nLab ten-fold way




Algebraic topology



Various structures in algebra and algebraic topology have a classification into 10 classes, more or less explicitly related to the 2 + 8 topological K-theory-groups of the point up to Bott periodicity: 2 for KU and 8 for KO, which are unified in KR.

The specific term “10-fold way” is a variation of the term “threefold way” used in Dyson (1962), which is referred to as inspiration by Heinzner, Huckleberry & Zirnbauer (2004) and Zirnbauer (2010) in reference to work going back to Altland & Zirnbauer (1997). While the term “10-fold way” is now often attributed to these authors, they may not actually have used it as a term (Zirnbauer (2010) finally speaks of the “10-way”, at least).

The term became rather popular in the wake of the seminal suggestion by Kitaev (2009) that “free” topological phases of matter, specifically free crystalline topological insulator-phases, are classified by some form of topological K-theory (though Kitaev (2009) does not use the term “10-fold way”, either): see at K-theory classification of topological phases of matter for more on this.

Kitaev’s suggestion was made more precise by Freed & Moore (2013) (who, however and ironically, doubted, on p. 57, its application to topological phases – but see pp. 2 of SS23 for resolution) and it is these authors who very much amplify (and may have actually coined) the term “10-fold way”. Moreover, they point out (pp. 75) that a 10-fold classification is already contained in Dyson (1962)(!), which the authors re-interpret as the classification of super division algebras.

This 10-fold way of super-division algebras? is further amplified in [Moore (2013), p. 129] and Geiko & Moore (2021)



Last revised on April 6, 2023 at 12:58:35. See the history of this page for a list of all contributions to it.