Non-commutative gauge theories with a non-constant NC-parameter are investigated. As a novel approach, we propose that such theories should admit an underlying L_∞ algebra, that governs not only the action of the symmetries but also the dynamics of the theory. Our approach is well motivated from string theory. We recall that such field theories arise in the context of branes in WZW models and briefly comment on its appearance for integrable deformations of AdS₅ sigma models. For the SU(2) WZW model, we show that the earlier proposed matrix valued gauge theory on the fuzzy 2-sphere can be bootstrapped via an L_∞ algebra. We then apply this approach to the construction of non-commutative Chern-Simons and Yang-Mills theories on flat and curved backgrounds with non-constant NC-structure. More concretely, up to the second order, we demonstrate how derivative and curvature corrections to the equations of motion can be bootstrapped in an algebraic way from the L_∞ algebra. The appearance of a non-trivial A_∞ algebra is discussed, as well.

The model of the position-dependent noncommutativity in quantum mechanics is proposed. We start with given commutation relations between the operators of coordinates $[{\widehat{x}}^i,{\widehat{x}}^j]={\omega }^{ij}(\widehat{x})$, and construct the complete algebra of commutation relations, including the operators of momenta. The constructed algebra is a deformation of a standard Heisenberg algebra and obeys the Jacobi identity. The key point of our construction is a proposed first-order Lagrangian, which after quantization reproduces the desired commutation relations. Also we study the possibility to localize the noncommutativity.

Deformation quantization is a formal deformation of the algebra of smooth functions on some manifold. In the classical setting, the Poisson bracket serves as an initial conditions, while the associativity allows to proceed to higher orders. Some applications to string theory require deformation in the direction of a quasi-Poisson bracket (that does not satisfy the Jacobi identity). This initial condition is incompatible with associativity, it is quite unclear which restrictions can be imposed on the deformation. We show that for any quasi-Poisson bracket the deformation quantization exists and is essentially unique if one requires (weak) hermiticity and the Weyl condition. We also propose an iterative procedure that allows one to compute the star product up to any desired order.

The problem of the consistent definition of gauge theories living on the non-commutative (NC) spaces with a non-constant NC parameter $$\Theta (x)$$Θ(x) is discussed. Working in the $$\hbox {L}_{\infty }$$L∞ formalism we specify the undeformed theory, 3d abelian Chern–Simons, by setting the initial $$\ell _1$$ℓ1 brackets. The deformation is introduced by assigning the star commutator to the $$\ell _2$$ℓ2 bracket. For this initial set up we construct the corresponding $$\hbox {L}_{\infty }$$L∞ 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 $$\Theta $$Θ . 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 $$\Theta $$Θ . 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.