Noncommutative scalar field theory in a curved background: Duality between noncommutative and effective commutative description

We study a noncommutative (NC) deformation of a charged scalar field, minimally coupled to a classical (commutative) Reissner-Nordstrom-like background. The deformation is performed via a particularly chosen Killing twist to ensure that the geometry remains undeformed (commutative). An action describing a NC scalar field minimally coupled to the RN geometry is manifestly invariant under the deformed U(1)(*) gauge symmetry. We find the equation of motion and conclude that the same equation is obtained from the commutative theory in a modified geometrical background described by an effective metric. This correspondence we call ''duality between formal and effective approach". We also show that a NC deformation via semiKilling twist operator cannot be rewritten in terms of an effective metric. There is dual description for those particular deformations.