Last Friday night when I was in Rome I received e-mails in quick succession from two science journalists asking what I thought about a new mathematical result, the “mapping of E8” that was going to be announced at a press conference on Monday. Information sent to journalists was embargoed until Sunday night at 11pm, but the first journalist sent me a copy of the brief press release and told me that there was a longer one available. Reading the press release left me still baffled about what this could be about: what was the “century old problem” that this group of 18 mathematicians had solved? The obvious interpretation of “mapping of E8”, mapping it as a geometrical object, didn’t make sense since that’s a well-understood problem. The group E8 is a 248 dimensional space, but its local geometry is the same everywhere and completely understood in terms of its Lie algebra. The global topology is interesting, but also well understood.
I wrote back to both journalists that the best person I knew to comment on this and its possible relation to physics would be John Baez, and asked to see the longer press release. It wasn’t much more enlightening, but it did have a link to a web-site with details. After spending a little time reading this I understood that “mapping of E8” was a calculation of the structure of representations of the split real form of E8, and decided that I was on vacation and not about to try and quickly write a blog posting about this.
Well, here are the press releases from MIT and AIM, and David Vogan did give a public talk about this yesterday at MIT. The media blitz was quite effective, getting the story into not just the usual suspects (there’s a good version of the story by JR Minkel at Scientific American), but also achieving a wide distribution in much less usual places such as today’s New York Times, the BBC, le Monde, and many, many others. I think this may be getting about as much attention as the proofs of Fermat’s Last Theorem and the Poincare Conjecture. There are also a huge number of blog postings, and I’m very pleased with myself to note that by far the best is the one by John Baez (crucially supplemented by the first comment there, from David Ben-Zvi), so I at least sent the journalists to the right place.
For mathematical details, John’s posting and the comments there are the best place to go besides the technical papers linked to from the AIM site.
While the calculation is a computational tour de force, and the computational methods may be useful elsewhere, the level of hype in the press releases, especially about the possible relations to physics, is somewhat disturbing. The AIM page on E8 and Physics contains statements such as
…once one adopts the basic principles of string theory, it can be argued that we live in the universe we live in because it is the only one that is possible.
as well as making the highly misleading claim that the new calculation has something to do with heterotic string theory.
What initially confused me about the press release is that, with the standard interpretation of what one means by “E8”, the “E8” that appears in heterotic string theory, there is no open problem to be solved. The group is well-understood, and so is its representation theory. As a compact Lie group, the representation theory of E8 is part of the standard Cartan-Weyl highest weight theory, and was worked out long ago. To read about this, there’s an excellent book by Frank Adams about the representation theory of E8 and other exceptional Lie groups, called Lectures on Exceptional Lie Groups. It is this representation theory that appears in the heterotic string story. For more about E8, and one of the stranger things I’ve seen in a math paper, you might want to look up a 1980 paper by Frank Adams called “Finite H-spaces and Lie Groups”, in the Journal of Pure and Applied Algebra.
What the new result is about is something quite different, the “split real form” of E8. The classification of compact Lie groups proceeds by classifying their Lie algebras, giving a well-known list, with E8 the largest of the exceptional cases. In doing this, one complexifies (works over the complex numbers), studying the complex semi-simple Lie algebras, which are the Lie algebras of the complexifications of the compact Lie groups. In the simplest example, one studies SU(2) by complexifying its 3d Lie algebra (R^3 with vector product), i.e. studying the Lie algebra of SL(2,C) instead. Finite dimensional unitary representations of SU(2) correspond to holomorphic representations of SL(2,C), and the same correspondence works in general between finite dimensional unitary reps of compact Lie groups and holomorphic representations of their complexifications.
Given the complexified group, one can ask if it has other “real forms”, i.e. subgroups other than the compact one which would have the same complexification. In the case of SL(2,C), there is another real form: SL(2,R). The representation theory of SL(2,R) is a vastly more complicated subject than the case of SU(2). One reason is that the group is non-compact. Geometrical constructions of representations like the Borel-Weil construction give infinite-dimensional irreducible unitary representations. The case of SL(2,R) is difficult enough (and a central topic in number theory), but the case of representations of general real forms of semi-simple Lie groups is extremely difficult and complicated. Representations are infinite-dimensional and labeled by “Langlands parameters” instead of highest weights. This theory has been pretty well worked out over the last 30-40 years or so, with the case of E8 one where it was known how to do calculations in principle, but they had so far been computationally intractable. Dealing with this is the new advance.
What actually is calculated are things called “Kazhdan-Lusztig” polynomials; for an explanation, see John’s blog. These tell one how to build arbitrary irreducible representations out of something simpler which one does understand, certain induced representations called “standard” representations. The numbers involved here also have a beautiful geometrical and topological interpretation. This is a generalization of what happens in the compact case, where the cell decomposition of the flag variety governs how irreducibles are built out of Verma modules.
So, this is a result about the structure of the irreducible representations of one of the real forms of E8 called the “split” real form. As far as I know it has nothing to do with heterotic string theory. The only thing I can think of that physicists have worked on that might make contact with this result is the work of people like Hermann Nicolai and Peter West trying to get physics out of Kac-Moody algebras like E10 and E11. I have no idea whether they have run into the split real form of E8 subalgebras and the representation theory of these in their work. In Pisa I had the pleasure of meeting blogger Paul Cook, a student of Peter West’s who is now a postdoc in Pisa and has worked on this kind of thing. Perhaps he would know about this.
Update: I hear from Jeffrey Adams that he has put together a web-page about this, aimed at mathematicians, and designed to explain the nature and significance of this result. It’s quite clear and does a good job of this, accessible if you have a bit of background in representation theory. If not, you may at least enjoy his comment on the media attention:
This leaves the question of why this story took off in the press. For us, that is harder to understand than the Kazhdan-Lusztig-Vogan Polynomials for E8.
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