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.