Mathieu moonshine
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Mathieu moonshine

By classical results due to Nikulin, Mukai, Xiao and Kondo in the 1980's and 90's, the finite symplectic automorphism groups of K3 surfaces are always subgroups of the Mathieu group M24. This is a simple sporadic group of order 244823040. However, also by results due to Mukai, each such automorphism group has at most 960 elements and thus is by orders of magnitude smaller than M24. On the other hand, according to a recent observation by Eguchi, Ooguri and Tachikawa, the elliptic genus of K3 surfaces seems to contain a mysterious footprint of an action of the entire group M24: If one decomposes the elliptic genus into irreducible characters of the N=4 superconformal algebra, which is natural in view of superconformal field theories (SCFTs) associated to K3, then the coefficients of the so-called non-BPS characters coincide with the dimensions of representations of M24.

An overview of our approach is contained in the final chapter of the expository article
Katrin Wendland,
Snapshots of Conformal Field Theory;
contribution to ``Mathematical Aspects of Quantum Field Theories", Mathematical Physics Studies, Springer;
preprint arXiv:1404.3108 [hep-th]
In joint work with Dr. Anne Taormina, first results of which are presented in
Anne Taormina, Katrin Wendland,
The overarching finite symmetry group of Kummer surfaces in the Mathieu group M24;
JHEP 1308:152 (2013); arXiv:1107.3834 [hep-th]
we develop techniques which eventually should overcome the above-mentioned "order of magnitude problem": For Kummer surfaces which carry the Kähler class that is induced by their underlying complex torus, we find methods that improve the classical techniques due to Mukai and Kondo, and we give a construction that allows us to combine the finite symplectic symmetry groups of several Kummer surfaces to a larger group. Thereby, we generate the so-called overarching finite symmetry group of Kummer surfaces, a group of order 40320, thus already mitigating the "order of magnitude problem". In
Anne Taormina, Katrin Wendland,
Symmetry-surfing the moduli space of Kummer K3s; preprint arXiv:1303.2931 [hep-th]
we extend these results and show how all symmetry groups of such Kummer surfaces can indeed be combined to a bigger group of order 322560, which is a maximal subgroup in M24. While these works address the leading order terms of the elliptic genus, which count massless states and for which there are some conceptual difficulties to make the Mathieu Moonshine phenomenon precise, we show that the first order massive contributions to the elliptic genus are indeed coverned by the Mathieu group M24 in
Anne Taormina, Katrin Wendland,
A twist in the M24 moonshine story; preprint arXiv:1303.3221 [hep-th].
Inspired by this work, in
Matthias Gaberdiel, Anne Taormina, Roberto Volpato, Katrin Wendland,
A K3 sigma model with Z28 : M20 symmetry;
JHEP 1402:022 (2014); preprint arXiv:1309.4127 [hep-th]
we study the non-linear sigma model on the tetrahedral Kummer surface with B-field chosen such that there is an extended symmetry. We determine its group of N=(4,4) preserving symmetries and prove that it is the group Z28 : M20, accounting for one of the largest maximal symmetry groups of K3 sigma models. The symmetry group involves also generators that, from the orbifold point of view, map untwisted and twisted sector states into one another.

A project which is probably related to Mathieu moonshine is the following. For SCFTs associated to K3 a "topological part" of all partition functions was given by Eguchi, Ooguri, Taormina and Yang and corrected in my thesis. This function is closely related to the branching function of the elliptic genus into N=4 superconformal characters which was mentioned above. It has interesting number theoretic properties. It can be expressed in terms of Appell functions, whose quasiperiodic behavior under modular transforms is possibly linked to automorphic forms on the moduli space. In fact, this function is an example of a so-called mock modular form. However, an explicit geometric and SCFT interpretation for it is lacking. I am studying this in collaboration with Dr. Anne Taormina, as well.

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