October 10, 1995

This Week's Finds in Mathematical Physics (Week 66)

John Baez

Well, I want to get back to talking about some honest physics, but I think this week I won't get around to it, since I can't resist mentioning two tidbits of a more mathematical sort. The first one is about π, and the second one is about the Monster. The second one does have a lot to do with string theory, if only indirectly.

First, thanks to my friend Steven Finch, I just found out that Simon Plouffe, Peter Borwein and David Bailey have computed the ten billionth digit in the hexadecimal (i.e., base 16) expansion of π. They use a wonderful formula which lets one compute a given digit of π in base 16 without needing to compute all the preceding digits! Namely, π is the sum from n = 0 to ∞ of

[(4/8n+1) - (2/8n+4) - (1/8n+5) - (1/8n + 6)] (1/16)n

Since the quantity in square brackets is not an integer, it requires cleverness to use this formula to get a given digit of π, but they figured out a way. Moreover, their method generalizes to a variety of other constants. If you can use the World-Wide Web, try the following sites:

1) The ten billionth hexadecimal digit of π is 9, by Simon Plouffe, http://groups.google.com/groups?selm=451p8p%24qcr%40morgoth.sfu.ca&output=gplain

2) David Bailey, Peter Borwein and Simon Plouffe, On the rapid computation of various polylogarithmic constants, available in postscript form from http://www.cecm.sfu.ca/personal/pborwein/PISTUFF/Apistuff.html

3) The miraculous Bailey-Borwein-Plouffe π algorithm, by Steven Finch, http://www.lacim.uqam.ca/~plouffe/Simon/Miraculous.pdf

The first one is an announcement that appeared on sci.math, and lists the billionth digits of π2, ln(2), and some other constants. The second one has the details. The third one gives a good overview of what's up.

Can we hope for a similar formula in base 10? More importantly, could these ideas let us prove that π is "normal", that is, that every possible string of digits appears in it with the frequency one would expect of a "random" number? It seems that this would be a natural avenue of attack.

Next, a tidbit of a more erudite sort concerning the elusive Monster manifold. Recall from "week63" and "week64" that the compact simple Lie groups can classified into 4 infinite families and 5 exceptions. I have always been puzzled by these "exceptional Lie groups", so I tried to explain a bit about how they are related to some other "exceptional structures" in mathematics, such as the icosahedron and the octonions. In physics, Witten has suggested that the correct theory of our universe might also be an exceptional structure of some sort. This idea has found some support in string theory, which uses the exceptional Lie group E8 and other structures I'll mention a bit later. In a more hand-waving way, one may argue that the theory of our universe must be incredibly special, since out of all the theories we can write down, just this one describes the universe that actually exists. All sorts of simpler universes apparently don't exist. So maybe the theory of the universe needs to use special, "exceptional" mathematics for some reason, even though it's complicated.

Anyway, as a hard-nosed mathematician, vague musings along these lines get tiresome to me rather quickly. Instead, what interests me most about this stuff is the whole idea of "exceptional structures" - symmetrical structures that don't fit into the neat regular families in classification theorems. The remarkable fact is that many of them are deeply related. As Geoffrey Dixon put it to me, they seem to have a "holographic" quality, meaning that each one contains in encoded form some of the information needed to construct all the rest! It thus seems pointless to hope that one is "the explanation" for the rest, but I would still like some conceptual "explanation" for the whole collection of them - though I have no idea what it should be.

Surely a clue must lie in the theory of finite simple groups. Just as the simple Lie groups are the building blocks of the theory of continuous symmetries, these are the building blocks of the theory of discrete - indeed finite - symmetries. More precisely "finite simple" group is a group G with finitely many elements and no normal subgroups, that is, no nontrivial subgroups H such that h in H implies ghg-1 in H for all g in G. This condition means that you cannot form the "quotient group" G/H, which one can think of as a "more simplified" version of G.

The classification of the finite simple groups is one of remarkable achievements of 20th-century mathematics. The entire proof of the classification theorem is estimated to take 10,000 pages, done by many mathematicians. To start learning about it, try:

4) Ron Solomon, On finite simple groups and their classification, AMS Notices Vol. 45, February 1995, 231-239.

and the references therein. Again, there are some infinite families and 26 exceptions called the "sporadic" groups. The biggest of these is the Monster, with

 246 320 59 76 112 133 17 19 23 29 31 41 47 59 71  = 

elements. It is a kind of Mt. Everest of the sporadic groups, and all the routes to it I know involve a tough climb through all sorts of exceptional structures: E8 (see "week65"), the Leech lattice (see "week20"), the Golay code, the Parker loop, the Griess algebra, and more. I certainly don't understand this stuff....

Even before the Monster was proved to exist, it started casting its enormous shadow over mathematics. For example, consider the theory of modular functions. What are those? Well, consider a lattice in the complex plane, like

               x          x
                   x          x 
            x          x
                x          x
         x          x
             x          x
      x          x   
          x          x

These are important in complex analysis, as described in "week13". To describe one of these you can specify two "periods" ω1 and ω2, complex numbers such that every point in the lattice of the form

                   n ω1 + m ω2.

But this description is redundant, because if we choose instead to use

                 ω'1 = a ω1 + b ω2

                 ω'2 = c ω1 + d ω2
we'll get the same lattice as long as the matrix of integers
                          a   b 

                          c   d
is invertible and its inverse also consists of integers. These transformations preserve the "handedness" of the basis ω1, ω2 if they have determinant 1, and that's generally a good thing to require. The group of 2x2 invertible matrices over the integers with determinant 1 is called SL(2,Z), or the "modular group" in this context. I said a bit about it and its role in string theory in "week64".

Now, if we are only interested in parametrizing the different shapes of lattices, where two rotated or dilated versions of the same lattice count as having the same shape, it suffices to use one complex number, the ratio

                       τ = ω12.
We might as well assume τ is in the upper halfplane, H, while we're at it. But for the reason given above, this description is redundant; if we have a lattice described by τ, and a matrix in SL(2,Z), we get a new guy τ' which really describes the same shaped lattice. If you work it out,
                   τ' = (a τ + b)/(c τ + d).
So the space of different possible shapes of lattices in the complex plane is really the quotient
Now, a function on this space is just a function of τ that doesn't change when you replace τ by τ' as above. In other words, it's basically just a function depending only on the shape of a 2d lattice. Now it turns out that there is essentially just one "nice" function of this sort, called j; all other "nice" functions of this sort are functions of j. (For experts, what I mean is that the meromorphic SL(2,Z)-invariant functions on H union the point at infinity are all rational functions of this function j.)

It looks like this:

           j(τ) = q-1 + 744 + 196884 q + 21493706 q2 + ....
where q = exp(2 π i τ). In fact, starting from a simple situation, we have quickly gotten into quite deep waters. The simplest explicit formula I know for j involves lattices in 24-dimensional space! This could easily be due to my limited knowledge of this stuff, but it suits my present purpose: first, we get a vague glimpse of where E_8 and the Leech lattice come in, and second, we get a vague glimpse of the mysterious significance of the numbers 24 and 26 in string theory.

So what is this j function, anyway? Well, it turns out we can define it as follows. First form the Dedekind eta function

              η(q) = q1/24   ∏    (1 - qn)
                          n = 1
This is not invariant under the modular group, but it transforms in a pretty simple way. Then take the E8 lattice - remember, that's a very nice lattice in 8 dimensions, in fact the only "even unimodular" lattice in 8 dimensions, meaning that the inner product of any two vectors in the lattice is even, and the volume of each fundamental domain in it equals 1. Now take the direct sum of 3 copies of E8 to get an even unimodular lattice L in 24 dimensions. Then form the theta function
                    Θ(q)   =   ∑   q<x,x>/2
                             x in L
In other words, we take all lattice points x and sum q to the power of their norm squared over 2. Now we have
                      j(τ) = Θ(q)/η(q)24
Quite a witches' brew of a formula, no? If someone could explain to me the deep inner reason for why this works, I'd be delighted, but right now I am clueless. I will say this, though: we could replace L with any other even unimodular lattice in 24 dimensions and get a function differing from j only by a constant. Guess how many even unimodular lattices there are in 24 dimensions? Why, 24, of course! These "Niemeier lattices" were classified by Niemeier in 1968. All but one of them have vectors with length squared equal to 2, but there is one whose shortest vector has length squared equal to 4, and that's the Leech lattice. This one has a very charming relation to 26-dimensional spacetime, described in "week20".

Since the constant term in j can be changed by picking different lattices in 24 dimensions, and constant functions aren't very interesting anyway, we can say that the first interesting coefficient in the above power series for j is 196884. Then, right around when the Monster was being dreamt up, McKay noticed that the dimension of its smallest nontrivial representation, namely 196883, was suspiciously similar. Coincidence? No. It turns out that all the coefficients of j can be computed from the dimensions of the irreducible representations of the Monster! Similarly, Ogg noticed in the study of the modular group, the primes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59 and 71 play a special role. He went to a talk on the Monster and noticed the "coincidence". Then he wrote a paper offering a bottle of Jack Daniels to anyone who could explain it. This was the beginning of a subject called "Monstrous Moonshine"... the mysterious relation between the Monster and the modular group.

Well, as it eventually turned out, one way to get ahold of the Monster is as a group of symmetries of a certain algebra of observables for a string theory, or more precisely, a "vertex operator algebra":

5) Igor Frenkel, James Lepowsky, and Arne Meurman, Vertex Operator Algebras and the Monster, Academic Press, Boston, 1988.

The relation of string theory to modular invariance and 26 dimensional spacetime then "explains" some of the mysterious stuff mentioned above. (By the way, the authors of the above book say the fact that there are 26 sporadic finite simple groups is just a coincidence. I'm not so sure myself... not that I understand any of this stuff, but it's just too spooky how the number 26 keeps coming up all over!)

Anway, now let me fast-forward to some recent news. I vaguely gather that people would like to explain the relation between the Monster and string theory more deeply, by finding a 24-dimensional manifold having the Monster as symmetries, and cooking up a field theory of maps from the string worldsheet to this "Monster manifold", so that the associated vertex operator algebra would have a good reason for having the Monster as symmetries. Apparently Hirzebruch has offered a prize for anyone who could do this in a nice way, by finding a "24-manifold with p1=0 whose Witten genus is (j-744) Δ" on which the Monster acts. Recently, Mike Hopkins at MIT and Mark Mahowald at Northwestern have succeeded in doing the first part, the part in quotes above. They haven't gotten a Monster action yet. Their construction uses a lot of homotopy theory.

I don't have much of a clue about any of this stuff, but Allen Knutson suggests that I read

6) Friedrich Hirzebruch, Thomas Berger, and Rainer Jung, Manifolds and modular forms, translated by Peter S. Landweber, pub. Braunschweig, Vieweg, 1992.

for more about this "Witten genus" stuff. He also has referred me to the following articles by Borcherds:

7) Richard E. Borcherds, The Monster Lie-algebra, Adv. Math. 83 (1990), 30-47.

Richard E. Borcherds, Monstrous Moonshine and monstrous Lie-superalgebras, Invent. Math. 109 (1992), 405-444.

For your entertainment and edification I include the abstract of the second one below:

We prove Conway and Norton's moonshine conjectures for the infinite dimensional representation of the monster simple group constructed by Frenkel, Lepowsky and Meurman. To do this we use the no-ghost theorem from string theory to construct a family of generalized Kac-Moody superalgebras of rank 2, which are closely related to the monster and several of the other sporadic simple groups. The denominator formulas of these superalgebras imply relations between the Thompson functions of elements of the monster (i.e. the traces of elements of the monster on Frenkel, Lepowsky, and Meurman's representation), which are the replication formulas conjectured by Conway and Norton. These replication formulas are strong enough to verify that the Thompson functions have most of the "moonshine" properties conjectured by Conway and Norton, and in particular they are modular functions of genus 0. We also construct a second family of Kac-Moody superalgebras related to elements of Conway's sporadic simple group Co1. These superalgebras have even rank between 2 and 26; for example two of the Lie algebras we get have ranks 26 and 18, and one of the superalgebras has rank 10. The denominator formulas of these algebras give some new infinite product identities, in the same way that the denominator formulas of the affine Kac-Moody algebras give the Macdonald identities.

© 1995 John Baez