Moonshine

Abstract

Abstract Monstrous moonshine relates distinguished modular functions to the representation theory of the Monster "Image missing". The celebrated observations that $$ 1=1,\ \ \ 196884=1+196883,\ \ \ 21493760=1+196883+21296876,\ \ \ \dots\dots (*)$$ 1 = 1 , 196884 = 1 + 196883 , 21493760 = 1 + 196883 + 21296876 , … … ( * ) illustrate the case of J(τ)=j(τ)−744, whose coefficients turn out to be sums of the dimensions of the 194 irreducible representations of "Image missing". Such formulas are dictated by the structure of the graded monstrous moonshine modules. Recent works in moonshine suggest deep relations between number theory and physics. Number theoretic Kloosterman sums have reappeared in quantum gravity, and mock modular forms have emerged as candidates for the computation of black hole degeneracies. This paper is a survey of past and present research on moonshine. We also compute the quantum dimensions of the monster orbifold and obtain exact formulas for the multiplicities of the irreducible components of the moonshine modules. These formulas imply that such multiplicities are asymptotically proportional to dimensions. For example, the proportion of 1’s in (*) tends to $$\frac{\dim(\chi_{1})}{\sum_{i=1}^{194}\dim(\chi_{i})}=\frac{1}{5844076785304502808013602136}=1.711\ldots \times 10^{-28}. $$ dim ( χ 1 ) ∑ i = 1 194 dim ( χ i ) = 1 5844076785304502808013602136 = 1.711 … × 1 0 − 28 . 2010 Mathematics Subject Classification: 11F11; 11 F22; 11F37; 11F50; 20C34; 20C35