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 more 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.
The ultimate aim of a project with Dr. Anne Taormina is to unravel this Mathieu moonshine mystery. I give an overview of our approach in the final section of the book chapter
First results of the joint work with Dr. Anne Taormina, are presented in
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 by what we call a Niemeier marking in our later work. The idea has inspired Nikulin to a general discussion of Niemeier markings. Our Niemeier markings enable us to give a construction, which we call symmtry surfing in later publications, 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".
The technique of symmetry surfing is fully developed for Kummer surfaces in
We show how all symmetry groups of Kummer surfaces as above can indeed be
combined to a bigger group of order 322560, which is a maximal subgroup in
M24. We also give evidence for the fact that the Mathieu moonshine
phenomenon is tied to M24 rather than M23, and we offer
an explanation to the fact that it seems not to be related to the full symmetry
groups of superconformal field theories (SCFTs) on K3.
While our previous 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 governed by the Mathieu group M24 in
We show that the leading order massive contributions to the elliptic genus originate from a subspace of the space of states in SCFTs on K3 which is generic to all Z2 orbifold CFTs on K3, and which transforms under the expected representation of M24. This is a highly nontrivial conformation of our program and provides direct evidence for the fact that M24 governs the elliptic genus of K3. Moreover, we find a new obstruction that this representation yields against an extension of symmetry groups to the full group M24, a result which amounts to an explicit calculation of effects of monodromy in the moduli space of conformal field theories on K3. Several research groups have employed our ideas and are studying their implications, e.g. Cheng and coauthors.
Inspired by this work, in
we study the non-linear sigma model on the tetrahedral Kummer surface with B-field chosen such that there is an extended symmetry, which I had already highlighted in my work "Orbifold Constructions of K3: A Link between Conformal Field Theory and Geometry" as particularly interesting from the viewpoint of mirror 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 Ph.D. 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.