Research Interests

My major interests lie in classical and quantum field theories, supersymmetry, string theory as well as the applications of methods and concepts from group theory, differential and algebraic geometry in theoretical physics. My research projects so far can be grouped into Twistor Geometry and Twistor String Theory, Non-(anti)commutative deformations of Field Theories, Geometric Quantization and M-Theory.

Below is a more detailed review of my research.

Non-anticommutative field theory.

Turning on a graviphoton background in superstring theory yields a so-called non-anticommutative deformation of spacetime. Field theories on thus deformed spaces have attracted much attention in the years after 2003. In hep-th/0401147, we developed the definition of non-anticommutative N=4 supersymmetric Yang-Mills (SYM) theory. Subclasses of the deformations we introduced (the beta-deformations) became subsequently popular in the AdS/CFT community.

Drinfeld twisting

One difficulty in studying such deformations is that they break many of the original symmetries of the underlying spacetime at first sight. From a different point of view, however, one can interpret these symmetries as merely hidden and even make them manifest in all formulas again. This technique is know as a Drinfeld- or Hopf-algebra twist. In hep-th/0506057, we defined such a Drinfeld-twist for arbitrary non-anticommutative field theories, thereby recovering useful structures such as chiral and anti-chiral rings of operators, Ward-Takahashi identities and non-renormalization theorems.

Fuzzy scalar field theories

Berezin- and Berezin-Toeplitz quantization are hybrid forms of geometric and deformation quantization. The most prominent example of a Berezin-quantized space is the fuzzy sphere, which received much attention in the non-commutative geometry community, but it also appears very naturally in certain D-brane configurations.

The general definition of scalar field theories on arbitrary fuzzy spacetimes was given in 0804.4555 [hep-th] (the papers hep-th/0611328, hep-th/0612173 contained preliminary results in the same direction). In this paper, we also clarified in detail the relationship between the mathematicians' Berezin-Toeplitz quantization and physicists' fuzzy spaces. The quantization of supermanifolds and its application to regularizing supersymmetric field theories was discussed in 0811.4743 [hep-th].

In hep-th/0606197, we used the above mentioned Drinfeld-twisting technique to write down an Einstein-Hilbert action on the fuzzy sphere. Moreover, our results in hep-th/0606197 suggest a way of overcoming obstacles in defining the Standard Model of elementary particles using fuzzy spaces.

Matrix models.

Field theories on fuzzy spaces are essentially special classes of matrix models. The latter have been studied extensively by mathematical physicists. We suggested and demonstrated the application of matrix model techniques to fuzzy field theories in 0706.2493 [hep-th] and made various new statements about such theories. These techniques were extended in the papers 1003.4683 [hep-th] and 1012.3568 [hep-th] to arbitrary even and odd dimensions leading in the latter case to matrix quantum mechanical models. We thus managed to reproduce analytically the theories' phase diagrams, which had been obtained previously by numerical methods.

Twistor string theory.

Edward Witten observed in 2003 that there are certain twistor geometries which are simultaneously suitable spaces for certain strings to live in; Twistor String Theory was born. In hep-th/0405123 we made the differential geometric background of classical twistor string theory explicit; hep-th/0410292 and hep-th/0505161 were variations on this theme. (Also in hep-th/0410292, various approaches to a new species of spaces, the exotic supermanifolds, were compared and analyzed.) A completely new twistor space for describing N=8 SYM theory in three dimensions was constructed in hep-th/0508137. It showed both interesting and strange properties: for example, this space is not a manifold.

Most promising, an explicit map between configurations of physical D-branes and topological D-branes was developed in hep-th/0511130. Pushed further, this might allow for applying powerful techniques from the description of the former (as e.g.\ derived categories) to the latter and vice versa.

M-theory

In 2007, Bagger and Lambert and independently Gustavsson proposed an effective description of stacks of M2-branes based on 3-Lie algebras. In 0807.0808 [hep-th], we gave a natural generalization of the notion of a 3-Lie algebra to overcome an obstacle in this description. We classified representations of these and other generalizations of 3-Lie algebras in terms of $\star$-algebras and gave formulations of the Bagger-Lambert-Gustavsson (BLG) model in $N=2$ and $N=4$ superspaces 0812.3127 [hep-th]. In 0901.3905 [hep-th], we pointed out the close connection between the $n$-Lie algebras used in the BLG model and strong homotopy (sh) Lie algebras. We also showed how to rewrite both the BPS equations and the equations of motion of the BLG model as Maurer-Cartan equations of a sh Lie algebra.

We performed a two-loop computation in the BLG model to identify and study exactly marginal deformations, in particular those relevant in the AdS/CFT context 0906.1705 [hep-th].

In 1001.3275 [hep-th], we proposed generalized quantization axioms for Nambu-Poisson manifolds, which allow for a geometric interpretation of n-Lie algebras and their enveloping algebras. We then derived a 3-Lie algebra reduced model from the BLG model, which contains the IKKT matrix model as a limit and which has quantized Nambu-Poisson manifolds as supersymmetric solutions 1012.2236 [hep-th].

My most important result so far is possibly the extension of the Nahm construction of monopoles to the case of abelian self-dual strings by reformulating the corresponding equations of motion using a transgression to loop space 1007.3301 [hep-th].