Cohomological Yang-Mills theory is formulated on a noncommutative differentiable four manifold through the θ-deformation of its corresponding BRST algebra. The resulting noncommutative field theory is a natural setting to define the θ-deformation of Donaldson invariants and they are interpreted as a mapping between the Chevalley-Eilenberg homology of noncommutative spacetime and the Chevalley-Eilenberg cohomology of noncommutative moduli of instantons. In the process we find that in the weak coupling limit the quantum theory is localized at the moduli space of noncommutative instantons. * This work is dedicated to Professor Alberto García on the occasion of his 60th birthday.

Consider a manifold endowed with the action of a Lie group. We study the relation between the cohomology of the Cartan complex and the equivariant cohomology by using the equivariant De Rham complex developed by Getzler, and we show that the cohomology of the Cartan complex lies on the 0 − th row of the second page of a spectral sequence converging to the equivariant cohomology. We use this result to generalize a result of Witten on the equivalence of absence of anomalies in gauged WZW actions on compact Lie groups to the existence of equivariant extension of the WZW term, to the case on which the gauge group is the special linear group with real coefficients.2010 Mathematics Subject Classification. 57R91, 57T10, 81T40, 81T70 .

The problem of computing systematically the gauge invariant extension of WZW term through equivariant cohomology is addressed. The analysis done by Witten in the two-dimensional case is extended to the four-dimensional ones. While Cartan's model is used to find the anomaly cancelation condition. It is shown that the Weil model is more appropriated to find the gauge invariant extension of the WZW term. In the process we point out that Weil's and Cartan's models are also useful to stress some connections with the abelian anomaly.

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