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Have adjusted the last sentence.
The classification of topological phases of matter is, at face value, a computation of twisted equivariant KR-theory classes. I don’t think it quite makes sense to say that the classification of super division algebras is the “basis” of the 2+8 Bott periodicity of KR-theory. At best it’s one of several perspectives on the phenomenon of Bott periodicity.
But Moore thinks the 2+8 is a red herring as I cited in the other thread:
A key point we want to stress is that the 10-fold way is usually viewed as 10 = 2+8, where 2 and 8 are the periodicities in complex and real K-theory. And then the K-theory classification of topological phases is criticized because it only applies to free systems. However, we believe this viewpoint is slightly misguided. The unifying concept is really that of a real super-division algebra, and there are 10 such. They can be parceled into 10 = 8+2 but they can also equally naturally be parceled into 10 = 7+3 (with the 3 referring to the purely even superdivision algebras).
Why “but”?
The logic of classifying topological phases of matter proceeds from the classification of filled Bloch bundles of electron states subject to CPT-equivariance. The class of these bundles is in KR-theory and that happens to have 10 graded groups of classes over point, up to Z/2-equivariant Bott periodicity. That is the 10-fold way in topological solid state physics.
Now one may dive into the general question how to see the Bott periodicity of topological K-theory. There are a variety of ways, some involving Clifford algebra, some involving loop space theory, some maybe involving super division algebras.
Ok, I thought Moore was intending something more with “The unifying concept is really that of a real super-division algebra”, as though he thinks that’s the conceptual underpinning of what’s going on in the classification of these phases.
I had another look at Moore 2013.
The section 25 announced to make contact to topological phases of matter is empty.
The section 14 about super division algebras seems to be that from appendix C of Freed & Moore (2013). I wouldn’t call this the “basis” of the results there (certainly it’s not advertised as such), it’s more a side remark highlighting the relation to Dyson’s work, thereby justifying the terminological allusion to his “threefold way”.
How about we sum this up as follows (have made that edit now, but feel free to complain):
The classification of the associative real division superalgebras by a “ten-fold way” is essentially attributed to Dyson (1962) by Freed & Moore (2013, Appendix C) (in the context of the K-theory classification of topological phases of matter), further expanded on in Moore (2013), Section 14 and Geiko & Moore (2021).
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