Magnetic fields, branes, and noncommutative geometry

We construct a simple physical model of a particle moving on the infinite noncommutative 2-plane. The model consists of a pair of opposite charges moving in a strong magnetic field. In addition, the charges are connected by a spring. In the limit of large magnetic field, the charges are frozen into the lowest Landau levels. Interactions of such particles include Moyal-bracket phases characteristic of field theories on noncommutative space. The simple system arises in the light cone quantization of open strings attached to D-branes in antisymmetric tensor backgrounds. We use the model to work out the general form of light cone vertices from string splitting. We then consider the form of Feynman diagrams in (uncompactified) noncommutative Yang-Mills theories. We find that for all planar diagrams the commutative and noncommutative theories are exactly the same apart from trivial external line factors. This means that the large N theories are equivalent in the ’t Hooft limit. Non-planar diagrams are generally more convergent than their commutative counterparts.