Branes, Duality and Supersymmetric Gauge Theories

Our knowledge of non-perturbative phenomena in field theory and string theory is rapidly improving. String techniques have proved to be very useful for the study of gauge theories. A particularly fruitful approach uses Dirichlet-branes as building blocks of gauge theories. In addition, it uses the duality between type IIA superstring and M-theory as a way of deriving nonperturbative information for four-dimensional theories. New Seiberg-Witten curves for four-dimensional gauge theories with N = 2 extended supersymmetry have been obtained in this way, together with many exact results for N = 1 theories. Many problems can still be analyzed following this approach. A first aim is to construct and study chiral gauge theories. Using string techniques we expect to gain a better understanding of central problems in gauge theories like supersymmetry breaking, chiral symmetry breaking and properties of domain walls. Although these techniques are more powerful when applied to supersymmetric gauge theories, non-supersymmetric theories could also benefit from them.

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.