Equivalence of quantum field theories related by the $\theta$-exact Seiberg-Witten map

The equivalence of the noncommutative U(N) quantum field theories related by the $\theta$-exact Seiberg-Witten maps is, in this paper, proven to all orders in the perturbation theory with respect to the coupling constant. We show that this holds for super Yang-Mills theories with $\mathcal{N}=0$, 1, 2, 4 supersymmetry. A direct check of this equivalence relation is performed by computing the one-loop quantum corrections to the quadratic part of the effective action in the noncommutative U(1) gauge theory with $\mathcal{N}=0$, 1, 2, 4 supersymmetry.