U(1) gauge invariant noncommutative Schr\"odinger theory and gravity

We consider the complex, massive Klein-Gordon field living in the noncommutative space, and coupled to noncommutative electromagnetic fields. After employing the Seiberg-Witten map to first order, we analyze the noncommutative Klein-Gordon theory as $c$, the velocity of light, goes to infinity. We show that the theory exhibits a regular “magnetic” limit only for certain forms of magnetic fields. The resulting theory is nothing but the Schr"odinger theory in a gravitational background generated by the gauge fields.