Kontsevich product and gauge invariance

We analyze the question of $U_{\star }\mathrm{}\left(1\right)\mathrm{}$ gauge invariance in a flat noncommutative space where the parameter of noncommutativity, ${\theta }^{\mu \nu }\mathrm{}\left(x\right),$ is a local function satisfying the Jacobi identity (and thereby leading to an associative Kontsevich product). We show that in this case both gauge transformations as well as the definitions of covariant derivatives have to be modified so as to have a gauge invariant action. We work out the gauge invariant actions for the matter fields in the fundamental and the adjoint representations up to order ${\theta }^2$ while we discuss the gauge invariant Maxwell theory up to order $\theta .$ We show that, despite the modifications in the gauge transformations, the covariant derivative, and the field strength, the Seiberg-Witten map continues to hold for this theory. In this theory, translations do not form a subgroup of the gauge transformations (unlike in the case when ${\theta }^{\mu \nu }$ is a constant) which is reflected in the stress tensor not being conserved.