Sunday, December 01, 2013

The Monster and the Leech


The Monster group is one of those jewels of finite group theory with still mysterious connections to disparate parts of the wide world of mathematics.  The Monster was constructed by Robert Griess as the symmetry group of an algebra structure (Griess algebra) in 196,884 dimensions. His work split the space into three subspaces, and his main task was to show there were symmetries intermingling these subspaces. The dimensions of the subspaces are:         
98,304 + 300 + 98,280 = 196,884
The first number 98,304 = 212 × 24 comes from the Golay code in 24 dimensions.
 Geometrically: The 398,034,000 vectors of norm 8 in the Leech fall into 8,292,375 'crosses' of 48 vectors. Each cross contains 24 mutually orthogonal vectors and their inverses, and thus describe the vertices of a 24-dimensional orthoplex. Each of these crosses can be taken to be the coordinate system of the lattice, and has the same symmetry of the Golay code, namely 212 × |M24|=4,096×24. 
Remark: The full automorphism group Co_0 of the Leech lattice has order 8,292,375 × 4,096 × 244,823,040. In the octonionic construction of the Leech, Co_0 is generated by F4 transformations.

The second number 300 = 24 + 23 + 22 + … + 3 + 2 + 1 is the dimension of the space of 24-by-24 
 symmetric matrices. 

The third number 96,280 = 196,560 / 2 comes from the Leech Lattice in 24 dimensions, where there are 196,560 vertices closest to a given vertex, forming 98,280 diametrically opposite pairs.
Geometrically: The 196,560 (norm four) vectors that span the Leech lattice are formed by three shells:
  1.  3x240=720                      (2lambda, 0, 0)
  2.  3x240x16=11,520           (lambda s*, +/-(lambda s*)j, 0)
  3.  3x240x16x16=184,320  ((lambda s)j,± lambda k,±( lambda j)k)
So geometrically, the splitting of 196,884 isn't so mysterious.  98,304 comes from symmetry of norm 8 vector crosses that can be taken as the coordinate system of the Leech lattice.  The third number also isn't so mysterious either as it's just half of the norm four vectors that span the Leech lattice.

The j-function is given by:

j(τ)=q −1+744+196,884q+21,493,760q 2+…

where q=exp(i2πτ), and famously 196,883+1=196,884, which is the dimension of the Griess algebra, which can mostly be understood from Leech vectors. Thompson (1979) noted the rest of the coefficients are obtained from the dimensions of Monster’s irreducible representations.

The automorphism group of the Leech 2.Co_1=Co_0 fits inside the monster as 2^24 Co_0, being the centralizer of an involution. Since Co_0 is generated by F4 transformations, the monster has F4 doing work on the 196,884 dimensional space.  In fact, 212 × M24 is a maximal 2-local subgroup of Co_0 which can still be expressed in terms of F4 transformations.

F4 performs rotations in BPS black hole charge spaces and its pretty easy to show Co_0 is generated by such U-duality transformations.  Geometrically, the whole Leech lattice itself can be seen as a lattice of BPS black holes by embedding it in the exceptional Jordan algebra.  So the Monster can be partly interpreted as describing Co_0 symmetries of BPS black holes in the Leech lattice that live in a 24+3=27 dimensional charge space, the exceptional Jordan algebra, which has F4 as its automorphism group.

Another approach via Witten, in a pure 3D gravity picture, 196,884 states are Virasoro descendants of the vacuum, where 196883 microstates of the black hole are primaries.  A minimal black hole in this pure three-dimensional gravity transforms as the minimal representation of the monster group.

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