well, okay, these spinors are built on the metric bundle, and we live in a base space, or at least we supposedly do, so what do these spinors look like from the perspective of the base space? Now the question about perspective from the base space is the same as a pullback operation in this instance. So that's what we do when we're allowed this

pullback operation, because we've already chosen a section of the bundle. Once this is pulled back, we then get what matches the standard model spinors. As for the gauge group, this one you can see because there's a reduction from spin 7,7 down to spin 1,3 cross spin 6,4. And from there, after moving to the maximal compact subgroup, the second factor goes down to spin 6 cross spin 4, which is Pati-Salam's Grand Unified Theory. In this way, the quantum numbers and the standard model gauge group actually have different origins, and the quantum numbers are fixed, while the

gauge group could have been different, or been broken down differently, theoretically, at least in geometric unity. Now as for the Higgs and other parts of the standard model, those can be seen as different connections on the full metric bundle, but also pulled back. In some ways, you can think of this as wondering about how we have these two different theories, one that describes particles, and one that describes gravity. Gravity is like a simple dove, in that it's beautiful and innocent, and the standard model is like this tortuous snake, in that the standard model is effective,

but it's convoluted. People have been trying to put these together for decades, and what Eric has found is that if you look at gravity itself and generalize it, the particles come out. You can visualize the 4-manifold as a chia pet, which grows fibers naturally, and that living on the blades of grass are the different particles we're looking for. Be as innocent as doves, and as wise as serpents. Firstly, we have to think about what's the goal. So to explain to a graduate student, I'm going to use Eric's frame of mind. What's the goal of Eric?

Eric is thinking, how do I resolve the chicken and egg problem of quantum gravity? That is, how can matter, which is described by spinors, exist between metric measurements if you require a metric in order to define the spinors? The answer, according to Geometric Unity, is that matter lives in the observer's, in this 14-dimensional metric bundle. And spinors can exist there, even though you're not making a canonical choice of spacetime metric. So, you start with a 4-manifold, X4, GU then constructs a metric bundle, which is on screen, and at each point, you have the decomposition in

the tangent space as follows. With the vertical space representing the metric variations. Eric then uses a certain construction, where if you choose a section of this metric bundle, it then gives you a connection, the Levi-Cevita connection. Now that metric on Y induces a connection itself. And this is what a choice of a horizontal subspace means, a connection. Now the chimeric bundle combines the vertical space, which has the signature 4,6, with the dual of the horizontal space. So the signature, 1,3, and that gives the total signature of 7,7. I should note once more that spinors are

derived on this chimeric bundle, which has a metric without requiring a connection. Now because this chimeric bundle is isomorphic to both tangent and cotangent bundles of Y, you get spinor bundles to exist prior to choosing a metric on the base space, X, or even a connection on Y. But once you do select a connection, you become canonically isomorphic. This is that Zorro construction, and it's a constitutional mechanism that powers geometric unity. The spinor bundle on the space then decomposes as follows. There is something else called the augmented torsion tensor, and that is supposed to resolve.

Again, we have to think, what is Eric thinking is the problem? What is the goal? Well, Eric's thinking, I'm looking for a map from the inhomogeneous gauge group to the space of connections, which is equivariant under this right action, and equivariant on the left-hand side as well. This is what becomes dark energy. Eric sees another problem, which he calls the gauge incompatibility problem. What that is, is that in Einstein's formulation, the Riemann curvature tensor is viewed as a 2-form taking values in a 2-form. So essentially, you have two copies of a 2-form, one from the

connection, and then one from the metric structure. When you contract this tensor to produce the Einstein tensor, you're treating both copies symmetrically. However, that makes it not transform properly under gauge transformations. This mismatch in the origins of the curvature components is what Weinstein, Eric Weinstein, refers to as the twin origins problem. So this augmented torsion tensor is one way of solving the problem with gravity that is not gauge covariant, but gauge theory is gauge covariant, so how do you make gravity gauge covariant? Eric then introduces a Shiab operator, which generalizes that contraction I referenced about

Einstein in a gauge covariant manner. The formula is on screen here, and then there's a Dirac-Rorita-Schwinger complex on this larger space. This generalizes the Diram complex, twisted by spinors in dimension 3. Now this complex gives three distinct sets of fermions. The scalar spinor, which gives the first generation, and then there's a vector spinor, which splits into two parts, a gamma trace part, which gives the second generation, and a gamma trace-less part, which gives the third. Each of these is 16-dimensional, and that matches the standard model's fermion structure. In particular, it matches spin 6,4, which is

an alternative real form of spin 10 complexified, and that's closely related to the well-known SO10 real form from grand unification theory. Hi, Curt here. If you're enjoying this conversation, please take a second to like and to share this video with someone who may appreciate it. It actually makes a difference in getting these ideas out there. Subscribe, of course. Thank you. Explaining geometric unity at the PhD level How about rather than quantizing gravity directly on x4, which gives renormalization problems, how about we work with fields on y14, that metric bundle. Then from there, we're going to

pull it back to x4 via sections of the metric bundle. Geometric unity's foundation is in constructing principal bundles over this y14, so principal bundles on principal bundles, this time with groups spin 1,3 and spin 7,7 and u64,64 as the structure groups, and we have a chain of inclusions on screen here. There's also something called inhomogeneous gauge group, which is on screen here, where the first factor is a genuine gauge transformation, and the second is actually just gauge potentials. Now this inhomogeneous gauge group calligraphic G combines the gauge transformations with the translation, and the space of

connections with a certain product structure given on screen as follows. Now let's say for any connection and any element inside this inhomogeneous gauge group, we also have a right action. That right action is defined as follows. The augmented torsion tensor, that is what gives a gauge covariant object that transforms correctly under both gauge and diffeomorphism symmetries, resolving that gauge incompatibility problem of the Riemann tensor. The Shiab operator generalizes Einstein's contraction in a gauge covariant manner, and I'll put the formula on screen here again. The action principle, because we all want to know what is the

action, what's the Lagrangian, that takes the form of a first order equation reminiscent or it rhymes with Einstein-Hilbert and even Chern-Simons theories, and that's on screen here. The field equations are derived from this action. The term involving the torsion T is what ensures the Euler-Lagrange current remains exact for an action function I, which is important to note because DI yields the Einstein replacement equation. This exactness property is what allows Eric the geometric formulation of the field equations. There's also a quadratic term, and that maintains the gauge covariance. The Dirac-Rarita-Schwinger operator takes the form on screen,

and it acts on regular spinors plus spinor-valued spinors. This operator on this space here comes from Eric thinking differently about supersymmetry. There was something pioneered in the late 70s by Salam and Stratti about constructing superfields, which are just fields defined on superspace. This construction, in Eric's mind, was erroneously applied to Minkowski spacetime, rather than the space of connections. To Eric, this explains why there's been so much time and money and effort wasted on finding superpartner particles on spacetime, because superpartners exist in connection space. The decomposition into three pieces occurs prior to the kernel operator, each

16-dimensional. And now the first generation is the regular spinor-valued function, so the zero forms, and the second and third come from the decomposition of the one forms on the spinor field, and those break up into gamma trace and gamma trace-less parts. The Higgs field comes from the vertical component of the gauge potential, so the variational pi, and when you pull it back to your base space, you get three parts. One that looks like a gauge potential downstairs, so that is to say an add-valued one form, interpretable as such on X4, the base space. Additionally, you

get a spin-zero piece downstairs, which gets interpreted as the Higgs field in Eric's framework, and you get a remainder piece that has not been identified or not seen and doesn't figure into our experimental observations in the physical universe so far. Just remember, there either is right now or is going to be an accompanying PDF to this Geometric Unity iceberg, so if you'd like more notes, such as expansions on these topics and proofs that I wasn't able to get to in this iceberg, then subscribe to my Substack, as that's where I'll publish it. It's CURTJAIMUNGAL.org, or

c-u-r-t-j-a-i-m-u-n-g-a-l.org. Open questions. Soon I'm going to do a full recap of the whole 30 steps of Geometric Unity, except in a flyby overview, just to hammer the point home further and show you that all of this initially abstruse math is now understandable by you. It's actually, in part, to congratulate you. At first you heard this foreign language, and now you're able to be somewhat conversant in it. You can order dinner. You can go back to your hotel. You can pick up a date. I'm taken, by the way, but thank you. Now first, let me talk

about what open questions I have. One of my questions is, what is the phenomenology of the theoretically predicted, but currently unobserved, decoupled sectors, quote-unquote. I put them in quotations because I've heard Eric call these dark sectors, but I find that terminology to be too evocative of dark matter or dark energy, so let's just call them decoupled sectors. Would gravitational interactions at high energies enable us to have direct or indirect observations of these? How? Okay, that's one question. My next is about how does Geometric Unity account for the matter-antimatter symmetry-slash-asymmetry. Is this observed asymmetry explained by

disconnected chiral sectors, where the antimatter is hidden, so it's a decoupled chiral sector? Or is the asymmetry still an initial condition? And another question I have is, given that Eric has squeezed plenty out of these numerical coincidences, particularly about spinor structures like SL10 and SU5, their connections to the Einstein field equations, the metric in four dimensions, I believe Wilczek actually remarked about this. So given that, and there's also the coincidences of the observed generations of fermions being precisely 16 fields, etc., how far do these numerical coincidences go? What other numerical coincidences are meaningful? So what

about Dirac's large number hypothesis? What is a red herring versus a smoking gun? Also, you should know that I'm going to be speaking with Eric directly on the podcast in the next couple weeks, so if you have questions, leave them below and feel free to subscribe for that conversation, as we'll go further in depth into Geometric Unity. At this point, I'd like to emphasize that I don't want you or other people to think that there's something negative given that they're open questions, as every single theory has open questions. For instance, here are some of my

notes about open questions in string theory, in loop quantum gravity, and in asymptotic safety. Eric's theory is a tour de force, and unless you have an understanding of physics, it's difficult to fully appreciate how many pieces there are in this one theory, despite the open questions that I mentioned, which this theory has been generated by a single person in isolation. Now believe it or not, what I've shown you for the past 3 hours or so still leaves maybe 30-40% of GU unexplored. Just so you know, if you have any questions, which you likely do, that's

alright, I'm going to have a large solo podcast with Eric just on Geometric Unity. It will be unlike any other podcast because we'll delve into the particularities of it, especially now that they've been explored in a fair amount of detail. Therefore, feel free to subscribe to be notified of this upcoming podcast with Eric Weinstein. Now, let me give a recap of the 30 steps of Geometric Unity for the interested viewer who cares about the steps, and this is a treat cause you'll be surprised how much of this finally makes sense. Maybe not all of it,

but initially maybe 5% made sense. So number 1, you begin with a smooth 4-manifold, say X4. You then construct its metric bundle Y14, where each fiber is the symmetric 2 tensors on X4, and that happens to be 14 dimensional. Number 2, you construct the frame bundle with the structure group GL4, then you lift to its double cover to enable finite dimensional spinor representations. Next, number 3, you define observation maps which are actually sections or local sections of this bundle. Next, you construct the tangent bundle of the metric bundle, and you also construct its dual, so

its code tangent bundle. Next, you split the tangent into its vertical bundle, which encodes the metric variations, and the horizontal part, which is just directions along the base space. Next, you use a certain construction which Eric happens to call the Zorro construction. It's a way of getting a section from the base space to the metric bundle, so you choose a section, you're choosing a different slice of this whole metric bundle. How do you get from there to a connection on the metric bundle? That's what this construction is about. Next, you define the chimeric bundle, which

is the vertical space directs on the dual of the horizontal. Next, step 8, you introduce the Frobenius inner product, and that acts on symmetric matrices to metrize the vertical fibers. Next, number 9, you decompose the symmetric 0,2 tensors into a trace and traceless part, and you choose a signature. This is one of the only parts in geometric unity where you make an actual choice that wasn't there implicit in the structure. In this case, you choose 4,6, and that's for the vertical part, and you choose 1,3 for the h part, the horizontal part. That gives an

overall 7,7. Number 10, you construct spinor bundles, but you do so on the chimeric bundle, and you use that exponential property as follows on screen. Next, you identify the structure group of spin 7,7, who has a real spinor representation of 2 to the power of 7, so 128 dimensional, and that splits, because you can divide that into two, into two chiral 64-dimensional real spaces, but you have to complexify. Now, number 12, from the principal bundle, with the structure group of U of the unitary matrices, 64,64, via the spinor representation here. Then, you define an inhomogeneous

gauge group, where the first part are actual bona fide gauge transformations, and the second are just gauge potentials. You just semi-direct product them together. Number 14, you specify a right action on the affine space of connections, and you do so with the following somewhat hairy formula, but not too hairy, and this formula has group associativity to it. Number 15, you define the augmented torsion tensor. This is the gauge equivariant analog of Einstein's contraction operator, and this addresses Einstein's quote-unquote gauge problem, or more accurately, it addresses Weinstein's rendition of Einstein's gauge problem. Number 16, you introduce

the Shiab operator. That acts on two forms, and it's defined on screen here, and that's the generalization of Einstein's contraction. Number 17, you formulate a first-order action. So, this is the action of geometric unity, as far as I can tell. Now, 18, you vary the action to derive field equations, and this is what gives you something that's analogous to Einstein's field equations, and also Yang-Mills. Number 19, you introduce fermion fields as a spinor-valued form. So, there's firstly a zero form, which is just a regular spin field, and then there's a spinor-1 form here, and that

gives a Dirac-Rarita-Schwinger operator, coupling bosonic and fermionic sectors, and that's why Eric sometimes refers to this as supersymmetry. I'm not a fan of that term, because supersymmetry means something in particular, but Eric is taking the analogy here of interchanging fermionic and bosonic degrees of freedom as supersymmetry, instead of the supersymmetry algebra. Number 20, construct the deformation complex here. This has a cohomology, which classifies gauge-inequivalent linearized fluctuations about a solution. Now, number 21, you see that there's a seesaw structure inside this Dirac-Rarita-Schwinger complex. And that's what explains mass hierarchies, mixing between light and heavy spinors. Number

22, you can do a reduction. So, you go from spin 7,7 to that spin 1,3, and you cross that with spin 6,4. You can do some further reductions, as I've described earlier on, and that recovers the standard model gauge group. Number 23, you decompose the Dirac-Rarita-Schwinger complex, and again, like I said, there's a zero form and there's a one form. The one form splits into two different parts, a gamma-traceless part and a gamma-trace part. One of those is the second generation, and the other one is the third generation, whereas the zero form is just the

first generation. Number 24, there's a Higgs field. Where is this Higgs field? Well, when you take the variational pi, the one form, on the principal bundle, it has a scalar part, and that corresponds to the Higgs field upon pullback. Number 25 is that the symmetric zero-two tensors themselves, which have a trace and a traceless component, there's a trace component here, which gets identified with the Higgs, and the traceless part becomes identified with a spin-2 graviton-like field. Number 26, you can derive a quartic potential when you do some expansions, and that was covered earlier, from the

Yang-Mills term, A wedge A squared, and that actually gets you a Mexican hat potential for the Higgs field. Number 27, you show that there's a minimal coupling in the Dirac operator, so on screen here, and that gives you Yukawa interactions when the A, the connection here, includes Higgs components. Number 28, there's a correspondence between the Higgs and the Yang-Mills sector, because you decompose this variational pi, and that gives you this unified origin of Higgs and Yang-Mills in this geometric structure of the metric bundle. This isn't even a step, it's more like a realization. Number 29,

you incorporate a quadratic term into the field equations, and this gives full gauge covariance analogous to Yang-Mills self-interactions. And finally, 30, you can decompose that gauge potential, like we talked about, under the pullback, and its constant component produces an effective term. This then gets identified with the cosmological constant. And that's all of fundamental physics. All right, thank you for coming along with me on this journey through fundamental physics, and its potential unification. Thank you to Eric Weinstein for providing this theory, and giving me plenty to chew on over the past few months. Thank you to

all the video editors who helped edit this together, even though all of this probably look like and sound like hieroglyphics to you, or at least look like hieroglyphics don't sound like anything. If you're interested in more icebergs like this, you should know I have a string theory iceberg, where I outline the math at the graduate level for string theory. I also have one where I cover different theories of consciousness. Soon, I'll be doing interpretations of quantum mechanics, as well as an iceberg on algebraic geometry. So feel free to subscribe. Again, thank you to Eric Weinstein.

It's an avant-garde and creative theory. Curt here, several months later, this has been so long in the making. Geez, you have no idea. Anyhow, I wanted to say that I mean what I just said. I may have said this before in the iceberg, and if I haven't, I should have because it bears repeating. I haven't seen a theory like this come from any single individual ever. Not one that's this fleshed out, or has this amount of unexampled connections within itself, as well as to what's known as the theoretical physics backbone that we talked about earlier.

And by the way, this is what I do for a living. I interview people on what theory they have of reality, whether it's of consciousness, or it's physics-based, or logic-based, or what have you. So again, thank you to Eric, and thank you to you for watching this. I hope you enjoyed it. I've received several messages, emails, and comments from professors saying that they recommend Theories of Everything to their students, and that's fantastic. If you're a professor or lecturer, and there's a particular standout episode that your students can benefit from, please do share. And as always,

feel free to contact me. New update! Started a Substack. Writings on there are currently about language and ill-defined concepts, as well as some other mathematical details. Much more being written there. This is content that isn't anywhere else. It's not on Theories of Everything. It's not on Patreon. Also, full transcripts will be placed there at some point in the future. Several people ask me, hey Curt, you've spoken to so many people in the fields of theoretical physics, philosophy, and consciousness. What are your thoughts? While I remain impartial in interviews, this Substack is a way to peer

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