In my doctoral studies, among other topics I have investigated the embedding of orbifold conformal field theories within the moduli space MK3 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
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 MK3 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 MK3. I prove that this suffices to explicitly determine the locations of all those G orbifold CFTs on K3 within MK3 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:
We describe mirror symmetry as an automorphism on the smooth universal cover of MK3 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 ZM 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:
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, and I emphasize the importance of this example. Indeed, later I have continued the investigation of this model in collaboration with Prof. Matthias Gaberdiel, Prof. Anne Taormina and Dr. Roberto Volpato: we show that its symmetry group is Z28:M20, the largest possible symmetry group among K3 sigma models, see here. The model is now known as Gaberdiel-Taormina-Volpato-Wendland (GTVW) model.