## December 1, 2021

### Mysterious Triality

#### Posted by David Corfield

When we started this blog back in 2006 my co-founders were both interested in higher gauge theory. Their paths diverged as Urs looked to adapt these constructions to formulate the elusive M-theory.

Over the years I’ve been following this work, which has taken up a proposal by Hisham Sati in Framed M-branes, corners, and topological invariants, Sec 2.5 that M-theory be understood in terms of 4-cohomotopy, culminating in what they call Hypothesis H. I even chipped in sufficiently to one article to be included with them as an author:

Philosophically speaking, I’ve been intrigued by the idea that the novel mathematical framework of twisted equivariant differential cohomology theory, required for Hypothesis H, may be formulated via modal homotopy type theory. This is the line of thought I mentioned a few weeks ago in Dynamics of Reason Revisited.

But I’ve also been thinking that you can’t add something important to fundamental physics without it causing ripples through mathematics. So I was interested to see appear yesterday:

The authors take mysterious duality which links del Pezzo surfaces with compactifications of M-theory, and throw in a third pole, iterated cyclic loop spaces of the 4-sphere, motivated by Hypothesis H. Now we have a mathematical mystery linking the algebraic geometry of del Pezzo surfaces with the algebraic topology of iterated cyclic loop spaces.

I’m sure there will be many further mathematical ramifications, but it’s intriguing to see in this paper two elements of an association I first heard of in a talk by John McKay that I reported here in which he related the triple of exceptional Lie groups $(E_8, E_7, E_6)$ to the triple (120 tritangents on sextic of genus 4, 28 bitangents on quartic, 27 lines on cubic). This was written up in

Since exceptional structures are so tightly inter-related, perhaps we should expect McKay’s triple of sporadic simple groups, (Monster, Baby monster, Fischer group $Fi_{24}$), to appear.

Posted at December 1, 2021 8:04 AM UTC

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### Re: Mysterious Triality

Everyone is aware that the diffeomorphism group of the 4-sphere is very interesting, cf eg

Watanabe, https://arxiv.org/abs/1812.02448

or for ex

Gay, https://arxiv.org/abs/2102.12890,

right?

Posted by: jackjohnson on December 1, 2021 10:29 AM | Permalink | Reply to this

### Re: Mysterious Triality

The nLab was aware of Watanabe’s article, but not Gay’s. That’s remedied now at least, thanks.

So the former is showing that the 4d analogue of Smale’s conjecture is false.

Posted by: David Corfield on December 1, 2021 4:54 PM | Permalink | Reply to this

### Re: Mysterious Triality

In the questions after Sati’s talk he remarks (around 1:36:40) that $S^4$’s appearance here is related to 4-manifold theory.

Posted by: David Corfield on December 8, 2021 8:35 PM | Permalink | Reply to this

### Re: Mysterious Triality

But I’ve also been thinking that you can’t add something important to fundamental physics without it causing ripples through mathematics. So I was interested to see appear yesterday:

It is very likely imo that the effects of [Urs’s efforts to formalise and make rigourous M-theory] on mathematics is going to be much greater than on physics, simply because, unlike the case with quantum field thoery, there isn’t really anything connecting M-theory and string theory in general to the real world, while there is a whole wealth of interesting and complex mathematical structures and techniques in M-theory and string theory that would be very useful in i.e. algebraic geometry.

### Re: Mysterious Triality

I don’t care to rerun the String Wars here, but the credit due to a physical theory for generating mathematical insight is something I haven’t taken up since my 2003 book, Chap. 5.

Posted by: David Corfield on December 1, 2021 4:33 PM | Permalink | Reply to this

### Re: Mysterious Triality

Looking again, it was my Chap. 6 ‘Uncertainty in mathematics and science’, which includes Michael Atiyah’s claim:

quantum field theory has had its credibility enhanced by its success in making correct mathematical predictions. Given the lack of rigorous foundations for quantum field theory, these successes provide great encouragement to physicists that their ideas are fundamentally sound. (Atiyah 1995: 6070)

1995, ‘Quantum Theory and Geometry’, Journal of Mathematical Physics,36(11),6069–72.

The discussion was largely about how to consider mathematical predictions turning out to be correct in a Bayesian framework when Bayesians typically don’t typically take into account mathematical learning.

But I think there are larger issues about the conceptual power of certain ways of thinking, as described by Atiyah, Dijkgraaf and Hitchin in Geometry and physics:

Whatever the successes and failures of string theory or supersymmetry in the experimental domain, it is clear that the impact on mathematics, and on geometry in particular, is permanent. Yet, there is no evidence of the existence of a hybrid between the theoretical physicist and the mathematician. It may be that string theorists can be accused of producing theories without any practical evidence, but what they feed to mathematics is firmly rooted in a particular view of the universe and what drives it, a view that is not yet part of a mathematician’s development.

Posted by: David Corfield on December 2, 2021 11:14 AM | Permalink | Reply to this

### Re: Mysterious Triality

There is a relation, due to Coxeter, between the 27 lines on a cubic surface and a maximal set of equiangular lines in $\mathbb{C}^3$. In short, one identifies the 27 lines with the 27 vertices of the “Hessian polyhedron” and then reads their coordinates projectively. There is also a more subtle relation between equiangular lines in $\mathbb{C}^8$ and the 28 bitangents to a general quartic; I say “more subtle” because you consider the pattern of orthogonalities between two conjugate sets of 64 equiangular lines. (I covered this in guest blog posts here and here, and in somewhat more polished form in this monograph.) One thing I was never able to figure out was whether the 120 tritangent planes had anything to do with complex equiangular lines.

Section 4 of the He and McKay paper linked above discusses the Horrocks–Mumford bundle, which is kind of the analogy in dimension 5 for the Coxeter set of lines in dimension 3.

Posted by: Blake Stacey on December 1, 2021 10:26 PM | Permalink | Reply to this

### Re: Mysterious Triality

How can a post on mysterious triality mention the exceptional Lie groups $E_6$, $E_7$, and $E_8$, but not the model of triality $D_4^{(3)}$? I’d mention it myself, but the subject matter is too foreign for me to make the connection ….

Posted by: L Spice on December 2, 2021 5:08 AM | Permalink | Reply to this

### Re: Mysterious Triality

The triality of the title is looking to connect del Pezzo surfaces, iterated cyclic loop spaces, and compactifications of M-theory, so not related to a Lie group possessing triality. U-duality has views on extending the $E$-series in both directions, but not to $D_4$.

The triality of $D_4$ or $Spin(8)$ does make a major appearance in Fiorenza, Sati and Schreiber’s Twisted Cohomotopy implies M-theory anomaly cancellation on 8-manifolds however.

Posted by: David Corfield on December 2, 2021 9:32 AM | Permalink | Reply to this

### Re: Mysterious Triality

It may be relevant that there is a unique exotic smoothing of the 8-sphere, cf \S 11 of

https://math.mit.edu/~araminta/Exotic.pdf ,

associated to the first element of the cokernel of $J$ in the stable homotopy groups of spheres.

Posted by: jackjohnson on December 2, 2021 2:43 PM | Permalink | Reply to this

### Re: Mysterious Triality

I certainly couldn’t identify any direct connection to $D_4^{(3)}$ … but no Lie theorist can hear of triality without thinking of forms of the spin group, and I thought perhaps, even if no connection was known, someone might speculate intelligently on whether there could be one or why there couldn’t.

Interestingly, extending the $\mathsf{E}$-series downwards gives $\mathsf{E}_5 = \mathsf{D}_5$, but $\mathsf{E}_4$ equals $\mathsf{A}_4$, not $\mathsf{D}_4$, so I guess that’s not the place to look for the connection.

Posted by: L Spice on December 3, 2021 6:31 PM | Permalink | Reply to this

### Re: Mysterious Triality

David wrote:

When we started this blog back in 2006 my co-founders were both interested in higher gauge theory. Their paths diverged as Urs looked to adapt these constructions to formulate the elusive M-theory.

It’s interesting that you start the post that way. I can’t resist a comment about this ‘divergence’.

I still retain some interest in higher gauge theory, but I’ve lost interest in theoretical work on fundamental physics because it’s unlikely that any such work will get experimental confirmation in my lifetime. Almost no new work of this sort has been confirmed by experiment since 1973. That’s almost a 50-year drought, and I don’t see any signs of it ending.

(By ‘fundamental physics’ I mean something very specific here. It’s hard to quickly define in an accurate way, but roughly: physics where you seek to find a few laws from which all others can, in principle, be derived. There has been a lot of work on physics since 1973 that’s ‘fundamental’ in the looser sense of transforming our world-view in broad and important ways.)

If one wants to do higher gauge theory that has a chance of experimental confirmation in our lifetimes, I think the place to look is condensed matter physics. It hasn’t happened yet, but it could. For example:

Very very roughly, this paper analyzes a certain class of ‘purely topological’ phases of 3-dimensional matter, and it shows that some correspond to higher gauge theories:

Here, we show that the 3+1D bosonic topological orders with emergent fermions can be realized by topological non-linear σ-models with $\pi_1(K) =$ finite groups, $\pi_1(K)= \mathbb{Z}_2$, and $\pi_n(K)=0$. A subset of those topological non-linear $\sigma$-models corresponds to 2-gauge theories, which realize and classify bosonic topological orders with emergent fermions that have no emergent Majorana zero modes at triple string intersections.

This raises the hope that we could make and study some of these kinds of matter in the lab, probably at low temperatures.

Of course, you can decide that the deep and beautiful mathematics of M-theory is so attractive that it’s worth pursuing even if the physics never gets confirmed — or disconfirmed! — in your lifetime. I can sort of understand that, though personally if I were willing to accept the lack of contact with experiment I’d probably go whole-hog and just do pure math.

To feel happy, I’ve had to go in the opposite direction. I’ve decided that even condensed matter applications of higher gauge theory are not practical enough to satisfy me. These days my ‘work’ is helping a team of people affiliated with the Topos Institute develop better models of infectious disease using ideas from category theory, while my ‘play’ consist of various projects in pure math, physics and chemistry. The first makes me feel I’m doing something useful; the second gives me my daily dose of beauty.

So that’s why our paths have diverged.

Posted by: John Baez on December 2, 2021 6:43 PM | Permalink | Reply to this

### Re: Mysterious Triality

Thanks for explaining this, John.

Regarding

If one wants to do higher gauge theory that has a chance of experimental confirmation in our lifetimes, I think the place to look is condensed matter physics,

as you’ll know, a number of people look to AdS-CFT for tools in condensed matter physics, including Xiao-Gang Wen, whose article you reference. See, for instance, his co-authored

As to what any success here would tell us about string theory for quantum gravity, whether this is anything more than a mathematical spin-off, we were already discussing this back in 2007.

Still, even if one sides with the mere spin-off idea, a movement in string theory may generate new spin-offs.

A talk Urs is giving next week claims that Hypothesis H may explain the appearance of chord diagrams in holographic entanglement entropy, a concept also studied in condensed matter physics.

Posted by: David Corfield on December 4, 2021 9:28 AM | Permalink | Reply to this

### Re: Mysterious Triality

Looking more closely at the paper I linked to, it appears to be about a more general holographic principle. The authors write:

We would like to mention that a structure similar to categorical symmetry was found previously in AdS/CFT correspondence, [63–65] where a global symmetry $G$ at the high-energy boundary is related to a gauge theory of group $G$ in the low-energy bulk. In this paper, we stress that the categorical symmetry encoded by the bulk $G$-gauge theory not only contains the $G$ symmetry at the boundary, it also contains a dual algebraic higher symmetry $\tilde{G}^{(n-1)}$ at the boundary.

I’m reminded of an nLab entry holographic principle of higher category theory.

Posted by: David Corfield on December 4, 2021 11:24 AM | Permalink | Reply to this

### Re: Mysterious Triality

David wrote:

Still, even if one sides with the mere spin-off idea, a movement in string theory may generate new spin-offs.

Somewhere, and I can’t find where, Polyakov once said something roughly like this:

Every sufficiently elegant theory of physics is in some sense true.

And he explained what he meant. The key phrase is “in some sense”. Sufficiently elegant theories of physics have a tendency to describe some physical system, even if not the one they were originally intended to describe.

Right now the place to take advantage of this idea is condensed matter physics. There are a couple of reasons. One is that high-energy particle physics experiments are vastly more expensive than condensed matter physics experiments. But a more fundamental one is that there are many different kinds of matter, but only one universe — well, to play it safe: only one that we can access experimentally right now. So, in condensed matter physics you are shooting your arrows at a much bigger target. You can even move your target to make the arrows more likely to hit it.

Posted by: John Baez on December 5, 2021 12:47 AM | Permalink | Reply to this

### Re: Mysterious Triality

Of course the idea of the post has been expressed before:

If history is a good guide, then we should expect that anything as profound and far-reaching as a fully satisfactory formulation of M-theory is surely going to lead to new and novel mathematics. (Gregory Moore, Physical Mathematics and the Future, p. 44)

The target is so tightly constrained, that success here must have profound mathematical consequences.

One might expect there to be a duality from physics underlying the mathematical one pointed out in ‘Mysterious triality’. And since there is a duality between M-theory and F-theory, an F-theoretic understanding of del Pezzo surfaces could be interesting, as in Global F-theory GUTs.

Posted by: David Corfield on December 8, 2021 4:20 PM | Permalink | Reply to this

### Re: Mysterious Triality

Urs remarks that the F-theory connection is hinted at on p. 7 of the article:

F-theory apparently wants to lift the $S^4$-coefficient of M-theory to its branched cover by $\mathbb{C}P^1 \times \mathbb{C}P^1$.

Posted by: David Corfield on December 9, 2021 3:20 PM | Permalink | Reply to this

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