From: baez@galaxy.ucr.edu Subject: Re: The Exceptionals Date: 19 Sep 2000 19:01:08 GMT Newsgroups: sci.physics.research Summary: [missing] Terry Pilling wrote: >I have noticed how important the exceptional Lie algebras >seem to be in physics. A more conservative assessment would be: lots of people *hope* the exceptional Lie algebra will be important in physics. It's always worth keeping in mind that no theory involving the exceptional Lie algebras has ever made an experimentally verified prediction. Thus the jury is still out. >For example Heterotic string theory >and even in monstrous moonshine which is supposed to have >something to do with CFT's (and I am dying to find out what >that's all about!) Briefly the idea goes like this: bosonic string theory likes to live in 26 dimensions, but we often describe it via a 2-dimensional conformal field theory (CFT) taking values in a 24-dimensional "target manifold". This target manifold must have some nice properties for things to work out. In particular, things will work out if the target manifold is a torus of the form R^24/L where L is a "self-dual even lattice". What this means is: 1) L is a lattice, i.e. a discrete set in R^24 closed under addition. 2) L is self-dual: if L* denotes the set of vectors v in R^24 such that v.w is an integer for all w in L, then we have L* = L. 3) L is even: v.v is an even integer for all v in L. (Here . denotes the usual Euclidean inner product.) Now, self-dual even lattices only exist in dimensions that are multiples of 8. In dimension 8 there is one, namely the E8 root lattice. In dimension 16 there are two, namely E8+E8 and something called D16+. (These give two versions of heterotic string theory, where we need to compactify 16 dimensions into a torus R^16/L to get the 26-dimensional bosonic string to talk to the 10-dimensional superstring.) In dimension 24 there are 24. Of these, the strangest and most beautiful is the Leech lattice. The symmetry group of the Leech lattice is a cousin of the Monster group, so it's perhaps not surprising that something very fun happens when we form a conformal field theory involving fields that take values in R^24/L where L is the Leech lattice. If you take this conformal field theory and tweak it slightly, you get a field theory which has the Monster group as a symmetry group! For details I recommend: Vertex Operator Algebras and the Monster, by Igor Frenkel, James Lepowsky, and Arne Meurman, Academic Press, 1988. Sphere Packings, Lattices and Groups, J. H. Conway and N. J. A. Sloane, second edition, Grundlehren der mathematischen Wissenschaften 290, Springer-Verlag, 1993. >This became even more interesting when I read that the exceptionals >can be constructed using the octonions. > >I am wondering if anyone can give me a reference, or show >me how this construction using the octonions is accomplished. There are *lots* of ways to construct the exceptional Lie algebras using the octonions. Some good starting points include: C. H. Barton and Anthony Sudbery, Magic squares of Lie algebras, preprint available as math.RA/0001083. Richard D. Schafer, Introduction to Non-Associative Algebras, Dover, New York, 1995. Feza G"ursey and Chia-Hsiung Tze, On the Role of Division, Jordan, and Related Algebras in Particle Physics, World Scientific, Singapore, 1996. These have lots of references to the older literature, including the classic works of Jacques Tits and Has Freudenthal, which are in French and German, respectively. BUT: if you're not in too much of a rush, you might wait for my review article on the octonions, which will explain this stuff in excruciating detail, along with possible applications to physics. This article is due next March so I'll certainly be done with it by then. In particular, this article will give 7 octonionic constructions of the exceptional Lie algebra E7, and 6 of the Lie algebra E6. I wish I knew 8 octonionic constructions of E8, but this one seems too simple to admit that many different constructions. I'll talk about this article in This Week's Finds when it's done.