John Baez has always been obsessed by octonions. He sees them everywhere. I love octonions but I realize that they don't play much role in most of physics - and not even in most of string theory.
In 2009, I wrote a related article
The Strangest Numbers in String Theory (demo; free version in 1 month)The first fact I find utterly crazy is that two people who manifestly and demonstrably don't understand string theory - not even at the undergraduate level - are writing articles for widely read journals pretending to be scientific magazines with "string theory" playing the role of one half of the title. As I will argue, they are really abusing the stellar brand of string theory to promote their idiosyncratic bullshit.
The SciAm article is a simplified version of this more technical article
Division Algebras and Supersymmetry II by John Baezposted at Jacques Distler's n-category and marijuana coffee shop. As sketched in my 2009 blog entry, Baez is obsessed by the observation that the classical string theories in D=3,4,6,10 - which use spinors with different reality and chirality projections - can be mapped to R,C,H,O, the four division algebras of dimensions 1,2,4,8.
In some sense, SL(2,O) may be interpreted as SO(9,1).
However, that's it. The actual structure of the octonions - their multiplication table whose characteristic automorphism group is G_2 - is not really used in SO(9,1) in a useful way. And if this table and the G_2 symmetry appears at a certain stage, it immediately disappears.
Moreover, the minimal superPoincaré algebras in 3,4,6,10 dimensions are far from being the only four superalgebras in their class. They're not even the most interesting or most symmetric ones. They're just algebras in four dimensionalities D such that D-2 - the number of physical polarizations of a gauge boson - is a power of two. It has to be a power of two in a minimal supersymmetric gauge theory because the number of bosonic polarizations has to match the number of fermionic polarizations and the latter come from a spinor; I don't need division algebras to prove that. Also, I don't need division algebras to prove that minimal super Yang-Mills theories can only exist in these four dimensions.
Also, the "other remarkable structures" that directly come from the R,C,H,O sequence are not that interesting. He also talks about membranes in 4,5,7,11 dimensions (not surprising that they have the same counting of physical degrees of freedom - it's just double dimensional reduction). Except for the last one, they're not terribly interesting theories or vacua. So the detailed data refute Baez's hypothesis; he clearly doesn't care. Also, the R,C,H,O argument doesn't really show why/that only the 10- or 11-dimensiona case is consistent at the quantum level (as a separate theory). This whole R,C,H,O perspective on the landscape of string/M-theory vacua is naive at the level of a kindergarten kid. You simply can't understand the secrets of string theory by learning the sequence 1,2,4,8.
In the slow comments under the 2009 blog entry, Robert Helling argued that there is a lot of interesting fog about the closure of the supersymmetry algebra etc. I find this whole approach to these issues irrational.
There's lots of fascinating, still poorly understood mysteries about string theory's internal mathematical consistency. But the topics that Baez, Huerta, Helling, and others are talking about are pretty much exactly those where the mystery has already been fully eliminated. The mysterious links to pure mathematics continue to exist but you must get much deeper to uncover them.
There are simple ways to prove that 3,4,6,10 are the relevant dimensionalities for the classical superstring - but only D=10 is the actual dimension that is allowed at the quantum level. However, counting of the dimensions is one of the most elementary facts about string theory. All the cancellations that Baez et al. hype can be easily proved - in many different ways, in fact. Joe Polchinski boasts that volume I of his book derives the critical dimension of the bosonic string theory in 7 different ways. So why is there so much ado about nothing? Our ability to derive results in 7 different ways shows that we kind of understand it. Those ways also connect different portions of mathematics - but it is the whole structure of string/M-theory, and not a division algebra, who unifies all this stuff.
Moreover, as hinted in the previous sentence, what I am really irritated by is Baez's obsessive tendency to reduce all the mathematical cleverness of string theory to the division algebras in general and octonions in particular. His way of looking at all these things shows that he is nothing else than an irrational numerologist who can never distinguish real insights from superficial distractions - and who apparently doesn't want to distinguish them.
The division algebra is just one way to look at all the issues linked to 7 imaginary units with the G_2 automorphism group, and all the associated algebraic structure. The octonions as an algebra are just one possible corollary or intellectual projection of the structure behind it. And even all these structures combined are just a totally minuscule portion of the string theory's wisdom, much like John Baez's knowledge of string theory is an infinitesimal fraction of the knowledge of a good graduate student.
So the article hyping a set of a few simple mathematical observations is just pathetic. It's not really demonstrably wrong - unlike Garrett Lisi's pseudoscientific "theories of everything" that can't agree with the most elementary facts of particle physics such as parity violation. But it's still morally wrong because it totally distorts what is understood and what remains mysterious about the remarkable mathematical structure we still call string theory.
This distortion shouldn't be unexpected from authors who don't have a clue about string theory. But this is no real excuse because Scientific American shouldn't be publishing stuff written by people who don't know what they're talking about.