On the Symmetry of Odd Leech Lattice CFT

We show that the Mathieu groups $M_{24}$ and $M_{23}$ 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 $2^{24}.M_{24}$ and $2^{23}.M_{23}$ 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 $\mathrm{Co}_0$ and the Leech lattice, which was already shown in [Griess 1973]. This study is motivated by the moonshine-type observation on the $\mathcal{N}=2$ extremal elliptic genus of central charge 24 by [Benjamin, Dyer, Fitzpatrick, Kachru arXiv:1507.00004]. We also investigate weight-1 and weight-$\frac{3}{2}$ currents invariant under the subgroup $2^{24}.M_{24}$ or $2^{23}.M_{23}$ of the automorphism group of the odd Leech lattice vertex operator algebra, and revisit an $\mathcal{N}=2$ superconformal algebra in it.