nLab ten-fold way

Idea

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)

Related concepts

• Bott periodicity

• K-theory classification of topological phases of matter

References

• Freeman Dyson, The Threefold Way. Algebraic Structure of Symmetry Groups and Ensembles in Quantum Mechanics, J. Math. Phys. 3 1199 (1962) [doi:10.1063/1.1703863]

• Alexander Altland and Martin R. Zirnbauer, Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures, Phys. Rev. B 55 (1997) 1142 [doi:10.1103/PhysRevB.55.1142]

• P. Heinzner, A. Huckleberry, M.R. Zirnbauer, Symmetry classes of disordered fermions [arXiv:math-ph/0411040]

• Martin R. Zirnbauer, Symmetry Classes [arXiv:1001.0722]

• Alexei Kitaev, Periodic table for topological insulators and superconductors, talk at: L.D.Landau Memorial Conference “Advances in Theoretical Physics”, June 22-26, 2008, In:AIP Conference Proceedings 1134, 22 (2009) (arXiv:0901.2686, doi:10.1063/1.3149495)

• Daniel Freed, Gregory Moore, Twisted equivariant matter, Ann. Henri Poincaré 14 (2013) 1927-2023 [arXiv:1208.5055, doi:10.1007/s00023-013-0236-x]

• Gregory Moore, Quantum symmetries and compatible Hamiltonians (2013) [pdf]

• Roman Geiko, Gregory W. Moore, Dyson’s classification and real division superalgebras, Journal of High Energy Physics 2021 4 (2021) 299 [doi: 10.1007/jhep04(2021)299]

Review:

• John Baez, The Tenfold Way (2020) [webpage]