### 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

$f(q) = (\Theta (q))^{3} - 720 q^2 \prod_{1}^{\infty} (1 - q^{2n})^{24}$

where $\Theta (q)$ 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 $E_{2}(\sqrt{z})$, $E_{4}(\sqrt{z})$ and $E_{6}(\sqrt{z})$ 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 $\zeta (2)$, $\zeta (4)$ and $\zeta (6)$. 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.

$f(q) = (\Theta (q))^{3} - 720 q^2 \prod_{1}^{\infty} (1 - q^{2n})^{24}$

where $\Theta (q)$ 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 $E_{2}(\sqrt{z})$, $E_{4}(\sqrt{z})$ and $E_{6}(\sqrt{z})$ 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 $\zeta (2)$, $\zeta (4)$ and $\zeta (6)$. 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.

## 1 Comments:

Hi Kea,

I briefly scanned the Crowell book 'Quantum Fluctuations of Spacetime' you referenced on google. I noticed chapter 3.6 "... Bose-Einstein Condensates", page 110-114.

I recently saw PBS NOVA 'Absolute Zero', Airdates 2008 January 8 and 15.

The transcript of this program BEC experiment reads:

"NARRATOR: Lene Hau [Harvard] created a cigar-shaped Bose-Einstein condensate to carry out her experiment. She fired a light pulse into the cloud. The speed of light is around 186,000 miles per second, but when the pulse hits the condensate, it slows down to the speed of a bicycle."

"The two-hour program video is available to view online": 2nd hour, chapter 10 from the beginning, has an outstanding movie or animation of the laser light slow-down.

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