Monstrous moonshine

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 initial numerical observation was made by John McKay in 1978, and the phrase was coined by John Conway and Simon P. Norton in 1979. The monstrous moonshine is now known to be underlain by a vertex operator algebra called the moonshine module (or monster vertex algebra) constructed by Igor Frenkel, James Lepowsky, and Arne Meurman in 1988, which has the monster group as its group of 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 proven 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 , with and τ as the half-period ratio could be expressed in terms of linear combinations of the dimensions of the irreducible representations of the monster group M with small non-negative coefficients. Let = 1, 196883, 21296876, 842609326, 18538750076, 19360062527, 293553734298, ... then, where the LHS are the coefficients of while the RHS are the dimensions of the monster group M. (Since there can be several linear relations between the such as , the representation may be in more than one way.) McKay viewed this as evidence that there is a naturally occurring infinite-dimensional graded representation of M, whose graded dimension is given by the coefficients of J, and whose lower-weight pieces decompose into irreducible representations as above. After he informed John G. Thompson of this observation, Thompson suggested that because the graded dimension is just the graded trace of the identity element, the graded traces of nontrivial elements g of M on such a representation may be interesting as well.

Almost impossible E_8 and Leech lattices

Almost impossible E_8 and Leech lattices