- #451

- 190

- 2

As I told elsewhere, in my case the hexagonal number pattern appears when you ask for the sBootstrap conditions, a coincidence between bosonic and fermionic degrees of freedom that happens in the QCD string. Half of this number (ie 3, 14, 33, 60, ... ) is the number of generations needed for the sBootstrap to exist.

I suspect that some quantisation of flavour will produce at least SO(32), if not E8xE8. This accounts for the 496. But no hint about Leech lattice.

I am not sure what sBoostrap is, and I suppose that the hexagonal number pattern is related to this. These generations, such as the 60, is half the number of spinor on the [itex]M_{12}[/itex] Mathieu group on the 120-cell or icosahedron. I am not sure if degeneracies are considered here, but 196560 is divisible by 32760 with 6 as the answer. I am not sure if there is any significance to this

I think a small verision of quantum gravity is the trio group [itex]S^3\times SL_2(7)~ \subset~M_{24}[/itex]. The Leech lattice contains [itex]M_{24}[/itex] as a quantum error correction code, which embeds three [itex]E_8's[/itex] --- an [itex]E_8\times E_8[/itex] for the graded heterotic supergravity field theory and the third for this configuration of all possible spacetimes. In the restricted [itex]S^3\times SL_2(7)[/itex] this is a thee dimensional Bloch sphere where each point on it is a "vector" in a three space spanned by the Fano planes associated with these three [itex]E_8[/itex]'s. [itex]S^3\times SL_2(7)[/itex] has 1440 roots and is itself a formidable challenge, but this represents a best first approach. [itex]M_{24}[/itex] has 196560 roots and clearly an explicit calculation of those is not possible at this time.

I am not sure if there is any magical numerology here, but with the 8 additional weights for each of the [itex]E_8[/itex]'s, which is then doubled to a total of 48 weights due to the double covering on the Bloch sphere, this gives 1488 as the size of the trio group, which when divided by 6 gives 248, or when divided by 3 gives 496. I am not sure, but this might have some relevancy to Carl Brennan's triality approach to things.

Lawrence B. Crowell