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24, the Monster, and quantum gravity

By Richard Green

Think of a prime number other than 2 or 3. Multiply the number by itself and then subtract 1. The result is a multiple of 24. This observation might appear to be a curiosity, but it turns out to be the tip of an iceberg, with far-reaching connections to other areas of mathematics and physics.

This result works for more than just prime numbers. It works for any number that is relatively prime to 24. For example, 25 is relatively prime to 24, because the only positive number that is a factor of both of them is 1. (An easy way to check this is to notice that 25 is not a multiple of 2, or 3, or both.) Squaring 25 gives 625, and 624=(24x26)+1.

A mathematician might state this property of the number 24 as follows:
If m is relatively prime to 24, then m^2 is congruent to 1 modulo 24.
One might ask if any numbers other than 24 have this property. The answer is “yes”, but the only other numbers that exhibit this property are 12, 8, 6, 4, 3, 2 and 1; in other words, the factors of 24.

The mathematicians John H. Conway and Simon P. Norton used this property of 24 in their seminal 1979 paper entitled Monstrous Moonshine. In the paper, they refer to this property as “the defining property of 24”. The word “monstrous” in the title is a reference to the Monster group, which can be thought of as a collection of more than 8x10^53 symmetries; that is, 8 followed by 53 other digits. The word “moonshine” refers to the perceived craziness of the intricate relationship between the Monster group and the theory of modular functions.

The existence of the Monster group, M, was not proved until shortly after Conway and Norton wrote their paper. It turns out that the easiest way to think of M in terms of symmetries of a vector space over the complex numbers is to use a vector space of dimension 196883. This number is close to another number that is related to the Leech lattice. The Leech lattice can be thought of as a stunningly efficient way to pack unit spheres together in 24 dimensional space. In this arrangement, each sphere will touch 196560 others. The closeness of the numbers 196560 and 196883 is not a coincidence and can be explained using the theory of monstrous moonshine.

It is now known that lying behind monstrous moonshine is a certain conformal field theory having the Monster group as symmetries. In 2007, the physicist Edward Witten proposed a connection between monstrous moonshine and quantum gravity. Witten concluded that pure gravity with maximally negative cosmological constant is dual to the Monster conformal field theory. This theory predicts a value for the semiclassical entropy estimate for a given black hole mass, in the large mass limit. Witten's theory estimates the value of this quantity as the natural logarithm of 196883, which works out at about 12.19. As a comparison, the work of Jacob Bekenstein and Stephen Hawking gives an estimate of 4π, which is about 12.57.

Relevant links
Wikipedia on the Monster group: http://en.wikipedia.org/wiki/Monster_group
Wikipedia on the Leech lattice: http://en.wikipedia.org/wiki/Leech_lattice
Wikipedia on Monstrous Moonshine: http://en.wikipedia.org/wiki/Monstrous_moonshine
A 2004 survey paper about Monstrous Moonshine by Terry Gannon: http://arxiv.org/abs/math/0402345