The n-Category Café

And on the summary slide of the second deck, the algebra for the complex qubit should be $\mathfrak{h}_2(\mathbb{C})$ instead of $\mathfrak{h}_2(\mathbb{O})$.

Thanks very much — I’ll fix these!

I want to improve my second batch of slides.

Right now they don’t explain why the subgroup of automorphisms of $\mathfrak{h}_3(\mathbb{O})$ preserving a copy of $\mathfrak{h}_2(\mathbb{O})$ is $Spin(9)$. I explained this in my series of posts.

The quick thing to say is that you can identify $\mathfrak{h}_2(\mathbb{O})$ with 10d Minkowski spacetime and think of $\mathfrak{h}_3(\mathbb{O})$ as the direct sum of 3 spaces:

• $\mathfrak{h}_2(\mathbb{O})$ (10d vectors)

• $\mathbb{O}^2$ (10d spinors)

• $\mathbb{R}$ (scalars).

Then the group of determinant-preserving transformations of $\mathfrak{h}_3(\mathbb{O})$ preserving $\mathfrak{h}_2(\mathbb{O})$ preserves this splitting and all the invariant operations involving vectors, spinor and scalars, so it’s $Spin(9,1)$. The automorphisms of $\mathfrak{h}_3(\mathbb{O})$ preserving $\mathfrak{h}_2(\mathbb{O})$ also preserve the ‘time axis’ in 10d Minkowski spacetime, so they form the subgroup $Spin(9)$.

I also don’t give any explanation of where $(SU(3) \times SU(3)) / \mathbb{Z}_3$ comes from. It’s not that hard to get an idea of where this comes from.

I don’t like how these important facts show up in my talk ‘as if by magic’, stated but not explained. But maybe explaining all this stuff would overload the talk. There’s a limit to how much information people can absorb in 50 minutes. I guess blogging and papers are better for detailed explanations. I want to continue my blog series and use it to expand on these talks.

I think any particular researcher’s optimism or pessimism is uninteresting, especially in fundamental physics where the chance of any one idea working out is very low, and the most optimistic people often come up with the worst ideas. So I’m not going to answer that question.

In my talk, though, I’ll say a bit more about what I think about this octonion business.

Thanks! I look forward to not teaching calculus and serving on university committees — it’ll give me more time to give talks and write.

It’s all well and good to realize $K = H_1 \cap_G H_2$ for some subgroups $H_1,H_2 \subset G$. But subgroups come in conjugacy classes. When you said that you had $SU(5) \subset SO(10)$ and $SO(4) \times SO(6) \subset SO(10)$, I think you had in mind some specific subgroups for a specific $SO(10)$, but a topologist or representation theorist would find it more natural just to declare the conjugacy classes of these inclusions. Now there is a problem: up to $SO(10)$-conjugacy, there is a unique $SU(5) \subset SO(10)$, but once you have chosen it, you have broken the $SO(10)$-symmetry. In particular, when you go looking for an $SO(4) \times SO(6) \subset SO(10)$, a priori you should look at all of these up to $SU(5)$-conjugacy, and there might very well be multiple orbits.

A typical thing that happens when there are multiple orbits is that there is a “generic orbit” and some “special” orbits. Actually, the “generic orbit” typically is not a single orbit. What it is is a family of orbits, but they are all qualitatively the same. In particular, you could imagine that, up to common $G$-conjugacy, perhaps there are lots of pairs of subgroups $H_1, H_2 \subset G$, but for most of them, the intersection $H_1 \cap H_2$ is always a copy of $K$.

Let’s do a trivial example. Let’s take $H_1 = H_2 = U(1)$ living inside $G = SU(2)$. Up to $SU(2)$-conjugacy, $H_1$ might as well be the standard $U(1) = diag(\lambda,\lambda^*)$. Then there is a 1-dimensional family of choices for $H_2$. There is a special choice: $H_1 = H_2$, in which case they intersect along a $U(1)$. All the other choices are generic, and the intersection is $\{\pm 1\} \subset SU(2)$.

In the Standard Model = Georgi-Glashow $\cap$ Pati-Salam example, what happens? Is this a generic intersection or a special one?

One could just as well contemplate the embedding $S(U(2) \times U(3)) \subset SU(5) \subset Sp(5)$.

Here my topologist is showing. Topologists tend to use the name $Sp(n)$ for the rank-$n$ compact group of $Sp$ type, with Dynkin diagram $C_n$. It is a thing which acts on $\mathbb{H}^n$. [Namely, the group which commutes with the left $\mathbb{H}^\times$ action is an $\mathbb{R}_{\gt 0}$ central extension of $Sp(n)$.] Representation theorists tend to write “$Sp(n)$” only when $n$ is even, in which case it is the split form of the $Sp$-type group with rank $n/2$, and is a thing which acts on $\mathbb{R}^n$ preserving a symplectic form.

Anyway, $Sp(5)$ has the same dimension as $Spin(11)$, but really it is morally the same size as $Spin(10)$. This is because, if you have an $10$-dimensional complex vector space, then to realize it as $5$ quaternionic dimensional requires choosing an antilinear square root of $-\mathrm{id}$, whereas to realize it as $5$-real-dimensional tensored with $\bC$ requires choosing an antilinear square root of $+\mathrm{id}$.

Do you have any interesting stories to tell about splitting $\mathbb{H}^5 = \mathbb{H}^2 \times\mathbb{H}^3$ analogous to the splitting $\mathbb{R}^{10} = \mathbb{R}^4 \times \mathbb{R}^6$? Is there a “quaternionic Pati-Salam model”?

Theo wrote:

Your talk this week started out with a very enticing question: Why does 5=2+3? As far as I can tell, your explanation is “Because 10=4+6”, which is less than satisfying.

That wasn’t supposed to be the explanation of anything yet, though string theorists have thought a lot about 10=4+6. I was just reviewing work people have done on massaging the Standard Model and its representation on fermions into a mathematically simple form, which is a prerequisite for understanding it in a more conceptual way. The second part of my talk will describe an octonionic “explanation” of the 10=4+6 split… but I’m not claiming this explanation really works: there are problems with it that I don’t know how to solve. It’s really just a vague sketch of the sort of thing that might eventually count as an explanation: something based on math that’s not just pretty but has some physical content.

Neat! I think I’ve been coming at some of the same concepts from a different direction (and a background of near-total ignorance).

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• R. L. Griess, Jr. “A Moufang loop, the exceptional Jordan algebra, and a cubic form in 27 variables,” Journal of Algebra 131 (1990), 281–93.

Terminological note: The “finite Heisenberg group” was conceived by Hermann Weyl in 1925, in the course of devising a finite-dimensional counterpart to the canonical commutation relation that was originally proposed by Max Born. It’s described in Weyl’s Gruppentheorie und Quantenmechanik (along with other neat things like an early recognition of the importance of entanglement). Accordingly, one will hear it called the “Weyl–Heisenberg group”; less frequently, in my experience, it is also termed the “generalized Pauli group”.

I just learned the other day that the normalizer in $SL_5(\mathbb{C})$ of the Weyl–Heisenberg group for dimension 5 is the symmetry group of the Horrocks–Mumford bundle. See He and McKay (2015), section 4. The Hesse SIC, which I care about for quantum-information reasons, has a symmetry group (the Hessian group) that is a semidirect product of the Weyl–Heisenberg group in dimension 3 with the binary tetrahedral group. It looks like the symmetry group of the Horrocks–Mumford bundle is analogous, except we go up to dimension 5 and use the binary icosahedral group instead.

Theo wrote:

A very basic reason is that it is the centralizer of an order-3 element in $F_4$.

Yes, that’s great! Yokota gives a nice account of this in terms of the exceptional Jordan algebra here:

• Ichiro Yokota, Exceptional Lie groups, Section 2.12.

and that’s the basis of Dubois-Violette and Todorov’s work, which I’ll be discussing on Monday. Yokota constructs the compact exceptional Lie groups using octonions, and then describes their maximal subgroups as centralizers of elements of finite order… all very octonionically.

Thanks for all the other nice information about this $(SU(3) \times SU(3))/\mathbb{Z}_3$ subgroup. It could come in handy! It’s nice to hear that the two $SU(3)$’s are intrinsically different.

After my second talk Michel Dubois-Violette made a long comment which I think could be important - you can hear it on the video. It’s not very easy to understand, but I think it offers some useful clues. I also enjoyed Kirill Krasnov’s talk this week. There could be an interesting picture emerging here, but I can’t really say exactly what it is yet, or whether it will really solidify.

Bertie wrote:

Dear Prof Baez, you introduced the idea early in the first talk that mathematicians are vitally interested in uncovering a mathematical structure which posseses the symmetries of the Standard Model.

No, I think few mathematicians are interested in this. I said that I was interested in this. I said something else: that while physicists are often happy to just work with a given group, the mathematical approach is to find a structure having that group as symmetries, and extract information from that structure. This is just a widespread mathematical tactic. But I think few mathematicians have taken this approach to the Standard Model gauge group. The Standard Model gauge group is not considered interesting by most mathematicians.

Also, I should add that I’m not looking for just any structure having the Standard Model gauge group as its symmetries. It’s easy to find dozens of them. I’m looking for a mathematically beautiful structure that has some significance for physics.

So, what I wasn’t sure about was why you seemed rather ‘low key’ about it (or were you ‘dancing on the inside’)?

I was extremely excited by this result when it first became clear, but by the time I gave my talk I’d spent a lot of time thinking about the next obstacles to be overcome, which seem even harder. Until all the puzzle pieces fit together, there’s a good chance we’re on the wrong track. That’s why I only work on particle physics as a kind of hobby. As a full-time job it would be too depressing, knowing that quite likely everything I did was a waste of time. I prefer to spend most of my time on projects where I’m definitely succeeding.