Higher Gauge Theory, Division Algebras and Superstrings
Posted by John Baez
I’m giving two talks at Hong Kong University this week:

Energy, the environment, and what mathematicians can do.
 Higher gauge theory, division algebras and superstrings.
These are roughly the first talk of my new life, and the last of my old. We’re chatting about talk 1 over on Azimuth, here and here. But the nCafé is the right place for chatting about talk 2!
I’m giving my talk Higher gauge theory, division algebras and superstrings as part of the Workshop on Geometry and Lie Groups organized by various folks including my old friend JiangHua Lu. It’ll be great to see her again. I met her back in the early 1990’s. She was doing at postdoc at MIT. I was on leave from U. C. Riverside, teaching for 2 years at Wellesley College to be closer to Lisa Raphals (who later came to Riverside and married me).
Lu was a student of Alan Weinstein, the guru of groupoids and classical mechanics. Around the time I met her, she did some great work on Hopf algebroids obtained by quantizing Poisson Lie groupoids. So, it’s quite fitting that this workshop features a talk by Peter Bouwknegt on ‘Leibniz algebroids’. Leibniz algebras are like Lie algebras without the antisymmetry of the bracket. They show up a lot more often than you might think. So if Lie algebras get to have Lie algebroids, Leibniz algebras deserve to be oidified too.
I’ll also be happy to see my old grad school friend Mathai Varghese, who was a student of Quillen and whose thesis became the MathaiQuillen formalism. He’s giving a talk on fractional index theory.
And I’ll be glad to see David Vogan, who was a professor at MIT back when Mathai and I were in grad school… and who still is! He’s one of the people who mapped E_{8}—as the journalists rather vaguely put it. As a student I was terrified of him because he knew infinitely more about Lie groups than I did. Somehow I’ve lost that fear, even though he still knows infinitely more about Lie groups. I guess I realized at some point that smart people don’t actually emit a ‘death ray’ that kills anyone stupid within a 3meter radius.
But enough gossip! What’s my talk about?
It’s all about John Huerta’s thesis. I love this thesis because it takes three old dreams of mine and shows how they fit together:
 Higher gauge theory: This is the dream that all the wonderful mathematics associated to gauge theory generalizes from point particles to higherdimensional objects (strings and branes). This wonderful mathematics includes Lie groups, Lie algebras, fiber bundles, connections on fiber bundles, and much more. The generalization proceeds by taking all the sets in sight, and replacing them by categories, 2categories and so on. A theory that simultaneously generalizes all this wonderful mathematics in such a natural way would be a magnificent thing even if it had no applications to physics—but a few people actually think the world might be made of higherdimensional objects like strings and branes!
 Division algebras: It’s a stark and stunning fact that there are only four finitedimensional normed division algebras: the real numbers, the complex numbers, the quaternions (which are noncommutative) and the octonions (which are also nonassociative). The first two of these are fundamentally important throughout mathematics, the third is useful in 3dimensional and 4dimensional geometry, and the fourth… who ordered that? A lot of ‘exceptional’ objects in mathematics, most notably all the exceptional Lie groups, are related to the octonions. But that doesn’t really answer the question of “what are the octonions good for?” It just shows that they can’t be brushed aside as an isolated curiosity. The dream is that the octonions are somehow relevant to our universe, and that their quirkiness somehow explains some of the quirky features of its fundamental laws.
 Superstrings: As a physicist I was never very fond of superstring theory, because there’s no evidence for “supersymmetry”, the idea that every boson has a partner fermion. But as a mathematician, I couldn’t help but want to learn a bit about it. It has a eerie way of making powerful connections between disparate ideas. And I couldn’t help being curious about one of the basic quirky facts about superstring theory: it works best in 10 dimensions. This is a very rough statement, which needs a lot of qualifications to become precise. But there’s something true about it, and part of why it’s true is that 10 = 8+2, with 8 dimensions corresponding to the octonions and 2 corresponding to the 2 dimensions of a string’s worldsheet. This fact, known for a long time, is something I’ve been itching to understand better, ever since I heard about it.
You can see already from my discussion that these three dreams fit together. That’s been known for quite a while, thanks to the work of many people. But John Huerta put the pieces together in a way that’s so clear that even I can understand it. Briefly:
Abstract: Classically, superstrings make sense when spacetime has dimension 3, 4, 6, or 10. It is no coincidence that these numbers are two more than 1, 2, 4, and 8, which are the dimensions of the normed division algebras: the real numbers, complex numbers, quaternions and octonions. We sketch an explanation of this already known fact and its relation to “higher gauge theory”. Just as gauge theory describes the parallel transport of supersymmetric particles using Lie supergroups, higher gauge theory describes the parallel transport of superstrings using “Lie 2supergroups”. Recently John Huerta has shown that we can use normed division algebras to construct a Lie 2supergroup extending the Poincaré supergroup when spacetime has dimension 3, 4, 6 or 10. He also used them to construct a Lie 3supergroup when spacetime has dimension 4, 5, 7 or 11. The 11dimensional case is related to 11dimensional supergravity, and thus presumably to “Mtheory”.
For a more detailed but still very gentle tour, try the slides of my talk!
If the talk is too technical for you, wait until May. Then you can read a Scientific American article that John Huerta and I wrote about the same subject. It’ll be called “The strangest numbers in string theory”.
(In case you’re wondering, the author doesn’t get to pick the title. We’d unimaginatively suggested “The octonions”, but to most people that’s about as alluring as “The szymidh”. Probably even less.)
A note for the experts: my talk slides don’t mention the fact that a somewhat more sophisticated Lie 2supergroup extension of the Poincaré supergroup seems required for anomaly cancellation, as Urs pointed out here. I’ll say that out loud, though…
Re: Higher Gauge Theory, Division Algebras and Superstrings
That does not sound like good reason not to be fond of superstring theory. The generic superstring vacuum gives an effective theory (which we might observe) that has no global supersymmetry. But the fact that the superstring is locally supersymmetric on its worldsheet implies that whatever the effective theory is, it will contain fermionic fields (not so for the bosonic string) and that is certainly something that there is strong evidence for.