The theta-exact Seiberg-Witten maps at the e3 order

We study wo distinct theta-exact Seiberg-Witten (SW) map expansions, (I) and (II) respectively, up to the e3 order for the gauge parameter, gauge field and the gauge field strengths of the noncommutative U*(1) gauge theory on the Moyal space. We derive explicitly the closed form expression for the SW map ambiguity between the two and observe the emergence of several new totally commutative generalized star products. We also identify the additional gauge freedoms within each of the e3 order field strength expansions and define corresponding sets of deformation/ratio/weight parameters, (kappa,kappa_i) and (kappa,kappa'_i), for these two SW maps respectively.