Characterizing Moonshine Functions by Vertex-Operator-Algebraic Conditions

TL;DR

This paper investigates the properties of moonshine functions associated with vertex operator algebras, showing that most non-monstrous functions cannot serve as trace functions on such algebras, thus characterizing moonshine functions.

Contribution

It provides a vertex-operator-algebraic characterization of moonshine functions, distinguishing them from non-monstrous completely replicable functions.

Findings

154 of 157 non-monstrous functions cannot occur as trace functions

Most non-monstrous completely replicable functions are excluded from vertex operator algebra traces

The work narrows down the possible functions associated with moonshine phenomena

Abstract

Given a holomorphic $C_2$-cofinite vertex operator algebra $V$ with graded dimension $j-744$, Borcherds's proof of the monstrous moonshine conjecture implies any finite order automorphism of $V$ has graded trace given by a "completely replicable function", and by work of Cummins and Gannon, these functions are principal moduli of genus zero modular groups. The action of the monster simple group on the monster vertex operator algebra produces 171 such functions, known as the monstrous moonshine functions. We show that 154 of the 157 non-monstrous completely replicable functions cannot possibly occur as trace functions on $V$.