Twisted Supersymmetry in Topological Field Models of Schwarz-Type

The aim of the proposed research program is to investigate systematically the twisting procedure of supersymmetry of topological field models of Schwarz type and its supersymmetric extensions on flat space-time and on curved space-time manifolds. In more detail we try to analyze Chern-Simons-models defined on odd space-time dimensions (flat and curved) and the 2,3 and 4-dimensional BF-model (flat and curved) in the presence of a twisted supersymmetry. All the investigations are based on the use of a Landau-gauge fixing procedure. In a final step we try to use also an axial gauge fixing. The discussion of the 3-dimensional supersymmetric Chern-Simons model will be done in a superfield formulation. In this context the topological supersymmetry is only realized in a non-linear manner. In order to study the models at the quantum level one applies the techniques of algebraic renormalization.

Due to certain circumstances (see final report, unabridged version) there was a shift of the focus of research towards noncommutative quantum field theory, which is an extraordinary active field of research at present. Noncommutative quantum field theory-NCQFT 1 The aim of noncommutative geometry, which is at the root and the source of NCQFT, is the formulation of familiar geometrical notions in algebraic terms and the subsequent replacement of commutative, associative algebras with non-commutative, associative algebras. A further motivation comes from string and M-theory, respectively, where noncommutative objects surfaced in certain ap- plications. The noncommutative algebras are represented by ordinary functions but with a special product (a so- called Moyal- or star-product), which involves the deformation parameter _ which arises from quantized space- time, i. e. noncom-muting coordinates. These products induce phase factors in field theory actions leading to a number of phenomena such as UV/IR-mixing and the distinction between planar and nonplanar graphs. In gauge theories a further interesting relation exists between noncommutative and commutative theories, namely the Seiberg-Witten map, mapping one onto the other. Various models with different properties (with and without supersymme-try, perturbative/non-perturbative in _, topological, gauge theories) were in-vestigated with respect to renormalization as part of the project with the em- phasis (at first) on _-expanded U(1)-NCQED, where the UV/IR-mixing is ab-sent, but despite some very encouraging results we had to shift our atten-tion to non-expanded theories since R. Wulkenhaar proved the one- loop non-renormalizability of U(1)-NCQED. These non-expanded theories do suffer from UV/IR-mixing; therefore, as a means to solve this problem, very general field re-definitions were applied with positive results hinting at a possible solution. Also, the Seiberg-Witten map was derived in the context of noncommutative Lorentz symmetry (after some preliminary work concerning dilatation symmetry and the energy-momentum tensor of noncommutative scalar theories) in another paper, whereas a different work was focused on the perturbative treatment of U(1) Super-Yang-Mills theory, which involved a huge amount of calculations and re-sulted in some evidence on gauge-dependence of the divergences coming from UV/IR-mixing. At the beginning, research was focused on Chern-Simons the-ory, a topological model with vector supersymmetry, on the one hand, and the Wess- Zumino model in the superfield formalism, on the other, yielding positive results concerning renormalizability in both cases. 1 In this abstract, we shall treat NCQFT only, since the bulk of the papers were con-cerned with it; also, research related to string theory/branes and topological field theory was conducted and papers published.