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

Taormina, Anne; Wendland, Katrin

A twist in the

M_{24}

moonshine story

Taormina, Anne ; Wendland, Katrin

Confluentes Mathematici, Tome 7 (2015), p. 83-113 / Harvested from Numdam

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Résumé

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 ${(ℤ_{2})}^{4} ⋊ A_{8}$ 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.

Publié le : 2015-01-01
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},
     volume = {7},
     year = {2015},
     pages = {83-113},
     doi = {10.5802/cml.19},
     language = {en},
     url = {http://dml.mathdoc.fr/item/CML_2015__7_1_83_0}
}

Taormina, Anne; Wendland, Katrin. A twist in the $M_{24}$ moonshine story. Confluentes Mathematici, Tome 7 (2015) pp. 83-113. doi : 10.5802/cml.19. http://gdmltest.u-ga.fr/item/CML_2015__7_1_83_0/

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