This week’s hype comes from an unusual source, John Baez and his ex-student John Huerta, who have a new article in Scientific American entitled The Strangest Numbers in String Theory.
The expository article about octonions by John (Baez) that appeared in the AMS Bulletin (copy here, a web-site here) is one of the best pieces of mathematical exposition that I have ever seen. The octonions can be thought of as a system of numbers generalizing the quaternions. As with the quaternions, multiplication does not commute, and things are even worse, it’s not associative either. So, probably best not to try and think of these as “numbers”, but they do give a very remarkable exotic algebraic structure, one that explains all sorts of other exotic structures occurring in different areas of mathematics. The article beautifully explains a lot of the intricate story of how octonions connect up surprising phenomena in algebra, geometry, group theory and topology.
If you’re a mathematical physics mystic like myself, you’re susceptible to the belief that anything this mathematically deep, showing up in seemingly unrelated places, must somehow have something to do with physics. The story of octonions is closely related to the story of Clifford algebras, which are definitely a crucial part of physics, but it seems to me we’re still a long ways from truly understanding the role in physics of Clifford algebras, much less the more esoteric octonions. One thing that is fairly well understood is that the sequence of division algebras explains some of the structure of low-dimensional spin groups in Minkowski signature, through the isomorphisms:
The octonion story is supposed to be the next in line, involving Spin(9,1), but made much trickier by the fact that SL(2,O) doesn’t really exist, since the octonions are non-associative.
Back in 1982, a very nice paper by Kugo and Townsend, Supersymmetry and the Division Algebras, explained some of this, ending up with some comments on the relation of octonions to d=10 super Yang-Mills and d=11 super-gravity. Baez and Huerta in 2009 wrote the very clear Division Algebras and Supersymmetry I, which explains how the existence of supersymmetry relies on algebraic identities that follow from the existence of the division algebras. Kugo-Townsend don’t mention string theory at all, and Baez-Huerta refers to superstrings just in passing, only really discussing supersymmetric QFT. There’s also Division Algebras and Supersymmetry II by Baez and Huerta from last year, with intriguing speculation about Lie n-algebras and what these might have to do with relations between octonions and 10 and 11 dimensional supergravity. For a nice expository paper about this stuff, see their An Invitation to Higher Gauge Theory.
In contrast to the tenuous or highly-speculative connections to string theory that appear in these sources, the Scientific American article engages in the all-too-familiar hype pattern. The headline argument is that octonions are important and interesting because they’re “The Strangest Numbers in String Theory”, even though they play only a minor role in the subject. It wouldn’t surprise me at all if octonions someday do end up playing an important role in a unified theory, but the rather obscure connection to the calculation of the critical dimension of the superstring that seems to be the main point of the Scientific American article isn’t a very convincing argument for such a role.
Somehow I suspect that those string theorists who were upset by Scientific American’s decision to publish speculation by Garrett Lisi about E8 and wrote in to complain, won’t be similarly upset to find this highly speculative material about the octonions appearing in the magazine.