A twist in the M 24 moonshine story

Abstract

Prompted by the Mathieu Moonshine observation, we identify a pair of 45-dimensional vector spaces of states that account for the first order term in the massive sector of the elliptic genus of K3 in every Z2-orbifold CFT on K3. These generic states are uniquely characterized by the fact that the action of every geometric symmetry group of a Z2-orbifold CFT yields a well-defined faithful representation on them. Moreover, each such representation is obtained by restriction of the 45-dimensional irreducible representation of the Mathieu group M24 constructed by Margolin. Thus we provide a piece of evidence for Mathieu Moonshine explicitly from SCFTs on K3. The 45-dimensional irreducible representation of M24 exhibits a twist, which we prove can be undone in the case of Z2-orbifold CFTs on K3 for all geometric symmetry groups. This twist however cannot be undone for the combined symmetry group Z2^4 : A8 that emerges from surfing the moduli space of Kummer K3s. We conjecture that in general, the untwisted representations are exclusively those of geometric symmetry groups in some geometric interpretation of a CFT on K3. In that light, the twist appears as a representation theoretic manifestation of the maximality constraints in Mukai's classification of geometric symmetry groups of K3.