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- > Prof. Dr. Katrin Wendland

In my doctoral studies, among other topics I have investigated the embedding
of orbifold conformal field theories within the moduli space M^{K3} of theories associated to K3. Afterwards,
I generalized these results to non-Abelian orbifold groups and found a simple
explanation in terms of the classical McKay correspondence, see

Consistency of Orbifold Conformal Field Theories on K3

- Adv. Theor. Math. Phys.
**5**(2001), 429-456 - hep-th/0010281

The paper provides a general explanation for the behavior of the B-field under
the orbifold procedure, completing the description of all subvarieties of the
moduli space M^{K3} which can be obtained
by orbifold constructions from toroidal theories. I use the known description
of the moduli space and the requirement that the spaces of orbifold CFTs must
be consistently embedded into M^{K3}. I
prove that this suffices to explicitly determine the locations of all those G
orbifold CFTs on K3 within M^{K3} which are
obtained from toroidal theories by dividing out an action of G without
translations.

These results serve as a basis for further joint work with Prof. Werner Nahm as well, where we investigate a version of mirror symmetry on K3 surfaces:

Mirror Symmetry on Kummer Type K3 Surfaces

We describe mirror symmetry as an automorphism on the smooth universal cover of
M^{K3} and investigate both geometric and
conformal field theoretic aspects of this automorphism.

We use the version of mirror symmetry due to Vafa and Witten, which has been
generalized to the celebrated Strominger/Yau/Zaslow construction. In
particular, we determine the action of mirror symmetry on (non-stable) singular
fibers of elliptically fibered Z_{M} orbifold limits of K3. We find a
map of finite order 4, 8, or 12 in the different orbifold limits, which can be
described by a transformation of discrete Fourier type in each fiber.

Our approach yields an explicit formula for the geometric counterparts of the twist fields in our orbifold CFTs. This formula can be viewed as a version of Yongbin Ruan's conjecture on the quantum cohomology of orbifolds. As such, and along with the idea to use mirror symmetry for the proof, our work has been a key ingredient to results by Perroni followed by Coates, Corti, Iritani, Tseng to prove "Ruan's conjecture" in the more general context of computations for genus zero twisted Gromov-Witten invariants.

The following work contains a summary and extension of our results on orbifold conformal field theories on K3, in particular an explicit comparison of various versions of mirror symmetry on K3:

Orbifold Constructions of K3:

A Link between Conformal Field Theory and Geometry

A Link between Conformal Field Theory and Geometry

- Orbifolds in Mathematics and Physics, Contemp. Math.
**310**(2002) pp. 333-358 - hep-th/0112006

This work contains a detailed discussion of an example which allows the
application of various versions of mirror symmetry on K3. I prove that all of
them agree in that point of the moduli space. Later, I have continued the
investigation of this model is in collaboration with
Prof. Matthias Gaberdiel,
Prof. Anne Taormina
and Dr. Roberto Volpato: We show that its symmetry group is
Z_{2}^{8}:M_{20}, one of the largest symmetry groups
among K3 sigma models, see here.