The Tenfold Way

Abstract: 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. These are finite-dimensional real $\mathbb{Z}/2$-graded algebras where every nonzero homogeneous element is invertible. 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 this expository article we give a quick proof that there are ten real super division algebras, and say a bit about applications of the tenfold way.