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There is a McKay-Thompson series for the Monster group, namely $T_{1A}$, responsible for,

$e^{\pi\sqrt{163}} \approx 640320^3 + 744$

Another one ($T_{2A}$) for,

$e^{\pi/2\sqrt{232}} \approx 396^4 -104$

And a third one ($T_{3A}$) for,

$e^{\pi/3\sqrt{267}} \approx 300^3 + 42$

It turns out, as proven by Conway, Norton, and Atkin, that this family of functions span a linear space of dimension 163. I found this so intriguing I had to write an article on it. See,

"The 163 Dimensions of the Moonshine Functions"

The Monster is the largest of the sporadic simple groups, and 163 is the largest d such that $Q(\sqrt{-d})$ has unique factorization. Do you think this is just a coincidence?

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    It is worth reading the Wikipedia page on Heegner Numbers: http://en.wikipedia.org/wiki/Heegner_number There are a lot of interesting things there. (What I said above, and the relation between those prime generating polynomials and Heegner numbers are in the article.)2011-07-04

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Gukov-Vafa Comm Math Phys 246, pp 181-110 has a remark in section 5.4 page 198.

I am following this up. It seems a knowledge (at least for physicists) of Ishibashi states and the Narain momentum lattice may be helpful.

Work by Connes, Consani, Marcolli, Ramachandran relates quantum statistical mechanics and class field theory.