The importance of the tenfold way in physics was only recognized in
this century. Simply put, it implies that there are ten fundamentally
different kinds of matter. But it goes back to 1964, when the
topologist C. T. C. Wall classified the associative real super
division algebras and found ten of them. The three 'purely even'
examples were already familiar: the real numbers, complex numbers and
quaternions. The rest become important when we classify
representations of groups on super Hilbert spaces. I explain this
classification, its connection to Clifford algebras, and some of its
implications for quantum physics.
You can also watch another version of the first talk, where I explain
this material to my friend James Dolan:
This 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 more patterns lurking in the tenfold way!
The tenfold way has many manifestations. It began as a tenfold
classification of states of matter based on their behavior under time
reversal and charge conjugation. Mathematically, it relies on the fact
that there are ten super division algebras and ten kinds of Clifford
algebras, where two Clifford algebras are of the same kind if they
have equivalent super-categories of super-representations. But Cartan
also showed that there are ten infinite families of compact symmetric
spaces! After explaining symmetric spaces, we show how they arise
naturally from forgetful functors between categories of
representations of Clifford algebras.