An electron moving on plane in a uniform magnetic field orthogonal to plane is known as the Landau problem. Wigner functions for the Landau problem when the plane is noncommutative are found employing solutions of the Schrödinger equation as well as solving the ordinary ⋆-genvalue equation in terms of an effective Hamiltonian. Then, we let momenta and coordinates of the phase space be noncommutative and introduce a generalized ⋆-genvalue equation. We solve this equation to find the related Wigner functions and show that under an appropriate choice of noncommutativity relations they are independent of noncommutativity parameter.

A formulation of (non-anticommutative) N=1/2 supersymmetric U (N ) gauge theory in noncommutative space is studied. We show that at one loop UV/IR mixing occurs. A generalization of Seiberg-Witten map to noncommutative and non-anticommutative superspace is employed to obtain an action in terms of commuting fields at first order in the noncommutativity parameter θ. This leads to abelian and non-abelian gauge theories whose supersymmetry transformations are local and non-local, respectively.

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 .

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