Duality invariance of non-anticommutative N = 1/2 supersymmetric U(1) gauge theory

A parent action is introduced to formulate (S-) dual of non-anticommutative N ≤ 1/2 supersymmetric U(1) gauge theory. Partition function for parent action in phase space is utilized to establish the equivalence of partition functions of the theories which this parent action produces. Thus, duality invariance of non-anticommutative N ≤ 1/2 supersymmetric U(1) gauge theory follows. The results which we obtained are valid at tree level or equivalently at the first order in the nonanticommutativity parameter C_μν.