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 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 Prof. Anne Taormina is to unravel this Mathieu moonshine mystery. I give an overview of our approach in the final section of the book chapter

Katrin Wendland
  • Mathematical Aspects of Quantum Field Theories, D. Calaque and Th. Strobl, eds.
  • Mathematical Physics Studies, Springer 2015, pp. 89-129
  • arXiv:1404.3108 [hep-th]

First results of the joint work with Prof. Anne Taormina, are presented in

Anne Taormina, Katrin Wendland
The overarching finite symmetry group of Kummer surfaces in the Mathieu group M24

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, on the one hand, to discuss generic symmetry groups of appropriate families of K3 surfaces simultaneously, and on the other, 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

Anne Taormina, Katrin Wendland
Symmetry-surfing the moduli space of Kummer K3s
  • Proceedings of the Conference String-Math 2012
  • Proceedings of Symposia in Pure Mathematics 90 (2015), 129-153
  • arXiv:1303.2931 [hep-th]

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. As yet, no other possible explanation for this fact is known to me. Our methods of describing symmetry groups of families of K3 theories by means of Niemeier markings have since been employed by several other groups, for example by Cheng, Harrison and their co-authors.

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

Anne Taormina, Katrin Wendland
A twist in the M24 moonshine story

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. Gaberdiel, Keller and Paul have generalized parts of our results to all massive contributions to the elliptic genus, providing further evidence in favor of symmetry surfing.

Further evidence in favor of symmetry surfing is obtained by a detailed study of the elliptic genus of K3. In particular, one may introduce the notion of a generic space of states which is shared by all K3 theories and which we have managed to show carries precisely the action of M24 predicted by Mathieu Moonshine. This action is compatible with symmetry surfing Kummer surfaces. To ultimately confirm symmetry surfing, it remains to prove compatibility with the action in general and with the structure of a vertex operator algebra.

Inspired by our investigations of large symmetry groups in K3 theories, in

Matthias Gaberdiel, Anne Taormina, Roberto Volpato, Katrin Wendland
A K3 sigma model with Z28 : M20 symmetry

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 and important K3 model. It is now known as the Gaberdiel-Taormina-Volpato-Wendland (GTVW) model. We determine its group of N=(4,4) preserving symmetries and prove that it is the group Z28 : M20, accounting for the largest possible symmetry group among K3 sigma models. The group also involves generators that, from the orbifold point of view, map untwisted and twisted sector states into one another.

Challenging our description of symmetries of K3 theories, in the first version of their manuscript An Uplifting Discussion of T-Duality, Harvey and Moore cast some doubts on our results. We disprove these doubts in our paper

Anne Taormina, Katrin Wendland
Not Doomed to Fail

In their publication Derived Equivalences of K3 Surfaces and Twined Elliptic Genera, Duncan and Mack-Crane discovered that the space of states underlying the above-mentioned so-called GTVW model, as a module of the diagonal Virasoro algebra at central charge 12, agrees with the space of states of the Conway Moonshine Module. In our work

Anne Taormina, Katrin Wendland
The Conway Moonshine Module is a Reflected K3 Theory
we introduce a procedure which we call reflection and which gives the precise relationship between the two models. Reflection transforms certain superconformal field theories into super vertex operator algebras plus their admissible modules. This builds a bridge between the two, which may be used in both directions, allowing vertex algebraists access to the realm of full-fledged superconformal field theories. For the K3 theory mentioned above, we show that it allows reflection, and that the resulting super vertex operator algebra plus admissible module is the Conway Moonshine Module. Our construction of reflection has since been applied in simpler examples as well, for instance by Anagiannis, Cheng, Duncan and Volpato, who however restrict their attention to toroidal superconformal field theories with c=6.

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 Prof. Anne Taormina, as well.