Monstrous Moonshine - History

History

Specifically, Conway and Norton, following an initial observation by John McKay, found that the Fourier expansion of j(τ) (sequence A000521 in OEIS), with τ denoting the half-period ratio, could be expressed in terms of linear combinations of the dimensions of the irreducible representations of M (sequence A001379 in OEIS):

where, and

\begin{align} 1 & = 1 \\ 196884 & = 196883 + 1 \\ 21493760 & = 21296876 + 196883 + 1 \\ 864299970 & = 842609326 + 21296876 + 2\cdot 196883 + 2\cdot 1 \\ & {}\,\,\, \vdots \end{align}

Conway and Norton formulated conjectures concerning the functions obtained by replacing the traces on the identity by the traces on other elements g of M. The most striking part of these conjectures is that all these functions are genus zero. In other words, if Gg is the subgroup of SL2(R) which fixes, then the quotient of the upper half of the complex plane by Gg is a sphere with a finite number of points removed, corresponding to the cusps of Gg.

It turns out that lying behind monstrous moonshine is a certain string theory having the Monster group as symmetries; the conjectures made by Conway and Norton were proven by Richard Ewen Borcherds in 1992 using the no-ghost theorem from string theory and the theory of vertex operator algebras and generalized Kac-Moody superalgebras. Borcherds won the Fields medal for his work, and more connections between M and the j-function were subsequently discovered.

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