

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 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 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}.
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
Katrin Wendland,
Snapshots of Conformal Field Theory;
in: "Mathematical Aspects of Quantum Field Theories",
Damien Calaque and Thomas Strobl, eds.,
Mathematical Physics Studies, Springer 2015,
pp. 89129;
arXiv:1404.3108 [hepth]
First results of the joint work with
Dr. Anne Taormina,
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 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 socalled 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,
Symmetrysurfing the moduli space of Kummer K3s;
to appear in the Proceedings of the Conference StringMath 2012,
Proceedings of Symposia in Pure Mathematics;
preprint arXiv:1303.2931 [hepth]
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 M_{24}.
We also give evidence for the fact that the
Mathieu moonshine phenomenon is tied to M_{24}
rather than M_{23}, 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 M_{24}
in
Anne Taormina, Katrin Wendland,
A twist in the M_{24} moonshine story;
to appear in Confluentes Mathematici;
preprint arXiv:1303.3221 [hepth].
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 Z_{2} orbifold CFTs on K3, and which
transforms under the expected representation of M_{24}. This is
a highly nontrivial conformation of our program and provides direct evidence
for the fact that M_{24} 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 M_{24}, 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
Matthias Gaberdiel, Anne Taormina, Roberto Volpato, Katrin Wendland,
A K3 sigma model with Z_{2}^{8} : M_{20} symmetry;
JHEP 1402:022 (2014);
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, 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
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 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 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.
