occasional meanderings in physics' brave new world

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Location: New Zealand

Marni D. Sheppeard

## Friday, February 22, 2008

### M Theory Lesson 160

Monstrous moonshine tells us that the 1-ordinal indexing of the j invariant by powers of $q$

$j = q^{-1} +744 + 196884 q + 21493760 q^{2} + \cdots$

is roughly associated to the 1-ordinal indexing of an operad, since the coefficients are dimensions of the Monster modules $V_{i}$, which form an operad algebra. When interpreted as a lattice theta function, this indexing corresponds to the lengths of lattice vectors.

This correspondence between distance from the origin and dimension crops up in many unexpected quarters. For instance, in the method of geometric quantisation, the representations of $SU(2)$ are given by discretely spaced spheres in the dual of the Lie algebra for the group, basically $\mathbb{R}^{3}$. In fact, root lattices are just like this, living in the dual space to the Cartan algebra.

Well, it's probably time to mention E8 again: not only does the j invariant label a single node of an E8 diagram, but as Gannon points out, the dimension of E8 also appears in a single term of the expansion

$j^{\frac{1}{3}} = q^{\frac{-1}{3}} (1 + 248 q + 4124 q^{2} + 34752 q^{3} + \cdots )$

which is the generating function for the modular congruence group $\Gamma (3)$. It's neat that $4124 = 1031 \times 2^{2}$ has a simple prime factorisation, just like $248 = 31 \times 2^{3}$ (as Kostant mentions) and $34752 = 181 \times 3 \times 2^{6}$. I wonder why?