Talk on the Tenfold Way

January 31, 2023

Posted by John Baez

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There are ten ways that a substance can have symmetry under time reversal, switching particles and holes, both or neither. But this fact turns out to extend far beyond condensed matter physics! It’s really built into the fabric of mathematics in a deep way.

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I gave a talk on this at Nicohl Furey’s seminar Algebra, Particles and Quantum Theory, and you can see a video of my talk here.

You can also watch another version, where I explain this stuff to my friend James Dolan.

I like the idea of being able to watch an official talk but also watch the speaker chatting about the talk with a friend. It gives another view of the material. I skim over stuff Jim already knows, explain things I didn’t have time to get into in the actual talk, and emphasize the things I don’t understand. And he points out lots more patterns lurking in the tenfold way!

You can see my slides and more notes here.

Posted at January 31, 2023 7:19 AM UTC

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Re: Talk on the Tenfold Way

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Presumably this is related to the four connected components of the Lorenz group O(1,3), cf

https://en.wikipedia.org/wiki/Wu_experiment

and maybe even axions

https://en.wikipedia.org/wiki/Peccei%E2%80%93Quinn_theory

oh my?

Posted by: jack morava on January 31, 2023 5:47 PM | Permalink | Reply to this

Re: Talk on the Tenfold Way

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Hi!

There’s definitely a connection to C, P, and T symmetry and the 4 components of the Lorentz group (containing 1, P, T, and PT)…

… but the point of my talk is that the 10-fold way doesn’t rely on any concept of spacetime: it shows up as soon as you study $\mathbb{Z}\!/\!2$ -graded Hilbert spaces. I’ll explain how any unitary representation of a group on a $\mathbb{Z}\!/\!2$ -graded Hilbert space that’s decomposable into irreducibles is automatically $\mathrm{X}$ -graded, where

$\mathrm{X} = \{0,1,2,3,4,5,6,7,\mathbf{0},\mathbf{1} \}$

is a 10-element set with a certain monoid structure. This monoid has the groups $\mathbb{Z}\!/\!8$ and $\mathbb{Z}\!/\!2$ as submonoids. It unifies real and complex Bott periodicity!

By the way, it’s the ‘Lorentz’ group, not the ‘Lorenz’ group. Lorenz was a physicist who worked on electromagnetism; so was Lorentz. The second was obtained from the first by a transformation that mixed the $t$ and $z$ coordinates.

Posted by: John Baez on January 31, 2023 7:03 PM | Permalink | Reply to this

Re: Talk on the Tenfold Way

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thanks, sorry, my bad. then there’s Weil and Weyl…

but… could $\pi_0 O(1,3)$ interact in interesting ways with representation theory graded somehow by this ten-element monoid?

Posted by: jack morava on January 31, 2023 11:58 PM | Permalink | Reply to this

Re: Talk on the Tenfold Way

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It certainly interacts in some way!
$\pi_0 O(1,3) \cong \mathbb{Z}\!/\!2 \times \mathbb{Z}\!/\!2$ is generated by time reversal T and space reversal, or ‘parity’ P. Physicists like to think of it as a subgroup of $\mathbb{Z}\!/\!2 \times \mathbb{Z}\!/\!2 \times \mathbb{Z}\!/\!2$ where the third generator, C, switches particles and antiparticles.

In physics this larger group often acts on a Hilbert space $\mathbf{H}$ in such a way that P acts as a unitary operator while C and T act as antiunitary operators. Sometimes $\mathbf{H}$ is $\mathbb{Z}\!/\! 2$ -graded in such a way that ‘particles’ are even and ‘antiparticles’ are odd. Then C acts as an odd antiunitary operator while T acts as an even antiunitary operator.

In my talk slides I discuss a more general situation where we have an irreducible representation $\rho$ of a group on a $\mathbb{Z}\!/\! 2$ -graded Hilbert space. This superficially has little to do with the story I’ve just been telling: yes there’s $\mathbb{Z}\!/\! 2$ -graded Hilbert space, and a group, but we don’t necessarily have $O(3,1)$ around. Nonetheless, for any such representation, we have:

The Tenfold Way. The irreducible unitary representations $\rho$ of a $\Z/2$ -graded group $G$ on a super Hilbert space $\mathbf{H}$ come in 10 types, based on the commutant: the real-linear operators that commute with $\rho(g)$ for all $G$ .

In 9 of these types the commutant contains:

an even antiunitary $T$ with either $T^2 = 1$ , $T^2 = -1$ , or no such $T$

and

an odd antiunitary $C$ with either $C^2 = 1$ , $C^2 = -1$ , or no such $C$ .

In the 10th type the commutant contains:

no such $T$ or $C$ , but an odd unitary $S$ ; we may assume $S^2 = 1$ .

In short, even when $T$ and $C$ aren’t put in via the Lorentz group O(1,3), they magically appear (or not) on their own, having the properties we expect in the O(1,3) case!

Posted by: John Baez on February 1, 2023 11:01 PM | Permalink | Reply to this

Re: Talk on the Tenfold Way

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IANAPhysicist, so there seems to be something funny about

https://en.wikipedia.org/wiki/CP_violation

that I don’t understand…

Posted by: jack morava on February 2, 2023 8:38 PM | Permalink | Reply to this

Re: Talk on the Tenfold Way

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I don’t know if I’m a physicist, but I’ve studied the subject for a long time, so tell me what seems funny about CP violation and maybe I can help you out.

CP violation is a big deal — and a big surprise, back around 1964.

Posted by: John Baez on February 3, 2023 12:12 AM | Permalink | Reply to this

Re: Talk on the Tenfold Way

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I guess it’s off-topic w.r.t. condensed matter physics but the strong CP problem

“… is a puzzling question in particle physics: Why does quantum chromodynamics (QCD) seem to preserve CP-symmetry?”

I don’t know the current status of the sci-fi story at

https://arxiv.org/abs/1610.08297

but think the idea of an axion with de Broglie wavelength of the order of kiloparsecs is hilarious.

Posted by: jack morava on February 4, 2023 2:58 PM | Permalink | Reply to this

Re: Talk on the Tenfold Way

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Yes, the strong CP problem is very interesting. It’s a ‘problem’ if you believe, as most particle physicists do, that you need a good explanation for any renormalizable term that you can cook up from the fields in the Standard Model that’s missing from the Standard Model Lagrangian. I.e. just setting a coupling constant equal to zero without explanation counts as ‘cheating’ for them.

The only such missing term is the CP-violating term $\tr(F \wedge F)$ where $F$ is the curvature of the $SU(3)$ connection describing the strong force. This is a ‘topological term’: if we treat spacetime as compact its integral gives the 2nd Chern class of the principal $SU(3)$ bundle over spacetime… up to a constant factor that I’m too lazy to recall.

Posted by: John Baez on February 7, 2023 5:34 AM | Permalink | Reply to this

Re: Talk on the Tenfold Way

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Hadn’t understood that - it seems very intriguing, thanks!

Posted by: jack morava on February 7, 2023 1:05 PM | Permalink | Reply to this

C is a unitary

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In physics this larger group often acts on a Hilbert space H in such a way that P acts as a unitary operator while C and T act as antiunitary operators.

In relativistic quantum field theory, C and P are implemented as unitary operators and T as an anti-unitary operator. (CPT is anti-unitary.)

Posted by: Jacques Distler on February 3, 2023 2:35 PM | Permalink | PGP Sig | Reply to this

Re: Talk on the Tenfold Way

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Jacques wrote:

In relativistic quantum field theory, $C$ and $P$ are implemented as unitary operators and $T$ as an anti-unitary operator.

Good point! A lot of condensed matter physics uses an antiunitary $C$ . See for example

Adolfo G. Grushin, Introduction to topological phases in condensed matter. Section III.2: Particle-hole symmetry.

For a more mathematical approach, see:

Gregory Moore, Quantum Symmetries and Compatible Hamiltonians. Section 16: Realizing the 10 classes using the CT groups.

I just noticed a discussion of this here, where Niall writes:

The interesting thing is that the particle hole operator is unitary and linear in the second quantised picture, but antiunitary and antilinear in the first quantised picture.

I think that’s important.

Posted by: John Baez on February 3, 2023 8:34 PM | Permalink | Reply to this

Re: Talk on the Tenfold Way

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You have “division algebra” defined in the slides to include associativity so as to limit the options to three, $\mathbb{R}$ , $\mathbb{C}$ , $\mathbb{H}$ .

Of course, elsewhere you had us think about the four normed division algebras (allowing non-associativity) and their relation to supersymmetry, (cf. nLab).

If the “super” version of the former gives us the tenfold way, is there a “super” version of the latter?

Posted by: David Corfield on February 9, 2023 4:55 PM | Permalink | Reply to this

Re: Talk on the Tenfold Way

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Good point! I think the reasonable definition of a normed division superalgebra $A = A_0 + A_1$ is one with norms on $A_0$ and $A_1$ such that

$\|a b\| = \|a\| \|b\|$

whenever $a$ and $b$ are homogeneous elements, i.e. $a \in A_i$ and $b \in A_j$ for some $i,j$ .

With this definition it’s easy to show that all the associative division superalgebras become normed division superalgebras, and that every associative normed division algebra is of this form. So, there are just 10 associative normed division superalgebras.

In 2005 Todd Trimble wrote up a classification of the associative division superalgebras. I asked him if he could classify the alternative division superalgebras, and he did. He found that besides the 10 associative ones there were a few coming from the octonions. Unfortunately I’ve lost his results. But by now I understand Todd’s arguments well enough that I could redo this classification in an hour.

Copying the arguments you can find at the link, you’ll get one alternative division superalgebra with $A_0 = \mathbb{O}, A_0 = \{0\}$ and one or more with $A_0 = \mathbb{O}, A_1 = \mathbb{O}$ . And, I’m pretty sure all of these will be normed division superalgebras.

Further, I predict that these will give us all the normed division superalgebras, except for the 10 associative ones we already know.

I would pursue this more assiduously if I could think of something interesting to do with alternative or normed division superalgebras.

Hmm.

Maybe we could use them to build one or more ‘exceptional Jordan superalgebras’ analogous to the usual one defined using octonions. I think someone has already looked into Jordan superalgebras.

Or maybe we could use them to build exceptional simple Lie superalgebras. I think Kac has classified those. I see that in 1998 he classified infinite-dimensional exceptional Lie superalgebras, finding exotic delights like “E(3,6)”. But that’s too fancy for me right now. Someone must have done the finite-dimensional ones earlier.

Posted by: John Baez on February 9, 2023 5:38 PM | Permalink | Reply to this

Re: Talk on the Tenfold Way

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Thanks!

I think someone has already looked into Jordan superalgebras.

Someone has even started an nLab page on them: Jordan superalgebra, where we learn that over an algebraically closed field of characteristic $0$

The only exceptional finite-dimensional example is the 10-dimensional Jordan superalgebra $K_10$ .

Posted by: David Corfield on February 10, 2023 9:23 AM | Permalink | Reply to this

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