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?

  • 1
    $\begingroup$ The Monster is the largest of the sporadic simple groups. There are arbitrarily large simple groups (there are infinitely many primes, for example!) $\endgroup$ – Mariano Suárez-Álvarez Jul 4 '11 at 21:38
  • $\begingroup$ Also related is Euler's famous polynomial $x^2+x+41$, which is prime for $0\leq x\leq 39$, factors as $\left(x+\frac{1-\sqrt{-163}}{2}\right)\left(x+\frac{1+\sqrt{-163}}{2}\right)$. $\endgroup$ – Eric Naslund Jul 4 '11 at 21:42
  • $\begingroup$ Yes, you plug $\tau = (1+\sqrt{-163})/2$ into the McKay-Thompson series T_1A with constant term 744, and you get -640320^3. Given the prime-generating polynomial P(n) = 2n^2+29 which is prime for n = {0 to 28} and plug the root of P(n) = 0 into T_2A, and you get 396^4. And so on. $\endgroup$ – Tito Piezas III Jul 4 '11 at 21:47
  • 1
    $\begingroup$ It is worth reading the Wikipedia page on Heegner Numbers: 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.) $\endgroup$ – Eric Naslund Jul 4 '11 at 21:48

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.