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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 so-called non-BPS characters coincide with the dimensions of
representations of M_{24}.

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

- 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

The overarching finite symmetry group of Kummer surfaces in the Mathieu group M_{24}

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

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
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. 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 M_{24} in

A twist in the M_{24} 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 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. 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 M_{24} 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

A K3 sigma model with **Z**_{2}^{8} : M_{20} 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
**Z**_{2}^{8} : M_{20}, 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

Not Doomed to Fail

- JHEP 2018:62 (2018); 10 pages
- arXiv:1708.01563 [hep-th]

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

The Conway Moonshine Module is a Reflected K3 Theory

- accepted for publication by Adv. Theor. Math. Phys.
- arXiv:1704.03813 [hep-th]

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.