Update: a lot of articles on this blog are dedicated to Witten's work on AdS3 gravity and monster group.If string theory is a real theory of everything, it should explain not only why the particles and forces are what they are, what happens with the information inside a black hole, how did the Universe begin, and why there are so many electrons and so many obnoxious crackpots in the world: it should also unify and "explain" all important structures in mathematics. No one can guarantee that this unification of physics with mathematics is what is going to happen except that it does seem that it will happen and it has already happened in various special cases.
Classification of simple finite groups
Groups are extremely important and natural mathematical structures and the finite groups seem to be the simplest subclass even though their understanding is not easy, as we will see instantly. A very important subclass of the finite groups seem to the simple finite groups: those that contain no normal subgroups i.e. subgroups that are invariant under the conjugation by any element of the large group as a set. All other groups can essentially be expressed as direct or semidirect products of the simple finite groups: they depend on the simple groups.
The classification of simple finite groups took many decades (1955-1983) and thousands of pages had to be filled with mathematics. The result of this collective enterprise is an "enormous theorem" as it is often called. Because the amount of knowledge that has to be verified and connected - and, arguably, has been verified and connected - is large, there remain skeptics who are not sure that the full proof has indeed been completed: Jean-Pierre Serre is a prominent skeptic.
Skepticism aside, the results say the following. A simple finite group must be isomorphic to a group in one of the following families:
- the cyclic groups "Zp" with a prime integer "p" - the simplest Abelian groups with "p" elements you can think about
- the alternating groups "An" of all even permutations of "n" elements for "n" greater than three (because "A2" is trivial and "A3" is "Z3"); note that the symmetric group "Sn" of all permutations is a semidirect product of "An" with "Z2"
- the groups of Lie type - essentially Lie groups with integer or rational entries of various kinds; this family includes both classical groups (projective, special unitary, symplectic orthogonal, and unitary over a finite field) as well as exceptional or "twisted" Lie groups over discrete fields
- Tits group that is sometimes included into the previous family because of its indirect relations with F4, but sometimes is counted as the 27th element of the next family
- 26 sporadic groups
Note that the word "sporadic" is "doubly exceptional" because the exceptional Lie groups only lead to "easy" finite groups of Lie type. At any rate, the sporadic groups are the real gems that were identified and listed by the classification of finite groups. They are named after various mathematicians and in most cases, they have been written in terms of a few generators once we specify which products of these generators are equal to the identity element - for example, "ap=1" for "Zp".
The largest sporadic group is the monster group M or F1 that has nearly 1054 elements: note that with such a huge number of elements, the crackpots would surely say that this group is not testable and it is not even wrong to study it. ;-) The second largest group is called the baby monster group or F2.
Six sporadic groups are however not subgroups of the monster group, unlike the remaining twenty (which includes M itself). They are called the pariahs: in our counting, they are "triply exceptional". This situation differs from the exceptional simple compact Lie groups: all these Lie groups are subgroups of "E8" and there are no further "Lie pariahs".
Because the monster group is the largest one, the group theorists often think that it is also the most interesting one. Its existence was predicted by Robert Griess and Bernd Fischer in 1973 and it was eventually constructed by Griess in 1980 as the automorphism group of the Griess (commutative, non-associative) algebra whose dimension is 196884.
Suddenly a lot of observations occured that may seem even crazier than the coincidences underlying the ADE classification. The term "monstrous moonshine" was picked to convey the feelings from the bizarre relations between seemingly unrelated structures. Two basic quasireligious observations were the following.
The dimension of the Griess algebra is 196884. One direction in this algebra must be simply the real numbers and the remaining 196883 directions form another irreducible representation of the monstrous group. That's analogous to the fact that the fundamental representation of G2, the automorphism group of the eight-dimensional algebra of octonions, is seven-dimensional.
Now consider an entirely different object: the j-function. The j-function is a one-to-one map from the fundamental domain of the SL(2,Z) modular group to the complex plane which is stereographically equivalent to the two-sphere. The expansion of j(tau) in terms of "q=exp(2.pi.i.tau)" - useful around "tau=i.infinity" where "q" is small - is
- j(tau) = 1/q + 744 + 196884 q + 21493760 q2 + ...
Now, the number 744 can be forgotten because a constant shift of j(tau) is an element of SL(2,C), the holomorphic maps from the sphere to itself, and these Möbius transformations are a matter of convention when you define the j-function first. The symbol J(tau)=j(tau)-744 is often used for the same function without the term 744; be careful because this shift also adds to the value of j(i), among other things, so that J(i) is less natural than j(i). But you see that the next coefficient is 196884 which is miraculously equal to one plus the dimension of the smallest non-trivial representation of the monster group.
Is it a coincidence? People like Simon Norton, John Conway, and John McKay gave a clear "no" answer once they realized that the other coefficients like 21493760 also seem to be the dimensions of nearly irreducible (a small number of components) representations of the monster group.
OK, so how is it possible that the expansions of modular functions know about the dimensions of the representations of the largest sporadic finite simple group?
CFTs with the monster group symmetry
The answer is, of course, string theory. Frenkel, Lepowsky, and Meurman have constructed the required Griess algebra from a vertex algebra - a chiral algebra with meromorphic OPEs - of a particular string theory. Inside the string theory community, I would recommend the paper
- Beauty and the beast: superconformal symmetry in a monstrous module
by Dixon, Ginsparg, and Harvey that also shows how to construct a weight 3/2 supercurrent in this setup. How do you construct this string theory? Take bosonic string theory and compactify it on the 24-dimensional torus that is given by the R24 divided by the Leech lattice. Finally, orbifold this background by a Z2 that reflects all the directions.
The Leech lattice, discovered by Ernst Witt in 1940 and re-discovered and published by John Leech 24 years later, is an even self-dual 24-dimensional lattice of the type that most string theorists don't know. It differs from the lattices Gamma8 and Gamma16 (and their obvious counterpart Gamma24) familiar from heterotic string theory because it has, in fact, no elements with squared length equal to two! Because of this property, it also defines the densest packing in 24 dimensions.
Recall that the elements of squared length equal to two would give you the roots and some enhanced gauge symmetry: there's nothing like that if you take the Leech lattice. At any rate, the discrete symmetry of this background is essentially the monster group and the partition function can be expressed using the j-function which is related to a ratio of powers of the E8 characters and Dedekind's eta-function. Note that the j-function is also an SL(2,C) transformation of the ratio "G43 / G62" of powers of the "elementary" modular forms of weights 4,6: the powers were chosen in such a way that the resulting weight is zero.
At any rate, I think it is fair to say that perturbative string theory, once again, provides you with an explanation why the modular j-function knows about the representations of the monster group - something that looked completely crazy at the beginning which is why Richard Borcherds deserved his 1998 Fields medal for his 1992 proof of the monstrous moonshine conjecture.
Jack Daniel's whiskey
But there exists another curious observation that is less understood from the CFT viewpoint even though I have some ideas how the explanation could look like. In the 1970s, Andrew Ogg was listening to a conference talk by his colleague about the monster group. He learned that the landscape-sized number of its elements could be decomposed as
- 246 x 320 x 59 x 76 x 112 x 133 x 17 x 19 x 23 x 29 x 31 x 41 x 47 x 59 x 71.
That was pretty interesting because Ogg had studied the genus of the quotients of the hyperbolic plane - that we will translate into the conformally equivalent upper half-plane - by the group "Gamma0(p)+". This group is generated by the subgroup of "SL(2,Z)" that is composed of elements "((a,b),(c,d))" where "c" is a multiple of "p", together with the matrix "((0,1),(-p,0))" that adds the sign "plus" at the end of the symbol.
Note that the determinant of the latter matrix generator is "p", not one, and it maps "tau" to "-1/(p.tau)". It turns out that the coset of the hyperbolic plane by the discrete group has genus equal to zero in some cases. It occurs iff "p" belongs to the following list:
- 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71
which are exactly the primes that appear in the decomposition of the order of the monster group: all primes up to 71 except for 37, 43, 53, 61, and 67. Ogg promised a bottle of whiskey to anyone who would explain this fact. This fact has been explained but as far as I know, the explanation doesn't depend on the stringy construction we mentioned previously.