Generalized Moonshine IV: Monstrous Lie algebras

TL;DR

This paper constructs infinite-dimensional Lie algebras associated with each element of the Monster group, proving that their characters are Hauptmoduln and resolving Norton's Generalized Moonshine Conjecture.

Contribution

It introduces a new Lie algebra construction linked to the Monster group and proves the Hauptmodul property for characters, confirming Norton's conjecture.

Findings

Characters of centralizers are Hauptmoduln for all Fricke elements

Constructed Lie algebras have a projective action of centralizers

Resolved Norton's Generalized Moonshine Conjecture

Abstract

For each element of the Fischer-Griess Monster sporadic simple group, we construct an infinite dimensional Lie algebra equipped with a projective action of the centralizer of that element. Our construction is given by a string-theoretic "add a spacetime torus and quantize" functor applied to an abelian intertwining algebra that is formed from a family of twisted modules of the Monster vertex operator algebra. We prove that for all Fricke elements in the Monster, the characters of centralizers acting on the corresponding irreducible twisted modules are Hauptmoduln. From these results, we resolve Norton's Generalized Moonshine Conjecture.