Consider the monster group, the largest representative of the sporadic groups in the classification of simple finite groups. It has nearly 10^{54} elements. It's such a mind-boggling number, in fact, that most critics of physics lose their mind when they see such large numbers.
Please believe me for a while that mathematics continues to be consistent even if the numbers are large! ;-)
The irreducible representations of this finite group have been linked to the expansion of the modular j-function, a unique map from the SL(2,Z) fundamental domain to the complex plane (up to a conventional SL(2,C) transformation that you must choose correctly). In fact, the expansion of the j-function,
- j(tau) = 1/q + 744 + 196884 q + 21493760 q^2 + ...
where q=exp(2.pi.i.tau) has the property that all coefficients starting from 196884 are dimensions of rather simple representations of the monster group. For example,
- 196884 = 196883 + 1
decomposes into the smallest faithful representation and a singlet. Similarly, the following coefficients are almost equal to the dimension of some large representation, plus much smaller ones. At any rate, the j-function is modular invariant - invariant under SL(2,Z) - and it can thus be the partition sum of a perturbative string theory. Below, this string theory will only be used as a boundary CFT.
The mysterious link between the modular functions and the monster group is referred to as monstrous moonshine.
Can you find the correct CFT? You bet. Open
which is written in a no-nonsense, string-theoretical jargon. The j-function is the partition sum of a chiral worldsheet theory of 24 bosons. Because of modular invariance, these bosons must be compactified on an even self-dual lattice. But don't expect a 24-dimensional counterpart of the E8 x E8 or SO(32) lattices that would give you an SO(48) symmetry.
There exists another, very different even self-dual lattice in 24 dimensions, the Leech lattice. It may be used to define the densest packing in 24 dimensions and it has the remarkable feature that the "l^2=2" sites - those that normally give you roots of a non-Abelian symmetry - are completely absent. Consequently, the compactification doesn't produce any enhanced symmetry whatsoever.
A basic introduction to the Leech group may also be found in the article under the first link ("monster group") in this text.
If you read the paper about the beauty and the beast, you will see that they define the compactification of the chiral bosons on the Leech lattice whose partition sum is the third power of the E8 lattice partition function, divided by the 24th power of the Dedekind eta-function. The following sections of their paper are dedicated to a construction of a Z2 orbifold. In fact, the only role of this orbifold is equivalent to an additive shift of the partition sum by a constant - a special example of an SL(2,C) transformation. The new partition sum becomes the capitalized J-function
- J(tau) = j(tau) - 744 = 1/q + 196884 q + ...
That's great. We have a partition sum of a two-dimensional CFT - a CFT that admits a natural action of the monster group. Well, whenever you see a CFT, you should wake up and scream "AdS" unless you want to look like a moron. ;-)
Fine. So what is the dual three-dimensional theory in an anti de Sitter space of this CFT? Note that by removing the constant term "744", we have eliminated all states with L_0=1. This value of L_0 corresponds to the holomorphic weight of operators associated with massless particles. That's great because it means that there are no massless particles in the theory.
Great. So we see that the CFT constructed from the Leech lattice is dual to a three-dimensional AdS theory without massless particles. Is that possible? It might be. Every bulk theory that enters a holographic duality must be a gravitational theory but this gravitational theory seems to contain no massless particles. Is it allowed? Yes, it is allowed because three-dimensional gravity has no physical graviton polarizations. It's because both the Ricci tensor R_{ab} and the Riemann tensor R_{abcd} have three independent components. The vanishing of the Ricci tensor - the Einstein equations - is thus equivalent to the vanishing of the Riemann tensor i.e. flatness. No waves can live in empty three-dimensional space.
But still, it is a theory of gravity so it must describe black holes. In fact, the numbers such as 196884 describe their degeneracies. More precisely, one of these 196884 states is a Virasoro descendant of the vacuum. That's why exactly 196883 microstates of the black hole are primaries. In other words, the minimal black hole in this pure three-dimensional gravity transforms as the minimal representation of the monster group!
Note that this literally monstrous discrete symmetry of the black hole arises exactly in the compactification of quantum gravity that has a minimal amount of low-energy fields. I feel that this might be a general principle: the less low-energy fields your vacuum has, the greater hidden discrete symmetries you might expect. It would be interesting to make a "new uncertainty principle" of this type more quantitative. It is plausible that a very large discrete group replacing the continuous symmetry is necessary to circumvent some unpleasant features of "pure gravity".
The central charge of the 24 bosons is "c_L=24". I suppose that one wants to consider a more general case of several - namely "k_L" - copies of the left-moving Leech CFT and several copies - namely "k_R" - of the right-moving Leech CFT. The corresponding partition sum will probably be
- J^{k_L} times Jbar^{k_R}
times the correct powers of "q,qbar" so that the expansion starts with the universal pole "1/q.qbar" unless the higher-k partition sums are completely different. ;-)
Note that the first subleading, constant term will still vanish if you take these powers. Witten shows that the integrality of k_L, k_R may be seen if you write the SO(2,2) connection in a Chern-Simons description of the bulk gravity (where SO(2,2) is the isometry of AdS3) as a combination of two SO(2,1) connections, and use separate levels for the two chiral Chern-Simons theories.
However, it's my understanding that the monster group is completely invisible in the gravitational and/or Chern-Simons description. Note that this factorization into the left-movers and right-movers is obviously a special property of AdS3/CFT2: in higher dimensions, nothing factorizes into two pieces. For example, SO(3,2) is simple.
Black hole with a monster symmetry
At any rate, the idea that the black hole in the "simplest" compactification of quantum gravity has a monster symmetry built in it is fascinating if it is true. In the quasinormal era four years ago, I was thinking about finding something like ln(248) in the quasinormal modes of 11-dimensional black holes, to confirm discrete-like counting of the black hole entropy that would moreover support the relevance of the Diaconescu-Moore-Witten E8 gauge field in the bulk. We could calculate that it was ln(3) that is actually associated with the Schwarzschild quasinormal modes in any dimension while other black holes have different numbers.
In some sense, the monstrous construction above is a working realization of the concept except that the unit of entropy - more precisely, the entropy of a minimal black hole - is ln(196883) instead of ln(248). ;-) The largest exceptional Lie group had to be replaced by the largest sporadic finite group. You should realize, however, that the counting of the entropy of larger black holes is not simply additive. More complicated representations of the monster group that describe heavier black holes can be found in the decomposition of powers of 196883 but many of them are rather small pieces in such decompositions. The degeneracies are not simple powers of 196883.
If you like numerology, note that "ln(196883)" is equal to "12.1904" which is extremely close to "4.pi = 12.5664", namely to the Bekenstein-Hawking entropy of the black hole with the same mass. (The values of simple black holes with quantized charges and masses in string theory tend to be simple multiples of 2.pi.) You can see that the higher-order corrections modify the entropy of this "tiny" black hole by a few percent only.
Olaf Dreyer done right
If you allow me to go a little bit further with heuristics, you can view the equation
- ln(196883) / (4.pi) = 1 + stringy corrections
to be a working realization of the loop quantum gravity concept arising from the Immirzi entropy discrepancy, namely
- ln(2) / (sqrt(3).pi) = 1 + loop quantum gravity corrections
except that the group SU(2) had to be replaced by the monster group M, the dimension of its fundamental representation 2 had to be replaced by the dimension of the smallest non-trivial representation of the monster group, 196883. Also, the really silly factor of sqrt(3) had to be replaced by 4, and loop quantum gravity had to be replaced by the correct theory of quantum gravity. ;-)
Note that the corrections in the loop quantum gravity case would have to be ten times larger than the leading contribution.
Monstrous monodromy
One should also ask whether the monster group is a local discrete symmetry or a global discrete symmetry. A lore says that every symmetry in quantum gravity is a local symmetry. That should mean that one can find counterparts of the cosmic strings - in 2+1D, they are cosmic particles - that exhibit a corresponding monodromy. In this theory, the monstrous black holes are the only particles.
How many of such objects would you expect? 8x 10^{53} which is the order of the monster group? Not really. I think that the right number of the cosmic strings/particles for the monster group should only be equal to the number of conjugacy classes of the monster group which is 194 (two of these 194 classes are classes of involutions if you care); it is also equal to the number of inequivalent irreducible representations, of course. ;-)
Homework: construct these 194 objects out of the black hole microstates or prove that they don't exist. ;-) Hint: the required cosmic strings/particles are not guaranteed to be unique.
Is this three-dimensional pure gravity a part of string theory? If it's consistent, it must be, by a definition of string theory. But it seems to me that there's no manifest way how to decompactify the 7 or 8 compact dimensions, to get to 10 or 11. In this fashion, the three-dimensional monster vacuum is the ultimate island - a compactification of M-theory on a generalized manifold we could call the Monster Manifold.
And that's the M(emo).
