

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 M_{24}. 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 M_{24}.
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 M_{24}: 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 socalled nonBPS characters coincide with
the dimensions of representations of M_{24}.
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 [hepth]
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
M_{24};
JHEP 1308:152 (2013); arXiv:1107.3834 [hepth]
we develop
techniques which eventually should overcome the abovementioned "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 socalled 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,
Symmetrysurfing the moduli space of Kummer K3s;
preprint arXiv:1303.2931 [hepth]
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 M_{24}.
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 M_{24}
in
Anne Taormina, Katrin Wendland,
A twist in the M24 moonshine story;
preprint arXiv:1303.3221 [hepth].
Inspired by this work, in
Matthias Gaberdiel, Anne Taormina, Roberto Volpato, Katrin Wendland,
A K3 sigma model with Z_{2}^{8} : M_{20} symmetry;
JHEP 1402:022 (2014);
preprint arXiv:1309.4127 [hepth]
we
study the nonlinear sigma model on the tetrahedral Kummer surface with Bfield 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
Z_{2}^{8} : M_{20}, 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 socalled 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.
