Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 030, 36 pages      arXiv:1710.00737      https://doi.org/10.3842/SIGMA.2019.030
Contribution to the Special Issue on Moonshine and String Theory

A Self-Dual Integral Form of the Moonshine Module

Scott Carnahan
University of Tsukuba, Japan

Received February 13, 2018, in final form April 06, 2019; Published online April 19, 2019

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.

Key words: moonshine; vertex operator algebra; orbifold; integral form.

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