Moonshine

TL;DR

This paper surveys the history and recent developments in moonshine, exploring connections between modular functions, the monster group, number theory, and physics, including quantum gravity and black hole entropy.

Contribution

It provides a comprehensive overview of moonshine research and introduces new formulas for multiplicities in moonshine modules, linking them asymptotically to dimensions.

Findings

Computed quantum dimensions of the monster orbifold.

Derived exact formulas for multiplicities of irreducible components.

Showed multiplicities are asymptotically proportional to dimensions.

Abstract

Monstrous moonshine relates distinguished modular functions to the representation theory of the monster. The celebrated observations that 196884=1+196883 and 21493760=1+196883+21296876, etc., illustrate the case of the modular function j-744, whose coefficients turn out to be sums of the dimensions of the 194 irreducible representations of the monster. 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.