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Библиогр.: 29 назв.

The problem of the consistent definition of gauge
theories living on the non-commutative (NC) spaces with
a non-constant NC parameter (x) is discussed. Working
in the L∞ formalism we specify the undeformed theory, 3d
abelian Chern–Simons, by setting the initial 1 brackets. The
deformation is introduced by assigning the star commutator
to the 2 bracket. For this initial set up we construct the corresponding
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 . 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 . For the particular case of su(2)-like NC space
we obtain an explicit all orders formulas for both NC gauge
transformations andNCdeformation ofChern–Simons equations.
The latter are non-Lagrangian and are satisfied if the
NC field strength vanishes everywhere.