John Baez

This page has videos and slides of two talks:
The tenfold way and Symmetric spaces and the tenfold way,
given at Algebra, Particles, and Quantum Theory, February 6 and May 15, 2023.

The Tenfold Way

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 see the slides here, and also a video:

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!

For more details, read this:

Symmetric Spaces and 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.

You can see the slides here, and also a video:

© 2023 John Baez