## Moonshine

Moonshine denotes a surprising connection between two very different areas of mathematics, those of

**modular forms**(functions that live on a torus) and**finite simple groups**. The original Moonshine observation was in the 1980's in the context of the largest sporadic finite simple group, the Monster group, and a modular form, called the j(τ) function; Mathieu Moonshine is a recent observation (2010) relating the sporadic finite simple group Mathieu M_24 to the elliptic genus of the K_3 manifold.## Monstrous Moonshine

The modular function in this case is the j(τ) function. Since a modular function is periodic in both directions around the torus, it has a Fourier expansion in terms of the complex structure τ of the torus (and q=exp(2πiτ). This expansion is given by
j(τ) = 1/q+744+196884 q+21493760 q^2+864299970 q^3+20245856256 q^4... |
The dimensions of the smallest irreducible representations of the Monster group are given by
1, 196883, 21296876, 842609326, 18538750076, 19360062527, 293553734298 ... |

The observation that 21493760 = 21296876 + 196883 + 1 and 196 884 = 196 883 + 1 by John McKay was the first hint that there was a deep connection between those two areas.

## Mathieu Moonshine

The modular form here is the elliptic genus of the K_3 manifold, which has been known for decades. But in 2010 it was noticed that via an inspired rewriting of this elliptic genus in terms of supercharacters of the N=4 superconformal algebra, a similar Moonshine phenomenon is displayed:
E_K3(τ,z)=−2Ch(0;τ,z)+20Ch(1/2;τ,z)+e(q)Ch(τ,z) where e(q)=90 q+462 q^2+1540 q^3+4554 q^4+11592 q^5+… |
The dimensions of the smallest irreducible representations of the Mathieu group M24 include
1, 45, 231, 770, 2277, 5796 ... |

The coefficients of the Fourier expansion in the elliptic genus are therefore twice the dimension of certain irreducible representations of the Mathieu group.