Non-commutative deformation of Chern-Simons theory

The problem of the consistent definition of gauge theories living on the non-commutative (NC) spaces with a non-constant NC parameter $\Theta(x)$ is discussed. Working in the L$_\infty$ formalism we specify the undeformed theory, $3$d abelian Chern-Simons, by setting the initial $\ell_1$ brackets. The deformation is introduced by assigning the star commutator to the $\ell_2$ bracket. For this initial set up we construct the corresponding L$_\infty$ structure which defines both the NC deformation of the abelian gauge transformations and the field equations covariant under these transformations. To compensate the violation of the Leibniz rule one needs the higher brackets which are proportional to the derivatives of $\Theta$. Proceeding in the slowly varying field approximation when the star commutator is approximated by the Poisson bracket we derive the recurrence relations for the definition of these brackets for arbitrary $\Theta$. For the particular case of $su(2)$-like NC space we obtain an explicit all orders formulas for both NC gauge transformations and NC deformation of Chern-Simons equations. The latter are non-Lagrangian and are satisfied if the NC field strength vanishes everywhere.