We show that the Mathieu groups M24 and M23 in the isometry group of the odd Leech lattice do not lift to subgroups of the automorphism group of its lattice vertex operator (super)algebra. In other words, the subgroups 224.M24 and 223.M23 of the automorphism group of the odd Leech lattice vertex operator algebra are non-split extensions. Our method can also confirm a similar result for the Conway group Co0 and the Leech lattice, which was already shown by Griess in [1]. This study is motivated by the moonshine-type observation on the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13130_2025_26524_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">N</mi> </math></EquationSource> <EquationSource Format="TEX">$ \mathcal{N} $</EquationSource> </InlineEquation> = 2 extremal elliptic genus of central charge 24 by Benjamin, Dyer, Fitzpatrick, and Kachru [2]. We also investigate weight-1 and weight-<InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13130_2025_26524_Article_IEq2.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="8" /> </InlineMediaObject> <EquationSource Format="MATHML"><math display="inline"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </math></EquationSource> <EquationSource Format="TEX">$ \frac{3}{2} $</EquationSource> </InlineEquation> currents invariant under the subgroup 224.M24 or 223.M23 of the automorphism group of the odd Leech lattice vertex operator algebra, and revisit an <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13130_2025_26524_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">N</mi> </math></EquationSource> <EquationSource Format="TEX">$ \mathcal{N} $</EquationSource> </InlineEquation> = 2 superconformal algebra in it.

On the symmetry of odd Leech lattice CFT

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2025年6月20日

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Masaki Okada

摘要

We show that the Mathieu groups M₂₄ and M₂₃ in the isometry group of the odd Leech lattice do not lift to subgroups of the automorphism group of its lattice vertex operator (super)algebra. In other words, the subgroups 2²⁴.M₂₄ and 2²³.M₂₃ of the automorphism group of the odd Leech lattice vertex operator algebra are non-split extensions. Our method can also confirm a similar result for the Conway group Co₀ and the Leech lattice, which was already shown by Griess in [1]. This study is motivated by the moonshine-type observation on the $N$ $ \mathcal{N} $ = 2 extremal elliptic genus of central charge 24 by Benjamin, Dyer, Fitzpatrick, and Kachru [2]. We also investigate weight-1 and weight- $\frac{3}{2}$ $ \frac{3}{2} $ currents invariant under the subgroup 2²⁴.M₂₄ or 2²³.M₂₃ of the automorphism group of the odd Leech lattice vertex operator algebra, and revisit an $N$ $ \mathcal{N} $ = 2 superconformal algebra in it.