  We analyze the question of $U_{\star} (1)$ gauge invariance in a flat
non-commutative space where the parameter of non-commutativity,
$\theta^{\mu\nu} (x)$, is a local function satisfying 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 modify 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, 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.

High Energy Physics - Theory