M Theory Lesson 151

L. B. Crowell has shown up at PF with some remarks about sphere packing, codes and quantum gravity. He seems quite interested in the Leech lattice and associated theta functions, as well as the familiar j-invariant. The triple of $E8$ lattices is most evident in the generating function for the Leech theta series, namely

$f\left(q\right)=(\Theta (q){)}^3-720q^2\prod _1^{\infty }(1-q^{2n}{)}^{24}$

where $\Theta \left(q\right)$ is the series for the $E8$ lattice. A triple $E8$ is suggestive from a traditional stringy point of view, extending the heterotic pair of $E8$ to a ternary logic. However, as Gannon explains, there is a more interesting triality involving affine $E8$, $F4$ and $G2$. One can label the $E8$ diagram by conjugacy classes of the Monster group! The $F4$ comes from a two folding of affine $E7$ and the $G2$ from a triple folding of affine $E6$. The $G2$ case corresponds to conjugacy classes for a Fischer group, which itself has a triple cover in one of the conjugacy classes of the Monster.

$\Theta$ is really the Eisenstein series $E_4$. The series $E_2\left(\sqrt{z}\right)$, $E_4\left(\sqrt{z}\right)$ and $E_6\left(\sqrt{z}\right)$ satisfy the triality, for $D=z\frac{d}{dz}$,

$D{E}_2=\frac{1}{12}(E_2^2-E_4)$
$D{E}_4=\frac{1}{3}(E_2{E}_4-E_6)$
$D{E}_6=\frac{1}{2}(E_2{E}_6-E_4^2)$

This triality uses the zeta values $\zeta \left(2\right)$, $\zeta \left(4\right)$ and $\zeta \left(6\right)$. The next Eisenstein series, $E_8$, corresponds to the theta series for $E8\oplus E8$. The investigation of trialities for mass generation always seems to come back to this very fundamental mathematics.