The M24 and M23 lattice symmetries of the odd Leech lattice do not lift to automorphisms of its lattice vertex operator algebra; the relevant group extensions are non-split.
Mathieu Moonshine and Symmetry Surfing
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abstract
Mathieu Moonshine, the observation that the Fourier coefficients of the elliptic genus on K3 can be interpreted as dimensions of representations of the Mathieu group M24, has been proven abstractly, but a conceptual understanding in terms of a representation of the Mathieu group on the BPS states, is missing. Some time ago, Taormina and Wendland showed that such an action can be naturally defined on the lowest non-trivial BPS states, using the idea of `symmetry surfing', i.e., by combining the symmetries of different K3 sigma models. In this paper we find non-trivial evidence that this construction can be generalized to all BPS states.
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On the Symmetry of Odd Leech Lattice CFT
The M24 and M23 lattice symmetries of the odd Leech lattice do not lift to automorphisms of its lattice vertex operator algebra; the relevant group extensions are non-split.