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 Θ(x), which satisfies two main requirements: it is gauge invariant and reproduces in the commutative limit, Θ → 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]★ ≈ {f, g}. We derive an explicit expression for both the NC deformation of Abelian gauge transformations which close the algebra [δf , δg]A = δ{f, g}A, and the NC field strength ℱ, covariant under these transformations, δfℱ = {ℱ, f}. NC Chern-Simons equations are equivalent to the requirement that the NC field strength, ℱ, should vanish identically. Such equations are non-Lagrangian. The NC deformation of Yang-Mills theory is obtained from the gauge invariant action, S = ∫ ℱ2. As guiding example, the case of su(2)-like non-commutativity, corresponding to rotationally invariant NC space, is worked out in detail.