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November 22, 2020

The Tenfold Way (Part 5)

Posted by John Baez

I now have a semiannual column in the Notices of the American Mathematical Society! I’m excited: it gives me a chance to write short explanations of cool math topics and get them read by up to 30,000 mathematicians. It’s sort of like This Week’s Finds on steroids.

Here’s the first one:

  • The tenfold way, Notices Amer. Math. Soc. 67 (November 2020), 1599–1601.

The tenfold way became important in physics around 2010: it implies that there are ten fundamentally different kinds of matter. But it goes back to 1964, when C. T. C. Wall classified real ‘super division algebras’. He found that besides ℝ,ℂ\mathbb{R}, \mathbb{C} and ℍ\mathbb{H}, which give ‘purely even’ super division algebras, there are seven more. He also showed that these ten algebras are all real or complex Clifford algebras. The eight real ones represent all eight Morita equivalence classes of real Clifford algebras, and the two complex ones do the same for the complex Clifford algebras. The tenfold way thus unites real and complex Bott periodicity.

In my article I explain what a ‘super division algebra’ is, give a quick proof that there are ten real super division algebras, and say a bit about how they show up in quantum mechanics and geometry.

For a lot more about the tenfold way, try this:

Posted at November 22, 2020 5:55 PM UTC

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Loop spaces

The ten-element set of families of compact real symmetric spaces comes with an action of ℤ\mathbb{Z}, called “take connected component of based loops”, with a size 2 orbit giving complex Bott periodicity and size 8 orbit giving real/quaternionic. What is the corresponding operation on super division algebras?

Also, a compact real symmetric space is slightly more info than a family – there’s one or two natural numbers involved, e.g. U(a+b)/U(a)xU(b) instead of just U/UxU. Is there a similar “destabilization” of the concept of super division algebra?

Posted by: Allen Knutson on November 23, 2020 1:27 AM | Permalink | Reply to this

Re: Loop spaces

“Good question”—as people always say right before they give a bad answer.

I’ll have to think about these. I plan to write more about the tenfold way, since right now the subject doesn’t cohere well enough, and you’ve given me some pointers to how it should be clarified.

For the first one: the operation of tensoring with the free superalgebra on an odd square root of -1 cycles us around the Morita equivalence classes of complex Clifford algebras, and similarly for the real ones; this is closely connected to looping in a way I could explain if it were earlier in the day. It’s all hiding in Milnor’s wonderful book Morse Theory.

This should also clarify the ‘unstable’ world. Clifford algebras depend on two parameters: the number of square roots of -1, and the number of square roots of +1. We can move through the Clifford algebras by two operations: tensoring free superalgebra on an odd square root of -1, and tensoring with the free superalgebra on an odd square root of +1.

Now the question becomes: what are the two operations on compact symmetric spaces?

Posted by: John Baez on November 23, 2020 3:16 AM | Permalink | Reply to this

Loop spaces

Do those operations commute? What is the group acting on this 10-element set, and what are the orbits? I only knew about the Z/8 action.

Posted by: Allen Knutson on November 23, 2020 4:08 AM | Permalink | Reply to this

Re: The Tenfold Way

I now have a semiannual column in the Notices of the American Mathematical Society!

Congratulations!

You also have a semidecadal custom of writing posts with this title :-)

The ten-fold way (Part 1)

The ten-fold way (Part 2)

Posted by: Tom Leinster on November 23, 2020 10:00 PM | Permalink | Reply to this

Re: The Tenfold Way

Thanks! And don’t forget:

So maybe this is Part 5! I’ll quit when I get to Part 10.

Posted by: John Baez on November 23, 2020 10:46 PM | Permalink | Reply to this

Re: The Tenfold Way

Ok, it is 10-fold because it is 8 and 2. Bug… Bott periodicity is not a invitation to fold the 8 into the 2?

Posted by: Alejandro on November 30, 2020 3:17 AM | Permalink | Reply to this

Bott folds

Ah now in the morning I remember… With Bott periodicity we have a sort of folding or projection of algebraic K-Theory into topological K-Theory, and somehow I was confusing this projection with some way to map real into complex Bott periodicity.

Posted by: Alejandro Rivero on November 30, 2020 11:57 AM | Permalink | Reply to this

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