### The Tenfold Way

#### 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 $\mathbb{R}, \mathbb{C}$ and $\mathbb{H}$, 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:

## Loop spaces

The ten-element set of families of compact real symmetric spaces comes with an action of $\mathbb{Z}$, 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?