On the Symmetry of Odd Leech Lattice CFT

TL;DR

This paper investigates the symmetry properties of the odd Leech lattice vertex operator algebra, showing certain Mathieu groups do not lift to automorphisms, and explores implications for moonshine phenomena and superconformal algebra structures.

Contribution

It demonstrates that specific Mathieu groups do not lift to automorphisms of the odd Leech lattice VOA and analyzes invariant currents and superconformal algebra structures.

Findings

Mathieu groups $M_{24}$ and $M_{23}$ do not lift to automorphisms.

Confirmed non-split extensions of automorphism groups.

Identified invariant currents and revisited superconformal algebra.

Abstract

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.