A twist in the $M_{24}$ moonshine story

Taormina, Anne; Wendland, Katrin

Confluentes Mathematici, Tome 7 (2015) no. 1, pp. 83-113.

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 ${\mathbb{Z}}_2$-orbifold CFT on K3. These generic states are uniquely characterized by the fact that the action of every geometric symmetry group of a ${\mathbb{Z}}_2$-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 $M_{24}$ constructed by Margolin. Thus we provide a piece of evidence for Mathieu Moonshine explicitly from SCFTs on K3.

The $45$-dimensional irreducible representation of $M_{24}$ exhibits a twist, which we prove can be undone in the case of ${\mathbb{Z}}_2$-orbifold CFTs on K3 for all geometric symmetry groups. This twist however cannot be undone for the combined symmetry group ${\left({\mathbb{Z}}_2\right)}^4⋊A_8$ 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.

Classification : 81T40, 81T60, 14J28

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