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...

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Published in:Symmetry, Integrability and Geometry: Methods and Applications
Date:2019
Main Author: Carnahan, S.
Format: Article
Language:English
Published: Інститут математики НАН України 2019
Online Access:https://nasplib.isofts.kiev.ua/handle/123456789/210192
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Journal Title:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Cite this:A Self-Dual Integral Form of the Moonshine Module / S. Carnahan // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 48 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine
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author Carnahan, S.
author_facet Carnahan, S.
citation_txt A Self-Dual Integral Form of the Moonshine Module / S. Carnahan // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 48 назв. — англ.
collection DSpace DC
container_title Symmetry, Integrability and Geometry: Methods and Applications
description 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.
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institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
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language English
last_indexed 2025-12-07T21:24:41Z
publishDate 2019
publisher Інститут математики НАН України
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spelling Carnahan, S.
2025-12-03T14:33:50Z
2019
A Self-Dual Integral Form of the Moonshine Module / S. Carnahan // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 48 назв. — англ.
1815-0659
2010 Mathematics Subject Classification: 17B69; 11F22; 20C10; 20C20; 20C34
arXiv: 1710.00737
https://nasplib.isofts.kiev.ua/handle/123456789/210192
https://doi.org/10.3842/SIGMA.2019.030
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.
I would like to thank Toshiyuki Abe for describing the constructions in [1] in detail at the "VOA and related topics" workshop at Osaka University in March 2017. I would also like to thank the anonymous referees for many helpful comments, and one referee in particular for their help with the proof of Lemma 2.13. This research was partly funded by JSPS Kakenhi Grant-in-Aid for Young Scientists (B) 17K14152.
en
Інститут математики НАН України
Symmetry, Integrability and Geometry: Methods and Applications
A Self-Dual Integral Form of the Moonshine Module
Article
published earlier
spellingShingle A Self-Dual Integral Form of the Moonshine Module
Carnahan, S.
title A Self-Dual Integral Form of the Moonshine Module
title_full A Self-Dual Integral Form of the Moonshine Module
title_fullStr A Self-Dual Integral Form of the Moonshine Module
title_full_unstemmed A Self-Dual Integral Form of the Moonshine Module
title_short A Self-Dual Integral Form of the Moonshine Module
title_sort self-dual integral form of the moonshine module
url https://nasplib.isofts.kiev.ua/handle/123456789/210192
work_keys_str_mv AT carnahans aselfdualintegralformofthemoonshinemodule
AT carnahans selfdualintegralformofthemoonshinemodule