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.

We show that the bosonic string theory quantized in the Beltrami parametrization possesses a supersymmetric structure like the vectorsupersymmetry already observed in topological field theories.

The Wess-Zumino consistency conditions for Lorentz and diffeomorphism anomalies are discussed by introducing an operator δ which allows one to decompose the exterior space-time derivative as a BRS commutator.

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We build up a class of Nϭ2 supersymmetric nonlinear models in an Nϭ1 superspace based on the Atiyah-Ward space-time of ͑2ϩ2͒-signature metric. We also discuss the gauging of isometries of the associated hyper-Kählerian target spaces and present the resulting gauge-covariant supersymmetric action functional.