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 -orbifold CFT on K3. These generic states are uniquely characterized by the fact that the action of every geometric symmetry group of a -orbifold CFT yields a well-defined faithful representation on them. Moreover, each such representation is obtained by restriction of the -dimensional irreducible representation of the Mathieu group constructed by Margolin. Thus we provide a piece of evidence for Mathieu Moonshine explicitly from SCFTs on K3.
The -dimensional irreducible representation of exhibits a twist, which we prove can be undone in the case of -orbifold CFTs on K3 for all geometric symmetry groups. This twist however cannot be undone for the combined symmetry group 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.
Anne Taormina 1; Katrin Wendland 2
@article{CML_2015__7_1_83_0, author = {Anne Taormina and Katrin Wendland}, title = {A twist in the $M_{24}$ moonshine story}, journal = {Confluentes Mathematici}, pages = {83--113}, publisher = {Institut Camille Jordan}, volume = {7}, number = {1}, year = {2015}, doi = {10.5802/cml.19}, language = {en}, url = {https://cml.centre-mersenne.org/articles/10.5802/cml.19/} }
TY - JOUR AU - Anne Taormina AU - Katrin Wendland TI - A twist in the $M_{24}$ moonshine story JO - Confluentes Mathematici PY - 2015 SP - 83 EP - 113 VL - 7 IS - 1 PB - Institut Camille Jordan UR - https://cml.centre-mersenne.org/articles/10.5802/cml.19/ DO - 10.5802/cml.19 LA - en ID - CML_2015__7_1_83_0 ER -
Anne Taormina; Katrin Wendland. A twist in the $M_{24}$ moonshine story. Confluentes Mathematici, Volume 7 (2015) no. 1, pp. 83-113. doi : 10.5802/cml.19. https://cml.centre-mersenne.org/articles/10.5802/cml.19/
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