A twist in the ${M}_{24}$ moonshine story
Confluentes Mathematici, Volume 7 (2015) no. 1, p. 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 ${ℤ}_{2}$-orbifold CFT on K3. These generic states are uniquely characterized by the fact that the action of every geometric symmetry group of a ${ℤ}_{2}$-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 ${ℤ}_{2}$-orbifold CFTs on K3 for all geometric symmetry groups. This twist however cannot be undone for the combined symmetry group ${\left({ℤ}_{2}\right)}^{4}⋊{A}_{8}$ 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.

DOI : https://doi.org/10.5802/cml.19
Classification:  81T40,  81T60,  14J28
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author = {Taormina, Anne and Wendland, Katrin},
title = {A twist in the $M\_{24}$ moonshine story},
journal = {Confluentes Mathematici},
publisher = {Institut Camille Jordan},
volume = {7},
number = {1},
year = {2015},
pages = {83-113},
doi = {10.5802/cml.19},
language = {en},
url = {http://www.numdam.org/item/CML_2015__7_1_83_0}
}

Taormina, Anne; Wendland, Katrin. A twist in the $M_{24}$ moonshine story. Confluentes Mathematici, Volume 7 (2015) no. 1, pp. 83-113. doi : 10.5802/cml.19. http://www.numdam.org/item/CML_2015__7_1_83_0/

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