A Self-Dual Integral Form of the Moonshine Module

TL;DR

This paper constructs a self-dual integral form of the moonshine vertex operator algebra, demonstrating its symmetry under the monster group and resolving key conjectures in modular moonshine theory.

Contribution

It introduces a new self-dual integral form of the moonshine module, confirming the last open assumption in the modular moonshine proof and linking it to Griess's original monster representation.

Findings

Existence of a self-dual integral form with monster symmetry

Resolution of the last open assumption in the modular moonshine proof

Griess's original monster representation admits a positive-definite self-dual integral form

Abstract

We construct a self-dual integral form of the moonshine vertex operator algebra, and show that it has symmetries given by the Fischer-Griess monster simple group. The existence of this form resolves the last remaining open assumption in the proof of the modular moonshine conjecture by Borcherds and Ryba. As a corollary, we find that Griess's original 196884-dimensional representation of the monster admits a positive-definite self-dual integral form with monster symmetry.