Magnetic fields, branes and noncommutative geometry

TL;DR

This paper models a particle on a noncommutative plane using charged particles in a magnetic field, revealing insights into string interactions and equivalences between commutative and noncommutative gauge theories at large N.

Contribution

It introduces a simple physical model to analyze noncommutative geometry effects and derives the form of lightcone vertices, demonstrating equivalences in large N limits.

Findings

Planar diagrams are identical in commutative and noncommutative theories.

Non-planar diagram convergence is improved.

Model connects string theory with noncommutative gauge theories.

Abstract

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 level. Interaction of such particles include Moyal bracket phases characteristics of field theory on noncommutative space. The simple system arises in lightcone quantization of open strings attached to D-branes in a.s. tensor background. We use the model to work out the general form of lightcone vertices from string splitting. We then consider Feynman diagrams in uncompactified NC YM theories and find that for all planar diagrams the comm. and noncomm. theories are the same. This means large N theories are equivalent in the 't Hooft limit. Non planar diagrams convergence is improved.