Moonshine theory

In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group M and modular functions, in particular, the j function. The term was coined by John Conway and Simon P. Norton in 1979.

It is now known that lying behind monstrous moonshine is a vertex operator algebra called the moonshine module (or monster vertex algebra) constructed by Igor Frenkel, James Lepowsky, and Arne Meurman in 1988, having the monster group as symmetries. This vertex operator algebra is commonly interpreted as a structure underlying a two-dimensional conformal field theory, allowing physics to form a bridge between two mathematical areas. The conjectures made by Conway and Norton were proved by Richard Borcherds for the moonshine module in 1992 using the no-ghost theorem from string theory and the theory of vertex operator algebras and generalized Kac–Moody algebras.

In 1978, John McKay found that the first few terms in the Fourier expansion of the normalized J-invariant (sequence in the OEIS),