Wednesday, October 26, 2011

Theory Update 118

Recall that Furey's Fano plane mixes the charged leptons and neutrinos, as in the projection of a charge cube on the $27$ qutrit paths.

The $21$ Fano paths remind us of the (triality labeled) 21 edges in the three dimensional associahedron polytope. Recall also how 16 paths (out of $27$) are placed on the associahedron. Space really wants to start out being three dimensional. Ribbon charge is specified using the quark path mixtures of the quandle plane. An $XXX$ charged lepton path obtains a unit charge from $1/3 + 2/3$. The six cyclic permutations of the neutrino source vertex themselves represent the hexagon of $6$ paths from source to target on a cube. The neutrino hexagon appears in the honeycomb dual to the tetractys brane, and may thus be considered as a surface of last scattering.


  1. So we see how electromagnetic charge is, in a sense, generated at its surface of last scattering, in much the same way that rest mass comes from its horizon. Here, the SLS marks off the neutrino vertex from the rest of the (octonion unit) cube.

  2. Fascinating. So you're essentially mapping the number of pairs of imaginary octonion units (given by 7 choose 2) to the edges of K_5. As each such pair generates a Grassmann algebra, each of the 21 edges acquires a fermionic interpretation.

  3. Which is what we want, yes? I was thinking first of the actual physics, but the algebra must follow ...


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