θ-exact Seiberg-Witten maps at ordere3

Abstract

We study two distinct $\ensuremath{\theta}$-exact Seiberg-Witten (SW) map expansions, (I) and (II), respectively, up to order ${e}^{3}$ for the gauge parameter, gauge field, and gauge field strengths of the noncommutative ${\mathrm{U}}_{\ensuremath{\star}}(1)$ 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 ${e}^{3}$-order field-strength expansions and define corresponding sets of deformation/ratio/weight parameters, $(\ensuremath{\kappa},{\ensuremath{\kappa}}_{i})$ and $(\ensuremath{\kappa},{\ensuremath{\kappa}}_{i}^{\ensuremath{'}})$, for these two SW maps, respectively.