The n-Category Café

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?

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.

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.

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 ….

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:

• Chenchang Zhu, Tian Lan and Xiao-Gang Wen, Topological non-linear $\sigma$-model, higher gauge theory, and a realization of all 3+1D topological orders for boson systems.

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.

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.