On the symmetry of odd Leech lattice CFT

A bstract 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 Co 0 and the Leech lattice, which was already shown by Griess in [1]. This study is motivated by the moonshine-type observation on the $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 2 extremal elliptic genus of central charge 24 by Benjamin, Dyer, Fitzpatrick, and Kachru [2]. We also investigate weight-1 and weight- $$ \frac{3}{2} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfrac> <mml:mn>3</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:math> 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 $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 2 superconformal algebra in it.