We show that the photon self-energy in quantum electrodynamics on noncommutative R 4 is renormalizable to all orders (both in θ andh) when using the Seiberg-Witten map. This is due to the enormous freedom in the Seiberg-Witten map which represents field redefinitions and generates all those gauge invariant terms in the θ-deformed classical action which are necessary to compensate the divergences coming from loop integrations.

We investigate the quantization of the θ-expanded noncommutative U(1) Yang-Mills action, obtained via the Seiberg-Witten map. As expected we find non-renormalizable terms. The one-loop propagator corrections are gauge independent, and lead us to a unique extention of the noncommutative classical action. We interpret our results as a requirement that also the trace in noncommutative field theory should be deformed.

Abstract. We reconsider the algebraic BRS renormalization of Witten's topological Yang-Mills field theory by making use of a vector supersymmetry Ward identity which improves the finiteness properties of the model. The vector supersymmetric structure is a common feature of several topological theories. The most general local counterterm is determined and is shown to be a trivial BRS-coboundary.

Abstract. For the noncommutative Yang-Mills field there exist two representations (primitive and covariant) of the (undeformed) group of rigid translations, rotations and dilatations. The SeibergWitten map is the equivalence between both representations on the level of classical noncommutative field theories. In the covariant representation the Yang-Mills field is θ-dependent according to a first-order differential equation and thus can be parametrized by its initial value at θ = 0, i.e. a gauge field living on commutative space-time.

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