A Self-Dual Integral Form of the Moonshine Module

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.