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John Baez (Nov 22 2020 at 19:14):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 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 and , 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:
There's a lot of category theory lurking in here - as the mention of Morita equivalence hints. But I haven't gotten around to explaining that anywhere.
Simon Burton (Nov 22 2020 at 20:05):"any chain complex, exterior algebra, or Clifford algebra is automatically a super vector space"
I'm curious how does this work, specifically for chain complexes. Don't you have to wrap up (fold modulo 2) the grades somehow? I don't see what a "fermionic" chain complex would be...
John Baez (Nov 22 2020 at 20:46):Yes, you fold it up: a chain complex is a -graded vector space with extra structure, but any -graded vector space automatically gives a -graded vector space using the homomorphism .
John Baez (Nov 22 2020 at 20:46):(Whenever you have a -graded vector space and a group homomorphism you can turn it into an -graded vector space.)
John Baez (Nov 22 2020 at 20:48):The point is, this "folding up" process is not an arbitrary silly thing: it's crucial for understanding chain complexes.