A novel approach to non-commutative gauge theory

We propose a field theoretical model defined on non-commutative space-time with non-constant non-commutativity parameter $\Theta(x)$, which satisfies two main requirements: it is gauge invariant and reproduces in the commutative limit, $\Theta\to 0$, the standard $U(1)$ gauge theory. We work in the slowly varying field approximation where higher derivatives terms in the star commutator are neglected and the latter is approximated by the Poisson bracket, $-i[f,g]_\star\approx\{f,g\}$. We derive an explicit expression for both the NC deformation of Abelian gauge transformations which close the algebra $[\delta_f,\delta_g]A=\delta_{\{f,g\}}A$, and the NC field strength ${\cal F}$, covariant under these transformations, $\delta_f {\cal F}=\{{\cal F},f\}$. NC Chern-Simons equations are equivalent to the requirement that the NC field strength, ${\cal F}$, should vanish identically. Such equations are non-Lagrangian. The NC deformation of Yang-Mills theory is obtained from the gauge invariant action, $S=\int {\cal F}^2$. As guiding example, the case of $su(2)$-like non-commutativity, corresponding to rotationally invariant NC space, is worked out in detail.