On the symmetry of odd Leech lattice CFT

Abstract

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 224.M 24 and 223.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 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- 3 2 $$ \frac{3}{2} $$ currents invariant under the subgroup 224.M 24 or 223.M 23 of the automorphism group of the odd Leech lattice vertex operator algebra, and revisit an N $$ \mathcal{N} $$ = 2 superconformal algebra in it.

Keywords

• Conformal Field Models in String Theory
• Conformal and W Symmetry
• Discrete Symmetries