Moonshine, in its original form, expressed a connection between a remarkable group of symmetries called the Monster, and an important sequence of numbers in number theory. Further investigation then led to the mathematical physics of string theory, as I explain in my book Symmetry and the Monster.
The original moonshine observation was made by John McKay. He knew that the first non-trivial dimension in which the Monster might operate was 196,883, and was astonished to find 196,884 appearing in a quite different branch of mathematics, as the first in a sequence of numbers expressing an important formula in number theory. This had to be more than a chance coincidence, and McKay wrote to John Thompson, who then found further numerical evidence, and was convinced something mysterious was going on. Thompson communicated his observations to John Conway who had complete data on the Monster at the Cambridge mathematics department. It appeared that the numbers in the sequence — which were coefficients in a formula for something called the j-function — were related to dimensions in which the Monster could operate, and in a 1979 paper with the title Monstrous Moonshine, Conway and Simon Norton proved there was a relationship, though no-one quite understood why.
The dimensions in which the Monster can operate are given in the first column of its character table, a square array of numbers that encodes a massive amount of data about the group in question, and Conway and Norton proved that the other columns generated similar functions in exactly the same way. This helped explain a coincidence, noticed in the mid-1970s by Andrew Ogg. He had attended a talk in Paris by Jacques Tits, who mentioned the recently-discovered Monster, and wrote down its size as a product of prime numbers. Ogg remarked that these prime numbers were precisely those for which functions similar to the j-function exist. No one had an explanation for Ogg’s observation, until Conway and Norton showed that these functions arise from columns in the Monster’s character table, one for each of the prime numbers concerned.
The Monster has 194 rows and columns in its character table, but some columns generate the same function, and the number of distinct ones is 171. These are not all linearly independent, and Conway and Norton showed they span a space of dimension 163. Since 163 has a very distinctive feature in number theory, this yields yet another coincidence, but one that remains completely unexplained.
Conway and Norton published their results in 1979, since when the Moonshine topic has blossomed into a field of study in its own right, drawing connections between the Monster and the mathematics of string theory. These connections stemmed from work of Igor Frenkel, James Lepowsky and Arne Meurman, who created something called the Moonshine Module. It is built up from subspaces stabilized by the Monster, whose dimensions are given by the coefficients of the j-function, one for each coefficient. And it is connected with the mathematics of string theory (a theory of elementary particles in physics). Richard Borcherds made great contributions to understanding this connection, and proved that the Moonshine module satisfies the Conway-Norton conjectures, work for which he was awarded a Fields Medal in 1998.