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?
$j = q^{-1} +744 + 196884 q + 21493760 q^{2} + \cdots$
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?
posted by Kea | 7:07 PM
2 Comments:
Hello Kea: Another fine M-theory lesson. (I couldn't reach your email again for some reason.)
You may read in the blog soon that I had dinner with someone very big in supernova research. No one has gone over to the dark side. It is more a case of them paying respect to us. If their spacecraft flies, it will just be more proof that c changes.
Good to hear from you Louise. I will email you.
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