John Baez and his student John Huerta have an article in Scientific American this month about octonions in string theory and M-theory. Peter Woit has given it a bit of a cynical review describing it as hype. The defence from John in the comments is worth reading. Here is a bit of what he says
“So, don’t try to make it sound like an obscure yawn-inducing technicality about “some supersymmetry algebra working out a certain way in a certain dimension”. It’s a shocking and bizarre fact, which hits you in the face as soon as you start trying to learn about superstrings. It’s a fact that I’d been curious about for years. So when John Huerta finally made it really clear, it seemed worth explaining — in detail in some math papers, and in a popularized way in Scientific American.”
The article entitled “The Strangest Numbers in String Theory” is about an early observation from the study of superstring theories that the four division algebras are related to four classical formulations of superstring theory in 3, 4, 6 and 10 dimensions. The four division algebras are the reals, complex numbers, quaternions and octonions with dimensions 1,2,4 and 8 and the dimensions of the superstring theories are 2 dimensions higher in each case.
Many of our readers will be familiar with the internet writings of John Baez where he has described these things so I wont attempt to cover any details. His student John Huerta has just finished off his thesis in which he clarifies these observations using higher dimensional lie algebras. The results extend to one dimension higher when strings are replaced by membranes. In the quantum theory only the highest dimensional versions related to the octonions hold up consistently giving us the 10 dimensional superstring theories and M-theory in 11 dimensions. Of course this is not complete since we still don’t know what the full formulation for M-theory is. Even these higher dimensional observations are not new, see for example Mike Duffs brane-scan from 1987 where the relationships were already plotted out. This new work clarifies these results using the concept of 3-lie-algebras from n-category theory.
The Scientific American version does not go into great detail but is a very well written introduction to the ideas. If you don’t have access to it don’t worry, John says he will be allowed to post an online version after a month or so. You can also explore what has been posted already starting here which is more advanced than the article but still very pedagogical.
Personally I find these algebraic ideas for M-theory very enticing. It is a major goal to formulate a complete non-perturbative version of string theory that encompasses all its forms and I think the purely algebraic approach is the best line of attack. It is especially intriguing that the octonions have such a direct relationship to the dimensions in which these theories work, but ultimately the algebraic structures we need to understand it fully are probably much more complex.
The work of Mike Duff and his collaborators which brings in the algebra of hyperdeterminants and qubits to understand a slightly different role of the octonions in string theory is one of the areas to follow. This work brings in the duality algebras found in string theory black holes. I know that several of our regular commenters are very familiar with this already so I need not give more details. Indeed it is the fact that the same algebraic structures keep appearing in different contexts that is so intriguing, yet so confusing, as if we are missing some principle of unification that relates these things.