The research results directly related to my Diploma thesis are joint work with Prof. Werner Müller. They are concerned with regularized determinants, Ray-Singer analytic torsion, and the Quillen metric, particularly their dependence on the metric.

Publications building immediately on my PhD project are devoted to orbifold techniques (partially published with Sayipjamal Dulat) and with superconformal field theories that can be given a geometric interpretation on a K3 surface. In particular, partially in joint work with Prof. Werner Nahm, I have investigated the structure of the moduli space of superconformal field theories on K3, orbifold conformal field theories on K3, and mirror symmetry on K3.

These works form the foundation for my math/physics activities since 2002:

SCFTs associated to smooth K3 surfaces

I have found and am investigating explicit constructions of SCFTs associated to smooth K3 surfaces.

SCFTs associated to Borcea-Voisin threefolds

Together with Dr. Madeeha Khalid we investigate methods to explicitly construct SCFTs on a certain class of Calabi-Yau threefolds, known as Borcea-Voisin threefolds.

Limiting processes for CFTs

A project with Dr. Daniel Roggenkamp is concerned with limiting processes in CFTs which lead to degeneration phenomena, yielding geometric structures and thus a conceptual approach to geometric interpretations to CFTs. We use ideas and techniques from noncommutative geometry in this project.

Elliptically fibered Calabi-Yau threefolds and string-string duality

With Dr. Anda Degeratu we are investigating aspects of the heterotic - type IIA duality from a mathematical point of view, particularly certain constraints on the geometric invariants of elliptically fibered Calabi-Yau threefolds arising from this conjectured duality.

On orbifolds and free fermion constructions

With Prof. Ron Donagi we join the venture to unravel the geometry underlying semi-realistic string models. Along with a new classification result we have discovered a rather unexpected connection between certain free fermion constructions and earlier work by Donagi, Ovrut, Pantev and Waldram which could lead to vast simplifications of existing techniques. It has already been applied to this effect by Vaudrevange and his coauthors in the construction of an MSSM with three generations.

Mathieu moonshine

Together with Prof. Anne Taormina we are investigating the role that finite symplectic symmetry groups of K3 surfaces play in a phenomenon discovered about a decade ago and known as Mathieu moonshine. We have developed a technique which allows us to combine the symmetry groups of different Kummer surfaces to larger groups, which vastly exceed the order 960 of the largest finite symplectic automorphism group of any K3 surface, an idea now known as symmetry surfing. We are first in obtaining representations of such large subgroups of M₂₄ directly from K3 theories, thereby providing direct evidence for the fact that M₂₄ governs the elliptic genus of K3. Our joint work with Prof. Matthias Gaberdiel, Prof. Anne Taormina and Dr. Roberto Volpato on symmetries of a special K3 sigma model is related.

The elliptic genus and its relatives

The as yet conjectural double life of the elliptic genus for a Calabi-Yau manifold X as regularized U(1)-equivariant index of a Dirac operator on the loop space of X, on the one hand, and as part of the partition function in every conformal field theory which describes superstring theory on X, on the other hand, is at the center of this project. Refinements of the elliptic genus lead to a number of insights regarding Mathieu Moonshine, as well.

Topological field theory, integrable systems and singularities

This is a project devoted to topological quantum field theories, which my entire research group was involved in during its funding through an ERC Starting Independent Researcher Grant The Geometry of Topological Quantum Field Theories. We investigate TERP structures as described by Hertling, and their role in the study of moduli spaces of supersymmetric quantum field theories in general and D-branes in particular. In joint work with Dr. Oliver Gray, we specifically work on a generalization of Cecotti and Vafa's ADE classification of superconformal field theories. The role of integrable systems in the context of ADE type singularities was a main focus of Dr. Florian Beck's Ph.D. project, supervised by me jointly with PD Dr. Emanuel Scheidegger. As a follow-up to the resulting thesis, in joint work with Dr. Florian Beck and Prof. Ron Donagi we have continued to investigate Hitchin and Calabi-Yau integrable systems of BCFG type.

Mathematical definition of conformal field theory

In this longterm project, I develop an axiomatic formulation of conformal field theory.

Other