From: baez@galaxy.ucr.edu (John Baez) Subject: M-theory, E8 and the octooctonions Date: 12 Sep 2000 23:29:35 GMT Newsgroups: sci.physics.research Summary: [missing] In article , Aaron Bergman wrote: >Interestingly, according to Witten, E_8 bundles are completely >determined by (1/16\pi^2) tr F /\ F which is a four form. Only if the base space has dimension less than 15, I suspect: The 1st homotopy group of E8 is trivial. (Since E8 has the marvelous property that its simply-connected form also has trivial center, we can't get nonsimply-connected Lie groups with Lie algebra E8. In this respect it's very different from the nonexceptional simple Lie groups.) The 2nd homotopy group of E8 is trivial. (This is true for all simple Lie groups, by a result of Whitehead.) The 3rd homotopy group of E8 is Z. (This is true for all simple Lie groups, by another result of Whitehead.) The 4th homotopy group of E8 is trivial. The 5th homotopy group of E8 is trivial. The 6th homotopy group of E8 is trivial. The 7th homotopy group of E8 is trivial. The 8th homotopy group of E8 is trivial. The 9th homotopy group of E8 is trivial. The 10th homotopy group of E8 is trivial. The 11th homotopy group of E8 is trivial. The 12th homotopy group of E8 is trivial. The 13th homotopy group of E8 is trivial. The 14th homotopy group of E8 is trivial. The 15th homotopy group of E8 is Z... ...and then I believe they're all trivial again until the 23rd. This means that if we're working with a manifold M of dimension < 16, a map from M to the classifying space for E8 bundles is the same as a map from M to K(Z,4) - the space with 4th homotopy group equal to Z and all higher ones trivial. Thus for such manifolds, E8 bundles are classified by the characteristic class you've described. But when you go to manifolds of higher dimension, this need no longer be true - maps from such manifolds can notice the higher homotopy groups of E8. >E_8 shows up all over the place with M-theory. I'm not sure it's quite >understood just what the deep relation is, however. Well, I have *my* theory: E8 is closely related to the octonions, being the isometry group of the octooctonionic projective plane. The octooctonions are the algebra O tensor O, where O is the octonions. Since this is not a division algebra, it's difficult to *define* the octooctonionic projective plane, and in fact right now nobody knows how to do it without explicit reference to E8, which makes it all annoyingly circular. Still, there is a nice Riemannian manifold whose isometry group is E8 and whose dimension is 2 x 8 x 8 = 128, exactly right for being the octooctonionic projective plane... and there are *other* constructions of E8 which involve the octooctonions, so it's only a matter of time before people figure out a better way to construct the octooctonionic projective plane "from scratch", without using E8. [1] The usual construction of the octooctonionic projective plane goes like this - it's sort of cute. At the Lie algebra level we have E8 = spin(16) + S+ where S is the space of right-handed spinors in 16 dimensions. This is not a Lie algebra direct sum, but spin(16) is a sub-Lie algebra. So at the group level we can form the quotient E8/Spin(16) and get a homogeneous space whose isometry group is E8 and whose dimension is that of S+, namely 128. There's a natural way to identify elements of S+ with pairs of octooctonions, so it's all very suggestive. By the way: I bet the "16" here has something to do with the point at which the homotopy groups of E8 stop being trivial, but I don't see precisely how that works. Anyway, since I believe that M-theory secretly relies on the octonions, the fact that E8 shows up all over is just grist for my mill. ......................................................................... [1] My hopes for a nice construction of the octooctonionic projective plane have been heightened by reading in Besse's book that Freudenthal found nice constructions of the bioctonionic and quateroctonionic ones, which have isometry groups E6 and E7, respectively. Besse refers to this paper: Hans Freudenthal, Bericht uber die Theorie der Rosenfeldschen elliptischen Ebenen [Algebraical and topological foundations of geometry (Proc. Colloq., Utrecht, 1959), pp. 35--37, which I have unfortunately been unable to obtain. Just now, I discovered that it was reprinted in this book: Raumtheorie. (German) [Theory of space] Edited by Hans Freudenthal. Wege der Forschung [Paths of Research], CCLXX. Wissenschaftliche Buchgesellschaft, Darmstadt, 1978. vi+408 pp. DM 79.00. ISBN 3-534-05604-3 Unfortunately the UC library does not have this book! :-( If anyone could send me a copy of the 2-page essay in question, they would have my undying gratitude. I'm convinced it holds the keys to the deepest mysteries of the universe. By the way, Freudenthal was a far-out dude. Not only did he do important work on Lie groups and topology, he also designed a language for communicating with extraterrestrials: Freudenthal, Hans, 1905-. Lincos; design of a language for cosmic intercourse. Amsterdam, North-Holland Pub. Co., 1960- v. 23 cm. (I assume he wasn't talking about sexual intercourse.)