occasional meanderings in physics' brave new world

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Location: New Zealand

Marni D. Sheppeard

Tuesday, May 22, 2007

M Theory Lesson 59

One of the most information packed 600+ page tomes in the library here is Sphere Packings, Lattices and Groups by J. H. Conway and N. J. A. Sloane. It looks at the 24 dimensional Leech lattice in numerous ways.

Chapter 1 covers the basics of lattice theory and sphere packings. On page 5 it is noted that for a hexagonal plane tiling, rather than basis vectors $(1,0)$ and $(0.5, 0.5 \sqrt{3})$ in two dimensions, it is useful to use the simple three dimensional coordinates $(1,-1,0)$ and $(0,1,-1)$, which lie on the plane $x + y + z = 0$. This basis gives a Dynkin diagram labelling of the $A_2$ root lattice. It corresponds to the densest sphere packing in two dimensions with a density of $\frac{\pi}{\sqrt{12}}$.

The theta function of an integral lattice is related to modular forms. For example, for the Leech lattice the series takes the form

$1 + 196560 q^4 + 16773120 q^6 + \cdots$

which follows from a term for the $E_8$ theta series minus another term which is 720 times the Ramanujan series

$q^2 - 24 q^4 + 252 q^6 -1472 q^8 + \cdots$

Defining $\theta (\tau) = \sum_{- \infty}^{\infty} e^{\pi i \tau n^{2}}$, one can use Riemann's observation that

$\theta (\frac{-1}{\tau}) = \sqrt{- i \tau} \theta (\tau)$

to prove the functional equation for the zeta function. Recall that in M theory this property of the zeta function is intimately related to the physical duality of Space and Time. In a logos style constructive approach to zeta functions it is therefore natural to view lattices and theta series as useful tools for the geometrization of operad polytopes.

Note that the currently popular j invariant may be expressed easily in terms of theta series as

$j (\tau) = 32 \frac{(\theta (\tau)^8 + \theta_{01} (\tau)^8 + \theta_{10} (\tau)^8)^{3}}{(\theta (\tau) \theta_{01} (\tau) \theta_{10} (\tau))^{8}}$

which comes from basic elliptic function theory, but smells of triality, I think.

kneemo said...

"...it is useful to use the simple three dimensional coordinates (1,-1,0) and (0,1,-1), which lie on the plane x+y+z=0."

The plane x+y+z=0 is especially interesting when using 3x3 Hermitian matrix coordinates. For then the eigenvalues yield coordinate triplets (x,y,z), where primitive idempotents serve as a 3D orthonormal basis. Requiring that our matrices be traceless is equivalent to the condition x+y+z=0.

When we allow our Hermitian matrices to have octonion entries, the off-diagonal degrees of freedom are promoted to 24 dimensions. We can rotate our 3D idempotent basis in 24 dimensional space using F4 transformations. Conveniently, the trace of an arbitrary 3x3 Hermitian matrix is an F4 invariant. Moreover, an irreducible representation of F4 is obtained by requiring that our Hermitian matrices be traceless, i.e., x+y+z=0.

So I suspect the honeycomb picture becomes really interesting in 24 dimensions. In such a context, the relationship to the Leech lattice should be more transparent.

May 22, 2007 7:24 PM
Hank said...

I am not sure if I got smarter or my eyes started to bleed, but I enjoyed it either way!

May 23, 2007 6:19 AM
CarlBrannen said...

In the preon model, my assumption has been that the three components are orthogonal. Might as well take them to be pointed in x, y, and z directions. The overall spin is then in oriented in the x+y+z direction.

This also implies preon speeds of c sqrt(3).

May 23, 2007 9:17 AM
kneemo said...

"In the preon model, my assumption has been that the three components are orthogonal. Might as well take them to be pointed in x, y, and z directions. The overall spin is then in oriented in the x+y+z direction."

Sounds good Carl. In the SUGRA framework, x+y+z=0 is probably equivalent to conservation of charge.

May 23, 2007 11:09 AM
Kea said...

especially interesting when using 3x3 Hermitian matrix coordinates...so I suspect the honeycomb picture becomes really interesting in 24 dimensions.

Why, kneemo yes, indeed, methinks this is all rather intriguing! Welcome to AF, Hank. Hi Carl.

May 23, 2007 11:22 AM
Doug said...

Hi Kea,
Thanks for the Conway and Sloane 'Sphere Packings, Lattices and Groups'.
I have skimmed through 3ed, 1999 quickly and find the following interesting:

1 - C_30 Borcherds is a coauthor of what appears to be a chapter perhaps written before his final proof of the Monster.

2 - C_29 Monster and 196884-D space with binary Golay Code and Leech Lattice.

3 - C_27 Automorphism Group of 26-D Lorentian Lattice, p531, figure 27.3 a pentagram within double penta-star representing set of 35 Leech roots for deep hole type A24.

4 - C_24 23 Construction for Leech Lattice, p511, figure 24.2 Hole diagrams for A11D7E6.

5 C_17 24-D Odd unimodular Lattices, p424, figure 17.1, Neighbor hood graphs in n=8, 16, 24-d [reminds me of Petri Nets and strategic-D rather than spatial-D].

6 C_11 Golay Codes and Mathieu Groups, p325, figure 11.40, turnaround permutations [reminds me of a reverse Sierpinski sieve].

7 C_4 Certain important lattices, p96, figure 4.1, 120 spherical triangles [10 per pentagon on dodecahedron].

8 C_1 Sphere packings and kissing numbers, p6, figure 1.3, hexagonal lattice.

May 23, 2007 12:09 PM
Kea said...

Goodness, Doug, you're a faster reader than I am! It's a cool book, heh?

May 23, 2007 12:30 PM
Doug said...

Hi Kea,