Thursday, April 08, 2010

K3 and Mathieu M24 group: a new moonshine

Click the picture to create the M24 Mathieu group out of a mutated Rubik snake. You will need to hire 56 kindergartens.

The first hep-th paper today is pretty fascinating. Tohru Eguchi, Hiroši Ooguri, and Yuji Tachikawa found a new robust numerological rule that is surely not just a coincidence:
Notes on the K3 surface and the Mathieu group M_24
They expand the elliptic genus of a K3 surface and they find the coefficients,
45, 231, 770, 2277, 5796, 13915...
to be dimensions of simple representations of the largest Mathieu group, M24. Now, I should say that in the classification of finite simple groups, there are various infinite sequences and 26 or 27 sporadic groups which are complete exceptions.

The monster group - the largest sporadic group - has been discussed many times but the oldest five known sporadic groups are the so-called Mathieu groups, M11, M12, M22, M23, M24: these remarkable subgroups of permutation groups S11, S12, S22, S23, S24 have been known since the 1860s and 1870s. The largest one, M24, seems to be relevant for the construction.

It has 2^{10}.3^{3}. = 244823040 elements and 26 conjugacy classes - yes, the number 26 appears pretty often in these discussions. ;-) It follows that it also has 26 irreducible representations whose dimensions are
1, 23, 45, 45, 231, 231, 252, 253, 483, 770, 770, 990, 990, 1035, 1035, 1035, 1265, 1771, 2024, 2277, 3312, 3520, 5313, 5796, 5544, 10395
The entries that appear twice are complex representations and their complex conjugates; close to the center of the list, 1035 appears thrice so there is a pair of complex representations as well as an independent real representation.

There is a clear similarity between this situation and that of the monstrous moonshine involving the monster group. Recall that the latter is explained by a vertex algebra - describing a CFT on a 24-dimensional torus - that happens to have the desired discrete symmetry group.

In this case, it's not known (yet) how the Mathieu group M24 would act on the K3 surface CFT. However, the Japanese authors actually manage to find a K3-Mathieu24 link through another 24-dimensional lattice, the extended binary Golay code (a way to write 12 bits as 24 bits so that any 3-bit error can be fixed and any 4-bit error can be detected). The chiral CFT on this 24-dimensional torus has M24 as its discrete symmetry.

At the same moment, the overall cohomology of K3 is also an even self-dual 24-dimensional lattice; but its signature is 4+20. By these relationships, they're at least able to find a subgroup of M24 that acts on a particular K3 surface. What does the whole M24 act upon, if anything, remains an enigma.

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