In the framework of my Ph.D. project, jointly with Prof. Werner Nahm we investigated the moduli space M of N=(4,4) superconformal field theories (SCFTs) with central charge c=6.
Our results can be found in
After a slight emendation of the previously known global description of M, we find the locations of various known models in the component MK3 associated to K3 surfaces. Among them are the Z2 and Z4 orbifold theories obtained from the torus component of M. Here, SO(4,4) triality is found to play a crucial role. Using our embedding of the space of Z2 orbifold theories in MK3, we give a direct proof for a widely spread but previously unproved statement about the behavior of the B-field under the orbifold procedure. We prove T-duality for Z2 orbifolds and use it to derive the form of M purely within conformal field theory.
Restricting to the bosonic parts of the theories, we also find two meeting points of Mtori and MK3, generalizing results by Kobayashi and Sakamoto. We correct and prove a conjecture by Eguchi, Ooguri, Taormina and Yang on possible meeting points of the moduli spaces of Z2 and Z4 orbifold CFTs. We moreover prove isomorphies of specific examples of orbifold CFTs to so-called Gepner and Gepner type models, one of which had been conjectured before by Eguchi, Ooguri, Taormina and Yang. This enables us to give the precise location of these models in the moduli space, in particular to prove the long standing and widely used conjecture that the Gepner model (2)4 admits a geometric interpretation on the Fermat quartic hypersurface. Later, I have clarified and extended the latter result, showing how on every "very attractive" quartic X a SCFT can be constructed explicitly.
With Sayipjamal Dulat I have made a contribution to the classification of two dimensional conformal field theories with central charge c=2:
We construct all the 26 non-exceptional non-isolated irreducible components of the moduli space of such theories that may be obtained by an orbifold procedure from toroidal ones, their parameter spaces, and their partition functions. The global structure of this part of the moduli space follows from our determination of all multicritical points and lines. The results of Dixon, Ginsparg and Harvey on the classification of superconformal theories (SCFTs) with central charge c=3/2 serve as a cross check. In particular, all their non-isolated orbifolds as well as orbifolds with discrete torsion gain geometric interpretations due to our results.
In my Ph.D. thesis, I had also corrected the claims made in Dixon, Ginsparg and Harvey on multicritical points among the orbifolds with central charge c=3/2. This discussion as well as the general construction and classification of all so-called superaffine, orbifold prime and super-M-orbifold models in arbitrary dimensions is given in my later publication