Abstract
Extended supersymmetries in D=4 topological Yang-Mills theory are studied. Their existence imposes constraints on the metric of the manifold, on which the topological Yang-Mills theory is defined. For irreducible manifolds we establish a 1-1 correspondence between extended supersymmetries and covariantly constant complex structures. In particular, the theory possesses one additional supersymmetry on a Kähler manifold. By analogy with a general riemannian case, this gives a way for the construction of invariants of complex structures. Ingredients, needed for this construction, such as the Donaldson map, are generalized for Kähler manifolds.
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