Last week I talked about asymptotic freedom - how the "strong" force gets weak at high energies. Basically, I was trying to describe an aspect of "renormalization" without getting too technical about it. By coincidence, I recently got my hands on a book I'd been meaning to read for quite a while:
1) Laurie M. Brown, ed., "Renormalization: From Lorentz to Landau (and Beyond)", Springer-Verlag, New York, 1993.
It's a nice survey of how attitudes to renormalization have changed over the years. It's probably the most fun to read if you know some quantum field theory, but it's not terribly technical, and it includes a "Tutorial on infinities in QED", by Robert Mills, that might serve as an introduction to renormalization for folks who've never studied it.
Okay, on to some new stuff....
It's a bit funny how one of the most curious features of bosonic string theory in 26 dimensions was anticipated by the number theorist Edouard Lucas in 1875. I assume this is the same Lucas who is famous for the Lucas numbers: 1,3,4,7,11,18,..., each one being the sum of the previous two, after starting off with 1 and 3. They are not quite as wonderful as the Fibonacci numbers, but in a study of pine cones it was found that while most cones have consecutive Fibonacci numbers of spirals going around clockwise and counterclockwise, a small minority of deviant cones use Lucas numbers instead.
Anyway, Lucas must have liked playing around with numbers, because in one publication he challenged his readers to prove that: "A square pyramid of cannon balls contains a square number of cannon balls only when it has 24 cannon balls along its base". In other words, the only integer solution of
12 + 22 + ... + n2 = m2,is the solution n = 24, not counting silly solutions like n = 0 and n = 1.
It seems that Lucas didn't have a proof of this; the first proof is due to G. N. Watson in 1918, using elliptic functions. Apparently an elementary proof appears in the following ridiculously overpriced book:
2) W. S. Anglin, "The Queen of Mathematics: An Introduction to Number Theory", Kluwer, Dordrecht, 1995.
For more historical details, see the review in
3) Jet Wimp, Eight recent mathematical books, Math. Intelligencer 18 (1996), 72-79.
Unfortunately, I haven't seen these proofs of Lucas' claim, so I don`t know why it's true. I do know a little about its relation to string theory, so I'll talk about that.
There are two main flavors of string theory, "bosonic" and "supersymmetric". The first is, true to its name, just the quantized, special-relativistic theory of little loops made of some abstract string stuff that has a certain tension - the "string tension". Classically this theory would make sense in any dimension, but quantum-mechanically, for reasons that I want to explain someday but not now, this theory works best in 26 dimensions. Different modes of vibration of the string correspond to different particles, but the theory is called "bosonic" because these particles are all bosons. That's no good for a realistic theory of physics, because the real world has lots of fermions, too. (For a bit about bosons and fermions in particle physics, see "week93".)
For a more realistic theory people use "supersymmetric" string theory. The idea here is to let the string be a bit more abstract: it vibrates in "superspace", which has in addition to the usual coordinates some extra "fermionic" coordinates. I don't want to get too technical here, but the basic idea is that while the usual coordinates commute as usual:
xi xj = xj xithe fermionic coordinates "anticommute"
yi j = - yj yiwhile the bosonic coordinates commute with fermionic ones:
xi yj = yj xiIf you've studied bosons and fermions this will be sort of familiar; all the differences between them arise from the difference between commuting and anticommuting variables. For a little glimpse of this subject try:
4) John Baez, Spin and the harmonic oscillator, http://math.ucr.edu/home/baez/harmonic.html
As it so happens, supersymmetric string theory - often abbreviated to "superstring theory" - works best in 10 dimensions. There are five main versions of superstring theory, which I described in "week74". The type I string theory involves open strings - little segments rather than loops. The type IIA and type IIB theories involve closed strings, that is, loops. But the most popular sort of superstring theories are the "heterotic strings". A nice introduction to these, written by one of their discoverers, is:
5) David J. Gross, The heterotic string, in "Workshop on Unified String Theories", eds. M. Green and D. Gross, World Scientific, Singapore, 1986, pp. 357-399.
These theories involve closed strings, but the odd thing about them, which accounts for the name "heterotic", is that vibrations of the string going around one way are supersymmetric and act as if they were in 10 dimensions, while the vibrations going around the other way are bosonic and act as if they were in 26 dimensions!
To get this string with a split personality to make sense, people cleverly think of the 26 dimensional spacetime for the bosonic part as a 10-dimensional spacetime times a little 16-dimensional curled-up space, or "compact manifold". To get the theory to work, it seems that this compact manifold needs to be flat, which means it has to be a torus - a 16-dimensional torus. We can think of any such torus as 16-dimensional Euclidean space "modulo a lattice". Remember, a lattice in Euclidean space is something that looks sort of like this:
x x x x x x x x x x x x x x x x x x x xMathematically, it's just a discrete subset L of Rn (n-dimensional Euclidean space, with its usual coordinates) with the property that if x and y lie in L, so does jx + ky for all integers j and k. When we form n-dimensional Euclidean space "modulo a lattice", we decree two points x and y to be the same if x - y is in L. For example, all the points labelled x in the figure above count as the same when we "mod out by the lattice"... so in this case, we get a 2-dimensional torus.
For more on 2-dimensional tori and their relation to complex analysis, you can read "week13." Here we are going to be macho and plunge right into talking about lattices and tori in arbitrary dimensions.
To get our 26-dimensional string theory to work out nicely when we curl up 16-dimensional space to a 16-dimensional torus, it turns out that we need the lattice L that we're modding out by to have some nice properties. First of all, it needs to be an "integral" lattice, meaning that for any vectors x and y in L the dot product x.y must be an integer. This is no big deal - there are gadzillions of integral lattices. In fact, sometimes when people say "lattice" they really mean "integral lattice". What's more of a big deal is that L must be "even", that is, for any x in L the inner product x.x is even. This implies that L is integral, by the identity
(x + y).(x + y) = x.x + 2x.y + y.yBut what's really a big deal is that L must also be "unimodular". There are different ways to define this concept. Perhaps the easiest to grok is that the volume of each lattice cell - e.g., each parallelogram in the picture above - is 1. Another way to say it is this. Take any basis of L, that is, a bunch of vectors in L such that any vector in L can be uniquely expressed as an integer linear combination of these vectors. Then make a matrix with the components of these vectors as rows. Then take its determinant. That should equal plus or minus 1. Still another way to say it is this. We can define the "dual" of L, say L*, to be all the vectors x such that x.y is an integer for all y in L. An integer lattice is one that's contained in its dual, but L is unimodular if and only if L = L*. So people also call unimodular lattices "self-dual". It's a fun little exercise in linear algebra to show that all these definitions are equivalent.
Why does L have to be an even unimodular lattice? Well, one can begin to understand this a litle by thinking about what a closed string vibrating in a torus is like. If you've ever studied the quantum mechanics of a particle on a torus (e.g. a circle!) you may know that its momentum is quantized, and must be an element of L*. So the momentum of the center of mass of the string lies in L*.
On the other hand, the string can also wrap around the torus in various topologically different ways. Since two points in Euclidean space correspond to the same point in the torus if they differ by a vector in L, if we imagine the string as living up in Euclidean space, and trace our finger all around it, we don't necesarily come back to the same point in Euclidean space: the same point plus any vector in L will do. So the way the string wraps around the torus is described by a vector in L. If you've heard of the "winding number", this is just a generalization of that.
So both L and L* are really important here (which has to do with the fashionable subject of "string duality"), and a bunch more work shows that they both need to be even, which implies that L is even and unimodular.
Now something cool happens: even unimodular lattices are only possible in certain dimensions - namely, dimensions divisible by 8. So we luck out, since we're in dimension 16.
In dimension 8 there is only one even unimodular lattice (up to isometry), namely the wonderful lattice E8! The easiest way to think about this lattice is as follows. Say you are packing spheres in n dimensions in a checkerboard lattice - in other words, you color the cubes of an n-dimensional checkerboard alternately red and black, and you put spheres centered at the center of every red cube, using the biggest spheres that will fit. There are some little hole left over where you could put smaller spheres if you wanted. And as you go up to higher dimensions, these little holes gets bigger! By the time you get up to dimension 8, there's enough room to put another sphere OF THE SAME SIZE AS THE REST in each hole! If you do that, you get the lattice E8. (I explained this and a bunch of other lattices in "week65", so more info take a look at that.)
In dimension 16 there are only two even unimodular lattices. One is E8 + E8. A vector in this is just a pair of vectors in E8. The other is called D16+, which we get the same way as we got E8: we take a checkerboard lattice in 16 dimensions and stick in extra spheres in all the holes. More mathematically, to get E8 or D16+, we take all vectors in R8 or R16, respectively, whose coordinates are either all integers or all half-integers, for which the coordinates add up to an even integer. (A "half-integer" is an integer plus 1/2.)
So E8 + E8 and D16+ give us the two kinds of heterotic string theory! They are often called the E8 + E8 and SO(32) heterotic theories.
In "week63" and "week64" I explained a bit about lattices and Lie groups, and if you know about that stuff, I can explain why the second sort of string theory is called "SO(32)". Any compact Lie group has a maximal torus, which we can think of as some Euclidean space modulo a lattice. There's a group called E8, described in "week90", which gives us the E8 lattice this way, and the product of two copies of this group gives us E8 + E8. On the other hand, we can also get a lattice this way from the group SO(32) of rotations in 32 dimensions, and after a little finagling this gives us the D16+ lattice (technically, we need to use the lattice generated by the weights of the adjoint representation and one of the spinor representations, according to Gross). In any event, it turns out that these two versions of heterotic string theory act, at low energies, like gauge field theories with gauge group E8 x E8 and SO(32), respectively! People seem especially optimistic that the E8 x E8 theory is relevant to real-world particle physics; see for example:
6) Edward Witten, Unification in ten dimensions, in "Workshop on Unified String Theories", eds. M. Green and D. Gross, World Scientific, Singapore, 1986, pp. 438-456.
Edward Witten, Topological tools in ten dimensional physics, with an appendix by R. E. Stong, in "Workshop on Unified String Theories", eds. M. Green and D. Gross, World Scientific, Singapore, 1986, pp. 400-437.
The first paper listed here is about particle physics; I mention the second here just because E8 fans should enjoy it - it discusses the classification of bundles with E8 as gauge group.
Anyway, what does all this have to do with Lucas and his stack of cannon balls?
Well, in dimension 24, there are 24 even unimodular lattices, which were classified by Niemeier. A few of these are obvious, like E8 + E8 + E8 and E8 + D16+, but the coolest one is the "Leech lattice", which is the only one having no vectors of length 2. This is related to a whole WORLD of bizarre and perversely fascinating mathematics, like the "Monster group", the largest sporadic finite simple group - and also to string theory. I said a bit about this stuff in "week66", and I will say more in the future, but for now let me just describe how to get the Leech lattice.
First of all, let's think about Lorentzian lattices, that is, lattices in Minkowski spacetime instead of Euclidean space. The difference is just that now the dot product is defined by
(x1,...,xn) . (y1,...,yn) = - x1 y1 + x2 y2 + ... + xn ynwith the first coordinate representing time. It turns out that the only even unimodular Lorentzian lattices occur in dimensions of the form 8k + 2. There is only one in each of those dimensions, and it is very easy to describe: it consists of all vectors whose coordinates are either all integers or all half-integers, and whose coordinates add up to an even number.
Note that the dimensions of this form: 2, 10, 18, 26, etc., are precisely the dimensions I said were specially important in "week93" for some other string-theoretic reason. Is this a "coincidence"? Well, all I can say is that I don't understand it.
Anyway, the 10-dimensional even unimodular Lorentzian lattice is pretty neat and has attracted some attention in string theory:
7) Reinhold W. Gebert and Hermann Nicolai, E10 for beginners, preprint available as hep-th/9411188
but the 26-dimensional one is even more neat. In particular, thanks to the cannonball trick of Lucas, the vector
v = (70,0,1,2,3,4,...,24)is "lightlike". In other words,
v.v = 0What this implies is that if we let T be the set of all integer multiples of v, and let S be the set of all vectors x in our lattice with x.v = 0, then T is contained in S, and S/T is a 24-dimensional lattice - the Leech lattice!
Now that has all sorts of ramifications that I'm just barely beginning to understand. For one, it means that if we do bosonic string theory in 26 dimensions on R26 modulo the 26-dimensional even unimodular lattice, we get a theory having lots of symmetries related to those of the Leech lattice. In some sense this is a "maximally symmetric" approach to 26-dimensional bosonic string theory:
8) Gregory Moore, Finite in all directions, preprint available as hep-th/9305139.
Indeed, the Monster group is lurking around as a symmetry group here! For a physics-flavored introduction to that aspect, try:
9) Reinhold W. Gebert, Introduction to vertex algebras, Borcherds algebras, and the Monster Lie algebra, preprint available as hep-th/9308151
and for a detailed mathematical tour see:
10) Igor Frenkel, James Lepowsky, and Arne Meurman, "Vertex Operator Algebras and the Monster," Academic Press, 1988.
Also try the very readable review articles by Richard Borcherds, who came up with a lot of this business:
11) Richard Borcherds, Automorphic forms and Lie algebras.
Richard Borcherds, Sporadic groups and string theory.
These and other papers available at http://www.pmms.cam.ac.uk/Staff/R.E.Borcherds.html; click on the personal home page.
Well, there is a lot more to say, but I need to go home and pack for my Thanksgiving travels. Let me conclude by answering a natural followup question: how many even unimodular lattices are there in 32 dimensions? Well, one can show that there are AT LEAST 80 MILLION!
Some of you may have wondered what's happened to the "tale of n-categories". I haven't forgotten that! In fact, earlier this fall I finished writing a big fat paper on 2-Hilbert spaces (which are to Hilbert spaces as categories are to sets), and since then I have been struggling to finish another big fat paper with James Dolan, on the general definition of "weak n-categories". I want to talk about this sort of thing, and other progress on n-categories and physics, but I've been so busy working on it that I haven't had time to chat about it on This Week's Finds. Maybe soon I'll find time.
Addenda: Robin Chapman pointed out that Anglin's proof also appears in the American Mathematical Monthly, February 1990, pp. 120-124, and that another elementary proof has subsequently appeared in the Journal of Number Theory. David Morrison pointed out in email that since the sum
12 + 22 + ... + n2 = m2,is n(n+1)(2n+1)/6, this problem can be solved by finding all the rational points (n,m) on the elliptic curve
(1/3) n3 + (1/2) n2 + (1/6) n = m2which is the sort of thing folks know how to do.
Also, here's something Michael Thayer wrote on one of the newsgroups, and my reply:
>John Baez wrote: >> In particular, thanks to the cannonball trick of Lucas, the vector >> >> v = (70,0,1,2,3,4,...,24) >> >> is "lightlike". In other words, >> >> v.v = 0 >I don't see what is so significant about the vector v. For instance, >the 10 dimensional vector (3,1,1,1,1,1,1,1,1,1) is also light like, and >you make no big deal about that. Is there some reason why the ascending >values in v are important?
Yikes! Thanks for catching that massive hole in the exposition.
You're right that there's no shortage of lightlike vectors in the even unimodular Lorentzian lattices of other dimensions 8n+2; there are also lots of other lightlike vectors in the 26-dimensional one. Any one of these gives us a lattice in 8n-dimensional Euclidean space. In fact, we can get all 24 even unimodular lattices in 24-dimensional Euclidean space by suitable choices of lightlike vector. The lightlike vector you wrote down happens to give us the E8 lattice in 8 dimensions.
So what's so special about I wrote, which gives the Leech lattice? Of course the Leech lattice is itself special, but what does this have to do with the nicely ascending values of the components of v?
Alas, I don't know the real answer. I'm not an expert on this stuff; I'm just explaining it in order to try to learn it. Let me just say what I know, which all comes from Chap. 27 of Conway and Sloane's book "Sphere Packings, Lattices, and Groups".
If we have a lattice, we say a vector r in it is a "root" if the reflection through r is a symmetry of the lattice. Corresponding to each root is a hyperplane consisting of all vectors perpendicular to that root. These chop space into a bunch of "fundamental regions". If we pick a fundamental region, the roots corresponding to the hyperplanes that form the walls of this region are called "fundamental roots". The nice thing about the fundamental roots is that the reflection through any root is a product of reflections through these fundamental roots.
[For more stuff on reflection groups and lattices see "week62" and the following weeks.]
In 1983 John Conway published a paper where he showed various amazing things; this is now Chapter 27 of the above book. First, he shows that the fundamental roots of the even unimodular Lorentzian lattices in dimensions 10, 18, and 26 are the vectors r with r.r = 2 and r.v = -1, where the "Weyl vector" v is
They all have this nice ascending form but only in 26 dimensions is the Weyl vector lightlike!
Howerver, Conway doesn't seem to explain why the Weyl vectors have this ascending form. So I'm afraid I really don't understand how all the pieces fit together. All I can say is that for some reason the Weyl vectors have this ascending form, and the fact that the Weyl vector is also lightlike makes a lot of magic happen in 26 dimensions. For example, it turns out that in 26 dimensions there are *infinitely many* fundamental roots, unlike in the two lower dimensional cases.
Just to add mystery upon mystery, Conway notes that in higher dimensions there is no vector v for which all the fundamental roots r have r.v equal to some constant. So the pattern above does not continue.
I find this stuff fascinating, but it would drive me nuts to try to work on it. It's as if God had a day off and was seeing how many strange features he could build into mathematics without actually making it inconsistent.
Yet another addendum (August 2001): now, with the rise of interest in 11-dimensional physics, there is even a paper on E11:
12) P. West, E11 and M-theory, available as hep-th/0104081.
© 1996 John Baez