A dialogue from X

@MPorter2025 

In GU, the physical metric on Y is not the canonical metric 

@MPorter2025

More precisely: you have a canonical metric on the fibers, but you also need to choose a specific metric on the base space, to define the "horizontal" part of the metric on Y that connects the fibers.

@YinMills

Hey, which part of @TimHenke9 's criticisms is that addressed to? The initial reply where he said that it's cool that Met(M) seems to come free with a canonical metric? I'm just trying to figure out what's going on here.

@MPorter2025

Mostly I'm stating my understanding of g_Y in Geometric Unity. You need a choice of g_X to determine the horizontal parts of g_Y... Incidentally, just to bring in physics, the Pati-Salam group is supposed to emerge in GU as a compact part of the isometry group of the fiber metric

@YinMills

Right, so you have the space Y of all metrics on X, and you are saying you need to pick a metric on X to get a metric of metrics on Y? Do I have that right? As an aside, why did he define ι like that? Why couldn't he just have it be any embedding of X into Y?

@MPorter2025

Y is metric bundle over X, which is organized differently to space of all metrics on X. Space of all metrics is ∞-D, each X-metric is just a point in that space. Metric bundle is 14-D, consists of fibers above the points of X, each X-metric is a "section" through all fibers...

@YinMills

I keep being imprecise with words, I'm sorry. By "the standard embedding of X in Y", I didn't really mean anything specific, I just meant why does he define this weird construction that takes various open sets in X to Y, what does he gain as opposed to just taking a section?

@MPorter2025

Possibly a nod towards the most general version of his overall idea. See 2021 paper, pages 16-17. (And compare with 01:06:19 in Oxford lecture.) In principle GU via the metric bundle is just one form, in practice it's the only one he talks about. "Open sets U" never show up again

@YinMills

That is, why doesn't he just say that the observerse is a space X and a bundle Y on X, and a section on that bundle ι?

@MPorter2025

In practice, that's what it is. Please see the references I gave, if you want to see him talking about how bundle-theoretic GU is just one possible version. If (like me) your main interest is the actual proposed theory, stick with the idea that ι is a section of the metric bundle

@YinMills

Alright, that makes things simpler. So in practice we just have a space X and a metric on it, that can be many things. What would you say the juice is? It's very confusing to read his paper because it's full of very math-y terminology and weird definitions like this.

@MPorter2025

I found it hard to get into too. My primary focus has been on forming a picture of what is coupled to what. You have Y, you have X which is a brane or a defect in Y, you have fields on Y. g_Y is a kind of uplift of g_X, and...

@MPorter2025

the 4-D non-gravitational fields are the bulk fields from Y restricted to X. I also had to suppose an Einstein-like coupling between those fields and g_X in order to close the loop. That's basically how I think of it at present...

@MPorter2025

and then it's just a matter of trying to fill in the details. GU's quantum theory is up in the air, but I think the classical theory of a GU bosonic sector could be defined and solutions studied, right now. Part of the interest is simply that this is an unfamiliar form of theory,

@MPorter2025

so I'd like to see how it fits into the broader world of field theories. The bulk theory on Y is like an uplift of Poincare gauge theory onto the metric bundle, the shiab operator is meant to imitate the Einstein tensor in a gauge-invariant way inspired by Chern-Simons theory,

@MPorter2025

there are various ways you can try to solve the problems like anomalies and nonrenormalizability, there's just lots to think about.

@YinMills

Alright, but does that actually give the fields of the SM with the correct interactions? If it does then it's interesting, otherwise I'm not sure what utility it has. I mean, anyone can decide to use any old space as base space for a field theory.

@MPorter2025

I can tell you some of how it's supposed to work. There's a gauge field on Y (with a large complexified gauge group), coupled to scalar, spinor, and spinor-vector fields. Part of the gauge field corresponds to spin group of the fibers above X, which have metric signature (6,4)...

@MPorter2025

... thus Spin(6,4). The spinorial fields are supposed to give rise to the known fermion generations when restricted to X, and the hope is that some mechanism reduces the spin group to its compact part Spin(6) x Spin(4), which is the Pati-Salam group.

@YinMills

Is it supposed to have a graviton too?

@MPorter2025

Maybe only in 4D EFT. The 14D gauge field is presumably more fundamental, e.g. in GU, the 4D Einstein tensor descends from the "augmented torsion" of that 14D field (augmented torsion being a gauge-covariant form of torsion).

@YinMills

Does the 14D theory have some notion of Poincare invariance?

@MPorter2025

GU is a 14D gravi-GUT theory in which the 14D gauge group includes Y's local Spin(7,7) symmetry, which splits into a local Spin(1,3) symmetry on X, and Spin(6,4) on the fibers. g_Y has (7,7) signature and is a block-diagonal product of g_X and the fibers' canonical metric...

@MPorter2025

The 14D gauge group is a semidirect product of GL(128,C) with translations in field space. This is done so that a gauge-covariant torsion can be defined. Apparently something similar is done with torsion in Cartan gravity...

@MPorter2025

This semidirect product structure is why I said it's like an uplift of Poincare gauge theory to the metric bundle, though as far as I can tell there's no relationship between the "translations in field space" and spatial translations on X.

@YinMills

I'm asking because there might be problems with Coleman-Mandula and the like.

@MPorter2025

There's Weinberg-Witten to think about too! But I'm still digesting the big picture in GU.

Invisible college

Two months ago, I mentioned that I had just been a guest on a GU-themed Youtube livestream. That channel is offline at the moment, but while it existed, it attracted quite a few people interested in GU, including some with mathematical knowledge. Notably, one pseudonymous commenter posted a reformulation of GU in terms of some higher-math concepts (derived geometry, shifted symplectic structures) that are also employed in the well-known "Geometric Langlands" research program. 

It is therefore interesting to see Langlands theorist Ed Frenkel stirring up physics twitter by praising his friend Eric's theory. Was that him lurking among the Youtube rabble? Probably not, lots of people know Langlands. In any case, he and Eric just spent a week going over GU together. We shall see if anything new comes of it. 

In related news, Eric had a public conversation with Elon's AI, Grok, about gauge-theoretic economics, which like Frenkel's declaration, attracted hostile attention from those who took for granted that Eric's theories were already refuted and buried. 

A while back, Eric said that, while SpaceX is the part of Elon's empire that is working on interplanetary travel, xAI is the part that is working on interstellar travel, because it's allowing fair discussion of GU to occur, despite the academic disinterest. I admire the creative chutzpah by means of which Eric thereby assimilated one of Elon's central ventures to his own agenda! 

I suppose I did something similar when I suggested that general relativity, string theory, loop quantum gravity, and geometric unity together form a natural quartet (metric-based and connection-based, theories of gravity and theories of everything). As he says in the conversation with Grok linked above, he has done "important work that has never been fairly digested by the fields in which it occurs", and that's my model of the proper context in which GU should be assessed. 

The big picture

Let's review the big picture... There are three bodies of formal work from Eric Weinstein that interest me: 

(1) his physical theory of everything, Geometric Unity, including the principles that motivate its form 

(2) a potential mathematical framework that includes variations on GU physical theory, and should include a gravitational derivation of the Seiberg-Witten equations

(3) coauthored with his wife Pia Malaney, a gauge-theoretic approach to economics. 

I also have some interest in his sociology of knowledge (DISC versus IDW), how his views compare to those of Peter Thiel, and so forth. But my main focus is on the formal work, and especially on the physics. 

The major source of information on GU is a 2021 draft paper. The 2013 Oxford lecture and its 2020 addendum also provide some context. The 2025 UCSD seminar on dark energy contains glimpses of how theory has evolved since 2021, and another scrap of information (about the fermionic sector) was recently revealed, in a public discussion between Eric and Elon Musk's "Grok" AI. 

The public discussion of the theory has been formal, in the sense that it focuses on motivating and constructing equations for the theory, but not on solutions to those equations. As models of what solutions might look like, I suggest the work in Eric's Harvard dissertation; the Seiberg-Witten moduli space but with the SU(2) gauge field interpreted as a gravitational connection; and the "BRSTQFTs" constructed in a 1997 paper by Baulieu et al, which build on Eric's dissertation. These are also potentially all part of the mathematical framework mentioned above. 

(Incidentally, in spaces online where the theory is discussed, I have run across efforts to reformulate the theory in terms of "derived geometry", which would be a significant broadening of the GU mathematical framework.) 

As a new starting point, I have suggested construction of cosmological vacuum solutions to the bosonic part of classical GU. The inclusion of fermions may require creative resolution of issues surrounding the quantum theory. 

Working in base time

My thinking about Geometric Unity has been centered on 14 dimensions for a long time. That is, I thought of the physics unfolding in 14 dimensions as primary, and of the 4-dimensional world as akin to a defect or a braneworld located in that background. 

However, I have found my way to a perspective centered on 4 dimensions. The reason is the multi-time metric signature of Y. Multi-time backgrounds usually spell trouble, and yet GU prefers a (7,7) signature for Y, so I thought I'd just have to deal with it. 

However, eventually I noticed that the preferred metric signature for X is (1,3), the usual Lorentzian signature with one time dimension. At the same time, I finally understood the dependence of Y's metric on X - you use a canonical metric within each vertical fiber, and then you stitch those fibers together horizontally, using X's Levi-Civita connection as the template. 

This opens the possibility of grounding everything in single-time evolution on X. Possibly the other timelike directions in the fibers can then be dealt with, through a combination of gauge-fixing and constraints. 

I've also been consolidating my understanding of other aspects of GU. For example, I've noticed that the equations of motion for the spin-1 component of the unified field on Y are basically "curvature = torsion", except that (1) the curvature is sheared of its Weyl-tensor-like component by the shiab operator (2) the resulting "swervature" and the torsion are then made gauge-invariant. 

At the other end of the scale in terms of complexity, I'm getting to know what happens to every component of the 14-dimensional spin-1/2 and spin-3/2 fields. Diagram 11.6 in the GU paper depicts how they end up in "F", "Q", and "Z" multiplets, but I want to know the implied interaction terms as well, e.g. 12.28 tells us that the 4-dimensional yukawa couplings are supposed to descend from the VEVs of certain 14-dimensional bifermion operators. 

History and analogy

There's a lot to talk about. But in this post I just want to mention two perspectives. 

The first is Geometric Unity's place in the spectrum of approaches to quantum gravity. I propose a fourfold way of thinking about it: that Geometric Unity is to Loop Quantum Gravity, as string theory is to general relativity. I like this as a way of thinking about the options in gravitational theory: metric-based versus connection-based approaches to relativistic gravity and the theory of everything. 

The second is to think of Weinstein versus Witten as similar to Schopenhauer versus Hegel. Hegel was the best-known philosopher in Europe, Schopenhauer was a young unknown, yet he dared to schedule his lectures for the same timeslot as Hegel. No-one came and he left academia, and it took thirty years for him to be noticed again; but in the end his work made its mark, not least by helping to create Nietzsche. 

A paradigm for GU's quantum theory

I have struggled to figure out how quantum GU works. By this, I don't mean the various technical challenges that hang over GU, but the more basic question of which parts of GU are "classical" and which parts are "quantum". 

Over the weekend, I was a guest on a GU-themed livestream by Youtuber "Uncompetative", and I tried to explain this problem. Now I think I have an approach. I'm not saying this is the only way to make GU quantum-mechanical, but it is something concrete enough that one can try to make it work. 

The field equations for GU are 14-dimensional, they describe fields living in a 14-dimensional manifold "Y". There is also a 4-dimensional manifold "X", on which we can choose a metric. The first step to connecting X and Y, is that Y is also the metric bundle of X, meaning that to choose a metric on X is to choose a 4-dimensional section of Y. 

So if we have a 14-dimensional solution to the GU field equations, this gives us a way to obtain something 4-dimensional: choose a metric, i.e. a 4-dimensional section of Y, and then consider the restriction of the 14D solution to that 4D submanifold. In effect, this tells us what all the non-metric fields on X (gauge fields, matter fields) are doing. 

The problem with this prescription is that the choice of metric is disconnected from the evolution of the non-metric fields. Everything non-metrical is figured out by dynamics on Y, and then you make an arbitrary choice of metric, to get physics on X. This is unlike general relativity, in which metric and non-metric are in constant mutual interaction. 

But I have omitted part of GU: the "Zorro construction"! 

When I first studied GU, I thought (in effect) that perhaps there was no metric on Y, and that the 14D theory was a topological theory. This is wrong. Also, there is actually a canonical metric for the whole of Y, arising from its interpretation as metric bundle of X. Theoretically, one could study the GU field equations on that background, which is the same no matter what the metric on X is. 

However, in GU, you're also not supposed to use the canonical metric on Y. Instead, you choose a metric for X, corresponding already to a particular section of Y. Then you will extrapolate the Levi-Civita connection on X to a Levi-Civita connection on Y, and from that construct the metric on Y. 

This establishes mutual interaction between X and Y. X determines the metric on Y, which is the background on which the GU field equations must be solved, and then Y determines via pullback, what the non-metric fields are doing on X. 

Now if we're trying to recover general relativity, one has to hope that the energy-momentum distribution of these 4D non-metrical fields, is consistent (via Einstein field equations) with the 4D metric we chose at the start. At present I don't see why that would be so. Maybe some of the GU cohomological complexes are meant to enforce such a relationship. But at least I now see how to "close the loop" and make the choice of metric relevant for the behavior of the non-metrical fields. 

As for the quantum theory, maybe it's still not fully fleshed out. There are still choices to be made. For example, you could treat X as completely classical, and have all the quantum uncertainty on Y. Or, you could have a wavefunction on X as well as on Y. What remains to be checked, is whether any version of this, gives us the desired consistency between metric and non-metric ingredients. 

Der Weinstein-Streit