## July 19, 2014

### The Tenfold Way (Part 1)

#### Posted by John Baez There are 10 of each of these things:

• Associative real super-division algebras.

• Classical families of compact symmetric spaces.

• Ways that Hamiltonians can get along with time reversal ($T$) and charge conjugation ($C$) symmetry.

• Dimensions of spacetime in string theory.

It’s too bad nobody took up writing This Week’s Finds in Mathematical Physics when I quit. Someone should have explained this stuff in a nice simple way, so I could read their summary instead of fighting my way through the original papers. I don’t have much time for this sort of stuff anymore!

Let me start by explaining the basic idea, and then move on to more fancy aspects.

### Ten kinds of matter The idea of the ten-fold way goes back at least to 1996, when Altland and Zirnbauer discovered that substances can be divided into 10 kinds.

The basic idea is pretty simple. Some substances have time-reversal symmetry: they would look the same, even on the atomic level, if you made a movie of them and ran it backwards. Some don’t — these are more rare, like certain superconductors made of yttrium barium copper oxide! Time reversal symmetry is described by an antiunitary operator $T$ that squares to 1 or to -1: please take my word for this, it’s a quantum thing. So, we get 3 choices, which are listed in the chart under $T$ as 1, -1, or 0 (no time reversal symmetry).

Similarly, some substances have charge conjugation symmetry, meaning a symmetry where we switch particles and holes: places where a particle is missing. The ‘particles’ here can be rather abstract things, like phonons - little vibrations of sound in a substance, which act like particles — or spinons — little vibrations in the lined-up spins of electrons. Basically any way that something can wave can, thanks to quantum mechanics, act like a particle. And sometimes we can switch particles and holes, and a substance will act the same way!

Like time reversal symmetry, charge conjugation symmetry is described by an antiunitary operator $C$ that can square to 1 or to -1. So again we get 3 choices, listed in the chart under $C$ as 1, -1, or 0 (no charge conjugation symmetry).

So far we have 3 × 3 = 9 kinds of matter. What is the tenth kind?

Some kinds of matter don’t have time reversal or charge conjugation symmetry, but they’re symmetrical under the combination of time reversal and charge conjugation! You switch particles and holes and run the movie backwards, and things look the same!

In the chart they write 1 under the $S$ when your matter has this combined symmetry, and 0 when it doesn’t. So, “0 0 1” is the tenth kind of matter (the second row in the chart).

This is just the beginning of an amazing story. Since then people have found substances called topological insulators that act like insulators in their interior but conduct electricity on their surface. We can make 3-dimensional topological insulators, but also 2-dimensional ones (that is, thin films) and even 1-dimensional ones (wires). And we can theorize about higher-dimensional ones, though this is mainly a mathematical game.

So we can ask which of the 10 kinds of substance can arise as topological insulators in various dimensions. And the answer is: in any particular dimension, only 5 kinds can show up. But it’s a different 5 in different dimensions! This chart shows how it works for dimensions 1 through 8. The kinds that can’t show up are labelled 0.

If you look at the chart, you’ll see it has some nice patterns. And it repeats after dimension 8. In other words, dimension 9 works just like dimension 1, and so on.

If you read some of the papers I listed, you’ll see that the $\mathbb{Z}$’s and $\mathbb{Z}_2$’s in the chart are the homotopy groups of the ten classical series of compact symmetric spaces. The fact that dimension $n+8$ works like dimension $n$ is called Bott periodicity.

Furthermore, the stuff about operators $T$, $C$ and $S$ that square to 1, -1 or don’t exist at all is closely connected to the classification of associative real super division algebras. It all fits together.

### Super division algebras

In 2005, Todd Trimble wrote a short paper called The super Brauer group and super division algebras.

In it, he gave a quick way to classify the associative real super division algebras: that is, finite-dimensional associative real $\mathbb{Z}_2$-graded algebras having the property that every nonzero homogeneous element is invertible. The result was known, but I really enjoyed Todd’s effortless proof.

However, I didn’t notice that there are exactly 10 of these guys. Now this turns out to be a big deal. For each of these 10 algebras, the representations of that algebra describe ‘types of matter’ of a particular kind — where the 10 kinds are the ones I explained above!

So what are these 10 associative super division algebras?

3 of them are purely even, with no odd part: the usual associative division algebras $\mathbb{R}, \mathbb{C}$ and $\mathbb{H}$.

7 of them are not purely even. Of these, 6 are Morita equivalent to the real Clifford algebras $Cl_1, Cl_2, Cl_3, Cl_5, Cl_6$ and $Cl_7$. These are the superalgebras generated by 1, 2, 3, 5, 6, or 7 odd square roots of -1.

Now you should have at least two questions:

• What’s ‘Morita equivalence’? — and even if you know, why should it matter here? Two algebras are Morita equivalent if they have equivalent categories of representations. The same definition works for superalgebras, though now we look at their representations on super vector spaces ($\mathbb{Z}_2$-graded vector spaces). For physics what we really care about is the representations of an algebra or superalgebra: as I mentioned, those are ‘types of matter’. So, it makes sense to count two superalgebras as ‘the same’ if they’re Morita equivalent.

• 1, 2, 3, 5, 6, and 7? That’s weird — why not 4? Well, Todd showed that $Cl_4$ is Morita equivalent to the purely even super division algebra $\mathbb{H}$. So we already had that one on our list. Similarly, why not 0? $Cl_0$ is just $\mathbb{R}$. So we had that one too.

Representations of Clifford algebras are used to describe spin-1/2 particles, so it’s exciting that 8 of the 10 associative real super division algebras are Morita equivalent to real Clifford algebras.

But I’ve already mentioned one that’s not: the complex numbers, $\mathbb{C}$, regarded as a purely even algebra. And there’s one more! It’s the complex Clifford algebra $\mathbb{C}\mathrm{l}_1$. This is the superalgebra you get by taking the purely even algebra $\mathbb{C}$ and throwing in one odd square root of -1.

As soon as you hear that, you notice that the purely even algebra $\mathbb{C}$ is the complex Clifford algebra $\mathbb{C}\mathrm{l}_0$. In other words, it’s the superalgebra you get by taking the purely even algebra $\mathbb{C}$ and throwing in no odd square roots of -1.

### More connections

At this point things start fitting together:

• You can multiply Morita equivalence classes of algebras using the tensor product of algebras: $[A] \otimes [B] = [A \otimes B]$. Some equivalence classes have multiplicative inverses, and these form the Brauer group. We can do the same thing for superalgebras, and get the super Brauer group. The super division algebras Morita equivalent to $Cl_0, \dots , Cl_7$ serve as representatives of the super Brauer group of the real numbers, which is $\mathbb{Z}_8$. I explained this in week211 and further in week212. It’s a nice purely algebraic way to think about real Bott periodicity!

• As we’ve seen, the super division algebras Morita equivalent to $Cl_0$ and $Cl_4$ are a bit funny. They’re purely even. So they serve as representatives of the plain old Brauer group of the real numbers, which is $\mathbb{Z}_2$.

• On the other hand, the complex Clifford algebras $\mathbb{C}\mathrm{l}_0 = \mathbb{C}$ and $\mathbb{C}\mathrm{l}_1$ serve as representatives of the super Brauer group of the complex numbers, which is also $\mathbb{Z}_2$. This is a purely algebraic way to think about complex Bott periodicity, which has period 2 instead of period 8.

Meanwhile, the purely even $\mathbb{R}, \mathbb{C}$ and $\mathbb{H}$ underlie Dyson’s ‘three-fold way’, which I explained in detail here:

Briefly, if you have an irreducible unitary representation of a group on a complex Hilbert space $H$, there are three possibilities:

• The representation is isomorphic to its dual via an invariant symmetric bilinear pairing $g : H \times H \to \mathbb{C}$. In this case it has an invariant antiunitary operator $J : H \to H$ with $J^2 = 1$. This lets us write our representation as the complexification of a real one.

• The representation is isomorphic to its dual via an invariant antisymmetric bilinear pairing $\omega : H \times H \to \mathbb{C}$. In this case it has an invariant antiunitary operator $J : H \to H$ with $J^2 = -1$. This lets us promote our representation to a quaternionic one.

• The representation is not isomorphic to its dual. In this case we say it’s truly complex.

In physics applications, we can take $J$ to be either time reversal symmetry, $T$, or charge conjugation symmetry, $C$. Studying either symmetry separately leads us to Dyson’s three-fold way. Studying them both together leads to the ten-fold way!

So the ten-fold way seems to combine in one nice package:

• real Bott periodicity,
• complex Bott periodicity,
• the real Brauer group,
• the real super Brauer group,
• the complex super Brauer group, and
• the three-fold way.

I could throw ‘the complex Brauer group’ into this list, because that’s lurking here too, but it’s the trivial group, with $\mathbb{C}$ as its representative.

There really should be a better way to understand this. Here’s my best attempt right now.

The set of Morita equivalence classes of finite-dimensional real superalgebras gets a commutative monoid structure thanks to direct sum. This commutative monoid then gets a commutative rig structure thanks to tensor product. This commutative rig — let’s call it $\mathfrak{R}$ — is apparently too complicated to understand in detail, though I’d love to be corrected about that. But we can peek at pieces:

• We can look at the group of invertible elements in $\mathfrak{R}$ — more precisely, elements with multiplicative inverses. This is the real super Brauer group $\mathbb{Z}_8$.

• We can look at the sub-rig of $\mathfrak{R}$ coming from semisimple purely even algebras. As a commutative monoid under addition, this is $\mathbb{N}^3$, since it’s generated by $\mathbb{R}, \mathbb{C}$ and $\mathbb{H}$. This commutative monoid becomes a rig with a funny multiplication table, e.g. $\mathbb{C} \otimes \mathbb{C} = \mathbb{C} \oplus \mathbb{C}$. This captures some aspects of the three-fold way.

We should really look at a larger chunk of the rig $\mathfrak{R}$, that includes both of these chunks. How about the sub-rig coming from all semisimple superalgebras? What’s that?

And here’s another question: what’s the relation to the 10 classical families of compact symmetric spaces? The short answer is that each family describes a family of possible Hamiltonians for one of our 10 kinds of matter. For a more detailed answer, I suggest reading Gregory Moore’s Quantum symmetries and compatible Hamiltonians. But if you look at this chart by Ryu et al, you’ll see these families involve a nice interplay between $\mathbb{R}, \mathbb{C}$ and $\mathbb{H}$, which is what this story is all about: The families of symmetric spaces are listed in the column “Hamiltonian”.

All this stuff is fitting together more and more nicely! And if you look at the paper by Freed and Moore, you’ll see there’s a lot more involved when you take the symmetries of crystals into account. People are beginning to understand the algebraic and topological aspects of condensed matter much more deeply these days.

### The list

Just for the record, here are all 10 associative real super division algebras. 8 are Morita equivalent to real Clifford algebras:

• $Cl_0$ is the purely even division algebra $\mathbb{R}$.

• $Cl_1$ is the super division algebra $\mathbb{R} \oplus \mathbb{R}e$, where $e$ is an odd element with $e^2 = -1$.

• $Cl_2$ is the super division algebra $\mathbb{C} \oplus \mathbb{C}e$, where $e$ is an odd element with $e^2 = -1$ and $e i = -i e$.

• $Cl_3$ is the super division algebra $\mathbb{H} \oplus \mathbb{H}e$, where $e$ is an odd element with $e^2 = 1$ and $e i = i e, e j = j e, e k = k e$.

• $Cl_4$ is $\mathbb{H}$, the algebra of $2 \times 2$ quaternionic matrices, given a certain $\mathbb{Z}_2$-grading. This is Morita equivalent to the purely even division algebra $\mathbb{H}$.

• $Cl_5$ is $\mathbb{C}$ given a certain $\mathbb{Z}_2$-grading. This is Morita equivalent to the super division algebra $\mathbb{H} \oplus \mathbb{H}e$ where $e$ is an odd element with $e^2 = -1$ and $e i = i e, e j = j e, e k = k e$.

• $Cl_6$ is $\mathbb{R}$ given a certain $\mathbb{Z}_2$-grading. This is Morita equivalent to the super division algebra $\mathbb{C} \oplus \mathbb{C}e$ where $e$ is an odd element with $e^2 = 1$ and $e i = -i e$.

• $Cl_7$ is $\mathbb{R} \oplus \mathbb{R}$ given a certain $\mathbb{Z}_2$-grading. This is Morita equivalent to the super division algebra $\mathbb{R} \oplus \mathbb{R}e$ where $e$ is an odd element with $e^2 = 1$.

$Cl_{n+8}$ is Morita equivalent to $Cl_n$ so we can stop here if we’re just looking for Morita equivalence classes, and there also happen to be no more super division algebras down this road. It is nice to compare $Cl_n$ and $Cl_{8-n}$: there’s a nice pattern here.

The remaining 2 real super division algebras are complex Clifford algebras:

• $\mathbb{C}\mathrm{l}_0$ is the purely even division algebra $\mathbb{C}$.

• $\mathbb{C}\mathrm{l}_1$ is the super division algebra $\mathbb{C} \oplus \mathbb{C} e$, where $e$ is an odd element with $e^2 = -1$ and $e i = i e$.

In the last one we could also say “with $e^2 = 1$” — we’d get something isomorphic, not a new possibility.

### Ten dimensions of string theory

Oh yeah — what about the 10 dimensions in string theory? Are they really related to the ten-fold way?

It seems weird, but I think the answer is “yes, at least slightly”.

Remember, 2 of the dimensions in 10d string theory are those of the string worldsheet, which is a complex manifold. The other 8 are connected to the octonions, which in turn are connected to the 8-fold periodicity of real Clifford algebra. So the 8+2 split in string theory is at least slightly connected to the 8+2 split in the list of associative real super division algebras.

This may be more of a joke than a deep observation. After all, the 8 dimensions of the octonions are not individual things with distinct identities, as the 8 super division algebras coming from real Clifford algebras are. So there’s no one-to-one correspondence going on here, just an equation between numbers.

Still, there are certain observations that would be silly to resist mentioning.

Posted at July 19, 2014 11:33 AM UTC

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### Re: The Ten-Fold Way

Of course, the real fun starts with the exceptional cases, i.e. G/H for G an exceptional group. The condensed matter guys haven’t really considered these cases but there already exists an interpretation within supergravity via toroidal M-theory compactifications.

Posted by: kneemo on July 19, 2014 4:21 PM | Permalink | Reply to this

### Re: The Ten-Fold Way

Posted by: John Baez on July 20, 2014 10:22 AM | Permalink | Reply to this

### Re: The Ten-Fold Way

Am I understanding right that with one involution, you get the threefold way, and with two commuting involutions, you get this tenfold way? If so, what about N commuting involutions?

(Or two noncommuting involutions, whose product has order 3…)

Posted by: Allen Knutson on July 20, 2014 3:45 PM | Permalink | Reply to this

### Re: The Ten-Fold Way

It’s not quite true that the threefold way is about a vector space with one involution. When you have an irreducible representation of a group on a complex vector space $V$, exactly one of these things happens:

1) there is an equivariant antilinear map $J: V \to V$ with $J^2 = 1$;

2) there is an equivariant antilinear map $J: V \to V$ with $J^2 = -1$;

3) there is no equivariant antilinear map $J: V \to V$.

The ten-fold way seems to be the generalization to $\mathbb{Z}_2$-graded vector spaces. So, it would be nice to generalize further to $G$-graded vector spaces, or other categories sufficiently resembling Vect.

I’ve been straightening this out a bit, so I’ll post another article in a while.

Posted by: John Baez on July 21, 2014 4:21 AM | Permalink | Reply to this

### Re: The Ten-Fold Way

Can’t help but note that the fermion/boson dichotomy doesn’t appear (which, for 2-d systems, I’m told, is inadequate anyway); is that perhaps related to the $\mathbb{Z}$s and $\mathbb{Z}/2$s? Maybe I should just glance at the papers… having now glanced at the AZ paper, I still can’t tell.

Posted by: Jesse C. McKeown on July 21, 2014 2:29 PM | Permalink | Reply to this

### Re: The Ten-Fold Way

I have been studying these things and have found that there are a few inequivalent versions of the “tenfold way”. The original AZ paper deals with symmetries of a fermionic “Nambu space” — a one-particle Hilbert space together with its dual. Their classification was based on the compact symmetric spaces — no group invariants yet! Other authors, like the Ryu et al refer to complex Hilbert spaces, and study gapped Hamiltonians compatible with T and C symmetries under a certain definition of the latter. In some cases, the two pictures could be interchanged. In principle, an interpretation of the tenfold way could be made for bosons, but I think that one will be dealing with the one-particle Hilbert space of charged bosons, which may not be physical. The point is that there are different physical interpretations (e.g. bosonic/fermionic) of the Hilbert space, the Hamiltonian, and the antiunitary operations like $T$ and $C$. This is not dissimilar to the various interpretations of the threefold-way of Dyson-Wigner, division algebras, etc.

As for the groups $\mathbb{Z}$ and $\mathbb{Z}_2$, they are the $K$-theory groups of a point. How does $K$-theory come in? Well, implementing a certain combination of $T$ and $C$ in the tenfold way, is more or less the same thing as finding a module for an associated Clifford algebra, (or a super-division algebra). Clifford modules are related to $K$-theory through the work of Atiyah, and especially Karoubi. Thus, the groups $\mathbb{Z}$ and $\mathbb{Z}_2$ are related to the ways in which $T$, $C$ may be implemented. However, the precise meaning of these groups is quite subtle (for instance, what does the inverse mean?), and there are as many physical interpretations of them, as there are formulations of $K$-theory.

In my opinion, much of the literature is not terribly clear with regards to the physical interpretation of these groups, and refers quite vaguely to “$K$-theory” and “classifying spaces”. Thus, one gets statements like “there are five possible types of topological insulators in each spatial dimension”, whereas there are quite a lot of assumptions that go into such a statement!

If I may do some advertising, I have attempted a rigorous treatment of these matters in http://arxiv.org/abs/1406.7366 in a general and unified manner. I will be happy for any feedback, and to discuss these things with anyone who is interested.

Posted by: Guo Chuan Thiang on July 21, 2014 5:28 PM | Permalink | Reply to this

### Re: The Ten-Fold Way (Part 1)

Many times, people have noticed a mathematical pattern or re-occurrence of specific numbers, and assumed it was profound. Sometimes, it’s because there is a common origin to various occurrences that might not be immediately apparent. Other times, it turns out to be a coincidence, such as in Plato’s “Timmeas” or Kepler’s “Mysterium Cosmographicum”. Is the re-occurrence of the number “10” in those examples due to a common origin, or is it a coincidence? In one of your previous papers, you explained how the ten dimensions of string theory are due to something called the “Three-Psi Rule”. The dimensions of the four normed division algebras are 0, 1, 4, and 8. A one-dimensional string sweeps out a two-dimensional worldsheet. 8 + 2 = 10, and thus superstring theory is only consistent in 10 spacetime dimensions. A two-dimensional M2-brane sweeps out a three-dimensional worldsheet. 8 + 3 = 11, and thus M-theory exists in 11 spacetime dimensions.

Posted by: Jeffery Winkler on July 29, 2014 9:46 PM | Permalink | Reply to this

### Re: The Ten-Fold Way (Part 1)

Jeffrey wrote:

In one of your previous papers, you explained how the ten dimensions of string theory are due to something called the “Three-Psi Rule”. The dimensions of the four normed division algebras are 0, 1, 4, and 8.

You mean 1, 2, 4 and 8: consecutive powers of two, thanks to the Cayley–Dickson doubling construction.

And yes, we get classical superstring Lagrangians in dimensions 2 higher — 3, 4, 6 and 10 — thanks to the 3-$\psi$’s rule. This rule is really a cocycle condition, and the resulting cocycle lets us extend the Poincaré supergroup to a ‘super-2-group’:

• John Huerta, Division algebras and supersymmetry III.

If you examine the argument you’ll see it relies on the fact that the 1, 2, 4 or 8 ‘extra’ dimensions, beyond those of the 2d string world-sheet, inherit features from one of the normed division algebras.

But only the 10d theory works well when you quantize it.

Similarly we get supersymmetric membrane Lagrangians in dimensions 3 higher — 4, 5, 7 and 11 — thanks an equation called the 3-$\Psi$’s rule,which is a cocycle condition allowing us to extend the Poincaré supergroup to a ‘super-3-group’.

John Huerta’s paper on this — “Division algebras and supersymmetry IV” — should hit the arXiv fairly soon!

All this is closely related to the brane bouquet. I like this portion because I like division algebras.

Posted by: John Baez on July 30, 2014 7:22 AM | Permalink | Reply to this

### Re: The Ten-Fold Way (Part 1)

Yeah, I’m sorry. Of course, I meant dimensions 1, 2, 4, and 8. When I wrote that, I was initially thinking about the number of imaginary units in the algebra, so the real numbers have 0 imaginary units, the complex numbers have 1 imaginary unit, and the quaternions have 3 imaginary units.

So, going back to this list.

There are 10 of each of these things:

Associative real super-division algebras.

Classical families of compact symmetric spaces.

Ways that Hamiltonians can get along with time reversal (TT) and charge conjugation (CC) symmetry.

Dimensions of spacetime in string theory.


Is there some fundamental, perhaps profound, reason why the number 10 reoccurs in each of these different cases?

Posted by: Jeffery Winkler on August 5, 2014 10:01 PM | Permalink | Reply to this

### Re: The Ten-Fold Way (Part 1)

Jeffrey wrote:

Is there some fundamental, perhaps profound, reason why the number 10 reoccurs in each of these different cases?

Yes: that’s what this post, and the references, explain.

Posted by: John Baez on August 6, 2014 4:09 AM | Permalink | Reply to this

### Re: The Ten-Fold Way (Part 1)

There’s now an online course that discusses this from the condensed matter physics perspective. Very cool!

Posted by: Simon Burton on April 30, 2019 12:47 PM | Permalink | Reply to this

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